Orders: Description and Roles

ORDERS: DESCRIPTION AND ROLES

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1982

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annals of Ccnerd Editor Peter L. H A M M E R , Rutgers University, New Brunswick, NJ, U.S.A. Advisory E ditors C. B E R G E , Universitk de Paris M. A . H A R R I S O N , University of California, Berkeley, C A , U.S.A. V. KLEE, Universityofwashington, Seattle,WA. U.S.A. J . H . VAN LINT, California Institute ofTechnology, Pasadena, C A . U.S.A. G.-C. ROTA, Massachusetts Institute ofTechnology, Cambridge, M A , U.S.A.

NORTH-HOLLAND -AMSTERDAM 0 NEW YORK

OXFORD

NORTH-HOLIAND MATHEMATICS STUDIES

99

Annals of Discrete Mathematics(23) General Editor: Peter L. Hammer Rutgers University, New Brunswick, NJ, U.S.A.

ORDERS: DESCRIPTION and ROLES in SetTheory, Lattices, Ordered Groups,Topology, Theory of Models and Relations, Combinatorics, Effectiveness, Social Sciences. Proceedings of the Conference o n Ordered Sets and theirApplication\ Chliteau dc IaTourcttc, I’Arbrcsle,JulyS-l I , 1982

ORDRES: DESCRIPTION et ROLES enTheorie des Ensembles, deslieillis, des Groupes Ordonnks: enTopologie,ThCorie des Modeles et des Relations, Cornbinatoire, Effectivite, Sciences Sociales. Actes de la Conference sur ies Enseniblcs Ordonnds et leur Applications Ch~teatidelaTourette. I’Arhresle. juillet S-11. 19x2

edited by

Maurice POUZET and

Denis RICHARD Laboratoire dMlgebre Ordinale Departement de Mathematiques Universite Claude Bernard Lyon I France

1984

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

ISBN: 0 444 87601 4

Pitblislirr:

ELSEVIER SCXENCE PUBLISHERS R.V. P.O. BOX I9YI 1000 BZ AMSTERDAM ‘THE NETHERLANDS .Solct/i.sir.ihiriot.~fi~r ilic

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ELSEVIER SCIENCE PUBLISHING COMPANY. INC 52 VAN I) E RB I LT AVENUE NEW YORK. N.Y. 10017 U.S.A.

Lattice adapted from Figure 16 of Cherlin and Rosenstein, Ho-categorical groups, J. Algebra 53 (1978), 188-226 .

Library of Congress Cataloglng In Publlcatlon Data

Conference on Ordered Sets and Their Applications (4th : 1982 : L’Arbresle, France) Orders--description and roles. (Annals of discrete mathematics ; 23) (North-Holland mathematics studies ; 99) 1. Ordered sets--Congresses. I. Pouzet, M. 11. Richard, Denis, 1942111. Title. IV. Title: Ordree--description et r8les. v. Series. VI. Series: North-Holland mathematics studies ; 99. QA171.48.C66 1982 511.3’2 84-13749 ISBN 0-444-87601-4

.

PRINTED IN THE NETHERLANDS

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vii

PREFACE

The 27 papers in this volume survey various aspects of the theory of order. These have been grouped into nine sections illustrating some of the main mathematical themes and applications in the theory. These papers were written for the “Conference on ordered sets and their applications” (I’ARBRESLE, july 1982). This international meeting, the fourth devoted to the theory of order (following BANFF, 1981; MONTEREY , 1959, CHARLOTTESVILLE, 1938) - will shortly be succeeded by two others (BANFF and LUMINY, 1984). This continuing activity, and other signs such as the appearance of the new journal ORDER, suggests that there is an increasing recognition of the importance of order and an acceleration in the development of its theory. All this calls for some interpretation of the role of order in the general landscape of mathematics. Editing the present work has also led us to consider this question, and we offer here some of our thoughts on this matter to the reader. If we imagine the theory of order as a river, we discover many contributory sources. The principal source is undoubtedly the ordinal arithmetic of G. CANTOR, but other important ideas come from the work on real analysis by G. CANTOR and R. DEDEKIND, the contributions to the theory of equations and groups by E. GALOIS and C. JORDAN (solvable groups, Jordan-Holder Series) and in the work on the theory of rings by E. NOETHER, to mention but a few important examples. From these diverse origins a number of modern current have developed such as lattice theory, boolean algebra, topology, etc. The theory of lattices which began with studying the subgroups of a group, has achieved several significant results that are now considered classical (e.g. the representation theorems of BIRKHOFF and STONE) and remains, after several decades, one of the principal preoccupations of mathematicians working in order theory. In a similar way set theory, even at the most primitive axiomatic level (the axioms of ZERMELO, KURATOWSKI, ZORN, ...), is intimately concerned with questions of order. The neighbouring disciplines of descriptive set theory, topology, measure theory, model theory, logic and combinatorics have all contributed to and enlarged the theory of order. For example, problems in model theory, from categoricity to stability, are already realised in the structure of chains, and conversely the models constructed by A. EHRENFEUCHT and A. MOSTOWSKI allows one to represent part of the complexity of chains in arbitrary structures. Even a description of the models of Peano arithmetic involves ordinal concepts (final or cofinal extensions, initial segments, the indicator functions of KIRBY & PARIS), and these sometimes completely determine the structure, for example the saturation of a model reduces to that of its order structure. It is quite possible that the real nature of the logical and combinatorial content of the independence results of J. PARIS and L. HARRINGTON may have an order-theoretic basis. Having crossed the paradise of the infinite, with its inaccessible summits (the problems of consistency) and fertile valleys (lattice theory, order groups, boolean algebra, noetherian rings, pointrset topology, problems concerning duality, representation and generation in Universal Algebra), the course of the river leads back to the realm of the finite. A frequent connecting link between these territories is compactness (e.g. from the finite to the infinite version of DILWORTH’s theorem and from the infinite to the finite form of

viii

Preface

RAMSEY’s theorems). Here, however, the soil is more difficult to cultivate. Due to the rigidity and boundedness of the objects there are very few general techniques available (consider, for example, the RAMSEY numbers or the many famous unsettled conjectures of number theory). However, it is here that we find the extensive theory of finite graphs which is very rich in applications (flows in networks, optimization, etc.). Today, the external world, with its social problems, technological advances and new sciences influences the course of our river, which the mathematician might naively have thought was simply there to be discovered, He is now required to add to his role of explorer that of engineer; he must help forge new tools. In so doing he has found himself in mathematical domains that he might not otherwise have considered. This is the case, for example, in the social sciences: the CONDORCET paradox is the beginning of the theory of social choice where one spectacular result is the theorem of ARROW. The rapid advance of computer science has led mathematicians to reconsider problems with a view to effectiveness. Recursion theory (K. GODEL, A. CHURCH) and complexity (M. RABIN) in dealing with these new problems from computer science (e.g. sorting) have extended into the new domains of algorithmic complexity (time and space), automata theory and formal languages. This dual activity as builder - as was J. Von NEUMANN - and discovered - W. SIERPINSKI referred to himself as “Explorateur de I’infini” - continues. But a panoramic view of the theory of order is still missing. A first survey (ORDERED SETS, D. REIDEL, 1982) was edited by I. RIVAL. In this volume we illustrate the appearance and the role played by order in set theory, lattice theory, topology, logic (model theory, theory of relations, Peano arithmetic), ordered groups, combinatorics, computer science and the social sciences. The editors intented that the reader of this book should pass from the infinite to the finite, from the descriptive view to the applications. Apart from the survey articles appearing in this volume, DISCRETE MATHEMATICS will separately publish the research papers presented a t the la TOURETTE conference which give the most recent advances in the theory of order. The texts of all these articles are written either in French or English together with an abstract o r introduction in the other language. This preface would not be complete without an expression of thanks to the many referees for their excellent criticisms and suggestions. Among these were some of the present authors and also the following colleagues: MM. A. ACHACHE ; R. ASSOUS ; J.P. AURAY ; B. BANASCHEWSKI ; H.J. BANDELT; P. CEGIELSKI ; Ch. CHARRETTON ; M. CHEIN ; R. DEAN ; J.P. DOIGNON ; P. DWINGER ; M.R. GAREY ; S. GRIGORIEFF ; L. HARPER ; S.S. HOLLAND ; J. ISBELL ; M. JAMBU-GIRAUDET ; H. KOTLARSKI ; D. LASCAR ; A. LASCOUX ; L. LESIEUR ; R. MAYET ; E.W. MAYR ; C. S t J. A. NASH- WILLIAMS ; E. NELSON ; G. Mc NULTY ; M.PARIGOT ; G. de B. ROBINSON ; L.SHEPP ; B.SIMONS; A. R. STRALKA ; L. SZABO ; T. TROTTER ; D. W. WEST ; S. WOLFENSTEIN, M. YASUHARA . Finally, a word of thanks to J.G. ROSENSTEIN for the motif appearing o n the cover. This is a reproduction of the needlepoint work he completed during the Conference.

Maurice POUZET and Denis RICHARD

PREFACE

Vingt sept textes Qcrits dans une perspective de synthdse constituent ce volume sur la thkorie de I’ordre. 11s sont regroupbs en neuf parties choisies parmi de grands thdmes mathbmatiques concernant les ordres ou le rble qu’ils y jouent. 11s ont ktk blaborbs i I’occasion de la “Confbrence sur les ensembles ordonnbs et leurs applications” (I’ARBRESLE, juillet 1982). Cette rencontre internationale - quatridme des grands congrds consacrks aux ensembles ordonnbs (i la suite de ceux de BANFF, 1981 ; MONTEREY, 1959 ; CHARLOTTESVILLE, 1938) - ainsi que les deux prochaines aujourd’hui annonckes (BANFF e t LUMINY, 1984) et d’autres faits - comme la parution de la revue ORDER tout semble indiquer I’importance croissante de la thkorie des ordres e t I’accelkration de son dkveloppement. Cette constatation appelle une interprktation du cours de la thkorie des ordres dans le paysage mathbmatique. L’bdition du prksent ouvrage nous conduisait aussi i une telle rbflexion; nous en soumettons quelques klbments au lecteur. Si I’on veut bien imaginer la thkorie des ordres comme un fleuve, on lui trouve de nombreuses sources. La thkorie des ordres vient en effet de I’arithmktique ordinale de G. CANTOR mais aussi des travaux en analyse rbelle de G. CANTOR e t R. DEDEKIND, e t encore de la thborie des kquations et des groupes avec E. GALOIS e t C. JORDAN (groupes rksolubles, suites de JORDAN - HOLDER) e t encore de la thborie des anneaux (E. NOETHER), ceci pour ne citer que quelques exemples importants. De ces diverses origines naissent plusieurs courants tels la thkorie des treillis, les alg&bresde Boole, la topologie, etc..., drainant les eaux vers le fleuve en formation. Issus de I’btude de I’ensemble des sousgroupes d’un groupe, la thborie des treillis, courant fkcond en rbsultats aujourd’hui classiques (e.g. les thkordmes de reprksentation de BIRKHOFF et STONE), est ainsi, depuis plusieurs dkcennies, une des pn5occupations principales des mathbmaticiens travaillant sur I’ordre. De mSme, e t ne serait-ce que dds I’abord axiomatique (Axiomes de ZERMELO, KURATOWSKI, ZORN, ...), les courants impbtueux de la thborie des ensembles ne pouvaient kviter d’affluer et de se mdler aux questions ordinales. Venus de contrees voisines, les apports de la thkorie descriptive des ensembles, de la topologie, de la thkorie de la mesure, de la thkorie des moddles, de la logique e t de la combinatoire grossissent la th6orie des ordres de leur flux: ainsi la problbmatique de la theorie des moddles - de la catkgoricitk i la stabilitk - se trouve dbji inscrite dans I’ktude des chahes; mais inversement, les moddles construits par A. EHRENFEUCHT et A. MOSTOWSKI permettent de reprbsenter partie de la complexitk des chaines dans des structures arbitraires; mSme en arithmktique, la description des moddles de Peano fait place aux notions ordinales (extensions finales ou cofinales, sections initiales e t indicatrices de KIRBY - PARIS), qui parfois gouvement compldtement les moddles puisque, par exemple, la saturation des moddles de I’arithmktique du premier ordre se reduit i celle de leur structure d’ordre (on peut mSme penser que le contenu logique et combinatoire des rbsultats d’indkpendance de J. PARIS et L. HARRINGTON est de nature ordinale). Ayant travers6 le paradis infinitiste oc se trouvent i la fois des sommets inaccessibles (e.g. tous les probldmes se ramenant a la consistance) e t des vallkes fecondes (les grandes structures: treillis, groupes ordonnks, algdbres de Boole, algdbre noethbrienne, topologie

Preface

X

ensembliste ... et ce qui se rattache i I’alggbre universelle avec la dualitk, les questions de roprbsentation et d’engendrement), le cows du fleuve se porte maintenant vers le territoire de la finitude. La liaison est parfois facilitbe par le canal de la compacitk (le passage de la version finitiste du thkoreme - si fondamental - de DILWORTH i sa version infinitiste et le passage de la version infinitiste du theorgme de RAMSEY i sa version finitiste). Cependant, ces terres sont, souvent plus difficiles i cultiver puisque I’on y substitue i la souplesse des concepts de I’infini e t i des outils bien blaborks, le caract6re born6 d’objets pour lesquels peu de mbthodes d’btude existent encore (que I’on pense aux nombres de RAMSEY), ou la rigiditk des nombres (dont les conjectures les plus cbl6bres de I’arithmbtique donnent une idbe). C’est dans ce domdne que I’on trouve I’immense thborie des graphes finis si riche d’applications (problemes de cheminement, rbseaux de transport, ...). A ce jour, le monde extkrieur par le biais des besoins sociaux, ou des avancbes technologiques, ou des sciences nouvelles, inflbchit le cours d’une rivigre dont le mathbmaticien pouvait narvement penser qu’elle n’btait la que pour qu’il la dbcouvre. On lui demande d’ajouter a son rble d’explorateur celui d’ingbnieur. I1 doit aider crber des outils; ce faisant, il se trouve, ou se retrouve, dans des domaines mathbmatiques qu’il n’imaginait pas forcbment. C’est le cas dans les sciences sociales oh la formalisation des situations proposbes conduit i des rbsultats e t des problemes d’ordre: le paradoxe de CONDORCET est l’origine de la modblisation des prbfbrences dont un rbsultat spectaculaire est le thboreme de ARROW, hi-m6me point de depart de toute une thborie. La dbcouverte du continent informatique a m h e le mathbmaticien constater le manque d’effectivitk de rbsultats qui lui semblent naturels ou simples. Le courant logique de la recursivitk (K. GODEL, A. CHURCH) et de la complexitk (M. RABIN) rejoint les problemes nouveaux poses par I’informatique au niveau le plus immbdiat (problgmes de tri, par exemple), et se prolonge dans les nouveaux domaines de I’algorithmique, de la complexitk concrete (temps et espace de calcul), de la thborie des machines et de celle des langages.

a

a

Cette double activitk de bitisseur - au sens ou le fut J. Von NEUMANN - et de dbcouvreur - W. SIERPINSKI ne se disaibil pas lui-mbme “Explorateur de I’infini” ? - se poursuit. Mais il manque encoreun panorama complet de la thborie des ordres. Une premiere synthese (ORDERED SETS, D. REIDEL, 1982) a btk bditke par I. RIVAL. Nous prbsentons ici une illustration de la prbsence e t du rble de I’ordre en thborie des ensembles, en thborie des treillis, en topologie, en logique (thborie des moddles, thborie des relations, arithmktique), dans les groupes ordonnbs, en combinatoire, en informatique thborique e t dans les sciences sociales. Les Qditeurs ont voulu, qu’au fur e t a mesure de son parcours, le lecteur aille de situations infinitistes i des situations de plus en plus finitistes, e t qu’il passe de la mdme faqon du point de vue descriptif des ensembles ordonnbs leurs utilisations. Les numbros spbciaux de la revue DISCRETE MATHEMATICS contiendront les articles de recherche des confbrenciers rbunis 6, la TOURETTE faisant &at de rbsultats les plus rbcents sur les ensembles ordonnbs et prolongeant les syntheses de I’ouvrage prbsentk ici. Tous ces textes issus de la Confbrence sont bcrits soit en franqais, soit en anglais, chacun rbdigb dans une langue b t a n t prbcbdb d’une introduction ou d’un rbsumk exprimb dans l’autre.

a

Cette preface serait incomplhte si nous ne remerciions pas tous les arbitres qui ont bien voulu nous aider, pour I’excellence de leurs critiques e t de leurs suggestions. Parmi ces arbitres, se trouvent certains des auteurs de ce livre, qui se reconnaftront ici, e t nos collegues Qtrangers ou franqais: MM. A. ACHACHE ; R. ASSOUS ; J.P. AURAY ; B. BANASCHEWSKI ; H.J. BANDELT ; P. CEGIELSKI ; Ch. CHARRETTON ; M. CHEIN ; R. DEAN ; J.P. DOIGNON ; P. DWINGER ; M.R. GAREY ; S. GRIGORIEFF ; L. HARPER ; S.S. HOLLAND ; J. ISBELL ; M. JAMBU-GIRAUDET ; H. KOTLARSKI ; D. LASCAR ; A. LASCOUX ; L. LESIEUR ; R. MAYET ; E.W. MAYR ; C. St J. A. NASH-WILIAMS ; E. NELSON ;

Preface

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G. Mc NULTY; M. PARIGOT ; G. de B. ROBINSON ; L. SHEPP ; B. SIMONS ; A.R. STRALKA; L. SZABO ; T. TROTTER ; D.W. WEST ; S. WOLFENSTEIN ; M. YASUHARA . Grand merci enfin a J. G. ROSENSTEIN pour le motif figurant sur la couverture, reproduisant la tapisserie qu’il a brod6e t o u t en kcoutant les confbrences.

Maurice POUZET

et

Denis RICHARD

xiii

TABLE OF CONTENTS

Preface Preface Concerning the Conference on Ordered Sets and their Applications Retour a la Conference sur les Ensembles Ordonnes et leurs Applications * List of Participants (Liste de Participants) * Conference Programme (Programme Scientifique) * Short Communications (Programme des Communications)

SOMMAIRE

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ix xv xvi xvii xxiv xxvii

PART I - ORDER AND SOME SET-THEORETICAL INVARIANTS Partie I - Ordre et Invariants Ensemblistes Eric C. MILNER, Recent results on the cofinality of ordered sets . . . . . . . . . . . . . . . . . . 1 J. Donald MONK, Cardinal functions on boolean algebras ........................ 9 PART I1 - ORDER AND LATTICES Partie II - Ordre et Treillis H.A. PRIESTLEY, Ordered sets and duality for distributive lattices . . . . . . . . . . . . . . . 39 Michael MISLOVE, When are order scattered and topologically scattered thesame? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Dwight DUFFUS, Maurice POUZET, Representing ordered sets by chains . . . . . . . . . . 81 Gunter BRUNS, Orthomodular lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 George GRATZER, David KELLY, The construction of some free m-lattices onposets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 3 I.G. ROSENBERG, D. SCHWEIGERT, Compatible orderings and tolerances oflattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

PART I11 - ORDER AND ORDERED GROUPS Partie I l l - Ordre et Groupes Ordonnes W. Charles HOLLAND, Classification of lattice ordered groups . . . . . . . . . . . . . . . . . . 151 A.M.W. GLASS, The isomorphism problem and undecidable properties 157 for finitely presented lattice-ordered groups ...........................

Table of Contents

x iv

PART IV - ORDER AND TOPOLOGY Partie IV - Ordre et Topologie Egbert HARZHEIM, On topological properties of Cartesian products of linearly ordered continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

Karl Heinrich HOFMANN, Order aspects of the essential hull of a topological To -space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193

PART V - ORDER AND THEORY O F MODELS AND RELATIONS Partie V - Ordre, Thkorie des Modeles et Thkorie des Relations 207 Wilfrid HODGES, Models built on linear orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . Claude FRASNAY, Relations enchahables, rangements et pseudo-rangements . . . . . . 235 James H. SCHMERL, H, -categorical partially ordered set ...................... Denis RICHARD, The arithmetics as theories of two orders ....................

269 287

Roland FRAISSE, L’intervalle en thkorie des relations; ses gknkralisations; filtre intervallaire et cl6ture d’une relation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313

PART VI - ORDER AND CHAIN CONDITION ON CLASSES O F STRUCTURES Partie VI - Ordre et Conditions de Chaine sur des Classes d e Structures

P.D.SEYMOUR, Neil ROBERTSON, Some new results on the well-quasi orderingofgraphs

...............................................

343

PART VII - ORDER AND COMBINATORICS Partie VII - Ordre et Combinatoire Ivan RIVAL, Linear extensions of finite ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Michel HABIB, Comparability invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Andrd BOUCHET, Codages et dimensions de relations binaries . . . . . . . . . . . . . . . . . . .387 Jerrold R. GRIGGS, The Sperner property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Gkrard VIENNOT, Chain and antichain families, grids and young tableaux . . . . . . . . . 409 PART VIII - ORDER AND EFFECTIVENESS Partie V I I I - Ordre e t Effectivite Joseph G. ROSENSTEIN, Recursive linear orderings . . . . . . . . . . . . . . . . . . . . . . . . . . Ir6ne GUESSARIAN, Maurice NIVAT, About ordered sets in algebraic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.K. LENSTRA, A.H.G. RINNOOY KAN, Two open problems in precedence constrained scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

465 477 509

PART IX - ORDER AND SOCIAL SCIENCES Partie I X - Ordre et Sciences Sociales Jean-Pierre BARTHELEMY, Bruno LECLERC, Bernard MONJARDET, Ensembles ordonnks et taxonomie mathkmatique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

523

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CONCERNING THE CONFERENCE ON ORDERED SETS AND THEIR APPLICATIONS This Conference, held under the auspices of the Centre National de la Recherche Scientifique (C.N.R.S.) and the Mathematical Society of France (S.M.F.), was organized by the Ordinal Algebra group a t the UniversiG Claude Bernard (LYON 1) and the group of French mathematicians comprising R.C.P. 6 9 8 of the C.N.R.S. in cooperation with I’Ecole Nationale Supkrieure des T616communications*. The Conference proceedings were dedicated to Professor E. COROMINAS founder of the Ordinal Algebra group - to mark the occasion of his election to Professor Emeritus.

-

The Conference brought together 125 participants from more than fifteen different countries. The meeting began with a lecture from Professor I. RIVAL - organizer of the 1981 BANFF meeting - and concluded with a memorable talk from Professor P. ERDOS. More than 9 0 papers were given at the conference including 34 invited addresses, 15 presentations a t the special sessions, 20 contributed papers and 20 problems given a t the Problem Sessions (Cf. The Scientific Programme). We express our thanks to the resident Dominicans for their warm welcome to the large group of mathematicians who for seven days invaded their serene setting a t the Chiteau de la Tourette in the hills above I’ARBRESLE near LYON. During the rare non-mathematical moments we were able to see GANCE’s magnificent production of Napolkon, and also to enjoy a fine concert by R. JAMISONWALDNER (cello) and G. BRUNS (piano). The Scientific programme and organization of the meeting was coordinated by M. POUZET with collaboration from R. BONNET and other members of the group, A. ACHACHE, R. ASSOUS, Ch. CHARRETTON, D. RICHARD, and the research students M. BEKKALI, M. BELHASSAN, D. MISANE, N. ZAGUIA, E. JAWHARI. Our colleagues from the logic group also gave much time and assistance - especially Marianne DELORME, whose efficiency and courtesy was widely appreciated. We d o not forget either the generous help from two other POUZET generations - Emile and Marc. Robert BONNET and Maurice POUZET

* We thank also the non-mathematical institutions that contributed to the success of the Conference especially the City of LYON and the Banque Nationale de Paris (B.N.P.).

xvi

RETOUR A LA CONFERENCE SUR LES ENSEMBLES ORDONNES ET LEURS APPLICATIONS Cette Confbrence, placbe sous les auspices du Centre National de la Recherche Scientifique (C.N.R.S.) et de la Socibtb Mathbmatique de France (S.M.F.), a Qtk organisbe par le laboratoire d’Alg8bre Ordinale de I’Universitk Claude Bernard (LYON 1)et le groupe de mathbmaticiens franqais constituant la R.C.P. 698 du C.N.R.S. avec le concours de 1’Ecole Nationale Supbrieure des Tblbcommunications*. Elle btait dbdibe au Professeur E. COROMINAS d’AlgGbre Ordinale - I’occasion de son Embritat.

- fondateur du laboratoire

Cette Confbrence rbunissait 125 participants d’une quinzaine de nationalitks. Ouverte par le Professeur I. RIVAL - organisateur du Symposium de BANFF en 1981 - elle s’est conclue par I’exposb du Professeur P. ERDOS, apr8s avoir donnb lieu a plus de 90 contributions comprenant 34 expods en sbance plbni6re, 15 exp o d s dans des sessions thbmatiques, plus de 20 contributions aux sessions de problGmes et 20 communications (Cf. Programme Scientifique). Dans le cadre du Chiteau de la Tourette, sur les hauteurs de la petite ville de I’ARBRESLE pr6s de LYON, les Dominicains ont accueilli chaleureusement les mathbmaticiens, sept jours durant, et nous les en remercions. Pendant les rares moments non mathbmatiques, certains ont eu I’occasion de dbcouvrir le Napolbon d’Abel GANCE e t tous se souviendront longtemps du concert donnb par R. JAMISON WALDNER (violoncelle) accompagnb par G. BRUNS (piano). L’organisation scientifique et rnat&rielle de la Confbrence btait coordonnbe par M. POUZET avec la participation de R. BONNET et de membres du laboratoire: A. ACHACHE, R. ASSOUS, Ch. CHARRETTON, D. RICHARD, des Qtudiants de recherche: M. BEKKALI, M. BELHASSAN, D. MISANE, N. ZAGUIA, E. JAWHARI. Nos coll6gues logiciens nous ont apportk leur aide - particuli8rement Marianne DELORME, dont tous ont apprbcib I’efficacitk e t la courtoisie mais aussi deux autres gknerations de POUZET puisqu’il faut citer a la fois Emile et Marc. Robert BONNET et Maurice POUZET

*

Nous remercions les Institutions non mathematiques qui ont apporte leur concours a cette Conference, en particulier la ville de LYON et la Banque Nationale de Paris (B.N.P.).

xvii

LISTE DES PARTICIPANTS

LIST OF PARTICIPANTS

A c h i l l e ACHACHE

DBpt. d e Math. - U n i v e r s i t C Claude B e r n a r d - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

Roland ASSOUS

Dept. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

J e a n - P i e r r e BARTHELEMY

Dept. d ' I n f o r m a t i q u e - E.N.S. Teli.communic a t i o n s - 46, r u e B a r r a u l t 75634 PARIS CEDEX 13 - F r a n c e

H.

F.B. Math. T e c h n i s c h e Hoschule D a r m s t a t D 6100 DARMSTAT - R.F.A.

BAUER

Mary K a t h e r i n e BENNETT

Dept. o f Math. U n i v e r s i t y of M a s s a c h u s s e t t s AMHERST - MA 01003 - U.S.A.

Mohamed BEKKALI

Dept. d e Math. - U n i v e r s i t e Claude B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Moulay BEL HASSAN

DBpt. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Claude BENZAKEN

I.M.A.G.

B.P. N i c o l e BLANCHARD

R o b e r t BONNET

53 X

-

U n i v e r s i t e de Grenoble - 38041 GRENOBLE CEDEX

-

France

Dept. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e DBpt. d e Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e 35

KASSEL

-

R.F.A.

B. BOSBACH

Vogelviesen,

Andre BOUCHET

1 3 , r u e Taragone -

O d i l e BOTTA

DQpt. d e Math. - U n i v e r s i t E ! Claude B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

72000

LE MANS

-

France

xviii

Gunter BRUNS

Jean-Claude CARREGA

List of participants Dept. of Mathematical S c i e n c e s Mc Master U n i v e r s i t y - HAMILTON, O n t a r i o L8S4K1 - Canada D e p t . d e Math. - U n i v e r s i t 6 Claude B e r n a r d - 4 3 , B l d du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

-

LYON 1

J o s e p h CHACRON

52, r u e C o z e t t e -

Maurice CHACRON

Dept. of Math. - C a r l e t o n U n i v e r s i t y OTTAWA, O n t a r i o K 1 5 5 ~ 6- Canada

C h r i s t i n e CHARRETTON

80000

AMIENS

-

France

DBpt. de Math. - U n i v e r s i t 6 Claude Bernard - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

Georges CHEVALIER

D6pt. de Math. - U n i v e r s i t g Claude B e r n a r d - 43, Bld d u 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

O l i v i e r COGIS

Dept. de Math e t I n f o r m a t i q u e - U n i v e r s i t 6 du Lanquedoc, P l a c e Eugene H a t a i l l o n 34060 MONTPELLIER CEDEX - F r a n c e

J u l i e n CONSTANTIN

688, r u e Vimy N . SHERBROOKE Quebec JlJ3N6 - Canada

Maria CONTESSA

Dept. de Math. - Italie

-

U n i v e r s i t e de Rome

ROME

E r n e s t COROMINAS

J e a n Louis COULON

Dept. de Math. - U n i v e r s i t e C l a u d e Bernard LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e Dept. de Math. - U n i v e r s i t 6 Claude B e r n a r d - 4 3 , B l d du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

R i c h a r d DEAN

Marianne DELORME

Dept. of Math. - CALTECH PASADENA, CA 91103 - U.S.A. DBpt. de Math. - U n i v e r s i t k C l a u d e B e r n a r d - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

LYON 1

-

Walter DEUBER

F a k u l t d t f u r Math. - U n i v e r s i t a t B i e l e f e l d D 4800 BIELEFELD - R.F.A.

Markus DICHTL

A b t e i l u n g Math. - 3 U n i v e r s i t a t Ulm 7900 ULM - R.F.A.

J e a n - P i e r r e DION

DPpt. d e Math. - U . Q . A . M . C a s e P o s t a l e 8888, SUC “ A “ - MONTREAL - P,Q. H3C3P8 Canada

Gernot DORN

-

F.B.

Math. T e c h n i s c h e Hochschule Darmstadt - R.F.A.

D 6100 DARMSTADT

Manfred DROSTE

Im Brinkmannsfeld 6 4 , D 4250 BOTTROP - R.F.A.

List ofparticipants

xix

P a u l DUBREIL

RPsidence F r k m i e t , Les Reaux 91840 SOISY-SUR-ECOLE - F r a n c e

Andre DUCAMP

14 A Dreve d e s E q u i p a g e s 1170 BRUXELLES - B e l g i q u e

Dwight DUFFUS

Dept. o f Math. and Computer S c i e n c e EMORY Univ. ATLANTA G e o r g i a 30322 - U . S . A .

I v o DUNTSCH

Inst.

f u r Math I1 Koningin-Luise s t r . 2426 33 - R.F.A.

-

1000 B E R L I N C h r i s t o p h e r EDWARDS

Dept. of E n g i n e e r i n g P r o d u c t i o n . U n i v e r s i t y o f Birmingham, PO BOX BIRMINGHAM B15 2TT Grande B r e t a g n e

P a u l ERDOS

Hungarian Academy of S c i e n c e . 1053 BUDAPEST - Hongrie

Marcel ERNE

I n s t . f u r Math. U n i v e r s i t a t Hannover W e l f e n g a r t e n l D 3000 HANNOVER - R.F.A.

Michel EYTAN

Dept. de Math. 12, r u e C u j a s -

Claude FLAMENT

L e s Blaques

Marc FORT

D6pt. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 1 1 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Roland FRAISSE

C a d e n e l l e - Cheverny, 122, r u e Cdt R o l l a n d 13008 MARSEILLE - F r a n c e

Claude FRASNAY

19, r u e d e l a p o s t e

J e r r o l d GRIGGS

Dept. of Mathematics and S t a t i s t i c s U n i v e r s i t y of South C a r o l i n a COLUMBIA SC 29208 - U . S . A .

-

Arnold G R U D I N

Dept. o f Math. - Denison U n i v e r s i t y GRANVILLE, Ohio 43020 - U.S.A.

-

I r h e GUESSARIAN

DBpt. de Math. - U n i v e r s i t e d e PARIS V I I T 45-55 ; 2, P l a c e J u s s i e u 75251 PARIS CEDEX 05 France

-

-

Realtanoda u .

U n i v e r s i t e Rene D e s c a r t e s 75005 PARIS - F r a n c e

04110 CERESTE -

31130

-

13-15

-

France

BALMA

- France

-

-

Marcel GUILLAUME

Math6matiques P u r e s - U n i v e r s i t e d e CLERMONT 2 Complexe S c i e n t i f i q u e d e s Cezeaux - B.P. 45 63170 AUBIERE - F r a n c e

C y p r i e n GNANVO

Departement de Math. - U n i v e r s i t e de COTONOU COTONOU - Rep. P o p u l a i r e du Benin

Michel HABIB

E c o l e d e s Mines de S a i n t E t i e n n e 158, Cours F a u r i e l 42023 SAINT ETIENNE CEDEX - F r a n c e

-

-

xx

List of participants

Geza HAHN

Dept. of Math. Quebec Canada

Mahmud HAIFAWI

Dept. of Math. ANKARA sity

Andras HAJNAL

Math. I n s t . Univ. 1088 Muzeum K r t . 6-8

Mc G i l l ,

-

-

MONTREAL

H3A2K6

- Middle E a s t T e c h n i c a l Univer- Turquie BUDAPEST

-

Honqrie

Rudolf H A L I N

Mathematisches Seminar UniversitatHamburg B u n d e s s t r . 55 D 2000 HAMBURG - R.F.A.

Georges HANSOUL

517 G r a n d ' r o u t e

E g b e r t HARZHEIM

P a l l e n b e r g s t r . 23,

R i c h a r d HENDERIKS

Econometrisch I n s t i t u t . Erasmus U n i v e r s i t e i t PO BOX 1738 - 3000 DR ROTTERDAM - Pays-Bas

W i l f r i d HODGES

Bedford C o l l e g e R e g e n t ' s Park. LONDON NWI 4NS- Grande B r e t a g n e

K a r l HOFMANN

B 4110

FLEMALLE

5 KOLN 60 -

-

Belgique

R.F.A.

Dept. of Math. Tulane U n i v e r s i t y La. 70118 - U.S.A.

NEW ORLEANS

C h a r l e s HOLLAND

Dept. of Math.

-

BOWLING GREEN Ohio

Bowling Green S t a t e U n i v e r s i t y 43403 - U.S.A.

L u i s a ITURRIOZ

DBpt. de Math. - U n i v e r s i t i . Claude B e r n a r d LYON 1 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Michele JAMBU-GIRAUDET

32, r u e de l a Reunion

Robert JAMISON-WALDNER

Math S c i e n c e s Dept. - Clemson U n i v e r s i t y CLEMSON SC 29631 - U.S.A.

E l Mostapha JAWHARI

Dept. of Math. U n i v e r s i t e Claude B e r n a r d LYON 1 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Gudrun KALMBACH

A b t . Math. 3 , OE, Univ. U l m D 7900 ULM - R.F.A.

Klaus KEIMEL

F a c h b e r e i c h Mathematik, T e c h n i s c h e Hochschule R.F.A. D a r m s t a d t , 6100 DARMSTADT

David KELLY

-

-

-

75020 PARIS

-

-

France

-

-

-

Dept. of Math.

- U n i v e r s i t y of Manitoba R3T 2 N 2 Canada

W I N N I P E G , Manitoba

Sabine KOPPELBERG

Mathematisches I n s t i t u t K o n i g i n - L u i s e - S t r . 24/26 1000

Henryk KOTLARSKI

U n i w e r s y t e t u Magellana 4m 19 02 777 VARSOVIE - Pologne

Germain KREWERAS

4 0 , r u e LacGpede

-

75005

BERLIN 33

PARIS

- France

- RFA

List ofparticipants

xxi

M a r t i n KRUSKAL

Program i n A p p l i e d Mathematics. P r i n c e t o n U n i v e r s i t - 7 - PRINCETON, N . J . 08544 - U.S.A.

Duro KUREPA

Zagrebacka 7 , 11000

Jean-Marie

LABORDE

BELGRADE

.

- Yougoslavie

I.M.A.G. Grenoble - B.P. 53 X 38041 GRENOBLE CEDEX - F r a n c e

R i c h a r d LAVER

Dept. o f Math. U n i v e r s i t y of C o l o r a d o BOULDER, C o l o r a d o -U.S.A.

Bruno LECLERC

C e n t r e de Mathematiques S o c i a l e s 5 4 , Bld R a s p a i l - 75270 PARIS CEDEX 06 France

-

- Martinsstr. - R.F.A.

K l a u s s LEEB

Informatik I n s t . 3 8520 ERLANGEN

P i e r r e LEFEBVRE

Dept. de Math. - U n i v e r s i t e Claude B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Domenico L E N Z I

Departement de Math. Italie

L&once LESIEUR

5 , A l l e e d e s Sophoras, 92330 SCEAUX - F r a n c e

G e r a r d LOPEZ

Departement de Math. Place Victor Hugo -

Henrik MARTENS

I n s t i t u t t f o r Matematikk N-7034 NTH TRONDHEIM -

-

-

U n i v e r s i t e d e LECCE,

-

U n i v e r s i t B d e Provence 13001 MARSEILLE France

-

-

Norvege

-

M a r i e - C a t h e r i n e MAURER-VILAREM

306, A l l e e du Dragon

A r l e t t e MAYET

Dkpt. d e Math. - U n i v e r s i t k Claude B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Renk MAYET

Dept. de Math. - U n i v e r s i t e Claude B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Michael MISLOVE

Dkpartement de Math. - T u l a n e U n i v e r s i t y NEW-ORLEANS - L a . 70118 - U.S.A.

E r i c MILNER

Dept. o f Math. a n d S t a t i s t i c s . U n i v e r s i t y of CALGARY - CALGARY - T2N1N4 - Canada

D r i s s MISANE

DBpt. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX France

91000

-

B e r n a r d MONJARDET

Donald MONK

EVRY

France

-

C e n t r e d e Mathkmatiques S o c i a l e s 54, Bld R a s p a i l - 75270 PARIS CEDEX 06

-

U n i v e r s i t y of C o l o r a d o , Dept. of Math. BOULDER, C o l o r a d o 80309 - U.S.A.

-

- France

xxii Noel MURPHY

List ofparticipants Dept. of Math. Trent University PETERBOROUGH - Ontario K9J7BB -

Canada

Evelyn NELSON

Dept. of Math. Science. Mc Master University HAMILTON - Ontario - Canada

Bernhard NEUMA"

Dept. of Math. Inst. of Adv. Stud. Australian University POB 4 CAMBERRA ACT 2600 - Australie

Peter NEVERMANN

AG1, FB4, Technische Hochschule Darmstadt 6100 DARMSTADT - R.F.A.

Serge OVCHINNIKOV

Dept. of Math. San Francisco State University 1600 HollowayAve SANFRANCISCOCA 94132 - U.S.A.

Michel PARIGOT

UER Math. T 45.55 - Univ. PARIS VII 75221 PARIS CEDEX 05 - France

E. PICHAT

C.N.A.M. - 292, rue Saint Martin 75141 PARIS CEDEX 03 - France

Jean-Marie PLA

20, rue Elis6e Reclus France

Werner PCGUNTZE

Fachbereich Mathematik Technische Hochschule 6100 DARMSTADT - R.F.A.

Norbert POLAT

30, rue Baldeyrou -

Bruno POIZAT

Departement de Math. - Universite Pierre et Marie Curie - PARIS VI - PARIS - France

Maurice POUZET

DBpt. de Math. - UniversitB Claude Bernard Lyon 1 - 43, Bld du 1 1 novembre 1918 69622 VILLEURBANNE CEDEX - France

H.A. PRIESTLEY

St. Anne's College Grande Bretagne

Karel PRIKRY

Dept. of Math. - University of Minnesota MINNEAPOLIS Minn. 55455 - U.S.A.

Robert QUACKENBUSH

Dept. of Math. - Universitg of Manitoba WINNIPEG, Manitoba R3T2N2 - Canada

Alain QUILLOT

Dept. of Math. - University South Carolina et COLUMBIA - U.S.A. 84, rue de Nohannent 62100 CLERMONT-FERRAND - France

Norman REILLY

Dept. of Math. Simon Fraser University, BURNABY British Columbia V5A 1S6 - Canada

Serge RIBEYRE

Dept. de Math. - Universite Claude Bernard LYON 1 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - France

-

-

42000 SAINT ETIENNE

42800 RIVE DE GIER - France

OXFORD

OX2

6H5 -

-

List of participants Denis RICHARD

A.H.G. RINNOOY KAN

xxiii

D6pt. de Math. - Universite Claude Bernard LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - France

-

Econometric Institute, Erasmus University ROTTERDAM - Hollande

PO BOX 3000 DR

Ivan RIVAL

Dept. of Mathematics and Statistics. University of Calgary - CALGARY, Alberta T2NlN4 - Canada

Neil ROBERTSON

Dept. of Math. Ohio State University COLUMBUS, Ohio 43210 - U.S.A.

IVO G. ROSENBERG

C. Rech. Math. Appl. Universite de Montreal CP 6128 Succ "A", MONTREAL QUE. H3C 3J7 - Canada

Joseph ROSENSTEIN

Dept. of Mathematics Rutgers University NEW BRUNSWICK, New Jersey, 08903 - U.S.A.

Matatyahu RUBIN

Dept. of Math. Ben Gurion University BEER SHEVA - Israel

Bill SANDS

Dept. of Math. - University of Calgary CALGARY, Alberta T2NlN4 - Canada

James SCHMERL

Dept. of Math. University of Connecticut STORRS CT 06268 - U.S.A.

Jochanan SCHONHEIM

School of Mathematics, Tel-Aviv University TEL-AVIV - Israel

Dietmar SCHWEIGERT

FB Mathematik. University of Kaiserslautern D 675 KAISERSLAUTERN - R.F.A.

Tom TROTTER

Dept. of Mathematics and Statistics, University of South Carolina, COLUMBIA SC 29208 - U.S.A.

William TUNNICLIFFE

Faculty of Mathematics - The Open University U.K. Walton Hall - MILTONKEYNES, MK76AA

Jules VARLET

Institut de Mathematiques - Universite de Liege Avenue des Tilleuls, 15 - B400 LIEGE - Belgique

Gerard VIENNOT

Departement de MathCmaticpes - Universite de BORDEAUX - 33400 TALENCE - France

Kurt WOLFSDORF

Wilhelmstr. 2, D-1 BERLIN 61

Robert WOODROW

Dept. of Mathematics and Statistics University of Calgary CALGARY, Alberta T2N1N4 - Canada

Nejib ZAGUIA

Dept. de Math. - Universite Claude Bernard LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - France

John ZELEZNIKOW

Dept. of Math. - Michigan State University EAST LANSING, Michigan 48824 - U.S.A.

-

R.F.A.

XXlV

CONFERENCE PROGRAMME

PROGRAMME SCIENTIFIQUE

_______________-__-_

Lundi 5 j u i l l e t 1982 Matinee:

..........

President de seance

E . COROMINAS

I . RIVAL ( C a l g a r y - Grenoble) , L i n e a r e x t e n s i o n s o f f i n i t e o r d e r e d sets. D. MONK ( B o u l d e r ) , Some c a r d i n a l f u n c t i o n s on Boolean a l g e b r a s . C . BENZAKEN ( G r e n o b l e ) , Decomposition Matroi'dale d'un s y s t e m e d'independance

......... R .

Aprks-midi: P r e s i d e n t d e s e a n c e

et application.

DEAN

H. PRIESTLEY ( O x f o r d ) , Ordered S e t s and d u a l i t y f o r d i s t r i b u t i v e l a t t i c e s . R. FRAISSE ( M a r s e i l l e ) , Abritement e t r a n g de non a b r i t e m e n t e n t r e c h a f n e s e t r e l a t i o n s . C . HOLLAND (Bowling G r e e n ) , C l a s s i f i c a t i o n o f l a t t i c e ordered groups.

S e s s i o n d e probl8mes:

.........

President de seance

M.

POUZET

S o i r e e : S e s s i o n s p e c i a l e q r o u p e s ordonnes: C. HOLLAND P r e s i d e n t de seance P a r t i c i p a n t s : M . JAMBU-GIRAUDET, L . LESIEUR, N . REILLY.

............

__ __

Mardi 6-- j-u _i l -l e -t -1982 --Matinee:

P r e s i d e n t d e seance

..........

B. N E U M A "

HODGES ( L o n d r e s ) , Models b u i l t on l i n e a r o r d e r i n q s . W.

E. HARZHEIM ( D u s s e l d o r f ) , On t o p o l o g i c a l p r o p e r t i e s of C a r t e s i a n p r o d u c t s o f l i n e a r l y o r d e r e d continua. C . FRASNAY ( T o u l o u s e ) , Chafnes G-compatibles, G-ranqement e t n o t i o n d e c l a s s e i n d i c a t i v e p o u r un sous-oroupe G du groupe s y m 6 t r i q u e S

m

Apr6s-midi: P r e s i d e n t d e seance J . ROSENSTEIN ( R u t g e r s )

.

..........

P. DUBREIL

,

Recursive l i n e a r orderings. R . QUACKENBUSH (Winnipeg) ,

Non modular v a r i e t i e s o f semi-modular l a t t i c e s . E . MILNER ( C a l g a r y ) ,

C o fi n a l it y

.

DUFFUS ( A t l a n t a ) , Retracts o f f i n i t e d i m e n s i o n a l l a t t i c e s . D.

xxv

Conference programme Session speciale conditions d e chaines President de seance Participants: N. ROBERTSON, M. POUZET.

....................

N.

ROBERTSON

Soiree: S e s s i o n s p e c i a l e th4orie d e s t r e i l l i s

President de seance

...........

L. LESIEUR

P a r t i c i p a n t s : H. BAUER, G . HANSOUL, K . H O F M A ” , M . M I S L O V E , J. VARLET.

__ _

-

K.

KEIMEL,

Mercredi l e t --1 -9 -8 -2 - -- - - -- 7 - j--u -i -l --

...........

Matinee: P r e s i d e n t d e s e a n c e

KELLY

D.

K.

PRIKRY

,

HAJNAL ( B u d a p e s t ) P a r t i t i on s r e 1 a t i ons . A.

(Winnipeg),

Some f r e e i n f i n i t a r y l a t t i c e s . I . GUESSARIAN

(Paris),

A p p l i c a t i o n d e la theorie d e s e n s e m b l e s o r d o n n e s d l a s e m a n t i q u e d e s schsmas d e p r o y r a m m a t i o n .

........

President de seance

AprGs-midi:

I.G.

ROSENBERG

D. KUREPA (Belgrade) On s e v e r a l k i n d of r a m i f i c a t i o n s . M.

RUBIN

(Beer Sheva)

C o n s i s t e n c y r e s u l t s on r e a l order t y p e s . J. SCHMERL ( S t o r r s )

C o m p l e x i t y and G.

,

N - c a t e g o r i c i t y o f p a r t i a l l y o r d e r e d sets.

0 BRIJNS ( H a m i l t o n )

Orthomodular l a t t i c e s . Session s p e c i a l e treillis orthomodulaires: P r e s i d e n t d e s e a n c e .................... G . BRUNS P a r t i c i p a n t s : L . I T U R R I O Z , G . KALMBACH, R . MAYET.

_______-_-__________

J e u d i 8 j u i l l e t 1982

Matinee: P r e s i d e n t d e s e a n c e J.P.

BARTHELEMY

...........

G.

KREWERAS

(Paris),

Ensembles o r d o n n e e s et t a x o n o m i e m a t h e m a t i q u e . I . ROSENBERG

(Montreal),

T h e l a t t i c e o f clones a n d t h e a s s o c i a t e d r e l a t i o n s . G.

VIENNOT

(Bordeaux),

C h a i n and a n t i c h a i n f a m i l i e s , g r i d s and Young t a b l e a u x .

- -__ __ __ __ _ -_

V e n d r e d i 9- j u i l _l e -t -1982 - --

Matinee:

P r e s i d e n t d e seance A.H.G.

RINNOOY K.’,J

.......... (Rotterdam)

P a r t i a l o r d e r e d sets i n s c h e d u l i n g t h e o r y .

B.

SANDS

xxvi

Conference programme J. GRIGGS (Columbia), The Sperner property.

M. HABIB (Saint Etienne), Une caracterisation p a r sous-qraphes exclus des qraphes sans c i r c u i t s ayant deux sauts. Aprgs-midi: President de shance

........

R . HALIN

R . JAMISON-WALDNER (Clemson), Order and convexity.

T. TROTTER (Columbia) The dimension o f the Cartesian product of p a r t i a l order.

Session speciale Combinatoire: President de seance T. TROTTER Participants: J.P. DION, P. ERDOS, J. GRIGGS, A. HAJNAL, G. KERWERAS, J.M. LABORDE, A. QUILLIOT, I. RIVAL, B. SANDS, J. SCHMERL, J. SCHONHEIM.

....................

~

~

~

~

Matin6e: President de seance

~

~

............ D. KELLY

-

A. BOUCHET (Le Mans) , Codaqes e t dimensions de r e l a t i o n s binaires. 0 . COGIS (Montpellier),

S u r l a dimension des qraphes e t des ordres.

A. DUCAMF' (Bruxelles), La bidimension e t ses applications.

D. SCHWEIGERT (Kaiserslautern), Clones of monotones functions. Apres-midi: President de seance

......... I. RIVAL

K. PRIKRY (Minneapolis) The Ramsey property and mesurable selections. E. COROMINAS (Lyon) S u r l e meilleur ordre de Nash-Williams.

Session de problemes: President de seance

.....................

D. DUFFUS

P. ERDOS (Budapest) Problems concerning the chromatic number o f f i n i t e and i n f i n i t e graphs.

*** Diner dans les jardins du Chlteau.

*

~

xxvii

SHORT COMMUNICATIONS

PROGRAMME DES COMMUNICATIONS

Yendredi-Xiuillet PrGsident: E . NELSON I. DUNTSH

PrBsident: R.W. B . MONJARDET

Axiomatique arrowienne de l a mediane dans l e s d e m i - t r e i l l i s medians.

Projective Stone A1 gebras.

S. KOPPELBERG

M.

Groups o f permutations with few fixed points. E.

PICHAT

ZAGUIA

V a r i e t y i n v a r i a n t s f o r ordered s e t s

M.

BEKKALI

S u r l e theorbme de Neumer-Fodor. une version topologique.

Deviation e t dimension de K r u l l des ensembles ordonnes. E . NELSON

EYTAN A doctrinal model for k i n s h i p .

P . NEVERMA”

Modele de donnees relationnel e t d’accbs e t ensembles ordonnes. N.

QUACKENBUSH

M.

HAIFAWI

Non archimedian a n a l y s i s and ordered sets.

Chain continuous algebras.

samedi-l2_iuillet President: R . WOODROW

President: R. HALIN C.

FRASNAY

D.

C . FLAMENT

H . KOTLARSKI

On elementary c u t s i n Models o f Ar i t h m e t ic

Comparability graphs with constraint . R.

DEAN A construction f o r large f a m i l i e s

RICHARD

Ordre nature1 e t ordre de d i v i s i b i l i t 6 dans l’arithmetigue de Peano.

Permutations b i v a r i a n t e s e t i n f i n i t ude.

.

R . LAVER

Theorems on i n f i n i t e t r e e s .

of k-element s e t s having t h e Erdos i n t e r s e c t i o n property. J. CONSTANTIN

C . CHARRETTON

Comparaison des structures engendrees p a r des chafnes.

Ordonnes escamotables e t p o i n t s fixes. M.C.

MAURER

B. P O I Z A T

S u r d e s travaux de M . Krasner.

Familles k - i n t e r s e c t a n t e s .

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This Page Intentionally Left Blank

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PART I ORDER AND SOME SET-THEORETICAL INVARIANTS

-

PARTIEI

-

ORDRE ET INVARIANTS ENSEMBLISTES

Eric C. MILNER Recent results on the cofinality of ordered sets. . . . . . . . . . , . . . . . . . P . 1 J. Donald MONK

Cardinal functions on boolean algebras. . . . . . . . . . . . . . . . . . . . . . . . P . 9

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 (1984) 1-8

1

0 Elsevier Science Publishers B.V. (North-Holland)

RECENT KESULTS ON THE COFINALITY OF ORDERED SETS ERIC C. MILNEK Department of Mathematics and S t a t i s t i c s The U n i v e r s i t y of Calgary Calgary, Alberta Canada

Dedicated t o E . COROMINAS Une p a r t i e A d'un ensemble p a r t i e l l e m e n t ordonn6 P = (P,<) e s t c o f i n a l e dans P si t o u t 6Gment 2 C 1' e s t major6 p a r au moins un 616ment t o F A . La c o t i v i a l i f 6 de P, n o t d e ef(p), e s t l e p l u s p e t i t c a r d i n a l K t e l que P c o n t i e n n e une p a r t i e c o f i n a l e de c a r d i n a l i t 6 K . Nous d i s o n s que P e s t i-o~inaZernc-~,t ,7@tcrmivli:s ' i l e x i s t e un o r d r e n a r t i e l , que nous n o t e r o n s R(P), t e l que t o u t e p a r t i e c o f i n a l e de P c o n t i e n n e une p a r t i e c o f i n a l e isomorphe pour l ' o r d r e . P a r exemple, une cha?ne e s t c o f i n a l e m e n t d6terminde p u i s q u ' e l l e c o n t i e n t une p a r t i e c o f i n a l e b i e n ordonne'e de c d r d i n a l i t 6 minimum. PI. Pouzet a montr6 que si P n e c o n t i e n t pas d ' a n t i c h a r n e i n f i n i e , a l o r s P e s t c o f i n a l e m e n t d6termind. En f a i t , dans c e c a s E(P) e s t une somme d i r e c t e d'ensembles dont chacun est isomorphe B un p r o d u i t d i r e c t de cardinaux r 6 g u l i e r s . I1 s ' e n s u j t que c f ( P ) e s t s o i t un c a r d i n a l f i n i , s o i t un c a r d i n a l r e g u l i e r i n f i n i . En d ' a u t r e s t e r m e s , si cf(P) e s t un c a r d i n a l s i n g u l i e r , a l o r s P d o i t c o n t e n i r d e s a n t i c h a f n e s i n f i n i e s . Nous p r e s e n t o n s dans c e t a r t i c l e d i f f g r e n t e s g 6 n 6 r a l i s a t i o n s du th6orbme de M. P o u z e t , o b t e n u e s p a r p l u s i e u r s a u t e u r s . IJn ensemble p a r t i e l l e m e n t ordonn6 n ' e s t g6n6ralement pas c o f i n a l e m e n t d6termin6 e t i l y a d e s exemples d'ensembles f i l t r a n t s P c o n t e n a n t un "grand" nombre de p a r t i e s c o f i n a l e s de c a r d i n a l cf(P) qui s o n t c o f i n a l e m e n t t r & s d i s s e m b l a b l e s en c e s e n s qu'aucun o r d r e p a r t i e l de c a r d i n a l cf(P) n e peut se p l o n g e r (isomorphiquement) dans deux d ' e n t r e eux. Wang e t L i o n t m o n t r s , qu'en d e p i t du f d i t que l e s p a r t i e s c o f i n a l e s peuvent g t r e t r h s d i s s e m b l a b l e s , c e s m h e s p a r t i e s c o f i n a l e s d'ensembles f i l t r a n t s o n t p o u r t a n t une s t r u c t u r e g d n e r a l e commune. C e l l e - c i p e u t 2 t r e d d c r i t e e n u t i l i s a n t un c e r t a i n p r 6 o r d r e s u r l e s p a r t i e s , d u t y p e s u i v a n t : A 5 R l o r s q u e t o u t 616ment d e A e s t major6 p a r au moins un Cl6ment d e B. D e c e nouveau p o i n t de vue - a o i n s r e s t r i c t i f - t o u t ensemble f i l t r a n t s e t r o u v e c o f i n a l e m e n t d6termjn6.

1.

PRELIMINARY RESULTS:

f f ( P , - ) i s a p a r t i a l l y o r d e r e d s e t , t h e n f o r any s u b s e t A C P w e d e n o t e by = {z C P : 3: a , f o r some a C A } t h e i n i t i a l segment g e n e r a t e d by A. A i s c o f i n a l i n P i f A = P and t h e c o f i n a l i t y of P i s

A

cf(P)

= fmjnl.41

: A = P}

.

W e s a y t h a t t h e p a r t i a l l y o r d e r e d s e t ( P , 5 ) i s cofinai!Zy determined i f t h e r e i s a p a r t i a l o r d e r A such t h a t e v e r y c o f i n a l s u b s e t A 5 P c o n t a f n s an order-isomorphic copy of A. Two e l e m e n t s a,b C P a r e compatible i f a < c and b < c f o r some c C P. ( P , 9 i s directed i f e v e r y p a j r of elements ? s c o m p a t i b l e . The s t a r t i n g p o i n t f o r many of t h e r e s u l t s d e s c r i b e d i n c h i s s u r v e y i s t h e f o l l o w i n g theorem of Pouzet [ l l ] :

E.C. Milner

2

(1.1) If thc iiirrcted p a r > t i a Z l yo r c l ~ ~ cset ~ 1 (p,<) i i ~ s170 irj;fiiziic nntii>hoiri, ilwi2 ever!{ cofinal subset of P contains (I cofirin7 slit st71 1 J h i c i i i:: the d i ? w c a f pro&+i @,f a r i n i t c nwnher of c h i n s . Theorem (1.1) i s a g e n e r a l i z a t i o n of H a u s d o r f f ' s c l a s s i c a l o b s e r v a t i o n t h a t a c h a i n has a minimal well-ordered c o f i n a l s u b s e t . Thus, we s e e from (1.1) t h a t i f (P,<)i s d i r e c t e d and h a s n o i n f i n i t e a n t i - c h a i n , then t h e r e is a f i n i t e number of regular cardinals K~,K),...,V~ such t h a t every c o f i n a l s u b s e t of P c o n t a i n s a c o f i n a l isomorphic copy of t h e d i r e c t product k 1 C3 K - bd C3 i T l .

...

A s u b s e t A i; P i s c a l l e d a

strong antiehairz i f no two elements of A a r e compatihle.

Erd8s and T a r s k i [4] observed t h a t

( 1 . 2 ) If ( P , < ) lias no infinite stroiig antichain, t l w n P i s t l z c J d i s , j o i n t u n i o n a " f i n i t e number of d i r e c t e d s e t s .

of

Combining (1.1) and (1.2) w e s e e t h a t , i f (P,<)c o n t a i n s n o i n f i n i t e a n t i c h a i n , then i t i s c o f i n a l l y determined and t h e common c o f i n a l s u b s t r u c t u r e i s a d i r e c t sum of a f i n i t e number of s e t s each of which i s a s i n g l e p o i n t o r a f i n i t e d i r e c t product of r e g u l a r c a r d i n a l numbers. S i n c e o n l y f i n i t e l y many c a r d i n a l s a r e i n v o l v e d , o n e immediate consequence of P o u z e t ' s r e s u l t i s t h a t , i f (P,<)h a s no i n f i n i t e a n t i c h a i n , t h e n t h e c o f i n a l i t y cf(P) i s e i t h e r f i n i t e o r a r e g u l a r c a r d i n a l number. E x p r e s s i n g t h i s i n a s l i g h t l y d i f f e r e n t way, we have:

(1.3)

I f ( P , S ) has singu7ar c o f i n a l i t y , thew (P,>) contains a n ',zj""rzfte a n t i -

chain. I t i s n a t u r a l t o a s k how l a r g e an a n t i c h a i n t h e r e must b e when t h e c o f i n a l i t y i s s i n g u l a r . If we c o n s i d e r t h e s i m p l e example where X i s a s i n g u l a r c a r d i n a l and P i s t h e union of cf(A) d i s j o i n t , u n r e l a t e d c h a i n s having l e n g t h s X m ( a c - f ( X ) ) , and where t h e c a r d i n a l numbers Xa a r e i n c r e a s i n g and have l i m i t X , we s e e t h a t IA < cf(X) f o r any a n t i c h a i n A C i". T h i s s u g g e s t s t h e f o l l o w i n g q u e s t i o n asked by Efilner and Sauer [ 9 ] :

1

(1.4)

If c . f ( P )

=

X

> cf(A), does ( P , 5 )

contain an a n t i c h a i n o f size c f ( h )

T h i s q u e s t i o n h a s been answered p o s i t i v e l y ( [ 5 ] , [ 7 ] , [ 8 1 ) under some a d d i t i o n a l s e t - t h e o r e t i c a l assumptions. For example, t h e g e n e r a l i z e d continuum h y p o t h e s i s i m p l i e s ( 1 . 4 ) . The s t r o n g e s t r e s u l t of t h i s kind t h a t we know a t p r e s e n t i s t h e f o r m u l a t i o n g i v e n by Milner and Pouzet [ 7 ] : (1.5)

If c f ( P )

=

X

> cf(X) and

if A"

=

X for 1 5

p <

\J,

t h r n P c o n t a i n s an a n t i -

chain of s i z e u. Of c o u r s e , t h e g e n e r a l i z e d continuum h y p o t h e s i s i m p l i e s t h a t w e can t a k e u = c.f(A) i n ( 1 , 5 ) , b u t i t would b e i n t e r e s t i n g t o know whether (1.5) i s b e s t p o s s i b l e o r whether t h e r e must always b e an a n t i c h a i n of s i z e cf(X). For any p a r t i a l l y o r d e r e d s e t ( P , 5 ) , l e t u o ( P ) ( u 1 ( P ) ) d e n o t e t h e s m a l l e s t c a r d i n a l number p such t h a t t h e r e i s no a n t i c h a i n ( s t r o n g a n t i c h a i n ) of s i z e v . C l e a r l y u l ( P ) 5 u 0 ( P ) . Erd& and T a r s k i [ 4 ] showed t h a t u,(P) cannot b e e q u a l t o N o o r a s i n g u l a r c a r d i n a l , b u t any o t h e r v a l u e is p o s s i b l e . S i n c e p l ( P ) 5 K g i m p l i e s t h a t y 1 ( P ) = k f o r some f i n i t e k , (1.2) f o l l o w s from t h e f o l l o w i n g s i m p l e observation: (1.6) If p,(P) = k < H ~ ,then P i s t h e union o f fefewer than k d i r e c t e d s e t s . T h i s i s r e l a t e d t o a n o t h e r ( l e s s obvious) decomposition r e s u l t f o r p a r t i a l l y ordered s e t s , t h e theorem of Dilworth [l]:

The cofinality of ordered sets (1.7)

If

u,(p)

=

Z: < ti",

3

then ? i s t h e union o f fewer than k chains.

I t i s well-known ( e . g . P e r l e s [lo]) t h a t ( 1 . 7 ) f a i l s h a d l v when t h e c o n d i t i o n = U, ( f o r any i n f i n i t e c a r d i n a l K t h e d i r e c t product K €3 K d o e s n o t c o n t a i n an i n f i n i t e a n t i c h a i n and i t i s n o t t h e union of fewer t h a n K c h a i n s ) . I n a s i m i l a r way ( f o r an example due t o Baumgartner s e e [ 8 ] ) , ( 1 . 6 ) f a i l s j u s t a s badly when t h e c o n d i t i o n p I ( P ) < U n i s r e p l a c e d by p l (?) = U 1 (we a l r e a d v noted t h a t u1 (P) = Un i s i m p o s s i b l e ) . More p r e c i s e l y , f o r any i n f i n i t e c a r d i n a l number K , t h e r e i s a p a r t i a l l v o r d e r e d s e t (P,<)such t h a t p l ( P ) = U 1 and P i s n o t t h e union of K d i r e c t e d s e t s . Milner and P r i k r y [ 8 ] e s t a b l i s h e d t h e f o l l o w i n g decomposition r e s u l t f o r h i g h e r c a r d i n a l s which i s something of a m i x t u r e of (1.6) and ( 1 . 7 ) :

u , ( P ) < K O i s weakened t o u , ( P )

(1.8)

I f po(P)

= p,

t h m ? i s t h e union o f fewer than

f(p)

d i r e c t e d s e t s , where

+

Here K d e n o t e s t h e n e x t c a r d i n a l g r e a t e r t h a n t h e c a r d i n a l K . We do n o t know the best value for f ( u ) i n (1.8). I f P i s a S u s l i n t r e e t h e n p n ( P ) = U 1 and so i t i s c o n s i s t e n t t h a t f(Ul.) = U 2 . As f a r a s w e know i t may be c o n s i s t e n t t h a t f(,) = p+ f o r every i n f i n i t e c a r d i n a l p . I t mav even b e c o n s i s t e n t t h a t f ( p ) = p T h i s i s t r u e f o r example i f p i s weaklv compact [ 8 ] , and r e c e n t l v TodorreviC [12] ( s t a t e d i n a l e t t e r t o t h e a u t h o r ) h a s proved t h e c o n s i s t e n c v of f ( N 1 ) = ul.

I n view of (1.8) i t does seem r e a s o n a b l e , i n s t u d y i n g c o f i n a l i t v p r o p e r t i e s of p a r t i a l l y o r d e r e d s e t s , t o r e s t r i c t a t t e n t i o n t o t h e c a s e of d i r e c t e d p a r t i a l o r d e r s . One g e n e r a l i z a t i o n of P o u z e t ' s f a c t o r i z a t i o n theorem (1.1) i s t h e f o l l o w i n g r e s u l t of M i l n e r and P r i k r y [ 8 ] : (1.9) Suppose P i s a d i r e c t e d s e t , tion relation K

+

cf(P)

= K,

u o ( P ) = p and suppose t h e p a r t i -

(KsV)*

holds. Then any c o f i n a l subset o f P contains a c o f i n a l subset which i s order isomorphic t o a d i r e c t product o f t h e form K 6a A, where A i s a d i r e c t e d s e t having c a r d i n a l i t y 1Al < K. I n s t a t i n g (1.9) w e have used t h e p a r t i t i o n symbol of Erd'6s and Rado, K -+ ( ~ , p ) ' , which means t h a t any graph on a s e t of s i z e K e i t h e r c o n t a i n s a complete subgraph of s i z e K o r an independent s e t of s i z e u . By a theorem of Dushnik, Miller and Erdl9s [ 2 ] , K -t ( K , u , , ) ~ h o l d s f o r any i n f i n i t e c a r d i n a l and so ( 1 . 9 ) e a s i l y y i e l d s (1.1). I n g e n e r a l , a p a r t i a l l y o r d e r e d s e t i s n o t c o f i n a l l y d e t e r n i n e d . The S i e r p i n s k i o r d e r i n g of t h e r e a l s ( i . e . t h e o r d e r r e l a t i o n < = < n < < , where < i s t h e u s u a l o r d e r i n g of W and < < i s a w e l l - o r d e r i n g i n t y p e Z U C ) i s a n example of a p a r t i a l l y o r d e r e d s e t C s a t i s f v i n g t h e f o l l o w i n g c o n d i t i o n s : ( i ) P is d i r e c t e d , ( i i ) p o ( P ) = N ~ , ( i i i ) cf(P) = c f ( 2 ' o ) and ( i v ) t h e r e i s a f a m i l v F of c o f i n a l s u b s e t s of C s u c h t h a t 1 FI > Z N O and whenever F1 ,F2 a r e d i s t i n c t member of F, t h e n ( A 1 < cf(ZNn) i f A i s order-embeddable i n b o t h F1 and F p . C o n d i t i o n ( i v ) s a y s t h a t t h e c o f i n a l s u b s e t s mav have v e r y d i f f e r e n t o r d e r s t r u c t u r e s .

2.

THE STRUCTURE OF THE COFINAL SUBSETS OF A DIRECTED SET.

D e s p i t e t h e d i s p l e a s i n g example mentioned a t t h e end of t h e l a s t s e c t i o n , i t i s p o s s i b l e t o g i v e some g e n e r a l c h a r a c t e r i z a t i o n o f t h e o r d e r s t r u c t u r e of t h e

E.C. Milner

4

c o f i n a l s u b s e t s of a d i r e c t e d s e t . T h i s i s t h e r e s u l t of some r e c e n t work of Wang and L i [13] which we now d e s c r i b e . A s w i l l become immediately c l e a r , i t i s j u s t a s convenient to c o n s i d e r q u a s i ordered s e t s . A b i n a r y r e l a t i o n < on Q i s a quasi-order i f i t i s r e f l e x i v e and t r a n s i t i v e ( b u t n o t n e c e s s a r i l y a n t i - s y m m e t r i c ) . We w r i t e q l 2 0 ) i f q 1 < q , and q 2 < q l , and q1 < q2 means ql < q7 and q2 $ q l . The q u a s i - o r d e r < on 0 Induces a quasi-order on P(Q), t h e s e t of s u b s e t s of 0, i n a n a t u r a l way. For P , P 0, we w r i t e A < B i f f o r e v e r y element a F A t h e r e i s some b C R such t h a t n < b . S i m i l a r l y we can d e f i n e a q u a s i - o r d e r on P'(0) = P(P(0)) e t c . Although we u s e t h e same symbol 5 t o d e n o t e t h e s e d i f f e r e n t q u a s i - o r d e r s , no c o n f u s i o n should a r i s e s i n c e we o n l y compare terms from t h e same P"(0). ~

I f N i s an o r d i n a l number and C = {q,,: v < a ) i s a c h a i n i n Q w i t h q, < q v l f o r v < v ' < N , we i n d i c a t e t h i s by w r i t i n g r = {qv:v < a ) , The c h a i n C i s extendable i n 0 i f t h e r e i s some q C 13 such t h a t qv c q f o r each qv C C. L e t N ( Q ) d e n o t e t h e s e t of a l l non-extendable c h a i n s of 0. We d e f i n e y(Q) = min{lAl: A C N(O)? and l e t D(Q) b e t h e s e t o f a l l nonextendable c h a i n s of I) having o r d e r type ~ ( 0 ) . C l e a r l y y(0) i s e i t h e r 1 o r a r e g u l a r i n f i n i t e c a r d i n a l depending upon whether o r n o t 0 h a s maximal elements. I t i s easy t o see t h a t i f A , B C N(0) and A 5 B, t h e n c f ( A ) = c f ( B ) .

.

The key i d e a of Wang and L i i s based upon t h e f o l l o w i n g s i m p l e , b u t v e r y u s e f u l , observation t h a t

(2.1)

If Q i s directed, then so a l s o i s D ( Q )

.

Thus D may b e r e g a r d e d a s an o p e r a t o r on d i r e c t e d sets which may b e i t e r a t e d , From now on w e c o n s i d e r o n l y d i r e c t e d quasi-ordered s e t s . Thus, f o r n < o , we d e f i n e Do(Q) = 4 and P+'(Q) = D(Dn(Q)). L e t yn+l = y ( D n ( Q ) ) . Wang and L i c a l l = F(Q) t h e n - t h c l a s s c h a i n s of &. Thus t h e 2-th c l a s s t h e elements of c h a i n s are j u s t elements o f &, t h e 1st c l a s s c h a i n s a r e non-extendable c h a i n s of ) ~ C o ( v l ) C Q, t h e 2nd c l a s s c h a i n s a r e nont h e form C1 = f C o ( v l ) : v 1 c y ~ where e x t e n d a b l e c h a i n s of t h e form C2 = { C 1 ( v 7 ) : v 2 < y2}< where each Cl(V2) = { C o ( v l , v 2 ) : v 1 < y l l i s a f i r s t c l a s s c h a i n , and s o on. In general, for 1 5 n < o, a n n-th c l a s s c h a i n i s a non-extendable c h a i n Fn = {Fn-l(vn): vn < y } of l e n g t h y, i n D n - I . For 0 5 for 0 5

Cn

C

Dn.

i i

5 n (
u

Ui: Dn

+

P(Dz)

We c a l l U . ( C n )

by s e t t i n g U n ( C n ) = {Cn) a n d , t h e i - t h c l n s s original of

u;(c~)= { ~ ~ ( v ~ + ~ , . . . , v ~ ) : vj < yj (i < j :n ) ) and, f o r i > 0 and f i x e d v < yj (i < j 5 n ) , ci (vi+l v n ) = {ci-l(vi,vi+l v n ) : vi < Yi) I n g e n e r a l , d i f f e r e n t v e c t o r s (v1, vn), (vl, A) ( v i , v i < "i) may correspond

,..., . ..., ..., t o t h e same element of t h e 0-th c l a s s o r i g i n a l set of a n n-th c l a s s c h a i n Cn, i.e. co(~1,...,vn) C o ( v i ,..., . I f (2.2) h o l d s o n l y when (vl,..., ,..., t h e n ? i s c a l l e d a pure n - t h c l a s s c h a i n . It can b e shown t h a t ? ,...,

(2.2)

=

v ) = (vi

(2.3)

v')

vA)

If An,Bn € 21" then An 5 Bn holds if and onZy if Uo(An) 5 U0(Bn). ( i i ) If An C Dn there is a pure c" C Dn such t h a t An 5 c". ) in and ( i i i ) ~f Q' is c o f i n a l i n Q, then f ( ~isfcofinaz (i)

YE(@')

= YYL(Q).

One of t h e main lemmas proved by Wang and L i i s t h e f o l l o w i n g r e s u l t :

The cofinality of ordered sets

5

T h u s , i n p a r t i c u l a r , a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a q u a s i o r d e r t o yl. have a c o f i n a l c h a i n i s t h a t y i = cf(0) o r , e q u i v a l e n t l y , t h a t y2

I t i s n o t d i f f i c u l t t o v i s u a l i z e t h e g e n e r a l s t r u c t u r e of t h e 0-th c l a s s o r i g i n a l s e t U , ( A n ) of an n-th c l a s s c h a i n . Diagram 1 i l l u s t r a t e s t h i s f o r t h e c a s e n = 4 where t h e o r d e r i n g between c h a i n s i s e i t h e r from l e f t o r r i g h t o r from bottom t o top

A3 ( 0 )

diagram 1 While w e c a n n o t d e s c r i b e t h e e x a c t o r d e r r e l a t i o n s i n 4 between t h e d i f f e r e n t elements of U 0 ( Q n ) , w e do have an o r d e r i n g between t h e v a r i o u s c h a i n s t r u c t u r e s . Thus f o r 1 5 i < n and f i x e d vci < y,i (1 i .= n ) w e have Ai-~ (O,vi+l,. ,vn) 5 Ai-l ( l , v i + l , . ,vn) c Ai-l(2 ,vi+l,. ,vn) 5 * ’ *

..

..

..

.

From ( 2 . 3 ) it f o l l o w s t h a t e i t h e r ( a ) t h e r e i s some k < o such t h a t In the y 1 < y2 < yl < y2 < < yp, = cf(4), y k c l = yk+? = = 1, o r ( b ) case when ( a ) h o l d s e v e r y c o f i n a l s u b s e t of 0 c o n t a i n s a c o f i n a l s e t which i s t h e 0 t h class o r i g i n a l a,(&) of some p u r e k-th c l a s s c h a i n Ck. I n c i d e n t a l l y , i t f o l l o w s from P o u z e t ’ s more p r e c i s e s t r u c t u r e theorem t h a t , i f Q h a s no i n f i n i t e a n t i c h a i n t h e n ( a) h o l d s and, i f y1
...

...

... .

E.C. Milner

6

Tf (b) above h o l d s , t h e n we p u t yo = s u p y, and d e f i n e Db t o be t h e s e t of a l l sets A" = {A1 , A 7 , . .}, where An i s an Y l - t h c l a s s c h a i n and

.

An

=

An+l(O)

(1

~

n

c (A),

i . e . An i s t h e f i r s t term of t h e ( n + l ) - t h c l a s s c h a i n A*+'. A' i s c a l l e d pure i f Bb i f and o n l y each An i s p u r e ( n < w ) . I f Do i s o r d e r e d by t h e r u l e t h a t A" i s a g a i n d i r e c t e d and i f UO(AW) U O(Bo), where U O ( A W ) = LI U O( A n ) , t h e n ~

n < ill

s o we can c o n t i n u e by d e f i n i n g y

W+l

= y(D"),

'oD

=

D(DLL') e t c .

Da+' = D(D") and, f o r l i m i t a , Dix i s t h e set of a l l s e t s of t h e form {A"+': v < a ) , where AV+' f Dv+' and f o r each v < a,AV = A"+'(O) (for l i m i t v , AV = {AP+': p < v } C D"). The c a r d i n a l numbers y v ( v ' 1 ) a s s o c i a t e d w i t h t h e c h a i n decomposition of 0 a r e d e f i n e d by Y,+~ = y ( D V ) and y, = s u p y o f o r l i m i t v . In g e n e r a l ,

A'

=

P
The main r e s u l t of Wang and L i i s t h e f o l l o w i n g t r a n s f i n i t e e x t e n s i o n of ( 2 . 4 ) :

I.f ( L l , - ) is a d i r e r t e d quasi-ordered s e t and h i s an ordinal numlicr such t h a t yA(n) # 1, then t h e foZlowinp statements are e q u i v a l e n t : THEOREM 2.5

(i) (ii) (iii) (iv) (v)

Y ~ + , = 1, yX+'

<

yX,

r.f(O)

=

Q'(0)

has a maximal element,

yX,

there i s a pure Xth-class chain YX i n 0.

6

x

x

4 such that Uo(C ) is ro f i v i a l

...

From t h i s r e s u l t we s e e t h a t t h e r e i s some o r d i n a l 1 such t h a t y1 y? < < Y? cf(Q) and y, = 1 f o r LI > A. Any c o f i n a l s u b s e t 6" of 0 c o n t a i n s , a s a c o f i n a l s u b s e t , a p u r e A-th c l a s s c h a i n .

=

A s a n i l l u s t r a t i o n of t h i s c o n s i d e r t h e awkward S i e r p i n i s k i o r d e r (I?,<) d e s c r i b e d i n 6 1 . While t h e o r d e r s t r u c t u r e of t h e c o f i n a l s u b s e t s may b e v e r y d i f f e r e n t i n d e e d , t h e r e i s a v e r y s i m p l e c h a i n s t r u c t u r e f o r t h e c o f i n a l s e t s . More p r e t h e n t h e r e i s a sequence of nonc i s e l y , i f S i s any c o f i n a l s u b s e t of (I?,<), e x t e n d a b l e o-chains C,(a < ~ f ( 2 ~ O ) ) i n S such t h a t CO < C1 < e and II Ca i s c o f i n a l i n (R,<).T h i s i s a consequence of t h e f o l l o w i n g s i m p l e lemma.

-

LEMMA 2.6 If S,T 5 R , S c o f i n a l i n (R,) and 12'1 non-extendable w-chain C _c S such t h a t T < C.

<

cf(ZKo),

then t h e r e i s a

F o r , i f S i s c o f i n a l i n (R,<), t h e n by t h e lemma w e may c o n s t r u c t by t r a n s f i n i t e i n d u c t i o n non-extendable w-chains C 2 S f o r c1 < c f ( 2 K 0 )so t h a t

Aa IJ U Cp 4 Ca , B
GENERALIZATIONS

The n o t i o n of c o f i n a l i t y a r i s e s q u i t e n a t u r a l l y i n o t h e r c o n t e x t s . I n d e e d , i f R i s any b i n a r y r e l a t i o n on a set E t h e n w e can d e f i n e a c o f i n a l ( o r dominating) R ( A ) , where s u b s e t t o be a s e t A C E such t h a t E 5 A

u

R ( A ) = { x C E : ( z , a ) C R ,for some a

f

A1

.

By analogy w i t h o u r e a r l i e r d e f i n i t i o n , we d e f i n e t h e c o f i n a l i t v o f (E,R) t o be

I

cf(E,R) = mint J A : E c A

u R (A))

.

Corresponding t o t h e n o t i o n of an a n t i c h a i n i n a p a r t i a l l y o r d e r e d s e t , we s a y t h a t a s u b s e t A C E i s R-independent i f A n R ( A ) = 0. A s i n t h e c a s e of a p a r t i a l

7

The cofinality of ordered sets o r d e r , i f t h e r e i s no i n f i n i t e .?-independent s e t , t h e n t h e c o f i n a l i t y c f ( E , R ) must be e i t h e r f i n i t e o r a r e g u l a r i n f i n i t e c a r d i n a l . I n f a c t , t h e f o l l o w i n g gener a l i z a t i o n of (1..5) i s g i v e n i n [6]:

(3.1) 11'

=

TLf

cf(i?,i!) = \ >

=

a f o (~1 7 7

~1

-' 1!,

thm

i + . f ( l ) arid if c.orittr;ns mi R-ii2~kpxcleni subset of sine

u.

The h y p o t h e s i s IAl = X i n (3.1) may a t f i r s t a p p e a r t o o r e s t r i c t i v e . But t o r e g a i n (1.5), we simply a p p l y ( 3 . 1 ) t o a c o f i n a l s u b s e t of t h e p a r t i a l l y o r d e r e d s e t (P,<). Tn f a c t , t h e h y p o t h e s i s IEl = X i s e s s e n t i a l . f o r (3.1) a s i s shown by t h e f o l l o w i n g r e s u l t f o r tournaments [ 6 ] : (3.2) (v.,R)

7.f

1. (iri mi!;i i i f i ? i i t r etr~~Ji'rznYs wz'th 3ui.h thcit c > f ( ~ , R )= h .

I.,

C ~ F I i:

h,

K

then there i s a tournmient

A tournament i s a r e l . a t i o n R such t h a t f o r each p a i r of d i s t i n c t e l e m e n t s x,y e x a c t l y one of t h e r e l a t i o n s (.z,p) F I? o r (:{,x)F 1'7 h o l d s , and so t h e r e i s n o t even an R-independent s e t of s i z e 2 . Note t h a t (3.1) shows t h a t t h e c o f i n a l i t y of a tournament on a s i n g u l a r c a r d i n a l X i s s t r i c t l y l e s s t h a n A .

Although we do n o t have any r e s u l t l i k e (1.5) f o r a r e l a t i o n a l s t r u c t u r e (EJ?), c ~ ( E , R )= A euf A , t h e f o l ~ o w i n gi n t e r e s t i n g r e s u l t due t o Pouzet when ( u n p u b l i s h e d ) s a y s something a b o u t t h e s i z e of independent s e t s :

1 ~ 1

(3.3) T! I c.f(x) a r k j 2" -. A .for p A nmj i~ ever3 c o f i n n l s u i i s e t o f ( E , I ~ ) co?ittr?'iis independent s p t s 0.f a r b i i ; r a q s i n c A ' c A, then (E,R) contains an indcpendent s e t of size A. Another s i t u a t i o n i n which t h e r e i s a n a t u r a l analogue t o c o f i n a l i t y i s t h e c a s e of a c l o s u r e r e l a t i o n . A elosur% r e l a t i o n on a s e t E i s a f u n c t i o n cy:P(C) - * P ( F ) such t h a t ( i ) :i sz v ( X ) , ( i i ) X C V + q ( X ) 2 cp(.y) and ( ) f$l(Q(X)) = Q(X). A (jcncrutinp s e t ( o r c o f i n a l s e t ) i s a s u b s e t X 5 E such t h a t Q ( X ) = E and we d e f i n e t h e dimension ( o r c o f i n a l i t y ! ) of ip t o he dim(@) = m i n i l x i : c r ~ ( X ) =

.

Fl

The analogue of a n a n t i - c h a i n i s a cq>-independent s e t , i . e . a s e t A 5 E such t h a t Of c o u r s e , i f (E,<-)i s a p a r t i a l l y o r d e r e d s e t , 3: f q(A \ 13:)) f o r each .?: f A . t h e n q(A) = A d e f i n e s a c l o s u r e r e l a t i o n on E. With Pouzet w e have shown [ 7 ] :

( 3 . 4 ) If q, is a closure r e l a t i o n on E and di:m(Q) = X cy-independent sets of arbitrarilg large f i n i t e size.

-,cf(X),

then E contains

It i s n o t p o s s i b l e t o o b t a i n a r e s u l t l i k e (1.3) i n t h i s more g e n e r a l s e t t i n g . Using a r e s u l t of Erdas and H a j n a l [3] on set-mappings one can c o n s t r u c t a c l o s u r e r e l a t i o n tp>t having s i n g u l a r dimension f o r which t h e r e i s no i n f i n i t e (0"independent s e t . I n f a c t , i t i s c o n s i s t e n t t h a t t h e c l o s u r e r e l a t i o n ip* c o n s t ructed i n [ 7 ] i s a l g e b r a i c , i.e. it i s f i n i t e l y generated o r s a t i s f i e s (O*(X)

= II{~~>*(F F ) 2:

x

unii

IFI

.

N ~ )

An open q u e s t i o n which remains h e r e i s whether t h e r e i s a n i n f i n i t e @"-independent cf(\) and i f '4" i s 2-generated, i . e . i f s e t i f dirn(in9:) = h
= I U { , ~ ~ : ( Fi;') :L

x

anii

IF'\

2)

-:

.

E.C. Milner

8

REFER EN CES

[I] D i l w o r t l i , R . P . , A d e c o m p o s i t i o n t heorem f o r p a r t i a l l v o r d e r e d s e t s , Ann. Ma t h .,

[2]

5 1 (1950)

of

161-766.

D u s h n i k , B . and ? f i l l e r ,

r.,

P a r t i a l l y o r d e r e d s e t s , h e r . J. M a t h . ,

6 3 (194l)

(1941) 600-610. [3]

Erd'ds, P. a n d l l a i n a l , A . , On t h e s t r u c t u r e o f s e t m a p p i n g s , A c t a Math. Acad. S c i . Hu n g a r ., 9 ( 1 9 5 9 ) 111-131.

141 E r d a s , P . and T a r s k i , A . , On f n m i l i e s

OF

m u t u a l l v e x c l u s i v e s e t s , Ann. of

Math., 44 ( 1 9 4 3 ) 315-329. [5]

H a j n a l , A . and S a u e r , N . , C o m p l e t e s u b g r a p h s of i n f i n i t e mu1 t i p a r t i t e g r a p h s and a n t i c h a i n s i n p a r t i a l l v o r d e r e d s e t s , ( s u b m i t t e d f o r p u b l i c a t i o n ) .

[6] M i l n e r , r.C. and P o u z e t , M . , in:

On t h e c o f i n a l i t y o f p a r t i a l l v o r d e r e d s e t s , R i v a l , I . ( e d . ) O r d e r e d S e t s , ( R e i d e l , D o r d r e c h t , 1 9 8 2 ) 279-298.

[7]

M i l n e r , F.C. and P o u z e t , PI., On t h e i n d e p e n d e n t s u b s e t s o f a c l o s u r e s v S t e m w i t h s i n g u l a r d i m e n s i o n , U. o f C a l g n r v R e s e a r c h P a p e r No. 516 ( 1 9 8 ? ) .

181

M i l n e r , I'.C. a n d P r i k r v , K . , The c o f i n a l i t v o f a p a r t i a l l y o r d e r e d s e t , Proc. London Math, S OC . ( 3 ) , 46 (198'3) 454-470.

[9]

M i l n e r , C.C. and S a u e r , N . , Remarks o n t h e c o f i n a l i t v o f a a p r t i a l l y o r d e r e d s e t a n d a g e n e r a l i z a t i o n of K i i n i g ' s l e m m a , D i s c r e t e Math. 3 5 ( 1 9 8 1 ) 165-171.

[lo]

P e r l e s , M.A., On D i l w o r t h ' s t h e o r e m i n t h e f i n i t e case, I s r a e l J. of Math. 1 ( 1 9 6 3 ) 108-109.

[ll] P o u z e t , M . ,

P a r t i e s c o f i n n l e s d e s o r d r e s p a r t i e l s ne contenant pas d ' a n t i c h a i n e s i n f i n i t e , J. London Math. S o c . ( t o a p p e a r ) .

[I21

T o d o r E e v i E , S . , Some r e s u l t s c o n c e r n i n g p a r t i a l l v o r d e r e d s e t s , J o u r n a l of Symb. L o g i c ( t o a p p e a r ) .

[I31

Wang S h a n g - ? h i a n d L i Ro-yu, On t h e m i n i m a l c o f i n a l s u b s e t s of a d i r e c t e d q u a s i - o r d e r e d s e t , Discrete M a t h , ( t o a p p e a r ) .

Annals of Discrete Mathematics 23 (1984)9-38 0 Elsevier Science PublishersB.V. (North-Holland)

9

CAKULNAL FUNCTIONS ON BOOLEAN ALGEBRAS

J . Donald Monk

University of Colorado Houlder

Colorado

U.S.A. T o Professor E . COROMINAS

RC sum6 Nous esquissons ici la thCorie des applications b valeurs cardiriales d6finit.s sur les algsbres booleennes. Nous btudions par exemple les applications : B est une image homomorphe infinie de A } , la longueur h A = min { I l l length A = s u p { I X I : X est une chaine de A} , et le nombre d e s ultrafiltres de A , Ult A Nous d6crivons complhtement en details 24 telles applications, et nous en mentionnons 16 autres. Nous pensons que l'6tude de c e s applications cardinales fournit un cadre commode pour analyser la structure des algkbres boolbennes et que c'est un bon terrain d'expbrimentation pour les mbthodes recentes de demonstrtltion en thPorie des ensembles. Beaucoup de ces applications -au moins 13- ont citt PtudiGes de facon approfondie sous leurs fornies topol.ogiquesduales ; l'Ptude de ces applications pour les olgsbres booleennes e s t un c a s particulier et important de ces recherches. Ainsi bien des problhmes que nous bnongons ont une forme topologique qui n'est resolue ni pour les espaces compacts les plus gCneraux, ni poiir les espaces rciguliers. L a plupart des rPsultats quc nous enoncons nc s o n t pas nouveaux ; les dbmonstrations que nous en donnons sont simples, et font souvent partie de ce qui ex-istc de p l u s classique. En gCn&ral, nous n'avons pas fait l'effort de dPcouvrir le premier auteur de chaque rCsultat mais n o u s l'indiquons dans certains cas. Nous mentionnons 66 problsmes non resolus.* Nous commencons par consid6rt.r quelques applications arithmbtiques. la cellularitb de A , cell A , est sup {lCl : C c A , o u C est form6 d'ClPments deux i deux disjoints}. Cette application a CtP CtudiCe de facon detaillee par COMFORT et NEGREPONTIS ( 1 9 8 2 ) . Les applications longueur et profondeur respectivement notees length A e t depth A = sup {ICI : C c A , oh C est une chalne bien ordonnbe} o n t etP btudibes Mc KENZIE et MONK ( 1 9 8 2 ) . L'application inc A = sup {IMI : M c A ob les blbments de M sont deux i deux incomparables) peut s'exprimer en fonction du nombre d'arbres contenus dans A . La densite est la m@me chose que le -ir-poids de l'espace boolben. La ramification est une application bien connue li6e 2 la cellularitb. Les applications algebriques que nous considbrons sont les suivantes : - l e nombre de sous-algkbres ; - l'irrPductibilit6 (irr A = sup {lXi : aucune partie propre de X n'engendre X}); - la profondeur des sous-algkbres ou fonction depth de A = sup { K : il existe une suite strictement dbcroissante B pour CY < K de sous-alghbres de A! ; - l a longueur des sous-alghbres Ydefinie partiellement) ; - l'indtpendance ( s u p { / B I : B est une sous-alghbre libre de A ) } ; - hA c o m e definie ci-dessus ; - le nombre d'endomorphismes ; - le nombre d'automorphismes ; Nous considbrons pour terminer quelques applications qui ont, au moins

.

*

7.3 I r t t r c s , lorsqu'elle suit le mot problhme,indique que la rbsolution de celui-ci fait probablement intervenir des rbsultats d'indbpendance de thborie des ensembles.

J.D. Monk i m p l i c i t e m e n t , une forme t o p o l o g i q u e : l e nombre d ' i d 4 a u x ; 1 ' 4 t a l e m e n t ( " s p r e a d " ) ; l e d e g r 6 h t r 4 d i t a i r e d e L i n d e l 6 f ; l a d e n s i t 4 h 6 r t d i t a i r e ; l a l o n g u e u r d e s i d 4 a u x ; l e nombre d ' u l t r a f i l t r e s ; l a d e n s i t 4 t o p o l o g i q u e ; l e n-rar,~s.L;re ; It, c a r a c t e r e ; 1 ' 4 t r o i t e s s e ("tightness").

'"4 s) L TRC t We g i v e a s u r v e y o f t h e most i m p o r t a n t c a r d i n a l f u n c t i o n s on B o o l e a n a l g e b r a s . I n a d d i t i o n t o known r e s u l t s a b o u t t h e s e f u n c t i o n s and r e l a t i o n s h i p s bet w e e n them, we formul a t e many open p r o b l e m s .

We s u r v e y c a r d i n a l number v a l u e d f u n c t i o n s d e f i n e d on Boole a n a l g e b r a s . Examples of such f u n c t i o n s a r e hA = m i n { l B I : B i s an i n f i n i t e homomorphic image o f A} , l e n g t h A = s u p { l X I : X i s a c h a i n i n A} , and I U l t A l = number o f u l t r a f i l t e r s on A. A l t o g e t h e r we d e s c r i b e i n some d e t a i l 24 s u c h f u n c t i o n s and m e n t i o n i n p a s s i n g some 16 more. We b e l i e v e t h a t t h e s t u d y o f t h e s e c a r d i n a l f u n c t i o n s g i v e s a c o n v e n i e n t framework f o r a n a l y z i n g t h e s t r u c t u r e of Boole a n a l g e b r a s and is a good p r o v i n g ground f o r r e c e n t s e t - t h e o r e t i c a l me thods . Many l e a s t 13--have been e x t e n s i v e l y s t u d i e d i n t h e i r t o p o l o g i c a l of our functions--at d u a l form; t h e i r s t u d y f o r B ool ean a l g e b r a s forms a n i m p o r t a n t s p e c i a l c a s e of these investigations. Thus many of t h e probl ems w e s t a t e have a t o p o l o g i c a l form t h a t i s open e v e n f o r a r b i t r a r y compact s p a c e s o r r e g u l a r s p a c e s . Most o f t h e r e s u l t s we s t a t e are not new; t h e p r o o f s we d o g i v e h e r e a r e s i m p l e , and a r e m o s t l y i n t h e f o l k l o r e . We have n o t , i n g e n e r a l , t r i e d t o t r a c e t h e f i r s t p e r s o n t o p r o v e each r e s u l t , but we g i v e s uc h c r e d i t s i n some c a s e s .

We me n t io n 66 open probl ems. The s u p e r s c r i p t on a proble m means t h a t i t s s o l u t i o n probably i n v o l v e s s e t - t h e o r e t i c a l independence r e s u l t s . S i n c e some p r o b l e m s have e v i d e n t l y not been c o n s i d e r e d b e f o r e , some of them may be r a t h e r e a s y . And b e c a u s e o f t h e s c o p e of t h e s u r v e y , w e may ha ve o v e r l o o k e d some work on t h e s e t o p i c s . T h i s p a p e r o v e r l a p s two o t h e r s u r v e y s : van Douwen, Monk, Rubin (801 a n d Arhangelskii [781. W e a r e g r a t e f u l t o E.K. van Douwen, P. Nyi kos and S. T o d o r z e v i ; f o r u s e f u l comments. The v a r i o u s c a r d i n a l f u n c t i o n s w e c o n s i d e r c a n be d i v i d e d i n t o t h r e e s o r t s : "sm all" o n e s , whose v a l u e s a r e bounded ( f o r example , h, d e f i n e d a b o v e , s i n c e hA < 2w f o r any BA A); " wi del y v a r y i n g " o n e s , whose v a l u e s c a n v a r y from w t o A ( f o r e x a m p l e , l e n g t h , d e f i n e d a b o v e ) ; and "big" o n e s , cardinals close t o

-

whose v a l u e s a r e a l w a y s c l o s e t o I A I ( f o r e x a m p l e , I A l < I U l t A ( < * I A ' ) . T h i s c l a s s i f i c a t i o n is h e u r i s t i c a l l y useful. For exam ple , it seems m o s t i n t e r e s t i n g t o i n v e s t i g a t e t h e r e l a t i o n s h i p s bet ween t h o s e c a r d i n a l f u n c t i o n s which a r e w i d e l y v a r y i n g . A l s o , f o r them i t is o f most i n t e r e s t t o d e t e r m i n e t h e i r behavi o u r u n d e r a l g e b r a i c o p e r a t i o n s s u c h a s s u b a l g e b r a s , homomorphisms, e t c . Of c o u r s e i t is a l s o p o s s i b l e t o c l a s s i f y t h e f u n c t i o n s by c ompa ring them w i t h e a c h o t h e r i n g e n e r a l , r a t h e r t h a n o n l y w i t h c a r d i n a l i t y . Thus c e l l u l a r i t y a n d i n d e p e n d e n c e a r e not w i d e l y v a r y i n g w i t h r e s p e c t t o e a c h o t h e r . A n o t h e r a p p r o p r i a t e q u e s t i o n c o n c e r n i n g many o f t h e s e f u n c t i o n s i s attainment. I f kA is d e f i n e d a s t h e supremum o f c a r d i n a l s K s a t i s f y i n g some p r o p e r t y P ( K , A ) , w e c a n a s k f o r kA a l i m i t c a r d i n a l w h e t h e r P(kA,A) itself h o l d s . For su c h a f u n c t i o n k i t is f r e q u e n t l y a p p r o p r i a t e t o c o n s i d e r a r e l a t e d function hold. with

Thus

kA

kSA,

defined a s the least

kSA = (kA)+

attained, while

if

kA

K

such t h a t

P(K,A)

f a i l s to

is a s u c c e s s o r c a r d i n a l , or a l i m i t c a r d i n a l

ksA = kA

for

kA

a l i m i t c a r d i n a l not a t t a i n e d .

11

Cordinal functions on boolean algebras There a r e s e v e r a l o t h e r c a r d i n a l €unctions r e l a t e d t o a given c a r d i n a l function k : (H+k)A = sup{kB: B an i n f i n i t e homomorphic image of A ] (H-k)A = min(kB: B an i n f i n i t e homomorphic image of A } (S+k)A = sup{kB: B an i n f i n i t e s u b a l g e b r a of A } (S-k)A = min{kB: B an i n f i n i t e s u b a l g e b r a of A }

.

We d i s c u s s t h e s e , when n o n - t r i v i a l , f o r many of o u r f u n c t i o n s k With e a c h f u n c t i o n k , w e c o u l d c o n s i d e r two s p e c t r u m f u n c t i o n s : k-hs(A) = k-ss(A) =

I

kB: B a n i n f i n i t e homomorphic image o f A } kB: B an i n f i n i t e s u b a l g e b r a of A )

To s h o r t e n t h i s s u r v e y we c o n s i d e r o n l y one of t h e many p o s s i b i l i t i e s , I - h s ( A ) = [ IBI: A + > B , B i n f i n i t e ) , i n s e c t i o n 1 2 . A n o t h e r r e l a t e d n o t i o n w i l l n o t be d i s c u s s e d h e r e , b u t we m e n t i o n it a s a Eood t o p i c f o r i n v e s t i a t i o n . Given a p r o p e r t y P o f s u b s e t s o f a BA A , w e s a y i f among any K e l e m e n t s of A t h e r e i s a s u b s e t of t h a t A h a s P-K-A-caliber power A s a t i s f y i n g P . T h i s n o t i o n h a s been s t u d i e d f o r p a i r w i s e i n t e r s e c t i n g s e t s ( s e e C o m f o r t , N e g r e p o n t i s [ 8 2 1 ) and for i n d e p e n d e n c e ( s e e Monk [ 8 3 1 ) . A t t h e end o f t h e p a p e r we g i v e a r a t h e r messy d i a g r a m showing t h e known r e l a t i o n s h i p s between o u r main c a r d i n a l f u n c t i o n s .

I

CONTENTS

0. A.

3. 4.

5. 6.

............................. .................................. ................................... ......................... ....................... ............................

Cellularity Length Depth Incomparability Algebraic density Ramification

13 14 15 16 18 20

Algebraic functions 7.

8. 9. 10. 11. 12. 13.

14. C.

.................12

Arithmetical functions

1. 2.

B.

Introduction, Notation

............................. 21 ............................ 22 ........................ 22 ....................... 23 ............................ 2 3 ..........25 ........................... 27 .................... . . .. .27

Subalgebras Irredundance Subalgebra depth Subalgebra l e n g t h Independence Homomorphism t y p e and s p e c t r u m Endomorphisms Automorphisms

T o p o l o g i c a l f u n c t i o n s : open sets o r i d e a

15. 16. 17. 18. 19.

............................ ............................ .................. ....................... ......................

Ideals Spread Ideal generation Ideal depth Ideal length

S

.....28 .. . . .29 .....29 . .... 3 0 .. ...31

J.D. Monk

12

Topological functions: points o r u l t r a f i l t e r s

D.

............................ ..................... ..................... ............................... ............................... Diagram .......................................... R e f e r e n c e s ....................................... 20. 21. 22. 23. 24.

31 32 32 33 33

Ultrafilters Topological density Ultrafilter density Character Tightness

35 36

means t h a t Our s e t - t h e o r e t i c a l n o t a t i o n is i n g e n e r a l s t a n d a r d . f : A - A means t h a t f i s one-one. f[X] i s t h e f-image o f f maps o n t o B ; f : A-B X , f[X ] = { f x : x E X ) For any i n f i n i t e c a r d i n a l K , dedK = sup{A : t h e r e i s a l i n e a r l y o r d e r e d s e t of s i z e A w i t h a d e n s e s u b s e t o f s i z e K } [a] is t h e e q u i v a l e n c e c l a s s of a u n d er some ( i m p l i c i t l y u n d e r s t o o d ) e q u i v a l e n c e

.

relation. X

.

<

K

we l e t

&K

=

K

.

[XIK i s t h e s e t of a l l s u b s e t s of

.

F or

and

MAK

MA

,

.

, In+ = l 2 ~

i s t h e s e t of a l l s u b s e t s o f

QX

of s i z e

X

; [XICK , t h o s e of s i z e

K

. Tree h as

A
s e e Kunen [ 8 0 ] .

t h e u su al

,

meaning. A p s e u d o - t r e e is a p a r t i a l l y o r d e r e d s e t P such t h a t f o r a l l x E P i s t h e h y p o t h e s i s t h a t t h e r e i s no {y : y < x} i s l i n e a r l y o r d e r e d ,

SHK

K-SOuSlin t r e e . F or B A ' s , 3 , --)),-

u s u a l l y mean homomorphisms.

.

For

X c

A

we l e t

.

X+ = X \{O} A t A is t h e c o l l e c t i o n of atoms o f A A r a = {x E A : x S a } . SgX i s t h e s u b a l g e b r a g e n e r a t e d by X ; IgX t h e i d e a l g e n e r a t e d by X , an d A FgX t h e f i l t e r g e n e r a t e d by X s A i s t h e S t o n e isomorphism: s a = I F E U l t A : a E F} , where U l t A i s t h e c o l l e c t i o n of u l t r a f i l t e r s on A For any BA A ,

.

-A

o r Ai s its completion. and c o f i n i t e s u b s e t s o f X

.

.

For an y s e t X , f i n c o x i s t h e BA o f f i n i t e f i p s t a n d s f o r f i n i t e i n t e r s e c t i o n p r o p e r t y . WCC

means weakly c o u n t a b l y c o m p l e t e : A s a t i s f i e s WCC i f f V X , Y E [A]-<w s u c h t h a t Vx E X Vy E Y ( x * y = 0 ) , t h e r e is an a € A s u c h t h a t V x E X Vy E Y ( x a FrK i s t h e f r e e BA on K f r e e g e n e r a t o r s x a < K If L and a * y = 0 ) a' i s a l i n e a r o r d e r w i t h f i r s t e l e m e n t , i n t a l g L is t h e RA o f s u b s e t s of L g e n e r a t e d by a l l s e t s [ a , b ) and [ a , - ) = {x : a I f T is a t r e e , t r e e a l g T is t h e BA o f s u b s e t s of T g e n e r a t e d by a l l sets T r t , where T t t = { s E T : t 5 6 ) , For T a p s e u d o - t r e e , p t r e e T is d e f i n e d s i m i l a r l y . Note t h a t homomorphisms on i n t a l g L a r e i n one-one c o r r e s p o n d e n c e w i t h convex I f A is h e r e d i t a r i l y a t o m i c , i t s c a r d i n a l e q u i v a l e n c e r e l a t i o n s on L U s e que nc e i s t h e s e q u en ce of c a r d i n a l i t i e s of t h e s e t of atoms i n i t s n a t u r a l (I,, = I 0 1 , Ia+l = { a : [ a ] is a f i n i t e sum o f s e que nc e of q u o t i e n t s A / I ,

.

.

5x1

<

.

.

atoms of of t h e

A/Ial , I A = (J a
IfEEIAi

A

limit). f E IIi

c o n s i s t s of t h o s e

f i n i t e l y many p l a c e s , and complements of such AXB

and

A*B

a r e used f o r only t w o f a c t o r s .

and C w i t h amalgamated s u b a l g e b r a A k-homogeneous i f kA = k(A r a ) f o r a l l

EiE IAi

.

If

0

f

f

is t h e d i r e c t product

which a r e

, *i

0

e x c e p t at

t h e i r free product;

i s t h e f r e e p r o d u c t of

B* C A

k i s any c a r d i n a l f u n c t i o n , a E A

.

B

A

is

Cardinal functions on boolean algebras

13

Arithmetical Functions

A.

We d i s c u s s f u n c t i o n s a s s o c i a t e d w i t h t h e a r i t h m e t i c of Boole a n a l g e b r a s : disjointness, chains, incomparability, e t c .

1. W e d e f i n e t h e c e l l u l a r i t y of c e l l A = s u p { l C I : C !z A

,

Cellularitv t o be

A

a f a m i l y of p a i r w i s e d i s j o i n t e l e m e n t s of

C

A}

This notion is e x t e n s i v e l y s t u d i e d , along with r e l a t e d notions not t r e a t e d h e r e , i n Co m f o r t, N e g r e p o n t i s [821 Note t h a t t h i s i s a w i d e l y v a r y i n g f u n c t i o n . By Erdos, T a r s k i ( 4 3 1 , c e l l A is always a t t a i n e d f o r s i n g u l a r c a r d i n a l s , but t h e r e

.

a r e c o u n t e r e x a m p l e s f o r weakl y i n a c c e s s i b l e c a r d i n a l s .

Thus

= (cellA)+

cellSA

u n l e s s c e l l A i s we a kl y i n a c c e s s i b l e and n o t a t t a i n e d , i n which c a s e c e l l S A = cellA A s a t i s f i e s t h e K-cc (K-chain c o n d i t i o n ) i f [ X I < K f o r e v e r y f a m i l y X o f p a i r w i s e d i s j o i n t e l e m e n t s of A I f A C B , c l e a r l y c e l l A 5 c e l l B . The c e l l u l a r i t y c a n d i f f e r a r b i t r a r i l y between A and B I f A+B , t h e r e i s n o r e l a t i o n s h i p , in g e n e r a l , be tw e e n c e l l A and c e l l B

.

.

.

.

C l e a r l y c e l l n i E I A i =I11 Usup. c e l l A i ; s i m i l a r l y f o r weak p r o d u c t s . IEI The c e l l u l a r i t y of an u l t r a p r o d u c t d e p e n d s on s a t u r a t i o n p r o p e r t i e s . Thus i f F is an y c o u n t a b l y i n c o m p l e t e u l t r a f i l t e r o n a set I , t h e n c e l l ( T l i E I A i / F ) o1

1.

A;

if all

are infinite.

Assuming

a 11 I+-good

countably incomplete u l t r a f i l t e r

2

cell(IT. A./F) 1 E I 1 i n f i n i t e set I

I

> 2 1.5.4.

11

I+

A.

if all

2w

f o r any

I

t h e r e is

cell(ni

on

F

are infinite.

t h e r e is an u l t r a f i l t e r

f o r a n y s y s t e m < A i : i E I>

2

F o r any i n f i n i t e set

w e have

MA

F ; see, e . g . McKenzie, Monk 1821 1 . 5 . 5 .

such

F

of i n f i n i t e

I

and h e n c e

On t h e o t h e r h a n d , f o r any A /F) L E I i

I

on

,

IA i/F)

such t h a t cell(Tl.

BA's;

see McKenzie, Monk [ 8 2 ]

We m e n t i o n h e r e t h e f o l l o w i n g v a g u e q u e s t i o n .

Problem 1.

If

F

is a III+-good

c o u n t a b l y i n c o m p l e t e u l t r a f i l t e r on

fli EIAi/F

have a p a i r w i s e d i s j o i n t s e t of power

what c o n d i t i o n s d o e s

,

I

under

2 II1?

Th e b e h a v i o r of c h a i n c o n d i t i o n s u n d e r f r e e p r o d u c t s h a s be e n e x t e n s i v e l y s t u d i e d . We s u r v e y t h e s e r e s u l t s . F i r s t w e t a k e t h e f r e e p r o d u c t o f two B A ' s ; t h e c a s e o f a f r e e p r o d u c t of i n f i n i t e l y many B A ' s reduces t o the f i n i t e case by a r e s u l t m e n t i o n e d bel ow, and t h e f i n i t e c a s e , o f c o u r s e , r e d u c e s by i n d u c t i o n

BA's

t o t h e case of two

,

((2K)+,K+)2

(1)

cel l ( A* B )

(2)

cellA

(3)

if

5

K

K+

+

The p a r t i t i o n r e l a t i o n s

,

(K+,CfK)2

-< 2ce11A*ce11B

2K

and

cellB

is s t r o n g l i m i t ,

5

for

strong l i m i t ,

K

+

(K )K +

,

(2K )+ +

give the following:

; K

cell(A*B)

cellA(

K

,

and

-<

cellA*cellB

5 c e l l( A * B) 5 ( c e l l A * c e l l B ) +

and

is a s u c c e s s o r c a r d i n a l ;

ZK ;

cellB

R e s u l t s (1) and ( 2 ) c a n be found i n Kurepa [ 6 2 1 . cellA

(2K)+

<

CfK

,

Thus unde r

; cel l (A*B)

= cellA

cell(A*B) = c e l l A

if

then GCH

if cellA

cell(A*B)

( K.

w e ha ve cellA

>

cellB

is a l i m i t

J.D. Monk

14

.

c a r d i n a l and c f ( c e l 1 A ) > c e l l R Thus even under c o v e r e d by ( 1 ) - ( 3 ) : cellA l i m i t with cf(cel1A) = c e l l B . The f i r s t c a s e a p p e a r s t o be open: Problem 2 . I f i s cell(A*B) =

is strong l i m i t singular,

K K

cellA =

(4)

Suppose

,

and C E K

5

<

cellR

,

K

The f i r s t r e s u l t we m e n t i o n i s d u e

K+-Souslin t r e e , and l e t

i s a normal

T

and

K

K

two c a s e s a r e not < c e l l A , and c e l l A

?

The second c a s e h a s been e x t e n s i v e l y s t u d i e d . t o Kurepa 1501:

cellA =

GCH

Ic e l l B

cell(A*B) =

K+

.

A = treealgT

Then

.

G a l v i n and Laver ( s e e G a l v i n [801) proved

(5)

if

2K =

cell(A*B) Todorfevi:

(6)

then there are

RA's

A

and

with

R

cellA = cellR =

and

K

= K+. [a]

proved

There a r e a r b i t r a r i l y l a r g e c a r d i n a l s

which s a t i s f y and

,

K+

while

K-cc

A*B

such that there e x i s t

K

A,A

RA's

does n o t ; t h i s h o l d s i n p a r t i c u l a r i f

K

5 2w

.

cfK = cf2'

F l e i s s n e r 1781 showed ( 7 ) I f K Cohen r e a l s are added t o M , t h e n i n M[G] t h e r e are c c c R A ' s A and B w i t h cell(A*B) K C o n s e q u e n t l y , i f M s a t i s f i e s CH a n d K , A , ~ are c a r d i n a l s of M w i t h K r e g u l a r and w' h , p 5 K , t h e n i n M[G] t h e r e

.

2

are

BA's

A

and

B

with

cellA = A

,

,

cellR = p

and

A f o l k l o r e r e s u l t , proved i n d e p e n d e n t l y by many p e o p l e , (8)

(MA + 2 O

>

w,)

F i n a l l y we n o t e t h a t

If

A

and

B

cell(*i

satisfy

,

ccc

2.

A*B

K

= 2w

.

.

: F E [I]

~

=

is

so does

= sup{cell(*. € 8 ; )

JuhLsz 1801, p. 107. The d e r i v e d f u n c t i o n s p r e a d , d i s c u s s e d below.

cell(A*B)

<w

1 ; see, e . g .

is the c a r d i n a l function

H'cell

Length

.

W e l e t lengthA = s u p { \ C l : C A , C a chain} This notion is extensively s t u d i e d i n McKenzie, Monk [821 I t is a w i d e l y v a r y i n g f u n c t i o n . A d i s c u s s i o n of maximal c h a i n s i n B A ' s c a n b e found i n J a k u b i k [581 , Day [701 and S . Koppelberg [ m l I f A h a s l e n g t h K and cfK = w , t h e n l e n g t h A i s a t t a i n e d ; i f K i s l i m i t w i t h CfK > w , t h e n t h e r e are c o u n t e r e x a m p l e s . It i s n o t c o m p l e t e l y c l e a r what happens t o l e n g t h u n d e r d i r e c t p r o d u c t . We have

.

.

max(dedlI1 ,supi E I l e n g t h A i )

5

lengthn

AAi

5

rIi

IlengthA i '

where t h e second i n e q u a l i t y c a n b e e q u a l i t y , a n d , c o n s i s t e n t l y , s o c a n t h e f i r s t . Problem 3 .

Does

l e n g t h J I i c IAi

depend o n l y on

11

I

The l e n g t h of u l t r a p r o d u c t s h a s not been i n v e s t i g a t e d .

and

< l e n g t h A i : i E I> ?

15

Cardinal functions on boolean algebras Problem 4 .

Discuss t h e length of u l t r a p r o d u c t s .

T h e r e i s a l s o a p r o b l e m c o n c e r n i n g t h e l e n g t h of f r e e p r o d u c t s , a l t h o u g h much is known. The h a r d e s t t h e o r e m i n McKenzie, Monk [ 8 2 ] i s t h a t i f CfK > w , A h a s n o c h a i n of power c f ~, and R h a s no c h a i n o f power K , t h e n A*R h a s no c h a i n o f power ic Hence l e n g t h a i = supi EIlengthAi However, t h e a b o v e

.

.

theorem is not b e s t p o s s i b l e . known : (1)

Suppose

Suppose

Then

A*R

<

CfK

5

<

11

,

K

11

< dedh , dedh -

A , B w i t h 11 = l e n g t h A not a t t a i n e d .

there exist length(A*R)

(2)

h

In fact, the following additional r e s u l t s are

Wh

<

CfK

(2

h

< CfK) , -

h a s a c h a i n of s i z e

( 3 ) I f CfK < K , IengthR = K , then

K

attained if

attained, lengthB =

.

cfK

<

1engthA

5

K

,

1engthB =

and

i s weakly compact, cfK = lengthA has a chain of s i z e K

CfK A*B

i ~ .= d e d h

.

Then

n o t a t t a i n e d , and

K

K

.

a t t a i n e d , and

.

( 4 ) I f CfK < K , A h a s a c h a i n o f s i z e c f K h a v i n g c f K p a i r w i s e d i s j o i n t i n t e r v a l s e a c h w i t h a t l e a s t two e l e m e n t s , and l e n g t h B = K , t h e n A*B has a c h a i n of s i z e K

.

A v e r y s i m p l e f o r m o f t h e q u e s t i o n s l e f t o p e n by t h e a b v o v e r e s u l t s i s a s €01lows. P r o b l e m 5 . L e t A be t h e i n t e r v a l a l g e b r a o f a S o u s l i n l i n e , NW = l e n g t h R n o t a t t a i n e d . Is l e n g t h ( A * B ) attained? 1 Note t h a t

lengthA

conjecture that

52

by t h e E r d o s , Rado t h e o r e m .

V K ~ [
<

2

K

TRA

A (cellA =

a

RA

with

It would b e n a t u r a l t o lengthA = A)]; t h i s

and

K

B

K

.

would f o l l o w f r o m i t s s p e c i a l c a s e VK 9BA A ( c e l l A = K a n d l e n g t h A = 2 ) T h i s is t r u e u n d e r GCH Rut a s M i t c h e l l [721 showed ( s e e a l s o J u h ' a s z [ 7 1 1 ) i f M GCH a n d G a d d s Hw Cohen r e a l s , t h e n i n M[G] e v e r y BA o f l e n g t h

.

k

1

->

u)

2

a t t a i n e d has c e l l u l a r i t y

2

2"

,

and

Z W = KW

,

Nw +1 1

I above f a i l s f o r The f u n c t i o n functions.

K = W

.

Thus t h e

1 '

H'length

d o e s not c o i n c i d e w i t h any o f our c a r d i n a l

P r o b l e m 6 . Is i t c o n s i s t e n t t o h a v e a RA A A h a s no i n f i n i t e homomorphic image o f power

<

such t h a t ]A! ?

w

<

lengthA

<

IAI

and

Some r e s u l t s of W. J u s t are r e l e v a n t t o t h i s p r o b l e m . P r o b l e m s 3-5 a r e e s s e n t i a l l y s t a t e d i n McKenzie, Monk [ 8 2 ] .

3.

Depth

By d e f i n i t i o n , d e p t h A = s u p { l C I : C i s a w e l l - o r d e r e d c h a i n i n A}. Again, t h i s is a n o t i o n e x t e n s i v e l y s t u d i e d i n McKenzie, Monk [ 8 2 1 , and is a w i d e l y v a r y i n g function. I f A h a s d e p t h K a n d CfK = w , t h e n d e p t h A is a t t a i n e d ; i f K is l i m i t w i t h CfK > w , t h e n t h e r e a r e c o u n t e r e x a m p l e s . d e p t h n ; = \I U

pi

supiEIdepthA i '

I

Concerning t h e depth o f u l t r a p r o d u c t s , t h e r e are s e v e r a l

J.D. Monk

16

i n t e r e s t i n g f a c t s , and one probl em. ultrafilter

on

F


system

<

K K>

such t h a t

For any

na
of i n f i n i t e

RA's.

t h e r e is a non-principal

K

h a s a c h a i n of t y p e

in a model of Woodin t h e r e i s a u n i f o r m u l t r a f i l t e r

that

2K

f o r every

On t h e o t h e r ha nd, L a v e r h a s shown

F

on

U,

such t h a t

w

I o l o / F \ = w1 ; h e n c e , of c o u r s e , By t h e E r d a s , Rado t h e o r e m , on

,

K

then

Pro b lem 7 .

on

,

K

depth(KA/F) If

A

is a

if

-< RA

d o e s it f o l l o w t h a t

2'

= w1

d e p t h ( 'A/F) dept hA

5

A

,

f o r every denumerable

A , and

K

RA

.

i s any u l t r a f i l t e r

F

. w i t h no c h a i n s of t y p e KA/F

A

h a s no c h a i n o f t y p e

and

F

ma x((2

is an u l t r a f i l t e r K +

,h) ?

The an s we r t o problem 7 is ( c o n s i s t e n t l y ) no i f w e r e p l a c e " u l t r a f i l t e r " by " f i l t e r " ; see McKenzie, Monk [ 8 2 1 , I f cfK > o , A h a s no c h a i n of t y p e cfK , and B h a s no c h a i n of t y p e K , then A*B h a s no c h a i n o f t y p e K ; but i f A h a s a c h a i n o f t y p e CfK a n d d ep th B = K , t h e n A*B h a s a c h a i n of t y p e K I f K i s i n f i n i t e and r e g u l a r , ] A 1 = K , and t h e r e i s a homomorphism from A o n t o a s u b a l g e b r a of K c o n t a i n i n g a l l s u b s e t s of power < K , t h e n t h e r e

.

exist

BA's

infinite algebras

R,C 2 A

satisfying

K + - CC

such t h a t

depth(B*AC)

2

.

K+

For any

i n a model M of GCH , t h e r e i s an e x t e n s i o n M ' of M w i t h B,C I A s u c h t h a t depth(B*AC) > max(de pthB, de pthC) ( t h i s is a result BA

A

o f S h e l a h found i n McKenzie, Monk [ 8 2 ] ) . Pro b lem 8. In ZFC i s i t t r u e t h a t f o r e v e r y i n f i n i t e B , C 2 A w i t h depth(BWAC) > max(dept hR,dept hC) ?

A

RA

there exist

Pro b lem 9 . F o r e v e r y i n f i n i t e BA A , i s t h e r e a c a r d i n a l K s u c h t h a t i f A and I B I , I C I K , then depth(BhAC) = max(de pthB, de pthC) ?

B,C

1.

2

5

d e p t hA 5 l e n g t h A 2 depthA , (H+depth)A c o i n c i d e s with t h e A ; see below. P r o b l e m s 7-9 a r e i n McKenzie, Monk [ 8 2 ]

We c l e a r l y have t i g h t n e s s of

4.

.

Incomparability

.

As w i l l incA = su p { l M l : M i s a s e t of p a i r w i s e i n c o m p a r a b l e e l e m e n t s o f A} be d e t a i l e d below, t h i s is a l a r g e f u n c t i o n . We l e t a b b r e v i a t e set o f p a i r w i s e i n c o m p a r a b l e e l e m e n t s . We f i r s t i n d i c a t e a n e q u i v a l e n t way of d e f i n i n g inc

.

tree,

Theorem 4 . 1 . T r A}

.

F o r any i n f i n i t e

BA

A

we have

No te t h a t f o r a t r e e T S A w e d o n o t i n s i s t t h a t incomparable i n T ; s e e s e c t i o n 6

.

incA = s u p { \ T I : T is a

s + t = 1

for

-

s

a nd

t

5

Proof. S i n c e any p i e is a t r i v i a l t r e e w i t h o n l y r o o t s , t h e p a r t of t h e e q u a l i t y is c l e a r . To show = , assume t h a t A h a s no p i e o f s i z e K , K r e g u l a r ; we show t h a t A h a s no t r e e of s i z e K , Suppos e t h a t T is a t r e e i n A of power K By R a u m g a r t n e r , Komjath [ S l ] , A h a s a d e n s e s u b s e t D o f power < K Now e a c h l e v e l of T i s a p i e , so h a s power < K Hence T h a s a t l e a s t K l e v e l s . L e t T' b e a s u b s e t of T of power K c o n s i s t i n g e x c l u s i v e l y o f e l e m e n t s of s u c c e s s o r L evel . For each d E D , l e t

.

Md =

It

.

ET'

.

: if

s

is t h e i mmedi at e p r e d e c e s s o r of

t

,

then

d

5 t*-s} .

17

Cardinal functions on boolean algebras

a

=udE D M d

T'

Thus

,

pie

,

d E D

so t h e r e is a

with

IM

d

I

=

.

K

But

is clearly

Md

contradiction.

.

5

2

I f A S B o r B-h'A , c l e a r l y incA incR I f A G B , t h e n inc (A xB) /A\ In f a c t , { ( a , - a ) : a E A} i s a p i e i n A X B Hence i f A i s c a r d i n a l i t y homogeneous and h a s no p i e of s i z e I A l , t h e n A is r i g i d . We ha ve inc(A*B) = m a x ( [ ~ ,\I n [ ) if \A\, 4 , s i n c e A*C ~ A X A i f Icl = 4

.

.

>

\RI

.

5

incA and \ A \ 5 Zinc* u s i n g t h e E r d o s , Rado t h e o r e m . Clearly cellA Some d e e p r e s u l t s and p r o b l e m s a r e found i n c o n n e c t i o n w i t h t r y i n g t o c o n s t r u c t a BA A w i t h no p i e of s i z e \ A \ ; w e c a l l such a BA n a r r o w . Bonnet and S h e l a h (Bo n n et

[m])

h a v e shown i n

t h a t t h e r e is a n a r r o w

ZFC

BA

of power

.

cf(Zw)

.

Bonnet [ m ] h a s shown a s s u m i n g GCH t h a t t h e r e i s a n a r r o w BA o f power K+ By a t h e o r e m o f A r h a n g e l s k i c [ 7 1 ] , i f I A l is singular strong l i m i t , then A i s n o t n arro w. B a u m g a r t n e r , Komjath and S h e l a h i n S h e l a h [ 8 3 ] ha ve shown t h a t i f A h a s no p i e of s i z e h , t h e n i t h a s a d e n s e s u s e t of s i z e < A So i f IAI i s s t r o n g l i m i t , t h e n A is not narrow. Even s t r o n g e r r e s u l t s a r e known i n which A a l s o does not have b i g c h a i n s . C a l l A c o n c e n t r a t e d i f A h a s no p i e a n d no c h a i n of s i z e \A\ S h e l a h h a s shown u n d e r GCH t h a t f o r e a c h h > w there

.

.

is a c o n c e n t r a t e d

BA

of s i z e

a A-complete

concentrated

Problem 10.

(CH)

BA

,

A+

A? o

and f o r e a c h

f

w1

t h e r e is

.

A+

of s i z e

A

with

Is t h e r e a c o n c e n t r a t e d

o-RA

w2 ?

of s i z e

Rubin [ 8 3 ] h a s shown t h a t i f B is a s u b a l g e b r a of a n i n t e r v a l a l g e b r a and is r e g u l a r , t h e n B i s not c o n c e n t r a t e d . Problem 11. I f B i s a s u b a l g e b r a o f an i n t e r v a c a n B be c o n c e n t r a t e d ?

a l g e b r a a nd

IBI

is singular,

IB[

IB I

No te by t h e a b o v e r e m a r k s t h a t t h e answer t o P rob em 11 is no f o r strong l i m i t singular. S h e l a h [ 8 0 ] and i n d e p e n d e n t l y van Wesep have shown t h a t i t is c o n s i s t e n t t o h a v e Zw a r b i t r a r i l y l a r a e and t o have a BA A of s i z e 2w w i t h c o u n t a b l e On t h e o t h e r h a n d , B a umga rtne r [801 h a s shown t h e l e n g t h and i n c o m p a r a b i l i t y . I

following consistent:

MA

+

Z w = w2

+ "every u n c o u n t a b l e

has an uncountable

BA

pie." S h e l a h h a s g e n e r a l i z e d t h i s , showi ng t h a t i t i s c o n s i s t e n t t o h a v e t h e continuum a r b i t r a r i l y l a r g e .

Is it c o n s i s t e n t t h a t every

P r o b l e m 12'.

BA

of power

w2

has a

pie

of s i z e

w2 ?

S h e l a h [831 h a s shown t h a t f o r any s i n g u l a r h w i t h c f A > w i t is c o n s i s t e n t t h a t t h e r e i s a BA A w i t h incA = A n o t a t t a i n e d . M i l n e r a nd P o u z e t ( u n p u b l i s h e d ) h a v e shown t h a t i f incA = h , w i t h w = c f h , t h e n incA is attained.

is a

BA

Problem 1 3 . cardinal K

Zw

Zw

with incomparability

not a t t a i n e d .

I s i t t r u e t h a t f o r e v e r y weakl y b u t not s t r o n g l y i n a c c e s s i b l e t h e r e i s a BA w i t h i n c o m p a r a b i l i t y K n o t a t t a i n e d ?

T o d o r r e v i t h a s shown t h a t i f t h e n A h a s a p i e of s i z e

We h a v e

i s wea kly i n a c c e s s i b l e , t h e n t h e r e

Zw

T o d o r c ' e v i i h a s shown t h a t i f

of s i z e

( H + i n c ) A = incA

cfh A

.

<

h

for a l l

A

and

A

.

is a

BA

Not e t h a t

w i t h a t r e e of s i z e H-inc

A

,

is i n g e n e r a l non-

J.D. Monk

18 t r i v i a l ; f o r e x a mp l e , following question. Pro b lem 1 4 . ni-IcA

(H-inc)A = 2

w

if

is

A

WCC

.

This gives rise t o t h e

Under any s e t - t h e o r e t i c a l a s s u m p t i o n s , is t h e r e a = w , and I B 1 > w f o r e v e r y homomorphic image

RA

R

Pro b lems 1 0 , 1 1 , 13 a r e ment i oned i n van Douwen, Monk, Rubin (801 r e l a t e d t o i n c o m p a r a b i l i t y has been c o n s i d e r e d by P. Nyikos: h -c o f ( A )

= min{K : e v e r y s u b s e t of

One can show t h a t

incA

5

h-cof(A)

[all has ideal.

.

w

An a l g e b r a

,

IAI = w

E

A

,

T

Clearly

Pro b lem 1 5 .

,

i s of power

CH

ow

c o n s t r u c t e d under

A

incA = w

Can one c o n s t r u c t

5

.

K)

.

well-founded)

w1

but has

by R a u m g a r t n e r , Komjath

1

( a E A : I A T a1

while

is n o t c o u n t a b l y g e n e r a t e d , s o

I

A function

and

The a l g e b r a c o n s t r u c t e d by S h e l a h [ 8 1 ] assumi ng h-cofinality

.

h a s a c o f i n a l s u b s e t of power

A

= sup{ [ T I : T

h - c o f(A)

with A ?

A of

in

ZFC

5.

a

BA

5 (d}

h-c of(A ) with

A

= I

i s a maximal

= w1

.

<

incA

?

h-c of(A )

Algebraic Density

.

nA = min{lDI : D d e n s e i n A} Again t h i s is a l a r g e f u n c t i o n . T h i s is t h e same a s t h e n-weight of u l t A , but we s h a l l u s u a l l y c a l l i t a l g e b r a i c density. I t h a s been e x t e n s i v e l y s t u d i e d in t o p o l o g y . F i r s t we d i s c u s s i t s behavior under a l g e b r a i c o p e r a t i o n s , then i t s r e l a t i o n s h i p s t o t h e p r e v i o u s f u n c t ions i n t r o d u c e d . Suppose A+B Then t h e r e i s no g e n e r a l l y v a l i d r e l a t i o n s h i p be tw e e n nA To g e t nA < nB , l e t A = PK Thus nA = K L e t C be a f r e e and nB

.

.

s u b a l g e b r a of

.

ZK

o f power

A

.

.

.

Id : C

E xt end

+

f : A+>

t o a homomorphism

C

.

C l e a r l y nB = zK To g e t nB < nA , t a k e F r ( 2 K ) + ) p ~ . One c a n s t i l l a s k , h o w e v e r , f o r what p a i r s K , A i t is t r u e t h a t e v e r y BA A o f power h a s a homomorphic image B w i t h nB = K Assuming GCH we c a n a ns w e r t h i s question fully (Corollary 5.5). B G C

A

.

Theorem 5 . 1 . f : in t a l g L- R

S uppose L i s a d e n s e l i n e a r o r d e r , Then *B 5 ID1

.

is d e n s e i n

D

.

L

,

a nd

We omit t h e e a s y p r o o f . Corollary 5 . 2 . such t h a t f o r every image

If B

Theorem 5.3. If B w i t h nB = A Proof.

K

,

<

A

A%B

<

dedSK , t h e n t h e r e i s a i m p l i e s nB K .

1x1 2 ( 2 K ) +

and

.

We u s e h e r e some r e s u l t s on

A

K+

,

then

A

BA

5

h a s a homomorphic

A

s = s p r e a d ; s e e below.

A r h a n g e l s k i : and S a p i r o v s k i ? ( s e e J u h e s z [ 8 0 1 , p. 56) we have Now i t is e a s y t o see t h a t s(U1t A) = s u p ( c e l 1 C : A - W } Hence c h o o s e

C

so t h a t

A-9C

and

wise d i s j o i n t , and l e t D = Sg(c)X Thus B i s as d e s i r e d .

.

cellC =

K+

Then e x t e n d

.

A

of power

By a t h e o r e m of

s(UltA)

2 K+ .

.

Let id : D

E

X +

D

[CIA with to

X

pair-

g : C-BBG

%.

19

Cardinal functions on boolean algebras

A

< K -

Theorem 5.4. I f K i s s i n g u l a r s t r o n g l i m i t and A h a s a homomorphic image B w i t h nB = A

Proof. -

,

K

t h e n f o r any

A s f o r 5 . 3 , u s i n g a l s o J u h > s z [801, 4 . 2 .

.

C o r o l l a r 5.5. (GCH) L e t '0 h < K K h a s Y a homomorphic image R wyth

power

I

[A

.

<

If h nB = h

.

c a r d i n a l , t h e n A i t s e l f h a s a-wei ght A If power K w i t h n o homomorphic image of n-weight

. = .

,

K

t h e n e v e r y BA A K is a l i m i t

of

h =

If

,

then t h e r e is a

BA

of

P r o b l e m 1 6 . D e s c r i b e c o m p l e t e l y t h e b e h a v i o r of n-weight u n d e r homomorp h ism s w i t h o u t GCH I n p a r t i c u l a r , is it c o n s i s t e n t t h a t t h e r e is a RA A

of

K

K

.

power

dedSol

such t h a t f o r every

B

,

A

implies

B

nB

-< o1

?

N e x t , s u p p o s e A 5 R , A g a i n , t h e r e i s no g e n e r a l l y v a l i d r e l a t i o n s h i p b etween .rrA a n d nA , as e a s y e x a m p l e s s h o w . The p r o b l e m o f s p e c i f i c a t i o n o f nA a r i s e s a s i n t h e c a s e of homomorphisms. Theorem 5 . 6 . homomorphic image

A has a s u b a l g e b r a s u c h t h a t nC = A

If

P r o o f . L e t f : A-C S u p p o a is dense i n C with

x

<

d -

.

d

S

G R

,

B

,

with

B

Corollary 5.7. such t h a t f o r e v e r y Theorem 5.8 nB = A

.

.

has a

.

B C l e a r l y TIC 5 A . c h o o s e 0 # xd E B

and l e t with

C

In f a c t , <w

).

A = Sg([w]

nC = o1

But no

. <

B

r

h < dedSK A -implies

IAl

>

(2")'

If B ,

A

contradiction.

h a s a homomorphic image

A

nB = h

If

then

t h e c o n v e r s e o f 5.6 d o e s not hold i n g e n e r a l .

I t is e a s i l y c h e c k e d t h a t

with

e x t e n d i n g t h e i d e n t i t y on < A . For a l l 0 f d E D

is dense i n

,

TIB = h

such t h a t

c o n s i s t of a l m o s t d i s j o i n t s e t s , I G I = wl,

[wIw

subalgebra

ID1

{xd : d E D}

Then

A s n o t e d by van Douwen,

let G

R

.

C

K

,

then t h e r e is a ITB - K

5 .

A

and

5 K+

,

A

then

BA

of power

A

A

has a subalgebra

B

Proof. I f A h a s a f a m i l y of p a i r w i s e d i s j o i n t e l e m e n t s of power K+ , l e t B = SgC , C a f a m i l y o f p a i r w i s e d i s j o i n t e l e m e n t s o f power A Otherwise,

-

has them.

A

A

5K

power A

.

(2K)*

i n d e p e n d e n t e l e m e n t s , and w e c a n l e t

is s i n g u l a r s t r o n g l i m i t and w i t h ITB = A

If K Corollary 5.9. A has a subalgebra

h

.

no s u b a l g e b r a of n-weight

If

K

]A]

.

B

C o r o l l a r y 5.10. (GCH) L e t w ( 'A K h a s a s u b a l g e b r a B w i t h nB =

i t s e l f h a s n-weight

b e g e n e r a t e d by

B

K

= p+

K

A.

,

.

If If

A

A

<

=

K

K

5 , K

A

of

t h e n f o r any

, t h e n e v e r y BA A o f is a l i m i t c a r d i n a l , t h e n

then there is a

BA

of power

K

with

,

P r o b l e m 1 7 . D e s c r i b e c o m p l e t e l y t h e b e h a v i o r of n-weight u n d e r s u b a l g e b r a s w i t h o u t GCH I n p a r t i c u l a r , is it c o n s i s t e n t t h a t t h e r e i s a RA A of power

.

dedSol

such t h a t f o r every

Clearly

x(ni

IAi) =

B G A

U.c ~

,

nB

5 o1 ?

l ~ ,i and s i m i l a r l y f o r weak p r o d u c t s ,

It i s

20

J.D. Monk

a l s o c l e a r t h a t f o r any u l t r a f i l t e r

F

on

,

I

n(ni cIAi/F)

<_ IIi c I n A i / F , b u t

n(IIi

=

w e do not know i f t h i s i s b e s t p o s s i b l e .

Problem 18.

For

I t i s well-known

an u l t r a f i l t e r on

F

that

is

uui

n ( * i E IAi) = 111

.

5

,

I

IAi/F)

ni

EInAi/F ?

~ ; is e e Juhisz 1801, 5 . 3 a ) .

I

5

Clearly cellA nA Baumgartner and Komjath [ 8 1 1 showed t h a t nA incA w h i l e S h e l a h [ 8 3 ] gave t h e s t r o n g e r r e s u l t t h a t f o r any i n f i n i t e c a r d i n a l K , i f nA K , then A h a s a n i r r e d u n d a n t p i e of power K ( s e e s e c t i o n 8 ) . A n o t h e r e a s y r e l a t i o n s h i p between o u r f u n c t i o n s i s g i v e n i n

,

1

IAI

Theorem 5 . 1 1 .

cellA

Proof. Let and a =

5 nAcellA .

.

b e d e n s e i n A , \Dl = 71 Then T h i s proves t h e theorem.

D

.

1 X)

F i n a l l y we n o t e t h a t

and

S-a

nA

5 (H+n)A < / A /

f o r any i n f i n i t e

C l e a r l y (H-n)A = w

RA

A

,

with

.

Va E A

<

D (1x1

3X

-<

p o s s i b l e i n both c a s e s .

Similar observations hold f o r

S'n

. 6.

A r a m i f i c a t i o n system i n

A

i s a subset

Vx,y E R ( x y or Vx R ((y E R : x see, e . g . ,

Horn,

. sup{ I R I

Tarski [48] ramA =

Ramification of

R

A

such t h a t

x o r x * y = 0) , 0 !$ R , is inversely well-ordered);

y

5 y}

We set : R

a r a m i f i c a t i o n system i n

.

A)

T h i s is a w i d e l y v a r y i n g f u n c t i o n . One can prove t h a t f o r any t r e e f o l l o w i n g two c o n d i t i o n s a r e e q u i v a l e n t :

( i ) T is i n v e r s e l y i s o m o r p h i c t o a r a m i f i c a t i o n s y s t e m i n ( i i ) t r e e a l g T is embeddable i n A

T

the

A ;

.

Thus we have t h e e q u i v a l e n t d e f i n i t i o n : ramA = sup{lBl ramA

: B

is a t r e e a l g e b r a and

is c l o s e l y r e l a t e d t o

,

ramA = c e l l S A = ( c e l l A ) '

is a cellSA-Souslin

tree.

cellA : cellA then

A

B

can be embedded i n

5 ramA 5 c e l l S A

,

Furthermore,

has a r a m i f i c a t i o n system

Conversely,

if

T

is a

A}

if

whose i n v e r s e

R

K+-Souslin

,

tree, then

+

+

.

Thus t h e r a m t r e e a l g T = I T / = K+ and r e l l ' t r e e a l g T = ( c e l l t r e e a l g T ) = K f o l l o w i n g problem h a s a p u r e l y s e t - t h e o r e t i c a l c h a r a c t e r : ( i t is equivalent t o T non-limit K ( S H ~ ) ) : Problem 19.

Is i t c o n s i s t e n t t h a t

VA(ramA = c e l l A ) ?

.

In fact, Another r e l a t i o n between o u r c a r d i n a l f u n c t i o n s is t h a t r a d i TCA o t h e r w i s e c e l l A 5 nA < ramA Say D is d e n s e i n A , 0 4 D , ID/ = %A For a l l d E D , l e t Md = { y E R : d L y } , where R i s a r a m i f i c a t i o n s y s t e m in A .

.

Then

R =

u d cDMd

,

s o t h e r e is a

inversely well-ordered

d E D

w i t h /Md

chain, contradiction.

I >

cellA

.

But

.

Md

is a n

21

Cardinal functions on boolean algebras We a l s o n o t e t h a t ramA and l e n g t h A a r e , i n g e n e r a l , n o t c o m p a r a b l e . In f a c t , i n i n t a l g R t h e r e i s no r a m i f i c a t i o n s y s t e m o f power wl For, suppose

.

T is such a system. For each r E Q l e t

Let

Lr = ( t E T ' Thus

Ur E Q L r

T' =

Lr

members o f

,

c o n s i s t o f a l l nodes of

T'

: if

is t h e p r e d e c e s s o r o f

s

r E Q

so t h e r e is a n

with

T

of successor l e v e l .

r E s

t , then

Lr

\

uncountable.

a r e pairwise d i s j o i n t , contradiction.

Conversely,

t)

But t h e if

is a n

T

A r o n s z a j n t r e e , t h e n l e n g t h ( i n t a 1 g T ) = w by a t h e o r e m o f B r e n n e r , Monk [ 8 3 1 . I t would be n a t u r a l t o d e f i n e a new c a r d i n a l f u n c t i o n u s i n g t h e n o t i o n of a pseudo-tree. T h i s d o e s n o t l e a d t o a n e s s e n t i a l l y new n o t i o n , h o w e v e r . In f a c t , Kurepa 1 7 7 1 showed t h a t i f T i s a p s e u d o - t r e e o f r e g u l a r s i z e K w i t h no c h a i n s of s i z e K , t h e n T c o n t a i n s a t r e e o f s i z e K Thus

.

sup{lT\ : T s A , T a pseudo-tree} = lengthA*incA ; s u p { l B \ : B i s a p s e u d o - t r e e a l g e b r a , B S A} = 1engthA-ramA

,

The a b o v e r e s u l t o f Kurepa d o e s n o t e x t e n d t o s i n g u l a r K , , a s h e e s s e n t i a l l y On t h e o t h e r h a n d , T o d o r C e v i c o b s e r v e d t h a t by o b s e r v e d i n t h e same p a p e r . adding N Cohen r e a l s t o a model o f GCH o n e c a n g e t a AA B s u c h t h a t w.

KW

B has a pseudo-tree 1 and no c h a i n o f s i z e 1 IBI =

Finally, note t h a t

of s i z e

,

N

b u t no t r e e or

pie

of s i z e

w1

K2

(H+ram)A ; t h e p o s s i b i l i t y o f e q u a l i t y i s

spreadA

r e l a t e d t o P r o b l e m 19. (H-ram)A is always w , although it is not complete Y t r i v i a l t o see t h i s . L e t A be an a r b i t r a r y i n f i n i t e BA Then A+>B f r some BA A w i t h f i n c o w S B c w S u p p o s e R is a n u n c o u n t a b l e r a m i f i c a t i o n Now R = U n E w { x E R : n E x } , so t h e r e i s a n n E w f o r which system i n B C = { x E R : n E x } is u n c o u n t a b l e . But t h e n C is a w e l l - o r d e r e d c h a i n , c o n t r a d i c t ion.

.

B.

BA's:

.

.

Algebraic functions

We now s u r v e y c a r d i n a l f u n c t i o n s h a v i n g t o d o w i t h a l g e b r a i c a s p e c t s of s u b a l g e b r a s , a u t o m o r p h i s m s , and homomorphisms.

7. We l e t

-< , I A 1 .

Subalgebras

be t h e set o f a l l s u b a l g e b r a s o f

SubA

A

.

\A\

Clearly

<

ISubAl

The f o l l o w i n g t o p o l o g i c a l e q u i v a l e n t o f t h e s u b a l g e b r a r e l a t i o n i s

well-known. Theorem 7 . 1 .

Let i~

B

Then

sB

A

be a

= {(G,F)

BA

.

If

: F,G E U l t A

is a n e q u i v a l e n c e r e l a t i o n on

B

is a s u b a l g e b r a of

and UltA

F

,

nB

and i f

= G

A

,

set

n B} .

FfB G

,

t h e n t h e r e is a

c l o s e d - o p e n s u b s e t U s u c h t h a t F E U and G f U , and U i s a u n i o n o f = B classes. C o n v e r s e l y , i f E i s a n e q u i v a l e n c e r e l a t i o n on U l t A s u c h t h a t TF,G E UltA (F G 3 9 c l o s e d open U w i t h F E U , G f U and U a u n i o n o f gc l a s s e s ) , l e t C, = { a E A : sa i s a u n i o n o f :-classes}. Then C -= i s a subs u b a l g e b r a of

A

,

and

1

c,

= F

.

Furthermore,

if

B

is any s u b a l g e b r a o f

A

,

J.D. Monk

22

.

= B R S h e l a h [791,

then

Cm

every regular

K

,

g e n e r a l i z i n g Rubin [ 8 3 ]

-> w

t h e r e is a

V = L

showed a s s u m i n g

w i t h \ A 1 = ISubAl =

A

BA

that for

.

K+

In section 8

.

we n o t e t h a t i f K i s a s t r o n g l i m i t c a r d i n a l and I A l = K , t h e n l S u b A \ = Z K T h e s e two f a c t s a r e e s s e n t i a l l y a l l t h a t is known a b o u t lSubA1. In p a r t i c u l a r , t h e following q u e s t i o n s a r e open. P r o b l e m 20.

Is l S u b A \ a l w a y s a power of

Problem 21. W A l L

Can o n e p r o v e i n w ? For

\A\ =

\SubA\

<

ZK ?

R+>A

,

then

If

and

A s; R

or

t h a t t h e r e is a

ZFC

A

RA

with

s i n g u l a r is i t c o n s i s t e n t t h a t t h e r e i s a

Problem 2 Z S . K

2 ?

K

\5

\SubA

8.

lSubR

BA

A

with

I .

Irredundance

.

A s u b s e t X o f A i s i r r e d u n d a n t i f Vx E X ( x f S g ( X \ {XI) We l e t irrA = sup{\X\ : X irredundant} T h i s is a l a r g e f u n c t i o n . S h e l a h 1 7 9 1 , g e n e r a l i z i n g Rubin [ 8 3 ] , showed a s s u m i n g V = L t h a t f o r e v e r y r e g u l a r K > w t h e r e i s a RA

.

.

A w i t h \ A \ = K + and irrA = K D e v l i n [731 showed t h a t i f K i s r e a l - v a l u e d m e a s u r a b l e , t h e n e v e r y a l g e b r a w i t h c o u n t a b l y many o p e r a t i o n s and w i t h K e l e m e n t s h a s a n i r r e d u n d a n t s u b s e t o f power K S h e l a h [ 8 3 ] showed t h a t i f nA K , then A h a s a n i r r e d u n d a n t p i e o f power K In particular, nA <

.

1.

,

irrA

is s t r o n g l i m i t , t h e n

\A\

and i f

t h a t it i s c o n s i s t e n t t o have

<

Zw

>

No

irrA = \ A 1

Kw

a n d e v e r y a l g e b r a o f power

o p e r a t i o n s h a s a n i r r e d u n d a n t s e t o f power

Problem 23.

Can one p r o v e i n

Problem 24s.

Is it c o n s i s t e n t t h a t

w

<

1

RA

A

BA

with

<

\A1 ?

irrA

with

a n d t h e r e is a

Zw

N(fl 1

*

t h a t t h e r e is a

ZFC

o f power

RA

w i t h no u n c o u n t a b l e i r r e d u n d a n t s e t ?

S h e l a h [ e l ] showed t h e r e i s a c o n c e n t r a t e d BA N1

S h e l a h [ 8 0 a ] h a s shown

1 1

Zw

.

.

, A

assuming with

.

CH

A o f power

irrA = w

an i n t e r v a l algebra.

,

\ A \ = incA = w1

Hence

lengthA

5

.

ramA

Problem 25.

Is

with irredundance

irrA

irrA

ZirrA

f o r any

BA

A

.

irrA = I A \

I t is c l e a r t h a t

BA A . irrA = \ A 1

f o r any

form t h e o r e m o f B r e n n e r , Monk [ 8 3 ] i t f o l l o w s t h a t a l g e b r a , so

N1

R u b i n ( u n p u b l i s h e d ) showed t h a t i t is c o n s i s t e n t t o h a v e a

Note a l s o t h a t

From t h e normal for

pirrA

9.

.

A

a tree

ISubA\

.

.

Subalgebra depth

s d e p t h A = sup{K : t h e r e is a s t r i c t l y d e c r e a s i n g s y s t e m A}

-<

- lSubA\ ?

P r o b l e m s 23 and 24 a r e m e n t i o n e d i n v a n Douwen, Monk, Rubin [SO]

s u b a l g e b r a s of

for

A g a i n t h i s is a l a r g e f u n c t i o n .


<

K>

F i r s t we g i v e some

Of

A

23

Cardinal functions on boolean algebras e q u i v a l e n t s of t h i s d e f i n i t i o n . Theorem 9 . 1 . For any RA A and any i n f i n i t e r e g u l a r c a r d i n a l K t h e following conditions a r e equivalent: ( i ) t h e r e is a s t r i c t l y d e c r e a s i n g s e q u e n c e < R : a < K > of s u b a l g e b r a s of A ;

( t i ) t h e r e is a sequence *fa < K ( b

4 Sg{hp:

<

a


<

Proof. -

Then all

(i)

<


Define

5 <

:


<

a

<

f Sg{bp: a

K(ba

K

=1

K>


and

(ii)

.

<

A

o f e l e m e n t s of

K>

fl

fSg{bp:

ba

Choose

ba

E

<

yz E

i s as d e s i r e d .

(iii)

K

3

(i) :

such t h a t

a <

for a l l

, y5 >

A

s u p { p : bB

K

E

Sg

B = Sg{b

Let

B

P

2

a}

for

.

From t h i s t h e o r e m i t i s c l e a r t h a t

irrA

5 sdepthA .

If

V = L

hence under

f o r every regular

irrA = s d e p t h A =

<

K

1.41 =

K+

.

K

-> o

is s t r o n g l y

A

c o n c e n t r a t e d i n t h e s e n s e o f v a n Douwen, Monk, Rubin [ 8 0 ] ,

with

such t h a t

.

a))

Ba\Ra+l

by i n d u c t i o n :

K>

of e l e m e n t s o f

K>

;

( i i i ) t h e r e is a sequence

va

<

t h e r e is a

then RA

1.21 ;

o f power

<


I n f a c t , suppose

<

sdepthA

A

h)

K+

is a

Let s e q u e n c e a s i n 9 . 1 ( i i ) , where h = \ A \ ; we s h a l l g e t a c o n t r a d i c t i o n . a,bl, b n > 0 be a s i n t h e d e f i n i t i o n o f somewhere d e n s e , a p p l i e d t o n ’ {c,: a < h } Choose a minimum s u c h t h a t a ca a + bl + , + b and

...,

+

bn

= c

.

<

Vi(ca*bi and

.

5

bi) y

<

<

B i ( c obi

P

Clearly

Problem 26.

.

h

a

<

p <

Then c h o o s e

bi) p,y

y

and

<

h

h

such t h a t

by t h e c h o i c e o f

5

a

,

<

ca

such t h a t

V i ( c *b

y

i

cP

<

.. 5a

b.)

+ by +

and

I

c

... *C

P v

contradiction.

i r r A = sdepthA ? 10.

Subalgebra length

.

s l e n g t h A = sup{lCLl : G i s a set of s u b a l g e b r a s o f A s i m p l y o r d e r e d by c} This is a large function. Thus s d e p t h A slengthA N o w , a s shown by Kurepa [ 5 7 ] , f o r any K w we have dedK = s u p { h : FK h a s a c h a i n o f s i z e A } Now X is l i n e a r l y o r d e r e d by i f X S A is i r r e d u n d a n t , X i n f i n i t e , and i f G S s , t h e n <SgY : Y E G > i s a n i s o m o r p h i s m from G i n t o SubA Thus s l e n g t l i A > dedlXl , hence

.

.

.

sup{dedK :

K

<

irrSA)

5

slengthA

.

Thus t h e most n a t u r a l q u e s t i o n c o n c e r n i n g t h i s c a r d i n a l f u n c t i o n i s : Problem 27.

s l e n g t h A = lSubA1 ? 11.

A subset generates,

X

of

A

Independence

is independent i f X f r e e l y g e n e r a t e s t h e s u b a l g e b r a i t - z $ 0 f o r any two d i s j o i n t f i n i t e rPI*II

or equivalently, i f

.

z € Z

s u b s e t s Y and 2 of X S e t indA = s u p { \ X ] : X c A , X i n d e p e n d e n t } . This i s a w i d e l y v a r y i n g f u n c t i o n . T h i s n o t i o n h a s been w i d e l y s t u d i e d and a d e t a i l e d s u r v e y c a n be f o u n d i n Monk [ 8 3 1 .

J.D. Monk

24

For e a c h l i m i t c a r d i n a l K , t h e r e is a BA A w i t h i n d e p e n d e n c e K n o t attained. I n d e p e n d e n c e i n s u b a l g e b r a s was c h a r a c t e r i z e d by S a p i r o v s k i ? [801; i f A 5 B , then indR = indA U s u p { i n d ( R / F g F ) C o n c e r n i n g homomorphisms, c l e a r l y

I indA

indB

: F

if

.

E UltA}

.

A4)B

If

A

h a s an

i n d e p e n d e n t set of power A w i t h AW = A , t h e n A + B f o r some R w i t h indB = A , and one c a n s p e c i f y w h e t h e r indB i s a t t a i n e d o r not u n d e r some m i l d c o n d i t i o n s ; m o r e o v e r , t h i s r e s u l t is i n a s e n s e b e s t p o s s i b l e . U s i n g a c o n s t r u c t i o n of T. Cramer [ 7 4 ] , f o r any

K

2

t h e r e is a

2w

hered-

i t a r i l y a t o m i c BA A (which t h u s h a s i n d e p e n d e n c e 0) such t h a t ind(WA) = T . C a r l s o n h a s shown t h a t i f i s a s y s t e m of i n t e r v a l a l g e b r a s ,

\AilL

and

i

for a l l

2

E

,

I

2,11 I Problem 2 8 . for all

If

i E I

,

then

2,II

i n d ( n i E I A;)

I .

E I > i s a s y s t e m of i n t e r v a l a l g e b r a s w i t h


: i

is

ind(n

.

K

\Ai

12 2

A ) = 3,111 ?

i E I i

S a p i r o v s k i ? [ 8 0 ] showed t h a t i f n e i t h e r A power K , t h e n n e i t h e r does A*B Hence

nor

.

B

has an independent subset of

.

Monk 1831 showed t h a t i f A i s WCC , t h e n ( i n d A ) W = indA B a l c a r and F r a n z k [ 8 3 ] showed t h a t any c o m p l e t e BA A h a s an i n d e p e n d e n t s u b s e t of power \ A 1 The r e l a t i o n s h i p b e t w e e n c e l l and i n d i s n o t f u l l y known, a l t h o u g h some s t r o n g r e s u l t s have been o b t a i n e d ; i n p a r t i c u l a r , t h e s i t u a t i o n is c l e a r i f V = L S h e l a h [ E O ] showed t h e f o l l o w i n g .

.

.

.


(1) L e t K = c e l l S A Suppose e i t h e r t h a t A i s r e g u l a r a n d V p < A ( p o r A = K i s weakly compact. Then among any A e l e m e n t s of A t h e r e a r e independent elements.

<

A), A

Assuming V = L , (1) i s b e s t p o s s i b l e f o r A r e g u l a r . Most of t h e e x a m p l e s needed t o show t h i s a r e e a s y , o r well-known, b u t t h e f o l l o w i n g example of A r g y r o s [ 8 2 ] is more i n v o l v e d : If

(2)

I-1

then t h e r e is a

BA

A

i n d e p e n d e n t s e t o f power Without

V = L

o f power

a

BA

Problem 31.

(In

satisfying the

p+

for a l l

(cfp)+-cc

K

<

cfp

,

w i t h no

t h e r e a r e s e v e r a l open p r o b l e m s ,

(In

of power

( c f p ) K = cfli

p+

P r o b l e m 2 9 . ( I n ZFC) I f i s t h e r e a BA of power K power K ? Problem 30.

,

is s i n g u l a r s t r o n g l i m i t , 2’ = p+

ZFC) A

K

i s s t r o n g l y i n a c c e s s i b l e b u t n o t weakly c o m p a c t , s a t i s f y i n g t h e K-cc w i t h no i n d e p e n d e n t set o f

Assume

p

<

satisfying the ZFC)

Assume

cfp

v

<

v+-cc

<

A r e g u l a r . Is t h e r e a BA o f power i n d e p e n d e n t set of power A ?

<

2’

p

A

<

A

-< 2 ’

, with

A

regular.

Is t h e r e

w i t h no i n d e p e n d e n t s e t of power

A

-< p c f p

and

Vp

<

p (pcfp

s a t i s f y i n g t h e (cfp)+-cc

<

p)

w i t h no

h ?

, with

25

Cardinal functions on boolean algebras A

For

s i n g u l a r , we f i r s t n o t e t h e f o l l o w i n g r e s u l t o f S h e l a h 1801:

( 3 ) Le t K b e r e g u l a r , and l e t A be s i n g u l a r w i t h the following conditions a r e equivalent: ( a ) e v e r y BA of power A s a t i s f y i n g t h e K-cc power A

Vp

fur a l l

A

, if

satisfies

A

K-cc , t h e n

<

A K: (

.

A)

Then

h a s an i n d e p e n d e n t set of

.

(b)

<

Ya

cfAA

The p r o p e r t y ( b ) h a s been w i d e l y s t u d i e d ; see C o m f o r t , N e g r e p o n t i s [ 8 2 ] . The i n i t i a l c a r d i n a l i t y c o n d i t i o n i n ( 3 ) is t r i v i a l u n d e r GCH ( t r u e i f K < A, false if K > A). W i t h o u t GCH , i t s f a l s i t y l e a d s t o t h e f o l l o w i n g q u e s t i o n . Pro b lem 3 2 .

(In

ZFC)

S uppose t h a t

K

i s u n c o u n t a b l e and w e a kly i n a c c e s s i b l e ,

2' < A f o r a l l v < K , and ZcK = A i s s i n g u l a r . Is t h e r e a s a t i s f y i n g K-cc w i t h no i n d e p e n d e n t s e t o f s i z e A ?

A

of power

BA

I f K is r e g u l a r , A i s s i n g u l a r , and Bp < A ( p c K < A ) , t h e n ( 3 ) a p p l i e s , a n d w e are c o n c e r n e d w i t h ( b ) . C o n d i t i o n ( b ) is f u l l y r e s o l v e d under V = L With o u t V = L t h e f o l l o w i n g probl em a r i s e s . We s a y t h a t A h a s p r e c a l i b e r K

.

Ba E K A

if

r}

R r E [ ~ ] ~ ( { a , :a E

( I n ZFC) I f Pro b lem 3 3 . f o r some p < A a n d v < K precaliber A ?

K

,

has

and A a r e r e g u l a r , K < A , A i s t h e r e a BA s a t i s f y i n g K-cc

P r o b l e m s 28-33 a r e m e n t i o n e d i n Monk ( 8 3 1 . see a l s o T o d o r z e v i k [ - I . 12.

.

fip )

>

2w , and without

pv

2A

I n c o n n e c t i o n w i t h t h e s e problems,

Homomorphism t y p e a n d s p e c t r u m

I2

L e t hA = m i n { \ B l : A+> B , IB w} for A infinite. Also l e t hsA = { \ B I : A+)B , I B I w} T h i s i s a d i f f e r e n t k i n d of f u n c t i o n from t h e hsA a l w a y s h a s a l a r g e s t preceding ones; i t s values a r e sets of c a r d i n a l s . element I A I and a s m a l l e s t one hA We are c o n c e r n e d w i t h t h e p o s s i b i l i t i e s We m e n t i o n some known r e s u l t s a b o u t H : f o r hA and hsA

.

L

.

.

(1)

( S . Koppelberg

(2)

(S. K o p p e l b e r g [ 7 7 1 )

(3)

(S. K o p p e l b e r g 1 771) a l g e b r a , t h e n hA = w

(4)

w

Problem 3 4 .

<

.

and

A

for

infinite

A

. .

WCC

w < I A I < 2 w 3 hA = w c a n b e embedded i n a f r e e

an d

A

or i n a i n t e r v a l

BA

C 0 n ( 2 ~ is a r b i t r a r i l y l a r g e and f o r e v e r y t h e r e is a BA A w i t h I A l = hA = A)

cfA

If

P r o b l e m 35".

,

MA If

(W. J u s t , u n p u b l i s h e d ) with

2w

hA = 2w

[751)

.

h a s a n i r r e d u n d a n t set of g e n e r a t o r s , i s

I s it c o n s i s t e n t t o have a

BA

A

with

A

-< 2w

<

A

hA = w ?

I A I = hA = A

,

w

<

cfA = w ?

N o w we t u r n t o

hs

.

A

If

t h e n by B a l c a r , F r a n e k [821 i n t e r v a l a l g e b r a s , we have: Theorem 1 2 . 1 . increasing in

L

if

, w h ere

is f r e e , t h e n

,

hsA = [ w , l A \ ]

n

{K

is a l i n e a r o r d r i n g and

L K

.

hsA = [ w , \ A \ ] : cw =


i s an i n f i n i t e c a r d i n a l , t h e n

<

A

If

.

K}

i s complete,

Concerning

K> [O,K]

is s t r i c t l y 8

hs(inta1gL)

.

J.D. Monk

26 P r o o f . I t s u f f i c e s t o show t h a t x , y E L and Va < K [ ( a a < x i f f a,

<

is a convex e q u i v a l e n c e r e l a t i o n on f o 1lows.

From Theorem 12.1 it f o l l o w s t h a t K

.

y)

hs(inta1gL . and ( x < a

L

with

E

K

S

[W,K']

y

x

.

a,)]

iff Then

[W,K]

11.1

f o r any i n f i n i t e c a r d i n a l

-> (ZK)',

then

.

hs(inta1gL)

.

h s ( i n t a 1 g I R ) = {w,Zo)

Theorem 1 2 . 2 .

<

c a s s e s , and t h e t h e o r e m

K

hs(inta1gr) =

Moreover, by t h e ErdGs, Rado t h e o r e m , i f

Define iff y

.

P r o o f . Suppose f : i n t a l g I R + ) A , where I A l > w Then f is deterby a convex e q u i v a l e n c e r e l a t i o n E on IR w i t h \ W E \ = \ A ) Now L ' = U { k : k is an E - c l a s s , \ k l > 1) i s B o r e l , so L" = l R \ L ' is also. Clearly

.

= \A1

IL"I

.

I A l = 2 w by t h e A l e x a n d r o f f , H a u s d o r f f t h e o r e m .

Hence

.

I f A is h e r e d i t a r i l y a t o m i c and i n f i n i t e , t h e n hA = w In fact, let [a] b e a n atom of A/IgAtA ; t h e n A 4 j A a s f i n c o K --)t f i n c o w f o r some K Also,

r

.

05 5

i t can be shown t h a t i f A i s h e r e d i t a r i l y a t o m i c , K A , then [ K , K c K ] On t h e o t h e r hand, JuhAsz, Nagy and Weiss [791 c o n s t r u c t e d u n d e r hsA f 0 V = L a BA A o f power N w l w i t h K, $? hsA van Douwen [ a ] c o n s t r u c t e d a n

.

n

.

A

h e r e d i t a r i l y a t o m i c BA the following question.

2w

of power

Problem 36. ( I n ZFC) I f A i s h e r e d i t a r i l y a t o m i c and A have a homomorphic image of s i z e K ? Problem 3 7 5 .

Con( VA(A

h e r e d i t a r i l y a t o m i c and i n f i n i t e hs :

We n o t e some o t h e r e a s y f a c t s a b o u t

5K

,

n

(5)

If

(6) (7)

I f A h a s a f r e e s u b a l g e b r a o f power hs(AxB) = hs(A*B) = hsA U hsB

w

\A\

then

hsA

[K,2K]

f

(8) I f

w

-< <

F o r , we t a k e

K

,

2O

then there is a

p w

A = FrK x

Theorem 1 2 . 3 .

(CH)

w

K

A

BA

\A\

9

van Douwen a s k e d

>

K = K

w

,

does

hsA = [ w , I A l ] ) ) ?

.

0

.

.

hsA = {w,Zw}

with

,

then

such t h a t

hsA

n

f

[K,K,]

0

.

hsA = [ w , ~ ]U {2w}

.

.

I f t h e r e is a

BA

A

such t h a t

hsA = {w,w

2

1

,

then

t h e r e is a Kurepa f a m i l y .

-

P r o o f . By f a c t ( 6 ) a b o v e , A h a s no u n c o u n t a b l e i n d e p e n d e n t s u b s e t . Hence A d o e s n o t s a t i s f y c c c , Hence A h a s a homomorphic image B s u c h t h a t f i n c o w l C B E 60, C l e a r l y , s t i l l hsB = {w,w 1 I f r is any c o u n t a b l e 2 f o r b E B is a homomorphism, and hence s u b s e t of w1 , t h e n b H b

.

.

nr

{b

nr

: b E B}

is countable.

I t is c o n s i s t e n t with

CH

Thus

B

is a Kurepa f a m i l y .

t h a t t h e r e i s no Kurepa f a m i l y , hence no

BA

as i n

12.3. Problem 38". I s it c o n s i s t e n t w i t h hsA = { w , w 1 ? 2

CH

that there is a

BA

A

such t h a t

21

Cardinal functions on boolean algebras 13.

Endomorphisms

.

EndA is t h e set o f a l l e n d o m o r p h i s m s o f A Since clearly IUltAI < lEndA\ , we a r e d e a l i n g h e r e w i t h a " l a r g e " c a r d i n a l f u n c t i o n , a n d t h e most i n t e r e s t i n g q u e s t i o n is t o c o n s t r u c t B A ' s A w i t h \ E n d A \ s m a l l . Theorem 1 3 . 1 .

A

2w

,

and

Suppose

I=

IEndCintatgL)

\

Proof. -

UlttintalgA)

=

IintalgL

is a c o m p l e t e d e n s e l i n e a r o r d e r i n g o f power

L

is a d e n s e s u b s e t o f

D

o f power K a n d f u n c t i o n s from X Recall that if

I-L

1x1

= into

.

K

A

=

X

i s any i n f i n i t e c a r d i n a l ,

A

RA

A

If

.

Then

continuous

wv

<

Bv

p ( p v = 11)

,

there

.

\ E n d ( i n t a l g 7R)

1

C o r o l l a r y 13.4.

(GCH)

i s i n f i n i t e and r e g u l a r , t h e n t h e r e is a

If

I A l = IEndAl =

such t h a t

K

p,

.

C o r o l l a r y 13.3.

= 2

>

w i t h a d e n s e s u b s e t of

is a n i n f i n i t e c a r d i n a l a n d

IL

\ A 1 = IEndAI = 2 "

such t h a t

A

i s minimum s u c h t h a t pv

v

and

.

C o r o l l a r y 13.2.

AK =

where

hK = A

Hence t h e r e a r e a t most as d e s i r e d .

t h e n t h e r e i s a c o m p l e t e l i n e a r o r d e r i n g of power Thus: power p

is a

,

K

is a l i n e a r l y o r d e r e d space w i t h a d e n s e subspace

X

,

of power

L

.

0

BA

.

'K

Problem 39. (GCH) For A a l i m i t c a r d i n a l o r t h e s u c c e s s o r o f a s i n g u l a r \ A \ = \EndA\ = A ? c a r d i n a l , i s t h e r e a BA A s u c h t h a t I t is e a s y t o s e e t h a t i f w

1

<

2O

\A1

2 w1

i m p l i e s t h a t t h e r e i s no

, A

.

-> 2w

then

IEndAI

with

\ A \ = IEndA

\

Thus t h e a s s u m p t i o n

w

=

1

P r o b l e m 40. I n ZFC c a n o n e show t h a t t h e r e a r e a r b i t r a r i l y l a r g e \A1 = \En&\ = K ? t h e r e i s a BA A w i t h Problem 41. BA A w i t h

Under a n y s e t - t h e o r e t i c a l a s s u m p t i o n s , I A l = A. a n d \EndAl = K ?

Problem 42.

Is

IEndA\

<

\SubA\

A

if

<

K

<

2A

K

,

f o r which

is t h e r e a

?

I

I

1

T h e d e r i v e d f u n c t i o n s H+ \End , H- lEnd , S' lEnd , S- !End u n r e l a t e d t o \End\ ; they have not been i n v e s t i g a t e d . 14.

\

appear to be

Automorphisms

.

AutA i s t h e g r o u p o f a u t o m o r p h i s m s o f A T h i s is a w i d e l y v a r y i n g T h e r e a r e many f u n c t i o n , n o t in g e n e r a l r e l a t e d t o most o f o u r o t h e r f u n c t i o n s . l A u t A \ = 1 (A i s papers studying t h i s group. There a r e i n f i n i t e BA's A w i t h t h e n c a l l e d rigid). I f AutA is f i n i t e , t h e n it i s i s o m o r p h i c t o a f i n i t e symmetric group. A

with

\A\ =

K

McKenzie, Monk [ 7 3 ] showed t h a t f o r any and

v a n Douwen [801 showed

\AutA\ = w ; assuming

,

MA

Con(ZFC + 2w = w2 + 3BA A ( I A \ = w1

McKenzie, Monk [ 7 3 ] it is a l s o shown t h a t if w i t h \ A ] = A a n d IAutAl = K For a n y A

.

w

<

K

there

K

2

\AutA\ = w

< A ,

7s

a

,

Zw

t h e r e is a

implies lAutA\ = w))

\A\

.

BA

1. 2w. In

t h e n t h e r e i s a BA BA A w i t h I A l = A

A

J.D. Monk

28 lAutA\ = 2

and

h

.

8,

In case

< A <

K

< 2 A .i t

t h e r e i s a BA A w i t h \ A \ = A and IAutAl = have been o b t a i n e d by Roitman and S h e l a h . Problem 4 3 s .

\A\

h

=

,

i s n o t c o m p l e t e l y c l e a r when K

.

Some c o n s i s t e n c y r e s u l t s

D e s c r i b e f u l l y when i t i s p o s s i b l e t o have a h No < A < K < 2 K , for

with

A

RA

.

(AutAl =

C o n s i d e r a t i o n of t h e f u n c t i o n s r i s e t o some n a t u r a l q u e s t i o n s :

H+

H-

and

associated with

/AutA\

Problem 4 4 . Is t h e r e a r i g i d RA A homomorphic image of A is r i g i d ?

such t h a t e v e r y i n f i n i t e a t o m l e s s

Problem 4 5 . Is t h e r e a BA r i g i d homomorphic image?

A

IAutAl = w

The f u n c t i o n s

S-IAutl

S+[Autl

and

with

h a s no i n f i n i t e

A

can vary widely.

Topological functions:

C.

such t h a t

gives

open s e t s o r i d e a l s

Next we d e a l w i t h f u n c t i o n s t h a t a t l e a s t i m p l i c i t l y a r e of a t o p o l o g i c a l nature. The f i r s t o n e s c o n c e r n open s e t s .

15.

Ideals

TdA = {I : I i s a n i d e a l of A) . Again we a r e d e a l i n g w i t h a " l a r g e " < \ I d A \ . We n o t e a l s o t h e f o l l o w i n g e a s y theorem o f function, since IUltAI Loats [771: Theorem 1 5 . 1 .

IIdAI

-<

\SubAl

. .

ProoE. -

L e t X = {I : I i s a p r o p e r non-maximal i d e a l of A} For a l l l e t f 1 = I U -I ; it is e a s i l y checked t h a t f i s one-one. To f i n i s h t h e p r o o f i t s u f f i c e s t o show t h a t l U l t A l 1x1 Fix F E U l t A . For any G E U l t A \ { F ) l e t gG = F n G Thus g maps U l t A \ {F} i n t o X , so i t s u f f i c e s t o show t h a t g i s one-one, which i s e a s y .

I

E

X

.

.

The c o n s t r u c t i o n o f S h e l a h 1791, g e n e r a l i z i n g Rubin [ 8 3 ] , y i e l d s a s s u m i n g f o r e v e r y r e g u l a r K 1. w a BA A w i t h i s a one-one c o r r e s p o n d e n c e between i d e a l s o f A problem c o n c e r n i n g what IIdAl c a n be i s r e l a t e d problem c o n c e r n i n g t h e number of open s e t s . Thus know t h a t lIdAl is never strong l i m i t s i n g u l a r ,

V = L

lIdA

Iw =

lIdAl

always h o l d s ( i t d o e s under

Problem 4 6 .

(In

Problem 4 7 . IIdAl = h ?

If

ZFC) w

<

K

h

K+ Since t h e r e and open s e t s i n U l t A , t h e t o t h e well-known t o p o l o g i c a l from JuhAsz [ 8 0 1 , 4 . 5 , 4 . 7 we and t h a t i t i s l i k e l y t h a t

GCH).

[ I d A I w = IIdAl ?

IS

<

.

I A I = lIdA\ =

<

2K

,

is t h e r e a

BA

A

with

\A1 =

and

K

V a r i o u s p o s s i b i l i t i e s i n Problem 4 7 w a r e e x c l u d e d by known r e s u l t s on t h e number o f open s e t s . Note t h a t [Id"\ = 2 whenever ( A \ = w Note t h a t

.

\ I d ( i n t a l g IR) Problem 4 8 .

=

Is

2O

,

\IdAl

ISub(inta1g R) IEndAl

I

=

for all

22w

,

I E n d ( i n t a 1 g R)

A

,

or

lEndA\

< -

I

=

IIdAl

2w always?

29

Cardinal functions on boolean algebras 16.

Spread

We l e t s p r e a d A = s u p { l X I : X is a m i n i m a l set o f g e n e r a t o r s o f IgX) There a r e s e v e r a l e q u i v a l e n t s of t h i s notion. For o n e o f them, we c a l l X

ideal-independent

Ym E w \

if

Vx

E "x

(x

atomic} = sup{cellB : A - - ? ) R }

To show of

A

K

I

< -

with

K

5 '

cellR

.

let

2

IgX

.

111

Let

E X

a,bo,cO,...,bm-l,cm-l

So

For

K~

< 1 -

K

5

K~

K

,

+ x

.

m- 1

X

K ~ , K ~ , K ~ , K ~ , K C ~ learly

J = 1gIa-b : a,b E X , a a E J

< bo*co -

a

diction.

...

.+

Kl = K2 be i d e a l i n d e p e n d e n t ; we f i n d a homomorphic image

X

loss o f g e n e r a l i t y , s a y

f X 1

i s i d e a l i n d e p e n d e n t , so

X

iff

B = A/J I t s u f f i c e s t o show t h a t we t h e n h a v e

where

x

<

Vi

.

)

.

Let t h e c a r d i n a l s i n q u e s t i o n be

Proof. -

minimal set o f g e n e r a t o r s of

=

we h a v e s p r e a d A = s u p { / X I : X i s i s d i s c r e t e } = s u p { l A t B I : A--P)B ,

Theorem 1 6 . 1 . F o r any i n f i n i t e BA A i d e a l i n d e p e n d e n t } = s u p { / D I : D C IJltA , D B

one-one

+

and

...

+ b

b.

1

m (a # bi)

m-1

1

a

Then

.

,

. B

and l e t

Assuming o t h e r w i s e ,

m-1

for all

c.

f

.

.C

b}

#

a E X

for all

is a

i bo

i

<

.+

...

m

.

Hence w i t h o u t

+ bm-l

,

contra-

5 ' suppose

A-)R

and

is a system o f p a i r w i s e d i s j o i n t

D

R ; w e f i n d a n a t o m i c homomorphic image

C of A with I D \ = l A t C \ . R L e t C = Sg D and by t h e S i k o r s k i e x t e n s i o n t h e o r e m l e t f : R + extend the i d e n t i t y on C Then r a n g e ( f ) is a s desired. So K < K 5 4 ' K~ K~ : l e t B b e a n a t o m i c homomorphic image o f A ; we f i n d a d i s c r e t e

elements of

.

5

subset of U l t A B l e t Fb = { a {Fb : b

6 AtR)

Finally, choose

.

IAtBl elements. Say fa] C l e a r l y Fb

5

f : A 4 ) B F o r e v e r y atom b i s a n u l t r a f i l t e r on A , and

.

of

is d i s c r e t e . for

aF E A

{ a : F E D} F

with

6A : b K

<

3-

so t h a t

K

1 saF

let

nD

be d i s c r e t e .

D E UltA = {F)

.


Then

F

Thus f o r e v e r y

: F E D>

i s one-one

F E D and

i s i d e a l i n d e p e n d e n t , as d e s i r e d .

Note t h a t a l l o f t h e e q u i v a l e n t d e f i n i t i o n s g i v e n i n 1 6 . 1 i n v o l v e s u p s , and t h u s g i v e r i s e t o a t t a i n m e n t p r o b l e m s . The p r o o f o f 1 6 . 1 s h o w s , h o w e v e r , t h a t a l l t h e s e s u p s a r e a t t a i n e d or n o t a t t a i n e d s i m u l t a n e o u s l y . Now by J u h e s z [SO], 4.2 , s p r e a d is a t t a i n e d f o r s i n g u l a r s t r o n g l i m i t c a r d i n a l s , f o r s i n g u l a r I f K is c a r d i n a l s o f c o f i n a l i t y w , and f o r w e a k l y compact c a r d i n a l s . s t r o n g l y i n a c c e s s i b l e and n o t w e a k l y c o m p a c t , a n d L i s a ~ - S o u s l i n l i n e , t h e n spread(inta1gL) = K not a t t a i n e d . P r o b l e m 4 9 s . Is it c o n s i s t e n t t o h a v e a s i n g u l a r and n o t a t t a i n e d ? A r e s u l t o f S a p i r o v s k i i [761 i m p l i e s t h a t

indA

< spreadA i incA

and

cellA 17.

5 spreadA

.

IAI

.

-<

such t h a t

ZSpreadA

spreadA

.

is

Note a l s o t h a t

Ideal generation

igA = min{K : e v e r y n o n - p r i n c i p a l elements]

A

BA

T h i s is a l a r g e f u n c t i o n .

i d e a l of

A

c a n b e g e n e r a t e d by

5

K

T h e r e a r e numerous e q u i v a l e n t s o f t h i s

J.D. Monk

30

n o t i o n , g i v e n in t h e n e x t t h e o r e m . I n p a r t i c u l a r , t h i s t h e o r e m shows t h a t iRA i s t h e same a s t h e h e r e d i t a r y L i n d e l o f d e g r e e o f U l t A , a well-known f u n c t i o n . Theorem 17.1. F o r A i n f i n i t e we h a v e igA = S U ~ { K : t h e r e is a s t r i c t l y i n c r e a s i n g s e q u e n c e of i d e a l s o f t y p e K ) = sup{K : t h e r e is a s t r i c t l y i n c r e a s i n g sequence of f i l t e r s o f t y p e K ) = S U P ( K : t h e r e is a s t r i c t l y i n c r e a s i n g s e q u e n c e of o p e n s e t s i n IJltA o f t y p e K ) = sup{K : t h e r e i s a s t r i c t l y d e c r e a s i n g s e q u e n c e o f c l o s e d s e t s i n L l l t A o f t y p e K } = sup(^ : t h e r e i s a r i g h t - s e p a r a t e d s e q u e n c e i n I l l t A o f t y p e K } ( s e e J u h i s z [ 8 0 ] ) = sup{LS : S s I J l t A ) ( s e e Juh'asz [ S O ] ) = r n i n { K : e v e r y S C U l t A h a s t h e p r o p e r t y t h a t a n y K ) = min{K : e v e r y c l o s e d s e t is t h e o p e n c o v e r h a s a s u b c o v e r o f power

I

'<

open s e t s } = min{u : e v e r y open s e t is a u n i o n o f < E KA Y U < K Tfr E [ a ] (Xa PE); i n t e r s e c t i o n of

K

K

f

5

TF

.

.

From t h i s t h e o r e m i t is c l e a r t h a t s p r e a d A igA The o n e - p o i n t c o m p a c t i f i c a t i o n o f t h e Kunen l i n e , c o n s t r u c t e d u n d e r CH , g i v e s a RA igA = w1 and s p r e a d A = o ; s e e J u h h s z , Kunen, Rudin [ 7 6 ] , Problem 50.

(In

Is t h e r e a

ZFC)

RA

A

with

<

spreadA

A

with

igA ?

.

On t h e o t h e r h a n d , Note t h a t i g ( i n t a l g IR) = (I) , w h i l e i n c ( i n t a 1 g lR) = Zo incA = o and igA = w1 f o r a n a l g e b r a A c o n s t r u c t e d by R a u m g a r t n e r , K o m j a t h

1811 u n d e r

0.

P r o b l e m 51.

(In

ZFC)

Is t h e r e a

RA

A

with

<

incA

igA ?

A l s o n o t e t h a t i f L i s a Souslin l i n e o b t a i n e d f r o m a n o r m a l Souslin t r e e i n t h e u s u a l way ( u s i n g b r a n c h e s ) , t h e n I r a m ( i n t a 1 g L ) = w1 w h i l e i g ( i n t a l g 1 , ) = fi) Since cellA igA f o r a l l A , w e see from t h e d i s c u s s i o n i n s e c t i o n 6 t h a t i t is n o t p o s s i b l e t o g e t i n ZFC a n a l g e b r a A w i t h igA < ramA

I

5

.

.

18.

Ideal depth

hdA = SUP{K : t h e r e i s a s t r i c t l y d e c r e a s i n g s e q u e n c e of i d e a l s o f A o f order type K } This large function, naturally called ideal depth, coincides

.

w i t h t h e h e r e d i t a r y d e n s i t y o f t h e S t o n e s p a c e , by t h e f o l l o w i n g t h e o r e m . Theorem 18.1. F o r A i n f i n i t e we h a v e hdA = S U ~ { K : t h e r e i s a s t r i c t l y d e c r e a s i n g sequence o f f i l t e r s of t y p e K ) = sup(K : t h e r e is a s t r i c t l y d e c r e a s i n g s e q u e n c e o f o p e n s e t s i n I l l t A o f t y p e K ) = sup{^ : t h e r e i s a s t r i c t l y i n c r e a s i n g s e q u e n c e o f c l o s e d s e t s i n U l t A o f t y p e K } = SUP{K : t h e r e is a l e f t - s e p a r a t e d s e q u e n c e i n U l t A o f t y p e K } ( s e e JuhAsz [ 8 0 1 ) = s u p ( d e n s i t y S : S E U l t A ] = min{lc : e v e r y S c U l t A h a s a d e n s e s u b s e t o f power < K} = sup{density U l t R : A B} = S U ~ { K : 'I< x ~ :a < K > B} = sup{ilR : A

.

The p r o o f o f t h i s t h e o r e m Eollows f r o m d u a l i t y t h e o r y and J u h A s z I801 C l e a r l y spreadA hdA A l s o , hdA i n c A ; i f A- 3)B t h e n nR

5

incA

.

by s e c t i o n 5 , s o hdA incA A l s o n o t e t h a t hdA sdepthA

5

5

by 18.1. by t h e p r o o f of 15.1.

5 i n c R '<

I t is a l s o c l e a r t h a t

< hdA. -

A g a i n t h e o n e - p o i n t c o m p a c t i f i c a t i o n of t h e Kunen l i n e ( J u h A s z , Kunen, w1 R u d i n [ 7 6 ] ) g i v e s u n d e r CH a RA A w i t h s p r e a d A = w and hdA

nA

2

MA + -CH

S z e n t m i k l o s s y [a01 showed t h a t u n d e r Problem 52.

(In

ZFC)

Is t h e r e a

RA

A

t h e r e i s no s u c h

such t h a t

spreadA

<

A

.

hdA ?

h a s i d e a l d e p t h ZK by 18.1 ( s i n c e i t h a s s p r e a d Z K ) b u t h a s a l g e b r a i c density K i n t a l g IR h a s i d e a l d e p t h w , b u t s u b a l g e b r a d e p t h a n d i n c o m p a r !?K

.

Cardinal functions o n boolean algebras ability 2" . S a p i r o v s k i ; [ 7 4 1 showed h d A ( u n d e r MA + 1 C H t h a t igA = w 3 hdA = w .

19.

31

(spreadA)+

.

J u h L s z [ 7 1 1 showed

Ideal leneth

We s e t i l e n g t h A = s u p { j X I : X i s a c h a i n u n d e r C o f i d e a l s o f A} ; t h i s i s i d e a l l e n g t h . C l e a r l y igA , hdA , l e n g t h A i l e n g t h A , which i s t h u s a

large function.

Note t h a t

ilengthA

-<

by t h e p r o o f o f 1 5 . 1 .

slengthA

I t is

.

p o s s i b l e t o h a v e i gA < i l e n g t h A , hdA < i l e n g t h A , a n d i l e n g t h A < s l e n g t h A Also n o t e t h a t i t i s c o n s i s t e n t t h a t t h e r e is a RA A w i t h i l e n g t h A < \ I d A l : l e t A b e t h e RA of f i n i t e a n d c o f i n i t e s u b s e t s of w1 Th e n i l e n g t h A =

.

W

de d wl

, while W

d e d w1 < 2

\idA(

.

By M i t c h e l l [ 7 2 1 i t i s c o n s i s t e n t t o h a v e

.

(In

Is t h e r e a

ZFC)

D.

in

.

I d e a l l e n g t h h a s b e e n s t u d i e d i n a g e n e r a l t o p o l o g i c a l s e t t i n g by

by G i n s b u r g [ m ] Problem 53.

= 2

A

BA

with

ilengthA

<

IIdAI ?

points or ultrafilters

Topological functions:

Ou r f i n a l g r o u p o f f u n c t i o n s a r e c o n c e r n e d w i t h u l t r a f i l t e r s , i . e . , p o i n t s IJltA

.

20.

It i s w 11-known t h a t

.

\A\

Ultrafilters

5 IUltAI .

Clearly also

IUltAl

IEndAl , 21AT Concerning t h e p o s s i b l e r e l a t i o n s h i p s between we m e n t i o n some w el l - kn own or e a s y f a c t s :

(1)

For

A

If

(3)

Con(2FC +

(4)

(See T a l l [80] ; t h e r e s u l t

Con( tlA(IA1 = w1 Problem 54. Th e

\ A 1 = w1

,

P r o b l e m 55.

-> 1 , w

B

,

1

1)

w1 [UltAl = 2

with

lUltRl =

K+

GCH

or

, K

A

BA if

++

K

Problem 5 6 .

.

B

2

and

.

= Xw2) 3

.

w),

.

IUltAI such t h a t

with

BA

A

c e l l A = indA = w,

2 ?

then

>

h a s a homomorphic

is i n f i n i t e a n d

A B

of

?

2ce11A.indA A

A ; also

WCC

,

then

lUltBIW =

IUltB

IUltB

.

with

IUltBl = w

lUltAI

5 lUltAI

If

v a n Douwen [ 8 1 ]

Is t h e r e a

(GCH)

ho mo mo r phi c i m age

>

and

0 is

with

++

f o r e v e r y i n f i n i t e hom om orphic im a ge

i n t h i s c a s e ; see, e . g . ,

K

. Is t h e r e a

( I n ZFC).

\UltA\

and

Con( 3 i n a c c e s s i b l e )

IAl

for

,

I l J l t A ] = Kel

+ 2 w = w1 + 2w1

Describe the possibilities

IAl =

with

is d u e t o K u n e n . )

= I I J l tAI E {wI,Zw

By a n e a s y a r g u m e n t , u n d e r i ma g e

I A l = Kw

such t h a t

A

c o n s t r u c t e d by F e d o r c u k [ 7 5 ] u s i n g

A

BA

A

BA

,

lIdAI

-> Z W .

infinite, l U l t A l = w or l U l t A 1 < 2 h < ZK , t h e n t h e r e is a RA

A < K h IUItAI = 2 .

(2)

5 \A\

IUltA

I>

w2

such that

A

h a s no

I

I

J.D. Monk

32 Pro b lem 5 7 . subalgebra

(CCH) with

R

RA

Is t h e r e a IIlltR[ = w

21.

tion.

-P r o o-f . < Thus

->

: S u p pose

A

.

Let F :

D

Proof. c e l l A = dA

ramA

2

[UltAI

w

Obviously

.

K)

T h i s a l a r g e func -

is i s o m o r p h i c t o a f i e l d of s u b s e t s of

.

a

F o r each

,

A

{Fa: a

and

.

UltA

5 dA

cellA

For e a c h

< <

let

K

K}

,

Fa = { a E A : a E a }

.

i s c l e a r l y dense i n

K}

UltA.

l e t f a = {F E D : a

a E A

E

F).

Then

; hence i f t h e theorem i s f a l s e , we have by s e c t i o n 6 .

ramA = ( c e l l A ) +

and

,

Clearly

5 +A .

A+.

o f power

A

is a w e l l - o r d e r e d

Fa

R = UaChFa dA

?

5 dA .

is a ramification system i n

.

has no

A-WPD.

Theorem 2 1 .2 .

x}

S@K

be d e n s e i n

A

such t h a t

(02

Topological density

dA = min{K : A

i s an u l t r a f i l t e r on

Fa

Clearly

E

(I) 3

dA = min{K : I ll tA h a s a d e n s e s u b s e t o f power A l e s s t o p o l o g i c a l € a r m f o r dA is: Theorem 2 1 . 1 .

a

>

lIlltAl

2 ?

(MA + 7 C H ) t A = c e l l A =

Pro b lem 5 8 .

with

A

c h a i n and

Say

A c '?A

For a l l

a

<

,

,

celLA = A

,

A = dA

R

Rut

contradiction. Sapirovski:

1 7 4 ) h a s shown t h a t

dA

L

.

(spreadA)+

.

I t is

e a s y t o see t h a t \ A \ 5 2dA C l e a r l y ramA < dA i n g e n e r a l ; dA < nA f r e e , i n g e n e r a l . We h a v e s p r e a d ( i n t a 1 g l R ) = d ( i n t a 1 g R ) = w Under t h e r e is a BA A w i t h spreadA = w and dA = w1 ( t h e Kunen l i n e ) . Pro b lem 5 9 .

.

h

IFa\

so

and

F a = (x E R :

let

h

(In

ZFC)

for

.

Is t h e r e a

BA

A

with

spreadA

<

A

CH

dA ?

.

R e c a l l from 1 8 . 1 t h a t (H+d)A = hdA A r h a n g e l s k i ; [ 7 0 ] prove d t h a t dA < 2 t A * c e l l A M a l yhi n and S a p i r o v s k i i [ 7 3 ] showed t h a t MA + -CH

.

-

tA,cellA = w

that

cellA = w spreadA

showed showed

5

,

5K

imp i e s dA = w ; u n d e r t h e same a s s u m p t i o n H a j n a l and Juhhsz 7 1 1 w +A < 2 * dA = w S a p i r o v s k i r ( s e e Arhangelski: [ 7 8 ) , t A < K 3 dA 5 K

.

<

Pro b lem 60s.

Con(dA

Pro b lem 61".

Con( VA W ( d A

Problem 62. IUltAI

<

(In

tA*cellA

ZFC)

<

for a l l

,

tA

Is t h e r e a

BA

2K

.

A) ?

< A

2K

,

cellA

with

<

K

5

dA

,

cellA = w

< dA

?

K)

>

w

a nd

2w ?

The i n t e r v a l a l g e b r a

A

on a S o u s l i n l i n e has

cellA = w

,

dA = w1

,

[A! = w

On t h e o t h e r h a n d , by t h e above r e s u l t o f H a j n a l and J uhLs z we h a v e , u n d e r MA + -CH , \ A \ = w1 and c e l l A = w 3 dA = w

.

22. F o r any u l t r a f i l t e r

F

on

Ultrafilter density

A

let

1'

Cardinal functions on boolean algebras ~ X F= m i n { l X I

The ultrafilter density of A

va E F 2x E x + ( x

:

33

5 a)].

is

nXA

sup{nxF : F E IJltA?

=

is

and the lower ultrafilter density of A

nx A = min{qF : F E IlltA} 0

.

Both of these are widely varying functions. Mainly we shall consider ultrafilter density. We mention about nxo only the useful result of Sapirovski; that indA = (H+rx')A

; see Sapirovski; [SO].

Sapirovskii [ 7 5 1 showed that IAl for

5~ K

nXA

5 tA .

Sapirovskii [741 showed

.

x A From various results it follows that uncountable and regular, nX(inta1gK) = K . 23.

ITA =

nXA-dA

.

Note that

Character

Let XA = min{u : every ultrafilter on A can be generated by 5 K elements}. This is a large function. Note that XA 5 igA The famous theorem

.

[691 implies that

of Arhangelski:

IUltA

I -< 2xA

.

One can have

XA

<

igA

example, if A i s the Alexandroff duplicate of FrK , then 2K = cellA while XA = K . The Kunen line (under CH) gives a BA with character

.

Szentmiklossy [80] showed under spread w implies XA = w Problem 63.

.

(In ZFC)

Is there a

]A] 5 xACellA

It is known that Problem 64. igA

=

XA'spreadA

BA

with

XA

>

that

spreadA

For

5 igA w1

,

and

= w

spreadA ?

.

?

Baumgartner and Komjath [81] assuming XA = w

A

MA + -CH

.

0

constructed a

RA

A

with incA = w

and

1 '

<

Problem 65.

XA

Problem 66.

(In

sdepthA ? ZFC)

Is there a

BA

A

24.

Our last function i s tightness.

tF = min{K : if

<

a

cellA

For any ultrafilter

F

.

there i s no G E

G G

f

dA ?

Tightness on

A E

we set

u Z)?.

There i s a useful equivalence of Arhangelskil

[71] i n terms of free sequences. A sequence

F,

and

Y C UltA and F E Y then BZ E [Y]LK(F tA = sup{tF : F 6 UltA?

This is a widely varying function. for every

XA = w

with

UltA

cu

rl< F

F,,)


5:

5

<

with

ncu F,,) F
a>

in

UltA

is

free i f

J.D. Monk

34

T h e n t A = s u p { l a \ : t h e r e is a f r e e s e q u e n c e of l e n g t h more a l g e b r a i c v e r s i o n of t h e d e f i n i t i o n :

is s t r i c t l y i n c r e a s i n g } = s u p { d e p t h ( A / I ) Proof. -

< 51-s K < 2-

r)

sag K

.

n

FF: E

choose

<

{F,:

q

<

a>


Let

1 '

n

sag

and

<

S

For a l l

.

an i d e a l i n

K ~ , K ~

.

,

.

a} = 0

and

a

E

C

. F,'

I t i s e a s i l y checke d t h a t

<

a>

a>

{F : rl

.

a)

5 < a

For a l l

5 <


<

n Y : F,

L e t
:

<

t;

be a s i n d i c a t e d .

Y

F,

such t h a t

A

Y = {F

Let

.

A)

5 K~

~ K,

K

t h e r e is an

a

There is a l s o a

a > € a A 9Y G U l t A < s a

: I

Let t h e t h r e e c a r d i n a l s be

be an y f r e e s e q u e n c e .

<

: ?I
t A = sup{lal

Theorem 24.1.

.

a}

is a f r e e

a>

sequence,

< K 3 : Let € a A and Y be a s i n d i c a t e d . L e t I = 2A : Y c s(-x)} C l e a r l y I i s an i d e a l i n A . Suppos e 5 < q < a Then s(at;*-a,) Y = 0 , so at; -an E I Hence [ a 1 < [ a 1 Choose K

{x

F

E

E

sa

a,

n Y\

r)

4

-aS K~

I K~

sequence i n then So

.

n

a5 sat ;

u

{-at;}

and

C,

.

sag

Then

.

Hence

:

Let

at;

nY

c sa

.

Then

F

\sat;

,

E sa

n

rl

has

E

Now

I

,

so

1

:

F,

T

.

Y p s(-(a,*-at;))

<[a

t;

<

a>

s(-x) . Suppos e x € I , so -at; + a n E F -at; f I

a,

fip

5 -

nY

.

.

Thus

.

[a,]

Y =

-an

.

n Y

<

b e an i d e a l ,

Set

and

F E s(a,*-at ; )

[ay] I

.

A/I

F

.

, so

F E G

.

Let

a s t r i c t l y increasing

t;

<

T

he nc e

<

.

a

If

EF

a

rl F = (x : -x

F E sa

and

C

nY,

F E sa

T'

E I} U {a,}

f o r some u l t r a f i l t e r

G

.

Thus

GEY

as desired.

The a c t u a l d e f i n i t i o n of t i g h t n e s s and t h e f r e e s e q u e n c e e q u i v a l e n t g i v e r i s e t o a t t a i n m e n t p r o b l e m s ; t h e two e q u i v a l e n t s i n 24.1 ha ve t h e same a t t a i n m e n t p r o p e r t i e s as t h e f r e e s e q u e n c e e q u i v a l e n t . For e a c h l i m i t c a r d i n a l K t h e r e is a BA A w i t h t i g h t n e s s K n o t a t t a i n e d ; f o r cfK > 0 t h e same i s t r u e f o r t h e f r e e sequence d e f i n i t i o n . I f K is s i n g u l a r , C ~ = K w , and A has t i g h t n e s s K , then A h a s a f r e e s e q u e n c e of l e n g t h K These f a c t s a r e e a s i l y proved u s i n g t h e methods of McKenzie, Monk [82]

.

.

W e have

t(AxB) = t A U t R

and

tn;

= supi

0 t h e r e i s a BA A w i t h t i g h t n e s s f o r each K Monk [83] I t f o l l o w s from a r e s u l t o f Mal yhi n [721 t h a t

.

C l e a r l y independence, d e p t h u l t r a f i l t e r density tightness.

L

S a p i r o v s k i ? [741 showed

hdA

5

tightness.

w

On t h e o t h e r h a n d ,

s uc h t h a t

t(@A ) = K ; see

= \I] Usup. tfti 1 E ItAi. S a p i r o v s k i t [751 h a s shown t h a t

Clearly tightness

5 spread,

spreadA*(tA)+ ,

We a l s o s h o u l d m e n t i o n t h a t

.

ItAi

t A = sup{nxB : A - + B }

.

character.

Cardinal functions on boolean algebras

depth

Possible equalities are indicated by ? . I n addition, there may b e relationships not shown; see the problems.

35

36

J.D. Monk REFERENCES

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37

Koppelherg, S. [751 Homomorphic images of o-complete Roolean algebras, Proc. AMS 51, 171-175. Koppelherg, S . (771 Boolean a l g ~ h r a s a s unions of chains of suhalgehras, Alg. llniv. 7, 195-204. Koppelherg, S . [ m ] Maximal chains in Roolean algebrds. Kunen, K. 1801 Set theory, North-Holland, 317 pp. Kurepa, D. [501 La condition de S u s l i n et une propriet; caracteristique des nombres rGels, Comp. Rendus (Paris) 231, 1113-1114. Kurepa, D. [57] Partitive sets and ordered chains, "Rad" de 1'Acad. Yougoslave 302, 197-235. Kurepa, D. [62] The Cartesian multiplication and the cellularity number, Publ. Inst. Math. (Brograd) 2, 121-139. Kurepa, D. (771 Ramified sets or pseudotrees, Publ. Inst. Math. (Beograd) 22, 149-163. Loats, J. 1771 On endomorphism semigroups of Boolean algebras and other problems, Ph.D. thesis, IJniv. of Colo., 75 pp. Malyhin, V. [72] On tightness and Souslin number in expX and i n a product of spaces, Sov. Math. Dokl. 13 (1972), 496-499. Malyhin, V., Sapirovski;, B. [731 Martin's axiom and properties of topological spaces, DAN SSSR 213, 532-535. McKenzie, R., Monk, J.D. [73] On automorphism groups of Boolean algebras, Erdjs Symposium, Colloq. Math. SOC. J. Bolyai 10, 951-988. McKenzie, R . , Monk, J.D. [82] Chains in Boolean algebras, Ann. Math. Logic 22, 137-175. Mitchell, M. [72] Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5, 21-46. Monk, J.D. 1831 Independence in Boolean algebras, Per. Math. Hungar. 14, 269-308. Rubin, M. [831 A Boolean algebra with few subalgebras, interval Boolean algebras, and retractiveness, Trans. AMS 278, 65-89. iapirovski;, 8. [74] Canonical sets and character. Density and weight in compact spaces, Sov. Math. Dokl. 1 5 , 1282-1287. iapirovskii, B. [75] On n-character and n-weight in bicompact spaces, DAN SSSR 223, 799-802. iapirovskii, B. [76] On tightness, n-weight, and related notions, Uc. Zap. Latv. Univ. 257, 88-89. iapirovskii, B. [80] On mappings onto Tychonov cubes, Ilsp. Mat. Nauk 35, 122-1 30. Shelah, S . [79] Boolean algebras with few endomorphisms, Proc. AMS 74, 135-142. Shelah, S. [SO] Remarks on Boolean algebras, Alg. Univ. 11, 77-89. Shelah, S. [80a] Independence of strong partition relation for small cardinals, and the free-subset problem, J . Symb. Logic 45, 505-509. Shelah, S . [all On uncountable Boolean algebras with no uncountable pairwise comparable o r incomparable sets of elements, Notre Dame J. Formal Logic 22, 301-308. Shelah, S. [83] Constructions of many complicated uncountable structures and Boolean algebras, Israel J . Math. 45, 100-146. Szentmiklossy, 2. (801 S-spaces and L-spaces under Martin's axiom, Colloq. Math. SOC. Janos Bolyai 23, 1139-1145. Tall, F. [80] Large cardinals f o r topologists, Surveys in general topology, Acad. Press, 445-477. Todorcevig, S . [m] Remarks on chain conditioins in products.

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-

PART I1 ORDER AND LATTICES

- PARTIEII ORDRE ET TREILLIS

H.A. PRIESTLEY Ordered sets and duality f o r distributive lattices. . . . . . . . . . . . . . . . p . 39 Michael MISLOVE When are order scattered and topologically scattered the same ? . . . p . 61 Dwight DUFFUS - Maurice POUZET Representing ordered sets b y chains. . . . . . . . . . . . . . . . . . . . . . . . .

p . 81

Giinter BRUNS Orthomodular Lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p . 99

George GRATZER - David KELLY The construction of some free m-lattices on posets. . . . . . . . . . . . . p . 103 I.C. ROSENBERG - D. SCHWEIGERT Compatible orderings and tolerances of lattices. . . . . . . . . . . . . . . . p . 11 9

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 (1984) 39-60 0 Elsevier Science Publishers B.V. (North-Holland)

39

ORDERED SETS AND DUALITY FOR DISTRIBUTIVE LATTICES

H . A. P r i e s t l e y Mathematical I n s t i t u t e U n i v e r s i t y of O x f o r d England

An a c c o u n t i s g i v e n o f t h e c a t e g o r i c a l d u a l i t y w h i c h e x i s t s b e t w e e n bounded d i s t r i b u t i v e l a t t i c e s a nd c ompa c t t o t a l l y o r d e r disconnected spaces. During t h e p a s t decade, a wide range of s t r u c t u r a l problems c o n c e r n i n g d i s t r i b u t i v e l a t t i c e s h a v e b e e n s o l v e d by t h e t o p o l o g i c a l and o r d e r t h e o r e t i c t e c h n i q u e s p r o v i d e d by d u a l i t y , and a r e p r e s e n t a t i v e s e l e c t i o n of these is presented. In addition, certain r e l a t e d d u a l i t i e s a r e b r i e f l y c o n s i d e r e d , as a r e c ompa c t t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s i n t h e i r own r i g h t . The paper ends w i t h a ' d i c t i o n a r y ' of mutually d u a l p r o p e r t i e s .

ENSEMBLES O R D O N N ~ S E T

DUAL IT^

POUR LES TREILLIS

DISTRIBUTIFS

Au c o u r s d e ces d e r n i P r e s a n n g e s , l e s d u a l i t 6 s du t y p e P o n t r y a g i n o n t p r o l i f 6 r 6 . L'u n e d ' e l l e s e s t c e l l e e x i s t a n t e n t r e l a c a t e ' g o r i e des treillis d i s t r i b u t i f s e t b o r n 6 s e t la c a t 6 g o r i e d e s e s p a c e s compacts e t t o t a l e m e n t s g p a r g s pour l ' o r d r e . (Un e s p a c e t o p o l o g i q u e X a v e c une r e l a t i o n d ' o r d r e p a r t i e l g e s t t o t a l e m e n t s 6 p a r 6 p o u r l ' o r d r e s i , q u e l s q u e s o i e n t l e s p o i n t s x,y d e X a v e c LI: $ y , i l e x i s t e un ensemble d g c r o i s s a n t ( c ' e s t - & d i r e , un i d g a l p o u r l ' o r d r e ) U q u i e s t o u v e r t e t ferm 6 e t t e l s q u e z U. y E U.) Le p r 6 s e n t d 6 v e l o p p e m e n t s ' i n t 6 r e s s e 2 l a d u a l i t 6 e n t r e 1) e t P , 2 ses r e p e r c u s s i o n s e t 2 s a p l a c e d a n s l e p a y s a g e m a t h 6 m a t i q u e .

-

-

4

Au n i v e a u d e s o b j e t s , l a d u a l i t 6 p e r m e t d ' i d e n t i f i e r un t r e i l l i s L 6 D a v e c l e t r e i l l i s d e s e n s e m b l e s d g c r o i s s a n t - o u v e r t s e t f e r m 6 s d ' u n e s p a c e XL p o u r XL, o n p e u t p r e n d r e l ' e n s e m b l e d e s i de' aux p r e m i e r s , o r d o n n 6 p a r i n c l u s i o n e t convenablement t o p o l o g i s 6 . On a donc l a g 6 n 6 r a l i s a t i o n n a t u r e l l e s i m u l t a n 6 e d e l a r e p r g s e n t a t i o n d e B i r k h o f f d e s t r e i l l i s d i s t r i b u t i f s f i n i s (oh l a t o p o l o g i e e s t d i s c r P t e e t ne j o u e aucun r 6 1 e ) e t l a r e p r 6 s e n t a t i o n d e S t o n e d e s a l g P b r e s d e B o o l e (oh l ' o r d r e e s t d i s c r e t ) .

6-z;

e t P e t permet d e t r a d u i r e les concepts de Une d u a l i t 6 c a t 6 g o r i q u e p a r f a i t e u n i t l a t h 6 o r i e des t r e i l l i s dans l a langue d e s e s p a c e s topologiques ordonngs. La r e p r s s e n t a t i o n imag6e q u e ces e s p a c e s f o u r n i s s e n t ( b i e n q u e c e l l e - c i ne s o i t p l u s g u P r e c o n s t i t u 6 e q u e d e di agrammes d e Venn s o p h i s t i q u s s ) e s t 6 t o n n a m e n t p u i s s a n t e . E l l e e s t i l l u s t r 4 e p a r un mbl ange d ' a p p l i c a t i o n s e t d ' e x e m p l e s q u i m e t t e n t B nu l a s t r u c t u r e d e s t r e i l l i s e t l a s t r u c t u r e d e s v a r i e t 6 s d ' a l g P b r e s dans

u.

c-f.

En e f f e t , c ' e s t l ' o i d r e q u i d i s t i n g u e L ' o r d r e j o u e un rsle c r u c i a l e n d u d l i t 6 cette d u a l i t 6 de l a d u a l i t 6 equivalente pour m e t t a n t e n j e u les espaces spectraux. L ' a c t i o n r k c i p r o q u e e n t r e l a t o p o l o g i e e t l ' o r d r e s u r un o b j e t d e p e u t 6 t r e a s s e z s u b t i l e . A l o r s qu'on a don& une r 6 p o n s e a c e r t a i n e s q u e s t i o n s c o n c e r n a n t l ' o r d r e ( i l y a , p a r exemple, d e s d e s c r i p t i o n s d i v e r s e s d e ceux d e s beaucoup de problsmes e n c o r e ensembles ordonn6s q u i peuvent d e v e n i r o b j e t s de e n suspens semblent a u s s i a t t i r a n t s qu'6pineux !

c),

H. A . Priestle y

40

1.

INTRODUC'I'LO?:

D u r i n g t h e p a s t t e n y e a r s , d u a l i t i e s o f P o n t r y a g i n t y p e ha ve p r o l i c e r a t e d . One such d u a l i t y i s t h a t between t h e c a t e g o r y o f bounded d i s t r i b u t i v e l a t t i c e s and the category o f compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s , w h i c h , a t t h e o b j e c t l e v e l , p r o v i d e s t h e n a t u r a l s i m u l t a n e o u s g e n c r a l i s a t i o n of B i r k h o f f ' s r e p r e s e n t a t i o n o f f i n i t e d i s t r i b u t i v e l a t t i c e s and S t o n e ' s r r p r e s e n L a t i o n o f Bo o lean a l g e b r a s , The f u l l c a t e g o r i c a l d u a l i t y whic h l i n k s a nd allows l a t t i c e t h e o r e t i c c u n c e p t s a n d p r o b l e m s t o b e t r a n s l a t e d i n t o t h e l a n g u a g e of o r d e r e d t o p o l o g i c a l s p a c e s . The f i n a l s e c t i o n of t h i s s u r v e y a s s e m b l e s f o r r e f e r e n c e 111 t h e e a r l i e r some of t h e most f r e q u e n t l y u s e d items i n t h e LLE ' d i c t i o n a r y ' . s e c t i o n s we d e s c r i b e tiow t h e d u a l i t y works and i l l u s t r a t e some of t h e ways i n which i t has b e e n u se d t o r e v e a l t h e s t r u c t u r e of l a t t i c e s i n a nd of c e r t a i n s u b c l a s s e s of In a d d i t i o n we b r i e f l y i n v e s t i g a t e t h e i n t e r a c t i o n b e t w e e n t h e t o p o l o g y and o r d e r i n R :-object .3nd, i n b a r e s t o u t l i n e , we c o n s i d e r t h e g e n e r a l s e t t i n g i n which d i s t r i b u t i v e l a t t i c e d u a l i t y s h o u l d b e p l a c e d .

2

!.

I t h a s n o t b e e n p o s s i b l e i n t h e s p a c e a v a i l a b l e t o p r o v i d e :I f u l l y c o m p r e h e n s i v e survey. I n s e l e c t i n g m a t e r i a l we h a v e t r i e d t o complement t h e a d m i r a b l e a c c o u n t o f d u a l i t y p r e s e n t e d by B . A . Uavcy and D . D u f f u s i n t h e i r p a p e r ' E x p o n e n t i a t i o n a n d d u a l i t y ' p u b l i s h e d i n t h e P r o c e e d i n g s of t h e Ba nff Symposium on O r d e r e d S e t s 1371; o u r a p p r o a c h i s somewhat more t o p o l o g i c a l and p e r h a p s l e s s i n f l u e n c e d by u n i v e r s a l a l g e b r a . We h a v e t r i e d t o make t h e b i b l i o g r a p h y c o m p l e t e i n r e s p e c t o f t h e E-E d u a l i t y a n d i t s a p p l i c a t i o n s t o p r o b l e m s c o n c e r n i n g d i s t r i b u t i v e - l n t t i r r o r d e r e d a l g e b r a s , b u t h a v e n o t s o u g h t t o g i v e full r e f e r e n c e s f o r t h e same problems t r e a t e d a l g e b r a i c a l l y .

We assu me f a m i l i a r i t y w i t h t h e r u d i m e n t s of t h e t h e o r y of d i s t r i b u t i v e l a t t i c e s a s s e t o u t i n 1 1 4 7 , 1 1 9 1 and C 5 4 1 . Our r e p r e s e n t a t i o n s p a c e s w i l l b e o r d e r e d s e t s , appropriately topologised. The n o t a t i o n ( X , C , d w i l l d e n o t e a s e t X e q u i p p e d w i t h a t o p o l o g y e a n d a p a r t i a l o r d e r <. The b a s i c p r o p e r t i e s of s u c h s p a c e s w e r e d e v e l o p e d i n 1 6 8 1 ; s e e a l s o C 2 4 1 , [ 4 9 1 and 1 7 1 1 . I t i s p e r h a p s w o r t h e m p h a s i s i n g now t h a t t h e p r o o f s u s i n g d u a l i t y of t h e v a s t m a j o r i t y of t h e a l g e b r a i c t h e o r e m s s t a t e d i n t h i s p a p e r r e q u i r e t o p o l o g i c a l e x p e r t i s e no g r e a t e r t h a n t h a t i m p a r t e d by a f i r s t ( u n d e r g r a d u a t e l e v e l ) c o u r s e i n g e n e r a l t o p o l o g y . If

iP,d i s

a n o r d e r e d s e t , we w r i t e , f o r :r E

4r= i y E P I y < 5 } and, f o r

12

I>,

t Z =

iyEz'I~{>al

& P,

I-& = U [ + x I 5 E Q I . A s u b s e t Q o f P i s s a i d t o b e decreasing ( i n c r e a s i n g ) i f +Q = Q (+Q = Q). 4Q

=

UiCZ

Ia

E

Q),

In t h e l i t e r a t u r e , d e c r e a s i n g s e t s are a l s o c a l l e d o r d e r i d e a l s , h e r e d i t a r y s e t s , i n i t i a l s e g m e n t s , down s e t s , l o w e r s e t s and l o w e r e n d s . I f t' i s a n o r d e r e d s e t we d e n o t e by Pop t h e o r d e r e d s e t o b t a i n e d by r e v e r s i n g t h e o r d e r .

2.

BASIC DUALITY FOR DISTRIBUTIVE LATTICES

The c a t e g o r y h a s a s o b j e c t s t h e bounded d i s t r i b u t i v e l a t t i c e s ; t h e u n i v e r s a l b o u n d s o f a n y L, t i, a r e d e n o t e d by O,1. Morphisms i n 0, a r e t h e 0 , l - p r e s e r v i n g l a t t i c e homomorphisms. W i t h i n a r e two i m p o r t a n t f u l l s u b c a t e g o r i e s : consists of t h e f i n i t e d i s t r i b u t i v e l a t t i c e s and of t h e B o o l e a n a l g e b r a s . A s l o n g a g o a s t h e 1 9 3 0 ' s k e y r e p r e s e n t a t i o n t h e o r e m s were p r o v e d f o r t h e s e s u b c a t e g o r i e s : THEOREM 2 . 1 ( G . B i r k h o f f 1181; o r see L l 9 1 ) . Let L € L&. Then I , i s isomorpJLic to t h c latticc, of decrea::lng .:uhsct:: of t h e S L L J ( L ) o f Join-irreducible elements

of L .

Ordered sets and duality for distributive lattices

41

1

13irIm tcl h e t h c s e t o f p r i m e i d e c i l s o l 1 , c , q u i p p d wit11 a n o r d c t r ( r ' f . Thc.clrc,nt 2 . 1 ) and 3 topology (i,J'. Theorem 2 . 2 ) iiiort'

c,),.

N

'

.

Accurdingly,

tci

eaclt 1, t 1 ' w e a s s o c i a t e i t s

H / i s t h c s e t or p r i m e idt,:ils and

, i

is s e t inclusion

k

has

:IS

.I

of

bast, t h c st'ts t,r t

Ti',

(Xi,,k, G) , where

/,,

.X/ I

cz!,

.I'

i s € A'),

I

3

.I'

( T I ,for

ii

t l,.

IJe c a n i d c n t i t y (k'L,P,2)

w i t h thr' honi-set p,(l,,z) of morphisnts o n t o t h e 2 - e l e m c n t w h e r e 2 d e n o t e s the, c h a i n , p o i n t w i s e o r d e r e d a n d t o p o l o g i s c d a s ;I s u h s p a c t ~o f 2-elfmcxnt ch,iin w i t h tlic d i s c r c t c t o p o l o g y . ?-'I,

The t o p o l o g y P i s compact. A l s o , the t h e c r u c i , i l p r o p e r t y o f Lctfil c ? r c / c ' 2 3 L 3: & y, t h e r e e x i s t s ;I & - c l o p e n d c c r r , p r o p e r t i e s o f compact t o t a l l y o r d e r P r o pi's i t i o t i 2 6

l o g y o f (Xl,:k,G) a r e l i n k e d by : g i v e n p o i n t s .I,,;/ i n XL w i t h t h a t :I.$ !/ € 1'. Basic nected spaces a r e c o l l e c t e d together i n

v,

..

We d e f i n e t o h e t h e c a t e g o r y o r conipnct t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s ( P r i e s t l e y s p a c e s ) and c o n t i n u o u s o r d e r - p r e s e r v i n g maps. F o r i' E we d e n o t e b y U ( P ) t h e l a t t i c e o f P-clopen d e c r c a s i n g s u b s e t s of P. We d e f e r u n t i l 5 7 t h e f o r m a l c a t e g o r i c a l s t a t e m e n t of t h e duality. F o r o u r i m m e d i a t e p u r p o s e s we n e e d o n l y t h e r e p r e s e n t a t i o n f o r o b j e c t s ( w h i c h giv6.s t h e common g e n e r a l i s a t i o n o f T h e o r e m s 2 . 1 and 2 . 2 ) and t h e basic f a c t s a b o u t d u a l m o r p h i s m s .

Q-E

THEOREM 2 . 3 ( 1 7 0 1 ) .

a +-+ in

E X L 11'

c.

( i ) Let L

+

€

c.

Y'/w??

,Y

(21.

( i i ) Lr:t T ' E Tt'/di O(/)) E or&l~ i s o ~ I ~ L i Y ~ism) pl1

triid

.

F'

2

Xi ) (,)

I,

E

I' nizd

I,

(2 1ic'r.c

We s h a l l h e n c e f o r t h i d e n t i f y i, w i t h O ( X ) and I' w i t h L*

2

O(.Y

T,

) 7iin tiit, n i q ,

nimning h o n ; c w r i ( iryii L:ni trntl

( L E Q, 1'

€

Ll.

The p r o o f of Theorem 2 . 3 i n v o l v e s a c o n s t r u c t i o n of p r i m e i d e a l s w h i c h n e c e s s i t a t e s i n v o k i n g snme f o r m of t h e Axiom of C h o i c e . In fact the duality is e q u i v a l e n t t o t h e Roolean Prime I d e a l Theorem, which i s s t r i c t l y theorem f o r weaker t h a n (AC). However t h e s t a t e m e n t t h a t e v e r y d i s t r i b u t i v e l a t t i c e w i t h 1 h a s a maximal i d e a l i m p l i e s ( A C ) ( s e e 1 1 7 1 ) . A f u l l e r d i s c u s s i o n of t h e s e p o i n t s can be found i n I h O l . The d u a l i t y t h e o r e m c a n e a s i l y b e a d a p t e d t o h a n d l e l a t t i c e s w h i c h l a c k e i t h e r z e r o o r i d e n t i t y ( s e e 1 7 1 1 ) . However i n p r a c t i c e i t i s u s u a l l y e a s i e r t o a d j o i n b o u n d s , s o as t o be a b l e t o work i n E . For m o r p h i s m s we h a v e PROPOSITION 2 . 4

f:L

+

Thcrc exi::ts a ii-

([70], 1 7 1 1 ) .

Id and ~-rr~orpl~i:;ins $:XM

+

X Lj givcn l ) ~ !

c Lion Oc h l L ' f 3 1 2 ~ - 1 m ~ p 1 1 i : ; i n s

E j"(u)

i f arid onZ!/ if $(!/) E n

H.A. Priestley

42

if trnd onl!j if Further, f is surtji.c2f ( a E L, y E XM). f is i n j e c t i v e if and only if $ is s u r j e c t i v c . COROLLARY 2 . 5 .

Let L,M E

c.

' I ' h m ( u p to isomorphism i n

$

7s an cmb
am!

p),

( i ) M i:: a ~ 0 , l l - s u h l a l t i r r of L if and only if X is i i i c i m a ~ 7X~ undcrz a M I, continuous order-prescJroing map; ( i i ) M = L / B f o r some congruence 8 on 1, if and on7y if XM i:; a clo;ed sitbsrt of *

We conclude t h i s s e c t i o n w i t h a p r o p o s i t i o n c o n t a i n i n g key elementary f a c t s about P-spaces. These r e s u l t s p r o v i d e a l l t h e b a s i c t o o l s used i n d u a l i t y p r o o t s o f a l g e b r a i c theorems.

PROPOSITION 2 . 6

disconnected.

( S e e [681, 1711).

l;rt (X,C,<)

b t compact and t o i a l l g order

l'izen

of < is closed i n X x X; is C-closed, +Q, +Q are r e s p c c t i w l y C-closed dccreasing, Z-c2oscd incwasing; ( i i i ) if Q i s C-clospd, Q contains a point idhich i s minimal (maximal) i n Q w i t / ) respect t o G; ( i ) t h e graph

(ii)

if

(iv)

if

3.

Q

LJ f o r all x E Q, y E R, then Q,R are C-closed .substts of X such t h a t x t h e r e e x i s t s a C-clopen decreasing subset V of X such that Q 5 X V , h' c V .

EQUATIONAL SUBCATEGORIES OF ALGEBRA

0: THE

INFLUENCE OF CATEGORY THEORY AND UNIVERSAL

This s e c t i o n b u i l d s on t h e b a s i c d u a l i t y t h e o r y i n 52 and f o c u s e s on c o n c e p t s and problems w i t h t h e i r r o o t s i n c a t e g o r y t h e o r y and u n i v e r s a l a l g e b r a . I n t e r a l i a , w e b r i e f l y c o n s i d e r f r e e o b j e c t s and coproducts i n and i t s s u b c a t e g o r i e s , and s u b d i r e c t l y i r r e d u c i b l e a l g e b r a s i n v a r i o u s e q u a t i o n a l s u b c a t e g o r i e s of

E.

Suppose ,C i s a s u b c a t e g o r y o f k . Provided w e can i d e n t i f y t h e P - o b j e c t s a r i s i n g a s d u a l s of C-objects and can d e s c r i b e t h e d u a l s of C-morphisms, we can r e s t r i c t t h e g-z d u a l i t y t o o b t a i n a d u a l i t y f o r ,C and t h e r e b y a t e c h n i q u e f o r a n a l y s i n g This approach has proved p r o f i t a b l e f o r , f o r example, d i s t r i b u t i v e p - a l g e b r a s and double p - a l g e b r a s , Stone a l g e b r a s , d e Morgan a l g e b r a s , Ockham a l g e b r a s and t u k a s i e w i c z a l g e b r a s ; r e f e r e n c e s can be found i n t h e b i b l i o g r a p h y .

C.

We i l l u s t r a t e w i t h a s i m p l e example. The c a t e g o r y R of ( d i s t r i b u t i v e ) p-algebras c o n s i s t s of t h o s e l a t t i c e s L E which p o s s e s s a unavy o p e r a t i o n , *, of pseudocomplementation s a t i s f y i n g c h a = 0 i f and o n l y i f c Q u*. The B -morphisms a r e "W t h e g-morphisms p r e s e r v i n g

*.

THEOREM 3.1 (C731). Let L E g, Then L E Bw if and only if XL is such that f U is open whenever U E O(XL); for a E L, a* = XL +a. Then f E & ( L , M ) if and only if $ maps Let f E D_(L,M) have duaZ $ E ~ ( X M XI,). , the minimal elements beZow y onto the minimal etements below $ ( y ) .

Theorem 3.1 p r o v i d e s a d u a l d e s c r i p t i o n of &-congruences from which one s e e s t h a t i s s u b d i r e c t l y i r r e d u c i b l e i f and o n l y i f L i s a Boolean a l g e b r a w i t h a new i d e n t i t y a d j o i n e d ( s e e [351, C371 and C731, o r , f o r an a l g e b r a i c approach, C641, C651).

L E

I

The s u b v a r i e t i e s of chain

&

were f i r s t d e s c r i b e d by K. B . Lee i n C651.

They form a

Ordered sets and duality for distributive lattices !-l

C

k0

(=

e) c El

( = Stone a l g e b r a s ) c R

-2

c

43

... c Rw;

for 1 n < w, B c o n s i s t s of t h o s e a l g e b r a s L i n H f o r which X i s such t h a t 7 2 -w L each of i t s elements m a j o r i s e s a t most n minimal p o i n t s . The c l a s s HW of d i s t r i b u t i v e double p - a l g e b r a s ( t h a t i s , a l g e b r a s 1, such t h a t L and Lop a r e w i n h' ) i s much l e s s t r a c t a b l e than However major advances have been made i n r e c 2 n t y e a r s by R . Beazer, B . A . Davey, T . Katriiidk and A . U r q u h a r t , u s i n g both a l g e b r a i c and t o p o l o g i c a l methods. I n p a r t i c u l a r , Urquhart h a s shown i n 1 9 7 1 t h a t RW h a s an uncountable l a t t i c e of s u b v a r i e t i e s ; h i s proof uses d u a l i t y backed up by-gome graph t h e o r e t i c i d e a s .

.

Anyone of a c a t e g o r i c a l b e n t would n o t f e e l f u l l y a c q u a i n t e d w i t h a g i v e n c a t e g o r y u n t i l he had d e s c r i b e d t h e f r e e o b j e c t s ( i f a n y ) , and i n v e s t i g a t e d l i m i t s and c o l i m i t s . Under o u r d u a l i t y , l i m i t s t r a n s f o r m i n t o c o l i m i t s , e t c . , we can a n a l y s e via

c.

so

THEOREM 3.2 (See 1371, 1 4 1 1 ) . The f r e e bounded d i s t r i b u t i v e l a t t i c e on K generators, F ~ ( K ) e, s i s t s and +as duaZ space ZK. For any ordwcd s e t Q, t h e free bounded d i s t r i b u t i v e Zatticc generated by Q, FED(&), e x i s t s and has dual spacP (2')OP (where i s t h c s p t of order-preservfR f u n c t i o n s from Q t o 2, ordered pointwise and topologisrd as u subspace o f 2 i.

zQ

there exists Here Fg(Q) i s r e q u i r e d t o p o s s e s s t h e f o l l o w i n g u n i v e r s a l p r o p e r t y : an order-embedding e : Q -+ FD(&) and, f o r e a c h 1, and e a c h o r d e r - p r e s e r v i n g map a:& -+ L , t h e r e e x i s t s a D-morphism G:FD(Q) -+ L such t h a t U 0 e = a. The embedding e of Q i n O ( ( 2 q P ) sends z E Q t o { h E 24 1 h ( z ) = 01. Coproducts e x i s t i n

0.

E x p l i c i t l y , we have

( i ) Let

THEOREM 3.3.

L A i s O( At A

be a fafarm:ly o f g-objects.

JTx, XEA

Then t h e coproduct

) (where t h e product o f t h e dual spaces i s ordered point-

X

wise i.

0. Then t h e coproduct L * M is O ( X L x X M ) , which i s isomorphic t o C G ( X L , M ) , t h e l a t t i c e of continuous order-preserving f u n c t i o n s from XL t o M, uhere M i s equipped with t h e d i s c r e t e topology.

( i i ) L e t L,M E

For a d i s c u s s i o n of t h e s e and o t h e r r e s u l t s on c o p r o d u c t s i n g, s e e 1 2 7 1 , 1321, L791, 1801. We c o n s i d e r f u r t h e r i n 54 t h e s p e c i a l c a s e of P o s t a l g e b r a s (coproducts B C, where B E and C i s a f i n i t e c h a i n ) . For c e r t a i n s u b c a t e g o r i e s ,C of _D, t h e C-coproduct c o n v e n i e n t l y c o i n c i d e s w i t h t h e _D-coproduct for (de Morgan a l g e b r a s p r o v i d e one example 1281). I n o t h e r s u b c a t e g o r i e s , i n s t a n c e , a s p e c i a l d e s c r i p t i o n of t h e coproduct i s needed ( s e e 1393).

*

&

An element of ,D i s a (weak) p r o j e c t i v e i f and o n l y i f i t i s a r e t r a c t of some F ~ ( K ) . Such l a t t i c e s have proved d i f f i c u l t t o d e s c r i b e e x p l i c i t l y ( s e e 191, C151). However, R. Balbes h a s proved i n 1131 t h a t e v e r y g - p r o j e c t i v e l a t t i c e i s D - c a t a l y t i c ( L i s g - c a t a l y t i c i f _D(L,M) i s a l a t t i c e under t h e p o i n t w i s e o r d e r i n g f o r e a c h M E ;E' s e e L631), and t h a t t h e converse a l s o h o l d s f o r c o u n t a b l e l a t t i c e s . ;-catalytic l a t t i c e s have been c h a r a c t e r i s e d a l g e b r a i c a l l y i n C131. We a l s o have

N

THEOREM 3.4

(1741).

Let L

E

.;

Then the following are e q u i v a l e n t :

(i) L i s 2-catalytic; ( i i ) X L i s a compact zero-dimensional

topologica2 l a t t i c e ;

( i i i ) L i s a r e t m c t of FD(Q), for some ordered s e t Q.

H.A. Priestley

44

This theorem i m p l i e s , a s i s well-known, t h a t t h e d u a l space of a i , - p r o j e c t i v c l a t t i c e i s a complete l a t t i c e , and t h a t L t g),, i s p r o j e c t i v e i f and only i f i t s More s p e c i f i c a l l y , an ordered s e t s e t of join-irreducibles J ( L ) i s a l a t t i c e . i s isomorphic t o t h e s e t of prime i d e a l s under i n c l u s i o n o f a U_-projective where both maps involved l a t t i c e i f and only i f i t i s a r e t r a c t o f some p r e s e r v e u p - d i r e c t e d s u p s and down-directed i d s ( C 7 4 1 ; s e e also 5 6 ) .

z",

a r e j u s t the complete Boolean a l g e b r a s (L9l; f o r a t o p o l o g i c a l R e l a t i n g t o completeness i n we have

The i n j e c t i v e s i n p r o o f , s e e C717). THEOREM 3.5.

Let I, E

g.

3'hen the following a r e eqiu7:vali?nt:

( i ) L is complete;

is ez i r emZ Zy order disconnectad (tiiut is, tho C-closwe: of U E 11 i : ; U-open, where C, U arc r c s p e c t i v d ! j Lhc C-opt?ii dccrca:;h g , i n c m m in9 stit:: in (XL, C,Q;

(ii) XL

( i i i ) t h e l a t t i c e CG(XLJR) of continuous ordcr-prcscw)ing function:: from XL t o W (with p o i n t u i s e ordar) 7::: c onditionally cornplc For a proof of ( i ) Q ( i i ) , s e e 1711; f o r ( i ) Q ( i i i ) s e e 1781, where f u n c t o r i a l p r o p e r t i e s of t h e assignment L *+ C ( X ,R) a r e a l s o c o n s i d e r e d . M . J . C a n f e l l i n < c h i s t h e s i s 1231 proves a r e s u l t ver? l i k e ( i i ) * ( i i i ) . On a h i s t o r i c a l n o t e , we remark t h a t i t was i n C231 t h a t compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s were f i r s t i n t r o d u c e d , i n c o n n e c t i o n w i t h a s t u d y of semi-algebras of c o n t i n u o u s functions.

4.

DUALITY I N A C T l O N :

POST ALGEBRAS AND BEYOND

S e c t i o n 5 w i l l be a p o t p o u r r i of a p p l i c a t i o n s of d u a l i t y , chosen f o r t h e i r d i v e r s i t y . I n t h i s s e c t i o n , by c o n t r a s t , we c o n s i d e r a r a t h e r r e s t r i c t e d , b u t n e v e r t h e l e s s i m p o r t a n t , c l a s s of l a t t i c e s : t h e P o s t a l g e b r a s and g e n e r a l i s a t i o n s of t h e s e . We b e g i n w i t h P o s t a l g e b r a s . These a r e a l g e b r a i c a l l y well-understood ( s e e , f o r example, 1141) and d u a l i t y can do no more than make s t a n d a r d r e s u l t s obvious. We t h e n c o n s i d e r chain-based l a t t i c e s , f o r which a d u a l s p a c e approach p r o v i d e s new s t r u c t u r a l i n f o r m a t i o n . A f t e r a c u r s o r y d i s c u s s i o n of o t h e r Post a l g e b r a g e n e r a l i s a t i o n s we f i n a l l y u n v e i l a p r e v i o u s l y e l u s i v e c l a s s : t h e d i s t r i b u t i v e l a t t i c e s of o r d e r < m. Our account i s based on work of W . G . Bowen (C201, c211, C221). We n o t e t h a t t h e d u a l s p a c e of a Boolean a l g e b r a i s , qua o r d e r e d s e t , an a n t i c h a i n , w h i l e t h e d u a l s p a c e of a c h a i n i s a c h a i n . For convenience we d e n o t e 1 and t h o s e of i t s d u a l s p a c e t h e elements of t h e z e l e m e n t c h a i n n by O , l , ,n-1. n;l ( t h e d i s c r e t e l y t o p o l o g i s e d (n-i)-element c h a i n ) by 1 , 2 , .

...$-

Let L = B

X

>k

n (where B E g) be a t y p i c a l Post a l g e b r a .

1;

t

n- 1

x2 x1

xB Figure 1

n-l

..

Then ,YL

=

can b e v i s u a l i s e d union of n clopen each homeomorphic Boolean space

XB

xn,

(z,i)

Q

x

nZ1, which

as t h e d i s j o i n t 'layers' Xi, t o the where

(y,j)

i f and o n l y i f z = y and i j, so t h a t each p o i n t of XL i s m a j o r i s e d p r e c i s e l y by t h e corresponding points i n higher layers (see figure 1 ) . Lattice elements a r e r e p r e s e n t e d by clopen d e c r e a s i n g s e t s . These a r e b u i l t

Ordered sets and duality for distributive lattices L e t co

u p from ' c h a i n e l e m e n t s ' a n d ' c e n t r a l e l e m e n t s ' . Then

=

0,

45 e.

=

u

j
X . (i > 1 ) . 3

O = t = 1 0 < c'l < * * * < c' n- 1 forms d c h d i n i n L i s o m o r p h i c t o ". The c e n t e r o f 1, ( i t s complemented e l e m e n t s ) i s {ij xn-1 I b E ii); i t f o r m s A Boolean a l g e b r a i s o m o r p h i c t o 8. F i g u r e 2 d e p i c t s these s p e c i a l elements.

chain element

c e n t r a l element Figure 2

a i n L may b e r e p r e s e n t e d d i a g r a m m a t i c a l l y a s i n F i g u r e 3 and i t i s e a s i l y seen t h a t i t has a unique r e p r e s e n t a t i o n i n each of t h e f o l l o w i n g forms.

A general element

D i s j o i n t r e p r e s e n t t i on : n- 1

Monotone r e p r e s e n t a t i o n :

n- 1

a

=

v

(cji

A

e.), c. E B,

i=o

L'

i

c

j

for

i

j.

I n F i g u r e 3 t h e d i s j o i n t r e p r e s e n t a t i o n i s i n d i c a t e d ; t h e monotone r e p r e s e n t a t i o n is o b t a i n e d i n a s i m i l a r f a s h i o n .

Figure 3

More i m p o r t a n t l y , t h e d u a l s p a c e makes t r a n s p a r e n t t h e r e p r e s e n t a t i o n o f e l e m e n t s ( s e e Theorem 3 . 3 and C801): o f L a s c o n t i n u o u s f u n c t i o n s f r o m XB t o the l e f t hand d i a g r a m i n F i g u r e 4 d e p i c t s a n e l e m e n t o f O(Xn x n--l); t h e r i g h t hand d i a g r a m shows t h e g r a p h o f t h e a s s o c i a t e d c o n t i n u o u s f u n c t i o n .

H.A. Priestle y

46

Figure 4

The coproduct s p e c i f i c way. fixed chain

n

k 2 i s g e n e r a t e d by i t s c e n t e r and an n-element c h a i n i n a very We now c o n s i d e r a more g e n e r a l s i t u a t i o n . Suppose t h a t C i s a

0 = co < e l <

... <

c

n- 1

=

1

i n a l a t t i c e L E 2. Then (L;C) ( o r where no c o n f u s i o n a r i s e s j u s t i.) i s s a i d t o be chain-based i f L i s g e n e r a t e d by ii U C, where I3 i s t h e c e n t e r of I,; t h e d i s t i n g u i s h e d c h a i n C i s c a l l e d a c i a i n basc f o r I,. We s h a l l c a l l I, E U_ a P 0- l a t t i c e i f some f i n i t e subchain of I, i s a c h J i n b a s e . Such l a t t i c e s were i n t r o d u c e d by T . Traczyk 1891 and have been s t u d i e d by G . E p s t e i n and A . Horn i n It is e a s i l y C441. Theorem 4 . 2 g i v e s t h e d u a l c h a r a c t e r i s a t i o n of P o - l a t t i c e s . proved from t h e o b s e r v a t i o n s c o n t a i n e d i n

LEMMA 4 . 1 . Lcl ( L ; C ) be chain-based and l e t v:XL + X b J a:XL + n-1 be thc duuls of t h e natural l'nclusion maps from U,C i n t o L . -1 ( i ) ( a ) a E B Q a = II ( h ) for some clopen subsel 0 o$ Xu; (b) each e

I

i

E

C, e i

f 0,

( i i ) { i r - l ( ~ i ) b clopen i n t h c Lopology on X L ; ( i i i ) (a) for

i

= 1,2,

(b) for z E (iv)

3: I-

XB,

(n(3:),a(x))

xB}

...,n-1,

i c c q r e s s i b ~ eas a -1 (ri);

u {a-'(i) a

-1

(i) is

I 7:

=

1 , 2 , ...,n- 11 forms a s h h a s e f o r

a n antic hain;

-1

(z) is a f i n i t r chain; is a homeomorphic order embedding of XI, i n t o XB

~i

THEOREM 4 . 2 (W.G.Bowen [20], [221). Let L only if f o r some Boolean space Z a n d some n

E

>

g.

x

nil.

Then L is a P0-lattice if and

1, XI, is a closed subspace of

2 x n-1. N

The d u a l space of a P O - l o t t i c e may be d e p i c t e d a s i n F i g u r e 5: prime i d e a l s l i e i n d i s j o i n t maximal c h a i n s and i n d i s j o i n t c l o p e n ' l a y e r s ' . The l a y e r i n g of t h e space depends on t h e c h o i c e of c h a i n b a s e ( s e e Lemma 4 . 1 ( i i ) ) . The d u a l s p a c e of L i s ( t o w i t h i n homeomorphism and o r d e r isomorphism) u n a l t e r e d i f a c l o p e n s u b s e t of a l a y e r i s moved t o occupy a vacant space i n a n adjacent l a y e r , b u t t h e c h a i n b a s e changes. ln:

-Ij

Y

Figure 5

Let C d e n o t e t h e c o l l e c t i o n of c h a i n b a s e s of a P o - l a t t i c e L of t h e maximal p o s s i b l e l e n g t h , o r d e r e d elementwise.

41

Ordered sets and duality for distributive lattices We say 1, i s a

E’l-l,atticc. i f C has a l a r g e s t e l e m e n t ,

a F 2-2utticc i f C has l a r g e s t and s m a l l e s t elements and a P:i-Lalticc i f /Ci = 1. The P . - l a t t i c e s a r e i n v e s t i g a t e d i n C 4 4 1 , 1901. Dual space diagrams make i t e a s y t o e s h n b l i s h many p r o p e r t i e s of P . - l a t t i c e s , i n c l u d i n g t h o s e s e t o u t i n t h e t a b l e below; t h c new r e s u l t s a r e provea i n [ 2 0 1 . We d e n o t e by t h e c l a s s of Heyting algebras: those L E f o r which 0

e x i s t s f o r a l l (l,b E /,.

+

b

1

c

A

a G 01

1c

v

a

Dually,

a

+

h

=

m l‘ n t c E i,

In t h e d u a l s p a c e a

where t h i s e x i s t s . a b by + ( b \ t i ) .

L

= maxic E

+

>

b},

11 i s r e p r e s e n t e d by X L

+ ( a -.b ) and

+

i s a chain-based l a t t i c e , w i t h C E C t h e c h a i n

I n t h c t a b l e (1,;L‘)

0

< d1

=

<

...

< en-, =

I,

I

and n i s t h e c e n t e r of L . F o r any a t L , ! iz = max{b t H b G u } , i f t h i s e x i s t s . Note t h a t Theorem 4 . 2 and C o r o l l a r y 2 . 5 ( i i ) imply t h a t ( L ; C ) i s (isomorphic t o ) t h e q u o t i e n t o f ii x n by a congruence.

A 1 geb r a i c equivalent

Dual equivalent

X L, X,

P -lattice

(L;C) a

e i + e

P -lattice 1

V

i-1

= e . 7,

i

x

n 1 -,

congruence Q u o t i e n t by a

decreasing i n

filter

x

nzl

(L;C) a

e

XL c l o p e n

Q u o t i e n t by a

P -lattice 2

and ! ei e x i s t s

decreasing i n

principal

V i

XB

filter

/I

i

+ c

i-1

I:

x

Heyting algebra equivalent

Q u o t i e n t by a

X L closed

XB = e

closed i n

R e l a t i o n of L Post a l g e b r a B * ~f

n-1

L,LOPE

and ! en-2 = 0

The d u a l s p a c e c h a r a c t e r i s a t i o n s make i t e a s y t o c o n s t r u c t examples of P . - l a t t i c e s . The f o l l o w i n g ones a l l have t h e same c e n t e r , viz. t h e Boolean a l g e b r a o f ’ f i n i t e and c o f i n i t e s u b s e t s of IN, whose d u a l s p a c e 2 i s t h e one-point c o r n p a c t i f i c a t i o n of t h e d i s c r e t e spacelN, w i t h t h e d i s c r e t e o r d e r . Each of t h e d u a l s p a c e s below g i v e s a l a t t i c e which is i n t h e i n d i c a t e d c l a s s b u t n o t i n t h e c l a s s above. A l l a r e s u b s e t s of X = 2 x 2, which i s t h e d u a l of a P - l a t t i c e ( P a s t a l g e b r a ) . 3

X

I I I 111:::1

H A . Priestley

48 (: 1d S s

1'. - l a t t i c e s

L

P 1- l a t t i c e s

I'"-lat t i c e s

Figure h

'I'hc f o l l o w i n g space p r o v i d e s a c o n t r a s t i n g e x a m p l e . Wc t a k e X < / j h l orrlercci s e t t o be A x 2 , w h e r e A i s t h e C a i i t u r s e t , d i s c r e t e l y u r d e r c d . 'This s e t c a n b c made i n t o a c o m p a c t t o t a l l y o r d e r d i s c o n n e c t e d s p a c e by t d k i n g a s ;i c L i ~ p e n s \ h b a s c > f u r ; I topology the s e t s I(.c,2) .I: E A ] U l i : x 2 (c i s c l o p e n i n A w i t h i t s u s u a l t o p o l o g y } . 'The r e s u l t i n g s p a c e i s made u p o f two c o p i t l s o f A f o r m i n g two l a y e r s , w i t h Lhe d i s c r e t e t o p o l o g y i n d u c e d o n t h e u p p e r l a y e r and t h e u s u a l t o p o l o g y o n t h e l o w e r o n e . The l a y e r s a r e , o f c o u r s c , n o t t o p o l o g i c a l l y d i s j o i n t a n d i t is p o s s i b l e t o show t h a t t l i e d i s t r i b u t i v e l a t t i c e X r e p r e s e n t s h a s no ch.iin hdsc; in f a c t i t i s n o t even a s u b l a t t i c e of I' - l a t t i c e .

I

1

0

The d u a l space o f a Post a l g e b r a L h i i s t h e f o l l o w i n g p r o p e r t i e s : (7:)

(-I-)

XI, i s o r d e r e d a s a d i s j o i n t u n i o n o f maximal c h a i n s ; XI, i s t o p o l o g i s e d iis ;1 f i n i t e d i s j o i n t u n i o n of B o o l e a n s p a c e s .

h a v e b e e n c o n s i d e r e d which s h a r e t o a g r e a t c ' r B e w i l d e r i n g l y many s u b c l a s s e s o f d e g r e e t h e a l g e b r a i c p r o p e r t i e s of P o s t algcsbras (n-valued C u k a s i e w i c z a l g e b r a s , S t o n e l a t t i c e s LIE o r d e r n , ) . A L L , d u a l l y , r e t a i n e i t h e r p r o p e r t y (") o r property (-1). A s y s t e m a t i c s t u d y O F t h e i n t e r a c t i o n b e t w e e n t h e s e c l a s s e s has b e e n u n d e r t a k e n by W . G . Bowen in 1 2 0 1 .

...

From among l a t t i c e s s a t i s f y i n g ( " ) we s i n g l e o u t f o r b r i e f m e n t i o n t h e n-valued f.ukasiewicz a l g e b r a s . The du al s p a c e X I of s u c h a n a l g e b r a i. l o o k s l i k e a Post a l g e b r a d u a l s p a c e 7 x l,=.n. i n w h i c h som; o f t h e l a y e r s h a v e b e e n p i n c h e d L o g e t h c r by a map & : ( : r , L ) ++ i\.(x:), where h . : Z -+ X1 ( i = l , 2 , Y L - ~ i) s c o n t i n u o u s , s u r j e c t i v e , o r d e r - p r e k e r v i n g and s & c h t l i a i r,!] a r e i n c o m p a r a b l e i t and on1.y i f rT ( s ), ii (!y) are incomparab I e

...,

.

.

The r i g l i t , i p p r o a c h Lo c 1 , i s s i F y i n g l , i t L i c e s w i t h p r o p e r t y ( I ) seems t o b e

Lo

mike

Ordered sets and duality for distributive lattices

49

Tire d u , i l s p : i c ~ > oI 'I 1 : i t t i c . c u i t h i s t y p c i s t h e d i s j o i n t u n i o n o f t h e I!oolenn sp.Ic<'s 'Y ( [ ' = 1,2, j/-l). 'I'lit, d i s t r i b u t i v e l a t t i c e s o f o r d e r ni f o r m

...,

"~;-l,* I ' I

c.

subcategory o f I t s n l g e b r ' i s Iicive b i n a r y o p e r a t i o n s , (I' = 1 , 2 : r t ) .ind n u l l a r y o p e r a t i o n s i'. ( i = 0 , 1 , a r e as i n D e t i n i t i o k 4 . 3 , i = 1 , i s n f o r d e r i i < m, t h e n , ( ] ' . 9 ' 'ii-l i

,...,

2'>

'1

,ind

(r

;t

.

..

i s t h e comptement

.

(.

UT

i- I v rr)

A

f

y

;

in

V,

A,

unary

...,m-1); (2

n-1

if

for

rt~j-i,~~,lll~

d i s t r i b u t i v e l a t t i c e L w i l l b e t o o s m a l l Tor 1, t o b e n i t e rlic3in i:. We may t h e n seek some o t h e r ' m i n i m a l ' e n c r a t e d b y bl u i?. I J ~assume !, rind L"p a r e t i e y t i n g i i i , I' 01 1, t o b e t h e s u b l a t t i c e g e n e r a t e d by a11

F r e q u e n t l y thr. cei1tc.r gcllel-.ltcd by ii ; I d '1 g s u b l a t t i c c bl s u c h t h a t n l g e b r n s .ind d c f i u e t h e C' leme1t t s

(11

--f

<2)

4

(c

+

/ I )

(b,c t I , ) .

F o r a n y i i E !i, (a ~ 2 ' ) ((2' n) = ii, so t h e c e n t e r i s t . u n t a i n e d i n the cxocenter. F u r t h e r n i o t i v a t i o n f o r the p e r h a p s c u r i o u s l o o k i n g d e f i n i t i o n i s p r o v i d e d by F i g u r e 8 , w h i c h i l l u s t r a t e s t h e l o o s e s e n s e i n w h i c h t h e e x o c e n t r a l ( ( 2 '- 1 1 ) c n n b e t h o u g h t o f as h a v i n g (c b) (fJ c) g i ' n c r r i t o r (1' + c) as an ' a p p r o x i t i l a t e coiiipteriren~' +

+

+

.

__

.

..

+

+

+

.

T h e .ippropri,itcness of the d c f i i i i t i o n i s c o n f i r m e d b y

The e x o c e n t e r w a s i n t r u d u c e d by C . Epstein 1 4 3 I ; h i s d e f i . n i t i o n is d i f f e r e n t irtm, b u t e q u i v a l e n t t o , t h e o n c g i v e n h e r e . ProperLics o f t h e e x o c e n t e r h a v e been i n v e s t i g a t e d i n d e p e n d e n t l y by W. G . Bowen i n 1 2 0 1 , I 2 1 I a n d by G . 1 7 p s t r i n a n d A . Horn ( s e e I 4 5 I ) .

5.

A M I S C E L I A N Y OF IIUALLTY A P P L I C A T I O N S

An i m p o r t a n t use t c ) w h i c h d u a l i t y tias b e e n p u t tins b e e n t h e p r o d u c t i o n o f

H.A. Priestley

50

counterexamples and of l a t t i c e s s a t i s f y i n g s p e c i f i e d c o n d i t i o n s . We g i v e some i l l u s t r a t i o n s , r e f e r r i n g t h e r e a d e r t o t h e c i t e d p a p e r s f o r t h e (sometimes i n t r i c a t e ) d e t a i l s of t h e c o n s t r u c t i o n s . A s a f i r s t example c o n s i d e r t h e problem of f i n d i n g which 1, E have t h e p r o p e r t y (*): e v e r y i d e a l ( f i l t e r ) i n L i s t h e i n t e r s e c t i o n o i maximal i d e a l s ( f i l t e r s ) . I t was p o i n t e d o u t by A . Monteiro t h a t (*) h o l d s i f L i s Boolean. Using d u a l i t y , M . E . Adams proved i n C21 t h a t i t h o l d s i f e i t h e r L E R, o r L i s s c a t t e r e d ( t h a t i s , t h e r a t i o n a l c h a i n does n o t embed i n L ) , He a l s o c o n s t r u c t e d an example of t h e f a i l u r e of (*); t h e d u a l space of a s u i t a b l e l a t t i c e was o b t a i n e d by t a k i n g t h e Cantor s e t i n i t s u s u a l topology and superimposing an a p p r o p r i a t e o r d e r . A d i f f e r e n t example was found by R . Balbes 1111 u s i n g a l g e b r a i c t e c h n i q u e s . T h i s is n o t t h e o n l y c a s e of a problem's b e i n g s o l v e d independently and almost s i m u l t a n e o u s l y by a l g e b r a i c and t o p o l o g i c a l methods which a r e n o t mut a l l y d u a l . Another i s provided by q u a s i v a r i e t i e s of p - a l g e b r a s : t h a t t h e r e a r e0'2 of t h e s e was shown a l g e b r a i c a l l y by A. Wroriski I 1 0 0 1 and t o p o l o g i c a l l y by M. E . Adanis c31. This r e s u l t was r e f i n e d by G . GrBtzer, H . Lakser and R. W . Quackenbush C551 who shnwed t h a t t h e q u a s i v a r i e t i e s w i t h i n form a non-modular l a t t i c e , of c a r d i n a l i t y 2% ; t h e i r t e c h n i q u e s i n c l u d e d u a l i t y , ?haugh a p p l i e d t o f i n i t e a l g e b r a s o n l y .

We now look a t s i i b l a t t i c e s . Fur M a { 0 , I ) - s u b l a t t i c e of 1, E M a r i s e s from (XL,E,4) as f o l l o w s . L e t

J"

=

I(z,y)

Iz

+ y,

(VU E

M)(y E a

=)

U,,

t h e d u a l space of

x E a)}

be t h e separating s e t f o r M. Then S U 4 i s a q u a s i o r d e r r e l a t i o n on X m d P given by x p y i f and o n l y i f (.z,y), (y,z) E S U 4 i s an e q u i v a l e n c e r e t a t i o n . By equipping t h e p - e q u i v a l e n c e c l a s s e s w i t h t h e obvious q u o t i e n t topology and o r d e r , we o b t a i n M ' s d u a l s p a c e . S e p a r a t i n g s e t s ( i n c l u d i n g t h e i r e x t e n s i o n t o n o t n e c e s s a r i l y bounded l a t t i c e s ) were f i r s t e x p l i c i t l y employed by M. E . Adams c11. The t e c h n i q u e i s perhaps n o t a s widely known a s i t d e s e r v e s t o be ( i t r e c e n t l y proved u s e f u l f o r s t u d y i n g t h e e x o c e n t e r (Theorem 4 . 4 ) ) . As a simple a p p l i c a t i o n we mention Adams' d u a l c o n s t r u c t i o n of a d i s t r i b u t i v e l a t t i c e w i t h no maximal p r o p e r s u b l a t t i c e s : t h e d u a l space c o n s i s t s of t h e n a t u r a l numbers s u i t a b l y o r d e r e d . The main r e s u l t s i n Cll concern t h e Frattin? s u b l a t t i c c @ ( L ) of a d i s t r i b u t i v e l a t t i c e L ; by d e f i n i t i o n , @ ( L ) i s t h e i n t e r s e c t i o n of t h e maximal p r o p e r s u b l a t t i c e s of L . I t i s shown, i n t e r a l i a , t h a t any d i s t r i b u t i v e l a t t i c e L i s Q ( L ' ) f o r some d i s t r i b u t i v e l a t t i c e I,'. Many y e a r s ago, A . T a r s k i posed h i s 'cube problem': does t h e r e e x i s t B E B, such B x B x B b u t B $ B X B ? I n a remarkable paper C621, J . Ketonen that B r e c e n t l y c h a r a c t e r i s e d t h e c o u n t a b l e Boolean a l g e b r a s and provided a ( c o u n t a b l e ) counterexample t o t h e cube problem. For c o p r o d u c t s of c o u n t a b l e Boolean a l g e b r a s , V. Trnkov5 C921 proved t h a t B p B * B * B does imply B B * B . By - o n t r a s t , M. E . Adams and V. Trnkovd have shown i n C81 t h a t i t i s p o s s i b l e t o f i n d 20' nonisomorphic c o u n t a b l e d i s t r i b u t i v e l a t t i c e s L E 0, such t h a t L 2 L * L * L b u t L $ L L ; f u r t h e r , given p r e s c r i b e d c o u n t a b l e bounded d i s t r i b u t i v e l a t t i c e s B,C, each of t h e l a t t i c e s L above can b e chosen t o have B a s a s u b l a t t i c e and C a s a homomorphic image. The c o n s t r u c t i o n u s e s d u a l i t y i n a n o n - t r i v i a l way.

*

Our l a s t example i s somewhat d i f f e r e n t and i n v o l v e s t h e i n t e r v a l t o p o l o g y . On any o r d e r e d s e t P, t h e i nt erval topology has a c l o s e d subbase o f t h e s e t s +x, +x f o r z E P. I t w i l l b e of some importance i n 56; h e r e we a r e concerned w i t h i t s appearance i n a r e c e n t paper by G . G i e r z and J . D . Lawson C5ll on s o - c a l l e d g e n e r a l i z e d continuous l a t t i c e s . They prove t h a t a d i s t r i b u t i v e l a t t i c e L i s t h e q u o t i e n t of a completely d i s t r i b u t i v e l a t t i c e L' under a map p r e s e r v i n g u p - d i r e c t e d supsanddown-directed i n f s p r e c i s e l y when t h e i n t e r v a l topology on L i s H a u s d o r f f . The proof of s u f f i c i e n c y i s o b t a i n e d by t a k i n g L' t o b e t h e l a t t i c e of o r d e r p r e s e r v i n g maps from (X )OP t o 2. D i s t r i b u t i v i t y of L i s used t o produce t h e F$?'e(4, r e q u i r e d q u o t i e n t map; i t i s unknown whether t h e theorem remains v a l i d w i t h o u t t h i s assumption.

Ordered sets and duality for distributive lattices 6.

51

TOPOLOGY AND ORDER IN PRIESTLEY SPACES

A P r i e s t l e y space i s s i m u l t a n e o u s l y an o r d e r e d s e t and a (Boolean) t o p o l o g i c a l s p a c e . I t must a l s o be t o t a l l y o r d e r d i s c o n n e c t e d . To s e e t h e s t r e n g t h of t h i s l a t t e r r e q u i r e m e n t we d e a l b r i e f l y w i t h t h e problem of t o p o l o g i s i n g an o r d e r e d s e t t o c r e a t e Li g - o b j e c t . E q u i v a l e n t l y , we a s k under what c o n d i t i o n s a g i v e n o r d e r e d s e t (I3,<) i s wprc.;cvitclDle, t h a t i s , i s isomorphic t o (X C ) f o r some L E A L’deep a n a l y s i s of r i n g s p e c t r a by M . tlochster [561 r e v e a l e d t h a t ( F , G ) i s r e p r e s e n t a b l e p r e c i s e l y when i t i s isomorphic t o t h e s e t of prime i d e a l s , o r d e r e d by i n c l u s i o n , o f a commutative r i n g w i t h i d e n t i t y . Thus r e p r e s e n t a b i l i t y i s of i n t e r e s t b o t h t o l a t t i c e t h e o r i s t s and t o r i n g t h e o r i s t s ; f o r a sample of t h e l i t e r a t u r e s e e [ l o ] , 1121, C311, C481, C561, 1661, C871, where f u r t h e r r e f e r e n c e s can a l s o be found.

t.

Any r e p r e s e n t a b l e s e t

(P,<) i s such t h a t

( i ) e v e r y u p - d i r e c t e d s u b s e t of P has a s u p , e v e r y down-directed s u b s e t h a s an inf; ( i i ) I’ i s weakly a t o m i c : i f .r < y , t h e r e e x i s t u,v such t h a t x rzd,ul = {u,u1 ( t h a t i s , L J . , ~ ] c o n t a i n s a c o v e r i n g p a i r ) ;

<

u

<

u

y and

( i i i ) t h e i n t e r v a l topology on P i s compact; ( i v ) Fop i s a l s o r e p r e s e n t a b l e . C o n d i t i o n s ( i ) - ( i v ) e s s e n t i a l l y a p p e a r i n C561. I t i s e a s y t o prove t h a t they hold i n t h e u n d e r l y i n g o r d e r e d s e t of any _P-object ( s e e 1701 and C491, 5VI.1). By ( i ) , any r e p r e s e n t a b l e s e m i l a t t i c e i s n e c e s s a r i l y a complete l a t t i c e . From ( i ) , ( i i ) we s e e immediately t h a t , f o r example, n e i t h e r o n o r w+w* i s r e p r e s e n t a b l e . I n f a c t , a c h a i n i s r e p r e s e n t a b l e i f and o n l y i f i t s a t i s f i e s ( i ) , ( i i ) . In general, even ( i ) , ( i i ) and ( i i i ) a r e i n s u f f i c i e n t t o g u a r a n t e e r e p r e s e n t a b i l i t y ( s e e 1661). Among o r d e r e d s e t s s a t i s f y i n g ( i ) and ( i i ) a r e c l e a r l y t h e a l g e b r a i c l a t t i c e s ( s e e 1491 f o r t h e d e f i n i t i o n and b a s i c p r o p e r t i e s ) . An a l g e b r a i c l a t t i c e P becomes a P r i e s t l e y s p a c e i f i t i s given t h e (Lawson) topology X ( P ) w i t h open subbase c o n s i s t i n g of t h e s e t s + k ( k E K(P), t h e compact e l e m e n t s i n P) and P 93: (J. i’) ( s e e [ 4 9 1 , C741, 1 7 5 1 ) . Among t h e d u a l s p a c e s of t h i s t y p e a r e t h e zero-dimensional t o p o l o g i c a l l a t t i c e s a r i s i n g i n Theorem 3.4. I n g e n e r a l a r e p r e s e n t a b l e o r d e r e d s e t can c a r r y many d i f f e r e n t t o p o l o g i e s which make i t a P r i e s t l e y s p a c e , and so r e p r e s e n t many non-isomorphic l a t t i c e s . A c l a s s i c example i s p r o v i d e d by M (a c o u n t a b l e a n t i c h a i n w i t h u n i v e r s a l bounds -objects. I f (P,C,s)E a d j o i n e d ) ; A(M ) and A(MwOP) yieYd non-isomorphic then LI c o n t a i n 2 t h e i n t e r v a l topology IV(P) on P (by P r o p o s i t i o n 2 . 6 ( i i ) ) ; t h i s i s how r e p r e s e n t a b i l i t y c o n d i t i o n ( i i i ) comes a b o u t . Thus C i s u n i q u e l y determined ( a s IV(P)) i f IV(P) i s H a u s d o r f f . When t h i s happens, we have ( P , Q) ( X L , ~ )f o r (up t o isomorphism) o n l y one L E Among l a t t i c e s u n i q u e l y determined i n t h i s way by t h e i r o r d e r e d s e t s of prime i d e a l s a r e t h e ; - c a t a l y t i c l a t t i c e s and i n p a r t i c u l a r any ;-object f r e e l y g e n e r a t e d by a n o r d e r e d s e t and any g - p r o j e c t i v e l a t t i c e . The i n t e r v a l topology on P i s Hausdorff whenever P h a s no i n f i n i t e a n t i c h a i n ; n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r IV(P) t o b e Hausdorff a r e given i n C461, C51l.

p,

0.

We have been c o n s i d e r i n g t o p o l o g i s i n g a g i v e n o r d e r e d s e t t o produce a P r i e s t l e y s p a c e . We remark i n p a s s i n g t h a t t h e problem of o r d e r i n g a Boolean s p a c e t o produce a _P-object may b e t r i v i a l o r h a r d depending on t h e t y p e of o r d e r i n g demanded. For example, no l i n e a r o r d e r e x i s t s on t h e Tychonov p l a n k (w+l) X (w,+l) T h i s i s e s s e n t i a l l y due t o R . D . Mayer and R . S . P i e r c e making i t a g - o b j e c t . [671; a l i n e a r o r d e r can b e imposed on a Boolean s p a c e (X,,t) i f and o n l y i f t h e Boolean a l g e b r a B h a s an o r d e r e d b a s e . We conclude t h i s s e c t i o n w i t h a c a u t i o n a r y example, due t o A. R . S t r a l k a [ 8 6 1 .

H.A. Priestley

52

C o n s i d e r t h e C a n t o r s e t A w i t h i t s u s u a l t o p o l o g y C a nd o r d e r Q d e r i n e d by 2: L/ i f and o n l y i f 5: = y o r J = 1, !/ = 0 o r i i !i c o v e r s .c i n t h e n a t u r a l o r d e r on A . 'The s p a c e (A,C,G) i s c a l l e d t h e L r : ; a L ? /w c / ~ si; t i s made up or a c o u n t a b l e c o l l e c t i o n o f 2 - e le me nt c h a i n s and an u n c o u n t a b l e a n t i c h a i n . Lt t u r n s o u t t h a t , g i v e n 5: $ y i n A , ( i ) 3 d i s j o i n t C-clopen

s e t s i l l , Vl w i t h .c t U 1 , !/ 6 V1,

( i i ) 3 d i s j o i n t C-open s e t s U V2, respectively increasing, decreasing, w i t h , 2' x E y E v2,

u2,

but (iii)

3

d i s j o i n t C - c l open s e t s 0, V , r e s p e c t i v e l y i n c r e a s i n g , d e c r e a s i n g , w i t h

5:

6

In f a c t

7.

u,

o(a)

E V. =

{$,a)!

DUALITY IN A WIDER PERSPECTIVE

? h e c r e d i t f o r i n i t i a t i n g t h e s t u d y o f ~ o p o l o g i c a l r e p r e s e n t a t i o n s of a l g e b r a i c o b j e c t s g o e s t o M . H . S t o n e , H i s c o n t r i b u t i o n i s p u t in h i s t o r i c a l p e r s p e c t i v e i n t h e i n t r o d u c t i o n t o 1'. T . J o h n s t o n e ' s book S t o n e S p a c e s [ 6 0 1 . We recommend t h i s s t i m u l a t i n g a n d w i d e - r a n g i n g a s s e s s m e n t a s comple me nta ry r e , i d i n g t o t h e duality. f o l l o w i n g s h o r t d i s c u s s i o n of d u a l i t i e s r e l a t e d t o t h e

!)-z

0. S t o n e ' s o r i g i n a l t o p o l o g i c a l r e p r e s e n t a t i o n e x t e n d s t o a d u a l i t y b e t w e e n f! a nd t h e c a t e g o r y ot s p e c t r a l s p a c e s ( s e e [ 1 4 1 , [ 5 4 1 , [ 8 5 1 ) . W . H . C o r n i s h p r o v e d i n [261 t h a t md S a r e i s o m o r p h i c ( n o t j u s t e q u i v a l e n t ) c a t e g o r i e s . I f t h e c - d u a l of L i s TXL,C,Q) t h e n i t s 2 - d u a l i s ( X L , C ) w h e r e C i s t h e t o p o l o g y o f C-open d e c r e a s i n g sets; conversely, C is constructed from a s p e c t r a l topology by t a k i n g d S a s u b b a s e t h e C-compact-open s e t s and t h e i r c o m p l e m e n t s , a nd Q i s g i v e n by s y to was p a r t i a l l y made by A . Nerode [691 a nd y E c l ~ { 3 c 1 . The t r a n s i t i o n f r o m M. H o c n s t e r [561. For p r a c t i c a l p u r p o s e s seems t o h a v e d i s t i n c t a d v a n t a g e s o v e r is S . However t h e r e a r e s i t u a t i o n s w h e r e a ' h u l l - k e r n e l ' a p p r o a c h ( a s w i t h a p p r o p r i a t e ; one s u c h i s t h e s p e c t r a l t h e o r y f o r d i s t r i b u t i v e c o n t i n u o u s l a t t i c e s r e c e n t l y d e v e l o p e d by K . H . Hofmann a n d J . D . Lawson ( [ 5 8 ] ; o r s e e [ 4 9 1 ) . One can recapture duality f o r from t h i s by r e g a r d i n g t h e p r i m e i d e a l s of 1, E as t h e p r i m e e l e m e n t s o f T d ( L ) , t h e i d e a l l a t t i c e of L , w h i c h i s a d i s t r i b u t i v e c o n t i n u o u s l a t t i c e ( s e e [ 4 9 1 , p . 3 2 5 ) . An e x t e n s i o n of d u a l i t y i n a q u i t e d i f f e r e n t d i r e c t i o n i s s u g g e s t e d by G.-C. R o t a i n 1761. We b e g i n w i t h some h i s t o r i c a l r e m a r k s on d u a l i t i e s f o r

2

Q

2

2)

and We now s e e k a more s y s t e m a t i c u n d e r s t a n d i n g of how t h e d u a l i t y b e t w e e n i n some s e n s e p r o t o t y p i c a l . We b e g i n w i t h a f o r m a l s t a t e m e n t o f t h e d u a l i t y between o u r c a t e g o r i e s

an d

i) :

d i s t r i b u t i v e l a t t i c e s w i t h 0,1, 0 , l - p r e s e r v i n g l a t t i c e homomorphisms

,F

c o mp a c t t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s c o n t i n u o u s o r d e r - p r e s e r v i n g maps.

:

2

P

is

2

We n o t e t h a t t h e o b j e c t "2" a p p e a r s i n two g u i s e s : as i t l i e s i n L l , as i t lies i n P. The n o t a t i o n h e r e i s d i f f e r e n t f r o m t h a t a d o p t e d i n § 5 ; 2 i s t o b e u n d g r s t o o d t o be t h e 2 - e l e m e n t c h a i n w i t h e l e m e n t s 0 , l . I t w i l l now b e more c o n v e n i e n t t o work w i t h p o i n t w i s e - o r d e r e d hom-sets r a t h e r t h a n p r i m e i d e a l s . DUALITY THEOREM.

varianl finetor;

T?w categ0ric.s

and ,P are d u a l l y equivalent u n d m t h e contra-

-D(-,L) :

+

p

(where Liic image c ( L , z ) of L has the pointwise o r d m and t o p o l o g y inherlited from

Ordered sets and duality for distributive lattices

53

2')

-P(-,?)

aiui

:'YlC,

JOit

i

:1

I J -+

f,

<(

4

r t ~ ' ~ g{Ii.h ' ri

( J h r l I, t

2,

i' E [) L J ; ~ f L J (E1,(/I))

a d

(f)=

(rtl,(x))(9)

=

11:

:

1

+

i'

- -I i ( -p ( - , ? ) ~

+

,2)

'3s

t-hc n l j i i ~ n c i i o r ~

I,,:

f(a)

o (.I-)

(.r

11, $ E C ( P , ? ) ) .

The P o n t r y a g i n d u a l i t y f o r comp;i~-t g r o u p s e x h i b i t s e s s e n t i a l l y t h e same f e a t u r e s duality. a s the I t a s s e r t s t h a t a n y A E /1 ( t h e v a r i e t y o f a b e l i a n g r o u p s ) i s i s o m o r p h i c t o t h e g r o u p or c o n t i n u o u s homomorphisms f r o m i t s c h a r a c t e r g r o u p A(/l,3') to h e r e t h e c i r c l e g r o u p 2' qim d i s c r e t e g r o u p T l i e s i n quo. c o m p a c t g r o u p 2' i t l i e s i n t h e d u a l c a t e g o r y . I n b o t h P o n t r y a g i n and F - t d u a l i t i e s t h e o b j e c t ; in e a c h of t h e m u t u a l l y d u a l c a t e g o r i e s c a n b e i d e n t i f i e d w i t h s t r u c t u r e d h o m - s e t s ( m o r p h i s m s i n t o Y,X, morphisms i n t o inherits a topological and o p e r a t i o n a l s t r u c t u r e - f r o m w h i l e i)(L,T) i n h e r i t s a t o p o l o g i c a l and r e l a t i o n a l s t r u c t u r e frum A g e n e r a l framework f o r d u a l i t i e s o f t h i s s o r t h a s b e e n s e t up by B . A . Ilavey a n d 11. Werner i n [ 4 2 1 . The t h e o r y h a s i t s r o o t s i n a v a r i a n t o f F r e y d ' s r e p r e s e n t a b i l i t y t h e o r e m and f o l l o w s e a r l i e r work b y , among o t h e r s , B . A . Davey [ 3 4 1 and I<. H . Hofmann a n d K . Keimel [ 5 7 1 . The t h e o r y d e v e l o p e d by Davey a n d Werner e n c o m p a s s e s , b e s i d e s t h e P o n t r y a g i n a n d d u a l i t i e s , M o r i t a d u a l i t y f o r modules, t h e Hofmann-Mislove-Stralka d u a l i t y f o r c o m p a c t z e r o - d i m e n s i o n a l s e m i l a t t i c e s [ 5 9 1 a n d d u a l i t i e s (some o l d , some new) f o r c e r t a i n o t h e r v a r i e t i e s which a r e g e n e r a t e d by a s i n g l e f i n i t e a l g e b r a . A s u b c l a s s o f U may h a v e two d u a l i t i e s , o n e by r e s t r i c t i o n of t h e {-f d u a l i t y , t h e o t h e r p r o v i d e d d i r e c t l y b y t h e Davey/Werner a p p r o a c h . Sometimes t h e s e d u a l i t i e s a r e e s s e n t i a l l y t h e same ( f o r S t o n e a l g e b r a s , f o r e x a m p l e ) , s o m e t i m e s t h e y a r e d i f f e r e n t ( as i s t h e c a s e f o r K l e e n e a l g e b r a s ) .

c-[

-

x;

4,

2".

TI,

2,2);A(A,r)

c-'

B r a n c h i n g o u t i n a d i f f e r e n t d i r e c t i o n , o n e n a t u r a l l y s e e k s a n e x t e n s i o n o f ,U-p duality t o l a t t i c e s i n general. Various r e p r e s e n t a t i o n s have been d e v i s e d , e a c h with i n t e r e s t i n g f e a t u r e s ([301, [501, [941). G . G i e r z a n d K . Keimel i n [ 5 0 ] e x t e n d t o t h e i n f i n i t e c a s e t h e n o t i o n o f scaffoZding i n t r o d u c e d f o r f i n i t e l a t t i c e s b y R . Wille [ 9 8 1 ; t h e i r r e p r e s e n t a t i o n , w h i c h i s b a s e d on c o m p a c t Hausdorff t o p o l o g i c a l p a r t i a l l a t t i c e s , h a s proved u s e f u l i n problems on d u a l i t y i s A. e x p o n e n t i a t i o n ( s e e [161, [991). Closest i n s p i r i t t o the U r q u h a r t ' s d u a l i t y [ 9 4 1 ; t h i s u s e s as r e p r e s e n t a t i o n s p a c e s c e r t a i n t o p o l o g i c a l s p a c e s equipped w i t h a p a i r o f q u a s i o r d e r s l i n k e d w i t h t h e topology and w i t h e a c h o t h e r . Any l a t t i c e c a n b e i d e n t i f i e d w i t h a c o l l e c t i o n of s u b s e t s o f s u c h a s p a c e u n d e r o p e r a t i o n s A , v, w h e r e A i s i n t e r s e c t i o n , b u t , a s i s i n e v i t a b l e f o r non-distributive l a t t i c e s , v i s n o t generally union. For non-surjective m o r p h i s m s t h e n o t i o n of a d u a l m o r p h i s m d o e s n o t e x i s t , b u t t h e d u a l i t y d o e s e x t e n d f a r enough t o e n a b l e Urquhart t o prove t h e independence of t h e congruence l a t t i c e and t h e automorphism group of a f i n i t e l a t t i c e .

C-z

U-P d u a l i t y

The exhibit special has the

8.

A

seems t o s t a n d i n a s t r a t e g i c p o s i t i o n . I t i s g e n e r a l e n o u g h t o many o f t h e s t r u c t u r a l f e a t u r e s t h a t p e r v a d e d u a l i t y t h e o r y , y e t i s e n o u g h t o p r o v i d e a p r a c t i c a l method o f s o l v i n g p r o b l e m s , a n d t h i s method merit of b e i n g p i c t o r i a l .

D-p

DICTIONARY

The t a b l e b e l o w s u m m a r i s e s some o f t h e m o s t commonly u s e d d u a l e q u i v a l e n t s . o f t h e s e now come i n t o t h e realm o f f o l k l o r e a n d w e h a v e n o t a t t e m p t e d t o

Many

H.A. Priestley

54

a t t r i b u t e them. Most a r e easy consequences of r e s u l t s i n 92 o r appear a s theorems elsewhere i n t h e t e x t . Throughout, L E and has d u a l space (XL,C,G); we i d e n t i f y L w i t h O(XL). The t o p o l o g i e s of &-open d e c r e a s i n g ( i n c r e a s i n g ) s e t s a r e denoted I n t h e t a b l e each s t a t e m e n t i n t h e f i r s t cslumn i s e q u i v a l e n t t o t h e by i: (U). c o r r e s p o n d i n g s t a t e m e n t i n t h e second column.

-D

-P

Notes/reference

L a t t i c e types

L finite

C discrete

Boolean

(Xu,<)

an a n t i c h a i n

a chain

(XL,G)a c h a i n

countable

C metrisable

comp 1e t e

ClUV

6

xL

-

f r e e on

K

generators

cv vE

Theorem 3 . 5

1:

Theorem 3.2

= ZK

S p e c i a l elements i n L Join-irreducible

(# 0)

M e e t - i r r e d u c i b l e (# 1)

Morphisms

f E g(L,M)

f injective f surjective

&-clopen s e t

+;c

t-clopen s e t X L

\

tr

@ E _P'XMJL' $ surjective $ a n embedding

I d e a l s , f i l t e r s , congruences Ideal i n L principal ideal

C-open d e c r e a s i n g s e t &-clopen d e c r e a s i n g s e t

prime i d e a l

set XL

maximal i d e a l

set XL

\

E XL)

fz

(3:

{XI

(X maximal)

Filter

C-open i n c r e a s i n g s e t

Congruence

t-open s e t

Lattices related to L Ideal l a t t i c e Filter lattice Congruence l a t t i c e LOP Minimal Boolean e x t e n s i o n Frattini sublattice

s e e [I]

Ordered sets and duality for distributive lattices

n

55

i'

Noteslreference

Categorical notions S u b l a t t i c e of L

Separating s e t for

Q u o t i e n t l a t t i c e of L

Closed subspace of

F i n i t e d i r e c t product

D i s j o i n t union

Coproduct

Direct product

XL XL

Tensor p r o d u c t

C o r o l l a r y 2.5 & 111 C o r o l l a r y 2.5

Theorem 3 . 3 s e e C771

Equational Subcategories

B

(p-algebras)

"W

Theorem 3.1

Hu

L

Pseudocomplement a*

R -morphism -W

R,"

(dual p-algebras)

L E

n,"

Dual pseudocomplement a+

2

(Heyting a l g e b r a s )

L E E

R e l a t i v e pseudocomplement

a+b

-H-morphism

XI, ' +(a 'b ) Dual $ such t h a t $ o + = + . $

Hop

( d u a l Heyting a l g e b r a s )

L E HOP

V E & * + V E C

R e l a t i v e d u a l pseudocomplement a b

+ ( b 'a )

HoP-morphism

Dual $ such t h a t

+

$ 0 4 = + 0 $

de Morgan a l g e b r a s L de Morgan

3 o r d e r - r e v e r s i n g homeomorphism g on X L w i t h g2 = 1

-a I n g e n e r a l , a unary homomorphic ( d u a l homomorphic) o p e r a t i o n on L c o r r e s p o n d s t o a c o n t i n u o u s o r d e r - p r e s e r v i n g ( o r d e r - r e v e r s i n g ) s e l f map on X L'

56

H . A . Priestley

REF I.:R K N C E S

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1711 P r i e s t l e y , H.A.,

O r d e r e d t o p o l o g i c a l s p a c e s and t h e r e p r e s e n t a t i o n o f d i s t r i b u t i v e l a t t i c e s , P r o c . London Mat h. S o c . ( 3 ) 24 ( 1 9 7 2 ) 507-530.

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( 1 9 7 4 ) 127-143. 1 7 3 1 P r i e s t l e y , H . A . , The c o n s t r u c t i o n of s p a c e s d u a l t o ps e udoc omple me nte d d i s t r i b u t i v e l a t t i c e s , Q u a r t . J . Math. Oxford ( 2 ) 26 ( 1 9 7 5 ) 215-228. 1 7 4 1 P r i e s t l e y , H . A . , C a t a l y t i c d i s t r i b u t i v e l a t t i c e s and c ompa c t z e r o dimensional topological l a t t i c e s , p r e p r i n t . r 7 5 1 P r i e s t l e y , H . A . , A l g e b r a i c l a t t i c e s as d u a l s o f d i s t r i b u t i v e l a t t i c e s , t o a p p e a r i n : P r o c e e d i n g s o f t h e C o n f e r e n c e o n T o p o l o g i c a l a nd C a t e g o r i c a l A s p e c t s o f C o n t i n u o u s L a t t i c e s , Rremen, 1 9 8 2 . The v a l u a t i o n r i n g o f a d i s t r i b u t i v e l a t t i c e , i n : P r o c e e d i n g s 1761 Rota, G.-C., o f t h e U n i v e r s i t y of H o u s t o n L a t t i c e T heory C o n f e r e n c e ( 1 9 7 3 ) . 1 7 7 ~1Schmi d , J . , T e n s o r p r o d u c t s of d i s t r i b u t i v e l a t t i c e s a n d t h e i r P r i e s t l e y d u a l s , A c t a Math. Acad. S c i . Hungar. 35 ( 1 9 8 0 ) 387-392.

[781 Schmid, J . , On l a t t i c e s o f c o n t i n u o u s o r d e r - p r e s e r v i n g f u n c t i o n s , Fund. Math. 1 1 6 ( 1 9 8 0 ) 19-30. 1 7 9 1 S c h m i d t , E . T . , Remark o n g e n e r a l i z e d f u n c t i o n l a t t i c e s , A c t a Ma th. Acad. S c i . Hungar. 34 ( 1 9 7 9 ) 337-339.

1801 S p e e d , ‘ T . P . , A n o t e on P o s t a l g e b r a s , C o l l o q . Math. 24 ( 1 9 7 1 ) 37-44. L811 S p e e d , T . P . ,

P r o f i n i t e p o s e t s , B u l l . A u s t r a l . Math. S o c . 6 ( 1 9 7 2 ) 177-183.

1821 S p e e d , T . P . , The o r d e r of p r i m e s , A l g e b r a Uni v. 2 (1972) 85-87. The t h e o r y o f r e p r e s e n t a t i o n s of Boole a n a l g e b r a s , T r a n s . A m e r . 1831 S t o n e , M . H . , Math. S OC . 40 ( 1 9 3 6 ) 37-111. r 8 4 1 S t o n e , M . H . , A p p l i c a t i o n s o f t h e t h e o r y of B o o l e a n r i n g s t o g e n e r a l t o p o l o g y , T r a n s . Amer. M a t h. S o c . 4 1 ( 1 9 3 7 ) 375-481. 1 8 5 1 S t o n e , M . H . , T o p o l o ~ i c a lr e p r e s e n t a t i o n of d i s t r i b u t i v e l a t t i c e s a n d B r o u w e r i a n l o g i c s , C a s o p i s P e s t . Mat. F y s . 67 ( 1 9 3 7 ) 1-25. 1861 S t r a l k a , A . R . , A p a r t i a l l y o r d e r e d s p a c e which i s n o t a P r i e s t l e y s p a c e , Sem i g r o u p Forum 20 ( 1 9 8 0 ) 293-297. 1 8 7 1 T a n , T . , On R e p r e s e n t a b l e P o s e t s , Ph.D. Canada ( 19 74)

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1881 Thomas, D . A . ,

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10

1 9 0 1 T r a c z y k , T . , L a t t i c e s w i t h g r e a t e s t ( l e a s t ) c h a i n b a s e , B u l l . Acad. P o l o n . S c i . S 6 r . S c i . Math. Astronom. P h y s . 23 ( 1 9 7 5 ) 935-938. 1911 T r a c z y k , T . , R e p r e s e n t a t i o n theorems f o r P o s t a l g e b r a s , Arta P o l y t e c h . P r i c e CVUT 1 3 ( 1 9 7 7 ) 57-62. 1 9 2 1 Trn k o v 5 , V . , I s o m o r p h i s m s o f sums o f c o u n t a b l e B o o l e a n a l g e b r a s , P r o c . A m e r . Math. S o c . 80 ( 1 9 8 0 ) 389-392. L931 U r q u h a r t , A . : F r e e d i s t r i b u t i v e pseudo-compl eme nte d 3 ( 1 9 7 3 ) 13-15.

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1 9 5 1 U r q u h a r t , A . , D i s t r i b u t i v e l a t t i c e s w i t h a d u a l homomorphic o p e r a t i o n , S t u d i a L o g i c a 38 ( 1 9 7 9 ) 201-209. 1961 Urquhart , A . , P r o j e c t i v e d i s t r i b u t i v e p - a l g e b r a s , 24 ( 1 9 8 1 ) 269-275.

B u l l . A u s t r a l . Math. S O C .

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Algebra

1 9 8 1 Wille, R . , S u b d i r e k t e P r o d u k t e v o l l s t Y n d i g e r VerbBnde, 3 . K e ine Angew. Math. 2 8 3 / 2 8 4 ( 1 9 7 6 ) 53-70. L991 Wille, R . , C a n c e l l a t i o n and r e f i n e m e n t p r o p e r t i e s f o r f u n c t i o n l a t t i c e s , H o u s t o n J . Math. 6 (1980) 431-437. C l 0 0 l Wrofiski, A . , The number of q u a s i v a r i e t i e s o f d i s t r i b u t i v e l a t t i c e s w i t h p s e u d o c o m p l e m e n t a t i o n , P o l i s h Acad. S c i . I n s t . P h i l o s . S o c i o l . B u l l . S e c t . L o g i c 5 ( 1 9 7 6 ) 115-121.

Annals of Discrete Mathematics 23 (1984) 61-80 0 Elsevier Science Publishers B.V. (North-Holland)

61

WHEN ARE ORUEK SCATTEKEI) AND TOPOLOGLCAI.I,Y SCATTERED THE SAME ?

M i c h a e l MISLOVE

De pa r t me n t of Ma t h ema t i c s Tulane U n i v e r s i t y New O r l e a n s , L A 70118

Nous d i s o n s q u ' u n e n s e m b l e o r d o n n 6 e s t d i s p e r s 6 a u s e n s de l ' o r d r e l o r s q u e s e s s e u l e s p a r t i e s denses (pour l ' o r d r e )

0 , l e s singletons e t l e s antichaInes. Par a i l l e u r s ,

sont

nous d i s o n s q u ' u n e s p a c e t o p o l o g i q u e e s t d i s p e r s e l o r s q u e chacun d e s e s sous-espaces

a au moins un p o i n t i s o l e pour

l a t o p o l o g i e i n d u i t e . La q u e s t i o n c o n t e n u e d a n s l e t i t r e e s t d e s a v o i r d a n s q u e l s c a s c e s n o t i o n s s e c o n f o n d e n t p o u r un e s p a c e t o p o l o g i q u e ( p a r t i e l l e m e n t ) o r d o n n e . Nous a p p o r t o n s une r b p o n s e p o u r l a c l a s s e d e s t r e i l l i s a l g b b r i q u e s e n mont r a n t que c e s n o t i o n s c o i n c i d e n t s i l e t r e i l l i s e s t d i s t r i b u t i f e t e n p r o d u i s a n t u n exemple d e t r e i l l i s a l g e b r i q u e nond i s t r i b u t i f q u i , s a n s G t r e d i s p e r s e , e s t d i s p e r s 6 au s e n s d e l ' o r d r e . C o m e t o u t compact ( o r d o n n e ) d i s p e r s e e s t a u s s i d i s p e r s i . pour l ' o r d r e , c e l a r P g l e l a q u e s t i o n pour l e s t r e i l -

l i s algbbriques. L ' a r t i c l e cornprend deux p a r t i e s . La p r e m i P r e ( s e c t i o n s I

B

IV) vise

2 p r C s e n t e r un r a p p e l d e l a t h e o r i e d e s t r e i l l i s

algebriques et de l e u r d u a l i t 6 avec l e s d e m i - t r e i l l i s

discrets.

Nous e x p l i q u o n s a u s s i comment l a t o p o l o g i e d o n t s o n t m u n i s

ces

t r e i l l i s e s t i s s u e d e f a c o n n a t u r e l l e d e l a d u a l i t 6 bvoqu6e. E n f i n , nous i n t r o d u i s o n s pour l e s t r e i l l i s a l g e b r i q u e s d i s t r i b u t i f s l a t h b o r i e s p e c t r a l e , q u i f o u r n i t une d u a l i t 6 e n t r e c e u x - c i e t l e s e s p a c e s ternpbres q u i s o n t l o c a l e m e n t q u a s i compacts e t n a n t i s d ' u n e b a s e d ' o u v e r t s quasi-compacts.

Le b u t

de c e t t e synthhse e s t de v a l o r i s e r l a r e l a t i o n e n t r e topologie e t algilbre ( r e l a t i o n s i prbcieuse, n o t a m e n t , pour l'approche de la s t r u c t u r e des t r e i l l i s algebriques e t de l e u r s applications

B

l a topologie e t

B

l'ordre).

M. Mislove

62

La s e c o n d e p a r t i e ( s e c t i o n s V i V I I ) p r G s e n t e ,

d a n s l e c a s d e s t r e i l l i s a l g & b r i q u e s , l e s r b s u l t a t s c onc e r n a n t l a connexion e n t r e l e s deux s u s d i t e s n o t i o n s

de

d i s p e r s i o n . On y e x p l o r e a u s s i l e l i e n e n t r e c c s n o t i o n s e t l e s n o t i o n s c o r r e s p o n d a n t e s du s p e c t r e d ' u n t r e i l l i s alg6brique d i s t r i b u t i f . Le r P s u l t a t e s s e n t i e l est l e th6orl.me 6 . 5 . e t s o n C o r o l l a i r e 6 . 6 .

0.

I NTKODUC T ION.

T h e q u e s t i o n w h i c h t h e t i t l e c o n t a i n s c a n b e more p r e c i s e l y p h r a s e d a s f o l l o w s . I f P i s a p o s e t , t h e n t h e s u b s e t A c P i s orcicr d m s t i f , g i v e n a , b E 4 w l t h a < h, t h e r e i s some c E A wiLh a < c < b. P i s s a i d t o b e ordcr scdtLcrJcd i f P c o n t a i n s n o o r d e r d e n s e s u b s e t s o t h e r t h a n fi, s i n g l e t o n s , and anti-chains. A t o p o l o g i c a l s p a c e X L S s a i d t o b e topologically scattered i f e v e r y non-empty s u b s p a c e o f X h a s a t l e a s t o n e i s o l a t e d p o i n t i n t h e r e l a t i v e t o p o l o g y . The o b v i o u s q u e s t i o n i s how t h e s e n o t i o n s r e l a t e o n a s e t I, w hic h i s a t o n c e a p o s e t a n d a t o p o l o g i c a l s p a c e . Of c o u r s e , some demands m u s t be made o n t h e t o p o l o g y a n d t h e o r d e r o n L i f a n y r e l a t i o n b e t w e e n them i s t o h e e x p e c t e d . For example, t h e unit interval i s clearly not order scattered, h u t i t is topologically scattered when endowed w i t h t h e d i s c r e t e t o p o l o g y . On t h e o t h e r ha nd, a n y non-empty p e r f e c t space ( i . e . , one without i s o l a t e d points) i s not topologically scattered, h u t i t i s order scattered i n t h e d i s c r e t e order. A n a t u r a l c l a s s o f o b j e c t s where t h i s q u e s t i o n makes s e n s e , a n d l e a d s t o n o n - t r i v i a l r e s u l t s i s t h e c l a s s o f t o p o l o g i c a l l a t t i c e s , o r , mo r e g e n e r a l l y , t h e c l a s s o f t o p o l o g i c a l s e m i l a t t i c e s . I n d e e d , e a c h a l g e b r a i c l a t t i c e h a s a n a t u r a l l y d e f i n e d t o p o l o g y r e l a t i v e t o w hic h i t i s a co m p act s e m i l a t t i c e . F o r t h e s e l a t t i c e s , we h a v e a s i m p l e a n s w e r t o o u r q u e s t i o n . Namely, i f L i s a d i s t r i b u t i v e a l g e b r a i c l a t t i c e , t h e n L i s o r d e r s c a t t e r e d i f and only i f L i s topologically scattered. On t h e o t h e r ha nd, w h i l e a n y c ompa c t pospace which i s t o p o l o g i c a l l y s c a t t e r e d m u s t a l s o b e o r d e r s c a t t e r e d , t h e r e a r e n o n - d i s t r i b u t i v e a l g e b r a i c l a t t i c e s which a r e o r d e r s c a t t e r e d b u t n o t topologically scattered, W h i l e we s h a l l d e v e l o p o u r r e s u l t s c o m p l e t e l y f o r a l g e b r a i c l a t t i c e s , m o r e general results a r e available. I n d e e d , a l l o f o u r r e s u l t s h o l d f o r t h e more g e n e r a l c l a s s of c o n t i n u o u s l a t t i c e s ( s e e ( M 2 1 ). However, w e r e s t r i c t o u r a t t e n t i o n t o a l g e b r a i c l a t t i c e s f o r two r e a s o n s . F i r s t , we c a n p r e s e n t t h e r e s u l t s we h a v e f o r t h e s e l a t t i c e s i n a c o m p l e t e l y s e l f - c o n t a i n e d f a s h i o n , w h l l e t h e r e s u l t s f o r c o n t i n u o u s l a t t i c e s i n g e n e r a l r e q u i r e some b a c k g r o u n d w h i c h i s not so easy t o duplicate. Second, w e w a n t t o u s e t h e o c c a s i o n o f t h i s p a p e r t o p r e s e n t some o f t h e t o p o l o g i c a l a s p e c t s o f a l g e b r a i c l a t t i c e s i n o r d e r t h a t t h e T h a t i s , t h e way i n w h i c h r e a d e r may s e e c l e a r l y how t h e s e r e s u l t s a r i s e . a l g e b r a i c l a t t i c e s may b e t o p o l o g i z e d , a n d t h e u s e s o f t h a t t o p o l o g y , a r e b a s i c t o s i m i l a r r e s u l t s f o r c o n t i n u o u s l a t t i c e s . The t e c h n i q u e s we e mploy h e r e s h o u l d b e u s e f u l i n s e t t l i n g o t h e r questions about continuous l a t t i c e s and a l g e b r a i c l a t t i c e s , and t h e s e q u e s t i o n s a r i s e n a t u r a l l y i n t h e a r e a of ordered sets, and a r e n o t e x c l u s i v e t o continuous l a t t i c e theory. Tnus, p a r t of o u r g o a l f o r t h i s paper i s t o p r o s e l y t i z e about a l g e b r a i c l a t t i c e s , continuous l a t t i c e s , and t h e i r In f a c t , t h e f i r s t p a r t o f t h i s p a p e r c a n b e v i e w e d a s a p r o l o g u e applications. t o t h e p a p e r [ M I ] , which p r o v i d e s a n i n t r o d u c t i o n t o t h e t h e o r y o f c o n t i n u o u s lattices. T h e s e c o n d p a r t o f t h e p a p e r , w h e r e we o b t a i n t h e r e s u l t s i n d i c a t e d a b o v e f o r a l g e b r a i c l a t t i c e s , p r o v i d e s a good e x a m p l e o f how t h e t o p o l o g y a n d t h e a l g e b r a o f a n a l g e b r a i c l a t t i c e i n t e r m i n g l e , a n d how o n e c a n u s e o n e a s p e c t t o settle questions about t h e other.

Order scattered and topologically scattered

63

T h e r e s u l t s o f t h i s p a p e r w e r e i n s p i r e d by a q u e s t i o n M a u r i c e P o u z e t p o s e d a t t h e NATO S.ynposiun o n O r d e r e d S e t s i n B a n f f , C a n a d a i n S e p t e m b e r , 1981. B a s i c a l l y , h e announced t h a t t h e L a t t i c e I ( P ) o f lower e n d s ( o r i n i t i a l segments) o f a p o s e t P i s o r d e r s c a t t e r e d i f and only i f i t i s t o p o l o g i c a l l y scattered. He t h e n a s k e d w h e t h e r t h i s r e s u l t c o u l d b e e x t e n d e d t o t h e m o r e general c l a s s of completely d i s t r i b u t i v e l a t t i c e s using t h e Rirkhoff-Priestley D u a l i t y . W h i l e we d o n o t u t i l i z e t h a t d u a l i t y , we o b t a i n t h e d e s i r e d generalization f o r t h e c l a s s o f d i s t r i b u t i v e algebraic l a t t i c e s i n t h i s paper, and o u r r e s u l t s i n [M2] g i v e t h e answer t o Pouzet's q u e s t i o n f o r t h e c l a s s o f continuous l a t t i c e s . T h i s w o r k was c o m p l e t e d w h i l e t h e a u t h o r was a f e l l o w o f t h e A l e x a n d e r v o n Humboldt S t i f t u n g a n d a g u e s t o f t h e F a c h b e r e i c h Mathematik d e r T e c h n i s c h e H o c h s c h u l e D a r m s t a d t , West Germany. T h a n k s a r e d u e t o b o t h f o r t h e i r k i n d h o s p i t a l i t y a n d g e n e r o u s s u p p o r t . T h e s e r e s u l t s were p r e s e n t e d a t t h e C o n f e r e n c e on O r d e r e d S e t s a t t h e C h a t e a u d e

1. -

l a T o u r e t t e i n J u l y , 1982.

ALGEBRAIC LATTICES

We b e g i n w i t h a d i s c u s s i o n o f a l g e b r a i c l a t t i c e s , how t h e y a r i s e , a n d t h e v e r y n a t u r a l t o p o l o g y w h i c h t h e y h a v e . T h e s e r e s u l t s a r e n o t new. Indeed, t h e y a r e L m p l i c i t l y c o n t a i n e d i n [ C ] (and, sometimes, e x p l i c L t l y ) , a n d e x p l i c i t l y gLven i n [ H M S ] , w h i c h i s d e v o t e d s o l e l y t o t h e t h e o r y o f a l g e b r a i c l a t t i c e s . However, we f e e l t h a t t h i s i s a g o o d t i m e t o r e v i e w t h e s e r e s u l t s , s o t h a t t h o s e who a r e interested can use t h i s paper as a f i r s t s t e p i n understanding t h e theory o f c on t i n u o u s 1a t t i c e s

.

A t t h e o u t s e t we p o i n t o u t t h a t we s h a l l b e d e a l i n g a l m o s t e x c l u s i v e l y w i t h c o m p l e t e l a t t i c e s . H e n c e , u n l e s s o t h e r w i s e e x p l i c i t l y s t a t e d , a22 l a t t i c e s are clsswned to be cornpzetc. T h a t i s , g i v e n a l a t t i c e L, we a s s m e t h a t e v e r y s u b e t o f L h a s a n i n f u n u m a n d a supremum. We s h a l l u s e w h a t h a s become f a i r l y s t a n d a r d n o t a t i o n a n d t e r m i n o l o g y . For a p o s e t P a n d d n e l e m e n t x E P, + x = { y E P : y 6 x } a n d f x = { y E P : x 6 y}. A c P i s a s u b s e t , t h e n J.A = U{+a : a E A} a n d + A = U { + a : a E A}. A is If a ZoLPr s e t ( a l s o c a l l e d a n i n i t i d 2 segment) i f A = J.A; A i s a n uppcr s e t i f A = 4 A . A non-empty s u b s e t D c P i s d i r e c t e d i f , f o r e a c h d , d ' E D, there is some d " E D w i t h d 6 d " a n d d o < d"; t h e nowempty subset F c P i s f i l t e r e d i f , f o r e a c h x , y E F t h e r e i s some z E F w i t h z 6 x a n d z 6 y .

1.1

DEFINITION.

Let

be a complete l a t t i c e . The e l e m e n t k E L i s c a l l e d A c L, w h e n e v e r k 6 s u p A , t h e r e i s some f i n i t e k 6 sup F. W e d e n o t e by K(L) t h e s e t o f a l l c o m p a c t

L

compact i f , f o r e v e r y s u b s e t subset F c A with e l e m e n t s o f L. rl

A l t e r n a t i v e l y , t h e element k E L i s compact i f a n d o n l y i f , f o r e v e r y d i r e c t e d s u b s e t D c L, w h e n e v e r k < s u p D, t h e n k C d f o r some d E D. I t may h a p p e n t h a t t h e o n l y c o m p a c t e l e m e n t o f a l a t t i c e L i s 0 ( f o r On t h e o t h e r hdnd, a n y example, c o n s i d e r t h e u n i t i n t e r v a l i n t h e u s u a l o r d e r ) . f i n i t e l a t t i c e h a s t h e p r o p e r t y t h a t e v e r y e l e m e n t i s compact.

1 . 2 DEFINLTLON. L e t L b e a c o m p l e t e l a t t i c e . x = s u p ( + x n K ( L ) ) f o r e v e r y x E L. n

L

i s alpbraic i f

As a s u n p l e e x a m p l e , l e t R b e a r i n g w i t h i d e n t i t y , a n d l e t L b e t h e l a t t i c e of (two-sided) i d e a l s o f R. Then L i s c l e a r l y a c o m p l e t e l a t t i c e w i t h g r e a t e s t e l e m e n t R a n d l e a s t e l e m e n t {O}. M o r e o v e r , i f r E R , then the p r i n c i p a l i d e a l w h i c h r g e n e r a t e s i s a c o m p a c t e l e m e n t i n t h i s l a t t i c e . I n d e e d , i f {Ai} i s a family o f i d e a l s o f R, a n d i f i s c o n t a i n e d i n t h e

M. Mislove

64

n ideal

A

which t h e

...,

generate, then

r =

C aI E o r s o m ~f i n i t e E a m i l y , A , , 1=I i = I, n , w h e r e aL E A,. Rut t h e n r , a n d h e n c e < r > , a r e contained i n t h e i d e a l g e n e r a t e d by A l , A,. T h u s t h e i d e a l w h i c h r g e n e r a t e s i s c o m p a c t 111 L . One c a n e a s i l y g e n e r a l i z e t h i s a r g u m e n t t o show t h a t e v e r y f i n i t e l y g e n e r s t c d i d e a l o f R i s c o m p a c t i n L, a n d c o n v e r s e l y . S ~ n c re v e r y i d e a l i s t h e s u p o f t h e f i n i t e l y generated i d e a l s which i t conta~ns, i t follows t h a t L is a n algebraic lattice. A i l s

...,

T h i s example c a n e a s i l y b e g e n e r a l i z e d t o t h e l a t t i c e o f c o n g r u e n c e s on a n y u n i v e r s a l a l g e b r a , w h e r e i n t h e f i n i t e l y g e n e r a t e d congruences form t h e compact elements of t h e lattice. These examples a r e t h e reason t h a t s u c h l a t t i c e s a r e c a lled a Igebraic. One p r o p e r t y o f K(L) w h i c h w e j u s t u s e d l e a d s t o a c h a r a c t e r i z a t l o n o f a l g e b r a i c l a t t i c e s . Namely, i f L i s a c o m p l e t e l a t t i c e a n d k and k ' are c o m p a c t e l e m e n t s o f L, then k V k' i s a l s o compact. Indeed, l e t D c L be S i n c e k i s c o m p a c t , t h e r e 1 s some d E T) w i t h d i r e c t e d w i t h k V k ' < s u p D. k < d; likewise, since k' i s c o m p a c t , t h e r e i s some d ' 6 D w i t h k ' < d ' . 'Then, s i n c e D i s d i r e c t e d , t h e r e i s some d " E D w i t h d , d ' < d", and i t H e n c e k V k" E K ( L ) . f o l l o w s t h a t k V k ' < d".

1.3 L

LEMMA. If L i s a c o m p l e t e l a t t i c e , t h e n c o n t a i n i n g 0. [1

i s a sup-subsemilattice of

K(L)

We w a n t t o r e t r i e v e L f r o m t h e s u p - s e m i l a t t i c e K ( L ) , b u t t h i s i s n o t a l w a y s p o s s i b l e ( r e m e m b e r t h e l a t t i c e [O,l]). However, i f L i s a l g e b r a i c , t h e n t h i s is possible. N a m e l y , f o r a n y x F_ L , we h a v e x = s u p ( C x n K ( L ) ) . If S i s a s u p - s e m i l a t t i c e , a f i l t e r i n S i s a l o w e r s e t F = CF c S w h i c h i s c l o s e d Thus every point u n d e r f i n i t e suprema. T h u s , Cx n K(L) i s a f i l t e r i n K ( L ) . o f L g i v e s r i s e t o a E i l t e r i n K(L), T h i s f i l t e r i s unique, s i n c e x f y i m p l i e s t h e r e i s some c o m p a c t e l e m e n t o f L b e l o w o n e o f x a n d y , b u t n o t t h e C o n v e r s e l y , s u p p o s e t h a t F c K(L) i s a o t h e r , s o Cx n K(L) f Cy n K ( L ) . f i l t e r , a n d c o n s i d e r x = s u p F ( i n L). T h e n c l e a r l y Cx n K(L) 3 F . On t h e x. S i n c e x = s u p F and k is compact, o t h e r h a n d , s u p p o s e k E K(L) w i t h k t h e r e i s some k ' E F w i t h k < k ' , as F is d i r e c t e d . Rut, t h e n k E F as F i s a l o w e r s e t i n K ( L ) . T h i s s h o w s t h a L F = Cx n K(L). 1 . 4 PROPOSITION. F o r a n a l g e b r a i c l a t t i c e L, t h e map x + Cx n K(L) : L + F i l t K(L) f r o m L t o t h e l a t t i c e o f f i l t e r s o f K(L) i s a l a t t i c e i s o m o r p h i s m . PROOF. We h a v e a l r e a d y shown t h a t t h e map i s a b i j e c t i o n . M o r e o v e r , i f x , y E L, t h e n i t L S o b v i o u s t h a t (Cx n K ( L ) ) A (Cy n K ( L ) ) = Cxy n K ( L ) . On t h e o t h e r Conversely, i f h a n d , c l e a r l y (Cx n K ( L ) ) V (Cy n K ( L ) ) c C(xVy) n K ( L ) . z = s u p ( ( C x n K ( L ) ) V (Cy n K ( L ) ) ) , t h e n x , y C z implies that x V y C z . Since k V k ' < x V y f o r e v e r y k E Cx n K(L) a n d k ' E Cy n K ( L ) , we h a v e z C x V y . Hence, F = (Cx n K ( L ) ) V (Cy n K ( L ) ) i s a f i l t e r i n K(L) w i t h K(L) s i n c e t h e map i s s u p F = x V y , s o (Cx n K ( L ) ) V (Cy n K ( L ) ) = C(xVy) one-to-one. n On t h e o t h e r h a n d , s u p p o s e t h a t w e s t a r t w i t h a s u p - s e m i l a t t i c e S w h i c h h a s a l e a s t e l e m e n t , 0. T h e n , t h e l a t t i c e L = F i l t S o f a l l f i l t e r s o f S i s c e r t a i n l y a c o m p l e t e l a t t i c e when o r d e r e d u n d e r i n c l u s i o n . Moreover, e a c h p o i n t s o f S g i v e s r i s e t o a c o m p a c t e l e m e n t o f L, namely 4s = I t E S : t C s ) . T h i s f i l t e r i s compact. F o r a d i r e c t e d f a m i l y { F i } o f f i l t e r s o f S , s u p F, = UF,, s o , i f Cs C s u p F i , t h e n s E F i f o r some 1, a n d s o C s c F, for that 1. Moreover, i f F i s a n y f i l t e r o f S, t h e n F = s u Cs r e a l i z e s F a s t h e S

s u p o f t h e compact e l e m e n t s

Cs, s

E

F.

€8

' h i s means t h a t

L

is algebraic.

Order scattered and topologically scattered Finally,

if

F

is any compact element of

f i n i t e l y many p o i n t s ,

sl,

...,sn

L,

then

F = su S

in

F

with

6.5

F = s u p Cs,.

Cs

means t h e r e a r e

,F S i n c e

F

is a filter,

1

we h a v e s = "1 V . . . V sn r e s u l t s so f a r a s follows:

1.5

is

also i n

F,

and t h e n

F = 4s.

We s u n m a r i z e o u r

THEOREM. (a) If Id i s a n a l g e b r a i c l a t t i c e , t h e n t h e map x + Cx n K(L) : L + F i l t K(L) i s a n i s o m o r p h i s m o f I, o n t o t h e l a t t i c e of f i l t e r s of K(L), t h r s u p - s u b s e m i l a t t i c e o f c o m p a c t e l e m e n t s o f L. (h)

If S i s a s u p - s e m i l a i t i c e w i t h 0, t h e n t h e map s + Cs : S + K(L) i s a n isomurphism o f S o n t o t h e s t ~ p - s e m i l a t t i c eo f compact e l e m e n t s of L = F i l t S, t h e l a t t i c e of f i l t e r s of S.

PROOF. W e h a v e p r o v e d e v e r y t h i n g e x c e p t t h e f a c t t h a t t h e map s + J.s isomorphism o f s u p - s e m i l a t t i c e s . R u t , t h i s i s c l e a r f r o m t h e o r d e r on = Filt S. fl

is an L

So f a r we h a v e e s t a b l i s h e d a n e q u i v a l e n c e b e t w e e n a l g e b r a i c l a t t i c e s , o n t h e o n e h a n d , a n d s u p - s e m i l a t t i c e s w i t h 0, on t h e o t h e r . Our n e x t g o a l i s L O g i v e a d u a l i t y between a c a t e g o r y o f a l g e b r a i c l a t t i c e s and a c a t e g o r y o f sup-semilatt i c e s w i t h 0. To d o t h i s , w e n e e d t o i n d i c a t e w h a t m o r p h i s m s w e w i s h t o u s e i n each case. For t h e algebraic l a t t i c e s , t h e r e a r e s e v e r a l p o s s i b i l i t i e s , but f o r s u p - s e m i l a t t i c e s w i t h 0, only one p o s s i b i l i t y i s reasonable. N a m e l y , we l e t S d e n o t e t h e c a t e g o r y w h o s e o b j e c t s a r e s u p - s e m i l a t t i c e s w i t h 0, a n d w h o s e With t h e s e morphisms i n m o r p h i s m s a r e s e m i l a t t i c e m o r p h i s m s p r e s e r v i n g t h e 0. h a n d , we now d e t e r m i n e w h a t m o r p h i s m s w e s h o u l d u s e b e t w e e n a l g e b r a i c l a t t i c e s .

Let S a n d T b e s u p - s e m i l a t t i c e s w i t h 0 , a n d l e t f : S + T b e a s u p s e m i l a t t i c e m o r p h i s n p r e s e r v i n g 0. Then, L = F i l t S and M = F i l t T are a l g e b r a i c l a t t i c e s , a n d we c a n d e f i n e F i l t f : M + L by ( F i l t f)(F) = f-l(F). S i n c e F is a f i l t e r and f p r e s e r v e s f i n i t e s u p s , i t follows t h a t ( F i l t f ) ( F ) i s a f i l t e r i n S . Moreover, F i l t f obviously preserves a l l i n t e r s e c t i o n s , which a r e i n f m a i n t h e f i l t e r l a t t i c e . While suprema i n t h e f i l t e r l a t t i c e a r e n o t g e n e r a l l y u n i o n s , c e r t a i n l y d i r e c t e d suprema are, a n d F i l t f c l e a r l y preserves these as well. Thus, we have determined t h a t , €or f : S + T a sups e m i l a t t i c e morphism b e t w e e n s u p - s e m i l a t t i c e s , t h e morphism F i l t € : F i l t T + F i l t S p r e s e r v e s a l l i n f m a a n d a l l d i r e c t e d suprema, a n d c l e a r l y ( F i l t f)(T) = S, so t h a t F i l t € p r e s e r v e s t h e i d e n t i t y a s well. T h u s , w e l e t AL denote t h e c a t e g o r y of a l g e b r a i c l a t t i c e s a n d maps p r e s e r v i n g a l l i n f m a , a l l d i r e c t e d suprema, a n d t h e i d e n t i t y . Suppose t h a t L and M a r e a l g e b r a i c l a t t i c e s a n d t h a t @ : L + M pres e r v e s a l l i n f i m a , a l l d i r e c t e d s u p r e m a , a n d t h e i d e n t i t y . We w a n t t o d e f i n e t h e If k E K(M), t h e n 4k i s a f i l t e r i n M w i t h t h e m a p K ( @ ) : K(M) + K ( L ) . p r o p e r t y t h a t D n 4k f 0 i f s u p D E 4 k , f o r e v e r y d i r e c t e d s u b s e t D c M. S i n c e $r p r e s e r v e s a l l i n f i m a a n d 4 k i s c l o s e d u n d e r a l l i n f m a , i t follows t h a t k ' = A@-l(4k) E @ - l ( 4 k ) . Moreover, s i n c e @ p r e s e r v e s d i r e c t e d suprema, i t i s a l s o t r u e t h a t , i f D c L i s d i r e c t e d w i t h s u p D E @-'(4k), then D n @ - l ( 4 k ) f I?. S a i d a n o t h e r w a y , t h i s m e a n s t h a t k ' = A @ - ' ( t k ) E K ( L ) . So, we d e f i n e K ( @ ) ( k ) = A @ - l ( + k ) . T h i s i s t h e n t h e smallest compact e l e m e n t o f L w h i c h @ maps a b o v e k .

To s how t h a t K ( @ ) i s a n S m o r p h i s m , we m u s t a l s o s h o w t h a t then f i n i t e suprema. B u t , i f k l , k g E K(M),

K(@) p r e s e r v e s

M. Mislove

66

and so

given

p r e s e r v e s f i n i t e suprema.

K(@)

I t i s now a r o u t i n e e x e r c i s e t o v e r Ey t h a t t h e f o l l o w i n g d i a g r a m s c o m n u t e , f : S + T i n S a n d @ : L + M i n A1 :

lS

jL

k

I

where

is(s) = +s,

and

i ( t ) = +t.

T

I

Of c o u r s e , t h e p o i n t we a r e maki ng i s s u n m a r i z e d by t h e f o l l o w i n g :

1.6 THEOREM. L e t S d e n o t e t h e c a t e g o r y o f s u p - s e m i l a t t i c e s w i t h 0 a n d semil a t t i c e maps p r e s e r v i n g 0, a n d l e t AL d e n o t e t h e c a t e g o r y o f a l g e b r a i c l a t t i c e s a n d maps p r e s e r v i n g a l l i n f u n a , d i r e c t e d s u p r e m a , a n d t h e i d e n t i t y a r e d u a l l y e q u i v a l e n t under t h e f u n c t o r s which a s s o c i element. Then S an d a t e t o a n S E S i t s l a t t i c e o f f i l t e r s , and t o a n L E AL i t s sup-semilattice o f compact e l e m e n t s . L1.

TOPOLOGLZLNG ALGEBRAIC LATTLCES

We now h a v e f o u n d a c o m p l e t e d u a l i t y b e t w e e n a c a t e g o r y o f a l g e b r a i c l a t t i c e s a n d a c a t e g o r y o f s u p - s e m i l a t t i c e s w i t h l e a s t e l e m e n t , b u t t h e maps b e t w e e n a l g e b r a i c l a t t i c e s w h i c h a r i s e i n t h i s s e t t i n g a r e somewhat s t r a n g e , a t l e a s t f r o m a p u r e l y a l g e b r a i c v i e w p o i n t . To g i v e a n a l t e r n a t i v e f o n w l a t i o n o f t h i s d u a l i t y , we i n t r o d u c e t o p o l o g y i n t o o u r s e t t i n g . F o r a n a l g e b r a i c l a t t i c e L, we h a v e s e e n t h a t L c a n b e v i e w e d a s t h e l a t t i c e of f i l t e r s on t h e sup-semilattice K(L) o f c ompa c t e l e m e n t o f L. A n o t h e r way t o s a y t h e same t h i n g i s t o r e g a r d e a c h c ompa c t e l e m e n t k o f t h e a l g e b r a i c l a t t i c e L a s g e n e r a t i n g a m o r p h i m @k : L + 2, w h e r e 2 d e n o t e s t h e tw-point l a t t i c e {O,l}. T h i s morphisn i s simply t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e u p p e r s e t o f k , 4 k . Now, t h i s map c l e a r l y p r e s e r v e s a l l i n f i m a , s i n c e 4k i s a p r i n c i p a l f i l t e r f r o m L. Moreover, @k a l s o p r e s e r v e s d i r e c t e d s u p r e m a s i n c e k i s c o mp a c t . F i n a l l y , i t is a l s o c l e a r t h a t = 1.

.

Order scattered and topologically scattered

61

We t h e r e f o r e h a v e t h e map e : L + zK(L) g i v e n by e ( x ) = ( @ k ( x ) )kEK(L), a n d t h i s map p r e s e r v e s a l l i n f i m a a n d a l l d i r e c t e d s u p r e m a . I f w e g i v e 2' the d i s c r e t e t o p o l o g y , a n d t h e n '2K(L) t h e T y c h o n o f f p r o d u c t t o p o l o g y , t h e n 2 K ( L ) i s a compact z r r o - d i m e n s i o n a l s p a c e , a n d t h e image e ( L ) i s c l o s e d i n 2 K ( L ) . R a t h e r t h a n v e r i f y t h i s l a s t d i r e c t l y , l e t u s approach t h e problem from a n o t h e r v iewpo i n t ,

We d e f i n e d t o p o l o g y o n L a 5 f o l l o w s . Let h d e n o t e t h e t o p o l o g y w h i c h a s x r a n g e s o v e r L, together h a s f o r d s u b b d s i s a l l s e t s o f t h e form L\+x, S i n c e 4 k l n 4kg = w i t h a l l s e t s o f t h e form 4 k , a s k r a n g e s o v e r K(L). n

4klvk2

f o r kl,k2

E

K(L), a t y p i c a l b a s i c open set o f L l o o k s l i k e

(4k)\(

U 1= I

fx,),

where k E K(L) d n d x 1 , . . . , x n E L. We w d n t t o d e r i v e t h e p r o p e r t i e s o f t h i s If x,y E L with x y, t o p o l o g y , a n d we b e g i n by n o t i n g t h a t i t i s H a u s d o r f f . t h e n s i n c e L i s a l g e b r a i c , t h e r e i s some c o m p a c t e l e m e n t k E L w i t h k < x but k y ( s i n c e x = s u p ( J . x (1 K ( L ) ) ) . Rut, t h e n x E 4k, which i s open, and y E L\4k, which i s a l s o open, a n d t h e s e s e t s a r e c l e a r l y d i s j o i n t . N e x t , we s h o w t h a t t h i s t o p o l o g y i s c o m p a c t , a n d t o d o t h i s , w e a p p e a l t o A l e x a n d e r ' s Theorem. T h i s m e a n s t h a t i t i s e n o u g h t o show t h a t e v e r y c o v e r o f L by s u b b d s i c o p e n s e t s So, l e t I, = ( U 4 k ) U ( U L \ 4 x ) , where H c K(L) and h a s a f i n i t e subcover.

#

4

XEA

kEH

be a cover of

A c L,

xo ( L\4x f o r any x x E 4kg f o r some k o

L

by s u b b a s i c o p e n s e t s . A, H.

E E

t h e r e m u s t b e some f i n i t e s u b s e t

u L\fxj = L\f(sup F) j=l n ( + u L \ f x j ) U f k o = L, J=l

L\+ko, s i n c e

3

ko

xo = s u

x.

Then

f o r e a c h x E A . T h u s , we m u s t h d v e x, a n d s i n c e ko i s c o m p a c t ,

F = {xl,

n

Let

XEI

s i n c e x < xo Now, xo =

<

...,x,} sup F.

c A

with

sup F

E

fko.

Then

It t h e n f o l l o w s t h a t

so we h a v e o u r f i n i t e sub-cover.

L a s t l y , w e n o t e t h a t e a c h map @k : L + 2 i s c o n t i n u o u s r e l a t i v e t o t h e 1topology: Indeed, @ k - l ( 2 ) = L, a n d @ k - l ( { l } ) = f k dre b o t h o p e n s e t s i n t h e A - t o p o l o g y , a n d h e n c e s o i s @k-'(O).

2.1

THEOREM, L e t L b e a n d l g e b r d i c l a t t i c e . Then ( a ) The A-topology, which h a s f o r a b a s i s a l l sets o f t h e form (4k)\1~14xi, k

on (b)

L

E

K(L),

XI

r e l a t i v e t o which

,...,x n L

E

L,

i s a compact Hausdorff t o p o l o w

is t o t a l l y disconnected.

'Ihe map e : L + 2 K ( L ) by e ( x ) = ( @ k ( x ) ) k E K ( L ) i s a n e m b e d d i n g o f L i n t o '2K(L) w h i c h p r e s e r v e s a l l i n f m a a n d a l l d i r e c t e d s u p r e m a . T h u s , t h e A - t o p o l o g y o n L i s t h e o n e L i n h e r i t s f r o m '2K(L) u n d e r t h i s embedding.

F o r ( a ) , we o n l y n e e d t o show t h a t L i s t o t a l l y d i s c o n n e c t e d i n t h e Atopology, b u t t h i s f o l l o w s from t h e o b s e r v a t i o n t h a t f k is b o t h open and c l o s e d f o r e v e r y k E K(L), and t h a t e a c h p o i n t h a s a neighborhood b a s i s o f sets o f t h e

PROOF.

form

n

(fk)\

U

4k1,

1=1

where

k,kl,

...,kn

a r e a l l compact e l e m e n t s o f

L.

F o r ( b ) , w e h a v e a l r e a d y n o t e d t h a t t h e map e : L + 2 K ( L ) i s o n e - t o - o n e , a n d e i s c o n t i n u o u s s i n c e e v e r y @k i s c o n t i n u o u s . S i n c e 2 K ( L ) 1s H a u s d o r f f 'Ihe a n d A i s compact a n d H a u s d o r f f , i t f o l l o w s t h a t e i s a n embedding. r e m a i n i n g p r o p e r t i e s f o r e f o l l o w f r o m t h o s e f o r e a c h o f t h e m a p s @k. fl T h e r e i s a f u r t h e r a s p e c t t o t h i s w h i c h w e h a v e n o t y e t m e n t i o n e d . Namely, i f we e q u i p L x L w i t h t h e p r o d u c t t o p o l o g y f r o m t h e A - t o p o l o g y , t h e n I t i s e a s y t o show t h a t t h e map ( x , y ) + i n f { x , y } : L X L + L i s C o n t L n u o u s . Thus, r e l a t i v e L i s a compact zero-dimensional s e m i l a t t i c e w i t h i d e n t i t y , t o t h e A-topology,

M. Mislove

a n d , i n f a c t , e v e r y p o i n t h a s a n e i g h b o r h o o d b a s i s oE s u b s e m i l a t t i c e s ( a s 1 s c l e a r f r o m t h e b a s i s w h i c h we e n m e r a t e d a b o v e ) . Semi l a t t i c e s w i t h t h i s l a s t p r o p e r t y a r e c a l l e d Lawson semilattices, s o we h a v e shown t h a t e v e r y a l g e b r a i c l a t t i c e i s a c o m p a c t z e r o - d i m e n s i o n a l Lawson s e m i l a t t i c e when e q u i p p e d w i L h t h e A - t o p o l o g y . M o r e o v e r , t h e c o n v e r s e h o l d s . ThaL i s , i f S i s a c o m p a c t z e r o - d i m e n s i o n a l s e m i l a t t i c e w i t h i d e n t i t y , t h e n S m u s t b e a Lawson s e m i l a t t i c e , s o S i s ( a l g e b r a i c a l l y ) a n a l g e b r a i c l a t t i c e , a n d t h e t o p o l o g y on S i s e x a c t l y t h e A-topology. F i n a l l y , a map f : S + T b e t w e e n s u c h s e m i l a t t l c e s w h l c h p r e s e r v e s f i n i t e i n f s i s a m o r p h i s n o f c o m p a c t s e m i l a t t i c e s p r e c i s e l y when f p r e s e r v e s a l l i n f m a a n d a l l d i r e c t e d s u p r e m a . T h u s , t h e c a t e g o r y o f a l g e b r a i c l a t t i c e s we h a v e found a s d u a l t o t h e c a t e g o r y s i s p r e c i s e l y t h e c a t e g o r y o f compact z e r o - d i m e n s i o n a l s e m i l a t t i c e s wiLh i d e n t i t y a n d c o n t i n u o u s s e m i l a t t i c e m o r p h i s m s . T h i s t o p i c was f i r s t t r e a t e d f r o m t h i s p e r s p e c t i v e i n [HMS], a n d t h i s t h e n l e d t o t h e more g e n e r a l t h e o r y o f c o n t i n u o u s l a t t i c e s i n [ C ] . T h e g e n e r a l l t y o f t h e l a t t e r i s most e a s i l y s e e n f r o m t h e f o l l o w i n g f a c t . Each c o n t i n u o u s l a t t i c e h a s a c o m p a c t H a u s d o r f f t o p o l o g y , c a l l e d t h e A - t o p o l o g y ( o r Lawson t o p o l o g y ) r e l a t i v e t o which t h e c o n t i n u o u s l a t t i c e i s a compact s e m i l a t t i c e i n which e a c h p o i n t h a s a neighborhood b a s i s o f subsemi l a t t i c e s ( n o t i c e t h a t "zero-dimensional'' i s missing). Moreover, e a c h compact s e m i l a t t i c e w i t h i d e n L i t y w h i c h h a s a neighborhood b a s i s o f s u b s e m l l a t t i c e s arises e x a c t l y i n t h i s fashion. F u r t h e r m o r e , t h e c o n t i n u o u s s e m i l a t t i c e maps a r e e x a c t l y t h o s e maps p r e s e r v i n g a l l i n f m a a n d a l l d i r e c t e d s u p r e m a . T h e i n t e r e s t e d r e a d e r i s d i r e c t e d t o [MI] a n d t o [C] f o r more d e t a i l s a b o u t a l l o f t h i s .

ILL.

DISTRIBUTIVE ALGEBRAIC LATTICES

We now t u r n o u r a t t e n t i o n t o a s p e c i a l c l a s s o f a l g e b r a i c l a t t i c e s , t h o s e It i s well-known t h a t a n y c o m p l e t e l a t t i c e i n w h i c h e v e r y which a r e d i s t r i b u t i v e . element i s a n infmum o f prime elements must b e d i s t r i b u t i v e . We f i r s t show t h a t t h e converse holds f o r algebraic lattices.

3 . 1 DEFINITION. Let L be a l a t t i c e . T h e e l e m e n t x E L i s irreducible i f , f o r e v e r y a , b E L, i f x = ab, t h e n e i t h e r x = a o r x = b. The element i f a b < x, then e i t h e r a < x or x E L i s prime i f f o r e v e r y a , b E L, b < x. W e d e n o t e by IRR(L) t h e s e t o f a l l i r r e d u c i b l e e l e m e n t s o f L, a n d by P r i m e L t h e s e t o f a l l p r i m e e l e m e n t s of L. F i n a l l y , Spec L d e n o t e s t h e s e t o f a l l p r i m e s o f L o t h e r t h a n t h e i d e n t i t y o f L. n N o t i c e t h a t P r i m e L c IRR(L) a l w a y s h o l d s , w h i l e P r i m e L = IRR(L) i f L i s distributive. Our f i r s t r e s u l t s h o w s t h a t t h e r e i s a l w a y s a n a b u n d a n c e o f i r r e d u c i b l e s i n any a l g e b r a i c l a t t i c e .

3 . 2 PROPOSITION. L e t L b e a n a l g e b r a i c l a t t i c e , a n d l e t x , y T h e n t h e r e i s some i r r e d u c i b l e r E LRR(L) w i t h y < r b u t x

4

E

4

L r.

with

x

4

y.

PROOF. S i n c e x y a n d L i s a l g e b r a i c , we c a n f i n d a c o m p a c t e l e m e n t w i t h k 6 x, but k y . T h i s means t h a t x E 4 k , b u t y E L\4k. By k E K(L) Z o r n ' s Lemma, w e c a n t h e n c h o o s e a m a x i m a l c h a i n C i n L \ 4 k w i t h y E C . We c l a m t h a t s u p C E L\4k. Indeed, s i n c e C i s a c h a i n , C i s t r i v i a l l y d i r e c t e d , s o t h a t s u p C E 4k w o u l d i m p l y t h a t C n 4k f 0, s i n c e k 1s compact. S i n c e C C L\4k, i t f o l l o w s t h a t s u p C E L\4k a s w e l l . Since C is a maximal c h a i n i n L\4k, s u p C i s a maximal e l e m e n t o f L\4k a b o v e y ( s i n c e I n f a c t , ~f a , b E L w i t h y E C ) . We now c l a i m t h a t c = s u p C i s i r r e d u c i b l e . a b = c, t h e n c 6 a,b. If a # c # b, t h e n s i n c e c i s maximal i n L \ 4 k , it f o l l o w s t h a t a , b E 4 k . B u t , 4k i s a f i l t e r , s o t h a t a b E 4 k , w h l c h c o n t r a d i c t s a b = c E L\4k. Thus, c is i r r e d u c i b l e , as claimed. This proves t h e Proposition.

4

n

3.3

COROLLARY.

If

y

=

L is a d i s t r i b u t i v e algebraic l a t t i c e , then i n f ( 4 y n S p e c L) f o r e v e r y y E L .

69

Order scattered and topologically scattered PROOF. W e h a v e j u s t shown t h a t , f o r x { y i n L , t h e r e i s some i r r e d u c i b l e r e IRR(L) w i t h y < r , but x r. Hence, y = i n E ( f y n IRR(L)) f u r e v e r y y e L, s i n c e t h i s inEunum c a n n o t be s t r i c t l y g r e a t e r t h a n y . If L is d i s t r i b u t i v e , t h e n Prime L = IRR(L), and t h e corollary then follows. fl

4

We s a y t h a t t h e s u b s e t A o f t h e l a t t i c e L order-generates L if x = inE(4x n A) € o r e v e r y x t L. T h u s , P r o p o s i t i o n 3 . 2 i m p l i e s t h a t 1RR(L) L f o r a n y a l g e b r a i c l a t t i c e L, w h i l e C o r o l l a r y 3.3 s t a t e s t h a t order-generates S p e c I, o r d e r - g e n e r a t e s L i f L i s a l s o distributive. T h e q u e s t i o n n a t u r a l l y a r i s e s how one m i g h t r e c o v e r a d i s t r i b u t i v e a l g e b r a i c l a t t i c e from i t s s e t o f primes. To a n s w e r t h i s , we m u s t r e v i e w t h e s p e c t r a l t h e o r y f u r t h e s e l a t t i c e s . To b e g i n , w e n o t e t h a t e v e r y e l e m e n t x o f t h e d i s t r i b u t i v e a l g 2 b r a i c l a t t i c e L i s u n i q u e l y d e t e r m i n e d by t h e s e t f x fl S p e c L . M o r e o v e r , t h e s e s e t s , { t x fl S p e c L : x E L } , Eorm t h e c l o s e d s e t s € o r a t o p o l o g y o n Spec L. For any f m i l y {xi}, we h a v e ? ( t x n S p e c L) = ~ ( V X , ) n S p e c I,, a s 1

1

,...,

is easily verified. M o r e o v e r , I € w e t a k e f i n i t e l y many x i , s a y XI x,, t h e n (4x1 n s p e c L ) u . . .U ( f x n Spec = { p t S p e c 1, : x i p f o r some i = 1, I}. S i n c e S p e c L c o n s i s t s o f p r i m e e l e m e n t s , we know t h a t I p E S p e c Id : XI xn < p } = I p E S p e c : x, < p t o r some i = I , n), so t h a t ( f x l n S p e c L ) (J. . . u ( t x n n S p e c L ) = + ( X I w,) n S p e c L . Thus, ( 4 x fl S p e c L : x E L} i s c l o s e d u n d e r a l l i n t e r s e c t i o n s a n d a l l f i n i t e u n i o n s , w h i c h m e a n s i t i s t h e s e t oE c l o s e d s e t s Eur a t o p o l o g y on S p e c L. We c a l l t h i s W e now e s t a b l i s h t h e r e l e v a n t p r o p e r t i e s o f t h e huZZ kernel topoZogy o n S p e c L . t h i s topology.

...,

. ..

...,

. ..

4

F i r s t , t h e t o p o l o g y i s To s i n c e , i f p , q E S p e c I, with p q, then (Spec L ) \ t p i s a n open s e t containing q , b u t n o t p. The topology i s a l s o sober; t h i s m e a n s t h a t a n y p r i m e o p e n s e t i s o f t h e Eorm ( S p e c L ) \ { p } - , where prime r e E e r s t o t h e l a t t i c e oE o p e n s e t s . This i s m o s t e a s i l y s e e n i n t h e l a t t i c e o f c l o s e d sets, where a prime element i s a l s o c a l l e d i F F e d U C i b k . The c l o s e d s e t A of Spec L i s i r r e d u c i b l e i f , whenever B and C a r e c l o s e d s e t s a n d A c B U C, then e i t h e r A c B or A c C. Clearly t h i s condition holds L E A = (p}-, f u r any p o i n t p o f Spec L. Conversely, suppose t h a t A c Spec L i s a n irreducible closed set. T h e n A = 4 x fl S p e c L f u r some x E L, since A is a c l o s e d s e t . We c l a i m t h a t x E S p e c L . If n o t , t h e n t h e r e a r e y , z E L w i t h but y x and z x. B u t , t h e n f y 0 S p e c L a n d f z fl S p e c L a r e y z 6 x, two c l o s e d s e t s , n e i t h e r o f w h i c h c o n t a t n s f x n S p e c I, (s ince Spec L orderx ) , b u t ( 4 y fl S p e c L ) U ( f z n S p e c L ) = + y z fl S p e c L 3 generates L and y , z f x fl S p e c L = A . H e n c e , L E x i s n o t p r i m e , t h e n A i s n o t p r i m e i n t h e s e t o f c l o s e d s e t s . T h i s s h o w s t h a t e v e r y p r i m e i n t h e l a t t i c e oE c l o s e d s e t s o f S p e c I, i s oE t h e f o r m f p n S p e c L ; b u t f p n S p e c L = ( p } - i n t h e h u l l k e r n e l t o p o l o g y .

4

9

4

To s m m a r L z e , we h a v e shown t h e f o l l o w i n g . If w e l e t L b e a d i s t r i b u t i v e a l g e b r a i c l a t t i c e , t h e n t h e map x + t x fl S p e c L : L + C ( S p e c L ) is a lattice isomorphism o f L o n t o t h e l a t t i c e o f h u l l - k e r n e l c l o s e d s u b s e t s of Spec L . I n d e e d , t h e m ap i s one -to-one s i n c e S p e c L o r d e r g e n e r a t e s L, and i t i s o n t o s i n c e e v e r y h u l l - k e r n e l c l o s e d s e t i s o f t h e f o r m f x fl S p e c L f o r some x E L. M o r e o v e r , s i n c e t h e h u l l - k e r n e l t o p o l o g y i s s o b e r , t h e map t a k e s S p e c L o n t o t h e s e t o f n o n - i d e n t i t y primes o f C(Spec L ) .

Of c o u r s e , n o t e v e r y s o b e r T o - s p a c e a r i s e s i n t h i s f a s h i o n , s o t o d e t e r m i n e e x a c t l y w h i c h o n e s do, we make t h e f o l l o w i n g o b s e r v a t i o n s . F i r s t , f o r L a n algebraic lattice, Spec L i s l o c a l l y quasi-compact i n t h e h u l l - k e r n e l topology. By t h i s we m e a n t h a t e a c h p o i n t o f S p e c L h a s a n e i g h b o r h o o d b a s i s o E q u a s i compact s e t s . Indeed, i f p E Spec L and x E L w i t h p E (Spec L)\tx, then, hut s i n c e L i s a l g e b r a i c , w e c a n f i n d a c o m p a c t e l e m e n t k E K(L) w i t h k < x,

M . Misloue

4

k p. We c o n s i d e r t h e o p e n s e t ( S p e c L ) \ + k . C e r t a i n l y p E ( S p e c L ) \ f k c (Spec L)\+x, s i n c e k C x. M o r e o v e r , s u p p o s e t h a t { x i } c I, w i t h ( S p e c L ) \ f k U(Spec I , ) \ + x i . As we p o i n t e d o u t a b o v e , U(Spec L ) \ + x , = ( S p e c L ) \ + ( V x i ) , so k p e c L)\I.k c (Spec L)\+(Vx,)

i m p l i e s tha:

1

means t h e r e a r e f i n i t e l y many i n turn,

implies t h a t

( S p e c L) \+k

k

<

Vxl.

Since

xi,

say n U

i=l

xi

,...,x,,

(Spec !>)\+xi.

i s ckmpact, t h i s

k n

1

(Spec I,)\fk c

C

with

k C , V xi, and t h i s , 1=1 We h a v e t h u s shown t h a t

i s quasi-compact.

T h u s , w e know t h a t S p e c L i s a l o c a l l y q u a s i - c o m p a c t s o b e r T o - s p a c e f o r a n y d i s t r i b u t i v e a l g e b r a i c l a t t i c e L. M o r e o v e r , i f we n o t e t h a t ( S p e c L ) \ 4 k is open f u r e v e r y k E K(L), t h e n we h a v e shown t h a t S p e c IA h a s a n e i g h b o r h o o d b a s i s o € quasi-compact open s e t s . C a n v e r s e l y , i f X I S any l o c a l l y quasi-compact s o b e r To-space w i t h a neighborhood b a s i s o f quasi-compact open s e t s , t h e n t h e l a t t i c e C(X) o f c l o s e d s u b s e t s o f X i s a d i s t r i b u t i v e a l g e b r a i c l a t t i c e , a n d S p e c C(Xl i s homeomorphic t o X u n d e r t h e m a p p i n g x + X\{x}-. Of c o u r s e , we h a v e n o t s a i d a n y t h i n g a b o u t m o r p h i s m s , a s s u n i n g we w a n t t o In f a c t , t h e morphisms b e t w e e n r e a l i z e a l l of t h i s i n a categorical setting. l o c a l l y quasi-compact s o b e r s p a c e s s h o u l d be t h e c o n t i n u o u s maps, a n d t h e s e d o g i v e r i s e t o a d u a l i t y w i t h t h e category o f d i s t r i b u t i v e a l g e b r a i c l a t t i c e s and morphisns which p r e s e r v e a l l i n f i m a a n d a l s o p r e s e r v e prime e l e m e n t s . T h i s a s s o c i a t i o n goes a s follows. If X and Y a r e l o c a l l y quasi-compact s o b e r s p a c e s e a c h h a v i n g a b a s i s o f quasi-compact open s e t s , and i f f : X + Y i s a for any c o n t i n u o u s map, t h e n w e c a n d e f i n e C ( f ) : C(X) + C(Y) by C ( f ) ( A ) = f ( A ) c l o s e d s u b s e t A of X. tt i s r o u t i n e t o v e r i f y t h a t C ( f ) preserves a l l infima f o r e a c h x E X, ( i , e . , c l o s u r e s o f unions) and t h a t C(f)({x)-) = {f(x)}whence C ( f ) p r e s e r v e s p r i m e s , a l s o . C m n v e r s e l y , i f @ : L + M i s a map p r e s e r v i n g a l l i n f i m a and prime e l e m e n t s , w i t h L and M d i s t r i b u t i v e a l g e b r a i c l a t t i c e s , t h e n Spec @ : S p e c L + Spec M which i s j u s t t h e r e s t r i c t i o n a n d c o r e s t r i c t i o n o f 4, i s e a s i l y s e e n t o b e a c o n t i n u o u s map w i t h r e s p e c t t o t h e hull-kernel topologies. We c a n s u n m a r i z e w h a t we h a v e j u s t shown a s f o l l o w s :

3 . 4 THEOREM. Let DAl denote t h e category o f d i s t r i b u t i v e algebraic l a t t i c e s a n d m o r p h i m s p r e s e r v i n g a l l i n f m a a n d p r i m e e l e m e n t s , a n d l e t LQSOB d e n o t e t h e c a t e g o r y o f l o c a l l y quasi-compact s o b e r TO-spaces h a v i n g a b a s i s o f quasi-compact o p e n s e t s , a n d c o n t i n u o u s maps. Then D A l a n d LQSOB a r e equivalent under t h e f u n c t o r s which assocLaLe t o a d i s t r i b u t i v e a l g e b r a i c l a t t i c e L i t s s p e c t r m i n t h e h u l l - k e r n e l t o p o l o g y , and t o a l o c a l l y quasi-compact s o b e r s p a c e X i t s 1 a t t L c e C(X) o f c l o s e d s u b s e t s , w i t h t h e e x t e n s i o n s t o morphisms i n d i c a t e d 9 above. The maps w h i c h a r i s e b e t w e e n d i s t r i b u t i v e a l g e b r a i c l a t t i c e s u n d e r t h i s equivalence a r e very s i m i l a r t o those we t a l k e d about before. Namely, t h e y d o p r e s e r v e a l l i n f m a , b u t n o t n e c e s s a r i l y a l l d i r e c t e d suprema, and, i n a d d i t i o n , t h e y preserve prunes. A c t u a l l y , t o o b t a i n t h e p r e s e r v a t i o n o f a l l d i r e c t e d s u p r e m a , w e m u s t make a E u r t h e r demand o n t h e maps b e t w e e n l o c a l l y q u a s i - c o m p a c t If E : X + Y is a s o b e r s p a c e s . Namely, w e m u s t r e q u ~ r et h a t t h e y b e proper. c o n t i n u o u s map b e t w e e n l o c a l l y q u a s i - c o m p a c t s o b e r s p a c e s , t h e n w e s a y f is proper i f , f o r a n y q u a s i - c o m p a c t s u b s e t Q c Y s u c h t h a t Q = n { U c Y : Q c U i s open), t h e n f-l(Q) i s q u a s i - c o m p a c t i n X. ( t f w e c a l l a s e t saturated p r e c i s e l y when i t i s t h e i n t e r s e c t i o n o f i t s o p e n n e i g h b o r h o o d s , t h e n w e c a n s a y f i s proper i f f - l ( Q ) is quasi-compact whenever Q is a saturated quasi-compact s e t . The r e a s o n f u r t h e i n c l u s i o n of t h i s t e c h n i c a l i t y i s t h a t t h e s p a c e s w i t h which we a r e d e a l i n g a r e n o t H a u s d o r f f . ) It h a p p e n s t h a t t h e c a t e g o r y o f l o c a l l y q u a s i - c o m p a c t s o b e r T o - s p a c e s h a v i n g a b a s e oE q u a s i - c o m p a c t o p e n s e t s a n d c o n t i n u o u s p r o p e r maps i s e q u i v a l e n t t o t h e c a t e g o r y o f d i s t r i b u t i v e

Order scattered and topologically scattered

71

d l g e b r a i c l a t t i c e s and morphims p r e s e r v i n g a l l infima, a l l d i r e c t e d suprema, and prime elements. IV.

COW LE'rELY D LSTRIRUTLVE ALGERRALC LATTICES

We now b r i n g t h e m o r e r e s t r i c t i v e c l a s s o f c o m p l e t e l y d i s t r i b u t i v e l a t t i c e s into the picture. R e c a l l i n g t h a t o u r i n s p i r a t i o n f o r t h i s work was t o g e n e r a l i z e Pouzet's result t h a t order scattered equals topologically scattered f o r L = I(P), t h e l a t t i c e o f l o w e r s e t s o f t h e p o s e t P, we f i r s t c l a r i f y t o w h a t c l a s s o f c u m p l e t e l y d i s t r i b u t i v e l a t t i c e s h i s r e s u l t a p p l i e s . Namely, we show t h a t i t i s e x a c t l y t h e completely d i s t r i b u t i v e a l g e b r a i c l a t t i c e s which a r i s e as I(P) for some p o s e t P. M o r e o v e r , we show t h a t t h e d u a l i t y b e t w e e n a l g e b r a i c l a t t i c e s a n d s u p - s e m i l a t t i c e s which we p r e s e n t e d i n S e c t i o n I a c t u a l l y i s h u t o n e o f s e v e r a l d u a l i t i e s a v a i l a b l e , a l l o f which r e l a t e t o c e r t a i n c l a s s e s o f a l g e b r a i c l a t t i c e s . We b e g i n by e s t a b l i s h i n g o u r c l a i m t h a t e v e r y c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c f u r some p o s e t P. l a t t i c e i s o f t h e Eorm I(P) Let L be a complete l a t t i c e . An e l e m e n t x E L i s completely a n y s u b s e t A c L, I€ i n € A = x, t h e n x E A. The element x i s completely p r i m e i f , f o r a n y s u b s e t A c L, i E i n f A < x, t h e n a < x f u r some a E A . D u a l l y , t h e e l e m e n t x E L i s completely co-irreducible i f f o r a n y s u b s e t A c L, i f x = sup A, t h e n x E A , a n d x i s completely co-prime L € , € o r a n y s u b s e t A c L, i f x < sup A, t h e n x < a € o r some a E A .

4.1

DEFLNLTLON.

irreducibk

I € , €or

We o n l y n e e d t h e s e t s o f c o m p l e t e l y c o - p r i m e a n d c o m p l e t e l y p r i m e e l e m e n t s , b u t w e h a v e i n c l u d e d t h e o t h e r d e f i n i t i o n s f o r c o m p l e t e n e s s s a k e . We d e n o t e by CoSpec L t h e s e t o f a l l n o n - z e r o cu-primes o f L ( i . e . , o f s u p p r i m e e l e m e n t s ) , a n d by CCoSpec L t h e s e t o f a l l n o n - z e r o c o m p l e t e l y c c r p r i m e e l e m e n t s o f L ; likewise, CSpec I, denotes t h e family of a l l non-identity completely prime e l e m e n t s o f L.

4 . 2 PROPOSITION. L e t L b e a c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e . L, a n d CCoSpec L o r d e r c o - g e n e r a t e s L. CSpec L o r d e r - g e n e r a t e s

Then

PROOF. Let x , y E L w i t h x y . As i n t h e p r o o f of P r o p o s i t i o n 3 . 2 , we c a n but k y , a n d we c a n t h e n t h e n f i n d a c o m p a c t e l e m e n t k E K(L) w i t h k < x, c h o o s e a m a x i m a l e l e m e n t r E L\4k w i t h y < r. We c l a i m t h a t r i s c o m p l e t e l y prune. I n d e e d , s u p p o s e t h a t A c L w i t h i n € A < r. T h e n r = r V ( i n € A) = i n f { r V a : a E A}, t h e l a s t e q u a l i t y following from t h e f a c t t h a t L i s completely d i s t r i b u t i v e . Rut, i f r < r V a €or every a E A, then s i n c e r is maximal i n L\4k, w e w o u l d t h e n h a v e r V a E 4k f o r e v e r y a E A , w h i c h w o u l d i n t u r n i m p l y t h a t r = i n f { r V a : a E A} > k , s i n c e t k i s c l o s e d u n d e r a l l infma. This w o u l d c o n t r a d i c t t h e f a c t t h a t r E: L \ 4 k , and so we c o n c l u d e t h a t r V a = r f o r some a E A . T h i s s h o w s t h a t r i s a c o m p l e t e p r i m e . The a r g u m e n t g i v e n t o p r o v e C o r o l l a r y 3.3 now a p p l i e s t o show t h a t CSpec L o r d e r g e n e r a t e s L. To p r o v e t h a t CCoSpec L o r d e r c o - g e n e r a t e s L, we a r g u e a s f o l l o w s : y . T h e n , a s w e h a v e j u s t s e e n , t h e r e i s a complete Again, l e t x , y E L w i t h x p r i m e r w i t h y < r, b u t x r. Let s = i n f ( L \ C r ) . Then s r, since r i s a c o m p l e t e prime. Hence s < y , b u t s x. Moreover, we c l a m t h a t s is a complete ccrprime. Indeed, i f A c L w i t h s < s u p A , then A c C r cannot H e n c e we m u s t h a v e h o l d , f o r Jr i s c l o s e d u n d e r a l l s u p r e m a , a n d s $ C r . A n L \ C r # 0. But s i n c e s = i n f ( L \ $ r ) , w e h a v e t h a t t s = L\Cr. Therefore A n 4s # 0, w h i c h s h o w s t h a t s i s a c o m p l e t e c o - p r i m e . The d u a l o f t h e a r g u m e n t g i v e n t o p r o v e C o r o l l a r y 3 . 3 now a p p l i e s t o s h a w t h a t CCoSpec L o r d e r L. co-generates

4

4

1

4

n

of

I f P i s a p o s e t , t h e n I ( P ) = { A c P : A = +A}, P, is a completely d i s t r i b u t i v e a l g e b r a i c l a t t i c e .

t h e l a t t i c e o f lower ends M o r e o v e r , f o r e a c h p E P,

M. Misloue

72

t h e i d e a l Cp i s a c o m p l e t e c o - p r i m e o f I ( P ) . T h i s f o l l o w s from t h e f a c t t h a t t h e supremum o f a n y f a m i l y o f l o w e r e n d s i s j u s t t h e u n i o n , a n d t h e u n i o n c a n and hence a l s o c o n t a i n 4p i f f o n e oE t h e members o f t h e f a m i l y c o n t a i n s p , c o n t a i n s 4 p . T h u s , t h e map p -f Cp : p + CCoSpec I ( P ) i s an o r d e r i s o m o r p h i s m o f P i n t o t h e s e t o f c o m p l e t e c o - p r i m e s o f I ( P ) . R u t , i f A i s c o m p l e t e coi m p l i e s t h a t A = 4 p f o r some p E A , prime o f t(P), t h e n A = s u p { l p : p E A) a n d s o e v e r y c o m p l e t e c o-prim e o f I ( P ) i s o f t h e f o r m Cp. H e n c e , t h e map p H Cp i s a n o r d e r i s o m o r p h i s m o f P o n t o t h e s e t CCoSpec I ( P ) . Olir inrxt r e s u l t s h o w s t h a t t h e converse holds:

4.3

THEOREM. ( a ) For a n y p o s e t order-isomorphism o f P (b)

If

L

P, t h e map p -f Cp : P + CCoSpec I ( P ) o n t o CCoSpec I ( P ) .

is an

i s a c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e , t h e n t h e map

x + Cx n CCoSpec 1, : L + I(CCoS pe c L )

i s a complete l a t t i c e

isomorphism. PROOF. Our com m ent s i n t h e p r e v i o u s p a r a g r a p h s e r v e t o p r o v e ( a ) . For ( b ) , l e t I, be a c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t t c e . According t o P r o p o s i t i o n 4 . 2 , t h e map x + Cx n P, w h e r e P = CCoSpec I,, i s one-to-one, s i n c e P o r d e r cog e n e r a t e s L. M o r e o v e r , i t i s e a s y t o s how t h a t t h i s map p r e s e r v e s a l l i n f i m a , s i n c e n(Cx, n P) = + ( A x i ) n P. On t h e o t h e r h a n d , we c l e a r l y h a v e U(Cx, n P) c 1

1

1

n P.

C(Vx;)

Conversely, suppose t h a t

1

p

E

C(vxl)

n P.

Th e n , s i n c e

p

is

1

c o m p l e t e l y c o - p r i m e , t h e r e m u s t b e s om e i w i t h p < xi, s o t h a t p E +xi n P c U ( J x; n P I . T h u s , t h e map a l s o p r e s e r v e s a l l s u p r e m a .

That i t

1

IS

onto

LS

obvious.

n

We c a n make t h i s i n t o a c o m p l e t e d u a l i t y i f w e t a k e f o r m o r p h i s m s b e t w e e n p o s e t s t h o s e maps w h i c h p r e s e r v e t h e o r d e r , a n d f o r morphisms b e t w e e n c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e s t h o s e maps which p r e s e r v e a l l i n f i m a a n d a l l suprema. In f a c t , t h i s d u a l i t y g e n e r a l i z e s t h e d u a l i t y between complete Roolean a l g e b r a s h a v i n g enough atoms t o c l r g e n e r a t e t h e a l g e b r a , a n d s e t s , which i s i m p l e m e n t e d by t h e a s s i g n i n g t h e p o w e r s e t t o a s e t , a n d a s s i g n i n g t h e s e t o f atoms t o a Boolean a l g e b r a . The d u a l i t y w e g a v e b e t w e e n a l g e b r a i c l a t t i c e s a n d s u p - s e m ~ l a t t i c e si n S e c t i o n I i s not a g e n e r a l i z a t i o n o f t h e s e o t h e r d u a l i t i e s , s i n c e t h e f o r m e r a r e n o t P o n t r y a g i n d u a l i t i e s w h i l e t h e d u a l i t y o f S a n d AL L S a Pontryagin duality. W e c a n s p e c i a l i z e t h e S-AL a s f o l l o w s . Certainly t h e c a t e g o r y oE l a t t i c e s w i t h 0 a n d 1 a n d l a t t i c e m a p s i s a s u b c a t e g o r y o f t h e I f we r e s t r i c t t h e P o n t r y a g i n d u a l i t y b e t w e e n S a n d A L t o t h i s c a t e g o r y S. L o f S, t h e n t h e c a t e g o r y w e o b t a i n a s i t s I m a g e i n AL i s t h e subcategory c a t e g o r y AR o f a r i t h e t L c l a t t i c e s , w h e r e a n a l g e b r a i c l a t t i c e L i s u r i t h m e t i c i f K(L) is a s u b l a t t i c e of L (i.e., K(L) i s closed under i n f s as well as sups)

.

T h i s ends t h e “review” p a r t o f t h i s paper. Almost none o f what we h a v e p r e s e n t e d s o Ear i s new, b u t i s e i t h e r e x p l i c i t l y o r i m p l i c i t l y i n p r e v i o u s w o r k . Namely, t h e d u a l i t y o f a n d AL i s d e s c r i b e d c o m p l e t e l y i n [HMS], a n d s k e t c h e d i n [ C ] . The d u a l i t y b e t w e e n p o s e t s a n d c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e s w h i c h we h a v e o u t l L n e d a b o v e i s i m p l i c i t l y i n [ L ] , w h e r e t h e g e n e r a l i z a t i o n i s g i v e n t o t h e c l a s s o f c o m p l e t e l y d i s t r i b u t i v e l a t t i c e s , on t h e o n e s i d e , a n d In f a c t , a l l o f t h e s t r u c t u r e t h e o r y a n d c o n t i n u o u s p o s e t s , on t h e o t h e r . s p e c t r a l theory f o r a l g e b r a i c l a t t i c e s which we have p r e s e n t e d i s t r u e for t h e more g e n e r a l c l a s s o f c o n t i n u o u s l a t t i c e s . These r e s u l t s a r e g i v e n i n d e t a i l i n [C], and are sketched f o r t h e c l a s s of continuous l a t t i c e s i n [ M l ] . The p o i n t ‘-ere h a s b e e n t o p r e s e n t t h e s e r e s u l t s i n t h e s i m p l e r s e t t i n g o f a l g e b r a i c l a t t i c e s , s o t h a t t h e more g e n e r a l t h e o r y o f c o n t i n u o u s l a t t i c e s c a n be more e a s i l y understood. T h a t we h a v e n o t d o c u m e n t e d who p r o v e d w h a t f i r s t i s m e r e l y a

s

Order scattered and topologically scattered

73

m e t h o d t o e x p e d i t e s om e wha t t h e p r e s e n t a t i o n . A very detailed list of historical remarks c a n be found a t t h e end o f e a c h s e c t i o n o f [C], and t h o s e remarks i n c l u d e t h e h i s t o r y o f t h e m a t e r i a l we h a v e j u s t p r e s e n t e d . THE CASE O F COMPLETELY DISTRLBIJTLVE A L G E B R A I C LATTICES

V.

A f t e r s o much p r e p a r a t o r y m a t e r i a l , w e r e c a l l t h e p o i n t o f t h i s w o r k . A s u b s e t A o f a p o s c t P i s order-dense i f , g i v e n a , b E A w i t h a < b, there i s some I: E A w i t h a < c < b. T h e p o s e t P i s t h e n c a l l e d order-scettered i E P h a s no o r d e r - d e n s e s u b s e t s o t h e r t h a n 4 , s i n g l e t o n s , o r a n t i - c h a i n s . A m o m e n t ' s t h o u g h t s h o w s t h a t t h i s i s e q u i v a l e n t t o t h e f a c t t h a t P h a s no o r d e r dense chains. f o r a t o p o l o g i c a l s p a c e X, we s a y X i s topologicdly A oE X h a s a t l e a s t o n e i s o l a t e d p o i n t i n t h e r e l a t i v e topology. The q u e s t i o n wc a r e a d d r e s s i n g h e r e i s t h e r e l a t i o n between t h e s e t w o c o n c e p t s i n a p o s e t L which a l s o h a s a topology. On t h e o t h e r h a n d ,

scattered i f e v e r y s u b s e t o f

DEFINITLON. T h e s p a c e X i s a Compact pOspace i f X is a compact Hausdorff s p a c e t o g e t h e r w i t h a p a r t i a l o r d e r which i s a ( t o p o l o g i c a l l y ) c l o s e d s u b s e t o f

5.1

x

x

x.

5.2 LEMMA. scattered.

A compact pospace

X

i s order-scattered i f i t is topologically

PROOF. S i n c e X i s a c o m p a c t p o s p a c e , w e know t h a t t h e t o p o l o g y o f X h a s a s u b b a s i s o f o p e n u p p e r s e t s a n d o p e n l o w e r s e t s ( s e e , e . g . , "1). This implies Now, suppose t h a t X is n o t t h a t J.x a n d 4 x a r e c l o s e d s e t s f o r e v e r y x E X. order scattered. Then X h a s a n order-dense c h a i n , C. Let Y = { x E X : x = i n f ( 4 x n C\{x})]. T h e n we h a v e Y # 0, a n d w e c l a i m Y h a s nu i s o l a t e d point. Lndeed, l e t x B Y , a n d l e t 11 c X b e a n o p e n s u b s e t oE X c o n t a i n i n g x. S i n c e X i s a c o m p a c t p o s p a c e , i t f o l l o w s t h a t e a c h f i l t e r e d s e t i n X c o n v e r g e s t o i t s i n f m u m , a n d s i n c e x E Y , we c o n c l u d e t h a t U n 4 x n C i s a n infinite set. It i s t h e n e a s y t o f i n d a p o i n t o f Y o t h e r t h a n x i n t h i s s e t , s i n c e C is order-dense. Thus Y h a s no i s o l a t e d p o i n t s , a n d s o X i s n o t topologically scattered. T h e r e s u l t f o l l o w s by c o n t r a p a s i t i o n .

n

We c a n c o n c l u d e f r o m t h i s Lemma t h a t a n y a l g e b r a i c l a t t i c e L i s o r d e r We s h o w t h a t t h e c o n v e r s e h o l d s i n scattered i f i t is topologically scattered. c a s e L i s completely d i s t r i b u t i v e .

5.3 THEOREM. A c o m p l e t e l y d i s t r i b u t i v e l a t t i c e only i f i t is topologically scattered.

L

i s order s c a t t e r e d i f and

PROOF ( P o u z e t ) . We h a v e a l r e a d y s e e n t h a t L i s o r d e r s c a t t e r e d i f L i s topologically scattered. We p r o v e t h e c o n v e r s e by c o n t r a p o s i t i o n . Suppose t h a t L is not topologically scattered. Then L c o n t a i n s a p e r f e c t s u b s p a c e ( i . e . , o n e w i t h o u t i s o l a t e d p o i n t s ) , a n d by c o n s i d e r i n g t h e c l o s u r e o f t h i s s u b s p a c e , w e h a v e a c o m p a c t p e r f e c t s u b s p a c e i n L, c a l l i t X. Now, a c c o r d i n g t o Theorem 4.3, L = I(P), w h e r e P i s t h e p o s e t o f c o m p l e t e c o - p r i m e s o f L. We recursively construct a n order-dense chain i n I(P) as follows. First, for l e t L( p) = { I E I ( P ) : p E I}, a n d l e t I ( - p ) = { I E I ( P ) : p # I}. p E P, R e c a l l t h a t J.p i s a c o m p l e t e c o - p r i m e o f I ( P ) , which means t h a t I ( p ) is a compact-open s u b s e t o f I ( P ) , as i s I(-p), f o r e v e r y p E P. Since the a n d a r e a l l o f t h e f o r m + p f o r some complete co-primes o r d e r co-generate Z(P), p E P, we c a n f i n d a p E P w i t h X n I ( p ) # 0 f X n I ( - p ) . L e t 1112 = J.p f o r t h i s p E P, a n d l e t X' = X n I ( p ) a n d X" = X n I ( - p ) . T h e n X' a n d XI' a r e a l s o compact p e r f e c t s u b s p a c e s o f I ( P ) , a n d s o t h e r e a r e p o i n t s p ' , p" E P w i t h X' n I ( p ' ) # 0 # X' n I ( - p ' ) a n d x" n I ( p " ) # 0 # X" n I ( - p " ) . We t h e n l e t 1314 = 1112 U J.p" a n d I114 = 1112 n + P I . We c a n c o n t i n u e t h i s a r g u m e n t

M . Mislove

74

recursively t o f i n d t h e desired order-dense chain i n K(P), which p r o v e s t h a t I ( P ) = L i s n o t o r d e r s c a t t e r e d . T h i s c o n c l u d e s t h e p r o o f o f t h e Theorem.

n

REMARK. T h i s proof i s due t o Maurice Pouzet. W e h a v e g e n e r a l i z e d i t a b i t by f i r s t i n v o k i n g t h e f a c t t h a t a n y c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e i s oE t h e form 1 ( P ) , where P i s j u s t t h e s e t o f c o m p l e t e co-primes o f t h e l a t t i c e . I t i s a simple matter t o s e t t l e Pouzet's o r i g i n a l question ( i . e . , t o determine t h a t o r d e r s c a t t e r e d i s t h e same a s t o p o l o g i c a l l y s c a t t e r e d ) f o r a l l c o m p l e t e l y distributive l a t t i c e s . We r e c a l l t h a t a n y c o m p l e t e l y d i s t r i b u t i v e c o m p l e t e l a t t i c e L i s e m b e d d a b l e i n a p r o d u c t o f u n i t i n t e r v a l s Ix u n d e r a map p r e s e r v i n g a l l Thus, t h e n a t u r a l i n f l m a a n d a l l s u p r e m a ( t h i s i s o r i g i n a l l y d u e t o Raney [ R ] ). compact H a u s d o r f f t o p o l o g y which s u c h a l a t t i c e c a r r i e s ( a n d r e l a t i v e t o which i t i s a compact t o p o l o g i c a l l a t t i c e ) i s t h e i n h e r i t e d t o p o l o g y from Lx. The l a t t i c e L i s t h e n a l g e b r a i c p r e c i s e l y when i t c a n b e s o embedded i n a p r o d u c t ZX, a n d , t o p o l o g i c a l l y , t h i s means precisely t h a t t h e t o p o l o g y o n L i s t o t a l l y d i s c o n nected. The p o i n t i s t h a t a c o m p l e t e l y d i s t r i b u t i v e complete LaLtice which i s n o t a l g e b r a i c m u s t h a v e a n o n - d e g e n e r a t e c o n n e c t e d c o m p o n e n t , a n d we c a n t h e n i n v o k e K o c h ' s A r c Theorern ( s e e [ C ] , p . 2 9 9 ) t o p r o d u c e a n o r d e r - a r c i n t h a t c o m p o n e n t . R u t , t h i s a r c i s a n order-dense s u b s e t , a s w e l l a s a p e r f e c t s u b s p a c e o f L, and s o L i s n e i t h e r o r d e r s c a t t e r e d n o r t o p o l o g i c a l l y s c a t t e r e d ~n t h i s c a s e .

V1.

THE QUESTLON FOR ALGEBRAIC LATTICES L N G E N E R A L

We h a v e j u s t s e e n t h a t a c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e i s o r d e r We now p r e s e n t a n e x a m p l e s c a t t e r e d i f and only i f i t i s t o p o l o g i c a l l y s c a t t e r e d . t o show t h i s i s n o t t h e c a s e f o r a l g e b r a i c l a t t i c e s i n g e n e r a l . To b e s u r e , Lemma 5 . 2 shows t h a t a n y a l g e b r a i c l a t t i c e w h i c h i s t o p o l o g i c a l l y s c a t t e r e d m u s t a l s o b e order scattered. T h e f o l l o w i n g e x a m p l e shows t h e c o n v e r s e i s f a l s e :

6 . 1 EXAMPLE. Let C A = {I l / n : n > I}

denote t h e u s u a l middle t h i r d Cantor s e t , and l e t { l } . We g i v e L' = C x A t h e f o l l o w i n g o r d e r : (cl,al) < ( c z , a 2 ) i f f a l , < a 2 o r a 1 = a 2 a n d c l = c 2 . We c a n t h e n d e f i n e t h e f o l l o w i n g F o r e a c h n > I , l e t R, = {((c~,l-l/n),(c2,l-l/n)) : f a m i l y o f r e l a t i o n s o n L'. ( c l - c 2 ( < 1/2n-1}. I f w e t h e n l e t R = U Rn U A ( L ' 1 , t h e n R i s a n e q u i v a l e n c e n r e l a t i o n on L', and L ' / R = L is a p a r t i a l l y ordered s e t under t h e q u o t i e n t t o L, then the resulting poset is order. I f w e a d j o i n a n i d e n t i t y e l e m e n t 1' t h e c o m p l e t e a l g e b r a i c l a t t i c e g i v e n i n t h e f o l l o w i n g diagram:

-

U

0

I'

6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c

0 0

L

The r e a s o n t h a t L U form [ c , a ] w i t h a

11')

<

1

00

A

U

{l'}

i s a l g e b r a i c i s t h a t each element o f t h e l a t t i c e o f t h e i s compact, a 6 i s t h e e l e m e n t 1 ' .

Order scattered and topologically scattered

I5

Now, i t i s c l e a r t h a t t h i s l a t t i c e i s n o t t o p o l o g i c a l l y s c a t t e r e d , s i n c e i t c o n t a i n s a C a n t o r s e t . However, t h e l a t t i c e i s o r d e r s c a t t e r e d , s i n c e a n y c h a i n i s mapped i s o m u r p h i c a l l y i n t o A U 1 1 ' ) by t h e p r o j e c t i o n map, a n d A U { I 1 } h a s n u order-dense chains. Mhile i t i s n o t t r u e i n general t h a t t h e notions of o r d e r s c a t t e r e d and topologically s c a t t e r e d coincide, t h e s e concepts d o c o i n c i d e f o r t h e c l a s s of d i s t r i b u t i v e a l g e b r a i c l a t t i c e s , a n d o u r n e x t g o a l i s t o p r o v e t h i s f a c t . We f i r s t n e e d t h e f o l l o w i n g r e s u l t , w h i c h g r e a t l y r e d u c e s t h e c l a s s o f l a t t i c e s we must s t u d y .

5.2

DEFINITION.

Let

non-empty s u b s e t o f

S S

be a n i n € - s e m i l a t t i c e . has an ~nfim~im.

We s a y

i s complete i f e v e r y

S

5 . 3 PROPOSITION. L e t S b e a c o m p l e t e i n € - s e m i l a t t i c e i n w h i c h S p e c S o r d e r g e n e r a t e s S ( i , e . , e v e r y e l e m e n t o f S i s t h e infimum o f p r i m e e l e m e n t s ) . T h e n either (a) for every

x

E

S,

there is a f i n i t e subset

F c Spec S

with

x = i n f F,

or ( b ) t h e r e i s a n i n j e c t i o n f : (2')' + S such that f preserves a l l infima, i s t h e l a t t i c e 21N m i n u s i t s a n d f ( S p e ~ ( 2 ~ ) 'c) S p e c S , w h e r e ( 2 " ) ' g r e a t e s t e l e m e n t , w h i c h w e v i e w a s t h e u n i o n s e m i l a t t t c e o f a l l non-empty s u b s e t s o f N. Moreover, t h e s e c o n d i t i o n s a r e mutually e x c l u s i v e . PROOF. S u p p o s e t h a t ( a ) f a i l s . T h e n t h e r e i s some and x = i n f A, then A is infinite.

x

E

S

so t h a t , i f A c Spec S

We m a k e two o b s e r v a t ~ o n s . F i r s t , s u p p o s e t h a t A C S p e c S i s f i l t e r e d . Then we c l a m i n f A E Spec S . Indeed, i € r,s E S w i t h rs < i n € A , then since A c S p e c S , w e h a v e A = A, U A s , where, f o r a n y t E S , we d e f i n e At = {a E A : t 6 a}. S i n c e A i s f i l t e r e d , i t t h e n f o l l o w s t h a t i n f A = i n € A, or i n € A = i n € A,, a n d s o e i t h e r r < i i i f A, = i n f A o r s i n f As = i n f A . In e i t h e r c a s e , we h a v e t h a t i n € A E S p e c S . Our s e c o n d o b s e r v a t i o n i s r e l a t e d t o t h e f i r s t . Namely, A c Spec S is i n f i n i t e , and suppose t h a t i n f A = i n € A f o r s u b s e t B c A. The , a g a i n we c l a i m i n € A E Spec S. Indeed, a n d a t l e a s t o n e o f A, o r As is o n c e a g a i n A = A, U A,, o u r h y p o t h e s i s i m p l i e s t h a t e i t h e r r < i n € A, = i n € A o r s t h a t i n f A E Spec S.

suppose t h a t every i n f i n i t e i f rs 4 i n € A , then infinite. But, then 4 i n f As = i n € A , s o

From t h e f i r s t o b s e r v a t i o n we c a n c o n c l u d e , by u s i n g Z o r n ' s Lemma, t h a t w h e r e M i s t h e s e t o f m i n i m a l p r i m e s a b o v e x; t h e p o i n t i s t h a t , s i n c e t h e i n f m u m o f a f i l t e r e d f a m i l y o f p r i m e s i s a g a i n a p r i m e , Z o r n ' s Lemma i m p l i e s t h a t e v e r y p r i m e a b o v e x d o m i n a t e s a m i n i m a l p r i m e a b o v e x, a n d t h e r e s u l t then follows. B u t , t h e p r i m e s i n M a r e i n c o m p a r a b l e , a n d s o t h e infunurn o f a n y f i n i t e s u b s e t F o f M d o e s n ' t f a l l below a n y e l e m e n t o f M\F, since M c o n s i s t s of primes. Note t h a t M i s infinite b y t h e h y p o t h e s i s a b o u t X.

x = i n € M,

We now c h o o s e a s e q u e n c e o f p r i m e s f r o m M r e c u r s i v e l y . F i r s t , s i n c e S p e c S , o u r s e c o n d o b s e r v a t i o n a b o v e i m p l i e s t h e r e i s some i n f M = x and x i n f i n i t e s u b s e t A1 c M w i t h x < i n € A l . Choose a n e l e m e n t pi E M\S(inf A l ) . Now, i n f A 1 i s n o t a p r i m e ( a s M c o n s i s t s o f m i n i m a l p r i m e s a b o v e x) a n d A 1 c M i s i n f i n i t e . Hence, o u r s e c o n d o b s e r v a t i o n a p p l i e s a g a i n t o imply t h e r e i s some i n f i n i t e s u b s e t A2 c A1 w i t h i n f A 1 < i n € A2. We t h e n p i c k p2 E A 2 \ 4 ( i n f A i ) . C o n t i n u l n g i n t h i s manner, we f i n d a s e q u e n c e o f p r i m e s {pn}

M. Mislove

16

such t h a t t h e primes are incomparable, and

pn

4

inE{p,

: rn

>

n).

We t h e n d e f i n e t h e map f : (ZN)' + S by f ( R ) = i n f { p m : m E M} f o r e v e r y Clearly f preserves a l l infima ( i . e . , unions) and s u b s e t R c lN w i t h R # (0. € ( S p e c (ZN)') c S p e c S . We c l a i m f i s a l s o one-to-one. Indeed, suppose B , C c TN a n d m E R \ C . Then, f ( B ) < pm. M o r e o v e r , p i . . . p m l * ( i n f { p n : n > in)) < f(C) and pi ...pml* (inf{pn : n > m pm, s i n c e n o n e o f p i pml or i n f { p n : n > m} i s b e l o w pm, a n d pm i s p r i m e . Thus f ( R ) f f ( C ) , and so f s a t i s f i e s (b).

4

,...,

To show t h a t t h e t w o c o n d i t i o n s a r e i n c o m p a t i b l e , s u p p o s e t h a t ( b ) h o l d s , a n d l e t x = i n € f((2')'). If ( a ) w e r e t o h o l d , t h e n we w o u l d h a v e x = p l . . . p m for f i n i t e l y many p r i m e s p i pm. F o r e a c h 1 = 1 m, we l e t A, = { f ( j ) : pi h f ( j ) a n d j E and we n o t e t h a t { f ( n ) : n E = A 1 U . . . U h. R u t , t h e n pi C i n f A, f o r each I , p1...pm = x = ( i n f A l l ( 1 ° F Am), a n d s o w e c o n c l u d e t h a t pi = i n € A f o r e a c h 1. T h i s m e a n s t h a t a t l e a s t o n e o f P I , . . . , pm c a n n o t b e p r i m e , s i n c e i n f A , i s n o t p r i m e f o r a t l e a s t o n e 1. Hence, ( a ) c a n n o t h o l d i f ( b ) d o e s n o t . n

m,

,...,

,...,

...

6 . 4 COROLLARY. L e t L b e a distributive a l g e b r a i c l a t t i c e , a n d s u p p o s e t h a t L c o n t a i n s a n element x s u c h t h a t x = i n f A , A c Spec L i m p l i e s A i s i n f i n i t e . Then L i s n e i t h e r o r d e r s c a t t e r e d nor t o p o l o g i c a l l y s c a t t e r e d .

PROOF.

If L s a t i s f i e s o u r hypotheses, t h e n L cannot s a t i s f y (a) o f t h e p r e v i o u s P r o p o s i t i o n , a n d s o t h e r e i s a n i n j e c t i o n f : (2')' + L preserving a l l i n f m a a n d a l l p r i m e s . R u t , i t i s well-known t h a t 2' contains an order-dense c h a i n , s o t h a t L m u s t a l s o . Hence, L is n e i t h e r o r d e r s c a t t e r e d nor topologically scattered.

n

Note t h a t , by C o r o l l a r y 3 . 3 , e v e r y d i s t r i b u t i v e a l g e b r a i c s a t i s f y a t l e a s t one of ( a ) o r (b) of P r o p o s i t i o n 6.3. Hence, t h a t t h e concepts of s c a t t e r e d c o i n c i d e f o r t h e s e l a t t i c e s , we they coincide f o r those l a t t i c e s satisfying (a) of Proposition

l a t t i c e L must i n o r d e r t o show o n l y h a v e t o show 6.3.

R e c a l l t h a t a s u b s e t A o f a t o p o l o g i c a l s p a c e X i s nowhere dense i f Ah a s no i n t e r i o r ; e q u i v a l e n t l y , A i s n o w h e r e d e n s e i f X\Ai s d e n s e i n X. T h e B a i r e C a t e g o r y Theorem s t a t e s t h a t , i f t h e s p a c e X i s e i t h e r c o m p l e t e l y r e g u l a r o r l o c a l l y compact H a u s d o r f f , t h e n a c o u n t a b l e u n i o n of nowhere d e n s e s u b s e t s o f X h a s no i n t e r i o r ; e q u i v a l e n t l y , t h e c o u n t a b l e i n t e r s e c t i o n o f o p e n , d e n s e s u b s e t s of X i s dense.

6 . 5 THEOREM. S u p p o s e t h a t L i s a d i s t r i b u t i v e a l g e b r a i c l a t t i c e i n w h i c h e v e r y e l e m e n t i s t h e infimum o f f i n i t e l y many p r i m e s . If L contains a perfect subspace, then K(L), t h e s e t o f c o m p a c t e l e m e n t s o f L, contaLns a n order-densc chain. PROOF. S u p p o s e t h a t X c L i s a p e r f e c t s u b s p a c e ( w h i c h w e may a s s m e i s a l s o S i n c e e v e r y e l e m e n t of L i s t h e c o m p a c t , by t a k i n g i t s c l o s u r e , i f n e c e s s a r y ) . L = U ( S p e c L)", a n d s o t h e Baire i n f m u m o f f i n i t e l y many p r i m e s , we h a v e

C a t e g o r y T h e o r e m i m p l i e s t h a t ( ( S p e c L)" n X ) ) h a s i n t e r i o r i n X f o r some n. If we t a k e n t o b e t h e m i n i m a l s u c h n, t h e n w e know ( S p e c L)"' n X i s n o w h e r e d e n s e i n X f o r m < n, and, by c o n s i d e r i n g o n l y t h o s e p o i n t s i n ( ( S p e c L)" n X)- w h i c h a r e n o t i n ( ( S p e c L l m n X I f o r a n y m < n, we c d n a s s u n e t h a t X h a s t h e p r o p e r t y t h a t e v e r y e l e m e n t o f X i s t h e l i m i t of p o i n S w h i c h a r e t h e inflmum o f e x a c t l y n p r i m e s , a n d n o f e w e r . We p r o c e e d b y i n d u c t ion on n:

n = I: ~-

H e r e we h a v e we can f i n d x,y E X

( S p e c L) with x

4

n X y,

i s d e n s e i n X. Then, s i n c e X i s p e r f e c t a n d , t h e n , s i n c e L i s a l g e b r a i c , we c a n f nd

Order scattered and topologically scattered

77

4

a k E K(L) w i t h k < x, but k y . IJ& l e t k 1 / 2 b e t h i s c o m p a c t e l e m e n t , a n d n o t e t h a t X' = X n f k a n d X \ t k a r e b o t h c o m p a c t p e r f e c t s u b s p a c e s o f L , and they each s a t i s f y t h e p r u p e r t y t h a t t h e i r i n t e r s e c t i o n w i t h Spec L is d e n s e i n them. T h u s , s i n c e 4k i s a d i s t r i b u t i v e a l g e b r a i c l a t t i c e , w e c a n f i n d a c o m p a c t e l e m e n t k 3 / 4 E K ( 4 k ) w i t h X' n 4 k 3 / 4 a n d X ' \ 4 k 3 / 4 b o t h c o m p a c t p e r f e c t subw e n e e d t o d o m o r e w o r k . N a m e l y , we s p a c e s o f f k . H ow ever, f o r X" = X \ 4 k , n o t e t h a t (Spec L)\fk i s d e n s e i n X", s i n c e f k is closed and Spec L is d e n s e i n X. M o r e o v e r , t h e map x + xk : I2 + Ck i s o n e - t o - o n e o n ( S p e c L ) \ f k . Indeed, i f p,q E (Spec L)\fk w i t h p q, then kp q s i n c e q i s prime, and k q . T h u s X"k i s a l s o a compact p e r f e c t s u b s p a c e o f Ck. F o r , i f s u kp z E X"k w e r e i s o l a t e d , t h e n t h e c o n t i n u i t y o f t r a n s l a t i o n by k w o u l d i m p l y with L k = { z } . R u t , t h e n pk = z f o r e v e r y t h e r e i s a n u p e n s u b s e t U o f X" p E S p e c I, n U, w h i c h c o n t r a d i c t s t h e f a c t t h a t t r a n s l a t i o n by k i s o n e - t o - o n e o n ( S p e c L ) \ + k . Thus, X"k i s a compact p e r € e c t s u b s p a c e o f Ck, a n d i s d e n s e i n X"k, s i n c e ( ( S p e c L ) \ f k ) k c S p e c Ck ( a s i s r e a d i l y S p e c Ck n X"k verified). R u t , t h i s m e a n s we may a p p l y t h e s a m e a r g u m e n t s t o X"k a s we d i d t o X, a n d t h u s f i n d a c o m p a c t e l e m e n t k 1 / 4 t K(Ck) w i t h X"k n + k 1 / 4 a n d X " k \ + k l / ~ , c o m p a c t p e r f e c t s u b s p a c e s o f Ck i n w h i c h e v e r y p o i n t i s a l i m i t o f p r i m e e l e m e n t s o f Ck. We c a n c l e a r l y c o n t i n u e t h i s p r o c e s s t o f i n d t h e d e s i r e d T h i s c o n c l l l d e s t h e p r o u E € o r t h e c a s e n = 1. order-dense c h a i n i n K(L).

4

4

4

S u p p o s e o u r r e s u l t s h o l d f o r m < n, and let L be a d i s t r i b u t i v e a l g e b r a i c l a t t i c e w i t h X c L a compact p e r f e c t s u b s p a c e i n which t h e s e t o f p o i n t s w h i c h a r e i n f u n a o f e x a c t l y n p r i m e s Eorms a d e n s e s u b s e t . We w a n t t o m i m i c t h e p r o o f we j u s t g a v e i n t h e c a s e n = 1. In f a c t , w e c a n s t i l l f i n d a compact e l e m e n t k o f L w i t h X n f k a n d X\4k b o t h compact p e r f e c t s u b s p a c e s O E L. So w e a g a i n Let k l / 2 b e t h i s c o m p a c t e l e m e n t . Moreover, 4k is again a d i s t r i b u t i v e a l g e b r a i c l a t t i c e w i t h X fl f k a p e r f e c t s u b s p a c e i n w h i c h t h e s e t o f e l e m e n t s which a r e i n f u n a o f e x a c t l y n primes i s s t i l l dense, s o t h a t p a r t of t h e proof c a r r i e s over as before. T h e p r o b l e m c o n c e r n s t h e s e t X" = X\fk. C e r t a i n l y X" i s a compact p e r E e c t s u b s p a c e o f L, a n d s i n c e t h e p o i n t s o f X" w h i c h a r e i n f i m a o f e x a c t l y n p r i m e s a r e d e n s e i n X", i t follows t h a t the w h i c h a r e i n f u n a o f a t m o s t n p r i m e s i n Ck a r e d e n s e i n X"k p o i n t s o f X"k ( s i n c e ( S p e c L)k c ( S p e c +k U { k } ) ) . Howe ve r, we d o n o t know t h a t X"k is a perfect space. So, s u p p o s e t h a t X"k h a s a n i s o l a t e d p o i n t , s a y z . Then, s i n c e t r a n s l a t i o n by k i s c o n t i n u o u s , t h e r e i s a n o p e n s u b s e t U ( w h i c h w e may a l s o with L k = {z}. a s s u n e t o b e c l o s e d s i n c e L i s t o t a l l y d i s c o n n e c t e d ) o f X" Now, U t o o h a s a d e n s e s e t oE p o i n t s D s u c h t h a t e a c h d E D i s t h e i n f i m u m oE e x a c t l y n p r i m e s o f L. M o r e o v e r , f o r s u c h a d E D, i f d = ...pn with s i n c e f k i s a subpi p r i m e , t h e n we c o u l d n o t h a v e pi E 4 k f o r e a c h i , l a t t i c e , and d E L\fk. Hence, a t l e a s t o n e p j E L \ 4 k . F u r t h e r m o r e , i f d' t D, t h e n we c l a m d ' < p j a l s o . Indeed, d ' k = dk = z ( a s Uk = { z ) ) , a n d s o S i n c e p j i s p r i m e , a n d w e know p j p' + k , w e m u s t h a v e d ' < p ' . IJe : n P i a c t g e n e r a l i z e t h i s t o s a y t h a t , € o r a l l d , d ' E D, f d fl ( S p e c L ) ( + k = fd' (Spec L ) \ f k , or, i n words, t h e set o f p r i m e s o f L i n L\fk above d is t h e sa me f o r e v e r y d E D. T h u s , i f we c h o o s e p j s o t h a t i t i s m i n i m a l a b o v e f o r a l l d ' t D. R u t , a s we h a v e j u s t s e e n , d, t h e n p j i s minimal above d' D c C p j, a n d s i n c e D i s d e n s e i n U, t h e same i s t r u e o f U. Since pj is p r i m e i n L b u t n o t p r i m e i n Cpj ( b y d e f i n i t i o n ) , w e h a v e U c + p j i s a compact p e r f e c t s u b s p a c e o f Cpt , w i t h t h e d e n s e s u b s e t D e a c h o f whose e l e m e n t s i s t h e inf i m um o f n-1 p r i m e s s i n c e w e n o l o n g e r n e e d p j i n C p . ) . Then o u r i n d u c t i v e h y p o t h e s i s a p p l i e s t o produce a n order-dense c h a i n i n K ( i p j ) c K(L), thus proving our result.

__

I n d u c t__ i v e- s t e p : __

Pt:

I _

.

H e n c e , w e may a s s u n e t h a t X"k is perfect. But, we have already pointed o u t t h a t X"k h a s a d e n s e s e t o f e l e m e n t s e a c h o f w h i c h i s t h e i n f i m u m o f a t most n primes. If X"k h a s a n o p e n s u b s e t w h i c h i t s e l f h a s a d e n s e s e t oE p o i n t s e a c h oE w h i c h i s t h e i n f u n u m o f a t most n - l primes, t h e n our i n d u c t i v e h y p o t h e s i s , a p p l i e d t o Ck, s h o w s t h e r e i s a n o r d e r - d e n s e c h a i n i n K(Ck) c K ( L ) , once again proving t h e result. H e n c e , w e may a s s u n e t h a t X"k is a compact p e r f e c t s u b s p a c e

M. Mislove

78

of Ck i n w h i c h t h e s e t o f p o i n t s w h i c h a r e l n f i m a u E e x a c t l y n p r i m e s i s dense. T h a t i s , we h a v e shown t h a t X"k c Ck a n d X' r: f k s a t i s f y t h e s a m e h y p o t h e s e s a s d u e s X c L. T h u s , we may c o n t i n u e t h e i n d u c t ~ v ep r o c e s s t o f i n d ( s i n c e K(+k), K(fk) c K(L)) w i t h k1/4 < k = k1/2 < k3/4 and k 1 / 4 , k 3 / 4 E K(L) X"k n f k 1 / 4 , X " k \ f k l / 4 , X ' n t k 3 / 4 , and X'\fk3/4 a r e a l l cumpact perEect s e t s . By r e c u r s i o n we c a n t h e n p r o d u c e t h e d e s i r e d o r d e r - d e n s e c h a i n i n K ( L ) , i n the c a s e t h a t t h e r e s u l t holds f o r m < n.

n

6.6

L,

COROLLARY. For a n y d i s t r i b u t i v e c o n t i n u o u s l a t t i c e is topologically scattered. s c a t t e r e d i f a n d u n l y i f I,

VLI.

n

L

i s order

LATTICES I N WHLCH E V E R Y ELEMENT I S 'THE L N F L M U M O F FLNL'rELY MANY PRIMES

I n t h i s l a s t s e c t i o n , w e e x p l o r e t h e wa ys i n w h i c h t h e s u b s e t s K(L) and S p e c L a c c i i r a t e l y r e f t e c t w h e t h e r I2 is scattered o r not, for distributive However, weTNpoint u u t t h a t we m u s t l i m i t o u r s t u d y t o a l g e b r a i c l a t t i c e s L. t h o s e l a t t i c e s having nu copy of 2 'The r e a s o n i s t h a t , w h i l e 2' is nut i s a d i s c r e t e anti-chaln, and hence is both scattered i n e i t h e r sense, S p e c (2') order s c a t t e r e d and topologically scattered; similarly, K(2N), w h i c h may b e h a s nu o r d e r d e n s e c h a i n s , a n d r e g a r d e d a s t h e f a m i l y oE c u E i n i t e s u b s e t s o f N , hence is order scattered. P r o p o s i t i o n 6.3 i m p l i e s t h a t t h e l a t t i c e s having no c o p y o f 2' a r e e x a c t l y t h o s e i n w h i c h e v e r y e l e m e n t i s t h e infimum o f f i n i t e l y many p r i m e s , a n d s o we r e s t r i c t o u r a t t e n t i o n t o t h e s e d i s t r i b u t i v e a l g e b r a i c We f i r s t ~ ( i n s i d e rt h e s i t u a t L o n a r o u n d K(L) f o r s u c h a l a t t i c e L. lattices.

.

If L is a d i s t r i b u t i v e a l g e b r a i c l a t t i c e i n which every element i s t h e in fu n u m o f f i n i t e l y many p r i m e s , t h e n T h e o r e m 6 . 5 i m p l i e s t h a t L i s s c a t t e r e d i f a n d o n l y i f K(L) c o n t a i n s nu o r d e r - d e n s e c h a i n ; i . e . , L i s scattered i f and i s o r d e r s c a t t e r e d (which i s t h e a p p r o p r i a t e n o t i o n of s c a t t e r e d , u n l y i f K(L) s i n c e K(L) is the dual of L i n t h e category s of d i s c r e t e sup-semilattices w i t h 0). Thus, L i s n o t s c a t t e r e d p r e c i s e l y wh e n K(L) c o n t a i n s a n o r d e r d e n s e c h a i n , w h i c h we may c e r t a i n l y a s s m e h a s nu l a r g e s t e l e m e n t , b u t t o w h i c h w e c a n a d j o i n 0. A m o m e n t ' s t h o u g h t s h o w s t h a t t h e o r d e r - d e n s e c h a i n %?a oE raLionals i n [O,l] i s t h e s e m i l a t t i c e K(C) f u r t h e C a n t o r c h a i n C i n t h e and s, the semiunit interval. And, u n d e r t h e c o m p l e t e d u a l i t y b e t w e e n A!l a t t i c e K(L) c o n t a i n s ma a s a s u p - s e m i l a t t i c e i f a n d o n l y i f L a d m i t s a q u o t i e n t o n t o t h e l a t t i c e C . ' the l a t t i c e s s a ~ i s f y i n gt h i s p r o p e r t y h a v e b e e n s t u d i e d , a n d , a s i s shown i n [HMS], t h e y a r e e x a c t l y t h o s e w h i c h , a s c o m p a c t zero-dimensional s e m i l a t t i c e s , admit a ccmpact s e m i l a t t i c e q u o t i e n t of p o s i t i v e d i m e n s i o n ( i n a n y oE t h e a p p r o p r i a t e n o t i o n s o f d i m e n s i o n € o r c o m p a c t s p a c e s ) . T h u s , we h a v e t h e f o l l o w i n g r e s u l t : 7.1 THEOREM. L e t 1. b e a d i s t r i b u t i v e a l g e b r a i c l a t t i c e i n w h i c h e v e r y e l e m e n t i s t h e i nf i m um o f E i n i t e l y many p r i m e s . T h e n t h e E o l l o w i n g a r e e q u i v a l e n t : (a)

L

c o n t a i n s a compact p e r f e c t subspace.

(b)

There i s a surmurphism o f compact s e m i l a t t i c e s t h e Cantor s u b l a t t i c e of t h e u n i t interval.

n

f : L

+

C,

Notice t h a t t h i s result i s not true i n general, f u r the l a t t i c e c o m p a c t p e r f e c t s u b s p a c e , b u t K(2') i s o r d e r s c a t t e r e d , w h e n c e ZN admit C a s a cumpact s e m i l a t t i c e q u o t i e n t .

where

C

2' has a dues not

We n w t u r n o u r a t t e n t i o n t o S p e c L . N o w , a s we d e s c r i b e d i n S e c t i o n ILL, t h e l a t t i c e L i s t h e t u p o l o g y o f S p e c L, i E w e endow S p e c L w i t h t h e h u l l k e r n e l t o p o l o g y f r o m L. T h u s , i t w o u l d a p p e a r t h a t t h e a p p r o p r i a t e n o t i o n oE s c a t t e r e d f u r S p e c L is t h a t oE t o p o l o g i c a l l y s c a t t e r e d . However, Spec L i s a l s o a p o s e t i n t h e i n h e r i t e d o r d e r f r o m L, s o t h e n u t i u n oE u r d e r s c a t t e r e d m a k e s p e r f e c t l y g o o d s e n s e f u r S p e c L , a l s o . A s we s h a l l s o c n s e e , t h e s e n o t i o n s

is

Order scattered and topologically scattered c o i n c i d e , L E w e endow

S p e c 1,

79

with t h e c o r r e c t topology.

Let L = { l / n : n > I } U {O}. T h e n F i r s t , c o n s i d e r t h e f o l l o w i n g example: d completely d i s t r i b u t i v e algebraic l a t t i c e , and Spec L = L\{l}. Moreover, : t h e b a s i s f u r t h e h u l l - k e r n e l t o p o l o g y o n Spec L i s t h e Eamily {[O,l.nl n > l}, and i t is immediately c l e a r t h a t t h e whole s p a c e Spec L h a s no i s o l a t e d point i n t h i s topology. T h a t i s , S p e c Id i s n u t t o p o l o g i c a l l y s c a t t e r e d i n t h e hul 1-kernel topology. Howe ve r, L i s s u r e l y s c a t t e r e d , s o we c u n c l u d e t h a t S p e c Id endow ed w i t h t h e h u l l - k e r n e l t o p o l o g y d o e s n o t a c c u r a t e l y r e E l e c t w h e t h e r L is s c a t t e r e d , e v e n f o r t h e l i m i t e d c l a s s O F l a t t i c e s i n w h i c h e v e r y e l e m e n t i s a n i n f ~ m u m o f f i n i t e l y many p r i e m s . 1,

1s

Ho w ever , t h e r e i s a n o t h e r t o p o l o p y o n S p e c L, t h e topology which Spec L i n h e r i t s f r o m t h e A - t o p o l o g y o n L. Moreover, t h i s topology a r i s e s n a t u r a l l y o n S p e c I>.

7.2 DEFLNLTLON. L e t X b e a t o p o l o g i c a l s p a c e , a n d l e t X1 denote X with an 1 adjoined. R e c a l l i n g t h a t a s u b s e t A c X i s saturated i E € I t isolated point i s t h e i n t e r s e c t i o n o f t h e o p e n s e t s c u n t a i n i n g i t , w e d e f i n e t h e patch topology o n X1 t o h e t h e t o p o l o w g e n e r a t e d by t h e o r i g i n a l t o p o l o g y oE X, together wherz Q r a n g e s o v e r t h e quasicompact s a t u r a t e d w i t h a l l s e t s o f t h e f o r m X’\Q, In t h e c a s e X i s l o c a l l y c o m p a c t H a u s d o r f f , t h i s a m o u n t s t o s u b s e t s o f X. E o r m i n g t h e o n e - p o i n t c o m p a c t i E i c a t i o n oE X. Now, f u r a d i s t r i b u t i v e a l g e b r a i c l a t t i c e L , i t t u r n s o u t t h a t Spec L in t h e p a t c h t o p o l o g y i s e x a c t l y S p e c L w i t h t h e i n h e r i t e d ?,-topology Erom L ( f o r d e t a i l s , s e e [ C ] , p.261). Since t h i s i s a n inherited topology, i t i s c l e a r t h a t I, cannot be s c a t t e r e d i f Spec L i s n o t s c a t t e r e d . We s h a l l s h o w t h a t t h e c o n v e r s e h o l d s , a n d , i n t h e p r o c e s s s how t h a t t h e n o t i o n s oE o r d e r s c a t t e r e d a n d t o p o l o g i c a l l y s c a t t e r e d c o i n c i d e o n S p e c I,, i f w e endow S p e c L w i t h t h e p a t c h t apology T h u s , throughout the rest of t h i s paper, we s h a l l consider Spec L

.

endowed with the patch topology. Now, s i n c e Spec L i s c l o s e d under f i l t e r e d inEima, a s i m p l e argument shows cannot b e t o p o l o g i c a l l y s c a t t e r e d i f Spec L c o n t a l n s a n ordert h a t S p e c I, dense chain. T h u s , i f S p e c L is n o t o r d e r s c a t t e r e d , t h e n n e i t h e r S p e c L n o r L i t s e l f is topologically scattered. To c o m p l e t e t h e e q u i v a l e n c e o f t h e s e n o t i o n s , we f i r s t n e e d t h e f o l l o w i n g l e m m a , w h o s e p r o o f i s a n e a s y e x e r c i s e :

7 . 3 LEMMA. Let endowed w i t h t h e t h e n s o d o e s Pi

Pi,

...,P,

be posets and let L f PI f u r some i n d e x I .

lexicographic order.

~

P I x P2 X . . * X Pn b e t h e p r o d u c t Pn h a s a n o r d e r - d e n s e c h a i n , 0

.

v

~

N o t e t h a t t h e lemma a c t u a l l y h o l d s i f t h e p r o d u c t o r d e r i s c o n s i d e r e d , t o o .

7.4 THEOREM. L e t L b e a d i s t r i b u t i v e a l g e b r a i c l a t t i c e i n w h i c h e v e r y p o i n t i s t h e i n f m u m o f f i n i t e l y many p r i m e e l e m e n t s . “ h e n t h e f o l l o w i n g a r e e q u i v a l e n t : (a)

I,

is scattered.

(h)

Spec L

is o r d e r s c a t t e r e d .

(c)

Spec L

is topologically scattered.

PROOF, W e have a l r e a d y s e e n t h a t (a) i m p l i e s (c) i m p l i e s ( b ) , s o we o n l y need show t h a t ( b ) i m p l i e s ( a ) . Suppose, t h e n , t h a t L h a s a n o r d e r - d e n s e c h a i n C . T h e n Cc o n t a i n s a compact p e r f e c t s u b s p a c e which must a l s o b e a c h a i n , a n d we c a l l t h i s C as well. F u r t h e r m o r e , t h e same a r g u m e n t w e u s e d i n t h e p r o o f o f T h e o r e m 6 . 5 s h o w s w e may a s s u n e t h a t e v e r y e l e m e n t o f C i s t h e i n f i m u m oE e x a c t l y n p r i m e s , a n d n o € e w e r . Now, l e t P = ( S p e c L) X * * * X ( S p e c L) d e n o t e

M . Mislove

80

t h e p r o d u c t o f S p e c I, w i t h i t s e l f n t i m e s , i n t h e l e x i c o g r a p h i c o r d e r . Wr s h a l l u s e t h e o r d e r d e n s e c h a i n C t o p r o d u c e a n o r d e r d e n s e c h a i n i n P, as f o l l o w s . Pick a n y p o i n t c E C s o t h a t t h e r e a r e p o i n t s c ' , c l ' E C w i t h c ' < c < c". T h e n c = P I . ..pn f o r e x a c t l y n p r i m e s , a n d we l e t o u r f i r s t p o i n t from P be t h e p o i n t ( p i pn). Then, c h o o s e c ' c C w i t h c ' < c. We knuw t h e r e a r e n p r i m e s , ql,..., qn i n L w i t h c 1 = q l . . . q n , a n d s i n c e c ' < c , we h a v e q 1 ...qn < p l . . . p n . S i n c e e a c h pi i s p r i m e , we c a n € i n d a q j w i t h q j < p L . qn) precedes (pi pn) T h u s , we c a n r e a r r a n g e ql,..., qn s o t h a t ( 4 1 i n t h e l e x i c o g r a p h i c o r d e r i n P, and c l e a r l y we c a n d o t h i s s o t h a t (qi, ...,q n) < ( p i , . . . , p n ) , s ~ n c e c' < c . ' h e n w e c a n s i m i l a r l y c h o o s e c " E C w i t h c < c", and t h e n a r r a n g e t h e n p r i m e s w h o s e infimum I S c" so t h a t ( p i , . ,pn) p r e c e d e s them s t r i c t l y i n P. C o n t i n u i n g i n t h i s rnanncr, we c a n u s e t h e i n P. R u t , o u r Lemma order-dense chain C t o produce a n order-dense chain C' 7 . 3 t h e n a p p l i e s t o show t h a t t h e r e i s a n o r d e r - d e n s e c h a i n i n Sp?c I,. T h i s shows t h a t (b) cannot h o l d i f (a) d o e s n ' t , a n d t h a t f i n i s h e s t h e proof o f t h e 'I Theorem.

,...,

,...,

,...,

..

REFERENCES

G

G i e r z , G . , e t a t , A Comprndiirn o € C o n t i n u o u s L a t t l c e s ( S p r i n g e r - V e r l a g , B e r l i n , H e i d e l b e r g , N e w York 1 9 8 0 ) .

HMS

Hofmann, K. H . , M . M i s l o v e , a n d A . R . S t r a l k a , T h e P o n t r y a g i n D u a l i t y o f Compact O - d i m e n s i o n a l S e m i l a t t L c e s a n d I t s A p p l i c a t i o n s , L e c t u r e N o t e s 111 Math. 396 ( 1 9 7 4 ) .

L

Lawson, J . D . , T h e d u a l i t y o f c o n t i n u o u s p o s e t s , H o u s t o n J o u r n a l o f M a t h . 5 ( 1 9 7 9 ) , 357-394.

MI

M i s l o v e , M . , An i n t r o d u c t i o n t o c o n t i n u o u s l a t t i c e s , i n : R i v a l , Z ( e d ) , O r d e r e d S e t s (D. R e i d e l P u b l i s h e r s , p . 379-406, D o r d r e c h t , 1 9 8 2 ) .

M2

Mislove, M . , Order s c a t t e r e d d i s t r i b u t i v e continuous l a t t i c e s a r e t o p o l o g i c a l l y s c a t t e r e d , Houston J o u r n a l o f M a t h e m a t i c s , t o a p p e a r .

N

N a c h b i n , L . , T o p o l o p y a n d O r d e r (Van N o s t r a n d , P r i n c e t o n 19651.

R

R a n e y , G. N . , C o m p l e t e l y d i s t r i b u t i v e c o m p l e t e l a t t i c e s , P r o c e e d i n g s oE t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y 3 ( 1 9 5 2 ) , 667-680.

Annals o f Discrete Mathematics 23 (1984) 81-98 0 Elsevier Science PublishersB.V. (North-Holland)

REPRESENTING

ORDERED

81

SETS B Y CHAINS

*

Dwight DUFFUS

Maurice POUZET

Emory University Atlanta, Georgia

Universite Claude Bernard LYON France

U.S.A.

DPdie avec affection au Professeur E . COROMINAS d l'occasion de son Emdritat.

RESUME.-

NOUS traitons ici de questions relatives B la notion de retraction dans les ensembles ordonnes (un ensemble ordonne Q est retract de l'ensemble ordonne P s'il existe deux applications croissantes, f de Q dans P, et g de P dans Q telles que g o f = Id ) . Notre travail

Q

se situe dans le cadre de la classification des ensembles ordonnes et de la theorie de leur representation proposees par D. DUFFUS et I. RIVAL en 1981. Cette classification est basee sur le concept de variete d'ordre (classe d'ensembles ordonnes close par retract et produit direct) tandis que la notion de representation s'inspire des representations sous-directes de l'alggbre universelle. (Une r e p r e sentation d'un ensemble ordonne P est une famille (Pi I i € I) d'ensembles ordonnes telle que P est retract du produit et P. retract de P pour tout

i E I

lorsque pour chaque representation moins un P ,)

;

1

T l (Pi i € I)

l'ensemble P est irreductible

(Pi I i € I)

il est retract d'au

.

Dans l'article de 1981 etaient decrites certaines varietes d'ordres remarquables ainsi que leurs elements irreductibles, en apportant une attention particuliere aux chaines irreductihles et aux varietes qu'elles engendrent. Une description similaire a et6 developp8e pour 1es graphes reflexifs par R. NOWAKOWSKI et I. RIVAL et s'Ctend maintenant aux graphes irreflexifs. A partir d'un point de vue quelque peu different, A. QUILLIOT a mis en evidence des proprietes de retraction communes aux ensembles ordonnes, aux graphes et aux espaces metriques. Des recherches mettent maintenant l'accent sur l'utilisation des concepts classiques de retract absolus, d'ohjet injectif et d'enveloppes injectives. Notre travail concerne la variete ch engendree par la classe des types d'isomorphie des chaines et apporte des el6ments de reponse aux questions suivantes posees par D. DUFFUS et I. RIVAL:

*

D. DUFFUS was supported by Emory University Research Committee Grant 8059-14 (Summer 82) and NSF Grant 8302054 (Summer 83).

D, Duffus, M. Pouzet

82

Quels s o n t les ensembles ordonnes appartenant 2 C h [ 3 , P r o b l e m 61 ? T o u t t r e i l l i s d e d i m e n s i o n f i n i e a p p a r t i e n t - i l 2 Ch [ 3 , P r o b l e m 91 ? L e s t h e o r e m e s 3 . 1 , 3 . 5 e t 4 . 2 c o n t i e n n e n t d e s r e p o n s e s part i e l l e s B ces d e u x q u e s t i o n s t a n d i s q u e le theori?me 5 . 2 e t l e c o r o l l a i r e 5 . 3 r e p o n d e n t e n t i e r e m e n t a u x d e u x q u e s t i o n s suivantes: T o u t e l e m e n t d e ch a - t - i l une r e p r 4 s e n t a t i o n p a r des chaines i r r 4 d u c t i b l e s [ 3 , P r o b l e m 71 ?

Y a - t - i l d a n s Ch des 4 1 6 m e n t s i r r e d u c t i b l e s a u t r e s q u e les chafnes i r r e d u c t i b l e s [ 3 , P r o b l e m 101 ?

1, I N T R O D U C T I O N The n o t i o n o f a n ( ~ d i l rv a r i e t y was f o r m u l a t e d by D u f f u s a n d R i v a l [ 3 ] a s a means o f c l a s s i f y i n : ; o r d e r e d s e t s .

An

order v a r i e t y

K

i s a c l a s s of

i s o m o r p h i s m t y p e s of u r d e r e d s e t s c l o s e d u n d e r the f o r m a t i o n o f r e t r a c t s

(K_(K) 5 K )

i s a r e t r a c t of

g

of

and

P

f

(p_(K)

and d i r e c t p r o d u c t s

Q

to

P

such t h a t

Q

K)

.

(For ordered sets

i f t h e r e a r e o r d e r p r e s e r v i n g maps g

as a c o r e t r a c t i r m . )

0

=

id

. 9'

also, r e f e r t o

g

of

I'

and

Q

to

Q ,

and

P

as a r e t r a c t i o n

K

I n f a c t , t h e o r d e r v a r i e t y g e n e r a t e d by a c l a s s

W(K)

of o r d e r e d s e t s i s j u s t

f

f

.

[3, P r o p o s i t i o n 4.21

I n a d d i t i o n t o c l ~ s s i f i c a t i o n , D u f f u s a n d R i v a l were c o n c e r n e d w i t h the r e p r e s e n t a t i o n of o r d e r e d sets.

They f o l l o w e d t h e model o f s u b d i r e c t r e p r e s e n -

t a t i o n i n u n i v e r s a l a l g e b r a , w i t h r e t r a c t and d i r e c t p r o d u c t a s t h e b a s i c constructions.

r e t r a c t of tion

P

P

A r e p r e s e n t a t i o n o f an ordered set

i s a r e t r a c t of

of o r d e r e d s e t s s u c h t h a t for all

( p i l i tl I )

of

Call

i t: I .

P,

I'

P

n(Pi1i

is a family E

I)

Pi

and

(Pili

&

I)

is a

irreducible i f f o r every representa-

i s a r e t r a c t of

Pi

f o r some

i

E

I .

They managed t o d e s c r i b e some t y p i c a l o r d e r v a r i e t i e s a n d t h e i r i r r e d u c i b l e elements, b r i n g i n g a p a r t i c u l a r a t t e n t i o n to t h e i r r e d u c i b l e chains and t h e o r d e r v a r i e t i e s g e n e r a t e d by t h e m

[ 3 , Proposition 5.7

and

theorem 7.11

A s i m i l a r p r o g r a m h a s b e e n r e c e n t l y c a r r i e d o n f o r r e f l e x i v e g r a p h s by R . NOWAKOWSKI a n d I . RIVAL

[El

a n d non e x t e n d e d t o i r r e f l e x i v e g r a p h s .

From a somewhat d i f f e r e n t a p p r o a c h

A . QUILLIOT [ 1 1 1

h a s p o i n t e d o u t some

c o n n e c t i o n s b e t w e e n t h e r e t r a c t p r o p e r t i e s o f o r d e r e d s e t s , g r a p h s a n d metric s p a c e s . The c u r r e n t

researches

c o n c e p t s of a b s o l u t e r e t r a c t s , p r e s e n t e d by

E.W.

KISS,

p u t some e m p h a s i s on t h e u s e o f t h e c l a s s i c a l i n j e c t i v e o b j e t s , and i n j e c t i v e enveloppes ( a s

L . MARKI,

P. PROHLE,

W. THOLEN [ 5 1 1 .

,

Representing ordered sets by chains Ch

Our work d e a l s w i t h t h e v a r i e t y

,

(C)

=

83 where C i s t h e c l a s s

o f a l l i s o m o r p h i s m t y p e s o f c h a i n s . We t r e a t t h e f o l l o w i n g q u e r i e s : Which o r d e r e d s e t s a r e m e m b e r s of

Ch

P r o b l e m 61

[3,

I s every f i n i t e dimensional l a t t i c e i n

Ch

Some p a r t i a l a n s w e r s i i r e p r o v i d e d b y Theorems 3 . 1 ,

?

[ 3 , P r o b l e m 91 3.5 a n d 4 . 1 .

H a s e v e r y member of c h a r e p r e s e n t a t i o n b y i r r e d u c i b l e c h a i n s [ 3 , P r o b l e m 71 ? A r e there a n y i r r e d u c i b l e m e m b e r s o f Ch i r r e d u c i b l e c h a i n s [ 3 , P r o b l e m 10; ? T h e s e a r e a n s w e r e d by Theorem 5 . 2

2.

a s i d e f r o m the

and C o r o l l a r y 5 . 3

.

PRELIMINARIES

ch a r e

R e t r a c t s o f p r o d u c t s of c h a i n s a r e l a t t i c e s , s o a l l members o f lattices.

Conversely, a l l complete l a t t i c e s belong t o

t h e two-element EXAMPLE 2 . 1

chain.

( R i v a l and Wille [ 1 3 1 ) .

--II__

w x w

1

(Pouzet c f . [ 3 ] ) .

where

2,

is

CIl.

Nonprincipal u l t r a f i l t e r s i n t h e lattice

of a l l s u b s e t s o f 3- c o u n t a b l e s e t d o n o t b e l o n g t o EXAMPLE 2 . 2

R_F({2,])

Ilowever. n o t a l l l a t t i c e s a r e members o f

Ck.

The l a t t i c e o f a l l p r o p e r i n i t i a l s e g m e n t s

Ilf

Ch.

does not belong t o

___

On t h e o t h e r h a n d , s e v e r a l c l a s s e s o f l a t t i c e s , d e s c r i b e d b y " f i n i t e n e s s "

ch.

c o n d i t i o n s , are c o n t a i n e d i n THEOREM 2 . 3

!Jelongs

THEOREM 2 . 4

and h a s a r e p r e s e n t a t i o n

( P o u z e t and R i v a l [ 9 ] ) .

and h a s 5 r e p r e s e n a t i o n

I

Every l a t t i c e of f i n i t e w i d t h

( R i v a l a n d W i l l e [ 1 3 , Theorem 2 1 ) .

to C h ,

_

&

irreducible chains.

Every c o u n t a b l e l a t t i c e b e l o n g s

to c k ,

irreducible chains.

We s h a l l b e c o n c e r n e d w i t h f i n i t e d i m e n s i o n a l l a t t i c e s [ 3 , P r o b l e m 9 1 Recall t h a t a n o r d e r e d s e t

k of

has dimension

P

k ,

written

dim P = k ,

.

if

i s t h e minimum number o f l i n e a r o r d e r i n g s w h o s e i n t e r s e c t i o n i s t h e o r d e r i n g

P.

In f a c t ,

p r o d u c t of

k

dim P

=

k

i f and o n l y i f

c h a i n s , b u t no f e w e r .

P

is o r d e r embeddable i n t h e

Thus, it is n a t u r a l t o a s k i f l a t t i c e s

w i t h t h i s property are a l s o retracts of products of chains. The p r o o f s o f Theorems 2 . 3 a n d 2 . 4 , on two n o t i o n s :

the =property

a n d o f o u r m a i n r e s u l t s , are b a s e d

and t h e s e l e c t i o n p r o p e r t y .

?

D. Duffus,M. Pouzet

84

A subset B

(B?:) .,.

=

where

(B,)",

I

defined dually. X*

1 8

S,

# 0

R(P)

Let

.

B" = ;x

A p a i r of sets

P

i

bound

i.s c a l l e d

(X,Y)

J

~x

of

P

X;':

I t i s e a s y t o see t h a t

for a l l

Y,

11

2 X2

Y2

and

The mal)

5 Yl ,

S(P)

and o r d e r

i

0

~l

and

P

by

(X,-,S,)

I

if

(B,,B*)

B

by

C

B(P)

5 I

P

f(B) c B

such t h a t

for all

B

t.

P

i n t r o d u c e d a more r e s t r i c t i v e v e r s i o n : i f t h e r e i s a n o r d e r p r e s e r v i n g map

f'(X,Y)

E

X*

11

equivalent [7,

Y,

A p a i r of s u b s e t s (I

B,

=

(X,S)

for all

P r o p o s i t i o n 31

d c f . [31.

t

of

G'(p)

.

P

of

(A,B)

P

has t h e

P

ordered sets, i f each gap of

P

t h e n each gap of

.

P

to

of

f

!j(P)

such t h a t

F o r l a t t i c e s , t i i e s e two a r e

is called a

g

of

K-gap

of

P

if

Q

p r e s e r v e s a gap

P

to

Q

property,

A

i s p r e s e r v e d by some

1'

P.

y B,

(A,B)

such t h a t

i s a r e t r a c t of

and of

P

( g ( A ) , g(B))

K

Tor some c l a s s

i s p r e s e r v e d by some member o f

of

K. I I ( P i l i i. I )

[ 3 , P r o p o s i t i o n 5.11. C-gap

In f a c t

and

.

C~I

TliEOREM 2 . 5 ( R i v a l a n d Wille [ 1 3 ] ) .

only i f it has t h e

1131,

Following

t h e f o l l o w i n g shows t h e i m p o r t a n c e of t h e

selection properties for

.

(C3<,C9c)

Nevermann and W i l l e [ 7 ]

l$'(P)

It is s t r a i g h t f o r w a r d t o prove t h a t i f

more c a n be s a i d :

if

- -

has t h e s t r o n g s e l e c t i o n property

f'

Say t h a t a n o r d e r e d s e t

Also,

Q.

.

denote

.

i f t h e r e i s an o r d e r p r e s e r v i n g map is a gap of

B(P)

is

(S,,,Y,,)

h a s t h e s e l e c t i o n p r o p e r t y i f t h e r e i s an o r d e r p r e s e r v i n g m a p

to

A?:

B,

B'(P)

in

Two v e r s i o n s o f t h e s e l e c t i o n p r o p e r t y h a v e b e e n u s e d .

P

and

G'(P)

and l e t

f3'(P)

i s t h e n a n o r d e r embedding o f

Bw(B,,B")

b

B #

is a bound s e t i f i t i s n o n e m p t y .

Order

P .

if

i s c a l l e d a b o u n d i n g pair i f

d e n o t e t h e s e t o f a l l bound s u b s e t s o f

t h e s e t o f a l l bounding p a i r s of XI

P

of an ordered set

B

C-=property

An o r d e r e d s e t i s a member of

C\L

i f and

and t h e s e l e c t i o n p r o p e r t y .

T h i s t h e o r e m p r o v i d e s a means o f s h o w i n g t h a t p a r t i c u l a r c l a s s e s o f l a t t i c e s belong t o

ck.

We r e t u r n t o t h i s s t r a t e g y i n s e c t i o n s

4 and 5 .

In

t h e n e x t s e c t i o n w e modify i t s l i g h t l y t o d e a l w i t h f i n i t e d i m e n s i o n a l distributive lattices.

3.

FINITE DIMENSIONAL DISTRIBUTIVE LATTICES E x a m p l e s 2 . 1 a n d 2 . 2 show t h a t n o t a l l d i s t r i b u t i v e l a t t i c e s b e l o n g t o

THEOREM 3.1.

E v e r y f i n i t e d i m e n s i o n a l d i s t r i b u t i v e l a t t i c e i s a member o f

Ch.

ch.

Our p r o o f r e q u i r e s t h r e e o b s e r v a t i o n s . L E W 3.2.

Let

D

d i s t r i b u t i v e l a t t i c e of d i m e n s i o n

n .

Then

D

&+

Representing ordered sets by chains

& 5 p r o d u c t uf

sublattice

chains.

11

I t s u f f i c e s t o show tliat S p e c D , t h e l a t t i c e o f p r i m e i d e a l s o f

Proof.

o r d e r e d by s e t i n c l u s i o n , i s t h e u n i o n of

u

Spec

of

c

=

D e f i n e a map

f

d

Ciid

t

...

c2

( 8

1

Cn.

Let

of

Jj

D

to

for all

D.

by

lri

,

A)IA::

X =

B

A -

If

implies

A .

For any

as

J. J

is prime.

X 2n

+1

j

t

?"+l.

j

~ ( x . l iE H)

n

chains.

5 x.

i

J

i

1'

This is a contradiction:

J n + l*

. * * ,

so

=

~ ( x . l ii B)

i

E

j

.

J.

E

J '

A

Thus,

an n - d i m e n s i o n a l

for all

x . E J. 1 J and

such t h a t

I3

2

~ ( x . l iE B)

Conversely,

J. 1'

x.

By o u r c h o i c e of

By D i l w o r t h ' s

Jl' J 2 '

and l e t

t h e r e must b e some

A ,

:t

=

,...,n + l : l .

.

~ ( x . l it B ) ~ ~ ( x , c l A)i

tlien

~ ( x ~ El A) i

I1,2

; ~ ( ~ . 6: l i

,

# i)

x . F Ji'\ U(J.Ij J

choose

f (d)

0

T h i s map i s a l a t t i c e e m b e d d i n g .

L e t u s s u p p o s e t h a t Spec D c a n n o t b e c o v e r e d by

i

is a chain.

I(Ci)

i ,

Theorem [ 21 t h e r e a r e p a i r w i s e n o n c o m p a r a b l e p r i m e i d e a l s For e a c h

let

b e t h e s e t of a l l i n i t i a l s e g m e n t s

1,2,...,n )

=

D

Suppose t h i s i s s o :

chains.

Then, f o r e a c h

![(I(Ci)li

d t

n

I(Ci)

o r d e r e d by s e t i n c l u s i o n .

Ci

iJ

85

5B

l a t t i c e cannot contain

..r

The n e x t o b s e r v a t i o n r e q u i r e s a m o d i f i e d s e l e c t i o n p r o p e r t y .

L,let

d e n o t e t h e s e t o f a l l n o n e m p t y s u b l a t t i c e s of

Sub L

binary r e l a t i o n

s

,

S

E

l(S) t S

T

if

s

t f S

A

and

s

&a

D

implies

5 f(T)

f(S)

sublattice

f

for all

In t h i s section we

f o r all

product

t E T

V

Sub L

of

For n l a t t i c e with the

to

S, T

E

L

such

Sub L

.

f i n i t e l y many c h a i n s .

Then

allows selection.

I t s u f f i c e s t o show t h a t a l l c h a i n s a l l o w s e l e c t i o n a n d t h a t i f

Proof.

D

S c1 T

and

Let

LEMMA 3.3.

D

(I

(This r e l a t i o n need n o t be t r a n s i t i v e , )

L a l l o w s s e l e c t i o n i f t h e r e i s a map

s h a l l say that

S

g i v e n by

c1

t t T .

L

1

X

D2 w h e r e

D1

D2

and

allow selection then

D

D

=

allows selection.

W e g i v e a n a r g u m e n t of D u f f u s , R i v a l , a n d S i m o n o v i t s [ 4 , Theorem l] t o show t h a t a c h a i n f o r each

<

W

.

Suppose t h a t

c2 = c1

so

A

f(S2)

selection.

C

c2 =

L

c2

S1

Let

allows selection.

let

S E Sub C

S

1

so

implies

CY

S2

in

f ( S ) = c1 1 c

<

<w

b e a w e l l o r d e r i n g of

b e t h e least element of

f(S)

cl.

Sub

C

implies Thus,

and

c1 = f(S1)

c1 cw c2 f(S1)

.

> f(S2)

But

5 f(S2)

and

C

with respect t o

S

c1 and

=

C

c2

=

c1

V

.

Then

c2

C

allows

S2

D. Duffus,M. Pouzet

86

Let

D2.

D1

X

s

= f

2

(1r

2

2

For

S

(S'))

,

S c1 T

in

wise,

nl(S)

any

f(S)

5

c1

f(T)

S ' = {(sl,s)l(sl,s)

Sub D1

in

and

(tl,t)

= (tl,sv t)

,

T' ,

.

Sl

f

D1

(sl,s) Thus,

TI

(S')

2

f(S)

f

2

(n (T)) 1

Let

.

For

S'

and

tl

=

(sl,sAt)

=

n2(T')

c1

and

.

S

E

=

a r e d e t e r m i n e d component-

5

(t,,t)

A

D

and l e t

s1 = f l ( n l ( S ) )

D2

X

1,2)

=

Then

s1 = f l ( n l ( S ) )

so

T'

E

E

D. ( i

f ( S ) = ( s1,s2) w h e r e

S i n c e j o i n s a n d meets i n

S'

C

let

where

nl(T)

(t,,t)

V

Sub D

E

Sub D .

(sl,s)

(bl,s)

b e a s e l e c t i o n map f o r t h e l a t t i c e

fi

C

Sub D 2 .

in

Hence,

.

Lemmas 3.2 a n d 3.3 show t h a t a l l f i n i t e d i m e n s i o n a l d i s t r i b u t i v e l a t t i c e s L e t us t u r n t o t h e

allow s e l e c t i o n .

Let

LEMMA 3.4.

and

D1

C-gap p r o p e r t y .

D2 & d i s t r i b u t i v e l a t t i c e s w i t h t h e p r o p e r t y

a l l t h e i r nonempty s u b l a t t i c e s h a v e t h e

_ _ I _

of

D1

Proof.

D

Let

c

-

~

D2

and l e t

D1

be a s u b l a t t i c e of

X

I

ideal

and f i l t e r I ,

chain

preserves

C

and

by a c h a i n

F

D.

i s a g a p of

i = 1,2.

(xl,s)

(ni(A),@)

D1

0,

I ri F =

(0,B) C ci3 C '

V

Say

(r,x2)

E

A*.

P r o o f of Theorem 3.1. By Lemma 3.2, Lemma 3 . 4 ,

D

D

is

(A,@)

i s preserved f o r

(A,B)

preserves

a chain

such t h a t

otherwise,

(r,x2)

ni(I)

E

i

D

Let

then there is a

to

ni(I)

E

I :

such t h a t

C

(A.0)

preserves

C

x

is a sublattice

ni(I) ni(I)

for

such t h a t

then

(xl,s)

This contradicts t h e f a c t t h a t

has t h e

D.

b e t h e map o f

and

and

V

(A,@)

x

i

I C

(r,x2)

via

ni(A)* E

I

D

i

E

I

(fi(Ai), to

let fi(Bi))

2

g i v e n by

f

of

Then t h e map

K =

I .

be a f i n i t e dimensional d i s t r i b u t i v e l a t t i c e .

C-gap p r o p e r t y .

For a l l

and

i s a gap of

i s a s u b l a t t i c e o f a p r o d u c t o f f i n i t e l y many c h a i n s .

a l l gaps of Ci

of

f

Obviously,

C.

(xl,s)

D2

X

i s a gap of

So w e may a s s u m e t h a t t h e r e i s

ni.

for

If

a n d a n o r d e r p r e s e r v i n g map

(f(ni(A)).O)

fx

u F,

I f w e d e m o n s t r a t e t h a t some

t h e n , by d u a l i t y ,

I

is a s u b l a t t i c e of

I

i = 1.2.

for

C

chain

o

for

= I

F.

I t f o l l o w s t h a t t h e l i n e a r sum

C' ,

The i d e a l

f

(A,@)

D

such t h a t

is a gap of

D .

there exist a (lattice)

D.

for

of

D

in

F

(6.B)

b e a gap of

(A,B)

We a p p l y a r e s u l t of P o u z e t a n d R i v a l [ 9 , Lemma 4 1 :

a gap of

Then a l l s u b l a t t i c e s

property.

c-m p r o p e r t y .

D2 h a v e t h e

X

that

n(;?_l x

D E

Let fi

{(Ai,Bi)(i

11

b e t h e set of

b e a n o r d e r p r e s e r v i n g map o f i s a g a p of

f x ( y ) = 1,

Ci.

if

y

to

D)

E

x n(cili

E

I)

,

2

For

x,

By

x and

E

D,

D

to

let

fx(y) = 0 ,

Representing ordered sets by chains d e f i n e d by

T

c c K

For each

U(c) If

c

U(c)*

I I

f(A.)*

t

L(c), 11

J

f(B.), J

5

Dic

t

0

=

71,

then

L(c)*

I I

,

i s an o r d e r embedding.

Tr.(c) J

E

B(c)

of

g

D

to

i s a gap of

r e t r a c t i o n because

it follows t h a t

B(c)

so

If

provided

dimensional d i s t r i b u t i v e l a t t i c e .

it

.

I

E

.

J

Since

This

Therefore,

j'

c'

i s order preserving.

g

j

C.

Sub D

5

c

.

Sub D .

E

of

to

and d e f i n e

D

in

then

K

Finally,

h(B(f(d)))= h({d}) = d .

=

Ch

An o r d e r e d s e t i s i n

h

.

g(c) = h(B(c))

f(d)

g

C

J

by

Sub D ,

in

B(c')

II

I(

in

J

Apply Lemma 3 . 3 t o o b t a i n a s e l e c t i o n map t h e map

f o r some

f . ( A . ) * ( 1 fj(Bj),< J

5 c'i

Dlf(d)

E

, Bj)

= (Aj

( f , ( A . ) , fj(Bj))

# 0 and

U(c),

L(c) = !d

and

( L ( c ) , U(c))

J =

fi

=

f(d)l

K ,

in

contradicts the fact that U(c)

f

0

let

'd

=

and

i

f =

0

87

g

is a

The proof i s complete.

i s a r e t r a c t of a f i n i t e

I n d e e d , we have no example of a f i n i t e

d i m e n s i o n a l l a t t i c e which i s n o t a r e t r a c t of a f i n i t e d i m e n s i o n a l d i s t r i b u t i v e We know t h a t two f a m i l i a r s e t s of l a t t i c e s a r e

o n e , and we a s k i f a l l a r e .

r e t r a c t s of f i n i t e d i m e n s i o n a l d i s t r i b u t i v e l a t t i c e s . F i r s t , h e r e i s a s h o r t proof of R i v a l and Wille's r e s u l t t h a t l a t t i c e s of

L

say

.

ch [ 1 3 1

f i n i t e w i d t h a r e members of

i s p a r t i t i o n e d by c h a i n s

S = (XI

=

iy

5x

t Lly

L

f o r some

x

f ( x ) = (XI given by

D

is a coretraction.

g(S) =

VS

I(Ci)

f

2

,

is a

L

L

of

X

be t h e l a t t i c e

o r d e r e d by s e t i n c l u s i o n .

L

of

g

g

i s o r d e r embeddable i n

i s t h e l a t t i c e of i n i t i a l segments of

D

to

T h i s i s b e c a u s e t h e map

D

S

D

Let

i s o r d e r p r e s e r v i n g and s a t i s f i e s

is f i n i t e dimensional:

where

Xi.

E

L ,

i s a d i s t r i b u t i v e l a t t i c e and t h e map

D

Recall t h a t

n

is there is a f i n i t e subset

o f a l l f i n i t e l y g e n e r a t e d i n i t i a l segments of Then

b e a l a t t i c e of w i d t h

L

...,Cn .

C1,C2,

f i n i t e l y g e n e r a t e d i n i t i a l segment of such t h a t

Let

given by

of 0

f

n ( I ( C i ) li

L

to

D

idL,

=

1,2,

=

Finally,

... ,n)

,

Ci.

Second, w e have t h i s f a c t . THEOREPI 3.5.

Every 2-dimensional modular l a t t i c e i s a r e t r a c t of 5 2-dimensional

distributive lattice. Recall t h a t a l a t t i c e

Proof. a,b,

i s modular i f f o r a l l

a,b,c

E

L

with

a A ( b V c ) = b V(ahc). Let

L

b e a 2-dimensional modular l a t t i c e and l e t

two l i n e a r e x t e n s i o n s of o t h e r words, t h e map o r d e r embedding. (x,x)

L

and

(y,y)

f

We t a k e

L1

and

L2

whose o r d e r s i n t e r s e c t i n t h e o r d e r o f

L

of

L

x'

to t o be

L1

X

L2

(x,x)

g i v e n by

,

in the distributive lattice

so L1

x' x

f ( x ) = (x,x)

v y' L2,

denote

L.

In

i s an

i s t h e j o i n of

and

x v y

is the

D.Duffus,M. Poutet

88 L ,

join in Let of

b e t h e s u b l a t t i c e of

D

L1

g e n e r a t e d by

L2

X

.

f(L)

Each e l e m e n t

i s some l a t t i c e p o l y n o m i a l e v a l u a t e d on t h e e l e m e n t s o f

D

.

f(L)

We

s h a l l show t h e s e h a v e a v e r y s p e c i a l form. F i r s t observe t h a t f o r

(min

, min

{x,r}

L1

{y,sj)

,

(x,y)

(r,s)

in

L1

(x,y)

L2,

X

with t h e j o i n given d u a l l y .

x

Let

(r,s)

A

is distributive,

x

is e q u a l t o a c o n j u n c t i o n of d i s j u n c t i o n s , e a c h i n

.

Also,

x1

=

(a'

V

b')

V

f(L)

x2

(c'

A

AX

and

If

(i).

= x3

d')

V

dim D = 2 x4

A

f o r some

a

a,b,c,d

x

=

a'

(c'

A

V

x = (c,a)

.

then

a'

x

=

If A

(c'

A

(c,d)

a' v b'

or

a

when

L.

E

and l e t

a'

5

d')

c'

c

a ' v b'

and

d'

V

and x

then

x

x = (a,a)

c,d

(a,b)

=

c

is cornparable w i t h

a'

=

.

x

If

c' = (a,c)

A

x

=

\

I v b

c'

V

(a,d)

=

c'

a'.

.

d' = (c,d)

then

or

d ,

x

x = (c'

Now

a'

V

d')

d'

A

/ I l : b v c )

x = a'

if

x

if

x = a'

A

if

x = a '

A

a' v b'

N o t e t h i l t t h e r u l e s of d e f i n i t i o n f o r It is obvious t h a t

g

f = idL.

0

s e v e r a l casec', a l l s t r a i g h t f o r w a r d .

(1). L e t

5

a' v b'

5 c'

x

5y V

d'

V

L.

a

We

x = (a,b) = or

a'

A

c

lf

x = a'

if

a

g

of

V

d'

1d . D

to

by:

L

a

is noncomparable t c

b

in

L,

b'

where

a

i s noncomparable t o

b

in

L,

(b'

v c')

where

g

d)'

a

1. b

in

L.

may b e assumed m u t u a l l y e x c l u s i v e . g

is order preserving requires

We g i v e a few o f t h e s e i n d e t a i l ; some L.

x = a' v b'

with

5 (c

.

where

To show t h a t

e x h i b i t t h e u s e of t h e m o d u l a r i t y of

a ' , b'

d'

x =

Either

=

V

,

if

=

c'

Then

L .

By s y m m e t r y ,

The p r e c e d i n g c a s e s show t h a t w e c a n d e f i n e a f u n c t i o n

P(X) =

d

where

in

b o t h b e noncomparable p a i r s i n and

i s noncomparable w i t h

d

a'

if

.

(c,d)

then

a

,

a'

=

A

w h i c h , b y symmetry, l e a v e s two cases. x = (a,d)

D ,

x =

Thus

b e noncomparable w i t h

.

a,b

5

b1e h a v e s e v e r a l c a s e s .

or

i s noncomparable w i t h V

X

L.

in

b

b

x = a'

and otherwise,

Let

may a s s u m e t h a t

c

If

, c

(iii).

(a,b)

.

d')

x = c' v d'

then

2

a

Let

x. t X .

i s comparable w i t h

i s noncomparable w i t h (ii).

implies t h a t f o r a l l f i n i t e

f o r some

t h e n , w i t h no less i n g e n e r a l i t y , a

D

L2

e l e m e n t s of

VX

=

Since

D.

C

SO

a, b

and

5C

y = c' v d' , V

d

in

L

.

Then NenCe

Representing ordered sets by chains g(x)

a v b - c v d = g(y)

=

(3).

a' , b '

'

x

Let

a' v b'

c

d' v e'

1. d

in

y

=

since so

b'

and

.

5y g(x)

-

5 c' , with

Then

so

x

=

d' x

a ' , b' =

a'

V

b'

v b'

a'

-<-

a ' v I)'

5c

y

A

c'

=

and

y

5c

-5 c ' ,

(d v e )

c'

=

d'

V

g(y)

=

.

d'

A

= a v b

g(x)

so

=

a v b

,

g(y)

.

(d' v e ' )

,

d

A

A

e'

Then

.

=

Also

.

A ( b ' v c ' ) , where a 2 b i n L , 5 a ' A c ' 5 d ' v e' 5 ( d v e ) ' . A l s o , D i s d i s t r i b u t i v e and a ' 2 b ' , b ' v ( a ' A c ' ) = x 5 y 5 (d v e ) ' 2 ( d v e ) ' . Tlius b v (a A c ) 5 d v e i n L . A s L i s m o d u l a r 2 b , g(x) = a A ( b v c ) = b v ( a A c ) 5 d v e = g ( y ) .

(5).

and

L

(d v e ) '

x

.

with

d'

- c' A

(4). Let where

5y

89

a

d'

Let

x

.

Then

v e'

We c o n c l u d e t h a t

-

y

I.

x = a'

with

(a

A

c)'

i s a r e t r a c t of

D .

I t remains t o be s e e n i f o t h e r c l a s s e s of f i n i t e d i m e n s i o n a l l a t t i c e s

a r e r e t r a c t s of f i n i t e d i m e n s i o n a l d i s t r i b u t i v e ones. for a lattice

L

L

f o r m u l a t e d wliicli a l l o w t h e r e t r a c t i o n o n t o

L

g e n e r a t e d b y t h e d i a g o n a l embedding o f

4.

GAPS

of t h e d i s t r i b u t i v e l a t t i c e

i n t h e p r o d u c t of l i n e a r e x t e n s i o n s .

IN LATTICES OF F I N I T E BREADTH Ch

it s u f f i c e s t o p r o v e i t

C-gap p r o p e r t y a n d t h e s e l e c t i o n p r o p e r t y .

For l a t t i c e s of f i n i t e

We h a v e s e e n t h a t t o show a l a t t i c e i s i n lias t h e

It i s q u i t e l i k e l y t h a t

of h i g h e r dimension, o t h e r a l g e b r a i c i d e n t i t i e s can b e

b r e a d t h , which i n c l u d e t h o s e of f i n i t e d i m e n s i o n , w e a r e h a l f w a y t h e r e .

THEOREM 4 . 1 .

E v e r y l a t t i c e of f i n i t e b r e a d t h has t h e

Recall t h a t a l a t t i c e X 2 L

finite L

has breadth

has s i z e k t o ,2

.

k .

L

lias b r e a d t h

there exists k

X' y X

k

k

if

lX'1

with

C - a property.

i s least such t h a t f o r a l l

5k

and

VX' =VX.

In fact,

i f and o n l y i f t h e l a r g e s t j o i n independent s u b s e t of

Thus, i f

In particular,

L

has breadth dim L

k

then

L

1,

has a subset isomorphic

k .

Our p r o o f o f Theorem 4 . 1 r e q u i r e s s e v e r a l d e f i n i t i o n s a n d a few p r e l i m i n a r y f a c t s (which s h a l l b e a p p l i e d i n t h e n e x t s e c t i o n a s w e l l ) . rhe c a r d i n a l i t y of a p a i r pair

(1A\,IB1)

.

(A,B)

o f s u b s e t s of a n o r d e r e d s e t

We call the pair regular i f

iAl

and

181

P

are b o t h

is the

D. Duffus, M. Pouzet

90

r e g u l a r , o r o n e is r e g u l a r and t h e o t h e r i s z e r o . subpair of

(A,B)

A' 2 A

if

a subgap of

(A',B')

a l l subgaps of

(A,B)

B' 5 B ;

and

.

(A,B)

A gap

Say t h a t

(A',B')

is a

i f both p a i r s are gaps, c a l l

(A,B)

P

of

i s s a i d t o b e minimal i f

h a v e t h e same c a r d i n a l i t y as

.

(A,B)

I t is straight-

f o r w a r d t o show t h a t e v e r y g a p h a s a s u b g a p w h i c h i s m i n i m a l . Call a gap

(A,B)

irreducible i f for a l l subpairs

,

(A',B')

i s a g a p i f a n d o n l y i f i t h a s t h e same c a r d i n a l i t y as

(A,B)

.

(A',B')

Irreducible

g a p s a r e j u s t m i n i m a l g a p s a l l o f w h o s e s u b p a i r s , of i t s c a r d i n a l i t y , a r e g a p s .

Every r e g u l a r , i r r e d u c i b l e g a p o f a n o r d e r e d set i s p r e s e r v e d

LEMMA 4 . 2 .

a chain. -~ Proof.

Let

(A,B)

be a r e g u l a r , i r r e d u c i b l e g a p o f a n o r d e r e d set

x

b e t h e s e t of a l l

A 2

is irreducible,

(A,B)

(A,B)

irreducibility oi -

course

A:

N

5A

A

AU

(CY

B

t h e same h o l d s f o r First, let of

A .

some

c1

<

<

K

and

,K

< K1

y

(A,B)

shows t h a t

for

let

x O = yo

i s least such t h a t

Now, l e t

yy

A

=

<

c(

y

,

B

a

A\Au

z.

Thus

A

P

Of

i n t o an

is t h e i n c r e a s i n g

for a l l

a ,

{yCYIcx<

let

{xolP < a1

and g i v e n

4 x Y- P

a E A ,

and f o r a l l

f3 <

f o r all

{x

t

Alx

k x B-

a)

f o r all

If

x

a.

A

=

UAU.

.

,

and t h a t

a

5 xa

K]

for

b e an

let

xa

=

yy

(The m i n i m a l i t y of

{xCYln< K

Finally, for a l l

Then

x

and

E

dominates a c o f i n a l subsequence of

E

# 0

P\A.

=

-

r e g u l a r s o such a subsequence has c a r d i n a l i t y x

B

i s a p a r t i t i o n of

A',Aa

,.

-

and

exists.)

i s a n i n c r e a s i n g s e q u e n c e of sets.

that

Let

P .

Since

b e a n o n d e c r e a s i n g , c o f i n a l s e q u e n c e of e l e m e n t s

xc $ xa

A,

where

x

-

A ,

-A = -A

Let

O b t a i n t h i s s e q u e n c e as f o l l o w s :

K .

e n u m e r a t i o n of

then

-

where

1A1)

=

.

d u a l l y and o b s e r v e t h a t t h e

P = A II B .

n,

< IAI

.

{xu1c1

T h i s means

Define

5 XI,

Ala

I t i s enough t o p r o v e t h a t

i n i t i a l and f i n a l segment. union of sets

- -n

A".

shows

'B c

B

,

1 {a t

such t h a t

P

E

,

! K)

Aa

.

{AalCY < for all However,

K} (1

K

< K

is

contradicting the fact c1

< b < K

,

xB

E

A\A,

,

so

# 0. Here i s t h e full r e l a t i o n s h i p b e t w e e n t h e

C-gap p r o p e r t y a n d r e g u l a r ,

i r r e d u c i b l e gaps. LEMMA 4 . 3 . g a p of

P

Proof.

Let

subgap of and, t h u s ,

An o r d e r e d set

P

has t h e

C-=property

contains 5 regular, irreducible b e a gap o f

(A,B) (A,B) (A,B)

.

P

i f and o n l y i f e v e r y

m.

and l e t

(A',B')

be a r e g u l a r , i r r e d u c i b l e

By L e m a 4 . 2 , t h e r e i s a c h a i n w h i c h p r e s e r v e s

.

(A',B')

Representing ordered sets b y chains Assume (A

0

,B ) 0

map o f

A

has the

P

C-gap

property. (A,B) ,

be a m i n i m a l s u b g a p o f t o a chain

1'

such t h a t

C

'(Ao)

A.

Let

L

of

f(Ao)

f(B,)

s u b s e t of

of t y p e

p ,

of t y p e

by t h e m i n i m a l i t y of Let

(A

2

,B

2

i s i n j e c t i v e on p

in

in

f(Bo)

.

of

P .

type

and

A2

M.

,

B2

Hence,

Thus,

f

f(A2)

(f(A,),

-

We c o n c l u d e t h a t

i s a g a p of

Let

C .

be a d u a l l y w e l l ordered, c o i n i t i a l

-1

,

(x)

f(Bo)

x c L,

.

is cofinal i n

A

f(Ao)

is a gap of

C

be

and d e f i n e

(A,B)

has type

A1

Now, l e t

Bl and i s minimal,

,

(x,u) .

of c a r d i n a l i t y

f(A2)

f(B2))

and l e t

and choose a w e l l o r d e r e d , c o f i n a l M

(Al,B1)

so

P

t o be an order preserving

.

(Ao,Bo)

b e a s u b p a i r of

)

f

is a r e g u l a r subgap of

(A1,B1)

Then

Take

t h e c o i n i t i a l i t y of

a s e t of r e p r e s e n t a t i v e s of t h e s e t s

similarly.

b e a gap of

(A,B)

(f(Ao), f(Bo))

be t h e c o f i n a l i t y of t h e c h a i n

subset

Let

91

in

L

and

and

and

Then

f

f(B2)

has

is c o i n i t i a l

f(B2)

is a gap

(A2,B2)

is a r e g u l a r , i r r e d u c i b l e gap.

(Al,Bl)

We s h a l l p r o v e Theorem 4 . 1 by s h o w i n g t h a t e a c h g a p o f a l a t t i c e o f

T h i s i s s i m p l i f i e d by t h e

f i n i t e b r e a d t h h a s a r e g u l a r , i r r e d u c i b l e subgap. following observation. Call a gap

(A',B') Let

i r r e d u c i b l e i f i t i s minimal and i f a l l s u b p a i r s

(A,B)

which s a t i s f y b o t h

left

IA'1

i s a gap and

(A,B')

= 1Al

,

are subgaps.

r e g u l - a r i t y o f a p a i r h a v e t h e o b v i o u s meaning a n d , d u a l l y , d e f i n e

right i r r e d u c i b i l i t y a n d

right

I t i s a s i m p l e m a t t e r t o show

regularity.

t h a t a r i g h t i r r e d u c i b l e subgap of a l e f t i r r e d u c i b l e gap is i r r e d u c i b l e . S i n c e a l a t t i c e h a s f i n i t e b r e a d t h i f and o n l y i f i t s d u a l h a s f i n i t e b r e a d t h ,

w e n e e d o n l y p r o v e t h a t e v e r y gap of a l a t t i c e of f i n i t e b r e a d t h h a s a l e f t regular,

left

i r r e d u c i b l e subpap.

P r o o f o f Theorem 4 . 1 . b e a gap of

L.

Let

L

b e a l a t t i c e of f i n i t e b r e a d t h a n d l e t

We may a s s u m e t h a t

(A,B)

I t i s c o n v e n i e n t t o b r e a k t h e proof

finite

,A. and

x 5 SI

.

There is a subgap A'

{xg\B <

i d e a l g e n e r a t e d by Proof o f

(A',B)

of

(A,B)

g e n e r a t e t h e same i d e a l of

,A.

and f o r a l l

Let

{y,la

a < /A1

,

a1

< \ A \3

let

n o t i n t h e i d e a l g e n e r a t e d by

i s minimal.

into several steps.

S i_ L

t h a t t h e ( l a t t i c e ) i d e a l g e n e r a t e d by

for all

is

{x

such t h a t

L,

and

x

E

Llx

F i r s t , recall

5 VX,

A ' = {x,la

f o r some

< IAI}

,

A

does n o t belong t o t h e

c1 < I A l

A.

b e a n e n u m e r a t i o n of

xa = yy

(A,B)

where

{xglB < a }

.

y

Let

xo = yo

is least such t h a t

Such a

yy

Yy

, is

e x i s t s as o t h e r w i s e

D. Duffus,M. Pouzet

92

,

( i x i i 1 3 ': ti} A'

B)

is a s u b g a p o f

{xaIcl < l A 1 )

=

A

and

,

(A,B)

contradicting i t s minimality.

g e n e r a t e t h e same i d e a l i n

and

L ,

Also,

(A',B)

is

a gap.

g.

If

i s singular then t h e r e e x i s t s

lA\

c o f i n a l i t y of

C.

For a l l

cf(lA1)

1 ~ 1 is Let

of c a r d i n a l i t y

i s a minimal gap, is n o t a gap.

(A1,B) \All

A;

xo

xk

E

D{

D;Z

.

Ixkl

B,

ri

. ii

.

B,

Then

g

(Ak+l,B)

c An\{~n-l)

.

and t h e f a c t t h a t

z

A~

(AO,B)

implies

IAhl

= ( A ; ) \ ( ~ ~ uI I)x o j

,

(Al,B)

~

- x ~ ~ v- z ~

given i n such t h a t

,R

.

~

+

x

xk

2

D{

II

.

i s minimal

.

Let

\Ail

,

(Dk,B)

x

~

>

have

is not a gap.

-( s i~n c e t h e j o i n o f

.

B*)

Dk

Now l e t

Ak+l

x

~

=

i s a g a p a n d , by t h e m i n i m a l i t y o f

\%+ll

=

\A(

.

-h a v~e b e e n o b t a i n e d . 1 in

Then

x

f

x

,- z n~

E

zn

~

For

~ v - z k~+

We now d e f i n e a j o i n

L.

~

.-

By ~ t h e p r o p e r t i e s of

w e can choose

An-l,

5

15 k

k ,

n - 1 ,

x ~ - ~z , ~

and t h e f a c t t h a t zk

= /A\

IDk\

Since

is again i n

n

An

i\i

=

We may a s s u m e

x

z

(Do(

Thus, and l e t

"1:.

and

Suppose t h a t

Z k c D~

,

B,

11

Let

Since

have b e e n o b t a i n e d and (A,,B) is a minimal k- 1 Apply s t a t e m e n t t o (\,B) t o o b t a i n t h e set

i t i s a minimal gap w i t h

Choose

,A.

g.

x

i n d e p e n d e n t s u b s e t of s i z e

Dk

DB

E

i s a minimal gap and

(Ai\(xk])

such t h a t

5A.

i s a s u b s e q u e n c e oT

C

p r o v i d e d by

cf(/AOI)

is minimal.

(A;),B)

and

.

IAl

and any e l e m e n t of

given i n

F

is singular.

t h e p r o p e r t i e s g u a r a n t e e d by

,

all finite

have t h e p r o p e r t i e s p r e s c r i b e d i n

Choose

\

=

(q,B)

We c h o o s e

of

m)* f o r

.

/A\

=

i\i

gap w i t h

(A,,B)

C

i s a g a p a n d , by t h e m i n i m a l i t y o f

Suppose t h a t

Then

,

]A]

F y A.

for all f i n i t e

lAl ,

]A\

and l e t

A

=

(Do,B)

with

W),

b e t h e s u b s e t of c a r d i n a l i t y

(AO,B)

Then

o f c a r d i n a l i t y the

D

regular.

A.

i A;

i A'

D

such t h a t

of c a r d i n a l i t y

A'

P r o o f of ,D. s u p p o s e t h a t

Do

,

T h i s i s o b v i o u s when o n e c o n s i d e r s t h a t

lxcllU c I A l j

,D.

2

C

P r o o f of ,C.

,

1Al

v

... v

z

.

+

z n-1

Dn-l tI Dn-1

we u s e t h e p r o p e r t i e s

..., ~ , zn

Finally, pick

% 20

t o choose

Dg

-

~

Representing ordered sets by chains

. ,. v z . I f z k 5 (z, v z1 v . . . z ~ - ~v )(zk+ ,.v z ~ D3; B, a n d xo 5 x1 2 ... - xk-1 , 'k 5 X k - l is a contradiction. Therefore, (zo.zl,. .. ,zn}

such t h a t

zo

p

z1 v z 2 v

v+

V

x.

:

1

This

has breadth a t least

L

+

n

1.

(A,B)

then because

"'

is j o i n i n d e p e n d e n t , and

1Al

Hence,

L .

Zn

n ,

is regular.

c o n t a i n s a l e f t i r r e d u c i b l e gap.

B.

Proof of

zn)

IJe c a n a c c o m p l i s h t h i s f o r a n y

c o n t r a d i c t i n g t h e f i n i t e b r e a d t h of

E.

.. .~ v

'k+ 1

1-1

1

93

Suppose t h a t

(A,B)

d o e s n o t c o n t a i n a l e f t i r r e d u c i b l e gap.

A

c o n t r a d i c t i o n i s o b t a i n e d by c o n s t r u c t i n g j o i n i n d e p e n d e n t s e t s o f a r b i t r a r y f i n i t e size. (Ao,Bo) = (A,B)

Let

and l e t

(A',B ) 0 0

(A',B ) i s n o t l e f t i r r e d u c i b l e t h e r e e x i s t

0

Co

0

that

lCol

,

=

xo t C* n Bl,( 0

(A;,B1)

and l e t

i s a gap.

(A1,B1)

A

1

Since

i s a minimal gap.

is a gap, and {xo}

.

il

(AC,Bo)

minimal,

Moreover,

y A;

and

Because lAll

that

(A,,Bk)

obtain and

Bk+l

( C k , B k + 1)

Let

A,+1

.

5 Bk

such

(Ab,Bl) and

lAOl

such t h a t

is n o t a gap. = (q\(xk])

lCkl

and

1x1, xk

Choose txk}.

1

=

Choose i s a gap,

(A1,Bl)

d e l e t e d from t h e " l e f t "

x

~

-h a v~e b e e n d e f i n e d s o

Apply

i\ t o

(Ak,Bk)

Since t h i s i s not l e f t irreducible there e x i s t

l e f t i r r e d u c i b l e subgap,

(Ai,Bk+l)

C* k

t

Since

II

w e may assume

xk

Suppose t h a t

,

E

with

lCn-ll

such t h a t

=

(A ,B ) n n

so

zn - l - # xn-*

n-1.

zk

statement zn.

V

E

Ck

Finally, pick

a s i n t h e p r o o f of

n,

{zo,zl

~ e c a u s e x n-2'

C

Once

~

An-l

zk

z o E Co

with

1

... z

+ z~k +, * ,

such t h a t

,... , z

zn

enables u s t o choose

z

x zo

~ - ~ .

h+lI

h a s t h e same p r o p e r t i e s a n d

h a s been d e f i n e d i n t h i s manner. xnTl.

,

lAn-ll

w e can again choose

15 k 5

zn

1x

i s a m i n i m a l g a p w i t h no

(A,',Bk+l)

(A,+l,Bk+l)

%

i s a g a p , and

;

Bk+l+

to

'k

=

-

b,I zn

(Ak,Bk)

c o n t a i n s no l e f t i r r e d u c i b l e gap.

(A,',Bk)

=

xo

.

(Ao,Bo)

Suppose t h a t t h e minimal gap

y Bo

B1

does not contain a l e f t i r r e d u c i b l e

(Al,Bl)

g a p as s u c h a g a p would b e a s u b g a p ( p o s s i b l y w i t h s i d e ) of

Since

is n o t a gap.

(Co,Bl)

= (A;)\(xo]) 16

A.

be a s i n statement

~ v- z k~ + l z1

V

z2

Choose and

Cn-1

z

E

n-1

5 AA-1

C

n-1

have been defined V

... v

z

V

...

z

is j o i n independent.

V

for

.

Just

D. Duffus,M. Pouzet

94

The proof of Theorem 4 . 1 i s complete. N.B. The p r o o f s of D and E a r e s i m i l a r and c o u l d be p a t c h t o g e t h e r . For a b e t t e r understanding w e prefered t h i s presentation. I t remains t o be s e e n i f a l l l a t t i c e s of f i n i t e b r e a d t h a l s o have t h e s e l e c t i o n p r o p e r t y . I n t h e n e x t s e c t i o n we p r e s e n t some consequences of t h e C-gap p r o p e r t y and t h e s e l e c t i o n p r o p e r t y .

5.

ch

REPRESENTATIONS I N

C h has a r e p r e s e n t a t i o n by i r r e d u c i b l e

Here we show t h a t every member of

ck.

c h a i n s , and t h a t t h e s e c h a i n s a r e t h e only i r r e d u c i b l e members o f For a s u b s e t

of an o r d e r e d set

X

segment g e n e r a t e d by A s above,

cf(lX\)

and

X ,

[X)

(XI

w e use

P

f o r t h e f i n a l segment g e n e r a t e d by

1x1 ,

and, a s

.

Also,

Pd

d e n o t e s t h e d u a l of

Let

let -

(A,B)

P

--

A

(iii)

C

(iv) Let

p = cf(lB1)

.

5 (C] 5 B,, =" c f ( / A ( ) ,

&z

(A,B)

such t h a t

and

l B B l < IB\

for a l l

The gap

(A,B)

Thus,

, sa

[D)

5

Sa

P

.

which p r e s e r v e s

and l e t

P

,

< Sa'

I n o t h e r words, d A @ p .

A) ,

A = U (Aala <

A

=

(A,B)

c f ( IAI)

ISc,,

B = U (BRIO < Fi)

S a = (Aa,B)

-

n B, # 0 and T R

= A;

sa

and

=

a r e bounding p a i r s .

TB

B'(P)

T R l < TB

TB(a < A

.

and {Aala <

,

\Aa\ < \ A \

,B

and

A* n B R A # 0

TO

=

for a l l

Say

B'(P)

to

f(Sa) = xa

{xa(a <

A1

P

such t h a t and

f(X,Y)

f ( T R ) = yB

i s t h e image of

A

=

E

< a' < A

i s a s u b c h a i n of

X* n Y,

for a l l cf( /A/)

,

a

for a l l

We have t h e f o l l o w i n g

of a l l bounding p a i r s of

< p}

(A,BB)

and

B'(P)

P :

l? < P ' < p . of t y p e

Use t h e s t r o n g s e l e c t i o n p r o p e r t y t o o b t a i n an o r d e r p r e s e r v i n g map of

A1

P < p.

< A,

CY

-

and

Then t h e r e

A* ;

cf(lB1)

a r e t r a c t of

c o m p a r a b i l i t i e s i n t h e o r d e r e d set < TO

5 ="

i s minimal, s o t h e p a i r s

a r e n o t gaps. t h a t i s ,

so

infinite.

IBI

There a r e s t r i c t l y i n c r e a s i n g sequences of s e t s

{BR(B < p )

6.

B Dd

b e a minimal gap of

(A,B)

and

\A\

r e g u l a r , i r r e d u c i b l e s a p then

&

C 03 D

and

and

with

P

satisfying:

P

i s g p ~ g P ;

(C,D)

(ii)

I n a d d i t i o n , if

Proof.

&

D

and

strong s e l e c t i o n p r o p e r t y

be an o r d e r e d set w i t h t h e

C

(i)

cf(lX()

P .

a minimal gap of

e x i s t chains

X.

i s t h e c o f i n a l i t y of t h e c a r d i n a l number

a c h a i n , it i s t h e i n i t i a l o r d i n a l o f c a r d i n a l i t y

LEMMA 5.1.

t o denote t h e i n i t i a l

f o r a l l bounding p a i r s

a < A,

B < p .

f

(X,Y)

Note t h a t

s o e i t h e r i t c o n t a i n s a maximum

.

Representing ordered sets b y chains

A.

element o r i s isomorphic t o a maximum element t h e n

x

E

The former i s i m p o s s i b l e :

An

Y

c 2

that

ud,

D ^=

A ,

<

Iy

!x

=

implies

12'

< A)

1U

if

x

, meaning

.

x E A" ri B, Y D = iy818 < p } such

and

,

xci < x

were such

Y

B < 8'

and

implies

.

Ye

I-

C

u <

for a l l

B,

I'

U

Now r e f i n e t h e s e s e t s t o o b t a i n

YI;'

95

P r o o f s of ( i i ) and ( i i i ) a r e now r o u t i n e . Assume t h a t and

1B/

-x P\A.

s e t of a l l -

and

B =

(A,B)

such t h a t

P

E

-

Then

2

(C]

P

a s i n t h e proof o f Lemma 4 . 2 .

1A

[I

-

and

A

A

segment; moreover,

1

(XI

A. ,

<

B

partition

B,

2

A

5

B

-

x u $ Ail

a

if

p,

C

and

# 0

A\xa

A =

for a l l

dually,

c

B

g(x) = xu for a l l

< A

a

i s l e a s t such t h a t

xcl.

x E

( g ( A ) , 0)

A

of

g

to

it w i l l follow t h a t

C dj D

i s a gap o f

< A,

i s a r e t r a c t of

Aa =

let

-

Aa

i s an i n i t i a l segment and xa

and

E

xB

increasing.

Aa

f o r some

Therefore, x

If

6

k x

if

With an i n c r e a s i n g

C .

then

a < A.

Since

A = U(xala <

{x

E

Aly

preserving

P

A\(xa]

E

A\Ta # 0

B

(A,B)

implies

x

and

D ,

.

2 y} .

-

Then

Aa

-

5 AB i f c1 < B . It i s a l s o obvious t h a t 8 - r e c a l l t h a t e l e m e n t s of C a r e s t r i c t l y

c1 <

E

-

A

).

1 (y c

x

A/y

E

5 x}I

Fa =

A:

y

E

.

< IAl

A\Ta # 0

A

Thus

{y E A l y

x

we know t h a t

B,,

fi

Finally,

a l l o w s u s t o choose

A* rr B,

Aa,

by

C

The c o n d i t i o n

sequence of f i n a l segments s a t i s f y i n g t h e d u a l c o n d i t i o n s f o r

c1

E

Suppose w e have such a

We can then d e f i n e a r e t r a c t i o n

insures that

For a l l

-

xu

-

A.

where ci

of i n i t i a l

< A) ,

-

sequence f o r

=

.

{Lala < A }

u (Aala

< A.

CL

-

5 A*

[D)

-

satisfying these conditions:

P

B

i n t o an i n i t i a l and f i n a l

P

and

=

Let

Y

define

It i s enough t o o b t a i n an i n c r e a s i n g sequence

segments of

-A1Ai be tAh e - -A , let A

i s r e g u l a r and i r r e d u c i b l e ; i n p a r t i c u l a r ,

We p a r t i t i o n

p .

=

E

5 XI 5

xa.

because t h e f a c t t h a t

y

such t h a t

$ xa

.

Clearly,

y

6

-

Aa.

Hence, ( i v ) h o l d s and t h e proof o f Lemma 5 . 1 i s complete. THEOREM 5.2. Proof.

Let

Every member of P

E

Ch.

o r d e r e d s e t s such t h a t r e t r a c t of

P ,

Ch

has 5

representation

&

We must show t h a t t h e r e i s a f a m i l y

P

and e a c h

i s a r e t r a c t of

P.

II(Pi/i

E

I)

,

i s an i r r e d u c i b l e c h a i n .

c h a r a c t e r i z e d by [ 3 , P r o p o s i t i o n 5.71 : 2, a , 6d , o r i f C has order type

1,

a chain CY

@

Bd

C

irreducible chains. (Pili each

E

P.

I)

of

is a

These c h a i n s a r e

i s i r r e d u c i b l e i f and o n l y

where

a

and

a r e regular

ordinals. Our proof c l o s e l y p a r a l l e l s t h a t given f o r Theorem 3.1 ( c f . [ 1 3 , Theorem 11) s o w e u s e t h e n o t a t i o n found t h e r e , w i t h

P

replacing

D ,

and s u p p r e s s t h e

D.Duffus, M. Pouzet

96 details.

Let As

P

i(Ai,B,)li

Clt,

E

P

I}

C

b e t h e set o f a l l r e g u l a r ,

i s a l a t t i c e w i t h t h e s e l e c t i o n p r o p e r t y (Theorem 2 . 5 ) ,

h a s t h e s t r o n g s e l e c t i o n p r o p e r t y [ 7 , P r o p o s i t i o n 31.

Ti

a retraction

g a p of Ci

P

of

i E I

for all

Ci

onto a chain

'i

. P

so

By Lemma 5.1, t h e r e i s (fi(Ai),

is a

fi(Bil)

P a r t s ( i i i ) a n d ( i v ) o f Lemma 5 . 1 show t h a t e a c h

is irreducible. x c P

For

f

P

of

K b e g i v e n by

IT

and

B(c)

L(c)"

=

I

IS.(c) 3

B(c)

E

rr.

L(c) in

P

P.

of

K

(21x

With to

E

P)

g

LJ

g

(Ci/i

E

U(c)

B(c) f 0

i s a gap of

. C

in

Then

c

B(c)

and

f(x)i,

is a

In t h e l a t t e r case, u s e

P.

f(L(c))"

E

5

ix c P I C

=

irreducible

f(U(c)),

I)

in

K ,

This i s impossible and, thus,

j '

c E K .

d e f i n e d by

P and

,

a n d Lemma 4 . 3 t o o b t a i n a r e g u l a r ,

By t h e s e l e c t i o n p r o p e r t y o f to

cl

Then e i t h e r

( L ( c ) , U(c))

for all

B(P)

'<

{x E Plf(x)

=

c I)

n(cili

x

f = f . .

0

P .

f . ( A . ) * n f.(B.), J I J J

E

P)

( L ( c ) , Ll(c))

of

( A , , Bj)

= II(&\X F

and

U(c), or

[x)

t o b e t h e c h a r a c t e r i s t i c f u n c t i o n of

to

let

p r o p e r t y of

C-gap

subgap

f

=

II

P,

bound s e t o f the

f

0

c t K,

For a l l

fx

we again define

a n d l e t t h e map

so

.

such t h a t

P

i r r e d u c i b l e gaps o i

0

g(c)

f = idp.

I)

P , =

t h e r e i s a s e l e c t i o n map

,

h(B(c))

P

Hence,

h

of

B(P)

i s a n o r d e r p r e s e r v i n g map

g

i s a r e t r a c t of

K

P

is t h e d e s i r e d r e p r e s e n t a t i o n of

and

by i r r e d u c i b l e

chains.

Ch h a s a r e p r e s e n t a t i o n by i r r e d u c i b l e c h a i n s

An i r r e d u c i b l e member of

a n d must b e a r e t r a c t of o n e of t h e s e c h a i n s . COROLLARY 5.3.

6.

of

Ch a r e t h e i r r e d u c i b l e c h a i n s .

REMARKS AND PROBLEMS For

P

E

minimum number dim,

The o n l y i r r e d u c i b l e members

P

Ch

,

k

d e f i n e t h e r e t r a c t dimension of such t h a t

P

P ,

dim

i s a r e t r a c t o f a p r o d u c t of

bounded a b o v e by some r e a s o n a b l e f u n c t i o n

of

P ,

k

D

is the

3

(dim D ) L .

For i n s t a n c e , l e t

P =

ith u n c o u n t a b l e o r d i n a l , a n d l e t

wl D =

X

P

w2 ti4

X

d

P

chains.

dim P ?

D with

It i s e a s y t o c o n s t r u c t examples of d i s t r i b u t i v e l a t t i c e s dim,

t o be the

...

.

X

wn,

where

D

i s an

Then

w

2

i

.

d i s t r i b u t i v e l a t t i c e whose r e t r a c t dimension is a t l e a s t n d T h i s i s b e c a u s e t h e r e a r e g a p s o f o r d e r t y p e ( a i , oj) f o r i , j = l,Z,.. . , n .

n-dimensional

Representing ordered sets by chains 3. :'in o r d e r y r c s c t - v L n g

i-,i~.\i

1)

crf

Also, any c o r r t r a c t i o n

gaps.

a l l g a p s 01 projcctions.

2

c h i n c a n p r e s e r v e a t most o n e ot t h e s e

to a

f

r;ll;i

of

I)

w i t l i its projvctiuns

U

In f a r t ,

(Lemmas 3 . 2 a n d 3 . 3 )

1) = n -'>

dim and

D

91

.

t o a * p r o d u c t of c h a i n s must p r e s e r v e lias a t l e a s t

hence,

i

s because

L)

c a n b e embed

7y,, ,!

in

.,f

1

s o t h a t a l l g a p s oi

D

n2

allows s e l e c t i o n

d

1 i,j

..n)

1, 2 , .

=

3

a r e preserved.

1t seems r e a s o n a b l c t o a s k :

&

D

dim

(dim

for a l l finite

d i m e n s i o n a l .d i s t r i b u t i s 1 a t t i c e s ' ? Tliere a r e u t l i e r i n t e r e s t i n g q u e s t i o n s c o n c e r n i n g f i n i t e d i m e n s i o n a n d

P o u z e t a n d R i v a l [ l o ] h a v e shown t h a t

retraction.

a l l ( l a t t i c e ) homomorphic

i m a g e s oi i i n i t e d i m e n s i u n a l c o m p l e t e l a t t i c e s a r e r e t r a c t s .

Can t h i s b e

I s e v e r y homomorphic _ image oi 5 ___ f i n i t e dimensional l a t t i c e _-

extended?

_ r e t_ ra_ ct of

I> 5

The u s u a l c o u n t e r e x a m p l e s a r e f r e t . l a t t i c e s , w h i c h a r e

L '?

inf iniLe dimensional. Tlic c l i a r a c t e r i z a t i o n o f o r d e r e d s e t s w i t h t h e

C-gap p r o p e r t y p r o v i d e d

by Lemma 4 . 3 l e a d s t o some i n t e r e s t i n g o b s e r v a t i o n s . F i r s t , c o n s i d e r t h i s r e s t a t e m e n t o f Example 2 . 1 :

i s an e a s y r e a s o n .

J

Let

d o e s n o t h a v e an i r r e d u c i b l e s u b g a p .

A'

gap and

is not irreducible.)

C-gap

property;

hence,

J f

Lemma 4 . 3

Here

b e t h e s e t of

J

J

which

t h e r e i s an

(A',@)

(A",@)

Thus,

.

ch

i s a g a p of

(A,@)

( F o r any subgap

A"

(A',@)

Then

such t h a t V A " C 5 .

i n f i n i t e subset

of

A

be s u c h a n i d e a l , a n d l e t

a l l s i n g l e t o n s u b s e t s of t h e countable s e t .

have t h e

n o n p r i n c i p a l maximal

t h e power s e t o f a c o u n t a b l e s e t , d o n o t b e l o n g t o

ideals i n

shows t h a t

is not a

does not

J

Ch.

T h i s c a n b e e x t e n d e d t o an a r b i t r a r y i n f i n i t e c a r d i n a l

ZK

i n t e r s e c t i o n o f n o n p r i n c i p a l maximal i d e a l s o f

no f i n i t e

:

K

h a s t h e gal' p r o p e r t y .

LK

leads u s t o ask for 5 simple description o k i d e a l s i n

This

which have t h e

C wp r o p e r t y . S e c o n d , l e t u s c o n s i d e r Example 2 . 2 : segments of 1'

w x wl

t l i e l a t t i c e of a l l p r o p e r i n i t i a l

Ch.

does not belong t o

'This c a n b e g e n e r a l i z e d .

T

b e a n u p d i r e c t e d o r d e r e d s e t w i t h n o maximum e l e m e n t a n d l e t

l a t t i c e of p r o p e r i n i t i a l segments of

P

i f and o n l y i f

P .

contains a cofinal chain. P

t h e l a t t i c e of i n i t i a l segments of

=

w

T

Then

X

has t h e

Let

be the

C-gap

property

T h i s c h a r a c t e r i z a t i o n shows t h a t

w1

does n o t have t h e

C-gap

property. I t is f a i r l y s t r a i g h t f o r w a r d t o prove t h e e q u i v a l e n c e . that

T

i s a meet-complete

Also, a p a i r that

P

(A,0)

is a gap o f

has a c o f i n a l c h a i n

regular ordinal.

Let

l a t t i c e so all g a p s o f

(A,@)

C ;

T

T

i f a n d only i f

w e may a s s u m e

b e a gap of

T.

F i r s t observe

a r e of t h e form

U(1lI

€

C = ix 1 ~ t For

u

C

A,

A)

X)

=

P .

(A,@)

Suppose

is a

choose

.

Iu

E

A

D.Duffus, M . Pouzet

98

xa

such that

C

Ia.

irreducible subgap of the

C-gap property.

With (A,@)

Aa

=

{(xg]/xa

.

= { Z U ( ; i ,:

xp)

41

,

(A',@)

is a regular,

To prove the converse, suppose that

Then the gap

regular, irreducible subgap let

A'

(A,@)

(A',@)

Then

.

,

Say

(Au,@)

with A' =

A =

{(XI

[x

i:

{(xLY]pY < A } .

is not a gap, so

elements o f

For

1 ix[i1

This observation makes it easy to construct a cofinal chain f o r

T

PI ,

P

has

has ti

':

-

'\,

'1

',

.

with

tx,/a < A}.

REFERENCES

B. BANASCHEWSKI and G. BRUNS, Categorical characterization of Mac Neille completion. Archiv. der Math. Base1 1 8

( 1 9 6 7 ) , 369-377.

DILWORTH, R.P., A decomposition theorem for partially ordered sets, Ann. Math. 51 ( 1 9 5 0 ) , 161-166. DUFFUS, D. and RIVAL, I., A structure theory for ordered sets, Discrete Math. 35 ( 1 9 8 1 ) , 53-118. DUFFUS, D., RIVAL, I., and SIMONOVITS, M., Spanning retracts of a partially ordered set, Discrete Math. 32 ( 1 9 8 0 ) , 1-7. E.W. KISS, L. MARKI, P. PROHLE, W. THOLEN, Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectvity. Seminarberich, no 1 1 - 1982, Fern Universitat. NEVERMANN, P., A note on axiomatizable order varieties Algebra Universalis Vol. 17 No I , ( 1 9 8 3 ) . NEVERMA", P. and WILLE, R., The strong selection property and ordered sets of finite length, preprint ( 1 9 8 1 ) . R. NOWAKOWSKI and I. RIVAL, The smallest graph variety containing all Discrete Math. 4 3 ( 1 9 8 3 ) , 223-234. paths

.

POUZET, M. and RIVAL, I., Every countable lattice is a retract of a direct product of chains, Alg. Univ. (to appear). POUZET, M. and RIVAL, I., Quotients of complete ordered sets, Alg. Univ. (to appear). A. QUILLIOT, Homomorphismes, points fixes, retractions et jeux de poursuite dans les graphes, les ensembles ordonnLe et les espaces mLtriques. ThSse de Doctorat d'Etat, juin 1983 - PARIS. RIVAL, I., The retract construction, in: RIVAL, I. (ed.), Ordered Sets (Reidel, Dordredcht, Holland) ( 1 9 8 2 ) . RIVAL, I., and WILLE, R., The smallest order variety containing all chains, Discrete Math. 35 ( 1 9 8 1 ) , 203-212.

Annals of Discrete Mathematics 23 (1984) 99-102 0 Elsevier Science Publishers B.V. (North-Holland)

ORTHOMODULAR

99

LATTICES

Giinter BRUNS Department of Mathematical Sciences Mc MASTER UNIVERSITY HAMILTON, Ontario Canada

Dedicated to Professor COROMINAS.

RESUME.L'article general [ 7 1 de S.S. Holland et le livre [81 de G. Kalmbach constituent actuellement une bonne introduction B la theorie des treillis orthomodulaires. Le present article est consacre B quelques points precis de cette theorie qui ne sont pas developpes dans ces deux ouvrages, en particulier au sujet des blocs et des classes equationnelles. Dans le dernier paragraphe sont presentees plusieurs conjectures voisines relatives aux classes equationnelles de treillis orthocompl6mentes modulaires. Nous faisons le point sur les resultats obtenus jusqu'ici sur ce sujet et nous donnons les directions de recherche actuelles. There are two sources available now to get a convenient introduction into the theory of orthomodular lattices. The first is the survey article by S. S. Holland [ 7 1 , the second is the book "Orthomodular Lattices" [81, which is about to appear. Holland's brilliant article gives an excellent description of the origins of the theory of orthomodular lattices in the theory of von Neumann algebras and the early development of the theory. Kalmbach's book gives an up-to-date account of practically all aspects of the theory. Because of this I will concentrate in this short article on very few aspects of the theory which have not been covered by these sources.

1.- DEFINITIONS AND EXAMPLE. An orthocomplementation on a lattice L with bounds 0 , l is a unary operation a -+ a' which satisfies a < b * b' < a ' a" = a a h a ' = O . a v a ' = l , ortholattice mentation as an where (L,v , A ) An orthomodular law: An

(abbreviated: OL) is a bounded lattice L with an orthocompleadditional operation, thus it is an algebra (L, V, A , I , 0, l), is a lattice with bounds 0, 1 and ' is an orthocomplementation. lattice (abbreviated: OML) is an OL satisfying the orthomodular then a v (a' A b) = b if a
.

The prime example of an OML is the lattice C ( H ) of all closed subspaces of a real or complex Hilbert space H. A s a pure lattice its structure does not seem to be rich enough to give rise to an interesting theory. It has none of the interesting properties commonly investigated in lattice theory. It is

distributive if and only if H is one-dimensional and it is modular if and only if H is finite-dimensional, both no doubt interesting properties, but not properties which reflect the specific difficulties which distinguish Hilbert space theory from pure linear algebra. The lattice C ( H ) becomes interesting 1 if one considers the additional unary uperation A + A which attaches to every

A E

f(H)

its orthogonal complement'A

. It follows from the very basics

of Hilbert space theory that this is an orthocomplementation. The fact that A v 'A

=

H

of the form

follows from the basic fact that every f

=

g

t

h

with

g E A

and

h E 'A

subspace B 3 A then h = f - g 6 B , hence 1 A v (A A B) = B , the orthomodular law.

f E H

has a representation

.

If f belongs to some closed 1 f E A V (A A B), which implies

A remark should be made about the choice of the word "orthomodular lattice". It is good! At first glance the orthomodular law seems to be a weakening of the general modular law , a < b * a v ( c A ~ =) ( a v c ) ~b from which it is obtained by replacing c by a' . For this reason it has been suggested to call OMLs "weakly modular lattices". It should be pointed out here that the orthomodular law is not a weakening of the modular law, it is a property of the orthocomplementation. It is easy to give examples of bounded lattices L which admit two orthocomplementations in such a way that it becomes an OML with one of the orthocomplementations but not with the other. Also, as G. Kalmbach [ 9 ] has shown, the orthomodular law does not imply any lattice equations, i.e. equations not involving the orthocomplementation.

2.BLOCKS. The early development of the theory of OMLs, as described in Holland's article, had close ties with it's "mother-theory", the theory of von Neumann algebras. The questions asked about OMLs were mainly those suggested by this theory. This changed abruptly with Greechie's work in the late ~ O ' S , in which he constructed many examples far removed from the theory of von Neumann algebras and therefore called "pathological". (They are not at all pathological if one does not restrict one's attention to van Neumann algebras. Greechie' examples include, for example, all projective planes which admit an orthocornplementation). Greechie's basic idea is the following. Every element a of an OML L is contained in the Boolean subalgebra {a, a', 0, 11 of L. Since the set of all Boolean subalgebras of L is clearly inductive, it follows that every a E L is contained in a maximal Boolean subalgebra or block of L. Thus every OML L is covered by the set A ( L ) of all it's blocks. Greechie's highly successful idea was conversely to build up 0.MLs from Boolean algebras. He developed a very suggestive graphical representation of the OMLs obtained by his constructions and was able to refute many earlier conjectures about OMLs. His methods are described in detail in Kalmbach's book to which the reader is refered for details. Greechie's studies of the interaction of blocks in an OML were taken up in two papers [ I , 2 1 by the author. The techniques developed in these papers were recently used by M. Roddy [ l o ] to prove the following remarkable result:

THEOREM -

Every OML with finitely many blocks is the direct product of a Boolean algebra and finitely many simple OMLs.

3.- VARIETIES. Early work on varieties of OMLs has been done by Bruns-Kalmbach [ 4 , 51 . Interesting recent work about varieties of OMLs, in particular the finite basis problem, has been done by R. Godowski. I would like to mention here a particularly puzzling open problem concerning varieties of

Orthomodular lattices

101

modular OLs. For n > 2 let MOn be the modular OL consisting of 2n pairwise incomparable elements and the bounds. Let M01 be the two-element Boolean algebra and let MOO be the one-element OL. It is then easy to see that the varieties [MOn] generated by the MOn form an increasing sequence and it follows from B. Jonsson's celebrated lemma that the variety [MO(n+l)] covers the variety [MOn]. It furthermore follows from a result by R. Baer regarding polarities in finite projective planes that the MOn (1 < n < w ) are the only finite, subdirectly irreducible modular O L s . This seems to suggest the following.

all

CONJECTURE I. [MOn] (0 < n < w )

N

Every variety k of modular O L s which is different from contains all MOn.

This conjecture can be broken up into a sequence of conjectures. CONJECTURE I in

Monl

.-

N

Every variety k of modular OLs which is not contained

contains MO(n+l)

.

For n = 0 this conjecture is trivially true, for n = 1 it was proven by Greechie (unpublished). The proof requires only basic knowledge of the commutator. In a recent paper [31 I proved the conjecture for n = 2 . It was shown in this paper that the conjecture I is equivalent with the following statement. (S ) If L is a subdirectly irreducible modular OL generated by three elements n a, b, c satisfying a < b and a A c' = a ' A c = b A c = b' A c' = a' A b A c' = b' A (a V c ) = 0 then MO(n+l) E [L] .

If in this statement the conclusion MO(n+l)F [ L ] is replaced by "[L] contains (the lattice of subspaces of) a projective plane", then the corresponding statement ( S ' ) is equivalent with the CONJECTURE 11.- Every variety of modular O L s which is different from all [MOn] (0< n < 0 ) contains a projective plane. It is not difficult to see that conjecture I1 implies conjecture I. It was pointed out to me by A. Day that ( S ) can be reformulated in such a way that it gets a purely geometric flavour. The relation between a, b, c in ( S ) express exactly the fact that the elements a, b', a' A b, c form a spanning 3-frame. By Frink's embedding theorem [61, every subdirectly irreducible modular OL L can be embedded into a projective geometry. If this projective geometry has (projective) dimension 2, L is a projective plane and hence the conclusion of ( S ' ) is satisfied. If it has dimension at least 3 then it is Arguesian and hence L is Arguesian. By Jonsson's coordinatization theorem there exists a (von Neumann) regular ring (with involution) R such that L is isomorphic with the lattice of all finitely generated submodules of the right R-module R3 . Thus in ( S ) and ( S ' ) it may be assumed that L is one of these more concrete lattices. M. Roddy has started investigating ( S ) and ( S ' ) under this new point of view, but final results are not yet available.

G. Bruns

102

BRUNS, A finiteness criterion fox orthomodular lattices, Canadian Journal of Mathematics 30, (19781,

1.-

G.

2.-

_-------, Block-finite orthomodular lattices, ibid. 31 (1979) 961-985.

3.-

_-__--__ , Varieties

4.-

__---_-and

5.-

-__--_-,

6.-

0. FRINK,

7.-

S.S.

8.-

G. KALMBACH,

9.-

-__--_---_, Orthomodular

10.-

of modular ortholattices, Mathematics ( 1 9 8 3 ) .

Houston Journal of

G. Kalmbach, Varieties of orthomodular lattices, Canadian Journal of Mathematics (1971),

Varieties of orthomodular lattices 11, ibid. (1972) 328-337.

Complemented modular lattices and projective spaces of infinite dimension, T r a n s . AMS 60 (1946), 452-467. HOLLAND, The recent interest in the theory of orthomodular lattices, in: Trends i n lattice theory (Ed. Abbott), Van Nostrand-Reinhold, New York (1970).

Orthomodular Lattices, Academic Press

(to appear).

lattices do not satisfy any special lattice equation, Arciv der Mathematik.

M. RODDY, An orthomodular analogue of the Birkhoff-Menger Theorem, Algebra Universalis (to appear).

Annals of Discrete Mathematics 23 (1984) 103-118 0 Elsevier Science Publishers B.V.(North-Holland)

103

THE CONSTRUCTION OF SOMF FREE m-LATTICES ON POSETS

George G G T Z E R and David KELLY Department of Mathematics University of Manitoba W i nn i peg, Man i t o ba CANADA Dedicated to E . Corominas

11 est bien connu que les treillis peuvent letre considgrgs, soit comme d e s ensembles ordonnes, soit comme des algsbres universelles. L'approche alg6brique

garantit l'existence d'un treillis libre sur tout ensemble ordonn6 ; c'est-2-dire du treillis engendr6 "le plus librement possible" B partir de l'ensemble ordonn6. Beaucoup des ensembles ordonn6s qui apparaissent naturellement dans les differrntes branches des math6matiques ant une structure de treillis. En fait ce sont habituellement des treillis complets. Cependant, quelques techniques alg6briques ne s'appliquent plus directement aux treillis complets. En particulier,

il n'y a pas de treillis complet libre sur un ensemble B trois 6lQments. Le concept

B la th6orie de l'ordre, tandis que la notion de treillis

de complgtude appartient

libre sur un ensemble ordonne provient de l'approche alggbrique. L'inexisterice de treillis complets libres montre qu'on doit-gtre prudent en essayant de combiner ces deux concepts. Nous rgsolvons cette difficult6 en affaiblissant le concept de compl6tude. Soit m

un cardinal r6gulier infini. Un treillis m-complet ou m-treillis

est un ensemble ordonnQ dans lequel toute partie non vide ayant mains de m.El6ments possede une borne supgrieure et une borne infgrieure. Chaque concept de la thgorie des treillis s'dtend en un concept correspondant dans la th6orie des m-treillis. On omet

m

quand

m

=

K,

car c'est de la th6orie des treillis. I1 est clair

qu'un treillis complet est m-complet pour tout m que l a cardinalit6 d ' u n ensemble ordonnh existe dans

P

. De

plus, si m

P , alors toute borne supgrieure qui

est la borne sup6rieure d'une partie non vide de

dinalit6 est inf6rieure 2

m

est plus grand

P dont la car-

; en bref toute borne sup6rieure est une m-borne

sup6rieure. Cependant, i l existe des m-treillis qui sont complets et dans lesquels toute borne sup6rieure n'est pas une m-borne supgrieure. Soulignons que n o u s prenons uniquement les bornes sup6rieures et inf6rieures d e parties non vides. Bien que les m-treillis aient des opgrations infinitaires, le nombre d e ces opgrations e s t born6 de sorte que les concepts et les mdthodes de l'alggbre universelle s'appliquent. Le point important est qu'il existe un ensemble (non une classe) qui indexe simultan6ment toutes les op6rations

G. Gratrer, D. Kelly

104

bornes superieures et inf6rieures dans chaque m-treillis. Une cons6quence particuliGre est l'existence, p o u r tout ensemble ordonne

P

de

Fm(P),

le

P. F,(P) est l e m-treillis qui contient P comme S O U S m-treillis libre sur ensemble ordonne pour lequel les m-bornes sup6rieures et m-bornes infcrieures Fm(P)

sont formees "aussi librement clue possible". (En particulier,

P

dr6 par

). Plus

application isotone d e longement B

F (P)

e s t m-engen-

F ( P ) est l'unique m-treillis pour lequel toute m dans n'irnporte quel m-treillis L a un unique pro-

prgcisgment, P

qui est un

m-homomorphisme.

Un ensemble ordonne est m r n s'il ne contient aucun d e s trois ensembles I 1 5 2 3 + + ; I , "v + 2. , 'L + 2. . I. RIVAL et R. WILLE ont dGcrit 'L le treillis libre sur tout ensemble ordonne mince. Nous avons Etendu ce resultat

ordonnes suivants

~

F (P) p o u r tout ensemble ordonn6 P . Comme d a m m l'article de I. RIVAL et R. WILLE l'argument cl6 dans la description est la d e s -

aux m-treillis en d6crivant

cription du m-treillis libre sur l'ensemble o r d o n n e

< a2 < a l ,

definie par .a denombrable, m

=

He)

.

F

m

bo

< b2 < b , ,

.a

H = {ao,a, ,a2,bo,bl,b2}

< bl

bo

< at .

Pour

m

non

(H) est un treillis complet (ce qui n'est p l u s vrai lorsque

Dans cet article de synthsse, n o u s decrivons F (H) m

et

Fm(H)

.

C o m e le m-treillis

est assez compliqu6, il ne peut pas stre appr6hend6 immgdiatement. Nous

donnons beaucoup de diagrammes pour expliquer l e s formules donnant les bornes

sup6rieures et inferieures dans

F,(H)

.

Une motivation intuitive est aussi donnee

p o u r beaucoup de ces formules. La preuve que le rn-treillis dgcrit est

Fm(H)

est

publi6e ailleurs.

Abstract

Let an

rn be an i n f i n i t e regular c a r d i n a l . We c a l l a Loset m-ccnplete l a t t i c e (or m l a t t i c e ) i f every nortenipty

subset with fewer thsn m elements has both a j o i n and a nieet. This concebt was introduceu by P. Crawley and h.A. Lean. For m - l a t t i c e s , enough of the a l g e b r a i c f l a v o r of the usual ( f i n i t a r y ) l a t t i c e case has been retained so t h a t Lie there are f r ee m - l a t t i c e s generated by any poset. s h a l l d e s c r i b e the f r e e m l a t t i c e on a p a r t i c u l a r 6-element poset 11, and i n d i c a t e w h y t h e d e s c r i p t i o n is c o r r e c t . l'he poset ii is a n a t u r a l example because i t s f r e e m-latt i c e can be used t o describe t h e f w e m - l a t t i c e on many other posets. IHlHOCUCTIOII

I t is well known t h a t l a t t i c e s can be considered e i t h e r a s ordered s e t s or a s universal algebras. The a l g e b r a i c approach guarantees the e x i s t e n c e of t h e f r e e l a t t i c e on any p s e t ; t h a t i s , the l a t t i c e t h a t is kenerated "as f r e e l y a s possibletr s t a r t i n g from t h e poset. ( I f no ordering is specif'ied on t h e gerlera t i n g p o s e t , 2n antichain is undtrstood. )

105

Some free m-lattices on posets

Eiany of t h e ordered s e t s t h a t n a t u r a l l y occur i n various f i e l d s of n:stkienl&t i c s have a l a t t i c e s t r u c t u r e . I n f a c t , they a r e usually com.~lete l a t t i c e s . llowever, sorile e l g ~ b r a i ctechniques no l o n t e r apply d i r e c t l y t o complete l a t t i c e s . I n p a r t i c u l a r , t h e r e is no f r e e complete l a t t i c e on t h r e e elen,ents. If such a complete l a t t i c e d i d e x i s t , then every coicblete l a t t i c e L w i t h t h r e e g e n e r a t o r s would be a complete homoniorpliic image of i t ; i n p a r t i c u l a r , t h e r e would be ari upper bound f o r t h e c a r d i n a l i t y of any such L. llowever, E'iLure 1 shows a 3-generated cornblete l a t t i c e of any given c a r d i n a l i t y . (In t h i s f i t u r e , the t h r e e conlplete generators a r e denoted by squares. ) O f c o u r s e , any diagrem of an i n f i n i t e poset r e q u i r e s i r i t e r p r e t a t i o n . Ari extreme p o s i t i o n is t h a t any such diaLran, is u s e l e s s . h'e a r e not of' t h i s opinion, b u t we do advise extreine c a r e , e s p e c i a l l y i n a proof, when r e a d i n t information fron, a ciisgran; of a i i n f i n i t e poset. b'or example, it is not obvious when an element is j o i n i r r e d u c i b l e . (F'or a f i n i t e l a t t i c e , j o i n i r r e d u c i b l e elements a r e those w i t h a unique lower cover.)

The left-hand chain C i n Figure 1 can be ariy well-orderec chain t h s t bias a l a r b e s t element, denoted by t . The r i ~ h t - h a n d choin is isomorphic t o C , arid t h e r e m i n d e r of t h e poset is a l i n e a r sunl of f i n i t e l s t t i c e s over C . For eacki i t i n C , t h e poset of Figure 1 tias an i-th l e v e l c o n s i s t i n g of' 6 elenients. i < t can only be generatea a f t e r , ari Observe t h a t an elenerit on any l e v e l element on every lower l e v e l has beeti t e n e r a t e d . The concebt of completeness belongs t o order theory. while f r e e l a t t i c e s or1

FIGURE 1 . An a r b i t r a r i l y l a r g e c o m p l e t e l a t t i c e

POSetS come from t h e a l g e b r a i c approsch.

The above example shows t h a t

One

must

G. Gratzer, D. Kelly

106

be c a r e f u l when attempting t o combine these two concepts. d i f f i c u l t y by weakening t h e concept o f completeness.

We s h a l l resolve this

We s h a l l always use t h e symbol m t o denote an i n f i n i t e r e t u l a r c a r d i n a l . An m-complete l a t t i c e , o r i i - l a t t i c e , is a poset i n which every nonempty s u b s e t with fewer than m elements has both a j o i n and a meet. We s h a l l pre fix each l a t t i c e concept with m t o denote t h e Corresponding concept f o r n - l a t t i c e s . Ue m i t m Some examples a r e in-join, ll.-hornornorphism and ni - s u b l a t t i c e . when m :to because t h i s i s t h e usual l a t t i c e case. Obviously, any complete l a t t i c e is an .i-lattice for any ( n . Moreover, P , then any join t h a t e x i s t s i n P i s a j o i n of a nonempty subset of P whose c a r d i n a l i t y is less than .it; i n s h o r t , any jo i n is an m-join. However, t h e r e a r e it-lattices t h a t a r e complete in which not every j o i n is an m j o i n . I f the left-hand chain C i n Figure 1 is t h e ( o r d i n a l ) successor of t h e i n i t i a l o r d i n al corresponding t o m, then t h i s f ig u r e provides such an example L. I n this cas e , note t h a t t h e top element t of C cannot be w ri t t en a s V S f o r a nonempty subset S of L with t L S and IS1 < m. In o t h er words, t is an m-join i r r e d u c i b l e eleuient of L. Consequently, t h e t h r e e squares i n Figure 1 do not m-generate L. if

n exceeds t h e c a r d i n a l i t y of a poset

We emphasize t h a t we only t ak e j o i n s and meets of nonempty s e t s . (In a complete l a t t i c e L , t h e j o i n of t h e empty s e t equals t h e meet of L.) Although m - la tti c e s have i n f i n i t a r y o p er at i o n s , t h er e is a bound on t h e number of such The operations so t h a t t h e concepts and methods of universal algebra apply. (not a c l a s s ) t h a t can simultaneously important point here is t h a t t h er e is a A particular index a l l t h e j o i n and meet o p er at i o n s of every m - l a t t i c e . consequence is t h e ex i s t en ce, f o r any poset P, of F,(P), t h e free m - l a t t i c e

set

on

P.

Fm(P)

is t h e

m-lattice

t h a t co n t ai ns

P

a s a subposet f o r which

m-joins and m-meets a r e formed "as f r e e l y a s possible". (In particular, F,(P) is al-generated by P . ) More p r eci s el y , F A P ) is t h e unique m - l a t t i c e f o r which every isotone map from P t o any m-la ttic e L has a unique extene s h a l l use n_ , a boldface sion t o an m-homomorphism from F AP ) t o L. W number, t o denote an n-element chain. A l i g h t f a c e number i n d i c a t e s an a n t i chain. A s u s u a l , P + Q denotes t h e d i s j o i n t union of t h e posets P and Q. We s h a l l c a l l a poset slender i f it does not contain any of t h e following th r e e p o s e t s a s a subposet: (i)

(ii) (iii)

3=1+1+1 2, + 1+

2 5

.

I. Rival and R. Wille [ I 6 1 described t h e f r e e l a t t i c e on any f i n i t e slender poset. I n a sequence o f papers "91, [ l o ] , [ill, [121), we have generalized t h i s f o r any slender poset. Our techniques a l s o work r e s u l t by describing F-(P) when m is countable, and so provide another proof of t h e Rival-Wille r e s u l t . However, since t h e rn A case d i f f e r s s i g n i f i c a n t l y from a l l o t h e r s , we

henceforth assume t h a t

ni

is uncountable.

S i m i l a r l y a s i n [16], t h e key ingredient i n describing t h e free m - l a t t i c e on any slender poset is a d es cr i p t i o n of t h e f r e e m -la ttic e on t h e poset H of Figure 2. The present paper is devoted t o describing Fm(H). We s h a l l se e t h a t

is a complete l a t t i c e (which would not be t r u e i f we allowed m F,(H) In g e n e r a l , we s h a l l only i n d i c a t e proofs. W e not only de sc ribe F,(R)

No). as a

Some free m-lattices on posets

107

p o s e t , but we L i v e t h e formulas !'or ( a r b i t r a r y ncnenpty) j o i n s and meets.

FIGURt 2. The Loset

I n order t o d e s c r i b e

11

Fm(fl), we mst f i r s t introduce sonle o t h e r

n-lat

t i c e s . I n Section 1 , we describe C ( m ) , t h e f r e e m - l a t t i c e on 2 + 2 , w i t h a new 0 and 1. We a l s o r e q u i r e t h e m - l z t t i c e s A and b oescribed i n Section 2. I n Section j , we d e s c r i b e D(m 1, which is Fm(H) with a new C, and 1 . Our paper [91 contains ttie proof's of a l l s t a t e n e n t s nade i r i t h e f'irst three sections. I n Section 4 , we d i s c u s s the p-oofs t h a t t h e m-lattices C ( m ) and C( m ) , a s d e s c r i b e d , a r e what we claimed. One approach w i l l be t o apply t h e S t r u c t u r e Theorem f o r f r e e m - l a t t i c e s on posets given by P. Crawley and Fi.A. Dean [21. Complete d e t a i l s f o r Section 4 can be found i n L91 x i a [ l o ] . Let P be a slender poset. I f F is a l s o l i n e a r l y indecoriiposable, then we show i n [ill t h a t F is countable, and t h a t Fm(P) can be embedded, as s n m-sublattice,

into

does not contain

F,,,(H).

Consequently,

a poset

F is slender i f f

Fm(P)

Fm(3) a s an m-sublattice.

W e conclude t h i s introduction by b r i e f l y mentioninb some f u r t h e r t o p i c s about m-lattices. Free m - l a t t i c e s , defined a s f r e e m - l a t t i c e s over a n t i c h a i n s , can a l s o be considerea a s f r e e ?.-products of one-element m-lattices. For f r e e products, we r e f e r t o G r a t z e r , Lakser and P l a t t [ I 4 1 arid Gratzer and Lakser [ l 3 ] . I n our paper 171, we Generalized sonJe o f t h e r e s u l t s o f ' t h e s e two papers, and a l s o studied what happens when m is v a r i e d . For example, we proved t h e S t u c t u r e Theorem f o r f r e e m-products, which is s i m i l a r t o t h e S t r u c t u r e Theorem f o r f r e e m - l a t t i c e s of Crawley and Dean [21. Another r e s u l t o f [ T I is: I f a covers b i n some f r e e m -product, then a continues t o cover b when m is increased. (Covers a r e of interest becsuse A. Cay [31 has shown t h a t every proper closed i n t e r v a l of a f i n i t e l y generated f r e e l a t t i c e contains a covering p a i r . ) We a l s o determined which f r e e m-products a r e complete i n [TI. I n p a r t i c u l a r , t h e f r e e " - l a t t i c e on t h r e e g e n e r a t o r s i s not complete ( s e e Whitnian [1&1 f o r t h e m = case). When r e p r e s e n t i n g an element of an m l a t t i c e i n terms of an m -generatink set (using m-Joins and m-meets), t h e r e is no general way t o s p e c i f y a unique r e p r e s e n t a t i o n , even i n a f r e e m - l a t t i c e . For example, t h e r e a r e many exaniples in

Fm(3)

of an

m-join

reducible element t h a t cannot be expressed a s an

irredundant m-join. In 171, we generalized t h e Normal Form Theorern t h a t 6 . Jdnsson [151 proved f o r t h e f r e e m - l a t t i c e on a poset. We used t h e t h e norm61 r e p r e s e n t a t i o n o f [7] t o g e n e r a l i z e a r e s u l t about f r e e products i n 1131.

A n a t u r a l question about an ordered s e t is: M a t is t h e l a r g e s t chain t h a t it c o n t a i n s ?

We say t h a t a poset P c o n t a i n s no chain of c a r d i n a l i t y n . r e g u l a r . M.E. Adams and 0 . K e l l y [ l ]

s a t i s f i e s the n-chain condition i f it Henceforth, l e t n 6 e uncountable and showed t h a t , i f t h e poset P s a t i s f i e s

G. Gratzer, D. Kelly

108

the n-chain c o n d i t i o n , then so does the f r e e l a t t i c e on P. However, f o r uncountable m, we contructed, i n our paper [61 with A . llajnal (for any n > m ) , a poset P with no uncountable chains such t h a t t h e f r e e m - l a t t l c e on P contains a chain of c a r d i n a l i t y n . I n Adam and k e l l y [ l ] , it was a l s o show t h a t t h e f r e e product construction preserves t h e n -chain condition. In our paper [61 with A. Hajnal, we a l s o showed t h a t a f r e e m - l a t t i c e cannot contain a chain with more than m elements. ( I n t h e previous sentence, we assumed t h e Generalized Continuum Hypothesis t o simplify t h e formulation.) If m gl and n is t h e successor c a r d i n a l t o t h e continuum, then we do n'ot know whether the f r e e

1.

m-product construction preserves t h e n-chain c o n a i t i o n .

THE COMPLETE LATTICE

C( m)

Recall t h a t m always denotes an uncountable r e g u l a r c a r d i n a l . The complete l a t t i c e C ( m ) of Figure 3 is t h e f r e e m-lattice on 2, + 2 with a new 0

and

1.

F(2 +

The diagram is a l s o v a l i d f o r

2)

( s e e Rolf [Yl or Chapter VI

of Gratzer [51). W e have denoted t h e 16-element l a t t i c e i n t h e middle of Figure 3 by C( A o ) . (The way t h i s f i n i t e l a t t i c e was used by I. Rival and R. Wille [I61 t o c o n s t r u c t F(H) is s i m i l a r t o t h e way we s h a l l use C ( m ) F A H ) .I I n t h i s s e c t i o n , we only d i s c u s s t h e complete l a t t i c e c( m). We leave u n t i l Section 4 t h e question whether C ( m ) is claimed. Observe t h a t a,, a;, b,, and b h generate C ( K O ) . a A b and f ' a ' v bh. t h e notation f, n m m m

t o construct s t r u c t u r e of what we have W e a l s o use

For every successor o r d i n a l j < in, t h e r e is a lower j-th l e v e l of 6 elements L. l a . , b . , c ., d ., e j , f j ) , and f o r every l i m i t o r d i n a l i < M J J J J J (including i 01, t h e r e is a lower i-th l e v e l of 7 elements Li = { a i , b i , c i , d i , e i , f i , g i ] . These elements a r e ordered a s shown i n Figure 3. There i s a l s o an upper i-th level

f o r each

Ui

i

m, defined d u a l l y and denoted by t h e safile

<

l e t t e r s with primes. For convenience, we a l s o l a b e l 6 elements of ah, p bO, p' bb, y go, y ' = gb. Greek l e t t e r s : a I ao, a '

I n a more formal approach, one d e f i n e s

c( so) uu ( LJ. I

j<m) u

C( m

t o be

LI ( uJ.

I j <m),

C(

m)

with

where each f i n i t e poset t h a t we have l i s t e d i s assumed t o be a s u b p s e t of C( T ) . Let i and j be o r d i n a l s l e s s than m The a d d i t i o n a l comparability u < v holds i n C ( m ) e x a c t l y when one of t h e following conditions is s a t i s f ied :

.

u = a i , v = a . and i < j ; J' u = bi, v = b . , and i < j ; J u E Li [ a i , b i l , v E L . , and i J u = ai and a, 5 v i n C ( % o ) ;

-

u

= b.1 and b,

-

5

v

in

C( E'o)

<

j ;

;

E Li { a i , bil and v E C ( g o ) or v E :U J' a previous condition holds w i t h u and v interchanged, L and U interchanged, primed elements, and r e v e r s a l of t h e i n e q u a l i t y i n

u

C(

so);

Some free m-lattices on posets

109

i limit

a:

Ago= y

FIGURE 3.

The complete lattice

C( m 1

G. Gratzer, D. Kelly

110

u

and

v

*

v

b!; J u = bi and v E U . but v :a!. J' J Clearly, y go is t h e least element of C ( m ) , and y' = pi, is t h e g r e a t e s t . In f a c t , C ( " ) is a complete l a t t i c e . To sketch t h e proof o f t h i s s t a t e m e n t , l e t u s e x p l a i n how t o c a l c u l a t e (nonempty) j o i n s . From Figure 3, one sees a n a t u r a l way to break C ( m ) up i n t o 7 p a r t s : C ( AO), the bottom l e f t chain [ a i I i 5 m I , t h e bottom middle p a r t from go t o k , t h e bottom a.

U.

E

J'

1

but

C( r i g h t chain bi j i 5 m I , and t h e corresponding t h r e e t o p p a r t s . has e x a c t l y one p o i n t i n common with each of t h e o t h e r six p a r t s . Observe t h a t each p a r t is a complete l a t t i c e , i n f a c t , a complete s u b l a t t i c e o f C( m ) . Since each i n f i n i t e p a r t is a l i n e a r sum over a complete c h a i n , it should be c l e a r how t o c a l c u l a t e t h e j o i n i n each p a r t . I t only remains t o determine t h e j o i n o f p a i r s of incomparable elements frorn d i f f e r e n t p a r t s . W e leave the d e r i v a t i o n of t h e s e formulas t o t h e r e a d e r .

8'.

Let u s i n d i c a t e why C( m ) We f i r s t g e n e r a t e elements of

-

[y, y ' ) is m-generated by a , a ' , C ( &o) i n t h e following o r d e r :

since 0 i d e a f o r t h e rest of t h e argument complete l a t t i c e of Figure 1. I f i < m , then it is easy t o generate

We can now generate

a l i m i t ordinal then

2.

a. J

C( X )

j

Li

m,

<

we have its four g e n e r a t o r s ,

We kave t h e

i n t h e i n t r o d u c t i o n when we discussed t h e Li has a l r e a d y been m-generated f o r some Li+l using f i n i t e j o i n s and meets. I f , for

has already been =-generated

b . can then be m-generated s i n c e J

and

.

p, and

a. =

( ai

J

f o r each

I i

<

j)

i

<

j,

.

THE LATTICE A

I n t h i s section, we d e f i n e t h e second b u i l d i n g block of D( m ) , t h e l a t t i c e of F i g u r e 4. This l a t t i c e was discovered by I . Rival and R . Wille L161. Let J be t h e set of dyadic r a t i o n a l s r t h a t s a t i s f y 0 5 r s 1. Every nonzero r E J has a unique r e p r e s e n t a t i o n , t h e normal form, r = a * 2-", where a is an odd

A

integer;

n

is t h e

order of

r; i n notation, n

= 0. Observe t h a t o r d ( r ) = o r d ( 1 maximum of o r d ( r ) and o r d ( s ) . We d e f i n e

A

a s a subposet of

- r). J2

Although

< r , s>

i n Figure 4.

E

A , then

v

o r d < r , s> t o b e t h e

-ord s - r = 2 I .

not

with t h e exception of J 2 , For example,

a s u b l a t t i c e of

<1/4, 1/2> = <1/2, I > A

can be redefined i n two ways:

A = I I r

<

s and

s

[ < r , s> I r

<

s

and

s

A

W e a l s o define

o r d ( r ) ,i ord(s)

is a l a t t i c e , it is

A

Ey convention, o r d ( 0 )

with t h e product o r d e r :

A = { < r , s > 1 r < sand Note t h a t i f

ord(r).

-

r = 2-",

n >. o r d ( r ) l

r = 2-",

n

2

ord(s)}

.

<0, l > .

111

Some free m-lattices on posets

We s h a l l use

and

nl

t o denote t h e f i r s t and second p r o j e c t i o n s of

n2

J2

J . For t E J , l e t us c a l l t h e set of a E A t h a t s a t i s f y n , ( a ) t x = t l i n e i n A , and d e f i n e t h e y z t l i n e similarly. The a l t e r n a t e s l i n e s i n A. In d e f i n i t i o n s of A y i e l d e x p r e s s i o n s f o r t h e x = r and y p a r t i c u l a r , < r , r + 2- Ord(') > is t h e l a r g e s t element on t h e x = r l i n e , and <s - 2- o r d ( s ) , s> is t h e s m a l l e s t element on t h e y = s l i n e .

onto

the

Each a formulas a r e :

For example, 1>1

denoted by

has a r i g h t lower cover

a*

and a r i g h t upper cover

a*.

The

< ( r + s ) / 2 , s> < r , ( r + s ) / 2 >.

< r , s>* < r , s>, {
A

-

<1/4, 1/2>* <3/8, 1/2> i n Figure 4. Moreover, i f a E A s a t i s f i e s o r d ( a ) = o r d ( n , ( a ) ) , then a has a l e f t lower c o v e r ,

*a, and d e f i n e d by

* < r , s> = < r

-

2-ord(r), s>.

-ord ( s ) > S i m i l a r l y , t h e l e f t upper cover * < r , s> e x i s t s and e q u a l s < r , s + 2 when o r d ( r ) < o r d ( s ) . Observe t h a t <0, 1> has no l e f t (lcwer or upper) c o v e r , b u t t h a t every o t h e r element o f A h a s e x a c t l y one l e f t cover. Moreover, each cover o f an element a E A l i e s on t h e x-line through a or t h e y-line through a .

FIGURE 4.

The l a t t i c e

A

G. Gratzer, D. Kelly

I12

2 I n Figure 4 , A seenis t o be a very s p a r s e subset of J For each a = < r , s> E A, l e t us c a l l the closed i n t e r v a l [ r , s ] ( i n e i t h e r t h e r e a l s or dyadics) t h e shadow of a . We say t h a t two closed i n t e r v a l s overlap i f they have an i n t e r v a l of nonzero length i n common, but n e i t h e r i n t e r v a l contains t h e o t h e r . The " r e l a t i v e sparseness'! of A is expressed by t h e following shadow p r i n c i p l e : The shadows of two elements of A cannot overlap. Let u s i n d i c a t e an easy way t o v i s u a l i z e shadows. W e add a v e r t i c a l l i n e ( t h e shadow l i n e ) t h a t passes through t h e i n t e r s e c t i o n of the x = 1/2 and y = 1/2 lines on t h e r i g h t s i d e of A i n Figure 4 ; index t h i s l i n e i n t h e obvious way w i t h 10, 11. I f one places an opaque r i g h t angle a t a E A (forming t h e x - l i n e and t h e y-line through a ) , then t h e i n t e r s e c t i o n of t h e r i g h t angle with t h e shadow l i n e g i v e s t h e shadow of a .

.

W e found t h e shadow p r i n c i p l e t o be a very n a t u r a l t o o l i n our proofs. Let u s apply it t o "explaint1 t h e emptiness of t h e l a r g e s t open square i n Figure 4: i f t h e r e was an element a i n t h i s square, then t h e shadows of a and <0, 1/2> would overlap.

I n Figure 4 , the d i s t a n c e of each element < r , s> E A from t h e shadow l i n e is proportional t o s r , t h e length of its shadow. Consequently, f o r two incomparable elements a , b E A , we can define a t o be t o t h e l e f t of b i f f

-

has a longer shadow than b. Observe t h a t a is t o t h e l e f t of b i f f < n,(b) iff n2(a) > n 2 ( b ) . With t h i s concept, we can v i s u a l i z e t h e 1 f i n i t a r y j o i n i n A. Let a and b be incomparable elements of A , with a t o t h e l e f t of b . The j o i n of a and b is t h e l e a s t element on t h e y - l i n e through a t h a t is g r e a t e r than b. Consequently, a V b an f o r ttie l e a s t n

a

n (a)

such t h a t an > b , where an is inductively defined by: a. T h i s shows t h a t A is a l a t t i c e .

3.

THE COMPLETE LATTICE D(

a , ai+l = ( a i ) *

.

t?t)

W e build up D( m ) from t h r e e d i s j o i n t s u b l a t t i c e s : l a t t i c e A was defined i n Section 2. We d e f i n e B = { < r , s> I < s , r >

E

A , B , and C.

The

A],

a subposet of J2. Clearly, B is a l a t t i c e and its diagram is obtained by r e f l e c t i n g Figure 4 about a v e r t i c a l l i n e . I n o t h e r words, mapping < r , s> E A to <s, r > E B d e f i n e s an isomorphism of A with B. Consequently, every element has

b

of

B

has l e f t covers

a r i g h t lower cover

b*

or

a

*b

and

*b,

and i f

r i g h t upper cover

b z < 1 , 0 > , then

b

b*.

F i n a l l y , l e t I be t h e r e a l i n t e r v a l [O, 11, and r e c a l l t h a t J denotes t h e s u b s e t of I c o n s i s t i n g of dyadic r a t i o n a l s . For each t E J , we t a k e a and generators a t , a ; , pt, copy Ct of C( m ) , w i t h bounds yt and y t ,

p i , For each t E I two-element chain with

t

E

I.

Since

I

which is riot a dyadic r a t i o n a l , Ct = [y,, y i ] is t h e yt < y.; W e define C a s t h e l i n e a r sum of t h e C t ,

is complete and each

Ct

is complete,

C

is a complete

lattice. To d e f i n e C ( m ) , ( s e e Figures 5 and 6):

we must describe t h e p a r t i a l ordering on

A u b u C

Some free m-lattices on posets

FIGURE 5.

The complete l a t t i c e

FIGURE 6.

Cetails of

fi( m )

113

D(

rn)

G. Gratzer, D. Kelly

114

If
S>

E

A,

< r , s> < r , s> < r , s> < r , s>

<

< t ,u> < t , u>

<

> <

>

>

< t , u> E B, v E I , and < t , u> i f f s < u; < t , u> i f f r > t ; p i f f s < v holds o r s p i f f r > v holds o r r p i f f t < v holds or t p i f f u > v holds o r u

I t is e a s i l y seen t h a t

p

= = = =

Cv,

then:

v v

and

av

_c

p

hold;

and

a' 2 V

p

hold;

v

and and

pv p{

5

p

2

c

hold; hold.

E

v

is a poset.

D(m)

However, since <0, 1/2> 1 < 1 , 1 / 2 > , v was assumed t o be a dyadic pv, o r p.; Tnus, i f v is not

2.

A u B is not a subposet of J Note t h a t r a t i o n a l wherever we used t h e notation a v , a ; ,

dyadic, then i n each of the l a s t four c a s e s , only t h e f i r s t c l a u s e can apply.

I t is not d i f f i c u l t t o show t h a t C( m ) is a l a t t i c e , and t h a t each of A , B, and C is a s u b l a t t i c e of D( m ) . For < r , s> E A , < t , u> E B , v E I , and p E C v , we give t h e formulas f o r joining p a i r s : (a)

< r , s> v p

is

( i ) as v p

E

(ii) < r , s>,

(b)

<s, r>

C , where t h e j o i n is formed i n

if

r

>

v

r

or

v

w

and

(iii)

t h e l e a s t <w, s> such t h a t w > v , i f

(iv)

t h e l e a s t <w, s> such t h a t w t v , i f

p

C, i f

r r

5 5

s

5

v v

<

s and

< s

and

< r , s> v < t , u> i s ( i ) < t , u>, i f s < u; ( i i ) < r , s>, i f t < r ; ( i i i ) the l e a s t <w, s> on t h e y s l i n e i n A such t h a t w ( i v ) t h e l e a s t < t , w> on t h e x = t l i n e i n B such t h a t w ( v ) as V ps, i f s = t , where t h e j o i n is formed i n Cs. There is an automorphism B, and interchanges a t and

E

@

pt

D( m )

of (a;

and p i )

v;

i n Cv;

5 a;

t h a t maps f o r any

p .k a; i n Cv; p 5 i n Cv;

> >

t , i f s >t; s, if s >t;

< r , s> E A t o t E J . In o t h e r

words, $ r e f l e c t s Figure 5 a b u t i t s c e n t r a l a x i s . With t h i s observation, t h e formulas f o r < t , u> v p follow from ( a ) . The pairwise meet formulas follow by duality. In order t o show t h a t D( m ) is a complete l a t t i c e , it s u f f i c e s t o f i n d X f o r any nonempty subset X of A. (The formula is s i m i l a r f o r B and we already know t h a t C is complete.) Let X, arid X2 be t h e f i r s t and t h e second p r o j e c t i o n s of X , and form u = v X1 and v = 'd X2 i n I .

v

If

u < v , then v E J , and v X is t h e l e a s t element o f y = v l i n e whose f i r s t coordinate is 2 u.

If

u

D(

m

1

v , then

-

{yo,

v yil

X

=

yu

if

uw v

ciu

if

u

is m-generated

To prove t h i s , one generates t h e four C,

-

{y,,

yi].

ao,

Po,

6

X2. <0, 1 > , < 1 , 0>, a { , p i .

m-generators of

Then t h e four elements of

on t h e

u L X2;

and

= v and u

by

A

A u B

Co

-

{yo,

ydl

and

of o r d e r 1 a r e generated.

One then m-generates C l I 2 - {y1,2, y11/21 2 L. W e alternately m-generate Ct {yt, y;]. A t each s t e p , we increment t h e order elements of A u B and

-

115

Some free m-lattices on posets

t

V(at

yi

t.

A u B, and the order of

of elements of

t

J

and

t

Finally, for any nonzero

In the remainder of t h i s section, we discuss m). An element a = < r , s> of A i s join reducible) i n C ( m ) i f f o r d ( r ) > o r d ( s ) . (Such lower cover a*, and a, 2 x whenever a > x i n element of A i s doubly m-reducible i n D( m ) .

x

Ct,

E

x = t

6

x x

E

x

E

yt

t

E

and

E

I,

i).

<

Ij(

If

i

some further properties of reducible (completely join an element a has a unique D( m).) I n particular, no

I , then x ism-join reducible i n t f 0 ; or

D( m )

J and x = eo or x E [ a i , b i , ei] for 0 is a join reducible element of C ( , t o ) ; or

a, or x = bm; or r c i , d i , f i ) for i

<

<

iff

i
m.

I n verifying that each of the above elements is m -join reducible, most cases If x is an element of Ct t h a t we have not l i s t e d , follow from Figure 3.

x

y i n C t : moreover, d < x implies d 5 y i n D ( m 1. Thus, x is completely join irreducible, and -a f o r t i o r i x is m-join irreducible. We leave the exceptional cases x y o and x = f t o the reader. The formulas for the m-meet reducible elements are m similar, One consequence of t h i s analysis is t h a t C( m ) contains no doubly m-reducible element. then, w i t h two exceptions,

4.

has a unique lower cover

JUSTIFYING THE DESCRIPTION OF

D'

Let

D' is

that

-

D( m )

denote

D( m )

F,(H).

Certainly, C '

[yo

yi).

We s h a l l comment on our two proofs

contains

H

as a subposet, and

t1 m -gen

v

erates D ' . Moreover, for any x E H, x # ( y E H I x # y ) i n D'. By the Structure Theorem of Crawley and Dean [21 for free m-lattices on posets, it only remains t o show that D' s a t i s f i e s the condition (W,). An m-lattice L s a t i s f i e s (Wm) i f f whenever S and T are nonempty m-subsets of L ( I S I ,

IT1 some

m

<

t

1, then /\S sV T

E

implies s

5

VT

for sane

s c S,

or A S s t

for

T.

Let u s c a l l an interval

[a, bl

of

D' a

(W )-failure i f there are non-

m -

S , T of D ' such that a = I \ S , b = V T , but S n [ a , b l empty m-subsets and T n [ a , b l are both empty. One way t o show that D ' is F,(H) is t o We have done t h i s i n [91. prove that D' does not contain any (W,)-failures. For example, suppose there is a

(Wm)-failure

[a, bl

w i t h both

a

and

b in

A. Let x be the greatest element on the x-line through a , and l e t y be the l e a s t element on the y-line through b. Since some element of S must be on the x-line through a, x L b. Consequently, n 2 ( x ) > n2(b) = n 2 ( y ) .

Similarly,

n,(y)

<

n,(a) = n , ( x ) .

Since t h i s means t h a t the shadows of

x

G. Gratzer, D.Kelly

116

and

y

o v e r l a p , such a

also satisfy or

b

in

(Wm). C

(WT)-failure is impossible.

Of course,

C( rn )

niust

I n f a c t , our proof t h a t t h e r e is no (W ) - f a i l u r e with a M C( m ) s a t i s f i e s (h). The A and b E B , required t h e most complicated argument.

is e s s e n t i a l l y an an al y s i s of why

f i n a l case, t hat

a

E

For our second proof i n [ l o ] , we assumed t h e de sc ription of F(H) given by I . Rival and R. Wille [16]. We then developed some elementary universal algeb r a ic lemmas t o show t h a t D' is Frn(H). The S t r u c t u r e Theorem was not used but we did r e q u i r e t h e f a c t t h a t C ( rn) - { y , y ' } is Fm(Z + 2). This proof ta ke s advantage of t h e f a c t t h a t t h e elements of f r e e l y rn-generate C {yo, y i l .

-

C

labeled with

a, a',

6, and

p'

Some free m-lattices on posets

117

Proof: We s h a l l p r o v e i t b y i n d u c t i o n on n = ( H I + ( K ( t h a t c o n d i t i o n s ( i v ) and ( v ) i m p l y t h e c o n d i t i o n ( i i ) o f theorem 3. L e t us now c o n s i d e r H,,H, K 1 /L K 2 ,

c'f

and K,,K,

e

f,such

that H15H2.

I K ~ ( + ( H (I = n , f ( H l ) = f ( H 2 ) and f ( K l ) = f ( K 2 ) .

a) I f K l i s n o t a s t a b l e , t h e n r i s i n q c o n d i t i o n ( i v )

,( H I I + I K l l < n

and t h u s b y i n d u c t i o n we have f i n i s h e d .

b ) If K l i s a s t a b l e . KI

L e t us c o n s i d e r G1=H1 al.

I f e v e r y q e n e r a l i z e d component o f

Comp(Gl)

then

is

a

stable,

transitive orientations.

(

Comp(G1)

admits

only

two

different

S H E V R I N and FILIPPOV have f i r s t n o t i c e d

t h i s f a c t i n 1181). And t h e n t h e r e s u l t i s g i v e n b y c o n d i t i o n ( i ) . E l s e Comp(G1) a d m i t s a q e n e r a l i z e d p r i m e component T w h i c h i s n o t a s t a b l e and t h e r e f o r e : HI(T)

GI =

H'1 bl

Since (H'I(

9

KI

, a1

KI = H ' i a1

Hl(T)

7

,bl

+ (Sf(HI(T))I
we can use t h e i n d u c t i o n h y p o t h e s i s .

a

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 (1984) 119-150 0 Elsevier Science Publishers B.V. (North-Holland)

119

COMPATIBLE O R D E R I N C S AND TOLERANCES OF LATTICES I. G. Kosenberg M.A. Universitb de Montrgal CP 6128 MoiitrCal H3 C3 57, Canada

D. Schwcigert FB Mathematik der llniversitat 675 Kaiserslautern, R . F . A .

.

C R.

DC.diP b E. Corominas,

a

l'occasion de son Embritat.

Ce trava 1 traite de questions qui interessent l'algkbre universelle

la thPorie des relations et la thCorie des ordres. I1

s'agit premihrement de probltmes d'ordre et de tolPrance dans les alg6bres ayant un polyn6me majoritb. N o u s abordons dans une deuxikme partie l'btude de certains cl6nes, surtout de cl6nes maximaux de primalit6 et de leurs analogues locaux, par des mbthodes purement relationnelles. La troisikme partie traite des polyn6mes locaux dans les treillis. Notre but est de faire un tour d'horizon et de prb-

senter certaines methodes relationnelles. Les treillis interviennent dans les trois domaines que nous venons de mentionner. MalgrC leur nature particulihre, ils posshdent une structure trks riche (et, de plus, apparaissent d'une facon naturelle dans beaucoup d'applications). I1 n'est donc pas surprenant que dans l'btude des polyn6mes dans les treillis on rencontre des resultats tr6s interessants. D'abord quelques definitions : Ctant donne un treillis

I-. =,

une relation binaire p sur L est compatible avec &- si p est un s o u s 2 treillis de 4 (i.e. si (x,,yl) Ep et (x2,y2)Cp impliquent b la fois (xl+x2,yl+y2) E p et (x,x2,yly2) E p ) . Une tolCrance de

L- est une relation

symCtrique et reflexive (i.e. un graphe avec boucles) compatible avec L- . Plusieurs resultats viennent du fait que la mediane m(x,y,z)=xy+xz+yz est une fonction majorit6 (i.e. satisfait m(x,x,y)=m(y,x,x)=m(x,y,x)=x) Dans certains cas, on peut aller un peu plus loin en ne supposant que la presence d'une fonct-ion " 2/3-majoritt5" (i.e. ne satisfaisant que m(x,x,y)=m(y,x,x)=x). Nous cornencons par des g6neralitCs sur les polyn6mes, les clcnes,

les cl6nes maximaux, leur completude et les variantes locales de ces notions (12). N o u s montrons qu'une alghbre finie A- ayant un polyn6me "2/3-majoritC1' esr polynomialement cornpl6te si elle est simple, ne possiide aucune tolerance centrale

(a2 telle que

aucun ordre born6 compatible avec A- .

Ia1.A

Fp pour un

a E A ) et

I.G. Rosenberg, D.Schweigert

120

Nous presentons aussi u n e extension du th6orhme de l o double

pro-

jection de Bergman pour des a l g h h r e s ayant une fonction "2/3-majorit4" qui satisfait une condition spbciale. Au paragraphe 95, nous faisons u n tour d'horizon de rPsultats asscz

d4finitifs concernant les ordres compatibles avec une fonction majoritb, un demi-treillis ou un treillis. Dans le dernier

les ordres sont l i 6 s

cas,

h d e s dbcompositions directes et sous-directes de treillis. Le paragraph(> § 6 contient quatre caracterisations des tolerances de treillis et leur des-

cription par des progressions (i.e. des inf-endomorphismes 6 d'un treillis complet

L- tels que x <- 6(x) pour tout x, qui sont additifs

s u r les s o u s -

ensembles filtrants supbrieurement de L ) . Les tol6rances (localement) centrales sont discutkes au paragraphe 1 7 . Le dernier paragraphe traite des treillis localement polynomialement complets pour l'ordre ( i . e . tels que les opbrations isotones pour l'ordre sont des polyn6mes du treillis).

51

INTRODUCTION

T h i s p a p e r d e a l s w i t h q u e s t i o n s which a r e t o p i c a l f o r t h r e e p r o b l e m a r e a s o f u n i v e r s a l a l q e b r a , l a t t i c e s o r d e r s and r e l a t i o n s . The f i r s t one t r e a t s t o l e r a n c e s o f a l q e b r a i c s y s t e m s ( a s u r p r i s i n g l y r i c h b i b l i o g r a p h y i s i n [ 6 , 7 ] ) . The s e c o n d one i s d e v o t e d t o t h e s t u d y o f c l o n e s , maximal c l o n e s and p r i m a l i t y and t h e i r l o c a l c o u n t e r p a r t s u s i n g r e l a t i o n a l methods. The t h i r d one i s c o n c e r n e d w i t h D o l y n o m i a l s o f l a t t i c e s . The p u r p o s e o f t h i s u a p e r i s t o s u r v e y t h e p e r t i n e n t res u l t s and t o draw a t t e n t i o n t o t h e p o s s i b l e u s e o f t h e s e c o n d a p p r o a c h . It is not coincidental t h a t the order-interpolation

p r o p e r t y of l a t -

tices h a s y i e l d e d s t r i k i n g r e s u l t s . A f t e r a l l , l a t t i c e s a r i s e n a t u r a l l y i n many areas a n d , though t h e v are q u i t e s p e c i a l a l g e b r a s , t h e y posess a r i c h s t r u c t u r e . F i r s t l e t us d e f i n e t h e b a s i c concepts o f t h i s paDer. For a l a t t i c e

L-

= (L;+,-)

a binary relation

P

on L i s c o m p a t i b l e w i t h

( x l , y l ) E P , (x2,y2) E P implies (x1+x2, y1+v2)

E P

L

if

and ( x , x 2 , y1y2) E P . & is a tolerance.

A s y m m e t r i c and r e f l e x i v e r e l a t i o n c o m p a t i b l e w i t h

A compatible o r d e r of

b

i s , of c o u r s e , a c o m p a t i b l e r e l a t i o n of

L

which i t s e l f i s a n o r d e r r e l a t i o n . Some o f t h e r e s u l t s depend e s s e n t i a l l y on t h e p r e s e n c e o f a m a j o r i t y o p e r a t i o n ( a t e r n a r y o p e r a t i o n m s a t i s f y i n g m ( x , x , y ) = m ( x , y , x ) = m ( y , x , x ) = x f o r a l l x and y ) p r o -

Compatible orderings and tolerances v i d e d by t h e median ( x y )

+

(yz)

+

121

( z x ) . We e v e n a t t e m p t t o go f u r t h e r

a s s u m i n g o n l y a 2 / 3 m a j o r i t y o p e r a t i o n ( i . e . dn o p e r a t i o n m s u c h t h a t m(x,x,y)

=

m(y,x,x)

=

x f o r a l l x and y ) . I n 5 2 w e s t a r t w i t h gene-

r a l i t i e s c o n c e r n i n g p o l y n o n i i d l s , c l o n e s , maximal c l o n e s a n d c o m p l e t e n e s s and t h e i r l o c a l v a r i a n t s . We a l s o b r i e f l y r e v i e w some g e n e r a l r e s u l t s on c o m p a t i b l e r e l a t i o n s . In 5 3 f i r s t w e point out t h a t a f i n i t e algebra with a 2/3 majority p o l y n o m i a l f u n c t i o n i s i n c l u d e d i n t h e maximal c l o n e s d e t e r m i n e d by t h e bounded c o m p a t i b l e o r d e r s , c o n q r u e n c e s and c e n t r a l r e l a t i o n s r e f l e x i v e and symmetric r e l a t i o n

such t h a t { a }

p

xA_cp

(a

f o r somea < - A )

o f t h e a l g e b r a and q i v e some h i n t s f o r f i n d i n g s u c h r e l a t i o n s .

Then

w e d e r i v e an e x t e n s i o n of Berqman's d o u b l e - p r o j e c t i o n theorem f o r a l gebras with a 2/3 majority operation s a t i s f y i n g a s p e c i a l condition. Much b e t t e r r e s u l t s a r e a v a i l a b l e f o r a l g e b r a s w i t h a ( l o c a l ) major i t y p o l y n o m i a l f u n c t i o n . We c a n ( l o c a l l y ) t r e a t i n f i n i t e a l g e b r a s and t h e c o m p a t i b l e bounded o r d e r s a r e now l a t t i c e s . However i n o u r l i s t w e m u s t i n c l u d e t o l e r a n c e s o f i n f i n i t e d i a m e t e r ( a r e f l e x i v e and symmetric r e l a t i o n

p

s u c h t h a t p n + A 2 f o r a l l n = 1,2,

...

while

2

U p = A , i . e . a c o n n e c t e d g r a p h w i t h loops w i t h b o u n d e d d i s t a n c e s ) . n=l From t h e B a k e r - P i x l e y t h e o r e m i t f o l l o w s t h a t t h e c o m p a t i b i l i t y w i t h

c e r t a i n b i n a r y r e l a t i o n s c h a r a c t e r i z e s l o c a l polynomial f u n c t i o n s . A polynomial f u n c t i o n o f an a l g e b r a t i o n s of

3- and t h e c o n s t a n t s o f

E

i s o b t a i n e d from t h e b a s i c o p e r a -

A v i a r e p e a t e d c o m p o s i t i o n . The s e t

of polynomial f u n c t i o n s o f 3 - i s denoted P ( & ) . An n - p l a c e

is a local polynomial f u n c t i o n of

function f

i f f o r every f i n i t e subset B of A

t h e r e s t r i c t i o n o f f t o B a g r e e s w i t h t h e r e s t r i c t i o n o f some p o l y n o mial function of

A. -

The c l o n e o f t h e l o c a l p o l y n o m i a l f u n c t i o n of

4

i s d e n o t e d by LOC P ( A - ) . An a l g e b r a J.s l o c a l p o l y n o m i a l l y c o m p l e t e i f e v e r y f u n c t i o n on A i s a l o c a l p o l y n o m i a l f u n c t i o n o f &. W e show: Let

4=

(A;F) h a v e a m a j o r i t y f u n c t i o n among i t s l o c a l p o l y n o m i a l

f u n c t i o n s and l e t R c o n s i s t o f

(i) lattice orders compatible with

( i i )l o c a l l y c e n t r a l t o l e r a n c e s o f

t o l e r a n c e s of

4

b,

(iii) congruences of

o f i n f i n i t e d i a m e t e r . Then

c o m p l e t e i f and o n l y i f R

=

4

is local polynomially

(3. F o r a l a t t i c e L - = (A;.')

A i s a l o c a l p o l y n o m i a l f u n c t i o n i f and o n l y i f f i s

preserves each tolerance of

&.

4,

and ( i v )

a f u n c t i o n on

monotone and

I.G. Rosenberg, D. Schweigert

122

I n 55 w e review t h e compatible o r d e r s f o r m a j o r i t y o p e r a t i o n s , s e m i -

l a t t i c e s and l a t t i c e s . F o r l a t t i c e s t h e y a r e c o n n e c t e d w i t h s u b d i r e c t and d i r e c t d e c o m p o s i t i o n s . A s u s u a l a n o r d e r i s up-

(down-) d i r e c t e d

i f e a c h p a i r o f e l e m e n t s h a s a n u p p e r ( l o w e r ) common bound. I f a c o m p a t i b l e o r d e r i n g of a l a t t i c e t h e n

is down-directed,

r

E is a join semilattice i f f

and d u a l l y

r

r e c t e d . Consequently C i s a l a t t i c e o r d e r i n g i f f

Let

and u p - d i r e c t e d .

&

and K- = ( A ; A , v )

= (A;+,*)

o r d e r s : and

s u c h t h a t (r i s

o r i e n t a t i o n of

&,

i?

i s up-di-

is down-directed

b e two l a t t i c e s w i t h

&.

compatible o r d e r of

&' 5 .

i n symbols

t is

i s a meet s e m i l a t t i c e i f f 1 7

The r e l a t i o n + i s

W e c a l l K an a symmetric

( a n d r e f l e x i v e ) r e l a t i o n on t h e s e t o f a l l l a t t i c e s on A. We show: L e t 1 b e a n o r d e r o n A w i t h a l e a s t e l e m e n t 0 and a g r e a t e s t e l e m e n t e .

Then

I i s an o r i e n t a t i o n o f a l a t t i c e 5

=

i f and o n l y i f & i s

(A;+,*)

b o u n d e d , 0 i s a n e u t r a l e l e m e n t and e i t s complement a n d E i s a l a t -

t i c e o r d e r s a t i s f y i n g x CI y follows t h a t a l a t t i c e i f and o n l y i f

&

&

Q

x0

yo and x + O

~.

y +O.

From t h i s

has only t h e t w o b a s i c o r i e n t a t i o n s

Y

and

'

i s d i r e c t l y indecomposable. A simple l a t t i c e h a s

only t r i v i a l o r i e n t a t i o n s . S e c t i o n 56 i s d e v o t e d t o t h e l a t t i c e o f t o l e r a n c e s o f a l a t t i c e and

i t s d e s c r i p t i o n by p r o g r e s s i o n s . The t o l e r a n c e s o f a l a t t i c e d e r e d by i n c l u s i o n ,

form a c o m p l e t e l a t t i c e To1

&.

4,

or-

There a r e f o u r

c h a r a c t e r i z a t i o n s o f t o l e r a n c e s i n terms o f b l o c k s (maximal s e t s B 2 s u c h t h a t B ' p ) a n d n e i g h b o r h o o d s y s t e m s ({x I ( x , a ) E P ~ a, E A ) . A p r o g r e s s i o n o f a complete l a t t i c e on A i s aninf-homomorphism that x 'otx)

and

G

x) =

(

v

G

s e t y c A . There i s a b i j e c t i o n between t o l e r a n c e s o f g r e s s i o n s of t h e i d e a l l a t t i c e of

such

(x) f o r e v e r y d i r e c t e d , non empty

x Ey

X EY

G

5

and t h e pro-

5.

I n 57 w e b r i e f l y look a t l o c a l l y c e n t r a l a n d c e n t r a l t o l e r a n c e s . I n

p a r t i c u l a r w e show: L e t p b e a p r o n e r t o l e r a n c e of a bounded l a t t i c e

&

on A

.

Then e i t h e r t h e t r a n s i t i v e h u l l o f P i s a p r o D e r c o n q r u e n c e

of L or t h e r e i s a number n , 0 c n

C

IA( s u c h t h a t

pn

i s a n r o p e r cen-

tral relation. 5 8 t r e a t s t h e o r d e r i n t e r p o l a t i o n n r o n e r t y and r e l a t e d p r o b l e m s . A s

u s u a l , a lattice has the order-interpolation order-nolynomiallv

comnlete i f every

nolvnomial function of

&.

proDerty o r i s l o c a l l y

-monotone f u n c t i o n i s a l o c a l

An unbounded l a t t i c e i s b r o a d i f i t h a s no

t o l e r a n c e o f i n f i n i t e d i a m e t e r , i t i s f l a t i f i t h a s no l o c a l l y c e n -

Compatible orderings and tolerances

123

t r a l t o l e r a n c e and s t a r f r e e i f i t h a s no c e n t r a l r e l a t i o n . An unbounded l a t t i c e h a s t h e o r d e r i n t e r p o l a t i o n p r o p e r t y i f and o n l y i f i t i s s i m p l e , f l a t and b r o a d . A bounded l a t t i c e h a s t h e o r d e r i n t e r w o l a t i o n

p r o p e r t y i f and o n l y i f i t i s s i m p l e and s t a r f r e e . A l a t t i c e

-

is

l o a f - c o m p l e t e i f e a c h monotone f u n c t i o n a d m i t t i n g a l l c o n g r u e n c e s o f

L-

i s a local polynomial f u n c t i o n of L . D i s t r i b u t i v e l a t t i c e s are l o a f -

complete. Now a l o a f - c o m w l e t e bounded l a t t i c e h a s t h e o r d e r i n t e r p o l a t i o n n r o p e r t y i f and o n l y i f i t i s s i m p l e . A p a r t i a l l y w e l l - o r d e r e d l a t t i c e w i t h 1 and 0 s u c h t h a t 1 i s t h e j o i n of f i n i t e l y many atoms

i s s t a r f r e e . I t i s i n t e r e s t i n g t o n o t e t h a t a modular l a t t i c e w i t h a t l e a s t o n e complemented i n t e r v a l ( e . 9 . o n e p r i m e i n t e r v a l ) h a s t h e o r d e r i n t e r p o l a t i o n D r o p e r t y i f f i t i s s i m p l e and r e l a t i v e l y complemented.

I n p a r t i c u l a r a modular l a t t i c e o f f i n i t e l e n g t h h a s t h e or-

d e r i n t e r p o l a t i o n p r o p e r t y i f f it i s a d i r e c t l y indecomposable proj e c t i v e geometry. To l e a r n a b o u t t h e s t r u c t u r e o f a l a t t i c e o r t o d e s c r i b e i t i n a s e n s i b l e way t o l e r a n c e s may r e v e a l more t h a n i t s e q u a t i o n a l t h e o r y bec a u s e u s u a l l y t h e y p r o v i d e good means o f decomposing t h e l a t t i c e i n t o simpler blocks. A tolerance y i e l d s a f a c t o r lattice i n a similar way as a c o n g r u e n c e d o e s and a l r e a d y t h e s u b d i r e c t p r o d u c t s o f f a c -

t o r l a t t i c e s o f d i s t r i b u t i v e l a t t i c e s i n c l u d e a l l l a t t i c e s . The conc e p t o f a t o l e r a n c e came from t h e t h e o r y o f measurements and t h e r e

a r e r e c e n t a p p l i c a t i o n s i n s o c i a l s c i e n c e s and r e l a t e d f i e l d s . Tolerances are i n t i m a t e l y connected t o t h e c l o n e o f l o c a l polynomial f u n c t i o n s . The ( f i n i t e ) o r d e r p o l y n o m i a l l y c o m p l e t e l a t t i c e s i n t h e v a r i e t y o f modular l a t t i c e s are indecomposable p r o j e c t i v e g e o m e t r i e s and t h u s q u i t e i n t e r e s t i n g c o m b i n a t o r i a l o b j e c t s . The p a p e r i s i n t e n d e d as a s u r v e y b u t i t c o n t a i n s a l s o new r e s u l t s . W e p r o v e o r g i v e a s k e t c h o f a p r o o f i n most cases b e c a u s e a mere

c o l l e c t i o n o f f a c t s i s u s u a l l y h a r d t o u n d e r s t a n d and g i v e s no i n d i c a t i o n of t h e proof techniques. W e l i k e t o t h a n k t h e r e f e r e e f o r s e v e r a l comments. The p a p e r w a s es-

s e n t i a l l y p r e p a r e d d u r i n g t h e a u t h o r s s t a y a t t h e U n i v e r s i t g ClaudeB e r n a r d i n Lyon. The p a r t i a l f i n a n c i a l s u p p o r t p r o v i d e d by t h e ENSERC g r a n t A 9 1 2 8 a n d t h e FCAC Q u e b e c g r a n t EQ-0539 i s g r a t e f u l l y acknowl e d g e d . T h e s e c o n d a u t h o r l i k e s t o thanktheDAAD f o r i t s f i n a n c i a l a i d .

I.G. Rosenberg, D. Schweigert

124

5 2 PRELIMINARIES

I n t h e s e q u e l A i s a f i x e d u n i v e r s e w i t h I A l > 2 and

2.1.

f i n i t a r y o p e r a t i o n s on A ( i . e . maps An --tA

- =CA;F> w h e r e F is a p a i r A

so. A

for n =O,l,

0

... ) .

a set of An a l g e b r a

t e r m ( o r a l g e b r a i c ) o p e r a t i o n of

&

( o r F ) i s a n o p e r a t i o n o b t a i n e d from F v i a ( r e p e a t e d ) c o m p o s i t i o n and < < F > > s t a n d s f o r t h e s e t of t e r m o p e r a t i o n s o f F . (The c o m p o s i t i o n c o n -

s i s t s o f p r o j e c t i o n s , p e r m u t a t i o n s and f u s i o n s of v a r i a b l e s a s w e l l a s , x n ) by g ( x . , , . ,xm) p r o t h e replacement of t h e i - t h v a r i a b l e of f ( x , ,

. ..

..

d u c i n g t h e o p e r a t i o n f ( x , , . . . , X ~ - ~ , ~ ( X ~ , . . . , X ,~X+~ ~+ -, , ~, ,). . . , X ~ + ~ - ~ ) ) . For i n t e g e r s n . i and t h e n - a r y for all xl,

'

1 and a r A t h e r e a r e t h e i - t h n - a r y p r o j e c t i o n e r

c o n s t a n t cn d e f i n e d by a ct(xl, ,xn) = a ,xn) =xi, er(x and 5 d e n o t e t h e s e t s o f a l l p r o j e c t i o n s xniA. Let

,,...

...,

...

and c o n s t a n t s . The s e t <
IJ

C>>

i s c a l l e d t h e set of polynomials

of fi - or F and d e n o t e d P ( A - ) o r P ( F ) . ( N o t e t h a t P ( F ) may b e r i c h e r t h a n <> b e c a u s e w e may u s e p r o j e c t i o n s and c o n s t a n t s ) . A c l o n e i s s u b s e t M of

0 such

that C<MUI>>

=

M ( i . e . i t i s composition c l o s e d

and c o n t a i n s a l l p r o j e c t i o n s ) . The s e t

&

o f c l o n e s , o r d e r e d by i n -

c l u s i o n , i s known t o b e a n a l g e b r a i c l a t t i c e . F o r A f i n i t e t h e l a t -

tice

5

h a s a f i n i t e number o f d u a l a t o m s , c a l l e d maximal

o r p r e c o m p l e t e ) c l o n e s and e a c h p r o p e r s u b c l o n e o f

(preprimal

0 extends

to a

maximal c l o n e . Our a p p r o a c h i s b a s e d o n t h e c o n c e p t o f p r e s e r v a t i o n of a r e l a -

2.2.

t i o n . F o r h p o s i t i v e i n t e g e r a n h-ary r e l a t i o n p i s j u s t a s u b s e t o f Ah.

I f f i s an n-ary o p e r a t i o n on A w e s a y t h a t f p r e s e r v e s

compatible with

p )

whenever a l l ( x l j

p

(or is

if

,..., x h j )

subalgebra of t h e h-th

( j =I,...,n) d i r e c t power < A ; f >h ) . Ep

(equivalently, i f P is a I n t h i s paper preserva-

t i o n w i l l be a p p l i e d almost e x c l u s i v e l y t o b i n a r y r e l a t i o n s o n l y . For example, f p r e s e r v e s an o r d e r ever x

c

yl,...,xn

2

if f(x,,

... , x n )

y n , i . e . i f f i s :-monotone.

f ( y l , . . . , y ,)

when-

Naturally

4 =

If A - Dreserves a r e f l e x i v e graph p (= symmetric b i n a r y r e l a t i o n c o n t a i n i n g o = { ( a , a ) : a C A I ) w e s a y

preserves p i f every f F F does. that

p

is a tolerance of

fi.

A t r a n s i t i v e t o l e r a n c e of

5

is a congru-

e n c e o f A. - I t i s a n e q u i v a l e n c e o n A s u c h t h a t t h e v a l u e s of e a c h

f r F s t a y w i t h i n t h e same b l o c k w h e n e v e r t h e v a r i a b l e s v a r y e a c h i n s i d e a block1.A r e f l e x i v e g r a p h p + A 2 i s l o c a l l y c e n t r a l

i f f o r each

Compatible orderings and tolerances

125

f i n i t e s u b s e t B o f A w e h a v e { a ] x B c p f o r some a C A .

I t is central

i f { a ) x A ~ from p some a F A . L e t _R d e n o t e t h e s e t of f i n i t a r y r e l a t i o n s on A . F o r p C E l e t

2.3. Pol

p

be t h e set of f C Q preserving p .

F o r R_'_R p u t P o l R := (-1

Polp.

I n t h i s p a p e r w e u s e o n l y f i n i t a r y r e l a t i o n s and t h e r e f o r e i t P C R by a c o a r s e r o n e .

i s c o n v e n i e n t t o r e p l a c e t h e c l o s u r e x ---t<.:x>> on An n - a r y

f i s a l o c a l t e r m o p e r a t i o n of & - i f f o r every f i n i t e subset

B of A the r e s t r i c t i o n f t i o n g o f &.

I' B

agrees with g

rB

f o r some t e r m o p e r a -

The s e t o f l o c a l t e r m o p e r a t i o n s o f

&

i s d e n o t e d LOC

9

o r LOC F. Thus f b e l o n g s t o LOC F i f f t h e f i n i t e " p i e c e s " o f f a r e a l l among t h e f i n i t e p i e c e s o f o p e r a t i o n s f r o m < C F > > . W e s a y t h a t F

i s a local c l o n e i f L OC F = F . W e a s k t h e r e a d e r t o a c c e p t t h e f o l l o w i n g f a c t s and n o t i o n s . 2.4. some

2.5.

Proposition.

F_cO i s a l o c a l c l o n e i f and o n l y i f F = P o l R f o r

RcR.

For F s G p u t Inv F := { @ E R : f

a nonemptv f a m i l y

p r e s e r v e s o f o r a l l f F F 1 . Given

: j C. LJ) o f h . - a r v

r e l a t i o n s !P E J 3 (jiI J ) , a n i n I I j a nositive integer h, i l f . . . , i€ h1 and mans

dex set I * @ ,

{cp.

,...,

n . : 11 h . ) -I (j € J ) o u t 7 I I u :={(a(il) cx(ih)) :a € A

,...,

j EJ]

v1

J

for all

1

The s i m n l e s t and q u i t e t y p i c a l example i s t h e r e l a t i o n a l

(2.1).

product

c ~ ( n ~ ( ..., 1 ) ) a~( n j ( h . ) )€ v .

01p2

of t w o b i n a r v r e l a t i o n s d e f i n e d by

{ ( x , y ) : ( x , u ) Ecp,,

ipl 0 i p 2

=

( u , y ) E ( 0 2 f o r some u l . To see t h a t i t i s o f t h e

form ( 2 . 1 ) p u t I = { l f 2 f 3 1 , i , = 1 , i 2 = 3 n 1 ( 2 ) = 2 , n ( 1 ) = 2, n ( 2 ) 2 2 be c a l l e d a r e s o l v e n t of { L P

=

j

J = 1 1 , 2 1 and n

1

(1) = 1,

3. A r e l a t i o n o f t h e form ( 2 . 1 ) w i l l

:j E J I .

For I f i n i t e a r e s o l v e n t i s ob-

t a i n e d t h r o u g h a l o q i c a l f o r m u l a b u i l t up e x c l u s i v e l y from t h e p r e d i c a t e s c o r r e s p o n d i n a t o @ . ( jt J ) , e q u a l i t y , c o n j u n c t i o n and '1 ( t h e use of v

7

,7

a n d V i s n o t a l l o w e d ) . W e a s k t h e r e a d e r t o accept t h e

f o l l o w i n g p r o p o s i t i o n ( c f [ 2 8 1 ) which is l i s t e d h e r e f o r comDleteness s a k e and whose a p n l i c a t i o n s w i l l b e q u i t e t r a n s n a r e n t . 2.6.

P r o p o s i t i o n . L e t R g J 3 and F = Pol R .

lently,

Pol

p

2 F ) i f and o n l y i f

p

Then P E I n f F l o r e q u i v a -

i s t h e union o f a d i r e c t e d f a m i l y

of r e s o l v e n t s o f s u b s y s t e m s o f R U { A } .

I.G. Rosenberg, D.Schweigert

126

2.1.

A local clone F i s l o c a l l y m a x i m a l

i f F c j ? and F c G c O f o r no

l o c a l c l o n e G and f i n a l l y H G O i s l o c a l l y p o l y n o m i a l c o m p l e t e , i f Loc(H U C ) = 0. F o r A f i n i t e c l e a r l y LOC F = t < F > > and t h u s w e s h a l l drop t h e a d j e c t i v e "local". A polynomially complete a l g e b r a i s also c a l l e d f u n c t i o n a l l y complete, p r i m a l w i t h c o n s t a n t s o r S h e f f e r w i t h

constants.

2/3-MAJORITY OPERATIONS

63

3.1. ( )

A t e r n a r y o n e r a t i o n ( ) on A i s a 2 / 3 - m a j o r i t y

operation i f

* e 32 and (xxy) = x = (yxx)

(3.1)

h o l d s f o r a l l x , y € A . A s w e s h a l l see t h e 2 / 3 m a j o r i t y o p e r a t i o n s a r e q u i t e common. The f o l l o w i n g i s o u r s t a r t i n g p o i n t . 3.2.

Theorem.

Let

a-

=

be a f i n i t e a l g e b r a w i t h a

2/3 m a j o r i t y

polynomial. Then

i s p o l y n o m i a l l y c o m p l e t e i f a n d o n l y i f fi - is s i m p l e ,

A

A-

is

f o r no b o u n d e d o r d e r 5 a n d A - h a s no c e n t r a l t o l e r a n c e (as 2 i s s i m p l e i f it has o n l y t h e t r i v i a l congruences w and A

S-monotone

&

usual, and

C

i s b o u n d e d i f i t h a s a l e a s t and a g r e a t e s t e l e m e n t ) .

P r o o f : W e a p p l y [ 271 Thm 2 . L e t < A ; +

, O> b e a n a b e l i a n g r o u p and

l e t m d e n o t e t h e t e r n a r y o p e r a t i o n on A d e f i n e d by s e t t i n g m ( x I y I z ) =

x -y

+z

for a l l x f y l z € A . Further put

mo = I ( x l y , z I x - y + z ) : x l y l z E A I . I t s u f f i c e s t o show t h a t ( ) does n o t p r e s e r v e m0. S u o a o s e i t d o e s . L e t x , y , z € A be a r b i t r a r y . Since a l l t h r e e quadruples ~Y-Z,OfX-Y+~fX~f~Y-Zf~fzfy~f~ololzlz~

b e l o n g t o m0 and ( ) i s a 2 / 3 m a j o r i t y o p e r a t i o n , w e g e t 0

( y - z l O , z f ( x y z ) ) E m , i . e . m ( x f y I z ) = y-z-O+z = y . Now t h e c h o i c e o f x f y r z w a s a r b i t r a r y and t h e r e f o r e ( = e3. T h i s c o n t r a d i c t i o n p r o v e s 2

t h a t ( ) d o e s n o t p r e s e r v e mo. Let & - b e as i n Thm 3.2.

0

The c o n d i t i o n s f r o m t h e o r e m 3 . 2 . may p r o v i d e

a r o u g h i n f o r m a t i o n on P ( F ) . W e c o n s i d e r them i n more d e t a i l . 3.3.

L e t 5 b e a bounded p a r t i a l o r d e r and ( ) a 2 / 3 - m a j o r i t y

t i o n w h i c h i s a l s o 2-monotone.

opera-

I f a r b r c r x f y l z l ta r e e l e m e n t s o f A

127

Cornpa t ib le orderings and tolerances

such t h a t

x

yI

a

x

b

f ies

y, z

t, z

b y; z

w

x

w

t h e n w := ( a b c ) s a t i s -

t

c

t.

In particular

or

a

b

c

a

c

b - c

w
b

c

a - b

w

b

a

c * b

w

c

a

b a a

w ' c

a

b

c - w = b

a,

c

I n s m a l l c o n c r e t e e x a m p l e s t h i s a n d s i m i l a r a r g u m e n t s may b e u s e d t o ( c f . a more g e n e r a l d i s -

d e t e r m i n e bounded o r d e r s c o m p a t i b l e w i t h ( ) cussion i n [ 2 7 1 ) . For a b i n a r y r e l a t i o n

Cp

{ x : Ix)

:=

p

put

A x { x i _c P I .A r e f l e x i v e g r a p h i s c e n t r a l

x A _c p ,

(cf.

i f f @ + C p cA. For c e n t r a l r e l a t i o n s w e have t h e f o l l o w i n g s u f f i -

2.2)

c i e n t c o n d i t i o n . The r e l a t i v e l y s t r a i g h t - f o r w a r d p r o o f i s o m i t t e d . 3.5. x

Lemma. L e t

f A and

(

) be a

each two-element

2/3-majority

{a

subset

0'

o p e r a t i o n such

that t o every

a 1 } o f A\{x] t h e r e e x i s t s y E A

and i t { O , 1 1 s u c h t h a t

(a.xy) = a Then < A ;

(

) > h a s no c e n t r a l

1 -i

or

(yxai) = al-i

(3.1)

tolerance.

The f o l l o w i n g i s a n e x t e n s i o n of B e r g m a n ' s d o u b l e - p r o j e c t i o n t h e o r e m 111 t o s p e c i a l 2 / 3 m a j o r i t y f u n c t i o n s . F o r a n n - a r y

and k

1

=

,...,h

p u t p r k p : = { ( x 1,..., ~

S i m i l a r l y f o r 1 Y i 'j s h p u t p r i j p

-

~

prkp

relation

p

on A

..., , xxn ) I~( x,,..., + ~ xn ~)

tp}.

: = ~ ( X ~ , X ~ ) ~ ( X ~ , . E. p. }~. X The ~ )

a r e c a l l e d t h e b i n a r y r e l a t i o n s of

relations prij

see t h a t b o t h

~

p.

It is easy t o

a n d t h e b i n a r y p r o j e c t i o n s of P a r e r e s o l v e n t s o f P .

For a 2 / 3 m a j o r i t y o p e r a t i o n ( ) on A a n d a , b € A l e t Bab be t h e l e a s t s u b s e t of A such t h a t i ) a E B w h i l e t l , t 3E { b l U Bab

go B a b ) . 3.6. tion

W e have:

Proposition. Let (

(e.g.

a n d i i ) ( t , t 2 t 3E)B a b w h e n e v e r t 2 E B a b ab ( b a b ) ,( b ( b a b )( b a b ) ) and ( a ( b a b ) b ) belong

)

A-

such t h a t b €Bab

relation p

(h

2

have a 2/3 m a j o r i t y l o c a l polynomial f u n c for a l l

a,b €A.

Then P ( A - ) p r e s e r v e s an h - a r y

3 ) i f and o n l y i f i t p r e s e r v e s a l l i t s b i n a r y p r o j e c -

t i o n s . If P ( A - ) p r e s e r v e s an h - a r y h a r e equal t o A2 t h e n p = A

r e l a t i o n p whose b i n a r y p r o j e c t i o n s

.

Proof: L e t h 2 3 and l e t P ( 8 ) u r e s e r v e an h-ary r e l a t i o n

p.

Put

u . : = P r . p (i = 1 , 2 , 3 ) a n d d e f i n e 1

T:=

{(XI,

..., I Xh)

(X2'

...,Xn ) E U l

(X1r

...I ... X3r

pXn)

Ea2, (X.,,x2,X4, . . . , X n ) t o 3 1.

I. G. Rosenherg, D. Schrueigert

128 Here

i s c l e a r l y a resolvent of

T

Clearly

PST.

such t h a t a l l h-tuples

(ulx2,.

( X ~ , X ~ , W , X ~ ~ . . . , bXe~l o) n g

. . ,x n ) ,

to p.

2

and f o r i = 1 , 2

..., x n )

t

(x, ,v,x3,.

7 ,

= P.

T

There a r e u,v,w

.. , x n ) ,

By i n d u c t i o n w e Drove t h a t

( x l , t l x3 , . . . , ~ n ) E p f o r a l l t t B v x ( x l ,t 2 , X 3 , .

Suppose t h a t

We p r o v e t h a t

{al,n2,u31.

For t h e c o n v e r s e l e t ( x , ,

.

Indeed t h i s holds f o r t

.., x n )

v. (3.1)

P

. . . ,x n )

e i t h e r t i = x 2 or ( x l , t i , x 2 ,

=

c p . Then t h e r e a r e

p and r s u c h t h a t

(P, . , , x 3 , . . . , x n )

E P ,

(x1,t3,r,x4,

..., x n )

(3.2)

t p

( P u t p = u i f t 1 = x2 a n d p = x1 o t h e r w i s e and s i m i l a r l y r = w i f

t 3 = x2 and r = x 3 e l s e . ) A p p l y i n q ( ) t o t h e f i r s t h - t u p l e from (3.2 I t h e h - t u p l e from 3 . 1 a n d t h e l a s t h - t u p l e from 3 . 2 w e q e t

.

Since ( ) is a ( t l t 2 t 3 )x,3 x 3 r ) , ( x 4 x 4 x 4 ),..., ( x n x n x n ) ) r p ( x , , ( t l t 2 t 3, x)2 , . ,xn) 1 p and h a v e by i n d u c t i o n ( x l , t , x 3 , ..., x n ) C p f o r a l l t C B v x By assump((PX X I )

I

..

2 / 3 majority operation we get the required

.

t i o n w e have x C B v x

and t h e r e f o r e ( x l , .

2

. ., x n )

2 i p proving T 5 0 .

By an e a s y i n d u c t i o n w e o b t a i n t h a t p i s a r e s o l v e n t o f t h e s y s t e m

I

tpr. .P 11

1 C i < j 5 h l p r o v i n g t h e f i r s t s t a t e m e n t . The o t h e r s a r e d i -

rect consequences.

54

0

MAJORITY OPERATIONS

4.1.

A 2 / 3 m a j o r i t y o p e r a t i o n ( x y z ) s a t i s f y i n g (xyx) = x f o r a l l

x l y t A i s c a l l e d a m a j o r i t y o p e r a t i o n . For example, e v e r y l a t t i c e h a s two m a j o r i t y p o l y n o m i a l s , c a l l e d m e d i a n s ,



(xyz), = x y + x z + y z ; (xyzIu = ( x + y ) ( x + z ) ( y + z ) ( w h e r e , as u s u a l x y + x z + y z s t a n d s f o r ( x s y ) + ( x 9 z ) (

)1 a n d

(4.1)

+

(y-z)). In fact

( j U a r e t h e l e a s t a n d g r e a t e s t m a j o r i t y monotone o p e r a t i o n s

on a l a t t i c e :

4.2.

F a c t . L e t < A ; + , - > b e a l a t t i c e and

Ii t s o r d e r .

I f

(

)

is a

<-monotone m a j o r i t y o p e r a t i o n

(4.2

(xyz f o r a l l x,y,z EA.

Proof: L e t x , y , z € A and i = x y ( X Y Z ) ~5

( x y z ) . S i m i l a r l y (xyz

Then i = ( i i z ) 2

(xyzIu.

0

I( x y z )

a n d by symmetry

129

Compatible orderings and tolerances

A l g e b r a s h a v i n g a m a j o r i t y o p e r a t i o n among i t s l o c a l p o l y n o -

4.3.

m i a l s h a v e much more s t r u c t u r e t h a n t h o s e w i t h a 2 / 3 - m a j o r i t y

(cf.

- 4.6.).

F i r s t w e need t w o more c o n c e p t s . W e s a y t h a t a r‘; 2 r e f l e x i v e graph has i n f i n i t e d i a m e t e r i f 11 p n = A 2 w h e r e a s p n + A n=1 1 ( h e r e p o u h a s been d e f i n e d i n 2.5., P := p a n d f o r a l l n =1,2,. n+l n D = p 0 is f o r n = 1 , 2 , . .. ) . A set R of ( f i n i t d r y ) r e l a t i o n s on A is Thm. 4 . 4 .

..

generic 0

for A

=

/A;F> i f t o e a c h p r o p e r l o c a l c l o n e C Z F t h e r e i s

( - Rs u c h t h a t C s P o l

p

‘0.

The f o l l o w i n g t h e o r e m i s a r e f o r m u l a t i o n o f [ 3 8 1 Thm 3 . S i n c e i t s p r o o f i n !381 i s b u t a d i r e c t v e r i f i c a t i o n o f t h e c o n d i t i o n s o f a l o -

c a l c o m p l e t e n e s s c r i t e r i o n 1 2 9 1 w e o m i t t h e p r o o f . The s e c o n d s t a t e ment i s i n 1281 w h e r e a s t h e l a s t i s a n i m m e d i a t e c o n s e q u e n c e o f t h e first. Theorem. L e t A - = h a v e a m a j o r i t y o p e r a t i o n among i t s l o c a l p o l y n o m i a l s . L e t R c o n s i s t o f ( i ) l a t t i c e o r d e r s c o m p a t i b l e w i t h A- , 4.4.

t o l e r a n c e s o f A, - ( i i i )n o n t r i v i a l c o n g r u e n c e s o f ( i v ) t o l e r a n c e s of A - of i n f i n i t e d i a m e t e r . T h e n R i s g e n e r i c .

(ii)i l o c a l l y c e n t r a l

A-

and

Moreover, mal.

f o r p listed in

Finally

A-

( i ) - ( i i i t)h e c l o n e P o l p i s l o c a l l y m a x i -

i s l o c a l l y c o m p l e t e i f a n d o n l y i f R = @.

F o r A f i n i t e c l e a r l y l o c a l l y c e n t r a l t o l e r a n c e s are c e n t r a l a n d ( i v ) d o e s n o t a p p l y . The r e l a t i o n s l i s t e d a b o v e are l i k e l y t o p l a y a n e m i n e n t r o l e f o r a l g e b r a s w i t h a l o c a l m a j o r i t y o p e r a t i o n . The f a c t

i s n o t a c c i d e n t a l . The f o l l o w i n g P r o p o s i t i o n i s a s p e c i a l c a s e o f a more g e n e r a l t h e o r e m [ I ] .

t h a t o n l y b i n a r y r e l a t i o n s f i g u r e i n Thm 4 . 4 .

F o r c o m p l e t e n e s s s a k e w e g i v e a s h o r t p r o o f . For a n h - a r y on A and 1 < k C h , P r k p := I ( x , ,

1 5 i 5 j c h put

.. .,xk-,

,xk+., , . . . , x h )

p r . . p : = I ( x i , x . ) : (x, 13

7

resolvents of I p l 4.5.

,..., x h )

Ep).

:

(x,,

. . ., x h ) t P 1

relation P

and

I t i s e a s y t o see t h a t b o t h a r e

(2.5.).

Proposition. L e t

A-

have a m a j o r i t y

local polynomial.

Then

Lac P ( A -) = Pol R f o r a s y s t e m R o f r e f l e x i v e b i n a r y r e l a t i o n s . I f a n 2 h - a r y r e l a t i o n p i s p r e s e r v e d b y - a n d a l l p r . .p = A ( 1 C i ‘ j 5 h ) t h e n 1 7 h p = A .

The p r o o f f o l l o w s f r o m P r o n o s i t i o n 3 . 6 .

I.G. Rosenberg, D.Schweigert

130

pn is the transitive n= 1 ( i . e . t h e least t r a n s i t i v e r e l a t i o n c o n t a i n i n g p ) . L e t p be

F o r a b i n a r y r e l a t i o n p t h e r e l a t i o n t r p := h u l l of

p

a b i n a r y r e f l e x i v e r e l a t i o n comDatible w i t h P ( & ) . Since a t h e u n i o n of a c h a i n o f r e s o l v e n t s p n o f { p l from ProD.

: = t r p

is

2.6. we i n f e r

t h a t a i s c o m p a t i b l e w i t h P ( f i ) . Now t r p i s c l e a r l y a n e q u i v a l e n c e and t h e r e f o r e a i s a c o n g r u e n c e of P ( & ) . W e h a v e : 4.6.

Proposition. Let

be a proper

A-

A-

have a m a j o r i t y l o c a l polynomial

and

let p

t o l e r a n c e o f A. - I f n e i t h e r t r p i s a proper congruence o f t h e n p n i s l o c a l l y c e n t r a l f o r some

nor p has i n f i n i t e diameter,

n >O. P r o o f . From t h e a s s u m v t i o n s w e g e t t h a t p k = A

2 f o r some k > 1 . L e t m

be t h e l e a s t i n t e g e r w i t h t h i s wrowerty and l e t n b e t h e l e a s t i n t e g e r >$n. Then a := p n i s a p r o a e r t o l e r a n c e s u c h t h a t a 2 = A 2 . : = { ( x , ,... , x h ) : ( x , , u ) € a ,..., ( x h , u ) E o f o r some u }

put Ah

.

For h ' 3 W e say

t h a t an h-ary r e l a t i o n i s t o t a l l y r e f l e x i v e i f i t c o n t a i n s each h-tup l e w i t h a t l e a s t o n e r e D e t i t i o n . From t h e s e c o n d s t a t e m e n t of P r o p .

4.5. we infer that for h

?

3 t h e r e l a t i o n Ah i s t h e s i n q l e t o t a l l y re-

f l e x i v e h - a r y r e l a t i o n from I n v P ( q ) From ci2 = A A 3 =A3.

.

2 w e o b t a i n t h a t A 3 i s t o t a l l y r e f l e x i v e of I n v P ( 4 ) . Thus

By t h e same t o k e n A 4 i s t o t a l l y r e f l e x i v e i . e .

t i n u i n g we qet Xh = A h

for all h

means t h a t a i s l o c a l l y c e n t r a l . I f P(A) c o n t a i n s t h e o n e r a t i o n s

h4 =A

4

.

Con-

3. Accordinq t o t h e d e f i n i t i o n t h i s

2

0

+

and

-

o f a l a t t i c e w e can r e s t r i c t

R t o t h e n a t u r a l o r d e r 2 o f t h e l a t t i c e and t h e t o l e r a n c e s of P ( & ) .

F o r t h e f i n i t e c a s e t h i s was p r o v e d i n [ 311 and i n g e n e r a l i n [ 2 1 . Our p r o o f f o l l o w s t h e i d e a s o f [ 2 1 . L e t A f l e x i v e c o m p a t i b l e b i n a r y r e l a t i o n s of r a n c e s o f &. 4.7.

(5) d e n o t e t h e s e t o f reg t h e s e t o f tole-

and To1

Theorem. L e t L_ - b e a l a t t i c e and 5 i t s o r d e r . Then = P o l ( 5 U T o 1 & ) , i e . an o p e r a t i o n on A i s l o c a l p o l y n o m i a l

LOC ),P.I(of

&

.

i f and o n l y i f f i s %'-monotone and p r e s e r v e s e a c h t o l e r a n c e o f

Proof. L e t L = < A ; + , - >

and l e t 6 € A ( & ) .

C l e a r l y 6 + and 6 ) a r e t o l e r a n c e s o f

&.

Set

+-

P u t p : = { ( x , y ) :x6+u6 y

for

&.

Compatible orderings and tolerances

some x

iu 1 y l .

Clearly p is a resolvent of

131

and t h e t w o t o l e r a n c e s

2

6 + and 6 + . I t r e n a i n s t o show t h a t p = 6 .

1. L e t ( x , y )

C=

6.

Setting u

(u,y) = (x+y,y+y) t 6 .

=

x+y w e h a v e ( x , u ) = ( x + x , x + y ) t 6 a n d

S i n c e x :u w e h a v e ( x , u ) C 6 + a n d from ( x , u ) F 6 +

and ( u , u ) t 6 + a l s o ( x , u ) C 6 + . I n a s i m i l a r f a s h i o n ( u , y ) E 6 + a n d therefore (x,y) E p .

2 . L e t ( x , y ) F p . Then t h e r e e x i s t s x r u " y r v ( x , v ) t6+, ( u , v ) C 6 + ,

(u,w) F 6 - ,

(y,w) t 6 - .

?

x + u and w Luy s u c h t h a t

Now

' x , u ) = ( x u , v u ) € 6 and

( u , y ) = (u+y,w+y) t & . F i n a l l y ( x , y ) = ( x u , u y ) E 6 which c o n c l u d e s t h e proofon 4.8.

Remark. L e t

r ( L- ) b e t h e s e t o f c o m p a t i b l e e x t e n s i o n s o f

h (L ) t h e set of compatible s u b r e l a t i o n s of s .

e.g.

$5

2,

Con

&

t h e t h r e e f i r s t are i s o m o r p h i c .

I n v i e w o f Thm 4 . 5 . s t u d i e d i n $96

and

I n ( 2 1 and [ 3 5 1 t h e r e

a r e s e v e r a l i n t e r e s t i n g i n t e r a c t i o n s b e t w e e n r(A),A(L),Tol a n d A(&),

~1

t h e set T o 1

&

i s of p r i m e i m p o r t a n c e . I t w i l l b e

- 8.

COMPATIBLE ORDERS I N ALGEBRAS W I T H LOCAL M A J O R I T Y POLYNOMIALS, SEMILATTICES AND LATTICES

5.1.

I n t h i s s e c t i o n w e c o n s i d e r t h e c o m p a t i b l e d i r e c t e d orders i n

a l g e b r a s w i t h a l o c a l m a j o r i t y p o l y n o m i a l ( ) . They t u r n o u t t o b e completely determined by ( ) .

For b e t t e r r e s u l t s we look a t algebras

h a v i n g l a t t i c e o p e r a t i o n s among i t s l o c a l p o l y n o m i a l s . A c c o r d i n g t o Thm 4 . 7 .

t h e orders compatible with a lattice

&

distinct f r o m its

o r d e r are n o t e s s e n t i a l ( i n t h e s e n s e t h a t t h e y d o n o t r e s t r i c t LOC P ( & ) i . e . may b e r e p l a c e d by t o l e r a n c e s ) . S t i l l i t may b e i n t e -

r e s t i n g t o know t h e s t r u c t u r e o f c o m p a t i b l e d i r e c t e d a n d d o w n - d i r e c t e d o r d e r s of a l a t t i c e

&.

T h i s h a s been completely s o l v e d i n [ 9 1

w h e r e t h e y a r e shown t o b e i n 1-1 c o r r e s p o n d e n c e w i t h d i r e c t decompo-

s i t o n s of & ( a n d t h u s f o r bounded l a t t i c e s w i t h t h e c e n t e r o f & ) i . e . t h e y are d e t e r m i n e d b y s p e c i a l c o n g r u e n c e s o n l y . Q u i t e s u r p r i s i n g l y , s i m i l a r a n d more g e n e r a l r e s u l t h o l d s f o r c o m p a t i b l e o r d e r s o f s e m i lattices [22]. F i r s t we study compatible orders f o r majority operat i o n s (5.2.

-

5.7.)

and t h e n f o r semilattices (5.8.

-

5.23).

The f o l l o w i n g i s i m p l i c i t i n [ 3 8 ] a n d [ 2 ] . W e s a y t h a t 5 i s t h e o r d e r of a p a r t i a l

u p p e r s e m i l a t t i c e < A ; v > i f a l e a s t u p p e r bound xvy e x i s t s

I. G. Rosen berg, D.Schweigert

I32

f o r e a c h p a i r x , y € A h a v i n g a common u p p e r bound. 5.2.

Lemma. L e t

o p e r a t i o n on A.

be a n o r d e r o n A a n d Then

i

Proof. L e t x

monotone m a j o r i t y

i s the o r d e r o f a p a r t i a l

x 5 z,y 5 z

a n d

) a

(

=$

upper semilattice

(xyz) = (xzy) = (zxy) =xvy

S e t x v y = (xyz).Then x = (xyx)

z and y 5 z .

and s i m i l a r l y y = ( x y y ) 5 x v y . Moreover, i f x

(5.1)

(xyu) = x v y

t and y - t , t h e n

x v y = (xyz) z ( t t z ) = t proving t h a t x v y = ( x y z ) i s t h e least upper bound of x and y . An a r b i t r a r y e x c h a n g e o f v a r i a b l e s i n ( ) l e a d s t o a .--monotone m a j o r i t y o p e r a t i o n ( 1 . From t h e a l r e a d y p r o v e d x v y = (xyz) we q e t (xzy) = (zxy) = x v y . 5.3.

COrOllarY. I f 2 i n L e m m a 5 . 2 .

o f a n u p p e r s e m i l a t t i c e

CI

i s directed,

then

i s the o r d e r

(5.1.) and

satisfying

(L1) ((xyu)zu) = (x(yzu)u) whenever

u is

an upper bound

o f

X,y a n d

Z.

(L, ) i s t h e a s s o c i a t i v e l a w .

Proof.

I n view o f Thm 4 . 4 .

w e a r e i n t e r e s t e d i n b o t h d i r e c t e d and down d i -

rected orders. 5.4.

C o r o l l a r y . L e t 2 be a d i r e c t e d a n d d o w n - d i r e c t e d o r d e r on A a n d

( ) a 6-monotone

lattice


m a j o r i t y o p e r a t i o n on A .

Then 5 i s the o r d e r o f

xlylz €A.

such that f o r a l l

(L!,)

(xyz) = (xzy) = (zxy) = x v y

i f

z 2

(L;)

(xyz) = (xzy) = (zxy) =

i~ f

z 2 xhy.

X

a

A

x vy,

Proof. Corollary 5.3. 5.5.

Corollary 5.3.

does n o t provide a c r i t e r i o n f o r t h e d e t e c t i o n

o f d i r e c t e d and d o w n - d i r e c t e d o r d e r s c o m p a t i b l e w i t h a g i v e n m a j o r i t y o p e r a t i o n . We may e x p r e s s t h e a s s o c i a t i v e l a w s as (L2) ((xyb)zb) = (x(ybz)b) i f b is a b o u n d a n d z and t h e a b s o r p t i v e

l a w s as ( L 3 )

( l o w e r or u p p e r ) o f

x,y

( ( x y u ) x l )= x = ( ( x y 1 ) x u ) i f

1 2 x 2 u a n d 1 S y 5 u ( t h e commutative l a w i s g u a r a n t e e d by ( L i ) and ( L y ) ) . C l e a r l y (L;), ( L i ' ) ,

( L 2 ) and ( L 3 ) p r o v i d e n e c e s s a r y c o n d i t i o n s

f o r t h e c o m p a t i b i l i t y of S and ( ) . T h u s t h e y may s e r v e f o r t e s t i n g o f c o m p a t i b i l i t y b u t w i l l n o t t e l l u s how t o f i n d o r d e r s c o m p a t i b l e

Compatible orderings and tolerances with

Proposition. Let

5.6.

let

and

Some o f t h e d i f f i c u l t i e s d i s a p p e a r i f

).

(

(

termined

) be

has a least element.

<-

o

be a d i r e c t e d order w i t h a l e a s t element

-monotone.

o and ( i i i )

by

133

(i)x

Then

y iff

(xyo) = x .

(ii)

i s de-

~

i s a l a t t i c e order with x h y = (xyo).

P r o o f . C o r o l l a r y 5. 3. F o r bounded o r d e r s t h e c o m p a t i b i l i t y may b e e x p r e s s e d e n t i r e l y i n

terms of 5.7.

(

)

and t h e b o u n d s .

Proposition. Let

greatest

e l e m e n t e and

2

let

(

) b e a m a j o r i t y o p e r a t i o n on A .

' - m o n o t o n e i f and o n l y i f f o r a l l

is

o

b e a n o r d e r on A w i t h a l e a s t e l e m e n t

and

Then

(

)

~ ~ x ' ~ y ~ yt A' ~andz , ~ '

i C I := { o , e ) (L1)

( x y z ) does n o t depend on h e o r d e r o f v a r i a b l e s whenever a t l e a s t o n e o f x , y and z b e l o n g s t o I

(L2)

((xyi)zi) = (x(yiz)i)

(L3)

((xyo)xe)

=

x = ((xye)xo

P r o o f : The c o n d i t i o n s assures that

5.8.

(

)

( L 1 ) - ( L 3 ) a r e t h e l a t t i c e axioms a n d ( M )

i s '-monotone.

0

W e t u r n t o semilattices.

Thm 4 . 1 .

and

I

S u r p r i s i n g l y t h e d e c o m p o s i t i o n from

works f o r compatible o r d e r s o f semilattices. L e t

a semilattice, t i b l e order of

i t s o r d e r ( d e f i n e d by x

S. -

A s i n Thm 4 . 1 . 6 + := [ n

5

= be

y i f xy = x ) and [ a compa-

put 6-

:=

[ n

1.

I t w i l l b e more c o n v e n i e n t t o u s e t h e t w o t o l e r a n c e s

.

: = 6 , 0 6;l (5.2.) :="I 0 6-; n2 u ' y , x [ u [ y f o r some u a n d ( x , y ) E n 2 i f f Ill

Thus ( x , y ) F q , x

u

pairs

5.9.

iff x

y I x [ u [ y f o r some u . The c h a r a c t e r i s t i c p r o p e r t i e s o f t h e (ql,n2)

t h u s o b t a i n e d are t h e f o l l o w i n g

Lemma. L e t

the associated

tolerances.

a s u b o r d e r o f [ Cl

(x,y)

Fni

[ b e a c o m p a t i b l e o r d e r of a s e m i l a t t i c e and

.,

and

Then:

E2:=n

n

( i ) n, n

n2

i s asuborder o f

2 w h e n e v e r x a n d y h a v e an u p p e r bound i n

n1

(ii)C1 =

= w I

1 rl ,

ci

and

q

11n2

fl-.

(iii)

( i= 1 , 2 ) .

i s

I.G. Rosenberg, D,Schweigert

134

Proof: L e t (x,y) E q l O n 2 .

Then t h e r e a r e u p p e r b o u n d s t , s o f x , y s u c h

t h a t x I t [ y and x [ s ] y . Due t o c o m p a t i b i l i t y w e h a v e x [ s t [ s x = x i.e.

x = s t . By symmetry s t = y p r o v i n q

show t h a t c ; , ~ [ . I n d e e d l e t ( x , y ) F C 1 . Then x

=

xt

q1 nq2 =

id.

We

y I x l u l y and

x < u ? y f o r some u . A p p l y i n g c o m p a t i b i l i t y w e h a v e x = xu [ xy = y

i.e.

( x , y ) C [ fie-.

.

Similarly

c2 E [

n . . I t follows t h a t 5 , is t r a n -

s i t i v e , i . e . a n o r d e r ( i = l , 2 ) . L e t x , y h a v e a n u p p e r bound z i n F 1 . Then from t h i s to

el

1 we

_c]

we get x

z

y, x ] z [ y, i.e.

c2.

get the result for

(x,y) € n l .

Applying

0

Lemma. L e t [ b e a c o m p a t i b l e o r d e r of s e m i l a t t i c e , n 1 and q 2 t o l e r a n c e s . Then [ e q u a l s u : = e l 0 C 2 = ( q l n c ) O ( n 2 1 1 ‘ ) .

5.10.

the associated

P r o o f . L e t x [ y . Then x = xx [ xy whence from x qet (x,xy) E

el.

S i m i l a r l v xy

xy and x [ x ] x y w e

x

(L.5.8.).

c2_c[

s e t of pairs ( n l , q 2 )

:=nl

n2

c[

and

o

L e t P be t h e s e t o f c o m p a t i b l e o r d e r s o f ( i i ) c;

L2.

y a n d x y [ v y = v l v shows ( x y , y )

y

T o q e t h e r ( x , ~ E) u and [ 5 0 . The c o n v e r s e f o l l o w s from E l

o f t o l e r a n c e s of

and C 2 : = n 2 n

2

2.

Further let T denote the

s u c h t h a t ( i )r l l

n n 2 = o, n i whenever

a r e o r d e r s and ( i i ) ( x , y ) E

x and Y h a v e a n u p p e r bound i n t i

( i = l , 2 ) . I t f o l l o w s from L . 5 . 9 .

t h a t f : + ( n l l n 2 ) maps P i n t o T. C o n v e r s e l y t o ( n l , n 2 ) E T w e associate q ( q l , n 2 ) = ( n , n l ) 0 ( n 2 n < ) . W e have:

Lemma. The m a p g maps T i n t o P. M o r e o v e r ,

5.11. (ql,n2)

E T we h a v e f ( q ( n l r n 2 ) )=

Proof. Obviously

< :=

symmetry l e t x i y 4 x

(n,

. By

f’ > )

0

(ni,ni)

( n 2 n.)

where

CJ

o f = i d p and f o r

nisni

(i=l,2).

i s r e f l e x i v e . To m o v e a n t i -

d e f i n i t i o n x > t‘ v ,

x n l t u 2 y and x ’ s

‘y,

xq s n y . From t = t y w e q e t t n l t s a n d s i m i l a r l y from t = t x w e g e t 2 1 t n 2 t s which c o u p l e d w i t h n 1 n n 2 = o p r o v e s t = t s . S i m i l a r l y t s = s ,

i.e.

t = s , and from x n l t , x n 2 t w e g e t x = t and s i m i l a r l y t = y . F i n a l -

l y t o p r o v e t h a t u i s t r a n s i t i v e l e t ( x , y ) E o and ( y , z ) F u . Then x c l t ~ 2 y c 1 s Cz f o r some t and s . A s a b o v e w e h a v e t c ; l t s S 2 s . S i n c e b o t h 2 F 1 and c 2 a r e o r d e r s w e h a v e a l s o x S l t s E 2 z p r o v i n g ( x , z ) E o . The i d e n t i t y q o f = i d p f o l l o w s from L . 5 . 1 0 . I t r e m a i n s t o p r o v e t h a t f ( ( ) := ( n { , q $ )

satisfies q j

sni

x n i y . Then t h e r e a r e t , u , s s u c h t h a t x ‘ u ? y , x > t‘ u ’ s

(i= 1,2). ( y ,

Let

Compatible orderings and tolerancor

135

x q 2 t q l u n 1 s n 2 y . From x = x u 0 x t = t w e g e t x = t . By t h e same t o k e n x = y 1 and from xS1u, yS1u w e g e t x11,y. The p r o o f f o r n 2 i s s i m i l a r . 0 5.12.

We b r i n q now a s i m i l a r c o n s t r u c t i o n which a s s o c i a t e s t o a com-1 a p a i r o f c o n g r u e n c e s on g . P u t e l := [ + 0 I+ and o [ - i . e . x e y i f f x:uiy, x [ u l y f o r some u , ( 5 . 3 . ) x e 2 y i f f

patible order [ of

e2=[I’

1

x ’ u ~ y x l u ~ yf o r some u. ( 5 . 4 . ) C l e a r l y 8,,e2 a r e t o l e r a n c e s o f

2.

P r o c e e d i n g a s a b o v e i t i s e a s y t o show t h a t t h e y are t r a n s i t i v e , i . e .

2. P u t e l n e e 2= w .

x i y i f xB1uB 2-v a n d x ? u 5 v f o r some u. I t may W e show t h a t i n t h i s case [ i s d e t e r m i n e d by

c o n g r u e n c e s of happen t h a t

L e t gi = S _ / O i ( i = 1 , 2 ) and = g, x E 2 . I t i s w e l l known t h a t t h e r e e x i s t s a s e m i l a t t i c e i n j e c t i v e h o m o m o r p h i s m

Y

e l a n d e 2 i n t h e f o l l o w i n g way. x + ( x 1 , x 2 ) from

s

into

u.

P u t x 4 y i f x1 > y l ( i n

-

gl)

and x2 c y 2 ( i n

g2).

Thus x 4 y i f f t h e image o f x i s smaller t h a n t h e image o f y i n d x S 2 (where S: i s t h e d u a l o f S 1 ) . W e s a y t h a t ( t h e d i r e c t p r o d u c t S1

stems f r o m t h e s u b d i r e c t r e p r e s e n t a t i o n . W e have:

5.13.

Theorem. T h e f o l l o w i n g a r e e q u i v a l e n t f o r a c o m p a t i b l e o r d e r [

o f a semilattice S =

(5.3.) and (i)

and

the congruences

and

€I1

B2

(defined by

(5.4.):

el

f l e 2 = a and

by

and

8

[ stems from the s u b d i r e c t r e p r e s e n t a t i o n g i v e n

2‘

(ii) Each i n t e r v a l

i n [ i s a convex s u b s e t o f 2 ,

( i i i )F o r a < b 5 c

S k e t c h of t h e p r o o f :

a [ c + b [ c ,

c [ a * c [ b .

( i )* ( i i ) .The i n t e r v a l [ a , b ] i n [ c o n s i s t s o f

t h e e l e m e n t s of A whose images f a l l b e t w e e n ( b l , a 2 ) and ( a , , b 2 ) . ( i i ) =5 ( i i i )D i r e c t .

( i i i )* ( i ) .Check O., n O 2 = w

and t h a t

<

from

5.11. equals [ .

In a lattice

&

:=

t h e c o n d i t i o n ( i i i )a l w a y s h o l d s .

Indeed, l e t

a s b < c . I f a [ c t h e n b = a + b [ c + b = c and i f c [ a t h e n = c + b [ a + b = b. S i n c e ConcA; * > = Con 4 w e o b t a i n from Thm 5 . 1 2 : 5.14

C o r o l l a r y . T h e map [

-B

(el,e2

from compatible orders o f a l a t t i c e

rect r e p r e s e n t a t i o n s o f

In particular,

L

and

2

f r o m 5.12.

L-

induces a bijection

and o r d e r s stemming from subdi-

&. a r e t h e s i n g e c o m p a t i b l e o r d e r s of a s u b d i -

r e c t l y i r r e d u c i b l e l a t t i c e . W e need t h e f o l l o w i n g :

I.G. Rosenberg, D.Schweigert

136

5.15.

Theorem. I f

i s a compatible ordering o f a lattice,

a m e e t s e m i l a t t i c e i f a n d only if [ is d o w n - d i r e c t e d ,

[ i s up-directed.

a j o i n s e m i l a t t i c e i f and o n l y i f

1

a l a t t i c e ordering i f f

i s down-directed

Proof. L e t [ be up-directed.

and

then

I

and d u a l l y

I

Consequently,

is I S

1

is

up-directed.

1

Let a,b,c r A s a t i s f y c

a , c ] b and l e t

t h e images i n t h e c o r r e s p o n d i n g s u b d i r e c t r e p r e s e n t a t i o n be ( a l l a 2 ) , ( b l , b 2 ) and ( c l , c 2 ) . Then c 1

a l b l and c 2

a 2 + b 2 . Then i n t h e imaqe

w e h a v e t h e e l e m e n t ( a l b l , c 2 ) ( a s t h e j o i n of ( a l b l , a 2 b 2 ) ) and

of

( c , , c 2 ) and t h e e l e m e n t ( a l b l , a 2 + b 2 ) ( a s t h e meet of ( a l + b l , a 2 + b 2 ) . The p r e i m a g e d of

( a l b l l c l ) ) dnd

( a l b l , a 2 + b 2 ) i s t h e n t h e l o i n of a

a n d b i n [ . The c o n v e r s e i s o b v i o u s .

5.16.

F o l l o w i n g 191 w e look more c l o s e l y a t t h e c o m p a t i b l e l a t t i c e

o r d e r s of l a t t i c e s . L e t with o r d e r s

L -

= (A;+,.>

g

and

b e two l a t t i c e s

= (A;v,A>

- W e call K s u c h t h a t [ i s a c o m p a t i b l e o r d e r o f &.

and

a n o r i e n t a t i o n of &, - i n s y m b o l s & -, g. F i r s t w e show t h a t + i s a symmetric ( a n d r e f l e x i v e ) r e l a t i o n o n t h e s e t o f a l l l a t t i c e s on A 1 9 1 . 5.17.

Lemma.

I f & -

+

-

K_ - t h e n K_ -

&. -

p r o o f . I t s u f f i c e s t o show t h a t t h e o r d e r r e l a t i o n t i b l e w i t h v and XI

A.

v x 2 = (x1x2) +

proof f o r

A

Let x. -yi

(XIZ)

+ (x2z)

is similar.

'

-

and l e t z = y

(i=1,2)

(Y1Y2) +

( u p )+

(Y2Z)

of

&

i s compa-

v y 2 . Then =

Y1 V Y 2 . The

0

F i r s t w e c o n s i d e r o r i e n t a t i o n s w i t h a l e a s t e l e m e n t 0 . Reca 1 t h a t a E A is n e u t r a l

i f f o r a l l x,y C A the set (a,x,y} generates a d i s t r i -

b u t i v e s u b l a t t i c e of

&.

Orientations with a least ( g r e a t e s t

a r e i n 1-1 c o r r e s p o n d e n c e w i t h t h e n e u t r a l e l e m e n t s o f

element

&. W i t h o u t

p r o o f t h i s i s made e x p l i c i t i n : 5.18.

Proposition.

An

order

[ on A w i t h a l e a s t e l e m e n t 0 i s a n o r i e n -

t a t i o n o f a l a t t i c e L_ - = (A;+,*> i f a n d o n l y i f (i) 0 i s a n e u t r a l e l e m e n t of - and

L,

(ii) [ i s a l a t t i c e o r d e r s a t i s f y i n g x [ y

c+

x 0 3yO

and x + O

< y

+O.

(5.5) W e t u r n t o t h e bounded o r i e n t a t i o n s o f

&

1 9 1 . Recall t h a t i n a l a t t i c e

L w i t h a l e a s t e l e m e n t 0 and a g r e a t e s t e l e m e n t 1 a n e l e m e n t a E A i s

Compatible orderings and tolerances complemented

137

i f a a ' = O and a + a ' = 1 f o r some a ' C A ( w h i c h i s u n i q u e

and c a l l e d t h e c o m p l e m e n t o f a ) . The s e t

N(5)

o f complemented n e u t r a l

elements i s c a l l e d t h e c e n t e r of L. W e need t h e f o l l o w i n g : Lemma. I f 0 C r N (-L ) , t h e n x = xv + x o + y o

5.19. a l l

y = xy

+XO'

+yo' for

x , y G L-.

The n e x t t h e o r e m [ 9 1 b r i n g s f o r w a r d a 1-1 c o r r e s p o n d e n c e b e t w e e n bounded l a t t i c e o r i e n t a t i o n s a n d t h e c e n t e r of 5.20.

Theorem. L e t

greatest

element

5.

[ b e an o r d e r on A w i t h a l e a s t e l e m e n t 0 and a

e. Then

i

i s an o r i e n t a t i o n o f a l a t t i c e

L i f and o n l y i f - =
(5.5).

The t h e o r e m mav b e p r o v e d d i r e c t l y from P r o p .

5.14 a n d L . 5 . 1 9 .

We

q u o t e an i n t e r e s t i n a r e s u l t from 191. Then L have i d e n t i c a l i n t e r v a l s ( c o n s i - and K d e r e d a s s e t s ) . If, m o r e o v e r , L - i s bounded, then t h e r e i s a unique

5.21.

Lemma. L e t L -

0 EN(L ) f o r which polynomials o f

4-

+

K- .

(5.5) holds,

loc P(&) = l o c P(5) f i . e . t h e l o c a l

and K - a r e t h e s a m e ) arid

+

i s transitive.

The t e r m o r i e n t a t i o n i s j u s t i f i e d by t h e f a c t t h a t t h e Hasse d i a g r a m s of

h

and

5

a r e i d e n t i c a l ( a s g r a p h s ) . F i n a l l y i t i s shown i n [ 9 ] :

Theorem. An o r d e r [ on A i s an o r i e n t a t i o n of a l a t t i c e L - if a n d o n l y i f [ s t e m s f r o m a d i r e c t d e c o m p o s i t i o n o f L. -

5.22.

P r o o f . W e show t h a t t h e e q u i v a l e n c e s commute. L e t x

2

t

C

u ? w 5 y and

x [ t ] u ] w [ y . I t w i l l be convenient t o have t h e followinq statement: (5.6)

I f a 2 c s b and a [ c t h e n a + b [ c . I f a s c l b and a [ c t h e n

ac [ c . I n t h e f i r s t case w e h a v e a + b [ c + b = b and i n t h e s e c o n d

similar

.

Applyinq r e p e a t e d l y ( 5 . 6 ) w e f i r s t g e t x 1 x + u [ u and t h e n x [ x u ] u hence w e may assume t h a t t =xu. S i m i l a r l y w e may assume w = u y and u [ y + u ] y , u ] u y [ y . Next x v u x v ( x + u ) = x shows x = x v u (x +u) v u = u and from x S u w e q e t xu = x. From x = x u l u ~ u yw e g e t by ( 5 . 6 ) t h a t x ] x y ] u y .

I.G. Rosenberg, D. Schweigert

138

P u t z = y v ( x y ) . I t s u f f i c e s t o show t h a t y

z

?

-,xy

z

x

because we

a l r e a d y know t h a t y [ z ] z I x y l x . Here y = y v y , y v x y = z and xy = xy v x y An a l g e b r a

y vxy =

5

4 is

2.

0

2

5

s i m p l e i f Con

= {w,A } .

The f o l l o w i n g i s p e r h a p s

worth s t a t i n g : lattice & - h a s o n l y t h e t w o b a s i c o r i e n t a t i o n s 'and 2 i f and o n l y i f L - i s d i r e c t l y indecomposable. A simple l a t t i c e

5.23.

Corollary.

A

has only trivial orientations.

56 6.1.

TOLERANCES I n view o f Thm 4 . 4 .

and 4 . 7 .

the set To1

4

of t h e t o l e r a n c e s of

a n a l g e b r a $ ( i . e . of r e f l e x i v e g r a p h s c o m p a t i b l e w i t h

4) p l a y s

anim-

p o r t a n t r o l e . From t h e g e n e r a l t h e o r y of c o m p a t i b l e r e l a t i o n s i t f o l -

lows t h a t the poset (To1

4:s) i s

a n a l g e b r a i c l a t t i c e i n which f i n i t e

and i n f i n i t e i n f i m a are s e t i n t e r s e c t i o n s , w = { ( a , a ) : a t A } t h e

l e a s t e l e m e n t and A'

t h e g r e a t e s t e l e m e n t . Note t h a t t h e l a t t i c e i s

c o m p l e t e and closed u n d e r s e t - t h e o r e t i c a l u n i o n s o f c h a i n s ( o r , e q u i v a l e n t l y , of d i r e c t e d s y s t e m s ) . F o r a r e l a t i o n a on A t h e r e l a t i o n To : = f l { p

€To1

9

:p ?a)

least t o l e r a n c e of

is the tolerance of

4 containing

4 generated

a).

Clearly To1 A - c o n t a i n s t h e c o n g r u e n c e l a t t i c e Con p i c a l l y Con

4

T

3 8 U 5 i n Con

5

5

of

5.

However, t y -

i s n o t e v e n a s u b l a t t i c e o f To1 4 ( e . g . g i v e n e , E EConA T : = ( T o 5 ) n ( 8 0 5 ) s a t i s f i e s T 2 0 iJ 5 b u t n o t n e c e s s a r i l y

the tolerance To1 A - = Con

by a ( i . e . t h e

4). F o l l o w i n g

[ 6 1 w e say t h a t

(a sufficient condition is t h a t

5 is tolerance 9 has a local

trivial if

2 / 3 mino-

r i t y polynomial: t h u s each A - i n a c o n q r u e n c e p e r m u t a b l e v a r i e t y i s tol e r a n c e t r i v i a l [ E l , a f a c t o f l i t t l e importance i n t h e s e q u e l ) . 6.2. Let

W e t u r n t o t h e t o l e r a n c e s o f a l a t t i c e . F i r s t some t e r m i n o l o g y .

5=


.> h a v e 0. We s a y t h a t

whenever x l y and xz = 0. Next

L

5

i s 0-modular i f x = y ( x + z )

is p s e u d o c o m p l e m e n t e d i f f o r every

a € A t h e s e t { x : x a = O } h a s a g r e a t e s t e l e m e n t a* ( c a l l e d t h e p s e u d o -

- i s s e c t i o n a l l y c o m p l e m e n t e d i f each c o m p l e m e n t o f a ) and f i n a l l y L i n t e r v a l [ O , a ] i s complemented. W i t h o u t p r o o f w e q u o t e [ 3 1 :

Compatible orderings and tolerances Theorem.

6.3.

L-

I f

is a lattice,

139

t h e n To1 L_ - i s 0-modular

and pseudo-

complemented.

If

4

i s d i s t r i b u t i v e , then so i s To1 L - (161, independent proof

[4!)

b u t t h e same d o e s n o t h o l d f o r m o d u l a r l a t t i c e s 1 3 1 . T o l e r a n c e s on d i s t r i b u t i v e l a t t i c e s are n-qenerated

4

An i d e a l I o f

i s a s u b l a t t i c e and down-set o f

i d e a l I such t h a t L\I

&

dual i d e a l of

by t h e f o l l o w i n q b i g t o l e r a n c e s .

L.

Prime i d e a l

i s an

i s a l s o a n i d e a l . L e t I a n d F b e a n i d e a l and a

such t h a t L

= I 11 F .

The r e l a t i o n

T~~

: = I 2IJ F 2 i s a to-

L. I n a d i s t r i b u t i v e l a t t i c e w e have t h e f o l l o w i n g s e u a r a -

l e r a n c e of

t i o n property [ 2 5 J : 6.4.

Lemma. L e t

to each

of

L-

(a,b) EL that u

such

p

2

o f a d i s t r i b u t i v e l a t t i c e L. - Then there a r e a p r i m e i d e a l I a n d a d u a l p r i m e i d e a l F

be a

\p

ab

tolerance

: = I 2 O F 2 i s a t o l e r a n c e o f L_ w i t h ( a , b ) ( u a b - @* t L2 \ p l .

C o n s e q u e n t l y p = ( > f a a b: ( a , b )

Before p r o c e e d i n g f u r t h e r w e need a c r i t e r i o n f o r t o l e r a n c e t e s t i n g . T o l e r a n c e s b e i n q n a t u r a l e x t e n s i o n s of c o n g r u e n c e s , c l e a r l y t h i s i s n o t a s i m p l e t a s k . We b r i n g t h r e e s l i g h t l y d i f f e r e n t a u p r o a c h e s . The p r o o f s a r e r o u t i n e and t h u s o m i t t e d . The f i r s t c r i t e r i o n i s somewhat

(or a b l o c k ) of a symmetric r e l a t i o n

global. A p - b l o c k

p

on A i s a

maximal c l i q u e ( = complete s u b g r a p h ) o f P i . e . B _ c A s u c h t h a t B while a x B

2

se

4

p f o r a l l a EA\B. L e t B denote t h e s e t of p-blocks of p P F o r s u b s e t s B , C o f A p u t B + C := { b + c : b E B , c E C ? a n d

BC = { b c : b E B , 'F

-

u

- FEF,

6.5.

c €C}.

Finally f o r a family F of subsets of A put

F2. W e have:

Proposition.

and a f a m i l y

( 1 ) F, = B

5

The following areequivalent f o r d

f o r a t o l e r a n c e p o f I,, -

"P (2) pFis a

tolerance o f

(3) (i)

L, -

i s a cover of A a d m i t t i n g no i n c l u s i o n a m o n g i t s m e m b e r s ,

(ii) each F E E i s a convex s u b l a t t i c e o f

(iii)i f F , G t F , (iv)

l a t t i c e L_ - =

o f s u b s e t s o f A:

L-

t h e n F G S H , F + G c I f o r s o m e H,I E F , a n d 2 i f L i s a s u b l a t t i c e of I -, a n d L s p F , t h e n L C J f o r s o m e

J

€x.

(The p r o o f r e q u i r e s Z o r n ' s lemma).

I.G. Rosenberg, D. Schweigert

140

A d a p t i n g a more l o c a l a p p r o a c h , f o r a C A c a l l Na := { x : x p a l t h e n e i g h borhood o f a. W e d e s c r i b e p i n terms o f i t s n e i g h b o r h o o d s y s t e m ( c f .

which e x t e n d s 1 1 8 1 I . e . L . 8 )

[3] L.I.I. 6.6.

P r o p o s i t i o n . L e t L- = < A , + , * > b e a l a t t i c e and

s y s t e m o f s u b s e t s o f A.

= { Fa : a r A } a

Then F i s t h e n e i g h b o r h o o d s y s t e m o f a t o l e -

r a n c e o f L_ - i f and o n l y i f For e a c h a E A t h e s e t F a i s a c o n v e x s u b l a t t i c e o f ( i i ) I f b , a E F a and b c ' a o r b + c a t h e n b E F c , (i)

( i i i ) I f z , x and y E F x f t h e n y

(iv)

I f

z

2

+z

L-

containlng a,

F F Z , and

x and y E P x f t h e n y z E F

.

The t h i r d and l a s t d e s c r i p t i o n i s b a s e d on t h e f o l l o w i n g u s e f u l ob- w e h a v e [ a , b I 2 s p whenever a < b and s e r v a t i o n : For p ETol L

apb.

Re-

c a l l t h a t an i n t e r v a l [ x , y ] i s w e a k l y p e r s p e c t i v e i n an i n t e r v a l l u , v l

i f y = v +x, x

2

u or u = yx,y

5

v . L e t Q be t h e s e t o f f a m i l i e s Q o f

i n t e r v a l s such t h a t (i)

Q convers A,

(ii) i f an i n t e r v a l I i s weakly p e r s p e k t i v e i n J t Q ,

some K

E

then I CK f o r

Q , and

( i i i ) i f 1 , J E Q h a v e a common e n d D o i n t , t h e n I

U

J s K f o r some K E Q .

The f o l l o w i n g i s a s l i q h t r e f o r m u l a t i o n o f [ 3 1 L . 1 . 2 . 6.7.

P r o p o s i t i o n . I f L_ - i s a l a t t i c e , t h e n To1

&

= {pQ

: Q €91

We c a n now d e r i v e a s u f f i c i e n t c o n d i t i o n f o r t o l e r a n c e t r i v i a l i t y ( c f [21 L.2.):

6.8.

Lemma. I f t o e v e r y t o l e r a n c e p o f a l a t t i c e L_ - and x < y < z w i t h

xpypz e i t h e r yw S x , y+w = z o r yw = x , y+w

2 z

f o r some w € A ,

t h e n L_ - i s

t o l e r a n c e t r i v i a l . Thus i f i n L_ - e a c h p a i r x < z i s i n c l u d e d i n a comp l e m e n t e d i n t e r v a l [ u , ~ ]o r [ x , u ] ( i n p a r t i c u l a r i f L_ - i s relatively o r s e c t i o n a l l y c o m p l e m e n t e d ) t h e n L_ - i s tolerance t r i v i a l .

P r o o f . Suppose yw < x and y + w = z . f o r e (x,z) = ( x + y w , y + w )

€ 0 .

Then (yw,w) = (yw,zw) t p and t h e r e -

Combining t h i s w i t h L . 6 . 6 .

and i s t h e r e f o r e a

t h a t p s a t i s f i e s t h e c o n d i t i o n s o f [ 181 1 . 3 . 1 . 8 . c o n g r u e n c e . The o t h e r case f o l l o w s by d u a l i t y .

we obtain

0

An i n t e r e s t i n g a p p r o a c h t o t o l e r a n c e s h a s b e e n r e c e n t l y p r o p o s e d i n [2,3].

If

h a s n o t 0 l e t I(&) d e n o t e t h e s e t o f a l l i d e a l s of

&.

If

141

Compatible orderings and tolerances

I? -

has 0 let I ( L - ) s t a n d f o r t h e s e t o f a l l non-empty i d e a l s o f k -. I t

i s well-known t h a t t h e i d e a l l a t t i c e L ) is an a l g e b r a i c - ( & ) : = ( I ( &,c) = < B ; v , A > i s a n A-enl a t t i c e . A p r o g r e s s i o n o f a c o m p l e t e l a t t i c e K_ domorphism G s u c h t h a t b (ii)o ( v x

i=

xx)

=

v

XFX

~ ( x €or ) every d i r e c t e d 0 e X 5 B . We have [ 3 1 :

P r o p o s i t i o n . For P

6.9.

a ( b ) f o r a l l b F B and

To1 L_ - and I C I ( L -) p u t p * ( I )

f o r some i C I ) . T h e n P + o *

a b i j e c t i o n from To1 _L - onto the s e t 0 El(&). -

is

S k e t c h o f t h e p r o o f : One h a s v e r i f y t h a t f o r p* ( I )

{x C A :xpi

n

s o f I__ (_L ) such t h a t sp) = p) i f

o f a l l proqressions

set

:=

(-1( L ) a n d t h a t I

-%

P* (I)

L- a n d I E I ( & ) t h e

P ETol

i s A-nreservinq,

satisfies I sp*(I)

and o r e s e r v e s d i r e c t e d f a m i l i e s . Conversely f o r s C I l p u t s*

(where ( a ] = ( x : x

'

:= { ( a , b :

a f : s ( b ] ,b t s ( a 1 )

a ) ) . By a d i r e c t c h e c k s* F T o l

somewhat t e d i o u s v e r i f i c a t i o n o f

** p

= p

and

** s

&.

I t remains t h e

= s for all p ETol

-

and s E n . 6.10.

T h i s a u o r o a c h i s e x p a n d e d i n [ 3 1 t o show t h e d i s t r i b u t i v i t y

o f To1

& f o r d i s t r i b u t i v e l a t t i c e s and t o p r o v e a d e s c r i p t i o n o f to-

l e r a n c e s o f f i n i t e l a t t i c e s by s p e c i a l + - e n d o m o r o h i s m s . W e a d d r e s s o u r s e l v e s t o s u c h a d e s c r i p t i o n f o r s l i g h t l y more g e n e r a l l a t t i c e s . A regression of a lattice L = <'A;+,*> i s a +-endomorphism r s a t i s f y -

inq r ( x )

x r f o r a l l x FA. The s e t of r e g r e s s i o n s o f

& is

denotedr(L).

I t i s n a t u r a l l y o r d e r e d c o m p o n e n t w i s e by s e t t i n q r-s i f r ( x ) ' s ( x ) f o r

all x

&. Note t h a t

,r ( & I ;

,

i s a n o r d e r e d monoid a n d t h a t

is a j o i n s u b s e m i l a t t i c e of t h e exuonent

L &=

(1( L ) ; '

)

( o f <-monotone s e l f m a p s

of A ) . A r e g r e s s i o n is sup-complete i f f o r each a F A t h e set

{x : r ( x )

a - x i has a g r e a t e s t element.

q r e s s i o n s of

& is

denoted

The s e t of s u n - c o m p l e t e re-

Ts(L).

R e g r e s s i o n s w i l l be l i n k e d w i t h s n e c i a l t o l e r a n c e s . W e s a y t h a t a t o l e r a n c e p of L - i s an i n f - t o l e r a n c e ( s u p - t o l e r a n c e ) i f e a c h n e i q h b o r hood h a s a l e a s t ( q r e a t e s t ) e l e m e n t . I f P i s b o t h a n i n f - t o l e r a n c e and a s u u - t o l e r a n c e w i l l b e c a l l e d a n i n t - t o l e r a n c e

(as each neigh-

borhood i s an i n t e r v a l ) . The s e t s o f i n f - t o l e r a n c e s Inft

&

and I n t t

2.

and i n t - t o l e r a n c e s

of

& are

d e n o t e d by

S i n c e each neighborhood i s a convex l a t t i c e , w e

have I n f t L - = To1 L - whenever

5

i s p a r t i a l l y well-ordered

(i.e. with-

I.G. Rosenberg, D.Schweigert

142

d e s c e n d i n g c h a i n s ) o r e a c h t o l e r a n c e of 2 i s a n inf-com2 S i m i l a r l y I n t t 4 = To1 & f o r l a t t i c e s w i t h o u t

out infinite

plete sublattice of

4.

i n f i n i t e c h a i n s o r whose t o l e r a n c e s a r e comDlete s u b l a t t i c e s o f L

2

1321.

A l r e a d y t h e w o r k of R . W i l l e 1 4 1 - 4 3 1 1-1

&

c o r r e s p o n d a n c e between T o 1

qives a hint t h a t there is a

I(&).

and

I n 1 3 7 1 t h i s r e s u l t was

proved f o r complete l a t t i c e s w i t h complete t o l e r a n c e s . W e s t a r t with t h e f o l l o w i n g o b s e r v a t i o n . L e t I, b e a t o l e r a n c e of L = < A ; + , * > . F o r a i A t h e infimum of t h e n e i g h b o r h o o d Na i s d e n o t e d l ( a ) ( p r o v i d e d i t e x i s t s ) . S i m i l a r l y f o r a , b C A t h e infimum of Na +Nb i s d e n o t e d by i

ab

( i f it e x i s t s ) .

6.11.

Lemma. L e t

p be

a tolerance o f a l a t t i c e L

l ( a ) e x i s t s f o r a l l a CA.

I f

x 'y

l(x)

and i

XY

s u c h t h a t

lfy)

i

I -

=

e x i s t s then

(6.1.)

XY

If f o r x , y E A a l l then

three i

XIXYl

i

YlX+Y

l ( x ) + l ( y ) = l ( x + y ) . ~ h u s ,i f i

XV

and

i

e x i s t and i XY

XY

= l ( X ) + l( y ) ,

e x i s t s and e q u a l s L ( x ) + l ( y ) f o r

t h e n 1 i s a r e g r e s s i o n o f L_. -

all x , y C A ,

y and l e t z F N x ,

Proof. L e t x

t CNy.

l ( x ) . I t follows t h a t

z t E N x and z t

Then ( x , z t ) = ( x y , z t ) 1 := i

XY

satisfies

w e have t l ( y ) , t x t 1 y i e l d i n g t Y l ( y ) + i and l ( y ) ' 1 ' l ( x ) p r o v i n g ( 6 . 1 . ) .

f o r every t F N shows l ( y )

1

i p

provinq

l ( x ) . Thus

l ( y ) + i . This

L e t x , y F A and z = x + y . Suppose t h a t i (6.1.)

i and i e x i s t . Then by YZ xy' xz w e h a v e l ( x ) + l ( y ) l ( x + y ) . F o r n E N x and v E N w e g e t n+v F N

-

and l ( x + y ) < i

XY *

I f , moreover, i

X+Y

Y X+Y = l ( x ) + l ( y )t h e n l ( x ) + l ( y ) = l ( x + y ) . o

The f o l l o w i n g p r o p o s i t i o n r e l a t e s i n f - t o l e r a n c e s

(int-tolerances) with

r e g r e s s i o n s ( s u p - c o m p l e t e r e g r e s s i o n s ) . The s k e t c h o f t h e p r o o f f o l lows t h e i d e a o f [ 3 7 ] . F o r p t I n f t L_ - and a E A l e t p ( a ) d e n o t e t h e

l e a s t e l e m e n t l ( a ) o f Na. 6.12.

Proposition. L e t L - be a lattice.

i s o m o r p h i s m and and c I n t t

&, -

0

o-isomorphism

,&> o n t o t h e o r d e r e d

Sketch of t h e proof: and I n t t

&

into

1( & )

Then p - p #

from t h e ordered monoids

( L ) ;0 ,5 > and

1 ) It is easy t o prove t h a t p - p # and

cs(L),

i s an o r d e r - a n t i -

m o n o i d s i I n f t I-, ; o , s >

(1s (I,)-

:0 , ~:>.

maps I n f t

4

Compatible orderings and tolerances 2 ) For s

f

T(&)

t h a t s # c To1

5

p u t s'

:=

143

{ (x,y) :s ( x ) + s ( y ) 'xy}.

&.

and t h e n t h a t s # C I n f t

If s C

F i r s t one proves

ys(G)

o n e shows t h a t

s C I n t t L- . 3) T o p r o v e t h a t

pg#

= P

and s"

I,

=

-+oY

and s - s #

s for all 0

Inft

(

a r e b i j e c t i o n s one h a s t o prove

&

(Intt

L)

and s F

L)). (L) - (rs(-

4) F i n a l l y o n e h a s t o p r o v e t h a t b o t h p - 0 4 and s - s q p --.D' i s a n o-homomorphism. o

are order

r e v e r s i n q and t h a t As noted before,

0

--to#

i s a l a t t i c e a n t i i s o m o r p h i s m o f To1

I(&)

f o r p a r t i a l l y well-ordered lattices. 6.13. 0'

Example.

[37] Let

= O and f o r a > 0 l e t

+ a

4 be

&

onto

l a t t i c e s and i n p a r t i c u l a r f o r f i n i t e

a modular l a t t i c e of f i n i t e l e n g t h . P u t

d e n o t e t h e meet o f t h e e l e m e n t s c o v e r e d by

a ( w h i c h e x i s t s s i n c e [ O , a ) i s d u a l l y atomic (1401 S.1.1., S.1.4.). I t i s shown i n [ 1 9 ] L . 6 . 1 . e

t h a t s ( a ) = a + i s a r e g r e s s i o n and t h e r e -

f o r e s# i s a t o l e r a n c e o f L. 6.14.

Remark. L e t

&

be a l a t t i c e d e t e r m i n e d by i t s c o v e r i n g r e l a t i o n .

Each r e g r e s s i o n i s t h e n d e t e r m i n e d by i t s v a l u e s on t h e s u p - i r r e d u c i b l e e l e m e n t s d i s t i n c t from 0 . A s e l f m a p o f A i s i d e m p o t e n t i f s 2 = s

( i . e . s ( s ( x ) )= s ( x ) f o r a l l x ( - A ) . F o r f i n i t e l a t t i c e s t h e f o l l o w i n g appears i n [31. 6.15.

P r o p o s i t i o n . An i n f - t o l e r a n c e

p

of L - is a c o n g r u e n c e o f L_ - if

and o n l y i f p # i s i d e m p o t e n t .

Proof. Necessity.

S i n c e p # ( x ) i s t h e l e a s t e l e m e n t o f Nx c l e a r l y

#

p

i s idemDotent. Sufficiency.

L e t s be an idempotent regression of

4. We

show t h a t

( x , y ) C s # i m p l i e s s ( x ) = s ( y ) . I n d e e d by t h e d e f i n i t i o n o f s # w e h a v e

s ( x ) ~ x -yx a n d t h u s by t h e - m o n o t o n i c i t y a n d i d e m p o t e n c y 2 s ( x ) = s ( x ) ' s ( x y ) / s ( x ) p r o v i n g s ( x y ) = s ( x ) . By symmetry s ( x y ) = s ( y ) whence s ( x ) = s ( y ) . T o p r o v e t h a t s' is t r a n s i t i v e l e t ( x , y ) E s# and ( y , z ) Es'.

By w h a t h a s b e e n shown s ( x ) = s ( z ) a n d h e n c e s ( x ) + s ( z )

=

= s ( x ) Lxy i n a s m u c h s ( x ) i s t h e l e a s t e l e m e n t o f t h e s u b l a t t i c e N x .

Thus ( x , z ) E s # a n d w e a r e d o n e . 6.16.

Example ( [ 2 ] P r o p .

s t r i b u t i v e lattice. L e t

p

3.2.).

0

L e t a be a cardinal

>H0

and

&

a di-

be t h e union o f t h e s q u a r e s o f i n t e r v a l s

I.G. Rosenberg, D.Schweigert

144

w h i c h a r e c l o s e d u n d e r j o i n s a n d meets of f a m i l i e s of c a r d i n a l i t i e s
&.

$ 7 LOCALLY CENTRAL TOLERANCES

R e c a l l t h a t a r e f l e x i v e qraDh o n A d i s t i n c t f r o m A 2 i s l o c a l l y c e n -

t r a l i f t o every f i n i t e s u b s e t B of A t h e qraph some a E A . S i m i l a r l y

p

is c e n t r a l i f a x A F

p

p

contains a x B f o r

f o r some a r A . L o c a l

c e n t r a l i t y i s e a s i l y d e s c r i b e d i n terms o f n e i a h b o r h o o d s . Lemma. A t o l e r a n c e p C A 2 i s l o c a l l y c e n t r a l i f and o n l y i f e v e -

7.1.

ry pair

c < d b e l o n g s t o some n e i g h b o r h o o d .

Proof: Necessity i s evident (take B = { c , d l ) . Sufficiency. Let b l , l o n q t o some N a .

...,b n

F A . Then c : = b ,

From c o n v e x i t y b l ,

...,b n

...b n E N

a'

a n d d = b 1 +...+ b n b e 0

F o r b o u n d e d l a t t i c e s t h e s i t u a t i o n i s s i m p l e r . The c e n t e r C P o f a ref l e x i v e g r a p h P i s t h e s e t [ a : a x A ~ p 1 .W e s t a r t w i t h :

7.2.

Lemma. I f p i s a t o l e r a n c e of a l a t t i c e w i t h 0 and

C p = No A N 1

proof:

Clearly C

P

a

=

Na y i e l d i n q C

a F A . From ( z a , a ) = ( z a , l a ) that z ECP.

1 then

.

€ 0

C N TIN,.

P -

0

Let z E N

0

n N 1 and

we q e t ( z , a ) = (ztza,O+a) t P proving

0

W e have

7.3.

Proposition.

The f o l l o w i n g a r e e q u i v a l e n t f o r a

a l a t t i c e w i t h 0 and

(1)

p

t o l e r a n c e p of

1:

i s locally central,

( 2 ) p i s c e n t r a l , and ( 3 ) No f l N 1 Proof: ProD. dent.

(1) 6.6.

*@. =$

( 2 ) : By L . 7 . 1 .

t h e bounds 0 and 1 b e l o n g t o s o m e Na.

( i )t h e n A _ c N a 1 . e . a E C P .

(2)

* ( 3 ) L.7.2.

(3)

=$

By

( 2 ) Evi-

Compatible orderings and tolerances Remark. Cp i s a convex s u b l a t t i c e of

7.4.

ment 1 t h e n p i s c e n t r a l i f f 1 No

h.

145

I f N1 h a s a l e a s t ele-

u f o r some u F N o .

has a g r e a t e s t element u, thenCp = r l , u l

In particular, i f

and P is c e n t r a l i f f 1

C o r o l l a r y . L e t r b e a r e g r e s s i o n of a l a t t i c e L - w i t h 0 and

7.5.

Then t h e c o r r e s p o n d i n g i n f - t o l e r a n c e

r ( a ) = O f o r some a

1.

is c e n t r a l i f and o n l y i f

~ ( 1 ) .

be a complete c h a i n w i t h a least element 0 and

Example. L e t

7.6.

r'

u.

g r e a t e s t e l e m e n t 1. A r e g r e s s i o n o f r o f

g- i s j u s t a '-monotone s e l f -

x f o r a l l x F C . SuDpose t h a t r - l ( 0 ) h a s a q r e a t e s t

map s a t i s f y i n g r ( x )

e l e m e n t i r . Then r i s c e n t r a l i f f r ( 1 ) : ir ( t o h a v e a c o n c r e t e e x a m p l e

l e t C be t h e r e a l i n t e r v a l [ O , l l and r any l e f t - c o n t i n u o u s s i n g real f u n c t i o n s a t i s f y i n g 0 , . r ( x )

~

x for all 0

1

non-decrea-

x 2 1).

I n f a c t , t h e c e n t r a l t o l e r a n c e p r o d u c i n g r e g r e s s i o n s s h a r i n g t h e same i r a r e i n 1-1 c o r r e s p o n d e n c e w i t h s i m i l a r f u n c t i o n s f r o m [ i , l ] i n t o [O,i].

For f i n i t e chains t h i s allows counting r e l a t e d t o t h e s i z e s of

t h e p o s e t L ( m , n ) ( o f d o w n s e t s of t h e d i r e c t o r o d u c t of a n m-chain and an n-chain). Now w e a p p l y P r o p .

4.6.

t o bounded l a t t i c e s a n d o b t a i n a s l i q h t i m -

m o v e m e n t o f [ 351 L . 3.

7.7. A.

Theorem. L e t p b e a p r o p e r t o l e r a n c e o f a b o u n d e d l a t t i c e _L -

Then e i t h e r t h e t r a n s i t i v e h u l l

or t h e r e i s 0 < n < IAl

P r o o f . By P r o p . 4 . 7 .

such

that pn

and Prop.

7.3.

of p i s a p r o p e r c o n g r u e n c e of

OR

L-

i s a proper central r e l a t i o n .

it suffices to consider a tole-

rance p o f i n f i n i t e d i a m e t e r . Obviously t h e r e i s a least Prop. n > O such t h a t ( 0 , l ) € o n . 2

g e t A _cpn

§8

8.1.

A w l y i n q Prop. 6 . 6 .

(i) t o t h e t o l e r a n c e

pn

we

i n contradiction t o t h e i n f i n i t e diameter of p .

ORDER COMPLETENESS

Now w e a p p l y t h e r e s u l t s o f t h e o r e c e d i n g s e c t i o n s t o l a t t i c e s

&

w i t h l a r g e LOC P ( & ) . The s t a n d a r d t e r m i n o l o g y seems t o b e q u i t e h e a v y and s o w e p r o p o s e t h e f o l l o w i n g . An unbounded l a t t i c e i s b r o a d i f i t h a s n o t o l e r a n c e of i n f i n i t e d i a m e t e r , i t i s f l a t i f i t h a s no locall y c e n t r a l t o l e r a n c e and s t a r f r e e i f i t h a s n o c e n t r a l r e l a t i o n . A s

I.G. Rosenberg, D. Schweigert

146

u s u a l , a l a t t i c e L, h a s t h e o r d e r - i n t e r p o l a t i o n

l o c a l l y o r d e r - p o l y n o m i a l l y c o m p l e t e i f LOC P(&) = P o l

e v e r y <-monotone o o e r a t i o n i s a l o c a l Dolynomial o f

- , i.e.

&

1 3 1 . Combining Thm 4 . 6 .

w i t h ProD. 4 . 7 .

if

(here "order"

means t h e l a t t i c e o r d e r ) . Such l a t t i c e s h a v e a u s e e . q . tiation [12,

(OIP) or is

property

i n eponen-

and Thm. 7 . 7 . w e

o b t a i n t h e following: 8.2.

Theorem. A n u n b o u n d e d l a t t i c e h a s t h e o r d e r i n t e r p o l a t i o n p r o -

p e r t y i f and o n l y i f i t i s s i m p l e , A

f l a t and broad.

bounded l a t t i c e h a s the o r d e r i n t e r p o l a t i o n p r o p e r t y i f and o n l y i f

i t i s s i m p l e and s t a r f r e e . 8.3.

F o r bounded l a t t i c e s t h i s i s i n a c e r t a i n s e n s e t h e b e s t re-

s u l t , s i n c e f o r a c e n t r a l t o l e r a n c e or a p r o p e r c o n q r u e n c e p o f L t h e clone Pol

p

i s l o c a l l y maximal. V a r i o u s v a r i a n t s and o t h e r O I P c r i -

t e r i a a p p e a r t h r o u g h o u t t h e l i t e r a t u r e . Thus o n e c r i t e r i o n i s t h e absence of proper tolerances [ 2 1 ]

4 simole

obviously implies

(which f o l l o w s from ProD.

2

is the set of reflexive sublattices of t e r e s t i n g are t w o c r i t e r i a from [ 3 1 :

ment:

{Z,>,u,A

N o s u b l a t t i c e of

L2 - i s p r o p e r l y between

o and 5

&

(meaning LOC

t h a t e a c h $-monotone

P(5) =

P o l { ' UCon

&2 .

(between

I t may happen t h a t t h e r e i s a redundancv i n Thm. 8 . 2 . generates To1

4.8.

4) o r ,

5

More i n -

L,)

2

and A 1 .

b e c a u s e Con

4

equivalently,

o p e r a t i o n admitting a l l congruences of

ready a local polynomial of order a f f i n e complete

and

f l a t and b r o a d ) o r t h e e q u i v a l e n t s t a t e -

&

is al-

I n t h i s case w e s a y t h a t L i s l o c a l l y

( b r i e f l y loaf-complete).

From Thm 7 . 7 .

and 5 . 2 .

we obtain: 8.3.

Proposition.

A

loaf-comp ete bounded l a t t i c e h a s the o r d e r i n -

t e r p o l a t i o n p r o p e r t y i f a n d on y i f i t i s s i m p l e .

Now a l a t t i c e may be l o a f - c o m p l e t e tolerance t r i v i a l i.e. To1

&

= Con

f o r t h e simple reason t h a t it i s

&.

i s loaf-complete;

c o n d i t i o n of L.6.8.

Thus a l a t t i c e s a t i s f y i n q t h e i n p a r t i c u l a r each r e l a t i v e l y

o r s e c t i o n a l l y complemented l a t t i c e i s l o a f - c o m p l e t e . Distributive lattice 8.4.

Proposition.

are l o a f complete ( f i n i t e [ 1 6 1 , i n f i n i t e [ 2 ] ) :

Distributive lattices are loaf-complete.

Proof. Consider a t o l e r a n c e a

ab

= I LU F L

from L.6.4.

and l e t

Compatible orderings and toleronces 8 : = I 2I! ( L \ I ) 2

and E : = F

and t h a t u a b = ( 8

0

so i s e v e r y

L.6.4.

2

5 ) n (5 p CTol

I t i s e a s y t o Drove t h a t 8 , E EConT, -

IJ( L \ F I 2 . 0

147

8). Thus a a b i s g e n e r a t e d by Con

L.

&

and by

o

Another s u f f i c i e n t c o n d i t i o n i s [ 4 3 ] .

8.5.

Lemma.

Let

- be a p a r t i a l l y w e l l o r d e r e d l a t t i c e w i t h 1 .

i s the j o i n o f f i n i t e l y many a t o m s ,

then

T, -

I f

1

i s s t a r f r e e and loaf-com-

plete.

b e a c e n t r a l t o l e r a n c e o f L, - l e t r = p and l e t 1 = a + . . . + a n where a l , , a n a r e atoms. W e may assume t h a t f o r some 1 0 i n w e have r ( a ) = r ( a i ) = O while r ( a . ) > O f o r j = i + l , rn. 1 1 From r ( a . ) a . w e q e t r ( a . ) = a . a n d r ( 1 ) = r ( a ) + ...+r ( a ) = a1+1 . ran. 7 1 1 1 1 By t h e same t o k e n r ( a l + . . + a . ) = O . By C o r . 7 . 5 . t h e r e i s a ? r ( 1 ) = Proof. L e t

p

...

...

=...

- -.

.

= ai+.,+.

. .+an s u c h

show a ' 1

i.e.

p

that r(a) =O.

Then a

= A2 i s t r i v i a l .

.

.

a l + . .+a. and a 1 a i + l + . .+an

0

Note t h a t a d i s t r i b u t i v e l a t t i c e w i t h more t h a n 2 e l e m e n t s i s n o t simple and hence does n o t p o s s e s s t h e O I P . For modular l a t t i c e s wehave

8.6.

Theorem. L e t L b e a m o d u l a r l a t t i c e w i t h a l e a s t o n e c o m p l e m e n -

ted interval. i f L i s simple

Then L h a s the o r d e r i n t e r p o l a t i o n p r o p e r t y i f and o n l y and r e l a t i v e l y complemented.

The c o n c l u s i o n h o l d s i f L h a s a t l e a s t o n e p r i m e i n t e r v a l ( i . e . nonempty c o v e r i n g r e l a t i o n ) . P r o o f . N e c e s s i t y . L e t p be t h e u n i o n o f t h e s q u a r e s o f complemented inervals of tolerance of

&. U s i n g P r o p . 6 . 7 . i t 2. By a s s u m p t i o n p + w

i s shown i n [ 2 1 Thm 3 t h a t hence P = A 2 , i . e .

p

is a

each i n t e r v a l of

L- i s complemented. S u f f i c i e n c y . By Lemma 6 . 8 .

the lattice

&

is tolerance t r i v i a l .

F o r modular l a t t i c e s o f f i n i t e l e n g t h w e h a v e t h e f o l l o w i n g remarka b l e r e s u l t [ 371 [ 321: 8.7.

Theorem. L e t L_ - be a m o d u l a r l a t t i c e of f i n i t e l e n g t h .

T h e n L_ - has

t h e o r d e r i n t e r p o l a t i o n p r o p e r t y i f a n d o n l y i f L_ - i s a d i r e c t l y indecomposable p r o j e c t i v e

geometry.

I.G. Rosenberg, D.Schweigert

148

P r o o f . N e c e s s i t y . The t o l e r a n c e s z f r o m E x . 6 . 1 2 .

1'

=O.

T h u s b y [ 51 Thm I V . 6 .

the lattice

&

i s t r i v i a l Droving

i s t h e n a complemented

s i m p l e l a t t i c e and hence a n indecomposable p r o j e c t i v e qeometry. S u f f i c i e n c y . C l e a r l y 1 is t h e j o i n o f a t o m s , h e n c e b y L . 8 . 5 .

tice

4

is loaf-complete.

the lat-

S i n c e i t i s a l s o s i m p l e i t h a s t h e OIP.

0

REFERENCES [

11 Baker K.A. and P i x l e y A.F., Polynomial i n t e r p o l a t i o n and t h e C h i n e s e r e m a i n d e r t h e o r e m f o r a l q e b r a i c s y s t e m s , Math. Z . , 1 4 3 ( 1 9 7 5 ) , 165-174.

[ 21 B a n d e l t H. J . , L o c a l p o l y n o m i a l f u n c t i o n s o n l a t t i c e s , H o u s t o n J. M a t h . 7 ( 1 9 8 1 ) , 3 1 7 - 3 2 5 . [

31 B a n d e l t H . J . , T o l e r a n c e r e l a t i o n s o n l a t t i c e s , B u l l . A u s t r . M a t h . SOC. 2 3 ( 1 9 8 1 ) , 3 6 7 - 3 8 1 .

[ 41 B a n d e l t H. print.

J.,

[ 51 B i r k h o f f G . ,

Toleranzrelationen als Galoisverbindungen, pre-

L a t t i c e Theory 3rd e d . A m e r .

Math. SOC. 1967.

[ 6 1 C h a j d a I., R e c e n t r e s u l t s a n d t r e n d s i n t o l e r a n c e s o n a l q e b r a s and v a r i e t i e s , P r o c . Colloq. F i n i t e a l g e b r a M u l t i p l e - v a l u e l o q i c ( S z e g e d 1979) C o l l . Math. SOC. I . B o l y a i 28, N o r t h H o l l a n d 1 9 8 1 , p p 69-95. [

71 C h a j d a I . , T h e o r y o f t o l e r a n c e s o n a l g e b r a i c s y s t e m s , b i b l i o g r a p h y , 1981- 19 8 3 .

[

81 C h a j d a I., D i s t r i b u t i v i t y a n d m o d u l a r i t y o f l a t t i c e o f t o l e r a n c e r e l a t i o n s , A l g e b r a U n i v e r s a l i s 1 2 ( 1 9 8 1 ) , 247-255.

[ 91 C z e d l i G . , Huhn A . P . , S z a b 6 , L . , On Compatible o r d e r i n g o f l a t t i c e s , C o l l . Math. SOC. I . B o l y a i 33, N o r t h H o l l a n d 1 9 8 0 , 87-99.

[ l o ] CzCdli G . , K l u k o v i t s L . , A n o t e on t o l e r a n c e s o f i d e m p o t e n t alg e b r a s , G l a s n i k M a t e m . 1 8 ( 3 8 ) ( 1 9 8 3 ) 35-38. 1111 C z C d l i G . , F a c t o r l a t t i c e s b y t o l e r a n c e s , A c t a S c i . M a t h . g e d ) 4 4 1-2 ( 1 9 8 2 ) 35-42.

(Sze-

[ I 2 1 Davey B . A . , D u f f u s D . , Q u a c k e n b u s h R.W. a n d R i v a l I . , E x p o n e n t s o f f i n i t e s i m p l e l a t t i c e s , J. London M a t h . S O C . , 1 7 ( 1 9 7 8 ) , 203-21 1 . [ 1 3 1 Davey B . A . a n d R i v a l I . , E x p o n e n t s of l a t t i c e - o r d e r e d a l g e b r a s , A l g e b r a U n i v e r s a l i s 1 4 ( 1 9 8 2 ) 87-98. [141 D o r n i n g e r , D., Nobauer W . , L o c a l p o l y n o m i a l f u n c t i o n s on l a t t i c e s a n d u n i v e r s a l a l g e b r a s , C o l l o q . M a t h . 42 ( 1 9 8 0 ) 8 3 - 9 3 . [ I 5 1 D o r n i n g e r D . , E i g e n t h a l e r G . , On c o m p a t i b l e a n d o r d e r - p r e s e r v i n g f u n c t i o n s o n l a t t i c e s , B a n a c h C e n t e r P u b l i c a t i o n s 9 , Warsaw 1 9 8 2 , 97- 1 0 4 . [ 1 6 1 G r a t z e r G., B o o l e a n f u n c t i o n s o n d i s t r i b u t i v e l a t t i c e s , A c t a M a t h . Acad. S c i . H u n g a r . 1 5 ( 1 9 6 4 ) 9 5 - 2 0 . [I71 Gratzer G., Lattice theory: F i r s t concepts anddistributive latt i c e s , Freeman ( S a n F r a n c i s c o ) 1971.

149

Compatible orderings and tolerances G e n e r a l l a t t i c e t h e o r y , (Akademie V e r l a q B e r l i n

I18

Gratzer G.,

i 19

Herrmann C . , S . - v e r k l e b t e ( 1 9 7 3 ) 255-274.

[ 20

H e d l i k o v h J . , B e t w e e n e s s i s o m o r p h i s m s of m o d u l a r l a t t i c e s A r c h . Math 37 ( 1 9 8 1 ) 1 5 4 - 1 6 2 .

Summen von V e r b a n d e n , Math. Z . ,

1978. 130

Kindermann M . , u b e r d i e X q u i v a l e n z von Ordnungspolynomvollstand i g k e i t und T o l e r a n z e i n f a c h h e i t e n d l i c h e r V e r b a n d e , C o n t r i b u t i o n s t o G e n e r a l Algebra ( e d . H. K a u t s c h i t s c h e t a l . ) , 1 9 7 9 , 145-149. K o l i b i a r M . , Compatible o r d e r i n g s i n s e m i l a t t i c e s , C o n t r i b u t i o n t o General Alqebra 2 , Proceedings o f t h e Klagenfurt Conference 1 982, (Teubner, S t u t t g a r t ) 215-220. K o l i b i a r M . , S e m i l a t t i c e s w i t h i s o m o r p h i c q r a p h s , C o l l o q . Math. SOC. J . B o l y a i 2 9 , Esztergom 1977 473-481. K o l i b i a r M . , I n t e r v a l s , c o n v e x s u b l a t t i c e s a n d s u b d i r e c t repres e n t a t i o n s of l a t t i c e s , U n i v e r s a l A l g e b r a a n d A p p l i c a t i o n s B a n a c h C e n t e r P u b l . v o l 9 , Warsaw 1 9 8 2 , 3 3 5 - 3 3 9 . N i e d e r l e J . , R e l a t i v e bicomplements and t o l e r a n c e e x t e n s i o n prop e r t y i n d i s t r i b u t i v e l a t t i c e s , C a s o p i s P e s t . Mat., 1 0 3 ( 1 9 7 8 ) , 2 50-25 4 . Rosenberg I . G . , F u n c t i o n a l l y c o m p l e t e a l g e b r a s i n congruence d i s t r i b u t i v e v a r i e t i e s , A c t a S c i . M a t h . ( S z e g e d ) 4 3 , 3-4 ( 1 9 8 1 ) 34 7- 3 5 2 . Rosenberg I . G . , F u n c t i o n a l completeness o f s i n g l e g e n e r a t e d o r s u r j e c t i v e a l g e b r a s , Proc. Colloq. F i n i t e Algebra Multiple-val u e d L o g i c , S z e g e d ( 1 9 7 9 ) , C o l l . Math. SOC. J . B o l y a i 2 8 , N o r t h H o l l a n d 1 9 8 1 , pp. 635-652. R o s e n b e r g I.G., S c h w e i q e r t D . , L o c a l l y maximal c l o n e s , E l e c t r o n i s c h e Informationsverarbeitung u . K y b e r n e t i k 1 8 ( 1 9 8 2 ) 7 / 8 p p . 389-401. Rosenberg I . G . , Szab6 L . , bra Universalis.

L o c a l c o m p l e t e n e s s I , t o a p p e a r Alge-

S c h w e i g e r t D . , Uber e n d l i c h e , ordnungspolynomvollstandige V e r b a n d e , Monatsh. Math. 7 8 ( 1 9 7 4 ) , 6 8 - 7 6 . S c h w e i g e r t D . , Some r e m a r k s o n p o l a r i t y l a t t i c e s a n d on o r t h o l a t t i c e s , P r o c e e d i n g s of t h e L a t t i c e T h e o r y C o n f e r e n c e , U l m ( 1 9 7 5 ) 254-256. S c h w e i g e r t D . , C o m p a t i b l e r e l a t i o n s of m o d u l a r a n d o r t h o m o d u l a r l a t t i c e s , P r o c e e d i n g s of t h e AMS, 81 ( 1 9 8 1 ) , 4 6 2 - 4 6 4 . S c h w e i g e r t D. , P o l y n o m v o l l s t a n d i g e a t o m i s t i s c h e P o l a r i t a t s v e r b a n d e , A b h a n d l u n g e n Math. Sem. U n i v . Hamburg 5 1 ( 1 9 8 1 ) 5 0 - 5 9 . Schweigert D., A f f i n e complete o r t h o l a t t i c e , Proceedings Amer. Math. SOC. 6 7 , 2 ( 1 9 7 7 ) 1 9 8 - 2 0 0 . S c h w e i g e r t D . , C e n t r a l r e l a t i o n s on l a t t i c e s , J . A u s t r a l . Math. SOC. ( S e r i e s A ) 35 ( 1 9 8 3 ) . S c h w e i g e r t D . , Szymanska M . , P o l y n o m i a l f u n c t i o n s o f c o r r e l a t i o n l a t t i c e s , A l g e b r a U n i v e r s a l i s 16 ( 1 9 8 3 ) 3 5 5 - 3 5 9 . S c h w e i g e r t D . , Szymanska M . , On c e n t r a l r e l a t i o n s o f c o m p l e t e l a t t i c e s , U n i v e r s i t a t K a i s e r s l a u t e r n , P r e p r i n t no. 6 4 , 1 9 8 3 . Szabb L . , T o l e r a n c e - f r e e a l g e b r a s w i t h a m a j o r i t y f u n c t i o n , Prep r i n t , Bolyai I n s t i t u t e Szeged.

I.G. Rosenberg, D. Schweigert

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[ 3 9 ] Szabb L . , C h a r a c t e r i z a t i o n o f compatible q u a s i o r d e r i n g s of l a t t i c e ordered a l g e b r a s , p r e p r i n t Bolyai I n s t i t u t e Szeqed. [ 4 0 ] W i l l e R.,

H a l b k o m p l e m e n t a r e V e r b a n d e , Math.

2.

94 ( 1 9 6 6 ) 1 - 3 1 .

[ 4 1 1 Wille R . ,

Eine C h a r a k t e r i s i e r u n g e n d l i c h e r , ordnungspolynomvolls t a n d i g e r V e r b a n d e , A r c h . Math. 2 8 ( 1 9 7 7 ) , 557-560.

[ 4 2 ] Wille R . , A n o t e o n a l g e b r a i c o p e r a t i o n s a n d a l g e b r a i c f u n c t i o n s of f i n i t e l a t t i c e s , H o u s t o n J . Math. 3 ( 1 9 7 7 ) 593-597.

[ 4 3 ] W i l l e , R . , uber e n d l i c h e , ordnungsaffinvollstandige V e r b a n d e , Math. 2. , 1 5 5 ( 1 9 7 7 ) 103-107.

- PART I11 ORDER AND ORDERED GROUPS -

PARTIEIII ORDRE ET GROUPES ORDONNES

W. Charles HOLLAND

Classification o f lattice ordered groups.

. . . . . . . . . . . . . . . . . . . . . p . 151

A.M.W. GLASS The isomorphism problem and undecidable properties f o r finitely presented lattice-ordered groups. . . . . . . . . . . . . . . . . . . . . . . . . . . p . 157

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 (1984) 151-156 0 Elsevier Science Publishers B.V. (North-Holland)

151

W . Clrirles llnlland

Depurtricnt 01' !kithemutics tind S t s t i s t i e s Ecowlinrf Green S t a t e University t i i v ~ l l n r : Green. Ohio U .S . A .

IJn ~:roupc r6ti,.!u16 est. un p o u p e dont les 6lCnients c o n s t i t u e n t un t r * e i l l i s cornpntible avec l ' o p 6 r a t i o n du Troupe. Nous d i s c u t e r o n s t r o I s iii6thodes d e classific!at,ion d e ces s t r u c t u r e s . Is premiPre est. c e l l e de:! v w i 6 t c s . q u i sont. !es c l a s s e s d 6 f i n i e s par les Pqii" I.ioiis. La c o l l e c t i o n tles variEt6r: c s t un t r e i l l i s coniplet, 4 61Pment:: qui a un plus p e t i t Plecieiit n o n - t r i v i a l e t un p l u s ::rcind Element n o n - t r i v i a l . IA deuxiPme mEthode est, c e l l e des c l a s s e s de t o r s i o n . J.es c l a s s e s d e t o r s i o n c o n s t i t u e n t iin t r e i i l i s cmnplet, mais w r i t t.rop nombr*euses pour c o n s t i - t u e r un ensemble. T.a t r o i s i t i n e m6thode est c e l l e d e s c l a s s e s Cl&nent.aires, qui s o n t d6firile:; par des enonces d u l a i c a f e dii premier o r d r e des groupes

PO

r6ticii16s. e i e n t l u ' i l n'y a i t qiie classes ~ ~ e m e n t n i r e s , besucoulj d e croupes r 6 t i cul6s bien connus peuvent etre caract e r i s d s dans l a classe d e s groupes d'aulomorphismes d e chnines p a r un s e u l 6noncE du l a r i p g e d u premier o r d r e . 0.

INI'HODUC'I'ION

:I Kroup whose elements a l s o have a I . a t t i c e s t r u c t u r e compatible with t h e croup t r a n s l a t i o n s , so t h a t x(y v z)w = (xyw) V (xzw) and d u a l l y . From t h e standpoint of ordered s e t s , e-groups m y be viewed a s l a t t i c e s endowed with a group s t r u c t u r e which f o r c e s them t o have n i c e l a t t i c e p r o p e r t i e s . For exarnple, they must be homogeneous (with r e s p e c t t o l a t t i c e outomorphisiris) and d i s t r i b u t i v e . There is a n o t h e r c l o s e r e l a t i o n s h i p between P-groups and ordered sets. The automorphism group of every t o t a l l y ordered set is an P-group under t h e point-wise o r d e r i w . Moreover, every P-group is n subgroup and s u b l a t t i c e of such an automorphism group [ 8 ] ,

A httict' drdereJ p w4 p (L-group) i s

Much of t h e 11 t e r a t u r e on !-groups i s concerned with s t r u c t u r e and r e p r e s e n t a t i o n theory. Good r e f e r e n c e s f o r backl:round m a t e r i a l a r e 111, 123, [ A ] , and [ > I . The emphasis i n t h i s paper, however, is on c l a s s i f ' i c a t i o n of 1-groups by various schemes, i n parLicular ( i ) e q u a t i o n a l c l a s s e s , o r v a r i e t i e s , ( i i ) t o r s i o n c l a s s e s , and ( i i i ) elementary c l a s s e s . The ninin i n t e r e s t h e r e l i e s not s o much i n t h e s t r u c t u r e of p a r t i c u l a r !-groups b u t i n t h e r e l a t i o n s h i p between d i f f e r e n t c l a s s e s of P-groups. For example t h e set of a l l equational c l a s s e s of !-groups i t s e l f forms a l a t t i c e with some remarkable p r o p e r t i e s . In t h i s paper, which corresponds roughly with a one-hour address e i v e n a t t h e Conference on Ordered S e t s a t t h e Chdtenu de l a Tourette i n 1982, I w i l l not t r y t o cover t h e s u b j e c t of f-groups i n g r e a t d e t a i l , nor even g i v e a g e n e r a l overview, but r a t h e r , I w i l l c o n c e n t r a t e on a few d e t a i l s of the c l a s s i f i c a t i o n problem with t h e hope of whettin(! t h e r e a d e r ' s i n t e r e s t , and I w i l l provide a guide t o f u r t h e r i n v e s t i f a t i o n f o r those i n t e r e s t e d . 1.

VAR1F:TIES

A w r i e t $ (of P-groups) is a c l a s s defined by any set of u n i v e r s a l l y q u a n t i f i e d

W.C. Holland

152

e q u a t i o n s , which may i n v o l v e t h e group or l a t t i c e o p e r a t i o n s or b o t h . For example, t h e c l a s s A of a b e l i a n !-groups d e f i n e d by t h e e q u a t i o n xy = yx i s a v a r i e t y . A more t y p i c a l example, because i t i n v o l v e s b o t h t h e group and l a t , t i c e o p e r a t i o n s , i s t h e representable v a r i e t y R d e f i n e d by e , where e i s t h e i d e n t i t y element, of t h e group. (x-l(y V e ) x ) A (y-l V e ) A c l a s s i c a l r e s u l t of Lorenzen [ l 4 ] shows t h a t t h e r e p r e s e n t a b l e L-groups a r e Just. t h o s e which a r e s u b d i r e c t p r o d u c t s of t o t a l l y o r d e r e d groups. Another v a r i e t y of s p e c i a l i n t e r e s t i s t h e norma2 vulued v a r i e t y N d e f i n e d by t h e law xy < y?x:‘ i f x , y > e . I n t h i s form, t h e law i s n o t a n e q u a t i o n , b u t i t i s e a s i l y s e e n t n 2 be e q u i v a l e n t t o t h e e q u a t i o n ( x v e ) ( y i/ e ) ( x V e ) - ( y v e ) - * V e - e . The c o l l e c t i o n of a l l v a r i e t i e s of L-groups i s i t s e l f a complete l a t t i c e o r d e r e d by containment w i t h l a r g e s t member i: ( = a l l [-groups, d e f i n e d by x = x) and s m a l l e s t member E ( = a l l one-element L-groups, defined by x = e ) . The f i r s t remarkable r e s u l t about t h e l a t t i c e of a l l v a r i e t i e s of L-groups was d i s c o v e r e d by Weinberg [26] who showed t h a t t h e a b e l i a n v a r i e t y A i s t h e o n l y cover of E; t h a t i s , any v a r i e t y which c o n t a i n s a n !-group w i t h more t h a n one element must c o n t a i n a l l a b e l i a n !-groups. Another dramatic way t o e x p r e s s t h i s r e s u l t i s t h a t i f a n f-group s a t i s f i e s xy = yx t h e n i t s a t i s f i e s every e q u a t i o n ( e x c e p t t h o s e which would f o r c e t h e group t o c o n t a i n o n l y e ) . Much of t h e e a r l y work i n v a r i e t i e s of L-groups was done by Martinez ( [151, [ 1 6 ] , [17]). He observed t h a t t h e normal valued v a r i e t y N i s v e r y l a r g e . I l a t e r showed [9] t h a t N i s t h e unique l a r g e s t proper v a r i e t y . That i s , i f an L-group s a t i s f i e s any n o n - t r i v i a l e q u a t i o n , i t must a l s o s a t i s f y xy < ~ 2 x 2 for x , y 2 e . I n a s e n s e , t h e law i n t h e p r e v i o u s s e n t e n c e i s t h e weakest e q u a t i o n a l c o n d i t i o n t h a t can b e r e q u i r e d of an !-group, w h i l e xy = yx i s t h e s t r o n g e s t .

I t f o l l o w s from u n i v e r s a l a l g e b r a i c c o n s i d e r a t i o n s t h a t s i n c e A i s d e f i n e d by one e q u a t i o n , A cannot b e a n o n - t r i v i a l i n t e r s e c t i o n of a tower of v a r i e t i e s . T h i s makes i t i n t e r e s t i n g t o look f o r covers of A . I n [ 2 3 ] , Scrimger d i s c o v e r e d a n i n f i n i t e c o l l e c t i o n of c o v e r s of A , one f o r each prime number p . Like most small v a r i e t i e s , each o f t h e s e i s g e n e r a t e d by a s i n g l e k-group

i s t h e L-group g e n e r a t e d by abp Let all

=

a.

a,b

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e

such t h a t

a

A

abi = e

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if

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And


S

P

and

Much l a t e r , Medvedev [ 2 l ] d i s c o v e r e d t h r e e r e p r e s e n t a b l e covers of

A.

M+ b e t h e t o t a l l y o r d e r e d group g e n e r a t e d by a , b > e w i t h an < ab for -I + n , and l e t fl b e the v a r i e t y g e n e r a t e d by M . S i m i l a r l y , !I i s defined

an < a b - l . F i n a l l y , Ill i s t h e v a r i e t y g e n e r a t e d by t h e f r e e n i l p o t e n t i j k c l a s s 2 group on a , b , o r d e r e d l e x i c o g r a p h i c a l l y on a b [ a , b ] Just before t h i s c o n f e r e n c e , G. Bergmann, f o l l o w i n g a s u g g e s t i o n of T . F e i l , found a n o t h e r r e p r e s e n t a b l e cover. These a r e a l l o f t h e p r e s e n t l y known c o v e r s o f A . The d i f f i c u l t y i n f i n d i n g more l e a d s one t o c o n j e c t u r e t h e r e may n o t b e many more.

with

.

F i n a l l y , c o n s i d e r t h e q u e s t i o n of t h e number and d i s t r i b u t i o n o f v a r i e t i e s . How many v a r i e t i e s a r e t h e r e ? How broad and how t a l l i s t h e l a t t i c e of v a r i e t i e s ? S i n c e a l l e q u a t i o n s may be c o n s i d e r e d t o be w r i t t e n i n a c o u n t a b l e a l p h a b e t , t h e r e a r e o n l y countably many e q u a t i o n s , t h u s

2

no

s e t s of e q u a t i o n s , and hence no more

t h a n 2’0 v a r i e t i e s . Kopytov and Medvedev [13] showed t h e r e a r e indeed ZHJ v a r i e t i e s . R e i l l y [22] gave a g e n e r a l method t o c o n s t r u c t v a r i e t i e s which produced a n uncountable c o l l e c t i o n o f mutually incomparable v a r i e t i e s , showing t h a t t h e l a t t i c e i s v e r y broad. hnd F e i l [ 3 ] c o n s t r u c t e d a tower of v a r i e t i e s whose o r d e r t y p e i s t h a t o f a r e a l i n t e r v a l , showing t h a t t h e l a t t i c e i s a l s o very tall.

Classificationo f lattice ordered groups

153

We have n o t d i s c u s s e d h e r e t h e very i n t e r e s t i n & a r i t h m e t i c s t r u c t u r e of t h e l s t t i - e of v a r i e t i e s . ? h e r e a d e r is r e f e r r e d t o t h e f a i r l y comprehensive paper

[6l. 2. 'I'!)K,'TON

CLSIXE:'

Mmy #:if the c l a s s e s o f i n t e r e s t i n f-groups a r e n o t e q u a t i o n a l l y d e f i n a b l e . Tn p a r t i c r i l x r , t h e r e a r e rimy c l a s s e s of i n t e r e s t a l r e a d y c o n t a i n e d i n t h e a b e l i a n c l a s s . F o r example, Archimedean L-iwoups must lie a b e l i a n (where Archimedean n:eans t h e r e a r e no n o n - t r i v i a l bounded c y c l i c s u b g r o u p s ) . Tn o r d e r t o make a Yiiier d i s t i r l c t i o n , t h e n , Martine:: [181, [l'3] i n t r o d u c e d t h e n o t i o n of a t o r s i o n c l a s s of k-t:roups: :I c l a s s c l o s e d under L-homomorphic images and under j o i n s of convex L-suljgrorips. I t i s c l e a r t h a t v a r i e t i e s a r e c l o s e d under L-homomorphic image". I t i s n o t so c l e a r t h a t v a r i e t i e s a r e c l o s e d under j o i n s o f convex f-suh$:roups. N e v e r t h e l e s s , i t i s t r u e , a s I showed i n [lo]. Other examples o f t o r s i o n c l a s s e s a r e the h,yperarchiirieiiean L-groups--those f o r which e v e r y k?-homomorphic image i s Archimedean, and t h e h y p e r i n i e g r a l L-groups--those for which e v e r y L-honmnorphic image i s a s u b d i r e c t p r o d u c t of c o p i e s of t h e L-group o f integers.

I t i s e a s i l y s e e n t h a t t h e i n t e r s e c t i o n of any c o l l e c t i o n of t o r s i o n c l a s s e s i s a t o r s i o n ~13:;s. Thus, l i k e v a r i e t i e s , the c o l l e c t i o n of a l l t o r s i o n c l a s s e s forms a complete l a t t i c e . Iicivever, t h e previous. s t a t e m e n t i s riot e n t i r e l y t r u e i n t h e usual s e n s e , for t h e " l a t t i c e " of t o r s i o n c l a s s e s i s a p r o p e r c l a s s , n o t a s e t . There simply a r e t o o many t o r s i o n c l a s s e s . Some f e e l i n g f o r t h e q u a n t i t y o f t o r s i o n c l a s s e s may be had by c o n s i d e r i n g t h e way i n which t h e l a t , t i c e of v a r i e t i e s i s eiiibedded i n t h e l a t , t i c e of t o r s i o n c l a s s e s . I t i s n o t a s u b l a t t i c e (Smith [24]) . Moreover, a s Martinez and I showed [ZO], no v a r i e t y c o v e r s a t o r s i o n c l a s s , s o e v e r y v a r i e t y i s the j o i n of a l l t o r s i o n c l a s s e s it, p r o p e r l y c o n t a i n s . B u t no n o n - t r i v i a l v a r i e t y i s t h e j o i n o f any s e t of t o r s i o n c l a s s e s i t p r o p e r l y c o n t a i n s . The irnport,ant word i n t h e p r e v i o u s s e n t e n c e i s " s e t . " T am n o t aware of any o t h e r n a t u r a l c o n t e x t i n which i t is i n t e r e s t i n g t o make a d i s t i n c t i o n between "complete" arid " s e t complete." That i s , a l a t t i c e may have t h e p r o p e r t y t h a t e v e r y s e t of i t s elements h a s an upper bound, y e t n o t be complete.

Many s i m i l a r phenomena o c c u r i n r a d i c a l c l a s s e s of groups ( w i t h o u t o r d e r ) a s s t u d i e d by Vovsi 1251.

3. bLEMENlARY CLASSES I n many ways the most n a t u r a l c l a s s i f i c a t i o n o f L-groups, o r any a l g e b r a i c s t r u c t u r e , i s by elementary c l a s s e s , t h a t is, c l a s s e s d e f i n e d by f i r s t o r d e r s t a t e m e n t s i n t h e language a p p r o p r i a t e t o t h e s t r u c t u r e . V a r i e t i e s a r e e l e m e n t a r y c l a s s e s , t o r s i o n c l a s s e s i n & e n e r a 1 a r e n o t . Other i n t e r e s t i n g c l a s s e s are ( i ) t h e c l a s s d e f i n e d by t h e s t a t e m e n t : i f x 2 = y 7 t h e n x - y , and ( i i )t h e c l a s s of d i v j s i b l e L-groiips, d e f i n e d by the i n f i n i t e s e t o f s t a t e m e n t s ( f o r each p o s i t i v e integer

n):

for a l l

x

there exists

y

such t h a t

y"

-

x.

The f i r s t o r d e r lanr:uage of o r d e r e d s e t s does n o t mnke very f i n e d i s t i n c t i o n s . I t is w e l l known, for example, t h a t any two dense t o t a l l y o r d e r e d s e t s w i t h o u t e n d p o i n t s a r e i n d i s t i n g u i s h a b l e i n t h e f i r s t o r d e r language. Because e v e r y L-group i s c o n t a i n e d i n an L-group A ( U ) of nutnniorphisms of a t o t a l l y o r d e r e d s e t $2, i t i s i n t e r e s t i n p t o c o n s i d e r the elementary t h e o r y o f such t-jiroups A(0). In 171, Gurevich and I i n v e s t i i 7 a t e d t h e q u e s t i o n of which e-groups a r e e l e m e n t w i l y e q u i v a l e n t t o t h e L-group A( B) of automorphisms of t h e r e a l l i n e . A t f i r s t g l a n c e , i t might h e expected t h a t s i i i c e 1 and Q, t h e r a t i o n a l s , a r e e q u i v a l e n t i n t h e f i r s t order lmipiage of o r d e r e d s e t s , perhaps A ( 1 ) and A ( Q )

W.C. Holland

154

are equivalent. Yhis i s fur from beinc the case, however, 9s con be seen f r m I.he following considerations. Iln element f E R ( Q ) i s said Lo be n ~?on:,es cyclrc if for some u € R, a < crf and i f R E Q and i3f # 13 then f o r :some intelTer n,

UP< P < a?’’.

Roughly speakinc, r i l l the points moved by

f

l i e In the

convexification or the o r b i t f n f n l of one point. This description of convex cycles i s not even i n t h e lanjp3,qe of‘ L-croups, l e t alone t h e f i r f i t order language. However, i t is seen t.0 be equivalent t o the followiw f i r s t order statement: f > e arid i f f = g V h and g A 11 e then E = e or h = e. I 1 can be shown [8] (and i s easy t o believe) thnt m y two convex cycles i n A ( 1 ) :ire condugate. ‘ihat i s , i f f and f ’ a r e convex cycles then there exisLs k such -1 t h a t f’ = k f’k. This f i r s t order stnten.ent is f a l s e for A ( Q ) bemuse a cmivex cycle whose interval of moved points hnu end points i n Q crmnot be con./u:nt.e to one whose i n t e r v a l of moved points has i r r a t i o n a l end pui!itn.

-

In 1’71 we showed the surprising r e s u l t t h a t there is R sentence + i n the first, order l w u a g e of L-groups such that A ( 1 ) satisfies $, and i f A ( R ) i s t r a n s i t i v e on R ( a technical requirement) arid s a t i s f i e s ti8 then 11 .is isoriiorphic t o P. That is, B i s completely characterized by a f i r s t order ststemen% concernint: i t s autoniorphism L-group. I t is not possi.ble t o chnracterixe Q i n t h i s way, since the L-eroup R(Q) is L-isomorphic t o A ( I I ), IT the i r r a t i o n n l s . B u t again i n [7] we showed there is u f i r s t order statement $1 i r i the 1aq:uabre of L-groups such thut A ( Q ) and A ( I I ) sntlvfy $ arid i f A ( R ) is t r a n s i l i v e arid s a t i s f i e s JI then R i s isomorphic tCJ e i t h e r Q o r IT. The sentences 4 and JI of the precedinf: pnrograph can be taken t o be purely i n the language of groups. And a s shown by .JRmbu-Ciraudet 1121 they cnn also be taken t o be purely i n the languwe of l a t t i c e s . The paper [l?] contains many more and deeper r e s u l t s along these l i n e s . REFERENCES Dieard, A., Keimel, K. and Wolfenstein, S., Groupee e t Anneaux RBticul6s (Lecture Notes i n Mathematics 608, Springer, 1077). Birkhoff, C., l a t t i c e Theory (Amer. Math. Soc. Colloquium Pub., vol. 25, 1969 1. Peil, T., An uncountable tower of L-group v a r i e t i e s , Algebra Universalis 11, 129-1 31.

( 1981)

Puchs, L., P a r t i a l l y rJrdered Algebraic Systems (Addison Wesley, New York, 19631. Class, A. M. W., Ordered Permutation Groups (London Math. Soc. Lecture Notes 55, Cambridge U. Press, 19H1). Class, A. M . W., Ilolland, W. C. und McCleary, Tr. H., v a r i e t i e s , Algebra Universalin 10 (1980), 1-20. Curevich, 7 . and Holland, W. C., hhth. SOC. 265 (1381), 527-534.

l?ie s t r u c t u r e of k-eroup

Hecoenixing the r e a l l i n e , Trans. Amer.

Hollnnd, W. C., The l a t t i c e ordered group of automorphisme of an ordered s e t , Mchlfzan Math. .T. 10 (1963), 399-408. Hollrind, W. C., The l a r g e s t proper variety of l a t t i c e ordered groups, Prvc. Amer. Math. Soo. 57 (19761, 25-28.

Classification of lattice ordered groups

155

I L+J I151

M a r t i n e z , .T., 'Torsion theor:/ Ti.r l a t t i c e ordered i:roups, ( 1.W5 ), ? g 4 - P ? .

Czech. Math.

.r. 25

Mart.iner,, :. , The fundarcental t,heorenl on torsion classes o f l a t t i c e o r d e r e d ):youps, '['rims. h e r . IMzith. :'oc . (19~?0),j11-317. M a r t i n e z , ,T. and Tiolland, iz'. f'. , A c c e s s i b i l i t y of' t . o r s i o n c l a s s e s , A lpe lirn U n i v e r s a l i s 9 197') ) , 19q-?Oh. Medvedev, N . ,Ta., 'The l a t t i c e s of v a r i e t i e s of l a t t i c e o r d e r e d p r o u p s and Lie a l g e b r a s , A1h;ehra and L n c i c 16 (1977), 27-30.

R e i l l y , N . R . , A s u b s e r ' i i l a t t . i c e tnf' t h e l a t t i c e of v a r i e t i e s of l a t t i c e o r d e r e d groups, Canad. ,T. Math. 33 (1981 ), 1109-1318.

Scrimger, E. 2,., A large class of small v a r i e t i e s of l a t , t i c e ordered groups, €'roc. h e r . Math. S o c . 51 (1975), 301-106. Smith, t T . , The l a t t i c e of !-group ( 11801, 347-?57.

v a r i e t i e s , Trans. h e r . hlath. S o c . 257

Vovsi, S. M . , i l n r a d i c a l and c o r a d i c a l classes of e-groilps, Algebra Universalis ( t o appear).

[26] Weinberg, E. C., F r e e l a t t i c e o r d e r e d groups, Math. Ann. 151 (1961), 187-18'4.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 (1984) 157-170 0 Elsevier Science Publishers B.V.(North-Holland)

THE ISOMORPHISM

157

PROBLEM AND UNDECIDABLE P R O P E R T I E S

FOR F I N I 'TELY PRESENTED L A T T I C E - ORDERED GROUPS A. M.

W. Glass

1

Department of Mathematics Bowling Green State University Bowling Green, Ohio 43403, U.S.A.

Resume

Nous donnons une version abstraite d'une technique gbnkrale permettant de montrer, s o u s des hypotheses convenables, que la plupart des problemes algorithmiques concernant l'ensemble rPcursif des algebres finiment prhsent6es ( d'une variete d'algkbres rkcursivement axiomatisee ) ne sont pas resolubles recursivement quand i l existe line algkbre finiment presentee avec un problhe des mots non rksoluble. Nous Gtablissons un lemme technique ( The messuage Lemma ) pour les groupes rkticulks, qui utilise Les groupes d'automorphisrnes de chaines, e t nous l'utilisons pour montrer, par exemple, qu'il n'y a pas d'algorithme pour determiner si oui ou non un groupe r6ticul6 finiiient present6 quelconque a un seul elbment, ou est abelien, o u est totalement ordonne.

A Zattice-ordered group G i s a group and a l a t t i c e such t h a t f ( g V h)k

=

{xi: i E I}

-1

,

under I f each

and

f(g A h)k

fgk A fhk

=

for a l l

{xi: i E 1) be a non-empty s e t of symbols.

Let in

fgk V fhk

is t h e s m a l l e s t s e t containing

the l a t t i c e operations V

r . ( x ) i s a word i n J

i s c a l l e d a presentation.

and A ,

{xi: i E I},

f,g,h,k E G.

The set of uords

{xi: i E I } closed and t h e group o p e r a t i o n .

then

(xi; r j ( s ) ) i E T

In t h e s p e c i a l case t h a t

I

,jcJ

and

J

are

EJ,

we form t h e

f i n i t e , w e o b t a i n a finite presentation. Given a f i n i t e p r e s e n t a t i o n

n

=

(xi; r j ( x ) ) i

finitely presented lattice-ordered group t h e f r e e l a t t i c e - o r d e r e d group

F

on

Gn

by taking the q u o t i e n t of

{xi: i E I} by t h e k e r n e l (convex

normal s u b l a t t i c e subgroup) generated by t h e subset F.

So

r.(z) J

=

e

in

G

n

for a l l

j E J.

{ r . ( L ) : j E J} J

(Throughout we w i l l use

f o r t h e i d e n t i t y element of an a r b i t r a r y l a t t i c e - o r d e r e d group. ) 'Research supported i n p a r t by a Bowling Green S t a t e University Faculty Research Grant.

of e

A.M. W. Glass

158

THEOREM (THE ISOMORPHILSM PROBLEM).

There i s no recursive algorithm

t o determine whether o r not two a r b i t r a r y f i n i t e l y presented l a t t i c r ordered groups are isomorphic; i . e . , t o determine whether or not thio arbitrary f i n i t e presentations give r i s e t o isomorphic f i n i t e l ? / presented Zattice-ordered groups. Let

P

be a property of ( f i n i t e l y p r e s e n t e d ) l a t t i c e - o r d e r e d groups

t h a t i s preserved under isomorphisms.

nl, the f i n i t e l y presented

( i ) f o r some f i n i t e p r e s e n t a t i o n l a t t i c e - o r d e r e d group

i s c a l l e d a Markov property i f

P

enjoys

G,

P,

and

1 (ii)

there i s a f i r r t e presentation presented l a t t i c e - o r d e r e d group

n2

such t h a t t h e f i n i t e l y cannot be er?,bedded i n

G,,

2 any f i n i t e l y presented l a t t i c e - o r d e r e d group which enjoys

THEOREM. Let

P

be a Markov property.

Then t h e r e i s no recursive

algorithm which determines, f o r every f i n i t e presentation n ,

not

G

P.

whether or

enjoys P.

COROLLARY. There i s no recursive algorithm which determines, f o r every f i n i t e presentation

TT,

whether o r not

G

TI

enjoys

P,

is any o f the fozlowing:

( i ) abelian

(ii) Archimedean (iii) t r i v i a 2 (ivl

free

(v) f r e e abelian (vil

l i n e a r l y ordered

( v i i ) l i n e a r l y orderable ( p r e s e m i n g the group s t m c t u r e l (viiil

representable

( i x ) normal valued

(21 unique e x t r a c t i o n o f a l l roots ( x i ) soluble word problem.

where

P

Isomorphism problem and undecidable properties

159

Clearly examples l i k e t h e s e can be m u l t i p l i e d till t h e cows come We leave it t o the reader t o a d j o i n h i s own f a v o l i r i t e s .

home.

Most of t h e above p r o p e r t i e s a r e h e r e d i t a r y - - i . e . ,

a finitely

presented l a t t i c e - o r d e r e d proup which can be enhedded i n one possessing

P a l s o enjoys

P.

However, unlike groups, being a f r e e l a t t i c e - o r d e r e d

group i s not a h e r e d i t a r y property t h o w h i t i s Markov as we w i l l s e e . Every f i n i t e l y presented l a t t i c e - o r d e r e d group i s e f f e c t i v e l y isomorphic t o one defined by a s i n p l e r e l a t i o n - - I f

n

T j ( 5 ) l i E I ,jcJ

(xi;

=

with

r ( 2 ) E V { l r . ( & ) l : j E J} where Then

G,

7

J G,

,

since

IyI

>e

and

I

IyI

(ii

=

{0,1,2,... 1 ,

f i n i t e and

y Vy

& Iy/ = e

only consider f i n i t e p r e s e n t a t i o n s subset of

=

J

(xi;

-1, l e t

only if y

= =

e.

(xi;

r(x))iEI.

Hence we need

r ( z ) ) i E I . Regarding

I

as a

we can e f f e c t i v e l y ( r e c u r s i v e l y ) a t t a c h a

n a t u r a l number t o each f i n i t e p r e s e n t a t i o n i n a one-to-one onto way.

If

denotes t h i s number, then t h e theorem, s t a t e d p r e c i s e l y , says t h a t if

P

i s a Markov property, then

recursive s e t .

{ # ( n ) : Gn

er.jojis P}

i s not a

However, s i n c e t h e i n t u i t i o n i s more informative and can

e a s i l y be converted t o formally c o r r e c t statements, we w i l l use t h e i n t u i t i v e words from now on and leave t h e conversion t o t h e f o r m a l i s t . Whereas t h e proofs of t h e corresponding r e s u l t s i n group theory r e l y heavily on f r e e products with amalgamation or t h e Higman-Neumann-Neumann theorem ( s e e [6] and

“71 ),

t h e s e c o n s t r u c t i o n s a r e not v a l i d f o r l a t t i c e -

ordered groups ( [l, Theorem lac] and [ 2 ] ) and we must r e s o r t t o .&permutation groups and r e s u l t s (proved using 8-permutation groups) whose group-theoretic analogues n a t u r a l l y belong t o combinatorial group theory.

The isomorphism problem and ( i ) , ( T i ) , (iv), ( v ) a n d ( x i ) of

t h e c o r o l l a r y can be proved q u i t e e a s i l y and we g i v e a s e p a r a t e s o l u t i o n f o r t h i s p a r t ; t h e main theorem r e q u i r e s more d e l i c a t e techniques which beside

giving a l l p a r t s of t h e c o r o l l a r y a l s o y i e l d

problem by ( i i i )of t h e c o r o l l a r y

( L e t nn

=

t h e isomorphism

(xo; xo).

I f we could

solve t h e isomorphism problem, we could r e c u r s i v e l y determine f o r an

A.M. W. Glass

160

a r b i t r a r y f i n i t e presentation whether or not

G,,

P

{el.

whether or not

TI,

G

*

E

G

i.e.,

;

no

This c o n t r a d i c t s ( i i i )of t h e c o r o l l a r y ) .

The key t o a l l t h e proofs i s t h e existence o f a f i n i t e l y presented (non-abelian) l a t t i c e - o r d e r e d group with i n s o l u b l e word problem ([l, Chapter 131 or, more f u l l y , i n

[&I).

We show t h e non-existence of

r e c u r s i v e algorithms by coding i n t h e word problem for t h i s l a t t i c e ordered group.

Most of t h e proofs a r e general a l g e b r a ; t h e exceptions

a r e t h e proofs f o r t h e Isomorphism Problem Theorem and t h e Messuage The Messuage Lemma i s p i v o t a l and i s proved by using l a t t i c e -

Lemma below.

ordered groups of automorphisms of ordered s e t r . To prove t h e isomorphism problem we use t h e followinp r e s u l t of Keith R . Pierce [l, Theorem 10B] which i s proved by using b?-perinutation groups ( i. e . , l a t t i c e - o r d e r e d groups of order-preserving permutations of l i n e a r l y ordered s e t s )

Lam 0. -

.

Every lattice-ordered group can be embedded in one in

which any two strictly positive Now l e t n o

I>

el elements are conjugate.

be t h e f i n i t e p r e s e n t a t i o n of t h e f i n i t e l y presented

non-abelian l a t t i c e - o r d e r e d group with i n s o l u b l e word problem; say

no

= ( x l , . , .,xn;

Let

= (xO,xl , . . . , x n ;

Ti

0,w

r(E),

G

-1

( x o ~ w l - ~ x ~ )](x i ( ) ) . BY Lema 0 , i=l

can be embedded i n a l a t t i c e - o r d e r e d group

G

. . , xn .

r ( 5 ) ) . Let w be an a r b i t r a r y word i n x l , .

H

(wI

i n which

and

,O n

v

\xi]

a r e conjugate ( i f

i=l

h-'/w(h

=

v

(xi( i=l

by t h e image of of

w # e

in

G

).

h E H be such t h a t

Let

n

and

G,,

Ho be t h e s u b l a t t i c e subgroup of together with

0 hold i n Ho i f

G,

xo

h.

image of

h,

G and t h e diagram

G,

generated

Since a l l t h e defining r e l a t i o n s

i s replaced by

0,w

H

Ho i s a homomorphic

-G noc

0,w

/

=o,w

HO i s commutative under t h e mapping

xi

M

xi (1< i

< n)

of

into

Gn

0

Isomorphism problem and undecidable properties

.

G n 0,w

If

,

e # g 6 Gn

then t h e image of

e.

0

Since t h e diagram i s commutative, t h e image of Hence t h e homomorphism of in

Ho i s not

in

g

161

.

G,,

into

G

n0 However, i f

w

e

=

in

in

i s not e . 0,W i s an embedding if w # e g

Gn

0,w G n O , then

w

=

G,,

e

i n any l s t t i c e -

0

r(2) = e .

ordered group i n which in

, xo-1Iw(xo

G,,

v

=

0,w IyI

e

=

only i f

y

=

e).

group of i n t e g e r s , Z ElZ

(l,-l); note t h a t

I-+

( x i ( ; hence

x

i=O

l a t t i c e - o r d e r e d group on

xo

In p a r t i c u l a r ,

w

=

e

in

Thus

xo

i

=

But,

.

cn

n

0,w ( / y (2 e

e (1 4 i S n )

with

is j u s t the free (abelian) 0,w ( I t i s isomorphic t o Z m Z where Z i s t h e Gn

i s ordered by:

(m,n) 2 ( 0 , O ) i f

m,n 2 0,

and

( 1 , O )= (l,-l)v (0,O)& ( 0 , l ) = -(l,-l) V (0,O)).

Now t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one g e n e r a t o r has s o l u b l e word problem [l, Theorem 1 1 . 5 1 .

I f t h e r e were a r e c u r s i v e algorithm

which determined whether t h e f i n i t e l y presented l a t t i c e - o r d e r e d groups given by two a r b i t r a r y f i n i t e p r e s e n t a t i o n s were isomorphic or n o t , we would be a b l e t o decide whether or not t h e f i n i t e l y presented l a t t i c e ordered group given by an a r b i t r a r y f i n i t e p r e s e n t a t i o n were isomorphic t o t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one g e n e r a t o r . could determine f o r an a r b i t r a r y word not

w

in

xl, ..., xn,

So w e

whether o r

were t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one 0,w i s non-abelian and has i n s o l u b l e word problem, g e n e r a t o r . Since G

Gn G

GTI

i s non-abelian and has i n s o l u b l e word problem whenever

w # e

in

0,w

.

(An algorithm f o r t h e word problem f o r

would imply t h e

0,W e x i s t e n c e of an algorithm f o r t h e word problem for G,,

.) 0

Hence

Gn

0,w

i s isomorphic t o t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one generator i f and only i f

w = e

in

GTT 0

.

Thus t h e r e c u r s i v e algorithm f o r t h e

ismorphism problem l e a d s t o a r e c u r s i v e algorithm f o r t h e word problem in

.

Gn

This l a s t i s a c o n t r a d i c t i o n and e s t a b l i s h e s t h e Isomorphism

0 Problem Theorem.

Since any f i n i t e l y generated f r e e l a t t i c e - o r d e r e d group

162

A.M. W. Glass

has s o l u b l e word problem [l, Theorem 1 1 . 5 1 and any Archimedean l a t t i c e ordered group i s a b e l i an,

i s a b e l i a n (Archimedean, f r e e , f r e e 0,w . a b e l i a n , has soluble word problem) i f and only i f w = e i n G

Gn

no

//

This e s t a b l i s h e s ( i ) , ( T i ) , ( i v ) , ( v ) and ( x i ) of t h e c o r o l l a r y .

The above proof a l s o e s t a b l i s h e s t h a t " f r e e " i s indeed a Markov property. I n o r d e r t o prove t h e Markov Property Theorem (and hence t h e e n t i r e c o r o l l a r y ) , we need t o do some work.

The proof hinges on t h e Messuage

Lemma which we now g i v e . Let word i n

d x ) ) i E I be

IT = (xi;

{xi: i

E I}. T I ( w )

f i n i t e p r e s e n t a t i o n such t h a t ( i i ) GTI can be embedded i n

THE MESSUAGE -

LEMMA.

a f i n i t e p r e s e n t a t i o n and

i s c a l l e d a w-rnessuage' of

Gn(w) i f

w # e

TI

w = e

{e} i f

( i )G n ( w )

in

be a

w

i f it i s a in

GTI and

G,.

There i s a recursive algorithm which, given an

a r b i t r a r y f i n i t e presentation

TI =

(xi; r l ~ ) l i f l and a word w i n

{xi: i E I}, constructs ( e f f e c t i v e l y ) a w-messuage of

K.

We temporarily postpone t h e proof of the Messuage Lemma (which a l s o appears as Theorem

D i n [ 3 ] where i t i s used t o prove t h a t every countable

l a t t i c e - o r d e r e d group can be embedded i n a 7 generator L-simple l a t t i c e ordered group) and show how t o deduce t h e Markov Property Theorem from i t .

x: Let

no

be t h e f i n i t e p r e s e n t a t i o n such t h a t

n1 and n 2

has

KO

i n s o l u b l e word problem.

no,

G

Let

nl, n 2 be f i n i t e p r e s e n t a t i o n s such t h a t

a r e d i s j o i n t s e t s of symbols,

enjoys

G,

P

and

1 G,

can not be embedded i n any f i n i t e l y presented l a t t i c e - o r d e r e d

2

'Messuage was " o r i g i n a l l y t h e p o r t i o n of land intended t o be occupied, o r a c t u a l l y occupied, a s a s i t e f o r a dwelling-house and i t s The word seems p a r t i c u l a r l y appurtenances" [Oxford English Dictionary] a p t f o r t h e property we a r e describing; " t e s t " was used i n [ 7 ] .

.

Isomorphism problem and undecidable properties

163

group that enjoys P. Let G be the free product (as lattice-ordered groups) of G

,

and G

TI

.

2 the union of those of n o and

Then G

=

G,

where

has as "generators"

0

where r(2) and s(y-)

and "relation" Ir(,x)l V I s ( x ) l

TI2,

are the "relations" of no

is effectively obtained from no

Note that

and n2

and n2.

respectively.

Since G

TI

0 has insoluble word problem, so does G,

can be (even effectively)

(Sn

0

embedded in GTI) .

L,et w be a word in the "generators" of

and G(w) be free product (as lattice-ordered

be the w-messuage of n g r o u p s ) of

G

,

and G

1

n(w).

Now if w

=

e

in G , , G(w)

and hence enjoys P. If w # e in G ,

isomorphic to G,

TI~W)

TI,

is

,

G

be embedded in G

2

n(w)

and so in G ( w ) ;

does not enjoy P.

thus G(w)

Consequently, G(w) enjoys P if and only if w G,

can

TI

1

=

e in GTI. Since

has insoluble word problem, the Markov Property Theorem follows. // We next note some consequences of the proof. A property P

of finitely presented lattice-ordered groups that is

preserved under isomorphisms is said to be incompatible x i t h f r e e ( d i r e c t ) productsiffor some finite presentation

(i )

TI

1

enjoys P,

the finitely presented lattice-ordered group G n 1

and (Ti)

the free (direct) product (as lattice-ordered groups ) of G 1 and any non-trivial finitely presented lattice-ordered group always fails to enjoy P.

THEOREM. I f

P

i s a property t h a t i s incompatible w i t h f r e e

( d i r e c t ) products, then t h e r e is no recursive algorithm t o determine, f o r every f i n i t e presentation

n,

whether or not

G n enjoys

P.

In

particuZar, there i s no recursive aZgorithm t o determine whether or not

a f i n i t e l y presented lattice-ordered group can be w i t t e n as t h e f r e e ( d i r e c t ) product ( a s Zattice-ordered groups) of two non-trivia2 l a t t i c e ordered groups.

A.M. W. Glass

164

prooS: Let n 1 be the guaranteed presentation and no as in the previous proof. Again we assume that the generators of no and are disjoint. For an arbitrary word

TI

1

w in the generators of n o ,

G(w) be the free (direct) product of G

);

and G

let

the presenta-

n1

tion n for G(w)

can be effectively given from that of

In either case, G(w)

enjoys P if and only if w

=

TI^

and n,(w)

.

e in G TI

This

0

proves the first part of the theorem. Let n1

=

(xl: x1 2 e);

f o r the “in particular. I’

so

G TI,

3

Z

has the desired requirements

//

Finally, we note that the above proofs have nothing to do with lattice-ordered groups. Hence: THEOREM (essentially Rabin [7]) ,

Assume a c l a s s of algebras has

f i n i t e signature and a one element t r i v i a l algebra contained i n a l l algebras.

Further suppose t h a t f r e e algebras e x i s t i n t h i s c l a s s , t h e

Messuage Lema h02ds and there e x i s t s a f i n i t e Z y presented algebra i n t h e c l a s s with insoluble word problem.

If, moreover, the c l a s s admits

f r e e ( d i r e c t ) products and every f i n i t e l y presented algebra can be embedded i n i t s f r e e ( d i r e c t ) product with any other f i n i t e l y presented algebra (and t h e d i r e c t product of t v o f i n i t e l y presented algebras can be given a f i n i t e presentation which i s e f f e c t i v e l y obtainable from that of t h e two swnmands), then t h e Markov Property Theorem holds a s does

t h e previous theorem.

In direct contrast to the above result, the proof of the Messuage Lemma that we give is very concretely rooted in lattice-ordered groups it is quite unlike Rabin‘s proof for groups (or the easiest proof given in [6,Section IV 4 1 ) and is clearly the pivotal result in this paper. Let

TI =

(xi; (r(z))ieIr

Assume, for ease of notation, that

xG,x1,x2,x3,xh j! {xi: i E I} and let w be a word in {xi: i E 11.

Isomorphism problem and undecidable properties

First, we assume that w

=

.

e in G ,

Then since r(5) = e in

-1

But xo lwlxo = V { l x i / : i E TI; hence -1 A lx41 = e; xi e for all i E I. Sin1.e w = e, lxcl = \x4 1 = ~wx/,~l -1 so x4 = e. Now x2 Ix Ix ixOI; thus x : e. By the last relation 4 2 6 i n the presentation of G x1 = e . Since 1x21 V ix3i Q xl, it (w)' f o l l o w s that x Therefore a11 generators of are 2 - x3 = T I (W)

w

G,(W)'

=

e

in

G n ( W ) '

equal to e in Gn(w);

i.e.,

G n L w ) 5 {el.

To prove the other part of the Messuage Lemma (viz.: in G,

, then G, can be embedded in G

if w # e

we need the following (w)' facts from [I]. The first is the Cayley-Holland Theorem (Appendix I), ll

the second comes from the proof of Lemma 2.2.1 and the third comes from the proof of Theorem 2E ( see the remark on page 58 ).

Lemmas 2 and 3

can be viewed as showing the considerable degree of "local" freedom in the choice of roots and conjugators. If R

for the lattice-ordered group of order-preserving permutations

i\(R)

of

is a linearly ordered set ( o r chain for short), we write

R

(where f Q g

. a

< al

(i

=

& Bo

O,l),

<

B1

if af Q ag

in R,

we say that R

lattice subgroup of A( R), group.

there is f € A ( R )

If for each

such that aif

is doubZy homogeneous.

=

Bi

If H i s a sub-

we say that (H,R) i s an t-permutation

let the support of h, supplhl = t o E R: oh # u}. can be uniquely extended to an automorphism of R, the

If h E A ( R ) ,

Each h E A ( R )

Dedekind completion of A(

f o r all a E R).

0. If the support of this extension of

h

to

5 ) is a single interval of 5 , we say that h is a bwnp. For any

other undefined terms, see [l]. ~LEMMA 1. If H such that

(H,T)

is a lattice-ordered group, there is a chain

is an L-permutation

grOUp.

T

A . M . W. Glass

I66

_L E_ l_M_ 2 . Let be bwnps.

Assume t h a t

k E A(R)

sup5(supp(fl)k

and similarly f o r

Then t h e r e is h E A l R ) supp(h)

W 3.

L

suppi-fl b s u p p ( g l ,

extending

h:

and

such tha t

h

6 E suppig),

afnh

fin].

[ n , a f J : [B,

=

be a doubly homogeneous chain and in

a3 = a$

and

R

g E AlQ)

then the re e x i s t s

0,1),

a E .supp(f),

-

E: A ( n i

s u p 5 ( s u p p ( g ) ) f o r some

=

Let

< f,q

E

@gn ( n E Z),

h-'j% = g.

Let

< 'a < a2 <

If no

(i =

and

(y,6)

--

inf.

t h e cZosure

contain

(v,6/

0 be a doubly homogeneous chain and

< f E A(Qi

e

qi: Iai,ai+ll

: bi+l,ai+21

with

such t h a t

g3 = f

gl[ai,ai+J = gi (i = 0,Z). If that and

i s a bump of

h

h

i s a bump and

h

g.

ah

Clearly

w # e

Now suppose t h a t

there i s a chain $n: T Let A(T) So

:T

T

Tnh*

g i v e n by:

with

.

G,,

if e

e

< g,h

E A(a)

G,, can b e embedded

By Lemma 0 ,

IwI and V I l x i l : i E I1 a r e

i n which

hgllwlho = V { / x i ( : i

E I).

By Lemma 1,

(H,T) is an f - p e r m u t a t i o n g r o u p .

such t h a t

ordered by: h

c-f

for a l l

Tn Tnf0

( n E Z ) and

E

via:

A(A) =

=

be orderrnorphisms o f a d i s j o i n t f a m i l y of c o p i e s of

A = U {Tn: n

in

0E H

in

H

i n a l a t t i c e - o r d e r e d group h

h Agh-'

g.

i s a bump of

c o n j u g a t e ; say

for all a E s u p p ( h ) , we w i l l cay

ag

=

=

fo

h*

n EX

.!n$nhog~~l (

Tn

where and T

< Tm

Tnh*

if

= T @

n n

h E A(T).

n

<m

T (n EX).

i n X.

We embed

( T~ E Tn; n E Z).

h$-' n

Let

Let

e

< fo

E~ 'rn; n E Z). Note t h a t

E A(A) TnfO

f i x e s no p o i n t o f t h e Dedekind completion

n

=

he Tn+l

of

A.

so, f g l l w * / f o = V I ~ ~ Zi IE : 11. We n e x t embed

A(h)

any bounded i n t e r v a l Theorem 10.2 of [l]).

(6

in

6 ) 1' 2

where

A(A')

of

A'

A

is cofinal i n

A'

and

i s a n a - s e t ( s e e t h e proof of

Now b u i l d an a - s e t

A"

with

A'

as a bounded

Isomorphism problem and undecidable properties interval. C

sup-,,A'

So

has c h a r a c t e r

a

coCc and

inf-,,A' A

Hence, by Lemma 10.3 of [l], we may assume t h a t

CYO'

supa(sup(f ) )

itself is a

A

supfi(sup?( w ) ) 0

belong t o the same A(R)-orbit and s i m i l a r l y f o r

0

w0 i s a bump of

where

has c h a r a c t e r

R i n which

bounded i n t e r v a l of a double homogeneous chain and

167

w'.

inf,

(The conclusion of t h i s paragraph can be

achieved without assuming t h e Generalised Continuum Hypothesis, but s i n c e t h e proof i s more d i f f i c u l t , we have omitted i t . ) Next we embed

{En:

n EZZ}

be sequences i n

in E ioA(n)

Gn E G o A ( R ) , bump with

A(R)

in

A(A)

Gnfl

6n+l

=

a s follows:

n EL.

for a l l

in+l f o r

& infl =

let

hi- E A ( R )

and

supp(h ) G U { ( ~ ~ ~ , B n~ €~ )Z:) .

be defined by:

=v{liil:i

w

t

fo

i s a bump of

0

has a unique bump of

p o i n t s i n t h e same

En <

< Gn+l

on

If21

g:

=

fl

and

6 0A ( R ) - -

7,8

E

ko:

[a0,6]P

of

A(R)

Go

[a,?]

with

and

v(fA)-*ko

orbit.

hl

=

g:

=

fl

( f:

v

gilf;gl

such t h a t

v

aohl

=

*t.

for

e

=

and each bump of

By Lemma 2, we can f i n d

-1 t f 2 (wolf2

E Z} and t -1 t supp(fOVfl fofl)

< 7 < Bo

n

kl:

[6,;]

=

;(fo)

;(fA)-*

v

gl E A ( Q ) =

= ko

Gl.

< 6 < y(f;)-'.

=

qf:

(A(n)

By Lemma 3 , t h e r e i s

gilfilf:flgl

fA.

=

n EZ}.

such t h a t

For l e t Let

be r e s t r i c t i o n s of elements

t -1 & y(fA)-lk,

z).

f2 E A ( R )

li { ( G n , B n ) :

=

gl-1f l-1foflgl)4 t

: [;,a,]

g,l[c(o,8]

f-1fOfl t

and

with

and

l4 E

gll[8,;] A( R).

zo(gl fl f o f l g l . f ~ . g ; l f A g l . ~ ; l f ~ ~ l )

i s doubly

gl 6 A ( R )

= kl.

Let

~n easy comuutation

-1 -1 t shows t h a t

(i2nji2n))

( n E Z;T~~ E

{on: n E Z}C R

Let

fY1f:flV

t r a n s i t i v e on a l l i t s o r b i t s i n with

h E A(A),

t contained i n i t with corresponding end-

V gi1f:glV

satisfy

be a

A(R)

wo

Observe t h a t

"(f:

E

fl

i s doubly homogeneous, by Lemma 3 t h e r e i s

R

Since

C f:.

<

and

{(a2n,E2n): n C Z } .

=

I ~ ( w ~ ) - A~ lwol ~ t

then

u f n = un (n € Z ) . 01

and

e

n E Z. For

all

s u p p ( fto )

supp(f 2 ) c U i(a2n-l,u2n):

such t h a t Clearly

A(R)

Let

We w i l l w r i t e

E 1) and

w*,

Gn < En <

A,

T2nf;2%fp

=

n

r ( i ) = e,

we have:

A(R)

(fA)-'lw1(f0) t If

T2nht

t

Now, i n

(60,go)=

with

i; : n E Z),

let

=

aL.

Since

fl

A.M. W. Glass

168

c o m u t e s with Gnhl

=

there i s

hl,

supp(hl)

( n E Z).

Since

< f 3 E.

e

f - l f f = hl 3 1 3 Thus t h e map x H

x.

inf3

I[Gnn'Gntl]: n E Z l

xG ++ wo

=

&n+l ( n E Z)--by Lemma 2.

=

of

fl

and

and

&"A(n),

H

Hence

f

fo, x1 ++ gl,

i s a homomorphism from

2 f l 2 If21,(f31. Since

A(n)

x2

such t h a t f

H

3 f

subgroup of

contains an isomorphic copy

{xi: i E 11 onto t h i s copy of G

"

. generated by (W)

t h e proof of t h e Messuage Lemma.

Ix

. i'

f l2.

2'

Gn and the homomorphism maps t h e s u b l a t t i c e subgroup of

generated by

<

A( R)--recall

into

"(W)

g:

supp(fl)

=

A ( n ) i s double t r a n s i t i v e on

Ci ( i E I ) , xo

++

f

f3,

u

having support U {[an,&n+l]: n E Zzl

A(R)

and

=

G

,

n

"(W)

the s u b l a t t i c e

i E I} i s

Gn

.

This completes

,$'

We conclude t h i s paper by obtaining an analogue of Theorem 1 of [O].

There i s a f i n i t e l y presented Zattice-ordered group

THEOREM.

C

such t h a t no r e c m s i v e ( e f f e c t i v e ) procedure e x i s t s t o determine whether or not an arbitrary word i n the generators of G n ( 1 ) An element o f t h e centre of

G

17

represents:

.

(2)

An element t h a t connnutes w i t h a given element of

(3)

An element whose conjugacy c l a s s i s f i n i t e .

(4)

An element d i s j o i n t from a given element of

(5)

An element of a given subZattice subgroup of

Proof:

Let

G

G

n

.

.

n

.

G

be a f i n i t e l y presented l a t t i c e - o r d e r e d group

Gn 0

r(5)).

with i n s o l u b l e word problem; say n o

n

=

- r(s),xo A e ) .

(xo,xl,...,xn'

l a t t i c e - o r d e r e d g r o u p s ) of

= ( X l,...,~n:

Then G n i s t h e f r e e product ( a s and a copy of

G

Let

Z.

Let

w be a word

" 0

i n no

and

vw

G

c e n t r e of

=

i f and only i f

and ( 2) ( t a k e xo If

w

=

-1 -1 [xo,w] = xo w xow.

e

in

w = e

We show t h a t in

a s t h e given element of

,

G, 0

then

vw

=

.

G

Gn

vw belongs t o t h e

This w i l l e s t a b l i s h (1)

1.

[xo,e] = e

which i s c l e a r l y

Isomorphism problem and undecidable properties

central i n

If w # e

G,.

,

in G,

let

169

be t h e homomorphism of

$

0 i n t o t h e wreath product

G,

G

wrZ

,O

y$

v

W

({Gnj,O) for y E G

=

({kn},O) where

=

kl

an d

,O =

w

in

w

=

where

f

2

r

0

if

n#O

. y

if

and

=

xo# = ( t e 1 , l ) and

k

w-l,

v

=

n fl

e

x,"vwx;

#

X

~

=

does not l i e i n t h e c e n t r e of

V(

~

Ixo/ A lvwl # e .

(4)follow.

n # 0,l.

if

and

w

G

W

The proof a l s o shows t h a t , provided i f~ mX and ~

Since

By using

v

W

(el

= e

so

n = O

( x o $ ) ~ l ( v w $ ) ( x o $#) vw$ and, t h e r e f o r e

i.e.,

G,;

=

n

-1, ko

( x o $ ) ~ l ( v w $ ) ( x o #=) ( { f n } , O ) n # 1,2. Thus

3;

given by:

TI

fn

Hence e

=

xo-1vwxo # vw

.

w # e

then i n in G n o ) a r e d i s t i n c t i n t e g e r s ) and

n

in

G,

if

w

=

e

G ,

G

( 3 ) and in

(or even

if

G,09

or t h e c e n t r e of

G

)

as

3

t h e s u b l a t t i c e subgroup of

G,

, we o b t a i n ( 5 ) .

//

I n [O], i t was a l s o e s t a b l i s h e d t h a t t h e r e e x i s t s a f i n i t e l y presented group

H

having a f i n i t e l y generated subgroup

not be f i n i t e l y presented.

G

which can-

The l a t t i c e - o r d e r e d group analogue was

e s t a b l i s h e d i n [ 5 ] - - i t can a l s o e a s i l y be derived from [ 4 ] . ACKNOWLEDGMNT.

I am most g r a t e f u l t o W . W. Boone and P . E.

Schupp f o r suggesting t h a t I look a t Ehrkov p r o p e r t i e s for l a t t i c e ordered groups.

Their encouragement while I was working on t h e problem

i s a l s o very much appreciated. NOTE. -

Recently we have shown t h a t t h e r e i s no r e c u r s i v e algorithm

which w i l l determine whether or not two a r b i t r a r y 10 generator 1 r e l a t o r ( a b e l i a n ) l a t t i c e - o r d e r e d groups a r e isomorphic [ A . M. W . Glass and J . J . Madden, The word problem versus t h e isomorphism problem, J . London Math. SOC.

( t o appear)].

170

A.M. W. Glass REFERENCES

0

G. Baumslag, W. W . Boone and B. H . Neumanri, Some unsolvable problems about elements and subgroups of groups, Math. S c a n d . , 7(195Q), 191-201,

1. A. M . W. Glass, Ordered Permutation Groups, London Math. Soc. Lecture Notes S e r i e s No. 55, Cambridge University Press, 1981. 2.

A. M. W. Glass, R e s u l t s i n p a r t i a l l y ordered groups, Comm. i n AZgebra, 3( 1975 1, 749-761.

3.

A. M. W. Glass, Countable l a t t i c e - o r d e r e d groups, Math. Proc. Cambridge PhiZosophicaZ Soc. ( t o appear ) .

4.

A . M. W. Glass and Yuri Gurevich, The word problem for l a t t i c e ordered groups, Trans. Amer. Math. Soc. ( t o a p p e a r ) .

5.

A . M. W. Glass and K. R . P i e r c e , E x i s t e n t i a l l y complete l a t t i c e ordered groups, Trans. Amer. Math. Soc. , 36( 1980), 257-273.

6.

R . C . Lyndon and P. E. Schupp, CombinatoriaZ Group Theory, Ergebnisse der Math. und i h r e r Grenzgebiete 89, Springer-Verlag, 1977.

7.

M. 0 . Rabin, Recursive u n s o l v a b i l i t y of group t h e o r e t i c problems, Annals of Math., 67( 1958), 172-194.

- PART IV ORDER AND TOPOLOGY

- PARTIEIVORDRE ET TOPOLOGIE

Egbert HARZHEIM On topological properties of Cartesian products of linearly ordered continua. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . p . 171 Karl Heinrich HOFMANN Order aspects of the essential hull of a topological To-space. . . . . . p . 193

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 ( 1 9 8 4 ) 171-192 0 Elsevier Science Publishers B.V. (North-Holland)

O N TOPOLOGICAL

171

PROPERTIES O F CARTESIAN PRODUCTS

OF L I N E A R L Y

ORDERED CONTINUA

Egbert

HARZHEIM

Mathematisches I n s t i t u t d e r U n i v e r s i t a t Dusseldorf UniversitZtsstraBe 1 4000 D u s s e l d o r d 1 F e d . Rep. of Germany Devoted t o E . COROMINAS on the o c c a s i o n of h i s E m e r i t a t .

The n o t i o n " l i n e a r l y o r d e r e d c o n t i n u u m " (LOC) g e n e r a l i z e s t h e s e t IR o f r e a l n u m b e r s . I n a n LOC one h a s t h e u s u a l o r d e r t o p o l o g y . A C a r t e s i a n p r o d u c t o f a f i n i t e number n of L O C ' s , e q u i p p e d w i t h t h e p r o d u c t t o p o l o g y , g e n e r a l i z e s t h e E u c l i d e a n n - s p a c e IR". Here we s t u d y w h i c h t h e o r e m s o f t h e t o p o l o g y o f IRn c a n b e g e n e r a l i z e d t o p r o d u c t s o f f i n i t e l y many L O C ' s . Those t h e o r e m s , o f c o u r s e , t h e n c a n b e i n t e r p r e t e d a s p u r e l y o r d e r t h e o r e t i c a l . I t t u r n s o u t t h a t s e v e r a l c l a s s i c a l invariance theorems a r e of t h i s k i n d .

RESUME.

-

Un e n s e m b l e C o r d o n n 6 l i n k a i r e m e n t , q u i n ' e s t p a s v i d e e t q u i n e p o s s h d e n i premier n i d e r n i e r Cl&nent, e s t appel6 continu l i n 6 a i r e s i pour t o u t e s e c t i o n i n i t i a l e S d e C s a t i s f a i s a n t 0 # S # C ou b i e n S a un d e r n i e r @ l @ m e n t ,ou b i e n C \ S a un p r e m i e r Clement ( e x c l u s i v e m e n t ) . Le p r o d u i t d ' u n nombre f i n i n d e c o n t i n u s I i n C a i r e s e s t a p p e l 6 n - c o n t i n u . Les c o n t i n u s 1 i n C a i r e s g 6 n C r a l i s e n t l ' e s p a c e R d e s nombres r 6 e l s e t l e s nc o n t i n u s l e s e s p a c e s IRn e u c l i d i e n s d e d i m e n s i o n n . La q u e s t i o n s e p o s e d e s a v o i r q u e l s s o n t l e s thCor&mes d e IRn q u i p e u v e n t 6 t r e g C n C r a l i s 6 s a u x n - c o n t i n u s . S i un t h e o r e m e d e R n admet une t e l l e g & n @ r a l i s a t i o n ,on p o u r r a d i r e q u ' i l e s t e s s e n t i e l l e m e n t un thCorhme s u r l ' o r d r e . L e s p r e m i & r e s r e c h e r c h e s d a n s c e domaine o n t C t C e n t r e p r i s e s p a r F . R i e s z e n 1904. Aprhs c e l a q u e l q u e s t h e o r h m e s d ' i n v a r i a n c e f u r e n t g b n 6 r a l i s C s a u x nc o n t i n u s p a r Wagner ( 1 9 4 9 ) ( i n v a r i a n c e d e l a d i m e n s i o n ) , L b ' t t g e n 1952 ( t h e o r h e d e J o r d a n s u r l e s c o u r b e s p l a n e s ) , Dyer ( 1 9 5 6 ) ( t h & o r & m ed e Brouwer s u r l e s p o i n t s f i x e s ) , Van D a l e n ( 1 9 7 2 ) ( i n v a r i a n c e d e s e n s e m b l e s o u v e r t s e n d i m e n s i o n 2 ) . J ' a i rCcemment p r o w 6 l ' i n v a r i a n c e d e l a p r o p r i C t 6 d ' s t r e u n e c o u p u r e p o u r l e s sous-ensembles compacts d e s n - c o n tin u s. C e la impl i que a u s s i l a gCn@ ral i sat i o n d u t h e o r b e d e l ' i n v a r i a n c e d e s e n s e m b l e s o u v e r t s p o u r l e s n - c o n t i n u s , oh n e s t un e n t i e r n a t u r e l a r b i t r a i r e . Ce thCor6me c o n t i e n t a u s s i l e t h b o r h m e d e l ' i n v a r i a n c e d e l a dimension d e s n-continus. Au p a r a g r a p h e 4 , n o u s p r C s e n t o n s une n o u v e l l e p r e u v e d u t h @ o r & m ed e (Brouwer-) Dyer s u r l e s p o i n t s f i x e s , p r e u v e q u i e s t B l a f o i s p l u s C l @ m e n t a i r e e t q u i g C n @ r a l i s e c e r 6 s u l t a t a u x p r o d u i t s d ' u n nombre a r b i t r a i r e d e f a c t e u r s . Dans c e t t e d i r e c t i o n n o u s @ t e n d o n s a u s s i un t h e o r 6 m e d e Van D a l e n a u x p r o d u i t s d ' u n nombre a r b i t r a i r e d e f a c t e u r s . Noti-e p r e u v e e s t b e a u c o u p p l u s c o u r t e . Au p a r a g r a p h e 5 , n o u s e t u d i o n s l e s s o u s - e n s e m b l e s d e s n - c o n t i n u s q u i s o n t f e r m b s e t c o n n e x e s . P l u s i e u r s t h & o r & m e s s u r l e s c o n t i n u s d e Rn ( i . e . l e s ensemb l e s fermes e t connexes) peuvent S t r e g 6 n 6 r a l i s 6 s aux n-cont i nus, en p a r t i c u l i e r c e u x d e J a n i s z e w s k i , S i e r p i n s k i e t PhragmCn. Nous o b t i e n d r o n s , e n t r e a u t r e s , l e thCor6me s u i v a n t : S o i t C un n - c o n t i n u , e t s o i t , p o u r t o u t nombre n a t u r e l V , un s o u s - e n s e m b l e fermC K d e C q u i n e s 6 p a r e p a s C . S i l e s Ku s o n t d e u x b deux d i s j o i n t s e t l e u r

v

union U e s t compacte, a l o r s U ne sCpare p a s C.

172

E. Harzheim

1. CONTINUA, LINEARLY ORDERED CONTINUA, AND PRODUCTS OF LINEARLY ORDERED CONTINUA

At first we recall some standard concepts resp. introduce some additional notations. A subset C of a topological space X shall be called a continuum (of X), if C is closed and connected in X. If C is open and connected it is called a region of X. In this paper we investigate a special kind of continua, namely linearly ordered continua and products of them. A linearly ordered set (C,$) shall be called a linearly ordered continuum, if C satisfies the following conditions: (i) C # 0, and C hast no first and no last element. (ii) C is dense: a , b E C and a < b implies the existence of an element x E C with a < x < b. (iii) If A is an initial segment of (C,s) with 0 # A # C , then A has a last o r C \ A has a first element. Evidently (iii) is equivalent to: (iii’) For every nonempty bounded subset S S C there exists the infimum inf S and the supremum sup S in C. So linearly ordered continua are generalizations of the real line R with its standard order.

It is easily verified that every linearly ordered set (L,s) can be embedded into a linearly ordered continuum: If a,b are elements of L and if a is an immediate predecessor of b then we put a linearly ordered set of the order-type “1 of the rationals between a and b. So L becomes a part of a dense linearly ordered set and then we apply the method of Dedekind cuts. In this context it turns out nearly immediately that every linearly ordered continuum contains a subset having the same ordertype as the real line IR. In a linearly ordered continuum ((2,s) we have the standard order topology, having as a basis the set of open intervals of C. It is trivial that a linearly ordered continuum with its order topology is a connected space. Further the closed intervals of (C,S) are compact subsets of the topological space C. The latter was already observed by F.Riesz [lS]. Of course, the linearly ordered continua have many features in

173

Topological properties of Cartesian products

common with the ordered set and topological space IR. But 61 has a lot of additional nice properties (countable base, metrizability) which linearly ordered continua in general donlt have. In this context there holds the theorem, that if a linearly ordered continuum is metrizable, it is homeomorphic to the real line IR. So IR is essentially the only metrizable linearly ordered continuum. (See e.g. [ll] ,p.78 and p.116 and for further information [20], 4 1.5 ! ) Now, if (Ci)iE I is a family of linearly ordered continua, each equipped with the corresponding standard order topology, then their Cartesian product ICili @ I] is equipped with the (Tychonoff) product topology. If here we have I = 11, ,n] for a positix C n shortly an ve integer n then we call the above product C1x n-continuum. Such a set will always be taken with its natural product topology. Now 1-continuum is the same as linearly ordered continuum. In the following n always denotes a positive integer and N the set of all positive integers.

x

... ...

It can be seen that an n-continuum C is a closed and connected subset (and thus a continuum) of C. The n-continua are generalizations of the euclidean n-space IR", - and this in an order-theoretical as well as in a topological sense. Similarly to the case n = 1 one can see that every n-continuum has a subset T = T1x xTn, where each Ti, i E {l,,..,n/,is order-isomorphic to LR. Then T is homeomorphic to Rn, and thus IRn turns out to be the simplest n-cont inuum.

...

2. HISTORICAL REMARKS The first appearance of products of linearly ordered sets in connection with topological considerations seems to be in the paper "Uber mehrfache Ordnungstypen 1" by F.Riesz L18) in 1905. Here he considered the question which theorems of the topology of IRn can be generalized to Cartesian products of finitely many linearly ordered sets. Of course he used quite another terminology in the pioneering stadium of the subject. He introduced some basic notions and investigated several fundamental properties of n-continua. In particular he seems to be the inventor of the concept of order-topology. Much later some classical topological theorems for Rn were generalized to n-continua:

1943 by K.Wagner the invariance of dimension [2l] , 1952 by U.Lottgen the Jordan curve theorem [12], [13],

174

E. Harzheirn

1958 by E.Dyer Brouwer's fixed point theorem [4], 1972 by J.van Dalen the invariance of domain for the dimension 2 [ZO]. In his thesis van Dalen deals with order-topological questions and gives a detailed account on this subject. 1964 I found another proof for Dyer's generalization of Brouwer's fixed point theorem which used the latter, but besides this only elementary methods, (I reported on it at the congress of the 0.M.G.in Graz 1964 but did not publish the proof until now.) Later

([7]) I also proved the invariance of the separation property for compact sets: If K is a compact subset of an n-continuum C which separates C an injective and continuous mapping, then also f(K) C and f: K + separates C. This generalizes parts of the Jordan-Brouwer theorem for all dimensions n E IN. Also this yields a generalization of the invariance of the open set (domain) for arbitrary n-continua with n E M (n = 2 is van Dalen's case). By the way, this contains the theorem on the invariance of dimension. A lot of other theorems of the topology of IRn can be generalized to n-continua. In many cases this is very easy, but sometimes it is difficult. We shall give here another and more elementary proof for Dyer's fixed point theorem. Then we generalize the theorems on continua in Rn of Janiszewski [S], Sierpinski [19] and Phragmgn [17] to closed and connected sets in n-continua. Thus a lot of theorems of Rn also hold for n-continua, that means, they can be interpreted as purely order-theoretical theorems. 3. SOME BASIC NOTIONS

The only metrizable n-continua are those which are homeomorx Cn is a metrizable n-continuum phic to Wn. Indeed, if C = C1x then each C,, , V = l,...,n, being homeomorphic to a subspace of C , is metrizable and hence homeomorphic to W as mentioned before. In view of the fact that most of the n-continua are not metrizable we have to replace all concepts defined via a metric by notions which can be defined in general n-continua. E.g. we replace balls by parallelotopes (boxes):

...

DEFINITION 3.1. Let I be a set, Ci a linearly ordered continuum for

Topological properties of Cartesian products

175

x

i E I , X their Cartesian product { Ci 1 i E It. For i E I let Bi = [ai,bi] be a closed interval of Ci; this shall imply ai < bi in the

x

order of Ci. Then the Cartesian product B : = { Bi I i E 11 is called a box o r parallelotope of X, also a closed box to distinguish it { (ai,bi)liE 11, where better from the following: The product ai < bi in Ci is called an open box, if I is finite. In this case indeed this product is an open set of X. For i E I we define the left i-face of the above set B to be Fi := \ ( x ~ > ~B \~x i = ai\ and the right i-face to be

x

Fi

:=

{ (xj)jc I c Blxi

=

bit. The sets Fi and Fi are called opposi-

te faces of B. are points of X with ai < bi If a = (aj)jE I and b = (b.) . J JEI for some i E I and a = b . for all j E I with j # i we define the J J segment S(a,b) = S(b,a) to be the set of all points ( c . ) . I ~ 6 X J which satisfy c . = a . for all j E I with j # i and ci E fai,bi]. J J Then S(a,b) is called a segment of type i. Of course, a and b are also the its endpoints, This concept goes back to F.Riesz [la], following: A segment chain in X is a set A 5 X which can be represented as the union of finitely many segments (of several types) S(al,a2), S(

a2,a3 ) ,

. . . ,S ( ak ,ak+l ) .

A subset T of a box B of X separates in B two opposite faces B which meets Fi F; of B if each closed connected subset K Fi, and F; also meets T.

In the following the concepts of connected set and connectivity component are used in the usual sense of the general topology. Besides this we have here the notion of S-component: If p E M C X we define the S-component of p in M to be the greatest subset of M containing p and all points x of M such that there exists a segment chain in M which links p and x. If X is an n-continuum and U an open subset of X, then for each point p E U the S-component of p in U and the connectivity component of p in U coincide. Here the connectivity components are open sets of X. This is quite analogous to the situation in Wn where in open sets for each point p the connectivity component and the path-connectivity component (and the S-component) are equal.

x

A subset M of X = {CiIiE 11 ( - here I can be infinite - ) is called bounded (in X) if there exists a box of X which contains M.

E. Harz heim

176

According to Tychonoffs product theorem boxes of X are compact subsets of X because, as we mentioned before, in linearly ordered continua closed intervals are compact subsets. S o boxes are also closed subsets of X. Thus we have the following statement:

x { C i l i E I ) of

3 . 2 REMARK. A subset of a product

linearly ordered

continua is compact iff it is bounded and closed. For the case of finite I this was already observed by F.Riesz. Further X is a regular Hausdorff space. For finite I it is also locally compact. In [ZO], Chapter I (Miscellaneous results on ordered spaces) van Dalen gives a detailed account on general topological properties of linearly ordered sets and products of such sets. Also Susl i n t s problem belongs to this part of mathematics. But in our following discussions it doesn't play an essential role. If not otherwise stated we always consider the space {CiliG I} with the usual Tychonoff product topology, which has as basis the set of all products 1 OiliE 11, where for i E I Oi is open in Ci and Oi = Ci for all i E I with exception of at most finitely many i.

x

x

Besides this one could also consider the Tietze-topology ( o r box topology) on { CiliE 11 which has as basis the set of all Oi is an open subset of { O i [ i 61 1 , where for i E I products Ci. In this connection still more topologies on the product set can be introduced, see e.g. [lo], p. 41-43. For finite sets I all these product topologies coincide.

x

x

Many of the following discussions will deal with boxes. In this connection we point out a significant difference between the Tychonoff- and the Tietze-topology and how they correspond to our intuition: PROPOSITION 3.3. Let Ci, i € I , be linearly ordered continua, B = [ai,bi]li€ 11 a box of X := {Cili€ 11. Then the boundary b B of B in X, where X is equipped with the Tychonoff-topology, is B , if I is infinite, whereas >B = U ( F i ~ F i I i E11 if I is finite. But if X is equipped with the Tietze-topology then for arbitrary I (finite or not) the boundary of B is the union of all faces of B, that means Fiu Fi(iE I ] ,

x(

x

u(

Topological properties of Cartesian products

177

4. THE GENERALIZATION OF BROUWER’S FIXED POINT THEOREM FOR PRODUCTS OF LINEARLY ORDERED CONTINUA 4 ] proved a fixed point theorem for chainable In 1956 E.Dyer [ continua which contained among others a generalization of Brouwer‘s fixed point theorem to n-continua. Here we shall give a new proof of this latter result which uses only elementary methods in addition to Brouwer‘s fixed point theorem. The idea of proof is to translate the theorem to an equivalent statement for which the euclidean situation is easier to transfer. By the way, we extend Dyer’s fixed point theorem for n-continua to products of arbitrarily many factors.

The following statement is equivalent to Brouwer‘s fixed point theorem (see e.g. [6J1p.161,Satz V.4.10 and also the proof of theorem 4.4 given below): STATEMENT 4.1. Let C = LO,l]” be the unit cube of Rn. For every n) let T, be a closed subset of C which separates in C V€{l, the opposite faces Fv and F; of C. Then IT,,/3 E { l , n)) # 0.

...,

...,

n

Now we transfer this statement to n-continua:

...

x [an,bn] be a box THEOREM 4 . 2 . Let B = [al,bllx . x Cn. For each V E { 1,. ,nt let Tv nuum X = C1x subset of B which separates in B the left v -face F, v-face F.; of B. Then we have n{T,,lVE i1,...,ntt

..

..

of an n-contibe a closed and the right # 0.

PROOF. We assume the contrary. Then T1nr\{TVIV = 2,...,n) = 0. Since the sets T9 are compact we can find finitely many open boxes of X such that their union V contains T1 but the union of their closures is disjoint to 1 T, I Y = 2,. ,n) Then also there exist 1 B2,.. 1 finitely many closed boxes B1, ,Bil which are subsets of B

n

.

.. .

such that their union V1 contains T1 and satisfies (1) V1n T2n

...

n

Tn

=

0.

...

Now T2 and V 1 n T 3 n T 4 n n Tn are closed, bounded and disjoint because of (1). Then there exist analogously finitely many closed 2 2 boxes B1, B 2 , . , , B I which are subsets of B , such that their union 2 V2 satisfies V 2 2 T 2 and V2n(VlnT3nT4n nTn) = 0.

.

...

So (1) remains valid if T2 is replaced by V2.

So

continuing we ob-

E. Harzheim

178

tain after n steps sets V,,=TV (2)

n{v,lv

=

~,...~nt =

for V E { ll...,nl with

PI,

9 where each Vv ,V = 1,...,n, is a union of finitely many boxes B1,.. ,Byv which are subsets of B. Let & = {B1,. ,Bk) be the set of all boxes B Y v = 1,. ,n,U , = 1,. , iy in an arbitrary ordering. P' For v = 0 we define Bo := B. Of course also Vv 2 TV separates the faces FV,F$ in B. And since 2& is a finite set we can accomplish our proof by a method of translating the situation into the euclidean space Rn. Let B, = 1 1 2 2 n n Lav,by] x Lav,] , b x x [a,, ] , b for v = 0 , k. Then we ascribe to these boxes By certain boxes

..

..

..

..

.. .

. ..

of IR", such that the coordinate relations between the boxes B, , V = O,...,k, are the same as those between the boxes B:,v= O,..,k. More precisely: For each,urz { I , n\ let Dpbe the set containing bgl ay, br,, ,ac, b t Then Dp is a finite suball elements set of C p . For each p~ { 1,. ,n] let D* be a set of real numbers

q,

...

.. ..

...,

.

P

,*!a b[*, , <*, *[b , where for x,y E Dp we have x < y (in CP ) exactly if x* < y* (in R ) . Then we define boxes Bg,..,,BC by ( 3 ) . To B ( = Bo) then there corresponds a box B* = Bg of IR", and to u BY* each v E (1,. ,nj we define V$ : = B:*u

..

. ..

iv

.

Now it can easily be seen that V," separates the V-th left and right faces FG, F;* of B* (as V,, did with they-th left and right faces F v l F; of B). For if we assume the contrary, there exists a continuum in B* and then also a segment chain Z * (in the sense of Def. 3.1, that means with segments parallel to the axes) which avoids V$ but has points in common with the opposite faces FS and F;*. To Z* we then could construct a segment chain Z in B avoiding V., but meeting the opposite faces Fv, Fd. But this contradicts the fact that Vy separates F,, FS. Now, according to Statement 4.1 we have n { V $ v = 1,. ,nt # 0 and in consequence of this ( Vv ( v = 1,. , n t # @ which contradicts (2).

n

..

..

Next we extend the last theorem to Cartesian products of arbitrarily many linearly ordered continua. Here we make use of the Axiom of Choice. So we assume that the index set I is well-ordered. In this case it makes no difference to presume 1 as an initial seg-

Topological properties o f Cartesian products

179

ment of the class of ordinal numbers. THEOREM 4 . 3 . Let I be a proper initial segment of the class of ordinal numbers. For i e I let Ci be a linearly ordered continuum, C := x {Cilic 11 and B [ai,bi] liE I) a box of C. For i g I let Ti be a closed subset of B which separates in B the opposite faces Fi and Fi of B. Then n { T i l i E 1) # $.

=x{

PROOF. If be the family

E

C and oc E I we define the Oc -segment of x to we put and denote it with oC(x). For S-C

a ( S ) := { a ( x ) l x E S } . Then we obtain: (4)

If A is a compact subset of C and a € I then = ( A ) subset in M(C).

is a closed

Indeed, the mapping which ascribes to every x E A its a-segment a(x) is continuous. So it preserves compactness and hence - ( A ) is compact and thus closed. Now our theorem foll.ows by transfinite induction on the cardinality of I: Let I be infinite and let the theorem be true for all cardinals < 111. We may assume that I is an initial ordinal CC, B ( - so I is the set of ordinals < W p -1. First we prove: For all ordinals i , IX with i < oc < I there holds: ( 5 ) =(Ti) separates oc(Fi) and o c ( F i ) in Or(B).

We assume the contrary, i.e.there exist ordinals i,cc with i < cx < I for which ( 5 ) fails. Then there exists a continuum K c =(B) which meets =(Fi) and cc(Fi) but avoids =(Ti). But then K* : = { x c B ( -(x) E K) is again a continuum. For K* is homeomorphic to KX 1 Cj I jc I A oc j), and the property of being connected resp. closed is productive. Trivially K* meets Fi and FI, but it avoids Ti, since each of its points differs from every point of Ti in one of the first acomponents. This yields a contradiction to the fact that Ti separates Fi, Fi in B. Thus ( 5 ) is valid for all i , a with i < a A*- > for 0 < o L < I. Be-

x

\.

z.... -

cause of the compactness of B there follows (6) E : = n { A , , , I O

< = < I)

0.

...

E. Harzheim

180

But now we have (7) E

G fXTiJiEI\.

For if x E E then for every i E I we have ~ ( x ) E -(Ti) for a l l a with i < oc < I , which gives x E Ti for all i E I since the Ti are closed, (6) and (7) now complete the proof. Using 4 . 3 we now derive the general fixed point theorem for boxes with arbitrarily many factors: THEOREM 4 . 4 . Let B be a box in a space are linearly ordered continua, f: B + B fixed point.

x {Cili€ I\ where

the Ci continuous. Then f has a

It can easily be seen that the sets Ti, i E I , are closed. Also we have for i E I: (8) Ti separates Fi and Fi in B.

Indeed, let i E I and K C B be a continuum satisfying K n Fi # $ K n Fi. If K n Ti would be void we would have K = K'u K" where K' : = \ x € Klxi < fi(x)i and Kff : = { x c K(xi > fi(x)l are closed sets and nonempty because of $ # K n Fi E; K' , $ # K n Fi E K". But this yields a contradiction to K being a continuum. So (8) is valid for all i E I. BY 4 . 3 the set n { T i l i E 11 of fixed points of f is nonempty.

#

A well-known theorem of Brouwer states that the fixed-point property of the euclidean unit n-ball Bn = x E IRn I 1x1 f 11 is equivalent to the statement that its boundary Sn-1 is not a retract of Bnl that means: There is no continuous mapping f: Bn Sn-l for which f(x) = x holds for all x E Sn-l. This theorem was generalized by van Dalen ([20] ,p.91-95) to boxes of n-continua. In his proof he applied a method of Dyer [4]. Using 4 . 3 we shall give a new (and shorter) proof and at the same time an extension of this theorem to boxes of arbitrarily many factors.

+

THEOREM 4.5. Let Ci, i g I , be linearly ordered continua. For i E I

Topological properties of Cartesian products

181

let cai,bi] be a closed interval of Ci. We put B : =

x {[ai,bi] IiE I}. The union u { Fiu Fi li€ It. Then there with f(x)

=

x for all x E

of all faces of the box B is U : = is no continuous mapping f: B + U

U.

PROOF. To construct a contradiction we assume that such a mapping f does exist. Then for each i E I we choose an element e i E Ci satisfying ai < e. < bi. We define for i E I Gi {Eji jG I] 1 where Ei : = [ai,ei) and E . * - C . for the indices j # i from I. J '- J If we replace here Ei = [ai,ei) by Ei = (ei,bi] , we obtain the definition for Gi. -1 -1 Further for i € I we put Oi : = f (Ci) and 0 ; := f ( G I ) . Of course, Oi and 0: are disjoint, and we have for i E I:

:=x

(9) Oi is open in B, Oi 2 Fi, O i n Fi

=

$

(9') 0 ; is open in B, O! 2 F i , 0; n Fi

=

$.

and

For i E I now Ti : = B \ (Oi u 0 ; ) is non-empty, because B is connected. Also T. is closed, and Ti separates Fi and FI in B. In1 deed, if K is a continuum -C B meeting Fi and F I we have K n Oi # 0 # K n 0 ; . Now if K n T i would be void we would have K G 0 i. u 0i' and thus K = (KnOi)d ( K n O I ) contradicting to the fact that K is connected. Hence, according to 4.3, D : = n { T i l i € 11 is non-empty. Now let z be an element of D. Then there exists an index j € I with f(z) E FjuF; 5 G . u G ! Thus z s f-l(Cj)u f-l(G!) = 0 u O! = B \ T J J' J J J j which contradicts to z € T . J'

In connection with 4.5 we mention the following: If p is any point of the interior of the unit n-ball Bn Wn, then its boundary Sn-l is a retract of Bn\ { p] Van Dalen ( [20] ,p.91-104) shows that an analogous statement is valid in certain product spaces X x X if X is a closed interval of a linearly ordered continuum satisfying a certain homogeneity condition. But in the general case such a statement is no longer valid. Van Dalen gives a counter-example in

.

[ZO], p.106, Theorem 10. There he proves: Let X be a segment [a,b] of a linearly ordered continuum, such that the corresponding open interval (a,b) is coinitial with -* and cofinal with , where L X , ~are regular initial ordinals with Let R be the boundary of X X X and p € X x X \ R , then R is cY>p not a retract of x x x \ { p}

.

p

.

In connection with 4.4 there arises a natural question, namely

182

E. Harzheim

whether the statement of 4.4 remains valid if the set )( 1 CiliG I) is equipped with the Tietze-topology (see 3 . 2 ! ) . But here the answer is negative in all cases where I is infinite. This can easily be shown using a theorem of Knight GO]: THEOREM 4.6. If I is an infinite set, Ci a linearly ordered continuum for i E I, [ai,bi] a closed interval of Ci, then the set B : = 1 [ai,bi] IiE I} , equipped with the Tietze-topology, is not connected, Then there exists a continuous mapping (with respect to the Tietze-topology) f: B + B which has no fixed points.

x

PROOF. The first part is a special case of Theorem 5 . 1 of [lo] according to which every two points of B which differ in infinitely many components, are in different connectivity components of B. Here one needs also that the spaces, where the components differ, are regular T1-spaces (see 3 . 2 ! ) . So there exists a decomposition B = B'LIB", where B' and B" are non-empty disjoint open sets of B. We choose points x'E B', E B" and put f(x) := X I if x E BIl and f(x) := XI! if x E B'. Then f has no fixed point but is continuous in the Tietze-topology. 5 . CLOSED CONNECTED SETS IN n-CONTINUA

In this section we shall investigate closed connected sets in n-continua. Here we have the opportunity to illustrate another technique of proving theorems concerning n-continua. In this context it turns out to be useful to generalize the notion of manifold to n-continua. In the euclidean case one defines an n-dimensional manifold to be a separable metric space, which is locally euclidean, that means: Each point has a neighborhood which is homeomorphic to Rn. Of course, we have to modify this concept now: DEFINITION 5 . 1 . Let n be a positive integer, C an n-continuum. We call a set Q S C a box union, if there exist finitely many closed boxes B1,...,Bm in C, such that the following condition is satisf ied: (10) For each two indices i,j E { l , . . , , m ) is void o r a box of C.

the intersection B i n B

j

If p e C \ Q we denote by R (Q) the boundary of the connectiP vity component of p in C \ Q. Evidently (as in the euclidean case)

Topological properties of Cartesian products

183

R ( Q ) is a subset of the union of the boundaries of the sets B1,.. P , Bm. Now a subset M C C is called a manifold, if M is connected and if there exists a box union Q and a point p E C \ Q such that M = R (Q). P What we have defined here corresponds roughly to the concept of a closed connected piecewise linear manifold of dimension n-1 which is embeddable into Rn.

..

Since there was only used a finite number of boxes to define a manifold and since this is required to be connected one can easily check the validity of the following theorem: THEOREM 5.2. If M is a manifold in an n-continuum C, then the set C \ M has exactly two connectivity components. One of them is bounded, the other not. The first one is called the inner region of M and the second one the outer region. We denote them by I(M) resp. A(M). We omit the proof of 5.2 which could be constructed analogously to the corresponding statement in the euclidean case or by reducing the above situation to the euclidean case (using the same Iltranslation principlettas in the proof of 4.2).But we give a sketch how the proof for the case C = Rn can be arranged: First the following analogon to the theorem of the Jordan arc is to be proved:

1) If M is a manifold (in the sense of 5.1) of IR", and if W is a subset of M which is isometric to an (n-1)-dimensional cube, then the set M \ W does not separate IR". 2) Let p be a point of IRn\ M. Then all rays starting at p and satisfying a certain condition of general position have an even (resp. odd) number of intersection points with M. In the first case p is called an even point, in the second an odd one. 3) If p~ IRn\ M is even (resp. odd) and if S is a segment containing p and avoiding M then all points of S are even (resp. odd). Thus all points of a connectivity component of Rn\ M are even or all are odd. 4) If p,q are in Rn\ M and one of these points is even and the other one odd then by 3) p and q are in different components of IRn\ M,

E. Harzheim

184

5 ) There exist points p,g E IRn \ M such that one of them is

even and the other one odd. So by 4 ) IRn\ tivity components.

M has at least two connec-

6) Let W be a subset of M which is isometric to an (n-l)-dimensional cube, c the center of W. Then for each p E IRn \ M there exists (because of 1)) a polygonal arc lT which links p with c P such that TT n M = 1 c) , Then TT arrives at c at one of the two siP P des of W. All points p J whose 7 arrives at the same side of W P then can be joined by a polygonal arc avoiding the whole set M. Thus IRn \ M has at most two connectivity components.

Next we need a lemma concerning the separation of disjoint closed sets by manifolds: LEMMA 5.3. If T is a closed bounded subset of an n-continuum C and if T is not connected then there exists a manifold M such that T n M = 0 , TnA(M) # 0 # T n I ( M ) holds. By assumption there exist disjoint nonempty closed sets T1, = T1u T2. For each x E T1 there exists an open box Bx containing x such that its closure Fx is disjoint to T2. There are finitely many open boxes of that kind such that their union covers TIJ say BlJ...,Bm. We can assume that their closures B I J . . .'Bm have pairwise an intersection which is void or a box. We divide the set of boxes B1,...,B, into classes such that two B , B , belong to the same class if they are subsets of the P same connectivity component of B 1 u . . u Bm' Let & be such a class. Then V := {ilB E& 1 is a box union. We choose a point p E T2. Then one can verify that M : = R p ( V ) (see 5.1!) is a manifold for which p and T1n V are situated in different connectivity components of C \ M. PROOF.

T2 with T

u

.

Now we generalize Janiszewski's theorem on continua ([8]) and We cannot with this the theorem of Sierpinski on continua ([IS]). take over Janiszewski's proof in which a continuum is constructed

185

Topological properties of Cartesian products

as limes superior of a sequence of point sets which are and wheref tends to 0. First we need a simple lemma:

-chains

LEMMA 5 . 4 . Let B be a box of an n-continuum, a a limit ordinal, and ( P A < A a transfinite sequence of points of B. Then there exists a point h E B such that for each ordinal o C < and each open box Q containing h there exists an index e with OC < e < such that p E

e

Q. Shortly spoken: In each neighborhood of h are cofinally many

points of (P,,),,<~. The proof follows immediately from B being compact. The next theorem treats a transfinite reduction process for certain sets. The generalization of Janiszewski’s theorem will be an easy consequence of it.

THEOREM 5.5. Let Q be a manifold in an n-continuum C and . A a closed subset of Q u I(Q). Let p be a point of I(Q) with the following property (11) Each manifold M C I(Q) with p E I(M) intersects Ao. Then there exists a continuum K C .A one point of Q.

which contains p and at least

PROOF, If .A is already a continuum there remains nothing to be proved because p E A 0 and Aon Q # g is valid as easily follows from the assumption. I n the other case according to 5 . 3 there exists a manifold M1 with Aon M1 = 0, Aon I(M1) # 0 # Aon A(MI). Because of p E .A we have p E I(M1) or p E A(M1). We assume the first case, the second is analogue. We put (12)

A1 : = Aon I(M1).

A1 is closed because of M 1 n A o

=

0. Further we have:

(13) Every manifold M C I(Q) with p E I(M) meets A1. Assume the contrary. Then there exists a manifold M s I(Q) with p E I(M) and M n A 1 = 0 . Let G be the set of all points q of I(Q) such that there exists a continuum which contains p and q and I(M1). avoids M1u M. Because of p E I(M1) then there holds G Without loss of generality we can assume that if B* is a box used for the construction of M1 and B** one for M then B*oB** is void or contains an open box. Then it follows that G is the inner

E. Harzheim

186

region I(M*) of a manifold M* with M* 5 MIu M. Now M1n A1 ( E M1n .A = ) @ , also M n A 1 = @ , and so we have M " n A 1 = 0. From (12) we conclude that M1u l(M1) is disjoint to Ao\ A1, By construction we have M A C M I U I(M1), and so also M*n,(Ao\ A1) = 0. Together we obtain that M" is disjoint to Ao. Further there holds (14) p

E

I(M*) and M*

I(Q).

-

Indeed, we have G C I(M) G I(Q) which gives M* G C_ M u I(M) G I(Q), Now (14), together with M*n A, = 0 gives a contradiction to (11). So (13) is proved.

A1 is a proper subset of A. because of AIC- I(M1) and AonA(M1) # 0. Now we iterate the process which lead from .A to A1. We use transfinite induction: If T is an ordinal > 0 and if for all ordinals (F
S<-C :

and A6is

and

closed,

(17) Every manifold M C I(&)

with p E I(M) meets As,

then we proceed as follows: If t has an immediate predecessor 5 - 1 and if AT-l is not connected we construct a subset A,G quite analogously to A I S Ao. Then (15) holds for c2= T and all Cl
k-l

n{

If T is a limit ordinal we put A,=:= A616 6 and p V E U. Then h is adherent to every As, S < t , so we have h E A& for all 6< f and thus h€AT.

Topological properties of Cartesian products

187

As an accumulation point of (p ) further h E M is valid, and v V
THEOREM 5.7. A bounded continuum K1 of an n-continuum is not representable as the union of countably many, but at least two, nonempty closed disjoint sets. PROOF, We can apply Sierpinski's method of proof after we have obtained 5.6. We outline the proof: Assume that K1 = u { A , , l V E I N } , where the A,, are pairwise disjoint nonempty closed sets. Then there exists a manifold Q satisfying the following three conditions: Q n (A1u A2) = 0 , I(Q)n A2 # 0, I(Q)n A1 = 0 . According to 5.6 there exists a continuum K 2 c K1 which contains a point a2 E A2 and a point q E Q which satifies K2 C I(Q)u Q. Then K2 intersects at least two of the sets A,, , V E I N . Otherwise we would have K2 S A2 and thus A2n Q # @ which gives a contradiction. B u t K2 is disjoint to A1. Now one can iterate this process and split away successively the sets A1, A Z , We obtain a decreasing sequence K I Z K2 2 of nonempty continua where K.d+l c\ A V = 0 holds for every V € IN. Then 1 K-,,l v E lN] is nonempty but also disjoint to every A V ,V E lN,

... .

n

..

E. Hurzheim

188

and thus also to K1 which gives a contradiction. Also the separation theorem of Phragmdn [17] can be generalized to n-continua. Proofs of this theorem which work with E: -chains of points cannot be taken over. But we can derive our generalization by means of the following lemma (see also 1153, Theorem 1 and [2] ,Theorem 3 ! 1: LEMMA 5.8. Let G1, G2 be two disjoint regions (connected open subsets) of an n-continuum, Bi the boundary b G i of Gi f o r i = 1 , 2 . Let B2 be bounded and B2 C B1. Then B2 is a continuum. PROOF. If B2 is not a continuum then there exists a manifold M with M n B2 = $ and (18) I ( M ) n B2 #

0 # A(M)n B2.

Since I(M) and A(M) are open sets and because of B2 = b G 2 we obtain from (18) that there exist points x i € I(M)nG2 and xa E A(M)nG2. Further there exists a polygonal arc T2 s G2 containing x i and xa. Of course, T2 meets M, and since we have M n B 2 = @ we conclude (19) M C G2.

Now B2 C_ B1 implies that (18) also remains valid if we replace the index 2 by 1. Then analogously to the above reasoning there exists a polygonal arc T l C G1 which meets M. This yields G1n M # 0. Together with (19) this contradicts G1n G2 = 0. /

Now the generalization of Phragmen’s separation theorem reads as follows: THEOREM 5 . 9 . If in an n-continuum C two points p1,p2 are separated by a closed bounded set A which does not contain p1,p2, then there also exists a continuum K = A which separates p1 and p2. PROOF, Let Gl be the union of all continua in C which contain p1 and are disjoint to A. Then B1 := b G 1 fulfils B I G A and separates p1 and p2. Further let G2 be the union of all continua which contain p2 and are disjoint to B1. Then we have B2 := b G 2 C _ B1, and B2 separates p1 and p 2 . Now, according to 5 . 8 , K := B2 satisfies the statement. A nice consequence of the last theorems is now (see also [14],

Topological properties of Cartesian products

189

THEOREM 5.10. Let I be a nonempty initial segment of IN and J v , v e I , pairwise disjoint nonempty closed subsets of an n-continuum C , such that no J,, V E I, separates C and that the union V : = Jvl I, E I) is bounded and closed. Then also V does not separate C.

u{

PROOF. We assume that there are two points p,q E C which are separated by V. Then by 5.9 V contains a continuum K which also separates p and q. Then we have K = U l K n J , , I v E It. At least two of the summands K n J, have to be nonempty, since each single J, does not separate p and q. But now we obtain a contradiction to 5.7. 6. ON THE GENERALIZATION OF SOME CLASSICAL INVARIANCE THEOREMS TO n-CONTINUA Recently ( [ 7 ] ) I proved the invariance of the separation property for compact subsets of n-continua: THEOREM 6.1. If K is a compact subset of an n-continuum C which separates C, and if f: K + C is continuous and injective then f(K) also separates C. Moreover, this statement can be generalized to continuous mappings f which satisfy only a weakened injectivity:

..

THEOREM 6.2. Let K be a compact subset of an n-continuum C = C 1 x ..xCn which separates C, and let (a,,b,) be an open interval of C v for V = l,...,n, such that the open box B = (al,bl)x . . x (an,bn) is a subset of a bounded connectivity component of C \ K. Let f: K j C be continuous. For v = l,...,n let A, := \(x,, xn) E Klx, 2 a, ) and B, : = {(x, xn) E KIx9 2 by). We suppose that for every v = l,...,n the sets f(Av) and f(By) are disjoint. Then f(K) separates C.

,...,

...,

The proof of 6.2 (and 6.1) makes among others use of Borsuk's separation theorem which characterizes the compact subsets of Rn which separate Wn. (See [ I ] , [ 3 ] ,p.357 or [6] ,p.203! ) Theorem 6.2 also generalizes the following theorem of [ 5 ] : n If W is an n-dimensional cube in IR , b W its boundary, and if f: b W 4 Rn is a continuous mapping such that the f-images of opposite faces of W are disjoint, then f ( b W) separates Rn.

E. Harzheim

190

An important notion which is used in the proof of 6.2 is the following: DEFINITION 6.3. Let = {BiliE 11 be a set of boxes and Sgr = (BZlie I a family of boxes of an n-continuum C. We say that the mapping which ascribes to every B i 6 9 the box BZ is a link homomorphism if the following condition is satisfied:

# $ then also B Z n B 4 # $. J We illustrate this notion by the following elementary state-

If i,j € I and Bin B

j

ment:

THEOREM 6.4. Let c be the cube [o,3]" of mn and let S = 1 ~ ~ = l i 1,. ,3"\ be the set of all 3" subcubes of C for which all vertices have integer components and f o r which the edges have length 1. Let = ( ~ ; ) ~ ~ { ~ , . . . , ~be n j any family of closed boxes of

..

a*

..

Rn such that the mapping Bi H B4 for a1 i E 11,. ,3"} i s a link { BZ I i = 1 homomorphism. Then V := 3n\ does not separate Rn.

u

.

PROOF. Let the contrary be assumed. Then there exists a point p = (pl,...'pn) in a bounded component of IRn\ V. We consider the line L, which contains p and is parallel to the xv-axis. L, must intersect V on both sides of p .

So there exists a maximal segment S v on L,, containing p such that S, n V is a two-element set \p-,p+\.There exist boxes BZ, B* with J p - G BZ, p + E B* From the construction it follows that BZn B* = $. j' J But then neither Bi nor B is the central box [1,2In =: B' of j Now Bin B' # 0 # B . n B ' and thus also BTnB'* # 0 # BtnB'*. This J J , . implies that the projection of B'* to the x,-axis must contain p Since this is valid for v = l,...,n, it follows that B'* contains p. This contradicts to p $ V.

.

Of course, Theorem 6.4 could also be stated in a more general form for boxes of n-continua instead of Rn. From Theorem 6.1 one can easily derive the thorem of the invariance of the inner point (resp. the invariance of the open set).

191

Topological properties of Cartesian products

Here the proof for the euclidean case (see e.g. p.153) can be taken over.

[3],p.358

or [ 6 ] ,

THEOREM 6.5. Let p be an inner point of a set M C C where C is an n-continuum, f: M j C continuous and injective. Then there exists an open box of C which contains f(p) and is a subset of f(M). PROOF. Let B C M be a closed box having p as an inner point. Then C \ f(dB) = (C \f(B))u(f(B) \ f(bB)). Here the summands on the right side are connected, C \ f(B) is so because of 6.1, and f(B) \f(bB) is so since continuous mappings preserve connectedness; are connected. According indeed, B \ b B and f(B \ b B ) = f(B)\f(bB) to 6.1 C \ f(bB) must have at least two connectivity components, and so f(B \ bB) = f(B) \ f(bB) is one of them and hence open. REMARK 6.6. From 6.5 one can derive the invariance of dimension for n-continua in the same manner as in the euclidean case (see e. g. [ 3 ] ,p.359 or ,p.153!):

161

If X is an n-continuum and Y an m-continuum with n # m , then X and Y are not homeomorphic.

REFERENCES: [ l ]

Borsuk, K., Uber Schnitte der n-dimensionalen euklidischen Raume, Math. Ann. 106 (1932) 239-248. [Z] Brouwer, L.E.J., Beweis des Jordanschen Kurvensatzes, Math. Ann. 69 (1910) 169-175. [3J Dugundji, J., Topology (Allyn and Bacon, Boston, 1967). [4] Dyer, E., A fixed point theorem, Proc. Amer. Math. SOC. 7 (1956) 662-672. [s] Harzheim, E., Eine Verallgemeinerung zum Jordan-Brouwerschen Satz, Math. Ann. 203 (1973) 175-182. [ 6 ] Harzheim, E., Einfuhrung in die kombinatorische Topologie (Wissenschaftliche Buchgesellschaft Darmstadt, 1978). [7J Harzheim, E., Die Invarianz des Zerlegens fur kompakte Teilmengen von Produkten lineargeordneter Kontinuen, Arch. Math. (19831, to appear. [8] Janiszewski, Z. , Th$se, Paris, 1911. [ g ] Kelley, J. L., General Topology (Van Nostrand, Princeton etc, 1960).

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192

110J Knight, C. J., Box topologies, Quart. J. Math. Oxford Ser ( 2 ) 15 (1964) 41-54. [I11 Kowalsky, H - J * , Topologische Raume (Birkhauser Verlag, Basel 1961). [12] Lottgen, U., Uber eine Verallgemeinerung des Jordanschen Kur-

vensatzes auf zweifach geordnete Mengen, Dissertation, Koln, 1952. [13] Lhttgen, U. und Wagner, K., Uber eine Verallgemeinerung des

Jordanschen Kurvensatzes auf zweifach geordnete Mengen, Math. Ann. 139 (1959) 115-126. [14] Mazurkiewicz, S., Remarque sur un the/or>me de M. Mullikin, Fund, Math. 6 (1924) 37-38. [151 Moore, R. L., Concerning the sum of a countable number of rnutually exclusive continua in the plane, Fund. Math. 6 (1924) 189-202. [16] Mullikin, Anna M.,

Certain theorems relating to plane connected point sets, Trans. Amer. Math. SOC. XXIV (1922) 148-154. 1171 Phragmgn, E., Uber die Begrenzungen von Continua, Acta Math. 7 (1885/86) 43-48. [IS]

Riesz, F., ifber mehrfache Ordnungstypen I, Math. Ann. 61 (1905)

406-421. [19] Sierpinski, W., Un the/or\eme sur les continus, Toh. Math. Journal 13 (1918) 300-303. [20] van Dalen, J., Finite products of locally compact ordered spaces, Doctoral dissertation, Amsterdam, 1972, iii + 115 pages. [21] Wagner, K. , Verallgemeinerung des Brouwerschen Invarianzsatzes

der Dimensionszahl mittels eines allgemeinen Stetigkeitsbegriffs von Abbildungen mehrfach geordneter Mengen, Math. Ann. 120 (1949) 514-532.

Annals of Discrete Mathematics 23 (1984) 193-206 0 Elsevier Science Publishers B.V. (North-Holland)

I93

ORDER ASPECTS OF' THE ESSENTIAL HULL OF A TOPOLOGICAL To-SPACE Karl Heinrich Hofmann Tulane University New Orleans, La.

and

Tectinische Hochschule Darmstadt

Rgsurni. Un plongernent i : X ->E d' espaces topologiques est appelg essentiel ssi pour n'importe quelle fonction continue f:E ->Y telle que fi soit un plongement, la fonction f est elle-mcme un plongement. On dit qu'un plongement essentiel est une c16ture essentielle s'il est maximal parmi tous Ies plongenients essentiels. En 1972, B. Banaschewski a construit une clbture essentielle pour des espaces topologiques arbitraires. Evidemment, l'idie de cl6ture essentielle s'applique 2 n'importe quelle catggorie et on constate que dans plusieurs catigories interessantes, les c1Etures essentielles sont automatiquement injectives. 15anaschewski remarquait que cela n'est pas vrai dans les catgyories d'espdces topologiques et de fonctions continues. Cette observation conduisait donc immhdiate_ment2 la question d'une caractgrisation des espaces dont la cloture essentielle est injective. A l'hpoque Dana S . Scott venait de caracthriser les espaces topologiques injectifs: en fait, un espace X est injectif s i et seulement si on a s u r X une structure de treillis

continu (terminologie de Scott) telle que la topologie donnge dhrive de la structure du treillis c o m e topologie de spicialisation. L'article de Banaschewski (qui paralt 5 ans aprgs sa rhdaction) est donc parmi les sources originelles de la thgorie des treillis continus sur lesquels on a maintenant plusieurs textes et monographies (par exemple IC-801). Le concept mEme de clcture essentielle amenait une suite de travaux recherchant les liaisons entre s e s aspects topologiques et ses aspects d'ordre et notamment ceux de R.-E. Hoffmann. Le nloment semble maintenant venu de risumer et de renouveler la thgorie de la clzture essentielle et l'injectivith pour les raisons suivantes: premisrement, on a maintenant une thhorie des trcillis continus trss bien developphe et des rhfirences bien accessibles. Cela permet de replacer trhs cormnodement la thiorie dans le cadre des treillis continus. Deuxismement, les idies de Banaschewski sont importantes et valables, mais son article est parfois difficile, et une consiquence de la proposition centrale (proposition 3 ) est inexacte : c'est sur cette consgquence que les travaux de Hoffmann sont fondhs dans une large mesure. Nous essayons donc de redevelopper la thhorie dans le cadre

des treillis continus et d 'approfondir notre comprghension des ph&nomGnes parfois difficiles qui se pr6sentent dans ce contexte.

K.H. Hofmann

194

Introduction. The numerous relations between the theories of partially ordered set.s and topological spaces are illustrated by the following examples. (a. Topology yields partial order) i) If X is a To-space, then the relation c given by x s y if and only if U(x)i U(y) (where U(x) denotes the filter of open neighborhoods of x introduces a partial order on X, called the s p a c i n Z i S Q ~ ~ Gorder. R It is also characterized by x < y iff x E{Y}-.

ii) For a T -space we set x < y if and only if the principal filter tx = {z L X: $ t { z l - ] is a neighborhood of y. This relation is transitive, but generally non-reflexive, and x.cy implies x < y . The relation < is called the Scott order. (b. Partial order yields topology) If P is a partially ordered set, then the collection of all upper sets U for which the relation s u p D c U with an arbitrary directed set D implies D n U # @, is a topology o(P), called the S ~ > o l t topozogy

.

Our terminology will adhere to that of the Compendium of Continuous Lattices rC-801. The concepts we have just touched are discussed in this source in considerable detail, In 1972, H.Banaschewski wrote an important article in which he constructs a unique essential hull for any T -space X. Indeed an embedding i:X * E of To-spaces is called e s s e r i t i u l iff for :ny continuous function f:E+Y for which fi is an embedding is itself an embedding. An e s s e n t i a l huZZ is maximal among all such embeddings. Banaschewski's construction is as follows: Let O ( X ) denote the topology of X (i.e. the lattice of all open subsets of X). We write Filt L for the algebraic lattice of all filters of a given inf-semilattice with identity. Then the function x-.U(x) : X -+ Filt O(X) is a topological embedding if Filt L is equipped with the Scott topology. The subspace E c F i l t O ( X ) containing all filters which are suprema in Filt O ( X ) of some set of neighborhood filters U(x) is precisely the essential hull of X rB-771. In many categories of interest, essential hulls are injective and thus provide injective hulls. A s Banaschewski n o t e s , the essential hull of a To-space is not always injective, and thus the question p o s e s itself for which spaces X the essential hull is injective in the category of To-spaces relative to the class of embeddings. At the time of Banaschewski's research of this problem, D.S. Scott had just given an order theoretic characterisation of injectivity for To-spaces rS-721. In fact Scott proved that a To-space X is injective iff the poset (X,.) obtained by equipping X with the specialisation order is what he then called for the first time a continuous lattice, and X carries the topology associated with (X,.) which has thereafter become known as the Scott topology. (The Compendium CC-801 provides much more backround.) Banaschewski gave a sufficient condition for a To-space X t o have an injective space as an essential hull, and the methods he introduces provide the platform for a precise characterisation. ( A recent example due to M. Mislove and the author shows that the original claim that Banaschewski's condition was at the same time necessary and sufficient was not correct.) The present note serves the purpose of describing severe1 equivalent necessary and sufficient conditions for a To-space to have an injective hull. One of these was recently established by R.E.Hoffmann in a different fashion. In fact we first establish a theorem on continuous lattices which does not depend on distributivity (which is involved in the case of topologies) nor on the property of being algebraic (which always applies to the case of filter lattices). Then we apply the results on continuous lattices to the problem of characterizing To-spaces whose essential hull is injective.

195

Essential hull of a topological To -space

Banaschewski's paper o n essential hulls is, next to Scott's seminal article on continuous lattices one o f the earliest sources on continuous lattices. In recent years it caused a whole line of research represented by several articles of R.-E. lioftmann. A review of and an alternative approach to the Banaschewski-Hoffmann theory may be timely for the following reasons: i ) We have today a well developed theory o f continuous lattices and accessible sources CC-80, C-811. ii) The fundamental approach in Banaschewski's paper is important and viable, but the paper is not easily readable in all places and one implication in rB-771, Proposition 3, on which some results in [Bl and numerous statements the papers of R.-E.Hoffmann are based, is incorrect. iii) It might be useful to take stock of the situation and to form a better intuition o f the sometimes delicate phenomena which arise. As a basis for such an intuition we will accept the reduction to the level of continuous lattices. 1.

Sup-closed in continuous lattices. ~ _ subsets _ _ _ -

We fix a continuous lattice L. From the theory of Galois connections ([C-801, p.18 ff.) we know that there is a canonical bijection between the set of supclosed subsets A o f L and the set o f kernel operators k:L->L. Indeed each kernel operator k determines a sup-closed subset A = k(L), and from each supclosed subset A we obtain a kernel operator k via k(x) = sup(ixnA) = max(+xnA). The two assignments invert each other. Our objective is to find conditions for A to be a continuous lattice in its own right. From [C-801, p.63 , Theorem 2.14 we know that A is a continuous lattice whenever the associated kernel operator k is continuous relative to the Scott topology on L. The converse may fail. (Simple example: k:[0,11-->[O,l~ with k(x) = x for x = 1 or x < 1 / 2 , and with k(x) = 1 / 2 otherwise.) However, there are cases when the continuity of k is sufficient and necessary for the continuity of A. Here arid ir, the foLloiJing, c o n t i n u i t y of a map belweeri p o s e t s always r e f e r s t o t h e S c o t t topologies. We shall use the notation of [C-801. E.g., according to rC-801, p.2, Definition 1.3 and p. 35 we set i x = { u f L: u a x} in a lattice L. The first part of the following Lemma is a continuous lattice formulation of ideas contained in Proposition 3 of Banaschewski's IB-771. 1.1 LEMMA. Let k be a kernel operator on a continuous lattice L. Then the following statements are equivalent with A = k(L):

(I) (2)

(3)

k is continuous. k(x) = sup k(+x) for all a = sup k($a) for all

xcL. A.

a c

Moreover, if G is a n arbitrary subset of L and if A is the set {xt L: x = sup(+xnG)] which is sup-generated by G, then the conditions ( I ) , and (3) are also equivalent to each of the following conditions: (4)

(5)

a g

= =

for all for all

sup(+anG) sup(+gnG)

(2)

aFA. g r G.

Proof. For the equivalence of ( I ) and ( Z ) , see CC-801, p. 112, Proposition 2 . 1 : ( I ) => (5). (A remark is in order on this reference: I n the Proposition we quote, the equivalence of the conditions (l),(2),(3), and (5) holds whenever S in continuous and T is merely complete. The implication ( I ) => (5) ostensibly uses only the continuity of the domain S, and a simple modification of the proof o f ( 5 ) => ( 4 ) dispenses with the hypothesis that the range T be also continuous.) (2) (3) 5

--

(3)

(2):

is trivial since k(a) = a for a c A . Let x t L and set a = k(x) FA. Then

sup k(+x) (since a

=

k(x)

5

x)

5

k(x)

= a.

Thus

a =

k(x)

=

sup k(+a) (by (3)) sup k(tx).

K.H. Hofmann

196

( I ) 3 ( 4 ) : I f k i s c o n t i n u o u s , t h e n A i s a c o n t i n u o u s l a t t i c e . L e t a i A a nd s u p p o s e t h a t x < < a . By t h e i n t e r p o l a t i o n p r o p e r t y (IC-801, p . 4 6 ) we f i n d an x ' I, w i t h x << x ' << a . S i n c e a = s u p ) a as A i s a c o n t i n u o u s l a t t i c e , and siiicc' I

A

by [C-SO],

+*a = tLa n A

p.

181, C o r o l l a r y I . 7 , w e f i n d a

b ,: A

x'

with

~

b

((

a.

Now b = s u p ( + b n G ) by t h e d e f i n i t i o n of A , S i n c e x " b , t h e r e i s a f i n i t e s e t F c + b rI C s u c h t h a t x <<sup F . We h a v e s u p F ,-cb <
(4) (5)

3

(5)

3

(4):

,- G : h

= sup {h

(4)

is t r i v i a l . a = sup(tanG)
= sup{&

f o r some

g

6

G}

G: g s a ) =

up^^^,^^^

sup(j.a n C) i a , hence

sup{h

e

G:

I1

c/g]

a = s u p ( + a il C ) .

( 3 ) : F o r a r A we h a v e a = s u p ( + a II C) ( b y ( 4 ) ) i s u p k ( t a ) s i n c e + a 11 G 5 k ( $ a ) as g <
I t i s u s e f u l t o r e c a l l from [ C - 8 0 1 , ~ . 6 0 , D e f i n i t i o n 2 . 6 t h a t a s u b s e t A o f a c o n t i n u o u s l a t t i c e L i s c a l l e d a suCaIgii.bra i f and o n l y i f i t i s c l o s e d u n d e r a r b i t r a r y i n f s and d i r e c t e d s u p s . Any s u b s e t X o f L ga~erutcsn subuLgdbrtz A, n am ely , t h e smallest s u b s e t of L c o n t a i n i n g X and b e i n g c l o s e d u n d e r a r b i t r a r y i n f s and d i r e c t e d s u p s .

1.2 LEMMA. L e t k:L->L be a k e r n e l o p e r a t o r o n a c o n t i n u o u s l a t t i c e a nd s e t A = k ( L ) . I f we assume t h a t L i s t h e s u b a l g e b r a g e n e r a t e d by A , t h e n t h e f o l l o w ing a r e equivalent: ~

(I)

k

(2)

A

i s continuous. i s a c o n t i n u o u s l a t t i c e and i s t o p o l o g i c a l l y embedded i n

L.

P r o o f . A s w e n o t e d b e f o r e ( I ) w i l l a l w a y s i mpl y ( 2 ) . (In f a c t more i s t r u e : S e e TC-801, p . 6 3 , Theorem 2 . 1 4 ! ) . We h a v e t o show t h a t ( 2 ) i m p l i e s ( I ) . I f A i s a c o n t i n u o u s l a t t i c e and t h e S c o t t topology of L i n d u c e s t h e S c o t t topology o f A , then A i s i n f a c t a continuous r e t r a c t , since A i s i n j e c t i v e i n the category of To -s p a c e s a n d w i t h i t s S c o t t t o p o l o g y i s embedded i n t o L w i t h i t s S c o t t t o p o l o g y . ( C f . IC-801, p p . 121 f f . ) Thus t h e r e i s a c o n t i n u o u s p r o j e c t i o n o p e r a t o r p:L -->L w i t h p ( L ) = A (Cf.[C-801, p . 2 1 , D e f i n i t i o n 3 . 8 . i . ) F o l l o w i n g IC-801, p . 2 2 , Lemma 3.11, a n d p . 6 3 , Theorem 2 . 1 4 w e d e f i n e Lk = [ x + L : p ( x ) <:x} Then Lk i s c l o s e d i n L u n d e r t h e f o r m a t i o n o f a r b i t r a t y i n f s a nd d i r e c t e d s u p s ( s e e [C-801, p . 6 3 , 1 o c . c i t . ) F o r a r A w e h a v e p ( a ) = a , whence a < Lk. S i n c e L i s t h e smallest s u b s e t c o n t a i n i n g A and b e i n g c l o s e d u n d e r a r b i t r a r y i n f s and d i r e c t e d s u p s we h a v e Lk = L . Thus p i s a k e r n e l o p e r a t o r w i t h p ( L ) = A = k ( A ) . Then k ( x ) i x i m p l i e s k ( x ) = p k ( x ) S p ( x ) , a n d l i k e w i s e p ( x ) S x i m p l i e s p ( x ) c k ( x ) . Thus p = k and t h u s k i s c o n t i n u o u s . U

.

T h i s Lemma, t o o , i s a n a b s t r a c t i o n a n d g e n e r a l i s a t i o n o f P r o p o s i t i o n 3 o f r B - 7 7 1 . We now come t o t h e m a i n r e s u l t o f t h i s s e c t i o n . 1 .3 THEOREM. L e t G be a subset of a coritiriuous latt,Lcp L and L P t A denoted t h e subset { x F L : x = s u p ( + x (I G ) } iJhich is sup-gewruted in L by C. I'oiisider t i i e foLZouiTig c o n d i t i o n s :

i s a continuous L a t t i c e i r i iis own right and i s t o p o l o g i c a l l y embedded into L r e l a t i v e t o the respective Scott topol o g i e s.

(I)

A

(2)

g

=

supL(fgnG)

(2) 3 ( I ) , ure equivulent. ! / ' h i

for) a l l

arid if L

g c C.

is t h e subaLgebra generated by

A, theri b o t h c o ~ i d i t i o n : :

P r o o f . By 1 . 1 , t h e c o n t i n u i t y of t h e k e r n e l o p e r a t o r k a s s o c i a t e d w i t h A i s e q u i v a l e n t t o ( 2 ) . B u t t h e c o n t i n u i t y of k i m p l i e s ( I ) ( s e e CC-801, p . 181, C o r o l l a r y 1 . 7 . ) Thus ( 2 ) i m p l i e s ( I ) . I f , however, L i s t h e smallest s u b a l z e b r a

197

Essential hull of a topological To -space generated by A, then by 1 . 2 , the continuity of k theorem is proved. 0 1.4 COROLLARY. I.f, i n Theorem 1.3, t h e l a t t i c e condition ( 2 ) o f 1.3 -is i n f a c t equivalent t o

(3)

g = sup [hc

H: there is a

c E K(L)

is equivalent to ( I ) . L

such that

Thus the

i s ever? algebraic, then h s c
g c G.

Remark. For any lattice the sup-subsemilattice of all compact elements will be denoted with K(L). (Cf. CC-801, p. 85) Proof. This is an immediate consequence of TC-801, p.86, Proposition 4 . 5 . 0 The gist of Theorem 1.3 and its Corollary is that the embedding of the generating set G into L alone determines whether or not A is a topologically embedded continuous lattice. If we define, for two elements g and h of G the relation g < h by g-xLh , then * g n G = + g and condition 1.3(2) then reads (2') g

=

supL Jg f o r a l l

g r G.

This condition implies (2") g

=

supG Sg

for a l l

g c G.

but the converse fails i n general, This is not so easy to see, but we shall have an example in Section 4 below. Algebraic lattices

2.

We recall first that in any lattice L we denote with Irr L the subset of all completely (meet-)irreducible elements (cf. [C-801,~. 9 2 , Definition 4 . 1 9 . ) 2.1 DEFINITION. Let L be an algebraic lattice. We say that a subset is K-generating if and only if the following conditions are satisfied:

G

of

L

Gc Irr L.

(i) (ii)

c

=

inf (+c nG)

for all

C E

K(L).

Remark. Notice that condition (2) is equivalent to (ii')

If c,c' E K(L)

and

cgc', then there is a

g c G with

cgg

and

c'sg.

2.2 LEMMA. A K-generating subset G of an algebraic lattice generates L as a subalgebra. Proof. Let L' be the subalgebra generated by G . By condition 2.I(ii) it follows that K(L)cL'. Since L is algebraic, for each x t L we have x = s u p ( + x n K ( L ) ) . But JxnK(L) is directed (see [C-801,p. 8 5 , 4 . 3 and 4 . 4 ) . Thus x6L'. Hence 0 x c L ' and thus L' = L. I

We may have K-generating subsets in an algebraic lattice which are not inf-generating. From the fact, that Irr L is the unique smallest inf-generating subset in an algebraic lattice (cf. tC-801,p. 9 3 , Theorem 4 . 2 3 ) we draw immediately the following remark: For a subset G of Irr L in an algebraic lattice we have x = inf(fxnG) for all X C L if and only if G = Irr L. In particular, for a K-generating subset G of an algebraic lattice we have G f Irr L if and only iff there is an X E L with x < inf(+xnG). REMARK.

For the formulation of the main result in our context the notation which follows will be convenient: NOTATION. (i) For a subset C

of a complete lattice L

h ( G ) = {xEL: x = sup(+xnG)}.

of subsets of

G.)

(In other words, X ( G )

we write is the set of all sups

K.H. Hofmann

198

(ii)

For

g,ht G we write

x
iff x <
We observe that h ( G ) is a complete lattice and is sup-closed in L. The relation on G depends on the embedding o f the poset G into the lattice L. The set +-g is the same thing as + g n G for gr G.

<

2 . 3 THEOREM. Let L be an aZgcbraia l a t t i c e avid Ihe?? t h e foZZowing statements are e q u i v a l e i i t :

G

a K -generati'vg s u b s e t of

(1)

X(G) is a continuous l a t t i c e which i s topologicailg mhedded i n t o L t o the respective Scott topologies.

(2)

For each

g t G we have

(3)

E'or each

gc G we have

(4)

(VgtG, rclrr L) g g r

g

-

g

=

sup iq

E

G : q'c5

g

for some

.

rclativr

c f K(L)}.

= sup&.

( 3 q f G , ccK(L))

(5) E'or euch gc G and each c E K(L) w i t h p,, .,pn c G arid a d t K(L) such t h a t

..

q$r

and

qScSg.

c< G c 5

t h e r e are f i n i t e l y many p I v.. .V pn 5 d i- g.

Proof. By 1 . 3 and 1.4 we know that ( I ) , (2) and ( 3 ) are equivalent. ( 2 ) (4): Let g c G and r e Irr L with g $ r . Since g i s the sup of all q c G with q 5 c < g for some c , there must be one of these q with q g r . Conversely, ( 4 ) =r (2): Let g c G and set g ' = s u p { ¶ c G : q 5 c i- g for some c E K(L)l. Assume g' # g. Then g $ g' and thus, since Irr L is order generating, there i s an r e Irr L with g'sr and g g r . By ( 3 ) we find a q c G and a ccK(L) such that q $ r , and q S c S g . By the definition of g', this q satisfies q < g ' z r , and this is a contradiction. ( 2 ) =3 (5) Let c 5 g with CEK(L) and gt G. Since c <
a=

...

...,

(5) ( 2 ) : Assume ( 5 ) and take an arbitrary g c G and an arbitrary c c K(L) with c S g, Let pj and d be as in ( 5 ) . Then c S p1V. ..Vpn,S sup{q E G: q s e 5 g) for some e EK(L) < g. Since g = sup (+gnK(L)) by the definition of an algebraic lattice (cf .[C-801,~. 85, Definition 4 . 4 ) , condition ( 2 ) follows. 0 Once again we remark that condition ( 3 ) in the previous g

=

sup +g G

for all

theorem implies

gc G ,

but, conversely, this condition does not suffice for ( 3 ) to hold. It is therefore important to observe whether the sup in question i s taken in L or in G. 3.

General topology

algebraic lattices

Let X be an arbitrary topological space. Then its topology O(X) is a complete Heyting algebra. We can form the algebraic Heyting algebra L = Filt O(X) of filters on O ( X ) . The map U + + U which associates with an open set U its neighborhood filter fU is an isomorphism O(X) -+K(L)OP. (Recall + U { V E O(X): UcVl ! ) Indeed L is even arithmetic: See CC-801,p. 8 6 , Definition 4 . 6 . The Scott topology u(L) of L is generated by the basic sets +(+U) E K(L) c_L. As a first step we record the identification of the completely (meet-)primes of

L = Filt O(X). Recall that a set in a topological space is called i r r e d u c i b z e if it is not the union of two proper closed subsets. 3.1

w. Let

Y

be a closed irreducible set. Then the filter f(Y) = L. Furthermore all completely prime

: U n Y # '$1 is completely prime in elements of L are so obtained.

{ U c O(X)

Proof. Among all filters on O(X) which do not contain X \ Y there is a unique largest one, namely, F(Y). Thus F(Y) is maximal in L\+(+(X\Y). Hence F(Y) i s completely (meet-)prime as the maximal element of the complement of a principal

Essential hull of a topological To -space

199

f i l t e r . But t h e c o n v e r s e i s a l s o t r u e : I f F i s a completely prime element i n L , then i t i s t h e maximal element i n t h e complement of a p r i n c i p a l f i l t e r g e n e r a t e d by a compact element + V of L . I f we d e f i n e Y = X ' \ V then Y i s a c l o s e d s e t f o r which F = { U c O ( X ) : U I I Y # 0) i s a f i l t e r ; then Y i s i r r e d u c i b l e . 0 One n o t e s t h a t i n any l a t t i c e L , t h e completely prime f i l t e r s a r e e x a c t l y t h e completely prime e l e m e n t s of F i l t L .

3 . 2 COKOLLAIIY. For each x e X , t h e neighborhood f i l t e r prime element of L = F i l t O ( X ) . P r o o f . Since

i n t h e terminology of 3 . I ,

U(x) = F ( { x l - )

u(x)

i s a completely

t h e c l a i m f o l l o w s from 3 . I .

u

I n p a r t i c u l a r , i f X i s a s o b e r s p a c e , i . e . a s p a c e i n which e v e r y c l o s e d i r r e d u c i b l e s e t h a s a unique dense p o i n t , t h e n t h e s e t of p o i n t neighborhood f i l t e r s u ( x ) , X L X i s p r e c i s e l y t h e s e t of c o m p l e t e l y meet prime e l e m e n t s of L = F i l t O(X). Every completely prime element i s completely i r r e d u c i b l e ; t h e c o n v e r s e f a i l s even i n d i s t r i b u t i v e a l g e b r a i c l a t t i c e s i n g e n e r a l ; i f L happens t o be j o i n Brouwerian, then t h e two a r e e q u i v a l e n t . A t any r a t e , we can r e c o r d t h e f o l l o w i n g f a c t .

3.3 and L = of For

3.4

LEMMA. (Banaschewski) I f we d e n o t e t h e s e t { u ( x ) : x i X) w i t h G & L e q u i p G w i t h t h e topology a ( L ) I G induced from t h e S c o t t topology of F i l t O(X), then t h e f u n c t i o n x + l l ( x ) : X-+G i s the universal To-reflection X and i s a homeomorphism i f f X i s a To-space. a proof s e e CB-771. L!

s. The s u b s e t

F

of

L

i s K-generating.

P r o o f . L e t c t K ( L ) . Then t h e r e i s a U E O ( X ) i n t h e f i l t e r l a t t i c e L = F i l t O(X) we habe =

c = S U . We o b s e r v e t h a t

u}.

t~ = n{Ll(x):

But then 2 . I ( i i ) i s s a t i s f i e d and

such t h a t

GcIrr L

0

h a s been observed i n 3 . 2 .

I n summarizing t h e p r e c e d i n g remarks we can s t a t e t h a t wc c o n si d e r every To-spaae as a K-generatirig subspace o f an a r i t h m e t i c l a t t i c e w i t h t h e S c o t t t o p o l o g y . I t is in f a c t a subspace o f the space of completely prime Elements arid agrees u i t h t h i s subspace iff i t is sober. We now c o n n e c t t h i s t r a i n of t h o u g h t w i t h t h e developments of S e c t i o n 2 . I n t h a t s e c t i o n we a s s o c i a t e d w i t h a K-generating s u b s e t G of an a l g e b r a i c l a t t i c e L t h e space X ( G ) = { x e L : x = s u p ( . l x n G ) ) w i t h t h e induced S c o t t t o p o l o g y . I n t h e c o n t e x t of a To-space X we must now c o n c e n t r a t e on t h i s o b j e c t . We t h e r e f o r e a l l o w o u r s e l v e s a s l i g h t abuse of language i n a g r e e i n g t o t h e f o l l o w i n g . CONVENTION. I f X i s a To-space we d e n o t e w i t h XX t h e subspace X ( G ) of L = F i l t O(X) w i t h t h e S c o t t topology o(L) A(G) of L . The e l e m e n t s of XX a r e t h o s e f i l t e r s on O(X) which a r e j o i n s of p o i n t neighborhood f i l t e r s u ( x ) . (Banaschewski c a l l s them join f i l t e r s . )

I

3.5 THEOREM. (Banaschewski) h u l l o f t h e To-space X . P r o o f . See [B-771. n

The map

x + u ( x ) : X +AX

is t h e unique essent7:aZ

NOTATION. We s a y t h a t a To-space X h a s a n i n j e c t i v e h u l l i f f t h e s p a c e AX i s i n j e c t i v e i n t h e s e n s e of S c o t t ( c f . [ S - 7 2 1 o r I C - 8 0 1 , ~ . 1 2 1 f f . ) . T h i s i s then t h e c a s e i f and o n l y i f AX i s a c o n t i n u o u s l a t t i c e w i t h u(AX) = u ( L ) XX w i t h L = F i l t O(X).

I

X t h e speciaZisation order I n t h e I n t r o d u c t i o n w e mentioned f o r any To-space x s y i f U(x) 2 U(y); t h u s i f we v i s u a l i z e X as embedded i n t o L , t h e s p e c i a l i z a t i o n o r d e r i s p r e c i s e l y t h e one induced from L . We a l s o i n t r o d u c e d t h e S c o t t order x < y i f f + x i s a neighborhood of y . We t h e n n o t e immediately:

K.R Hofmann

200

3.6 REMARK. For elements x are equivalent:

and

(1) X4Y. (2) There is an open set V

(3)

There is an open set V

(4)

U(x) <
L

(with

y

in a To-space X, the following conditions

of

X

such that y < V

of

X

such that

and

c

U(x)

Vcfx.

+V 2 U(y).

= Filt O(X)).

Thus the Scott order < on X is induced by the way below order << on L. I n this sense, the notation is consistent with the one we used at the end of Section 1 and in 2.3. For X E X and Y & X we write x = supxxY iff U(x) = supL{U(y): y E Y1. A simple translation of the information contained i n Theorem 2.3 now allows u s t o formulate the characterisation theorem for spaces with injective h u l l s :

3.7

THEOREM. For a T -space X

(I)

X

, the

f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t :

has an i n j e c t i v e h u l l .

(2) For each x E X , t h e neighborhood f i l t e r U(x) i s the sup in t h e f i l t e r l a t t i c e Filt O(X) o f a l l U ( y ) w i i h yqx. (3) (4)

(5)

For each

x we

xE

have

x

sup +x. XX For each X E X and each completely i r r e d u c i b l e f i l t e r U n o t containing t h e r e i s a p o i n t z w i t h U ( z ) C U such t h a t f o r some open neighborU(x) hood V o f x we have U(z)c+V. =

For each p o i n t x t X and each open neighuorhood neighborhood V o f x and f i n i t e l y many p o i n t s V c + x , n...n

fx

neighborhoods

Wj

wI

and of

n.

. ,n

Wn

c

(i.e., such t h a t

xj

W of xl,

x

...,x

t h e r e is an open such t h a t

f o r suitable

W

W E U(x,) v . . . ~ U(xn)c_fV.)

Proof. The five conditions above are translations of the five corresponding conditions of Theorem 2.3. ( I n the case of (3) we use 3.6 above and the fact that AX is embedded into L under preservation of arbitrary sups.) Thus Theorem 2.3 proves Theorem 3.7. 0 The c o n d i t i o n s of Theorem 3 . 7 i m p l y each of t h e

3.8 COMPLEMENT TO THEOREM 3.7. f o 1 lowing c o n d i t i o n s : (6) For each (7)

XE

X we have

The topology o f

(8) For set

sup +x = sup +x. X X AX is its o m S c o t t topozogy.

U , V E AX, we have V E V such t h a t U

Proof. (6): The relation x Moreover, x
x

=

U<
supxx+x

of


o(x) V

i f f t h e r e i s an open

(3) implies the relation x

=

supXfx.

implies x <
is a direct consequence of ( 1 ) .

(8): By IC-801,p.181, Corollary 1.7 we have U
+VEK(Filt O(X))

for with

U,VEAX iff U<< Filt

u

5 fV

c V.

o(x)V.

n

Condition (6) is formulated completely in terms of the space X and its specialisation order alone. Examples which we will discuss in Section 4 show that (6) does not suffice for ( I ) . It is important that in condition (3) the sup is taken in AX. At any rate let u s extract the most concise order theoretic characterisation of the existence of an injective hull from Theorem 3.7 as follows:

20 1

Essential hull of a topological To -space SCHOLIUM I .

1 To-spcrce

has i(n i n j e c t i v e h u l l

X

= sup

x ItL

r ' u ~ e , tlie Scott

tJ1L.i

<<

if and o n l y if t h e S c o t t order

X sc2ti;fids the cor7d;tion

+oil

<

OY&Y

of

&'s induced from t h e way below relatiori

X

AX.

0s

The e q u i v a l e n c e o f 3 . 7 ( 1 ) and 3 . 7 ( 5 ) was i n d e p e n d e n t l y shown by R.-E.Hoffmann an e l a b o r a t i o n o f B a n a s c h e w s k i ' s o r i g i n a l m e t h o d s .

by

We r e c a l l a t t h i s p o i n t t h e d e f i n i t i o n o f a c o n t i n u o u s p o s e t ([C-801,~. 5 2 , D e f i n i t i o n 1.26) 3.9 DEFINITION. A poset P following three conditions:

i s up-complete

P

(1)

i s s a i d t o be m?itinuous p r o v i d e d i t s a t i s f i e s t h e

(i.e.,

d i r e c t e d sups e x i s t ) .

(ii)

For a l l

xt P

we h a v e

x = s u p +x.

(iii)

For a l l

X E

P

the set

$x i s d i r e c t e d .

I f X i s t h e To-space o b t a i n e d by e q u i p p i n g a c o n t i n u o u s p o s e t P w i t h i t s S c o t t t o p o l o g y , t h e n x < y i f f x<< y i n X ( s e e [ C - 8 0 1 , ~ . 110, E x e r c i s e 1 . 2 4 and p . 5 2 , E x e r c i s e 1 . 2 7 ) . Thus C o n d i t i o n ( 6 ) a b o v e i s s a t i s f i e d . But [C-801,p. 110, E x e r c i s e 1.22 shows t h a t C o n d i t i o n ( 5 ) o f Theorem 3 . 7 i s s a t i s f i e d w i t h n = 1 . Thus we h a v e 3.10

every rontinuozis p o s e t with i t s

( B a n a s c h e w s k i - K.-E.Hoffmann).

COKOLLARY.

A w t t topology has

UH

injective hull.

The c o n v e r s e f a i l s i n g e n e r a l ; i n t h i s r e g a r d t h e p r o o f of C o r o l l a r y I and C o r o l l a r y 2 o f P r o p o s i t i o n 3 of [B-771 and t h e p a p e r s [Hff 8 1 1 and 8123 n e e d t o be corrected accordingly. I t was o b s e r v e d by R.-E.Hoffmann t h a t , u n d e r s u i t a b l e c i r c u m s t a n c e s , one may c l a i m t h a t c e r t a i n subspaces of spaces with i n j e c t i v e h u l l s have i n j e c t i v e h u l l s , too. Some c a r e i s i n o r d e r . A t a n y r a t e we h a v e t h e f o l l o w i n g : 3. I I

PROPOSITION.

LGt

be a subspuce of a T -space

Y

IS x = sup

i.e. if

Since

is

Y

Proof.

Let Y

(J.XOY) AX s u p AX- dense i n X,

y i Y . Then

i s sup

AX

ztY

and

-dense i n

zSx<

Ux(z) 5 Ux(v) z

E

for a l l

X =

X F

y

=

sup

X

5

y

with

X E

X.

we c o n c l u d e sup A x ( ~ x n Y )

SUPAYSYY 5 Y

Then Theorem 3 . 7 ( 3 ) i s s a t i s f i e d f o r

Y

V

of

X

with

Uy(z)c_~y(v) for all

V . But t h e n a l s o

Y

,

has an i n j e c t i v e h u l l .

Y

y , t h e n t h e r e i s a n open s e t v

w i t h an i n j e c t i v e h u l l .

by 3 . 7 ( 3 ) . L e t

y . Thus w e h a v e , i n v i e w o f t h e embedding

conclusion

3.12

theri

y = supxx s X y

x If

for a l l

X

U - t { L I E O(X):

yrV

and

v c V n Y , i.e.

U n Y E U l : AY-t AX, t h e

*

n

and t h e a s s e r t i o n f o l l o w s .

COROLLARY. Every sup-deiise subset X o f a continuous Z a t t i e e o ( L ) ( A X has an i n j e c t i v e h u l l .

L

endOlJed

w i t h thd topo2ogy

P r o o f . The s p a c e ( L , o ( L ) ) i s i t s own i n j e c t i v e h u l l t h e n f o l l o w s d i r e c t l y from 3 . 1 1 . 0

X ( L , o ( L ) ) . The a s s e r t i o n

If we t a k e t h i s c o r o l l a r y t o g e t h e r w i t h t h e o r i g i n a l f a c t t h a t e v e r y To-spaces X w i t h a n i n j e c t i v e h u l l i s s u p - d e n s e l y and S c o t t t o p o l o g i c a l l y embedded i n t o t h e continuous l a t t i c e A X w e have a p r e t t y eood p e n e r a l i n t u i t i o n

K.H. Hofmann

202

which To-spaces have an injective hull- assuming, of course, that we have a satisfactory intuition of continuous lattices. We formulate this as follows: SCHOLIUM 11. Aiig J,-,-paoe X blitii a11 ir'juctiue h u l l is abiLi?')itd c1.s n clp-
+

We can thus easily manufacture spaces X which have an injective hull but which are not locally quasicompact. We will show an example in Section 4. In the opposite direction we record the following observation: 3.13

1'1 i t

PROPOSITION. is discrete.

A I ' I - s p c o hns mi ivrjed,ive huZZ if

(Banaschewski)

u?id

01

ly

Proof. Condition 3.7(9) applies to prove the claim: Indeed in a Tl-spdce the relation x < y holds iff x = y and x is isolated. Thus every non-discrete compact Hausdorff space gives an example of a sober locally quasicompact space without injective hull. In the absence of sobriety, local quasicompactness should be replaced by the condition that O(X) is a continuous lattice. Such spaces are often called core compact. In order to juxtapose the two conditions of having an injective hull ( L H ) and of being core compact (CC) we write these two conditions next to each other in an appropriate form:

\3/

..., xn)

(IH)

(VXEX)(VWEU(X))(JV~~(X))(~X,,

(CC)

(t/xc X)(W< U(x))(3V<

U(x))(3H)

Wc

W I J = 1 U(xj)

H c

+ V

and

5 tV.

H is

Scott open in O(X).

(Cf. [C-801, p.131) It thus follows that a To-space with an injtctive h u z l core compact whenever any finite sup of poirit neighborhood filters is a Scott open set on O(X)

.

In order to derive a sufficient condition, let us make the following remark: 3.14

(I)

REMARK. For a To-space

X, the following conditions are equivalent:

Relative to the specialisation order, X

(2) The embedding x->U(x): X ->L

=

is a topological sup-semilattice.

Filt O(X)

is a sup-semilattice morphism.

Proof. Condition (2) is equivalent t o (3)

U(xv y)

=

U(x) v U(y)

Since we always have (3')

U(xvy) 5 U(X)V

for all

U(x)vU(y) U(y)

x,y

E

5 U(xVy),

for all

X. condition (3) is equivalent to

x,y~X.

However, condition (3') expresses exactly the continuity of the operation (x,y) x v y : X x X 4 X. Thus ( 3 ' ) 3 (I). U -f

Since every point neighborhood filter is Scott open in O ( X ) the following result:

we obtain immediately

3.15 PROPOSITION. If a lo-space with an injective h u l l is a topological scrmlutice relative to the b i n u q sup-operation f o r the speciaZisation order, then it is core compact. If, in addition it is sober, then it is l o c a l l y quasicompact. 0

4.

Examples

We know from Scholium I1 that we obtain To-spaces with injective hulls if we take sup-dense subsets of a continuous lattice L and induce the Scott topology from L. Most phenomena can be studied by taking for L the square [ 0 , 1 l 2 with the componentwise lattice structure; for the remainder o f this Section, L will denote the square.

Essential hull of a topological To -space

203

4.1 EXAMPLE. Let X be the boundary of the square L. Its own Scott topology relative to the induced order is the induced Scott topology of L. Since X is sup-dense in L and is topologically embedded, L is the injective hull of X. A s a poseL, X is not a continuous poset, since the set +x(l,l) = {(x,y):x = 0 and y < I , or y = 0 and x < I 1 is not directed (cf.3.9(iii)). The set U = {(x,y): x = I and y > O , or y = I and x > O > is open in X but fails to have an injective hull, since condition 3.7(3) fails for x = ( 1 , l ) . (This shows that Corollary 4 in [B-771 is false.) Notice that Y is locally quasicompact and sober.

By contrast one should mention a result of R.-E.Hoffmann's according to which a dZo;cu subspace of a To-space with an injective hull has itself an injective hull. (Indeed, if Y L X S A X with injective A X and Y closed in X, then there is a Scott closed subset S of A X with Y = X n S . If S E S, then s = sup(+snX) since X is sup-dense in AX. But S is a lower set, whence J s n X c S n X = Y so that s = sup(JsnY). Set e = sup S and A = S u {el. Then A is a subalgebra of + e , hence is a continuous lattice, in which Y is sup-dense. The Scott topology 01 A is that induced from the Scott topology of A X , whence Y is topologically embedded into A . Hence Y has an injective hull by 3.11.)

4.2 EXAMPLE. belongs to X (a)

x

(b) (c)

1/2 < x

x

= =

Let C be the following subset of the square L: A point (x,y) if at least one of the following conditions is satisfied:

0 or

y

or

= 1/2

y

=

0,

1/2
.

We equip X with the topology induced from the Scott topology of the square L. In particular, the point m = (l/2, 1 / 2 ) 6 X has a basis of neighborhoods W(a,b) = {(x,y): a < x and b 'yl with a, b < 1/2. ( A picture helps!) Since X is sup-dense in L, i t has the square L as injective hull. One verifies the following properties:

(i)

X is sober. (All directed closed sets have sups.) X is not locally quasicompact in m. (Exercise!)So the poset X is not continuous. (iii) The point m is isolated from below but is not compact (i.e. does not satisfy m < < m in X). (iv) The Scott topology of X is properly finer than o ( L ) I X. (If U is a Scott open subset of L which is contained in C and contains +m\{m}, then U u {m)< o(X). One easily obtains such neighborhoods which contain none of the neighborhoods W(a,b).) (ii)

Remarks. Since X has an injective hull, we have m = SUPAX +m. If we denote with X' the space (X,o(X)), then we have x < x y if X < ~ for I ~all x,ycX. In particular, m = S U P ~ J . ~ m = sup ,+x, m. But by property (1v) we have

not have an injective hull. This provides the example announced at the end of Section 1 ; it verifies the claim of the last paragraph of Section 2 and substantiates the claim that condition (6) of 3.8 does not imply condition (3) of 3.7. It is interesting to notice that as a poset, X barely fails to be continuous. In fact, if we define a binary relation <
204

K.H.Hofmann

(A)

X

has an i n j e c t i v e h u l l .

(B)

X

is sober.

(C)

X

i s core compact ( i . e . ,

O(X) i s a c o n t i n u o u s l a t t i c e ) .

(D) X

is l o c a l l y quasicompact.

(E)

is a continuous poset with i t s Scott topology.

X

. C

A

lB

Essential hull of a topological To - y a c e

205

REFERENCES B-77

Banaschewski, B., Essential extensions of T -spaces, Gen.Topology submission 28 Feb. 1972). Appl. 7 (1977), 233-246 (date'of

C-8 I

Banaschewski, B. and R.-E.Hoffmann editors, Continuous Lattices, Lecture Notes in Mathematics 871 (1981).

C-80

Gierz, G . , K.H.Hofmann, K.Keime1, J.D.Lawson, M.Mislove and D.S.Scott, A Compendium of Continuous Lattices, Springer Verlag Berlin-Heidelberg-New York,(l980).

Hf f-77

Hoffmann, K.-E., Essentially complete T -spaces, Manuscr.Math. 2(1977), 401-452. Essentially complete T -spaces 11. A lattice theoretical approach, Math.Z. 179 ?1982), 73-90. Projective sober spaces, in C-81 above, 125-158.

Hf f-82

Hff-81, Hff-8I2

-

Continuous posets, prime spectra of completely distributive complete lattices and Hausdorff compactifications, in C-81 above, 159-208.

Hff-831

-

Hf f-83*

-

The &-compactification and the injective hull of a continuous poset, manuscript. The Fell compactification revisited, Preprint 1982.

ti f -M- 8 2

Hofmann, K.H. and M. Mislove, A continuous poset whose compactification is not a continuous poset. SCS-Memo 5-28-82.

Hf-82

Hofmann, K.H. The essential hull revisited, SCS-Memo 6-8-82. Scott, D.S., Continuous lattices, Lecture Notes in Math. 274 (1972), 97-136.

S-72

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- PART V ORDER AND THEORY OF MODELS AND RELATIONS -

PARTIEV-

ORDRE, THEORIE DES MODELES ET THEORIE DES RELATIONS

Wilfrid HODGES Models built on linear orderings. . . . . . . . . . . . . . . . . . . . . . . . . . .

p . 207

Claude FRASNAY Chainable relations, rangements and pseudo-rangements. . . . . . . . . p . 235 James H. SCHMERL H, -categorical partially ordered set. . . . . . . . . . . . . . . . . . . . . . . . .

p . 269

Denis RICHARD The arithmetics as theories of t w o orders. . . . . . . . . . . . . . . . . . . . p . 287 Roland FRAISSE L 'intervalle en theorie des relations; ses ge'neralisations; filtre intervallaire e t clbture d'une relation. . . . . . . . . . . . . . . . . . . p . 313

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23. ( 1 9 8 4 ) 207-234

0 Elsevier Science Puhlishers B.V. (North-Holland)

207

MODELS B U I L T ON L I N E A R OKDERINGS W i l f r i d Hodges Hedford C o l l e g e R e g e n t' s Park London N W l 4NS

KC s um6 En 1956, A n d r z e j E h r e n f e u c h t e t A n d r z e j Mostowski o n t d C c r i t un p r o c e d e permettant d'engendrer,;

p a r t i r d e c h a I n e s , d e s m o d s l e s d ' u n e t h C o r i e donnCe :

Ces rnodkles s o n t o b t e n u s comme e n v e l o p p e , p a r d e s f o n c t i o n s d e Skolern, d ' u n e chai^ne d ' e l t m e n t s ( o u d e u p l e s ) i n d i s c e r n a b l e s , q u i l e u r s e r t d ' C p i n e d o r s a l e . J'c,xpose i c i l e s p r i n c i p a l e s a p p l i c a t i o n s d e c e t t e c o n s t r u c t i o n 2 l a l o g i que e t i l ' a l g h b r e u n i v e r s e l l e c o n t e r n p o r a i n e s : c e c i cornprend l ' e x i s t e n c e d e modhles d ' E h r e n f e u c h t - M o s t o w s k i p o u r l e s l a n g a g e s f i n i s e t i n f i n i s ; l a c o n s t r u c t i o n d e b e a u c o u p d e rnodeles d e t h e o r i e s i n s t a b l e s ; l a c o n s t r u c t i o n d e b e a u c o u p de groupes localement f i n i s u n i v e r s e l s ; l e s i n d i s c e r n a b l e s d e S i l v e r e t l e rCel 0'

e n t h b o r i e d e s e n s e m b l e s ; l e t h e o r e m e d e L6wenheim-Skolern

( ce1ui"vers

le

h a u t " ) p o u r les l a n g a g e s i n f i n i s ; l e thCor6rne d e s d e u x c a r d i n a u x e t q u e l q u e s autres.

I n 1956 A n d r z e j E h r e n f e u c h t and A n d r z e j Mostowski p u b l i s h e d o n e of t h e s e m i n a l p a p e r s o f model t h e o r y [ 1 9 ] .

T h i s paper d e s c r i b e d a

way o f t u r n i n g l i n e a r o r d e r i n g s i n t o m o d e l s o f a g i v e n t h e o r y ,

so t h a t

p r o p e r t i e s of t h e o r d e r i n g were r e f l e c t e d i n p r o p e r t i e s of t h e model. d e v i c e i s known a s t h e E h r e n f e u c h t - M o s t o w s k i

construction.

Their

Today model

t h e o r i s t s u s e i t a t l e a s t o n c e a week, and t h e r e a r e s t i l l a number of f u n d a m e n t a l t h e o r e m s o f model t h e o r y w h i c h h a v e no known p r o o f s w i t h o u t t h i s construction.

I n 551,Z

b e l o w I d e f i n e E h r e n f e u c h t and M o s t o w s k i ' s c o n s t r u c t i o n

a s a c l a s s of f u n c t o r s s a t i s f y i n g c e r t a i n a x i o m s , and I p r o v e some b a s i c p r o p e r t i e s of s u c h f u n c t o r s .

I n 53 I q u o t e two e x i s t e n c e t h e o r e m s w h i c h

s a y t h a t we c a n f i n d f u n c t o r s s a t i s f y i n g t h e a x i o m s . a r i c h c r o p o f e x a m p l e s and a p p l i c a t i o n s ;

i n 554-7

These theorems y i e l d I s k e t c h some a p p l i c a t i o n s

i n model t h e o r y , s e t t h e o r y , u n i v e r s a l a l g e b r a and g r o u p t h e o r y .

208

W. Hodges The f o l l o w i n g n o t a t i o n w i l l be i n f o r c e t h r o u g h o u t .

'Chain'

means ' l i n e a r o r d e r i n g ' , and q , 5 , 5 a r e v a r i a b l e s f o r c h a i n s .

'Increasing'

n - t u p l e s from a c h a i n q w i l l always be understood t o be s t r i c t l y i n c r e a s i n g .

R i s a s i m i l a r i t y t y p e , i . e . a s e t of c o n s t a n t s , f u n c t i o n symbols and r e l a t i o n symbols, finite arity.

The f u n c t i o n and r e l a t i o n symbols a r e assumed t o have

I n t h e a p p l i c a t i o n s I s h a l l assume f o r s i m p l i c i t y t h a t R i s t h i s i s n o t t o h i d e a mad a u n t i n t h e a t t i c , b u t simply

a t most c o u n t a b l e ;

t o avoid having t o mention t h e c a r d i n a l i t y o f S2 i n theorems. i s an e x p r e s s i o n b u i l t up from v a r i a b l e s f u n c t i o n symbols;

x with

-

x i s a f i n i t e t u p l e (x,,

term

A

t(X)

t h e a i d of c o n s t a n t s and

...,x n ) .

S t r u c t u r e s of s i m i l a r i t y t y p e R ( R - s t r u c t u r e s f o r s h o r t ) a r e w r i t t e n A , B, C ; the tuple means:

a

dom(A) i s t h e s e t of e l e m e n t s of A .

of elements of A s a t i s f i e s t h e formula $(;)

t h e sentence $ i s t r u e i n A .

"A

\

$(a)'' means:

in A.

"A

k@''

" t A ( a ) " i s t h e v a l u e of t h e term t ( x )

when t h e f u n c t i o n symbols and c o n s t a n t s a r e i n t e r p r e t e d by A and t h e v a r i a b l e s

x a r e r e p l a c e d by

t h e elements

a

of A .

'Formulas' a r e always assumed t o have j u s t f i n i t e l y many f r e e variables. disjunctions

But sometimes I u s e i n f i n i t e c o n j u n c t i o n s first-order

and d i s j u n c t i o n s a r e f i n i t e .

A iEI$i

and i n f i n i t e

formulas a r e t h o s e i n which a l l c o n j u n c t i o n s

The l o g i c i n which c o n j u n c t i o n s and d i s j u n c t i o n s

of a r b i t r a r i l y l a r g e s e t s of f o r m u l a s a r e allowed i s c a l l e d L r e a d a b l e a c c o u n t of LW and i t s u s e f u l n e s s i n Barwise [ 41.

.

mw'

there is a

Logic i n which

t h e c o n j u n c t i o n s and d i s j u n c t i o n s have t o have c a r d i n a l i t y l e s s t h a n c a l l e d LKW. Thus f i r s t - o r d e r

logic is L

WW

.

(The second s u b s c r i p t

t h a t we o n l y q u a n t i f y f i n i t e l y many v a r i a b l e s a t a t i m e . )

K

is

means

209

Models built o n linear orderings 1.

Ehrenfeucht-Mostowski f u n c t o r s d e f i n e d

We s h a l l say t h a t t h e s t r u c t u r e A c o n t a i n s t h e c h a i n e v e r y element of n i s a n element of A ;

t h e o r d e r i n g r e l a t i o n on

q

iff

q

need n o t

An Ehrenfeucht-Mostowski f u n c t o r , o r EM f u n c t o r f o r

be r e p r e s e n t e d i n A .

s h o r t , i s d e f i n e d t o be a f u n c t i o n F which t a k e s each c h a i n

n

t o an Q-structure

F ( q ) s o t h a t t h e f o l l o w i n g t h r e e axioms a r e s a t i s f i e d :

n,

t h e s t r u c t u r e F ( q ) c o n t a i n s n a s a s e t of g e n e r a t o r s

I.

For e a c h c h a i n

2.

For e a c h o r d e r - p r e s e r v i n g map f : q F ( f ) : F(q)

3.

-+

( I n ' ]F(n)

+

<,

,€ t h e r e i s a n embedding

F(S) which e x t e n d s f .

F is functorial; g: 5

+

i . e . f o r a l l o r d e r - p r e s e r v i n g maps f :

q

+

5 and

F ( g f ) = F ( g ) . F ( f ) , and f o r e v e r y c h a i n q , F ( l ) = 1

n

a r e r e s p e c t i v e l y t h e i d e n t i t y maps on

q,

F(n)'

F(n).)

By Axioms 2 and 3 a n EM f u n c t o r i s a f u n c t o r from t h e c a t e g o r y of c h a i n s and o r d e r - p r e s e r v i n g maps t o t h e c a t e g o r y of R - s t r u c t u r e s and

So f o r example, i f f : q

embeddings. F(f): F(q)

n

+

+

n i s a n automorphism t h e n

F( n ) i s a n automorphism e x t e n d i n g f .

The automorphism group of

i s n a t u r a l l y embedded i n t h e automorphism group of F ( r ( ) . By Axiom 1 , F ( q ) c o n t a i n s

says skeleton

-

q.

We c a l l

n

the

spine of

but skeletons a r e not l i n e a r l y ordered.)

g e n e r a t e s F ( q ) , e v e r y element of F ( q ) i s of form

and some i n c r e a s i n g t u p l e a from

%(rd

(a)

F(q) (Shelah

Since the spine f o r some t e r m t ( x )

q.

The two c e n t r a l p r o p e r t i e s of EM f u n c t o r s a r e known i n t h e t r a d e a s s l i d i n g ( i . e . we c a n s l i d e e l e m e n t s up and down t h e s p i n e w i t h o u t n o t i c i n g ) and s t r e t c h i n g ( i . e , we c a n p u l l o u t t h e s p i n e i n t o a s l o n g a c h a i n a s we l i k e ) . The f i r s t two lemmas w i l l make t h i s p r e c i s e .

W.Hodges

210

LEMMA I

(Sliding).

Let F be an EM f u n c t o r , and l e t

i n c r e a s i n g n - t u p l e s from c h a i n s q , 6 r e s p e c t i v e l y . f r e e formula $ ( x , ,

...,xn)

we have F ( q )

1 $(;)

0

b.

F(S)

5 5.

Assuming t h a t F ( q )

1 $(a)

5 5 5 and t h e r e i s

Consider t h e diagram

1 $(a)

5 and

by t h e l e f t - h a n d embedding.

s i n c e F ( f ) i s a n automorphism and F ( f )

Hence l i k e w i s e F ( S )

5

and n o t i n g t h a t embeddings p r e s e r v e

q u a n t i f i e r - f r e e f o r m u l a s , we have F ( 5 ) Then F ( 5 )

be

k$(b).

where t h e embeddings on l e f t and r i g h t come from t h e i n c l u s i o n s n

6

b

Then f o r e v e r y q u a n t i f i e r -

To prove t h i s , f i n d a c h a i n 5 such t h a t n _C 5, a n automorphism f of 5 such t h a t f a =

a.

1 $ ( i )p,r o v i n g

(a)

=

f

(a)

=

b.

the l e m a .

Lemma 1 makes t h e f o l l o w i n g d e f i n i t i o n s s e n s i b l e . s t r u c t u r e c o n t a i n i n g a c h a i n II and l e t $ ( x , ,

...,x

L e t A be a

) be a f o r m u l a .

We s a y

- -

t h a t q i s $ - i n d i s c e r n i b l e i n A i f f f o r any two i n c r e a s i n g n - t u p l e s a , b from 17, A

k$(a)

a A

k$(b).

a l l q u a n t i f i e r - f r e e $.

By Lemma I ,

17

i s $-indiscernible i n F(q) f o r

A c h a i n which i s $ - i n d i s c e r n i b l e

for a l l first-order

formulas $ i s s a i d t o b e a n i n d i s c e r n i b l e sequence.

We d e f i n e t h e t h e o r y of q i n A , Th(A,n), t o be t h e c l a s s of a l l formulas $ ( x , ,

...,x

) such t h a t A

1 $(a)

f o r every increasing n-tuple

a

from q .

The t h e o r y of t h e EM f u n c t o r F , T h ( F ) , i s d e f i n e d t o be t h e c l a s s of a l l formulas $ ( x , ,

a

from n, F ( n )

...,x

) such t h a t f o r e v e r y c h a i n q and e v e r y i n c r e a s i n g n - t u p l e

$(a).

S i n c e e v e r y formula h a s j u s t f i n i t e l y many f r e e

v a r i a b l e s , Lemma 1 t e l l s us t h a t Th(F) c o n t a i n s e x a c t l y t h e same q u a n t i f i e r - f r e e

Models built on linear orderings f o r m u l a s a s T h ( F ( n ) , U ) f o r a n y i n f i n i t e c h a i n n. from Th(F(w),w).

So T h ( F ) i s r e c o v e r a b l e

I n f a c t t h e w h o l e of F i s r e c o v e r a b l e u p t o i s o m o r p h i s m

from F ( o ) : LEEMA 2 ( S t r e t c h i n g ) .

formulas

0,

containing the

If w i s @-indiscernible i n A f o r a l l

c h a i n w a s a s e t of g e n e r a t o r s . quantifier-free

Let A be any R - s t r u c t u r e

t h e n t h e r e i s a n EM f u n c t o r F s u c h t h a t A = F ( w ) ;

F i s unique u p t o n a t u r a l isomorphism.

To c o n s t r u c t F i n t h e lemma, t a k e a n y c h a i n s e t o f e x p r e s s i o n s t ( a ) w h e r e t i s a t e r m and 0

of t h e r i g h t l e n g t h t o match t .

two e x p r e s s i o n s t ( a ) , s ( b ) i n X ,

n,

i s a n i n c r e a s i n g t u p l e from

the sentence R(t(a),s(b))

can be w r i t t e n a s

-

-

f o r m u l a $(x) and some i n c r e a s i n g t u p l e c f r o m r l .

We d e c r e e t h a t R ( t ( a ) , s ( b ) ) s h a l l h o l d i f f r e l a t i o n symbols of

and l e t X b e t h e

F o r a n y b i n a r y r e l a t i o n symbol R and a n y

$ ( c ) f o r some q u a n t i f i e r - f r e e

a

0

$(x)

E Th(A,w).

Likewise f o r a l l

and l i k e w i s e f o r t h e e q u a l i t y symbol = .

This defines

a n R - s t r u c t u r e F ( Q ) whose e l e m e n t s a r e t h e e q u i v a l e n c e c l a s s e s of e l e m e n t s t ( a ) , s(<)

of X u n d e r t h e r e l a t i o n " t ( a ) = s ( b ) h o l d s " .

Lemma 2 r e c a l l s a well-known f a c t of u n i v e r s a l a l g e b r a :

the

f r e e a l g e b r a s i n a v a r i e t y a r e c o m p l e t e l y d e t e r m i n e d by t h e f r e e a l g e b r a

on c o u n t a b l y many g e n e r a t o r s . Lemma 2 .

A c t u a l l y t h i s f a c t i s a s p e c i a l c a s e of

For any v a r i e t y V, d e f i n e F ( n ) t o be t h e f r e e a l g e b r a i n V w i t h

b a s i s t h e e l e m e n t s o f rl.

Then F i s a n EM f u n c t o r .

This i s an unilluminating

e x a m p l e of a n EM f u n c t o r , b e c a u s e i n t h i s c a s e t h e o r d e r i n g r e l a t i o n o n

n

p l a y s no r o l e w h a t e v e r .

I s h o u l d comment t h a t i n d i s c e r n i b l e s e q u e n c e s a r e i m p o r t a n t i n model t h e o r y f o r many r e a s o n s b e s i d e s t h e i r c o n n e c t i o n w i t h EM f u n c t o r s . They w e r e f i r s t i n t r o d u c e d by F r a i s s ;

[ 2 2 1 i n t h e g u i s e of ' e n c h a i n a b l e

r e l a t i o n s ' , and t h e y p l a y a l e a d i n g p a r t i n S h e l a h ' s a n a l y s i s of s t a b l e t h e o r i e s [551.

W. Hodges

212 2.

P r e s e r v a t i o n theorems

Category-minded r e a d e r s w i l l have n o t i c e d t h a t EM f u n c t o r s always p r e s e r v e d i r e c t l i m i t s .

I n model t h e o r y , f u n c t o r s which p r e s e r v e

d i r e c t l i m i t s u s u a l l y p r e s e r v e some o t h e r t h i n g s t o o , and s o i t happens h e r e A s e n t e n c e i s a formula w i t h no f r e e v a r i a b l e s ;

a u n i v e r s a l s e n t e n c e i s one

of form v & ( i ) where J, i s q u a n t i f i e r - f r e e .

LEMMA 3 .

L e t F be a n EM f u n c t o r t a k i n g c h a i n s t o R - s t r u c t u r e s .

Then: (a)

For e v e r y two i n f i n i t e c h a i n s

~ ( n )1 $

s e n t e n c e 4,

(b)

Let

be a n i n f i n i t e c a r d i n a l

e q u i v a l e n t c h a i n s , t h e n F ( n ) , F(C) a r e L

F preserves L

mu

€, and e v e r y u n i v e r s a l

+.

F(S) K

n,

t K

W

2 IRI.

If

n, 5

are L

-equivalent 0-structures.

example t h a t $ i s Vyz$(y,z).

1$

If F(q)

-

F(n)

Now $(t(a),s(b))

f i x e d t and s , f i n i t e l y many p o s i t i o n s of

x

x

can be w r i t t e n a s x ( c ) f o r some

-

x

a

and

in

n.

x

may occur h e r e , a c c o r d i n g t o t h e r e l a t i v e

So a s t and s range over a l l terms of R , a t most

a r e needed, where K = IRl+w.

L e t @ b e t h e s e t o f such f o r m u l a s

n.

S i n c e rl i s i n f i n i t e , i t f o l l o w s t h a t 0

5 Th(F).

But t h e n

E 0 i s a l s o s a t i s f i e d i n F(€,) by a l l i n c r e a s i n g t u p l e s from 5 .

Running t h e whole argument backwards, we f i n d t h a t F(S) 91;

For a

E 0 i s s a t i s f i e d i n F ( n ) by a l l i n c r e a s i n g t u p l e s of a p p r o p r i a t e

l e n g t h from each

Suppose f o r

t h e n f o r e v e r y p a i r of terms

q u a n t i f i e r - f r e e formula x ( x ) and some i n c r e a s i n g t u p l e c from n.

Each

Hence

and a l l i n c r e a s i n g t u p l e s a , b of a p p r o p r i a t e l e n g t h from n,

b$(t(a),s(b)).

formulas

-

-

-

K

O

-equivalence.

The f o l l o w i n g proof of ( a ) a d a p t s t o g i v e ( b ) t o o .

t ( i ) , s(;)

t

K

I$ too.

Eklof [ 2 0 1 p r o v e s similar r e s u l t s by back-and-forth

( C f . Chang

arguments.)

x.

Models built on linear orderings Lemma 3 i s weaker t h a n one might hope. f i r s t - o r d e r l o g i c , LWw. @(;)

F(n)

=

Part (b) can f a i l f o r

A l s o p a r t ( a ) s u g g e s t s t h a t i f a u n i v e r s a l formula

i s s a t i s f i e d by some i n c r e a s i n g t u p l e

be s a t i s f i e d by a l l .

213

Not so.

a

from rl i n F ( r l ) , t h e n i t should

The s i m p l e s t counterexample i s t o t a k e

rl f o r e v e r y c h a i n rl, and c o n s i d e r t h e u n i v e r s a l formula Vy(y

LX).

I n t h e s t r u c t u r e w = F ( w ) t h i s formula i s s a t i s f i e d by 0 b u t n o t by 1 . I n g e n e r a l t h e b e s t one c a n s a y i s t h a t i f rl and

5 a r e i n f i n i t e chains then

F ( n ) and F ( S ) a g r e e on u n i v e r s a l s e n t e n c e s (by Lemma 3 ( a ) ) , and i n c r e a s i n g t u p l e s from rl and 5 a g r e e on q u a n t i f i e r - f r e e

f o r m u l a s (by Lemma 1 ) .

However, t h e r e i s a u s e f u l d e v i c e i n v e n t e d more t h a n h a l f a c e n t u r y ago by Skolem.

I t e n a b l e s u s t o r e d u c e any s e t of f o r m u l a s t o

a s e t c o n s i s t i n g of j u s t q u a n t i f i e r - f r e e f o r m u l a s and u n i v e r s a l s e n t e n c e s , a t t h e c o s t of a d d i n g some e x t r a symbols t o t h e s t r u c t u r e s i n v o l v e d :

LEMMA 4 . t y p e R.

L e t 0 be a s e t of

Then t h e r e i s a s e t 1 of

general using

K

(a)

2

K

5

K

f o r m u l a s of L + K

O

in similarity

u n i v e r s a l s e n t e n c e s of L + K

W

,

in

new r e l a t i o n o r f u n c t i o n symbols, such t h a t Every R - s t r u c t u r e c a n b e expanded t o a model of C by a d d i n g i n t e r p r e t a t i o n s of t h e new symbols, and

(b)

Modulo C , e v e r y f o r m u l a i n

i s equivalent t o an atomic

formula. I f @ i s a s e t of f i r s t - o r d e r f o r m u l a s , t h e n Z c a n a l s o b e chosen t o be first-order.

(For t h e proof of t h i s lemma, s e e 83.3 o f Chang and K e i s l e r

[ 1 1 1, t o g e t h e r w i t h Chang [ 101. )

We s h a l l s e e i n a moment t h a t Lemma 4 a l l o w s u s t o f i n d EM f u n c t o r s F s o t h a t e v e r y s t r u c t u r e F ( q ) i s a model of a g i v e n s e n t e n c e @, even when @ i s n o t u n i v e r s a l .

W. Hodges

214

3 . E x i s t e n c e the ore ms

T h e r e a r e two main ways of f i n d i n g EM f u n c t o r s . for first-order

logic, the other i s for infinitary logics.

EXISTENCE THEOREM 5 ( f o r f i r s t - o r d e r R-structure containing a chain

n

of o r d e r - t y p e

logic).

One i s Here t h e y a r e :

L e t A be an

Then t h e r e i s a n EM

W.

f u n c t o r F ( p o s s i b l y u s i n g a l a r g e r s i m i l a r i t y t y p e t h a n f2) whose t h e o r y contains a l l first-order

f o r m u l a s i n Th ( A,q ) .

by i n d u c t i o n o n t h e

1 B

when 6 i s a l i m i t .

EXISTENCE THEOREM 6 ( f o r L K t w ) . F o r e a c h i n f i n i t e c a r d i n a l

i s a l e a s t o r d i n a l 6 ( ~ ) ,K + I f 0 i s a n y s e t of a well-ordered

K

5 6(~)

c h a i n rl o f l e n g t h

(2 )

<

f o r m u l a s of L + K

K

W

+

, with

K

there

t h e following property.

and A i s a n y $ ] - s t r u c t u r e c o n t a i n i n g

1

t h e n t h e r e i s a n EM f u n c t o r F

6(K)'

(possibly using a l a r g e r s i m i l a r i t y type than

n) whose t h e o r y c o n t a i n s

a l l f o r m u l a s from 0 w h i c h a r e i n Th ( A,n ) .

The p r o o f s o f t h e s e t h e o r e m s r u n as f o l l o w s .

Su p p o se f i r s t t h a t

t h e s e t Y of f o r m u l a s t h a t we want t o h av e i n t h e t h e o r y o f F c o n s i s t s o n l y of q u a n t i f i e r - f r e e f o r m u l a s and u n i v e r s a l s e n t e n c e s .

Then by Lemmas 1-3,

i t s u f f i c e s t o f i n d a n a - s t r u c t u r e B c o n t a i n i n g a w e l l - o r d e r e d c h a i n 5 of

l e n g t h w, such t h a t ( i )

5 is @-indiscernible f o r a l l quantifier-free

@,

and

( i i ) e v e r y f o r m u l a i n 't' i s s a t i s f i e d by a l l i n c r e a s i n g t u p l e s f r o m 5 . Now i n t h e f i r s t - o r d e r

c a s e we g e t ( i i ) t o g e t h e r w i t h ( i ) f o r a n y f i n i t e

s e t of f o r m u l a s

@,

does t h e rest.

F o r L K + w t h e r e i s a n a n a l o g u e o f t h e c o m p a c t n e s s t h e o r e m , and

by u s i n g Ramsey's th eo r em t o t h i n o u t

n;

then compactness

a p a r t i t i o n r e s u l t o f ErdUs and Rado s e r v e s i n p l a c e o f Ramsey's t h e o r e m .

Models built o n linear orderings

215

Suppose n e x t t h a t Y c o n t a i n s f o r m u l a s which a r e n o t e i t h e r q u a n t i f i e r - f r e e formulas o r u n i v e r s a l sentences.

Then a t t h e s t a r t o f t h e

argument we r e d u c e a l l t h e f o r m u l a s of Y t o t h e s e two forms by u s i n g Lemma 4 . T h i s i s where t h e s t r u c t u r e s F ( 5 ) may a c q u i r e new f u n c t i o n symbols b e s i d e s t h o s e used i n

iy.

D e t a i l s of t h e proof of Theorem 5 can be found i n t h e proof of Theorem 3 . 3 . 1 0 i n Chang and K e i s l e r [ I l l ;

t h e argument comes from t h e o r g i n a l

paper of Ehrenfeucht and Mostowski [191. There a r e o t h e r

approaches which avoid

Ramsey's theorem; c f . Gaifman [251, Oniarov [441, Taimanov [571 and K e i s l e r [ 3 4 a ] Theorem 6 i s due t o Morley [411 and t h e r e i s a proof i n s e c t i o n V I I 4 . of Barwise [ 51.

The o r d i n a l

w e l l - o r d e r i n g number of We have 6 ( w )

= wl,

K,

&(K)

i n Theorem 6 i s sometimes known a s t h e

f o r r e a s o n s which a r e made c l e a r i n Barwise [ 51.

b u t 6 ( ~ )>

K+

whenever

K

has uncountable c o f i n a l i t y .

The

f u l l e s t i n f o r m a t i o n a v a i l a b l e on w e l l - o r d e r i n g numbers 6 ( ~ i) s i n Theorem V I I . 5 . 5 of S h e l a h 1551 ( S h e l a h w r i t e s 6 ( ~ , 1 ) f o r 6 ( ~ ) ) .

S e t t h e o r i s t s know some o t h e r ways of f i n d i n g EM f u n c t o r s . For example, i f cardinality <

K,

K

i s a m e a s u r a b l e c a r d i n a l and R i s a s i m i l a r i t y type of t h e n e v e r y R - s t r u c t u r e of c a r d i n a l i t y

K

s u b s t r u c t u r e which i s of form F ( K ) f o r some EM f u n c t o r F. due t o Erdt(s and H a j n a l ;

contains a (This r e s u l t i s

d e t a i l s of t h i s and r e l a t e d theorems a r e i n

s e c t i o n 29 of J e c h [ 3 2 ] and s e c t i o n 7 . 3 of Chang and K e i s l e r [ I l l . )

There i s a n o t h e r e x i s t e n c e theorem i n a r a t h e r d i f f e r e n t s t y l e . I f R i s a r e l a t i o n symbol, t h e n {R} i s t h e s i m i l a r i t y t y p e c o n s i s t i n g o f j u s t R.

An { R l - s t r u c t u r e A i s s a i d t o be k-monomorphic i f f any two k-element

s u b s t r u c t u r e s of A a r e isomorphic.

A i s s a i d t o be e n c h a i n a b l e i f f A = F ( n )

f o r some EM f u n c t o r F and some c h a i n rl c o n s i s t i n g of t h e e l e m e n t s o f A; t h i s i s e q u i v a l e n t t o s a y i n g t h a t dom(A) can be o r d e r e d a s a c h a i n n which

i s $-indiscernible i n A f o r a l l q u a n t i f i e r - f r e e formulas 9 .

W.Hodges

216 THEOREM 7 .

L e t R be a n m-ary r e l a t i o n symbol w i t h m

l e t A be a n i n f i n i t e { R } - s t r u c t u r e .

k

2 2m-2,

5 , and

I f A i s k-monomorphic f o r some

t h e n A is e n c h a i n a b l e .

The proof i s t o o i n v o l v e d even t o s k e t c h h e r e .

By Theorem 1 2 . 1 . 1

of F r a s n a y [231, i t s u f f i c e s t h a t A should be k-monomorphic f o r some

g (m)+m, where g (m) i s t h e ' s e u i l d e r e c o l l e m e n t ' d e f i n e d on p. 84

k

1

of 1231.

I

For m

1. 5

one c a n e x t r a c t t h e bound g (m) I m - 2 from Hodges,

Lachlan and Shelah 1311;

1

F r a s n a y had a weaker bound.

C f . Theorem 15 below

for a group-theoretic application.

4.

A p p l i c a t i o n s of t h e f i r s t k i n d :

non-isomorphic models

Henceforth a l l s i m i l a r i t y t y p e s a r e assumed t o be c o u n t a b l e .

Most a p p l i c a t i o n s of Ehrenfeucht-Mostowski f u n c t o r s f i t w i t h i n the following format:

INPUT STEP.

We show t h a t i f t h e r e e x i s t s t r u c t u r e s w i t h some p r o p e r t y

P , t h e n t h e r e i s a n EM f u n c t o r F whose t h e o r y c o n t a i n s a c e r t a i n set 4 of f o r m u l a s .

OUTPUT STEP.

We show t h a t i f F i s any EM f u n c t o r whose t h e o r y c o n t a i n s

t h e s e t 4 , t h e n t h e s t r u c t u r e s F ( n ) have c e r t a i n p r o p e r t i e s Q.

U s u a l l y t h e purpose o f t h e e x e r c i s e i s t o b u i l d s t r u c t u r e s w i t h p r o p e r t i e s Q ; t h e c l e v e r p a r t i s t o f i n d s e t s of f o r m u l a s 4 and p r o p e r t i e s P which make t h e whole argument work.

The argument of t h e I n p u t S t e p always r e s t s on one of t h e E x i s t e n c e Theorems of 13 above, and t y p i c a l l y i t i n v o l v e s a p a r t i t i o n theorem of Ramsey o r of Erd8s and Rado.

(Morley [421 s u r v e y s t h e r e l e v a n t

p a r t i t i o n theorems and how t h e y a r e a p p l i e d . )

When i n f i n i t a r y c o m b i n a t o r i a l

Models built on linear orderings

217

arguments a r e i n t h e o f f i n g , i t i s o f t e n u n p l e a s a n t t o s t a t e r e s u l t s i n b e s t p o s s i b l e form.

Most o f t h e r e s u l t s quoted below a r e g i v e n i n s t r o n g e r

b u t more complicated forms i n t h e o r i g i n a l p a p e r s .

Our f i r s t a p p l i c a t i o n s a r e a b o u t c o n s t r u c t i n g many non-isomorphic models of a t h e o r y .

S h e l a h [541 s e e s t h i s work a s a p a r t of a l a r g e r p r o j e c t :

t o c l a s s i f y c l a s s e s of s t r u c t u r e s i n t o two t y p e s , v i z . t h o s e c l a s s e s which have ' s t r u c t u r e theorems' and t h o s e which d o n ' t .

A ' s t r u c t u r e theorem' f o r

a c l a s s s a y s t h a t each s t r u c t u r e i n t h e c l a s s can be c h a r a c t e r i s e d by a system of i n v a r i a n t s , a s f o r example e a c h a l g e b r a i c a l l y c l o s e d f i e l d i s determined by i t s c h a r a c t e r i s t i c and t h e s i z e o f a t r a n s c e n d e n c e b a s i s . One way t o show t h a t a c l a s s h a s no s t r u c t u r e theorem i s t o prove t h a t t h e r e a r e too many non-isomorphic s t r u c t u r e s i n t h e c l a s s .

The most i m p o r t a n t r e s u l t of t h i s t y p e i s due t o S h e l a h ( [ 5 0 ] , s e e a l s o [551 Ch. V I I I):

THEOREM 8 . Q

Suppose A i s a s t r u c t u r e c o n t a i n i n g a n i n f i n i t e c h a i n

whose o r d e r i n g i s d e f i n e d by a f i r s t - o r d e r f o r m u l a $ ( x , y ) ( i . e . f o r a l l

a # b i n Q, a c Q b < * A k $ ( a , b ) ) .

Then i n e v e r y u n c o u n t a b l e c a r d i n a l i t y

K

t h e r e a r e Z K p a i r w i s e non-isomorphic s t r u c t u r e s which a r e a l l e l e m e n t a r i l y equivalent t o A.

Theorem 8 s a y s f o r example t h a t i f T i s a f i r s t - o r d e r t h e o r y whose models are a l l p a r t i a l l y o r d e r e d , and some model of T c o n t a i n s a n i n f i n i t e c h a i n , t h e n i n e v e r y u n c o u n t a b l e c a r d i n a l i t y , T h a s t h e maximum c o n c e i v a b l e number of p a i r w i s e non-isomorphic models.

The proof of t h e theorem i s hard work, b u t h e r e i s a b r i e f g u i d e . Without l o s s we c a n suppose t h a t t h e c h a i n Q h a s o r d e r - t y p e

W.

The formula

218

W. Hodges

+ ( x , y ) need not l i n e a r l y o r d e r t h e whole o f dom(A); ji(x,y)

A

but r e p l a c i n g i t by

- r $ ( y , x ) i f n e c e s s a r y , we c a n suppose w i t h o u t l o s s t h a t i t does

d e f i n e a n asymmetric r e l a t i o n on dom(A).

Adding Skolem f u n c t i o n s by Lemma 4 ,

we can suppose a l s o t h a t cL(x,y) i s a n atomic formula " x 4 y". s e t of f o r m u l a s { x 1: x 2 , V x y ( x 4 y

+

1

y 4 x)}.

(Input step:)

Let 0 be t h e By E x i s t e n c e

Theorem 5 t h e r e i s a n EM f u n c t o r F w i t h 4 U ( f i r s t - o r d e r t h e o r y of A) (Output s t e p : )

5 Th(F).

Using some q u i t e e l a b o r a t e s e t t h e o r y , we can c o n s t r u c t f o r

each u n c o u n t a b l e c a r d i n a l

K

a family ( 0

i

: i < Z K ) of c h a i n s of c a r d i n a l i t y c ,

such t h a t i f F i s any EM f u n c t o r whose t h e o r y c o n t a i n s

i < j < ZK, t h e s t r u c t u r e s ( d o m ( F ( r t . ) ) , A 1

a r e not isomorphic.

F(ni)

a, t h e n f o r a l l

) and ( d o m ( F ( n . ) ) , A ) J F(nj)

A f o r t i o r i t h e r e d u c t s of t h e F ( n . ) t o t h e s i m i l a r i t y

t y p e of A a r e n o t isomorphic e i t h e r .

The h y p o t h e s i s i n Theorem 8 can be weakened t o s a y t h a t f o r some f i n i t e n, t h e r e i s an i n f i n i t e s e t of n - t u p l e s of e l e m e n t s of A which

i s l i n e a r l y ordered by some f i r s t - o r d e r formula $(;,?);

t h e same p r o o f works.

T h i s weaker h y p o t h e s i s i s e q u i v a l e n t t o s a y i n g t h a t t h e complete f i r s t - o r d e r t h e o r y of A i s u n s t a b l e i n S h e l a h ' s s e n s e .

(Cf. Shelah [551.

Using a n o t h e r

p r o o f , Shelah was a b l e t o weaken t h e h y p o t h e s i s s t i l l f u r t h e r and assume o n l y t h a t t h e t h e o r y of A i s u n s u p e r s t a b l e ;

c f . Shelah [531 o r Ch. VIII

of [551.)

S e v e r a l a p p l i c a t i o n s of Theorem 8 f a l l o u t of t h e f o l l o w i n g o b s e r v a t i o n of Baldwin and Lachlan [ 2 1 :

LEMMA 9 .

L e t $ ( x , y ) b e an e q u a t i o n and l e t A b e a s t r u c t u r e

w i t h e l e m e n t s a , b such t h a t J l ( a , a )

A

$(b,b)

A

$(a,b) A-r$(b,a) holds.

Then f o r e v e r y o r d i n a l a , t h e C a r t e s i a n power Aa c o n t a i n s a c h a i n of e l e m e n t s which i s o r d e r e d i n o r d e r - t y p e a by t h e formula - I $ ( Y . X ) .

219

Models built o n linear orderings For l e t a . ( i < a ) b e t h e element of Aa d e f i n e d by:

if k

2 i,

ai(k) = if k > i.

For example, l e t A be t h e symmetric group S3, l e t a and b be the cycles ( I x-2yx2 = y .

2 ) and ( I 2 3 ) r e s p e c t i v e l y , and l e t $ ( x , y ) be t h e e q u a t i o n Then t h e h y p o t h e s i s of Lemma 9 h o l d s .

From S h e l a h ' s theorem

we i n f e r t h a t e v e r y v a r i e t y of groups which c o n t a i n s S3 c o n t a i n s ZK p a i r w i s e non-isomorphic g r o u p s i n e a c h u n c o u n t a b l e c a r d i n a l i t y

K.

Baldwin and

McKenzie [ 3 1 u s e t h e same method t o show among o t h e r t h i n g s t h a t i f V i s a congruence-modular non-abelian v a r i e t y , t h e n i n e v e r y u n c o u n t a b l e c a r d i n a l i t y K,

V c o n t a i n s ZK p a i r w i s e non-isomorphic a l g e b r a s .

(Hodges [ 3 0 a ] s t r e n g t h e n s

this last result.) What happens i f w e want t h e 2'

s t r u c t u r e s i n Theorem 8 t o be

e q u i v a l e n t t o A i n some i n f i n i t a r y language?

The answer i s t h a t we u s e

e x a c t l y t h e same argument a s f o r Theorem 8 , b u t w i t h E x i s t e n c e Theorem 6 i n p l a c e of E x i s t e n c e Theorem 5.

The h y p o t h e s i s i n Theorem 8 h a s t o be

strengthened t o say t h a t t h e chain enough o r d e r - t y p e . an i n f i n i t a r y s e n s e ;

n is

well-ordered

by $ ( x , y ) i n some l o n g

(Such c h a i n s e x i s t i f t h e t h e o r y of A i s ' u n s t a b l e ' i n see Shelah 1511.)

M a c i n t y r e and S h e l a h [381 used t h e i n f i n i t a r y v e r s i o n of Theorem 8 t o answer a q u e s t i o n of Kegel and W e h r f r i t z [331 a b o u t t h e number of non-isomorphic u n i v e r s a l l o c a l l y f i n i t e g r o u p s i n a n u n c o u n t a b l e c a r d i n a l i t y (A group i s l o c a l l y f i n i t e i f f e v e r y f i n i t e l y g e n e r a t e d subgroup i s f i n i t e ;

K.

W.Hodges

220

i t i s u n i v e r s a l l o c a l l y f i n i t e i f f i t i s l o c a l l y f i n i t e , i t c o n t a i n s a copy

of every f i n i t e group, and e v e r y isomorphism between f i n i t e subgroups

e x t e n d s t o a n i n n e r automorphism of t h e whole g r o u p . ) They showed t h a t t h e number i s always Z K .

T h i s r e s u l t has somehow gained a

q u i t e undeserved r e p u t a t i o n f o r b e i n g o b s c u r e ( p r o b a b l y b e c a u s e t h e a u t h o r s r e f e r t o [511,

which i s a r e d h e r r i n g ) .

In f a c t t h e proof f a l l s s t r a i g h t o u t L e t me s k e t c h i t .

of r e s u l t s a l r e a d y used above, t o g e t h e r w i t h f a c t s i n [331.

The f i r s t s t e p i s t o check t h a t t h e c l a s s of u n i v e r s a l l o c a l l y f i n i t e groups i s a x i o m a t i s e d by a s i n g l e s e n t e n c e $ of L immediate from t h e d e f i n i t i o n .

f i n i t e and hence e v e r y power of i t i s l o c a l l y f i n i t e . ordinal

.

This i s I Next we n o t e t h a t t h e symmetric group S3 i s

(Iw s u f f i c e s , given t h e value

6(w)

= w

1

1

w w

L e t a b e some l a r g e

i n 13).

By Lemma 9 t h e

l o c a l l y f i n i t e group ( S )a c o n t a i n s a c h a i n rj of e l e m e n t s o r d e r e d i n t y p e a 3 by a q u a n t i f i e r - f r e e f o r m u l a .

Now (S )a c a n b e extended t o a u n i v e r s a l 3

l o c a l l y f i n i t e group G (by Theorem 6.5 i n [ 3 3 1 ) .

S i n c e G i s a model o f @ and

r~ is o r d e r e d by t h e same q u a n t i f i e r - f r e e formula i n G a s i n (S )a, E x i s t e n c e 3

Theorem 6 f i n d s a n EM f u n c t o r F j u s t a s r e q u i r e d i n t h e proof of Theorem 8 above.

The ' o u t p u t s t e p ' of t h a t proof g i v e s u s ZK p a i r w i s e non-isomorphic

models of $ i n e a c h u n c o u n t a b l e c a r d i n a l i t y

K.

A c t u a l l y most of t h i s proof h a s n o t h i n g t o do w i t h g r o u p s . A g e n e r a l s t a t e m e n t of t h e r e s u l t i s a s f o l l o w s .

L e t V be a v a r i e t y

such t h a t ( i ) V c o n t a i n s a f i n i t e a l g e b r a s a t i s f y i n g t h e h y p o t h e s i s of Lemma 9 f o r some e q u a t i o n J l ( x , y ) , and ( i i ) f o r any embeddings f : A g: A

-+

-+

B and

C between f i n i t e a l g e b r a s i n V, t h e r e i s a f i n i t e a l g e b r a D i n V

w i t h embeddings f ' : B + D and g ' : C uncountable c a r d i n a l i t y

K

+ .

D such t h a t f ' f = g ' g .

Then i n e v e r y

t h e r e a r e ZK p a i r w i s e non-isomorphic e x i s t e n t i a l l y

c l o s e d l o c a l l y f i n i t e V-algebras. Can w e e x t e n d Theorem 8 by making t h e ZK s t r u c t u r e s non-isomorphic i n some s t r o n g e r s e n s e ?

For example, c a n they b e chosen t o have o n l y a

small overlap with each o t h e r ?

Models built o n linear orderings For a l l u n c o u n t a b l e r e g u l a r t h e f a m i l y of Z K models of c a r d i n a l i t y embeddable i n any o t h e r .

S h e l a h [551 was a b l e t o choose

K,

K

22 1

s o t h a t none was e l e m e n t a r i l y

I n any r e g u l a r non-weakly-compact

cardinality

K,

Hodges [301 found two s t r u c t u r e s B, C e l e m e n t a r i l y e q u i v a l e n t t o any g i v e n i n f i n i t e A , such t h a t i f D i s e l e m e n t a r i l y embeddable i n b o t h B and C t h e n D c o n t a i n s no c h a i n of c a r d i n a l i t y

(This e x t e n d s E h r e n f e u c h t [ 1 7 1 . )

K

which i s o r d e r e d by a f i r s t - o r d e r f o r m u l a .

For some t h e o r i e s s u c h a s ZFC, one c a n

c o n s t r u c t p a i r s of a r b i t r a r i l y l a r g e models B , C such t h a t a n y t h i n g e l e m e n t a r i l y embeddable i n b o t h B and C i s c o u n t a b l e (Hodges [ 2 9 1 ) . used EM f u n c t o r s .

A l l these constructions

R e c e n t l y C h a r r e t t o n and Pouzet [ I 2 1 showed:

THEOREM 10.

(GCH) L e t T be a f i r s t - o r d e r t h e o r y whose models

a r e p a r t i a l l y o r d e r e d by a formula $, and suppose some model of T c o n t a i n s a n i n f i n i t e c h a i n which i s o r d e r e d by I$. Then i n e v e r y s u c c e s s o r c a r d i n a l i t y K,

T h a s a f a m i l y of 2 K models such t h a t any s t r u c t u r e e l e m e n t a r i l y

embeddable i n two of them h a s c a r d i n a l i t y <

K.

The proof o f Theorem 10 h a s two i n g r e d i e n t s . Bonnet's c o n s t r u c t i o n o f s u p e r - r i g i d

chains

61;

The f i r s t i s

given the generalised

continuum h y p o t h e s i s (GCH), t h i s p r o v i d e s i n e a c h s u c c e s s o r c a r d i n a l i t y

K

a

f a m i l y of 2K c h a i n s such t h a t any c h a i n which i s embeddable o r r e v e r s e embeddable i n two c h a i n s of t h e f a m i l y must have c a r d i n a l i t y <

K.

The second i n g r e d i e n t i s a lemma a b o u t how t h e o r d e r - t y p e of t h e s p i n e rl c o n t r o l s t h e o r d e r - t y p e s of d e f i n a b l e o r d e r i n g s i n F(rl), f o r any EM f u n c t o r F . published;

( I t was used a l s o i n t h e o r i g i n a l proof of [ 2 9 1 .

not

a p p a r e n t l y i t f i r s t appeared i n Arrow [ 1 1 , camouflaged a s a

theorem a b o u t s o c i a l w e l f a r e ! )

-

a l l elements t f r e e formula.

F(n)

L e t t ( x ) be a term, and suppose t h e s e t of

-

(a) with a increasing i n

n is

p r e - o r d e r e d by a q u a n t i f i e r

Then t h e o r d e r - t y p e of t h i s p r e - o r d e r i n g is embeddable i n a

f i n i t e l e x i c o g r a p h i c p r o d u c t of c o p i e s of

n and i t s r e v e r s e c h a i n

rl

*

.

W.Hodges

222

I n a l l t h e r e s u l t s of t h i s s e c t i o n , t h e i d e a was t o show t h a t i n c o m p a t i b i l i t i e s between c h a i n s rl c a r r y over t o i n c o m p a t i b i l i t i e s between the corresponding s t r u c t u r e s F ( Q ) . T

C h a r r e t t o n and Pouzet [ I 3

1

show t h a t i f

i s a c o u n t a b l e f i r s t - o r d e r t h e o r y w i t h t h e s t r i c t o r d e r p r o p e r t y (Shelah [551

p.69)

,

t h e n t h e e m b e d d a b i l i t y s t r u c t u r e of c h a i n s i s r e f l e c t e d i n models of T

i n t h e f o l l o w i n g s t r o n g s e n s e . For each u n c o u n t a b l e c h a i n ~l t h e r e i s a model E(n) of T , of t h e same c a r d i n a l i t y a s

rl,

such t h a t E(n) i s e l e m e n t a r i l y

embeddable i n E(S) i f and o n l y i f rl i s embeddable i n 6 .

5.

A p p l i c a t i o n s of t h e second k i n d :

o r d e r - t y p e s of models

The f o l l o w i n g r e s u l t of S i l v e r ( [ 5 6 1 , e x t e n d i n g e a r l i e r work of Gaifman and Rowbottom) r e v o l u t i o n i s e d s e t t h e o r y i n t h e l a t e 1960s:

THEOREM 1 1 .

Suppose t h e r e e x i s t s a m e a s u r a b l e c a r d i n a l .

Then

t h e r e i s a c l a s s of o r d i n a l s which i s a n i n d i s c e r n i b l e sequence i n G u d e l ' s c o n s t r u c t i b l e u n i v e r s e L , g e n e r a t e s L , and c o n t a i n s a l l ( r e a l - w o r l d ) uncountable c a r d i n a l s .

The c o n c l u s i o n of S i l v e r ' s theorem s a y s t h a t L i s a n e x t r e m e l y t h i n s l i c e of t h e r e a l world.

S i n c e a t l e a s t some ( g e n u i n e ) u n c o u n t a b l e

c a r d i n a l s are r e g u l a r i n L , t h e c o n c l u s i o n i m p l i e s t h a t t h e y a l l a r e ;

and

s i m i l a r l y a l l uncountable c a r d i n a l s a r e strong l i m i t c a r d i n a l s i n L.

Hence

t h e r e a l w 1 i s a s t r o n g l y i n a c c e s s i b l e c a r d i n a l i n L.

S i l v e r ' s theorem i s proved and d i s c u s s e d i n 530 of J e c h [321; s e e a l s o s e c t i o n 7.4 of Chang and K e i s l e r [I

11.

What S i l v e r showed i s t h a t

i f c e r t a i n ' E r d u s c a r d i n a l s ' e x i s t , t h e n t h e r e is a n EM f u n c t o r F such t h a t G u d e l ' s L i s of form F ( q ) where

i s t h e c h a i n of a l l o r d i n a l s , and

moreover t h e s p i n e I- a p p e a r s i n L a s a c l a s s of o r d i n a l s which i n c l u d e s a l l (genuine) uncountable c a r d i n a l s .

These o r d i n a l s a r e known a s t h e S i l v e r

223

Models built on linear orderings # i n d i s c e r n i b l e s , and t h e f i r s t - o r d e r p a r t o f t h e t h e o r y of F i s c a l l e d 0

.

Sometimes t h e c o n c l u s i o n of Theorem i 1 i s pronounced "0* e x i s t s " .

S i l v e r ' s proof f a l l s i n t o t h e format d e s c r i b e d a t t h e b e g i n n i n g of 54 above. formulas.

S i l v e r w r i t e s down what he c a l l s a ' r e m a r k a b l e ' s e t 0 o f

His Output s t e p c o n s i s t s i n showing (among o t h e r t h i n g s ) t h a t i f

F i s a n EM f u n c t o r whose t h e o r y c o n t a i n s 0, t h e n f o r e v e r y w e l l - o r d e r e d 'i,

F ( q ) has a d e f i n a b l e w e l l - o r d e r i n g which e x t e n d s t h a t on q .

chain

For h i s

I n p u t s t e p he u s e s a n ErdUs c a r d i n a l t o f i n d a n EM f u n c t o r F whose t h e o r y c o n t a i n s Q t o g e t h e r w i t h Zermelo-Fraenkel s e t t h e o r y and t h e axiom o f constructibility.

I n f a c t any argument which f i n d s such a f u n c t o r w i l l l e a d

t o t h e c o n c l u s i o n t h a t 0% e x i s t s .

So f o r example we now know t h a t i f some #

s i n g u l a r c a r d i n a l becomes r e g u l a r i n L , t h e n 0

e x i s t s (by J e n s e n ' s c o v e r i n g

lemma - t h e r e i s a modern proof i n Dodd [ 1 5 1 ) ;

and i f a n a l y t i c games a r e

determined t h e n 0* e x i s t s ( H a r r i n g t o n [ 2 6 1 ) .

See a l s o D e v l i n and P a r i s [ I 4 1

f o r a n o t h e r s u f f i c i e n t c o n d i t i o n i n terms of ' f r e e s u b s e t s ' of a n a l g e b r a .

P a r i s 1461, 147 1 s t u d i e d t h e S i l v e r i n d i s c e r n i b l e s .

For example

he examined how t h e y t h i n o u t i f we r e p l a c e L by L[xl f o r a r e a l number x .

S i l v e r ' s theorem i s one of a group of r e s u l t s a b o u t o r d e r - t y p e s of models.

L e t T be a f i r s t - o r d e r t h e o r y which i m p l i e s a s t a t e m e n t " $ ( x , y )

l i n e a r l y o r d e r s t h e s e t of a l l e l e m e n t s x such t h a t Px".

For i n s t a n c e T can

be s e t t h e o r y , s i n c e i t i m p l i e s t h a t t h e r e l a t i o n E l i n e a r l y o r d e r s t h e ordinals.

I f A i s a model of T , we can d e f i n e t h e o r d e r - t y p e of A t o b e

t h e $-order-type

of t h e e l e m e n t s of A s a t i s f y i n g Px.

We s a y t h a t t h e c h a i n 0 i s

K-%

i f f 0 has c a r d i n a l i t y

e v e r y p r o p e r i n i t i a l segment of q h a s c a r d i n a l i t y <

K.

K

and

Mac Dowel1 and Specker

[ 3 7 ] showed t h a t i f T h a s a model of o r d e r - t y p e w t h e n T h a s A-like models

W. Hodges

224 for a l l h

o.

T h e i r proof was ad hoc.

But t h e r e i s a proof u s i n g EM

f u n c t o r s , and t h i s proof was a d a p t e d t o y i e l d t h e f o l l o w i n g r e s u l t s . has a model which i s K-like f o r some weakly compact c a r d i n a l A-like models f o r a l l u n c o u n t a b l e c a r d i n a l s h ( H e l l i n g [ 2 7 1 ;

K

> o then T has

Schmerl and

Shelah [491 weaken t h e h y p o t h e s i s t o one i n v o l v i n g Mahlo c a r d i n a l s ) . has a model which i s K-like f o r some s t r o n g l i m i t c a r d i n a l

K

If T

If T

t h e n T has

A-like models f o r a l l s i n g u l a r c a r d i n a l s A ( K e i s l e r [ 3 4 1 ) .

There a r e f u r t h e r

r e s u l t s of t h e same s o r t i n Schmerl [ 4 8 a l and S h e l a h [ 5 2 1 .

Fuhrken [ 2 4 1

showed how t o u s e such r e s u l t s t o g e t i n f o r m a t i o n about t h e q u a n t i f i e r s “There e x i s t a t l e a s t h e l e m e n t s such t h a t

. . . ‘ I .

S i l v e r [ 5 6 ] and Hodges [ 2 9 1 u s e EM f u n c t o r s t o c o n s t r u c t models w i t h a bound on t h e l e n g t h of d e s c e n d i n g sequences i n t h e o r d e r - t y p e .

A l l t h e s e r e s u l t s were proved i n t h e format g i v e n a t t h e

b e g i n n i n g of 54 above.

When T i s s e t t h e o r y o r number t h e o r y , t h e p a r t i t i o n

theorem used i n t h e I n p u t s t e p may i t s e l f b e a theorem of T, so t h a t we c a n c o n s t r u c t t h e EM f u n c t o r ‘ i n s i d e ’ a model of T.

T h i s t r i c k c a n be

used t o g e t v a r i o u s k i n d s of e n d - e x t e n s i o n of models and o t h e r i n f o r m a t i o n about o r d e r - t y p e s .

See f o r example K e i s l e r and Morley [ 3 5 1 f o r some

set-theoretic applications.

Gaifman and o t h e r s used t h i s d e v i c e t o s t u d y

m o d e l s of a r i t h m e t i c ( e . g . P a r i s [ 4 5 1 ) .

There i s a l a r g e r e c e n t l i t e r a t u r e

on p a r t i t i o n theorems and models of a r i t h m e t i c . b u t I a m n o t competent t o say which of t h e r e s u l t s c a n u s e f u l l y be p u t i n terms of EM f u n c t o r s .

6.

A p p l i c a t i o n s of t h e t h i r d k i n d :

upward Luwenheim-Skolem theorem6

The a p p l i c a t i o n s i n t h i s s e c t i o n a r e about c o n s t r u c t i n g b i g models.

For example, h e r e i s t h e upward Luwenheim-Skolem theorem f o r t h e

logic L + : K W

225

Models built on linear orderings L e t $ be a s e n t e n c e of L +

THEOREM 1 2 . card n a l i t y L

K

1 ( 2 ~( )o r+even 2

where

&(K)

W

w i t h a model A of

i s a s i n Theorem 6 ) .

Then $ has a r b i t r a r i l y l a r g e models.

For t h i s , well-order

resulting chain. $

E Th(F).

dom(A) i n any way and l e t rl be t h e

By E x i s t e n c e Theorem 6 t h e r e i s a n EM f u n c t o r F s u c h t h a t

Then f o r e a c h c a r d i n a l A , F(X) i s a model of $ w i t h c a r d i n a l i t y

a t least A.

Morley [431 a d a p t e d t h i s proof t o g i v e a n a n a l o g o u s r e s u l t f o r s t r u c t u r e s which have a d e f i n a b l e p a r t isomorphic t o a g i v e n s t r u c t u r e . Flum 121 1 a d a p t e d i t t o o t h e r l a n g u a g e s .

I f V i s a v a r i e t y , t h e n t h e c l a s s of s u b d i r e c t l y i r r e d u c i b l e a l g e b r a s i n V and t h e c l a s s of s i m p l e a l g e b r a s i n V a r e e a c h d e f i n e d by

.

Walter T a y l o r showed t h a t i f V c o n t a i n s a I s u b d i r e c t l y i r r e d u c i b l e a l g e b r a of c a r d i n a l i t y > 2w t h e n i t c o n t a i n s s i n g l e s e n t e n c e s of L

w w

a b r i t r a r i l y l a r g e s u b d i r e c t l y i r r e d u c i b l e a l g e b r a s (158

1, see

a l s o [59

1).

McKenzie and S h e l a h 1391 a d a p t e d T a y l o r ’ s argument t o p r o v e t h e same f o r simple algebras.

T a y l o r , McKenzie and S h e l a h d i d n ’ t u s e EM f u n c t o r s , b u t

t h e y came c l o s e t o them.

The f o l l o w i n g r e s u l t was f i r s t proved by Vaught [ 6 0 1 w i t h o u t t h e a i d of EM f u n c t o r s o r a n y t h i n g l i k e them.

We d e f i n e

1 (K)

inductively

I n ( K )

by:

10( = K, ~l )n + = 2 ](~)

THEOREM 13.

L e t T be a f i r s t - o r d e r t h e o r y whose l a n g u a g e Suppose t h a t f o r e a c h n

i n c l u d e s two I-ary r e l a t i o n symbols P and Q. T h a s a model A

i n f i n i t e cardinals A

K

I

1)

Then f o r any two > In(lQA + w. n n t h e r e i s a model B of T w i t h IP = h and B

such t h a t lP,

w,

I

IQBI

=

K.

226

W. Hodges To prove t h i s by Ehrenfeucht-Mostowski methods (Morley [ 4 1

we f i r s t have t o p u t a n e x t r a t w i s t i n t o E x i s t e n c e Theorem 5.

I),

I t i s not

n e c e s s a r y t o have a s i n g l e s t r u c t u r e A c o n t a i n i n g rl a s i n t h a t theorem. compactness theorem l e t s u s g e t by w i t h a f a m i l y (A contains a chain n

s u c h t h a t each A

: n < w)

of o r d e r - t y p e ) n ;

The

of s t r u c t u r e s

the conclusion

i s t h a t t h e r e e x i s t s a n EM f u n c t o r whose t h e o r y c o n t a i n s a l l f o r m u l a s

...,x

) s a t i s f i e d by e v e r y i n c r e a s i n g t u p l e i n rl

@(xl,

Theorem 6 has a s i m i l a r s t r e n g t h e n i n g : (Ai : i <

1

f o r a l l l a r g e enough n .

i n p l a c e of A we have a sequence

) of s t r u c t u r e s , and e a c h A . c o n t a i n s a c h a i n

n.

of l e n g t h

6(K)

> i. -

Using t h i s s t r e n g t h e n e d form of Theorem 5, t h e proof of Theorem 13 goes a s f o l l o w s . well-ordered

(Input step:)

i n some a r b i t r a r y way.

Rado we can f i n d i n e a c h A length

Think of e a c h A

lQA I+,

n formulas $ ( x i ,

i n t h e theorem a s b e i n g

Using a p a r t i t i o n theorem of Erd8s and

a chain q

of elements s a t i s f y i n g Px. of

which i s $ - i n d i s c e r n i b l e f o r a l l n - v a r i a b l e f i r s t - o r d e r

...,x

),

2

Then f o r e v e r y n

2m, a l l formulas o f t h e

f o l l o w i n g form a r e s a t i s f i e d by a l l i n c r e a s i n g 2m-tuples from r~ :

For o t h e r w i s e some formula ( I ) f a i l s on a l l i n c r e a s i n g 2m-tuples from rl

.

I+ c o n s e c u t i v e m-tuples; f o r e a c h of t h e s e n f; a r e two of t h e m-tuples then t m-tuples l i e s i n Q , but i f An An I+ d i s t i n c t elements - which i s i m p o s s i b l e t A (2) # tA ( b ) , so t h a t Q, has n n n An Now l e t @ be t h e s e t c o n s i s t i n g of T , Px and a l l t h e f o r m u l a s ( I ) . By Then we can chop

a,

n

i n t o IQ,

a,

(a)

IQ

1

t h e s t r e n g t h e n e d E x i s t e n c e Theorem 5 , t h e r e i s an EM f u n c t o r F w i t h @ (Output s t e p : )

I f h i s any i n f i n i t e c a r d i n a l , then F(X) i s a s t r u c t u r e of

c a r d i n a l i t y h whose s p i n e c o n s i s t s of elements s a t i s f y i n g Px; I P F ( h ) I = A.

Th(F).

so

But by ( I ) , f o r e a c h term t ( G ) , e i t h e r a l l t h e e l e m e n t s

221

Models built on linear orderings

-

-

t F ( A ) ( a ) w i t h a i n c r e a s i n g i n A a r e e q u a l t o e a c h o t h e r , o r none of them a r e in Q F(1)'

So Q F ( A ) h a s no more e l e m e n t s t h a n t h e number of terms of t h e

s i m i l a r i t y t y p e , and hence i t i s c o u n t a b l e . when

K

1 Q F(A)l

The argument t o g e t

=

> w i s s i m i l a r b u t more c o m p l i c a t e d .

S h e l a h 1521 proved some s t r e n g t h e n i n g s of Theorem 13. example he b u i l t s t r u c t u r e s B w i t h IP

B

I

=

A,

I QB I

= K

For

u

and ( R B I =

whenever

A ' K , L J L W .

7.

Sporadic a p p l i c a t i o n s

( I ) Automorphisms.

E h r e n f e u c h t and Mostowski [ I 9 1 a d v e r t i s e d

t h e i r c o n s t r u c t i o n a s a way of f i n d i n g models w i t h many automorphisms. Ebbinghaus [ I 6 1 combined EM f u n c t o r s w i t h some o t h e r t r i c k s t o g e t s t r u c t u r e s which have many automorphisms and a r e models of s e n t e n c e s w i t h q u a n t i f i e r s "There a r e a t l e a s t A e l e m e n t s such t h a t (2) R e a l i s i n g few t y p e s .

. . . ' I .

of a n element b of a

The 0-

s t r u c t u r e A i s t h e s e t of f o r m u l a s $ ( X I E 0 s u c h t h a t A

1$(b).

We s a y t h a t

A r e a l i s e s t h e @-type p i f f p i s t h e @-type of some element of A .

Suppose F i s a n EM f u n c t o r , rl i s a c h a i n and t ( x ) i s a term. Then Lemma I s a y s t h a t f o r any two i n c r e a s i n g n - t u p l e s e l e m e n t s t F ( r l ) ( a ) and t

F(rl)

a,

b

from n , t h e

( b ) have t h e same q u a n t i f i e r - f r e e t y p e .

We a r e

s t i l l assuming t h a t t h e s i m i l a r i t y t y p e of F ( n ) i s c o u n t a b l e , so t h a t t h e r e a r e j u s t c o u n t a b l y many d i f f e r e n t terms t

THEOREM 1 4 .

of c a r d i n a l i t y

K.

(x).

Hence:

L e t F b e a n EM f u n c t o r and rl a n i n f i n i t e c h a i n

Then:

(a)

F ( n ) r e a l i s e s o n l y c o u n t a b l y many q u a n t i f i e r - f r e e

(b)

For e v e r y 1-ary r e l a t i o n symbol P , t h e number of e l e m e n t s of F(n) s a t i s f y i n g Px i s e i t h e r

zw

or =

K.

types.

W.Hodges

228

When Th(F) c o n t a i n s t h e a p p r o p r i a t e Skolem t h e o r y Z a s i n Lemma 4 , we can replace 'quantifier-free' formula Ji (x)

by ' f i r s t - o r d e r ' ,

and ' P x ' by any f i r s t - o r d e r

. E h r e n f e u c h t [I81 n o t i c e d t h a t p a r t ( a ) of t h i s theorem i s

r e l e v a n t t o t h e c a t e g o r i c i t y problem f o r f i r s t - o r d e r t h e o r i e s .

In f a c t

Morley's improvements of t h i s r e s u l t played a v i t a l p a r t i n h i s s o l u t i o n of Lo6's problem [ 4 0 1 , and they l a i d t h e f o u n d a t i o n s of s t a b i l i t y t h e o r y . Rowbottom [48] adapted p a r t (b) of t h e theorem t o show t h a t i f t h e r e i s a measurable c a r d i n a l , t h e n f o r e v e r y c a r d i n a l X c o n s t r u c t i b l e s u b s e t s of A .

(3)

Heirs.

and P o i z a t [361.

t h e r e a r e a t most X

T h i s preceded S i l v e r ' s r e s u l t , Theorem 1 1 above

T h i s a p p l i c a t i o n p r e s u p p o s e s knowledge o f L a s c a r

These a u t h o r s i n t r o d u c e t h e n o t i o n o f t h e

They show (Theorem 4 . 9 of p a i r of models M -4 over N.

'> w

heir of

a type.

[361) t h a t i f T i s a s t a b l e t h e o r y t h e n f o r e v e r y

N and e v e r y t y p e p over M ,

t h e r e i s a u n i q u e h e i r of p

The f o l l o w i n g argument p r o v e s t h e c o n v e r s e :

i f h e i r s a r e unique

then T i s s t a b l e .

Assume T i s u n s t a b l e , so t h a t some formula $ ( x , y ) l i n e a r l y o r d e r s a n i n f i n i t e c h a i n of n - t u p l e s i n some model. be 1 ( t h e proof a d a p t s t o t h e c a s e n > I ) . EM f u n c t o r F w i t h T

u

{$(xl,x2)

Skolem t h e o r y from Lemma 4 .

of t h e Skolem t h e o r y 2,

A

For s i m p l i c i t y l e t n

By E x i s t e n c e Theorem 5 t h e r e i s a n

z

- I $ ( X ~ , X ~u ) } 5 Th(F), where C i s t h e

L e t q be t h e c h a i n w+{a}+{b}+{c}+w*.

* i n c l u s i o n F(w+w )

the

+

F(n) i s elementary.

Because

Let p

*

be t h e type of b o v e r F(w+w ) , and l e t q , r be r e s p e c t i v e l y t h e t y p e s of a , c over F (w+{b)+w*). F(q)

F(q)

b X(c,t(7,b,46*))

Then q , r a r e b o t h h e i r s of p: t h e n by s l i d i n g , F ( n )

x(b,t(7,&3,46*)) by a second s l i d e .

s i n c e Jl(x,b) i s i n q b u t n o t i n r .

f o r example i f

1 x(c,t(7,8,46*)),

and s o

However, q and r a r e d i s t i n c t

229

Models built o n linear orderings A s i m i l a r proof shows t h a t i f

t h e formula

has the order

@

p r o p e r t y then t h e r e a r e u n d e f i n a b l e $-types o v e r models. r e s u l t of S h e l a h - s e e f o r example Theorem 1 1 . 2 . 2 i n [ 5 5 ] .

T h i s i s a well-known But t h e proof

by EM f u n c t o r s has one d i s t i n c t a e s t h e t i c a d v a n t a g e over t h e p r o o f s i n p r i n t , namely t h i s .

The p u b l i s h e d p r o o f s u s e c a l c u l a t i o n s w i t h u n c o u n t a b l e

c a r d i n a l s i n o r d e r t o prove a s t a t e m e n t a b o u t c o u n t a b l e t h e o r i e s ;

t h e proof

by EM f u n c t o r s u s e s o n l y e l e m e n t a r y arguments and t h e compactness theorem f o r countable logic.

(4) P e r m u t a t i o n g r o u p s . the s e t X.

L e t G be a group of p e r m u t a t i o n s of

We s a y t h a t G i s k - s e t - t r a n s i t i v e

i f f f o r any two k-element

s u b s e t s of X, t h e r e i s a n element o f G which t a k e s t h e one s u b s e t t o t h e o t h e r .

THEOREM 15.

Let m

2

5, and l e t t h e p e r m u t a t i o n group G on t h e

c o u n t a b l e s e t X be k - s e t - t r a n s i t i v e

f o r some k 2 2 m - 2 ,

b u t not m - t r a n s i t i v e .

Then t h e r e i s a l i n e a r o r d e r i n g < of X s u c h t h a t e v e r y element of G p r e s e r v e s t h e r e l a t i o n “ e x a c t l y one of a , c i s i n t h e i n t e r v a l from b t o d i n t h e <-ordering”.

( E q u i v a l e n t l y , e v e r y element of G which i s n o t t a e i d e n t i t y

e i t h e r r e v e r s e s t h e o r d e r i n g , o r t a k e s a n i n i t i a l segment t o t h e e n d , o r b o t h . )

With a s t r o n g e r h y p o t h e s i s t h i s was proved by Cameron [ 7 1 i n d e p e n d e n t l y of Theorem 7 above.

Then s e v e r a l p e o p l e saw t h e c o n n e c t i o n ,

and Theorem 15 i s now a c o r o l l a r y of Theorem 7 .

1

81;

(Cf. Higman [ 2 8 1 , Cameron

t h e l a t t e r p a p e r h a s i n f o r m a t i o n a b o u t t h e c a s e m < 5.)

W.Hodges

230 REFERENCES

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Annals of Discrete Mathematics 23 (1984) 235-268 235

0 Elsevier Science Publishers B.V. (North-Holland)

CHAINABLE RELATIONS, R A N G E M E N T S AND P S E U D O - R A N G E M E N T S Claude Frasnay L a b o r a t o i r e d'Alg6bre et C o m b i n a t o i r e Universiti? P a u l - S a b a t i e r , T o u l o u s e France

F r o m 1963 to 1 9 8 3 , s o m e f i f t e e n n o t e s a n d p a p e r s w e r e p u b l i s h e d a b o u t the theory o f c h a i n a b l e o r (more g e n e r a l l y ) m o n o m o r p h i c r e l a tions. T h e s e r e l a t i o n s p r o v i d e d L o g i c (and C o m b i n a t o r i c ) w i t h t h e f i r s t n o n - t r i v i a l e x a m p l e s of u n i v e r s a l c l a s s e s i n the s e n s e used (since 1953-54) by T A R S K I - V A U G H T a n d F R A T S S E . F o r e a c h n < w , a n m - a r y r e l a t i o n r (or a r e l a t i o n a l s y s t e m ) . ) i s c a l l e d ~ - t n u n o m o k p h i c if a l l its n - r e s t r i c t i o n s r l X i ~ E I (for IXl=n) a r e i s o m o r p h i c (and r is c a l l e d mononiofiphic i f r is

r=(r

n - m o n o m o r p h i c f o r a l l n
I f t is a l i n e a r o r d e r (or C h a i n ) and if

is a m o n o m o r p h i c b i r e l a t i o n , the m-ary r e l a t i o n r is c a l l e d

c h a i ~ n b f e (by t).

group S

m'

F o r i n s t a n c e , if G is a s u b g r o u p o f t h e s y m m e t r i c

a n e x a m p l e f o r r is the G - f i a n g c m e n t s a t i s f i e d b y t h e m-

u p l e s (xo(o)'xo(])' O€G.

)

.'*'Xa(m-])

so

. . .< xm- I

that x < x I <

( m o d . t)

and

F o r e a c h s u b g r o u p G o f S , w e h a v e p r o v e d that : "the c l a s s o f m

C - r a n g e m e n t s is u n i v e r s a l " . T h i s r e s u l t i s e q u i v a l e n t on o n e s i d e to o u r " G - f i e c u f f e m e n t t h e o r e m " ( 1 9 6 3 , a b o u t G-compatibLt c h a i n s ) a n d o n the o t h e r s i d e to t h e fact that t h e r e e x i s t s a m i n i m a l f i n i t e c a r d i n a l i t y v(G)

(which w e c a l l t h e w a f e n c e o f G) s u c h t h a t : " f o r n>,v(G)

and a n m-ary r e l a t i o n r o f c a r d i n a l i t y

> n , if a l l n - r e s t r i c t i o n s

r ( X a r e G - r a n g e m e n t s , t h e n r is a G - r a n g e m e n t " . If v m is t h e m a x i m a l v a l e n c e of t h e s u b g r o u p s G of S (first v a l u e s : v = I , v = 3 , v = 4 , m 1 2 3 2 v =6), w e h a v e p r o v e d i n 1963 : v m 6 9 m . I n t h i s p u r p o s e , w e u s e d 4 l o c a l a u t o m o r p h i s m s of the G - r a n g e m e n t r c h a i n e d by the n a t u r a l -f

c h a i n w : i f G(')=Aut(r(Xn)

with X = [ O , l , .

and (for a n i n d i c a t i v e c f a h h y ( G )

. . ,m+n-1 ,

a n d n.?y(G))

G

then G ( O ) = G (n) is a w e l l k n o w n

i n d i c a t i v e g f i u u p H m + n (of t y p e H = s , T , D , I ~ ' ~o r J P ) f o r w h i c h t h e n a t u r a l c h a i n plays a s p e c i a l role. F o r the m a x i m a i n d i c a t i v e c l a s s y m , w e o b t a i n e d v m 6 m + l + ym 6 9 m

2

.

236

C. Frasnay

In t h i s paper,

using a sharp result of

( 1 9 7 6 ) w h i c h g i v e s y hm-3 We a l s o r e c a l l

properties, a)

If

f o r a l l m>5,

(without proofs,

for instance

>vm, and

if

r e l a t i o n s which have

(But,

b)

are

isomorphic

: then

m

chainable.

m-ary

and any two n - r e s t r i c t i o n s

relation of

p >,v

m

such t h a t ,

m

r e l a t i o n of

r e l a t i o n s of

infinite cardinalities).

1967) : t h e class

o f monomorphic m-ary

fm

r e l a t i o n s are universal.

c h a i n a b l e m-ary

The f a m i l y o f

the

semi-lattice

the involution

indicative subgroups of

by t h e c y c l e ( 0 ,

1,

(1,

...,

permuting respectively 0,

m-2)

m-1)

),

1,

...,

The o t h e r s i n d i c a t i v e g r o u p s a r e : ( = S

m

if

p + q

> m)

and

(1954)

Corollary

(by

r e l a t i o n s and

the

is the smallest

1

...

1, Tm ( o f o r d e r m , g e n e r a t e d

I P T 1a n d

1'"

(of

p-1

and

m-p,

m-p+l,

1

Tm ,

Dm

=

J p = J A 1'"

m

S,"

a

Jm (of o r d e r 2 , g e n e r a t e d by

t o which belong:

(0, m-I)

rlX,

c a r d i n a l i t y 2pm

class

upper

and

in

r is

if

theorem

JEAN,

of

S

then r is

c a r d i n a l i t y )p,,

( s t r o n g e r t h a n t h e i n i t i a l FRAISSE's

on monomorphic

isomor-

relations r, s, of cardinality

Hence, a n y m o n o m o r p h i c m-ary

is chainable

are

sly

s i s c a l l e d a maximal IJ4~eudU-(;-~arigYtiiC111).

There e x i s t s a f i n i t e cardinality

v -monomorphic

t h e same f i n i t e

rlX,

f o r each subgroup G of

t h e r e e x i s t s non-isomorphic

n + l , a l t h o u g h r i s a G-rangement sly

f o r a l l ma3.

m

a n y two v - r e s t r i r t i o n s

then r, s a r e isomorphic

phic,

n+l=v(G),

we p r o v e v = 2 m - 2

r e f e r e n c e s o n l y ) many o t h e r s

:

s a r e c h a i n a b l e m-ary

r,

cardinality

but

HODGES-LACHLAN-SHELAH

m

m

Jm

.

IP,(I m

o r d e r p!

=

,

...,

m-1).

IP91

11*(1

m

m

Relations enchaina bles, rangements et pseudo-rangements

237

KELATIONS E N C H A ~ N A B L E S , RANGEMENTS ET PSEUDO-RANGEMENTS Claude Frasnay L a b o r a t o i r e d'Alg6bre et C o m b i n a t o i r e Universit6 Paul-Sabatier, Toulouse France

I 1 n o u s est a g r 6 a b l e d e p r g s e n t e r c e t e x p o s 6 c o m m e u n cordial hommage au Professeur E. COKOMINAS : ses recherches et celles qu'il a i n s p i r s e s ( n o t a m m e n t s u r le b e l o r d r e et le m e i l l e u r o r d r e ) o n t contribuii 2 f a i r e d u L a b o r a t o i r e d ' A l g S b r e o r d i n a l e ( U n i v e r s i t g C l a u d e - B e r n a r d , Lyon) le p i v o t d'une r 6 c e n t e R.C.P. ( * )

.

" E n s e m b l e s ordonniis et a p p l i c a t i o n s " .

0.

INTRODUCTION. Le cara2tSi-e c o m b i n a t o i r e d s l'expos6

qui suit est assez

6 v i d e n t . D e s g r o u p e s H m d e p e r m u t a t i o n s (de degriis f i n i s m) s o n t e n g e n d r 6 s par la c o m p a r a i s o n l o c a l e d'ordres t o t a u x (ou chainen)s,t de m@me s u p p o r t i n f i n i E e t , de c e t t e f a G o n , o n o b t i e n t d e s s u i t e s

H

=

(Hm)mco

d e g r o u p e s c o m p l g t e m e n t r 6 p e r t o r i G s d e p u i s 1963 : l e s

T m , D . P a r a i l l e u r s , l'6tude d e s m relations m-aires dont la morphologie est la p l u s simple ( 2 savoir

g t u u p e n i n d i c a t i d s S m , I P > q, J:,

les r e l a t i o n s monomotphen p a r m i l e s q u e l l e s f i g u r e n t l e s r e l a t i o n s l i b r e - i n t e r p r g t a b l e s p a r une c h a i n e , d i t e s r e l a t i o n s e n c h a i n a b l e n ) fait apparaItre des nombres

TT m ,

dm, vm, ym, 6

m

dont le r3le combi-

n a t o i r e c o m m e s e u i l s f i n i s e s t a n a l o g u e 5 c e l u i d e s n u m b t r ~ 6 de R A M S E Y pL(n). 0.1.

L a p r e m i s r e p a r t i e d e l'expos6

dgveloppe un fragment de la

th6orie des relations dont quelques notions g6nGrales (teAt4ictiOK et i n u m u t p h i e , i n t e t p k P t a b i L i t G , d Z d u t m a b i l i t Z ,

clanne uniwehnelee

de relations) s u f f i s e n t p o u r G t u d i e r les r e l a t i o n s m o n o m o r p h e s et les r e l a t i o n s e n c h a l n a b l e s . a) En u t i l i s a n t l ' u l t r a f i l t r a t i o n e t le t h 6 o r S m e de R A M S E Y , ~

~

(*) Recherche coop6rative sur programme n o 698, associge au C.N.R.S., 2

laquelle contribue la pr6sente publication.

C. Frasnay

238

R. FRAYSS;

(1954,

p o u r les c a r d i n a l i t E s infinies) p u i s nous-msme

(1964, pour les c a r d i n a l i t s s finies) a v i o n s montrE q u e "toute r e l a t i o n m-aire m o n o m o r p h e d e c a r d i n a l i t s 2n

est e n c h a P n a b l e " . m P l u s f o r t e m e n t , n o t r e t h s s e [ I 1 1 mettait e n G v i d e n c e u n e waeeklce

m-aice maximal!e v

( c a r d i n a l i t s m a x i m u m d e s r e l a t i o n s m-aires finies m e n c h a P n a b l e s et d G f o r m a b l e s , n o t a m m e n t v 1 = I , v 2 = 3 , v 3 = 4 , v 4 =6) e t un d e g t i P o p t i m a l ! d m d e m o n o m o h p h i e m-u4tie (notamment d l = l , d 2 = 3 , d = 4 , et m+l$d 3

'm

< v <9m2 pour m > 4 ) . m

b) D a n s la lign6e d e s t r a v a u x s u r l e b e l o r d r e i n s p i r E s par E. C O R O M I N A S , n o u s E v o q u o n s E g a l e m e n t d a n s cette p r e m i s r e p a r t i e la d 6 c o u v e r t e par M . P O U Z E T (1972) d'une c l a s s e

%

(de R t ? h

betees

s t a b l e p a r l i b r e - i n t e r p r E t a b i l i t 6 , c o n t e n a n t la c l a s s e

kc?l!Utionb)

des cha'ines (donc a u s s i la c l a s s e des r e l a t i o n s encha'inables) e t t e l l e q u e : les r e s t r i c t i o n s f i n i e s d g f o r m a b l e s d ' u n e r e l a t i o n r e

o n t u n e c a r d i n a l i t 6 m a x i m u m v(r)

l'existence d e n o s s e u i l s v parcourt la classe

- c e qui permet d e r e t r o u v e r

( m a x i m u m d e s uai?encen v(r)

tm des m r e l a t i o n s m - a i r e s

lorsque r

enchaPnables), m a i s

q u i n e d o n n e a u c u n e i n f o r m a t i o n s u r l e u r v a l e u r . I 1 n'en r e s t e pas m o i n s qu'on d i s p o s e de d e u x m E t h o d e s tres d i f f g r e n t e s ( [ I l l ,

1201)

g a r a n t i s s a n t le r 6 s u l t a t c i - a p r S s (premier t h E o r G m e n o n trivial d'indP6otrnabiLitP o u de k e c v n 6 , t h u c t i o n i s o m o r p h i q u e e n t h E o r i e d e s r e l a t i o n s - p o u r l e q u e l il est c o m m o d e d'adopter l a t e r m i n o l o g i e s u i v a n t e : d e u x r e l a t i o n s m-aires m o n o m o r p h e s r e t s de c a r d i n a l i t 6 a n sont dites

l!ocaLement n - i b o r n o h p h e b l o r s q u e leurs n - r e s t r i c t i o n s

r l X , s l y s o n t isomorphes) : "Si d e u x r e l a t i o n s m-aires e n c h a P n a b l e s , de msme c a r d i n a l i t 6 f i n i e >,v,,

sont localement v -isomorphes, alors m

ces d e u x r e l a t i o n s s o n t i s o m o r p h e s " . 0.2.

L a s e c o n d e p a r t i e de l'expos6, c o n s a c r E e 2 la t h 6 o r i e d e s

c h a 7 n e s p e r m u t 6 e s e t a u x g h o u p e 6 d ' a u t o m o k p h i n m e n l!ocaux d e s relations e n c h a y n a b l e s , v a c o n d u i r e p r o g r e s s i v e m e n t a u c a l c u l d e s nombres v

m

.

a) E n p r e m i e r l i e u , n o u s f a i s o n s i n t e r v e n i r des r e l a t i o n s m-aires e n c h a P n a b l e s p a r t i c u l i s r e s a p p e l 6 e s G-kaRgtrnentb (pour la s i m p l e r a i s o n qu'une

t e l l e r e l a t i o n e s t s u s c i t 6 e p a r u n e c h a y n e t et r e l i 6 e

5 un s o u s - g r o u p e G d u g r o u p e s y m s t r i q u e S ) , a i n s i q u e l e u r s d g f o r m m a t i o n s a p p e l E e s pbeud0-G-hangernent6. P a r e x e m p l e , e n n o t a n t b r i s v e -

Relations enchaina bles, rangements et pseudo-rangements m e n t (abc)=

{(a,b,c),

(b,c,a),

239

(c,a,b)j, o n peut d 6 f i n i r d e u x r e l a -

tions t e r n a i r e s r et s de s u p p o r t { O ,

I ,

2 , 3 1 par leurs e n s e m b l e s

c a r a c t c r i s t i q u e s (123) u (023) I: ( 0 1 3 ) u ( 0 1 2 ) et (321) u (023)

L)

(310) ~ ( 0 1 2 ) :

d2s lors (pour lc s o u s - g r o u p e i n d i c a t i f T 3 de S ) r e s t u n T 3 3 + r a n g e m e n t et s e s t u n p s e u d o - T 3 - r a n g e m e n t . La chai'ne u s u e l l e o (dont le s u p p o r t est l'ensemble w d e s n a t u r e l s ) s u s c i t e des G - r a n g e m e n t s " n a t u r e l s " G dont la v a l e n c e peut s t r e n o t 6 e s i m p l e m e n t v(G).

En

1 9 7 2 - 7 3 , l e s t r a v a u x d e D . CLARK et P . KRAUSS d'une p a r t , d e M . P O U Z E T d'autre p a r t , d o n n a n t u n r a l e m a j e u r 2 l a s e u l e e x i s t e n c e (maximum des v a l e n c e s de t 0 n f e . s les r e l a t i o n s m-aires m enchaTnables), n o u s c o n d u i s e n t 5 r e v o i r e n c e s e n s l a t e c h n i q u e d e

des nombres v

hecaeYemehif d e s G - r a n g e m e n t s c o m p a t i b l e s

( o u de

G - 4 e c o L l e m e n t des

c h a r n e s G - c u r n p U t i b l ? c h q u i l e s suscitent) G l a b o r c e e n 1 9 6 3 - 6 4 ,

et B

est a u s s i la v a l e n c e m a x i m u m d e s A c u e b r a n g e m e n t s m m-aires. L a r 6 u s s i t e d e c e t t e t e c h n i q u e t i e n t a u f a i t q u e (pour tout

montrer que v sous-groupe

que

G

de Sm) : "la c l a s s e d e s G - r a n g e m e n t s e s t u n i v e r s e l l e "

plus pr&cisZment,

v(G)

est le plus petit des naturels n tels

"si t o u t e s l e s n - r e s t r i c t i o n s d'une r e l a t i o n m - a i r e r

:

> n ) sont des G-rangements, alors r est un

(de c a r d i n a l i t 6 G-rangement".

A s s e z c u r i e u s e m e n t , c e t t e m u l t i t u d e de c l a s s e s uni-

v e r s e l l e s n o n t r i v i a l e s (li6e a u x g r o u p e s de p e r m u t a t i o n s ) est r e s t 6 e m c c o n n u e jusqu'en

1976 - p a r c o n t r e , e n u t i l i s a n t le d e g r 6

d e m o n o m o r p h i e m - a i r e , M. J E A N a v a i t d E m o n t r 4 d 2 s 1967 m que : "la classe d e s r e l a t i o n s m-aires m o n o m o r p h e s et l a classe

optimal d

d e s r e l a t i o n s m - a i r e s enchai'nables s o n t d e u x c l a s s e s u n i v e r -

s e 11 es" b)

.

D6signons par

support w )

rm

l'ensemble

(fini) d e s r e l a t i o n s m - a i r e s (de

libre-interpr6tables par la charne usuelle

p o u r toute r e l a t i o n r e

t i o n (C(n))ne

r m'

d . En

1964-65,

nous avions envisag6 la suite de d i e a t a -

d e r d a n s l a q u e l l e G(n)

d 6 s i g n e le s o u s - g r o u p e d e

Sm+n c o n s t i t u 6 p a r les a u t o m o r p h i s m e s d e la r e s t r i c t i o n rl (m+n) support { O , I , 2 ,

. . . , m+n-l]).

r = 6 p o u r l e q u e l G(O)=G

(de

D a n s le c a s p a r t i c u l i e r d u G - r a n g e m e n t

(et e n u t i l i s a n t l e s u p . d e m i - t r e i l l i s d e s

s o u s - g r o u p e s i n d i c a t i f s d e Sm' c e q u i a t t r i b u e 1 G une d i c h e i n d i c a t i u i ? H d e t y p e S , I p t q , J p , T , D t e l l e q u e Hm s o i t l e g r o u p e i n d i catif maximum contenu dans G ) ,

c l a o b e i n d i c a t i v e y(G)

nous avions mis e n Svidence une

a u s e n s s u i v a n t : le g h o u p e d i P a t P G(n)

n o n i n d i c a t i f p o u r n
p a r c o n t r e G(n)

=

est

Hm+n p o u r t o u t n > , y ( G )

2 40

C. Frasnay

(de sorte que y(G)=O

caractgrise G comme groupe indicatif).

dehors d u groupe indicatif .J:

En

(bien connu comme groupe d'invariance

+ x )=2(x 0 x 1 + x 2 x 3 ) , 3 par exemple sur le corps Q ) pour lequel v(Ji)=6, tous l e s s o u s de la relation quaternaire harmonique (x + x )(x 0

1

2

groupes G de Sm vsrifient l'inegalitt? : v(G)cm+y(G)+l.

La classe

fournit ainsi la majoration :

indicative m-aire maximum y

m v <m+l+y et cette technique nous avait permis d'obtenir en m m' 1 1964-65, grzce B une inggalitd y
.

avons trouvt? (pour tout m>3) un sous-groupe G de Sm verifiant y(G)=m-3 tO,I,

et v(G)=2m-2,

. . . , m-11

fixant

o

:

h savoir le groupe S 1 des permutations de m et m-1. Connaissant les valeurs y , = y 2 = y 3 = 0 ,

y 4 = 2 et v l = l , v 2= 3 , v 3 = 4 , v 4 = 6 , l'int?galitt? y ' m <m-3 pour m>5 (de HODGES-LACHLAN-SHELAH) combinse avec notre inggalitg vm<m+l+y et m I 1 nos deux ggalitgs y ( S )=m-3, v(S )=2m-2 pour m23, conduit aux deux m m rssultats dEfinitifs : y =m-3 pour tout m25, et v =2m-2 pour tout m m m23. c) Nous achevons la seconde partie de l'expost? e n justifiant un lemme (de la thdorie des chafnes permutdes) qui apporte u n argument simplificateur dans la ddmonstration par J.P. CAMERON (1976) d'un thborsme sur les groupes h a m o g P n c 4 de permutations d'un ensemble infini. Nous mentionnons ggalement l'extension B toute relation

r e T m par

G.

HIGMAN (1976) d u resultat que nous avions obtenu e n

1964 pour l e s G-rangements r=G : toute relation r c T une cJ!?adAe i n d i c a t i w c 6(r) locaux Aut(rl(m+n))

admet aussi m telle que le groupe d'automorphismes

soit un groupe indicatif pour n>S(r).

Bien

pour la classe indicative maximum 6 m (mais les valeurs 6 n e semblent pas connues en m m dehors des valeurs triviales S,=S,=O).

entendu y(G)=G(G),

et y,
des relations r r r

* * * En concJ!?u.6ion, la Combinatoire des relations monomorphes

o u enchaf-

nables (aprss le thdorsme inaugural de R. FRAPSSi e n 1 9 5 4 ) a connu durant l a periode 1963-83 des

S U C C ~ S

nombreux et spectaculaires.

Parmi les problSmes ouverts, la determination des nombres d (rbpondant aux questions : d = v m ? , m l'attention des chercheurs.

* * *

ym

=

Am

?)

m semble mdriter

et 6 m

24 1

Relations enchaina bles, rangements et pseudo-rangements

I . U N FRAGMENT DE LA T H ~ O R I I : DES RELATIONS. Dans s e s aspects relationnistes prgcisant quelques notions bien d6gagGes depuis 1 9 5 3 - 5 4

( 2 savoir :

(a) r e l a t i o n r ~ o c o m o h p h e ,

r e l a t i o n t b t c / ~ n i M c L b t C ? r, e l a t i o n { c k ~ & b i l ' L ~ t P E ,(b) c l a s s e relationnelle de caractPre

I ~ J L ~ o P ' L ~ ~le P ) f, r a g m e n t

de th6orie E v o q u g ci-aprss

s'inspire l a r g e m e n t d e s t r a v a u x d e R . 1201-r2.21)

FRATSS[:

et M .

d ' u n e p a r t , A. T A R S K I e t R.L. V A U G H T

POUZET

([4J-r8],

( [ 2 3 1 [ 2 4 1 ) , M.

JEAN

1 1 7 1 , D . C L A R K et P . K R A U S S ( 1 2 1 r 1 9 1 ) d'autre part. 1 . 1

Dans u n e p r g s e n t a t i o n de t y p e B E R N A Y S - F R A E N K E L

soit

21

o u KELLEY-MORSE,

la c l a s s e d e s e n s e m b l e s et s o i t w l e p r e m i e r o r d i n a l i n f i n i

(ensemble d e s n a t u r e l s , o u o r d i n a u x f i n i s : O = d ,

I={OJ,

Z = { O , l } , ...).

P o u r tout m c o , le g'tOupC? 5 q t t l ~ f f i ( q I A t ? S m e s t l e g r o u p e d e s p e r m u t a tions de m={O,l,

. . . , m-I]

: n o u s designons par

T m le t r e i l l i s d e s

s o u s - g r o u p e s d e 'rn* En

1963-64

( 2 l'instigation de R .

F R A Y S S E qui pressentait

qu'une " t h E o r i e d e s c h a E n e s p e r m u t 6 e s " p e r m e t t r a i t d ' a p p r o f o n d i r 1 ' E t u d e d e s r e l a t i o n s m-aires monomorphes),

nous a v o n s e x t r a i t de

1 un s u p . d e m i - t r e i l l i s C' c o n s t i t u E p a r c e r t a i n s s o u s - g r o u p e s d e m m S m (les g h i u u p t h 4 V l d 4 C U t h d 4 ) p o u r l e s q u e l s la c h a f n e 0 < 1 < 2 < . . . < m - I joue un r6le privilEgi6. Mais, avant d'expliquer l'interfgrence des et I;, m r e l a t i o n s m-aires e t l a s o u s - c l a s s e

d e u x p r o b l g m a t i q u e s c o n c e r n a n t d'une p a r t C classe

f~~d e s

d'autre p a r t l a

I t r ndes

rela-

t i o n s m-aires m o n o m o r p h e s , i l s e r a i t u t i l e d e b i e n p r 6 c i s e r c e r t a i n s c o n c e p t s de l a "thgorie d e s r e l a t i o n s " . 1.2

Pour tout

~ € 1 1et

tout m e w , o n note $ ( E ) ,

r e s p e c t i v e m e n t : la c l a s s e d e s i n j e c t i o n s E

P(E),

,

P,(E),

E~

l'ensemble d e s

parties de E ,

l'ensemble d e s r n - p a k t i e h d e E ( e n s e m b l e s X g E d e c a r -

d i n a l IXl=m),

l'ensemble d e s m - u p l e s (x o , x

, , . . . , x m-

1

)

d'6lEments

de E . 1.2.1

I1 e s t c o m m o d e d ' i d e n t i f i e r l a c l a 4 4 t d C 4

classe

bL

d e s t r i p l e t s (m,E,p)e

id

XUXu

h d U t i O n h

t e l s q u e pc.Em : d a n s

c e t t e c o n c e p t i o n , p o u r u n e h C e U t i O n rn-aihC r=(m,E,p) / E l est la c a h d i n a L i t 8 de r et p est l'ensemble ) d e s m - u p l e s (x o , x l , .. . ,x m- I

s a n t m o u (m,E),

Rrn et &

m (E)

et la

de b u p p o 4 t E ,

(cahact~ni4tique)

Em q u i "satisfont" r. E n particularid e s i g n e n t r e s p e c t i v e m e n t : la c l a s s e

d e s r e l a t i o n s m-aires et l'ensemble d e s r e l a t i o n s m - a i r e s de support E .

2 42

C. Frasnay

1.2.2

t o u t e r e l a t i o n m-aire

A

r=(m,E,p),

deux f a m i l l e s de r e l a t i o n s m-aires

on a s s o c i e t o u t d ' a b o r d

e t une s o u s - c l a s s e

Crl d e

6,m

:

( a ) l a f a m i l l e ( r I x ) x 6 p ( E ) des kebXhiCRi.OMb de r , d 6 f i n i e par m rlX=(m,X,pn X ) ( s i X e P n ( E ) , on d i t que r i X e s t une n - h P 4 Z h i C f i U ~ de r ) . (b)

l a famille

alors

t$(r)

6 c r i t encore : r

y:

@(El y

).

'pe

y(r)=(rn, y < E > ,

[rl=

:

( $(r))

'L

d 6 f i n i e par

21

de.s r e l a t i o n s i b o m u k p h e . 4 B r , La C L a b b e d ' i b o m a k p h i e d e r e s t

: y e O ( E ) I e t , pour t o u t e r e l a t i o n r ' k r r l , on r'.

%

S i r'= y(r),

E'= y<E>, p'= y

, la bijection

e s t un ibomokphibme e n t r e r=(m,E,p) e t r ' = ( m , E ' , P ' ) , 9 q u ' o n n o t e e n c o r e : r 'L r ' . E

-f

ce

Pour rERm, s 6 R m , l o r s q u e r e s t isomorphe 2 une r e s t r i c t i o n

1.2.3 r'

E'

d e s , o n d i t q u e r b'abki-te danb s , o u que s a b h i t e r , e t l ' o n

n o t e : rss. Dans l e c a s c o n t r a i r e r q s , o n p e u t d i r e q u e s P v i Z e r et

( p l u s g6n6ralement)

lorsque s

Pvite chacune des r e l a t i o n s appar-

d e @Lm, o n p e u t d i r e q u e s B v i t e

t e n a n t 1 une s o u s - c l a s s e

L a formule rss d 6 f i n i t dans chacune d e s c l a s s e s k

m

d'abfiitement m-aire.

l e pkfio4dhe

21.

un p r 6 o r d r e :

On a p p e l l e z g e d e s l a c l a s s e B d e s

-3 e s t

r e l a t io n s r de c a r dinalit6 f i n i e v 6 r i f i a n t r<s. S i

l'ensern-

b l e d e s p a r t i e s f i n i e s du s u p p o r t de s , a l o r s E e s t l a r e u n i o n d e s classes d'isomorphie [SIX]

l'on note s(n)

(pour n e w )

n-restrictions

de s (grossisrement

-

on d 6 f i n i t une s u i t e s

: w

.

A cet egard, s i 9 s l e nombre d e s c l a s s e s d ' i s o m o r p h i e d e s

lorsque X parcourt

-f

w

-

: s ( n ) < 2 (nm) pour

appel6e

pkodid d e

tout

sE Q L ~ ) ,

s

Une r e l a t i o n s t e l l e q u e s ( n ) < l e s t d i t e n - t t t o n o m o t p h e ( c e q u i r e v i e n t 1 d i r e q u e , p o u r t o u t c o u p l e (X,Y) d e n - p a r t i e s de s : SIX dit

'L

sly).

L o r s q u e s e s t n-monomorphe

( b r i s v e m e n t ) q u e s e s t manomo4phe. S i l ' o n n o t e

des r e l a t i o n s m-aires d'exemple)

que

monomorphes, on p e u t rernarquer

]\t2c o n t i e n t

du s u p p o r t

pour t o u t n e w ,

]\^em

(5 t i t r e

l a classe des relations binaires

(ou ChaintA).

d'ordre

total

1.2.4

E t a n t donne une r e l a t i o n r = ( k , E , P ) , t o u t isomorphisme

rlX

'L

local

on

la classe

r l Y e n t r e d e u x r e s t r i c t i o n s d e r e s t a p p e l 6 un i o o m o 4 p h i ~ m e d e r. P o u r d e u x r e l a t i o n s

lorsque tout

et

s e R S md e mEme s u p p o r t E ,

isomorphisme l o c a l d e r e s t un isomorphisme l o c a l d e s ,

on d i t que s e s t Libfie-intehpkZtabLe p a r r .

Relations enchaina bles, rongements et pseudo-rangements a)

l o r s q u e r est u n e

Par e x e m p l e ,

~ € ( 8E )

libre-interprgtablr

un

IXI=lYl,

l'injection

croissante

isomorphisme e n t r e S I X e t sly". p r o p o s i t i o n 5.1.)

( [ I l l ,

les couples

pour b)

r e s t d i t e C ~ i c h u i x a b ~ (Cp l u s

par

2 d i r e : " P o u r t o u t c o u p l e (X,Y) d e p a r t i e s f i n i e s d e

ce13 revient que

c h a f n e d e s u p p o r t E, u n e r e l a t i o n

C ~ I C ~ I C ( T I I G pC a r r e t o n m o n t r e f a c i l e m e n t q u e

pr@cis&rnent, e l l e est

E tel

(X,Y)

enchaTnables,

n a b l e est m o n o m o r p h e ) .

dSmontr6 "Pour

18,

§

il rGsulte En

lY64,

r)

de vsrifier

cette condition

de E).

rm

la classe des

:

3

t m ]I[,

relations

(toute r e l a t i o n enchaf-

l'instigation

99 - v o i r a u s s i I l l 1 ,

p.

de X s u r Y est

I E [ > m + l , on p e u t p r o u v e r

partielle

de

R. F R A T S S E ( q u i

: "Toute

de cardinalit6 i n f i n i e est enchafnable"

monomorphe annonci.

suffit

1954 une r g c i p r o q u e

dGs

a v a i t demontre

qu'il

(mod.

Lorsque

de m-parties

s i l'on n o t e

De c c s r e m a r q u c s ,

m-aires

243

proposition

(r41,

relation r6sultat

I l.2),

le tti6orGme s u i v a n t ( n o t e [ I 0 1 e t t h s s e 1 1 1 1 , t h .

tout

m e o, i l e x i s t e u n c o u p l e ( n , p ) e

r e l a t i o n m-aire

n-monomorphe

UXU

pour

nous avons 12.1.1.):

lequel

toute

2p est encha4nable

de cardinalit6

(donc

le plus p e t i t des n a t u r e l s n apparaism ( n , p ) , nous disons que d est le d@ghc m < ~ l ~ t i m c cde e m o n o m o r p h i e m - a i r e . P l u s p r S c i s G m e n t , on p e u t m o n t r e r

S i l'on n o t e d

monomorphe)". sant dans de

qu'il

couples

e x i s t e deux s u i t e s de n a t u r e l s

: "Pour

que

tels

t o u t n a t u r e 1 n2d

de c a r d i n a l i t 6

enctiafnable)"

>pn (resp.

(P,),,~

m' de c a r d i n a l i t 6 si p

(bien entendu,

sont

: d =0, d = I , I

c)

tout

n a t u r e 1 m22,

Pour

phes

d

2

('n'n2d

m-aire

telles

m

n-monomorphe

>q ) e s t monomorphe

e t qn s o n t p r i s

male : pn.
et

toute relatio!

(resp.

de manisre mini-

l e s d e g r s s o p t i m a u x d e mo n o -

= 3 , d = 4 , d =5 ou 6 ( ? ) . 3 4

i l e x i s t e d e s r e l a t i o n s m-aires

monomor-

non e n c h a T n a b l e s e t , c o m m e c o r o l l a i r e d u t h s o r c m e p r G c S d e n t ,

obtient

( [ I l l ,

m-aires

monomorphes

proposition

12.2.1)

: "Pour

m22,

Ies r e l a t i o n s

n o n e n c h a T n a b l e s o n t u n e c a r d i n a l i t s maximum

" ( I 1 e n r s s u l t e un r e n f o r c e m e n t p o u r l e s c a r d i n a l i t s s m f i n i e s d u t h 6 o r S m e d e R. F R A T S S E c i t 6 p l u s h a u t , 2 s a v o i r : " T o u t e

finie k

r e l a t i o n m-aire

Par e x e m p l e connue).

monomorphe

2

m

> km e s t e n c h a E n a b l e " ) . (mais dEjS l a v a l e u r k3 n e semble p a s

de cardinalit:

: k =3 e t k > m + l

on

244

C. Frasnay

1.3

Si

&.

est u n e s o u s - c l a s s e de

(autrement dit : (vu)('dr)

c l o s e par a b r i t e m e n t

((uCr et rs?t)

=j

--

o u encore :

(ue-A)),

est une r 6 u n i o n de c l a s s e s d ' i s o m o r p h i e t e l l e q u e , pour t o u t e r e l a t i o n r de s u p p o r t E et t o u t X c E :(re&)

.h

d i s o n s , pour a b r g g e r , q u e

(rlXc2k ) ) , nous

est u n e c l a s s e

i n 4 t i U 8 t

DSs l o r s , n a t a n t X c Y l'inclusion s t r i c t e ( X & Y

n o u s a p p e l o n s p J ~ u d ~ - P l ? $ m e fdlet

ft. ( o u

"bu4ne" de

de r e l a t i o n s et X # Y ) ,

.k ,

a u s e n s de

R. F R A ' L S S E ) t u u t e r e l a t i o n m-aire s d e s u p p o r t fini Y t e l l e qut? ( s + . f t ) et

(1x1

:

( s l x t ~ ) .

X I Y Bien entendu, si

s

s'

et s i s est p s e u d o - 6 l C m e n t d e

a l o r s s'est a u s s i p s e u d o - 6 1 6 m e n t de

h .

S i b(h)

s u p 6 r i e u r e d e s c a r d i n a l i t s s d e s p s e u d o - 6 l g m e n t s de b(h)
( o u b(k) E

id)

des pseudo-616ments

h

h

, l a condition

Q q u i v a u t h d i r e q u e l c s c l a s s e s d'isomorphie de

sont e n n o m b r e f i n i ( o u e n c o r e q u e

n'admet, 2 l'isomorphie p r S s , qu'un n o m b r e f i n i d e "bornes") dit a l o r s q u e l a c l a s s e i n i t i a l e 1.3.1

,

d 6 s i g n e la b o r n e

est

4c : on

{iMd-b(J&RBt.

S i r est u n e r e l a t i o n m - a i r e d e c a r d i n a l i t 6 p, s o n S g e E; est

u n e c l a s s e i n i t i a l c et l e c a r d i n a l b(P)

e x p r i m e une p r o p r i 6 t 6 d e l a

f a m i l l e d e s r e s t r i c t i o n s "finies" d e r . B i e n e n t e n d u , s i p est f i n i , a l o r s : b(^r),
(et o n peut

pour t o u t e r e l a t i o n e n c h a T n a b l e

De m a n i P r e g G n g r a l e , l o r s q u e b(T)

(autrement dit : s i l'sge F e s t fini-bornE),

I

est f i n i

o n dit simplement que

la r e l a t i o n r est d i n i - b o t n C t . a) En 1 9 6 3 ,

n o u s a v i o n s o b t e n u u n e proposition(d6not6e

Cll1)donnant

13.1.2

dans

c o m m e c o r o l l a i r e i m m 6 d i a t : "Toute r e l a t i o n e n c h a P n a -

b l e est fini-born6e". P l u s p r G c i s E m e n t , s i r est u n e r e l a t i o n m-aire encha?nable de c a r d i n a l i t 6 p(finie

o u non),

o n peut lui

a s s o c i e r les pseudo-616ments d e ^r a y a n t u n e c a r d i n a l i t 6 < p , p u i s n o t e r v(r)

la b o r n e s u p 6 r i e u r e des c a r d i n a l i t 6 s de c e s pseudo-

6 1 6 m e n t s (bien e n t e n d u v(r)

=

b(?)

s i p e s t i n f i n i , et v(r)
s i p est fini) : n o u s d i s o n s q u e v(r) e n c h a 9 n a b l e r . En p o s a n t : v v

m

E iii

(pour tout m t u ) .

m

est la

= s u p v(r),

re

't:m

W U ~ ~ M Cde C

la r e l a t i o n

n o u s obtenions :

245

Relations enchaina bles, rangements et pseudo-rangements v =3, v = 4 , v = 6 et a u - d e l 5 , n o u s n e 1 2 3 4 p o u v i o n s a v a n c e r (5 c e t t e 6 p o q u e ) qu'une m a j o r a t i o n a p p a r e m m e n t

P a r e x e m p l e : v =0, v = I ,

trss g r o s s i s r e v < 9 m 2 p r o v e n a n t de l'utilisation d u th6orSrne d e s 'm "bichail141" (P. E R D B S et G . S Z E K E R E S , 1 3 1 ) : " P o u r n e o , s i d e u x cha'ines s , t o n t u n &me une (n+l)-partie

2

s u p p o r t E tel q u e I E l > n , a l o r s i l e x i s t e

X d e E pour l a q u e l l e les cha'ines r e s t r e i n t e s S I X ,

tlX s o n t i d e n t i q u e s o u o p p o s 6 e s " . b) A l o r s q u e l a f o r m u l e v mE O ( e x p r i m a n t le c a r a c t s r e f i n i - b o r n 6 d e t o u t e r e l a t i o n m-aire encha?nable) n ' o c c u p a i t d a n s n o t r e t h s s e d e 1965qu'une p o s i t i o n m a r g i n a l e , u n r e n v e r s e m e n t d e t e n d a n c e ( d e p u i s 1970) a r e n f o r c 6 le r c l e de c e t t e p r o p r i E t 6 : c e t t e n o u v e l l e

o r i e n t a t i o n est a p p a r u e d a n s les t r a v a u x d e D. C L A R K et P .

KRAUSS

d'une p a r t , R. F R A T S S i et M. P O U Z E T ( C 8 1 C 2 0 1 ) d'autre p a r t

(C21C191)

( t r a v a u x d o n t t i e n t c o m p t e la n o t e C 1 3 1 s u r les v a l e n c e s m - a i r e s m a x i m a l e s v ) . A i n s i e n 1972, d a n s u n e a p p r o c h e p u r e m e n t q u a l i t a t i v e m (note r 2 0 1 , s a n s i n c i d e n c e s u r les v a l e u r s v m ) , M. P O U Z E T a d g f i n i (pour tout m s o) la c l a s s e

firn d e s

b e P P ~ 0h e t a t s o n 0 r n - u L h e i ,

X420

de telle sorte que : ( I )

P o u r tout

m g IW,

s i re

(2) P o u r tout ( k , m ) t o u o , - i n t e r p r & t a b l e par u , Comme

(2) :

q)2

rm,/Rm

bm,a l o r s si

~

6

r est fini-born6e. est libreet % s i~ re'% m

alors r t 6 m .

c o n t i e n t la c l a s s e d e s cha'ines, il r 6 s u l t e d e ( I )

et

et toute r e l a t i o n encha'inable est f i n i - b o r n 6 e .

c) L'6tude q u a n t i t a t i v e e f f e c t u 6 e e n 1963-65 d a n s l e s n o t e s [ 9 l [ l O l et la thSse [ I l l u t i l i s a i t la " t h 6 o r i e d e s cha'ines p e r m u t 6 e s " et l e

.

I' des sous-groupes indicatifs de S Elle fournissait m m (pour le d e g r 6 o p t i m a l d m d e m o n o m o r p h i e m-aire et l a v a l e n c e m-aire

demi-treillis

m a x i m a l e v ) les i n 6 g a l i t 6 s : d m < v m (pour tout m e w ) et m+l,2). L ' e s p o i r d'une m a j o r a t i o n d e l a f o r m e v m < c x m + B s e r 6 a l i s e r a e n 1 9 7 6 - 7 7 , par a p p l i c a t i o n d'un n o u v e a u r e s u l t a t s u r I' m dCi 1 W. H O D G E S , A.H. L A C H L A N et S. S H E L A H

1161.

E t a n t d o n n 6 ( m , n ) ~W X W , o n a p p e l l e o p P 4 a t c u t L i b f i e

1.3.2

les a r i t s s m , n )

une application f

r o l a t i o n m-aire r d e s u p p o r t E ) , ( I )

f(r)

:

Rm +%

(pour

v 6 r i f i a n t (pour t o u t e

les t r o i s c o n d i t i o n s s u i v a n t e s :

est u n e r e l a t i o n n - a i r e d e s u p p o r t E.

246

C. Frasnay

(2) P o u r tout (3)

P o u r tout

X E

P(E)

Y=@(E)

: f ( r J ~ )= f(r)

IX

%

f(j(r))

:

=

y(f(r))

a) Etant d o n n e u n tel o p 6 r a t e u r l i b r e f (pour les a r i t 6 s m,n), s o i t

h

la c l a s s e des r e l a t i o n s r=(m.E, p)

k

On dit q u e

t e l l e s q u e f(r)=(n,k.,E").

est la c e u 5 5 e a n c u e 4 6 e e f c (au s e n s de A .

h e p h P n e n t P e par l ' o p e r a t e u r libre f . Le d c g r l g de

petit d e s n a t u r e l s n tels q u e libre f :

fLm

h

e s t le plus

s o i t r e p r 6 s e n t a b l e par un o p G r a t e u r

(Par e x e m p l e , la c l a s s e des c h a i n e s est u n i v e r -

!?L

+

h

T A R S K I 1231)

s e l l e de d e g r E 3 ) . T o u t e c l a s s e u n i v e r s e l l e est i n i t i a l e (close par abritement),

et on peut m o n t r e r q u e la c o n c e p t i o n de A .

TARSKI

e q u i v a u t 2 d e u x a u t r e s c o n c e p t i o n s q u i n e n e c e s s i t e n t pas de f a i r e appel 2 la n o t i o n d'opgrateur l i b r e : ( I )

de

Au s e n s e q u i v a l e n t d e R . L .

Rm est

V A U G H T 1241, une s o u s - c l a s s e

u n i v e r s e l l e s i e l l e est i n i t i a l e et s'il e x i s t e n t

(I

v E r i f i a n t l a c o n d i t i o n s u i v a n t e : " L o r s q u e r est une r e l a t i o n m - a i r e d o n t t o u t e s les r e s t r i c t i o n s d e c a r d i n a l i t 6 alors r appartient 2

.k

"


Lk

,

est le p l u s petit n a t u r e 1

(Le d e g r e de

n v G r i f i a n t c e t t e condition). (2)

A u s e n s e q u i v a l e n t de R . F R A Y S S E 1 5 1 , u n e s o u s - c l a s s e

A

Lr

de fL e s t u n i v e r s e l l e s'il e x i s t e u n e n s e m b l e fini de r e l a t i o n s m est l a c l a s s e d e s m-aires de c a r d i n a l i t e s f i n i e s tel q u e : "

ft

r e l a t i o n s m-aires q u i G v i t e n t et l e d e g r 6 de

$.

pseudo-Glements d e

lr

"

(Alors

.k

est u n e c l a s s e i n i t i a l e ,

e s t la b o r n e s u p e r i e u r e des c a r d i n a l i t e s d e s

.&

).

b) E n u t i l i s a n t n o t r e t h e o r s m e s u r le degrg o p t i m a l d e m o n o m o r p h i e m-aire, M. J E A N a o b t e n u e n 1967 C 1 7 1 ,

pour chaque m e

(11,

les d e u x

p r e m i e r s e x e m p l e s non t r i v i a u x de c l a s s e u n i v e r s e l l e , e n d e m o n t r a n t : "La c l a s s e

cm

d e s r e l a t i o n s m - a i r e s e n c h a i n a b l e s et l a c l a s s e

des r e l a t i o n s m - a i r e s m o n o m o r p h e s s o n t d e u x c l a s s e s u n i v e r s e l l e s " . c) S i g n a l o n s ici u n l i e n e n t r e l a n o t i o n d e l i b r e - i n t e r p r 6 t a b i l i t 6 et c e l l e d'opgrateur l i b r e

151

:

"Pour qu'une r e l a t i o n s e !J,n

l i b r e - i n t e r p r e t a b l e par u n e r e l a t i o n r e

a

am,il

soit

f a u t et il s u f f i t

an

qu'il e x i s t e un o p E r a t e u r l i b r e f : + tel q u e f(r)=s". m Comme l e n o m b r e d e s o p s r a t e u r s libres f (pour d e s a r i t G s m,n donnges) est f i n i , il e n r E s u l t e q u e (pour r et n donn6s) l e n o m b r e d e s

Relations enchaina bles, rangements et pseudo-rangements

247

relations n-aires libre-interprgtables par la relation r est fini. P a r e x e m p l e , s i r est u n e cha'ine de s u p p o r t E tel q u e ( E / > n ,o n p e u t m o n t r e r q u e le n o m b r e d e s r e l a t i o n s n - a i r e s de s u p p o r t E e n c h a f n k e s par r est 2 " i ( n ) , oC v(n)

est le n o m b r e d e s p r E o r d r e s t o t a u x s u r

~ = { o , I .,. . , n - I I . 1.3.3 E,

L o r s q u e r et s s o n t d e u x r e l a t i o n s m-aires de m@me s u p p o r t d i s o n s qu'elles sont une d h d o h m u t i o h l l'une d e l'autre (ou

nous

q u e c h a c u n e est d ' ? < o 4 w l U b t e e n l'autre)

si :

( I )

r et s n e s o n t p a s i s o m o r p h e s ,

(2)

pour tout

X C

E , les r e s t r i c t i o n s s t r i c t e s r

/ et~

SIX

sont

isomorphes. a) Si m = I , une t e l l e s i t u a t i o n e x i g e I E I C 1 .

O n trouvera dans " 2 1 1

les r 6 f s r e n c e s a u x t r a v a u x de G. L O P E Z (1972, p o u r m=2) et de

M. P O U Z E T (1975, p o u r m>3) s e l o n l e s q u e l s : l e s r e l a t i o n s b i n a i r e s "finies" d6formable.s o n t u n e c a r d i n a l i t 6 m a x i m u m G g a l e B 6 - t a n d i s q u e , pour t o u t m > 3 et tout p e

'0,

d g f o r m a b l e s de c a r d i n a l i t 6 f i n i e

i l existe des relations m-aires >p.

b) N o t r e p r o p o s i t i o n 13.1.2 de [ I l l s ' e x p r i m e suivante : "Pour tout m e w ,

e n c o r e d e la m a n i s r e

les r e l a t i o n s m - a i r e s e n c h a f n a b l e s de

cardinalit6 finie > v

sont indsformables". Bien entendu, puisque m les r e l a t i o n s m - a i r e s m o n o m o r p h e s n o n e n c h a P n a b l e s o n t u n e c a r d i n a l i t 6 m a x i m u m f i n i e k m , il e n r 6 s u l t e q u e "toute r e l a t i o n m o n o m o r p h e

est f i n i - b o r n 6 e " e t e n c o r e : " P o u r t o u t m e w ,

il e x i s t e p c w

lequel toute r e l a t i o n m-aire m o n o m o r p h e d e c a r d i n a l i t 6 f i n i e

pour >p

es t indEf o r m a b le" .

1.4

Etant donn6 u n ensemble I et une famille

u=(mi)itI

r e l s , o n a p p e l l e b t ' Z u c - t u h ( ! h d u , t i u n n t L L e u - u i h e de

[muLtihei!ution

ou

de natu-

Auppohf E

k-tclatiu~ p-aire d e s u p p o r t E s i I est f i n i ,

/ I / = k ) t o u t e f a m i l l e (ri)ie I t e l l e q u e , p o u r tout i t I , r i

soit

u n e r e l a t i o n d'arit6 m i et d e s u p p o r t E. S i n e w e s t t e l q u e mi$" pour t o u t i r I , o n dit a u s s i q u e l'arit6 p

e s t b v t t n a e et q u e

(ri)ie I est u n e s t r u c t u r e r e l a t i o n n e l l e u u p l u h n - u i h e .

1.4.1

T o u t e l a t e r m i n o l o g i e i n t r o d u i t e B p r o p o s d e s r e l a t i o n s (en

1.2 et 1.3) s'Btend a i s 6 m e n t a u x s t r u c t u r e s r e l a t i o n n e l l e s . P a r

248

C. Frasnay et tout y c Q(E),

exemple, pour tout X t P(E)

chaque structure rela-

de support E admet une restriction i it1 % 'L L'isoet une image isomorphique y(R)=(y(ri))ik I.

tionnelle p-aire R=(r ) RI X=(ri 1 X)

E I

morphisme explicite (R

?,

R') ou implicite ( R

'L

R'),

l'abritement

l'zge R , de m t m e que les propri6tGs

R
(pour une structure R) de n-monomorphie, monomorphie, enchafnabilitd, dgformabilit6, . . . se conqoivent aisdment. Par exemple, pour

. . . rk)

qu'une k-relation ( r l , r 2 ,

de support E soit libre-

interergtable par une chaPne r

de support E , i l faut et i l suffit

( r o , r l , r 2 ,. . . , rk) soit monomorphe. Une

que la (k+l)-relation

m u t t i c h u i n e de support E est une famille finie

de chafnes

t i de m2me support E.

1.4.2

D e nombreux r6sultats 6tablis pour les relations m-aires

s'6tendent (par une juste adaptation) aux multirelations et aux structures relationnelles d'arit6 p born6e. Donnons quelques exemples : a) T h Z o h P m e d e b m u e t i c h a i n e b ( [ 9 l [ l 11) gdngralisant le th6orSme des bichafnes (k=l) de P. ERDUS et G . SZEKERES C31: Pour (k,n)&

w

XW,

le naturel n (2k) est le plus petit des

naturels p vgrifiant la condition suivante : "Pour toute(k+l)-chafne

...,

t ) de support E tel que / E l > p , il existe une (n+l)-partie (to,tl, k X de E pour laquelle les chafnes restreintes t IX, t,lX , . . . , tklX

sont deux B deux identiques o u oppos6es". b) ThEofiZme d e monomohphi,c o p t i m a t e (I I 2 1 I 2 1 1) :

I une arit6 born6e.

Soit p=(rni);

I1 existe alors u n couple ( n , p ) e

vdrifiant la condition

0x0

suivante : "Toute structure relationnelle p-aire n-monomorphe de cardinalit6 p)

est enchafnable (donc monomorphe)".

Si l'on note d

v

le plus petit des naturels n apparaissant dans

de tels couples (n,p),

o n peut dire que d

est le d e g t t o p t i m a ! . de

1-I

monomorphie p-aire. Corollairement, i l existe u n naturel k

(mini11 mal) tel que : "Toute structure relationnelle p-aire monomorphe de cardinalit6 >k

LJ

est enchafnable".

c) ThEafiPmed d ' i n d E ~ a h m u b i t i t E ([20l-C221) A toute arit6 bornde p=(m;)ie

:

I , o n peut encore associer deux

Relations enchaina bles, rangements et pseudo-rangements

249

( v u k e n c e U-uihe maximale) e t w tels que : "Toute U P structure relationnelle U-aire enchaTnable (resp. monomorphe) de

naturels v

cardinalit6 finie >v

(resp. > w ) est ind6formable".

u

u

Ces 6nonc6s (pour toute arit6 P born6e) imposent un caractSre finiB toute structure relationnelle p-aire enchaPnable o u

born6

g6 n 6 r a 1 eme n t ) mo nomo rp h e Remarques : ( I )

(plus

.

Signalons aussi ( G . LOPEZ, 1 9 7 6 ) que les multirela-

tions au plus binaires et de cardinalit6 finie ,212

sont ind6for-

mables.

(2) Les m6thodes adopt6es dans C l l l , C 1 2 1

et 1 2 2 1 , pour

une relation o u une multirelation, valent aussi pour une structure relationnelle d'arit6 born6e. Pour toutes l e s arit6s p des structures relationnelles au plus m-aires, o n obtient ainsi les in6galit6s : dP 6 vu 6 vm. La dgtermination de la valence m-aire

v

m

prend donc une importance accrue : lorsqu'une relation m-aire "finie" r , enchainable et dGformable, aura la cardinalit6 maximum v

nous dirons que r est une relation enchainable c h i t i q u e .

N'

C'est par une dtude combinatoire du treillis C m des sous-groupes S m que nous obtiendrons'un exemple de telles relations m-aires ainsi que la valeur v m = 2m-2 (pour tout m>3).

G du groupe sym6trique

2.

L A T H ~ O R I ED E S CHAINES

PERMUTEIES.

Un groupe de permutations s'associe tout naturellement 1 une relation r=(m,E,p) (1)

dans deux circonstances :

Si S E d6signe le groupe des permutations de E , o n note Aut(r)

(et o n appelle g h o u p e d t n u u t o m o h p h i n r n t A

y~

S E telles que

%

(p(r)=r.

de r) le groupe des permu-

P l u s ggngralement, lorsque X parcourt

les g h o u p e n d'automokphiomen e a c a u x de r sont les groupes

P(E),

Aut (rlX) associ6s aux restrictions r ( X de r. Par exemple, une relation r de support w admet (parrni ses groupes (Gn)npw Si

B:E

+

est un isomorphisme entre deux ?elations r et r'

E'

de supports respectifs E et E ' ) , tion Aut(r)

locaux) une suite

de groupes G n € En d6finie par : G =Aut(rl n).

9-0

o

Y

o

en Aut(r').

0-I)

i l lui correspond (par transmuta-

un isomorphisme de S E sur S E l qui transforme

250

C. Frasnay

(2) L e s p e r m u t a t i o n s o ez S m c o m p o s d e s a v e c les m-uples x c E d e s m-uples x o x

0

m

e E

0

0 6

donnent

( p l u s e x p l i c i t e m e n t , p o u r x=(xO,x l , . . . , ~ m - , ) : )).

=(XO(o)’xa(l)’.”’xo(m-l)

permutations

m

S i Inv(r)

S m t e l l e s q u e : (\dx)(x&p

d6signe l e groupe des

-y x o

(?

C

o), o n d i t

q u e I n v ( r ) ~ Zm est le gkaupe d ’ i i i u a k i a n c e m a x i m u m d e la r e l a t i o n r=(m,E,p).

Tout isomorphisme r

.?,

r ’ i m p l i q u e : Inv(r)

=

Inv(r’).

P l u s g B n g r a l e m e n t , o n peut a s s o c i e r B tout g r o u p e G 6 C m et 2 toute relation re&, l a r e l a t i o n G x r( ddfinie comme suit :

km

si r=(m,E,p),

alors G

x

r = (m,E,p’)

avec p ’ = { x o o : x ~ pet

0 6 GI.

P u i s q u e r et G x r o n t m 8 m e a r i t 6 m , m 8 m e s u p p o r t E , et q u e l e u r s e n s e m b l e s caract6ristique.s p . G

x

p’

vdrifient

p g p ’ ,

o n peut d i r e que

r e s t u n e e x p u n h i O n de r : plus p r B c i s E m e n t , G x r est la

G - e x p a n b i o n d e r. B i e n e n t e n d u (si E

m d s s i g n e la p e r m u t a t i o n i d e n t i q u e , B l g m e n t n e u t r e de S ) , le g r o u p e t r i v i a l I = { E ~ } est u n a g e n t m m n e u t r e : I~ x r = r (pour t o u t r e P l u s g B n B r a l e m e n t , p o u r G E Cm ’

Rm).

la condition G x r=r Bquivaut 1 la condition G

C_

Inv(r)

et (dans ce

cas) o n dit q u e l a r e l a t i o n r est G - i n u a k i a n t e , o u e n c o r e q u e G est u n g h a u p e d’inuakiance de r. L e s r e l a t i o n s m-aires hymEtkiqucA s o n t les r e l a t i o n s r E @(E),

Rm telles

q u e Inv(r)

=

sm .

p o u r rt(R,(~),

X ~ ZP(E),

les f o r m u l e s :

G x r E (?Lm(E),

G

X

(r(X)

que la G-expansion r

I--+

=

(G

G

X

X

r)(X, G x

y% (r)=

%

~ ( G x r ) , montrent

r d 6 f i n i t u n o p 6 r a t e u r l i b r e de

(k m

d a n s lui-m8me. E n f i n , p o u r tout G t C m , o n p r o u v e a i s s m e n t q u e l a c l a s s e d e s r e l a t i o n s m-aires G - i n v a r i a n t e s e s t u n e c l a s s e u n i v e r selle. N o u s a l l o n s a d a p t e r ces c o n s i d 6 r a t i o n s g 6 n B r a l e s ( p o r t a n t s u r

les r e l a t i o n s et l e s permutations) a u x c h a f n e s et a u x r e l a t i o n s enchafnables,

2.1

+ E t a n t d o n n 6 u n n a t u r e 1 m (support d e l a c h a f n e m) et une c h a f n e

t de s u p p o r t E , l ’ e n s e m b l e p d e s i n j e c t i o n s c r o i s s a n t e s de

m={O,l,

. . . ,m - 1 )

+ (ordonn6 p a r m) d a n s E ( o r d o n n 6 par t ) d s f i n i t u n e

r e l a t i o n m-aire tCm)=(m,E,p) uples x=(xo,xI, (mod. t). t(m)

. . . ,xm- 1 )

d e s u p p o r t E , s a t i s f a i t e par les m-

d ’ B l 6 m e n t s d e E t e l s q u e : x < x C...<x 0

1

e s t le h u n g e m e n t m - a i k e t k i u i a k ? s u s c i t B (dans E) par l a

c h a f n e t.

m- I

P a r a l l u s i o n a u g r o u p e t r i v i a l I m = { ~ m } , o n peut d i r e q u e

Relations enchainables, rangements et pseudo-rangementr

25 1

Le premier acte de la thgorie "des chafnes permut6es"

2.1.1

consiste 2 introduire, pour chaque groupe de permutations G E C le G - h a n g c m e n t t G tG =

=

est donc la G-expansion du rangement trivial t(m)

(m,E,D')

s o n ensemble caractgristique

('a(o)

9

UGG

'n(l)

*

1

e t x <x 0

m'

G X ~ ' ~ suscit6 ) (dans E) par l a chafne t :

1

'Xo(m-~)) C...<x

p'

e Em

vsrifiant les deux conditions :

(mod. t).

m- I

et

est l'ensemble des m-uples

Bien entendu, si / E l C m , les

G-rangements suscit6s dans E coincident avec la relation m-aire "vide" : t G = (m,E,d) (qui admet S comme groupe d'invariance). Par m contre, si IEl a m , les G-rangements suscit6s dans E sont des relations m-aires t G " n o n vides" dont l e groupe d'invariance maximum est prgcis6ment

: Inv (t

G

)=G.

Lorsqu'un G-rangement t G abrite une relation m-aire

2.1.2

alors s est a u s s i u n G-rangement

s,

(la preuve s'effectue d'abord dans

le cas trivial G = I m , puis dans le cas gEn6ral G E E ) . Autrement m la classe des G-rangements est une m' classe initiale de relations m-aires". Si l'on appelle p h e u d u - G -

d i t : "Pour tout groupe G e i :

h a n g e m e n f tout pseudo-616ment de cette classe initiale, et si l'on tient compte des deux remarques suivantes : ( I )

tout ensemble E supporte u n G-rangement,

( 2 ) deux G-rangements tG,th de m8me cardinalit6 finie sont

implique t'=%(t G Y

isomorphes (puisque t'=;(t)

alors i l r6sulte de la terminologie g6n6rale ( 1 . 3 )

G) ) .

qu'un pseudo-

G-rangement est une relation m-aire r de cardinalits finie n+l verifiant l'une o u l'autre des d e u x conditions suivantes : (I)

r n'est pas un G-rangement, mais toute n-restriction de r est un G-rangement.

(2)

r est une dsformation d'un G-rangement.

Par ailleurs, tout G-rangement t

G

est enchalnable ( p l u s pr6ci-

sEment : t G est libre-interprgtable par la chafne t qui le suscite) et tout pseudo-G-rangement 2.2

est monomorphe.

Etant donn6 u n groupe G c C

respectifs A,B, on dit que

s

m

et deux chafnes s,t de supports

et t sont G - c o m p a t i b L Q d d S s que les

G-rangements s G et t G ont la mEme restriction de support A n B : s

G/ A n B

= t

G

[ A n B. Cette condition est trivialement r6alis6e

C. Frasnay

252

lorsque ( A n B ( c m e t , dans le cas ggn6ra1, elle revient 5 dire que les deux chafnes s ( A n B , t l A n B (de support commun A n B )

suscitent

l e mgme G-rangement. Si (s,t) est une bichafne de support E , la G-compatibilitG

2.2.1

des chaPnes s,t peut s'exprimer plus directement en associant 1 toute m-partie X de E une permutation

x

=

0

e

X ~ , . . . , X ~ - ~x} ,< x <...<x

cx

0

1

Sm

m- 1

de la faGon suivante : (mod.

s)

(mod. t). Une telle permutation est et x o(m-I) u ( 0 )< x a ( 1 ) < - * unique, d'oG une application X e P ( E ) k u E S (on dit parfois que m X m ox est la permutation permettant le p a o h a g e de la chafne S I X 1 la chafne tlX. Dhs lors, pour que les cha'ines

s

et t soient G-compati-

bles, i l faut et i l suffit que u X E G pour tout Xe Pm(E).

Ou encore,

si H m ( s , t ) ~ C m d6signe le groupe engendr6 par les permutations lorsque X parcourt Pm(E),

0

X

la G-compatibilit6 de s,t Gquivaut 1

l'inclusion H ( s , t ) ~ G (on peut dire que H (s,t) est le ghoupe d t m m

compatibiLitP minLrnum des cha'ines s,t pour le degr6 m). Si

T,

ddsigne l'ensemble des cha'ines de support E , il e n

rdsulte (pour tout groupe G E E m ) une relation binaire d'gquivalence de support dite h e L a t i o n d e G-compntibiLitP, encore not6e :

f,,

s E t

(mod. G) (formule logiquement Squivalente 1 l'identit6

s G = t G des G-rangements

suscit6s par les chafnes s,t).

2.2.2

Pour tout groupe G E Z le G-rangement suscit6 par la m' + m-I}) n'est autre que la chafne usuelle m (de support m={O,l,

...,

relation (m,m,G).

Pour simplifier, nous noterons G le G-rangement + de support w suscit6 par la chafne usuelle w : ce G-hangeme-nt

6

natuheL

+ par w,

est l'unique relation m-aire de support w ,

de restriction ilm=(rn,m,G).

bijective G H

6

enchafn6e

I1 en r6sulte une correspondance

entre le treillis C

m

et l'ensemble des rangements

m-aires naturels. Pour n e w , une permutation

~

E

lorsque la chafne associ6e (appartenant h

S est ~ +dite ~ G-admionibLe

<

e(O)<e(l)<

et pour laquelle : m+n avec la chafne usuelle

(pour

( 8 7.1.),

. . .
nous avons

8 e S m c n constituent un montr6 que les permutations G-admissibles groupe G (n)E 'm+n : ce groupe est caractsrisd par le fait que, pour

Relations enchaina bles, rangements et pseudo-rangements t o u t e b i c h a i n e ( s , t ) de c a r d i n a l i t 6 N o u s d i s o n s q u e G(n)

entendu

est le

ghOurJP

e t , p o u r (p,q)&wXw

G(O)=G

253

m+n, ~~t(mod.G)~s~t(rnod.G(~)), d i B U t E k?-i?tnc d u g r o u p e G . B i e n

, G(p)(q)=G(p+q).

E n r e v e n a n t a u C - r a n g e m e n t n a t u r e 1 G , o n peut m o n t r e r est a u s s i la s u i t e d e s g r o u p e s d ' a u t o m o r p h i s m e s

q u e (G("))nEw

l o c a u x : G(n)=Aut(hlm+n). 2.2.3

A

t o u t e b i c h a E n e ( s , t ) de c a r d i n a l i t 6 i n f i n i e , a s s o c i o n s la

suite ( H ) d o n t c h a q u e t e r m e H e 1 est le g r o u p e d e c o m p a t i b i m mCW m m l i t 6 m i n i m u m de s,t p o u r le d e g r 6 m . N o u s d i s o n s q u e c e t t e s u i t e H est une 5 u i t c i n d i c n f i v e et q u e t o u t g r o u p e H

figurant dans une m t e l l e s u i t e est u n g h v u p e i k t d ~ c a t i f i . N o u s n o t o n s 1; l ' e n s e m b l e d e s

s o u s - g r o u p e s i n d i c a t i f s de S

m

.

a) D a n s [ I l l , n o u s a v o n s r s p e r t o r i 6 t o u t e s l e s s u i t e s i n d i c a t i ves. O n peut les n o t e r : S,T,D,IP'q,Jr >I).

p,q,r

sont des naturels

L a s u i t e S est c e l l e d e s g r o u p e s s y m g t r i q u e s Sm (on peut d i r e

qu'elle est de hang

0 ) .

So,SI,...,Sh-I, puis H m)h

(oG

m

Toute suite indicative H # S commence par est un s o u s - g r o u p e s t r i c t de S

(on p e u t d i r e a l o r s q u e le r a n g de H est h). La suite J

I

m

p o u r tout

Plus pr6cissment :

e s t d e r a n g 3 e t , p o u r m 2 3 , J/ est le g r o u p e

(d'ordre 2) e n g e n d r g par l e r e t o u r n e m e n t (0,m-I)

(l,m-2)

...

L a s u i t e T est de r a n g 3 e t , p o u r m 2 3 , T m est l e g r o u p e (d'ordre m) e n g e n d r s par la p e r m u t a t i o n c i r c u l a i r e ( O , l , 2 , . . . , m-I).

La suite D

est d e r a n g 4 e t , p o u r m 2 4 , D m est l e g r o u p e p r o d u i t d i r e c t (d'ordre

2m) : D m = J A Tm . P o u r p k l , q a l , l a s u i t e "'1

est d e r a n g p+q e t , p o u r m)p+q,

est le g r o u p e (d'ordre p!q!) vant l'intervalle initial {m-q,

. . . , m-2,m-I}

c o n s t i t u g p a r les p e r m u t a t i o n s c o n s e r -

{ o , I , ...,p-

I } , l'intervalle final

e t c h a c u n des n a t u r e l s p , p + I ,

l a s u i t e J r e s t d e r a n g 2r e t , p o u r m > 2 r , J: d i r e c t (d'ordre 2(r!) b)

2

)

: J:

=

. . . , m-q-I.

Pour r22,

e s t le g r o u p e p r o d u i t

J A I:'r.

On p e u t c o n s t a t e r q u e t o u t g r o u p e i n d i c a t i f H

m

# Sm appar-

t i e n t 2 u n e s u i t e i n d i c a t i v e u n i q u e et q u e 1' e s t un s u p . d e m i m il t r e i l l i s (si G ' e t G " s o n t d e u x s o u s - g r o u p e s i n d i c a t i f s de S m' e n e s t d e m t m e p o u r le g r o u p e G ' B G " , t a n d i s q u e G' A G " = G ' n G" n'est pas n 6 c e s s a i r e m e n t un g r o u p e indicatif).

Tout groupe G e C

m

contient donc un groupe indicatif maximum H E C ' qui dgtermine sans m m

C. Frasnay

254 ambiguit6 la diche

indicative H de G .

Les dilatss successifs d'un groupe indicatif sont des groupes indicatifs de msme fiche. Notamment, p o u r les groupes symgtriques (de fiche S )

:

(Sm)(n)=Sm+n.

Pour toute suite indicative H P S , de

rang h , o n d6montre sgalement : (Hm)(n)

Hm+n p o u r tout m>h.

=

sont indicatifs : c e m (pour le degr6 I ) , S 2 et 1;" (pour

c) Pour m<3, tous les sous-groupes de S sent S

(pour le degr6 0), S l

le degre 2), S3, I:*', 11", I:*', J 31 , T 3 (pour le degr6 3). 3 Le nrouae . S . contient 30 sous-groupes dont I I sont indicatifs, 2 l , 1 1,2 12,1 - 2 , 2 1 , 3 13,1 1 2 savoir : S 4 ' I 4 9 I4 , 4 , 14 I4 3 4 , J 4 , T4, D4* 4m2 groupes indicatifs. Plus ggnsralement, pour m24, Sm contient 3 . 1 7 1 9

2.3.

9

9

L'un des rEsultats-cl6s de la thsorie des chaines permutges

(note C 9 1 et thPse [ I l l )

a 6t6 la mise e n evidence, pour chaque

sous-groupe G de S

d'un nature1 minimum y ( G ) qu'on peut appeler m' la c L a b h e i n d i c a t i v e de G pour les raisons suivantes : si H est la fiche de G (et H m le groupe indicatif maximum contenu dans G ) ,

alors la suite ( G ( ~ )

)nt w

des dilatgs de G v6rifie H m + n ~ G(n)

(inclusion stricte) pour O < n < y ( ~ )et H

m+n

= G ( ~ ) pour n

Corollairement (pour tout n e u et tout G

2 y(G).

: n > y ( ~ ) M G ( ~ )est

indicatif. 2.3.1.

En 1 9 6 5 ,

n o u s avions explicit6 [ I l l

les classes indicatives

Em).

AprSs y= ,O

y(G)

les calculs concernant

et leur maximum y m (quand G parcourt

pour m63, n o u s obtenions y4=2 en calculant y ( G )

pour l e s 1 9 sous-groupes non indicatifs de S 4

: y(A

4

) = 2 pour le

groupe altern6 et y ( G ) = l pour les 18 autres groupes. Pour m 5 5 , la technique employ6e (utilisant le th6oreme des bichafnes C31) donnant une majoration assez grossisre y

m

6 am'+bm+c

dont les

coefficients ( a = 9 par exemple) importent peu.

2.3.2

Plus rgcemment, 1 l'occasion de recherches concernant le

degrs d'homog6nEitE et le degr6 de transitivits des groupes de permutations, certains mathsmaticiens (P.J. CAMERON [ I ] ,

C l 5 1 , W. HODGES

-

A.H. LACHLAN

-

S.

G.

HIGMAN

SHELAH 1 1 6 1 ) ont retrouv6 ind6-

pendamment de n o s travaux ant6rieui-s (et, de ce fait, dans leur terminologie propre) la notion de "gnoupe indicatif" et s o n rSle

Relations enchainables, rangements e t pseudo-rangements

255

d a n s 1 ' g t u d e d e s r e l a t i o n s c n c h a l n a b l e s . On d o i t a i n s i a u x t r o i s d e r n i e r s c i t P s 1 1 6 1 une a m g l i o r a t i o n d g c i s i v e d e l a m a j o r a t i o n d e s

ob t e n i r c e t e m a j o r a t i o n , concernant

r de support E ,

m-aire CLI 12 5

P cu i c 6 .s

(mod.

:

~ ( a ) = be t

d'une

(mod.

t)

y(x)= x pour tout x E

rl(E-{a})

9:

la bijection

e t a,<x&b ( m o d . t ) ,

ble par r , modulo r

alors

(E-{a,b?).

On & t e n d a i n s i

DCs

(h,h+l,.

lors,

et

. . ,k) e

,

O
si r

notons c i , j m,n

Sm t e l s q u e O
pour t o u t groupe G c 1

l e lemme d e c o n s C -

m'

c u t i v i t g p e u t s ' g n o n c e r de l a faqon s u i v a n t e

:

i { i , j l e s t une

"

;In,

alors

Gqq.

Une r e l e c t u r e d e l ' a r t i c l e c l a s s e i n d i c a t i v e y(G)

1161 f a i t a l o r s comprendre que l a

peut dspendre de l a f i c h e H de G ( c 'e s t - l -

d i r e d u g r o u p e i n d i c a t i f maximum H p o u r H=T o u D ,

y(G)&Max(2,m-p-q)

Dans l e c a s d e l a f i c h e

H=Jr).

mieux p o u r m25

: y(G)$m-3.

par W.

A.H.

6

(si

s i s est libre-interprgta-

p a i r e d'ElCments c o n s 6 c u t i f s modulo l e G-rangement

m

telle

{a,b}eP2(E)) l a cons6cutivitg

(pour toute paire

des cycles

O
m >, 5

l'image de (E-{a})

i m p l i q u e l a c o n s P c u t i v i t 6 modulo s .

l'ensemble

-

est

(E-{b})+

a l o r s x = a o u x = b ) . En o u t r e ,

E t a n t d o n n P d e s n a t u r e l s n>m>,l

m,n

fin

{ a , b ? t P 2 ( E ) e s t formCe d ' C l 6 r n e n t s

e t s s o n t deux r e l a t i o n s de support E e t

c i ,j

trss

relation

r e l a t i o n une notion c l a s s i q u e pour une chafne t E f E

B toute

acb

une p a i r e

par

Pour

I e s a u t e u r s u t i l i s e n t u n lemme

lorsque l a relation

r)

l a relation rl(E-{b}) que

6 m-3 p o u r t o u t m 3 5 .

c ~ n ~ ~ c u f i w i f c V? i.s - s - v i s

n o t i o n de

a

y,

2 savoir :

classes indicatives,

HODGES,

C G ) , p a r exemple : y(G),<2

m-

pour H=Ip'q

( e t pareillement pour

triviale H=I"',

o n o b t i e n t &me

Finalement,

LACHLAN e t

S.

l e b i l a n de l a mgthode s u i v i e

SHELAH e s t

:

y,<

m-3

pour tout

( i n f o r m a t i o n 2 complCter p a r nos v a l e u r s y4=2 e t y =O pour

m

3).

2.3.3.

DCbut

de f i c h e

1983,

I"',

obtenue en

nous avons trouvE

t e l que y(G)=m-3

1976-77

( p o u r m > 3 ) un g r o u p e G c Z m ,

(d'oii i l r 6 s u l t e que l a majoration

par l e s auteurs prCcit6s 6 t a i t

l a meilleure

possible). P o u r m=2p o u m = 2 p + l , e t n > l , n o t o n s Sn l e s o u s - g r o u p e

m

d 6 f i n i de l a faqon s u i v a n t e ( I )

Si

I < n < p , S:

est

de S

m

:

l e g r o u p e (non i n d i c a t i f )

des permutations

256

C. Frasnay

conservant chacun des Elfments O , l , 2 , (au t o t a l

2n GlEments s u r m,

. . . , n-I

e t m-n,

m-n+l,

. . . ,m - 1

c e q u i l a i s s e s u b s i s t e r au moins

2 o u 3 ClSments i n t e r c a l a i r e s ) . S i nDp,

(2)

S t (d'ordre

S:

=

11"

I

=

m

(groupe i n d i c a t i f

et

l a fiche triviale II'I,

f a c i l e (s:)(')

=

Sn+ 1 m+l

'

I

trivial).

s i 2n)m)

i r i t e h c a t a i k e i d e Sm

appelEs l e s hOuh-ghoupe4 ve e s t

m

s i 2n&m, e t d ' o r d r e

(m-2n)!

Les groupcs

peuvent g t r e

: leur

fiche indicati-

ils s e d i l a t e n t s u i v a n t l a r 6 g l e

DBs l o r s , portons n o t r e a t t e n t i o n s u r l e (pour m>3)

premier groupe i n t e r c a l a i r e S'

m

:

a)

S i m = 3 , a l o r s S 1 = 1;" (groupe i n d i c a t i f t r i v i a l ) e t 3 1 S i m=4, a l o r s S 4 n ' e s t p a s i n d i c a t i f , m a i s s o n p r e m i e r y(S3)=0. I

dilatE (Si)(l)=S:

=

b ) P l u s gGnGralement, dilatEs successifs sont

(S;)(n)

=

I donc y ( S ) = I

est indicatif,

1;"

p o u r m 2 5 , S;

4

(mais y 4 = 2 ) .

n ' e s t pas i n d i c a t i f e t s e s

:

( S m1 ) (n) = S m - 3 2m-4

Sn+'. A i n s i , p o u r n = m - 4 , m+n

(laissant

s u b s i s t e r 2 ElEments i n t e r c a l a i r e s ) n ' e s t i n d i c a t i f . P a r c o n t r e , p o u r n=m-3 indicatif et

(finalement)

I

pas encore un groupe Sm-2 I,1 est : (SA)(n) = 2m-3 - '2m-3

y(Sm)=m-3.

En t e n a n t c o m p t e d e c e g r o u p e p a r t i c u l i e r t i o n obtenue dans [16],

S 1 e t de l a majora-

m

nous o b t e n o n s donc b i e n

: y,=m-3

comme

c l a s s e i n d i c a t i v e maximum p o u r t o u t m 2 5 .

2.4

De 1 9 6 3 1 1 9 7 6 , u n a u t r e r E s u l t a t - e l 6

c h a i n e s permutEes G-recollement

(prEsentC i n i t i a l e m e n t

des ordres totaux")

de l a t h 6 o r i e des

1 9 1 comme " t h E o r B m e d e

e s t apparu sous d i v e r s e s formes

S q u i v a l e n t e s . La f o r m e l a p l u s r E c e n t e de l a SociEtE f r a n q a i s e de Logique,

(annoncEe a u Colloque

S c i e n c e s , mais d o n t l a p u b l i c a t i o n a Ct6 r e t a r d s e problBmes d ' E d i t i o n ) G&Cm,

e s t a u s s i l a plus brPve

l a c l a s s e d e s G-rangements

1976

Philosophie e t MEthodologie des

1141 par des

: "Pour

est universelle".

tout

groupe

Le d e g r 6 d e

c e t t e c l a s s e ( c a r d i n a l i t 6 maximum d e s p s e u d o - G - r a n g e m e n t s )

e s t Egal

1 l a v a l e n c e v(G) du G-rangement n a t u r e 1 ( s u s c i t C p a r l a c h a ? n e -+

u s u e l l e a) : p a r a b u s d ' 6 c r i t u r e e t d e l a n g a g e , n o u s n o t e r o n s s i m -

Relations enchahables, rangements et pseudo-rangements

p l e m e n t v(G) y4OUpC

2.4.1

251

cette valence et nous l'appellerons la w u ! e ~ c e du

G. E x p l i q u o n s c o m m e n t c e t t e v a l e n c e v(G)

i n t e r v i e n t d a n s le

thGorSme de G-recollement. E t a n t d o n n 6 u n n a t u r e 1 n et u n e n s e m b l e E , o n dit qu'une f a m i l l e (Ai)icI d ' e n s e m b l e s e s t u n ~ - 4 C C 0 u w h e m e n t d e E l o r s q u e , il e x i s t e u n i n d i c e i t 1 tel q u e X & A i

p o u r tout X,Pn(E),

n o t i o n u s u e l l e de r e c o u v r e m e n t c o r r e s p o n d a u c a s n=l).

DSs

la c o n j o n c t i o n d e s d i f f 6 r e n t s t r a v a u x e f f e c t u g s d e 1 9 6 3 - 6 5

1 1972-73

(121C131)

f a i t a p p a r a i t r e l a v a l e n c e v(G)

(la lors,

(C911111)

c o m m e G t a n t le

plus p e t i t d e s n a t u r e l s n v g r i f i a n t la c o n d i t i o n s u i v a n t e : "Pour tout e n s e m b l e E d e c a r d i n a l 1El2n et p o u r t o u t e f a m i l l e d e c h a l n e s d e u x 1 d e u x G - c o m p a t i b l e s d o n t les s u p p o r t s f o r m e n t u n n-recouvrement de E ,

il e x i s t e u n e c h a f n e s d e s u p p o r t E q u i e s t

G - c o m p a t i b l e a v e c c h a c u n e d e s c h a f n e s t . ( p o u r i eI)". S i deux relations m-aires r , r '

d e s u p p o r t r e s p e c t i f s A,A',

sont d i t e s c o r n p a t i 6 L e h d S s q u e rl ( A n A')=r'l ( A n A'),

o n obtient une

v a r i a n t e ( t r s s proche) d e l a c o n d i t i o n p r 6 c G d e n t e e n p r e n a n t u n e f a m i l l e (ri)it I d e G - r a n g e m e n t s d e u x 1 d e u x c o m p a t i b l e s , c e q u i p e r m e t d ' i n t r o d u i r e u n e r e l a t i o n r l e s a d m e t t a n t c o m m e restrictions: r

i

rlAi.

=

D S s lors, l a valence v ( G )

est le p l u s petit des naturels

n t e l s q u e : "Si les n - r e s t r i c t i o n s d'une r e l a t i o n m-aire r (de c a r d i n a l i t s )n) 2.4.2

s o n t d e s G - r a n g e m e n t s , a l o r s r e s t u n G-rangement".

L e s t r a v a u x c i t e s prec6dernment ( n o t a m m e n t C 1 1 1 1 1 3 1 )

rnontrent a u s s i q u e la v a l e n c e m - a i r e m a x i m a l e v m ( c a r d i n a l i t s m a x i m u m d e s r e l a t i o n s m-aires "finies" e n c h a l n a b l e s et d g f o r m a b l e s ) e s t l a p l u s g r a n d e d e s v a l e n c e v(G)

l o r s q u e G p a r c o u r t Cm. Autrement

d i t , p a r m i l e s r e l a t i o n s m-aires e n c h a l n a b l e s et d s f o r m a b l e s d e cardinalits finie maximum v o n est assurg de trouver certains m' G - r a n g e m e n t s . N o u s d i r o n s qu'un g r o u p e G c 1m e s t u n g4Oupe C 4 i t i g u e s'il e x i s t e u n p s e u d o - G - r a n g e m e n t d e c a r d i n a l i t s m a x i m u m v m (autrement d i t : s i v(G) = vm). a) Nos c a l c u l s 1111 et c e u x d e P. J U L L I E N [ I 8 1 o n t m o n t r 6 q u e l a v a l e n c e d e s g r o u p e s i n d i c a t i f s G c C m' e s t e n g e n 6 r a l v(G)=m+l. S e u l s f o n t e x c e p t i o n les g r o u p e s s y m s t r i q u e s S m et l e g r o u p e 2

J 2 : v(S )=m et v(J4)=6. 4 m

C Frasnay

258 b) L o r s q u ' u n g r o u p e G c E

m rnontrg [ I l l que s a v a l e n c e v(G)

n ' e s t pas i n d i c a t i f , n o u s a v o n s e s t li6e 2 s a c l a s s e i n d i c a t i v e

p a r l'insgalitg : v(G)<m+y(G)tI.

C e t t e i n 6 g a l i t E (valable 2 e n c o r e p o u r les g r o u p e s i n d i c a t i f s a u t r e s q u e J 4 ) , j o i n t e B la y(G)

m a j o r a t i o n d e y m (HODGES tion : v

m

<

-

LACHLAN - S H E L A H ) ,

c o n d u i t 5 la m a j o r a -

2m-2 p o u r tout m 3 5 .

c) P a r a i l l e u r s , n o s c a l c u l s d i r e c t s r l I I a v a i e n t fait conna'itre tous les g r o u p e s c r i t i q u e s d e d e g r 6 m C 4 et les v a l e u r s : 2 v =0, v l = l , v 2 =3, v 3 = 4 , v 4 =6. P a r e x e m p l e , le g r o u p e i n d i c a t i f J 4 et le g r o u p e a l t e r n 6 A 4 s o n t d e s groupes c r i t i q u e s d e d e g r 6 4 . F i n a l e m e n t , la m a j o r a t i o n v < 2 m - 2 v a u t p o u r tout m23. 'm 2.5

Nous allons dsmontrer l e th6orSme suivant (nouveau, 2 notre

connaissance)

Lai4e SA 2.5.1 11, I

v(s,)

I

:

e h t un

' ' P O U ~t o u t natu4e.l m > 3 , L e p4emiefi gficrupe i n t e . f i c a g h o u p ~c f i i t i q u e . de. v a l e n c e v m - 2 m - 2 " . O n s a i t (2.3.3) q u e S mI e s t un g r o u p e d e f i c h e

Preuve :

,

I

d e c l a s s e i n d i c a t i v e y ( S )=m-3 ( m a x i m u m s i m2,5), d e v a l e n c e m I I m+y(Sm)tl = 2m-2. P u i s q u e v(S ) v 6 2m-2 ( 2 . 4 . 2 ) , i l m m

<

s u f f i t d e p r o u v e r q u e 2m-3 < v(S,)

1

: n o u s a l l o n s le f a i r e e n u t i -

l i s a n t le t h 6 o r b m e d e r e c o l l e m e n t des cha'ines et e n r a i s o n n a n t par l'absurde. S i v(SA)<2m-3,

o n pourrait appliquer la condition de

1

S - r e c o l l e m e n t ( g n o n c 6 e e n 2.4.1)

m

E

=

2m-2

=

{O,l,2,

. . . ,2111-31

a u x n a t u r e l s n = 2 m - 3 et -f

=

ntl. Soit r=wlE la chaine usuelle :

vis-5-vis d u g r o u p e i n t e r c a l a i r e G r Z (G=Szrn-2 m-2

=

lit6 d e r d a n s

(m-Z,m-I)}),

cE.

d e degri? 2m-2 et d ' o r d r e 2 ntl s o i t {r,sl la c l a s s e d e G - c o m p a t i b i -

I n t u i t i v e m e n t , o n peut d i r e q u e l a c h a i n e s se d E d u i t d e l a cha'ine u s u e l l e r e n i n t e r v e r t i s s a n t les d e u x 6 1 6 m e n t s m g d i a n s m-2, m-l

:

. . <m-3<m-2<m-l<m< . . . < O
O<1 <.

Sur chacune des n-parties Ai u n e cha'ine t i (de s u p p o r t A . )

2m-3

(mod. r)

2m-3

(mod. s)

=

E-{i}

e n posant :

d e E (OCiCn), d g f i n i s s o n s

259

Relations enchaina bles, rangernents et pseudo-rangenzer~ts pour 0 6

i : m-3

t.

=

rl A;

L.

=

r l A i = s j ~ .p o u r i = m - 2 et i=m-l

t. = s ( A i pour

m 6 i 6 2m-3

O n o b t i e n t d e s p a i r e s i t . , t . ! dt. c h a l n e s c o m p a t i b l e s (donc 1 . 1

S I - c o m p a t i b l e s ! ) p o u r : 0 6 i f j < m - l et m - 2 < i c j < 2 m - 3 ( n o t a m m e n t (i,j)

=

(ni-2,m-I)

d a n s c h a c u n de c e s d e u x cas).

avec t . = r l A i , t . = s l A J j' A i n A . = E - { i , j 'i, I A . fl A . = 2 m - 4 , et d G s l o r s t i ( A . f ) A , ) , t . ( A i (1 A . ) 1 l J 1 1 J J m- 3 ni- 3 e s t aussi le dilate sont S 2 m - 4 - c o m p a t i b l c s . Comiiie le g r o u p e S2m-4

iieste le cas

: OciGm-3 e t

m;ji2m-3,

I

I

(s

i1)

)

(

1

, o n e n d 6 d u i t e n c o r e q u c t 1. , tJ. s o n t S mI - c o m p a t i b l e s . I

n (et p u i s q u e ( A 1. ) o
D a n s l ' h y p o t h c s e v(Sm)

n - r e c o u v r e m e n t "exact" d e E=n+l),

le thi?ori!me

est un

d e r e c o l l e m e n t des

d'une c h a f n e 0 d e s u p p o r t E q u i

c h a l n e s i m p l i q u e r a i t l'existence

s e r a i t S I - c o m p a t i b l e a v e c cliacune d e s c h a f n e s t i . O n a u r a i t : m 1 elAi t i (mod. S ) p o u r O s i G n , a v e c I A i / = 2 m - 3 . C o m m e l e dilate d e m ~

et les c h a l n e s t i s e r a i e n t d e u x 5

(pour Oci(n),

aurait : t. = B I A i

d e u x c o m p a t i b l e s , e n c o n t r a d i c t i o n a v e c m-2rm-I

(mod. t . )

et

m-lcm-2 (mod. t.) p o u r 0 6 i C m - 3 , m < j < 2 m - 3 . J

I 1 e n r e s u l t e : 2m-3Cv(S bien apparaftre

2.5.2 dente),

I

1

)<2m-2, d o n c 2m-2=v(S )
Corollairement

( d a n s les n o t a t i o n s d e la p r e u v e p r 6 c 6 -

p u i s q u e l e s cha?nes t . s o n t S ' - c o m p a t i b l e s , e l l e s s u s c i t e n t m I

s u r l e s n - p a r t i e s A; d e E d e s l S - r a n g e m e n t s c o m p a t i b l e s SLxtjm).

m E u n e et u n e s e u l e r e l a t i o n m-aire 11 t e l l e q u e

I1 en rgsulte SUP I $ ( A i = Smxtim) ( p o u r Osil'n).

P o u r O<,k*n, t o u t e s l e s k - r e s t r i c t i o n s

1

d e UJ s o n t d e s S - r a n g e m e n t s : p a r c o n t r e $=(m,E,p) (de c a r d i n a l i t 6 m n + l = 2 m - 2 ) n ' e s t p a s un S I - r a n g e m e n t , p u i s q u ' a u c u n e c h a l n e 0 (sur m E) n'est S 1 - c o m p a t i b l e a v e c c h a q u e cha'ine t i . D o n c $ e s t un p s e u d o m r a n g e m e n t m - a i r e d e c a r d i n a l i t 6 m a x i m u m v m = 2m-2 ( p o u r m>3). S o n g r o u p e d ' i n v a r i a n c e e s t Inv((l)=Sm caractgristique est

101

I

=(m+i)(m+2)

et l e c a r d i n a l d e s o n e n s e m b l e

. . . (2m-2).

260

C. Frasnay

2.5.3

La proposition selon laquelle des rangements m-aires

(notamment des S1-rangements) figurent toujours parmi les relations m m-aires enchafnables critiques se laisse prgciser, pour tout groupe G € C m , de la manisre suivante ( 1 2 1 1 1 4 1 )

:

a) Soit (t,r) une birelation monomorphe de support E , de cardinalit6 ( E l >/ m, comprenant une chaine t et une relation m-aire r (autrement dit, r est une relation encha?nable et t est l'une des chafnes par lesquelles r est libre-interpretable).

ex

partie X de E , soit t / X ) sur m={O,l,

l'injection croissante de X (ordonnge par +

. . . ,m-I}

(ordonng par la chafne usuelle m ) .

Lorsque

'L

X parcourt P,(E), r'

Pour toute m-

la relation €I (rlX) garde une valeur constante X e x : X + m est l'unique isomorphisme

(m,m,p) et la bijection

=

-f

de la birelation (tlX, rlX) vers la birelation (m, r'). groupe d'automorphismes G=Aut(r')

est un sous-groupe de S m dont la

connaissance redonne le groupe d'automorphismes Aut(r/X)

{zi o

=

u

8x

o

*

a

u

E

Puisque le

locaux :

G j , o n peut dire (par abus de langage)

que G est le " g t r a u p e l?acaf?l' (mod. t) de la relation enchainable r. La relation r' est l'une des 2v(m) m={O,l,

. . . , m-I})

relations m-aires (de support +

libre-interprgtables par la chaine usuelle m ( u ( m )

d6signe le nombre des preordres totaux de support m : v(O)=u(l)=I, v(2)=3,

u(3)=13,

v ( 4 ) = 7 5 , u(5)=541,

G d Cm, le G-rangement r=Gxt(m)

. . .) .

Par exemple, pour tout

suscit6 par la chafne t admet G

comme "groupe local" (mod. t) puisque r'=(m,m,G). b) D S s lors, l a preuve de ( [ I l l , v(G)

13.1.2)

montre que la valence

est la cardinalit6 maximum des relations m-aires "finies'',

enchafnables et dgformables, admettant G comme "groupe local". Le th6orSme 2 . 5 concerne les cas oii ce "groupe local" est le premier groupe intercalaire S;

v(S

= S l ,

1

3

I

S2

=

I:",

s: . Pour mi3, ce groupe est indicatif S:

=

I;",

:

1 1 de valence v ( S 1 ) = l = v l , v(S 2 )=3=v2,

)=4=v3. D ' o C les deux propositions : (I)

Pour

1 LOU^ m>,l, Sm est un groupe critique (autrement dit, sa

1

valence v ( S )=v est la cardinalit5 maximum des relations m-aires m m "finies" enchaynables et dEformables). (2) v = I , 1

v = 3 et v =2m-2 2 m

pour tout m a 3 .

Relations enchainables, rangements et pseudo-rangements c) B i e n e n t e n d u , les g r o u p e s S ;

n e s o n t pas n E c e s s a i r e m e n t les

seuls groupes critiques. En dehors de S

I

l

l

S 2 , S 3 , il y a a u s s i 5 1 2 , I32,l , T 3 , .J3, .I4.

autres groupes indicatifs critiques : I3 le d e g r 6 4 , le g r o u p e [ c 4 , (1,3)1

261

Pour

(de f i c h e i n d i c a t i v e I"')

et

1

le g r o u p e a l t e r n e A 4 (de f i c h e i n d i c a t i v e J ) s o n t d e s g r o u p e s criI

tiques non indicatifs autres que S 4

=

{ t 4 , ( l , 2 ) } .

Pour m > 4 , 1'inE-

\< m+y(G)+I

g a l i t 6 v(G)

e t 1 ' 6 g a l i t E v = 2 m - 2 m o n t r e n t qu'un g r o u p e m c r i t i q u e G n o n i n d i c a t i f doit a v o i r la c l a s s e i n d i c a t i v e m a x i m u m y(G)=m-3,

m a i s c e t t e c o n d i t i o n n k c e s s a i r e n'est p a s s u f f i s a n t e

(par e x e m p l e , p o u r le g r o u p e G v(G)=5).

=

{r,,

1

m

)=2m-2,

de d e g r E m=4:y(G)=l

et

I

- y ( S )=m-3 n ' i m p l i q u a i t pas m une d s m o n s t r a t i o n d u t h G o r E m e 2 . 5

Pour cette raison, puisque

n g c e s s a i r e m e n t v(S

(O,3)}

s ' i m p o s a i t !.. D e m z m e , l e s d e u x e x e m p l e s d e H O D G E S - L A C H L A N - S H E L A H [I61

(montrant q u e l e u r m a j o r a t i o n y m 6 m - 3 e s t "pointue", s o i t -clans I 1 d o n n e n t seulement:v(A ),<8,v(S & l o . 5 6 En reprenant la mgthode de L 1 6 1 donnant pour y(G) une majoration qui depend de

n o s n o t a t i o n s : y(A5)=2,y(S6)=3)

la f i c h e i n d i c a t i v e H de G , o n p e u t c o n j e c t u r e r q u e H=I'"

ou J

1

pour t o u t g r o u p e c r i t i q u e G de d e g r E m > 6 . Les g r o u p e s a l t e r n E s A =I1", 2

2

A =T3 e t A 4 s o n t d e s g r o u p e s c r i t i q u e s (et i l e s t p o s s i b l e 3

q u e A 5 le s o i t Ggalement). C o m m e l a f i c h e i n d i c a t i v e de A v a u t m I J , D , 1'" , T r e s p e c t i v e m e n t p o u r m F O , 1 , 2 , 3 (mod. 4 ) , il e n ) ~ l 2 orsque m est impair : ainsi A n'est pas u n g r o u p e m m c r i t i q u e p o u r m i m p a i r > 7 , p a r c o n t r e i l est p o s s i b l e ( ? ) q u e A m

r 6 s u l t e y(A

s o i t u n g r o u p e c r i t i q u e p o u r tout m p a i r > 2 .

2.6

P o u r t e r m i n e r c e t t e E t u d e , n o u s a l l o n s examiner(d'apr2s

M. P O U Z E T , 1221) c o m m e n t l a t h e o r i e d e s c h a f n e s p e r m u t g e s p e u t c o n t r i b u e r 1 l a d g t e r m i n a t i o n d e c e r t a i n s g r o u p e s de p e r m u t a t i o n s d'un e n s e m b l e E , d i t s g r o u p e s h o m o g P n e n . D a n s l a d g m o n s t r a t i o n d'un t h 6 o r S m e 1 c e s u j e t p o u r E inf i n i , d G 5 P . J . C A M E R O N (II ] ,

§

6 ) , les

1

, J m , T m , D m (ainsi q u e les c o m p a t i b i g r o u p e s i n d i c a t i f s S m' 1''' m l i t g s a s s o c i E e s d a n s l'ensemble des chafnes sur E ) intervien-

XE

nent d a n s u n e t e r m i n o l o g i e v o i s i n e d e l a n 6 t r e - t o u t e f o i s l'auteur n ' u t i l i s e p a s l a p r o p o s i t i o n 8.3.2. de r i l l r s p e r t o r i a n t t o u t e s l e s suites indicatives ( S ,

IP q , J r , T , D) e n g e n d r g e s p a r les b i c h a f n e s 9

(s,t) d e s u p p o r t E i n f i n i .

C. Frasnay

262 2.6.1

l e s 8 r e l a t i o n s binaircls

I'armi

et

l e s 8192 r e l a t i o n s

"

un r 6 l e t e r n a i r e s l i b r e - i n t e r p r 6 t a b l e s p a r une chaEne t c 'C' (dont l'ensemble

p r i v i l6 g i 6 e s t accord6 2 l a chafne Upl~uSCct ' c a r a c t 6 ri s t i q u e e s t celui des

(x , x , ) e E2 t c l s q u e x 1 6 x o ( m o d .

e t 2 l a r e l a t i o n C ~ h c u ~ ! u c h ce d I 5 r i v g e d e t

x 1 <, x 2 6 x o o u x 2

<

xo

<

la relation circulaire c' (1)

y

(t)

t

(2)

4'

y(t)=t

Pour 2 $ m < l E l ,

6 SE transforme

E

(t)

On l e u r a d j o i n t g g a l e m e n t

de t e s t

-2 E'

l a chaene t e n une c h a r n e

l a c l a s s e de I 1 ' l - c o m p a t i b i l i t G de t e s t { t i .

m

'f

l e groupe des permutations

: ces permutations

y

(ou

de t e s t { t , t ' ) .

yc

( t ) l e groupe des permutations

SE t e l l e s que

E 'L t r ( t ) = t ' , ce groupe e s t a u s s i c e l u i des permutations

%

y ( t ) = t ou

t-monotonen

que

Aut(t).

=

(3) P o u r 3 < m < l E I , l a c l a s s e d e J ' - c o m p a t i b i l i t 6 m note J

SE t e l l e s

s o n t l e s automorphismes de t

t - ~ K o i n o n n f e o ) ,d o n c IE(t)

permutations

S i l'on

011

( m o d . Sm).

On n o t e e n c o r e I rv

t)).

x2

dSriv6e de l a chafne t ' .

I<m$lEl, l a c l a s s e d e S - c o r n p a t i h i l i t 6 m

Pour

e t toute permutation

-.

x 1 (mod.

<

xI

: xo

( x o , x I , x 2 ) e L3 v g r l f i a n t

t g r i s t i q u e e s t c e l u i des

t))

(dont l'ensemble cnrac-

(autrement d i t : t-croissantes

ou t-dEchol46nntel).

( 4 ) Pour 3<mhlEl, l a c l a s s e de T - c o m p a t i b i l i t G d e t e s t consm t i t u g e p a r l e s c h a ' i n e s s u r E a d m e t t a n t c comme r e l a t i o n c i r c u l a i r e

ye

d6rivGe. Pour t o u t e permutation

SE,

la relation circulaire

Y

%

~ ( t n) ' e s t a u t r e q u e

d6riv6e de l a c h a i n e

l e groupe des permutations

T E ( t ) = A u t( c ) .

y

(c).

S i TE(t) d6signe

'L

Y

sE

que

y,(c)=c,

alors

:

(5) Pour 4,<m$lEl, l a c l a s s e de D - c o r n p a t i h i l i t & de t e s t consm t i t u s e p a r l e s c h a f n e s s u r E a d m e t t a n t c o u c ' comme r e l a t i o n c i r c u l a i r e d 6 r i v 6 e . On n o t e DE(t)

y

g

SE t e l l e s que

-

'L

~ ( c ) = co u

Par abus d ' c c r i t u r e ,

'L

y

l e groupe des permutations (c)=c'.

lorsque l a chayne t e t l a r e l a t i o n

c i r c u l a i r e dGriv6e c s o n t f i x s e s une f o i s p o u r t o u t e s , on n o t e s i r n p l e m e n t I E , J E , T E , DE l e s g r o u p e s p r 6 c 6 d e n t s .

On p e u t d i r e a l o r s

q u e I E ( r e s p . T E ) e s t l e g r o u p e d e s p e r m u t a t i o n s d e E q u i C(JnA@KW C I ? ~l a c h a i ' n e t ( r e s p . l a r e l a t i o n c i r c u l a i r e c ) , e t q u e J ( r e s p .

E

D ) e s t l e g r o u p e d e s p e r m u t a t i o n s d e C q u h conneKwen-t E ntnt la chaIne t (resp. l a relation circulaire c ) .

o u Ketouh-

Relations enchainables, rangements et pseudo-rangemento 2.6.2

So1t

c

sb . Pour n

a)

treillis des sous-groupes

$1,

un g r o u p e G e

~ o u p l e( X , Y )

tout

'f

le

de n-parties

y

2 G t c l l c que

on d i t

simplement

X'=

que G est

permutation e t

iil

(ce qui

i

I!J

sE

de

E,

S i G

dit

est

E

du g r o u p e s y m 6 t r i q u e

~i-/ioniug?uie d S s

que,

pour

tout

infini),

on d i t qu'une

e s t ccdhPtrcntc 2 u n g r o u p e ~t z E s i , p o u r m x t 1; , i l e x i s t e y 6 G t e l q u e \)ox=

2 d i r e que

neiri,

hort~cigCnp.

t o u t m-uple

revient

pour

i l e x i s t e une p e r m u t a t i o n e s ~t n - ~h o m o g s n e

(essentiellement pour E

Par a i l l e u r s

m

Y.

il:

263

y

e t

h/J

o n t mzme

tout

Y

Ox

sur la

restriction

...,

p a r t i e f i n i e X={xo,xl, x 1 de E ) . L'ensemble d e s p e r m u t a t i o n s m- I d e 1: a d h 6 r e n t e s a u g r o u p e G e ), e s t a l o r s un g r o u p e G e IE ( q u ' o n E p e u t a p p e l e r ( c t n ~ c t u k cde G ) c t , l o r s q u e G=G, on p e u t d i r e q u e l e

$etitni'.

groupe G e s t b)

Lorsque

l'ensemble

des chaTnes s u r E c o n t i e n t

E est

i n f i n i dgnombrable,

un s o u s - e n s e m b l e

d e s c h a E n e s d Q n b c 5 sans m i n i m u m n i m a x i m u m chafne

u s u e l l e du corps ordonn6 Q des

peut Gtablir G

soit

&

IiomogPne,

et une

t e l telles

quc Aut(R)

-

= G

indicative

(H

DSs

: "Pour

-

Lemme

Sc A

a P u e 5 La elllYle

A

U

(s,t) )

de support

infini E,

on peut est

c t t A u n t d c u x c h a i n e s d c mPnic ~ ~ p p o f i .Et e t ~

n4

iCn d t c a t ( w e

dc4 5 Ju4tt5 En v e r t u d e

et

suppose que t

s'il

pour

identiques

m>2r,

s, r 1 J

H e n g e n d k h e pati La J , J 1 , T , D.

la proposition 8.3.2

( p a r exeniple)

montrons qu'alors

ou opposEes

par

t

associer

le groupe

5 t

L'une

maxcmum ( c e q u i impose que E e s t i n f i n i ) , bcchaine

de [ I l l ,

(4,X)

ebt

la preuve consiste

I p ' q ou J r p o u r r = M a x ( p , q ) 3 2 .

n'a

n i minimum n i maximum d a n s

e x i s t e un n a t u r e 1 r < 2 t e l q u e s e t

tout

on

groupe

e x i s t e une chazne

dont chaque

2 Gcarter toute suite indicative l'on

lors, qu'un

libre-interprGtable

m m e 111 tfrme H m G m ' d e c o r n p a t i h i l i t 6 minimum d e s , t p o u r l e d e g r E m .

d ' e t ! c ) A n ' a nc mcnmuni

e,

: celui

."

c ) A toute bichafne la suite

"22 I

il f a u t e t il s u f f i t q u ' i l

structure relationnelle R

'

+

isomorphes 1 l a

(toutes

rationnels).

la caractcrisation suivante

l'ensemble

remarquable

t

soient

Si E,

Jr-compatibles

m 1 s e t t sont J3-compntihles

: s = t ou s = t ' ) . P r e n o n s m=2r+3

: pour

(donc tout

264

C. Frasnay

triplet t n'a

u

( x ~ , x ~ , x ~t e)l ~q uEe ~x < x < x 2 ( m o d . 0

n i minimum n i maximum)

= xo,

sait (d'aprss

Jr m

I

=

Jm

0

t e l l e que u

(7

1

I r , r) que

* '

t).

< x,

< x2 o u x 2 < x

l o J3), c e qui gquivaut

< x

1

(d'aprPs

m

s ) , on

(r,r+l,r+2)

:

s ) . Finalement 6.2.3)

[ I l l ,

.Jr

IrPr e t m

le triplet (mod.

t e l que

u e

Si (mod.

.<' o(m-I)

"conserve ou retourne"

0

m

(mod,

(mod.

m- 1

(o)
,. . . , u m- 1 ) e E m

l a d6finition des groupes i n d i c a t i f s

il e n r h s u l t e x s :t

(uo , u l

~ + ~ e= t xu ~< u < . . . < u

U ~ + ~ = X ] u ,

la permutation

est

u n m-uple

il existe (puisqe

t ) ,

1

1 : s = t

ou s = t ' .

Remarque

: En m o d i f i a n t

lgggrement l a r6daction de la preuve,

nous

a u r i o n s pu prendre m = 2 r + l . d ) L o r s q u e E e s t un e n s e m b l e i n f i n i d g n o m b r a b l e , structure relationnelle chafne t t

2,

(de support E)

( c h a f n e d e n s e sans m i n i m u m n i m a x i m u m ) .

c o u r t l e groupe des automorphismes de R, prGc6dent 1 l a bicha7ne c e t t e technique,

: "Lorsqu'une

support E i n f i n i dgnombrable,

,2E,a l o r s

5 groupes SE, I E ( t ) , J E ( t ) ,

2.6.3

Rappelons qu'un

tout couple tation O
ye :

=

yi).

'p

par-

~ ( t ) .E n u t i l i s a n t la propo-

structure relationnelle

(de ses automorphismes)

R,

de

p a r u n e chai'ne est l'un

des

TE(t), DE(t)".

groupe G

(x,y) de n-uples G t e l l e que

'f(xi)

=

est libre-interprgtable

l e groupe Aut(R)

y

Quand

21

t e l l e que s

(s,t)

p a r une

o n p e u t a p p l i q u e r l e lemme

POUZET d s m o n t r e a l o r s t r e s s i m p l e m e n t

M.

sition suivante r22J

t t

s o i t K une

libre-interprgtable

e s t d i t tz-Zfiun4ibi6 s i , p o u r

):E

i n j e c t i f s dans E, o x = y

il e x i s t e une permu-

(autrement d i t ,

pour

Un t e l g r o u p e e s t 6 v i d e m m e n t n - h o m o g s n e ,

mais p a r e x e m p l e ( l o r s q u e l e c o r p s Q d e s r a t i o n n e l s e s t o r d o n n 6 p a r sa cha'ine u s u e l l e )

I-transitif 2-transitifs 3-transitif

: l e g r o u p e homogsne

s a n s Gtre 2 - t r a n s i t i f ,

s a n s letre 3 - t r a n s i t i f s , sans S t r e 4-transitif.

q u e SQ e s t n - t r a n s i t i f Rgciproquement, relationnelles

I

Q

est

les groupes homogenes J

l e g r o u p e homog6ne D Bien entendu,

Q

Q

et T

Q

sont

est

le groupe symstri-

pour t o u t n t w.

l a p r o p o s i t i o n de M.

POUZET s u r l e s s t r u c t u r e s

de s u p p o r t i n f i n i dhnombrable ( s u s c e p t i b l e d ' u n e

preuve simple u t i l i s a n t

l e lemme de l a t h 6 o r i e d e s cha'ines

permutges

Relations enchainables, rangements et pseudo-rangements

265

cit6 prGc6demment) donne comme corollaire une version du beau t h 6 o r S m e de P.J. C A M E R O N ( r l l ,

6) : So4f G u n g k u u p e AekrnP de

§

pe+tmutatcon5 d l u n e n b e m b t e E c n d i n c dPnornbkable. l o k d q u e G e d t n-tfian4ct4d p o u k t v u t n e u , n l o f i b G = S E .

e t n-tfian54t4d

ban5

e x 4 5 t e une c h a i n e t e tetPc que : ( I )

o u bien

(2)

OU

(3)

ou b i e n

2.7

2,

G e b t hornog@ne

i c h a i n e d e n J e nanb m-cn-cmum n c max-cmum)

et G

n

=

1,

b ~ e n n

=

2, e t G

S u r le m g m e

Lokdrjue

C t a e [n+l)-tfianbct4d, a t o m n = 1 , 2 , 3 e t it

n = 3,

et G

=

IE(t),

= J E ( t ) 011 =

T,(t),

DE(t)."

thsme, signalons 6galement un article d e

G. H I G M A N 1 1 5 1 d a n s l e q u e l l'auteur r e t r o u v e (dans u n e t e r m i n o l o g i e propre) p l u s i e u r s t h 6 o r S m e s i m p o r t a n t s c o n c e r n a n t : d'une p a r t les r e l a t i o n s m o n o m o r p h e s (qu'il

a p p e l l e r e l a t i o n s homogcneb),

d'autre

part l e s g r o u p e s d ' a u t o m o r p h i s m e s l o c a u x d e s r e l a t i o n s e n c h a f n a b l e s (la t h s o r i e d e s c h a i n e s p e r m u t g e s e s t , e n f i n d e c o m p t e , l'6tude d e c e s groupes). 2.7.1.

P a r e x e m p l e les p r o p o s i t i o n s 2.1 et 2 . 3 de 1 1 5 1 s o n t les

v e r s i o n s i n f i n i t i s t e s des p r o p o s i t i o n s 12.2.1 et 5 . 4

d e 1111 s e l o n

l e s q u e l l e s ( 2 p a r t i r d e s e u i l s f i n i s c o n v e n a b l e s n ( m ) et 5 ( n ) m a s s o c i 6 s a u x c a r d i n a u x m < w , m l i ( m )

e s t e n c h a f n a b l e " et (2) ( l e m m e

d ' e n c h a f n a b i l i t 6 r e s t r e i n t e ) "toute b i r e l a t i o n de c a r d i n a l i t 6 >/c,(n),

f o r m g e d'une c h a f n e et d'une

n - r e s t r i c t i o n monomorphe".

relation m-aire, admet une

L a m 6 t h o d o l o g i e de la p r e u v e d e ( 1 ) ( 2 )

s e m b l e a u s s i s'imposer a v e c u n e s o r t e d e n 6 c e s s i t 6 (puisqu'il s'agit de p r o b l s m e s c o m b i n a t o i r e s "1 s e u i L s num6riques")

: l'utRka6iLtna-

t i o n , c o m m e t e c h n i q u e d e p a s s a g e d u f i n i 2 l'infini e t r g c i p r o q u e m e n t ( [ I l l , 1.3 et 3 . 4 ) , qu'on a i t d 6 m o n t r 6 ( 2 )

intervient dans la preuve de ( I )

k

r e l a t i f a u x a p p l i c a t i o n s f : P,(E)

(nornb4en de R A M S E Y

aprSs

e n f a i s a n t a p p e l a u t h 6 o r S m e d e F.P. R A M S E Y +

F p o u r (F(,
: il e x i s t e a l o r s u n e n - p a r t i e X d e E p o u r

l a q u e l l e la r e s t r i c t i o n d e f 1 P (X) est constante). m 2.7.2.

P a r a i l l e u r s (sous les n o t a t o n s

retrouve les suites indicatives H

=

s,

Hm)m 6 w

A"~,

B',c,D)

G. H I G M A N

d o n t il c o n f i r m e

266

C. Frasnay

e n montrant: que l e u r s

l'intGri5t

apparaissent

toujours,

c e r t a i n rang)

de manicre asymptotique

dans des

"order-coherent

2 nos s u i t e s d e d i l a t a t i o n

titre

quc

(G("))n,

travaux de M.

les

"pour

toute

nalitG

Ji,

TE, D E "

Pour m <

seuils [inis

de

Aut(r)

de

c a r d i n a l 2\J(m))

est

par

son " g r o u p e

on peut

dire aussi

t o t a u x d e s u p p o r t m=[O,l,

quc H e s t

l a "fiche

Pour m
r e rm(n)

<-I

-' r

' t

HIGMAN,

r

(n')

une "classc

indicative"

( v n ) (rn+C'j(r)
(maximum d e s indicative

4m

r m ( i L i ) ) . I1

est

: 'A4

l'6galiti.

(qui

indicative"

a(r)

de

la

:

relation

canonique l ~ m ( n ) ,r m ( n ' ) )

En v e r t u d u t l i G o r S m c d e

(de fiche

qui est

indicative

H)

admct

aussi

l e n a t u r e l minimum t e l q u c

-.:+ A u t ( r / n ) = H n ) naturel

G,

on r e t r o u v e

y,
facile d'obtenir

% 2 e t =

rs

15

1

=

ii2

= 0,

i n a i s nous i g n o r o n s

( c l a s s e i n d i c a t i v e maximum d e s 8 1 9 2 r e l a t i o n s

encha9nables

f,

fm ( f o )

l'inGgalit6

d G j 2 l a v a l e u r <S3 ternaires

r i m

m e n t r e l a c l a s s e i n d i c a t i v e y, y(G) l o r s q u e l e g r o u p e G p a r c o u r t 1. ) c t l a c l a s s e m (maximum des 6(r) lorsquc la relation r parcourt

d'oti

y(C)=tS(d),

. . . ,m - 1 1 ) .

sa r e s t r i c t i o n

bijective

: r=r'In.

s i r e s t l e G-rangement

Bien entendu,

par

( e n t r e les dcux ensembles

relation r e

toute

(on r a p p e l l e que .'(m)

11

1 o c a 1 " G = ~ u t ( r J m ) ,d e f i c h e i n d i c a t i v e tl

est definic par restriction

ment

usuelle

m

fournit

indieatif

CAMERON ! I ] ,

notons I '

I' (n) e s t d G t e r m i n P e

6

de cardi-

(n) l'cnsemble f i n i de m 1 e s r e l a t i o n s m - a i r e s ( d c s u p p o r t n)

111,

l a cha'ine

l e nombre d e s p r e o r d r e s

Toute r e l a t i o n r

r).

C

constitug par

libre-interprgtables

t c l s clue

!I

SE e s t d e t y p e

(dans

e t m ,C n

[,I

I15 I renuu-

l a t h e o r i e d e s chaEncs permu-

l e "main theorem" d e P . J . r: I , 1 s e u l s i n t e r v i e n n e n t l e s t y p e s S E , Tli , J E1 , T E , D E ) .

2.7.3.

A u InSme

57 et 8 ) .

enchaPnable r de s u p p o r t E c t

le sous-groupe

m'

d'un

un peu a n a l o g u e s

d e 6 . HlGMAN

travail

le

l'existence

r e l a t i o n m-aire

1El>m+6

SE, I S ' t ,

(I I I I ,

st!

v e l l e e t approfondit avantageusemcnt : il montre cn e f f e t

(2 partir

sequences"

POUZET ( 1 2 1 1 1 2 2 1 ) e t d e W . H O D G E S -

A.H. L A C H L A N - S. S H E L A H 1 1 6 1 , tGes

(les groupes i n d i c a t i f s )

tcrmes H

r t 1 7 3 ( t ~ ~ ) ) . Notre calcul de

f i m >, m - 3 ?

pour

tout m 3 5.

A-t-on

y ni d o n n e s e u l e (ou non)

:

Relations enchaina bles, rangernents et pseudo-rangernents

267

B I B L LOGRAPH I E

I I 1 CAMERON, ordered

I:!I C L A R K ,

I).

EKDOS, try,

sets,

P.,

and

R.,

relations,

I 5 I FRATSSE, formule

R.,

des

par

R.,

(1971),

1 9 i FRASNAY,

Alger,

et

e t

1974),

M.,

POUZET,

C.R.

Acad.

I'OUZET,

Sc.

Pf.,

Paris,

Sur une

C.R.

de permutations

finies

C.,

Groupes

de cornpatihilit;

thGorie

A-259

Quelques

totaux et

(Grenoble,

1 1 2 I PRASNAY, C . , tntaux

(1964),

les

problcmes

1624-1627.

relations

Sc.

e t

Paris,

faniilles

relations,

de deux o r d r e s

d'orcires

Acad.

C.R.

r e l a t i o n m-aire,

combinatoires

d e G-recollcnient

: Permutations

in

Villars

1974),

mcnt d ' u n 865-868.

(1971),

Acad.

monomorphes,

; application aux relations

C.,

d'une relation

Sc.

totaux C.R.

;

Acad.

concernant

Ann.

Tnst.

les

Fourier,

1965) 415-524.

multirelations,

I I 3 I FRASNAY,

Gauthier-

3910-3913.

relations

'I'h6orGme

(Paris,

des

d'une

a p p l i c a t i o n a u x n-morphismes

15-2

A-272

f i n i de bornes,

2944-2947.

C.,

en th6o-

1972),

275-278.

(1963),

ordres

et

102-113.

relations

(Paris,

c l a s s e de

A-257

I I I FRASNAY,

(Relation

1971),

Interpr6tabilitC

Paris,

Paris,

t. I

35-182.

211-228.

; a p p l i c a t i o n '2 l a

Sc.

,

les

: Permutations

in

systsmes d e

(1954),

(Paris,

totaux

1 1 0 1 FRASNAY,

des

A-I

P r o b l 6 m a t i q u e a p p o r t 6 e par

Groupes

C.,

classifications Univ.

Gauthier-Villars

q u ' u n nombre

n'ayant

A-273

Scient.

i n geome-

463-470.

Cours de Logique math6matique

u n e ctia'ine,

I 8 1 FKATSSE,

A combinatorial problem

G.,

Sur quelques

(Paris,

I FKAYSSE, R . ,

127-139.

(1976),

( I 935).

2

permutations,

Villars

I 7

R.,

on un-

groups

, Monomorphic r e l a t i o n a l s y s t e m s ,

SZEKERES,

Publ.

permutation

229-235.

Math.,

logique),

r 6 1 FRATSSE,

148

Z.,

, a n d KKAUSS, i'.

Compos.

1 4 1 FRATSSi,

r i e

Math.

Z . , 18 ( 1 9 7 2 ) ,

Math. r 3 l

T r a n s i t i v i t y of

P.J.,

d'une

famille d'ordres

monomorphes, (Paris,

1972),

extension aux Gauthier-

229-237.

InterprGtation

groupe nnturel,

relationniste

C.R.

Acad.

Sc.

du seuil

Paris,

de

A-277

recolle-

(1973),

2 68

C. Frasnay

[ I 4 1 F R A S N A Y , C., S u r q u e l q u e s c l a s s e s u n i v e r s e l l e s d e r e l a t i o n s

m - a i r e s , Publ. D E p . M a t h . (Lyon), 1 6 ( 1 9 7 9 ) , n o 3 - 4 ,

21-32.

1 1 5 1 H I G M A N , G . , H o m o g e n e o u s r e l a t i o n s , Q u a r t . J . M a t h . Oxford

S e r . ( 2 ) , 28 ( 1 9 7 7 ) , 1 1 6 1 HODGES, W . ,

31-39.

L A C H L A N , A . H . , and S H E L A H , S . , P o s s i b l e o r d e r i n g s

o f a n i n d i s c e r n i b l e s e q u e n c e , Bull. L o n d o n Math.

SOC., 9 ( l 9 7 7 ) ,

21 2-2 15. 1 1 7 1 J E A N , M., R e l a t i o n s m o n o m o r p h e s et c l a s s e s u n i v e r s e l l e s , C . R .

A c a d . Sc. P a r i s , A - 2 6 4

(1967),

591-593.

[ I S ] J U L L I E N , P . , S u r q u e l q u e s p r o b l s m e s d e l a t h g o r i e des c h a T n e s

p e r m u t s e s , C.R. A c a d . S c . P a r i s , A - 2 6 2

(1966),

205-208.

[ I 9 1 K R A U S S , P., U n i v e r s a l l y c o m p l e t e u n i v e r s a l t h e o r i e s , 2. M a t h .

L o g i k G r u n d l a g . M a t h . , 17 ( 1 9 7 1 ) , r 2 0 1 P O U Z E T , M.,

351-370.

Un belordre d'abritement et ses r a p p o r t s avec les

b o r n e s d'une m u l t i r e l a t i o n , C . R . A c a d . S c . P a r i s , A - 2 7 4

(1972),

1677-1 680.

[ 2 1 1 P O U Z E T , M . , A p p l i c a t i o n d'une p r o p r i E t 6 c o m b i n a t o i r e des part i e s d'un e n s e m b l e a u x g r o u p e s et a u x r e l a t i o n s , Math. Z . , 150 (1976),

117-134.

1 2 2 1 P O U Z E T , M., A p p l i c a t i o n d e l a n o t i o n d e r e l a t i o n presque-

e n c h a l n a b l e a u d s n o m b r e m e n t d e s r e s t r i c t i o n s f i n i e s d'une relation, 2 . Math. Logik Grundlag. Math., 2 7 ( 1 9 8 1 ) ,

289-332.

1231 TARSKI, A . , Universal arithmetic c l a s s e s o f mathematical syst e m s , Bull. A m e r . M a t h . S O C . , 5 9 ( 1 9 5 3 ) ,

390-391

and C o n t r i b u -

t i o n s to the theory of m o d e l s , I n d a g . M a t h . , 1 6 ( 1 9 5 4 ) , 588,

ibid. 1 7 ( 1 9 5 5 ) ,

572-

56-64.

1 2 4 1 VAUGHT, R.L., Remarks o n universal classes of relational s y s -

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589-591.

***

3 9 1 a n d Indag. M a t h . ,

Annals of Discrete Mathematics 23 (1984) 269-286 0 Elsevier Science Publishers B.V. (North-Holland)

269

H,-CATEGORICAL PARTIALLY ORDERED SETS James H. Schmerl Department of Mathematics U n i v e r s i t y of C o n n e c t i c u t S t o r r s , CT 06268, U . S . A .

To Professor Emzest Corominas RCsumC: Un ensemble dBnombrable p a r t i e l l e m e n t ordonnB (A; < ) e s t .-cat&orique l o r s q u e , pour t o u t ensemble g dcnombrable q u i e s t p a r t i e l l e m e n t ordonne e t q u i p o s s l d e l a mcme t h B o r i e du p r e m i e r o r d r e que A ( i . e . Th(A) = T h ( B ) ) , A 8 . Nous c o n s i d g r o n s l a q u e s t i o n g 6 n 6 r a l e s u i v a n t e :

A

=

Q u e l l e complexit; d o i t a v o i r un ensemble A k o - c a t 6 g o r i q u e p a r t i e l l e m e n t ordonng pour que Th(A) s o i t complexe? Les deux emplois du terme "complexe" d a n s l a q u e s t i o n d o i v e n t e t r e vagues e t d i f f 6 r e n t s . L e p r e m i e r "complexe" f a i t r'efgrence aux p r o p r i g t 6 s s t r u c t u r e l l e s de A t e l l e s que l a h a u t e u r , l a l a r g e u r , l a dimension ou l e plongement d e c e r t a i n e s c o n f i g u r a t i o n s . Le second emploi du mot f a i t r g f g r e n c e aux a t t r i b u t s l o g i q u e s t e l s que l a d B d c i d a b i l i t 6 ou l ' a x i o m a t i s a t i o n f i n i e . Cet a r t i c l e , q u i s ' a d r e s s e au l e c t e u r a y a n t peu d e c o n n a i s s a n c e s dans l e domaine d e l a l o g i q u e , p r & e n t e un risum; d e s r g s u l t a t s e t l e s t e c h n i q u e s p e r m e t t a n t d e les o b t e n i r , c e q u i a p p o r t e une rgponse l a q u e s t i o n posge e t aux q u e s t i o n s a d j a c e n t e s . Un exemple t y p i q u e d e r ' e s u l t a t partiellement e s t c e l u i d e t o u t ensemble H .-cat;gorique ordonng de l a r g e u r f i n i e d o n t l a t h B o r i e n ' a q u ' u n nombre f i n i d'axiomes. Quelques thBorlmes nouveaux, complets e t accompagnBs de p r e u v e s , s o n t e'galement p r g s e n t b s . Par exemple, chaque ensemble K .-cat&gorique p a r t i e l l e m e n t ordonn; de dimension 2 e t de h a u t e u r f i n i e p o s s l d e une t h g o r i e d g c i d a b l e . Par c o n t r e , il e x i s t e d e s ensembles H .-cat&orique p a r t i e l l e m e n t ordonnBs, de dimension 3 e t de h a u t e u r 2. L ' a r t i c l e c o n t i e n t Bgalement d e s q u e s t i o n s o u v e r t e s .

e

1.

INTRODUCTION

Probably t h e e a r l i e s t theorem on K . - c a t e g o r i c a l

p a r t i a l l y o r d e r e d sets ( o r itself, i s C a n t o r ' s theorem c h a r a c t e r i z i n g t h e o r d e r t y p e of t h e r a t i o n a l numbers.

p o s e t s ) , proved l o n g b e f o r e t h e i s o l a t i o n of t h e n o t i o n of H . - c a t e g o r i c i t y THEOREM 1.1. (A; R ) is order-isomorphic t o t h e r a t i o n u l s foZlowing two conditions hold:

endpoints.

I&;

<)

i f f the

is countable; is a dense, Linearly ordered s e t without f i r s t and l a s t

(1)

A

(2)

(A; R )

J. H. Schtiterl

270

Let u s e x a m i n e ( 2 ) more c a r e f u l l y . The p r o p e r t i e s of ( A ; R) w h i c h a r e specified i n ( 2 ) a r e a l l first-order properties; that is, there a r e sentences has t h e s e o f f i r s t - o r d e r l o g i c w h i c h a r e t r u e i n ( A ; R) j u s t i n c a s e ( A ; K ) properties. I t i s a s i m p l e m a t t e r t o w r i t e down t h e f i r s t - o r d e r s e n t e n c e s i n volved. transitivity:

Vx Vy Vz (xRy

antisymmetry:

Vx Vy (xRy

c o n n e c t e d n e s s : Vx Vy ( x

# y

denseness:

Vx Vy (xRy

endpointless:

Vx 3y (yRx)

A

-

yRz

xRz);

yRx);

+

--t

xRy

A

yKx);

V

3z(xRz

+

A

zRy));

Vx 3y ( x K y ) .

(A; R) is c o u n t a b l e and i f e v e r y f i r s t - o r d e r s e n t e n c e T h u s , w e see t h a t i f t r u e i n ( Q ; <) i s t r u e i n ( A ; R ) , t h e n (9; < ) a n d ( A ; R ) are i s o m o r p h i c . T h i s i s j u s t what is r e q u i r e d f o r (0; < ) t o be .-categorical; we now g i v e a p r e c i s e d e f i n i t i o n of t h i s n o t i o n . F o r o u r p u r p o s e s , a ~ i r ~ u c t u rics a s e t A t o g e t h e r w i t h f i n i t e l y many r e l a t i o n s R 1 , Rq, Rk o n A , w h e r e i < i i s n i - a r y ( i . e . R i E Ani) f o r some itk). p o s i t i v e i n t e g e r n i . We w i l l w r i t e a s t r u c t u r e A as ( A ; R 1 , R 2 , F a m i l i a r e x a m p l e s o f s t r u c t u r e s a r e p o s e t s and g r a p h s . 'The t/iccira:j o f A , d e n o t e d by Th(A) , i s t h e s e t of f i r s t - o r d e r s e n t e n c e s t r u e i n A . When we s a y t h a t A i s c o u n t a b l e , w e w i l l mean t h a t i t s u n ~ l t 9 r ~ $ sz ek t~ ~A i s c o u n t a b l e , i n c l u d i n g t h e p o s s i b i l i t y t h a t A i s f i n i t e . S i m i l a r l y , A is f i n i t e i f A is f i n i t e . We c a n now d e f i n e t h e n o t i o n o f i-4 . - c a t e g o r i c i t y .

...,

...,

DEFINITION 1 . 2 . A structure c o n d i t i o n s are m e t :

(1) A (2)

A

is

Ko-onti'~i"r'i('"2 p r o v i d e d t h e f o l l o w i n g t w o

is countable;

whenever

6

i s c o u n t a b l e and

Th(6) = Th(A), t h e n

6

:A

.

(9io')

ti us, is H,-categorical. I t i s v e r y e a s y t o see t h a t e v e r y f i n i t e structure is categorical. In a sense, No-categoricity generalizes f i n i t e n e s s ; Theorem 2 . 1 w i l l make t h i s e v e n more e v i d e n t . We now p r e s e n t some o t h e r e x a m p l e s o f K o - c a t e g o r i c a l p o s e t s . Boolean a l g e I t is w e l l known t h a t a n y two c o u n t b r a s a n d l a t t i c e s w i l l be v i e w e d as p o s e t s . a b l e , atomless B o o l e a n a l g e b r a s a r e i s o m o r p h i c , a n d t h a t a n y two f i n i t e B o o l e a n It e a s i l y follows t h a t a l g e b r a s w i t h t h e same number o f a t o m s a r e i s o m o r p h i c . a n y two c o u n t a b l y i n f i n i t e B o o l e a n a l g e b r a s w i t h t h e same f i n i t e number of a t o m s a r e i s o m o r p h i c , t h e r e b y i m p l y i n g t h e f o l l o w i n g t h e o r e m from t h e f o l k l o r e . THEOREM 1 . 3 .

E v e q c o u n t u b k BooLenn ulgebr?~w i t h o n l y J i n i t e l y mung izlorns i::

Uo-cnti!~or~?:cuI. We e n d t h i s i n t r o d u c t i o n w i t h some m o d e l - t h e o r e t i c r e m a r k s o n c a t e g o r i c i t y . The r e a d e r who i s s e e k i n g more i n f o r m a t i o n s h o u l d c o n s u l t t h e s t a n d a r d t e x t 111 by Chang and K e i s l e r . A f i n i t e s t r u c t u r e A i s n o t o n l y X o - c a t e g o r i c a l , i t i s cutcgorical ( m e a n i n g t h i s : € o r a n y s t r u c t u r e 8, i f T h ( B ) = T h ( A ) , t h e n B z A ) The c o n v e r s e i s a l s o t r u e : i f A is c a t e g o r i c a l , then A i s f i n i t e . The r e a s o n f o r t h i s i s o n e o f t h e most f u n d a m e n t a l t h e o r e m s o f model t h e o r y , t h e I.~wenheim-Skolem Theorem.

.

THEOREM 1 . 4 .

wid i/,y

K K

(The LEwenheim-Skolem Theorem)

If A is t m g i n f i n i t e structure ( I structur2t. B 01' cnudiricil-

nn!j i n f i n i t e cnrd'inal nurribc:rd, t h e n thi'rs is such LIut l'hiBI = Th(AI.

No a l e g o r i c a l partially ordered set

27 1

The f o l l o w i n g d e f i n i t i o n , g e n e r a l i z i n g D e f i n i t i o n 1 . 2 , becomes n a t u r a l i n l i g h t of t h e p r e v i o u s theorem. L)l~I~lN17’I0N 1.5.

is ~ , 6 - A .

Let

K

b e an i n f i n i t e c a r d i n a l and 8 cardinality K

- ~ ~ ~ I ~ , ~ i ~f f[ ,whcnever ’ ~ , / ~ ~ ~ ~ ~has ~

A and

a s t r u c t u r e . Then A T h ( 6 ) = T ~ ( A ) ,t h e n

S i n c e f o r each i n f i n i t e c a r d i n a l K t h e r e is a correspondi ng n o t i o n of c a t e g o r i c i t y , t h e s i t u a t i o n seems c h a o t i c . However, t h e f o l l o w i n g b e a u t i f u l t h e o r e m of M. Morlcy [ 7 ] h a s t u r n e d t h i s c h a o s i n t o o r d e r , a t l e a s t p a r t i a l l y . THEORDI 1 . 6 .

untob7i- /-.:riyiinuIs, tiicit A t l n l Th(A) = T h l B ) . Then

L

8

ilnj

A

is

M o r l e y ’ s t h e o r e m r e d u c e s t h e number o f t y p e s o f c a t e g o r i c i t y t o j u s t two: & - c a t e g o r i c i t y and H 1 - c a t e g o r i c i t y . The s t u d y o f H 1 - c a t e g o r i c i t y h a s l e d t o some of t h e d e e p e s t and most r e v e a l i n g work i n model t h e o r y , a s w i t n e s s [ 9 ] . Howe v e r , H l - c a t e g o r i c a l p o s e t s a r e n o t s o i n t e r e s t i n g a s p o s e t s , s i n c e f o r any H lc a t e g o r i c a l p o s e t t h e r e i s a f i n i t e u p p e r bound on t h e l e n g t h s o f i t s c h a i n s .

2 . ORBITS Why s h o u l d a n y o n e , o t h e r t h a n a l o g i c i a n , b e i n t e r e s t e d i n H 0 - c a t e g o r i c a l p o s e t s ? One o f t h e most c o n v i n c i n g a r g u m e n t s i s t h a t t h e r e e x i s t s a n a l g e b r a i c c h a r a c t e r i z a t i o n of n o - c a t e g o r i c i t y . For a s t r u c t u r e A = (A; R 1 , R p , ..., R k ) , l e t Aut(A) b e i t s g r o u p o f a u t o m o r p h i s m s . For e e c h n a t u r a l number n , a n nor.bit o f A i s a s e t o f t h e form { (a(al),

a(a2),

...,

.(an)>

:

CY

E

Aut(A)I

f o r some a l , a 2 , ..., a n E A . The f o l l o w i n g t h e o r e m i s o n e form o f t h e fundament a l c h a r a c t e r i z a t i o n of H . - c a t e g o r i c i t y p r o v e d by R y l l - N a r d z e w s k i [16] i n 1 9 5 9 .

Supposc A i s muntabLe. Then t h e r e are o n l y f i n i t e l j mang n-orJbits.

THEOREM 2 . 1 .

n

A

i s H o-cntegoriraZ iff for tach

H e r e i s a n a p p l i c a t i o n o f Theorem 2 . 1 . We m e n t i o n e d i n Theorem 1 . 3 t h a t e a c h c o u n t a b l e B o o l e a n a l g e b r a w i t h a t most f i n i t e l y many a t o m s i s H . - c a t e g o r i c a l . Theorem 2 . 1 c a n b e u s e d t o d e m o n s t r a t e t h e c o n v e r s e t o t h i s : I f a Boolean a l g e b r a i s H o - c a t e g o r i c a l , t h e n i t h a s o n l y f i n i t e l y many a t o m s . F o r , s u p p o s e t h e r e w e r e i n f i n i t e l y many a t o m s . Then f o r e a c h n t h e r e i s a n e l e m e n t a n which i s t h e j o i n o f n + l a t o m s b u t i s n o t t h e j o i n o f n a t o m s . Then d i s t i n c t an and am a r e i n d i f f e r e n t 1 - o r b i t s , s o t h a t t h e r e a r e i n f i n i t e l y many 1 - o r b i t s .

...,

A r e d u c t o f a s t r u c t u r e A = ( A ; R 1 , R2, Rk) i s a s t r u c t u r e o b t a i n e d by i g n o r i n g some of t h e r e l a t i o n s of A . For e x a m p l e , i f m < k , t h e n (A; R 1 , R2, R,,,)), i t follows G) i s a r e d u c t o f A . S i n c e Aut(A) c A u t ( ( A ; R 1 , R 2 , from Theorem 2 . 1 t h a t i f A i s P o - c a t e g o r i c a l , t h e n so a r e a l l o f i t s r e d u c t s .

...,

...,

Let u s s a y t h a t a s t r u c t u r e i s n-determined i f f t h e f o l l o w i n g h o l d s f o r each p a i r < a l , a2, , am> , < b l , b g , bm> o f m - t u p l e s : and a r e i n t h e same m - o r b i t i f f whenever 1 c il _< i2 _< _ and < b i l , b i , b i > a r e i n t h e same n - o r b i t . then
...,

THEOREM 2 . 2 .

. ..

...,

...,

...,

For each

but w i t h i n f i n i t p l y many

n

..., ...

t h e r e is a countable p o s e t w i t h f i n i t e Z 3 many n-orbits in+LI-orbits.

J.H. Schmerl

212

I t i s a l s o p o s s i b l e t o c o n s t r u c t an Ho-categorical poset which i s not n-determined f o r any c h o i c e of n. A d d i t i o n a l l y , f o r each n t h e r e i s an H o - c a t e g o r i c a l p o s e t which i s n o t n-determined b u t i s (n+l)-determined. One more t y p i c a l a p p l i c a t i o n of Theorem 2 . 1 w i l l now be p r e s e n t e d . Given a p o s e t A , l e t i t s chain s p e c t r m be t h e set of a l l p o s i t i v e i n t e g e r s n such t h a t A h a s a maximal c h a i n of l e n g t h n. The f o l l o w i n g theorem c h a r a c t e r i z e s t h o s e sets which can occur a s c h a i n s p e c t r a of N o - c a t e g o r i c a l p o s e t s . THEOREM 2.3.

(I)

M

(2)

m

2

b, m

For a s e t

M

of p o s i t i v e i n t e g e w , the foLZowing are e q u i v a l e n t :

there i s an Ho-categorical poset whose chain s i J e c t r m i s M; E

M

i s eventuaZly p e r i o d i c ( i . e . there are iff m + p E Ml.

b

PROOF: Only t h e proof of (1) => ( 2 ) w i l l be g i v e n now. d i r e c t i o n w i l l b e d e l a y e d u n t i l a f t e r Theorem 3.4.

and

rl

suck t h a t for. a l l

The proof of t h e converse

What w e w i l l prove i s t h e f o l l o w i n g , which by Theorem 2 . 1 s u f f i c e s : Every p o s e t w i t h only f i n i t e l y many 1 - o r b i t s h a s an e v e n t u a l l y p e r i o d i c c h a i n spectrum. Suppose A h a s e x a c t l y k 1 - o r b i t s , where k is f i n i t e . We w i l l show t h a t we can t a k e p = k ! , b u t w e w i l l d e t e r m i n e o n l y t h e e x i s t e n c e of b by proving t h a t whenever k < m E M , t h e n m k! E M.

+

...

So, suppose k < m E M , and l e t a 1 < a2< < a,,, be a maximal c h a i n . Since m > k , t h e r e are i < j 5 m such t h a t a i and a j are i n t h e same 1 - o r b i t . Let a be an automorphism of A such t h a t a ( a i ) = a j . Then a1 < a 2 < < aj < a(ai+l) < < .(am) i s a maximal c h a i n of l e n g t h j i m - i , s o t h a t m ( j - i ) E M. elements of t h i s c h a i n , Also, n o t i c e t h a t a j and a ( a j ) , t h e j - t h and ( 2 j - i ) - t h are i n t h e same o r b i t . Thus, w e can s i m i l a r l y show t h a t (m+(j-i)) + ( @ j - i ) - j ) = m + 2 ( j - i ) E M. Proceeding i n d u c t i v e l y , f o r any n a t u r a l number r , m + r ( j - i ) E M. S i n c e 1 I j - i 2 k w e g e t , i n p a r t i c u l a r , t h a t mfk! E M. 0

... +

...

Corresponding t o c h a i n s p e c t r a i s t h e analogous n o t i o n f o r a n t i c h a i n s . Given a p o s e t A , l e t i t s a n t i c h a i n s p e c t r m be t h e s e t of a l l p o s i t i v e i n t e g e r s n such t h a t A has a maximal a n t i c h a i n of c a r d i n a l i t y n . We know of no p o s s i b i l i t i e s f o r a n t i c h a i n s p e c t r a o f )4 o - c a t e g o r i c a l p o s e t s o t h e r t h a n f i n i t e s e t s of p o s i t i v e i n t e g e r s , and t h e s e can e a s i l y be r e a l i z e d i n f i n i t e p o s e t s . On t h e o t h e r hand, w e know of no s e t which cannot be a n a n t i c h a i n spectrum of an H o - c a t e g o r i c a l poset. QUESTION 2.4. For which s e t s c h a i n spectrum M ?

3.

M

i s t h e r e an

H

.-categorical

p o s e t which h a s a n t i -

HOMOGENEITY

Theorems 1.1 and 2 . 1 t o g e t h e r imply t h a t t h e r a t i o n a l s ( 9 ; <) have o n l y f i n i t e l y many n - o r b i t s f o r each n - h a r d l y a s u r p r i s i n g f a c t . Moreover, t h e numb e r of o r b i t s i s i n some s e n s e as small a s p o s s i b l e . T h i s w i l l b e made more prec i s e a f t e r f i r s t s t a t i n g a d e f i n i t i o n . Two n - t u p l e s ( a l , a 2 , . , an) and < b l , b2 bn) from A are isomorphic i f {(ai, b ) : i = 1, 2 n l i s an isomorphism from t h e s u b s t r u c t u r e o f A induced by t a l , a 2 , . , an} o n t o t h e one induced by { b l , b2, bn}. C l e a r l y , i f two n - t u p l e s a r e i n t h e same o r b i t , t h e n t h e y are isomorphic. I n t h e c a s e of (Q; <), t h e proof of C a n t o r ’ s theorem a c t u a l l y shows, c o n v e r s e l y , t h a t isomorphic n - t u p l e s a r e i n t h e same o r b i t . This suggests a method of o b t a i n i n g H o - c a t e g o r i c a l s t r u c t u r e s f o r which a d e f i n i t i o n i s needed.

..

,...,

..

...,

A countable s t r u c t u r e DEFINITION 3.1. t u p l e s are i n t h e same n - o r b i t .

A

,...,

i s homogeneous i f any two isomorphic n-

Ho -categorical partially ordered set

273

C l e a r l y , isomorphism of n - t u p l e s i s an e q u i v a l e n c e r e l a t i o n w i t h o n l y f i n i t e l y many e q u i v a l e n c e c l a s s e s . Applying Theorem 2 . 1 , w e t h u s o b t a i n t h e followi n g consequence.

If

COROLLARY 3.2.

A

i s homogeneous, then

A

i s Ho-categoricaZ.

The t e c h n i q u e f o r c o n s t r u c t i n g homogeneous s t r u c t u r e s evolved o u t of work of Fra’iss;? 1 2 1 . I f A = (A; R 1 , R2 ,..., Rk) and 8 = ( B ; S 1 , S2 ,..., Sk) are s t r u c t u r e s , and a: A + B is an isomorphism of A w i t h a s u b s t r u c t u r e of B, t h e n we say t h a t a i s an embedding of A i n t o B, o r t h a t A i s embeddable i n B. The s t r a t e g y f o r c o n s t r u c t i n g homogeneous s t r u c t u r e s i s t o f o c u s on t h e c l a s s of f i n i t e s t r u c t u r e s which a r e embeddable i n t h e homogeneous s t r u c t u r e , r a t h e r t h a n on t h e homogeneous s t r u c t u r e i t s e l f . I t i s c o n v e n i e n t h e r e t o p e r m i t s t r u c t u r e s t o have u n d e r l y i n g s e t s which are empty. DEFINITION 3.3.

(1) K beddable i n (2) K a l : A, + A1

6 1 : A,

+

8

Let

K

be a c l a s s of f i n i t e s t r u c t u r e s .

A

has t h e hereditary property ( H F ) i f whenever

E

B

and

K

i s em-

A , then

B E K. h a s t h e amaZgamation property ( A F ) i f whenever and and

“2:

A0

+

A2

62: A2

+

B

A,,

a r e embeddings, t h e n t h e r e a r e 8 such t h a t B l a l = B2a2.

E

A 1 , A2 E K and K and embeddings

For a s t r u c t u r e A w e l e t H(A) d e n o t e t h e c l a s s of f i n i t e s t r u c t u r e s which a r e embeddable i n A . The c o n n e c t i o n between homogeneous s t r u c t u r e s and t h e prop e r t i e s i n D e f i n i t i o n 3 . 3 i s t h e c o n t e n t of t h e n e x t theorem, which is t h e method of Frays&.

THEOREM^.^.

(1)

If

A

is a homogeneous s t r u c t u r e , then

HlA)

has b o t h

HF

and

AP.

(2) If K is a c l a s s o f f i n i t e s t r u c t u r e s which has both HP and AP, then there i s a unique (up t o isomorphism) countable homogeneous A such t h a t H ( A I = K. Here i s a s i m p l e example of how Theorem 3.4 i s implemented. Let K be t h e c l a s s of a l l f i n i t e p o s e t s . Then i t i s q u i t e c l e a r t h a t K h a s b o t h HP and A P , so t h a t t h e r e i s a u n i q u e homogeneous p o s e t A such t h a t H(A) = K. T h i s p o s e t i s c a l l e d t h e universaZ homogeneous p o s e t . I n g e n e r a l , i f A i s any homogeneous s t r u c t u r e and B t u r e such t h a t H ( B ) C H ( A ) , t h e n 8 i s embeddable i n A .

is any c o u n t a b l e s t r u e

A s a n o t h e r d e m o n s t r a t i o n of t h e u s e of Theorem 3 . 4 , w e w i l l complete t h e proof of Theorem 2 . 3 , by p r o v i n g (2) => (1). We w i l l c o n s i d e r o n l y t h e c a s e t h a t M consists o f a l l p o s i t i v e i n t e g e r s ; f o r a r b i t r a r y e v e n t u a l l y p e r i o d i c M, e i t h e r t h e f o l l o w i n g proof o r t h e r e s u l t i n g s t r u c t u r e can b e a p p r o p r i a t e l y m o d i f i e d .

Consider t h e c l a s s K of f i n i t e s t r u c t u r e s ( A ; <, B , T , I ) , where ( A ; < ) i s a p o s e t , B and T a r e 1-ary r e l a t i o n s , I i s a 2-ary r e l a t i o n , and t h e f o l lowing s e n t e n c e s are i n Th((A; <, B , T, I ) ) : Vx Vy (B(x) Vx VY (T(x) VX

+

-f

vy ( I ( X , y )

l y < x), i x < Y). -f

X

y

A VZ l ( X

<

2 A 2 <

y)).

The i n t e n t i s , of c o u r s e , t h a t o n l y minimal e l e m e n t s are i n B , o n l y maximal e l e m e n t s a r e i n T , and ( a , b ) E I i m p l i e s t h a t b i s a cover of a It i s e a s i l y s e e n t h a t K h a s HP, and o n l y s l i g h t l y l e s s e a s i l y s e e n t h a t K h a s AP. By Theorem 3 . 4 ( 2 ) t h e r e i s a homogeneous B s u c h t h a t H ( B ) = K , and C o r o l l a r y 3 . 2 implies t h a t B is No-categorical. Let A be t h e p o s e t which i s t h e r e d u c t of 8; an o b s e r v a t i o n made a f t e r Theorem 2 . 1 i m p l i e s t h a t A i s H o - c a t e g o r i c a l . To

.

J.H. Schnierl

214

see t h a t that for B = {l], dable i n

t h e c h a i n s p e c t r u m o f A i s t h e s e t of a l l p o s i t i v e i n t e g e r s , j u s t n o t i c e e a c h n , t h e s t r u c t u r e (Cn; < , B , T , I ) i s i n K , w h e r e Cn= 1 1 , 2 ....,n t , T = { n ] and I = { 1 , 2 , 2 , 3 n - l , n 1. T h i s s t r u c t u r e i s embed8 a s a ma x i mal c h a i n of l e n g t h n .

,...,

I t i s o f t e n i n t e r e s t i n g and u s e f u l t o c h a r a c t e r i z e t h e homogeneous s t r u c t u r e s o f a p a r t i c u l a r k i n d ; however, i t may n o t a l w a y s b e p o s s i b l e t o do s o , e s p e c i a l l y when t h e r e are u n c o u n t a b l y many. Henson [ 3 ] showed t h i s i s j u s t t h e s i t u a t i o n f o r d i r e c t e d graphs. THEOREM 3 . 5.

Them

U Y J ~

homvgicni:ous (i%rv:r.tc,Ll ~ ~ r u p l ~ s .

2 "I

H From t h i s t h e o r e m , a s Henson n o t e d , t h e e x i s t e n c e of 2 graphs and p o s e t s c a n b e o b t a i n e d . COROLLARY 3 . 6 .

There an? 2

K

No-categorical

~ o - c o t i : ~ i "1 ~L ~~ ir ~~ ~ u ~~ ~i h s .

PROOF. From a n y g i v e n d i r e c t e d g r a p h w e c a n c o n s t r u c t a g r a p h i n t h e f o l l o w i n g m an n er. I f a a n d b a r e a r b i t r a r y v e r t i c e s of t h e d i r e c t e d g r a p h w i t h a d i r e c t e d e d g e from a t o b , t h e n r e p l a c e t h a t d i r e c t e d e d g e by t h e f o l l o w i n g c onf i g u r a t i o n , a d d i n g 7 new v e r t i c e s .

I t i s c l e a r t h a t n o n i s o m o r p h i c g r a p h s a r e c o n s t r u c t e d from n o n i s o m o r p h i c d i r e c t e d graphs. F u r t h e r m o r e , i t i s e a s y t o see from R y l l - N a r d z e w s k i ' s Theorem 2 . 1 t h a t e a c h H o - c a t e g o r i c a l d i r e c t e d g r a p h p r o d u c e s a n H o - c a t e g o r i c a l g r a p h , S i n c e , by Theorem 3.5 t h e r e a r e 2 ' 0 r( 0 - c a t e g o r i c a l d i r e c t e d g r a p h s , w e ha ve j u s t r o n s t r u c t e d 2 ' 0 nonisomorphic N o - c a t e g o r i c a l g r a p h s . 0 COROLLARY 3 . 7 .

There are

2

H

Ho-cateyorical p o se ts.

PROOF: From a n y g i v e n g r a p h , w e c a n c o n s t r u c t a p o s e t i n t h e f o l l o w i n g m a n n e r . L e t (A; E ) b e a g r a p h . Then l e t F = { { a , b ) : a , b and l e t B = A u F. D e f i n e < on B so t h a t x < y i f f x E A , y E F an: y . Then i t f o l l o w s fro m Theorem 2 . 1 t h a t (B; <) i s K o - c a t e g o r i c a l i f (A; E) i s . A l s o , n o n i s o (A c u r i o u s a nd u n i q u e morphic p o s e t s are c o n s t r u c t e d from nonisomorphic g r a p h s . e x c e p t i o n t o t h i s l a s t s e n t e n c e is t h a t t h e complete graph on 2 e l e m e n t s and t h e empty g r a p h on 3 e l e m e n t s y i e l d i s o m o r p h i c p o s e t s . ) 0

['"x

The p r e c e d i n g c o n s t r u c t i o n of p o s e t s i s f r o m [ 2 3 ] . a d d i t i o n a l p r o p e r t i e s whi ch w i l l b e c o n s i d e r e d l a t e r . COROLLARY 3.8.

There m e

2

H 0

T h e s e p o s e t s h a v e some

Ho-cateyorical l a t t i c e s .

PROOF: C o n s t r u c t p o s e t s ( B ; d ) w i t h t h e same u n d e r l y i n g s e t s as t h e p o s e t s cons t r u c t e d i n t h e p r o o f of C o r o l l a r y 3 . 7 , b u t now d e f i n e x -!y i f f x E y I f we

.

275

No -categorical partially ordered set

a d j o i n a new minimal element 0 and a new maximal element 1 , t h e n w e o b t a i n a l a t t i c e which i s H,-categorical provided t h e graph ( A ; E ) was. A l s o , nonisomorp h i c l a t t i c e s a r e c o n s t r u c t e d from nonisomorphic g r a p h s . 0 In c o n t r a s t t o Theorem 3.5 a r e t h e n e x t two r e s u l t s , each of which have remarkably l o n g and i n t r i c a t e p r o o f s . THEOREM 3 . 9 .

(1)

Lachlan-Woodrow [ 4 ] )

There are only countably many homogeneous

gru+. (2) UYW

(Lachlan [ 5 ] )

The1.e a r e onZy five homogeneous tournaments, two of which

j-inite.

I n each of t h e p a r t s of Theorem 3 . 9 t h e homogeneous s t r u c t u r e s a r e e x p l i c i t l y g i v e n . There a r e no s u r p r i s i n g examples; t h e d i f f i c u l t y is i n showing t h a t t h e r e a r e none b u t t h e obvious o n e s . Much more a c c e s s i b l e i s t h e c o r r e s p o n d i n g r e s u l t f o r p o s e t s proved i n [ 2 1 ] .

There are onZy countab2y many homogeneous p o s e t s .

THEOKEM 3 . 1 0 .

Again, t h e r e i s a n e x p l i c i t l i s t of t h e homogeneous p o s e t s . There a r e t h r e e i n f i n i t e c l a s s e s which a r e q u i t e e a s i l y d e s c r i b e d . Let n r a n g e o v e r c a r d i n a l numbers, 1 5 n 5 % . The f i r s t c l a s s c o n s i s t s of a n t i c h a i n s w i t h n e l e m e n t s ; t h e second c l a s s c o n s i s t s of n d i s j o i n t c o p i e s of (Q; < ) ; and t h e t h i r d c l a s s c o n s i s t s of p o s e t s which r e s u l t by r e p l a c i n g each p o i n t i n Q w i t h a n a n t i c h a i n of n e l e m e n t s . There i s one l a s t homogeneous p o s e t - t h e u n i v e r s a l homogeneous poset.

4.

THEORIES

C a n t o r ' s theorem s a y s more t h a n j u s t t h a t (Q; <) i s N o - c a t e g o r i c a l , f o r i t a l s o g i v e s an e x p l i c i t f i n i t e set C of s e n t e n c e s (namely t h o s e f i v e s e n t e n c e s l i s t e d i n 5 1 ) which a r e i n c l u d e d i n Th((Q; < ) ) w i t h t h e p r o p e r t y t h a t whenever 8 i s c o u n t a b l e and C C Th(B), t h e n 8 (Q; < ) . Suppose A i s a s t r u c t u r e and C C Th(A). Then C i f f o r any B , Th(B) = Th(A) i f f Z C Th(B). I f n i t e a x i o m a t i z a t i o n , t h e n A i s f i n i t e 2 3 miomatizab2e. DEFINITION 4.1.

A

matization f o r

i s a n axiohas a f i -

A

F i n i t e a x i o m a t i z a b i l i t y of a s t r u c t u r e A has i m p o r t a n t consequences conc e r n i n g Th(A). We make t h e f o l l o w i n g d e f i n i t i o n which by n e c e s s i t y i s somewhat vague. A s e t T of s e n t e n c e s i s decidable i f t h e r e i s a computer program ( i . e . an e f f e c t i v e a l g o r i t h m ) which can i d e n t i f y whether o r n o t a n a r b i t r a r y s e n t e n c e i s i n T. More p r e c i s e l y , T i s d e c i d a b l e i f i t is r e c u r s i v e i n t h e s e n s e of t h e t h e o r y of r e c u r s i v e f u n c t i o n s [ 1 2 ] . One of t h e common t y p e s of problems i n l o g i c is t o d e t e r m i n e whether o r n o t a s p e c i f i c Th(A) i s d e c i d a b l e . We w i l l s a y t h a t t h e s t r u c t u r e A i s decidable whenever i t s t h e o r y Th(A) i s d e c i d a b l e . The f o l l o w i n g fundamental c o n n e c t i o n between f i n i t e a x i o m a t i z a b i l i t y and dec i d a b i l i t y i s a consequence of t h e Lowenheim-Skolem Theorem 1 . 4 and t h e Completen e s s Theorem of l o g i c . THEOREM 4 . 2 .

If

A

is finitely uxiomatizabZe, then A

is decidab2e.

Thus, C a n t o r ' s theorem i m p l i e s t h a t (Q; < ) i s d e c i d a b l e . Is i t p o s s i b l e t h a t e v e r y H o - c a t e g o r i c a l p o s e t A i s d e c i d a b l e ? Far from i t , s i n c e t h e r e are o n l y c o u n t a b l y many nonisomorphic N o - c a t e g o r i c a l d e c i d a b l e s t r u c t u r e s A , and as we have s e e n i n C o r o l l a r y 3 . 7 , t h e r e are 2 '0 K o - c a t e g o r i c a l p o s e t s . However, i t f r e q u e n t l y happens t h a t i f f u r t h e r c o n d i t i o n s a r e imposed on an N o - c a t e g o r i c a l A , t h e n A i s d e c i d a b l e . For example, e v e r y f i n i t e s t r u c t u r e i s f i n i t e l y a x i -

J.H. Schmerl

216

o m a t i z a b l e . Also, of t h e homogeneous p o s e t s ( s e e Theorem 3 . 1 0 ) , a l l a r e d e c i d a b l e and o n l y f o u r of t h e s e a r e n o t f i n i t e l y a x i o m a t i z a b l e . A l l of t h e homogeneous graphs and tournaments a r e a l s o d e c i d a b l e . I t should be a p p a r e n t t h a t e v e r y r e d u c t of a d e c i d a b l e s t r u c t u r e i s d e c i d a b l e . I n p a r t i c u l a r , every r e d u c t of a f i n i t e l y a x i o m a t i z a b l e s t r u c t u r e i s d e c i d a b l e . Many of t h e d e c i d a b l e N o - c a t e g o r i c a l s t r u c t u r e s r e f e r r e d t o l a t e r i n t h i s paper a r e d e c i d a b l e by v i r t u e of t h e i r b e i n g r e d u c t s of f i n i t e l y a x i o m a t i z a b l e s t r u c t u r e s . The r e d u c t of an u n d e c i d a b l e s t r u c t u r e may be d e c i d a b l e . However, i f A i s a r e d u c t of 8, Aut(A) = Aut(B) and A i s d e c i d a b l e and N,-categorical, No-categorical. then 8 i s a l s o d e c i d a b l e and, of c o u r s e ,

5.

EXPANDABILITY

Many c l a s s e s o f p a r t i a l l y o r d e r e d s e t s are c l a s s i f i e d by b e i n g r e d u c t s of c e r t a i n t y p e s o f s t r u c t u r e s , o r , from a n o t h e r v i e w p o i n t , by t h e i r e x p a n d a b i l i t y . For example, a p o s e t (A; <) h a s dimension a t most n i f f t h e r e a r e l i n e a r o r cn e x t e n d i n g < such t h a t < = n < 2 n . . . n
...,

We begin w i t h a n H o - c a t e g o r i c a l v e r s i o n of t h e p r i n c i p l e a s s e r t i n g t h a t e v e r y s e t can b e l i n e a r l y o r d e r e d . A s p e c i a l c a s e w a s f i r s t proved i n [ 9 ] ; t h e theorem t h a t f o l l o w s was proved i n [ 2 1 ] . THEOREM 5 . 1 . Suppose A = (A; R1, R2,. . . , R k ) is No-categorical. Then t h e r e is a linear order < o f A such that (A; R1, R2, ..., Rk, < I is N o - c a t c y o r i c u l . Moreover, if A is i n f i n i t e , then < can be chosen so t h a t ( A ; < ) 2 (Q; < ) .

The n e x t two theorems, b o t h proved i n [ M I , u t i l i z e Theorem 5 . 1 i n t h e i r proofs. THEOREM 5.2.

t h a t (A; El such t h a t :

. ., R k ) is an No-categorical s t r u c t u r e such i s a cornparubiZity graph, then t h e r e is a partia2 order < o f A If (A; E, R I , R2,.

(1)

(A; E, ~ 1 RZ , ,..., ~ k ,< I

(2)

(A; El

is

N0-categorical;

i s t h e comparability graph o f

(A;

<).

S i n c e t h e i n c o m p a r a b i l i t y graph of a 2-dlmensional p o s e t i s a l s o a comparab i l i t y g r a p h , i t f o l l o w s q u i t e e a s i l y from Theorem 5.2 t h a t H o - c a t e g o r i c a l 2d i m e n s i o n a l p o s e t s have N o - c a t e g o r i c a l r e a l i z a t i o n s . THEOREM 5 . 3 .

If

i s an

(A;

and

are l i n e a r orders (1)

(A;

(2)

< =

<,

<,I

<el i s

<2

No-categoricaZ 2-dimensionu2 poset, then there o f A such t h a t :

Ho-categorical;

n <2.

I t i s unknown whether t h e r e i s a n analogue o f Theorem 5.3 f o r a l l f i n i t e - d i mensional H o - c a t e g o r i c a l p o s e t s . Even t h e c a s e of dimension 3 i s s t i l l u n s e t tled. QUESTION 5 . 4 . For e v e r y H,. - c a t e g o r i c a l 3-dimensional p o s e t (A; < ) , are t h e r e l i n e a r o r d e r s < I , < 2 , <3 of A such t h a t (A; <,
Ho -categorical partially ordered set

277

One of t h e b a s i c theorems about p o s e t s i s D i l w o r t h ’ s theorem c h a r a c t e r i z i n g p o s e t s of f i n i t e w i d t h . The next theorem i s t h e H o - c a t e g o r i c a l v e r s i o n of D i l w o r t h ’ s theorem, which i s i m p o r t a n t i n t h e proof of Theorem 6 . 4 . I t i s proved i n [ 2 2 ] , and a l s o i n [ 2 6 ] where an e r r o r i s c o r r e c t e d . THEOREM 5.5.

thew

m e ( I )

If

ehniris ( A ; <,

(A; < I is un H,-cutcgur~cuL posct of f i n i t e width C1, C;?,.. . Cn -C A such that:

CI, C,,

. . .,

CrL) i s

n, then

H,-cutegor>iea’L;

(2) A = CI u Cz u... u C,. The Order Extension P r i n c i p l e s t a t e s t h a t e v e r y p o s e t can be extended t o a l i n e a r l y o r d e r e d s e t . Theorems 5 . 1 , 5 . 3 and 5 . 5 each imply some i n s t a n c e of an N,-categorical v e r s i o n of t h i s p r i n c i p l e . l H E O R E M 5.6. Suppose (A;
Ho-cutegorical posct which has f i n i t e Then it has a linear extension < such

No i n s t a n c e of t h e f a i l u r e of t h e H o - c a t e g o r i c a l v e r s i o n of t h e Order Ext e n s i o n P r i n c i p l e i s known, y e t Theorem 5.6 r e p r e s e n t s j u s t about t h e e x t e n t of o u r knowledge. We do n o t know i t s v a i l d i t y even f o r 3-dimensional p o s e t s . T h i s s u g g e s t s t h e f o l l o w i n g q u e s t i o n f i r s t asked i n [ 2 2 ] . QUESTION 5 . 7 . < such t h a t

Does e v e r y Ho-categorical poset (A; < , < ) remains H,-categorical?

(A;<)

have a l i n e a r e x t e n s i o n

We end t h i s s e c t i o n w i t h a r e l a t e d q u e s t i o n which does n o t e x p l i c i t l y mention ( A ; E) i s a n H o - c a t e g o r i c a l n-colp o s e t s . Theorems 5.2 and 5.5 imply t h a t i f o r a b l e graph which is t h e complement of a c o m p a r a b i l i t y g r a p h , t h e n i t c a n be nc o l o r e d w i t h c o l o r c l a s s e s C1, C 2 , Cn such t h a t (A; E , C 1 , C 2 , C,) is Ho-categorical. still The same c o n c l u s i o n can be o b t a i n e d j u s t from Theorem 5.2 i f ( A ; E) i s a n H o - c a t e g o r i c a l , n - c o l o r a b l e c o m p a r a b i l i t y graph.

...,

...,

Can e v e r y QUESTION 5.8. g o r i c a l l y n-colored?

6.

H o - c a t e g o r i c a l n - c o l o r a b l e p e r f e c t graph b e

Ko-cate-

F I N I T E AXIOMATLZABILITY

C a n t o r ’ s proof of t h e H,-categoricity of ( Q ; <) a l s o d e m o n s t r a t e s t h a t (Q; i)i s f i n i t e l y a x i o m a t i z a b l e . R o s e n s t e i n [14] h a s shown t h a t t h i s phenomenon H o - c a t e g o r i c a l l i n e a r l y o r d e r e d sets. extends t o a l l THEOREM 6.1.

Every

uo-categorical l i n e a r l y ordered s e t i s f i n i t e l y axiomatizable

.-cateA c t u a l l y R o s e n s t e i n proved more by s t r u c t u r a l l y c h a r a c t e r i z i n g t h e g o r i c a l l i n e a r l y o r d e r e d s e t s . Another i n s t a n c e of t h i s phenomenon of f i n i t e a x i o m a t i z a b i l i t y o c c u r s w i t h Boolean a l g e b r a s , as w e a l r e a d y have s e e n b u t n o t made e x p l i c i t . Macintyre and R o s e n s t e i n [ 6 ] extended t h i s to augmented Boolean a l g e b r a s , which a r e s t r u c t u r e s of t h e form ( B ; <, 11, 1 2 , I k ) w i t h (B; < ) a Boolean a l g e b r a and e a c h I i a n i d e a l .

...,

THEOREM 6.2.

mat i z a b 1e .

Every

Ho-categoricaI augmented Boolean algebra i s f i n i t e l y axio-

S i n c e , as p r e v i o u s l y n o t e d , l i n e a r l y o r d e r e d sets a r e 2-determined, Theorem 6 . 1 can be r e s t a t e d t o s a y t h a t e v e r y l i n e a r l y o r d e r e d s e t w i t h f i n i t e l y many 2o r b i t s i s f i n i t e l y a x i o m a t i z a b l e . M . Rubin [ 1 5 ] improved upon t h i s , even though t h e r e are c o u n t a b l e l i n e a r l y o r d e r e d sets which have o n l y f i n i t e l y many 1 - o r b i t s

J. H. Schm erl

278

but a r e not THEOREM 6 . 3 .

m i o m a t i z u bl e

U0-categorical.

Every ZineuYly ouder>ed S P L m‘th f i n i t e z u man$

.

I-or>bits is j’init( LIJ

Theorem 6 . 1 was improved upon i n a n o t h e r d i r e c t i o n i n [ 2 3 ] . some of t h e i d e a s o c c u r r i n g i n t h e proof of Theorem 5 . 5 . THEOREM 6 . 4 .

Eueray

The proof u s e s

i$,-categor,ic*aZpo.’eL of f i n i t e width is f i n i t e l y miornuti--

able. The proof of Theorem 6 . 4 proceeds by i n d u c t i o n on t h e w i d t h . I n o r d e r t o m a i n t a i n t h e i n d u c t i o n f o r t h e c o u r s e of t h e p r o o f , a s t r e n g t h e n e d form of Theorem 6.4 must b e proved. T h i s s t r o n g e r theorem t u r n s o u t t o be q u i t e u s e f u l i n t h e I n o r d e r t o s t a t e t h e s t r o n g e r v e r s i o n i n Theop r o o f s of Theorems 7 . 3 and 7 . 1 3 . I f ( A ; < ) i s a p o s e t and R 5 A2 a b i n a r y rem 6 . 5 , w e w i l l need a d e f i n i t i o n . i f whenever < x , y ) E R , r e l a t i o n , t h e n R i s monotone ( w i t h r e s p e c t t o (A; < ) ) x ’ 5 x and y 5 y ‘ , t h e n ( x ’ , y ’ ) E R. THEOREM 6 . 5 . Suppose A = (A; <, R I , (A; <,J has f i n i t e width and each Ri tizabZe.

R?, ..., R k l

The b a s i s s t e p of Theorem 6 . 5 , when f i r s t proved i n [ 1 9 ] .

(A; < )

.-cateyoricaZ, wher>e A is j’initclg miornu-

is

zs monotone.

Then

i s a l i n e a r l y o r d e r e d s e t , was

One might wonder what happens i f t h e r e s t r i c t i o n t o f i n i t e width on t h e posets of Theorem 6.4 i s r e l a x e d t o j u s t f o r b i d d i n g any i n f i n i t e a n t i c h a i n . An e a s y consequence of t h e Compactness Theorem of l o g i c i s t h a t f o r each p o s e t (A; < ) having a r b i t r a r i l y l a r g e f i n i t e a n t i c h a i n s , t h e r e i s a c o u n t a b l e poset (B; <) w i t h a n i n f i n i t e a n t i c h a i n such t h a t Th((B; < ) ) = Th((A; < ) ) . Thus, i f an H o - c a t e g o r i c a l p o s e t has no i n f i n i t e a n t i c h a i n s , t h e n i n f a c t i t a l r e a d y has f i n i t e width. I t i s n o t p o s s i b l e t o improve Theorem 6 . 4 i n t h e d i r e c t i o n of Theorem 6 . 3 . This i s demonstrated i n t h e n e x t two theorems, which s o l v e Problem 5 . 5 of [ 2 3 ] . THEOREM 6.6. There arc? 2 % countable posets which have pairwise d i s t i n c t t h e ories such t h a t each posst has width 3 und o n l y one 1-orbit.

PROOF: The r e s t r i c t i o n t h a t t h e p o s e t s be c o u n t a b l e can be i g n o r e d because of t h e Lowenheim-Skolem Theorem 1 . 4 . L e t R be t h e s e t of r e a l s , l e t Z3 = ( 0 , 1, 21, and l e t A = R x 2 3 . For each i r r a t i o n a l a , w i t h 1 < a < 2 , d e f i n e a p o s e t A, = ( A ;
x < y;

(y, i + l )

iff

x+l 5 y ;

( x , i )
iff

x+a 5 y.

c,

There i s o n l y 1 - o r b i t s i n c e f o r e a c h r t R and ( x , i ) I+ ( x + r , i+k) ‘is an automorphism of A,.

k

E

23, t h e f u n c t i o n

I t remains t o check t h a t whenever a and f3 a r e i r r a t i o n a l s , where 1< a < < 2 , t h e n Th(A,) # Th(A6). N o t i c e t h a t each a = (x, i ) t A h a s two L e t u s d e n o t e t h e s e by c l ( x ) c o v e r s i n A,, namely ( x + l , i + l ) and (x+a, i - 1 ) . and c 2 ( x ) r e s p e c t i v e l y . These two c o v e r s a r e d i s t i n g u i s h a b l e i n t h e f o l l o w i n g s e n s e . Independently of a , t h e r e a r e f i r s t - o r d e r formulas JI1(x,y) and J12(x,y) and y = c 2 ( x ) r e s p e c t i v e l y . To o b t a i n such which hold j u s t i n c a s e y = c l ( x ) formulas we make u s e of t h e f o l l o w i n g two formulas which are t r u e i n A, : c 1 ( c 2 ( x ) ) = c Z ( c l ( x ) ) and c l ( c l ( c l ( x ) ) ) < C ? . ( C ~ ( C ~ ( X ) ) L) .e t C(x, Y ) b e t h e f i r s t - o r d e r formula a s s e r t i n g t h a t y i s a c o v e r of x Now l e t $ l ( x , y) be t h e f o l l o w i n g formula:

.

Ho -categorical partially ordered set

3y2, Y39 21, Z2r $ ( Z l , 22) Y # Zl

A

w [ $ ( x , Y)

239

A

$(22, 23)

$(w, Y3)

A

A

A

$ ( y , y2)

$ ( y , W)

$(W, 23)

h

A

A

279

$(Y2, Y3)

$ ( Z 1 , W)

Y3 < 231

.

A

W

A

#

$(x, z1) y2

h

W

# 22

.

Then $ 2 ( x , y ) i s j u s t $ ( x , Y ) A - $ l ( x , y ) For p o s i t i v e i n t e g e r p d e f i n e $P,(x, y ) so t h a t $ & ( x , Y ) = $ e (X, Y ) and ~ i ~ + ' ( x ,Y ) = l z [ r / l E ( x , Z ) A $ e ( ~ y, ) ~ . Then, c l e a r l y , t h e s e n t e n c e

vx v Y , vY2[$y(X' Y1) A 0 i f f a > p/q

.

holds

$q(X.

Y p ) - + Y1 < Y21

THEOREM 6 . 7 . Thcr7e is a countable poset of width 2 but which is not j-ifinitely ax-iomatizable.

which has onZy one

1-orbit

We w i l l d e s c r i b e s u c h a p o s e t . L e t {Q1,Q2} b e a p a r t i t i o n of t h e set Q o f r a t i o n a l s i n t o two s e t s e a c h o f w h i c h i s d e n s e . F i x some i r r a t i o n a l a > 0 . D e f i n e 4 o n Q so t h a t x < y i f f e i t h e r x u. < y o r e l s e x < y and x , y a r e i n t h e same Q i . The p o s e t (Q;<) h a s t h e p r o p e r t i e s demanded by t h e t h e o r e m . N o t i c e t h a t t h e p o s e t i s , u p t o i s o m o r p h i s m , i n d e p e n d e n t o f t h e c h o i c e o f a. It c a n b e shown t h a t t h i s p o s e t i s n o t f i n i t e l y a x i o m a t i z a b l e , b u t t h a t i t i s decidable.

+

If w e form a p o s e t by r e p l a c i n g e a c h e l e m e n t o f (Q; < ) by a n i s o m o r p h i c copy o f ( Q ; 4 ) , t h e n t h e r e s u l t i n g p o s e t s t i l l i s c o u n t a b l e , has w i d t h 2 , h a s o n l y o n e 1 - o r b i t , and i s n o t f i n i t e l y a x i o m a t i z a b l e . T h i s p o s e t i s t h e example o f P o u z e t [lo] o f a c o u n t a b l e p o s e t o f w i d t h 2 . C u r i o u s l y , t h e p o s e t s o c c u r r i n g i n t h e p r o o f o f Theorem 6 . 6 c a n b e u s e d t o d e m o n s t r a t e t h e n o n e x i s t e n c e of a c o u n t a b l e p o s e t o f w i d t h 3 which embeds e v e r y c o u n t a b l e p o s e t o f w i d t h 3 , t h u s a n s w e r i n g i n t h e n e g a t i v e a q u e s t i o n o f P o u z e t ( 7 . 1 5 o n p a g e 833 o f [ll]). For e a c h i r r a t i o n a l a, w i t h 1 < LY < 2 , l e t B, be a countable poset such t h a t Th(B,) = T h ( ( A ;
A c o m p a r i s o n of Theorems 6 . 6 and 6.7 s u g g e s t t h a t w i d t h 2 m i g h t b e a much more s e v e r e r e s t r i c t i o n t h a n w i d t h 3 . T h i s i s t h e g i s t o f t h e f o l l o w i n g q u e s t i o n .

QUESTION 6 . 8 . Does t h e r e e x i s t a n u n d e c i d a b l e p o s e t o f w i d t h l y many 1 - o r b i t s ?

2

with only f i n i t e -

Theorem 6 . 4 h a s b e e n e x t e n d e d i n [ 2 6 ] , s t a t e d as Theorem 6 . 9 b e l o w , i n a way Theorem 6 . 9 i m p l i e s Theorem 6 . 4 which i s i n t e r e s t i n g l y compared w i t h Theorem 7.2. q u i t e r e a d i l y , b u t t h e r e a p p e a r s t o b e no s u c h r e a d y c o n v e r s e i m p l i c a t i o n . THEOREM 6 . 9 . Every Ho-categorical, finite-dimensionaz, f i n i t e l y miornatizable.

d i s t r i b u t i v e l a t t i c e is

We end t h i s s e c t i o n w i t h a v e r y i n t e r e s t i n g o p e n q u e s t i o n w h i c h i s q u i t e n a t u r a l a f t e r t h e a p p e a r a n c e o f Theorems 6 . 2 and 6 . 9 . QUESTION 6 . 1 0 . able?

Is e v e r y

vo-categorical

d i s t r i b u t i v e l a t t i c e f i n i t e l y axiomatiz-

I t i s e v e n n o t known t h a t t h e r e are n o t

2

%-categorical

d i s t r i b u t i v e lat-

tices.

7.

DECIDABILITY Theorem 6 . 2 i m p l i e s t h a t t h e r e a r e o n l y c o u n t a b l y many s , , - c a t e g o r i c a l

posets

J.H. Schmerl

280

of f i n i t e width. No such r e s u l t f o l l o w s i f e i t h e r t h e dimension o r t h e h e i g h t (which i s t h e number of elements i n t h e l o n g e s t c h a i n , i f oIie e x i s t s ) i s s i m i l a r l y r e s t r i c t e d . The f i r s t of t h e f o l l o w i n g two theorems i s a q u i t e easy consequence of Theorem 3.5; i n d e e d , a l l t h e p o s e t s c o n s t r u c t e d i n t h e proof of C o r o l l a r y 3 . 7 have h e i g h t 2 . The second theorem, which i s proved i n [22], r e q u i r e s more c a r e f o r i t s p r o o f . A somewhat d i f f e r e n t c o n s t r u c t i o n can be used t o o b t a i n 2 " r ( o c a t e g o r i c a l l a t t i c e s of dimension 2 , t h u s r e n d e r i n g as u n l i k e l y a s a t i s f a c t o r y s o l u t i o n t o Problem 2 o f [ 2 8 ] .

H

THEOREM 7 .l.

There are

2

THEOREM 7 . 2 .

There are

2

'Ko-categorical

p o s e t s o f height

Ho-cateyoricaZ p o s e t s o f dimension

Theorems 7 . 1 and 7.2 cannot be merged i n t o one theorem. which i s a new o b s e r v a t i o n , g i v e s t h e r e a s o n f o r t h i s . THEOREM 7.3.

2.

Every Ho-cateyorical poset of dimension

2

2.

The next theorem,

and f i n i t e height i s

decidable. PROOF: Let (A; <) b e an . - c a t e g o r i c a l p o s e t having dimension 2 and f i n i t e and < 2 o f A,* such t h a t h e i g h t . By Theorem 5 . 3 t h e r e are l i n e a r o r d e r s < = n < 2 and (A; <,
Therefore, is d e c i d -

I n c o n t r a s t t o t h e p r e c e d i n g theorem i s t h e c a s e of p a r t i a l l y o r d e r e d s e t s of dimension 3 . The f o l l o w i n g i s a l s o a new theorem. TFEOREM 7.4.

There are

2

H

Ho-categorical p o s e t s o f dimension

3

and height 2.

PROOF: Let (A; R) b e one of t h e 2 H o - c a t e g o r i c a l p o s e t s of h e i g h t 2 which e x i s t a c c o r d i n g t o Theorem 7 . 1 . L e t A1 b e t h e s e t of minimal elements of (A; R), and A2 t h e s e t of maximal elements We can assume w i t h o u t l o s s of gene r a l i t y t h a t A 1 n A2 = 0. By Theorem 5 . 1 t h e r e i s a l i n e a r o r d e r < of A such t h a t (A; R, <) i s s t i l l H o - c a t e g o r i c a l . We can t h e n assume t h a t A L Q n (0,l). We w i l l c o n s t r u c t a p o s e t & = ( B ; 4 ) of h e i g h t 2 . The set of maximal e l e ments of B w i l l be A, and t h e s e t of minimal e l e m e n t s w i l l be t h e s e t Bo = R u { < O , l > , < 1 , 0 > } . Define 4 on B = B o u A so t h a t <x, y> a i f f one of t h e f o l l o w i n g h o l d s :

<

a

E

A1

and

x < a;

a

E

A2

and

y < a.

It f o l l o w s from an a p p l i c a t i o n of categorical.

Ryll-Nardzewski's

Theorem 2 . 1 t h a t

8

is

We w i l l now d e m o n s t r a t e t h a t t h i s c o n s t r u c t i o n y i e l d s 2 nonisomorphic 8 by showing t h a t i f w e s t a r t w i t h p o s e t s (A; R) which are n e i t h e r isomorphic n o r a n t i - i s o m o r p h i c , t h e n w e end up w i t h nonisomorphic 8. This w i l l be done by "recovering" e i t h e r (A; R) o r i t s d u a l from &. N o t i c e t h a t { A i r A21 i s t h e unique p a r t i t i o n IP1, P 2 } of t h e set of maximal elements of B f o r which t h e r e are e l e m e n t s p i , p2 E B such t h a t P i = { x E B : p i 4 X I W e get t h i s p a r t i t i o n

.

No -categorical partially ordered set

28 1

. .

by s e t t i n g p i = <0,1> and p2 = <1,0> Define < i on A i by: x c i y i f f x The o r d e r i n g < on i Ai coincides f o r some a E B , a < y b u t n o t a with t h e o r d e r i n g < o n A i . Now, f o r x E A 1 and y E A2 t h e f o l l o w i n g h o l d s : xRy i f f f o r some a E B ,

<

{b c B : a Thus

R

<

bl = i b

E

A1 : x

b ) u {b

E

A 2 : y c 2 b}.

is recoverable.

8 h a s dimension 3. We c l a i m t h a t 4 = <1n.(2,,<3 t h e iia r e l i n e a r o r d e r s of B d e f i n e d so t h a t i f < x , y > , E Bo, then t h e following hold:

I t remains t o show t h a t

where

or

al ( 2

bl

iff

a1 < bl;

a2 42

b2

iff

a2 < b2;

< x , y > <3

iff

(x = z

and

y < w);

d 2 < x , y > ;

< x , Y > < ~a l ; < x , Y ><3

a2;

a1 <3

9;

a1 <3

bl

a2 <3

bg

iff iff

b l < al; b2 < a2.

It i s v e r y e a s y t o check t h a t

4

=

( 1

n

(2 n ( 3

.

0

Let P = ( P ; <) be a f i n i t e p o s e t , and t h e n l e t Kp b e t h e class of p o s e t s which have no s u b p o s e t s isomorphic t o 9. Thus, (A; < ) b e l o n g s t o K t j f f Z h o KO-cateP H((A; < ) ) . We w i l l c o n s i d e r t h e q u e s t i o n of whether t h e r e are g o r i c a l p o s e t s i n KP. When t h i s f a i l s f o r a p a r t i c u l a r P, i n a l l known c a s e s , i t i s because e v e r y N o - c a t e g o r i c a l p o s e t i n Kp i s d e c i d a b l e . N o t i c e , by way of comparison, t h a t i f P i s a f i n i t e a n t i c h a i n , t h e n Theorem 6 . 4 asserts t h a t e v e r y B o - c a t e g o r i c a l p o s e t i n Kp is f i n i t e l y a x i o m a t i z a b l e . Conversely, i f P i s n o t an a n t i c h a i n , t h e n t h e c o u n t a b l y i n f i n i t e a n t i c h a i n i s i n Kp, and t h i s poset is No-categorical but not f i n i t e l y axiomatizable.

4

A p o s e t ( P ; <) i s a weak a n t i c h a i n i f t h e r e i s e x a c t l y one element of P which is n o t maximal and e x a c t l y one element which i s n o t minimal. A p o s e t h a s weak width 5 n i f f i t i s i n Kp where P i s t h e weak a n t i c h a i n w i t h n+2 elements. Thus, a p o s e t h a s weak w i d t h 5 1 j u s t i n c a s e i t i s a weak o r d e r . Not i c e t h a t a c o u n t a b l e a n t i c h a i n h a s weak w i d t h 0 , y e t i t i s tZ - c a t e g o r i c a l and n o t f i n i t e l y a x i o m a t i z a b l e . The f o l l o w i n g r e s u l t i s proved i n 7231.

THEOREM 7 . 5 .

Every

&-categorical

poset w i t h f i n i t e weak width is decidable.

It w a s proved i n [ 1 8 ] , and i n [17] by somewhat d i f f e r e n t means, t h a t e v e r y c a t e g o r i c a l t r e e i s d e c i d a b l e . T h i s r e s u l t can be improved upon i n two d i f f e r e n t

J.H. Sc hmerl

282 ways.

The n e x t theorem, proved i n [ 2 3 ] , i s one way; t h e o t h e r i s C o r o l l a r y 7 . 9 .

We have d e p i c t e d below t h e Hasse diagrams of 1 0 f i n i t e p o s e t s , PO, P1, We w i l l w r i t e Ki f o r Kpi.

THEOREM 7 . 6 .

Every X3-categorical poset i n KO

DEFINITION 7 . 7 .

Let

(A; < )

..., p9.

i s decidable.

be a f i n i t e p o s e t .

(1) A s u b s e t I S A is an i n t e r v a l i f f o r every a b o t h of t h e f o l l o w i n g hold: a < b i f f a < c , and b < a (2) (A; <) i s indecomposable i f f whenever e i t h e r I = A o r ( I ( 5 1.

I S A

E

A \ I and i f f c < a.

b,c

E

I,

is an i n t e r v a l , then

The f o l l o w i n g i s a new r e s u l t which i s proved by an e x t e n s i o n of t h e t e c h n i q u e used i n [25] t o prove C o r o l l a r y 7.9 below. The p r o o f , which will n o t b e g i v e n h e r e , i s based on r e s u l t s i n a forthcoming paper [ 2 7 ] . THEOREM 7.8. For each n 2 4 , every undecidable N,-categoricaZ decomposable subposet w i t h exact Zy n elements.

poset has an i n -

An example of a n indecomposable p o s e t i s P i , which i s t h e o n l y indecomposab l e p o s e t w i t h e x a c t l y 4 e l e m e n t s . I n f a c t , it i s proved i n [24] t h a t e v e r y i n decomposable p o s e t w i t h a t l e a s t 4 elements h a s a subposet isomorphic t o Pi. The f o l l o w i n g c o r o l l a r y i s immediate. COROLLARY 7 . 9 .

Every H,-categoricaZ

poset i n K 1

i s decidable.

The n e x t theorem y i e l d s v a r i o u s t y p e s of examples of u n d e c i d a b l e Ho-categ o r i c a l p o s e t s . The f i r s t two c a s e s a r e proved i n [ 2 2 ] . The p o s e t s c o n s t r u c t e d i n t h e proof of C o r o l l a r y 3.7 can b e used t o prove t h e l a s t c a s e . THEOREM 7.10.

classes:

There are

2

H,-eategorical

p o s e t s i n each of t h e following

Kz n K3, K2 n Kg, and K 5 n K6.

I t can e a s i l y be checked f o r e a c h f i n i t e p o s e t ing t h r e e conditions holds:

P e x a c t l y one of t h e follow-

H, -categorical parlia1I.v ordered set

in

(1)

P

i s a n a n t i c h a i n o r a weak a n t i c h a i n ;

or

(2)

?

has height

p2, P 3 , P4, P t .

Pg,

P

i s citiier

23;

or

P. (3)

p 7 , P:,

7-83

P

Po, P i , o r

is

$, o r

Pg

P1;

is e m b e d d a b l e

~ 8 .

or

I t f o l l o w s i r o m l'lieorems 7 . 5 , 7 . 6 a n d C o r o l l a r y 7 . 9 t h a t i f (1) i s s a t i s f i e d , t h e n e v e r y H - c a t e g o r i c a l p o s e t i n Kp i s d e c i d a b l e . C o n v e r s e l y , i t follows from T h e 0 r c n i s ~ 7 . 1 and 7 . 1 0 t h a t i f ( 2 ) i s s a t i s f i e d , t h e n t h e r e a r e 2 NO k - ' a t e g o r i r a l p o s e t s i n Kp. T h i s l e a v e s as u n s e t t l e d w h a t t h e s i t u a t i o n i s i n t h e c a s e of ( 3 ) . T h u s , Lhe f o l l o w i n g two q u e s t i o n s , o r i g L n a l l y a s k e d i n [ 2 1 ] , a r e s t i l l unanswered.

QUESTION 7 . 1 1 .

Is t h e r e an u n d e c i d a b l e

Ho-categorica7

poset i n

QUESTION 7 . 1 2 .

Is t h e r e a n u n d e c i d a b l e

Ho-categorical

p o s e t i n Kg?

K7?

We end w i t h a new o b s e r v a t i o n . A ' i s a p o s e t t h a t c a n b e represent e d by u n i t i n t e r v a l s . Equivalently, t h e semiorders are p r e c i s e l y t h e p o s e t s i n the c l a s s K3 n Kg. THEOREPl 7 . 1 3 .

L'LICY2

is dcciJabk,

Ncl-~'Ot

PROOF: L e t ( A ; <) b e a n W o - c a t e g o r i c a l s e m i o r d e r . We w i l l u s e t h e n o t a t i o n x / y t o d e n o t e t h a t x and y a r e incomparable. F o l l o w i n g Theorem 5 . 1 , l e t < be a l i n e a r o r d e r of A such t h a t (A; < , < ) i s still Ho-categorical. Define t h e b i n a r y r e l a t i o n 4 on A so t h a t x 4 y i f f a t l e a s t one of t h e f o l l o w i n g holds :

(1) x

<

y;

(2)

XIy A 3 z ( X

<

(3)

Xly

A32(7.

< y

(4)

X/y

A

X

Z A y I Z ) ; A

ZlX);

< y AvZ[(X

2

y <

2) A (2

<

X

f-t

2

<

y)].

I t f o l l o w s f r o m R y l l - N a r d z e w s k i ' s Theorem 2 . 1 t h a t (A; < , 4 ,4 ) i s (\',-categori c a l , s i n c e A u t ( ( A ; < , < , a ) ) = Aut ( ( A ; < , < ) ) . An e a s y e x e r c i s e now i s t o show t h a t ( A ; Q ) i s a l i n e a r l y o r d e r e d s e t . It i s a l s o a n e a s y e x e r c i s e t o show t h a t t h e r e l a t i o n < i s m o n o t o n e w i t h r e s p e c t t o a. T h u s , by Theorem 6 . 5 , t h e s t r u c t u r e ( A ; a , <) is f i n i t e l y a x i o m a t i z a b l e , and t h e r e f o r e d e c i d a b l e . (A; < ) i s a l s o d e c i d a b l e . 0 Thus, i t s r e d u c t

J.H. Schmerl

284

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Recursive models, in: Aspects Yarra Glen, 1981).

Co.,

[25] Schmerl, J.H., Arborescent structures, 11: Interpretability in the theory of trees. Trans. Amer. Math. SOC. 266 (1981), 629-643. [26]

Schmerl, J . H . , Ho-categorical distributive lattices of finite breadth, Proc. Amer. Math. SOC. 3 (1983), 707-713.

[27] Schmerl, J . H . and Trotter, W.T., Indecomposable Graphs, Posets, Tournaments and other Binary Relational Structures (Tentative title, in preparation). [28]

Smith, K.W., Stability and categoricity of lattices, Can. J. Math. 1380-1419.

2 (1981),

[29] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, (North-Holland, Amsterdam, 1978).

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 (1984) 287-312 0 Elsevier Science PublishersB.V. (North-Holland)

THE ARITHMETICS

OF

TWO

287

AS

THEORIES

ORDERS

Denis RICHARD Universitg Claude Bernard (LYON I ) Dedie a Monsieur le Professeur COROMINAS qui crba le laboratoire d'Alg6bre ordinale de LYON.

ABSTRACT. We study properties of the natural order and the order of divisibility in arithmetic. After a brief survey of some classical results on endextensions and cofinality for non-standard models of Peanoarithmetic, we l o o k at the incompleteness of arithmetic and present the undecidability of Th(W, S, 1)and several related theories. Next we discuss saturation: in particular a model is saturatedifandonly if one of its order structures is saturated. Then, we give a survey, new proofs and new results of the definability of arithmetic as a theory of two orders and answer two questions of J . ROBINSON. RESUME. Nous ?Studions des proprigtss de l'ordre nature1 et de l'ordre de divisibilits en arithmstique. AprPs un bref survol de rssultats classiques sur les extensions finales et cofinales des modsles nonstandards de l'arithmgtique de Peano, nous passons 21 l'incomplctude de celle-ci et b l'indscidabilit.5 de Th(lN, S, 1) et de plusieurs autres thgories. Ensuite nous parlons de saturation: en particulier un modzle est satur6 si et seulement si l'une de ces deux structures d'ordre l'est. Nous terminons par une synthPse, avec de nouvelles preuves et de nouveaux r?Ssultats, sur la dsfinissabilitg de l'arithm6tique comme thCorie de deux ordres et r6pondons b deux questions de J. ROBINSON. INTRODUCTION In the theory of numbers there are many results on approximations and asymptotic behaviour of classical functions needed for the investigation of integers (see for instance [HW] or [EMF]). This involves more the study of the connection between analysis and arithmetic than the connection between theory o f orders and arithmetic. There is also a combinatorial approach (see [EG]). Our aim is to demonstrate the importance of order structures in arithmetic. We start from the following simple observations. The usual axioms of arithmetic of second order (this and other logical and mathematical terms are defined in section 0 ) can be formalized only with two types of objects (the non negative integers and the sets of non negative integers) and the successor function S. Also the Peano arithmetic PA is usually described in the language { 0 , 1 , S , < , + , 0 but, already long ago, Julia ROBINSON [JR] showed that PA can be perfectly well defined in the language I S , I ) where I is the divisibility relation. In other words, the arithmetics can be considered as theories of two orders < and I and therefore the (distributive) lattice order I linearly extended by < Firstly we give interesting examples of rich order theories. Among them Th(lN, S, 1) and obviously also Th(lN, S, I ) - where I denotes coprimeness - are shown to be undecidable in $ 3 (we have recently proved that Th(Z, S, 1 ) is also undecidable [OR31 ) .

.

D.Richard

288

Secondly, we prove certain deep properties of saturation which may be generalized to "biordered sets"; for example we prove that the non-standard models of PA are w -saturated if and only if their reducts to < , or to I are 1

w

1

-saturated (true also for w -saturation [JFP] )

. Thirdly certain interesting

conjectures originally motivated by logic and order are equivalent to purely [JRI is number-theoretical ones; e.g. "Is Th(W) = T n ( H , S, 1) ?" equivalent to "Is-it possible to { S , 1) - define an < - increasing sequence of primes ?I' ( § 3) and also equivalent to "Is a universal k such that x = y whenever x + i and y + i have the same prime divisors for all 0 6 i < n ? " [AW] The last conjecture is implied by a conjecture due to ERDOS [ER]

.

.

First in section 0, we give all the necessary definitions and facts on nonstandard models, the two countable non-isomorphic order structures of nonstandard models of arithmetic and the classical theorems of Mac DOWELL and SPECKER, GAIFMAN, FRIEDMAN and WILKIE which are all very closely related to the natural order. In section 1, before treating problems of decidability, we concisely give the famous theorem of incompleteness of PARIS and HARRINGTON which is both combinatorial and order-theoretical. We recall some known theorems on the decidability of Th(R, S) , T h ( I N , < ) , Th(N, 1 ) . We conclude by proving the undecidability of Th(TN, S , I ) and announce the same for Th(Z, S, I )

.

Section 2 treats the saturation of Peano arithmetic and hence the possibility of defining elements by types ([CK] p.97) which are a kind of infinite existential definition. Surprisingly, everything that we can insert into a model without destroying it, can be added just using types in the languages I<) or [ I ) . We prove it for o -saturation (already published in [DRZ] and [JFP]) and we 1

.

The other results also note that these results are not extendable to Th(2L) (the extension to w -saturation, the preservation of saturation by end and cofinal extension) arelistedwithout proofs, which may be found e.g. in [JFP], [HKI,

[KSI)

.

Section 3 presents a survey of definability based on the orders < and 1 or their "parts", namely S and 1 All the proofs are given with the exception of three results which are to appear in [DR3] and [DR4]. The first proof of theorem 3 . 3 . (the definability of + and 0 from S and I ) is the original proof of Julia ROBINSON [JR] and the theorem 3.6. and its proof (the definability of + and from < and I ) are due to A . WOODS [AW] All the other proofs are new and situated in the frame-work of a coding method based on the ZSIGMONDY - BIRKHOFF VANDIVER theorem ( I 0). Theorem 3.5. gives our I + , 11 - definition of multiplication. We answer positively a question posed by Julia ROBINSON [JR] : are there I S , I ) - definitions of + , 0 and < for integers (including negative ones) ? We note that { + , 1 1 is a weaker language than { <, 11 and { S , I } which are both synonymous with I t , , < I over integers (including negative ones). The last theorem of the section is a step in the direction of the arithmetical definability of + and 0 in terms of S and 1 (which, in its turn, is equivalent so the { S , 1 1 - definability of <) since it asserts that both natural order and divisibility orders are I S , L}- definable over the powers of primes.

.

.

.

.

Sections 0-1-2 presents an overview of the domain whereas section 3 contains both a survey and new results. We must apologize to logicians for section 0 which presents, besides very known classical theorems, such an elementary result as the proposition 01 giving the order type of the countable non-standard models of Peano, but a good understanding of such very simple facts in Peano arithmetic is necessary for what follows.

Arithmetics as theories of two orders 5 0-

289

DEFINITIONS, NOTATIONS, RESULTS OF THEORY OF NUMBERS; CLASSICAL THEOREMS ON NON-STANDARD MODELS OF PEANO. (§

0.1.) -

The aritbmetics -______________

-

In a reasonable theory of sets, say ZF, the set IN of the non negative integers can be defined as a set, with a distinguished element 0 and an injective selfmap S called the sucLvssur funct7'on such that S ( W ) = IN \{O} , verifying the second order induction V A GIN

(0 € A

A V

x ( x € A + S(x)€

A))

+ A

=

.

IN

All the first order statements (i.e. those expressed by formulae in terms of two quantifiers, all logical connectives, + and 0 and variables ranging only over elements) satisfied in IN, constitute the t r u e a r i t h m e t i c Th(IN) Similarly for the set 22 of negative and non negative integers, we have the true arithmetic of arbitrary integers Th(lL). The Peano arithmetic PA is the L = {=, 0 , 1 , S, <, +, . I given by the first order theory in the language axioms 1 = S(0) ; V x ( x t 0 = x) ; V xy (x + S(x) = S(x+y)) ; V x (x.0 = 0) ; V xy (x.S(y) = x.y t x) ; x < y f-t 3 z ( z # 0 A x + z = y) ; and the infinite list of axioms

.

v

XI

...

[(

x

XI,

V(0,

..., x

) A V X(V(X, X I ,

V(S(X),

+

XI'

...,

Xn)

..., xn)))+

+

v

x V(X,

XI'

**.,

xn)]

where 7 ranges over a11 formulae in L. As usual, from the primitive symbols of L we define x = u x' ( q (x', x l , .., x,)) by and 9 (x, x l , xn) A (y < x + lv(y, x l , ..., x,))

.

x

=

max

XI

..., ( xi, x l , ...,

For an integer

I

x y and

bv

mpr.iie'

a

, we

put

,

x,)) x

Z

y(a)

whenever it exists, in an analogous way. iff

3

k Ix-yI

=

ka

; next we define

k (y = kx) ; the usual symbol P(x) stands for "x is a p r i m e " means "x and y have no common prime divisor", i.e. "x and y are iff we put 3 ! x q(x, X I , ..., xn)

3

i

q (x, x I ,

. .., xn)

(y # x

A

+

1 q

(y, x l ,

.

. . ., x,))

; obviously x


is

The symbol Supp(x) stands for the s e t of equivalent to x < y V x = y t h c 2 p r i m e dii>isors of x Now the f u Z l Inngucqe L(P) will be precisely the language we can obtain by adjoining to the basic language L all the symbols definable by formulae of L with or without induction in PA. A sublanguage of Peuno is a subset of L ( P ) The full language L(Z) is obtained by adjoining to the language L' = { + , 0 , < I all symbols definable by formulae of L' in Th(Z) A subZanguage of Th(Z) is a subset of L ( Z ) . We call a sublan(resp. L(Z)) if in guage K of PA (resp. of the Th(Z) synmyrnous with L(P) Th(IN) (resp. in Th(Z) ) + and 0 are definable by formulae of K. Actually, the synonymity of {t , 0 ) with L(P) is provable in PA (hence in Th(IN)) using the representation theorem of exponential function via the GODEL function f? (see [HE] p.245).

.

.

.

(§

0.2.)

- ------________________________________ Some key-results in theory of numbers

-

First of all is the BERTRAND'S postulate, namely the

TCHEBYCHEV's theorern such that

(see [HW] p.343). For every n >, 1, there is a prime p n < p 6 2n ; that is, if pr is the r-th prime

prtl < 2 pr

for every positive integer r

A s is customary in this fiels we u s e the

.

D. Richard

290 ('li2'ni ; L w r m i ~ n ' k t the=. ~ bl,

...,

For a r b i t r a r y i n t e g e r s

bn

0 < i

with

at,

...,

and f o r

a

p a i r w i s e c o p r i m e t h e s y s t e m of c o n g r u e n c e s

6

n

tins 'In i n f i n i t y o f s o l u t i o n s i n

x = ai (bi) iz \ Td 1 .

IN (a nd i n

The n e x t i m p o r t a n t t h e o r e m i s t h e

DTN~C'l!I,iT's t h ~ ~ r ~ rFno .r a and b c o p r i m e t h e r e a r e i n f i n i t e l y many p r i m e s o f ( c f . [HW] p . 1 3 o r [EMF] p . 2 4 2 ) . t h e form a n + b Combining t h e two l a s t t h e o r e m s i t f o l l o w s t h a t a s y s t e m of c o n g r u e n c e s s a t i s f y i n g t h e h y p o t h e s i s o f t h e C h i n e s e r e m a i n d e r t h e o r e m h a s an i n f i n i t y of p r i m e s o l u t i o n s when f o r e v e r y i (0 < i < n ) a i and b . a r e c o p r i m e . F i n a l l y t h e most i m p o r t a n t t o o l i n 8 3 o f t h i s p a p e r ( u s e d f o r c o d i n g ) i s t h e b e a u t i f u l ZSICMONDY's t h e o r e m r e d i s c o v e r e d by B I R K H O I ~ F a nd V A N D I V E R ( c a l l e d b elo w t h e "ZRV-theorem" [BV] ) . %I:lI;MOi)RDY - F~IRKlIOF1~'-VANDlVi?/f thcnrJt,iri. L e t a, b be c o p r i m e i n t e g e r s ( p o s i t i v e F o r a ny 1 ' 0 , t h e r e i s a t of n e g a t i v e ) s u c h t h a t 7 a l > Ibl

.

l e a s t a prime d i v i s o r of

m

for

<

an - bn

dividing n o n h f the

( s u c h d i v i s o r s a r e c a l l e d p~'~;rr'/,'ti~)c for

n

,

except i n the following cases

n = I

a+b=21X (azl);n - 3 ,

a = t 2 , b = ; I

t 2 a = -

,

b = t 1

a-b = I ; ;

am - bm

an - b n )

,

n = 2

n = 6 ,

( f o r m u l a t i o n d u e t o SCHINZEL i n [ S C l ) .

Note t h a t i t i s e a s y t o e x p r e s s them w i t h S . The f o l l o w i n g c o r o l l a r y i s e f f i c i e n t f o r c o d i n g t h e powers of t h e p r i m e s ( s e e t h e A ppe ndix) : COROLLARY o f t h e ZRV-Theorem.

i Proof. Supp(Z6

pa

of

Note t h a t -

> 2

F o r e v e r y p r i m e p and e v e r y p o s i t i v e i n t e g e r

I ) = {3,71

i s c h a r a c t e r i z e d by

p

26 - I

.

is the only

Supp(p" - I )

2a - I

(a

> I)

such t h a t

General s i t u a t i o n i n t h e ZBV-theorem.

n = 1

T h e r e i s n o t h i n g t o s a y f o r t h e e x c e p t i o n w he re

.-.-

.-

t h e power

.

because Supp(2 - I ) = 8 c h a r a c t e r i z e s Zn - 1 i n t h i s c a s e . t h e r e i s no e x c e p t i o n s i n c e p and b h a v e t h e same s i g n .

FIGURE 1

,

I

I

For

n

=

3

Remdrk. We illustrate the situation described in the corollary by some diar,ramms where a , h means "a divides b" in the lattice of divisibility and a ' h means "a i13" in the 1aLtice of natural order (see figures I and 2).

5 means I ; means < ; italic niimbers 'irk- primitive divisors; Supp(x) is in the corresponding circle t o x . +

f The order

:

< o f a non-standard model of

PA

is a chain of order type

where w , w:: and n are the usual order types of the non negative integers, of the negative integers and of the rationals.

(*I

+ (u" + u ) n

D. Richard

292

therefore eachnon-standard integer is greaterthan everyn ~ W . H e n c efora fixed nonstandard a and for every n e R , the integer a + n and a - n are also non-standard. Denote the set a + 21. in M . Note that a = if for a E M\W and

a

a

5

E M \W

z t 2L

there is an integer

equivalence relation. Now define in x

=

2y v x

a -+ z

the element

y

=

P

, which

is an

=

x] by With this, note that there are countably many classes

.

2y + 1

=

such that

PA

in M , each of which is of order type

iii"

+

id

.

CI

We prove that classes themselves

a dense countable order without a least or a greatest element which is therefore of <-order type q Put a << B iff a t M \ W , 5 t M\TN , form

.

a

< B and

.

a#

This relation <<

< by the equivalence that << 3 implies

I:] <<

a

<<

:

.

-a

is an order which is the quotient of

<< is dense, it is enough to check << 7 (a + 5 ) ] << which is easy to do. Similarly To prove that

#

I'

#

prevents the existence of least and greatest elements.

Remark. Roughly speaking, a countable non-standard model of IN followed by countably many copies of z! ordered a s Q : FIGLIRE 3 . -

21.

21.

~...+...+....+ O

W

21.

PA is a copy of

..........

I

n is the <-order typeyf the copies of So all countable non-standard models of

PA

0

21.

I

.

have the same order type

w + (to" + w ) q

but they need not be isomorphic; in fact, one can show the existence of continously many pairwise not isomorphic such models.

Now we quickly review (without proof) some deep results closely connected to the natural order. First we must define an end e x tension of a model M of PA V b E M (b < a) as a model M ' of the same theory such that V a E M ' \ M In the usual way we define that a model M is < - c o f i n a l in a model M ' . An elementary extension (see [CK] p.82) of a model M of PA is a model M' 3 M of n PA such that for every formula ~p EL(P) and for every (al, an) E M ,

.

...,

the formula

~p

is satisfied in M

iff it is satisfied in M ' .

The following results are classical: I ) . Theorem of Mac DOGJELL and SPECKER

[MS]. proper elementary end extension.

Every countable model of PA has a

2 1 , G A P M A N ' S theorem

[HG]. Every model M of Peano is an elementary extension of every model M ' o f Peano which is cofinal in M . Every extension of M is obtained by an end extension followed by a cofinal one.

31.

FRIEDMAN'S theorsern [ H F ] .

If M is a countable non-standard model of PA, then M is isomorphic to an initial segment of itself.

4 ) . CJLLKIE's theorem

[ALW]. For every countable non-standard model M of Peano, there is a model M ' of Peano Which is an end isomorphic extension of M and such that M' solves a diophantine equation with coefficients in M , that is not solvable in M.

Arithnietics as theories of two orders

293

Concerning the final segments of models of Peano, we answered [DRII the following question of JENSEN and EHRENFEUCHT (Problem 3 in [ J E l p.242):

"If M I and M2

(models of Peano) have final segments that are isomorphic, what properties of M M

Actually all properties are preserved since M I and 1 are preserved in M2 ?". are isomorphic as Peano models. To see this, let q be an isomorphism from a

2

final segment I1 of M

1

onto a final segment I2 of M2

E M I and for 1 This defines a mapping from M

m

€

MI , put

into M 2

.

It can easily be shown that

any

$(m)

.

Fix m

- q(m,). I $ is a bijection and an additive isomor-

= q(m+ml)

phism. The proof that it is multiplicative follows from a calculus for which we recall that ~ ( x ) is a prioz'i o n l y defined for x & m l ~ ( x - y )= q ( x ) - q(y)

that a-b

=

=

a

c means

@ ( m '+ m l + m l

§ 1

-

2

.

=

b+c

We have

-

ml ) - q ( m l )

2

=

. Moreover, note

x 2 m l , y 2 m l , x - y 2 m l and

whenever

$(m)

$(mm')

$(m')

,

so

2

= q ( m m ' + m (m+m') + m l ) 1

we are done.

LNCOYPLETENESS AND UNDECIDABILITY IN ARITHMETIC. (§

1.1.)

-

hsmpletenesp

-

In 1931, GODEL showed that if a theory T is strong enough (for instance if it contains the Peano arithmetic) and is recursively axiomatisable, then there is a sentence which is both t r u e and not provable from the axioms of T. In 1977, PARIS and HARRINGTON provided a combinatorial theorem, closely connected to the order structure of the positive integers, which is not provable in PA For this they used the notion of dense set: X is dense iff Card(X) >,min(X) Mac ALOON notes (IKMI p.04) : "la notion d'ensemble dense donne une mesure de la taille d'un ensemble fini d'entiers qui ne dspend pas uniquement de la cardinalit6 mais de "l'ordinalit6" de l'ensemble, c'est-%-dire de sa distribution dans la suite des entiers finis

.

.

.

A basic tool of the argument is the concept of KIRBY-PARIS i n d i c a t o r function which we have to define. Take a L(P)-definable function Y of two variables. Take a model M of Peano, and the set IN(M) of all the initial segments of M which are models of Peano. Suppose that for every I E IN(M) the interpretation of Y in I is exactly the restriction of the interpretation of Y in M, (this happens when Y is a b s o l u t e ) . Moreover, suppose that we can prove in PA that y' Y(x', Y) d Y(x, Y') x' d x 6 y

<

-f

Then every function Y satisfying these conditions is an i n d i c a t o r if and only i f for every model M, YM(x, y) > IN tt 3 I E IN(M) (x < I < y)

.

THEOREM ( K I R B Y - P A R I S - H A R R I N G T O N )

i

a) There is an indicator function such that N satisfies V x V y 3 z [Y(x, z ) > y] ; such an indicator is called admissible.

D.Richard

294

I

b) If Y is an admissible indicator function then the sentence V x V y 3 z [Y(x, y ) > 21 is not provable in PA although true in 7 N .

The proof uses the RAMSEY's theorem to build up an admissible indicator by coding as usual the finite sets of positive integers (see [PHI and [ K M ] )

.

-

( I 1.2.) - Undecidability

A classical result of PRESBURGER MP asserts that Th(lN, +) is complete and decidable. The proof is an example of the method of the elimination of the quantifiers. SKOLEM began the work for Th(N, .)and CEGIELSKI in [PC] gives the proof of the decidability of Th(IN, a ) and axiomatizes this theory. Then

,

Th(lN, S)

Th(N, <)

and < are obviously

,

Th(N,

1)

(+) -definable

and and

Th(N,

I

and

1

1)

are decidable since S

are of course

(a)

-definable.

For theories with two predicates we first prove the LEMMA 1.2.

Proof.

i

If K is a synonymous sublanguage of undecidable.

To every formula 9 of L ( P )

,

then Th(N, K)

is

with n variables, we can recursively

of K also with n variables such that for every

associate a formula

..., xn)

L(P)

...,

xn) is satisfied in IN the formula p(xl, x2, iff N realises q(xl, x2, In particular, if we only take sentences xn) is recursively reducible to Th(N, K) , as formulae 9 , we see that Th(W) (xl, x2,

E

INn

...,

.

Proof. -

The sublanguages concerned are shown to be synonymous with section 3 .

L(P)

in 0

THEOREM 1 . 4 .

1

The theory Th(IN, S,

1)

is undecidable.

Proof. With theorem 3.9. below, it is easy to build up in IN an inner model isomorphic to IN on the powers of 2. Then an adaptation of lemma 1.2. leads t o undecidability of Th(N, S, 1). Remark. I . )

In fact, part a) of the corollary 1 . 3 . also follows from theorem 1.4. Moreover, we do not know whether IS, 11 is synonymous with L(P) (see § 3 ) .

.

2 . ) Our first result was that Th(N, = , S, 1) is undecidable when simultaneously and independently A. WOODS proved and told us that Th(IN, S, 1) i s undecidable. Actually our proof was independent of the identity in virtue of the corollary of the ZBV-theorem.

Arithmetics as theories of two orders 8 2 -

295

SATURATION OF PEANO MODELS.

Let IY. be a cardinal. A model M of Peano is a-saturated iff for every subset X C M with fewer than n elements, the model M realizes every type E(v) of the language L(P) U Icx I x E X i which is consistent with Th(M). A model is 8 / i t r m i t r f liff i t is

(see [CKI p . 2 1 4 ) . Let Lo be a sublanguage

181-saturated

is a-snturnting for a cardinal a (or that PA is

of Peano. We say that

Lo iff for every model M of Peano, the u-saturation of the

a - ~ t 7 t z i ~ ~ ~ zover i,~d L ) 0

(M) of the structure M to Lo implies the a-saturation of M LO (see [CK] p . 4 2 5 ) . In other words, L is a a-saturating sublanguage iff whenever

r'ciizdct

Red

0

the only types expressed in Lo are realized in M , then all the positive types expressed in the full language are also realized in M.

Now we show that the

Peano arithmetic is w -saturated over order and divisibility. 1

LEMMA 2.1.

(w -saturation criterion). 1

1 A model M of PA of cardinality w-sequence

(ai)

ul

is w -saturated iff every 1

of M is an initial segment of a coded

ic w sequence of M. Proof. Let M be a non-standard model of P and I a proper initial segment of M with no last element. We show that there is no formula in L(P) for expressing x € I . Suppose we have a formula 7(x, a l , ..., an) with parameters in M such that for all b E M it is true that b E I iff M satisfies q

..., an):

(b, a l ,

Consider

X

=

{x E M

IM

satisfies

X has a first element, say xo

By the axioms of Peano,

As I is closed under successor, xo t I :

l q ( x , a],

, but

contradiction.

..., an)) .

then xo

-

1 E I ,

With this we know

that if a formula*$ is satisfied by all the x t IN in M, then there is at least an element y € M \ IN such that M satisfies $(y) This property is called oVerspiZT,. Now take a coded sequence which extends the sequence

.

(r(i))it

such that

r(i)

Let

u +qk(r(y)))

.

iew V y

(k

6y<

realizes the formulae

.

E(x) = (qi)

type

$,(u)

7. for

j

4 i of a given

be the formula

Since for every

n

t

IN

the model M satisfies

, there is an element ak E M\IN such that M satisfies $,(a,) . Since M satisfies v 6 u -+ ($,(u) + $ ,(v)) , the existence of the a; s allows us to lLk(n)

define a decreasing sequence s such that k t IN.

O(v)

V ij (i

by

every n

t

<

j


-t

v < s(j)

<

s(i)).

is true in M for every

Hence

IN and, therefore, there is an element

is true in M for every every

Jlk(s(k))

With a code of a sequence of M which extends s we can define a formula

r(y)

where

k

€

IN. Consequently the type

IN < y < a

.

e(n)

a E M\IN

E(x)

is true in M for such that

$,(a)

is realized by 0

D.Richard

296

THEOREM 2.2.

1 The Peano arithmetic is

Proof.

Take a sequence

substructure c

E

M

Red

M

w -saturated over

1

a = (a(i))ito is w -saturated.

(1) 1 realize the following (I)-type

Then the sequence of the exponents

.

Suppose that for a model M of P A , the Let

p

of the

a.

(1) .

be the n-th prime.

pi

Let

in the usual decomposition

of c form a coded sequence s (by the code c) in M.

Since a is an initial

segment of s, the previous lemma achieves the proof.

U

The same strong result is also true for < : THEOREM 2.3.

1 The Peano arithmetic is

w

I

-saturated over order.

Proof. We associate to every sequence (ai) of a model M of Peano another i is w (bi) given by bi = Z (a, + I ) . By assumption, the order structure of M j=o

iE u

J

c t M

is w -saturated so that there is an element I (ci)

c +I izCO ,< x < 2 O

n+l ;

c

c. 2 l < X <

n

c

c

.

so

-1

c 2i+2"+1

i=O


(bi)

ie u is strictly decreasing and

where c. = c - b i ir w we can code it by realizing the following type: The new sequence

such that

I

.

i=O

Then the theorem 2.3. follows from the lemma

.

2.1.

,

0

The classic results on order types, proposition 0 . 1 . and theorem 2 . 3 . lead to the characterization of the w -saturated models of Peano by their order 1 structure: COROLLARY

Proof.

1

A model M of Peano of cardinality its order type is

w + (w"

+ w)

wI

nl

Left to the reader familiar with orders.

.

is saturated if and only if

0

When we replace in the definition of the a-saturation the condition of cardinality of the type C(v) by the condition of being recursive then we get recursive saturation. The previous results are not extendable t o recursive saturation. THEOREM 2.4. [DR2].

I

In every non-standard model M of Peano, the order structures of M for < and I are recursively saturated. Consequently there is no recursive saturation of Peano arithmetic over [<) and over I 1 1 .

Arithnzetics as theories of two orders

297

Another impossibility of extension of the theorems 2.2. and 2.3. is asserted by the 1HliOREM 2.5.

I DR2 1 .

The arithmetics of arbitrary integers (and Th(iZ) in particular) are not o -saturated over the sublanguages {<) and [ I ) J I This last theorem combined with theorem 3 . 8 . b) shows that divisibility gives weaker structures when negative integers appear. More positively, the wl-saturating character of the natural order and of divisibility is preserved

.

by increasing the cardinal

wl

*

THEOREM 2 . 6 . I JPP].

i

Let a he a cardinal greatest than a-saturated with respect to

{<}

w,

.

Every model of PA which is

(or to

)

is

a-saturated.

To conclude this section, we list some results connecting cofinality and satura-

tion for the models of Peano. First SMOKYNSKI and STAVI proved

I SS I .

THEOREM 2 . 7 .

1

Every cofinal extension of a recursively saturated (resp. o-saturated) model of Peario i s itself recursively saturated (resp. o-saturated).

sirvp7c ,'.rfrn.sion of a model M of PA is the SKOLEM closure (see [CK] p . 1 4 2 ) of M U id} where d @ M, namely the smallest model of PA containing both M and d. In [HK], KOTLARSKI showed the following

A

THEOREM 2 . 8 .

i

[HK]. I f a model M of Peano is

w

extension of M, then M' is

1

-saturated and M' is a simple cofinal w

1

-saturated.

Also in [HK], KOTLARSKI asked whether the last theorem can be extended t o larger

cardinals. The answer was recently given by KAUFMANN and SCHMERL who show THEOREM 2 . 9 .

1 5

[KS].

Suppose that a model M of Peano is X+-saturated. Then M has a .. proper, proper, simple simplecofinal cofinalextension extension which which is is 1'-saturated A+-saturated hut but not not

. tt ++ -saturated. X X -saturated.

3 - DEFINABILITY FROM THE NATURAL AND DIVISIBILITY ORDERS IN ARITHMETIC.

It is well known that the full language L ( P ) of Peano arithmetic consisting of all relations and functions arithmetically definable from 0 , I , S , < , + , 0 (either with induction or without induction) is actually definable by formulae of the sublanguage { + , S ) (see [HE] p. 246) using the Godel function fi In this section, we try to find out what is definable from the natural order < and the divisibility order I . This study of the strength of expression of the sublanguages based only on two orders leads to the investigation of the on the other side. links between < , S , I , 1 on the one side and + and This point of view allows us to consider the true arithmetic Th(N) o r the Peano arithmetic PA of the first order as a theory of two distinct orders, one

.

.

I).Richard

298

b e i n g a l i n e a r e x t e n s i o n o f t h e o t t i e r . F i r s t we h a v e t o compare p a i r w i s e .:, S , ~, and 1 t o f i n d o u t w h e t h e r t h e y d i f f e r r e l a t i v e t o d e f i n a b i l i t y . S e c o n d l y we s l ~ o w 1 , ' S , [ t , ' , + , l1 and t h e synonymity ( a s d e f i n e d § O ) of t h e sublanftuagcs T<,l 1 w i t t i t h e f u l l l a n g u a g e I , ( P ) , In t h e c o n c l u s i o n o f s e c t i o n I we g i v e p a r t i a l r e s u l t s c o n c e r n i n g t h e o p e n p r o b l e m of w h e t h e r i S , l 1 i s a w e a k e r s u b l a n g u a g e of L(1'). Tn p r o p o s i t i o n I we compare t h e d e f i n i n g powers o f .:, S , and 1 u s i n g m o s t l y the PADOA's method: i f t h e r e i s a model M i n t e r p r e t i n g a r e l a t i o n K a nd a n R-autornorphism o f M n o t p r e s e r v i n g a n o t h e r r e l a t i o n K ' , t h e n K' i s c e r t a i n l y n o t R'-def i n a b l e .

'.:,I

~

PROPOSITION 3 . 1 . ( C o m p a r i s o n lemma) 1 i).

i t

P + y = S ( X ) 4 - r ( x < yA v z ( x < z P k- vx v y ( x l y *--' v z ( z I x A z I y

+

y

6 z));

->

v

t (z

I

t))).

I

i i ) . L e t R and R ' s y m b o l s t a k e n i n !<, S , , 1 1 ; w i t h t h e two e x c e p t i o n s above and i n t h e t r u e a r i t h m e t i c T h ( N ) , no symbol R is R'-definable i f R # R ' .

P r o o f . We d i s t i n g u i s h n i n e c a s e s ( s e e t h e t a b l e w h e r e I means R i s R ' - d e f i n a b l e and 0 means K i s n o t R ' - d e E i n a b l e ) . i ) . Obvious. are n o t i i ) . To show t h a t < a n d S - d e f i n a b l e c o n s i d e r a non-standard model M of 'I'h(N) and l e t a and b be d i s t i n c t elements of M \ N such that a b .

I

I

Now d e f i n e a map f f r o m M o n t o M by s e t t i n g f ( x ) = x f o r any X E M \ ( ~ + Z ) U ( ~ + Z a)n d f o r a n y ~ F Z f, ( a + y ) = b + y and f ( b + y ) = a + y : t h e o r d e r s < and a r e n o t S - d e f i n a b l e because f i s an S-automorphism p r e s e r v i n g n e i t h e r < n o r . Next t a k e N and d e f i n e b b a g : N u N by s e t t i n g g(Za 3 x) = 2 3 x when we h a v e a Z 0 , b 2 0 and x 1 6 A s g i s an (a)-automorphism, g is also a ( 1 ) - a u t o m o r p h i s m and a 1 - a u t o m o r p h i s m b u t I: p r e s e r v e s n e i t h e r S n o r < , so t h a t < and S a r e n e i t h e r ( I ) - d e f i n a b l e n o r 1 - d e f i n a b l e . Take a g a i n A n o n - s t a n d a r d model M o f T h ( N ) a n d map e v e r y n o n - s t a n d a r d e l e m e n t x o f M o n t o x + I by a f u n c t i o n h C l e a r l y ti i s a <-automorphisrc and a S-automorphism b u t p r e s e r v e s n e i t h e r [ n o r 1 F i n a l l y c o n s i d e r t h e mapping

I

I

.

.

k : N--, I

@

N

{ I , 21

,

.

such t h a t f o r every

.

k ( Z i i x ) = 2" x

x 1 2,

k ( 2 x ) = 4x

,

k ( 4 x ) = 2x

and f o r

Obviously, k respects 1 without preserving

I .

Now we show t h a t a r i t h m e t i c c a n b e c o n s i d e r e d as a t h e o r y o f two o r d e r s < a n d .'ll1e.urem 3.2 i s a c t u a l l y a c o r o l l a r y o f t h e f o l l o w i n g t ~ i e o r e m 3 . 3 . , ttte most c l a s s i c r e s u l t i n t h e a r e a of a r i t h m e t i c a l d e f i n a b i l i t y f r o m o r d e r s . However we p r e s e n t a n i n d e p e n d e n t p r o o f b e c a u s e i t i s t h e s i m p l e s t i n s t a n c e o f t h e c o d i n g method f o r t h e powers o f p r i m e u s e d l a t e r b a s e d o n t h e ZBV-theorem.

1

THEOREM 3.2.

1

The s u b l a n g u a g e

[

<,

I

1 i s synonymous w i t h L ( P ) .

I t i s c l e a r t h a t t h e g r e a t e s t common d i v i s o r ( g . c . d . ) a n d t h e l e a s t a r e t h e i n f and s u p i n t h e ( I ) - l a t t i c e of i n t e g e r s and s o ( I ) - d e f i n a b l e . The i d e n t i t y r e l a t i o n x = y i s ( 1 ) - d e f i n a b l e by

Proof.

G

n m u l t i p l e (1.c.m.)

Aritlirnelics as theories of two orders

1

299

1

V z (z x z y ) . The s u c c e s s o r f u n c t i o n i s c l e a r l y < - d e f i n a b l e . I f we c a n d e f i n e t h e m u l t i p l i c a t i o n i n {i, 11 , t h e n wc a r e done b e c a u s e i n PA t h e sum o f a a n d b i s t h e e l e m e n t c s a t i s f y i n g S ( a c ) S ( b c ) = S ( c c S(ab)) ( s e e l J R l t h e o r e m 1 . I . ) . 'To d e f i n e the p r o d u c c i n { c , 11 , f i r s t o b s e r v e t h a t t h e p r e d i c a t e s P ( x ) and P ( p , y ) meani ng " x i s a rime " a n d "p i s a - d e f i n a b l e as atoms prim e an d x a power of p d i f f e r e n t f r o m I " are We show t h a t t h e p r e d i c a t e a n d e l e m e n t s c o v e r i n g a s i n g l e atom i n t h e o r d e r 2tx E P(p , y ) me a n in g "p i s p r i m e and y = p f o r some a , 0" i s a l s o i < , 11 - d e f i n a b l e by t h e i < ,11 - f o r m u l a : I'(p, y ) ,\ 3 z P ( p , 2 ) A v q ( p ( q ) (q Y-1 -(q 2-1 v q z+l))) I

(7)

I .

-

1

I

.

1

Now we p r o v e t l i a t tl ie l o g i c a l e q u i v a l e n c e a b o v e i s a d i r e c t c o n s e q u e n c e o f t h e ZHV-thcorem ( I 0 ) . I n d e e d i t i s enough t o o b s e r v e t h a t , i f S u p p ( a ) d e n o t e s t h e s e t o f p r i m e d i v i s o r s o f a , we h a v e f o r a p r i m e p a nd p o s i t i v e i n t e g e r s 8 1

and

12

:

I)

Supp(p'-

Supp(p'{- I ) U Supp(p'+ I )

=

, 8

I

except f o r the case

a = 2

left to the reader.

Uith the p r p d i c a t e

product of

x

a nd

pa

=

y = pp

x

l e a s t primes g r e a t e r than g r e a t e s t power o f if

q = r ,

a s s e r t s tiiat x+p z = p or

p

clearly pcx < q

z

<

where t h e EP

{<,

,

y qr

.

.

p

<

.

(<,

If

q

respectively, then q r = 1.c.m.

As

ii, 1 1 - d e f i n a b l e . Rut

is

~ p ( ' pCi+'

11 - d e f i n i t i o n

we s h a l l

f o r any prime

and

less than

,a+[:+ 1

z = p

=

i f a nd o n l y i f

pB

and

I}

r

2

.

is

the

d e n o t e tlie

denotes t h e

z

(qr)

of p

-define

a nd

,

ru = 28

if

q # r

a nd

q

2

TCHEBYCHEV's t h e o r e m ( § 0 )

< r < pB+'

a nd h e n c e

z

A c t u a l l y w e know how t o e x p r e s s t l i e f a c t t h a t {< ,

a power o f e v e n e x p o n e n t o r a power of odd e x p o n e n t i n s i d e o f

.

1)

,

is

using

o n l y EP and a c c o r d i n g t o t h e s t a t u s of x and y C o n s e q u e n t l y , p a+% i s t h e g r e a t e s t power of e v e n e x p o n e n t o f p smaller t h a n q r i f E P ( p , x ) i s l o g i c a l l y e q u i v a l e n t to E P(p, y ) and t h e g r e a t e s t power o f odd e x p o n e n t o f p s m a l l e r t h a n q r i f I:P(p, x) i s n o t l o g i c a l l y e q u i v a l e n t t o E P ( p , y ) It can e a s i l y be seen t h a t t h e d e f i n i t i o n is a {<, -formula. I n s i d e I < , 11 w e c a n now e x p r e s s t h e p r o d u c t o f two a r b i t r a r y p o s i t i v e i n t e g e r s a a nd b , as t h e l e a s t p o s i t i v e i n t e g e r s u c h t h a t f o r a l l p , x a n d y s u c h t h a t P ( p , x ) and P ( p , y ) , w e h a v e x a A y b xy c C l e a r l y x = ah , concluding the proof. n

1

I

-

.

I}

I

.

Remark. T h i s p r o o f i s n a t u r a l b e c a u s e t h e p r o d u c t o f a a nd b i s b u i l t up fro m i t s s i m p l e s t f a c t o r s . A l s o t h e t o o l - t h e ZBV-theorem - a n d t h e c o d i n g t y p e p r o o f method w i l l l a t e r y i e l d s t r o n g e r r e s u l t s .

.

1)

We p r e s e n t two We weaken t h e s u b l a n g u a g e { < , by t a k i n g S i n s t e a d of < proofs; t h e f i r s t one is an a d a p t a t i o n of t h e o r i g i n a l proof while t h e o t h e r ( i n d e p e n d e n t o f t h e p r e v i o u s ) i s i n t h e m a i n s t r e a m of t h e ZBV-theorem. THEOREM 3.3. A d d i t i o n a nd m u l t i p l i c a t i o n o f p o s i t i v e i n t e g e r s a r e a r i t h m e t i c a l l y d e f i n e d i n t e r m s o f S and ( t h i s t h e o r e m i s dtir i n t h e s e t e r m s t o J u l i a ROBINSON ( s e e L J R l t h e o r e m 1 . 2 . ) ) .

I

Proof I . - L e t a , b , c be a r b i t r a r y p o s i t i v e i n t e g e r s and l e t m a n i n t e g e r coprime t o ah U s i n g t h e C h i n e s e r e m a i n d e r t h e o r e m , t h e r e a d e r may v e r i f y t h e e x i s t e n c e o f x a n d y s u c h t h a t x , y and a h a r e p a i r w i s e c o p r i m e a nd a x : by z - l ( m ) . A s i n t h e p r o o f o f t h e o r e m 2 , i t i s e nough t o I S , -define t h e m u l t i p l i c a t i o n . N o t i c i n g t h a t z = 1 + + V t (z t) , we c l a i m t h a t a b = c

.

I

1)

D. Richard

300

is logically equivalent to:

( a = I ~ b = --tc=l)v l mlS(I.c.m.(a,

x))A

V x y m [ ( x l y ~ x l a ~ x l c~ y l b~

mlS(I.c.m.(b,

y

l

c

~

I .

y)) - - , 3 u ( m ~ u - - , S ( u ) = I . ~ . m . ( c , x , y ) )

To verify the logical equivalence, we note that if ah = c then, for every convenient pair (x, y), the integer u = abxy-1 will do. Conversely, the Further from existence of the postulated u leads u s to cx 5 I ( m ) abxy : I ? c(m) for every m coprime to ab ; ax ? bx : -I(m) , we get hence c = ab

.

.

U

Proof 2. - This relies on the ZBV-theorem and on the following lemma which is a device implicit in J. ROBINSON and appears in A . WOODS "W], lemma 2.1 (11) 1 for the language I S , 1 1

.

LEMMA 3 . 4 .

Proof. -

(COMPARISON OF CARDINALITIES OF SUPPORTS)

I

There is an { S , 1 ) - formula INJ(x, y) defining tlie relation there is an injection from supp(x) into Supp(y) "

.

"

We shall first analyse the case where

I)

x

and

y

are odd and coprime.

Let A be the binary relation l'x and y are odd and coprime and there exists an injection from Supp(x) into Supp(y) ' I . Suppose (x, y) A and let F be an injection from Supp(x) into Supp(y). We shall associate to F a set of integers "coding" it. To this purpose, let us consider, for every p in Supp(x) and q in Supp(y) the system of congruences X(p, q , x, y) :

\ ==I -- 1I 2

z

z 5

I

(p) (q) (r)

for a11

r

E

has an infinite set

Observe that if p and p' are in Supp(x) and q and q ' in Sup!(y) if (P, 9) f (P', 4 ' ) then X(p, q, x, y) and X(P', q , x, Y) are disjoint: this is so because for all p SUPP(X) u SUPP(Y ) we have - 1 : 1 (p) since p # 2 We shall consider integers f "coding F" i.e. such that for every p in Supp(x) and every q in Supp(y) the set X(p, q , x, y) intersects Supp(f) if and o n l y if F ( p ) = q codes obviously exist.

are in

.

2)

.

~upp(x) u ~upp(y) \ tp, q~

By corollary 1.3, the system C(p, q, x, y) X(p, q , x, y) of prime solutions.

We are now in a position to introduce a set "3, 1 1 -definable and equal to A .

B

.

Such

which is both

We let B be the set of couples (x, y) t IN' such that: 'lx and y are odd and coprime and there exists an intezer f such that ( * ) " where ( * ) is "for every p in Supp(x), there is an unique q in Supp(y) such that the set Supp(f) intersects X(p, q, x, y) ; and also for every q in Supp(y) , there is a most one p in Supp(x) such that the s e t Supp(f) intersects X(p, q, x, y ) "

.

.

The previous analysis shows the inclusion A C B Conversely, if (x, y) E B then every f such ( x , y , f) satisfies ( * ) defines This shows a natural injection from Supp(x) into Supp(y)

.

A = B .

Arithmetics as theories of two crders

30 1

At last, let u s give an { S , I 1 -definition of B. To this effect, note that the relation z t X(p, q, x, y) is { S , I 1 -defined by the following formula ~ ( p ,q , x, y, z) which is: p

supp(s(z))

v r [(r

q E supp(s(z))

A

supp(x) u supp(y)

E

A

A

p(Z)

p 1r

A

A

q 1r)

-

r

.

~upp(z - I ) 1

E

A convenient { S , l 1 -formula INJ'(x, y) meaning (x, y) t B is 2 C sUPP(X) A 2 t SUpp(y) A x 1 y A 3 f [MAP(f, x, y) A OTO(f, x, y)] where MAP(f, x, y) is: P

suPP(x)

3q

sUPP(Y)

3

71

t

-

[ P(P, q, X, y,

Supp(f)

?T)

A

[~(p, r, x, y, a) q 1 r I1 and means that one can define a functional correspondence from the set Supp(x) into the set Supp(y) ; V r E SUPP(Y)

V q E Supp(f)

- which asserts that the above correspondence one-to-one- is:

While OTO by f ) is

v q

t

SUPP(Y)

{[ 3

i7

e sUPP(f)

v p'

'd p E SUPP(X) 3 71'

Supp(x)

-

e Supp(f) ( P(p, q, X, y,

PPP'. q', x, Y, a))] 3) For the general case we shall replace x constructed from x integer z which is A convenient formula 3X'

3y'

. )

(SUPP(X)

(2 E sUPP(Y) sUPP(Y') INJ'(x',

p I p' 1

. .

INJ(x, y)

(2 E sUPp(X) SUPP(X')

7) A

where x or y can be even or not coprime, and y by odd integers x' and y' simply and y , and compare x' and y' to another odd and coprime to x' and y' is the following:

{[21X-X'=X]

3 2

(coded

A[21yYyy'=y]A 3

u

--C

P

[P(p)

A

pl

AP # 2

X

SUPP(P))

\

SUPP(2)l)

3q [p(q)

A

ql Y

A

INJ'(y',

z) AINJ'(z,

A

9 # 2A

A

(sUPP(Y) u sUPp(q) \ SUpp(2)l) z)

A

X'l

A

.

y')}

2 A

y'l

Z A

Remark. Actually, the lemma holds with I S , 1 1 instead of I S , I 1 because P is { S , 1) -definable (see [DR3] or [ D R 4 ] ) and since for every prime p and every integer x, p 1 x is logically equivalent to 1 (pl x)

.

Now return to the promised Proof 2 of the theorem 3.3.

-

As

in the theorem 3.2., it is enough to

11 -define the multiplication. For this (and always, as in the theorem 3.2.) it is sufficient to { S , 11 -define the product pa. pB for all powers pa and p8 of a given prime p. We treat the general case p # 2 (there is a {S,

slight difference in the { S , I } -definitions of 2n.2B = 2a+8 due to the exceptional cases in the ZBV-theorem. We leave the straightfoward adaptation of the proof to the reader). The idea of the proof is illustrated in figure 4 : to every

Ea

and

Fa

such that

c1

# 8

implies that

Ea

, Fa ,

E8

pa 9

we associate

F

8 are

D.Richard

302

T

pairwise disjoined. Then we define two transversals figure. It is clear that a + B

=

T

and

fi,a

as in the

Y

if and only i f there is a bijection from

y

Y E

1 f

c1

y

=

x E E

meaning

r

+p-l+p

Y

f

+ B = y iff there is a bijection from

Now we have to express the situation in given power

pa

of

: P(p,y)

PWA

v

z

where

z

< y stands for

p

I1.

{S,

{S,

I

(p(q)

[(qlz-1 --DxZ~(~)) A((q1y-1

1y

1(y 6

A

T Y E

-define

. for a

Codl(x, y)

A q ~ 2 ) A

1 z) .

2

pY

f

onto

--

A

r

py-I

5

Tcx,

we use the following predicate

(~(p, z ) ~ Z < y ) - v q z

To

p

<-

2

E2

E

p-l

l(q1z-1))-

~~-1(q))ll]

I

IS,

Next we want to

.

1 -define

.

the height of a given power z = p of p For this consider the predicate HI(c, z ) so defined that c has exactly 6 prime divisors. Formally we put H1(c, z ) if and only if : P ( p , z ) A v q [ [ q E SUPP(C) 3 t (P(p, t) A t +

v Cod-l(x, y) 1.-

t)

A

t

I

2)

-(3

Codl(q, t))] q (Codl(q, t)

and

of

6

0)

q E

-1

z =

p6

.

;

I

SUPP(C)) I]

.

1 -define IS,

I

}-define

We note:

By the Chinese remainder theorem, for every power y at least one x such that Cod (x, y), proving that I Fa #

A

in Cod (x, y) and H1(c, z ) we is, I to get our previous Fa and H - I ( ~ , z ) which also 1

By permuting the height

[(P(P,

t

I ZA

=

pa

of

Ecl # 0

p there is (similarly

Arithmetics as theories of two orders 2.-

3.-

By t h e ZBV-theorem E,l

H ( c , z) o r 1 Supp(c) and

H - l ( ~ , z)

Ei

4.-

y = p

a'

,

Codl(x, y ' ) )

i

Fa n Fa, # 0 ) ;

(similarly

i m p l i e s t h a t t h e r e i s a b i j e c t i o n between

...,

i x l , x2,

( lCodl(x, y ) v

xSl

where e v e r y

x.

for

<

1

i ,< S

is in

; H ( c , z) A H - I ( ~ ' , z ' ) 1

Also z'

Vx

Ea, # 0 i f

proving t h a t

f?

--

y # y'

303

=

p6'

,

then

implies

c 1 c'

i s r e p r e s e n t e d by

TSS,

,

and i f

Supp(cc').

z = p6

and

The p r o d u c t of t h e

powers y and z of a f i x e d prime p i s x i f and o n l y i f : P ( p , x) v H - I ( ~ , z)

"for a l l

(a, b, c )

such t h a t

t h e r e i s a b i j e c t i o n between

Following lemma 3 . 4 . ,

H l ( a , x ) , Hl(b, y) Supp(a)

and

yz

is,

t h e l a s t formula b e l o n g s t o t h e sublanguage

From t h e f a c t s 1 , 2 , 3 and 4 we s e e t h a t t h e only i n t e g e r t h i s formula i s

and

Supp(bc)".

x

1

I

.

satisfying

.

0

..

Remark. Proof 2 i s l o n g e r t h a n proof 1, b u t on t h e o t h e r hand t h i s proof 1 i s e x t e n d a b l e t o an { S , } - d e f i n i t i o n de + , and f o r n e g a t i v e i n t e g e r s as asked i n [ J R ] p. 102. This i s done i n [DR3]

I

<

The n e x t theorem answers a n o t h e r q u e s t i o n of J u l i a ROBINSON'S ( i n [ J R ] p. 102) who asked whether i n weakening i n 1 and s t r e n g t h e n i n g < by + , we s t i l l g e t a synonymous sublanguage of L(P). J u l i a ROBINSON ( a c c o r d i n g t o A . WOODS) and A. WOODS h i m s e l f gave t h e r e s u l t and t h e i r p r o o f s a r e based on t h e SCHN1RELMA"'s theorem. ( T h i s one a s s e r t s t h a t t h e r e i s a n a t u r a l number n such t h a t e v e r y number x 2 2 i s t h e sum of fewer than n p r i m e s ) . Our proof i s d i f f e r e n t and works i n coding i n t e g e r s , r e l a t i o n s and f u n c t i o n s by means of primes.

I

THEOREM 3 . 5 .

1

Proof. -

The sublanguage

{ + , 11

The i d e n t i t y r e l a t i o n i s

i s synonymous w i t h {+, 1 1 - d e f i n a b l e .

L(P).

A convenient

{+, l ) - f o r m u l a meaning x = y i s f o r i n s t a n c e t h e f o l l o w i n g : v z v t [ ( x + z 1 t ) tf (y + z 1 t ) l

s i n c e t h i s formula i m p l i e s t h a t , f o r e v e r y prime p, w e have x y (p) . Of course the r e l a t i o n s x = 0 , x < y , y = S(x) , x = 1 , x = 2 , x = 3 , where S i s t h e s u c c e s s o r f u n c t i o n a r e r e s p e c t i v e l y d e f i n a b l e by t h e f o l l o w i n g x + x = x i+, 11-formulas: 3 2 [ Z # O A X + Z = y ] ; x < y A v z {x < 2 [ ( y < 2 ) v (y = z ) ] ) ; = I tf = s ( o ) i ; [:< = 2 f-f = s(1) i ; iX = 3 tf = s ( 2 ) l -+

rx

The unary p r e d i c a t e (X

# 0)

.

P ( x ) of p r i m a l i t y i s c l e a r l y { + , 1 ) - d e f i n e d by t h e formula: (X # 1) vy [ ( ( y # 0 ) A (y < X)) * y 1 X I

.

I

For a g i v e n prime p , t h e prime Suc(p) , t h e r e l a t i o n p x , x :y (p) a r e r e s p e c t i v e l y { + , 1 ) - d e f i n a b l e by t h e formulas: ; sUC(P) = U P ' ( p ( P ' ) A P < P ' ) ; P ( p ) A 1 (P 1 x ) v z ~ ( x + z = y v x = y + z ) + p ~ z l

the relation

.

The Chinese remainder theorem and t h e D I R I C H L E T ' s theorem a l l o w u s t o {+, 1 ) - d e f i n e , f o r e v e r y prime p > 2 and f o r e v e r y i n t e g e r x such t h a t

D.Richard

304 26x
, the prime

q ( p , x)

by the following I + , 1)-formula

A (2 p) A (2 < X ) (X < P) y = u p i ~ ( p ' ) ~ ( <x p') A b'q [(P(q) A q < We note that, if q(p, x) = q ( p ' , x') with p < and q(p',xf) : 1 ( p ) so that x E 1 ( p ) with contradiction: x = 1 , Similarly if q(p, x! then p divides lx- x'1 < p so that x = x one-to-one.

:

P(p)

.

p' l(q)lA p ' : x ( p l l . then q ( p , x) :x (1,) 2 < x < p , leading to a (p', x') with p = p ' , Therefore the function is

p)

-t

p'

Now we shall code the map x -p(x) where p(0) = 2 and p(x) is the x-th prime using the integers c(x) which we { + , 1)-define by the following formula: C(X) = UC{( 9(3,2) 1 C) A VP vy [((2,(Y)A (Y<X)A p(P) A ( V(P, Y) 1 C)) (Y(SUC(P), Y + l ) Ic)l Now it is provable by induction that p = p(x) is equivalent to [(x=o) A(p=2)]V[(x= 1) A ( p = 3 ) ] V [ 2 < x x ( q ( p 1 X+l)lc(x+l)] Similarly a code d(x, y) for multiplication is I + , 1 }-defined by d(x, Y) = v d I ( q(p(x), 2) 1 d) A (t
.

+

. .

-f

With such a code d(x, y) we can {+,I }-define the usual multiplication because it is provable by induction that xy = z is logically equivalent to [ y = o ++z=O]A[y=l-z=x]A ( 2 < y A t = m a x t ' [3q(9(q, t')Id(x,y))l A ( V(P, t) I d(x, y)) p = p(z) 1 0 + +

.

Immediately we reach the {<,l}-definability of L(P). We can only use incomparable elements for divisibility, but we have the full linear extension of the divisibility provided by the natural order. The theorem 3.6. was at first proved by A . WOODS who gives two different proofs of it in his thesis. This result could also be shown using our theorem 3.10. below. THEOREM 3.6.

i

(A. WOODS)

The sublanguage {<,I 1 is synonymous with L(P). Consequently {S,l} is synonymous with L(P) if and only if < is { S , 11-definable.

Proof. - Denote by Q(a, b) and R(a, b) for b # 0 the unique quotient and remainder under the usual euclidian division. It is enough to { S , 1 }-define x + y for given x and y We will show that z = x + y if and only if "x < y and y 4 z , x + y :z (6) , and for every prime p < z , p # 7 , Q(z, p) 5 Q(x, p) + Q(y, p) + i , with i = 0 or i = 1 I' This condition is obviously necessary. To prove that it is also sufficient, suppose that z < x + y and take p # 7 such that 2p \< x + y - z 6 4p : this is possible by TCHEBYCHEF's theorem from the fact that x + y - z = 6k # 0 The calculus proves that p has the forbidden property Q(z, p ) E Q(x, p)+ Q(y, p ) + j with j # 0 and j # 1 Now it remains to see that this condition is I<, 1 }-definable.

.

.

.

.

This follows from the logical equivalence in true arithmetic: z = p Q(x, p) [ P ( p ) A z = max z ' ( z ' < x A i(p 12')) 1 Moreover for O C i 6 6 , t 5 i ( 7 ) and t I i (6) (the last congruence is equivalent to t i ( 2 ) A t 5 i(3) ) are clearly { S , 1 }-definable. From t 3 i (7) and u 3 j ( 7 ) for some i and j (0 6 i \ < 6 , 0 \< j < 6) , one can {S, 1 ]-define without using + and 0 the congruences t + u I k (7) and ++

.

Arilhnietics as theorips of two orders

305

.

tu C ( 7 ) with 0 < k < 7 , 0 6 9. 6 7 Hence Q ( z , p ) E Q(x, p ) + Q(y, p) + i with i = 0 or i completing the proof.

= 1

is

{<, 1

}-definable, 0

Remark. This proof can be presented in such a way that we need only a {<,l )-definition o f binary predicates meaning Q(x, p ) E 1 (3) , Q(x, p ) 1 - I ( 1 ) , for every integer x and every prime p . This leads to the equivalence of the synonymity of { S , l 1 and L(P) (studied with more details below) with the I S , l 1-definability o f the remainder modulo 3 of Q(x, p) for every integer x and every prime p

.

The previous theorem asserts that the successor function S coming from < and the entire relation of divisibility are together sufficient for the description of arithmetic. Now are the "successors" defined from divisibility and the entire relation < also sufficient to the complete description o f arithmetic ? the answer is contained in the

KE:MARK __

3 . 7.

Let S ( p , x) the binary predicate meaning that " p is a prime and there is an y such that x = p y " , i.e. that x is a successor in the (I)-lattice for some y ; then the sublanguage I S ( , ) , < } is synonymous with the full language L(P)

.

-

Proof. In virtue of the theorem 6 , it is sufficient to I S ( , 1, < I -define the predicate 1 . This results from the following { S ( , )}-definition: x 1 y V p [ S ( p , x) -+ lS(P, Y) 1 0

.

At this step, one can ask what are the previous results on positive integers which are extendable to arbitrary integers. Julia ROBINSON posed the question (in [JR] p.102) noting that t h e diffic2nZty i n tkz's casp reduces to tiitrt of d e f i n i n g the n o t i o n of a p o s i t i v e integer) i n terms o f I anJ S

.

: a) The full language L(1) of Th(l) I < , 1 }-definable but is neither I + , I }-definable. 1: b) The language

is

L(Z)

IS,

1

is {+

,

I

{<,I

l-definable, l-definable nor

}-definable.

Remark. One can observe that, curiously and for all theories with negative and positive integers, I+,1 (and consequently { + , 1 } too) is a weaker 1 , since { S , 1 is synonymous with L(Z) sublanguage than [ S ,

I

I

Proof of a ) . - Recall that L(I) is synonymous with show the negative part of the theorem. Let f : Z -+ this map is an { t , I }-automorphism (and hence an the model 2 of Th(l) , but does not respect < s o I t , l } are synonymous with L ( 2 )

.

For {<, 11 and Indeed x I-+ 1x1

.

I

.

{<, +, 0 ) Firstly we 2 be such that f(z) = - z

only complete the proofs of theorem 3 . 2 . and 3 . 6 . l-definable by

{ 1 i 1. we

is

I<,

I

y = 1x1 * 'dt (t 1 y * t I X ) A (y > O V y = 0 ) V t (t I y) For non negative integers the formula x I y y = 0 ++ {<,I ]-definable (theorem 3 . 6 . ) , by a formula 7 (x, y). Replacing every

with

.

;

{+,l}-automorphisme) of that neither {+, 1 1 nor

is

.

D. Richard

306

variable v and every constant c by v 3 0 and c >/ 0 , we define easily i p ' ( x , y) meaning 1x1 1 IyI for x and y arbitrary integers. So x + 1x1 is I S , 1 }-definable by a formula analogous t o one of the case I S , 1 Similarly, as with the definitions o f x + 1x1 , we can e x p r e s s 1x1 IyI = ( z ( either in I<, 1 1 or {<, 1 ) , s o that :

.

1

The synonymity of I S , 1 1 with L ( P ) is an open and certainly very difficult question. We give below some equivalent problems and we favour those which are in the area of order theory. However some of the equivalent questions in other fields of mathematics are actually beautiful. For instance, A . WOODS shows in his thesis (see [AW]) that this question is another formulation of the following open problem in the theory of numbers: is there a universal positive k such that for every pair (x, y) of positive integers, the identity x = y is equivalent to the condition 'I X + i and y + i have the same set of prime divisors for 0 < i < k ?

.

Our investigation begins with the

I

A successor function S and a divisibility relation C$ are { S , 11-definable over Fowers of 2 so that: = S

2s(x)

(2x)

and

'2

$

'2 * x I y

Consequently an addition H and a multiplication : { S , l }-definable over powers of 2 , so that:

are

.,

2x H '2

ZX+'

=

and

2x

.

'2 '=' 2

Proof. - First we note that being a power of a s t u n r h d prime (as 2 instance) is a { S , 1)-definable property : P(2, If

X)

t--f

vyz [ y 1

z

(x 1 y V X 1

-+

f

is a mapping we write sometimes i i+l Now we define 5 We put 2 2 = 2

fx

.

v

put

t t'

[ SSt' = ?(St)

]

2)

A

instead of

for 0 6 i 6 6

[P(2, St)

i-+

P(2, SSt')

A

f(x)

.

for

convenience. 6 t > 2 - 1 we

; for A

v u ( u l S S 0 + (u 1 t

Actually this definition is equivalent in Th(lN)

l ( x 1 SS(0))

is, for

u 1 t'))

f-t

to V x

(ZSx

] =

1

,

.

S(2'))

SSt' = 2" , then for every prime x'-1 t' - 1 =q the relation q I 2 x - 1 is equivalent to q 2 2 . Conversely, assume that for every prime q this equivalence holds. By the

To prove this, note that if

St

=

2x , and

I

ZBV-theorem there is a primitive prime q 2x'-l = 1 (q) it follows that x I x ' - 1 x' - 1

1x

Next we

. Therefore

{ S , l ]-define

V t t ' [ St $ St'

3

1 (4)

and

=

2x $ 2x' x

I x'

.

.

Hence from

Similarly, it can be shown that

SSt'

and

1

=

2x' =''2

= S(St)

=

.

S(2')

by

@

* [St'

and we show that 2x

x' = x + 1

dividing -'2

1

V

iff

( P ( 2 , St)

x llx'

implies '2

.

A

P(2, St')

A

V U (U

1 t' +

Clearly, for every prime

: 1 (4)

and by definition

U

1 t))]

]

q

2x $ 2x'

for

.iiinietics as theories of t w o orders

.:,

S t = 2' then

and

St

St'

St'

theorem 3.3.,

=

we c a n

we h av e a l s o But

are

5

and

2"'

an d

and 2'

is, $1

t i o n s of

ZXy

and

+

2

by

S

Conversely i f a p r i m i t i v e prime

zX' = 1 ( 4 ) and c e r t a i n l y (5, @ ) - d e f i n e two o p e r a t i o n s

implies

formulae d e f i n i n g symbol

.

2"

I

from

by

2'

Q

@ =

.

2"'

from

and

2sx

and

2'

,

1 =

x

q

I x' . a nd

[9

ZX- I

divides

,

I f w e u s e now by t h e "same"

replacing respectively every

S(2x)

a nd

ZXy = 2x

2'

9 2'

x

iff

.

2'

1

y

,

( i n s p e c t t h e formulae used f o r t h e d e f i n i -

Obviously,

'2

and

S

Since

IS, 1)-definable j, ) .

307

13 and

{ S , 1 1 - d e f i n a b l e as a r e

are

.

Remark. T h i s t h e o r e m '3.9. i s u s e d t o p r o v e t h a t T h ( l N , S , 1) i s u n d e c i d a b l e ( s e e t h e o r e m l . 4 . ) , w hi ch i s a n e c e s s a r y c o n d i t i o n f o r h a v i n g a c h a n c e t o g e t t h e d e s i r e d synonymy b e t w e e n { S , 13 a n d L ( P )

.

To c o n c l u d e t h i s s e c t i o n , we show t h a t t h e o p e n q u e s t i o n o f t h e synonymy o f { S , 1 1 w i t h L(P) h e a v i l y depends o n t h e p o s s i b i l i t y of {S, 1 ) - d e f i n i n g e x p o n e n t i a l f u n c t i o n s o r one-t o-one ( f o r i n s t a n c e i n c r e a s i n g ) s e q u e n c e s o f p r i m e s o r o f p o we r s o f p r i m e s . To show t h i s w e need a t h e o r e m t h e p r o o f o f wh ich i s o m i t t e d b e c a u s e o f l a c k o f s p a c e .

THEOREM 3.10.

i

The r e s t r i c t i o n o f x 2x divisor is {S, l)-definable.

t o i n t e g e r s having e x a c t l y a prime

The p r o o f wo r k s by c o d i n g d i f f e r e n t k i n d s o f c r d e r b a s e d o n t h e ZBV-theorem a n d o t h e r r e s u l t s o f t h e o r y o f numbers. The i n t e r e s t e d r e a d e r w i l l f i n d i t i n IDR41. From t h e t h e o r e m s 3 . 9 . and 3.10 we g e t

,

{ S , 1 ) - d e f i n a b l e i f and o n l y i f o n e o f t h e f o l l o w i n g cond t i o n s holds:

:

a) t h e r e l a t i o n

< (or the relation

I

is

)

IS, 11-definab e,

1

::

c ) t h e r e i s a I S , 1 ) - d e f i n a b l e one-to-one ( f o r i n s t a n c e i n c r e a s i n g ) linearly s e q u e n c e u o f p r i m e s (a s u b c h a i n o f t h e atoms o f o r d e r e d by < s h o u l d b e p e r f e c t a n d t h e map x p ( x ) also),

.. c ' )

Proof.

I

Same as c ) w i t h "powers o f p r i m e s "

The condition a)

follows from theorems

3 . 6 . and 3.3.

The condition b2 follows easily theorem 3.9. p > 2

,

an obvious analogous theorem with

is necessary and sufficient. Clearly

c)

p

and

i n s t e a d of p r i m e s .

.

.

For a standard prime

2 holds so that b ) P are necessary conditions.

instead of c')

Firstly we show the sufficiency of c) : for every integer n , n +-t {S,

1)-definable by theorem 3.10. above and the prime character o f

2'(n) n(n)

is

. Rut

D.Richard

308 there is an @ which is {!3

I+,

, a )-definable

)-definable in replacing

+

and

by a formula by which

by

and

Q

. We

2'(n) = @ (2") , leading to a I S , 1)-definition of nproving that c') is sufficient, an analogous argument holds.

is

have Zn

. For CI

We have just seen that I S , 1) is synonymous with L ( P ) if and only if < is I S , 1)-definable. The corollary 3 . 1 2 . gives a necessary condition for that, since it proves that < is I S , 1)-definable over integers having no more than one prime divisor. COROLLARY 3 . 1 2 .

i

Over the powers of 2, it is natural to

Proof. 3' 2

(ZX

Now, let then

The restriction of the natural order to all the powers of all the primes is I S , 1 1-def inable.

pa

2' pa

<

= )'2

and

q8 -[(r

and it is clear that

2x

IS,

l)-define

@ '2

2x

@ '2

if and only if

by x
qB be two powers of the primes p and q respectively; a R = '2 ) A ( s = 2q ) A r @ s ] provides the desired

I S , l)-definition since the right side of this logical equivalence is, by theorem 3.10. expressed in terms of S and 1 0

.

Remark. To try to extend corollary 3.12. to all the integers is a posssible way for proving a positive answer to the Julia ROBINSON'S question. On the other hand, any eventual construction of a I S , l)-automorphism of a model M of Peano which is not an autornorphisrn of M will have to respect the natural order on the powers of primes.

.

Arithmetics as theories of two orders

309

APPENDIX.

I t a l i c n u mb e r s d e n o t e the pinitivc divi-

sors. I n s t a n c e s of an- I f o r a=2, a - 3 , n < 2 2 a n d f o r a=5, b = 7 , nd12 f o r t h e ZBVt h e o r e m . Ex a mp l e s of t h e t h e o r e m o f EULERFERMAT c a n b e c h e c k e d on t h i s t a b l e . A l s o e v i d e n t i s t h e prop e r t y of t h e l a t t i c e ( p r o v e d by CARMICHAEL) a n d b y wh ich

(1)-

1 ::1

2423.3851 2 -5.7.13.73

i s an (/)-hornorno mrphism.

1

2.3

2

2 .3 2 2.3 .19 5 2 2 .3.5

3

4 5 6 7

a

4

2.3.2801 4 2 2 . 3 .19.43 2.3.29.4733 6 2 2 .3.5 .izoi I)

2.3”. 1 9.37.1063

2 2 2 2 . 3 . 5 .13.19.43.181

1

D.Richard

310

REFERENCES:

[AW]. A . WOODS, Some problems in Logic and number theory, and their c o n n c c t i ~ m s , Thesis, University of Manchester (1981), 51-70 ; 121-122. [ALW].

A. WILKIE, On models of a r i t h m e t i c . Answers t o two problems raised by N. GAIFMAIV, J.S.L. Vol. 40, Num 1 , (March 1975), 41-47.

[BV].

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[CK].

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bn,

[DRI]. D. RICHARD, Thbse de 3Pme Cycle, UniversitG Claude Bernard LYON I (1979). [DRZ]. D . RICHARD, De l a s t r u c t u r e a d d i t i v e 2 l a s a t u r a t i o n des modzles de Peano e t n' une c Z a s s i f i c a t i o n des sous-langages de l'arithm4tique, in MODEL THEORY AND ARITHMETIC, Proceeding Paris 1979-1980, edited by C. Berline, K. McAloon and J.P. Ressayre, Springer-Verlag Lecture Notes in mathematics (1981), 270-296. [ DR3

.

[ DR4

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D. RICHARD, Answer t o a problem raised by J . ROBINSON: The a r i t h m e t i c o f p o s i t i v e and negative i n t e g e r s i s definable from successor and d i v i s i b i l i t y - (Submitted) RICHARD, A l l arithmetical s e t s o f powers o f primes are f i r s t order definable in terms o f t h e successor f u n c t i o n and t h e coprimeness predic a t e . (To appear in DISCRETE MATHEMATICS).

[EMF]. W. J . ELLISON et M. MENDES-FRANCE, Les nombres premiers, Hermann (1975). [EG].

P. ERDOS and R.L. GRAHAM, Old and new problems and r e s u l t s i n combinat o r i a 2 number theory. Publications de 1'Enseignement Mathgmatique, Genbve ( I 980).

[ER].

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[HE].

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[HF].

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[HG].

H. GAIFMAN, A n o t e on models and submodels o f a r i t h m e t i c , Conference in mathematical logic, London, Springer Lecture notes in mathematics, no 255 (1970), 128-144.

[HKI. H. KOTLARSKI, On c o f i n a l extensions of models o f a r i t h m e t i c , (Preprint, to appear in the J.S.L.). [HW].

G.H. HARDY and E.M. WRIGHT, An Introduction t o t h e Theory o f Numbers, Oxford at the Clarendon Press, (fourth edition 1959).

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a n - b n , Proceedings of the Cambridge Philosophical Society the Journal of the London Mathematical Society (oct. 1962), 555-562.

[ssl. C. SMORYNSKI and

J . STAVI, C o f i n a l e x t e n s i o n p r e s e r v e s r e c u r s i v e saturat i o n , Proceedings, Karpacz, Poland 1979, Springer Lecture Notes in Mathematics N o 834 ( 1980), 338-345.

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Annals of Discrete Mathematics 23 (1984) 313-342 0 Elsevier Science PublishersB.V. (North-Holland)

313

L' INTERVALLK EN THEORIE DES RELATIONS ; SES GENERALISATIONS FILTRE INTERVALLAIRE ET CLOTURE D'UNE RELATION Roland FraissC - UniversitG de Provence B Marseille A Ernest Corominas Abstract

<

Given a chain, or total nrdrring

, an interval can be defined in two I of the base, such

ways : absolute and relative. An ;iAsolutr intervdl is a subset x, y E I

that if

{a,bl where

and

x


, then

.

A relative interval with bound

, is the set of elemrnts x such thdt a < x < b

a'b

a relative interval with bound

F

, or all satisfy

. More

generally

(subset of the base) is a set, maximal with res-

pect to inclusion, among those sets D x
I

z E

t E F

such that, given

, all

x E D satisfy

.

x > t

A key for generalization to an arbitrary relation A , is the notion of local automorphism. A bijective function

f

is a

10Cdl

tion A/Dom f

onto

A/Rng f

. Then a

subset

A-interval, if any local automorphism f

f

map is still a local automorphism of

A

(A1

and

is an absolute

of the restriction A/I

. Any

f

is an isomorphism of the restric-

I of the base

IAl-I ; i.e. the union of

by the identity map on

A , if Dom

automorphism of

(range) are subsets of the base, and if

Rng f

f

is extensible

and of the latter identity

intersection of

A-intervals is an

A-interval. A rharacterization is given for the exterval (complement of a n interval). A subset if

D

of the base is a relative A-interval with bound

D

is disjoint from F

sets D

.

(subset of the base),

and is maximal, with respect to inclusion, among those

such that any local automorphism of

F

map o n

F

A/D

is extenseible by the identity

Every absolute A-interval i s a relative A-interval. Both notions are

identical for a chain A , but already differ for a partial ordering A. Two related notions are introduced : those of finite-val and subval. The finite-val is a boolean notion : union, intersection, complement of finite-vals are finite-vals. For a circular ordering

x


then

I

(mod A) is a

C

defined from a chain A

by

C

(x,y,z)

= +

iff

or a n y condition obtained from this by a circular permutation ;

C-subval iff

I is an A-absolute interval or exterval ; intuitively

the subval is the circular segment, or cake-portion. Every interval or exterval is a subval ; every subval is a finite-Val, but not conversely. An

A-filter (A-ultrafilter) is defined as usually, by replacing sets by

A-absolute

intervals. A P3rrpdCtdble relation is defined in two equivalent ways : ( 1 )

I , the complement of

A-interval

each A-ultrafilter tely many Finally

F

F

, associate

I is

a

finite union of

an element

for every

A-intervals ; (2) to

h ( F ) E F ; then there exist fini-

such that the union of corresponding

h (F)

covers the base.

A-ultrafilters and usual ultrafilters are used to extend to arbitrary

relations : ( I )

the closure of rationals by real numbers ; (2) the Stone closure

of a boolean lattice.

314

R. Fraisse

INTRODUCTION

P a r t o n s tle d e u x d 6 f i n i t i C i n s u s u t ~ l l t ~dse 1 ' i n t e r v a l l r d , l n s ou o r d r c t o t a l . PremiPremcnt,

I'intcrvalic~ .ibsuiii c s t un c n s e r n h l ~ c l o s p a r ii1tt. r-

mi.diarit6 : si

x r t y > x appnrtienncnt % I ' i n t c r v a l l e ,

termsdiaire

< z < y)

(x

011

F

appartient % I'intervnlle.

line p a r t i e a r h i t r a i r r dtz

de hornr

a1~11-st o u t i.li.rnt,nt

z in-

~ e u x i G m c m e n t 1 ' i i i t c ? r v a j l c >r c ' -

E l a b a s e d r 1'1 r,haint. ( r ~ i i s i ~ m h ldr r s e l6rntJ nts o r d o n n i . p a r I n c l i a i i i C ' ) ,

latif : soit et

u n c d ~.hnint7,

.

F

iin

r s t un t ~ n s r m h l e , maximdl p a r i n c l u s i o n ,

ayant I n proprigti. suivante : si

tout x de D rst

a p p e l 6 b n r n i ~ .A l o r s

F

> t.

intervslle relatif

s t a

,

parmi 1t.s 1 m r t i e s I) de L:-I'

t E F , uu b i e n t o u t x d r D e s t

P a r exrmple l o r s q u r F

F


ou h i e n

r P d u i t a u s i n g I e t i , n dc, 1'616mt-nt t ,

i l e x i s t e d r u x i n t r r v a l l e s r r l a i f s , e n s e m b l e d e s x < t e t enscmhlc, d e s x > t . I.orsque F = { t . u ) avec t

< 11,

iI

rxistr

t r o i h i n t e r v a l Jes r c l a t i f s ( p o s s i b l c m e n t v i -

dcs). T o u t i n t e r v a l l r a b s o l u D e s t u n i n t e r v a l l r r e l a t i f , l o r s q u ' o n p r e n d K-11 po u r b o r n e . I n v e r s e m r n t d a n s u n e c h a P n e , t o u t i n t e r v a l l e r e l a t i f e s t c l o s par inte rmi. d i a r i t g , donc r s t un i n t r r v a l l e a h s o l u . Ccpendant c ' r t t t l e q u i v a l e n c e nc s u h s i s t e r a p a s l o r s q u e noiis 6 t e n d r o n s l a n o t i o n d ' i n t e r v a l l e 2 u n e r e l a t i o n a r b i t r a i r e : p o u r l e c a s d ' u n o r d r e non t o t a l ,

l ' i n t e r v a l l e r r l a t i f s e r a d P j l une notion p l u s v a s t e

que l ' i n t e r v a l l e a b s o l u : v o i r $

3 ci-dessous.

La g g n g r a l i s a t i o n d e l ' i n t e r v a l l e a u x r e l a t i o n s , c o n d u i t a u x n o t i o n s d e f i n i v a l l e e t d e s u b v a l l e ( v o i r 5 4 e t 5 ) . Au § 6 , n o u s d g f i n i s s o n s l c f i l t r e e t l ' u l t r a f i l t r c i n t e s v a J l a i r e s , c e q u i permet d ' g t e n d r e a u x r e l a t i o n s l a n o t i o n d e c l o t u r e , c a l q u s e s u r l e p a s s a g e d e l a c h a i n e d e s r a t i o n n e l s 5 c e l l e d e s rGels. E n f i n a u § 7 , l ' u l t r a f i l t r e localerncnt i n t e r v a l l a i r e p e r m e t d ' 6 t e n d r e a u x r e l a t i o n s I n c l o t u r e dc'

St-onc.

ou r e p r g s e n t a t i o n p a r i n c l u s i o n d e s l a t t i s h u o l g e n s .

La n o t i o n d ' i n t e r v a l l e d ' u n e r e l a t i o n r e m o n t e aux a n n 6 e s 1950. E l l e e s t e x p o s e e chez Frai'ssE,

1971 ( v o l 1 ) p . 1 1 1 - 1 1 2 ,

r t 1977. Le p r g s e n t a r t i c l e r e p r e n d l e

c o n t e n u d e 1 9 7 7 , a v e c un c o m p l e t remani ement d e l a r G d a c t i o n , d e nouve a ux r E s u l t a t s e t de n o u v e l l e s n o t i o n s ; p a r exemple, l a g g n g r a l i s a t i o n de l a c l o t u r e de S t o n e rg p o n d 2 une c r i t i q u e c o n s t r u c t i v e d e M . Y a s u h a r a (Math. Re vie w s j u i n 1979, n o 16082, a n a l y s a n t I ' a r t i c l e dtl 1 9 7 7 ) . Parmi l e s a u t e u r s q u i o n t u t i l i s E de m a n i P r e i n t g r e s s a n t e l ' i n t e r v a l l e d e r e l a t i o n , c i t o n s F o l d P s 1973, G i l l a m 1 9 7 8 e t 1979, Lopez 1 9 7 8 , P o u z e t 1979 e t 1981. Le p r g p r i n t du p r 6 s e n t a r t i c l r a 6 t E examin6 p a r un r e f e r e e , q u i a propo-

s6 d e n o t a b l e s a m s l i o r a t i o n s ; j e l ' e n r e m e r c i r v i v e r n e n t .

I- INTERVALLE ABSOLU ; EXTERVALLE

R a p p e l o n s q u ' 6 t a n t donn6 d e u x r e l a t i o n s t i o n f d'une p a r t i e F de l a base

I

A1

A,A'

d e m@me a r i t g , u n e b i j e c -

s u r une p a r t i e F ' d e l a b a s e

IA'I,

est d i t e

3 15

L'interualle en thborie des relations

un i s v r i l o r p h i s m e l o c a l de A vers A ' , lorsque f est un isomorphisme de la restriction A / F sur A ' / F ' . A'

. . . ,f(xn)

(f(xl),

l'arit6 de A et A').

Plus pr6cis6ment lorsque =

A

(x i,...,x ) pour tous x i ,..., x

de F (l'entier n Ptant

Pour que f soit un tel isomorphisme local, i l suffit q u e f,

restreintc 5 chaque ensemble d'au p l u s n 616ments de F, soit un isomorphisme local d e A vers A ' . Lorsque A '

=

A,

nous disons que f est un a u t o m o r p h i s m e l o c a l

de A . Kappelons encore qu'une r n ~ l t i r e l ~ i t i oAn de base E est une suite finie de relations A . de base E , dites c o i a p o s a n t e s de A . Un isomorphisme local de la multirelation A vers A' est dEfini c O m e une bijection q u i , pour chaque composante A i d e A et la composante correspondante A' de A ' , est un isomorphisme local de A .

i

vers A ! ; mZme gSn6ralisation pour l'automorphisme. 1

Interva 1l e Etant donnP une relation A de base E, une partie D de E sera dite un A-intervalle

a b s o l u , ou simplement un A-intervalle, lorsque tout automorphisme

l o c a l d e la restriction A/D, 6tendu par l'identit6 sur E-D,

donne un automorphisme

local de A ; disons alors que tout automorphisme local de A/D est e x t e n s i b l e par l'identit6 sur E-D. Soit n l'aritb de A (ou le maximum

des arit6s des compo-

santes de A , lorsque A est une multirelation). Par c e qui prGcSde, pour que D soit un A-intervalle, il suffit que tout automorphisme local de A / D dEfini sur un domaine de p <--I

616ments de D, soit extensible par l'identit6 sur toute partie

1 n-p ElSments dans E-D. Si A est une cha?ne, nous retrouvons l'intervalle usuel, ou ensemble clos par intermgdiarit6. Si A

est

un ordre non total de base E, alors D est un

intervalle lorsque, pour chaque PlSment t de E-D, ou bien tous l e s 616ments de D sont

< t,

ou tous sont 1.1.

> t,

ou enfin tous sont

I

t (incomparables 1 t, modulo A ) .

Pour chaque relation, l'ensernble v i d e , l a b a s e e n t i e r e e t l e s i n -

g l e t o n d e c h a q u e g l e r n e n t d e l a b a s e , sont d e s i n t e r v a l l e s a b s o l u s . I1 exi.ste des

relations dont ce sont les seuls intervalles ; par exemple la consEcutivit6 C sur les entiers naturels : C (x,y) = + lorsque y

=

x + I , valeur - dans les autres cas.

Disons qu'une multirelation est u n a i r e lorsque s e s composantes sont des relations unaires.Si A e s t u n a i r e , t o u t e p a r t i e d e l a b a s e e s t u n A - i n t e r v a l l e . 1.2.

E t a n t d o n n e u n e r e l a t i o n A, t o u t e i n t e r s e c t i o n d e A - i n t e r v a l l e s e s t

un A - i n t e r v a l l e . 0

Partons d'un ensemble de A-intervalles D. et notons C leur intersec-

tion. Soit f un automorphisme local de A/C : i l suffit de voir que f est extensible par l'identitg sur une partie finie quelconque H de I A1 -C que Card (H) <arit6 de A ) .

Parmi les Di, soit D,,

une intersection disjointe de H. Pour chaque i

=

...,Dh ...,h

I,

(il suffit m2me

(en nombre fini h) donnant appelons u . les 616ments

de H qui n'appartiennent pas 1 D i : de sorte que H est l'ensemble des u.

R. Fra'isse'

316

...,

(i = I , h). Puisque D 1 est un intervalle, f est extensible par 1'identitG sur les u l . Puisque D2 est un intervalle, f dGj2 Gtendu par 1'identitG sur les u , de D2, est extensible par 1'identitG s u r les autres u , et sur les u 2 ; donc par l'identit6 sur tous les u 1 et u 2 ; etc. 1.3. E t a n t d o n n e u n e n s e m b l e d e A - i n t e r v a l l e s ,

f i l t r a n t par inclusion,

l e u r r e u n i o n est un A - i n t e r v a l l e . 0 Soit D les intervalles considGrEs, C leur rgunion. Soit f un autoi morphisme local de A / C , dont nous pouvons toujours supposer qu'il est de domaine

et codomaine finis. I1 existe un D. incluant 1 la fois le domaine et le codomaine ; donc f est extensible par l'identit6 sur I A1

- D i'

donc par l'identit6 sur I A1 - C .

0

1 . 4 . Soit A une multirelation, n le maximum de son aritG. S o i t F , G d e u x A-intervalles

s u p p o s o n s q u e t o u t e restriction d e A / ( F U G ) , d e c a r d i n a l

;

< n-I,

a d m e t t e u n e i s o m o r p h e p a r m i l e s r e s t r i c t i o n s d e A / ( F 17G ) . A l o r s l a r e u n i o n F U G

est un A - i n t e r v a l l e .

Dans le cas d'une chalne A , ou plus g6nsralement d'un ordre, ou mgme d'une relation binaire rGflexive, nous voyons que la r6union de deux intervalles ayant au moins un 616ment commun, est un intervalle (n'oublions pas qu'il s'agit d'intervalles absolus, notamment dans le cas d'un ordre non total). Soit f un automorphisme local de A / ( F U G ) . I1 suffit de supposer que les domaine et codomaine sont de cardinal m

f

n-I, et de prouver que f est

extensible par l'identit6 sur un ensemble de n-m 616ments arbitraires pris hors

...,

de F U G. Notons u l , . . . , u m les Gl6ments du domaine de f, et v l , vm leurs transformss par f. Par hypothlse il existe w l ,...,w appartenant 5 l'intersection m F n G , chacune des transformations des u

en wi (i = I ,

...,m)

et des w. en v i

( o u inversement) Stant un automorphisme local de A . I1 suffit donc de prouver que

la premisre transformation est extensible par l'identit6 sur n-m 616ments arbitraires hors de F U G. Nous pouvons toujours supposer qu'il existe un p < m avec les u l , 616ments de F, et les u . en w.

(i

=

U~+~,...,U

m

...,u P

Qlsments de G et non de F . La transformation des

I , ...,p) est de domaine et codomaine inclus dans F . Elle est donc

...,

extensible par l'identit6 sur les u (j = p+l, m) et sur un ensemble de n-m 616j ments arbitraires hors de F U G . Par ailleurs la transformation de w I ,...,w P' en les w. (i = I,...,m) est un automorphisme local : il suffit pour ~p+l'...,u m w ~ , u ~ ..., + ~ u, en les le voir, de la dgcomposer en la transformation des w I , m u . (i = I , m), suivie de la transformation des u . en w. La transformation

...,

...,

.

consid6r6e est de domaine et codomaine inclus dans G : elle est donc extensible par l'identit6 hors de F U G. Finalement,par composition, la transformation des u . en wi

(i

= I,

1.5.

...,m)

est extensible de mZme.0

Intervalles disjoints ; repr6sentant

S o i t A u n e r e l a t i o n , F,G d e u x A - i n t e r v a l l e s d i s j o i n t s , f u n a u t o m o r p h i s m e

l o c a l d e l a r e s t r i c t i o n A / F , e t g u n a u t o m o r p h i s m e local d e A / G .

317

L'interualle en theorie des relations A l o r s 12 r e u n i o n f U g 0

e s t u n a u t o m o r p h i s m e local de A .

Etendre les automorphismes par identiti.. p u i s composer.

0

S o i t A une multirrlatic>n,E sa b a s e , n I f maximum de I'aritP. Partageons

E en intervalles disjoints

L).

de rSlinion E . A chaque D i associons une partie finie

< n-I

d . telle que chaque restriction de A/Di 2

5lGments admette une isomorphe

restriction de A/di. Unc telle partie d i sera dite un r e p r G s e n t a n t de D i'

1.6. Considgrons une relation A de base E, une partition de E en intervalles disjoints D . et pour chaque D. un reprgsentant d.. Alors t o u t e r e l a t i o n d e 1

1

b a s e E, s d m c t t a n t l e s D . comme i r i t e r v d l l e s a v e c l e s m6mes r e s t r i c t i o n s A/Di, e t a y a n t l a m&e

r e s t r i c t i o n q u c A 9 l a r e u n i o n d c s r e p d s e n t a n t s di,

est i d e n t i q u c

B A. O

Soit A ' une relation vsrifiant les hypothPses de l'Snonc6. I1 suffit

de prouver que, pour chaque ensemble F de


616ments de E, nous avons A ' / F

=

A/F.

ou dans la rgunion des di. Supposons i F fl D i de cardinal < n-I. Par hypothPse i l existe un isomorphisme dc A/F. sur une restriction de A/di. Les isomorphismcs ainsi d6finis sont des automorphismes locaux p o u r A ' comme pour A , et leur rgunion est un auto-

Cela est 6vident s i F est inclus dans un D donc chaque intersection Fi

=

morphisme local pour A' comme pour A : l'ggalit6 A ' / F I . 7 . E t a n t d o n n e u n e r e l a t i o n A, A-intervalle

=

A/F en rgsulte.

0

un s o u s - e n s e m b l e D d e sa b a s e et un

F, 1 ' i n t e r s e c t i o n D n F e s t u n ( A I D ) - i n t e r v a l l c .

Notons qu'gtant donng un (A/D)-intervalle G, il n'existe p a s forcgment un A-intervalle F tel que G = D

f l

F. Par exemple, prenons pour A un ordre partiel,

lattis bool6en reprEsent6 par un cube usuel tridimensionnel dont le minimum u et le maximum v sont deux sommets o p p o s s s . Prenons pour D l'ensemhle d c s trois sommets x,y,z immgdiatement antsrieurs 2 v, et soit G

=

{x.y}.

Alors un A-intervalle

incluant G comprend forc6ment 1'6lGment immGdiatement antgrieur 2 x et z , donc incomparable 1 y ; donc i l comprend encore z : contradiction. E t a n t d o n n e A e t u n A - i n t e r v a l l e D, l e s ( A I D ) - i n t e r v a l l e s sont e x a c t e m e n t

l e s A - i n t e r v a l l e s i n c l u s d a n s D, o u e q u i v a l e m m e n t l e s i n t e r s e c t i o n s a v e c D d e t o u s les A - i n t e r v a l l e s .

Cons6quence immgdiate de l'dnoncE pr6cEdent.

1.8. LEMME. S o i t A u n e r e l a t i o n a d m e t t a n t comme i n t e r v a l l e s t o u t e s l e s p a i r e s d ' e l e m e n t s d e l a b a s e . C o n s i d e r o n s p e l e m e n t s x,,

h l a permutation cyclique

:

h(x.)

=

...,xP

d e l a b a s e , e t soit

(i
=

x l . S i h est

u n a u t o m o r p h i s m e l o c a l d e A , i l en e s t d e m&ne d e c h a q u e t r a n s p o s i t i o n (xi,xi)

(i,j 0

< p).

C o m m u n i q u e , a i n s i q u e 1 'enonce s u i v a n t , p a r DELEGLISE,

1983.

I1 suffit d e prouver par exemple que la transposition (x,,x2) est un automorphisme

local de A. La transformation de x2 en x,, celle de x 1 en x

x

donc celle de x 2 en P' sont des automorphismes locaux extensibles par l'identit6. Composons avec la

P' transformation simultan6e de x ,xl en xI,x2, qui est aussi un automorphisme local : P nous obtenons la transposition (xl,x2). 0

R. Fra'isst

318

Etant don&

une relation A, si toute paire d'elPments de la base est un A-lnterval-

If, alors toute partie de l a base est un A-intervalle. oSoit E la base, n l'arite de A et h un automorphisme local de A. Notons Dom h

le domaine, et Cod h le codomaine de h (transform6 par h du domaine). I1 suffit de montrer que h est extensible en un automorphisme local de A, par l'identite s u r

E

-

(Dom h U Cod h). Par les remarques du d6but du pr6sent paragraphe I , i l suf-

fit de consid6rer le cas oS Dom h et Cod h sont de cardinal G n - I . Ainsi nous pouvons supposer finie la r6unjon Dorn h U Cod h ; rGpartissons donc ses 616ments d'une

...,

x avec x = h(xi) (i < p) et X , = h(Xp) ; P i +1 d'autre part en suites maximales finies d'iteration : memes conditions mais avec

part en cycles finis d'it6ration x l ,

x 1 & Cod h et x 4 Dom h. Par le lemme prgcsdent, dans chaque cycle d'itEration, P la transposition entre deux elements quelconques d u cycle est un automorphisme local

de A, par hypothPse extensible par l'identite. Par composition, h elle-msme, restreinte au cycle, est un automorphisme local extensible. Dans une suite maximale d'iteration x I ,*..'XP' la transformation de x

en x est un automorphisme local P-1 P extensible ; composons-la avec la transformation de xp-2 en xp-I' elle-msme automorphisme local extensible, et ainsi de suite en remontant jusqu'8 la transformation de x1 en x2 : nous voyons que h, restreinte aux 616ments de la suite d'it6ration sauf x est extensible par l'identite. Finalement h toute entiere est extensible P' par l'identit6 sur E - (Dom h U Cod h).o 1.9.

Extervalle

Etant donne une relation A, appelons A-extervalle le complGment, par rapport 1 la base, d'un A-intervalle. Une partie D de la base I A I est un A-extervalle ssi, pour tout automor-

*

phisme local f de AID et toute partie G de D

=

I A I - D, ou bien f n'est pas ex-

tensible par l'identite sur G, ou f est extensible par tout automorphisme local de

*

A I D .de domaine G .

OSupposons q w D soit un extervalle, donc D*

=

IA

I - D est un intervalle. Soit

g

un automorphisme local de A/D*. de domaine G . Supposons que l'automorphisme local f de AID soit extensible par l'identit6 Ie sur G : donc f U IG est un automorphisme

local de A. Puisque D* est un intervalle, g est extensible par l'identit6 sur le codomaine F' de f, soit IF'

. Donc

la r6union f U g est un automorphisme local,

compos6 de f U IG et IF' U g . Inversement supposons que D ne soit pas un extervalle, donc D*

*

ne soit pas un in-

tervalle. Alors il existe un automorphisme local g de A/D et une partie F de D, tels que I U g ne soit pas un automorphisme local. Alors l'identit6 I est 6videmF F ment extensible par l'identit6 sur G, sans stre extensible par g, contre la condition de 1'6nonce.

0

3 19

L'interualle en t h b r i e des relations

2- LNTERVALLE ET LIBRE-INTERPRETABILITE

Rappelons qu'une relation B de mzme base que A est dite l i b r e - i n t e r p r e table en A lorsque tout automorphisme l o c a l de A est un automorphisme local de B ; s i n e s t l'arit6 d e B, i l suffit que tout automorphisme local de A , sur < n SlEments,

soit un automorphisme local de B. Equivalemment, il existe une formule logique libre (sans quantifieurs) representant B lorsque son pr6dicat est remplace par A. 2 . 1 Soit A , B deux m u l t i r e l a t i o n s de base commune, chacune l i b r e - i n t e r -

pr.c

B-intervalles sont l e s m&nes.

En consGquence, s ' i l r x i s t e uric m u l t i r e l a t i o n unaire B t e l l e que A e t B soient chdcune libre-intc?rpr&tablc en l ' d u t r e , alors toutes l e s p a r t i e s de la base sont des A-intervalles.

Exemple, la concatenee d'une relation unaire U et de 1'Equi-

valence (binaire) d6finie par les deux classes des x tels que U (x)

=

+ et U (x) =

-.

L'inverse est faux : l o chaine sur deux 6lGments admet c o m e intervalles toutes les parties de sa base, sans vErifier l a condition donnGe. De msme l'extension de cette chaync 2 une infinit6 d'6lements nouveaux x, avec valeur - pour les couples diagonaux (x,x).

2.2. Equivalence

transposition

Etant donn6 une multirelation A , disons que deux dlements u,v de sont Pguivalents par transposition (mod A),

1A I

lorsque la transposition (u,v) Gtendue

par l'identit6, est un automorphisme de A . Pour q u ' i l e x i s t e une m u l t i r e l a t i o n unaire en l a q u e l l e A s o i t l i b r e i n t e r p r e t a b l e , i l f a u t e t s u f f i t que l e s c l a s s e s d'equivalence p a r transposition

(mod A) soient en nombre f i n i . Plus prEcis6ment si ces classes sont en nombre fini, et si 5 chacune nous associons la relation unaire valant + sur la classe et

-

ailleurs, l a s u i t e

de ces r e l a t i o n s forme une m u l t i r e l a t i o n unaire en laquelle A e s t l i b r e - i n t e r p r e tab1 e .

Supposons A libre-interprgtable en une multirelation unaire U. Alors deux Glements quelconques u,v qui donnent la mEme valeur B chaque relation unaire composante de U, sont equivalents par transposition (mod A ) : les classes d'equivalence sont donc en nombre fini. Inversement supposons ces classes en nombre fini, et associons 1 chacune une relation unaire valant + s u r cette classe et

-

ailleurs.

Soit n le maximum de l'arit6 de A ; tout automorphisme local f de U, s u r un domaine F de cardinal

< n,

est la restriction au domaine F d'une permutation de F U f (F),

composee d'un nombre fini de transpositions qui conservent U, donc A : f est donc un automorphisme local de A. Done A est libre-interprstable en U.0 Notons que, si une relation B est libre-interpretable en A , i l est possible soit que t o u t B-intervalle soit un A-intervalle, soit l'inverse, soit qu'aucun des deux ensembles, des A-intervalles et des B-intervalles, ne soit inclus dans l'autre.

R. Fra'issi

320

Exemple du premier valeur A (x)

t =

A

cas :

A est une relation unaire prenant au moins deux fois la

et une fois - , et B est In relation d'gquivalence correspondante. Si (XI)

= t

et A ( y )

=

-

, la paire {x,y] est un A-intervalle, ainsi

que

tout sous-ensemble de la base ; mais cette paire n'est pas un B-intervalle, puisque la transformation de x en y est un autumorphismr local de B, non extensible par l'identitg sur

{XI).

Exemple du druxisme cas : soit A l a chaine des entiers naturels, et B l a relation unaire toujours + sur ces entiers. Exemple du troisisme cas : soit A

= (N.0)

oii

N est la chaine des entiers naturels

et 0 la relation unaire singleton de zero. Soit B la relation d'gquivalence avec deux classes, qui sont le singleton de zero

et

l'ensemble des entiers positifs.

Alors B cst libre-interprstable e n 0, d o n c e n A. La paire { O , l } est un A-intervalle mais non un B-intervalle. La paire { 1 , 3 }

est un B-intervalle mais non un A-interval -

le.

2.3. Soit A u n e multirelation de base E. Disons que deux 616ments x,y de E sont 6quivalents par isornorphie des sinyletons lorsque les restrictions A/Ix}

et A/{y) sont isomorphes. Autrement dit lorsque la transformation de x en y est un automorphisme local d c A . Si A est une relation, il ne peut exister que 2 classes pour cette gquivalence : valeur + ou valeur

-

pour A(x,

...,x).

Si A est une multi-

relation avec h composantes, il existe au plus 2h classes. 1,'Equivalence par transposition d6finie en 2.2., est identique ou plus fine que l a pri5sente gquivalence par isornorphie des singletons. Par exemple si A est une cha?ne, tous l e s 61Ements d e la base donnent d e s singletons isomorphes. Cependant aucune transposition entre d e u x 616ments distincts n'est un automorphisme local de la chaine A : donc il existe autant de classes p a r transposition que d'glgments dans la b a s e . (1)

Soit A d e b a s e E ; s i toute p a i r e , donc toute partie de E e s t un A-intervalle

e t s i x,y,z sont t r o i s dlkments d i s t i n c t s , equivalents par isornorphie des s i n y l c -

t o n s , a l o r s x,y,z sont encore equivalents p a r transposition (mod A ) . ( 2 ) En consSquence, si A e s t une multirelation form6e de h composantes, et s i toute

partie de la base est un intervalle, alors i l e x i s t e au plus Zh+l classes d'equivalence par transposition. Plus prscisbment chaque classe d'isomorphie des singlet o n s , ou bien donne une unique classe par transposition, o u donne deux de ces c l d s s e s , reduite chacune a un singleton. 0

(DELEGLISE, 1983).

Par hypothese, l a transformation d e y en z est un automorphisme local de A . La

paire { y , z > Stant un intervalle, la transformation de x,y e n

x,z

est encore un auto-

morphisme local. De mEme la transformation d e x,z en y.z puis celle de y,z en y,x. Par composition d e c e s trois transformations, la transposition (x,y) est un automorphisme local de A, extensible p a r l'identit6. 2.4.

0

Soit A une multirelation de base E. Notons UA l a rnultirelation unaire

d'isomorphie des sinqletons : chaque composante d e U

A

prenant la valeur + sur une

et

32 1

L'interualle en th6orie des relations une seule des classes d'gquivalenc? par isomorphie des singletons. Notons

Ui l a

r n u l t i r e l a t i o n u n a i r e a s s o c i e e d e m&me d 1 ' e q u i v a l e n c e p a r t r a n s p o s i t i o n (mod A). Pour q u ' i l existe une m u l t i r e l a t i o n u n a i r e U telle q u e A et

U soient c h a c u n e l i b r e -

- i n t e r p r e t a b l e en l ' a u t r e , i l f a u t e t s u f f i t q u e l e s d e u x e q u i v a l e n c e s , p a r i s o m o r p h i e d e s s i n g l e t o n s e t p a r t r a n s p o s i t i o n , se c o n f o n d e n t

;

nous pouvons alors prendre

U = U 0

= Uf; (DELEGLISE, 1983). A Supposons qu'il existe une multirelation unaire U avec A et U chacune libre-inter-

pr6table en l'autre. Alors U et A ont l r s rnzmes automorphismes locaux ; donc U se confond ci la fois avec UA et avec U o

A '

Inversement supposons U A

=

Ui. Alors les classes d'gquivalence par transposition

sont en nombre fini. Par 2.2, la multirelation A est libre-interprgtable en Uf;

.

Par ailleurs i l est 6vident que UA est libre-interprgtable en A : i l suffit donc de poser U

=

UA - -U ; ; . "

2.5. Soit A de base E ; si t o u t e p a i r e , donc toute partie de E est un A-intervalle,

(I)

alors :

i l e x i s t e u n e m u l t i r e l a t i o n u n a i r e en l a q u e l l e A est l i b r e - i n t e r p r e t a b l e

;

(2) si de plus i l existe une seule, o u aucune classe d'6quivalence par transpoun singleton, alors il e x i s t e u n e m u l t i r e l a t i o n u n a i r e U t e l l e q u e sition r6duite .i A e t U s o i e n t c h a c u n e l i b r e - i n t e r p r e t a b l e en l ' a u t r e (POUZET, 1981). Le ( I ) rgsulte de 2.2. et 2.3. (2). Le (2) r6sulte de 2.3. (2) et 2.4. 3- INTERVALLE RELATIF ; BORNE Soit A une relation de base E, et F une partie de E. Alors une partie D de E-F sera dite un ( A , F ) - i n t e r v a l l e , ou A - i n t e r v a l l e relatifadmettant F comme borne, lorsque D est maximal, par inclusion, parmi les parties D'

de E-F telles

que tout automorphisme local de A / D ' soit extensible par l'identitg sur F (donnant ainsi un automorphisme local de A ) . Si A est une chaine, les ensembles D' possibles ont, pour chaque 61Ement t de F, tous leurs 616ments


ou tous

>t

; prenant pour D un D ' maximal par in-

clusion, nous obtenons un ensemble clos par intermgdiaritg, donc un intervalle usuel. Tout intervalle absolu D est un intervalle relatif, admettant entre autres la borne E-D. L'inverse est faux, d6j2 pour un ordre non total A . Par exemple prenons a et b > a

(mod A) ; I'ensemble des x tels que a < x < b est un intervalle re-

latif admettant la borne {a,b}, entre autres. Si maintenant i l existe x,y de l'intervalle D consid6r6, et t de E - D ,

avec a < x


et t 1 y et t I b, alors l'auto-

morphisme local d u singleton {x} sur Iy} est extensible par l'identitg sur {a,b} mais non par l'identit6 sur {t}

:

l'intervalle D n'est pas absolu.

Puisque tout intervalle absolu rst relatif, nous avons, pour chaque relation, les intervalles relatifs vide, base entisre et singletons ; nous allons voir que certaines relations n'admettent p a s d'autres intervalles relatifs.

R. Fra'isse

322

A chaque cha'ine A associons la relation ternaire de cyclordre, o u ordre c y c l i q u c ? rnqendre par A : relation C de mGme base q u e A, dEfinie par C (x,y,z) x S y S z (mod A) o u lorsqu'on

a

= +

lorsque

l'une des permut6es circulaires de cette condition.

Par exemple, si A est la cha'ine des entiers naturels, le cyclordre engendrP par A est encore engendr6 par toute cha'ine form6e des entiers successifs 2 partir d'une valeur p donnge, soit p , p + l , p + 2 , Soit

C

>3

un cyclordre sur

...

suivis pour terminer de 0, I , 2, ..., p - I .

616ments, D une partie stricte de la base E. Prenons

u , v distincts appartenant 1 D : La transposition (u,v) est tin automorphisme l o c a l

de C, non extensible, &me

pas par l'idrntit6 sur u n singleton dans E-D. Donc un

tel cyclordre n'admct pour intervalles relatifs q u e le vide, la base et l e s singlctons. Par contre, l a consgcutivit6 des entirrs naturels, qui n'admet c o m e intervalles absolus que le vide, l a base et les singletdns (voir l . l ) , admet bien d'autres intervalles relatifs : par exemple l a paire { O , l } en prenant c o m e borne l'ensemhle

> 3.

des entiers

La base entisre est l e s e u l intervalle qui admette c o m e borne l'ensemble

vide (consgquence de l a condition de maximalitg). Soit D et D' 3 D deux A-intervalles relatifs ; si F est une borne de D , ni F ni aucun ensemble incluant F n'est une borne de D ' (autre consgquence de I s maximalit&). Si n est I'aritE de A, i l suffit, c o m e pour l'intervalle absolu, de prendre un automorphisme local arbitraire de A/D' sur un domaine de p S n-l

616ments

de D', et d'exiger qu'il s o i t extensible par I'identitE sur tout (n-p)-emble inclus dans F ; un intervalle relatif D 6tant encore un D' maximal par inclusion.

3.1. Etant donng

une

relation A et une partie F de sa base, l a d u n i o n

des (A,F)-intervalles est E-F. En effet, pour chaque 6lEment u de E-F, le singleton de u est un ensemble D' au sens du paragraphe 3 : il existe donc au moins un intervall e relatif qui inclut ce singleton.

Dans certains cas particuliers, c o m e le cas d'une relation binaire r6flexive A , les (A,F)-intervalles sont mutuellement disjoints. En effet ici n-l

= I

i l suffit de considgrer les automorphismes locaux dgfinis sur un unique ElEment, c'est 5 dire toutes l e s transformations d'un 6lGment de E-F en un autre, et d'exiger qu'une telle transformation soit extensible par I'identitE sur chaque Slsment de F. Autrement dit, si a dEsigne u n 616rnent de F, et x un Elsment quelconque de l'intervalle, ledit intervalle est dEfini par les valeurs A(a,x) et A(x,a)

qui

doivent rester les mEmes lorsque x varie dans le (A,F)-intervalle. 11 en rEsulte qu'un msme x ne peut appartenir 1 deux (A,F)-intervalles distincts. Passons au c a s u n peu plus gEnEral d'une relation ou multirelation bim i r e (arit6 maximum 2 ) A et d'un ensemble F. En ce cas r6partissons l e s 6lEment-s de l a base en classes

d'isomorphie des singletons (voir 2.3).

Cela donne au plus

2 classes pour une relation (A(x,x) = + ou - ) , un nombre fini de classes pour une

;

323

L'intervalle en theorie des relations

multirelation. Chaque classe s e subdivise en sous-classes, en mettant x et y dans une mzme sous-classe lorsque l'automnrphismr local de A q u i transforme x en y , est extensible par 1'identitP sur F : donc chaque sous-classe est dSfinie par les valeurs A(a,x) et A ( x , a ) qui pour a 616ment de F, doivent rester constantes lorsque x varie dans une soi~s-classe.Alors chaque (A,F)-intervalle est obtenu en prenant dans chaque classe non vide, une de s e s sous-classes non vides, p u i s la r6union des sous-classes ainsi choisies. En consgquence, Stant donnt? line relation binaire A et une partie F de sa base ; si deux (A,F)-intervalles ont un BlPment commun dans chaque classe d'isomorphie des sinyletons, ces intervalles sont identiques. ProblPme. Contre-exemple de l'Pnonc6 ci-dessus avec une relation ternaire. 3.2. Soit A une relation, F,G deux parties de sa base avec G C F ; alors c h a g u e (A,F)-intervalle est inclus dans un (A,G)-intervalle.

Dans le cas d'une relation binaire A avec G

c F,

chaque (A,G)-interval-

le non vide inclut un (A,F)-intervalle non vide. 0

Rgpartissons les ElPments du (A,G)-intervalle considErE, en classes d'isomorphie

d e s singletons (voir 2.3).

Cela donne au plus 2 classes si A est une relation, un

nombre fini de classes dans le cas gEn6ral d'une multirelation. Prenons un 61Ement t reprgsentant de chaque classe. Pour assurer la maximalit6 qui fait partie de la

dEfinition de l'intervalle relatif (voir paragraphe 3), joignons 2 chaque reprgsentant t les El6ments t' de la m@me classe, pour lesquels la transformation de t en t', qui est d6j2 extensible par tit6 sur

1'identitE sur G, est de plus extensible par l'iden-

F.0 Problsme. Existence d'une relation ternaire A avec un F et G

c F,

et un

(A,G)-intervalle qui n'inclut aucun (A,F)-intervalle. 3.3.

L'intersection de deux intervalles relatifs distincts mais ayant

une borne commune F, ne peut Stre un intervalle relatif de borne F, en raison de la maximalite. Soit A une relation binaire ; consid6ron.s un ensemble de A-intervalles relatifs Di, chacun admettant entre autres une borne Fi. Alors l'intersection des

D. est incluse dans un A-intervalle relatif ayant p o u r borne la reunion des F.. D6j2 pour une relation binaire, l'intersection de deux intervalles re-

latifs n'est p a s forcement un intervalle relatif. Prenons une base E de 6 El6ments a,b,c,c',d,e avec A(x,x) x

=

a,b,d,e, et

A(e,a)

=

- ,

tions (b,e),

-

pour x=c,c'. Orientons les paires suivantes : A(a,e)

= +

pour

= +

et

ce que nous nommons l'orientation (a,e) ; prenons de mgme les orienta(e,d),

(c,d), (d,c') ; les autres paires prenant par exemple la valeur

+ d a m les deux sens. L'ensemble A = {a,c,c') est un intervalle admettant la

borne F

=

{b,e} : on ne peut lui ajouter d en raison des orientation (a,e) et (e,d).

L'ensemble B

=

{b,c,c'} est un intervalle admettant la borne G

=

{a,e] : on ne peut

lui ajouter d en raison d e s orientation (b,e) et (e,d) : la maximalit6 est bien

R. Fraisse'

324 satisfaite. L'intersection A n B

=

{c,~'} n'est pas un intervalle.

En effet la borne ne peut pas comprendre d, en raison des orientations (c,d) et (d,c'). Donc cette borne Bventuelle est incluse dans {a,b,e). Alors la maximalit6 n'est pas satisfaite, puisqu'5 c et c' n o u s pourrons toujours ajouter d (de valeur A (d,d)

= +). 0

3 . 4 . Soit A une r e l a t i o n binaire d e base E , e t s o i t D un A-intervalle r e l a t i f ; a l o r s la reunion des bornes de D e s t une borne de D. 0

Soit u,v deux Qlgments de D, avec isomorphie des resteictions de A aux

singletons de u et de v. Alors pour chaque 616ment c de la rBunion des bornes, l'automorphisme local transformant u en v est extensible par l'identitg sur c. De plus D est maximal par inclusion, parmi les ensembles qui vBrifient ce qui prBcPde. ProblZmes. Contre-exemple P ce qui pr&cZde, dans le cas d'une relation ternaire. Existence d'une relation ternaire A, d'un A-intervalle D de borne F , d'un A-intervalle D' de borne F' avec un automorphisme local de AID' n D' inextensible par l'identit6 sur F U F'.

3.5. Contrairement 5 l'intervalle absolu (voir 1.7),

l'intervalle re-

latif n'est pas prCservC par restriction : si D est un sous-ensemble de la base de A, et si F est un A-intervalle relatif, l ' i n t e r s e c t i o n D n F n ' e s t p a s forcement un (AID)-intervalle r e l a t i f .

lorsque y

Prenons la cons6cutivit6

=

x+l. Alors l'ensemble des entiers > 2 est un intervalle, admettant

A

siir les entiers naturels : A(x,y)

= +

0

pour borne le singleton de 0. Supprimons l'ClCment 0 : nous obtenons la consscutivit6 sur les entiers positifs, pour laquelle l'ensemble des entiers un intervalle.

>2

n'est plus

0

3 . 6 . Etant donne une r e l a t i o n binaire A e t une p a r t i e f i n i e F de sa base, l e s (A,F)-intervalles sont en nombre f i n i . 0

Disons que deux ClCments u,v de I A I - F sont equivalents lorsque la

transformation de u en v est un automorphisme local de A, extensible par l'identit6 sur F. Puisque F est fini, les classes de cette 6quivalence sont en nombre fini,

ce qui entraPne l'bnonc6. L'Bnonc6 prCc6dent ne s'Ctend pas au cas ternaire. Par exemple prenons pour A un cyclordre infini ; pour prgciser, le cyclordre des entiers naturels, et prenons pour F le singleton de 0 . Chaque singleton autre que 0 , est un (A,F)-intervalle ; en effet la maximalit6 est vCrifibe, puisque l'addition d'un second entier v B l'entier u qui constitue notre intervalle, autoriserait 5 considsrer la transposition (u,v) qui est un automorphisme local non extensible par l'identite sur 0. ProblSme, suggerd par 1 . 3 . Etant donn6 un ensemble, filtrant par inclusion, d'intervalles relatifs, leur rdunion est-elle t o u j o u r s un intervalle relatif.

0

325

L'interualle en thiorie des relations 4- LE F I N I V A L L E , UNE N O T I O N BOOLEENNE

Soit A une relation, D une partie de sa base. Pour chaque entier posirt v i ( i = I , ...,p) dans D sont e q u i v a l e n t s i lorsque la transformation qui change chaque u . en v. est bijective

tif p, disons que deux p-uplrs u modulo ( A , D ) ,

et est un automorphisme local de A extensible par l'identit6 sur I A I -D. L'ensemble D est d i t un A - f i n i v a l l e

lorsque, pour chaque p, i l n'existe qu'un nombre

fini de classes d'6quivalence mod(A,D) entre p-uples. Si n est l'arit6 de A (le maximum de l'arit6, pour une multirelation), il suffit de limiter la longueur d e s suites aux p

< n-I,

et de limiter l'extensi

bilit6 1 n-p 616ments de I A I - D . L'6nonc6 1 . 7 reste valable pour les finivalles : e t a n t d o n n e u n e r e l a t i o n A et un sous-ensemble

D d e s a b a s e , t o u t A - f i n i v a l l e , intersecte p a r D

donne un

(AID)-finivalle. 4.1 . T o u t e r e u n i o n f i n i e d ' i n t e r v a l l e s d i s j o i n t s e s t u n f i n i v a l l e (la

condition de disjonction sera inutile aprSs le 4 . 4 ci-dessous). 0

Deux p-uples sont Cquivalents s i la transformation d e l'un en l'autre

est un automorphisme local, et si de plus les premiers termes sont dans un mzme intervalle, les deuxismes termes sont dans un mGme intervalle, et ainsi de suite. En particulier, pour toute relation, t o u t e n s e m b l e f i n i e s t u n f i n i v a l l e , puisque r6union finie de singletons, qui sont des intervalles ; cela se voit

aussi immgdiatement sur la dgfinition. P o u r u n e c h a i n e , l e s f i n i v a l l e s sont e x a c t e m e n t l e s r e u n i o n s f i n i e s d ' i n t e r v a l 1es

.

A chaque chaine A nous avons associ6 la relation ternaire de c y c l o r d r e

( o u ordre cyclique) voir paragraphe 3 . Nous avons vu alors que les seuls interval-

les (absolus ou relatifs) du cyclordre sont le vide, la base et les singletons. Soit alors A la chaine des entiers relatifs, C son cyclordre engendrG : l'ensemble D des entiers positifs n'est ni un C-intervalle ni un extervalle ni une r6union ou intersection finie d'entre eux ; mais D est un C-finivalle, deux p-uples 6tant Gquivalents mod(C,D) lorsque la transformation d'un en l'autre pr6serve l'ordre A . 4 . 2 . L e c o m p l e m e n t a i r e d ' u n f i n i v a l l e est u n f i n i v a l l e . 0

Soit A la relation, n s o n aritc, E sa base, D un finivalle.

Pour chaque entier positif p, il n'existe qu'un nombre fini d e classes d'Gquiva-

lence mod(A,D).

Pour chaque p-uple (p

< n-1)

dans D, prenons un unique reprGsen-

tant appartenant P la mzme classe, et appelons H la partie finie de D, rEunion de ces

reprcsentants. Pour chaque entier positif r. disons que deux r-uples dans E-D

sont Gquivalents, lorsque la transformation de l'un e n l'autre est un automorphisme local, extensible par l'identit6 sur H. Puisque H

est fini, i l n'existe, pour

chaque r, qu'un nombre fini de classes pour cette Equivalence.

R. Fra'isse

326

Pour voir que E-D est un finivalle, i l suffit maintenant de

prouver que, si g

est un automorphisme local de A/(E-D), extensible par 1'identitE sur H, alors g est encore extensible par l'identitE sur D. I1 suffit mcme de prouver que, s i g est extensible par l'identit6 sur H, alors g est extensible par l'identit6 sur toute partie F de D, de cardinal

< n-1.

Par hypothsse, il existe une partie F ' de H

et un automorphismc local f de domaine F et codomaine F'. extensible par l'identi-

t6 sur E-D, donc aussi par l'identit6 IG sur G

=

Dom g. Donc f U I et g G

u

IF'

(06 I F ' est l'identitE sur F') sont deux automorphismes locaux de A ; il en est de

mGme de f U g par composition. Puisque f est extensible par l'identit6 s u r E-D,

il l'est aussi par I G l qui est l'identit6 sur G '

=

-1

Cod g ; donc f

U I G l est u n

automorphisme local de A , et par composition I u g est un automorphisme local. 0 F En particulier, pour toute relation, l e complementaire d'une p a r t i s f i n i e quelconque de l a base e s t un f i n i v a l l e . 4.3. (1)

S o i t A une r e l a t i o n , B une autre de msme base, libre-interpr4-

table en A ; alors t o u t A - f i n i v a l l e e s t un B - f i n i v a l l e . 0

Prenons deux suites finies de mzme longueur et telles que la transfor-

mation de l'une en l'autre soit un automorphisme local de A , donc encore de B. Ces suites 6tant dans une partie D de la base, si la transformation est extensible, pour A, par l'identitE hors de D, elle est extensible de mzme pour B .

0

En particulier, s ' i l e x i s t e une m u l t i r e l a t i o n unaire en l a q u e l l e A s o i t l i b r e - i n t e r p r e t a b l e , a l o r s t o u t e p a r t i e de l a base e s t un A - f i n i v a l l e . ( 2 ) Etant donne une m u l t i r e l a t i o n B e t un B - f i n i v a l l e D , i l e x i s t e

une m u l t i r e l a t i o n A de m6me base, en l a q u e l l e B s o i t l i b r e - i n t e r p r e t a b l e e t t e l l e gue D s o i t un A-intervalle

(POUZET, 1975, non publi6).

Soit n l'arit6 de B . Pour chaque p

< n,

considgrons 1'6quivalence dE-

finie en mettant deux p-uples extraits de D dans une mzme classe lorsque la transformation de l'un en l'autre est un automorphisme local de B , extensible par l'identitb sur 1 B 1 - D. Pour chaque classe U, prenons la relation p-aire encore appel6e U, valant + pour les p-uples qui appartiennent 1 la classe et - pour tous autres p-uples dans la base. Nos classes Gtant en nombre fini, la suite formge de B et des U pour p = I ,

...,n, constitue une multirelation A

en laquelle B est

libre-interprgtable. De plus si deux p-uples dans D sont transformgs l'un en l'autre par un automorphisme local de A , ils appartiennent 1 une mzme classe, donc ladite transformation est extensible par l'identitg hors de D, pour A c o m e pour B.n ( 3 ) En consEquence, e t a n t donne une r e l a t i o n B e t une p a r t i e D d e

sa base, D e s t un B - f i n i v a l l e s i e t seulement s'il e x i s t e une A en l a q u e l l e B s o i t l i b r e - i n t e r p r e t a b l e , avec l a condition que D s o i t un A-intervalle.

Notons qu'en g6n6ra1, B gtant donnbe, il n'existe pas de multirelation A en laquelle B soit libre-interprgtable, avec la condition que t o u t B-finivalle soit un A-intervalle. En effet, tout ensemble fini 6tant un B-finivalle, il faudrait que tout ensemble fini, donc toute partie de la base, soit un A-intervalle (voir 1 . 8 ) .

327

L'interualle en thiorie des relations I1 faudrait donc que A , donc B, soit librr-interprdtable en une multirelation

unaire (voir 2.5).

4.4. (I)

Tout finivalle, augmente ou diminue d'un nombre fini d'elements

de l a base, donne un finivalle. 0

A chacun des El6ments ajoutEs, associons sa relation unaire singleton,

puis ajoutons ces relations singletons 5 la multirelation A de l'dnoncE ( 3 ) cidessus. On passe ensuite au cas d e 1 3 suppression d'un nomhre fini d'616ments, par passage au complEmentaire.

(2)

La

0

reunion et 1 'intersection de deux finivalles quelconques

c s t un finivalle (POUZET,

1976, non publiE ; ut. ax. choix dgnombrable ; ZF suf-

fisant lorsque la base est dEnombrahle). 0

Prouvons l'i.nonc6 pour la r6union ; on passera ensuite au compl6men-

taire. Rrtisonnons par l'absurde

en

supposant q u e U et V sont deux finivalles, mais

non leur rEunion. Soit p le plus petit entier pour lequel existe une infinit6 de classes de p-uples, deux p-uples dans U U V Etant dits Equivalents lorsque la transformation de l'un en I'autre est un automorphisme local extensible par l'identit6 hors de U U V. Prenons une o-suite de p-uples mutuellement non 6quivalents u les termes appartenant 5 U , et non

(ax. choix d 6 n . ) . Pour chaque p-uple, notons

1 V ; notons v l e s termes appartenant 5 V et non 1 U ; enfin w les termes appartenant 1 ]'intersection U

n V. Puisqu'il y a une infinit6 de p-uples en considEra-

tion, n o u s pouvons supposer que, pour chaque indice r

< p,

l e r-6me terme est

toujours un u , ou toujours un v, o u toujours un w. En restant dans le cas gEn6ra1, nuus pouvons supposer p

=

3 avec chaque suite formEe d'un u , un v et un w : pour

chaque entier i, nous avons donc la suite d e s trois termes u.,v.,w.. 1

1

1

Puisque U est un finivalle, et que chaque suite u i wi est formCe de deux ElEments de U, ces suites se r6partissent en un nombre fini de classes, pour 1 ' 6 quivalence d6finie par un automorphisme local extensible par l'identits hors de U. Nous pouvons donc supposer que toutes Les suites u . wi appartiennent i une m&ne

classe. Donc pour chaque i, la transformation d e u

vi wi en uo vi wo est un auto-

morphisme local extensible par Z'identitE hors d e U. Par ailleurs puisque V est un finivalle, i l en est d e &me

de V diminus d e 1'61Ement wo : voir le ( 1 ) pr6c6dent.

Les termes v appartenant Z ce nouveau finivalle, et les termes uo et wo 6tant en dehors, nous pouvons supposer que, pour chaque paire d'entiers i , j , la transformation de u

vi wo en uo vj wo es t un automorphisme local extensible par l'identi-

ti3 hors de V. Finalement pour tots i,j, nous avons un automorplisme local tralsformant u . v. w. en u 1

1

1

vi wo, puis en u

O

v. wo, puis en u J

vj v.,

avec 1 chaque 6tape

l'extersitilit6 par l'identitg h o r s de U U V : contradiction avec la d6finition de la suite infinie d e p -uple non Cquivalents 4.5.

.

0

Soit A une relation, E s a b e , D une partie de E . A l o r s pour que D

soit un finivalle, il faut et suffit qu'il existe une partie finie F du complementaire E-D,

telle que chaque automorphisme local de A I D , ou bien n'est pas extensible

R. Fra'isse

328

p a r l ' i d e n t i t k s u r F . o u b i e n e s t e x t e n s i b l e p a r l ' i d c n t i t d sur E-D t o u t e n t i e r (6nonc6 communiqu6 p a r POUZET, 1 9 7 8 ) . 0

S u p p o s o m d ' a b r d que F e x i s t e . Alors pour chaque e n t i e r p . r 6 p a r t i s -

s o m l e s p - u p l f f i d a m D e n un n o m k e f i n i d e c l a s s ~ s ,deux p ? ~ p I p s 6 t a n t d i t s

6 q u i v a l e n t s l o r s q u e l a t r a m f o r m a t i o n d e l'un e n l ' a u t r e , r 6 u n i e 3 l ' i d e n t i t s s u r F , donne un a u t o m a r h i s m e l o c a l d e A . La c o n d i t i o n d e l ' h o n c e c o n c e r n a n t F

e n t r a ' i n e a l o r s que D s o i t un f i n i v a l l e . I n v e r s e m e n t s o i t D un f i n i v a l l e ; n o t o m n l'arit6 d e A . Pour c h a q u e e n t i e r p o s i t i f p < n , r 6 p a r t i s s o f f i Iffi p i p l r s e n leiits c l a s s e s d ' P q u i v a l e n c e mod(A,D) : v o i r d 6 t u t d u p r k e n t p a r a g r a p h e 4 ; ces c l a s s -

s o n t e n nombre f i n i .

# C i , p r e n o r s un p j Gl&ments d e E - D , t e l que la

P o u r c l a q u e p e t c h a q u e p a i r e d e c l a s s e s d e p - u p l ~C ~. e t C

d ' a u p l u s n-I

u p l e dam c h a c u n e , e t u n e r s e m b l e Fi

c h o i s i pour C ne s o i t p"s e x i ' e n 'j j' L a r 6 u n i o n f i n i e F des F p o u r t o e les p < n , ij* ij v 6 r i f i e l ' 6 n o n c 6 . En e f f e t 6 t a n t donn6 deux p l l p l f f i u ' . d e C e t u' d e C la i j j' t r a m f o r m a t i o n d e u i e n u ' est p a r h y p o t h k e e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E-D ; t r a n s f o r m a t i o n du p - u p l e u

i

chois i p o u r C

teffiihle par l ' i d e n t i t 6 s u r F

i

d e mEme l a traffi f o r m a t i o n d e

11

j

en u '

j'

Donc s i l a t r a m f o r m a t i o n d e u! e n u ! 6 t a i t J

e x t e n s i b l e p a r l ' i d e n t i t 6 s u r F , i l e n s e r a i t d e mZme d e l a t r a n s f o r m a t i o n de u i en u . : c o n t r a d i c t i o n . F i n a l e m e n t F v 6 r i f i e 1 ' 6 n o n c 6 p o u r l e e n t i e r s p < n , e t i l

J en r f f i u l t e que F v E r i f i e e n c o r e l ' 6 n o n c 6 p o u r u n automorphisme l o c a l q u e l c o n q u e d e AJD.

4 . 6 . En v u e des 6noncffi s u i v a n ' s , ciaprk

l e l e c t e u r r e t r o u v e r a les t r o i s lcmmffi

( d o n t l e d e r n i e r remonte 1 LOPEZ, 1 9 6 9 ) :

(I)

S o i t A une r e l a t i o n n a i r e , u , v deux 616ments d e l a h e ; s i l a

t r a n s p o s i t i o n ( u , v ) m o d i f i e A ( i . e . c e t t e trampos i t i o n , 6 t e n d u e p a r l ' i d e n t i t 6

s u r 1 s 616ments a u t r e s que u , v , n ' f f i t pas u n a u t o m o r p h k m e d e A ) , a l o r s i l e x i s t e u n e n s e m b l e H d ' a u p l u s n+l e l e m e n t s , p a r m i l e s g u e l s u e t v , t e l q u e ( u , v ) m o d i f i e A/H. ( 2 ) S o i t E un e m e m t l e , f une p e r m u t a t i o n d e E e t F une p a r t i e f i n i e d e

E ; i l e x i s t e u n e s u i t e s a n s r k p e t i t i o n d ' P l 6 m e n t s u I , ...,u,,

d e F, t e l l e que p o u r

c h a q u e Plernent x d e F , l e t r a n s f o r m e f ( x ) s o i t i d e n t i q u e a u t r a n s f o r m 4 d e x p a r l a composee d s s t r a n s p o s i t i o n s s u c c e s s i v e s (u

, f ( u I ) ) , . . . , (y,,f (r,) ) .

( 3 ) Soit A une r e l a t i o n d e b a s e E, et f une p e r m u t a t i o n d r E g u i m o d i f i e A ; a l o r s i l existe un e l e m e n t u d e E t e l g u e l a t r a n s p o s i t i o n ( u , f ( u ) ) m o d i f i e A .

4.7.

(I)

S o i t E l ' e m e m t l e dffi e n t i e r s n a t u r e l s , N l a c h a i n e u s u e l l e s u r

ces e n t i e r s . S i A e s t l i b r e i n t e r p r e t a b l e en N e t s i , p o u r c h a q u e p r o g r e s s i o n arithmetique D de raison

> 2,

i l e x i s t e d e u x e l e m e n t s x e t y > x d e D t e l s q u e la

t r a n s p o s i t i o n (X,y) s o i t u n a u t o m o r p h i s m e l o c a l d e A, e x t e n s i b l e p a r 1 ' i d e n t i t 8

sur E-D,

a l o r s t o u t e p e r m u t a t i o n d e E e s t u n a u t o r n o r p h i s m e d e A (6nonc6s ( I )

et

( 2 ) dus 1 POUZET e t YASUHARA). 0

S u p p o s o n s qu'il e x i s r e une p e r m u t a t i o n d e E qui m o d i f i e A . P a r

lffi

lemmes p r G c 6 -

329

L'interualle en theorie des relations

d e n t s , i l e x i s t e un e n s e m b l e f i n i F d ' e n t i e r s e t d e u x e l 6 m e n t s a , l a t r a n s p o s i t i o n ( a , t) m o d i f i e A / F .

Fixiins

b de F tels que

t > a . Soit u =

1 6 idges en posant

Max(2, b a ) ; s o i t v l e n o m l r e d e s 616ments < a e t w c e l u i d e s 616me nts Puisque A ffit l i b - e - i n t e r p r 6 t a b l e

rn N,

>

k da ns F .

t o u t e i n j e c t i o n c r o i s s a n t e d e doma ine F

> x+u,

e s t un a u t o m o r p h i s m e l o c a l d e A . Donc p o u r t o u s e n t i e r s x 2 a e t y

l a trans

-

p o s i t i o n ( x , y ) m o d i f i e A . P l us p r G c i s 6 m e n t e l l e m o d i f i e l a r e s t r i c t i o n d e A 1 t o u t e n s e m t l e q u i comprend a u moins v e l e m e n t s

<x

et

wi.16ments > y

e t u-l

616ments

e n t r e x e t y . En p a r t i c u l i e r G t a n t donne l a p r o g r e s s i o n a r i t h m 6 t i q u e D = { a , a+u, a+2u,.

. . 1,

p o u r d e u x 6l Gmrnt s q u e l c o n q u e s x e t y

>x

de D, l a trans -

p o s i t i o n ( x , y ) e t e n d u e p a r l ' i d e n t i t e s u r E-D, n ' e s t p a s un a u t o m o r p h i s m e l o c a l de A. 0

( 2 ) S o i t D un e r s t . m t l e i n f i n i d ' e n t i e r s , d o n t l e c o m p l g m e n t a i r e E - D e s t i n f i n i . S i t o u t e p a r t i e d e D est un A - f i n i v a l l e , a l o r s il e x i s t e d e u x elements x et y

>x

d e D,

t e l s q u e l a t r a n s p o s i t i o n (x,y) soit un automorphisme

local d e A, e x t e n s i b l e p a r l ' i d e n t i t e s u r E - D . Co rsCq u e n c e de ( I )

e t (2) : s i A e s t l i b r e - i n t e r p r e t a b l e

en N e t s i t o u t e n s e m b l e

d ' e n t i e r s est u n A - f i n i v a l l e , a l o r s t o u t e p e r m u t a t i o n d e E est u n a u t o m o r p h i s m e d e A. 0

P r e u v e du ( 2 ) . A c h a q u e p a i r e d ' 6 1 6 m e n t s x , y d e D a s s o c i o n s l e c o u p l e o r d o n n d

(x,y

> x),

c e q u i nous p e r m e t d e r E p a r t i r l e s p a i r e s e n un n o m I r e f i n i d e c l a s s e s

d ' g q u i v a l e n c e mod (A,D). P a r l e t h e o r s m e d e RAMSEY, i l e x k t e u n e p a r t i e i n f i n i e D* d e D d a n s l a q u e l l e t o u t e les p a i r e s a p p a r t i e n n e n t 1 une m t m e c l a s s e . Donc p o u r tous x < y

e t x' < y '

616ments do D*, l a t r a n s f o r m a t i o n d e x e n x ' e t y e n y ' e s t

un a u t o m o r p h i s m e l o c a l d e A , e x t e r s i t l e p a r l ' i d e n t i t g s u r E - D . P r e n o n s d a n s D* les 6lGments d e r a n g s p a i r s : s o i t D*. P u k q u e ce D* e s t un A - f i n i v a l l e , p r e n o n s P P u n e p a r t i e i n f i n i e D** d e D* d a m l a q u r l l e l e s s i n g l e t o n s s o i e n t t o u s g q u i v a l e n t s

P

avec x < y < z , l a mod ( A , D * ) . Donc d t a n t d o n n b x , z d a n s D** e t y d a m D*-D* P P t r a n s f o r m a t i o n d e x e n z e t d e y e n l u i m z m e , e s t un a u t o m o r p h i s m e l o c a l d e A ,

e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E-D.

P u i s q u e x , y , z a p p a r t i e n n e n t 1 D*,

p a r composi-

t i o n l a t r a n s p o s i t i o n (x,y) e s t un a u t o m o r p h i s m e l o c a l d e A , e x t e n s i k l e p a r l ' i d e n t i t 6 s u r E-D. 4.8.

0

S o i t A u n e r e l a t i o n , E s a t a s r . Supposom q u e t o u t e p a r t i e d e E

s o i t un A i i n i v a l l e ; a l o r s ( c o m e en 2 . 3 ( 2 ) e t e n 2 . 5 ( I )

(I) fini

) :

L e s c l a s s e s d ' e q u i v a l e n c e p a r t r a n s p o s i t i o n (mod A ) sont en n o m b r e

;

(2) I 1 e x i s t e u n e m u l t i r e l a t i o n u n a i r e en l a q u e l l e A e s t l i b r e - i n t e r p r e t a b l e ( u t i l i s e l ' a x i o m e d e c h o i x d G n o m l r a t l e , ZF s u f f k a n t l o r s q u e l a h e E e s t

d6 n o m b ra b l e ; GnoncCs dus 5 POUZET). P a r c o n t r e i l n ' e x i s t e pas f o r c E m e n t u n e m u l t i r e l a t i o n u n a i r e

u

telle

que A e t U s o i e n t chacune l i t r e i n t e r p r 6 t a U e e n l ' a u t r e . Prenons e n e f f e t pour A une r e l a t i o n

t i n a i r e d ' g q u i v a l e n c e B un n o m t r e f i n i d e c l a s s e s . P o u r c h a q u e e n t i e r

R. Fra'issB

330

pour n ' i m p o r t e q u e l l e p a r t i e D d e l a

p, d e u x p - u p l e s s o n t 6 q u i v a l e n t s mod(A.D) bse qui

lffi

c o n t i e n t , p o u r v u q u ' i l s s o i r n t t r a n s f o r m & p a r une E j e c t i o n e t quc

d e u x termes de &me

r a n g d a m les d e u x p - u p l e s a p p a r t i e n n e n t 2 u n e m6me c l a s s e .

A i n s i t o u t e p a r t i e de l a Lase e s t un f i n i v a l l r ; mais une m u t t i r e l a t i o n U e n l a q u e l l e A s e r a i t l i l r e i n t e r p r g t a t l e , c o m p r e n d r a i t p a r exemple une r e l a t i o n u n a i r e valant

+ s u r une c l a s s e d e

A et

-ailleurs,

e t c e l a p o u r c h a q u e c l a s s e . Nous voyons

q u ' a l o r s U ne s e r a i t pas l i h - e - i n t r r p r 6 t d b l e 0

en A .

( I ) R e p a r t i s s o n s l e s 616ments d e E e n c l a s s e s d ' f q u i v a l e n c e p a r t r a n s p o s i t i o n

(mod A) : v o i r 2 . 2 .

Supposons q u ' i l e x i s t e une i n f i n i t 6 d e c e s c l a s s e s . P a r

l ' a x i o m e d e c h o i x d 6 n o m k r a U e , p r e n o n s un e n s e m t l e d 6 n o m l r a ' d e D d'6lGments d e l a Lase, m u t u e l l e m e n t non g q u i v a l e n t s . C o n s t r u i s o n s s u r l a Lase D u n e isomorphe N d e l a cha'ine des e n t i e r s n a t u r e l s . P a r l e thilorsme d e RAMSEY, now pouvons t o u j o u r s s u p p o s e r que l a r f f i t r i c t i o n A/D e s t l i h - e - i n t e r p r E t a b l e

e n N . Plus p r S c i s g m e n t ,

m e t t o n s dans une msme c l a s s e d e u x p a i r e s d ' E l 6 m e n t s x , y e t x ' x
e t x' < y

I ) ,

, y ' de ' es t

lorsque l a transformation de x en x ' e t y en y

D (avec

un a u t o m o r

-

phisme l o c a l d e A . A l o r s nous pouvons e x i g e r q u e t o u t e s les p a i r e s a p p a r t i e n n e n t 2 une &me

classe.

Disons q u e d e u x s i n g l e t o n s dans D s o n t G q u i v a l e n t s , o u q u e d e u x p a i r e s d ' 6 l P m e n t s d e D s o n t g q u i v a l e n t e s , l o r s q u e l ' u n i q u e automorphisme l o c a l d e N q u i t r a n s f o r m e l e premier s i n g l e t o n e n l e s e c o n d , ou l a p r e m i c r e p a i r e e n l a s e c o n d e , e t q u i e s t

e n c o r e un automorphisme l o c a l d e A , e s t de p l u s e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E - D . P u k q u e D f f i t un A - f i n i v a l l e , l e s c l a s s e s d e c e t t e g q u i v a l e n c e s o n t e n n o m l r e f i n i . E n c o r e p a r RAMSEY, o t t e n o n s une p a r t i e d 6 n o m l r a U e D , d e Do = D , dans l a q u e l l e t o u s les s i n g l e t o m i o n t e q u i v a l e n t s , e t t o u t e s les p a i r e s g q u i v a l e n t e s .

It6rons en

r e m p l a p n t Do p a r D I , a v e c l ' e x t e n s i k i i l i t 6 s u r E - D l : e t a i n s i d e s u i t e n f o i s , o b t e n a n t l e s ensemUes e m b ? t & s u c c e s s i f s D Soit

3 D

0 -

2

1 -

... _> D n .

u , v d e u x glements d i s t i n c t s q u e l c o n q u e s d e D

.

Par c e q u i prscsde,

5 n+l

l a t r a n s p o s i t i o n ( u , v ) m o d i f i e A . P a r 4 . 6 , i l e x i s t e une p a r t i e H d e

d e E , parmi l e s q u e l s u e t v , t e l l e q u e ( u , v ) m o d i f i e A / H . un e n t i e r h ( I 5 h a u c u n des

5

n-l

5 n)

t e l que l a d i f f e r e n c e e n s e m b l i s t e

Glements d e H

- Iu,v}.

6lEnents

Donc i l e x i s t e a u moins

4,

- 4,

n e comprenne

Donc c e t t e d i f f B r e n c e s t d k j o i n t e d e H .

P a r l ' G n o n c 6 4 . 7 , l a t r a n s p o s i t i o n ( u , v ) e s t un a u t o m o r p h i s m e d e A / D ,

puisque toute

p a r t i e d e D e t un (A/D) X i n i v a l l e

:

donc ( u , v ) automorphisme d e A / D h . Mais ( u , v )

n ' e s t p a s un automorphisme de A / ( E

-

(Dh-I-Dh)),

Autrement d i t , B t a n t donnE u , v d e D n , phisme l o c a l d e A / D h ,

non e x t e n s i b l e p a r 1 ' i d e n t i t B s u r E

D ' a u t r e p a r t B t a n t donnB u , v , u ' , v ' la t r a m formation de u en u' e t v en v '

phisme l o c a l d e N / 9 , donc d e A / \ r a i s o n d e l a d g f i n i t i o n msme d e u,v,u',v'

de D

d u f a i t que H C E

avec u < v e t u '

,

-

(Dh-,-Dh).

i l e x i s t e u n h t e l q u e ( u , v ) s o i t un a u t o m o r -

ffi

-?I - 1 '

d e Dn a v e c u < v e t u ' < v '

t pour chaque h

(1

< h 5 n)

(mod N ) ,

un a u t o m o r -

e x t e n s i b l e p a r l ' i d e n t i t s s u r E - Dh -1 ' en p a r c o m p o s i t i o n , q u ' E t a n t donne

4.11 en r b u l t e , > v'

(mod N ) ,

i l e x i s t e un h t e l que l a t r a n s f o r -

33 1

L'intetvalle en theorie des relations

, v ' s o i t un automorphisme l o c a l d e A/Dh, non

mation de u,v respectivement en u ' extens g

H e par l'identiti. s u r E -

t un a u t o m o r p h i s m e l o c a l d e A / D

4,

n'

A pl16

f o r t e raison, c e t t e transformation

lion e x t r n s i

F i n a l e m e n t , 6 t a n t donn6 u , v , w d e D

Ue par l ' i d e n t i t e s u r E - D

avec u

< w
.

l a trans f o r -

(mod N ) ,

m a t i o n du s i n g l e t o n d e u e n r e l u i d e v e s t u n a u t o m o r p h i s m e l o c a l d e A / D n ,

non

e x t e m i hl p p a r l ' i d e n t i t i . s u r ( E - D ) U iwl. I1 e n r f f i u l t e q u e , p a r e x e m p l e , 1 ' e r s e m U e d e s GlGments d e D contradiction.

d e r o n g s p a i r s (mod N/Dn).

n ' s t p a s un A - f i n i v a l l e :

0

L ' C n o n c 6 (2) r P s u l t e du ( I ) e t d e 2 . 2 .

5- LE SUBVALLE, N O T I O N I N T E R M E D I A I R E ENTRE L'INTERVALLE ET LE FINIVALLE ; CAS DU CYCLORIIRE AVEC LE SECTEUR-PAKT-DE-GATEAU

E t d n t donne u n e r e l a t i o n A d e b

E-D,

en

ou

e E , une p a r t i e D d e E s e r a d i t e un

l o r s q u e , pour t o u t automorphisme l o c a l f d e A/D e t t o u t e p a r t i e G d e

A-suhvalle

f n ' e s t pas e x t e n s i b l e p a r l ' i d e n t i t 6 s u r G , ou f e s t e x t e n s i b l e p a r

l ' i d e n t i t e s u r c h d c u n e d e s p a r t i e s d e E-D o b t e n u e s d e G p a r un a u t o m o r p h i s m e l o c a l de A.

A i n s i q u ' i l a d 6 j h Gt6 d i t p o u r l ' i n t e r v a l l e e t l e f i n i v a l l e , i l s u f f i t q u e f s o i t d e f i n i s u r un d o m a i n e d e p s o i t de c a r d i n a l

5

n-l

616ments

(n

=

a r i t E d e A) e t que G

n-p.

Si A est u n e c h a i n e , no16

v q r o n s q u e les A - s u b v a l l e s

s o n t exactement les

i n t e r v a l l e s et les e x t e r v a l l e s . R a p p e l o l s q u e l e c y c l o r d r e engendrE p a r une chaTne e s t d 6 f i n i a u p a r a graphe 3. La n o t i o n d e s u b r a l l e d ' u n c y c l o r d r e est t r b

i n t u i t i v e : reprEsentant

l e c y c l o r d r e p a r un c e r c l e , nous r e t r o u v o n s p o u r l e s u l u a l l e l e s e c t e u r c i r c u l a i r e ,

ou part-de-g8teau.

Rigoureusement

C : les A-subvalles

s o i t A u n c y c l o r d r e e n g e n d r 6 p a r une c f i a i n e

s o n t exactement les C - i n t e r v a l l e s et les C - e x t e r v a l l e s ,

l e c a s d ' u n e b e d e 4 616ments 0

:

except6

( p o u r l e q u e l t o u t e p a r t i e est un s u b r a l l e ) .

I1 s u f f i t d ' 6 t u d i e r d e u x cas : ou l i e n f d E f i n i s u r u n d o m a i n e s i n g l e -

t o n e t G p a i r e d ' s l G m e n t s , ou f d P f i n i s u r u n e p a i r e e t G s i n g l e t o n . Dans l e p r e m i e r cas d i s o r s q u e f t r a n s f o r m e 1 ' 6 1 6 m e n t u d e D e n u ' d e D , e t s o i t G = { v , w j inclls

d a m E-D.

S u p p o s o r s D i n t e r v a l l e ou e x t e r v a l l e d e l a c h a i n e : a l o r s f e s t

evidemment e x t e n s i b l e p a r l ' i d e n t i t i . s u r C . Dans l e s e c o n d cas, f t r a n s f o r m a n t u,v de D en u ' , v '

d e D e t G = s i n g l e t o n d e 1 ' G l E m e n t w d e E-D.

c e t t e v a l e u r s u b i s t e p o u r t o u t w d e E-D A ( u ' , v ' , w) = A ( u , v , w) p o u r t o m

S i A(u,v,w)

; d e mzme a v e c la v a l e u r

-.

= +,

Donc o u h i e n

w d o n c f e x t e n s i H e S t o w les s i n g l e t o n s . Ou

A ( u ' , v ' , w) # A ( u , v , w) e t f n ' f f i t e x t e n s i H e d a u c u n s i n g l e t o n d a m E-D.

11 res t e h v o i r q u e s euls les C - i n t e r v a l l e s

ou e x t e r v a l l e s conviennent.

E x c e p t a n t l e c y c l o r d r e s u r 4 E l g m e n t s , s u p p o s o n s q u ' i l e x i s t e a u moins 5 ElEments

R. Frdisse

332

d i s t i n c t s a , b , c , d , e dans l ' o r d r e d ' u n e des c h a i n e s q u i l ' e n g e n d r e n t . A l o r s nous v q o m q u e l a p a i r e { a , c } n ' e s t pas un s u b u a l l e , l a t r a n s f o r m a t i o n d e a e n c E t a n t e x t e n s i b l e p a r l ' i d e n t i t e s u r { d , e ) mais n o n s u r { b,d}.

Par a i l l e u r s { a , c , d l

n ' e s t p a s non p l u s un s u l v a l l ~ , l a t r a f f i f i i r m a t i o n d e a e n c e t d e d e n lui-mzme e t a n t i n e x t e n s i b l e p a r l ' i d e n t i t 6 s u r l e s i n g l e t o n { b } , mais e x t e n s i k l e p a r l ' i dentit6 s u r

{el.

0

L ' e x e m p l e du c y c l o r d r e p r o u v e clue l'intersection ou la reunion d c deux subvalles n'est pas forckment un subvalle. 5.1.

T o u t intervalle ou extervalle est un subvalle.

E v i d e n t d a m l e c a s d ' u n i n t e r v a l l e . S o i t D un e x t e r v a l l e d e l a r e l a t i o n A d e b e E : donc E-D e s t un i n t e r v a l l e . S o i t f un a u t o m o r p h i s m e l o c a l d e d e domaine F e t codomaine F ' , e t s o i t g u n a u t o m o r p h k m e l o c a l d e A/(E-D),

A/D,

d e domaine G e t codomaine G ' . Supposons f e x t e n s i b l e p a r 1 ' i d e n t i t E IG s u r G :

il s u f f i t d e v o i r que f e t e n c o r e e x t e n s i b l e p a r l ' i d e n t i t e I G l f U I

G

g U I F , e s t e n c o r e un a u t o m o r p h i s m e l o c a l , donc f De p l u s g

avec f

.

La r e u n i o n

E t a n t un a u t o m o r p h k m e l o c a l d e A , e t E-D e t a n t un A i n t e r v a l l e , l a r g u n i o n

u

u g,

u g en est un, par composition. -1

IF e s t un automorphisme l o c a l , donc a u s s i g

nous o l t e n o n s f

u

IGl

U IF : p a r c o m p o s i t i o n

.

I1 e x i s t e des s u h r a l l e s q u i n e s o n t n i un i n t e r v a l l e ni un e x t e r v a l l e .

P a r exemple prenons l e q c l o r d r e d 6 j 5 6 t u d i 6 p r E c S d e m e n t , e t r a p p e l o n s q u e les s e u l s i n t e r v a l l e s ( a b o 1 u s ou r e l a t i f s ) s o n t l e v i d e , l a b e e n t i s r e e t les s i n gletons : v o i r paragraphe 3 .

5.2. Le complPmentalre d'un subvalle n'est pas forcement un subvalle. P a r t o n s d e d e u x c h a f n e s A , B e t cons t r u i s o m s u r l a r e u n i o n E d e l e u r s b f f i ,

s u p p o s 6 e s d i s j o i n t e s , l ' o r d r e daffi

incomparable B chaque elgment d e

1 Bl

l e q u e l c h a q u e BlEment d e l a h

e 1.4

I ffit

.

Notons B' un i n t e r v a l l e m e d i a n d e B ; p u i s c o n s i d e r o n s l ' e n s e m l l e D ,

reunion des lases I A

I

et

I B'I

. D'une

p a r t D e s t u n s u b a l l e ; e n e f f e t un a u t o -

morphisme l o c a l dans D , ou l i e n t r a n s f o r m e a u moins un e l e m e n t d e

I B'I ou i n v e r s e m e n t

: alors il n'est

I A I en

un d e

pas e x t a n s i kle p a r a u c u n e i d e n t i t 6 h o r s d e

D . Ou n o t r e automorphisme l o c a l e s t La r e u n i o n d ' u n automorphisme l o c a l d e A e t

d ' u n d e B ' , e t a l o r s i l e s t e x t e n s i U e p a r l ' i d e n t i t e s u r E-D.

D ' a u t r e p a r t E-D

n ' s t pas un s u b a l l e ; e n e f f e t l a t r d n s f o r m a t i o n d ' u n e l e m e n t d e B, a n t e r i e u r l ' i n t e r v a l l e B',

e n un e l e m e n t p o s t B r i e u r h B ' ,

s u r l e s i n g l e t o n d ' u n GlBment d e I A

I B'l

.

I mais

e s t extensible par l'identite

non s u r l e s i n g l e t o n d ' u n e l e m e n t d e

0

Dam l e c a s d u c y c l o r d r e ,

l e c o m p l 6 m e n t a i r e d ' u n s u h r a l l e e s t un s u t -

v a l l e . De m t m e dans l e c a s d ' u n e c h a f n e , oh les s u h r a l l e s s o n t e x a c t e m e n t les i n t e r v a l l e s e t l e s e x t e r v a l l e s . Cela now c o n d u i t a u problBme s u i v a n t . ProUBme. S i A est l i h - e - i n t e r p r e t a b l e d ' u n A s u h r a l l e e s t - i l t o u j o u r s un A - s u b a l l e .

e n une c h a i n e , l e c o m p l e m e n t a i r e

L'interualle en thiorie des relations

333

5.3. Tout subvalle est un finivalle. 0

S n i t A une r e l a t i o n n - a i r e

i l s u f f i t q u e pour chaque p < n

de

h s e E . P o u r q u e D s o i t un A - f i n i v a l l e , dans l a l a s e d o n t l a l o n g u e u r e s t p ,

Ics s u i t e s

s e r e p a r t i s s e n t e n un n o m t r e f i n i dts c l a s s e s d ' e q u i v a l e n c e mod(A,D) p a r a g r a p h e 4.

: v o i r d e b u t du

R a p p e l o ns q u e deux s u i t e s d e l o n g u e u r p d o i v e n t d e f i n i r une trans -

f o r m a t i o n d e l ' u n e e n l ' a u t r r q u i s o i t un a u t o m o r p h i s m e l o c a l d e A , e x t e n s i t l e p a r l ' i d e n t i t e s u r t o u t rnsemtle de

_<

n-p e l e m e n t s d e E-D.

5 n,

S u p p o so n s m a i n t r n a o t qcte I) s o i t u n A s u l v a l l e . P o u r q deux ensembles G , G '

d e q 61Ements d e 13-D,

disons que

s o n t Gquivalents, l o r s q u e A/G e t A/G'

i s o m o r p h e s . P r e n o n s un r r p r e s e n t a n t d a n s s h a q u e c l a s s e , e t s o i t H l ' e n s e m l - l e f i n i , reunion de ces representants

pour tous les q

< n.

-

D i s o n s q u e d e u x s u i t e s da ns D ,

n, s o n t P q u i v a l e n t e s l o r s q u e l a t r a n s f o r m a t i o n

s o i t u e t v , d ' u n e mEme l o n g u e u r p

d e u en v e s t un a u t o m o r p h i s m e l o c a l d e A , e x t e n s i b l e p a r l ' i d e n t i t i . s u r H. P u i s q u e H e s t f i n i , les c l a s s e s d e c e t t e G q u i v a l e n c e s o n t e n nomh-e f i n i . I1 s u f f i t d e

n o t e r q u e , D G t a n t un s u h r a l l e , I n t r a n s f o r m a t i o n d e u en une s u i t e G q u i v a l e n t e v de longueur p , s i e l l e est e x t e n s i b l e p a r l ' i d e n t i t e s u r H, est encore e x t e n -

s i t l e par l ' i d e n t i t i . s u r t o u t ensemtle de

< n-p

GlGments d e E-D.

L e f i n i v a l l e P t a n t une n o t i o n l o o l e e n n e , d ' a p r 2 s

0

l'enonce precedent il

e x i s t e d e s f i n i v a l l e s non s u h r a l l e s . Un f i n i v a l l e n ' e s t pas f o r c e m e n t o b t e n u 2 p a r t i r de s i n t e r v a l l e s e t e x t e r v a l l e s p a r c o m p l e t i o n t ool Genne ( u n i o n e t i n t e r s e c t i o n f i n i e s , e t c o m p l g m e n t a t i o n ) : i l s u f f i t d e p r e n d r e l e c a s des c y c l o r d r e s , 05 c e t t e c o m p l g t i o n EoolEenne n e d o n n e r a i t q u e les e n s e m t l e s f i n i s e t l e u r s c o m p l e m e n t a i r e s ,

donc n e d o n n e r a i t

pas t o u s l e s s u h r a l l e s , e t 2 p l u s f o r t e r a i s o n les f i n i v a l l e s .

P r o ' d S me . E x i s t e n c e d ' u n f i n i v a l l e non o k e n u p a r c o m p l e t i o n b o l G e n n e 2 p a r t i r d e l ' e n s e m b l e des s u l x a l l e s .

5.4. La libre-interpr6tabilite ne conserve p a s les subvalles ( a l o r s q u ' e l l e c o n s e r v e les f i n i v a l l e s : v o i r 4.3 ( I ) ) . 0

S o i t C une cha?ne, U,V deux i n t e r v a l l e s d e C , d i s j o i n t s , l a i s s a n t e n t r e

e u x un i n t e r v a l l e non v i d e , e t a v e c u n i n t e r v a l l e non v i d e p o s t e r i e u r a u s e c o n d . La r e u n i o n D un C - s u h r a l l e .

=

U U V n ' e s t n i un C - i n t e r v a l l e n i un C - e x t e r v a l l e ,

d e mOme V l a r e l a t i o n u n a i r e v a l a n t + s u r V e t (C,U,V),

d o n c n ' e s t pas

Appelons e n c o r e U l a r e l a t i o n u n a i r e v a l a n t + s u r l ' i n t e r v a l l e U ; -ailleurs.

Pour la m u l t i r e l a t i o n

l ' e m e m k l e D e s t un s u h r a l l e , e t mOme u n i n t e r v a l l e . En e f f e t s o i t f u n

a u t o m o r p h i s m e l o c a l d e (C,U,V)

d e domai ne e t d e codoma ine i n c l u s da ns D . A l o r s f

t r a n s f o r m e f o r c e m e n t u n E l ement d e U e n un e l e m e n t d e U ; d e mOme p o u r V ; donc f

est e x t e n s i h l e p a r l ' i d e n t i t s s u r

I

C

I

- D.

0

5.5. R a p p e l o n s , p a r l e g r a p h e s u i v a n t , l e s r a p p o r t s e n t r e les d i f f e r e n t s "valles".

R. Fra'isse

334 Intervalle a k o l u

>-

In~ervallerelatif

d ' i n terva I 1 es

(bolEen et conserv6 par la l i h-e-inter prEtakilit6)

6- CLOTURE D'UNE RELATION ; RE1,ATION COMPACTABLE 6.1. Filtre intervallaire Ainsi qu'il a EtE d E j P dit. i l s'agit d'gtendre aux rela ions la c 1 8 ure classique de la chaIne des rationnels par celle des rEels. Soit A une relation de lase E ; dEfinissons un A-filtre intervallaire, o u simplement A-filtre, c o m e un ensemtle non vide

F de A-intervalles alsolus non

vides, v6rifiant les deux conditions : (1)

tout A-intervalle qui inclut un ElEment de F est un ElEment de F ;

(2) l'intersection de deux 616ments quelconques de

F est un ElEment de

F (c'est un A-intervalle, par 1.2). C o m e pour les filtres usuels, un A-filtre sera dit p l u s fin qu'un autre lorsqu'il l'inclut. Un A-ultrafiltre sera u n A-filtre maximal par inclusion. Un A-ultrafiltre sera dit trivial lorsqu'il est l'ensemble de tous les A-intervalles qui comprennent un ElGment donnE de la b s e E. (Rappelom que tout singleton est un A-intervalle). Si un A-ultrafiltre est non trivial, l'intersection de ses 616ments est vide, et chacun de ses 6lEments est une partie infinie de E. Etant donnE un A-intervalle D et un A-ultrafiltre chaque 616ment de

F , et alors D appartient P F

; nu

F,

ou E e n D intersecte

i l existe un El6ment de F , dis -

joint de D. En consgquence, Etant donnE deux A-ultrafiltres distincts F et existe un El6ment D de F et un D ' de

F', il

F' avec D no' vide.

6.1.1. Si une r6union finie de A-intervalles D. est 6lEment du A-ultra1

filtre

F, alors l'un au moins des

Di ts t 616ment de

F.

6.1.2. Tout A-filtre admet un A-ultrafiltre plus fin (i.e. surensemlile du A-filtre donn6) ; &once

ottenu P l'aide de l'axiome usuel de l'ultrafiltre.

Lorsque A est la multirelation r6duite P sa b s e E , ou encore une relation valant toujours +,

nu

plus g6nEralement une relation admettant toute permutation de E

c o m e automorphisme, alors les A-filtres et ultrafiltres s e rgduisent aux filtres et ultrafiltres ensemllistes usuels, sur E. 6.2. C18ture d'une relation selon un ensemble d'ultrafiltres interval laires. Partors d'une relation A de lase E ; identifions chaque A-ultrafiltre

335

L'intervalle en theorie des relations t r i v i a l avec l ' e l e m e n t de E qiii l ' e n g r n d r e : 1 ' e r s e m E l e de t o m

les A-ultrafiltres

d e v i e n t un s u r e n s e m b l e d e E . A c h a q u e R-

u l t r a f i l t r e non t r i v i a l F , a s s a c i o n s u n e n s e m b l e a r t i t r a i r e E ( F ) ,

f i n i ou i n f i n i ,

G v e n t u e l l e m e n t v i d e ; p u i s u n e r r l d t i u n A ( F ) d e base E ( F ) e t d e mGme a r i t 6 que A,

a v e c l e s e x i g e n c e s s u i v a n t e s , q u i p e u v r n t t o u i o u m G t r e s a t i s f a i t e s . Nous s u p -

F d i s t i n c t s c o r r e s p o n d e n t deux E ( F ) d i s j o i n t s

posors qu'2 deux u l t r a f i l t r r s p l a chaque E

: de

( F ) e s t d i s j o i n t d e E . Pour c h a q u e p a r t i e f i n i e X d e E ( F ) e t c h a q u e

e l e m e n t D d e F, e x i g e o r s q u ' e x i s t e un i s o m o r p h i s m e a u moins d e l a r e s t r i c t i o n A ( F ) / X SUK une r e s t r i c t i o n d e A / D .

Dans l e c a s g e n g r a l a v e c E

(F)

infini, cette

e x i g e n c e e s t s a t i s f a i t e e n u t i l i s a n t It. lemme d e c o h e r e n c e , g q u i v a l e n t 1 l ' a x i o m e d e l ' u l t r a f i l t r e ( v o i r Note I e n f i n d ' a r t i c l e ) . Suppcsors c h o i s i e l a r e l a t i o n A ( F ) pour chaque A u l t r a f i l t r e F , e t n o t o n s E+ l a r g u n i o n d e E e t des E ( F ) . A l o r s l a c l o t u r e A+ d e A s e r a d 6 f i n i e s a n s a m b i y u i t e comme s u i t , s u r l a Lase E + . S o i t n l ' a r i t 6 d e A , e t x ] ,

de

.'E

...x

des 61Ements

S i x . a p p a r t i e n t 1 E , c o n s e r v o n s l e ( i = 1 ou...ou n ) . Groupom e n s e m b l e

l e s x . q u i a p p a r t i e n n e n t 5 un &me

E

isomorphisme l o c a l de A ( F ) v e r s A / D ,

F donc l e s e n s e m t l e s

( F ) , e t prenons l e u r s t r a n s f o r m e s x! p a r un oil D e s t un e l e m e n t d e F. Les u l t r a f i l t r e s

par n o t r e s u i t e x , ,

D concern&

...,x

s o n t en nomke f i n i :

nous pouvors donc e x i g e r q u e c e s D , q u i a p p a r t i e n n e n t 1 des F d i s t i n c t s , s o i e n t m u t u e l l e m e n t d i s j o i n t s , e t s o i e n t d i s j o i n t s des s i n g l e t o n s des x . q u i a p p a r t i e n -

,..., x

n e n t 2 E . A l o r s posons A+ ( x l

) = A(x;

,...,x:),

en posant x! = xi l o r s q u e

x . a p p a r t i e n t 1 E . La v a l e u r a i n s i d e f i n i e e s t i n d k p e n d a n t e d u choix d e s i n t e r v a l l e s d i s j o i n t s D,

du f a i t que des automorphismes l o c a u x d e A , d o n t les domaine e t

codornaine s o n t i n c l u s , p o u r c h a c u n , d a m un i n t e r v a l l e D , d o n n e n t une r e u n i o n q u i at elle-mEme un a u t o m o r p h i s m e l o c a l d r A : v o i r 1 . 5 . Notons q u e c h a q u e

A

(F)

P s t

un i n t e r v a l l e d e l a c l d t u r e A+.

R e t r o u v o r s c o m e s u i t l a c l G t u r e de l a c h a i n e Q des r a t i o n n e l s , ou p l u t s t l ' u n e des c l z t u r e s , q u i es t l a c h a i n e des r e e l s . Les u l t r a f i l t r e s i n t e r v a l l a i r e s d e Q s o n t d e c i n q s o r t e s : p o u r d7aque r a t i o n n e l u , l ' u l t r a f i l t r e t r i v i a l e n g e n d r e p a r u ; l ' u l t r a f i l t r e des i n t e r v a l l e s B d r o i t e d e u , d e l a f o r m e l u , x ] 02 x > u

;

l ' u l t r a f i l t r e dffi i n t e r v a l l e s 1 g a u c h e d e u ; e n f i n 1es u l t r a f i l t r e s 1 d r o i t e e t 2 g a u c h e d e c h a q u e i r r a t i o n n e l . I1 s u f f i t d e p r e n d r e E

(F)

v i d e pour l ' u l t r a f i l t r e

1 d r o i t e ou 2 gauche d e chaque r a t i o n n e l , a i m i q u e , p a r exemple, pour l ' u l t r a f i l t r e 1 gauche d e chaque i r r a t i o n n e l , e t E l ' u l t r a f i l t r e B d r o i t e de

(F) r g d u i t B u n u n i q u e e l g m e n t p o u r

chaque i r r a t i o n n e l .

C e t t e n o t i o n de c l s t u r e , a i m i que l a n o t i o n de r e l a t i o n compactable exposee c i - a p r k ,

r e m o n t e 5 FRAYSSl!,

deux p r o blPmes l o g i q u e s s u i v a n t s ProblSme I .

F,

une A

(F)

; e l l e donne l i e u

aux

E t a n t d o n n e une r e l a t i o n A e t un A - u l t r a f i l t r e non t r i v i a l

associons l a r e l a t i o n de b

existe-t-il

1972 ( 2 ) p . 164-165

.

e v i d e aux autrffi A - u l t r a f i l t r e s

non t r i v i a u x . Alors

d e b e n o n v i d e d o n n a n t une e x t e n s i o n l o g i q u e ( e n c o r e d i t e

R. Fraksi

336 extens i o n 616mentaire) dc A .

ProUPme 2 . S o i t F e t G deux A - u l t r d f i l t r r s

non t r i v i a u x , A ( F ) e t

A (G) deux r e l a t i o n s donnant chacune une c l 6 t u r e q u i s o i t une e x t e n s i o n l o g i q u e de A (les a u t r e s

u l t r a f i l t r e s 6 t a n t a f f e c t & d'une

o k e n u e en prenant h l a f o i s A ( F j e t A

h s e v i d e ) . Alors l ' e x t e n s i o n

(G) e s t - e l l e une e x t e n s i o n l o g i q u e .

6 . 3 . Ensemble compact de f i l t r e s , r e l a t i o n compactable Partons d'un ensemble E , e t d ' u n r n s e m l l e de f i l t r e s F s u r E . Disons qu'un t e l ensemtle de f i l t r r s e s t compact l u r s q u e , pour chaque f o n c t i o n de c h o i x h s u r les F , t e l l e que h ( F ) s o i t un 6lEment de F pour chaque F , i l e x i s t e un nomh-e f i n i de

F t e l s que l a r6union de l e u r s t r a n s f o r m &

h

( F ) recouvre E .

E t a n t donn6 un ensemble E , l'ensemble de tous les ultrafiltres sur E, est compact ( u t . ax. u l t r a f i l t r e ) . 0

Supposons l e c o n t r a i r e : i l e x i s t e une f o n c t i o n de c h o i x h t e l l e que,

pour chaque ensemUe f i n i U d ' u l t r a f i l t r e s s u r E , 1 ' e n s e m U e E U Egal B E diminu6 de l a r6union des h ( F ) pour tous les F de U, s o i t non v i d e . Alors l e s s u r e n s e m t l e s des E

U

forment un f i l t r e s u r E . Prenons un u l t r a f i l t r e plus f i n : i l e s t d i s t i n c t

de t o w l e s

F

; contradiction.

0

Revenons aux r e l a t i o n s , e t d i s o n s qu'une r e l a t i o n A es t compactable l o r s

-

que l ' e n s e m t l e des A - u l t r a f i l t r e s e s t compact. Disons qu'une r e l a t i o n A de l a s e E e s t intervallement compactable l o r s q u e , pour chaque A - i n t e r v a l l e D , l e compl6ment a i r e E-D e s t une r6union f i n i e de A - i n t e r v a l l e s .

Par exemple, t o u t e r e l a t i o n o u

m u l t i r e l a t i o n u n a i r e , o u e n c o r e t o u t e c h a i n e e s t i n t e r v a l l e m e n t compactable. Par c o n t r e l a c o n s 6 c u t i v i t Q s u r l e s e n t i e r s n a t u r e l s n ' e s t pas i n t e r v a l l e m e n t compac

-

table (voir 1 . 1 ) . Dire que A e s t i n t e r v a l l e m e n t compactakle Gquivaut 2 I ' u n e ou 2 I ' a u t r e des c o n d i t i o n s s u i v a n t e s :

( I ) pour chaque ensemble fini de L-intervalles Di, E

-

l e complementaire

(U Di) est une rgunion finie d e A-intervalles ; ( 2 ) pour chaque couple de A-intervalles C , D avec D

C-D est une reunion finie de A-intervalles

2

C, l a diffgrence

: mgme 6nonc6 en r e m p l a p n t

D p a r une

r6union f i n i e d ' i n t e r v a l l e s D . C C . 1 -

0

o l a c u n e des c o n d i t i o n s ( I ) ,

( 2 ) e n t r a i n e Qvidemment que A s o i t i n t e r -

v a l l e m e n t compactable. L ' i n v e r s e e s t v r a i pour l a c o n d i t i o n ( I ) ,

du f a i t que t o u t e

i n t e r s e c t i o n d ' i n t e r v a l l e s e s t un i n t e r v a l l e . Par a i l l e u r s pour l a ( Z ) , une r d u n i o n f i n i e d ' i n t e r v a l l e s , il en e s t de mgme pour C-D = C aux i n t e r v a l l e s i n t e r s e c t i o n s .

(E-D),

s i E-D e s t

en p a s s a n t

0

6.4. Soit A une relation intervallement compactable filtre sur la base E de A,

n

;

alors tout ultra-

une fois reduit a ses e1Qments qui sont des A-interval-

les, donne un A-ultrafiltre.

Par c o n t r e , pour l a c o n s Q c u t i v i t 6 C s u r les e n t i e r s n a t u r e l s , 1es s e u l s C - i n t e r v a l l e s Q t a n t l e v i d e , l a lase e t les s i n g l e t o n s , t o u t u l t r a f i l t r e non t r i -

L'interualle en thiorie des relations

337

vial sur la b e I C I, une fois r6duit aux C-intervalles, donne le singleton de la lase : ce n'est pas un C-ultrafiltrr. Soit F un ultrafiltre sur E, et A l'ensemkle des A-intervalles 616ments

0

de F . Supposons que A ne soit pas un A-ultrdfiltre. Alors i l existe un A-intervalle D, non elt5ment de A mais q u i intersec-te C h d q U e 6lCment de A . Ce D n'est pas element

de l'ultrafiltre F , donc E-D est el6ment de

F. Par hypothke E-D est une

reunion finie de A intervalles : l'un d'eux, soit D*, est 616ment de F, donc de A . Finalement D et D* sont disjoints, mdis D intersecte chaque 616ment de A , donc D intersecte D

:

contradiction.

0

6 . 5 . Les notions de r e l a t i o n compactable e t intervallement compactable, se confondent, moyennant l'axiome de choix. O

Soit A une relation intervallement compactable, E sa h e . Supposons A

non compactable : il existe une fonction de choix h telle que, pour chaque ensemble fini U de A-ultrafiltres, l'ensemkle E 6gal B E diminu6 de la r6union des h ( F ) U pour tous les F de U, soit non vide. En ce cas les surensemkles des EU constituent un filtre sur E. Prenom un ultrafiltre plus fin, puis r6duisons le & ses A-intervalles. Cela nous donne un A-ultrafiltre, par l'6nonc6 pr6c6dent : appelons-le H et notons que h ( H ) est 616ment de H. Pour chaque ensemble fini U de A-ultrafiltres, EU est par hypothsse une r6union finie de A intervalles (voir 6 . 3 ( 1 ) ) : l'un d'eux est 616ment de H (voir 6.1. I ) . Prenons U = s ingleton de H ; nous o kenons E = U E - h (H) ; il existe un blsment de H qui soit inclus dans E - h (H), donc disjoint de h (H)

:

contradiction prouvant que A est compactahle. Inversement supposons A compactable, et supposons qu'il existe un A-in-

tervalle D, tel qu'aucune r6union finie de A-intervalles ne donne E-D. Alors 1 chaque A-ultrafiltre F, associons h (F)

=

D lorsque D est 616ment de F ; sinon

choisissons h ( F ) 616ment de F et disjoint de D. Avec cette fonction de choix h , aucune r6union finie de h ( F ) ne recouvrr E : contradiction.

0

ProNSme. Existence d'une reldtion intervallement compactahle de lase E, et d'un intervalle D tel que E-D ne

S U ~ Lpas

une r6union finie d'intervalles m u t u e l -

lement d i s j o i n t s cela exclut les chafnes et les multirelations unaires.

7 - ULTRAFILTRE ET FAMILLE LOC-INTERVALLAIRE ; CLOTURE D'UNE RELATION PAR CES

ULTRAFILTRES ; CAS DE LA REPRESENTATION DE STONE.

7.1. Ultrafiltre loc-intervallaire Soit A une relation de b e E ; un ultrafiltre

F

sur E sera dit A-loc-

i n t e r v a l l a i r e (ahrgviation de "localement intervallaire"), soit lorsqu'il es t tri -

vial, soit lorsque, pour chaque partie finie H de E, il existe un Pigment D C E-H Hde F, tel que tout automorphisme local de la restriction A/D soit extensible par H l'identitb sur H. Par exemple, Ptant donnb un A-ultrafiltre intervallaire, au sens d u

§

6 prbcedent,

R. Fra'isse

338

tout ultrafiltre plus fin est A-loc-intervallaire. Cependant il existe bien d'autres exemples, en raison de l'6noncQ suivant. Si A est une relation binaire, tout ultrafiltre sur la base I A I est A-loc-interVal1aire. 0

Pour l'arite 2, i l suffit, dans la definition ci-dessus, que tout automorphisme

local transformant un seul QlQment de DR en un autre, soit extensible par l'identit6 sur H. Mettons dans une msme classe d'gquivalence deux GlQments quelconques u,v de

E-H, lorsque la transformation du seul u en v est un automorphisme local extensible par l'identitg sur H. Les classes ainsi dEfinies Etant en nombre fini, l'une d'elles et une seule est QlQment de l'ultrafiltre. 0 Par contre, etant donne la relation ternaire de cyclordre A de base infinie E (paragraphe 3 ) , seuls les ultrafiltres triviaux sur E sont A-loc-intervallaires. En effet, prenons pour H un singleton, et reprenons la notation DH prQcQdente. La transposition qui Qchange deux QlEments quelconques de DH est un automorphisme local du cyclordre A, inextensible par l'identitg sur H. 7.2.

Famille loc-intervallaire d'ultrafiltres

Soit A une relation de base E ; un ensemble, ou famille d'ultrafiltres dont chacun est A-loc-intervallaire, sera dit famille A-loc-intervallaire, lorsque, pour chaque partie finie H de E, et chaque suite finie d'ultrafiltres F , ,

. .. , F

de la famille (il suffit de prendre n au plus Bgal P l'aritE), on peut associer 1 chacun l'un de ses SlQments, soit D, E F , ,

.. ., Dn E Fn,

les D Stant mutuellement

disjoints, de sorte que toute reunion d'automorphismes locaux de A/D ,,..., de A/Dn, soit un automorphisme local de A, extensible par 1'identitQ sur H. Dans le cas oii certains ultrafiltres sont triviaux, on prendra Qvidemment 1'QlQment correspondant D qui se rQduit 1 un singleton (Eventuellement singleton d'un SlQment de H) ; et pour les

non triviaux, on prendra un D disjoint de H.

Exemple de la representation de Stone. Soit A un lattis boolden. Plus prQcisEment A est une relation binaire d'ordre (rgflexive, transitive, antisymgtrique) ; pour deux QlQments quelconques u,v de la base, il existe un infimum u n v tel que les x 1 l a fois Q u et Q v soient exactement les x Q u n v ; de mgme il existe un supremum u

u v,

avec distributivitg entre supremum et infimum ; de plus il existe

un minimum 0 et un maximum I et, pour chaque Cl&ent u

nlu

=

u , un complEment

7 u tel que

0 et u U l u = I . Supprimons le minimum 0, et disons qu'un ultrafiltre

ensembliste F sur la nouvelle base (aprSs suppression du 0) est stonien lorsque, pour chaque Q16ment u initial < 7 u ,

f

0, ou bien l'intervalle initial Q u , ou l'intervalle

est un Gldment de

F.

En particulier, un ultrafiltre trivial est

stonien ssi il se rQduit au singleton d'un QlEment minimal (irmngdiatement supdrieur au 0 qui a QtQ supprim6). Disons que deux ultrafiltres stoniens F,G

sont stone-

distincts, lorsqu'il existe un u pour lequel l'intervalle initial Q u est QlQment de

F, et l'intervalle initial < l u est QlQment de G.

Tout ensemble d'ultrafiltres triviaux ou stoniens, l e s stoniens etant mutuellement

339

L'interualle en the'orie des relations s t o n e - d i s t i n c t s , es t u n e f a m i 1l e l o c - i n t e r v a l l a i r e . u

L'aritg du lattis boolcien A 6tant 2, tous nos ultrafiltres sont A-loc-interval-

laires. I1 suffit donc de considGrer le cas de deux ultrafiltres non triviaux F,

G. Par hypothsse il existe un u avec l'intervalle initial F des G u , 6lGment de F , et l'intervalle initial comparable (mod

A)

G

des S l u , GlEmrnt de G. Tout ElEment de F est donc in-

2 tout Gl6ment de G. Donc la r6union d'un automorphisme local

quelconque de A / F et d'un de A / G est un automorphisme local de A. Finalement on tient compte d c l'ensemble fini H de 7.2 en remplaqant F par F pour G .

-

(F

fl H)

; de insme

0

Bien que, p o u r L ' a r i t k 2, tout ultrafiltre soit loc-intervallaire, il est faux que t o u t e n s e m b l c d ' u l t r a f i l t r e s soit l o c - i n t e r v a l l a i r e .

Par exemple, partons de la

cha'ine Q des rationnels, qui est dense, sans minimum ni maximum ; prenons deux ultrafiltres F,G

ainsi dsfinis. Pour chaque partition de l'ensemble des ration-

nels e n un intervalle initial non vide et un intervalle final, prenons, dans F c o m e dans G , l'intervalle initial. Plus ggnEralement, pour chaque partition, l'une des parties compl6mentaires au mains est coinitiale B Q, et nous la prenons si elle est la seuls des deux qui soit coinitiale. Nsanmoins nous assurons la diff6rence entre F et G en prenant par exemple l'ensemble des rationnels binaires dans F et son complgment dans G. I1 est alors Gvident que deux 616ments F de F et G de G sont coinitiaux de Q , donc qu'il existe un automorphisme local de Q/F transformant un Glement u de F en v < u ,

et un w de G intermsdiaire : la transformation de u,w

en v,w n'est pas un automorphisme local de Q. 7.3. Cl6ture d'une relation selo n une famille loc-intervallaire

Partons d'une relation A de base E et d'une famille A-lac-intervallaire d'ultrafiltres. Nous pouvons alors reprendre l'argumentation du 6.2 ; autrement dit, associer B

chacun des ultrafiltres non triviaux de la famille, soit F , un

ensemble E ( F ) et une relation A ( F ) de base E ( F ) ; les E ( F ) stant mutuellement disjoints, et disjoints de E. Comme en 6.2 il y a ambiguit6 pour le choix du cardinal de chaque E ( F ) (qui est arbitraire, nul, fini ou infini). I1 y a encore une ambiguits pour le choix de chaque A

(F), bien que toute restriction finie de A ( F )

doive n6cessairement figurer parmi l e s restrictions communes aux A / F pour tous les 616ment F de F . Mais, c o m e en 6.2, u n e f o i s l e s A ( F ) c h o i s i e s , i l ne s u b s i s t e p l u s aucune a r n b i g u i t b p o u r a c h e v e r d e c o n s t r u i r e l a c l d t u r e A

+

sur la r6union de E et

des E ( F ) . Tout c o m e en 6.2, les A (F) deviennent des intervalles de la clature A'. Bien que cela ne semble pas essentiel, notons qu'il est toujours possible, par le th6orSme de RAMSEY, d'exiger que chaque A ( F ) soit enchalnable, i.e. libre-interpr6table e n une chaTne arbitrairement donn6e sur la base E ( F ) . 7 . 4 . Pour achever la jonction avec le th6orSme de reprgsentation de Stone,

associons, 1 un lattis bool6en donn6 A , sa cl6ture A+ dEfinie c o m e suit. Nous prenons tous les ultrafiltres triviaux, p l u s les ultrafiltres non triviaux stoniens

R. Fra'isse

3 40

F.

Chaque

F est d6fini en dQcidant, pour chaque 616ment

u # 0 , si l'intervalle

initial ,< u , o u l'intervalle initial < l u , rst 616ment de

F.

Un unique

F 6tant

choisi c o m e reprssentant pour chaque fonction possible de dgcision, les F sont mutuellement Stone-distincts. A chaque F stonirn non trivial, associons un E ( F ) , donc un A ( F ) singleton, les A (F) 6tant mutuellement incomparables, c o m e il r6sulte de 7.2.

Par contre, entre un A ( F ) non trivial rt un 616ment u de E, nous

avons soit A ( F ) < u (si, au dgpart, l'intervallr initial


6tait un 616ment de

F ) , soit A ( F ) incomparable 5 u (si, au dgpart, l'intervalle initial < l u 6tait un 616ment de F ) . Cela achsve de d6finir la cl8ture At de A. Evidemment, dans la repr6sentation classique de Stone, on supprime de A+ tous les 616ments de E, except6 les 616ments minimaux (immsdiatement sup6rieurs au 0), s'il en existe. Chaque G16ment u ainsi supprim6 est remplac6 par l'ensemble des A ( F ) < u,et des 616ments minimaux < u avec la relation d'inclusion c pour remplafer

< (mod ,)'A

la rQunion

pour remplaGer le supremum, l'intersection pour remplafer l'infimum. Note 1 . Enonce du lemme de coherence (voir par exemple FRA'ISS6 1972 ( 1 ) Soit I un ensemble d'ensembles F sur chacun desquels nous avons un ensemble p. 52). fini non vide UF de relations de base commune F , et d'arit6 commune, avec les deux hypothsses suivantes : (I)

I filtrant : si F , F '

(2) si F,G E I et F

2

E

I

il existe un G E I avec G

_>

F U F' ;

G , toute relation 6lQment de UG, une fois restreinte B F ,

donne un Qldment de UF ; alors il existe une relation R de l'arit6 commune donnQe, bas6e sur la reunion des F , et telle que pour chaque F , la restriction R/F EUF.

Ce lenime 6quivaut 2 l'axiome de l'ultrafiltre. dans le cas gQn6ral ; mais ZF suffit lorsque les F sont finis et leur rsunion dsnombrable. Note 2. La notion usuelle d'intervalles contigus

se

g6n6ralise de manisre

naturelle : deux A-intervalles sont dits c o n t i g u s lorsque leur r6union est un A-intervalle. Le lecteur d6finira ainsi 1'616ment adhPrent cl6ture topologique d'un A-intervalle.

D un A-intervalle, donc la

34 1

Istvan FOLDES, 1973

-

R e l a t i o n s denses et d i s p e r s e e s

;

extension d ' m t h b o r e m e d e H a u s d o r f f .

C o m p t e s r e n d u s A c a d . S c i . P a r i s , vol 2 7 7 A , p . 2 6 9 - 2 7 1 .

Roland FRAYSSA, 1971-72Cours d e l o q i q u e mathematique, ( 1 ) R e l a t i o n et f o r m u l e l o g i q u e , 197 p . : ( 2 ) T h e o r i e d e s m o d e l e s , 1 7 7 p . : P a r i s ( G a u t h i e r - V i l l a r s ) : trad.

1973-74-

C o u r s e of m a t h e m a t i c a l l o g i c , D o r d r e c h t ( R e i d e l ) .

1977- P r e s e n t p r o b l e m s a b o u t i n t e r v a l s i n r e l a t i o n - t h e o r y

and l o g i c .

Non c l a s s i c a l l o g i c s , m o d e l t h e o r y a n d c o m p u t a b i l i t y , vol 8 9 d e S t u d i e s

i n L o g i c , Amsterdam (North-llolland).

David GILLIAM, 1978 A

-

concrete r e p r e s e n t a t i o n t h e o r e m f o r i n t e r v a l s o f m u l t i r e l a t i o n s .

Z e i t s c h r i f t M a t h . L o g i k vol 2 4 p . 4 6 3 - 4 6 6 . 1 9 7 9 - I n t e r v a l s o f b i n a r y r e l a t i o n s . I b i d e m vol 2 5 p . 5 7 - 6 0 .

Gerard L O P E Z , 1978 L ' i n d e f o r m a b i l i t e d e s r e l a t i o n s et m u l t i r e l a t i o n s b i n a i r e s . Z e i t s c h r i f t M a t h . L o g i k vol 2 4 p . 3 0 3 - 3 1 7 .

Maurice POUZET, 1979

-

R e l a t i o n m i n i m a l e p o u r son d g e . Z e i t s c h r i f t M a t h . L o g i k vol 2 5 p . 3 1 5 - 3 4 4 . 1981- R e l a t i o n s i n p a r t i b l e s . D i s s e r t a t i o n e s Mathematica

, vol 1 9 3 , p . 1 - 4 3 .

This Page Intentionally Left Blank

-

PART VI ORDER AND CHAIN CONDITION ON CLASSES OF STRUCTURES

- PARTIE VI ORDRE ET CONDITIONS DE CHAINE SUR DES CLASSES DE STRUCTURES

P.D. SEYMOUR

-

Neil ROBERTSON

Some new results on the well-quasi ordering of graphs.

.... . ... .

p . 343

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Annals of Discrete Mathematics 23 (1984) 343-354 0 Elsevier Science PublishersB.V. (North-Holland)

343

SOME NEW RESULTS ON THE WELL-QUASI-ORDERING OF GRAPHS

P.D. SEYMOUR* and Neil ROBERTSON Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, Ohio 43210 U.S.A.

D e d i c a t e d t o Professor E . COROMINAS.

RESUME.-

Quelques resultats nouveaux sur la relation de be1 ordre des graphes finis.

Designons par 9 l'ensemble des graphes finis, et, lorsque G, H € 9 , Bcrivons H E G si H est isomorphe a un mineur de G, c'est-8-dire a une contraction d'un sous-graphe partiel de G . Dans les annees 1960, K. WAGNER a fait la conjecture (non publiee) que 9 est belordonne par cette relation d'inclusion. Pour des graphes finis, cela signifie que dans tout sousensemble infini un certain graphe sera isomorphe a un mineur donne d'un autre graphe. Recemment, les auteurs de cet article ont demontre des theorsmes qui laissent fortement presager que la conjecture de WAGNER est correcte. Notre methode consiste ?I montrer que l'ensemble des graphes G qui ne contiennent pas un certain graphe fixe H comme mineur a une structure definie, et de montrer alors, qu'etant donnee cette structure, l'ensemble des graphes qui la possedent est belordonnd. Si on savait le faire pour tout graphe H, il en decoulerait que ( 3 5 ) est lui-meme belordonnd, donc que la conjecture de WAGNER est vraie. La structure de graphes mentionnde p l u s haut est construite en trois Btapes: (1) plonger un graphe dans une surface S ; ( 2 ) adjoindre au plus n sommets, qui peuvent etre adjacents a n'importe quels des autres sommets, et (3) joindre les unes aux autres des copies disjointes de tels graphes en une structure d'arbre. Ici, S et n sont determines par le graphe H exclu, et les graphes adjacents dans la structure d'arbre formant G sont joints en identifiant des sousensembles de leurs sommets d'une maniere univoque. De tels sous-ensembles ne peuvent intersecter la surface de plongement S q u e trivialement, c'estA-dire au p l u s en trois sommets contenus dans la frontisre d'une face du plongement. Ce programme a maintenant BtB execute pour des classes de graphes G qui ne contiennent pas c o m e mineur un graphe planaire fixe H; c'est un cas oh il

n'est pas necessaire d'utiliser la structure de la surface. Pour tout graphe planaire H, il existe un entier w tel que tout graphe G qui ne

*

Partially supported by National Science Foundation grant number MCS 8103440

.

P.D. Seymour, N. Robertson

344

contient pas H comme mineur peut &tre construit en joignant en une structure d'arbre des graphes avec au plus w sommets. De plus, on montre que les classes de graphes G qui peuvent etre construites de cette mani6re sont belordonnees; dans cette demonstration on utilise une generalisation du th6orhe de KRUSKAL sur la structure d'arbre, que l'on prouve pour des structures d'arbre de certains hypergraphes structures (familles d'ensembles) de dimension bornee. Les consequences de ce travail sont:

(1) que toutes les anti-chaines (ensembles d'elements non relies deux B deux) dans ( 9 , C_ ) qui contiennent au moins un graphe planaire H sont finies, et ( 2 ) qu'il existe un algorithme pour determiner la relation

H

5G

(H est isomorphe B un mineur de GI lorsque H est un graphe planaire fixe et G E 9 , avec temps de calcul born6 par des polyndmes de la dimension de G . Nous pensons que nos methodes nous permettront de demontrer la conjecture generale structurelle ainsi qu'un theoreme de be1 ordre, avec les deux memes consequences qu'auparavant sans la condition que H soit planaire, c'est-5dire de demontrer la conjecture de WAGNER et d'obtenir un bon algorithm pour determiner la relation H 5 G pour tout G E 9 . ABSTRACT.It has been conjectured by K. WAGNER that finite graphs are wellquasi-ordered by minor inclusion, i.e. being isomorphic to a contraction of a subgraph. A method is reported on here that shows promise of settling this conjecture. We have proved (1) that all graphs G not including a fixed planar graph H as a minor can be constructed by piecing together graphs on a bounded number of vertices in a tree-structure, and (2) by elaborating the KRUSKAL tree theorem that the class of graphs formed by piecing together graphs of bounded size in treestructures is well-quasi-ordered. It follows from this that no infinite antichain of finite graphs can include even one planar graph and that there is a "good" algorithm for testing the presence of a fixed planar graph as a minor.

Background. Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation < is defined. If x , y E Q and x < y , we say that y includes x . We write x < y if x < y % x . A subset S of Q is (i) an antichain if there do not exist distinct elements, x, y E S such that x < y , (ii) an upper order ideal if y E S whenever x < y for some x E S , (iii) a lower order ideal or lower ideal if y E S whenever y < x f o r some x E S . An upper order ideal S is finitely generated if it has a finite subset F such that S = {x E Q : f < x for some f E F} . It is an easy exercise to prove that the following conditions on a quasi-ordered set Q are equivalent: (a) all upper order ideals of I) are finitely generated, (b) in Q there is neither an infinite antichain nor an infinite descending chain (i.e. an infinite ), sequence x2, such that x > x2 > x3 > (c) for every infinite 1.

'

sequence and

XI'

xi < x.

. ..

x*,

.

...

...

1

of elements of

Q

there exist

i, j

such that

i< j

A quasi-ordered set satisfying any of the equivalent conditions

1 (a), (b), (c) is said to be well-quasi-ordered.

Well-quasi ordering of graphs

345

2.

Graph I n c l u s i o n R e l a t i o n s . Examples o f w e l l - q u a s i - o r d e r e d s e t s a r i s e n a t r i r a l l y i n a l g e b r a , g r a p h t h e o r y and l o g i c . S i n c e t h e 1940's, a b r a n c h of o r d e r e d s e t t h e o r y h a s developed from t h e s t u d y of w e l l - q u a s i - o r d e r i n t h e s e f i e l d s . The a u t h o r s of t h i s p a p e r have r e c e n t l y made p r o g r e s s on t h e open g r a p h t h e o r e t i c a l problems and r e p o r t h e r e on t h e r e s u l t s t h e y have o b t a i n e d . Denote by 9 t h e c l a s s o f f i n i t e g r a p h s . Under subgraph c o n t a i n m e n t 9 i s n o t w e l l - q u a s i - o r d e r e d , a s i s shown by t h e i n f i n i t e a n t i c h a i n o f f i n i t e c i r c u i t s . Most o f t h e t h e o r y o f g r a p h w e l l - q u a s i - o r d e r i n g c o n c e r n s t h r e e c l o s e l y r e l a t e d q u a s i - o r d e r s on 9 o f a t o p o l o g i c a l n a t u r e . To d e f i n e t h e s e r e l a t i o n s w e i n t r o d u c e some n o t a t i o n . Suppose G E 3 and X 5 V ( G ) A simple path i n G i s a p a t h w i t h a t l e a s t one edge and no r e p e a t e d v e r t i c e s , e x c e p t p o s s i b l y i t s e n d v e r t i c e s . The g r a p h of a s i m p l e p a t h ( i . e . t h e subgraph o f G d e t e r m i n e d by i t s v e r t i c e s and e d g e s ) w i t h i t s e n d v e r t i c e s i n X and no o t h e r v e r t i c e s i n X i s c a l l e d a n X-join, and i s a n x - l i n k o r an X-loop i f t i s endvertices a r e , r e s p e c t i v e l y , d i s t i n c t or e q u a l . Then P(G, X ) d e n o t e s t h e g r a p h w i t h vertex-set X , e d g e - s e t t h e s e t o f X - j o i n s o f G , and i n which e a c h X-join i s i n c i d e n t with i t s e n d v e r t i c e s i n X

.

.

X-joins i n G t h e n a c h o i c e f u n c t i o n f o r P i s a mapping s u c h t h a t f ( P ) E E ( P ) f o r a l l P E !J' Given such a c h o i c e f : f' + E ( G ) w i t h e n d v e r t e x x E X , d e n o t e by P f ( x ) t h e segment o f f u n c t i o n and P E 9) If

P

from

9' i s a s e t o f

x

.

t o t h e f i r s t vertex of

may g e n e r a t e t o t h e s i n g l e v e r t e x

i n c i d e n t with

P

f(P)

.

Note t h a t

Pf(x)

.

x

The g r a p h P ( G , X) o f X-joins can be used t o def i ne t h e i ncl usi on r e l a t i o n s f o r w e l l - q u a s i - o r d e r i n g i n a u n i f i e d manner. H ' 5 P(G, X ) f o r any X 5 V ( G ) i s embedded i n G when t h e X-joins o f E ( H ' ) meet p a i r w i s e o n l y a t common e n d v e r t i c e s . Then a g r a p h H i s c o n t a i n e d i n G by embedding i n c l u s i o n , w r i t t e n H <e G

A graph

,

when H i s i s o m o r p h i c t o a g r a p h H' embedded i n G. A graph

H'

5 P(G,

X)

f o r any

X

5 V(G)

i s immersed i n G when t h e

X-joins of E ( H ' ) have p a i r w i s e no common e d g e s . Then a g r a p h H i s c o n t a i n e d i n G by immersion i n c l u s i o n , w r i t t e n H
i s isomorphic t o a g r a p h A graph

H'

5 P(G,

X)

H'

immersed i n

f o r any

X

5 V(G)

G.

i s a minor of G when t h e r e

i s an i n j e c t i v e choice f u n c t i o n f : E ( H ' ) + E ( G ) such t h a t i f are i n c i d e n t w i t h P, Q E E ( H ' ) , r e s p e c t i v e l y , t h e n p, q E V(H') p # q i m p l i e s P f ( p ) , Q (9) are d i s j o i n t . Then a g r a p h H i s f c o n t a i n e d i n G by minor i n c l u s i o n , w r i t t e n H <m G , when H i s isomorphic t o a minor

H'

of

G

.

I n a l l t h r e e o f t h e above d e f i n i t i o n s t h e i n j e c t i v e mapping onto V ( H ' ) d e t e r m i n e d by t h e isomorphism between H and H' i s c a l l e d vertex-mapping of t h e i n c l u s i o n r e l a t i o n . I t is r o u t i n e , b u t n o t a l t o g e t h e r t r i v i a l , t o v e r i f y t h a t t h e s e t h r e e i n c l u s i o n r e l a t i o n s are q u a s i o r d e r s . An e q u i v a l e n t , and more s t a n d a r d way t o d e f i n e t h e minor i n c l u s i o n r e l a i s as f o l l o w s . F o r two g r a p h s , G , H w e s a y t h a t H 4 G i f f o r t i o n <m

a

: V(H) + V(G)

m

each x E V ( H ) w e c a n s e l e c t a c o n n e c t e d subgraph f ( x ) of G and f o r e a c h e E E ( H ) w e c a n select a n edge f ( e ) o f G i n s u c h a way t h a t ( i )t h e s u b g r a p h s f ( x ) ( x E V ( H ) ) o f G are p a i r w i s e d i s j o i n t , ( i i )t h e e d g e s f ( e ) ( e E E ( H ) ) of G are d i s t i n c t and ( i i i ) i f x and y a r e t h e v e r t i c e s j o i n e d by a n

P.D. Seymour, N Robertson

346

edge e in H , then f(e) joins a vertex of f(x) to a vertex of f(y). The equivalence of these two definitions of minor inclusion is also easy to check. It is worth pointing out that the most usual way of producing a minor of G is by a succession of three elementary operations (a) single-edge deletion, (b) single-edge contraction, and (c) removal of an isolated vertex. Indeed, to verify a property is inherited by the minors of a graph it is only necessary to check that it is preserved by these simple operations. We make a final observation that embedding inclusion is the special case of both immersion and minor inclusion where the X-joins P E E ( I 1 ' ) meet only at common endvertices. In practice our three inclusion relations are often extended to apply to graphs with vertices labelled from some well-quasi-ordered set ( Q , < ) . Let 9 5 and ( Q ) = {(G, !L) : G E 9 and !L : V(G) Q is a function). -f

If <x

denotes one of our graph inclusion relations then

means that G <x G' v -+ v'

such that

behaved under

<,

(G, !L) <x

(GI, ! L ' )

and an inclusion isomorphism exists with vertex-mapping k?(v)

<

!L'(v') for all

v E V(G)

.

We say that 3 is well-

when for every well-quasi-ordered set

(Q,<)

F ( Q ) is well-quasi-ordered under the extensed inclusion <x

the family

. Note

that

because the graphs involved are finite and their vertex-labels are taken from a well-quasi-ordered set , all three extended inclusion relations are without infinite descending chains. Thus graph well-quasi-ordered follows from the finiteness condition on antichains alone.

3.

The Context. The standard theorems about graph well-quasi-ordering concern embedding inclusion. It seems (see Krustal [31) that the subject of wellquasi-ordering began with two conjectures of A. Vazsonyi, that under embedding inclusion: (A) finite trees are well-quasi-ordered, and (B) graphs with vertex-valencies < 3 are well-quasi-ordered. The major theorem in this context, which settles Vazsonyi's conjecture (A), is Kruskal's theorem [21 that finite trees are well-behaved under embedding. A similar deep theorem for infinite trees, under the stronger better-quasiordering property, was proved by C. St. J. A. Nash-Williams [7] and Richard Laver [ 4 1

.

W. Mader [ 5 1 generalized Kruskal's theorem to sets of finite graphs with bound k on the number of pairwise edge-disjoint circuits that the graphs can contain. Erdds and Posa [l] had earlier shown that for some integer k' , any such graph can be reduced to a forest by deleting at most k' vertices. Mader proved that any family of finite graphs whose members differ by a bounded number of vertices in this way from the graphs in a well-behaved family must also be well-behaved. By Kruskal's theorem forests are well-behaved, and so the result of Mader follows. However, Mader's theorem cannot be greatly extended, at least for families of graphs containing graphs with arbitrarily many vertices of valency > 4 because of a counterexample. The set of graphs obtained from circuits by doubling each edge in parallel is an infinite antichain in ( 3 , %e'

-

The two main conjectures to avoid this difficulty work by weakening the inclusion relation to allow more inclusions so that the above counterexample does not apply. Nash-Williams conjectured in 171 that: (C) ( '$2,
Wellquasi ordering of graphs

(D)

(

9 ,4 m )

347

is well-quasi-ordered.

I n c l u s i o n s a r e t h u s c r e a t e d between doubled c i r c u i t s of d i f f e r e n t s i z e by a l l o w i n g , i n two d i s t i n c t ways, t h e p a t h s c o r r e s p o n d i n g t o t h e embedded edges t o i n t e r t w i n e . I t i s n a t u r a l t o s t r e n g t h e n c o n j e c t u r e s ( C ) and (D) t o a s s e r t t h a t 9 i s well-behaved under and <m , r e s p e c t i v e l y . Embedding
and immersion i n c l u s i o n do n o t d i f f e r on t h e c l a s s of f o r e s t s , w h i l e minor i n c l u s i o n i s a somewhat s t r o n g e r r e l a t i o n . For example, t h e t r e e shaped l i k e t h e l e t t e r X i s t h e minor o f t h e t r e e shaped l i k e t h e l e t t e r H formed by c o n t r a c t i n g t h e c e n t r a l e d g e . Thus K r u s k a l ' s theorem v e r i f i e s t h a t t h e c l a s s of f o r e s t s i s well-behaved under a l l t h r e e i n c l u s i o n r e l a t i o n s . The i n c l u s i o n r e l a t i o n a l l a g r e e on t h e c l a s s of g r a p h s w i t h v e r t e x - v a l e n c i e s < 3 i n v o l v e d i n V a z s o n y i ' s open c o n j e c t u r e (B), which t h u s forms a common s p e c i a l c a s e of c o n j e c t u r e s ( C ) and (D).

4.

Embedding S t r u c t u r e s . I n o u r o p i n i o n , t h e s e somewhat d i f f e r e n t c o n j e c t u r e s o f Nash-Williams and Wagner a r e e q u a l l y l i k e l y v a l i d and l i k e l y b o t h c o r r e c t . One p o s s i b l e approach t o t h e i r p r o o f i s t o show t h a t a l l f i n i t e g r a p h s G which do n o t i n c l u d e a f i x e d g r a p h H admit a p a r t i c u l a r t y p e o f s t r u c t u r e , and t h a t g r a p h s h a v i n g t h i s s t r u c t u r e do n o t form i n f i n i t e a n t i c h a i n s . Of c o u r s e , t h e s t r u c t u r e f o r c e d on G by e x c l u d i n g H would v a r y w i t h t h e i n c l u sion r e l a t i o n involved. a graph

An example o f s u c h a s t r u c t u r a l t y p e r e s u l t i s K u r a t o w s k i ' s theorem t h a t G embeds i n t h e p l a n e i f and o n l y i f n e i t h e r K n o r K5 embed i n

3,3

G. W e p o i n t o u t t h a t t h e c l a s s of p l a n a r g r a p h s i s c l o s e d under t a k i n g m i n o r s , from embedding i n G i s e q u i v a l e n t and t h a t t h e dominant e x c l u s i o n o f K

3,3 t o e x c l u d i n g i t a s a minor. I n f a c t , a c o n n e c t e d g r a p h

which d o e s n o t have

G

a s a minor e i t h e r i s p l a n a r or i t c a n be s t r u c t u r a l l y e x p r e s s e d a s 3,3 Given H _C G , d e f i n e and c o n s e q u e n t l y o b v i o u s l y h a s an embedded K5

K

W ( G , H) = {x E V ( H )

:

x i s i n c i d e n t w i t h some edge o f

t h a t G can b e structurally expressed a s

x2

, x3 ,

H. , 1.1

x4

,

x5

of

K5

n o t i n HI

G

.

'

Kg

.

W e say

i f there e x i s t five vertices

x

1 '

and t e n p a i r w i s e e d g e - d i s j o i n t c o n n e c t e d s u b g r a p h s

G

such t h a t

( a ) W ( G , Hi (b)

,) I3

= {xi,

U H. l < i < j < 5

for

1

<

i

<

j

<

,

5

and

. = G

The e f f e c t o f e x c l u d i n g

K

a s an embedded g r a p h i s more c o m p l i c a t e d

K5

< 5

since i t l e a v e s a l l g r a p h s w i t h some n o n p l a n a r g r a p h s l i k e

,

x.) 7

4,4

v e r t i c e s of valency

>

4

,

as w e l l

which have enough v e r t i c e s of v a l e n c y

accommodate t h e embedding. Moreover, e x c l u d i n g

K5

as

>4

a s a n immersed g r a p h l e a v e s

t h e c l a s s o f g r a p h s w i t h n o t enough v e r t i c e s o f v a l e n c y

>

4

,

b u t it d o e s n o t

l e a v e a l l p l a n a r g r a p h s . F o r example, any p l a n a r g r a p h w i t h t h e o c t a h e d r o n K a s an immersion. I n b o t h t h e s e cases t h e s t r u c t u r e o f t h e 5 g r a p h s n o t i n c l u d i n g K5 is w e l l hidden.

embedded h a s

On t h e o t h e r hand, i f

K5

to

i s e x c l u d e d as a m i n o r , t h e n a t h e o r e m - o f

P.D. Seymour, N. Robertson

348

Wagner 1161 s t a t e s t h a t t h e graphs which remain c a n be b u i l t up from d i s j o i n t (formed be adding t o an p l a n a r g r a p h s and c o p i e s o f a s p e c i f i c graph V8

octagon t h e f o u r edges j o i n i n g o p p o s i t e v e r t i c e s ) by r e p e a t e d clique-summing. I n clique-summing, t w o graphs a r e j o i n e d by choosing complete subgraphs of t h e same s i z e i n each, i d e n t i f y i n g t h e i r v e r t i c e s i n a one-to-one f a s h i o n , and d e l e t i n g t h e i r edges. T h i s produces a " t r e e - s t r u c t u r e ' ' of t h e corresponding p l a n a r graphs and V ' s , w i t h each component graph s e p a r a t i n g t h e component 8 graphs l y i n g i n d i f f e r e n t branches o f t h e " t r e e - s t r u c t u r e " . We w i l l r e t u r n t o t h i s s t r u c t u r e l a t e r i n a more p r e c i s e way, b u t remark now t h a t t h i s theorem i s a p r o t o t y p e of t h e s t r u c t u r a l r e s u l t s we o b t a i n and are c o n j e c t u r i n g . I t i s t h u s n a t u r a l t o postpone work on t h e s t r u c t u r e r e s u l t i n g from embedding o r immersion e x c l u s i o n of a f i x e d graph u n t i l t h e s i m p l e r s t r u c t u r e a r i s i n g from minor e x c l u s i o n i s w e l l understood. Wagner's c o n j e c t u r e (D) s t a t e s t h a t any upper o r d e r i d e a l i n

( 9 , <m)

i s g e n e r a t e d by a f i n i t e l i s t of minimal graphs. These graphs are a l s o c a l l e d t h e minimal excluded minors f o r t h e complementary lower o r d e r i d e a l . From t h e a v a i l a b l e evidence it seems r e a s o n a b l e t o f u r t h e r c o n j e c t u r e t h a t :

(E) any lower o r d e r i d e a l o f

(

9 ,< m )

i s c h a r a c t e r i z e d by a f i n i t e

list o f "embedding s t r u c t u r e s " which e x a c t l y g e n e r a t e i t i n t h e way p l a n a r graphs a r e g e n e r a t e d by t h e 2-sphere. A s a t i s f a c t o r y s o l u t i o n of c o n j e c t u r e s (D) and ( E ) should i n c l u d e e f f e c t i v e

a l g o r i t h m s f o r t e s t i n g minor i n c l u s i o n and s t r u c t u r e embedding. That i s : (F) given a graph H € 9 ( o r r e s p e c t i v e l y , a n embedding s t r u c t u r e .Z) t h e r e s h o u l d b e c o n s t a n t s c , r depending on H (or X ) a l o n e , such t h a t f o r any G E 9 i t can b e determined i n


Ir

IV(G) elementary s t e p s whether o r n o t G embeds i n X I

.

H

is a minor o f

G

Although we can v e r i f y many s p e c i a l c a s e s o f t h e above "exact embedding s t r u c t u r e c o n j e c t u r e " (E), and t h u s have a v a r i e t y of examples and an a p p r e c i a t i o n o f t h e i r g e n e r a l f e a t u r e s , w e have n o t y e t been a b l e t o c o n s t r u c t o r d e f i n e t h e embedding s t r u c t u r e s a p p r o p r i a t e t o c h a r a c t e r i z e a r b i t r a r y p r o p e r lower ideals i n ( 9 , <m) W e imagine an "embedding structure" 7. t o consist of a

.

s p e c i f i c a t i o n of c e r t a i n s t r u c t u r a l p r o p e r t i e s and t h e embedding r u l e s under which a graph h a s t h e s e p r o p e r t i e s . S e v e r a l e s s e n t i a l f e a t u r e s are c l e a r : ( i ) t h e class o f graphs

be a lower i d e a l

9, which embed i n a given s t r u c t u r e

1

should

( i . e . t h e lower i d e a l g e n e r a t e d by t h e s t r u c t u r e ) ;

(ii) a s t r u c t u r e s h o u l d be s u f f i c i e n t l y homogeneous t h a t t h e r e are o n l y

f i n i t e l y many e s s e n t i a l l y d i f f e r e n t ways t o embed a given g r a p h , and (iii) a s t r u c t u r e s h o u l d n o t simply embed a l l of

9

;

some graph s h o u l d

n o t embed. These t h r e e c o n d i t i o n s would allow us t o s t u d y t h e p r o p e r lower i d e a l s i n a c o n s t r u c t i v e f a s h i o n . Regarding c o n d i t i o n (iii) w e note t h e c u r i o u s f a c t ( s e e page 52 of [ 1 7 ] ) t h a t a l l G E 9 embed t o p o l o g i c a l l y [as w i t h p l a n a r embedd i n g s ) i n t h e s t r u c t u r e formed by i d e n t i f y i n g t h r e e 2-dimensional d i s k s a l o n g a common arc of t h e i r b o u n d a r i e s . An important development i n t h i s s t u d y w a s t h e r e a l i z a t i o n t h a t many o f t h e a l g o r i t h m i c and t h e well-quasi-ordering consequences e x p e c t e d from a

349

Well-quasi ordering of graphs

( 3 ,< m )

s t r u c t u r a l c h a r a c t e r i z a t i o n of t h e p r o p e r lower i d e a l s i n

would

s t i l l f o l l o w p r o v i d e d enough embedding s t r u c t u r e s c o u l d b e c o n s t r u c t e d t o c o n t a i n ( p e r h a p s p r o p e r l y ) a l l t h e p r o p e r lower i d e a l s . Suppose t h a t s u c h a f a m i l y S = ( E : i E I } of embedding s t r u c t u r e s h a s been g i v e n ; and f o r e a c h i

,

i E I

that:

a r e shown t o be w e l l - q u a s i - o r d e r e d

by

<m

’Zi

the

(’)

(2) t h e r e i s an e f f e c t i v e a l g o r i t h m t o t e s t whether

G

1

€9

over a l l

‘i G

(3)

,

E 9

g i v i n g an embedding when one e x i s t s , and

f o r each whether

E 9

H

t h e r e i s an e f f e c t i v e a l g o r i t h m f o r t e s t i n g ‘i G over a l l G f 9

.

<

H

‘i

I t i s t h e n s t r a i g h t f o r w a r d t o v e r i f y Wagner’s c o n j e c t u r e (D), and c o n j e c t u r e (F) t h a t t h e r e a r e e f f e c t i v e a l g o r i t h m s t o t e s t ( a ) f o r any f i x e d H E 9 whether H <m G , o v e r a l l G € 9 , and ( b ) f o r any f i x e d lower i d e a l

9 5 9 whether

9=

G

€7 ,

over a l l

E

G

.

9

.

To o b t a i n (D) suppose 0 5 9 i s a n a n t i c h a i n and choose H E 0 Let Now 0 - { H I 5 9 by t h e d e f i n i t i o n o f an a n t i c h a i n , € 9 : H 1;, G )

.

{G

5 5 9

and

s

on

f o r some i E I b e c a u s e 9 i s a lower i d e a l . Then c o n d i t i o n (1) ‘i i m p l i e s t h a t (3 - {H} i s f i n i t e and it f o l l o w s t h a t (3 i s f i n i t e .

3

.

E 9 : H +m G I

= {G

i E I

A s before,

9

H E

To o b t a i n ( F ) c o n s i d e r f i r s t p a r ( a ) . L e t

e x i s t s with

and

. Then

95 9 =i

G C % we t e s t whether

9

G E

then

G E

,

H <m G

,

and H e 9 ‘i ‘i by d e f i n i t i o n o f

9

u s i n g c o n d i t i o n ( 2 ) on

5

.

If

G

E 9

‘i

then

m

,

G E

If

obviously.

G

9 ‘i

.

s

u s i n g c o n d i t i o n ( 3 ) on

and

p a r t (b). For any p r o p e r lower i d e a l

9

.

5 9

G

EY

f 5

.

Suppose

H

9

E

G E

. If

H

E 9

and

‘i This gives p a r t ( a ) .

3

t h e r e i s an G €

and t e s t whether

9

then

‘i a finite list

HI

G

,

H2

, ...,

When

G

E

9,

,

‘i

w e can t e s t whether

‘i

If

s

‘i

H 4

H <m G

given

i

E

9, L .

I

Consider then such t h a t using condition ( 2 ) .

condition (1) t e l l s u s there is

i Hh

of graphs i n

9

g e n e r a t i n g t h e upper

‘i order i d e a l of

, <m)

(g

complementary t o 3

.

But t h e n c o n d i t i o n ( 3 )

‘i

a l l o w s u s t o t e s t e f f e c t i v e l y whether d e c i d e whether

G E

9

.

Hi

<m G

for

i = 1, 2 ,

..., h

and t h u s

These r e d u c t i o n s make i t p o s s i b l e t o work w i t h w e l l - d e f i n e d s t r u c t u r e s w i t h o u t f o r m u l a t i n g a g e n e r a l d e f i n i t i o n o f embedding s t r u c t u r e . I n o u r i n v e s t i g a t i o n s w e add s t r u c t u r e t o a g r a p h G € 9 i n f o u r b a s i c ways:

(a)

by l a b e l l i n g t h e v e r t i c e s of

G

w i t h e l e m e n t s from som w e l l - q u a s i -

P.D. Seymour, N . Robertson

350

(el<),

o r d e r e d set

t o a g r a p h t o p o l o g i c a l l y enbedded o n a

(b)

by an isomorphism from surface S ,

(c)

by s p e c i f y i n g a s u b s e t X 5 V ( G ) t h e s u b g r a p h G - X , and

(d)

by decomposing (T, . F ) .

G

of bounded s i z e , and hence f i x i n g

i n t o subgraphs according t o a t r e e - s t r u c t u r e

G

For t h e most p a r t it i s q u i t e o b v i o u s how t h e s e s t r u c t u r e s behave under t h e t a k i n g o f minors so t h a t t h e c l a s s o f s t r u c t u r e s i s p r e s e r v e d .

tree-structure ( T , 9 ) on

A

E V(T)} o f subgraphs

9 = {G.

: i

(2) i f

i, j , k

rates

E V(T)

from

and

G c o n s i s t s o f a t r e e T and a f a m i l y G such t h a t ( 1 ) G = U G . ( i E V ( T ) ) and

of

l i e s on t h e

k

(i, j)-path i n

.

i E V(T)

in G Normally, t h e Gi for 3 t r i c t e d and a r e g i v e n s t r u c t u r e o f t h e i r own and t h e G.

G.

T

Gi

n

G

k sepaa r e further res-

for

G.

1 satisfy conditions relative t o t h i s structure.

{ i , j j E E(T)

then

The s t r u c t u r e theorem o f Wagner d i s c u s s e d e a r l i e r i s a r e l a t i v e o f K u r a t o w s k i ' s theorem and makes a g o o d i l l u s t r a t i o n o f t h e u s e o f g r a p h s t r u c t u r e . When (T, 9 )i s a t r e e - s t r u c t u r e on G w i t h F = {G, : i E V ( T ) } , d e n o t e by

t h e g r a p h formed from

G#,

of v e r t i c e s i n

# fl

that

Gi

G!

E V(T)

i, j

3

n

Gi

become a d j a c e n t f o r a l l

G ,

1

and copies o f

E

i

V(T)

K5

$,

# 17 G!

for 3 ensures t h a t

{m

G

K

5

i4

G,

B

$m G

K

when

G

.

This ensures

G

is planar.

n

Gj)

I

it f o l l o w s t h a t

, or

(2)

included i n

<

3

,

G I K3,3

i E V(T)} )

G

E 3

and

K5

bm G

,

such t h a t , f o r e a c h

#

embeds i n t h e p l a n e with e a c h Gi a face-boundary. The l a s t c o n d i t i o n

and so when

,

Wagner's

i s a t r e e - s t r u c t u r e of planar graphs

(T, {Gi :

V8

{ i , j ) E E;T) /V(Gi

5

then

and G

i n a strong sense. S p e c i f i c a l l y , if

e i t h e r (1)

Gi

K5

V8

, fim

V8

h a s a tree-structure

,

E E(T)

.

Clearly,

G

jl

{i,

i s a l w a y s ( a t l e a s t ) a complete g r a p h f o r a l l d i s t i n c t

theorem s t a t e s t h a t i f

then

by a d d i n g e d g e s so t h a t any n o n a d j a c e n t p a i r

G.

G

G

or

V8

is internally-4-connected G

and

is planar.

Here, i n t e r n a l l y - 4 - c o n n e c t e d means t h a t G i s a s i m p l e 3-connected g r a p h h a v i n g no e d g e - d i s j o i n t d e c o m p o s i t i o n s G = H U Ha i n t o s u b g r a p h s 1 H I , H2 w i t h /V(H1 n H = 3 and IE(H1) > 3 < IE(H2)I Thus G i s

2

)I

I

.

4-connected e x c e p t f o r p o s s i b l e t r i v a l e n t v e r t i c e s , and h a s no c i r c u i t s s m a l l e r than a t r i a n g l e .

Our s t r u c t u r a l c o n j e c t u r e , which we c a l l t h e " a p p r o x i m a t e embedding s t r u c t u r e c o n j e c t u r e " h a s a s t a t e m e n t s i m i l a r t o Wagner's theorem: (G)

Suppose H € 9 is g i v e n . Then a s u r f a c e S and a n i n t e g e r n e x i s t s u c h t h a t t h e f o l l o w i n g h o l d s t r u e . For any G € 3 w i t h H %m G t h e r e e x i s t s a tree-structure (T, {Gi : i E V(T)) ) such t h a t f o r each

i

E V(T)

there exist

(a)

Xi

5 V(Gi)

with

Well-quasi ordering of graphs

lXil

#

(G. 1

< 11

and ( b ) an embedding o f

- X.) n

G.

3

1

for

# -

G.

{ i , j } E E(T)

35 1 Xi

into

S

with each

i n c l u d e d i n a face-boundary.

We can e n s u r e t h a t H d o e s n o t embed on S , and so an i n t e r e s t i n g s p e c i a l case o c c u r s when H i s p l a n a r . The s u r f a c e d i s a p p e a r s and t h e c o n j e c t u r e r e d u c e s t o t h e f o l l o w i n g : If H i s a p l a n a r graph t h e r e e x i s t s a n i n t e g e r n such t h a t f o r any G €9 with H < G t h e r e e x i s t s a t r e e - s t r u c t u r e

m

(T,

{Gi

:

i E V(T)I )

a tree-structure

G

with

IV(Gi)

I
for all

i E V(T)

.

Thus

G

is just

of g r a p h s w i t h bounded s i z e .

We d e f i n e t h e t r e e - w i d t h of admits a t r e e - s t r u c t u r e (T, (G,

G : i

t o be t h e s m a l l e s t i n t e g e r n so t h a t with jV(Gi) < n + 1 for

I

E V(T)} )

.

a l l i E V(T) T h i s convention e n s u r e s t h a t f o r e s t s a r e t h e graphs w i t h treew i d t h < 1 , which seems t o be t h e n a t u r a l number t o c h o o s e . A t r e e - s t r u c t u r e i n which t h e t r e e i s a p a t h i s c a l l e d a p a t h - s t r u c t u r e . The p a t h - w i d t h of G i s defined s i m i l a r l y t o tree-width b u t using p a t h - s t r u c t u r e s i n p l a c e of t r e e structures.

5.

N e w theorems.

I n t h e p r e s e n t s t a t e o f t h i s program ( A p r i l , 1983) t h e work on embedding s t r u c t u r e s n o t i n v o l v i n g s u r f a c e s i s c o m p l e t e d . We p r o v e d i n t h e autumn of 1981 t h a t ( 1 ) ( s e e [ 9 1 ) i f a f i x e d f o r e s t is e x c l u d e d a s a minor then t h e r e m a i n i n g g r a p h s h a v e bounded p a t h - w i d t h , and ( 2 ) g r a p h s o f bounded path-width a r e w e l l - q u a s i - o r d e r e d . I n s p r i n g of 1982 w e gave ( 3 ) e f f e c t i v e a l g o r i t h m s f o r f i n d i n g a p a t h - s t r u c t u r e of bounded width and f o r t e s t i n g minor i n c l u s i o n o v e r c l a s s e s of g r a p h s with such s t r u c t u r e . T h i s showed t h a t it w a s r e a l i s t i c t o hope t o prove o u r b a s i c c o n j e c t u r e s . The c o r r e s p o n d i n g r e s u l t s f o r p l a n a r g r a p h s were by and l a r g e a l s o o b t a i n e d e a r l y i n 1982. Of c o u r s e , c h a r a c t e r i z a t i o n s o f p l a n a r i t y and p l a n a r embedding a l g o r i t h m s a r e well-known. W e showed ( 4 ) ( s e e [ l l ] ) t h a t e x c l u d i n g a p l a n a r g r a p h H a s a minor o f a p l a n a r g r a p h G implies t h a t G h a s treewidth bounded by a f u n c t i o n o f H , and i n t h e s p r i n g o f 1982 t h a t ( 5 ) t h e r e i s an e f f e c t i v e a l g o r i t h m f o r t e s t i n g embedding i n c l u s i o n (and h e n c e minor i n c l u s i o n ) w i t h i n any c l a s s of g r a p h s h a v i n g bounded t r e e - w i d t h . F i n a l l y , i n autumn o f 1982 w e gave (6) a n o n c o n s t r u c t i v e p r o o f t h a t a n e f f e c t i v e a l g o r i t h m e x i s t s f o r t e s t i n g g r a p h s G € 3 f o r a f i x e d bound on t h e i r t r e e - w i d t h s . These During autumn o f 1982 and w i n t e r o f algorithmic r e s u l t s a r e collected i n [ l o ] 1983 w e o b t a i n e d ( 7 ) well-behavedness r e s u l t s f o r p l a n a r g r a p h s under minor i n c l u s i o n , and ( 8 ) good a l g o r i t h m s f o r t e s t i n g embedding i n c l u s i o n H < G f o r

.

a f i x e d graph

H

, and

with

G

i n t h e class o f p l a n a r g r a p h s .

During t h e w i n t e r o f 1983 w e showed t h a t ( 9 ) (see [1211 sets of g r a p h s with bounded t r e e - w i d t h a r e w e l l - q u a s i - o r d e r e d . I n March o f 1983, t h e r e s u l t w a s o b t a i n e d t h a t ( 1 0 ) (see [ 1 3 ] ) i f H i s a f i x e d p l a n a r graph, t h e n any G € 3 with H < G h a s t r e e - w i d t h bounded by a f u n c t i o n o f H Thus w e had b o t h t h e

.

w e l l - q u a s i - o r d e r i n g r e s u l t t h a t no i n f i n i t e a n t i c h a i n o f f i n i t e g r a p h s c a n i n c l u d e a p l a n a r g r a p h , and t h e a l g o r i t h m i c r e s u l t t h a t any f i x e d p l a n a r g r a p h c o u l d b e found a s a minor i n a p o l y n o m i a l l y bounded number o f s t e p s . The main d i f f i c u l t y i n p r o v i n g t h e s t r u c t u r a l theorem ( 9 ) i n v o l v e d a lemma g i v i n g c o n d i t i o n s for t h e p r e s e n c e o f l a r g e g r i d s a s m i n o r s . A 0 - g r i d i s a simple graph with v e r t e x - s e t { C i , j) : i , j E 2:,i.., €I} } such t h a t ( i , j ) i s a d j a c e n t t o ( i ' , j ' ) i f and o n l y i f 1- 1 + Ij- j'I = 1 . W e were a b l e t o show t h a t ( 1 1 ) f o r a n y p o s i t i v e i n t e g e r 0 t h e r e e x i s t s an

{i:

P.D. Seymour, N . Robertson

352 integer

8' such t h a t i f

, G2

G1

a r e subgraphs o f

G E 9

c o n n e c t e d components a p i e c e , and a r e such t h a t each c o n n e c t e d component o f

meets each c o n n e c t e d component o f

G

2

then

G

8'

with a t l e a s t G

1

c o n t a i n s a & g r i d a s a minor.

Once t h i s theorem was e s t a b l i s h e d w e showed t h a t g r a p h s of s u f f i c i e n t l y l a r g e t r e e - w i d t h c o n t a i n subgraphs G 1, G2 each w i t h 8 ' connected components which p a i r w i s e i n t e r s e c t and t h u s produce a 8 - g r i d minor. Taking l a r g e enough any p l a n a r g r a p h H c a n be produced a s a minor, t h u s p r o v i n g ( 1 0 ) t h a t e x c l u d i n g H as a minor l e a v e s o n l y g r a p h s w i t h a f i x e d bound on t h e i r t r e e - w i d t h .

An i n t e r e s t i n g c o r o l l a r y t o t h e s e r e s u l t s was a g e n e r a l i z a t i o n of t h e Erdos-Posa theorem t h a t f o r any p o s i t i v e i n t e g e r k t h e r e i s a n i n t e g e r k ' such t h a t any graph w i t h o u t k d i s j o i n t l o o p s a s a minor (i.e. k disjoint polygons as s u b g r a p h s ) d i f f e r s by a t most k ' v e r t i c e s from a g r a p h w i t h o u t a s i n g l e l o o p a s a minor ( i . e . from a f o r e s t ) . W e can show, u s i n g t h e s t r u c t u r a l theorem (10) and p r o p e r t i e s of f a m i l i e s of s u b t r e e s of t r e e s , t h a t (12) f o r such t h a t any graph w i t h o u t any p o s i t i v e i n t e g e r k t h e r e is an i n t e g e r k ' k p a i r w i s e d i s j o i n t c o p i e s o f a f i x e d p l a n a r graph H a s a minor d i f f e r s by a t m o s t k' v e r t i c e s f r o m a graph w i t h o u t H as a minor. W e can e a s i l y embed 2 k ' t 1 c o p i e s of any n o n p l a n a r g r a p h on i t s s u r f a c e of minimum genus so t h a t t h e embedded g r a p h s meet p a i r w i s e o n l y a t ( f i n i t e l y many) i n t e r n a l p o i n t s o f t h e embedded e d g e s and no such i n t e r n a l p o i n t i s i n more t h a n t w o o f t h e c o p i e s . Then t h e union o f t h e embedded g r a p h s forms a g r a p h G on S , w i t h t h e common i n t e r n a l p o i n t s v e r t i c e s , which d o e s n o t embed t w o d i s j o i n t c o p i e s o f H ( b e c a u s e H i s n o n p l a n a r and S i s i t s genus s u r f a c e ) and r e q u i r e s removal v e r t i c e s t o e l i m i n a t e a l l embeddings o f H I t f o l l w s t h a t theorem of > k ' (12) f a i l s when H i s n o n p l a n a r .

.

6.

P o s s i b l e Consequences. W e a r e o p t i m i s t i c a b o u t s e t t l i n g t h e c o n j e c t u r e s a b o u t s t r u c t u r e (E), w e l l - q u a s i - o r d e r i n g (G) and a l g o r i t h m s (F) s t a t e d above. These would imply Wagner's c o n j e c t u r e ( D ) and V a z s o n y i ' s c o n j e c t u r e ( B ) . Another c o r o l l a r y would be t h e Lovasz. and Milgram [ 1 5 ] i n t e r t w i n i n g c o n j e c t u r e , a t l e a s t i n t h e form i n v o l v i n g minor i n c l u s i o n . T h i s c o n j e c t u r e s t a t e s t h a t :

(H) If

G1

and

many minimal

G2

a r e f i x e d graphs then t h e r e a r e o n l y f i n i t e l y

G E

3

which i n c l u d e b o t h

GI

and

G2

*

Another c o n j e c t u r e , o r t r i p l e o f c o n j e c t u r e s , which f o l l o w s from Wagner's c o n j e c t u r e r e g a r d s g r a p h c o n n e c t i v i t y . W e u s e t h e term vertex-kc o n n e c t e d t o mean k-connected i n t h e u s u a l s e n s e , c y c l i c - k - c o n n e c t e d t o mean t h e m a t r o i d d u a l t o k-connected (see [ E l ) and d u a l l y - k - c o n n e c t e d t o mean b o t h v e r t e x - k - c o n n e c t e d and c y c l i c - k - c o n n e c t e d (see [14], c h a p t e r 10). D e f i n e c o n t r a c t i o n i n c l u s i o n t o mean H < G when H i s isomorphic t o a graph o b t a i n e d from G by c o n t r a c t i n g e d g e s and a l l o w i n g e d g e - d e l e t i o n s o n l y i f a p a r a l l e l edge e x i s t s . Then:

(I) t h e r e a r e o n l y f i n i t e l y many minimal (a) (b) (c) Thus maximal c h a i n s

vertex-k-connected g r a p h s under c o n t r a c t i o n i n c l u s i o n , c y c l i c - k - c o n n e c t e d g r a p h s under embedding i n c l u s i o n , and d u a l l y - k - c o n n e c t e d g r a p h s under minor i n c l u s i o n . H < H < < Hh of "k-connected" g r a p h s under t h e O Z # above t h r e e v e r s i o n s of "k-connected" and i n c l u s i o n r e l a t i o n s < , can be s a i d t o be f i n i t e l y b a s e d ( i . e ; have o n l y f i n i t e l y many p o s s i b l e i n i t i a l t e r m s HO), and i n t h i s s e n s a l l "k-connected" g r a p h s c a n be c o n s t r u c t e d from t h e

...

f i n i t e set of minimal "k-connected" g r a p h s .

Well-quasi ordering of graphs The problem of showing ( 9

, <m)

353

i s well-behaved,

which i s t h e v e r s i o n

of Wagner's c o n j e c t u r e t h a t o u r methods s h o u l d s e t t l e , i n v o l v e s d e t e r m i n i n g t h e e x c l u d e d minor s t r u c t u r e f o r ( G J ( Q ) , Gm) , where (Q,<) is an a r b i t r a r y w e l l - q u a s i - o r d e r e d s e t . Our hope i s t h a t overcoming t h e r e m a i n i n g o b s t a c l e s i n t h e program w i l l l e a d toward a d e e p e r s t r u c t u r e t h e o r y t h a t w i l l s e t t l e t h e open immersion q u e s t i o n s and b e t t e r e x p l a i n t h e f a i l u r e o f w e l l - q u a s i - o r d e r i n g on embeddings. W e c o n j e c t u r e t h a t :

(J)

t h e r e a r e e f f e c t i v e a l g o r i t h m s f o r t e s t i n g embedding and immersion inclusion i n y ( Q ) , where ( Q , -4 is any w e l l - q u a s i - o r d e r e d set which i t s e l f c a n b e r e c o g n i z e d by an e f f e c t i v e a l g o r i t h m .

This is an extremely s t r o n g c o n j e c t u r e , e s p e c i a l l y r e l a t i v e t o t h e evidence a p a r t from t h e i n v e s t i g a t i o n w e r e p o r t on h e r e . I t i s d e b a t a b l e even whether or h a s an e f f e c t i v e a l g o n o t o r d i n a r y embedding o r immersion i n c l u s i o n of K5. r i t h m , f o r example. I n k e e p i n g w i t h t h e c o n j e c t u r e s i n ( J ) we s u p p o r t NashW i l l i a m ' s immersion c o n j e c t u r e (C) and a l s o v e r s i o n s of t h e Erdds-Posa-type theorem 4 , where embedding or immersions r e p l a c e minor i n c l u s i o n .

[l]

ERDOS, P. and POSA, L . , On t h e i n d e p e n d e n t c i r c u i t s c o n t a i n e d i n a graph, Canad. J. Math. 17 ( 1 9 6 5 ) , 347-352.

[2]

KRUSKAL, J. B . , W e l l - q u a s i - o r d e r i n g , t h e tree theorem, and V a z s o n y i ' s c o n j e c t u r e , T r a n s . A m e r . Math. SOC. 95 (19601, 210-225.

131

A f r e q u e n t l y discovered KRUSKAL, J . B . , The t h e o r y of w e l l - q u a s i - o r d e r i n g : c o n c e p t , J . C o m b i n a t o r i a l Theory, S e r . A 13 (19721, 297-305.

141

LAVER, R . ,

On F r a i s s e ' s o r d e r t y p e c o n j e c t u r e , Ann. of Math. 93 ( 1 9 7 1 ) ,

89-111. 151

MADER, W . , Wohlquasigeordnete K l a s s e n e n d l i c h e Graphen, J . C o m b i n a t o r i a l Theory, S e r . B 12 (19721, 105-122.

[6]

NASH-WILLIAMS, C . S t . J . A . , On w e l l - q u a s i - o r d e r i n g Camb. P h i l . S O C . , 59 ( 1 9 6 3 ) , 833-835.

f i n i t e trees, P r o c .

[71

NASH-WILLIAMS, C . S t . J . A . , On w e l l - q u a s i - o r d e r i n g Camb. P h i l . SOC., 61 ( 1 9 6 5 ) , 697-720.

i n f i n i t e trees, P r o c .

[8]

ROBERTSON, N., t o appear.

[9]

ROBERTSON, N . and SEYMOUR, P . D . , Graph m i n o r s . I . E x c l u d i n g a f o r e s t , J. C o m b i n a t o r i a l Theory, S e r . B . 35 ( 1 9 8 3 ) , 39-61.

[lo]

ROBERTSON, N . and SEYMOUR, P . D . , tree-width, submitted.

[I11

ROBERTSON, N . and SEYMOUR, P . D . , Graph minors. J. C o m b i n a t o r i a l Theory, S e r . B , t o a p p e a r .

Minimal c y c l i c - 4 - c o n n e c t e d

g r a p h s , T r a n s . A m e r . Math. S O C . ,

Graph m i n o r s . 11. A l g o r i t h m i c a s p e c t s o f

111. P l a n a r t r e e - w i d t h ,

P.D. Seymour, A? Robertson

354 [I21

ROBERTSON, N . and SEYMOUR, P . D . , quasi-ordering, submitted.

Graph minors.

I V . Tree-width and w e l l -

[13]

ROBERTSON, N . and SEYMOUR, P.D., graph, submitted.

Graph minors. V . E x c l u d i n g a p l a n a r

1141

TUTTE, W.T., C o n n e c t i v i t y i n Graphs, Mathematical E x p o s i t i o n s , No 15 (Univ. of T o r o n t o P r e s s , T o r o n t o , O n t a r i o ; Oxford Univ. P r e s s , London, 1966).

[15]

UNGAR, P . , A d i s s e c t i o n problem and a n i n t e r t w i n i n g problem, i n : Bondy, J . A . and Murty, U.S.R., ( e d s . ) , Graph Theory and R e l a t e d T o p i c s . (Academic P r e s s , N e w York, 1 9 7 9 ) , 370-371.

[16]

WAGNER, K . ,

[171

WHITE, A . T . , Graphs, Groups and S u r f a c e s , Mathematical S t u d i e s , No 8 (North-Holland, Amsterdam; American E l s e v i e r , N e w York, 1973)

B e w e i s e i n e r Abschwachung d e r Hadwiger-Vermutunq, 153 ( 1 9 6 4 ) , 139-141.

Math. Ann.

.

- PART VII ORDER AND COMBINATORICS -

PARTIEVII ORDRE ET COMBINATOIRE

Ivan RIVAL Linear extensions of finite ordered sets. . . . . . . . . . . . . . . . . . . . . .

p . 355

Michel HABIB Comparability invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p . 371

Andrk BOUCHET Codages e t dimensions de relations binaires. . . . . . . . . . . . . . . . . . . p . 387 Jerrold R. GRIGGS The sperner property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p . 397

GBrard VIENNOT Chain and antichain families grids and young tableaux. . . . . . . . . . p . 409

This Page Intentionally Left Blank

Annals of Discrete Mathematics 23 (1984) 355-370 0 Elsevier Science PublishersB.V. (North-Holland)

355

LINEAR EXTENSIONS OF F I N I T E ORDERED SETS

Ivan RIVAL

Department of Mathematics and Statistics The University o f Calgary Calgary, Canada T 2 N 1 N 4 Dedicated to E.Corominas R e s urn&

La theorie des extensions lineaires des ensembles ordonnes est fkconde. Dans le cas infini les rdsultats principaux sont souvent ensemblistes tandis que les resultats et les methodes dans le cas fini s o n t typiquement algorithmiques. Les rdsultats les plus connus d'extensions lindaires traitent du cas infini - les m@mes rgsultats dans le cas fini sont souvent triviaux. Malgr6 tout, il y a eu recement des r6sultats concernant surtout les ensembles ordonngs finis. En informatique la raison d e l'importance d e s extensions lindaires d'ensembles ordonnes finis est Claire.

11 y a correspondance

evidente entre l'id6e d'extension lineaire et l'idhe de tri topologique. Dans cet article nous allons considerer trois themes qu'on rencontre dans l'6tude des extensions lin6aires des ensembles ordonnks finis :

(i)

Q u e l l e e s t l a t a i l l e de 1 'ensemble des extensions l i n e a i r e s d'un ensemble ordonnP f i n i ?

(ii)

Duel e s t l e p l u s p e t i t nombre d'extensions l i n 6 a i r e s d'un ensemble ordonnt? f i n i dont 1 ' i n t e r s e c t i o n redonne c e t ensemble ordonne ?

(iii) Comment construit-on une extension l i n e d i r e optimale

The work reported here was supported in part by N . S . E . R . C .

A4077.

?

I. Rival

356

CONTENTS How many l i n e a r e x t e n s i o n s a r e t h e r e ? The o r d e r dimension Optimal l i n e a r e x t e n s i o n s

INTRODUCTION The t h e o r y of l i n e a r e x t e n s i o n s of o r d e r e d sets i s r i c h and v a r i e d : i n t h e i n f i n i t e c a s e t h e main r e s u l t s and methods a r e t y p i c a l l y s e t - t h e o r e t i c and modelt h e o r e t i c i n c h a r a c t e r ; i n t h e f i n i t e c a s e t h e r e s u l t s and methods a r e c h a r a c t e r i s t i c a l l y i n s p i r e d by a l g o r i t h m i c c o n s i d e r a t i o n s .

A linear extension L of an o r d e r e d s e t P i s a t o t a l l y o r d e r e d set on t h e same s e t a s P such t h a t z < y i n L whenever z < y i n P. For i n s t a n c e , t h e t o t a l l y o r d e r e d s e t of n a t u r a l numbers 1 < 2 < 3 .; 4 < 5 is a linear e x t e n s i o n of t h e o r d e r e d s e t of n a t u r a l numbers w i t h r e s p e c t t o d i v i s i b i l i t y 1.; 2 < 4 1< 3 < 6 e t c . The p o i n t about a l i n e a r e x t e n s i o n i s t h a t i t "extends" t h e o r d e r r e l a t i o n t o a t o t a l o r d e r r e l a t i o n , w i t h o u t " e n l a r g i n g " t h e u n d e r l y i n g s e t . L e t L ( P ) s t a n d f o r t h e s e t of a l l l i n e a r e x t e n s i o n s of t h e ordered s e t P .

...

...,

...,

The b e s t known r e s u l t s about l i n e a r e x t e n s i o n s of o r d e r e d s e t s a r e encountered i n the i n f i n i t e case - the so-called "general" case. I n the f i n i t e case t h e s e same r e s u l t s a r e o f t e n a b a r e t r i v i a l i t y . Even t h e s t a r t i n g p o i n t of t h e s u b j e c t b e a r s o u t t h a t s e n t i m e n t : L ( P ) i s nonempty f o r every o r d e r e d s e t P (E. SZPILRAJN [ 1 9 3 0 ] ) . A more r e c e n t q u e s t i o n , whether o r n o t an o r d e r e d s e t h a s a complete l i n e a r e x t e n s i o n , i s w i t h o u t any t h e o r e t i c a l i n t e r e s t f o r f i n i t e o r d e r e d s e t s ; t h e " g e n e r a l " c a s e , on t h e o t h e r hand, i s q u i t e engaging (M. POUZET and I . RIVAL [ 1 9 8 l ] ) . There i s much t h e same dichotomy f o r t h e problem of c h a r a c t e r i z i n g t h e o r d e r e d sets f o r which t h e r e i s , up t o isomorphism, j u s t one l i n e a r e x t e n s i o n (R. ASSOUS [ 1 9 8 2 ] ) . Many of t h e s e and r e l a t e d q u e s t i o n s a r e t a k e n up i n t h e survey paper of R. BONNET and M. POUZET [1982]. There i s a l s o t h e r e c e n t book of J . ROSENSTEIN [1982] which s y s t e m a t i c a l l y t r e a t s of t o t a l l y o r d e r e d sets - almost always i n t h e " g e n e r a l " c a s e . A l l of t h i s evLience n o t w i t h s t a n d i n g , t h e r e i s a s u b s t a n t i a l body of r e s u l t s r e c e n t l y emerging j u s t about l i n e a r e x t e n s i o n s of f i n i t e o r d e r e d s e t s . A t a f i r s t g l a n c e t h i s i s an u n l i k e l y c l a i m e s p e c i a l l y as t h e r e a p p e a r t o be s o few approaches t o t h e very c o n s t r u c t i o n of a l i n e a r e x t e n s i o n . T y p i c a l l y , a l i n e a r < z 1 of a n n-element o r d e r e d s e t P i s c o n s t r u c t e d e x t e n s i o n Izl < r 2 <

...

a c c o r d i n g t o an a l g o r i t h m which s t a r t s w i t h a minimal element zlof P , p r o c e e d s

z2< ... < zn' z2,..., I of t h e elements of P d e t e r m i n e s a l i n e a r e x t e n s i o n of P, i f each xi+, i s chosen from one element a t a t i m e , t o d e l i n e a t e t h e t o t a l o r d e r i n g z1 <

ending w i t h a maximal element

z of P.

A p e r m u t a t i o n zl,

among t h e minimal elements of t h e "up s e t "

i

So x1 E min(P) = min(Ul) and, f o r

i

=

l,Z,...,n-l,

C min(Ui+l).

Figure 1

357

Linear extensions o f finite ordered sets

N

i l l u s t r a t e s a four-element o r d e r e d s e t t o g e t h e r w i t h a l l ( f i v e ) of i t s linear extensions. I n t h e o r e t i c a l computer s c i e n c e c i r c l e s t h e s e i d e a s have come t o be known as " t o p o l o g i c a l s o r t i n g " ( c f . A.B. KAHN [ 1 9 6 2 ] , D . E . KNUTH and J . L . SZWARCFTTER [19741).

a

One a l g o r i t h m f o r c o n s t r u c t i n g a l i n e a r e x t e n s i o n of a n o r d e r e d s e t P can b e d e s c r i b e d u s i n g t h e depth f u n c t i o n 6 of P :

6(x)

=

b

P

max{lCI/C chain i n P and z = i n f C}.

The d e p t h - f i r s t l i n e a r e x t e n s i o n chooses, a f t e r e a c h s t e p i, an element of " g r e a t e s t depth" i n Ui+l, t h a t i s , z E min(Ui+l)

i+ 1

Ll

L2

such t h a t

L3

L4

L5

Figure 1 =max{&(x)12 ~ m i n ( U ~ + ~ ) l .

For i n s t a n c e , each of t h e l i n e a r e x t e n s i o n s L , , L 2 , L 3 , L , i s d e p t h - f i r s t .

Only

L s i s n o t s i n c e , a f t e r x1 = b 6 min(U1) = min P , min(Uz) = { a , d } and 6 ( a ) = 2 ,

6(d)

=

1.

Another a l g o r i t h m c o n s t r u c t s a greedy l i n e a r e x t e n s i o n : a f t e r each s t e p i, ziflC min(Ui+l) which, i n P, i s g r e a t e r

i t i s p r e d i s p o s e d t o s e l e c t i n g an element

E min(Ui+l)Ix xi+l E min(Ui+l) i f {x C min(Ui+l) l z > 2.1 than z..

L1, L2, L

x1

=

b

That i s , xi+l C

5

i s greedy.

-+ x2

=

f:free lattices

L

3

13:

> 2.) i f t h e r e i s one, o r , any =

0.

Each of t h e l i n e a r e x t e n s i o n s

i s n o t greedy s i n c e min(U ) = { a , d } and x1 = b 2

i

d , but

a.

g:partition lattices

h h t t i c e s of preorders

Of c o u r s e , t h e s e a r e n o t t h e only algorithms f o r t h e construction of a l i n e a r e x t e n s i o n of a f i n i t e o r d e r e d s e t . They a r e r a t h e r common though. Imagine t h e p r o p o s a l t o w r i t e a monograph on o r d e r e d s e t s and suppose t h a t t h e c h a p t e r themes and t i t l e s a r e a l r e a d y s e l e c t e d . I n which ( t o t a l ) o r d e r s h o u l d t h e chapt e r s b e a r r a n g e d ? Some c o n s t r a i n t s o r i g i n a t e from t h e need t o p r e c e d e c e r t a i n c h a p t e r s by o t h e r s which contain the prerequisite s k i l l s . These precedence c o n s t r a i n t s cons t i t u t e an o r d e r on t h e u n d e r l y i n g s e t of c h a p t e r s . An i n t e r e s t i n g example i s t h e i n t r o d u c t o r y t e x t on o r d e r e d s e t s by M. BARBUT and B. MONJARDET [ 1 9 7 0 ] . I t c o n s i s t s of e i g h t c h a p t e r s which ( f o l l o w i n g the authors' preface) a r e linked

by t h e precedence c o n s t r a i n t s i n d i c a t e d i n t h e F i g u r e 2 . There may w e l l be d i f f e r e n t b e n e f i t s from d i f f e r e n t l i n e a r e x t e n s i o n s . I f t h e r e a d e r i s meant t o a c q u i r e a working knowledge of a v a r i e t y of t o p i c s , a t t h e expense of e a r l y s p e c i a l i z a t i o n , then a d e p t h - f i r s t l i n e a r e x t e n s i o n such as L 1 may be a p p r o p r i a t e . I f , i n s t e a d , i t i s p r e f e r a b l e t o r e a c h some s p e c i a l i z e d t o p i c s e a r l y , a t t h e expense of d i v e r s i t y , t h e n a greedy l i n e a r e x t e n s i o n L , may b e b e t t e r . And of course, there a r e other l i n e a r extensions too, such a s L 3 . c r i t e r i a , too. text?)

There a r e o t h e r

(What do you suppose was t h e a c t u a l t o t a l o r d e r i n g used i n t h a t

The aim o f t h i s paper i s t o s u r v e y b r i e f l y t h e t h r e e p r i n c i p a l themes encountered i n t h e s t u d y of l i n e a r e x t e n s i o n s of f i n i t e o r d e r e d s e t s . (i) (ii)

BOW

IL(PI I ?

big i s

What i s the l e a s t number k of linear extensions k L1, L,, Lk C L ( P I such t h a t fl Li = P?

...,

(iii)

i= 1

How i s an optimal member o f L(PI constructed?

HCW MANY LINEAR

ARE THERE?

EXTENSIONS

Counting i s h a r d ! And c o u n t i n g t h e l i n e a r e x t e n s i o n s of a n o r d e r e d s e t i s no e x c e p t i o n . T h i s c o u n t i n g problem i s t r a c t a b l e o n l y f o r some s p e c i a l and v e r y simple c l a s s e s o f o r d e r e d s e t s . An n-element a n t i c h a i n h a s n ! l i n e a r

L e t L ( P ) = Il(P)I f o r an o r d e r e d s e t P. e x t e n s i o n s . I n symbols,

n

+

L(

L) = n !

,

i=1 The c a r d i n a l sum of an n-element

chain

2 has

n+m l i n e a r extensions.

The d i r e c t p r o d u c t of t h e c h a i n s L(2

*

x

n) IV

2 and m, h a s

=

n+l

l i n e a r e x t e n s i o n s . The n-element fence ("zig-zag") F = { a l c a,, a 2 > a 3 , a 3 < a4,. 1 i s s a i d t o s a t i s f y

..

i=O and from t h i s , t h e 2n-element c?om C2n = { a l < b,, b , > a 2 , a , < b ,

pi:]

,...,

L ( F * i ) L(Fn'n-1-2i )

an < bn, bn

>

a 11 s a t i s f i e s

There i s l i t t l e hope f o r a u n i f y i n g formula e x p r e s s i n g L ( P ) i n terms of some s t r u c t u r a l i n v a r i a n t s o f an o r d e r e d s e t P. N e v e r t h e l e s s , in r e c e n t y e a r s , t h e r e have appeared s e v e r a l a r t i c l e s which d a r e t o c o n f r o n t t h i s c o u n t i n g problem fairly directly.

359

Linear extensions o f finite ordered sets

Let P he an ordered set. For elements 2 , y of P such that y $ z let P ( z < y ) stand for the transitive closure of P with the additional comparability 3: < y , that is, u < u in P ( z < 9) if u < u in P or if u < z in P and y < v in P. How likely is it that a "typical" extension L of P contains z < y? One way to find out is by a "simple majority" criterion. Write zBy if

L(P(z < Y)) > L(P(y < z)). Observe that L(P(x < y ) ) = L ( P ) if 3: < y in P and in this case just put L ( P ( y < 2 ) ) = 0. For instance, suppose the elements of P represent candidates for some elective office and the order u < U in P stands for v is unanimously preferred to u. Now suppose each elector ranks all of the candidates in a total order. If there are L ( P ) electors each selecting a different linear extension then the binary relation ii summarizes the outcome according to simple majority rule. Notice that R need not impose a total ordering on P: if P is an antichain then L ( P ( z < y ) ) = L ( P ( y < 2)) for each z # y s o D leaves P totally unordered. What is surprising though is that R need not even he a transitive order on P. A n example to illustrate this (see Figure 3) was first') put forward by FISHBURN [ 1 9 7 4 ] . This 31-element ordered set P has a three-element "6-cycle": L ( P ) = 6270; L ( P ( z iy ) ) = 3218 so zBy; L ( P ( y < z ) ) = 3192 so yBz: L ( P ( z < x)) = 3150 so Z E X . (Recently both M. AIGNER [1982] and P.M. WINKLER [1982] have remarked that 6 is a transitive ordering of P if and only if li decomposes P into a linear sum of antichains.) P.C.

A more encouraging direction is proposed in the series of articles R.L. GRAHAM, A . C . YAO and F.F. YAO [1980], L.A. SHEPP [1980], [1982], P.M. WINKLER [1982a], [1982b]. At this writing the most important result is this so-called "transitivity inequality": f o r elements a, b, c of G f i n i t e ordered set P

X

Figure

L P ( G < b ) ) . L ( P ( a < c)) 5 L ( P ( a < b , a < c)).

L(P)

(L.A. SHEPP [1982]). Why is this "transitivity inequality" interesting? One motivation is an algorithmic one. It is concerned with the problem of sorting numbers a , b , c,... by binary comparisons " a : b" to construct successively "finer" orders P on { a , b , c, ) until a total order is finally established. It is said that a quantity fundamental in deciding the expected efficiency of such algorithms is

....

the "probability" that the result of a : b is a c b , assuming that all linear extensions of P are equally likely. (In this model P is thought to be some intermediate state of order (disorder) of { a , b , c, . . . I and a , b are not yet + as in Figure 4 , then sorted.) For instance, if P

22 2

1) The well known "voting paradox" example of CONDORCET [1785] will not do. Suppose there are three candidates x, y, z and three voters ranking the z < x. Evidently y is "preferred" candidates by z < y < z, z < z < y and y to x, z is "preferred" to y , but x is "preferred" to z . Nevertheless, if a l l of the linear extensions of a three-element antichain are considered then, by symmetry, B still leaves all pairs noncomparable.

360

I. Rival

'1 1:

Q

P

while

Pr(d < b(P)

=

0.

Moreover, "conditional probabilities" can also be considered. For instance

C

Pr(a c blP(a < d ) )

9(P) Figure 4

=

L(P(a < b , a < d)) L(P(a < d ) )

=

2 5

and

Pr(a

<

d / P ( b < c))

=

7. 4

Probably a more profound motivation of this "transitivity inequality" is concerned with the very character of transitivity among all of the binary relations on a set { a , b , c , . . . ) . We can express the "transitivity property'' in this way: for elements a; b , c

a < c and c < b impZy a < b. In particular, every linear extension of an ordered set P = {a, b , c , which a < c and c < b , must include the inequality a < b , that is,

. . . I in

P r ( a < b l P ( a < c , c < b ) ) = 1. And, unless a < c and c < b are both satisfied in P itself, we will, in general, ju s t have

o

5 P r ( a < b l ~ 5)

1.

However, what if we have "partial" information, say a < c? This reasoning in part led I. RIVAL and B. SANDS [1982] to conjecture the transitivity inequality. The proof of the transitivity inequality is itself interesting (L.A. SHEPP [1982]). There are three highlights and the last is ingenious! The first step is an item of background, the "down set inequality" for distributive lattices ( D . J . KLEITMAN [1966], cf. C.M. FORTUIN, P.W. KASTELEYN, and J . GINIBRE [19711). For down s e t s A , B of a f i n i t e distributive Zattice D IAl

- 1BI

5

\A fl BI

*

(DI

In Figure 5 , the shaded elements enumerate an example of the corresponding subsets

A

An 0

0 Figure 5

D

36 1

Linear extensions o f finite ordered sets The second s t e p i s on t h e s u r f a c e a m a t t e r of formalism

Pr(a < b l ~ =) l i m

I,PcQ < b , 1 If1

N-

3 is

t h e d i s t r i b u t i v e l a t t i c e of a l l o r d e r - p r e s e r v i n g maps of t h e o r d e r e d s e t P i n t o the chain F = { O < 1 < 2 < < N - 1 1 o r d e r e d (componentwise) by f 5 g i f f(z) C g(x) f o r each x C P. The p o i n t about t h i s a s y m p t o t i c e s t i m a t e i s t h a t t h e t r a n s i t i v i t y i n e q u a l i t y can now b e c a s t i n t h e language of d i s t r i b u t i v e l a t t i c e s . I t i s enough t o p r o v e , f o r any i n t e g e r N ,

...

w

I t seems we a r e done by merely o b s e r v i n g t h a t D = that

f

is a d i s t r i b u t i v e l a t t i c e ,

i y ( - b , a < b L A " , w

where

A = $(a

< b,

N

and

and a p p l y i n g t h e down s e t i n e q u a l i t y .

There i s a h i t c h .

Neither

$ ( a .= b ) N

nor

$ ( a c c) need be down s e t s i n

$.

w

This i s i l l u s t r a t e d i n Figure 6, using three-element a n t i c h a i n .

=

{0 < 1) and P = {a, b , e} a

Figure 6 The f i n a l s t e p i n t h e proof of t h e t r a n s i t i v i t y i n e q u a l i t y i s t h e n o v e l i d e a t o change t h e o r d e r on t h e u n d e r l y i n g s e t of

and

s o t h a t both

I. Rival

362

a r e down s e t s and y e t distributive!

$ remains

I n f a c t , t h i s can

be accomplished as f o l l o w s .

f, g E

(1,0,0)

$, d e f i n e f a

g

For

i f , (as

integers with the usual ordering)

f(a) 5 g(a) and, f o r

z# a,

f ( a ) - f(z)5 g ( a )

-

lDIC

g(d.

'L

(componentwise order)

An example i s i l l u s t r a t e d i n

(O,I,?)

datbtc ' ( aorder)

'L

F i g u r e 7.

Figure 7

And F i g u r e 6 i s then transformed i n t o F i g u r e 8.

2{o
,-u

2b
2b
,-L

,-L

da,b,cl ,-L

Figure 8

The down set i n e q u a l i t y i s t h e f o r e r u n n e r of s e v e r a l g e n e r a l i z a t i o n s commonly denoted by t h e acronym "FKG i n e q u a l i t y " ( c f . C.M. FORTUIN, P.W. KASTELEYN, and J . G I N I B R E [1971], P.D. SEYMOUR and D . J . A . WELSH [1975], D . E . DAYKIN [19791, R. AHLSWEDE and D . E . DAYKIN [ 1 9 7 9 ] , D.B. WEST [ 1 9 8 2 ] . The most i m p o r t a n t o u t s t a n d i n g q u e s t i o n i s whether t h e t r a n s i t i v i t y i n e q u a l i t y i s a c t u a l l y a s t r i c t i n e q u a l i t y i n a l l b u t t h e t r i v i a l c a s e s , t h a t i s , i f a, b , c are pairwise %on-

comparable i n a f i n i t e ordered s e t P then i s L(P(a < b ) )

- L(P(a

< c)) < L ( P ( a < b , a < c))

- L(P)?

THE ORDER DIMENSION Suppose a and b a r e noncomparable elements of an o r d e r e d s e t P. Then t h e r e a r e l i n e a r e x t e n s i o n s L1, L, of P such t h a t a < b ( L 1 ) and b < a ( L p ) , s o , i n

L1

n L,, a

and b a r e noncomparable and y e t

z < y ( L 1 f' L2) i f z < y ( P ) .

How many

of a l l of t h e l i n e a r e x t e n s i o n s of P are "needed" t o determine P? The (order) dimension dim P of P i s t h e l e a s t number Ll,L2,...,Lk of members of L(P) s u c h t h a t

k

n

i=l that is,

3: iy ( P )

i f and only i f

3:

L.

'

=

P,

< y ( L i ) f o r each 5 = 1, 2 ,

..., k .

How a r e

Linear extensions o f finite ordered sets

363

such "minimal" s e t s of l i n e a r e x t e n s i o n s c o n s t r u c t e d ? These are q u i t e o l d q u e s t i o n s and t h e r e i s a l r e a d y a n e x t e n s i v e l i t e r a t u r e and many deep r e s u l t s about them. Perhaps t h e d e e p e s t r e s u l t i n t h e dimension t h e o r y of o r d e r e d s e t s i s t h e c o n s t r u c t i o n of a l l of t h e 3 - i r r e d u c i b l e o r d e r e d s e t s . An o r d e r e d s e t P i s n-irreducible i f dim P = n and i f , f o r each

5

E P,

dim(P -

{z})< n

.

The p o i n t a b o u t n - i r r e d u c i b l e o r d e r e d s e t s i s t h a t a n o r d e r e d s e t P h a s dimension a t l e a s t n i f and o n l y i f i t c o n t a i n s a n n - i r r e d u c i b l e s u b s e t . For i n s t a n c e , dim P 3 2 i f and o n l y i f P c o n t a i n s a two-element a n t i c h a i n , (which i s t h e only 2 - i r r e d u c i b l e o r d e r e d s e t ) . The d e s c r i p t i o n of a l l of t h e 3 - i r r e d u c i b l e o r d e r e d s e t s h a s t u r n e d o u t t o be a much more d i f f i c u l t t a s k (D. KELLY [ 1 9 7 7 ] , W.T. TROTTER, J r . and J.I. MOORE [19761). The 3 - i r r e d u c i b l e o r d e r e d s e t s w i t h a t most seven elements a r e , up t o d u a l i t y , i l l u s t r a t e d i n F i g u r e 9. But t h e r e a r e 3 - i r r e d u c i b l e o r d e r e d s e t s of a l l s i z e s , b i g and t a l l .

??$dk&k& Figure 9

There a r e two p a r a l l e l and q u i t e d i f f e r e n t approaches t o t h e c o n s t r u c t i o n of t h e complete l i s t . ( " I t i s t h e r e f o r e q u i t e u n l i k e l y t h a t any 3 - i r r e d u c i b l e h a s been missed!") The one approach of D. KELLY [1977] i s based on a c h a r a c t e r i s a t i o n of p l a n a r l a t t i c e s . The l i n k t o l a t t i c e t h e o r y i s f o r g e d by t h e normal completion (completion by c u t s ) L(P) of an o r d e r e d s e t P. The p o i n t i s t h a t dim P = dim k ( P ) and dim L ( P ) 5 2 i f azd only i f L ( P ) i s a p l a n a r l a t t i c e . Now i f P i s 3 - i r r e d u c i b l e t h e n L(P7 i s a n o n p l a n a r l a t t i c e - a n d s o L ( P ) must c o n t a i n one of t h e "minimal" nonpl%ar l a t t i c e s ( c f . D. KELLY and I . %IVAL [ 1 9 7 5 ] ) . Moreover, t h e u n d e r l y i n g s u b s e t of j o i n i r r e d u c i b l e o r meet i r r e d u c i b l e elements of a "minimal" n o n p l a n a r l a t t i c e t u r n s out i t s e l f t o be 3-irreducible. The o t h e r approach of W.T. TROTTER, J r . and J . I . MOORE [1976] i s based on a c h a r a c t e r i s a t i o n of c o m p a r a b i l i t y g r a p h s . For a n o r d e r e d s e t P, dim P 5 2 i f and only i f t h e n o n c o m p a r a b i l i t y graph of P i s i t s e l f a c o m p a r a b i l i t y graph. I f P i s 3 - i r r e d u c i b l e t h e n i t s n o n c o m p a r a b i l i t y graph i s not a c o m p a r a b i l i t y g r a p h and s o i t must c o n t a i n one of a l i s t of "minimal" f o r b i d d e n subgraphs ( c f . T . GALLAI [ 1 9 6 7 ] ) . Moreover, i f t h e complement of a "minimal" f o r b i d d e n graph i s i t s e l f a c o m p a r a b i l i t y g r a p h t h e n , up t o d u a l i t y , i t h a s a unique o r i e n t a t i o n and t h i s o r i e n t a t i o n y i e l d s a 3-irreducible ordered s e t . The c o m p a r a b i l i t y graph h a s p r o v i d e d t h e s e t t i n g f o r a n o t h e r q u i t e s t r i k i n g

I. Rival

364

r e s u l t about dimension. I t i s t h i s . I f o r d e r e d s e t s P and Q have graph isomorphic c o m p a r a b i l i t y graphs t h e n dim P = dim Q . T h i s r e s u l t i s e s t a b l i s h e d by many a u t h o r s ( c f . J . C . ARDITTI [ 1 9 7 8 ] , J . C . ARDITTI and H.A. JUNG [ 1 9 7 9 ] , R. G Y S I N [ 1 9 7 7 ] , W.T. TROTTER, J r . , J . I . MOORE and D . P . SUMNER [ 1 9 7 6 ] ) . A r e l a t e d r e s u l t , c r e d i t e d t o R.P. STANLEY, i s t h i s . I f P and Q have isomorphic c o m p a r a b i l i t y graphs t h e n L ( P ) = L(Q). I n f a c t , t h e v a l u e of t h e s e r e s u l t s i s t h a t they highl i g h t t h e b a s i c r e s u l t which t i e s t h e enumeration of a l l of t h e o r i e n t a t i o n s of a c o m p a r a b i l i t y graph t o t h e i d e a of l e x i c o g r a p h i c decomposition ( c f . L.N. SHEVRIN and N.D. F I L I P P O V [ 1 9 7 0 ] , M. GOLUMBIC [198O]).

Q b e a n o r d e r e d s e t and l e t (P Iq E Q) b e a f a m i l y of o r d e r e d s e t s 4 Q. The lex
indexed by

is

1exicographicaZly nondecornposable. In F i g u r e 1 0 w e i l l u s t r a t e t h r e e p a i r s of o r d e r e d s e t s . E a c h i n t h e f i r s t p a i r

Figure 10 is l e x i c o g r a p h i c a l l y nondecomposable and t h e one i s t h e d u a l of t h e o t h e r . Each i n t h e second p a i r h a s a l e x i c o g r a p h i c decomposition w i t h i n d e x s e t t h e f i v e element c h a i n and b l o c k s which are e i t h e r two-element a n t i c h a i n s o r s i n g l e t o n s ; t h e one i s o b t a i n e d from t h e o t h e r b y "permuting" t h e b l o c k s o v e r t h e i n d e x s e t . Each i n t h e t h i r d p a i r h a s a l e x i c o g r a p h i c decomposition w i t h i n d e x s e t t h e 4-fence and b l o c k s which a r e s i n g l e t o n s e x c e p t f o r t h e "lower endpoint" of t h e 4-fence where t h e b l o c k i s a 3-fence; t h e one i s i n t h i s c a s e o b t a i n e d from t h e o t h e r by d u a l i z i n g a n o n t r i v i a l b l o c k . Each member of a g i v e n p a i r h a s i d e n t i c a l dimension and t h e same number of l i n e a r e x t e n s i o n s . No wonder t h e i r c o m p a r a b i l i t y g r a p h s ( s e e F i g u r e 11) are graph isomorphic. And t h i s i s t h e h e a r t of t h e m a t t e r . I f a n o r d e r e d s e t P i s l e x i c o g r a p h i c a l l y nondecomposable t h e n s o i s e v e r y o r i e n t a t i o n of i t s c o m p a r a b i l i t y graph: up t o d u a l i t y , i t h a s p r e c i s e l y one orientation.

If P =

1

4 FQ graph of P ( q € Q) and Q ' , 4

Pq, and P ' i s an o r i e n t a t i o n of t h e c o m p a r a b i l i t y 4 t o o , i s a n o r i e n t a t i o n of t h e c o m p a r a b i l i t y graph of Q

Linear extensions of finite ordered sets

(a 1

365

(b) Figure 11

(C)

1 P' has the same comparability graph as P. All orientations of q€Q the comparability graph of P are constructed in this way. I proposed this as a conjecture to R. Wille recently. He has since shown that it does follow from a careful analysis of lexicographic decompositions (see R. WILLE [1982]).

then P'

=

Recently, M. HABIB [1981] (cf. M. HABIB [1983]) reports that a real-valued function defined for ordered sets is "invariant" over all orientations of a given comparability graph provided that it is invariant with respect to its dual and it is invariant over all orientations in which the index set is a chain. The dimension of an ordered set is such an invariant. A definitive account of the dimension theory of ordered sets is found in D. KELLY and W.T. TROTTER, Jr. [1982].

OPTIMAL LINEAR EXTENSIONS Among all of the linear extensions of an ordered set how is an "optimal" one constructed? Such questions arise in the design of a "schedule" to process a set of jobs; technological or other constraints on the jobs may impose precedence constraints on the jobs. The "schedule", too, must comply with them. Whether a schedule is "optimal" depends, of course, on the objectives o f the schedule: earliest completion time, minimum maximum lateness, etc. The literature on this topic is graving rapidly; an important introduction to it is i n E.L. LAWLER and J . K . LENSTRA [19821. We shall consider here just one such problem. Suppose a single machine is to perform a set of jobs, one at a time; a set of precedence constraints prohibits the start of certain jobs until someaother jobs are already completed. Any job which is performed immediately after a job which is not constrained to precede it, however, requires a "setup" or "jump" entailing some fixed additional cost. Our problem is this: scheduZe the

jobs t o minimize the nwnber of j w s . The problem seems to be posed first in J. KUNTZMA" and A . VERDILLON [1971] (cf. M. CHEIN and P. MARTIN

N P

b

I. Rival

366

[1972] and a l s o , G. CHATY, M. C H E I N , P . MARTIN and G . PETOLLA [19741). I n t h e language of o r d e r e d s e t s t h e problem i s commonly r e n d e r e d a s f o l l o w s . Let P b e a f i n i t e o r d e r e d set and f o r L C L ( P ) s e t

1 { (a,b)

s(P,L) =

C PxP(

a c o v e r s b i n L and a

%

b i n PI

I

and S ( P ) = min { s ( P , L ) ( L

c L(P)~

t h e jump number of P. The problem t o c h a r a c t e r i z e s ( P ) i s c o n s i d e r e d i n s e v e r a l r e c e n t a r t i c l e s ( e . g . G. CHATY and M. CHLIN [1979], M. C H E I N and M. HABIB [1980], D. DUFFUS, I . RIVAL and P.M. WINKLER [1982], G . G I E R Z and W . POGUNTKE [ 1 9 8 2 ] , M . HABIB [19811, W.K. PULLEYBLANK [ 1 9 8 2 ] ) . But t h i s "jump number problem" h a s a b a s i c a p p e a l even a p a r t from any supposed c o n n e c t i o n t o s c h e d u l e s . T h i s i s because of i t s a p p a r e n t l i n k t o t h e "chain decomposition" theorem of R . P . DILWORTH [1950]: i n a f i n i t e o r d e r e d s e t P t h e minimum number of d i s j o i n t c h a i n s whose s e t union is a l l of P e q u a l s t h e maximum number of p a i r w i s e noncomparable elements of P . Now, if C,, C2, C i s any minimum f a m i l y of d i s j o i n t c h a i n s of a f i n i t e o r d e r e d s e t

...,

m P whose set union i s P then t h e l i n e a r sum C, eR C2 @ ... CH Cm of t h e s e c h a i n s need not b e a l i n e a r e x t e n s i o n of P . On t h e o t h e r hand I any L E L(P) can b e expressed as t h e l i n e a r sum L = C, N C, @ @ C of c h a i n s C. i n P ( s o k chosen t h a t s u p Ci $ i n f p Ci+, f o r each i) and while i=1 Cz. = P , t h i s f a m i l y of c h a i n s need not

...

b

be t h e minimum number whose s e t union i s P ( s e e F i g u r e 1 3 ) . The "jump number problem'' i s p e r h a p s t h e f i r s t one t o s e e k a r e c o n c i l i a t i o n between t h e " c h a i n decomposition" theorem and t h e " l i n e a r e x t e n s i o n " theorem ( s e e I . RIVAL [ 1 9 8 2 ] ) .

)

Figure 13

The jump number of a n o r d e r e d s e t satisfies

s ( P ) 2 w(P)

-

1

where w ( P ) s t a n d s f o r the w i d t h of P . At l e a s t i n t h e case t h a t P contains no 2n-crown C2n(n 3 2)

s(P)

=

w(P)

-

1

(D. DUFFUS, I . R I V A L , and P.M.

WINKLER [19821 and c f . M. C H E I N and P . MARTIN 119721, M. HABIB [1975], M. C H E I N and M. M. HABIB HABIB [1982]). [1982]). Another Another wide wide ccllaassss of of oorrddeerreedd sets sets ffoorr which which tthhee jump jump number number is is known known iiss tthhee ccllaassss of of series-parallel o r d e r e d s e t s . An ordered ordered sseett PP iiss series-parallel, s e r i e s - p a r a l l e l , iiff iitt can can bbee decomposed decomposed iinnttoo ssiinngglleettoonnss u s i n g o n l y t h e c o n s t r u c t i o n s of l i n e a r sum and c a r d i n a l sum ( s e e F i g u r e 15). I n f a c t , t h e jump number of a seriesp a r a l l e l o r d e r e d s e t can b e computed by c o u n t i n g t h e number of + ' s i n i t s unique s e r i e s - p a r a l l e l decomposition ( c f . 0. COGIS and M. HABIB [ 1 9 7 9 ] ) .

b2" 03''

bih a1

a2

a3

02"

Linear extensions o f finite ordered sets

367

More useful is the fact that s(P)

=

s(P,L)

for any greedy linear extension of a series-parallel ordered set.

Actually

this follows from this more general result.

Call any ordered set isomorphic

to the ordered set P of Figure 1 or Figure 12 an N ("en").

A f i n i t e ordered s e t P whose diagram contains no N s a t i s f i e s s(P)

=

s(P,L)

for any greedy linear extension L of P (I. RIVAL [19821). This result generalizes the series-parallel class rather substantially. First of all the ordered set illustrated in Figure 16(a) is not series-parallel, yet it contains (in its diagram) no N . (A finite ordered set P is series-parallel just if its directed comparability graph (a) contains no N . ) Second, any ordered set can be embedded in one which contains, in its diagram, n o N (see Figure 16(b)). The theme of N ' s in the diagram of P figures in a result of P. GRILLET [1969] (cf. B. LECLERC No N's and B. MONJARDET [19731). A c6 finite ordered set contains no N (b) in its diagram if and only if Figure 16 each maximal antichain meets each maximal chain.

N

M-M

REFERENCES M. Aigner (1982) Private communication. J.C. Arditti (1978) Partially ordered sets and their comparability graphs, their dimension and their adjacency, in ProblDmes combinatoires e t t k b o r i e des grupkes, Orsay 1976, Co1oq.int.C.N.R.S. No. 260, pp. 5-8. J.C. Arditti and H.A. Jung (1980) The dimension of finite and infinite comparability graphs, J . London Math. Soc. (2) 21, 31-38. R. Assous (1982) Private communication. M. Barbut and B. Monjardet (1970) Ordre e t CZassification, AZgDbre e t Combinatoire. Paris, Hachette.

368

I. Rival

R. Bonnet and M. Pouzet (1982) Linear extensions of ordered sets in Ordered S e t s (I. Rival, ed.) D. Reidel Dordrecht, pp. 125-170. C. Chaty and M. Chein (1979) Ordered matchings and matchings without alternating cycles in bipartite graphs, U t i Z i t a s h'atlz. 16, 183-187. (;.

Chaty, M. Chein, P. Martin, C. Petolla (1974) Some results about the number of jumps in acircuit digraphs, U t i l i t a s !lath., Cong. Nwn. X , 267-279.

M. Chein and M. Habib (1980) T h e jump number of dags and posets: an introduction, Annals of Discrete Math. 9, 189-194. M. Chein and M. Habib (1982) Jump number of dags having Dilworth number 2, preprint. M. Chein and P. Martin (1972) Sur le nomhre de sauts d'une forst, C.R. Acuii. Sci. Paris 275 A . 159-161. 0.

Cogis and M. Habib (1979) Nombre de sauts et graphes sgrie-parallsles, R.A.I.R.O. I n f . Th. 13, 1 , 3-18.

M. De Condorcet (1785) I:ssai sur l'application de l'analyse h la probabilit6 des decisions rendues B la pluralit6 des voix, Paris (Chelsea), New York (1974). R.P. Dilworth (1950) A decomposition theorem for partially ordered sets, Ann. oj' Math. 51, 161-166. D. Duffus, I. Rival and P. Winkler (1982) Minimizing setups for cycle-free ordered sets, Proc. h e r . Math. Soc. P.C. Fishburn (1971) On the family of linear extensions of a partial order, J . Combin. Th. ( B ) 17, 240-243. C.M. Fortuin, P.W. Kasteleyn and J. Ginibre (1971) Correlation inequalities on some partially ordered sets, Corn. Math. Phys. 22, 89-103. T. Callai (1967) Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18, 25-66. C. Gierz and W, Poguntke (1982) Minimizing setups for ordered sets: a linear algebraic approach, SIAM J . Algebraic Discrete Methods. M. Golumbic (1980) Algorithmic Graph Theory and P e r f e c t Graphs, Academic Press, New York. R.L. Graham (1982) Linear extensions of partial orders and the FKG inequality, in Ordered S e t s (I. Rival, ed.) D. Reidel, Dordrecht, pp. 213-236. R.I..

Graham, A . C . Yao and F.F. Yao (1980) Some monotonicity properties of partial orders, S I N J . Algebraic Discrete Methods 1, 251-258.

P.A. Grillet (1969) Maximal chains and antichains, Fund. Math. 65, 157-167. R. Gysin (1977) Dimension transitiv orientierbarer Graphen, Acta Math. Acad. Sc. Hungar. 29, 313-316. M. Habib (1975) Partitions en chemins des sommets et sauts dons les graphes sans circuits, Thsse, Universite P. et M. Curie, Paris. M. Habib (1981) Substitution des structures combinatoires, th6orie et algorithmes, ThPse, Univ. Pierre et Marie Curie, Paris.

Linear extensions of finite ordered sets

369

M. Habib (1983) Comparability i n v a r i a n t s , p r e p r i n t . B . J 6 n 5 s o n (1982) A r i t h m e t i c of o r d c r c d s e t s i n Ordered S e t s ( I . R i v a l , e d . ) I ) . R e i d e l , D o r d r e c h t , p p . 3-41.

A.R.

Kahn (1962) T o p o l o g i c a l s o r t i n g o f l a r g e networks, Comnurzications A.C.M. 558-562.

D . Kclly (1977) 'I'he 3 - i r r e d u c i b l e p a r t i a l l y o r d e r e d s e t s , 367-383.

5,

Canad. J . Math. 29,

D. K e l l y and I . R i v a l (1975) P l a n a r L a t t i c e s , Canad. J . !lath. 27, 636-665. D . Kelly and W.T.

T r o t t e r , J r . (1982) Dimension t h e o r y f o r o r d e r e d s e t s , i n

Orderad S e t s ( I . R i v a l , e d . ) D. R e i d e l , D o r d r e c h t , pp. 1 7 1 - 2 1 1 . D . I . Kleitman (1966) F a m i l i e s of n o n - d i s j o i n t

s e t s , J . Combin. T k . 1, 153-155.

D.E.

Knuth (1973) T h e A r t of Computer Programming, Vol. 1 , V o l . 3, Addison-Wesley.

D.E.

Knuth and J.L. S z w a r c f i t e r (1974) A s t r u c t u r e d program t o g e n e r a t e a l l t o p o l o g i c a l s o r t i n g a r r a n g e m e n t s , Tnfor. Proc. L e t t e r s 2 , 153-157.

J . Kuntzmann and A. V e r d i l l o n (1971) Recherche d ' u n o r d r e t o t a l minimal c o m p a t i b l e

avec un o r d r e p a r t i e l donn5, Sdm. I n s t . !lath. Grenoble. E.L.

Lawler and J . K . L e n s t r a (1982) Machine s c h e d u l i n g w i t h precedence c o n s t r a i n t s , i n Ordered S e t s ( I . R i v a l , e d . ) D . R e i d e l , D o r d r e c h t , pp. 655-675.

B . L e c l e r c and B . Monjardet (1973) Orders " C . A . C . " , M.

Fund. i"bth. 79, 11-22.

Pouzet and I. R i v a l (1982) Which o r d e r e d s e t s have a complete l i n e a r e x t e n s i o n ? Canad. J . !lath. 33, 1245-1254.

W.R.

P u l l e y b l a n k (1982) On minimizing s e t u p s i n precedence c o n s t r a i n e d s c h e d u l i n g ,

Discrete Applied Math. I . R i v a l (1982) Optimal L i n e a r e x t e n s i o n s by i n t e r c h a n g i n g c h a i n s , Proc. Amer.

!lath. Sac. I . R i v a l and B. Sands (1982) Problem 3.1, R e i d e l , D o r d r e c h t , p . 806.

i n Ordered S e t s ( I . R i v a l , e d . ) D .

J . R o s e n s t e i n (1982) Linear Orderings, Academic Press, N e w York. L.A.

Shepp (1980) The FKG i n e q u a l i t y and some m o n o t o n i c i t y p r o p e r t i e s of p a r t i a l o r d e r s , SIAM J . AZgebraic Discrete Mathods 1, 295-299.

L.A.

Shepp (1982) The xyz c o n j e c t u r e and t h e FKG i n e q u a l i t y , Annals of

L.N.

S h e v r i n and N . D . F i l i p p o v (1970) P a r t i a l l y o r d e r e d s e t s and t h e i r comparab i l i t y g r a p h s , Siberian Math. J . 11, 497-509.

Probability.

E . S z p i l r a j n (1930) S u r l ' e x t e n s i o n de l ' o r d r e p a r t i e l , Fund. Math. 1 6 , 386-389.

W.T. T r o t t e r , J r . and J.I. Moore (1976) C h a r a c t e r i z a t i o n problems f o r g r a p h s , p a r t i a l l y o r d e r e d s e t s , l a t t i c e s , and f a m i l i e s of s e t s , Discrete Math. 1 6 , 361-381. W.T.

T r o t t e r , J r . , J.I. Moore and D . P . Sumner (1976) The dimension of a c o m p a r a b i l i t y g r a p h , Proc. Amer. Math. Soc. 60, 35-38.

I. Rival

370

R . W i l l e (1982) Lexicographic decomposition of o r d e r e d s e t s , p r e p r i n t .

P.M. Winkler (1982) P r i v a t e communication.

P.M. Winkler (1982a) Average h e i g h t i n a p a r t i a l l y o r d e r e d s e t , Discrete Nuth. 3 9 , 337-341.

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Annals of Discrete Mathematics 23 (1964) 371-386

37 1

0 Elsevier Science PublishersB.V. (North-Holland)

COMPARABILITY

INVARIANTS

Michel HABIB Ecole des Mines de SAINT ETIENNE 158, Cours Fauriel 42023 SAINT ETIENNE CEDEX

C e t a r t i c l e e s t d e d i e au p r o f e s s e u r E .

RESUME.-

COROMINAS.

Nous presentons ici une caracterisation des invariants de comparabilit6 (i.e. des fonctions qui donnent la meme valeur aux graphes transitifs ayant, a l'isomorphie prPs, le meme graphe de comparabilit6 sous- jacent)

.

Cette caracterisation a l'air tres interessante car elle permet l'unification de plusieurs resultats connus: - un theoreme de DILWORTH 161 sur l'indice de partition des sommets en chemins, - un theoreme dfi 2 ARDITTI [ I 1 et 2 TROTTER, MOORE, SUMNER [ I 9 1 sur la dimension d'un ordre partiel, notion introduite par DUSHNIK et MILLER 171, - un theoreme de STANLEY [lo] sur le nombre d'ordres totaux superieurs 2 un ordre partiel donne. En outre, nous obtenons un resultat analogue pour le nombre de sauts d'un ordre partiel (nous avons present6 ce dernier invariant dans [31 1 .

TIETRACT -_ _ - - - - - : we present here a characterization of comparability invariants. Let us recall that a comparability invariant is a directed graph invariant which keeps the same value over the whole class of transitive directed graphs that have the same underlying comparability undirected graph. This characterization seems to be a very practical tool since it allows the unification of several results: - a theorem of DILWORTH [61 on the path-partition number, - a theorem of ARDITTI [ l ] , and TROTTER, MOORE, SUMilER [191 on the dimension of partial orders, as defined by DUSHNIK and MILLER [ 7 ! , - a theorem of STANLEY [ l o ] on the number of linear extensions of a given partial order. Eesides, we obtain a similar result for the jump number of a partial order (we have introduced this invariant in [ 3 1 ) .

I -

INTRODUCTION

We deal in this paper with finite simple graphs (i.e. without loops and multiple edges). We u s e standard graph terminology ( s e e for example BERGE [ 2 ] 1 .

M . Habib

312 Therefore f o r a given graph

G = (X,

t h e induced subgraph a s f o l l o w s : d e n o t e by

IGI =

1x1

U) , and f o r e v e r y

G ( A ) = (A, V)

,

with

A

5X ,

V = U

n

we define

, and w e

A2

t h e number o f v e r t i c e s of t h e g r a p h G .

A d i r e c t e d graph G = ( X , U) i s c a l l e d t r a n s i t i v e l y o r i e n t e d i f U i s a t r a n s i V x, y , z 6 X , ( x , y ) E U and ( y , z) E U t i v e b i n a r y r e l a t i o n on X : i . e . implies ( x , z) E U ( S i n c e l o o p s are n o t a l l o w e d , w e j u s t o n l y c a r e w i t h t r i p l e s made up w i t h distinct vertices).

.

S i m i l a r l y a t r a n s i t i v e l y o r i e n t e d g r a p h which h a s no c i r c u i t i s c a l l e d a p a r t i a l order o r a poset. L e t u s d e n o t e by (resp. (resp. p a r t i a l orders),.

9

)

t h e c l a s s of a l l t r a n s i t i v e l y o r i e n t e d g r a p h s

For a g i v e n u n d i r e c t e d g r a p h G = (X, E) , w e d e f i n e an o r i e n t a t i o n $ of G , a s a d i r e c t e d g r a p h H = ( X , U) , d e n o t e d by $ ( G ) , such t h a t t o any edge $ ( [ a , b l ) 5 { ( a , b ) , (b, a ) } and U = $(e) [ a , bl E E , we a s s o c i a t e eeE

u

A c o m p a r a b i l i t y graph i s an u n d i r e c t e d g r a p h which h a s a t r a n s i t i v e o r i e n t a t i o n

(which i s n o t n e c e s s a r i l y a p a r t i a l o r d e r ) . For any graph.

G Ez

, we

d e n o t e by

Comp(G)

t h e a s s o c i a t e d undirected comparability

-

Two g r a p h s G and G ' , e l e m e n t s of t , are e q u i v a l e n t ( i n symbol G G ' ) i f and o n l y i f t h e i r c o m p a r a b i l i t y g r a p h s are isomorphic ( i . e . Comp(G) w Comp(G') ) . Note t h a t i f t h e r e i s several isomorphisms from Comp(G) t o Comp(G') , w e s h a l l choose one o f them (which we c a l l " t h e " isomorphism). O u r r e s u l t s do n o t depend on t h e c h o i c e d isomorphism. F o r u s g r a p h - i n v a r i a n t i s a mapping f from 7; t o R E 7; , G w H implies f(G) = f(H)

.

G, H

such t h a t f o r every

F u r t h e r m o r e , w e now r e a c h t h e main d e f i n i t i o n . DEFINITION:

The g r a p h - i n v a r i a n t f : 7; -B R ( r e s p . f : 9 -t R ) * i s a c o m p a r a b i l i t y 7; - i n v a r i a n t (resp. 3 - i n v a r i a n t ) i f and only i f : (resp. G, G' E ) t h e condition For e v e r y G , G ' E 6 G G' implies f (G) = f ( G I )

-

For any d i r e c t e d g r a p h

.

w e d e f i n e i t s i n v e r s e , G-

G = ( X , U)

by r e v e r s i n g a l l t h e a r c s , i . e . w i t h t h e above d e f i n i t i o n .

U- = {(x, y )

I

( y , x)

E

U]

=

(X, U-) o b t a i n e d

. Therefore

G

-

G-,

A t l e a s t , w e d e f i n e t h e t w o f o l l o w i n g t r i v i a l g r a p h s on p v e r t i c e s :

*

S = ( [ I , p ] , 0) P p-independent;

*

K

P '

t h e e m p t y graph,

t h e complement o f S

P

,

a l s o c a l l e d p - s t a b l e or

a l s o c a l l e d t h e symmetric complete g r a p h .

L e t u s r e c a l l two famous examples o f c o m p a r a b i l i t y i n v a r i a n t s t!iat in literature.

can b e found

-

The p a t h - p a r t i t i o n number o f a g r a p h G ( i . e . t h e minimum c a r d i n a l i t y of a s e t of e l e m e n t a r y p a t h s t h a t p a r t i t i o n s t h e v e r t i c e s of G ) , d e n o t e d by p ( G ) .

*

T h i s h y p o t h e s i s i s n o t c r u c i a l , we c o u l d e a s i l y g e n e r a l i z e t h e f o l l o w i n g r e s u l t s t o i n v a r i a n t s d e f i n e d from z ( r e s p . ? .' ) t o o t h e r s t r u c t u r e s t h a n R .

Comparability invariants

373

The well known theorem of DILWORTH [61 which states the equality between p(G) and a(G1 (the stability number) of any transitive graph G enables us to class p as a comparability %-invariant.

-

The dimension of partial orders defined by DUSHNIK and MILLER [ 7 1 , has also been classed as a comparability ?-invariant, nearly at the same time by ARDITTI [11, and by TROTTER, MOORE, SUMNER [191.

In the following, we shall prove a general result (theorem 3 ) that allows the classification of many comparability invariants and in particular of the two previous ones. Our approach is derived from the study of substitution decompositions of directed graphs, HABIB [ll]. For other studies on closely related decompositions see also: CUNNINGHAM, EDMONDS [5], MAURER [ I s ] , [161, and MOHRING, RADERMACHER [17]. Other authors have found similar results when studying comparability graphs, for a complete bibliography see GOLUMBIC [lo], chapter 5. We end this paper by using this characterization to prove that the jump number is a comparability .Ip -invariant, we have introduced this invariant in 131.

I1 -

A CHARACTERIZATION OF COMPARABILITY INVARIANTS

Let us first recall the definition of the substitution operation on directed graphs. DEFINITIONS:

Let

H = (Y, V ) , K = (Z, W)

be two directed graphs with

~ n z = 0 . We define

x u

G = (X,

=

(Y - {a])

=

wu

=

U)

H

for any

a E Y

,

as follows:

uz

t(x,y) E

VI

x

=

a, y = a1

u

u

i(x,y)

1 XEZ, ~

E Y ,(a,y) E

I(X,Y) 1 X E Y , Y E Z , (x,a) E

vl

VI

.

This operation is called an elementary substitution as we can consider the graph G obtained from the graph H by replacing the vertex a by the graph K. Furthermore Z is called an externally related subset of G. For a complete study and a bibliography on the substitution operation see HABIB [ l l ] , or more recently MOHRING, RADERMACHER [17]. We just notice that 9 is closed under this operation, but not

z .

Obviously there exist trivial elementary substitutions which are: For any graph G = ( X , U) and any a E X

,

Thus we can now define: G is irreductible when it admits only the above trivial decompositions by elementary substitution, otherwise G is called decomposable. L e t us notice that this definition implies that all the graphs with strictly less than 3 vertices are irreducible. W e define now a general substitution operation.

G

=

(..

.

(H :l) 1

...)

Gm m

where xi for i = 1,

-..,m

are distinct vertices of H.

M . Habib

314

Using t h e a s s o c i a t i v i t y of t h i s o p e r a t i o n , t h e p a r e n t h e s e s can be o m i t t e d and

When f o r

P

=

{XI,

...,

X

m

)

graph H whose v e r t i c e s a r e

t h a t p a r t i t i o n s t h e v e r t i c e s of G , t h e r e e x i s t s a

al,

...,

a

. ..

G = H G(X1) ('m) then al m ( T h e r e f o r e P is made up w i t h e x t e r n a l l y

such t h a t

m

P i s c a l l e d a c o h e r e n t p a r t i t i o n o f G. r e l a t e d s u b s e t s of G ) .

DECOMPOSITION RESULTS: L e t u s r e c a l l a w e l l known decomposition r e s u l t ( d i s c o v e r e d by many a u t h o r s ) . "Every d i r e c t e d g r a p h h a s a unique s u b s t i t u t i o n decomposition i n t o s u b g r a p h s , e a c h of whose members i s an i r r e d u c i b l e g r a p h , a complete graph, a s t a b l e o r a t o t a l o r d e r . " (For a p r o o f of t h i s r e s u l t , s e e f o r example CUNNINGHAM, EDMONDS [ 5 1 , MAURER [ 1 5 ] , [ 1 6 1 or HABIB [ I l l ) . I n o t h e r words, f o r e v e r y d i r e c t e d g r a p h G o n a v e r t e x s e t X , unique c o h e r e n t p a r t i t i o n d e n o t e d by P ( G ) = {X1, Xk}

...,

0

every i ,

1

<

i


G(Xi)

there e x i s t s a such t h a t f o r

i s a n i r r e d u c i b l e g r a p h , a complete g r a p h , a

s t a b l e or a t o t a l o r d e r . Therefore i, i

<

i

PO(G) i s t h e minimal non t r i v i a l c o h e r e n t p a r t i t i o n o f G ; f o r e v e r y

< k,

t h e subgraph

G(XI)

w i l l be c a l l e d a p r i m e component o f G . W e c a n

a p p l y t h i s p r o c e s s r e c u r s i v e l y and d e f i n e t h e f o l l o w i n g f i n i t e sequence of g r a p h s : Go = G , G1 = G / P O ( G ) , , G i + l = Gi/Po(Gi) ,

...

... .

Then we have c a n o n i c a l sequence of prime components and t h e i r p r o j e c t i o n on X y i e l d what we c a l l g e n e r a l i z e d prime components o f G.

W e must n o t i c e t h a t i n t h e above d e f i n i t i o n such a prime component K i s whether minimal and non t r i v i a l w i t h t h i s p r o p e r t y when K i s i r r e d u c i b l e o r maximal i n t h e t h r e e o t h e r cases. Of c o u r s e , a l l t h e s e n o t i o n s ( s u b s t i t u t i o n o p e r a t i o n , e x t e r n a l l y r e l a t e d s u b s e t , c o h e r e n t p a r t i t i o n , i r r e d u c i b l e , prime component, ) can b e d e f i n e d f o r u n d i r e c t e d g r a p h s , and e x a c t l y si m i l a r r e s u l t s h o l d . F u r t h e r m o r e , i n t h i s p a r t i c u l a r c a s e , prime component c o r r e s p o n d s t o m u l t i p l e x d e f i n e d by GOLUMBIC i n [ 9 1 .

...

RELATIONSHIP BETWEEN DECOMPOSITION OF G AND COMP(G) Of c o u r s e t o e v e r y s u b s t i t u t i o n d e c o m p o s i t i o n of a d i r e c t e d g r a p h , w e c a n a s s o c i a t e t h e same s u b s t i t u t i o n d e c o m p o s i t i o n o f t h e u n d e r l y i n g u n d i r e c t e d graph. The c o n v e r s e i s i n g e n e r a l f a l s e , b u t f o r t r a n s i t i v e g r a p h s , w e have some p a r t i a l r e s u l t s r e l a t e d w i t h prime components a s d e f i n e d p r e v i o u s l y . B u t b e f o r e g o i n g f u r t h e r , w e must mention a q u i t e s t r a n g e r e l a t i o n s h i p between s u b s t i t u t i o n decomposition and f o r c i n g c l a s s e s o f e d g e s . L e t u s r e c a l l t h e d e f i n i t i o n of t h e f o r c i n g classes o f e d g e s . L e t G = ( X , E ) b e a n u n d i r e c t e d g r a p h , we s a y t h a t [ a , b ] E E forces [b, c ] E E i f and o n l y i f [a, c ] E E or a = c

.

The t r a n s i t i v e c l o s u r e of t h i s b i n a r y r e l a t i o n d e f i n e d on t h e e d g e s o f G , p a r t i Ek t i o n s E i n t o f o r c i n g classes of edges: El ,

...,

.

When l o o k i n g a t t h e t r a n s i t i v e o r i e n t a t i o n s of G , we see t h a t t h e o r i e n t a t i o n of t h e whole f o r c i n g c l a s s i s c o m p l e t e l y d e t e r m i n e d ( f o r c e d ) by t h e o r i e n t a t i o n of one o f i t s e l e m e n t s .

375

Comparability invariants

,

i E [l, k l

Furthermore for every

an externally related subset of G

X. =

{x E

X

1

3 e E Ei

, e n x

=

01

is

,

This relationship between forcing classes of edges and substitution decomposition was first noticed by GALLAI [ 8 1 , and rediscovered by GOLUMBIC 191. This relationship can be strengthened, for details see GOLUMBIC [ l o ] , and this yields 3

an O(n ) algorithm for the substitution decomposition of undirected graphs, see HABIB, MAURER [12]. In the following we only need a partial result, although generalized in the directed case.

-

THEOREM 1

If G E , having strictly more than 2 vertices, is is reduced to G, irreducible then the class of G modulo

-

and G-

.

Hints for t h i s p r o o f : In 1111, I have proposed a generalization of the .................... forcing relation, let us say the "directed f o r c i n g relation". This new relation keeps the same property about externally related subsets. Besides an irreducible transitive graph G with n > 2 vertices is necessarily a partial order. At least for the class of partial orders the forcing relation and its generalization are identical and thus we can apply the similar result for undirected graphs, see for example GOLUMBIC [ l o ] . THEOREM 2 -

Let

G E

then

G

=

, and K be a prime component of H G (K)

Comp(G),

(Where G(K) is the subgraph of G associated with K and where H is the simple graph obtained by contracting the subgraph G(K) of G in a single vertex a E GI.

x C H- a

Proof: ----- Let us suppose there exists

edge of

Comp(H)

.

Since

such that

[x, a]

, necessarily all the vertices

Comp(G) = Comp(H)E

of G(K) are connected with x. This yields a partition P = {XI, X2, X 3 j that : X1 = {y E K I (x, y) E G , (y, x) 4 GI

X2

=

{y E K

X j = {y E K

1 I

G

(x, y)

(x, y) F G

is an

, (y, x) E , (y, x) E

such

GI

.

GI

IKI > 3 . Necessarily G ( K ) is also irreducible. Since there is no irreducible partial order having 3 vertices, we can suppose IK/ > 4 . Using the transitivity of G and G(K), the partition P yields necessarily a substitution decomposition of G ( K ) , as we have supposed G(K) irreducible IKI > 4 implies IPI = 1 this decomposition must be a trivial one. But G (K) And thus we have: G = H

1 ) When K is irreducible and

.

2) When K is a stable, therefore G(K) is also a stable. Then using the transitivity we have:

*

*

If \ X 3 \ # 0 then 1X 1 = 1 and IX I = IX 1 = 0 and so 3 1 2 reduced to a single vertex and the result is trivial. If

IX

have:

3

I

=

0

G = H

,

then whether

G (K)

IX

2

I

=

0 or

IX

1

I=

3 ) When K is a complete graph. If I P / = 1 the result is obtained, let us now suppose:

0

IPI

G(K)

and thus we

>

2

.

is

M.Habib

316

As K is a prime component of Comp(G), Ktx is not a prime component of Comp(G). Thus there exists y E H - a - x such that [x, 21 is an edge of Comp(H), and [y, 21 is not an edge of Comp(H) for some z E K . /PI = 2 and using transitivity there exists necessarily some vertex E G(K) connected in G with y , and thus using the substitution decomposition of Comp(G), we can see that K+x is a prime component of Comp(G) , which is excluded. and G = H G (K) And thus ( P I = 1

As z'

Since (KI = 2 and K irreducible implies that K is a stable or a complete graph, we have considered all the cases. From this result we can easily deduce the very useful following result. COROLLARY

-

Let G, H, K E ^ e

and

G = H

K

with

a E H

.

If Comp(K)

is a prime component of Comp(G) , then for any G' E % G'

H'

-

-

, , there exist H' and K' E % , a' E H' such that: K' (where a' corresponds H , K' K , and G' = H' a ,

G

-

to a in the isomorphism from Comp(H') to Comp(H)

)

.

In other words if K is a prime component of an undirected comparability graph G, then every transitive orientation $(GI admit a substitution decomposition "around" $(K) Of course this result is still valid for any generalized prime component of Comp(G) .

.

Now we can express the main result of this paper. THEOREM 3 -

Let f : % + R (resp. ? + R ) a graph-invariant then f is a comparability Z-invariant (resp. ? -invariant) if and' only if f satisfies the following conditions (O), (i) and (ii):

*

.

t ,

(Where T and K2 denote respectively 2 the total order and the symmetric complete graph on two vertices).

( 0 ) f (T2 = f(K2)

(i) For every G E Z (ii) For every H, H' E^G For every K, K' E t

(resp. 3'

)

(resp. 3 (resp. 9

) )

f(G) such that such that

=

H K

.

f(G-)

--

H' K'

, ,

For a E H and a ' its image . by -the isomorphism from Comp(H) to Comp(H') , the condition f(H) = f(H') and f(K) = f(K') implies K K' f(H a) = f ( H ' a,)

.

pof:

These conditions are obviously necessary. Let us consider their sufficiency, first in the case of comparability %-invariants.

The proof goes by induction on the number of vertices, and since there exists only one simple directed graph having one vertex, the result is obvious in this case. Let u s suppose for every have: f ( G ' ) = f (G")

.

W e consider now

G',

GI, G2 E ?f,

G" E %

such that

GI

-

G"

and

(GI(

< n , we

two transitive orientations of the same comparabi-

Comparability invariants

lity graph having n vertices

(i.e. Comp(G

)

1

Comp(G2)

=

Comp(G1) ,

Let us take K a prime component of

377

)

.

IKl > 1

with

,

(there exists

at least one such prime component, Comp(G1) itself). Then using theorem 2, we have: Gl(K) and

rily we have: 1") If

GI a(K)

Gl(K)

K # Comp(G

-

G2(K)

,

)

1

and

l

H1

H

N

IG1(K)I < n

then

2") If K

H2 G2 (K) where a2 associated with K. Necessa-

,

=

f(H2)

=

Comp(G)

2 '

.

By induction hypothesis we know that

f (H 1

G2 =

and 1 G2(K) are the subgraphs of resp. G1, G2 G1 = H

f (GI(K)) = f (G2(K)) and

hence we conclude with the condition

,

H1 = ({a,}, p )

then

H2

fi:

,

(ii).

I

and

/G1(K)

=

n

.

a) If K is irreducible and IKI > 3 using theorem 1 and condition (:)

,

b) If K is a stable, then G1 and

are also isomorphic stables.

Therefore G

1

G

M

and

2

G2

.

then the result is obtained by

f(G1) = f(G2)

since f is a graph-

invariant. c) If K is a complete graph. We notice that any transitive orientation of Comp(G1) can be obtained by substituting complete graphs in a total order.

cl, Thus

G1

- O1

xl,

..., c ..., xP

D1, and

is a total order on p vertices

...,

D

. . . I

Yq

G2 = 0

2 Y1'

(resp. q ) , and

,

C . D. I' I'

where

1 < i

O1 (resp. 0 2 )

1

<

j


are complete graphs. Let us choose x G2 = F

J2 x:

,

E

C1 and

where

Of course, we have

x2

F 1 , F2

F1

-

F2

E

D1

,

then we may write:

have only two vertices namely

and

J

-

J2

J1 F 1 x, , and 1 {x,, xi} and

G1

=

and hence using the induction hypo-

thesis and conditions ( O ) , (ii), we have finished. And exactly similar proof gives the corresponding result for comparability

9 -invariants.

1.-

Conditions ( 0 ) and (i) are obviously satisfied by many graphinvariants, and there is no need of condition ( 0 ) for comparability P -invariants.

2.- To establish condition (ii), we shall prove, in each particular application, a formula to compute

f(G)

when

G = G

G2 a

,

and

show that this formula yields condition (ii), this can be written as corollaries.

M . Habib

378

Indeed we have written the two corollaries that follows for the purpose of classification of three graph-invariants: path-partition number, dimension and jump-number. These results give the flavor of similar properties that have to be obtained from theorem 3 in order to classify a particular graph-invariant.

COROLLARY 1. If f : 6 -+ B (resp. 9 satisfies (01, (i) and:

+

(iii) There exists a mapping V G1, G2 € 6

G2 f (G1 a ) ^c;

Proof: _----

=

We just have to show

)

(v) There exists a mapping G1, G2 E^c; G1

,

-+

R

such that:

,

a E GI

then f is a comparability

(iii) implies (ii)

(iv) V G E ^c; (resp. 3 f(G) < \GI .

E

2

v

f (G2))

-+

V a

R

:

(resp. 9 -invariant).

COROLLARY 2. If f : Tt; N (resp. P ! and satisfies (01, (i) and:

v

$

(resp. .'p ) ,

$ (f( G 1 ) ,

-invariant

is a graph-invariant and

R )

N

-+

, which

is a graph-invariant

)

which is not a stable $ : NL

3

N

such that:

(resp. 3 ) ,

f(G1

G2 a)

f' =

$ (f(G2),f(G1

then f is a comparability invariant).

K1

-

,

K2

If we take Comp(H2)

,

IK1( + IH1( = n

al E H1

and

a2

and

,

K 1 , K2 € ' G

f(H ) = f(H2) 1

(G2)

and

such that

(resp. 9 -

that conditions H1

f(K1) = f(K2)

its image by the isomorphism from

1

a

r:-invariant

Proof: _---- We shall prove it by induction on n = \HI + IKI (iv) and (v) imply the condition (ii) of theorem 3. Let u s now consider H1, H2 E 6

is obvious

-

H

.

Comp(H1) to

= $

(f(K1), f(H1

1

'f(K1) a 1 1

$ (f(K2), f(H2

f ' (KZ) a 1

I

and

H

,

f(H2

=

2

2

a) Let K1 is not a stable, then using condition (iv), \ H I + IK1 I < n thus by induction we have finished.

K

1

,

and then apply (v) we have:

f(H1

b ) If

H2

is a stable, let us consider

component of

Comp(G1)

K1 G1 = HI al

is a stable, then

.

If every generalized

Comp(G1) admits only two different

transitive orientations. (SHEVRIN and F I L I P P O V have first noticed this fact in [ 1 8 1 ) . And then the result is given by condition (i)

.

Else Comp(G1) admits a generalized prime component T which is not a stable and

Comparability invariants

K1

H1(T)I K 1 therefore:

G1 =

HI1

'

bl

H'l a

=

al

379

, HI(")

.

Hence applying (v), we

1 ' bl

K

IH'll

Since

I

ISf(H1(T))

+

I11 -

,

< n

we can use the induction hypothesis

APPLICATIONS

A) Path-partition number This invariant p : 3 + N satisfies trivially conditions ( 0 ) and (i). In order to obtain condition (ii) we may use the following result. PROPOSITION 1 -

Let G, G1, G2 E Z

such that

G = G

l

G1 a

a E G

and

1 '

S P(G2)

then

p(G)

=

Proof: -_-- The inequality

p(G)

< p(Gl

p(G

l S

-

change each vertex of

.

h

> p(G2 )

Let us suppose these paths are

If

h

p(G2)

If

h > p(G2)

=

, ,

P(G2) is obvious as we can

)

a

Let us consider now the other inequality. Let

G2

.

by a path of a path-partition of G

S P(G2)

partition of G. There exist

1

a

we have finished we contract in

P

{Cl,

=

...,

2'

Ckj

a path-

paths of this partition P that cross cl,

. . . I

.

'1'

' * . I

.

h'

the vertices of

'p(G2)

G

2

to o n e

vertex (for example the first one of the path in G2). Thus we obtain CI1

, ...,

Sp (G2) C'

PtG2)

paths

Of

Furthermore, we delete from possible as C'

p(G2)+1 ,

Thus Since

...,

Cp(G2)+1

. , ...,

Ck

the vertices of

G

2

(thus is

and we obtain

new paths.

Cth

S

...,

IPI = IP' I

,

=

a

GI was supposed to be transitive),

iC'l,

P'

G1

Ctk}

is a path-partition of

G1

P(G-) 2 a

we have the desired result. m

Thus by proposition 1, it is not very hard to notice that we can apply the corollary 2 of theorem 3, with mapping @ equal to the second projection mapping. So the path-partition number is a comparability 3 -invariant. An

exactly argument shows that the path-covering number is a comparability

6 -invariant.

M . Habib

380

Note -

I n t h e f o l l o w i n g examples, we s h a l l now o n l y c o n s i d e r p a r t i a l o r d e r s and hence c o m p a r a b i l i t y 9 - i n v a r i a n t s .

DEFINITION:

L e t P be a p a r t i a l o r d e r , we d e n o t e by C ( P ) t h e s e t of a l l t o t a l ord ers g reat er than P ( a l s o c a l l e d t h e s e t of a l l l i n e a r extensions of P )

.

P a r t i a l o r d e r dimension

W e r e c a l l t h a t t h e dimension of a p a r t i a l o r d e r P , d e n o t e d by d e f i n e d by DUSHNIK, MILLER i n [ 7 ] , a s t h e minimum i n t e g e r h ,

n

P =

with

T~

Ti

d i m ( P ) , was such t h a t :

c(P)

E

1 6 i c h For a r e c e n t s u r v e y on p a r t i a l o r d e r dimension t h e o r y , s e e KELLY, TROTTER [141 W e j u s t r e c a l l t h e p r o o f s made by A R D I T T I [ l ] and TROTTER, MOORE, SUMMER 1191 a s t h e y i n s e r t t h e m s e l v e s v e r y w e l l i n o u r frame.

.

HIRAGUCHI f i r s t noticed i n t 1 3 1 :

,

V P1, P2 E .’P

V a

E P1

dim(P p 2 ) l a

:

.

= max { d i m ( P 1 ) , d i m ( P 2 ) )

With t h i s formula and t h e p r e v i o u s c o r o l l a r y 1 o f t h e theorem 3 , w e can t r i v i a l l y show t h a t t h e dimension i s a c o m p a r a b i l i t y 9 - i n v a r i a n t , by i d e n t i f y i n g $ w i t h t h e a p p l i c a t i o n max. Number of l i n e a r e x t e n s i o n s of a p a r t i a l o r d e r

I

1

.

L e t u s adopt t h e following n o t a t i o n L(P) = f (P) From GOLUMBIC [ 101, ?-invariant. w e know t h a t STANLEY h a s proved t h a t L ( P ) i s a c o m p a r a b i l i t y S i n c e it i s very e a s y t o be c o n v i n c e d of t h e f o l l o w i n g formula P V P1, P2 E 9 and V a E P1 , L ( P 1 a 2 ) = L ( P 2 ) . L(P1

”,2

f o r any

o2 E

c

(P2)

,.

t h e n w e c a n p r o v e , a s i n c o r o l l a r y 2 of theorem 3 , t h a t t h i s formula i m p l i e s t h e c o n d i t i o n ( i i ) o f theorem 3 . Here t h e argument g o e s by i n d u c t i o n on t h e number of uncomparable p a i r s of v e r t i c e s of t h e p a r t i a l o r d e r . Hence w e have a n o t h e r p r o o f of STANLEY’S r e s u l t .

E ~ t g-

T h i s r e s u l t c a n n o t b e g e n e r a l i z e d t o t h e number of t o t a l p r e o r d e r s g r e a t e r than a given p r e o r d e r .

D) J u m p number

For

P E9

(x., x .

and

1

<

T =

i

xl,


..., x

such t h a t

C

E xi

(P)

< xi+l

,

a “jump“ of T i s a c o u p l e

.

P

W e d e f i n e a l s o t h e j u m p number o f T , d e n o t e d by u ( 7 , P ) a s t h e number o f a ( P ) = m i n o (T, P) , t h e jump number o f J . such c o u p l e s . S i m i l a r l y we d e f i n e T E f (P) W e have f i r s t b e l i e v e d t h a t : V P 1 , P2 E 9 and f o r any x E P1 , P o ( P I x 2 ) = IJ ( P 1 ) + 0 ( P 2 )

.

And we wrote it i n

[111

.

38 1

Comparability invariants N e v e r t h e l e s s , COGIS, POGUNTKE and R I V A L [ 4 ] showed u s a counter-example ( c f . f i g u r e 1). n

r 2 ) , u ( P 1 ) and a ( P 2 ) i s a l i t t l e more 1 x s o p h i s t i c a t e d a s w e s h a l l see i n t h e f o l l o w i n g theorem 4 . T h i s theorem means r o u g h l y t h a t t h e p o s e t d e s c r i b e d i n f i g u r e 1 i s t h e " o n l y " counter-example o f t h e above formula. Indeed t h e formula between a ( P

To prove t h i s theorem, w e need two lemmas and u n t i l t h e end of t h e p r o o f w e now assume

P = P

p2 1 x

w i t h P 1 , P 2 E .'P

.

x E P1

and

For any T E C ( P ) w e use i t s c a n o n i c a l d e c o m p o s i t i o n i n t o maximal s e q u e n c e s , (resp. P 2 ) , d e n o t e d by a . ( r e s p . h i ) , made up w i t h v e r t i c e s o f P1

,

Furthermore t o e a c h o f t h e s e s e q u e n c e s a . or b .

1

w e c o n s i d e r two d i s t i n g u i s h e d

v e r t i c e s , which are r e s p e c t i v e l y t h e l e a s t and t h e g r e a t e s t e l e m e n t w i t h r e s p e c t t o the t o t a l order T W e d e n o t e t h i s e l e m e n t s by s u p ( l ) , i n f ( 1 ) for such a sequence 1 .

.

LEMMA 1

-

...

... E

a. bi+l T = b. 1 1+1 ( s u p ( b i ) , i n f ( a i + l ) ) and

If

jumps o f T

f

then

(P)

(sup(ai+l), inf(bi+l))

are

.

-----

P r o o f : By t h e d e f i n i t i o n o f e l e m e n t a r y s u b s t i t u t i o n a l l t h e v e r t i c e s of t h e s e q u e n c e s b . and b . have t h e same neighbourhood o u t s i d e o f P 2 ' 1+1 Since T E f (P) , necessarily the v e r t i c e s of a. are n o t comparable w i t h 1+1 t h o s e o f b . and b . 1+1 *

1

-

LEMMA 2

a (P ) 1

+ u

more, when

-

(P ) 2

1< a ( P ) < o ( P 1 ) +

u (P2)

>

1

, u

and o n l y i f t h e r e e x i s t s

u(.rl, P

that

)

= a(P1)

(P) =

T~

,

=

1

a x a a

1 2 3 sup(a ) < x

1

and f u r t h e r -

0(P2)

u(P ) + o(P2)

-

1

f (P1)

E

,

<

x

if

such

inf(a3)

,

l P

1 p1 and x i s n o t comparable w i t h t h e v e r t i c e s o f a2 ( s u p ( a 2 ) , i n f ( a 3 ) ) i s a jump o f T

,

and

E f

(P2)

.

Emof: Let T E 1 Thus w e make

1") The r i g h t i n e q u a l i t y i s n e a r l y o b v i o u s . W e c a n decompose

that satisfies O(P) 2 O ) Let

< U(P1) T(P)

't1

U(T,

+ U(PJ

= T 1' x T 1"

P) = U ( T l ,

. P1)

+ a ( ?2 , P 2 )

f (P1) , T = T

'

T

2

T2 TI"

and w e have :

.

be an optimal l i n e a r e x t e n s i o n

(i.e.

U(T,

P) = U ( P ) )

E

. c (P)

M. Habib

382 a)

T

= al bl

+

o(P1)

... b9

o(P2)

<

a

q+l

o(P)

.

q

If

.

O (T,

P) 4

= al b

T'

.

P)

U(T,

b

1

, when

T

a

2

2

a

T

=

a l b l a 2 b2 a 3 E

.

f(P)

a l b l b2 a 2 a j E

=~

'

C (P)

>

a

...

3 b3

q

Thus we can a p p l y t h e same p r o c e s s u n t i l T~

>

q

.

2

(sup(bl), inf ( a 2 ) ) , (sup(a2), inf ( b 2 ) ) ,

( s u p ( b 2 ) , i n f ( a 3 ) ) a r e jumps of

that

then obviously

L e t u s c o n s i d e r t h e c a s e where

From lemma 1 , we know t h a t

Then we can make:

,

1

=

>

.

3

3

C

E

aq + l

( P ) , such

and o b t a i n

I n t h i s c a s e , we can always make

.

But i f s u p ( a ) 4 i n f ( b l ) , 1

p1

s u p ( b ) 4 i n f ( a 2 ) and s u p ( a 2 ) , i n f ( a ) 2 3 p1 then O ( T [ ' ~ ,P ) = U ( T ~ ,P ) t 1 , e l s e

a r e n o t comparable i n P

P)

U(TI0,

= U ( T ~ , P)

1'

.

(The f i r s t c a s e can o c c u r , s e e f i g u r e 1 ) . Anyway, we always have: O(P)

>

O(P1) +

U P 2 )

+

1

= U(P )

+

P)

P) 4 O(To,

"(TIo,

and t h e r e f o r e

.

- 1

L e t u s c o n s i d e r t h e two o t h e r c a s e s .

@

)

~

= ba

...

-

when

q = 1

-

when

q

>

2

such t h a t y ) ~ = ab l

...

a

q

, ,

b

.

q+l

we t r i v i a l l y have

u s i n g lemma 1 , we make

U(T'

,

<

P)

o(P)

i s n e c e s s a r i l y a jump o f

...

T

i

2

...

= al bl

T'

.

a or b q q o(P1) + o(P2)

=

=

o(P )

b

9

a

9

E

f ( P

P).

O(T,

z

where

z

two s e q u e n c e s t h e n

T' = a l b l b 2

O(P)

I f T i s made w i t h o n l y

.

Else

(sup(bl), inf(all

and we c o n s t r u c t :

z E f (P)

u ( T ' , P)

with

<

O(T,

P)

.

Hence i n c a s e 6 o r y , w e b u i l d a n o t h e r l i n e a r e x t e n s i o n on which w e c a n a p p l y t h e r e s u l t o f case a Thus, we have proved t h e f i r s t i n e q u a l i t i e s o f lemma 2 .

.

L e t u s now prove t h e e q u i v a l e n c e s t a t e d i n lemma 2 . 1 ) I f t h e r e e x i s t s such a l i n e a r e x t e n s i o n

T~

E f (P1), since

o(P2)

>

1

,

t h e r e e x i s t s a n o p t i m a l l i n e a r e x t e n s i o n of P 2 ,

T = bl b2 such t h a t 2 Therefore T = a l b a b a T ( s u p ( b l ) , i n f ( b ) ) i s a jump o f 1 2 2 3 E 1 ( P ) 2 and s i n c e ( s u p ( a 2 ) , i n f ( a 3 ) ) i s a jump o f T , w e h a v e :

.

U(T,

P)

< O(P1) +

O(P2)

- 1

. o(P) = o(P )

2 ) C o n v e r s e l y , i f w e suppose

1

+

o(P2) - 1

we n o t i c e t h a t as i n

t h e p r e v i o u s p r o o f o f t h e l e f t i n e q u a l i t y o f lemma 2 , we must n e c e s s a r i l y find a linear extension: sup(al)

< P

inf(bl)

,

T~

sup(b ) 2

b a b a E C(P) 1 1 2 2 3 i n f ( a 2 ) , and s u p ( a 2

= a

< P

,

with

n o t comparable w i t h

Comparability invariants

383

i n f ( a 1.

3

Thus t h e r e e x i s t s and

sup(a2)

= al x

T~

a2 a j E

i s n o t comparable w i t h

t h e v e r t i c e s of a

Proof of theorem 4 : -----__-----_---__

=

o(P ) 1

t

o(P2)

inf(a ) 3

2 '

I t j u s t r e m a i n s t o p r o v e f o r any

o(P)

sup(a ) < x , x < i n f ( a ) , 3 I P P and x i s n o t comparable w i t h

with

C(P1)

- 1

P2 E

, with

3'

u(P2)

>

1

S,

i f and o n l y i f

L e t u s look a t t h e e q u i v a l e n c e s t a t e d i n lemma 2 , we n o t i c e t h a t t h e r i g h t c o n d i t i o n depends o n l y on P1 and t h u s f o r any P 2 E ,? , w i t h o(P2) > 1 , the

c o n d i t i o n i s t h e same and t h u s we can t a k e t h e s m a l l e s t p a r t i a l o r d e r w i t h u(P2) > 1 , i.e. s2 . From t h i s theorem 4 , w e deduce: COROLLARY 1 : L e t P

1'

P

2

E .'P

and

x € PI

,

then

o(P )

2

>

1

implies

Proof: _ _ _ _ _ T h i s r e s u l t i s o b v i o u s l y o b t a i n e d by a p p l y i n g t h e above theorem 4

so (P2)

P

t o the posets:

p1 x

COROLLARY 2 :

and

P

1

x

The jump number i s a c o m p a r a b i l i t y

3 -invariant.

Proof: _---- S i n c e V P E 9 , o ( P ) < /PI , t h e p r e v i o u s c o r o l l a r y 1 e n a b l e s u s t o u s e t h e r e s u l t c o n t a i n e d i n t h e c o r o l l a r y 2 o f t h e theorem 3 w i t h a mapping I$ e q u a l t o t h e second p r o j e c t i o n mapping minus o n e .

AS-n-coNczusLo!! _______________ : I t seems t h i s approach h e l p t o c l a s s i f y a wide c l a s s o f c o m p a r a b i l i t y i n v a r i a n t s , f u r t h e m o r e it y i e l d s some o t h e r p r o b l e m s . L e t u s go back t o t h e f i r s t example c i t e d : t h e p a t h - p a r t i t i o n number. W e know from a theorem o f DILWORTH [ 6 ] , t h a t t h i s number i s e q u a l t o t h e c a r d i n a l i t y of a maximum i n d e pendent s e t i n t h e a s s o c i a t e d c o m p a r a b i l i t y g r a p h .

T h u s t h e r e i s no need o f t h e o r i e n t a t i o n t o compute t h e v a l u e o f t h e p a t h p a r t i t i o n number. T h i s f a c t c o u l d be g e n e r a l i z e d f o r o t h e r c o m p a r a b i l i t y i n v a r i a n t s and t h e r e f o r e , w e c a n a s k t h e f o l l o w i n g q u e s t i o n . How i n t e r p r e t o r compute a c o m p a r a b i l i t y i n v a r i a n t on t h e a s s o c i a t e d u n d i r e c t e d comparability graph ? Is t h i s p r o p e r t y always r e l a t e d w i t h min-max p r o p e r t i e s ? ( A s f a r as we know, t h e r e i s no such r e s u l t f o r t h e dimension o r t h e jump number)

.

@-O".EEEMEWs_ __________-----

:

We c o u l d l i k e t o t h a n k M. C H E I N ,

U. FAIGLE, R. SCHRADER, W. POGUNTKE and an anonymous r e f e r e e f o r t h e i r c a r e f u l r e a d i n g and h e l p f u l c r i t i c i s m s o f t h e f i r s t d r a f t s of t h i s p a p e r .

M . Habib

38 4

:Hi; 1

p1

2

Figure 1

[l]

-

[2]

- C. BERGE, Graphes e t h y p e r g r a p h e s , Dunod. Paris ( 1 9 7 3 ) .

[3]

-

M.

[4]

-

0.

[5]

-

W. H. CUNNINGHAM, J. EDMONDS, A c o m b i n a t o r i a l d e c o m p o s i t i o n t h e o r y . Canad. 3. Math., Vol. 3 2 , N o 3 , (1980) 734-765.

[61

- R.P. DILWORTH, A d e c o m p o s i t i o n theorem f o r p a r t i a l l y o r d e r e d s e t s ,

J.C. ARDITTI, G r a p h e s d e c o m p a r a b i l i t . 4 e t d i m e n s i o n d e s o r d r e s , Note de Recherches, CRM 607. Centre de Rech. Math. Univ. Montreal ( 1 9 7 6 ) .

CHEIN, M. HABIB, T h e j u m p number of d i g r a p h s and p o s e t s a n i n t r o d u c t i o n . Annals of Discrete Mathematics, 9 (1980) 189-194. COGIS, W. POGUNTKE, I. RIVAL, P e r s o n a l Communication.

Ann. Math. Ser. ( 2 )

51

(19501 161-166.

[71

-

[81

- T. GALLAI, T r a n s i t i v o r i e n t e r b a r e g r a p h e n . Acta Math. Acad. Sci. Hangar.

B. DUSHNIK, E.W. MILLER, P a r t i a l l y o r d e r e d s e t s . h e r . J. Math, 6 3 , (1941) 600-610.

18

(1967)

25-66.

[91

-

M.C. GOLUMBIC, C o m p a r a b i l i t y g r a p h s and a n e w m a t r o i d . J. Combin. Theory (B), 22 (1977) 68-90.

[lo]

-

M.C. GOLUMBIC, A l g o r i t h m i c g r a p h t h e o r y and p e r f e c t g r a p h s . Academic Press, N.Y. ( 1 9 8 0 ) .

1111

- M. HABIB, S u b s t i t u t i o n d e s s t r u c t u r e s c o m b i n a t o i r e s , theorie e t a l g o r i t h m e s . These Sci. Math. Universite P. et M. Curie

1121

(1981).

- M. HABIB, M.C. MAURER, On the X - j o i n d e c o m p o s i t i o n f o r u n d i r e c t e d g r a p h s . Discrete Applied Math. 1 ( 1 9 7 9 ) .

201-207.

Comparability in variants [I31

- T. HIRAGUCHI, On t h e d i m e n s i o n of p a r t i a l l y o r d e r e d s e t s . Sci. Rep. Kanazaws Univ. 1, (1351) 77-94.

[I41 - D. KELLY, W.T. TROTTER Jr., D i m e n s i o n t h e o r y f o r o r d e r e d s e t s , i n O r d e r e d s e t s . I. RIVAL ( e d . ) D. REIDEL Publish. Comp. (1980) 171-211. [I51

- M.C. MAURER, U n i c i t e d e l a d e c o m p o s i t i o n d ' u n g r a p h e en j o i n t s u i v a n t un g r a p h e j o i n t i r r k d u c t i b l e d ' u n e f a m i l l e d e ses s o u s - g r a p h e s . C.R.A.S., Paris, t. 282, serie A (1976) 289-292.

(161 - M.C. MAURER, J o i n t s e t d 6 c o m p o s i t i o n . s p r e m i i . r e s , d a n s l e s q r a p h e s . These de troisieme cycle. Universite P. et M. Curie, (1977). [I71

-

R.H. MOHRING, F.J. RADERMACHER, S u b s t i t u t i o n d e c o m p o s i t i o n f o r d i s c r e t e s t r u c t u r e s a n d connections w i t h c o m b i n a t o r i a l o p t i m i z a t i o n . Preprint.

[I81 - L.N. SHEVRIN, N.D. FILIPPOV, P a r t i a l l y o r d e r e d s e t s a n d t h e i r c o m p a r a b i l i t y g r a p h s . Siberian Math. J. 11, (1970), 497-509.

[I91

- W.T. TROTTER, J.I. MOORE, D.P. SUMNER, T h e d i m e n s i o n o f a c o m p a r a b i l i t y g r a p h . Proc. Amer. Math. SOC. 60, (1976) 35-38.

385

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Annals of Discrete Mathematics 23 (1984) 387-396

387

0 Elsevier Science Publishers B.V. (North-Holland)

C O D A G E S E T DIMENSIONS D E RELATIONS B I N A I R E S A . IiOUCHET Dkpartement d e MathGmatiques e t I n f o r m a t i q u e U n i v e r s i t t ? d u Maine R o u t e d e L a v a 1 - 72017 L E MANS C e d r x FRANCE ABSTRACT.-

5 E 1 X E 2 be a b i n a r y r e l a t i o n . The means t h a t ( e l , e . ) v e r i f i e s t h e r e l a t i o n .

L e t E l a n d E 2 be two s e t s a n d p notation

tf

,

p e2

-

2

A c o d i n g of p i n t o t h e complete l a t t i c e El T , (p2: E 2 + T s u c h t h a t e l p e 2

(pl:

+

T i s d e f i n e d by two maps (pl(el)

< W2(e2)

.

I f E l and E 2

a r e f i n i t e , t h e d i m e n s i o n o f p i s t h e m i n i m a l v a l u e o f d s u c h t h a t p c a n be coded i n t o a p r o d u c t o f d c h a i n s . I t i s shown t h a t d i s e q u a l t o t h e d i m e n s i o n o f D u s h n i k a n d M i l l e r when and p d e f i n e s a p a r t i a l o r d e r . A s a n a p p l i c a t i o n o f t h e main r e s u l t w e El E2 d e r i v e a s h o r t p r o o f o f Mac N e i l l c ' s c o m p l e t i o n t h e o r e m . The c o d i n g s i n t o B o o l e a n l a t t i c e s a r e more e s p e c i a l l y s t u d i e d .

7

INTRODUCTION Chaque r e l a t i o n b i n a i r e c o n s i d e r k e s e r a d b f i n i e s u r un p r o d u i t c a r t b s i e n E X E d e d e u x e n s e m b l e s q u i p o u r r o n t G t r e d i s j o i n t s ou n o n , f i n i s ou i n f i n i s . 1 2 Une t e l l e r e l a t i o n s c r a n o t C e [ E , , p, E 2 1 , p b t a n t l e s y m b o l e r e l a t i o n n e l . On Ccrira alors pour s i g n i f i e r que I e couple ( e l , e ) v b r i f i e l a r e l a t i o n . 2 clpe2 A i n s i , u n e r e l a t i o n d ' o r d r e p a r t i e l s u r u n e n s e m b l e E p o u r r a Ctre n o t e e [ E , <, E l . S o i t une r e l a t i o n b i n a i r e = (pl

[T, <, TI

: E

1 + T

et

.

Un c o d a g e (p2 : E 2

+

R = [ E l , p, E21

(Q,, T

(p,)

: R

t e l l e s que

+G

e t un t r e i l l i s c o m p l e t

est c o n s t i t u b p a r deux f o n c t i o n s

e l P e2

Q

vl(el)

< v21e2) . <, E l

c'=

Considerons m a i n t e n a n t une r e l a t i o n d ' o r d r e p a r t i e l [E, i m m e r s i o n d e o d a n s e e s t u n e a p p l i c a t i o n u, : E + T t e l l e q u e e l < e2 v ( e 1 ) < w(e,). L e c o u p l e ((p, cp) e s t a l o r s u n c o d a g e d e Q

. Une fl d a n s Z .

A i n s i l ' e x i s t e n c e d ' u n e i m m e r s i o n i m p l i q u e c e l l e d ' u n c o d a g e mais l a r e c i p r o q u e n ' e s t pas bvidente. Le r e s u l t a t p r i n c i p a l d e c e t r a v a i l c o n s i s t e e n un t h b o r h m e q u i admet p o u r c o n s k q u e n c e l ' b q u i v a l e n c e e n t r e c o d a g e s e t i m m e r s i o n s . On d C f i n i t a l o r s l a dimens i o n d ' u n e r e l a t i o n b i n a i r e R c o m e l e p l u s p e t i t e n t i e r d t e l q u ' i l e x i s t e un c o d a g e d e R d a n s Nd Le t h 6 o r b m e e s t a l o r s a p p l i q u b p o u r m o n t r e r qiie c e t t e n o t i o n d e d i m e n s i o n g 4 n b r a l i s e l a d i m e n s i o n d b f i n i e p a r D u s h n i k e t Miller [51 p o u r l e s o r d r e s p a r t i e l s . Finalement on s ' i n t e r e s s e aux codages c o n t r a c t b s q u i s o n t l a g r i n b r a l i s a t i o n l a p l u s f i d 6 l e d e s i m m e r s i o n s e n c e sens q u e t o u t c o d a g e c o n t r a c t 6 (cp 1 ' cp2): 0 +l5 d ' u n o r d r e p a r t i e l d a n s un t r e i l l i s c o m p l e t e s t t e l q u e tPl = cp2

.

.

Tous c e s r b s u l t a t s f o n t p a r t i e d e n o t r e t r a v a i l d e t h h s e p r e s e n t 6 e n 1971. L ' e x p o s e e n e s t i c i s i m p l i f i b . Les a l g o r i t h m e s d e r e c h e r c h e d e l a d i m e n s i o n q u i b t a i e n t a u d e p a r t l a m o t i v a t i o n p r i n c i p a l e o n t b t e l a i s s b s d e cste. En e f f e t , s i c e s a l g o r i t h m e s e t a i e n t p l u s p e r f o r m a n t s que l e s a l g o r i t h m e s connus 2 l ' e p o q u e , notamment g r z c e a u f a i t q u ' i l s t r a v a i l l a i e n t s u r une s o u s - r e l a t i o n g C n b r a t r i c e a u l i e u d e t r a v a i l l e r s u r t o u t un o r d r e p a r t i e l , i l s r e s t e n t n o n p o l y n o m i a u x , e n a c c o r d a v e c un r e s u l t a t r e c e n t d e Y a n n a k i s . L e l e c t e u r i n t b r e s s b p o u r r a d o n c c o n s u l t e r [2] p o u r u n e 6 t u d e p l u s c o m p l k t e .

A . Bouchet

388

Cette etude traite egalement l a d4termination des codages contractes dans N d alors que le present travail se limite h Zd. L'essentiel de la methode est inchange, il s'agit de couvrir la relation binaire complGmentaire par des paves maximaux. Signalons l'etude recente de Cogis [ 3 ] sur la dimension des relations binaires. Cette etude e s t dans son esprit plus combinatoire que notre methode algebrique qui donne des resultats valides dans les cas infinis. Le lecteur trouVera des applications aux sciences humaines et l'extension de certaines proprietes au cas infini dans le travail de Doignon, Ducamp et Falmagne [ 4 1

.

2 - DEFINITIONS ET NOTATIONS

D'une facon genbrale nous a w n s conservb les notations de 121 notions de base sont exposees dans 1 1 1

. Les

.

Soit une relation d'ordre partiel O = [ E , 5 , E]

.

tine section initiale

principale sera notee 2 (ensemble des y 5 x ) et une section finale principale sera

"

notee x (ensemble des y

5

x). Un sous-ensemble U C E (resp. I c E ) sera dit V-gcne-

rateur (resp.A -g&nerateur) si tout element x de E est borne supbrieure (resp.inferieure) d'un sous-ensemble contenu dans U (resp.1). On verifie aisement que U est V-ggnerateur (resp.1 est A-gengrateur) si et seulement si x (resp. x

Inf

=

(y n I))

rateur, on dit que : g Un

p1

X J &

n U)

pour tout x de E. Si U est V-gknerateur et I est A- gene=

[U, < , I] est une sous-relation generatrice de 0 ,

p2 d'une relation binaire R

X

Sup ( 2

=

=

[E1,p , E ] est constitub par

2

un sous-ensemble p, C E 1 et un sous-ensemble pg C E2 tels que e, € p1 et e2 e p2 e l P e2. =)

3 - THEOREME DU CODAGE

- Soit [U, 5 , I] une sous-relation generatrice d'une relation d'ordre 5 , E l . Alors pour tout x dans E, (2 n U) x (y n I) est un pave maximal de [U, 5 , I1 . ./ Preuve : Posons A : 2 n U et B = x n I. Si on considere a € A et b 8 B , on a : 3 . 1 .LEMME

partiel [E,

a

L

x

2

b. Ainsi A

X

B est bien un pa&

de [ti,

5,

11

.

S i ce pavC n'est pas maxi-

mal on peut trouver u € U \ A ou i € I \ B de telle fason que (A A

X (B

u {u}) x

B ou

lJ {i}) soit encore un pave. Dans la premiere eventualit&, u estun niinorant de B. Mais d'autre part

x est borne inferieure de B puisque I est A-generateur. I1 vient alors u contredit u € U \ A .

utilisant le fait que 3.2.THEOREME

-

u

5

x qui

i 8 I\B

Dans la seconde eventualit6 on contredit de &me

en

est V-gCn&rateur..

Soit un codage

( v l , v2)

: [U,

5 , I1

+

[T,

est une sous-relation generatrice d'une relation d'ordre

5,

TI o h g

0=

[E,

=

[U,

5 , El

5 , I]

Codages et dimensions de relations binaries

=

[T, 5

,

TI est u n treillis complet. Soient 0, : E

g l ( x ) = sup ql(ii

rl

+ 0

u),

des immersions de (I dans

(XI =

Inf

u,(X' n I).

Preuve : Montrons d'abord que l'application

(u")

u2(v).

codage.Ainsi

(il

x'

(1

I

=

fi I C

<" n

T definies par

0 , et 0, sont

est necessairement injective. (i

1

( u ' ) = , q l ( u " ) , et montrons que

(i

1

Cette derniere inkgalit6 implique u"

x'

-+

fl I.

Considbrons un Cl&ment v dans

5

: E

Les applications

On a alors u ' s v avec u ' € U et v € I. Ceci implique (cl

T et 0,

.

Soient donc u ' et u" deux elkments de U tels que u ' = u".

+

389

3'fl I.

(v). D'oh : 2 est un v puisque (ql, (u')

5 q

v2)

Nous etablirions de m6me l'inclusion inverse. Donc

I. Mais on a u '

=

Borne inf

puisque I est kgBn6rateur. Ainsi u '

=

(y' n I)

et u"

=

(x" fl I)

Borne inf

u".

On montrerait de facon duale que q2 est injective. Ainsi, si on pose

Q

P = ql(U),

=

q2(I)

et R

5 , Q] ,

= [P,

( q l , q2) est un codage bijectif de g s u r R.

Nous allons maintenant montrer que pour tout x dans E, on a 1'CgalitC : fi (A) q 1 ( ? nu) = 0 , ~P

n

n U) x cc 1 ( 2 n U)

L'ensemble (P

W

( x , n I) est un pave maximal de g d'apres le lemme (3.1 .).Donc v

n

x cc ( x I) est un pave maximal de R puisque ( e l , v2) est un 2 A v codage bijectif. D'autre part, l'ensemble ( 0 , ( x ) n P ) x ( g l ( x ) Q) est aussi un l'ensemble

-n

pave de R car on a a s 0 1 (x) 5 b pour tout a dans 0 1 (x) v Q. Montrons l'egalit6 de ces deux paves.

@,(x)

n

P et tout b dans

n

D' u n e

part, on sait que 0 , ( x ) est la borne superieure de q ( 2 (1 U). I1 1

en resulte :

i)

n (x)

0,

Ti P z.q 1 ( a n u ) .

D'autre part, le fait que q l ( n

ii)

v

x q2(x

n I)

est un pav6 de R signifie

I) majorent 0 (x) qui est la borne sup de q (? 1 1

que tous les elements de q2(? I1 en rCsulte :

n U)

w 0 , (x) fl Q

v 2 q2(x

n I). A

n U).

w

n P)

x ( 0 (x) ( 1 Q) 1 (1 I). Puisque le second de ces pavCs est maximal

Les inclusions i) et ii) montrent que le pave ( 0 (x), 1

contient le pave q (? 11 U) x q (x' 1 2 les deux sont kgaux. L'egalite (A) resulte alors de l'Bgalite de ces paves. Soient x et y dans E. L'inCgalitb x s y equivaut h 2

U est V-gbnerateur pour la relation d'ordre [E,

v1 ( 2 n U)

5,

El

n U _C

. L'inclusion

9

nU

puisque

precddente

C_ ( 9 U ) puisque q1 : U P est bijective. Cette dernihre inclusion implique g1 (x) = Borne sup q ( a n U ) s Borne sup 9, (9 n U ) = 0, (y). 1 P. D'oh : 'Inversement si on a 0 , (x) O 1 ( y ) , il vient @ , ( X I n P _C 0 , ( y ) v (P II U) _C ql (?( n U ) grsce h l'egalite (A). On remonte alors h x 5 y . Ainsi x y Bquivaut h

-+

1 Bquivaut h 0 , ( x ) s g l ( y ) , et 0 , est bien une immersion. La &me

montrerait de facon duale pour

g2.

propriCt6 se dB-

A. Bouchet

390

3 . 3 . COROLLAIRE - En conservant les notations du theorkme, s'il existe un codagc

4

e

(9 dans

Ze

-

, alors il existe une immersion

.

de (5 d a n s Z

APPLICATIONS Le theorbme prCcCdent permet une dtmonstration rapide du thearkme de

Mac Neille. Soit une relation d'ordre partiel (9 = [ E ,

5,

I:].

Soit M le sous-ensemblc

d e s parties d e E defini par P 6 M si et seulement s i P = E ou il cxiste X

que P

?. Le treillis complet M ( (9)

I-

=

=

[K,

5,

5" tel

t i ] est l'extension minimale de

XBX

0 en treillis complet. L'immersion y : x

-t

? est l'immersion canonique de 0 dans

M ( 0 ) .

e , alors

THEOREME (Mac Neille) - Si 0 est immergeable dans un treillis complet M( d ) est immergeable dans 0

.

Preuve : Considerons l'immersion canonique y : 0

M ( 0 ) , Tout Glement de M

est par definition une intersection d'el4ments de y(E) ; ainsi y(E) est A - g t n b rateur pour M(

0 1. Par ailleurs, tout bl4ment

de M est une section initiale de

8 , et une telle section initiale est union de sections initiales principales. Etant donne que y(E) est constitue de l'ensembledes sections initiales principales

6 ,y(E) est V-genGrateur pour M( 0 M( f i ) sur Y(E) est generatrice. de

) . Ainsi la sous-relation g induite par

Considerons maintenant une immersion(c:fl+ G immersion de g dans G

. Puisque

.

Alors

$

o y

-1

est une

g est une sous-relation gbneratrice de M( 0 ) , il

suffit d'appliquer le corollaire.. Supnosons maintenant quc 0 est une relation d'ordre partiel sur un ensemble fini. La dimension de 0 est le plus petit entier d tel qu'il existe une immersion

(c

:

0

-f

INd. Par extension, nous appelons dimension d'une relation bi-

naire finie R le p l u s petit entier d tel qu'il existe un codage

((c

1

, V ) de R dans 2

N d , Le coroIlaire du theorbme nous permet d'affirmer que ces deux notions de dimension coincident lorsque l'on considkre un ordre partiel. 5

-

CARACTEKISATION DE LA DIMENSION D'UNE RELATION BINAIRE

Soit une relation binaire finie R =[E 1 , p , E2]. Pour tout e l 8 E 1 on pose L(ei) = {e, 6 E2 : e l p e2} et on dit que L(e ) est une ligne de R. Pour tout I e 2 8 E2, on pose C (e ) = { e € E, : e l p e21 et on dit que C ( e , ) est une colonne 2

de R.

La relation R est une relation de Ferrer si a l

p

b2, c,

p

b2, c 1 p d g im-

pliquent a l p d2.

5.1. PROPOSITION [ 6 ] - Les quatre propriCtes suivantes sont equivalentes :

i) ii)

R est une relation de Ferrer, Les lignes de R sont totalement ordonnbes par inclusion,

Codages et dimensions de relations binaries iii) iv)

39 1

Les colonnes de R sont totalement ordonnCes par inclusion,

R est de dimension 1.

En fait, seules l e s trois premitres bquivalences sont Ctablics dans [ G I . L'Gquivalence de iv) avec ii) ou iii),est B peu prPs Cvidente. Si on reprksente

?

R par une matrice de 0 et de 1 et si on ordonne les lignes et colonnes selon les ordrcs totaux dCcrits en ii) et iii), on obtient la forme d'un escalier ; d ' o h le nom quc nous donnons en [ l ]

3. c e s relations.

-

Une relation binaire finie R =[IJ1,p , E 2 1 est codable dans N si e t seulemcnt si elle est Cgale B l'intersection de d escaliers.

5.2. PROPOSITION

Preuve : Soit un codage ( U considgrons q . j : E. Soit ij

p j , E2] la

I

, q,L )

: R

Nd

.

Pour chaque i

=

1

1,2 et j 2

=

N de telle facon que q.(e.) = (V. (e.),V. (e.) 1

1

1

1

relation binaire dbfinie par e l p j e 2

1

1

1 ,2,.

d

. . ,d

,..., qid (ei)).

y , j ( e l ) s q2'(ez).

Le couple (qlj,q2j) est Cvidemment un codage de 6' dans N ; ainsi [ j est un escalicr. Par ailleurs q (e ) 9 q2(e2) e s t vCrifibe si et seulement si (ilJ(el)s q (e 1 1 2 2 est vkrifiP pour tout j = l , Z , ..., d. Ainsi R est l'intersection des escaliers *,- 1 1 r, 2 , . . . , r , d

.

Rkciproquement, si R est donnee comme intersection d e d escaliers 6 1, F 2 ,...,Ed, on construit pour chaque j = 1,2 ,..., d un codage (U,', U),' E J + N e t on en dCduit un codage ( q 1 , V2) : R +Nd.

:

5.3. COROLLAIRE - La dimension d ' u n e relation binnire est l e nombre minimal d'escaliers d o n t elle est l'intersection.

5.4. COROLLAIRE - La dimension d ' u n e relation binaire est le nombre minimal d'escaliers dont sa complbmentaire est l'union. Pour l e second corollaire, i l suffit de remarquer que l e compl6mentaire d'un escalicr est un escalier d'aprPs l a caractkrisation ii) des escaliers. Cette caractsrisation de la dimension par la complCmentaire d e R est particulihrement intCressante d a n s le cas de la 2-dimension dont nous faisons maintenant une etude directe. 6 - CODAGES BOOLEENS

L'ensemble des parties d'un ensemble D est not6 P(D). Soit unc relation binaire R

= [E,,p

, E2] qui n'est plus nkcessairement

' P (D )

ct une application V2 : E 2 - + P ( D ) . Pour chaque d dans D, on note P l d le sous-cnsemble d e s 61Cments e, de E l tels que finie, un ensemble D, une application

(4

: El

de Ez tels que d k? V2(e2).

La

A. Bouchel

392 Si

( v1'

( c ) est un codage de R dans P(D) muni de l'ordre induit par

2 l'inclusion, on dira que c'est un codage booleen de R. La dimension booleenne ( o u 2-dimension) de R est le cardinal minimal de D tel que le codage existe. 6.1.

PROPOSITION

-

Le couple de fonctions (VlLq2)

est un codage booleen si et

seulement si le domaine de definition de la relation complementaire de R est @gal B l'union des sous-ensembles de la famille ( Pd 1 ~ Pd2 ) d

.

.

d d Supposons que ( v ,(c ) est un codage et montrons U P 1 x P2 1 2 d€ D que P est le domaine de definition de la relation complementaire de R. Soit (el,e2) Preuve : Posons P =

dans P, et raisonnons par equivalences successives. d d (el,e2) € P 9 d € D : el 6 P 1 et e2 € P2 Q

,

3 d 8 D : d € ql(el) et d G I q(e2), Qt

tCl(el)

v2(e2),

ca

e l p e2 non verifie.

Reciproquement, supposons que P est le domaine de definition de la relation complementaire de R. On reprend le raisonnement par equivalences successives en partant de la dernigre assertion. On arrive alors B l'avant-dernikre assertion. Ainsi ( q ,q est bien un codage.. 1 2 6.2. COROLLAIRE

-

La dimension booleenne de R est egale au cardinal minimal d'une

famille de paves couvrant la relation complementaire de R. 7

-

CODAGES CONTRACTES Soit un treillis completT( et une sous-relation [ P , s

'P

, Q]

de t

. Notons

l'ensemble des bornes superieurcs des sous-ensembles contenus dans P et Q'

l'ensemble des bornes inferieures des sous-ensembles contenus dans Q. La relation [P,

2

, Q ] est contractee si P CQ' et Q CP'.

p , E 1 et un codage ( ( c 1 % ) : R + T ( . Le codage (V1,V2) 1' 2 1 2 est contract6 si la sous-relation image [(c (E ) , s , cc. ( E ) ] est contractee. 1 1 2 2

Soit R = [ E

-

Le codage (CC. , q ) : R +T( est contracte si et seulement s i les ega1-2 lites suivantes sont verifiees pour tout e dans E et tout e dans E2 : 11 -2 ql(el) = inf v2(L(el)),

7.1. LEMME

q2(e2)

=

sup v1(C(e2)).

Preuve : Posons P = v,(El) et Q

=

v2(E2).

Les egalites de l'bnonc6 sont evidem-

ment suffisantes pour avoir P CQ' et Q CP'.

Reciproquement, supposons P cQ.et

considerons un bl&nent e, de El. Les sous-ensembles q2(L(el)) egaux car (Vl,V2) est un codage surjectif de R s u r [P,s , Q ]

et

.

\ I (e )

c+

1

1

nQ

sont

Codages et dimensions de relations binaries

393

Pour demontrer la premiere Cgalite du lemme, il suffit donc de montrer que vl(el)

=

v

-

inf (Vl(el) nQ). L'616ment %,(el) est dans P et l'on a PCQ' ; il

(4 (e ) = Inf K. Le sous-ensemble K est contenu dans 1 1 v v V1(el) n Q ; on a donc Inf K 5 Inf (vl(e ) nQ). D'oL v (el) 2 Inf ($,(el) flQ). 1 1 L'inPgalite opposee s'obtient en remarquant que v (e ) est un minorant de

existe donc K C Q tel que

\ I

v 1 (e1 ) n Q .

1 1 On a donc bien l'egalitb desir6e. La deuxikme partie de 1'Cnonce s'ob-

tiendrait de facon duale en partant de Q C P ' . Soit une relation d'ordre partiel &ratrice

g

[U,

=

5,

o = [E, 5 , Elet

une sous-relation g6-

I1 de cet ordre partiel. Considerons toujours le treillis

( v1, ( 4 2)

complet 5 et un codage

prolonge l e codage ( q , V 1 2 % i I.

: g

+

Z

.

On dira qu'une immersion cc : 0

si q1 est la restriction de

(i

+

5

B U et (i2 est la restric-

7.2. PROPOSITION - Si l e codage (Ul,v2) est contract6 alors les immersions 0 et 1theoreme 3. 2. sont des prolongements de ce codage.

g2 du

Preuve : Posons P = vl(U) et Q

o1(x) =

Sup

vl( 2

V2(I).

=

Soit x un element de E. Rappelons que v

U) et 0,(x)

Inf v2(x

=

un pave de la sous-relation [U,

5,

n I).

L'ensemble ( 2

I]. L'ensemble ql(?

n U)

x

n U)

(z

x

v2(v" n I)

(I I) est est donc

un pave de la sous-relation [ P , i _ , Q ] puisque (v1,v2) est un codage. I1 en rksulte sup q1( 2

ri U)

Inf

2

v 2 G fi I)

; soit :

0,

0, (x)

(A)

(x)

Consid6rons maintenant un element e, dans U. L'element q (e ) appartient B

vl(^el fl U)

1

(B)

Vl(el)

=

vl(el)

5

@,(el).

Puisque ( ( 4 , v ) est contracte, on sait par le l e m e que : 1 2 v Inf v2(L(el)). Dans ce cas L(el) est egal B e l n I. L'CgalitG pr6cPdente

est donc equivalente B V. ( e ) = @,(e,).

1

Vl(el)

=

1

; d'oL :

@,(el) = @,(el).

En combinant avec (A) et (B) il vient alors

1 On montrerait de fason duale 9 (e ) 2

2

=

0 (e 1

) =

2

02 (e2)

pour tout e2 dans 1.m Definition : Soient deux relations binaires R = [El, p , E2] et R'

=

Un isomorphisme de R' sur R est un couple de bijections i, : El1

+

[El1 , p ' , E >

1.

El,

i2 : El2 + E2, telles que e' p ' el2 a il(e' 1) p i2(er2). La relation R ' peut ttre un ordre partiel - et alors E ' = El2 - alors que la relation isomorphe R peut 1 stre definie entre deux ensembles distincts - El # E2 -

.

7 . 3 . COROLLAIRE

-

Soit une relation binaire R =[El, P , E21, un treillis completz

et un codage contract6

( v1 2v2)

: R +'7?

. La relation binaire R est

une relation d'ordre partiel si et seulement si tC1&q telles que vl<E1)-q

(E ) 2-2--'

isomorphe

2 sont deux injections

,

A . Bouchet

394

-

Preuve : La condition suffisante est 6vidente puisque (q1,(r2)

.

[(c (E ) , s , V (E ) I 1 1 2 2 K&ciproquement, consid6rons un ordre partiel 61 = [E,

phisme de K sur l'ordre partiel phisme (il,i2) : 0

R. Le couple de fonctions (i o

-f

avec ce codage contract6 et (!/ =

i

o q1

i2 o

=

< , Llet un isomor-

i2 o CC2) est un codage Appliquons le thCor&me 3.2. et la proposition prtcedente

.

contract6 de c." dans 5 que 8

Ptablit un isomor-

v2.

=

(cl,

g. On trouve alors une immersion 0 : 0

+

6 tellc

Une immersion d'un ordre partiel dans un treillis est

necessairement une injection. Ainsi 0 est une injection et il en est nbcessairepour (cl et v2. Enfin on a q ( E ) = tF (E ) = 0 ( E ) . W 1 1 2 2 Ainsi les codages contractbs g6nCralisent parfaitement les immersions

ment de &me

car pour tout ordre partiel

,

6 et tout treillis complet

( V ,q ) :

1

2

0

+

G est

un codage contract; s i et seulement si Y1

= 'p et cette fonction est une immersion. 2 Revenons au cas d'un codage booleen ( ( c l , q 2 ) : R P ( D ) . Nous avons vu que la donn6e d'un tel codage equivaut B celle d'une famille (P, d x P2)deD d de -f

pav6s couvrant l a relation complementaire de I?. 7.4.

PROPOSITION

-

Le codage booleen (vl,q2)

chacun des pav6s de la famille (P:zPi)dt?.D

est contract6 si et seulement si est un pave maximal de la relation

compl6mentaire de R. Preuve : Rappelons qu'un pave P

,

x P2 d'une relation R est maximal si et seulement

si : P1

=

{el 6 E l : P2 C L (e,)]

P2

=

{ e , 6 E2 : P1

et C

C

(e,))

Nous nous intbressons ici aux pav6s de la relation compl6mentaire de R. Pour cette relation compl6mentaire de R , notons donc L'(e un 616ment e , t?. E

1

et C'(e

2

) l a colonne relative

1

ment L'(e 1 ) = E2 \L(el) et C'(e2) = El\ C(e2). Un pave Pd 1 pour l a relation compl6mentaire de R si et seulement si : P: = {el 6 E~ : p2d c ~ ' ( e , ) ~ et

d

) la ligne relative 5

B un 616ment e 2 6 E2 .On a 6videmd X

P2 sera donc maximal

d

P 2 = {e, € E g : P, cC'(e2)}

Appliquons le lemme et consid6rons l a premikre condition caractbristique d'un codage contract6 : V e l 6 E , : tcl ( e l ) = Inf p2 (L (el))

Raisonnons par equivalences successives B partir de cette condition. Nous obtenons successivement :

e 1 € E l , V d t?. D : (d 6 vl(el)CV

e2 6 L(el) : d 8 q2(e2))

Codages el dimensions de relations binaries V el 6 El, V d 6 D Ve

: P

: (el 8

G V e2 6 L(el) : e 2 !6 P2d )

€ E , , V d € D : (el € Py

1

L ( e l ) c E 2 \ P 2d

V e l € E , , V d 6 D : ( e l 8 P, -P;

V d

€

D : P;'

=

{el 6 El

395

CL'(e,))

: P2d C L ' ( e l ) }

D e facon duale, la deuxibme condition caracterisant un codage contract4

donnerait par equivalences successives la deuxibe condition caractbrisant un pavb maximal de la relation complbmentaire de K. W La proposition prbc6dente peut Stre gbnkralis4e dans le cas d'un codage

dans INd ; ceci est fait dans [ 2 1

.

La proposition suivante montre que les codages

contractCs ne sont pas moins g4nciraux que les codages.

7.5. PROPOSITION - S i une relation binaire K admet un codage dans un treillis complet Z

,

alors elle admet aussi un codage contract6 d a n s Z

Preuve : Soit [P,

5,

.

Q ] la sous-relation image par le codage de R dans

5,

allons montrer qu'il existe un codage de [ P , relation contractke de Z

. La

'e . Nous

Qldont l'image soit une sous-

composition des deux codages donnera le codage con-

tract6 d4sirC. Soit f la fermeture dCfinie s u r e ayant pour ensemble de fermes l'ensemble Q ' d e s bornes infkrieures d e s sous-ensembles de Q. Pour tout blkment x de % et pour tout Clkment y invariant par f, la relation x s y est v4rifi4e si et seulement s i f ( x ) d e [P,

5 , Ql

5

y. P.insi le couple de fonctions (f,l) permet de definir un codage

sur [P,

5,

(A)

Ql avec ? P C Q'

=

f (P). On

n

alors :

Considerons maintenant la fermeture duale g dCfinie sur 5 ayant pour ensemble d'invariants l'ensemble P+ des bornes superieures des sous-ensembles d e P. De &me que ci-dessus, le couple (1,g) permet de dkfinir un codage de [P,5 ,Q] sur

[P,5 ,q]

avec

3 (B)

=

g ( Q ) . On a alors :

Q

5 Pt

La composition des deux codages precedents donne un codage sur Compte-tenu de (B) il reste 5 montrer q u e

FQ'pour

4tablir que [ P ,

5

[P, 5 ,a]

,Ti] est

une

sous-relation contractee. D'aprks (A) tout 614ment z 8

P est

borne inferieure d'un sous-ensemble

A C Q . Soit z' la borne inferieure d e g ( A ) . L'GlCment z minore A et c'est un in-

variant de l a fermeture duale g ; ainsi z minore g(A) et z

5

z'. Chaque element

de A majore son image dans g ( A ) puisque g est une fermeture duale ; ainsi la borne inferieure de A majore la borne infkrieure de g(A) et z est donc borne inferieure de g ( A ) , et l'on a bien

CG'.

2'.

L'elCment z

.

396

A . Bouchet REFERENCES

[I

1

G.BIRKOFF, Lattice theory, A.M.S.Colloquium Publications, 1948.

[2] A.BOUCHET, Etude combinatoire des ordonnes finis, Applications, ThPse de Doctorat d'Etat, Universite Scientifique et Medicale de Grenoble, 1971. [31 O.COGIS, On the Ferrers dimension of a graph, Discrete Mathematics 38 (1982) 47-52. [41 J.P.DOIGNON, A.DUCAMP, J.C.FALMAGNE, On realizable biorders and the biorder dimension of a relation, 1 paraftre. [5] B.DUSHNIK, E.W.MILLER, Partially ordered sets, Amer.J.Math.63 (1941) 600-610. [6]

J.RIGUET, Les relations de Ferrers, C.R.Acad. Sci. Paris 232 (1951) 1729-1730

Annals of Discrete Mathematics 23 (1984) 397-408 0 Elsevier Science Publishers B.V. (North-Holland)

397

THE SPERNER PROPERTY

*

Jerrold R. Griggs

Department of Mathematics and Statistics University o f South Carolina Columbia, S.C. 29208 USA Dedicated to Professor Corominas A finite ranked poset has the Sperner property if no antichain is larger than the largest rank in P. This was first shown to hold in the lattice of subsets of an n-set by Sperner in 1928, Efforts to prove analogues of Sperner's theorem for other posets have led to the emergence of an entire theory. A big advance has been the introduction into the field of techniques from linear algebra, category theory, and even algebraic geometry. This is a survey of progress made in this area over the past few years.

R6 s um6 Soit Ai @ A

j

{Al,

A2,.. Akl

d5s que

une famille de sous-ensembles de

i # j. Pour n

,

{l,

...,n}

telle que

fix6, quelle est la taille maximum d'une telle

),;,(

famille? En 1928, Sperner [Sr] a montr6 que k

I;

.

Cette borne est la est Ai El,.. .,nl ou de tous

meilleure possible et il y a 6galite' si et seulement si la famille des la famille de tous les sous-ensembles les sous-ensembles 2

[I

61gments.

a'

E]

6lgments de

En termes d 'ensembles partiellement ordonngs (posets) le thgorgme de Sperner se traduit par: dans l'algkbre de Boole Bn de tous les sous-ensembles de {l,, ,n] (ordonne's par inclusion), aucune antichaine n'est plus grande que le rang maximum. Quelques quarante ans plus tard, Rota [ R o ] 6mit une cglsbre conjecture:le treillis n des partitions de 11, n1 (ordonngs par raffinement) 9 aurait la meme propriet; ou plus prgcisement, aucune antichaine ne contiendrait plus d'glgrnents que le maximum (pris pour h variant del'an) du nombre de partitions de {l,. ..,n1 en h parties. Cela est vrai pour des petites valeurs de n, mais, il y a presque dix ans, Canfield [CL] a montrg que cette conjecture est Cet ordre de grandeur fausse lorsque n est suffisamnent large, n ?I 6 x 5X1O6 [Sh, p. 3 1 . a e't6 abaissc depuis 'a n

..

...,

Bien que cette conjecture soit fausse, les essais pour la prouver ont entrain6 la dgcouverte de toute une thCorie s u r les "posets" qui satisfont la proprigth du thdorbme de Sperner. D e rgcentes contributions 5 ce domaine tr6s vivant ont utilisg, avec succSs, des outils empruntgs $ l'alggbre linGaire, la

* Supported

in part by the National Science Foundation and by a U.S.C. Research and Productive Scholarship.

J.R. Griggs

398

thgorie des catGgories, la programmation liniaire et mgme 5 la g8omGtrie alggbrique. Ici notre principal intsrgt portera sur les relations entre la proprigti. de Sperner et les conditions d'existence de certaines familles d'ensembles totslement ordonnis dans un "uoset" et de certaines transformations lingcires dansl'espace vectoriel associg au"poset". Une synth'ese plus complgte doit etre diff4r6 mais la bibliographie couvre tout l e domaine. Pour de larges synthhses couvrant entiirement l e domaine de la thgorie des ensembles extrgmaux, voir [ C K 3 , We21.

1,

INTRODUCTION Suppose A1 ,...,4, are subsets of

i=j.

{l,.

.. ,n)

such that

How large can such a collection of sets be, for given

[Sr] showed that only if the A.'s

k

5

(Ln;2J),

are all the

Ai a; Aj n?

unless

In 1928 Sperner

This bound is sharp, with equality holding if a n d Ln/2l-subsets of

...,

11,

111,

or all the

rn/2~1-

subsets. In the language of posets, Sperner's theorem states that in B the Boolean algebra of all subsets of {l,..,,nl (ordered by inclusion), no anpichain is larger than the largest rank. Some forty years later Rota [no] made a famous conjecture that nn! the lattice of partitions of {l,...,nl (ordered by refinement), would have a similar property: No antichain should contain more elements than the nl into k parts. This maximum, over k, of the number o f partitions {l, is true for small n , but it was almost ten years before Canfield [Cl] showed that the conjecture fails for sufficiently large n, n 6 X l o z 4 , This has since been lowered to n T, 5 x l o 6 [Sh,P3].

...,

Although the conjecture failed, attempts to prove it inspired the discovery of an entire theory about posets which satisfy analogues of Sperner's theorem. Recent contributions to this lively field have successfully employed tools from linear algebra, category theory, linear programming, and even algebraic geometry. Here we shall focus on connections between the Sperner property and conditions on the existence of certain chain collections and linear transformations. A more complete survey on the Sperner theory must be postponed, but the list of references covers the whole area. For broad surveys of the entire field of extremal set theory, see [ G K 3 , We2]. 2.

DEFINITIONS

All posets in this paper are assumed to be finite and ranked (graded). A poset P has rank n if it is the disjoint union of nonempty subsets Po,Pl,:..,P n' called ranks, where Po is the set of minimal elements of P and for all 1 Pi+l' whenever x E P. and y covers x, then Y Each rank P. is an antichain (totally unordered set). P has the Sperner property if no antichain in P is larger in size than the largest rank. Thus Bn has the Sperner property, but En does not in general. A k-family in P is a subset of P such that no (k+l) elements lie on the same chain. Equivalently, a k-family is the union of k (or fewer) antichains. For example, the union of k ranks of P is a k-family. In 1945 Erdb's [ E l generalized Sperner's theorem by showing that f o r all k the union of the k is a k-family of maximum size. Motivated by this example, largest ranks in B we say P is k-Speker if no k-family is larger than the union of the k largest ranks. P is strongly Sperner if it is k-Sperner for all k. This notion of strongly Sperner posets is important in the theory now emerging.

The Sperner property

399

Many r e s u l t s a l s o d e p e n d c r i t i c a l l y on p r o p e r t i e s o f t h e s e q u e n c e o f W h i t n e y ( r a n k s i z e s , \ P i \ ) . A p o s e t P of r a n k n i s s a i d t o b e r a n k - u n i m o d a l i f , f o r some j ,

~numbers

I i....5

[ p [ i: [ P 0 1

(P.1 J

s*..<

\Pn/.

P is rank-symmetric i f f o r a l l k \ P k / = IPn-kl. P i s Peck i f i t i s r a n k s y m m e t r i c , r a n k - u n i m o d a l , a n d s t r o n g 1 . y S p e r n e r . Many i n t e r e s t i n g e x a m p l e s of S p e r n e r p o s e t s t u r n o u t t o be P e c k . 'These i n c l u d e B ; D , t h e l a t t i c e o f th:! l a t t i c e o f s u b s p a c e s o f d i v i s o r s o f n ( o r d e r e d by d i v i s i b i l . i t y ) ; a n d I. ((1): a n n - d i m e n s i o n a l v e c t o r s p a c e u v e r (:F(q) ( o r d e r E d by c o n t a i n m e n t ) . A c h a i n i n P i s s a t u r a t e d i f f o r some i 5 j i t meets c o n s e c u t i v e r a n k s i , i + l , .. . , j a n d meets no o t h e r r a n k s . A c h a i n i s s y m m e t r i c i f i t i s s a t u r a t e d and j = n - i . m denotes a chain with m elements.

The ( d i r e c t ) p r o d u c t o f two p o s e t s , I' a n d Q , i s d e n o t e d by P x Q a n d h a s i f p 5 p' and e l e m e n t s { ( p , q ) : p c P , q t Q } o r d e r e d by ( p , q ) 5 ( p ' , q ' ) q 5 q'. The q u o t i e n t i f a r a n k e d p o s e t P u n d e r a g r o u p o f ( o r d e r - p r e s e r v i n g ) a u t o m o r p h i s m s C i s d e n o t e d by P/C. I t s e l e m e n t s are t h e o r b i t s of P u n d e r G , o r d e r e d by A S B i n P/C i f t h e r e e x i s t a E A , b E B s u c h t h a t a 5 b i n P .

3.

C H A I N CONDITIONS

For any c h a i n C and k-tamily t i t i o n C o f P i n t o c h a i n s c:,

The C r e e n e - K l e i t m a n (ranked o r not) has h o l d s above. Thus t h a t t h e sum o n t h e

I C n FI S M i n ( l C / , k ) ,

F,

Thus f o r a n y p a r -

theorem [ C K l ] a s s e r t s t h a t f o r a l l k e v e r y f i n i t e p o s e t P a k - s a t u r a t e d p a r t i t i o n , which i s a C s u c h t h a t e q u a l i t y P is k-Sperner i f and o n l y i f t h e r e i s a p a r t i t i o n C s u c h r i g h t e q u a l s t h e sum o f t h e k l a r g e s t W h i t n e y n u m b e r s o f P .

An e s p e c i a l l y n i c e t y p e of p o s e t i s a s y m m e t r i c c h a i n o r d e r . This is a poset P which c a n b e p a r t i t i o n e d i n t o symmetric c h a i n s . T h i s p a r t i t i o n s u f f i c e s t o g u a r a n t e e t h a t P i s k-Sperner f o r a l l k . Indeed, P i s Peck. Some e x a m p l e s o f s y m m e t r i c c h a i n o r d e r s i n c l u d e B n , D n a n d I,,(q). A18cther i n t e r e s t i n g p o s e t i s i n t e g e r s i n t o p a r t s 3 = (al,a2, ordering is

-f

-+

a 5 a'

if

a . 2 a! 1

1

L(m,n),

..., a m ) ,

t h e l a t t i c e of a l l p a r t i t i o n s of w h e r e n 2 al 2 a2 t...> a t 0 . The

for all

-f

i.

a

m

has rank

Cai.

Equivalently,

L(m,n) i s t h e l a t t i c e o f o r d e r i d e a l s o f 2 x 2 ( o r d e r e d by i n c l u s i o n ) . L(m,n) c a n a l s o b e v i e w e d as t h e F e r r e r s d i a g r a m s w h i c h f i t i n t o a n mxn r e c t a n g l e . -f L(m,n) is rank-symmetric s i n c e f o r any a of r a n k k t h e r e c o r r e s p o n d s t h e n-al) o f r a n k mn-k. I n t h e a b s t r a c t [ S t l ] S t a n l e y announced e l e m e n t (n-a m i s S p e r n e r f o r a l l m a n d n . T h e r e i s a c h a i n c o n d i t i o n , (;t), w h i c h t h a t L(m,n) S t a n l e y showed i s t r u e f o r L(m,n) and which i m p l i e s t h e S p e r n e r p r o p e r t y . A r a n k - s y m m e t r i c p o s e t P o f r a n k n s a t i s f i e s c o n d i t i o n (") i f f o r e a c h i=O,l, I.n/ZJ, t h e r e exist \ P i \ d i s j o i n t s a t u r a t e d c h a i n s g o i n g f r o m Pi U P to P n-1

,...,

..., ..

The c o l l e c t i o n s o f c h a i n s i n (") c a n b e u s e d t o c o n s t r u c t a p a r t i t i o n c of I c h a i n s . To do t h i s , t a k e t h e s u b c h a i n s b e t w e e n P . a n d i n t o only IpLn/2J a n d P . f r o m t h e c h a i n s g o i n g b e t w e e n Pi a n d P P. a n d b e t w e e n Pn-i-l n-i' 1+l 11-1 no t h i s f o r e a c h i < Ln/21 a n d c o m b i n e t h e s u b c h a i n s t o f o r m c. From c i t

P

400

J.R. Griggs

follows that P is Sperner and rank-unimodal. I noticed [Gg3] that in fact P must be k-Sperner for all k, for in the same spirit one may construct, for each k, a chain partition Ck such that C min(k, C. ) is the sum of the k largest Thus we may conclude that if' P is rank-symmetric and satisWhitney numbers. fies ( * ) , then P is Peck. Conversely, if P is Peck, the k-saturated partitions of P (which exist by the Greene-Kleitman theorem) can be used t o generate the chains needed for ( A ) , and P is rank-symmetric because it is Peck.

I I

is Peck. We can now see that Stanley's (") condition implies that L(m,n) It remains an open problem t o determine whether it is a symmetric chain order. This has been confirmed after considerable effort for Min(m,n) 5 4 [LnZ], Wel, Ri], but it may require some "global" technique t o prove this in general. Note that for different values of i, the chains in (A) may differ considerably, s o that a Peck poset need not be a symmetric chain order. (Figure 1 is an example of such a poset.)

Fig. 1 The equivalence above between (*) and Peck for rank-symmetric posets inspired me to probe for connections between the strongly Sperner property and chain conditions without restrictions on the Whitney numbers [Gg3]. Here is an appropriate way to extend (*) to arbitrary ranked posets. P satisfies condition T if for all k there exists a collection of disjoint chains in P which each intersect each of the k largest ranks and which cover the kth largest rank. This reduces to (*) if P is rank-symmetric and rank-unimodal. (What I originally called "condition S" is now called the strongly Sperner property. I named condition T after Stanley.) Here are the results which generalize the equivalence of the Peck and ( * ) conditions. If P is strongly Sperner, then it satisfies condition T. (This again can be proven by the existence of k-saturated partitions [Gg3]. Another proof employs network flows [GSS].) The converse is false, in general. For example, the poset in Figure 2 satisfies T but is not even Sperner.

Fig. 2 However, if P is rank-unimodal and satisfies condition T, then it is strongly Sperner. The original proof [Gg3] required the existence of simultaneously k- and (k+l)-saturated partitions, but a simpler, elementary proof has been found [GSS].

40 1

The Spernerproperty 4.

LINEAR ALGEBRA

In a landmark paper [St3], Stanley introduced linear algebra as a tool in studying the Sperner property. This has created a new wave of activity in Sperner theory. A s he announced earlier in the abstract of [Stl], the fact that L(m,n) is Sperner follows from its satisfying the chain condition ( * ) . This condition is implied in turn by a third condition involving linear transformations. Here is the actual theorem from [St3], which included the equivalence of (") and Peck, from [Gg3], discussed in the last section. denotes the complex vector space Pi with basis P..

Theorem 1. Let P be a finite rank-symmetric poset of rank following three conditions are equivalent: (i)

P

is Peck.

(ii)

P

satisfies

(iii)

I-

i

oi: Fi

+

ii+l

(a)

0i i

For

<

i

n, there exist linear transformations

For each

with the following two properties:

Ln/2], the composite transformation

-

...

$n-i-l @n-i-2 (b)

The

( 9 ~ ) ~

0

For

n.

i, 0

@i+l @i : Pi

-

+

Pn-i

i i n, and each

i

x

E

is invertible. Pi, Qi(x) =

C

cYy Ycpi+l

c = 0 unless x < y. Y Condition (iii) permits the introduction of algebraic tools to prove posets are Peck. Stanley employed techniques from algebraic geometry to prove that certain posets QX, derived from a class of algebraic varieties X, are Peck. An important instance of this is when QX is defined in terms of the Bruhat order of a Weyl group. These include, in turn, L(m,n) and M(n). M(n) is the poset with elements being the subsets of 11,. ,n], ordered as follows: where

..

,... ,a.] J

Suppose A = {a, with

al <...<

for 0 M(n)

=

2

i

2

a

j'

b

and bk.

B

1 For instance,

j-1.

where

<...<

P

=

Z1.

{bl

=

,...,bk 1

Then A

2

B

are subsets of

if

{1,3) i {1,2,4}.

j 5 k

and

{l,.. .,nl

a. 3-i

bk-i

One may also define

,

Now consider a seemingly unrelated conjecture of Erdzs and Moser: For any set S of 2L+1 distinct real numbers, the maximum number of subsets of S whose element sums are equal is attained by taking S = {-L, -L+l,. , L l and all subsets of S whose element s u m s are zero. Lindstrh [Lnl] noticed a connection between this conjecture and M(n), and he conjectured that M(n) is Sperner. Stanley came across M(n) as one of the posets QX and proved it is Sperner, indeed Peck, without any knowledge o f the work of LindstrEm or Erdgs and Moser. Harper pointed out the connection, and Stanley went on to prove the Erdgs-Moser conjecture and generalize it. Thus an elementary number theory problem was solved via algebraic geometry and Sperner theory!

..

Stanley's solution of the ErdEs-Moser problem can be reduced to using linear algebra in place of algebraic geometry but this proof is still motivated by deeper considerations [Prl, P3]. A more elementary proof that L(m,n) is Peck is described in the next section.

J.R. Griggs

402

Another n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a p o s e t P t o b e Peck has b e e n d i s c o v e r e d by P r o c t o r [ P r 2 ] . The c o n d i t i o n i s t h a t P c a r r i e s a c e r t a i n r e p r e s e n t a t i o n o f t h e L i e a l g e b r a sR(2.C). This condition l e a d s to s h o r t e r p r o o f s , f r e e o f a l g e b r a i c g e o m e t r y , t h a t some B r u h a t o r d e r s a r e P e c k .

W e s a w i n t h e l a s t s e c t i o n t h a t t h e e q u i v a l e n c e of (") a n d P e c k c a n b e gene r a l i z e d t o t h e e q u i v a l e n c e of s t r o n g l y S p e r n e r and T f o r rank-unimodal p o s e t s . [ P S S ] o b s e r v e d how t o u s e l i n e a r a l g e b r a a n d t h i s r e s u l t t o g e n e r a l i z e t h e t h e o r e m a b o v e . A s b e f o r e , 7, i s t h e s p a c e w i t h b a s i s Pi. Let be the s p a c e w i t h b a s i s P. A r a i s i n g ( r e s p e c t i v e l y , l o w e r i n g ) o p e r a t o r o n P i s a

p:

linecir transformation x

E

covers

is

+

(resp.,

A:

is

=

Theorem 2 . L e t a r e equivalent.

P

-t

F)

such tlmt f o r a l l

implies

be a f i n i t e r a n k e d p o s e t .

c

Y

=

0

(i)

P

is rank-unimodal

and s t r o n g l y Sperner.

P

i s rank-unimodal

and s a t i s f i e s c o n d i t i o n

There e x i s t a r a i s i n g o p e r a t o r

X

operator

and

on P

(a)

lPil

5 1p.1

3

(b)

lPil

2 IPjI

=$

J

p

such t h a t f o r a l l pj-i

I Pi

xj-i

unless

y

The €allowing t h r e e c o n d i t i o i - s

(ii) (iii)

5.

is

C cyy ( r e s p . , A(x) = C c y ) Yep y€P x i n P (resp., x cuvers y i n P).

P, p ( x )

T.

and a l o w e r i n g i < j,

is i n j e c t i v e , is injective.

PRODUCTS AND QUOTIENTS

Many i n t e r e s t i n g p o s e t s c a n be formed by a p r o d u c t of p o s e t s o r by t h e (See s e c t i o n 2 f o r d e f i n i q u o t i e n t o f a l a r g e r p o s e t by a n a u t o m o r p h i s m g r o u p . tions.) F o r e x a m p l e , B a n d D a r e p r o d u c t s o f c h a i n s a n d L(m,n) is a q u o t i e n t of

Bmn

under an a p p r o p r i a t e group.

Some o f t h e m o s t p r o f o u n d S p e r n e r

t h e o r e m s c o n c e r n how S p e r n e r p r o p e r t i e s a r e p r e s e r v e d u n d e r p r o d u c t s and q u o t i e n t s . C a n f i e l d [C2] a n d P r o c t o r , Saks, a n d S t u r t e v a n t [PSS] i n d e p e n d e n t l y d i s c o v e r e d t h a t t h e p r o d u c t o f Peck p o s e t s is Peck. T h e i r p r o o f s i n v o l v e l i n e a r a l g e b r a , w h i c h seems t o b e t h e p r o p e r " b o o k k e e p i n g " t o o l t o s t o r e a l l t h e i n f o r m a t i o n n e c e s s a r y t o show t h a t PXQ i s s t r o n g l y S p e r n e r when P a n d Q a r e P e c k . (PXQ i s e a s i l y s e e n t o b e r a n k - s y m m e t r i c a n d r a n k - u n i m o d a l . ) Proctor [Pr2] has f o u n d t h e s h o r t e s t known p r o o f o f t h e p r o d u c t t h e o r e m , u s i n g t h e r e p r e s e n t a t i o n condition. T h e r e i s n o known p u r e l y c o m b i n a t o r i a l p r o o f o f t h i s c o m b i n a t o r i a l result. [PSS] g e n e r a l i z e d t h e Peck p r o d u c t theorem, a n d Saks [SaZ] f u r t h e r s t r e n g t h e n e d t h i s as f o l l o w s . P a n d Q a r e s a i d t o b e corn a t i b l e i f t h e r e e x i s t s a t s u c h t h a t f o r a l l i a n d j , l P i l < IPj o n l5 y* i IQ T h i s means, t-1 t-J e s s e n t i a l l y , t h a t f o r a l l k, the l o c a t i o n of t h e kth l a r g e s t rank i n P r e l a t i v e t o t h e l a r g e s t r a n k c o r r e s p o n d s o p p o s i t e l y t o t h e l o c a t i o n o f t h e kth l a r g e s t Saks' theorem s t a t e s t h a t t h e p r o d u c t rank i n Q r e l a t i v e t o t h e l a r g e s t rank. o f two c o m p a t i b l e , s t r o n g l y S p e r n e r p o s e t s i s S p e r n e r . E x a m p l e s show t h a t t h i s r e s u l t c a n n o t b e improved. The P e c k p r o d u c t t h e o r e m i s n o t a c t u a l l y c o n t a i n e d i n t h e t h e o r e m , b u t i t f o l l o w s from i t by an e l e g a n t s h o r t a r g u m e n t [ P S S ] .

I

. I.

The Sperner property

403

The p r o d u c t o f s y m m e t r i c c h a i n o r d e r s i s a s y m m e t r i c c h a i n o r d e r , w h i c h i s e a s y t o show [ K a Z ] . P e r h a p s t h e m o s t u s e f u l p r o d u c t t h e o r e m i s d u e t o H a r p e r [ H l ] w i t h a n o t h e r p r o o f d u e to H s i e h a n d K l e i t m a n [HK]. I t s t a t e s t h a t i f 1' a n d (1 h a v e a c e r t a i n p r o p e r t y , c a l l e d the LYM p r o p e r t y , a n d h a v e l o g a r i t h m i c a l l y conc a v e IJhitney numbers ( t h i s i s s t r o n g e r t h a n r a n k - u n i m o d a l i t y ) , t h e n PYQ h a s t h e same p r o p e r t i e s . S e e [ G H 3 J f o r t h i s a n d more on t h e LYM p r o p e r t y , w h i c h i s stronger than t h e strongly Sperner property. For Harper's "global" category t h e o r y t r e a t m e n t of t h i s , see [ H 3 ] . [Gg4] a p p l i e d t h e p r o d u c t t h e o r e m s a n d o t h e r techniques t o o b t a i n c o l l e c t i o n s of s u b s e t s w i t h t h e Sperner p r o p e r t y . The m a i n r e s u l t t h e r e i s g e n e r a l i z e d v i a c a t e g o r y t h e o r y i n [Wa]. T h r e e p a p e r s from 1 9 7 6 a n d b e f o r e [Kn, LW, P o ] i n c l u d e r e s u l t s a b o u t a permut a t i o n group a c t i n g on a f i n i t e s e t which c a n be u s e d t o s h m tli:it B /C, t h e rankB o o l e a n a l g e b r a modulo a p e r m u t a t i o n g r o u p C o n i t s g e n e r a t o r s , i s u n i m o d a l a n d S p e r n e r . T h i s i s d o n e u s i n g l i n e a r t r a n s f o r m a t i o n s on t h e s p a c e gene r a t e d by t h e e l e m e n t s o f B,/G. I n f a c t , simple l i n e a r a l g e b r a i c techniques, inc l u d i n g Theorems 1 a n d 2 o f t h e l a s t s e c t i o n o f t h i s p a p e r , c a n b e e m p l o y e d t o prove t h a t Bn/G i s P e c k [H4, PR, S t 4 ] . T h e r e are two p a r t i c u l a r l y n i c e a p p l i c a t i o n s o f t h i s r e s u l t . The f i r s t o n e i s t o o b s e r v e t h a t t h e r e i s a n e l e m e n t a r y ( l i n e a r a l g e b r a i c ) p r o o f t h a t L(m,n) is P e c k [H4, PR, S t 4 1 . T h i s f o l l o w s f r o m t h e o b s e r v a t i o n t h a t L(m,n) is t h e q u o t i e n t of Bmn, c o n s i d e r e d as t h e B o o l e a n a l g e r s o f a l l s u b s e t s o f a n mxn a r r a y o f s q u a r e s , modulo tile g r o u p C g e n e r a t e d by t h e p e r m u t a t i o n s o f t h e e l e m e n t s o f e a c h row a n d t h e p e r m u t a t i o n s o f t h e r o w s w i t h e a c h o t h e r . The s e c o n d a p p l i c a t i o n i s t h a t t h e p o s e t o f all s i m p l e u n l a b e l l e d g r a p h s on n v e r t i c e s , o r d e r e d by containment (embedding), i s Peck. T h i s f o l l o w s b e c a u s e t h i s p o s e t i s t h e q u o t i e n t of t h e p o s e t B(;) o f a l l s i m p l e l a b e l l e d g r a p h s o n n v e r t i c e s modulo t h e p e r m u t a t i o n s of t h e v e r t i c e s [H4,PR]. F o r w h a t r a n k e d p o s e t s P a r e a l l q u o t i e n t s P/G P e c k ? H a r p e r [H4] a n d S t a n l e y [ S t 4 1 f o u n d t h a t P/C i s P e c k i f P i s u n i t a r y P e c k , w h i c h m e a n s t h a t the "unitary" l i n e a r transformations $ i ( x ) : Fi+ Pi+1 d e f i n e d by bi(x)

=

C y, Y€P. 1+1

for

x

E

P.

Y>X

t o g e t h e r s a t i s f y c o n d i t i o n ( i i i ) ( a ) o f Theorem 1 i n s e c t i o n 4 . T h a t i s , P i s u n i t a r y P e c k i f i t i s e n o u g h t o t a k e all c = 1 f o r x < y i n c o n d i t i o n ( i i i ) (b) i n o r d e r t o o b t a i n t h e mappings t o p r o y e t h a t P is Peck. Not all P e c k p o s e t s are u n i t a r y P e c k , e . g . , F i g u r e 3 .

Fig. 3 T h e r e are e x a m p l e s known w h i c h show t h a t n e i t h e r t h e P e c k p r o p e r t y n o r t h e u n i t a r y Peck p r o p e r t y i s p r e s e r v e d u n d e r q u o t i e n t s . The p r o o f s of t h e P e c k p o s e t t h e o r e m c a n b e e x t e n d e d t o p r o v e t h a t t h e p r o d u c t o f u n i t a r y P e c k p o s e t s i s u n i t a r y P e c k [H4, S t 4 ] . Thus Bn a n d D,,, w h i c h a r e T h a t L n ( q ) , t h e l a t t i c e s of s u b s p a c e s , p r o d u c t s o f c h a i n s , are u n i t a r y Peck. i s u n i t a r y P e c k w a s shown i n e f f e c t by K a n t o r i n 1 9 7 2 [ K n ] . I t t h e n f o l l o w s by t h e p r o d u c t t h e o r e m t h a t m o d u l a r g e o m e t r i c l a t t i c e s are u n i t a r y P e c k [H4]. S t a n l e y ' s a l g e b r a i c g e o m e t r y a r g u m e n t s [ S t 3 1 show t h a t L(m,n) a n d M(n) are u n i t a r y P e c k , a l t h o u g h q u o t i e n t m e t h o d s a l o n e o n l y e s t a b l i s h t h a t L(m,n) i s Peck.

J.R. Griggs

404

Pouzet and Rosenberg [PR] derived the theorem that Bn/G is Peck as part of a more general study o f relation systems, permutation groups, and posets. Independently, Harper [H4] obtained the more general unitary Peck quotient theorem as part of an even more general study using category theory. Stanley's derivation [St41 of the unitary Peck quotient theorem is more compact, requiring only some linear algebra. More recently, Proctor [Pr3] noted that there is another short proof of the quotient theorem which u s e s the sR(2,C) representation characterization of Peck posets. I n conclusion, we have made great progress in recent years in studying the Sperner property and building an elegant theory, The introduction of algebraic tools has been particularly fruitful, Some nagging questions remain open, and it is hoped that efforts to answer them will lead to even deeper insights.

ACKNOWLEDGEMENT Thanks to JKrgen Stahl for editorial assistance, and to Madeleine Paoli for the French translation. M. Aigner, "Lexicographic matching in Boolean algebras, 14 (1973), 187-194.

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17

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Annals of Discrete Mathematics 23 (1984) 409-464 0 Elsevier Science Publishers B.V. (North-Holland)

409

CHAIN AND ANTICHAIN F A M I L I E S

G R I D S AND YOUNG TABLEAUX G . Viennot U.E.R.

de Math6matiques e t Informatique UniversitC de Bordeaux I 33405 - Talence - FRAXCE

Dedicated t o Professor Corominas

A b s t r a c t . - G r e e n e ' s T h e o r e m r e l a t i n g c h a i n and a n t i c h a i n f a m i l i e s with m a x i m a l cardinalities i n a poset (partially o r d e r e d s e t ) w a s motivated by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e . T h i s c o r r e s p o n d e n c e i s a b i j e c t i o n b e t w e e n p e r m u t a t i o n s a n d p a i r s (P,Q ) of s t a n d a r d Young t a b l e a u x having t h e a m e s h a p e . T h i s b i j e c t i o n o r i g i n a t e d i n t h e r e p r e s e n t a t i o n t h e o r y of the s y m m e t r i c group. The underlying combinatorics i s v e r y d e e p and a l s o takes r o o t s i n t h e t h e o r y of s y m m e t r i c f u n c t i o n s a n d a l g e b r a i c g e o m e t r y . T h e p u r p o s e of t h i s t a l k i s t h r e e f o l d . F i r s t we g i v e a brief s u m m a r y of t h e p r i n c i p a l c o m b i n a t o r i a l p r o p e r t i e s of the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e , e s p e c i a l l y t h o s e having a n o r d e r - t h e o r e t i c a l f l a v o r . Second we s h e d s o m e l i g h t on i t s r e l a t i o n s h i p with p o s e t t h e o r y a n d Greene's Theorem. T h e t h i r d p u r p o s e of t h i s t a l k i s t o s o l v e t h e following o p e n p r o b l e m : give a n i n t e r p r e t a t i o n of t h e v a l u e l o c a t e d i n t h e ( i , j ) c e l l of t h e Young t a b l e a u x P a n d Q . T h i s "local" c h a r a c t e r i z a t i o n of t h e c o r r e s p o n d e n c e is c o m p l e t e l y s y m m e t r i c i n r o w s a n d c o l u m n s a n d r e q u i r e s t h e c o n c e p t of g r i d s a n d e x t e n d a b l e g r i d s . T h e s e l a s t r e s u l t s c a n be e x t e n d e d t o a r b i t r a r y p o s e t s . C o m p l e t e p r o o f s w i l l be g i v e n e l s e w h e r e . R k s u m 6 . - L e th6or'eme de G r e e n e s u r l e s c a r d i n a u x m a x i m a u x d e s f a m i l l e s de c h a r n e s e t d ' a n t i c h a r n e s e x t r a i t e s d ' u n e n s e m b l e ( p a r t i e l l e m e n t ) ordonnk a s o n o r i g i n e d a n s la c o r r e s p o n d a n c e de R o b i n s o n - S c h e n s t e d . C e t t e c o r r e s p o n d a n c e e s t une b i j e c t i o n e n t r e l e s p e r m u t a t i o n s e t l e s p a i r e s ( P , Q ) de t a b l e a u x s t a n d a r d s d e Young de meme f o r m e . C e t t e b i j e c t i o n p r o v i e n t e n f a i t de la t h k o r i e d e s r e p r C s e n t a t i o n s du g r o u p e s y m k t r i q u e . L a c o m b i n a t o i r e s o u s - j a c e n t e e s t f o r t r i c h e e t p r e n d Cgalement r a c i n e d a n s la t h 6 o r i e d e s f o n c t i o n s s y m 6 t r i q u e s e t e n geomktrie algebrique. L e but de c e t e x p o s 6 e s t t r i p l e . D ' a b o r d n o u s donnons un bref a p e r s u d e s propriCtCs c o m b i n a t o i r e s l e s p l u s c l a s s i q u e s de c e t t e c o r r e s pondance, n o t a m m e n t c e l l e s e n r e l a t i o n a v e c la m6thodologie d e s e n s e m b l e s ordonnCs. L e deuxi'eme but de cet e x p o s 6 e s t de f a i r e s e n t i r ?I u n public motivC p a r l e s e n s e m b l e s ordonne's l ' i n t k r s t de m i e u x c o n n a i t r e

G. Viennot

410

c e t t e m e r v e i l l e u s e c o r r e s pondance . Enfin une t r o i s i ' e m e p a r t i e e s t c o n s a c r k e 'a l a rCsolution d ' u n p r o bl'eme o u v e r t : donner une i n t e r p r k t a t i o n s y m k t r i q u e e n l i g n e s e t c o l o n n e s de l a v a l e u r s i t u 6 e d a n s l a c a s e ( i , j ) d e s t a b l e a u x de Young P e t Q . C e t t e dkfinition "locale" de la c o r r e s p o n d a n c e de RobinsonS c h e n s t e d u t i l i s e l e s nouveaux c o n c e p t s de g r i l l e e t de g r i l l e p r o l o n geable. C e s d e r n i ' e r e s i d k e s peuvent Etre 6 t e n d u e s a u x e n s e m b l e s o r donnks q u e l c o n q u e s , L e s p r e u v e s c o m p l e t e s de c e s nouveaux r 6 s u l t a t s s e r o n t donnkes a i l l e u r s .

4

1 - Introduction. In 1938, G . de B . Robinson i n t r o d u c e d [77] a c o r r e s p o n d e n c e between

the

n !

p e r m u t a t i o n s of the s y m m e t r i c g r o u p

6n

and p a i r s of c e r t a i n c o m b i -

n a t o r i a l o b j e c t s c a l l e d s t a n d a r d Young t a b l e a u x . T h i s c o r r e s p o n d e n c e g i v e s in f a c t a bijective ( c o n s t r u c t i v e ) proof of the following identity 2

n ! = 13 f X ,

x

w h e r e the s u m m a t i o n extends o v e r all i r r e d u c i b l e r e p r e s e n t a t i o n s ( o v e r t h e f i e l d of c o m p l e x n u m b e r s ) of t h e s y m m e t r i c g r o u p

'5

and

fh

denotes the d e g r e e

of the c o r r e s p o n d i n g r e p r e s e n t a t i o n . T h i s identity i s c l a s s i c a l i n the r e p r e s e n t a t i o n t h e o r y of finite g r o u p s , w h e r e the left h a n d - s i d e i s the o r d e r of the g r o u p .

6

, e a c h of t h e i r r e d u c i b l e r e p r e n s e n t a t i o n s i s i n b i j e c t i o n with a v e r y s i m p l e c o m b i n a t o r i a l o b j e c t : a p a r t i t i o n of In t h e c a s e of t h e s y m m e t r i c g r o u p

the i n t e g e r

n

below) with

n

.

Such a p a r t i t i o n i s v i s u a l i z e d a s a F e r r e r s d i a g r a m ( s e e f i g u r e 2 c e l l s . In a s e r i e s of p a p e r s ( s e e the c o l l e c t e d p a p e r s [118]) ,

Young i n t r o d u c e d his f a m o u s t a b l e a u x a s a c o m b i n a t o r i a l tool i n t h e study of the r e p r e s e n t a t i o n s of t h e s y m m e t r i c g r o u p . T h e s e s o - c a l l e d Young tableaux a r e l a belings of the F e r r e r s d i a g r a m s with i n t e g e r s s u c h t h a t t h e n u m b e r s a r e i n c r e a sing in r o w s and c o l u m n s ( s e e definition b e l o w ) . The dimension

fA

i s i n f a c t t h e n u m b e r of s t a n d a r d Young t a b l e a u x

a s s o c i a t e d with a g i v e n p a r t i t i o n

A

. The

r i g h t - h a n d s i d e of (1) i s i n t e r p r e t a t e d

a s t h e n u m b e r of p a i r s of s t a n d a r d Young t a b l e a u x having the s a m e shape

(i.e.

F e r r e r s diagram). T h e p r o b l e m to find

fA

b e c o m e s a p u r e l y c o m b i n a t o r i a l p r o b l e m , and

41 1

Chain and antichain families i s i n f a c t a p o s e t p r o b l e m : f i n d t h e n u m b e r of l i n e a r e x t e n s i o n s of t h e p o s e t defined by the c e l l s of the F e r r e r s d i a g r a m . C o m b i n a t o r i c s of Young t a b l e a u x a p p e a r s a l s o in many o t h e r a r e a s ,

. , .), i n v a manifolds, . . . ) . The

a s f o r e x a m p l e the t h e o r y of s y m m e t r i c f u n c t i o n s ( S c h u r f u n c t i o n s , riant theory o r a l g e b r a i c g e o m e t r y (Schubert calculus, flag i n t e r e s t e d r e a d e r w i l l s e e t h e books [72],[80], recent

t h e s u r v e y p a p e r [7] a n d s o m e

w o r k of L a s c o u x a n d S c h u t z e n b e r g e r . We s h a l l r e s t r i c t o u r s e l f to t h e

c o m b i n a t o r i a l point of v i e w . T h e c o r r e s p o n d e n c e i n t r o d u c e d by Robinson w a s r e d i s c o v e r e d by S c h e n s t e d [89] a n d defined m o r e c l e a r l y i n p u r e l y c o m b i n a t o r i a l way by a r e c u r s i v e a l g o r i t h m : the "bumping p r o c e s s " . Since t h e n , t h i s c o r r e s p o n d e n c e h a s been known a s t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e a n d a p p e a r s a l s o t o be of p u r e l y c o m b i n a t o r i a l i n t e r e s t . Following S c h u t z e n b e r g e r , m u c h c o m b i n a t o r i a l w o r k h a s b e e n done. A good s u r v e y of the s t a t e of t h e a r t i n 1972 c a n be found in K n u t h ' s book ;52],

s e c t i o n 5 . 1 . 4 , a n d m o r e r e c e n t l y i n t h e book [13] e d i t e d

by D . F o a t a a n d r e l a t e d t o the S t r a s b o u r g " t a b l e r o n d e " in A p r i l 1976. The Robinson-Schensted correspondence l i n k s with p o s e t t h e o r y . With e v e r y p e r m u t a t i o n

0

U

*

(P(O),

Q(u))

has strong

, one c a n a s s o c i a t e a p o s e t

having d i m e n s i o n G 2 , a n d c o n v e r s e l y a n y s u c h p o s e t c a n be o b t a i n e d i n t h i s way. Increasing subsequences c o r r e s p o n d to chains and d e c r e a s i n g subsequences c o r r e s p o n d t o a n t i c h a i n s . S o m e c o m b i n a t o r i a l p r o p e r t i e s of t h e c o r r e s p o n d e n c e a r e i n f a c t p r o p e r t i e s of t h i s p o s e t . F o r e x a m p l e t h e well-known S c h e n s t e d p r o p e r t y g i v e s t h e length of the l o n g e s t i n c r e a s i n g ( r e s p . d e c r e a s i n g ) s u b s e q u e n c e a s t h e n u m b e r of e l e m e n t s of t h e f i r s t r o w ( r e s p . c o l u m n ) of t h e F e r r e r s d i a g r a m common to

P(u)

and

Q(o) . A g e n e r a l i z a t i o n h a s b e e n g i v e n by G r e e n e

[35], i n t e r p r e t a t i n g t h e e n t i r e s h a p e ( F e r r e r s d i a g r a m ) of

P(0) a n d

Q(0)

i n t e r m s of a f a m i l y of c h a i n s a n d a n t i c h a i n s with m a x i m u m c a r d i n a l i t i e s . T h i s w a s t h e s t a r t i n g point f o r t h e d e e p a n d now c l a s s i c a l G r e e n e ' s T h e o r e m [37] : t h e duality between f a m i l y of c h a i n s and a n t i c h a i n s c o m i n g f r o m the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e ( p o s e t s with d i m e n s i o n C 2 ) i s v a l i d for any poset. F r o m a p o s e t point of view, i t m a y be f r u i t f u l t o h a v e a good u n d e r standing of t h e o r d e r - t h e o r e t i c a l p r o p e r t i e s of t h e R o b i n s o n - S c h e n s t e d c o r r e s pondence ( t h i s c a n be a l r e a d y v e r y d i f f i c u l t ) a n d t h e n t o l o o k f o r p o s s i b l e e x t e n s i o n s t o a r b i t r a r y p o s e t s . In p a r t i c u l a r s o m e g e n e r a l i z a t i o n s of t h e c o r r e s p o n -

G. Viennot

412

dence h a v e been p r o p o s e d f o r a r b i t r a r y p o s e t s by F o m i n [16] a n d G a n s n e r [ 2 7 ] . G r e e n e ' s proof of h i s t h e o r e m r e l i e s on s o m e j o i n t w o r k w i t h K l e i t m a n [39] g e n e r a l i z i n g t h e well-known D i l w o r t h ' s T h e o r e m . T h i s w o r k i s p a r t of the s o - c a l l e d " e x t r e m a l p r o p e r t i e s " a n d a s u r v e y of t h i s f i e l d c a n be found i n W e s t [113]. Since the p a p e r s of G r e e n e a n d Kleitman, e x t e n s i v e r e s e a r c h h a s b e e n done a n d s e v e r a l o t h e r p r o o f s have been given ( s e e b e l o w ) . In p a r t i c u l a r , G r e e n e ' s T h e o r e m c a n a l s o be d e d u c e d f r o m i n t e g e r p r o g r a m m i n g t e c h n i q u e s , using the m i n i m a l c o s t flow a l g o r i t h m of F o r d a n d F u l k e r s o n [23], [47]

.

F o r a s u r v e y of

the u s e of l i n e a r p r o g r a m m i n g i d e a s i n poset t h e o r y , s e e Hoffman [ 4 5 ] . T h i s p a p e r is a r r a n g e d i n the following way. F i r s t in s e c t i o n 2, we r e c a l l G r e e n e ' s T h e o r e m ( f o r a r b i t r a r y p o s e t s ) . T h e n we give d i f f e r e n t v e r s i o n s of the Robinson-Schensted c o r r e s p o n d e n c e : the o r i g i n a l definition of S c h e n s t e d in s e c t i o n 3, a " p l a n a r i z a t i o n " given by Viennot [lo91 in s e c t i o n 5, t h e n the s y n t h e s i s m a d e by S c h i i t z e n b e r g e r a n d L a s c o u x with t h e 'ljeu de taquin" and c u l m i nating i n the plactic monoid [62], [97] in s e c t i o n 6 , S o m e c l a s s i c a l c o m b i n a t o r i a l p r o p e r t i e s of the c o r r e s p o n d e n c e a r e given in s e c t i o n 4 . In s e c t i o n 7 we i n t r o duce the new c o n c e p t s of g r i d s and extendable g r i d s . We a r e t h u s a b l e t o give a n i n t e r p r e t a t i o n of t h e s h a p e of t h e Young t a b l e a u x

P(0) a n d

Q(a)

that i s

c o m p l e t e l y s y m m e t r i c in r o w s a n d c o l u m n s . M o r e o v e r , we give a n e x p l i c i t m i n Q. .(u)) l o c a t e d i n t h e ( i , j ) c e l l 'J 1J ( r e s p . Q(0)). In s e c t i o n 8 we extend t h e s e i d e a s to a r b i t r a r y p o s e t s .

m a x f o r m u l a f o r the v a l u e of

P(a)

P..(u) ( r e s p .

T h i s e x t e n s i o n r e l i e s on s o m e t h e o r e m s of F r a n k [23], p r o v e d by l i n e a r p r o g r a m m i n g t e c h n i q u e s , f r o m which G r e e n e ' s T h e o r e m c a n be d e d u c e d . An o p e n p r o b l e m is t o prove d i r e c t l y the r e s u l t s of s e c t i o n 8 without

the u s e of l i n e a r

programming. In t h i s p a p e r , we do not p r e t e n d t o give a n e x h a u s t i v e s u r v e y of t h i s huge s u b j e c t a n d apologize f o r o m i s s i o n s a n d unquoted p a p e r s . A l s o we c o n f e s s t h a t we c h o s e s o m e of the m o s t s p e c t a c u l a r p r o p e r t i e s r e l a t e d t o the high u s e of t r a n s p a r e n c i e s in the t a l k . Unfortunately, t h e s i m p l i c i t y of the c o m b i n a t o r i a l c o n s t r u c t i o n s , t o g e t h e r with t h e m a g i c of t h i s v e r y beautiful c o r r e s p o n d e n c e , cannot be w r i t t e n down i n a p a p e r a s e a s i l y a s i t c a n be d e s c r i b e d in a n o r a l c o m m u n i c a t i o n with a f r i e n d o r using s u p e r p o s i t i o n of p i c t u r e s with t r a n s p a r e n c i e s . T h i s i s the r u l e i n c o m b i n a t o r i c s , N e v e r t h e l e s s , we hope t h a t the r e a d e r , not having a t t e n t e d t h e t a l k c o r r e s p o n d i n g t o t h i s p a p e r , will a g r e e with Knuth's s t a t e m e n t ([52], page 6 0 , l i n e 21) : " T h e unusual n a t u r e

Chain and antichain families

413

of t h e s e c o i n c i d e n c e s m i g h t l e a d u s t o s u s p e c t t h a t s o m e s o r t of w i t c h c r a f t i s o p e r a t i n g behind the s c e n e s !

4

2 -

".

Greene's Theorem. Let

P

be a f i n i t e p o s e t ( p a r t i a l l y o r d e r e d s e t ) . An a n t i c h a i n i s a

P

s u b s e t of p a i r w i s e i n c o m p a r a b l e e l e m e n t s of t a l l y o r d e r e d by t h e induced o r d e r of k - f a m i l y ) i s a s u b s e t of c h a i n s ) . We denote by chain f a m i l i e s Example :

P

P

.

(resp.

P

Let

k anti-

d (P)) the m a x i m u m c a r d i n a l i t y of

k

1,2, . . ., 9

.

P

.

F o r t h e r e a d e r u n f a m i l i a r with p o s e t t h e o r y , we

4

P

of

t i v e c l o s u r e of t h e following r e l a t i o n : x d y y

chains ( r e s p .

be t h e p o s e t defined by t h e d i a g r a m d i s p l a y e d i n f i g u r e 1.

r e c a l l that t h e ( p a r t i a l ) o r d e r r e l a t i o n

i s below

k

( r e s p . a n t i c h a i n f a m i l i e s ) of t h e p o s e t

The vertices a r e

x

A chain i s a s u b s e t that i s to-

A chain k-family ( r e s p . antichain

t h a t i s t h e union of

ck(P)

.

i s d e f i n e d a s t o be t h e t r a n s i -

iff a n d e d g e l i n k s

.

F i g u r e 1 : A poset

.

x

and

y

and

414

For

G. Viennot

t h e following s u b s e t s a r e c h a i n k - f a m i l i e s with m a x i m u m c a r -

k = 1,2, 3

dinality

The subset

Y2

= [1,2,5,8,93

k = l ,

Y

k = 2 ,

Y2 = { l , 2, 3 , 4 , 6 , 7 , 8 , 9

k = 3 ,

Y = P = [ l ., . . , 9 ) ,

1

3

3

is a union of two c h a i n s ,

i s a union of t h r e e c h a i n s y3 c2(P) = 8 , c3(P) = 9 .

Y2 = [ 1, 2, 3 , 4 ) U { 6 , 7 , 8 , 9

1,

while

Y3 = { l , 2 , 3 , 4 3 U { 5) U ( 6 , 7 , 8 , 9 ] . T h u s c l ( P ) = 5 ,

Note t h a t , i n t h i s p a r t i c u l a r e x a m p l e , t h e r e e x i s t s only one c h a i n k - f a m i l y with m a x i m u m c a r d i n a l i t y , f o r For

k = 1, 2 , 3 , 4, 5

k = 1, 2 , 3

,

, t h e following s u b s e t s a r e a n t i c h a i n k - f a m i l i e s

with m a x i m u m c a r d i n a l i t y . k = l ,

61 = { 3 , 5 , 7 3 ,

k = 2 ,

6 = {1,61 U { 3 , 5 , 7 1 , 2

k = 3 ,

6

k = 4 ,

6 4 = (1,6] U {2,7) U r3,8)

k = 5 ,

6

We h a v e w r i t t e n e a c h Thus

dl(P) = 3

Here, for each

bk

3

5

= (1,6) U {3,5,71 U ( 4 , 8 ) ,

= P = {1,67

a s a union of

k

u

U { 2 , 71 U f 3 , 8 7 U [ 4 , 97 U [ 5 ? .

antichains.

, d2(P) = 5 , d3(P) = 7 , d4(P) = 8 , d5(P) = 9 k , 1G k

with m a x i m u m c a r d i n a l i t y

<4

,

(4,9)

.

, t h e r e e x i s t s s e v e r a l distinct antichain k-families

dk(P)

.

In t h e l i t e r a t u r e , c h a i n k - f a m i l i e s a r e often c a l l e d k - c o f a m i l i e s , a n t i c h a i n k - f a m i l i e s a r e a l s o c a l l e d k - f a m i l i e s , while a n t i c h a i n k - f a m i l i e s with maximum cardinality dk(P) Usually

dk(P)

are c a l l e d S p e r n e r k - f a m i l i e s . T h e notation

i s in honor of Dilworth (Saks c a l l s t h i s t h e kth D i l w o r t h n u m b e r ) . ck(P)

i s denoted by

throughout t h e p a p e r ,

ck(P)

2k ( P ) ,

We have c h a n g e d t h i s notation b e c a u s e

i s a s s o c i a t e d with t h e "shape"

X

of the Young

t a b l e a u x obtained by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e , while a s s o c i a t e d with the "conjugate s h a p e " , usually d e n o t e d by A p a r t i t i o n of a n i n t e g e r

n

^x

or

A*

dk(P)

is

.

i s a n o n - i n c r e a s i n g s e q u e n c e of non z e r o

Chain and antichain families

415

's, \ . . . 3 k > O s u c h t h a t n = X t . . . t X . Usually a p a r t i t i o n 1 2 P 1 P A i s v i s u a l i z e d by a " F e r r e r s d i a g r a m " F ( s e e f i g u r e 2 , i n " F r e n c h notation") th T h e n u m b e r of e l e m e n t s i n t h e i row i s X i . If :1 d e n o t e s t h e n u m b e r of J e l e m e n t s i n t h e jth c o l u m n , we obtain a n o t h e r p a r t i t i o n X = (X 2 . . . > X ) , 1 9 c a l l e d the con,jugate p a r t i t i o n of 1 . We have d e n o t e d by q t h e n u m b e r of e l e -

X

integers

P .

m e n t s of the f i r s t row of

With t h e s e definitions we c a n s t a t e t h e well-known G r e e n e ' s T h e o r e m

a s follows T H E O R E M 1.

(Greene)

Let

P

be a f i n i t e p o s e t with

n

elements,

q

( r e s p . p) t h e length of t h e l a r g e s t c h a i n ( r e s p . a n t i c h a i n ) . We define t h e n u m bers

k i ( P ) = ci(P) - c.

1-1

&ri(P) = di(P) - di-l(P)

(P) and

(with t h e convention

c , ( P ) = d o ( P ) = 0 ) . T h e n we have t h e following p r o p e r t i e s (i)

A1(P)3 . . . 1 1 ( P ) ,

(ii)

y1(P)I

(iii)

t h e p a r t i t i o n s d e f i n e d by ( i ) a n d ( i i ) a r e c o n j u g a t e .

P

. . . 31 u

9

(P) ,

T h e F e r r e r s d i a g r a m a s s o c i a t e d with t h e p a r t i t i o n defined by ( i ) w i l l b e c a l l e d t h e G r e e n e d i a g r a m of

Example :

P

G(P)

a n d d e n o t e d by

.

T h e G r e e n e d i a g r a m a s s o c i a t e d with t h e p o s e t defined by f i g u r e 1

i s displayed i n figure 2 .

a

= (5,3,1)

*

h = (3,2,2,1,1)

Figure 2.

T h e F e r r e r s d i a g r a m of

X

= (5, 3 , l ) .

416

G. Viennot

Note t h a t r e l a t i o n s ( i ) a n d ( i i ) of G r e e n e ' s T h e o r e m a r e not a t a l l o b v i o u s . In p a r t i c u l a r , a c h a i n k - f a m i l y ( r e s p . a n t i c h a i n k - f a m i l y ) with m a x i m u m c a r d i n a l i t y i s not n e c e s s a r i l y obtained by adding a c h a i n ( r e s p . a n t i c h a i n ) f r o m a c h a i n (k-1)-family ( r e s p . a n t i c h a i n ( k - 1 ) - f a m i l y ) with m a x i m u m c a r d i n a l i t y . T h e poset of f i g u r e 1 h a s a unique c h a i n of length 5 a n d a unique c h a i n 2 - f a m i l y of c a r d i n a l i t y 8 which i s a union of two c h a i n s of length 4. O r i g i n a l l y , G r e e n e p r o v e s h i s r e m a r k a b l e t h e o r e m [37] by using s o m e w o r k done with K l e i t m a n [39] t h a t e x t e n d s t h e c l a s s i c a l D i l w o r t h ' s T h e o r e m [ l l ] , d ( P ) of the l a r g e s t a n t i c h a i n in t h e p o s e t P i s the 1 m i n i m u m n u m b e r of c h a i n s which c o v e r P . Given a n y c h a i n p a r t i t i o n C of

which s t a t e s that the s i z e

m ( C ) = C m i n ( k , c ) . Since e a c h c h a i n c o n t r i b u t e s a t m o s t k e l e CEC m e n t s t o a k - f a m i l y , dk(P)4 m ( C ) . If equality holds then C i s c a l l e d a k k - s a t u r a t e d partition. Greene and Kleitman proved that f o r any k t h e r e e x i s t s

P , let

a chain partition

C

t h a t i s both

k

and ( k t 1 ) - s a t u r a t e d . T h i s f a c t i s known

a s t h e t - p h e n o m e n o n , a n d i m p l i e s condition ( i i ) of t h e o r e m 1. Since t h e n , m a n y o t h e r p r o o f s have been g i v e n : S a k s c o n s i d e r e d a n t i c h a i n k - f a m i l i e s i n a p r o d u c t of two p o s e t s [86], €881 a n d gave a s h o r t proof of the e x i s t e n c e of k - s a t u r a t e d p a r t i t i o n s [87]. T h e t o o l s of l i n e a r a l g e b r a c a n be u s e d : S a k s [86] and G a n s n e r [27] p r o v e d that t h e n u m b e r s

ck(P)

c a n be c o n s i -

d e r e d a s the i n v a r i a n t s of a nilpotent m a t r i x , t h a t i s the block s i z e s of i t s J o r d a n c a n o n i c a l f o r m ; thus t h e y p r o v e d condition ( i ) of t h e o r e m 1. Dual i n t e r p r e t a t i o n s d (P) . A n o t h e r kind of proof c o m e s f r o m l i n e a r p r o g r a m k m i n g using the m i n i m a l c o s t flow a l g o r i t h m of F o r d a n d F u l k e r s o n : Hoffman and

exist for the numbers

S c h w a r t z [47], F o m i n [16],

F r a n k [23],

s e e a l s o H o f f m a n ' s s u r v e y p a p e r on the

u s e of l i n e a r p r o g r a m m i n g i n p o s e t t h e o r y [ 4 5 ] ,

$ 3

-

T h e Robinson-Schensted a l g o r i t h m . In t h e c a s e of a p o s e t of d i m e n s i o n 2 , t h e r e is a v e r y convenient way of

computing the G r e e n e d i a g r a m : j u s t apply t h e R o b i n s o n - S c h e n s t e d a l g o r i t h m . We denote by group on

[n]

the s e t

[n]. F o r e v e r y p e r m u t a t i o n

t o be t h e s e t of points

{ l ,2 , 0

6 = { ( i , O ( i ) ) ,i E [n] )

o r d e r of t h e u s u a l p r o d u c t o r d e r :

., ,.n)

a n d by

En

the s y m m e t r i c

6 we a s s o c i a t e a p o s e t P o s ( ~ ) n o r d e r e d by t h e i n d u c e d c [n] x [n]

of

Chain and antichain families ( x , y ) & (XI,y ' )

(2)

E v e r y p o s e t of d i m e n s i o n G 2

iff

A

and

y

< y'

i s a s u b p o s e t of the p r o d u c t of two c h a i n s , and

c a n in f a c t be identified with a p o s e t Let

x 4 x'

417

be a p a r t i t i o n of

g r a m . A Young t a b l e a u of s h a p e

A.

-

f o r a certain permutation

0

n

and

0

.

the associated F e r r e r s dia-

Fx

i s a l a b e l i n g of t h e c e l l s of

with i n t e -

F

x

g e r s s u c h t h a t t h e s e n u m b e r s a r e s t r i c t l y i n c r e a s i n g in t h e c o l u m n s (down-up i n t h e " F r e n c h notation") a n d a r e weakly i n c r e a s i n g i n t h e r o w s ( l e f t t o r i g h t ) . If the e n t r i e s of t h e Young t a b l e a u a r e d i s t i n c t i n t e g e r s and a r e the i n t e g e r s 1,2,

. . .,n

, t h e n we h a v e a s t a n d a r d Young t a b l e a u on

[n]

( s e e f i g u r e 3 ) . In

o t h e r w o r d s , a s t a n d a r d Young t a b l e a u i s a n e m b e d d i n g in t h e c h a i n poset a s s o c i a t e d with

5

of t h e

[n]

( w h e r e the o r d e r i s the i n d u c e d o r d e r of t h e p r o d u c t

order).

m

h 1

2

2

3

m

4

A Young t a b l e a u , a n d a s t a n d a r d Young t a b l e a u o n [9] with s h a p e A. = ( 5 , 3 , l ) .

Figure 3 .

We define now r e c u r s i v e l y a p r o c e s s t h a t i n s e r t s a n i n t e g e r Young t a b l e a u tinct f r o m

x If

I(T,x)

T

.

.

We s u p p o s e t h a t t h e e n t r i e s of

x

If not, t h e n l e t

T

.

Then

x

.

Let

I(T,x)

I(T,x).

i s g r e a t e r t h a n e v e r y e l e m e n t of t h e f i r s t r o w of

z

x

in a

a r e all d i s t i n c t a n d d i s -

T h e r e s u l t i s a Young t a b l e a u d e n o t e d by

i s t h e Young t a b l e a u o b t a i n e d by adding

t e r than

T

x

T , then

a t t h e end of t h i s f i r s t r o w .

be t h e s m a l l e s t v a l u e of t h e f i r s t r o w of

T

grea-

be t h e Young t a b l e a u o b t a i n e d by d e l e t i n g the f i r s t r o w of T1 i s t h e t a b l e a u o b t a i n e d f r o m T by r e p l a c i n g z by x

418

G. Viennot

a n d by i n s e r t i n g ( r e c u r s i v e l y ) t h e v a l u e

in the tableau

z

T1

T h e a r r a n g e m e n t i s a Young t a b l e a u a t e a c h s t a g e b e c a u s e the r e p l a c e m e n t of

by

z

x

s a t i s f i e s the r e q u i r e d i n e q u a l i t i e s i n i t 5 r o w a n d c o l u m n .

Note t h a t t h e shape of shape of

T

(T,x)

. An e x a m p l e of

i s obtained by adding a c e l l on the b o r d e r of the t h i s "bumping p r o c e s s ' ' i s given i n f i g u r e 4 .

I(T,3)

F i g u r e 4.

Now if tableaux

P(O)

u and

The tableau

P = 6 , P1 = ( a ( l ) ) , . 0 pitl = 1 ( P i , ~ ( i + 1 ) ) .

T h e bumping p r o c e s s .

is a p e r m u t a t i o n of

Q(o)

6, . , . , Q n = Q(u)

6 , we define t w o s t a n d a r d Young

n by t h e following.

P(u) i s o b t a i n e d by s u c c e s s i v e i n s e r t i o n s :

. ., Pn ( 0 ) = P(o)

T h e tableau Qo=

=

Q(O)

where, for

it1

i=l,

. , . , n - 1 , Qitl

,

is obtained f r o m

on the b o r d e r of t h e t a b l e a u

s a m e p o s i t i o n as t h e unique c e l l t h a t is in the s h a p e of s h a p e of

. . ., n-1

i s o b t a i n e d b y c o n s t r u c t i n g a s e q u e n c e of t a b l e a u x

such that, f o r

by a d d i n g a new c e l l l a b e l e d

i = 1,

Qi

, in the

Pi+1 but not i n t h e

.

pi In o t h e r w o r d s , the t a b l e a u

Q(u)

s h a p e s o c c u r i n g i n t h e s e q u e n c e of i n s e r t i o n s

i s a coding of t h e s u c c e s s i v e

P1,

. . . , Pn .

i'

Chain and antichain families

419

u - 6 1 7 2 5 8 3 9 4.

Example:

~

-.

IT

I j

r6-1

3

Q

J.

I /

/

Figure 5.

T h e Robinson-Schensted a l g o r i t h m .

It i s e a s y t o s e e t h a t t h i s p r o c e s s c a n be r e v e r s e d a n d the c o r r e s pondence

u + ( P ( u ) , Q ( u ) ) is a b i j e c t i o n . T h i s is t h e R o b i n s o n - S c h e n s t e d c o r -

respondence. The tableau

P(u)

is t h e P - s y m b o l , w h i l e

Q(u)

i s the

Q- symbol. THEOREM 2 -

(Robinson-Schensted)

The correspondence

defined above i s a bijection b e t w e e n t h e n

I p e r m u t a t i o n s of

d

* (P(o),Q(u))

en

and the

G. Viennot

420 p a i r s of s t a n d a r d Young t a b l e a u x on

[n]

having the s a m e s h a p e .

T h i s c o r r e s p o n d e n c e g i v e s a bijective proof of identity (1). The number

fX

of s t a n d a r d Young t a b l e a u x of s h a p e

X

c a n be c o m -

D i f f e r e n t f o r m u l a e have been found and h a v e r e c e i v e d m u c h

puted e x p l i c i t l y . investigations.

h. = ( A , > .

Let let

X'. = X i

i-p-i

. . 5 X P)

be a p a r t i t i o n of

n

and f o r

i , 1 6i 6 p

,

Using c o m p l i c a t e d a l g e b r a i c m e t h o d s ( g r o u p c h a r a c t e r s , s y m m e t r i c p o l y n o m i a l s ) , Young [I181 and F r o b e n i u s [ 2 5 ] independently gave the f o r m u l a

A c o m b i n a t o r i a l proof of ( 3 a ) , using d i f f e r e n c e m e t h o d s , w a s found by Mac Mahon, i n t e r m s of a " n - c a n d i d a t e ballot p r o b l e m " [7 4 ], of " l a t t i c e p e r m u t a t i o n s " [73]

o r equivalently

, page 133.

Another way t o prove ( 3 a ) i s to d e r i v e t h i s f o r m u l a f r o m the d e t e r m i nantal e x p r e s s i o n (3b)

= 0

(with t h e convention

when

X.-itj

0). See f o r example

(A, -itj) Z e l e v i n s k y [120], page 9 i . A v e r y s i m p l e "bijective" proof h a s b e e n g i v e n by

G e s s e l a n d Viennot [30] a s p a r t of a m o r e g e n e r a l w o r k i n t e r p r e t a t i n g m a n y d e t e r m i n a n t s of c o m b i n a t o r i c s a s t h e n u m b e r of c e r t a i n n o n - c r o s s i n g configur a t i o n s of p a t h s ( a n d a bijection between Young t a b l e a u x and s u c h c o n f i g u r a t i o n s ) . T h e f o r m u l a ( 3 a ) c a n be r e a r r a n g e d i n t o a m o r e s i m p l e and e l e g a n t form. Let

x

be a c e l l of the F e r r e r s d i a g r a m

define t h e hook length of x , including

Fx

a s t h e n u m b e r of c e l l s of

x

a s s o c i a t e d with

X

FX l o c a t e d above

. We or a t

itself ( s e e f i g u r e 6 ) . Using t h e f a c t t h a t , f o r any th i , 1 G i G p , the p r o d u c t of the hook length of t h e i r o w is Xi ! / n (,I\h.; ), jbi we o b t a i n f r o m ( 3 a ) t h e v e r y well known a n d c l a s s i c a l f o r m u l a of F r a m e ,

the r i g h t of

Robinson a n d T h r a l l [22]

x

42 1

Chain and antichain families

(3c)

(hooklength formula)

f

X

n! = n h ' x x

w h e r e the p r o d u c t i s t a k e n o v e r a l l hook l e n g t h s

Hook length

h the ( 1 , ~ cell )

of

FX .

Hook lengths of the Ferrers diagram ( 5 , 3 , 1 )

= 5 of

Figure 6 .

hx

Hook l e n g t h s .

A p r o b a b i l i s t i c proof of ( 3 b ) h a s been g i v e n by G r e e n e , N i j e n h u i s and Wilf [41]. B i j e c t i v e p r o o f s of ( 3 c ) h a v e been g i v e n by R e m m e l [76] a n d Z e i l b e r g e r [119]. In f a c t , R e m m e l c o m b i n e d the n o n - c r o s s i n g c o n f i g u r a t i o n s of p a t h s m e n t i o n e d above a n d i n t e r p r e t a t i n g ( 3 b ) , with a bijective d e r i v a t i o n of ( 3 a ) a n d ( 3 c ) f r o m ( 3 b ) . using t h e "involution p r i n c i p l e " i n t r o d u c e d by G a r s i a a n d Milne [29] i n t h e i r bijective proof of t h e c e l e b r a t e d R o g e r s - R a m a n u j a n i d e n t i t y . N e v e r t h e l e s s , no s i m p l e proof of ( 3 c ) i s known a n d n o d i r e c t c o m b i n a t o r i a l c o r r e s p o n d e n c e e x p l a i n s the r o l e of t h e hook l e n g t h s . A s f o r t h e R o g e r s - R a m a n u j a n i d e n t i t i e s , i t i s s t i l l a n open p r o b l e m to know w h e t h e r t h e r e e x i s t s a " s i m p l e " a n d " n a t u r a l " bijective d e r i v a t i o n of ( 3 c ) . Perhaps,

the b e s t c o m b i n a t o r i a l i n t e r p r e t a t i o n of t h e hook l e n g t h i s

due t o G r a s s l a n d H i l l m a n [31]. T h e hooks a p p e a r n a t u r a l l y i n a c o m b i n a t o r i a l c o r r e s p o n d e n c e involving the s o - c a l l e d p l a n e p a r t i t i o n s . T h i s s u b j e c t i s v e r y c l o s e to o u r t o p i c , but w i l l not be t o u c h e d i n t h i s p a p e r . F r o m G r a s s l a n d H i l l m a n ' s c o r r e s p o n d e n c e , f o r m u l a ( 3 ) c a n be d e r i v e d using a n a s y m p t o t i c argument. K r e w e r a s [53] gave a n e x t e n s i o n of f o r m u l a (3b) t o "skew Young

422

G. Viennot

tableaux" ( s e e definition i n

9 6

b e l o w ) . Canfield and W i l l i a m s o n [I171 gave a n

" o p e r a t o r " f l a v o u r f o r hook l e n g t h s . T h e n u m b e r s

fX

a r e a l s o related t o the

( w e a k ) B r u h a t o r d e r ( s e e the end of t h i s p a p e r ) . T h e p r o b l e m of computing p r o b l e m i n poset t h e o r y . If

P

of

P

i s a p a r t i c u l a r c a s e of a c l a s s i c a l

is a poset with

n

( o r o r d e r - p r e s e r v i n g m a p ) i s a bijection

V(X)~.Cp(Y)

if

X 4 Y

1,2,

. . . ,n

e l e m e n t s , a l i n e a r extension

cp : P

+

[n]

such that

( i n P ). T h i s is a l s o c a l l e d topological o r d e r i n g ( o r

topological s o r t i n g ) . T h e i n t e g e r s are

fX

rp(x)

c a n be c o n s i d e r e d a s l a b e l s (the l a b e l s

a n d e a c h n u m b e r a p p e a r s once a n d only o n c e ) . S o m e t i m e s ,

s u c h l a b e l i n g with l a b e l s i n c r e a s i n g along t h e c h a i n s of

P , a r e also called

" n a t u r a l l a b e l i n g " . Such e x a m p l e i s given in f i g u r e 1. The F e r r e r s diagram

Fx c a n be c o n s i d e r e d a s a l o w e r i d e a l of

Nx lN ( o r d e r e d by t h e r e s t r i c t i o n of the p r o d u c t o r d e r ( 2 ) ) . T h e n u m b e r the n u m b e r of l i n e a r e x t e n s i o n s of t h i s p o s e t

FA

.

fX

is

The problem for general

p o s e t s i s connected with m a n y o t h e r c o n s i d e r a t i o n s ( s e e f o r e x a m p l e S t a n l e y

[ 9 9 ] ) a n d i s widely o p e n . Two o t h e r kinds of p o s e t s a r e known, giving r i s e t o a n analogous f o r m u l a f o r t h e n u m b e r of l i n e a r e x t e n s i o n s a s a r a t i o of

n 1

by a p r o d u c t of

n u m b e r s playing the r o l e of "hook l e n g t h s " . One kind i s t h e s h i f t e d F e r r e r s d i a g r a m s , that is the induced p o s e t s obtained by deleting the c e l l s above t h e diagonal f r o m F e r r e r s d i a g r a m s ( T h r a l l [108]). A n o t h e r f a m i l y of p o s e t s i s t h e f a m i l y of t r e e s ( w h e r e t h e o r d e r m e a n s "to be a d e s c e n d a n t of" ). It i s e a s y t o prove t h e analog of f o r m u l a ( 3 ) w h e r e the "hook length" i s t a k e n a s t h e s i z e of a s u b t r e e ( s e e Knuth [52], e x e r c i s e 20, s e c t i o n 5 . 1 . 4 . ) . It is a m a j o r p r o b l e m t o find a n o t h e r f a m i l y of p o s e t s having a "hook length" f o r m u l a ( s e e Sagan [82]). T h e c o r r e s p o n d e n c e of t h e o r e m 2 c a n e a s i l y be g e n e r a l i z e d t o s e q u e n c e s with r e p e t i t i o n of l e t t e r s , t h a t i s w o r d s . T h e t a b l e a u t a b l e a u while t h e t a b l e a u

Q

P

i s a Young

i s a s t a n d a r d Young t a b l e a u . Knuth gave i n [51]

a n u l t i m a t e g e n e r a l i z a t i o n w h e r e both i n d i c e s

i

and v a l u e s

o(i)

c a n be

repeated.

5

4 -

S o m e c l a s s i c a l p r o p e r t i e s of t h e Robinson-Schen,r3ted c o r r e s p o n d e n c e . The dihedral group

en

D4

( s y m m e t r i e s of t h e s q u a r e ) a c t s on t h e s e t

of p e r m u t a t i o n s ; i n o t h e r w o r d s , p e r m u t a t i o n m a t r i c e s c a n be f l i p p e d o r

42 3

Chain and antichain families rotated t o obtain other permutation m a t r i c e s . T o r e l a t e this to the RobinsonS c h e n s t e d c o r r e s p o n d e n c e , it s u f f i c e s t o view a p e r m u t a t i o n

6

embedding

= {(i,O(i)))

by the two s y m m e t r i e s :

0

(1

a s t h e obvious

in the s q u a r e [n] x [ n ] . T h e a c t i o n c a n be g e n e r a t e d -1 *u ( i n v e r s e ) a n d u + U* with U * = O(n). . , ~ ( 1 )

( m i r r o r i m a g e ) . T o study t h e a c t i o n , l e t u s e x a m i n e the e f f e c t of the o p e r a t i o n s

u*

u + U-1 and

on the P - s y m b o l a n d Q - s y m b o l .

6*

E v e n though the c o n s t r u c t i o n of the

P

and Q-symbol a r e completely

d i f f e r e n t , we have t h e s u r p r i s i n g l y s i m p l e (and not t r i v i a l ) p r o p e r t y ( S c h u t z e n b e r g e r [91]) :

PROPOSITION 3 -

Q(5) ,

P(m-') =

Q(u-') = P ( u )

F o r the e f f e c t of t h e m i r r o r i m a g e we n e e d t o define two o p e r a t i o n s

Y

on s t a n d a r d Young t a b l e a u x . T h e t r a n s p o s e of a s t a n d a r d Young t a b l e a u t h e t a b l e a u o b t a i n e d by r e v e r s i n g r o w s a n d c o l u m n s a n d i s d e n o t e d by second operation, called

dual, w a s

YT

is

.

The

i n t r o d u c e d by S c h u t z e n b e r g e r [91] a n d i s

more elaborate,

Y

Let trace

Tr(Y)

lowing : x

1 1 4 p < k , if

.

be a s t a n d a r d Young t a b l e a u o n [n]

to be the l o n g e s t s e q u e n c e

. ..

{x,

F i r s t we define t h e

defined by t h e f o l k is t h e value of the (1.1) c e l l ( t h a t i s t h e s m a l l e s t e n t r y of Y ) ; for x

P

(i, j ) , then

i s t h e value of t h e c e l l

of t h e two e n t r i e s l o c a t e d in the c e l l s

(i,j t l )

Y , let

x

and

is the s m a l l e s t P+l ( i t l , j ) , (if one of t h e s e x

be the value i n t h e c e l l that i s ) . By P " l o n g e s t s e q u e n c e " , we m e a n t h a t t h e v a l u e x i s l o c a t e d i n a c e l l (i, j ) k s u c h t h a t t h e c e l l s ( i , j t l ) a n d ( i t l , j ) a r e not i n t h e s h a p e of Y . Such a c e l l s i s not i n the s h a p e of

c e l l of

Y

x

will be c a l l e d a c o r n e r c e l l of t h e F e r r e r s d i a g r a m of We define the t a b l e a u

replacing each value t h e c e l l with v a l u e

x

x

P

d(Y)

Y

. Y

by

of

Tr(P)

2,

by the v a l u e

x

. . . ,n , and shape contained in the

.

LEMMA 4 - (Schtitzenberger) T

Y

, deleting P+l and keeping i n v a r i a n t all o t h e r v a l u e s . Obviously, d ( P ) (1 d p < k )

k i s a Young t a b l e a u with d i s t i n c t e n t r i e s s h a p e of

as the tableau obtained f r o m

Let

u

be a s e q u e n c e of d i s t i n c t i n t e g e r s , and

be t h e s e q u e n c e o b t a i n e d by deleting t h e m i n i m u m e l e m e n t . T h e n we have

P ( T )= d ( P ( 0 ) ) .

G. Viennot

424 Y

Let

YJ

be a s t a n d a r d Young t a b l e a u . We define t h e t a b l e a u

the following l a leling. T h e c e l l containing the g r e a t e s t value

a

xk

of

by

Tr(Y)

is

r e l a b e l e d 1 . F o r 1 , 1 G 4 n , the c e l l which i s in d ( Y ) , but not i n Rt1 a ( Y ) ( t h a t is the c e l l containing t h e g r e a t e s t v a l u e of T r ( d ( Y ) ) ) , i s r e l a d beled

Y

k+l

.

The tableau

with t h e i n t e g e r s

1, 2 ,

YJ

i s a labeling of t h e F e r r e r s d i a g r a m u n d e r l y i n g

. . .,n

such that rows and columns a r e s t r i c t l y d e -

c r e a s i n g , Such a t a b l e a u i s c a l l e d a r e v e r s e s t a n d a r d Young t a b l e a u a n d i s n a m e d the

dual of

.

Y

An e x a m p l e is shown i n f i g u r e 7 .

d(Y) =

b,, , m,,, 5

6

8

4

2

m

& a

7

Figure 7.

5

4

2

6

8

5

4

7

5

4

IZBn 4

5

6

8

2

2

[q,, 9

m

1-51

2

T h e dual of a s t a n d a r d Young t a b l e a u , obtained by "vidage - r e m p l i s s a g e " ,

PROPOSITION 5 - ( S c h e n s t e d , S c h i i t z e n b e r g e r ) . F o r any p e r m u t a t i o n u of T J T p(0") = p ( 0 ) , QC(o*) = (Q ) ( 0 ) , w h e r e QC denote the r e v e r s e e n , t a b l e a u o b t a i n e d f r o m Q by r e p l a c i n g e a c h v a l u e i by n t l - i .

425

Chain and antichain families

T h e r e a d e r i s u r g e d t o t e s t t h i s p r o p e r t y w i t h t h e p e r m u t a t i o n of f i g u r e 5 u s i n g t h e c o n s t r u c t i o n of f i g u r e 7 . By r e v e r s i n g t h e o r d e r , o n e c a n d e f i n e t h e d u a l of a r e v e r s e s t a n d a r d Young t a b l e a u . T h e a b o v e p r o p o s i t i o n i m p l i e s t h e r e l a t i o n

(4)

J J (Y ) = Y

f o r a n v s t a n d a r d Young t a b l e a u

.

In [ 9 4 ] , S c h u t z e n b e r g e r h a s e x t e n d e d t h e c o n s t r u c t i o n t r a r y posets. Let

P

be a p o s e t a n d

PL

s t a n d a r d Young t a b l e a u x , w e c a n d e f i n e

Y

+ YJ

a linear embedding in

for arbi[n] . As f o r

T r ( P L ) , a n i n c r e a s i n g s e q u e n c e of l a -

b e l s c a l l e d t h e t r a c e of

Using the o p e r a t o r analogous to d(Y) , it i s posPL . s i b l e t o d e f i n e a d u a l " r e v e r s e n a t u r a l " l a b e l l i n g of PL . A d e e p r e s u l t i s t h a t relation (4) i s still valid. Knuth's transpositions. W e s t u d y t h e c l a s s of p e r m u t a t i o n s h a v i n g t h e s a m e P - s y m b o l . T h e n u m b e r of s u c h p e r m u t a t i o n s i s o b v i o u s l y

fk

where

X

i s t h e s h a p e of

P

.

We m a y

a s k how to g e n e r a t e all t h e s e p e r m u t a t i o n s f r o m o n e of t h e m , u s i n g e l e m e n t a r y t r a n s f o r m a t i o n s . T h i s i s d o n e by t h e s o - c a l l e d Knuth t r a n s p o s i t i o n s ( K n u t h [51]). Let

DEEn

If t h e v a l u e or

z =

and U(i-1)

x = u(i) ,

or

z =

y = O(it1)

O(it1)

be t w o c o n s e c u t i v e e l e m e n t s of

is b e t w e e n

y < z C x ) , t h e n t h e t r a n s p o s i t i o n of

x

and

y

( t h a t is

u .


x

and

y

i s c a l l e d a Knuth t r a n s -

x < z

position. T h e P - s y m b o l i s invariant under Knuth's transpositions. Conversely, any permutation

T

such that

P(0) = P ( T ) c a n be o b t a i n e d f r o m

u

by a s e q u e n c e

of K n u t h ' s t r a n s p o s i t i o n s .

A p e r m u t a t i o n i s c a l l e d r o w c a n o n i c a l ( r e s p . c o l u m n c a n o n i c a l ) if t h e s e quence

~ ( 1 ). .. ~ ( n ) c a n b e o b t a i n e d by r e a d i n g t h e e n t r i e s of t h e P - s y m b o l i n

t h e f o l l o w i n g w a y : r e a d e a c h r o w f r o m l e f t to r i g h t w i t h r o w s o r d e r e d t o p - d o w n ( r e s p . r e a d e a c h c o l u m n top-down with the c o l u m n s o r d e r e d f r o m l e f t to r i g h t ) . In e a c h class h a v i n g t h e s a m e P - s y m b o l t h e r e i s o n e a n d o n l y o n e c a n o n i c a l p e r mutation. Example -

The permutation

5

=

6 3 581 247 9

i s row canonical. Possible

K n u t h ' s t r a n s p o s i t i o n s a r e t h e t r a n s p o s i t i o n s of c o n s e c u t i v e v a l u e s

( 6 ,3)

G. Viennot

426 and (8,l).

Of c o u r s e , w e c a n d u a l l y d e f i n e K n u t h ' s t r a n s p o s i t i o n s o n t h e v a l u e s i n s t e a d of t h e p o s i t i o n s a n d k e e p t h e Q - s y m b o l i n v a r i a n t . G r e e n e ' s i n t e r p r e t a t i o n of t h e s h a p e . We c o m e b a c k t o t h e m a i n m o t i v a t i o n of t h i s t a l k .

PROPOSITION 6 with t h e p o s e t P(o) a n d

-

For any permutation

Pos(u)

Q(u)

( d e f i n e d in

8

the Greene diagram associated

3

2 ) i s t h e s a m e a s t h e F e r r e r s d i a g r a m of

o b t a i n e d by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e .

In t h e o t h e r w o r d s , t h e n u m b e r of e l e m e n t s of t h e f i r s t c o l u m n s ) i s the m a x i m u m c a r d i n a l i t y of s u b s e q u e n c e s of k

k

rows ( r e s p .

w h i c h a r e union of

0

increasing (resp. decreasing) subsequences. The particular case

k=1

i s known a s S c h e n s t e d ' s T h e o r e m [89] a n d w a s

t h e m o t i v a t i o n of S c h e n s t e d ' s o r i g i n a l c o n s t r u c t i o n . G r e e n e p r o v e d p r o p o s i t i o n 6 in [35] by s h o w i n g i t s i n v a r i a n c e u n d e r K n u t h ' s t r a n s p o s i t i o n s , a n d noting t h a t t h e p r o p e r t y is t r i v i a l f o r r o w c a n o n i c a l p e r m u t a t i o n s . We s h a l l s e e b e l o w a d i r e c t g e o m e t r i c c o n s t r u c t i o n of s u c h s u b s e q u e n c e s ( t h a t i s , u n i o n s of

k

increasing (decreasing) subsequences with m a x i -

m u m c a r d i n a l i t y ) , a n d a l s o a c h a r a c t e r i z a t i o n of t h e s h a p e of t h e P - s y m b o l c o m p l e t e l y s y m m e t r i c with r e s p e c t t o r o w s a n d c o l u m n s .

A n i c e c o r o l l a r y of S c h e n s t e d ' s t h e o r e m is a p r o p o s i t i o n of E r d 6 s a n d S z e k e r e s [12] : a n y p e r m u t a t i o n c o n t a i n i n g m o r e t h a n

n2

elements h a s mono-

t o n i c s u b s e q u e n c e ( i n c r e a s i n g o r d e c r e a s i n g ) of l e n g t h g r e a t e r t h a n

n

(this

p r o p o s i t i o n i s a l s o a c o r o l l a r y of D i l w o r t h ' s T h e o r e m , a s a p p l i e d to p o s e t s of d i m e n s i o n 2 ) . N o t e t h a t t h e r e e x i s t p e r m u t a t i o n s with

.

m o n o t o n i c s u b s e q u e n c e s of l e n g t h g r e a t e r t h a n

n

e x a c t l y t h o s e having a P - s y m b o l with a s q u a r e

nxn

n2

elements having no

These permutations are shape.

Many i n v e s t i g a t i o n s h a v e b e e n m a d e of t h e n u m b e r of p e r m u t a t i o n s h a v i n g a m a x i m u m - l e n g t h i n c r e a s i n g s u b s e q u e n c e . No e x a c t f o r m u l a i s known ( s e e [791

f o r e x a m p l e ) . T h i s p r o b l e m is e q u i v a l e n t t o e n u m e r a t i n g s t a n d a r d Young t a bleaux having

k

r o w s , a n d is r e l a t e d t o p r o b l e m s i n a l g e b r a (see R e g e v [ 7 5 ] ) .

*.

It h a s b e e n p r o v e d by H a i n m e r s l e y [43] t h a t t h e m e a n of t h e l e n g t h of maximal increasing subsequence is asymptotically

c n

L o g a n a n d S h e p p [71]

proved

Chain and antichain families c

2 , while K e r o v a n d V e r s h i k [5O] p r o v e d

c Q2

427

.

c = 2 , as was sug-

Thus

g e s t e d by B a e r and B r o c k [122]. A l s o F r e d m a n [24] gave a n o p t i m a l a l g o r i t h m in nlogn-nloglogn c o m p a r i s o n s t o c o m p u t e t h i s m a x i m a l l e n g t h .

"Line-of - r o u t e " and up-down s e q u e n c e An i m p o r t a n t notion i n t h e e n u m e r a t i v e t h e o r y of p e r m u t a t i o n s i s t h e up-down s e q u e n c e of a p e r m u t a t i o n with two l e t t e r s

t

,

-

.

It i s defined a s a w o r d of l e n g t h n-1

a s follows with

UD(0) = w l ' . .Wn-l

(5)

UEEn

w,1 =

+

~-

iff

o(i) < o(it1) ( r i s e )

if

o ( i ) 2 o ( i + l ) (fall)

T h e dual notion of a r i s e ( r e s p . f a l l ) i s t h a t of a d v a n c e ( r e s p . r e t r e a t ) : x = o(i)

i s a n a d v a n c e ( r e s p . r e t r e a t ) if x t l = O ( j ) with i j -1 y i e l d s t h e a d v a n c e s and r e t r e a t s ( r e s p . i b j ) . T h e up-down s e q u e n c e of 0

the value

.

a

of

S h i i t z e n b e r g e r s h o w e d t h a t the value

u

iff t h e v a l u e

tableau with

xtl

is a n a d v a n c e of t h e p e r m u t a t i o n

x

i s l o c a t e d t o the S o u t h - E a s t of the v a l u e

P(o) (that is

x

i s in the cell

(i, j )

and

xtl

x

i n t h e Young

in t h e cell

(i',jt)

it* i , j ' 4 j ) . Note t h a t if

W e s t (that is

x+l

i s not S o u t h - E a s t of

h a s b e e n c a l l e d t h e l i n e of r o u t e by F o u l k e s . -1 Dually, by taking t h e i n v e r s e u of of r o u t e of

x

, then

xtl

i s a t the North-

i ' 4 i , j ' > j ) . T h i s s u c c e s s i o n of S o u t h - E a s t o r N o r t h - W e s t s t e p s

Q(u)

a

a s m e n t i o n e d a b o v e , the l i n e

g i v e s the up-down s e q u e n c e of the p e r m u t a t i o n ( s e e e x a m p l e

on figure 8 ) . Example -

For

u = 6 1 7 2 5 8 3 9 4 , UD(u)

= -

1

+-

- + - and - -+t+ .

t t

UD(o- ) = t t t

T h e l i n e s of r o u t e of t h e P - s y m b o l a n d Q - s y m b o l a r e t h e following

G. Viennot

428

Figure 8

-

L i n e s of r o u t e of t h e P - s y m b o l a n d Q - s y m b o l

F o u l k e s gave beautiful a p p l i c a t i o n s of t h i s i n t e r p r e t a t i o n to e n u m e r a t i v e p r o b l e m s of p e r m u t a t i o n s [19], [ZO], [Zl] duality

J

Y+Y )

k-matchings

P

in the

-

. Other applications

( u s i n g a l s o the

c a n be found i n F o a t a , S c h u t z e n b e r g e r [15].

A m a j o r q u e s t i o n i s t o give a d i r e c t i n t e r p r e t a t i o n of t h e v a l u e s

a n d Q - s y m b o l . G r e e n e g a v e a n i n t e r p r e t a t i o n of t h e s e t of v a l u e s

which a r e above t h e kth r o w i n the P - s y m b o l . F o r t h a t p u r p o s e , he i n t r o d u c e d the new c o n c e p t of k - m a t c h i n g s . A k - m a t c h i n g of a p e r m u t a t i o n

o

is a n array

(aij)lG i G p , 1 4 j 4 k

i n t e g e r s s u c h t h a t e a c h r o w i s a d e c r e a s i n g s u b s e q u e n c e of of e a c h c o l u m n a r e d i s t i n c t . T h e s e t

all,

. . . , a Pl

u

of

a n d the e l e m e n t s

i s c a l l e d the s o u r c e of t h e

k - m a t c h i n g . F i r s t , t h e n u m b e r of e l e m e n t s i n t h e r o w s above a n d including t h e kth r o w of

P ( u ) is e q u a l t o t h e m a x i m u m s i z e ( a m o n g k - m a t c h i n g s , k f i x e d )

of the s o u r c e of a k - m a t c h i n g of t h e p e r m u t a t i o n [ 3 8 ] . F o r the s e c o n d r e s u l t , we o r d e r s u b s e t s of [n] A


a ppe a r s in

A

t h a t is

by l e x i c o g r a p h i c o r d e r ,

iff t h e s m a l l e s t e l e m e n t t h a t o c c u r s i n only one of

.

A

and

B

G r e e n e [38] showed t h a t t h e s e t of e l e m e n t s l o c a t e d i n the r o w s

above a n d including t h e kth r o w of

P ( u ) is the l e x i c o g r a p h i c a l l y m i n i m u m

s o u r c e a m o n g the s o u r c e s of m a x i m u m - s i z e d k - m a t c h i n g s of t h e p e r m u t a t i o n . G a n s n e r [27] g a v e p r o o f s of t h e s e two f a c t s using t o o l s f r o m l i n e a r a l g e b r a , a n d gave a n a n a l o g of t h e f i r s t p r o p e r t y f o r c o l u m n s , by introducing t h e c o n c e p t of k - s c a t t e r s of a p e r m u t a t i o n . We s h a l l d i s c u s s below the e x t e n s i o n of t h i s t o a r b i t r a r y p o s e t s .

Chain and antichain families § 5 -

429

Planarization In o r d e r t o give a s i m p l e e x p l a n a t i o n of p r o p o s i t i o n 3 ( a n d o t h e r p r o p e r t i e s

such a s G r e e n e ' s i n t e r p r e t a t i o n s ) , Viennot i n t r o d u c e d [lo91 a g e o m e t r i c v e r s i o n of the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e , t h a t is a p i c t o r i a l d e s c r i p t i o n with "shadows" a n d " b o r d e r s of shadows" i n t h e p l a n e . We c o n s i d e r t h e p o s e t

TT = Z x

Z

o r d e r e d by t h e u s u a l p r o d u c t o r d e r ( 2 ) .

We i m a g i n e s o m e light coming f r o m t h e S o u t h - W e s t . T h e shadow of a point

.

x2 i , y B j

(i, j )

i s the s e t of points

F

T h e shadow of a f i n i t e s u b s e t

dows of e a c h point of

F

.

of

(x,y)

of

IRx

IR

with

i s the union of the s h a -

IT

T h e b o r d e r of t h i s shadow ( i n t h e t o p o l o g i c a l s e n s e )

i s a " z i g - z a g l i n e " s u p p o r t e d by the points t h a t a r e "in f u l l light", t h a t i s the

F

points which a r e not i n t h e shadow of a n o t h e r point of

A finite subset

F

of

11

.

i s s a i d t o be a q u a s i - p e r m u t a t i o n iff n o two

d i s t i n c t points a r e o n the s a m e v e r t i c a l o r h o r i z o n t a l l i n e . T h i s t e r m i n o l o g y i s not c l a s s i c a l . S e v e r a l o t h e r n a m e s a r e u s e d , i n p a r t i c u l a r independent s e t . When

TI

i s considered a s a poset ( d i r e c t product) such s e t s a r e called

s e m i a n t i c h a i n s . We i n t r o d u c e h e r e t h e t e r m " q u a s i - p e r m u t a t i o n ' ' b e c a u s e i t w i l l c o r r e s p o n d below t o a s u b s e q u e n c e of a p e r m u t a t i o n viewed a s a s e q u e n c e , i n which the i n f o r m a t i o n a b o u t t h e p o s i t i o n s h e l d by t h e e l e m e n t s i s r e t a i n e d . Let L1,

. . ., L k

shadow of

F

be a q u a s i - p e r m u t a t i o n of

.

We define a s e q u e n c e

of b r o k e n l i n e s by t h e following p r o c e s s :

F , and f o r

L1,. . . , Li

o u t s t a n d i n g of

F

i,

1 6i

k ,

Li+l

i s t h e b o r d e r of t h e L1 i s t h e b o r d e r of the shadow of t h e

Fi+l o b t a i n e d by deleting f r o m

quasi -pe rmutation the l i n e s

n

.

(see figure 9 ) . The lines

L1,

F

t h e points t h a t a r e on

. . .,Lk

w i l l be c a l l e d t h e

G. Viennot

430

-I !;:.+-I Figure 9

-

T h e outstanding l i n e s a n d t h e s k e l e t o n of t h e permutation u = 6 1 7 2 5 8 3 9 4 .

In t h i s c o n s t r u c t i o n , s o m e r e m a r k a b l e points a p p e a r . Following the

F , one c h a n g e s d i r e c t i o n f o r e a c h point of

outstanding l i n e s of

a l s o a change of d i r e c t i o n f o r o t h e r points t h a t a r e not i n points i s a q u a s i - p e r m u t a t i o n of

S(F)

Z x

0

be a P e r m u t a t i o n of

= ( ( i , o ( i ) ) , i E [n]

3

,

Z c a l l e d t h e s k e i e t o n of

. T h e y a r e m a r k e d with c r o s s e s i n f i g u r e Note t h a t IS(F)I t k = IF I . Let

F

Bn

,

F

. There

is

T h e s e t of t h e s e

F

and denoted

9.

We c o n s i d e r t h e q u a s i - p e r m u t a t i o n

a n d apply the c o n s t r u c t i o n

F + S (F) a s many t i m e s as

we need t o obtain the e m p t y q u a s i - p e r m u t a t i o n . We t h u s c o n s t r u c t a s e q u e n c e

s

( a ) = 6 , . . . , sp-l(3) w h e r e f o r

i , 1d i

ep ,

Si+l(6) = S(Si(6)) , a n d

Sp(G))= $

(see figure lo).

An outstanding l i n e of a q u a s i - p e r m u t a t i o n s e g m e n t s (joining a point of

F

with a point of

F

is f o r m e d with f i n i t e

S(F))

and with two infinite

h a l f - l i n e s , one i s v e r t i c a l a n d t h e o t h e r i s h o r i z o n t a l . T h i s v e r t i c a l ( r e s p .

Chain and antichain families horizontal) half-line h a s coordinate x ' s axis

v(L)

(resp. h(L))

43 1 with r e s p e c t with the

(resp. y's axis).

We a r e now r e a d y t o s t a t e t h e " g e o m e t r i c " v e r s i o n of t h e R o b i n s o n Schensted correspondence :

PROPOSITION 7 of the P - s y m b o l h(Lki)

(Viennot) P(0)

Let

0

be a p e r m u t a t i o n of

( r e s p . Q-symbol

Q(u))

G n . The ith row

.

h(L1), . . ,

i s the sequence

( r e s p . v(L1), . . . , v ( L k , ) ) of c o o r d i n a t e s of t h e h a l f - l i n e s c o m p o s i n g 1

t h e outstanding l i n e s of

si-l(8).

9

a 7

Outstanding lines of

6

-

5

- 0 -

C S(0)

--_- s2

(Q-)

4 3 2 1

F i g u r e 10 - P l a n a r i z a t i o n of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e

432

G. Viennot

The permutation

0

c a n be r e c o n s t r u c t e d when knowing, f o r e v e r y

the c o o r d i n a t e s of the h a l f - l i n e s a p p e a r i n g i n the outstanding l i n e s of the i

iM, th

S.(6). T h i s c o m e s f r o m t h e following g e o m e t r i c p r o p e r t y .

skeleton

F , let

F o r a quasi-permutation

I

and

on t h e two c o o r d i n a t e a x i s , T h e d i r e c t p r o d u c t

F

s u p p o r t of

and i s denoted by

J

F

be t h e p r o j e c t i o n s of

Zx Z

Ix J

i s c a l l e d the

S u p p ( F ) . A f u n d a m e n t a l l e m m a is t h a t t h e

F + ( S u p p ( F ) , S ( F ) ) i s a n i n j e c t i o n f r o m the s e t of q u a s i - p e r m u t a t i o n s of

map

into the s e t of p a i r s

[n] x [n] the f o r m

Ix J

, and

B

(A, B )

where

A

i s a s u b s e t of

i s a q u a s i - p e r m u t a t i o n included in

it d o e s not s e e m obvious to give a c o n s t r u c t i o n of

F

A

when

[n]x [n] ,

of

At f i r s t s i g h t ,

(Supp(F), S ( F ) )

i s known. N e v e r t h e l e s s , t h e r e e x i s t s a n e a s y g e o m e t r i c c o n s t r u c t i o n ( s e e Viennot [109]).

F

F = T ( S ( F )U C ) , w h e r e

is given by

(with 2 k points if

S(F)

has

k

C

i s a c e r t a i n s u b s e t of

outstanding l i n e s ) a n d

T

Zx

Z

i s the notation

f o r a n analogous definition of the s k e l e t o n of a q u a s i - p e r m u t a t i o n , but with t h e "light" c o m i n g f r o m t h e N o r t h - E a s t . T h u s , by r e p e t i t i v e a p p l i c a t i o n of t h i s r e l a t i o n , the p e r m u t a t i o n

u

can

be r e c o n s t r u c t e d when knowing the ( d e c r e a s i n g ) s e q u e n c e of s u p p o r t s of the

. , .,S

(8). T h i s s e q u e n c e is c o m p l e t e l y defined by P-1 t h e c o o r d i n a t e s of the h a l f - l i n e s a p p e a r i n g i n the outstanding l i n e s of S o ( 6 ) , . . . , different skeletons

So(6),

T h e r e a d e r i s u r g e d t o t a k e a r e c t a n g u l a r p i e c e of p a p e r , hide t h e points of

6

f o r the p e r m u t a t i o n d i s p l a y e d o n f i g u r e 10, t h e n move t h i s p i e c e of p a p e r

f r o m l e f t to r i g h t , a n d look c a r e f u l l y what happens e a c h t i m e a new c o l u m n of the [n] x [n]

g r i d a p p e a r s , When p a s s i n g f r o m c o l u m n

i

to

it1

, the c h a n g e of

the outstanding l i n e s c r o s s i n g t h e l e f t - h a n d s i d e of the p i e c e of p a p e r i s nothing but a g e o m e t r i c coding of the "bumping p r o c e s s " i n s e r t i n g of the w o r d

it1

i n the P - s y m b o l

~ ( 1 )... u ( i ) . A new v e r t i c a l h a l f - l i n e a p p e a r s i n c o l u m n

c o r r e s p o n d i n g outstanding l i n e belongs t o a c e r t a i n s k e l e t o n

S,(U)

it1

.

.

The

T h e index

i s a coding of the " Q - p a r t " of t h e R o b i n s o n - a l g o r i t h m . T h i s i s e s s e n t i a l l y the proof of p r o p o s i t i o n 7 . It would be p o s s i b l e to m o v e t h e piece of p a p e r f r o m t h e b o t t o m t o t h e t o p and t h u s d e s c r i b e d a dual v e r s i o n of the bumping p r o c e s s . In f a c t t h i s duality i s -1 , which c o r r e s p o n d s t o exchanging r o w s nothing but t h e t r a n s f o r m a t i o n 0 + 0

Chain and antichain families

433

a n d c o l u m n s i n t h e p i c t u r e . V e r t i c a l c o o r d i n a t e s of t h e h a l f - l i n e s of t h e o u t standing l i n e s a r e e x c h a n g e d with t h e h o r i z o n t a l c o o r d i n a t e s . T h u s p r o p o s i t i o n 3 -1 is m a d e c r y s t a l c l e a r : t h e duality 0 3 0 c o r r e s p o n d s t o exchanging t h e

P - and Q -symbol s. The symmetry

U

3

u*

( m i r r o r i m a g e ) c o r r e s p o n d s t o m a k i n g the "light"

c o m e f r o m the S o u t h - E a s t . In f a c t , t h e r e a r e f o u r d i f f e r e n t p o s s i b l e d i r e c t i o n s f o r the "light" a n d t h u s f o u r d i f f e r e n t c o n s t r u c t i o n s of s e t s of o u t s t a n d i n g l i n e s a n d s k e l e t o n s . Unfornately, t h e r e do not s e e m t o b e s i m p l e g e o m e t r i c r e l a t i o n s b e t w e e n t h e s e f o u r c o n s t r u c t i o n s . In p a r t i c u l a r , p r o p o s i t i o n 5 d o e s not s e e m t o have a si mpl e explanation in t h i s g eo met r i c model. N e v e r t h e l e s s , G r e e n e ' s T h e o r e m (proposition 6 and k-matching i n t e r p r e t a t i o n ) c a n be p r o v e d in a c o n s t r u c t i v e g e o m e t r i c way, without changing t h e p e r m u t a t i o n t o a c a n o n i c a l one with Knuth's t r a n s p o s i t i o n s , T h e r e a d e r will f i n d in [lo91 a g e o m e t r i c c o n s t r u c t i o n of a m a x i m a l s i z e d a n t i c h a i n k - f a m i l y , using a n o p e r a t o r , s o m e w h a t " i n v e r s e " t o t h e o p e r a t o r

F

*

S ( F ) , sending e v e r y s u b s e t of

S(F)

t o a s u b s e t of

F

according to some

"light" c o m i n g f r o m the N o r t h - E a s t . A k - m a t c h i n g with l e x i c o g r a p h i c a l l y m i n i m u m m a x i m u m - s i z e s o u r c e c a n be o b t a i n e d in t h e following w a y . We s t a r t with t h e s e t of y - c o o r d i n a t e s of the S k - l ( ~ ). Below e a c h of t h e s e p o i n t s , t h e r e i s a unique point of

points of Sk-2(u)

.

T h e s e c o n d c o l u m n of the k - m a t c h i n g will be f o r m e d with t h e y - c o o r -

dinate of t h e s e p o i n t s , (which a r e d i s t i n c t v a l u e s ) . Going down f r o m s k e l e t o n t o

6 g i v e s v a l u e s t h a t c a n be a r r a n g e d i n a r e c -

s k e l e t o n t i l l r e a c h i n g points of

t a n g u l a r a r r a y having t h e k - m a t c h i n g p r o p e r t y .

F o a t a ' s m a t r i x c h a r a c t e r i z a t i o n a n d k e r n e l of t h e i n v e r s i o n p a i r g r a p h . When using t r a n s p a r e n c i e s , t h e d i f f e r e n t s k e l e t o n s c o l o r e d black, r e d , blue, g r e e n , points of We h a v e a

by t h e v a l u e

S.(U)

nxn

Si(U)

, iS 0 , a r e

. . . and e v e r y t h i n g is c l e a r . H e r e , w e l a b e l t h e . All o t h e r points of [n] x[n] a r e l a b e l e d 0 .

it1

m a t r i x with i n t e g e r e n t r i e s . A q u e s t i o n i s t o f i n d a c h a r a c t e -

r i z a t i o n of t h i s m a t r i x . T h i s h a s b e e n done by F o a t a i n [14]. We need t o define t h e "hook" joining two p o i n t s . L e t be two points of

Z x Z


or

and

(XI,

y')

s u c h t h a t t h e s e c o n d is at t h e S o u t h - E a s t of t h e f i r s t

o n e . T h e h o o k joining t h e s e two points i s t h e s e t of points x

(x, y )

i = x' , y 3 j

> Y'

( s e e f i g u r e 11).

(i, j )

with

j = y

,

434

G. Viennot A m a t r i x with i n t e g e r e n t r i e s h a s the f o r m defined above (coding of t h e

d i f f e r e n t s k e l e t o n s of a p e r m u t a t i o n

IJ

en )

of

iff i t s a t i s f i e s the t h r e e f o l -

lowing conditions : (i)

t h e r e e x i s t s one a n d only one 1 i n e a c h r o w and column

(ii)

for every entry

i t 1 2 2 , t h e r e e x i s t s a n e n t r y i both on the l e f t a n d

below the e n t r y

it1 ,

(iii)

f o r e v e r y hook joining two points l a b e l e d it1

i 3 1 , there exists an entry

l o c a t e d on t h e hook between t h e m .

Such m a t r i c e s h a v e been i n t r o d u c e d by F o a t a [14] u n d e r t h e n a m e of Viennot's m a t r i c e s . Example -

0 0 0 1 0 0 0 0 0

0 0 0 2 0 0 0 0 1

0 0 1 0 0 0 0 0 0

0 0 2 0 0 0 0 1 0

0 0 0 0 1 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 0 3 2 0 1 0 0

1 0 0 0 0 0 0 0 0

0 2 0 0 0 1 0 0 0 F i F " 11

- A hook

joining two p i n t s .

T h e above c h a r a c t e r i z a t i o n c a n a l s o be w r i t t e n in a n i c e way i n t e r m s of graph theory. Let

0

t a t i o n . We denote by [n] x [n]

6 and 5 [n] x [n] i t s p l a n a r r e p r e s e n n the g r a p h whose v e r t i c e s a r e a l l t h e points of

be a p e r m u t a t i o n of

GI(o)

s u c h t h a t t h e r e e x i s t s a point of

6

o n the l e f t a n d below t h e m ( t h a t

i s t h e union of all the c o r n e r s of t h e hooks joining points of edge b e t w e e n t h e v e r t i c e s y = y'

and

pairs

(i. j )

x > x'

.

(x.y)

and

(x', y')

iff

x = x'

5

). We put a n

and

y > y'

or

T h e n u m b e r of v e r t i c e s of t h i s g r a p h i s t h e n u m b e r of

such that

i

<j

n u m b e r of t h e p e r m u t a t i o n

and 0

.

o ( i ) 3 O(j) , i . e . t h e s o - c a l l e d i n v e r s i o n s

Such d i a g r a m s have been i n t r o d u c e d by Rothe

[8l] i n o r d e r t o give a n i c e ( g e o m e t r i c ) proof of t h e f a c t t h a t t h e i n v e r s i o n

435

Chain and antichain families

n u m b e r i s invariant under the t r a n s f o r m

0

=;b

u

-1

.

In t h e i r r e c e n t w o r k r e l a t i n g c o m b i n a t o r i c s (Young t a b l e a u x , p l a c t i c m o noid, S c h u r f u n c t i o n s , . . . ) a n d a l g e b r a i c g e o m e t r y (flag m a n i f o l d s ,

. . .)

[611,

1651, r66], L a s c o u x and S c h u t z e n b e r g e r showed that t h e s e " i n v e r s i o n p a i r s " g r a p h s play t h e s a m e r o l e f o r S c h u b e r t f u n c t i o n s a s F e r r e r s d i a g r a m s f o r S c h u r f u n c t i o n s . T h e y c a l l e d t h e s e g r a p h s Riguet d i a g r a m s .

F i g u r e 12

-

T h e i n v e r s i o n p a i r s g r a p h (on Riguet d i a g r a m ) of a p e r m u t a t i o n .

A c l a s s i c a l notion i n t h e t h e o r y of d i r e c t e d g r a p h s i s t h e k e r n e l of a g r a p h , that i s a s e t of points s u c h t h a t e v e r y v e r t e x of the g r a p h i s t h e s o u r c e of a n edge ending i n the k e r n e l , a n d e v e r y edge having i t s s o u r c e i n t h e k e r n e l h a s i t s end not i n t h e k e r n e l ( s e e B e r g e [5]). Such a s e t i s unique if i t e x i s t s . It i s a l s o t h e s e t of "winning p o s i t i o n s " of a " N i m g a m e " played on t h e g r a p h . F r o m [lo91 t h e s k e l e t o n

S(0)

of t h e p e r m u t a t i o n

CJ

is e x a c t l y the

G. Viennot

436

k e r n e l of i t s i n v e r s i o n p a i r s g r a p h . Writing t h i s p r o p e r t y f o r all s k e l e t o n s S.(U) leads to Foata's characterization. F r o m F r e d m a n [24] a n d Viennot [lo91 t h e c o n s t r u c t i o n of the l o n g e s t s u b -

s e q u e n c e of a p e r m u t a t i o n

u

n e e d s the c o n s t r u c t i o n of the s k e l e t o n . T h u s ,

t h e c o n s t r u c t i o n of the s k e l e t o n c a n be done by a n o p t i m a l a l g o r i t h m ( f o r t h e w o r s t c a s e i n "on-line" r e a d i n g ) i n

nlogn-nloglogn

comparisons, For more

u n d e r s t a n d i n g about t h e s e t h e o r e t i c a l c o m p u t e r s c i e n c e c o n c e p t s , s e e f o r e x a m p l e Knuth [52] o r Aho, H o p c r o f t , Ullman [2]

6

6

-

.

Plactic monoid. We have s e e n t h a t t h e Robinson-Schensted c o r r e s p o n d e n c e i s r e l a t e d t o

t h e r e p r e s e n t a t i o n t h e o r y of t h e s y m m e t r i c g r o u p . T h e c o m b i n a t o r i c s of Young t a b l e a u x i s a l s o i n t i m a t e l y c o n n e c t e d with t h e t h e o r y of s y m m e t r i c f u n c tions. A b a s i s of s y m m e t r i c f u n c t i o n s h a s b e e n i n t r o d u c e d by J a c o b i [48] a s a quotient of a n t i s y m m e t r i c functions ( e x p r e s s e d a s d e t e r m i n a n t s ) . Following F r o b e n i u s [25), S c h u r [9O] d i s c o v e r e d t h e i r r e l e v a n c e t o t h e r e p r e s e n t a t i o n t h e o r y of t h e s y m m e t r i c g r o u p s a n d t h e g e n e r a l l i n e a r g r o u p s . T h e s e f u n c t i o n s h a v e b e e n c a l l e d S c h u r f u n c t i o n s ( o r S - f u n c t i o n s ) by Littlewood and R i c h a r d s o n [TO),

A l s o they a r e r e l a t e d t o s o m e w o r k i n i t i a t e d by P i e r i i n 1873, f o l l o w e d by

S c h u b e r t , G i a m b e l l i a n d o t h e r s , a n d which i s nowadays n a m e d a "cohomology r i n g of G r a s s m a n n v a r i e t i e s " . We s h a l l not t o u c h t h i s d e e p a n d huge s u b j e c t . We s h a l l only give t h e w e l l known c o m b i n a t o r i a l definition of t h e S c h u r f u n c t i o n s ( s e e f o r e x a m p l e Littlewood [ 6 9 ] ).

X

Let

be a p a r t i t i o n a n d

be a Young t a b l e a u of s h a p e

Y

X

(with

e n t r i e s s t r i c t l y i n c r e a s i n g i n c o l u m n s a n d w e a k l y i n c r e a s i n g i n r o w s ) . We d e i xp" , w h e r e i.!, i s t h e n u m b e r of e n note by m(Y) t h e m o n o m i a l

4'. . .

' trie s equal to

.!,

in the tableau

s u m of all m o n o m i a l s

m(Y)

and e n t r i e s taken among Example

-

s ( 2 , 1) (xl,

X2'

1,2,

Y

. T h e S c h u r function

Sx (xl,

. . . ,xn)

e x t e n d e d o v e r all Young t a b l e a u x with s h a p e

is t h e

X

. . ., n .

x 3 ) = x12 x 2 + x 2 x3+ x3 2 xl+ x12 x3+ x 2 XI+ x 2 3 x2 + 2x1 x2 x 3 *

Chain and ontichain families

437

c o r r e s p o n d i n g t o the e i g h t Young t a b l e a u x d i s p l a y e d i n t h e following f i g u r e .

F i g u r e 13 -

A Schur function.

L a s c o u x a n d S c h u t z e n b e r g e r i n t r o d u c e d the p l a c t i c monoid a s a n o n - c o m m u t a t i v e c a l c u l u s , unifying a n d extending p r e v i o u s p r o p e r t i e s of the R o b i n s o n S c h e n s t e d c o r r e s p o n d e n c e a n d of the t h e o r y of s y m m e t r i c f u n c t i o n s [62]. F o r example, the r a t h e r m y s t e r i o u s , s o - c a l l e d "Littlewood-Richardson rule" (giving t h e p r o d u c t of two S c h u r f u n c t i o n s a s s u m of S c h u r f u n c t i o n s ) i s m a d e clear. T h e c o m b i n a t o r i a l p a r t of t h i s t h e o r y is b a s e d on t h e "jeu d e taquin", [62] ,[97].

Y

+

d(Y)

T h i s r u l e f o r moving l a b e l s i n d i a g r a m s g e n e r a l i z e s t h e o p e r a t i o n m a d e i n t h e c o n s t r u c t i o n of t h e d u a l

YJ

of a t a b l e a u

Y

.

We s h a l l only g i v e h e r e a f e w h i n t s of t h i s beautiful t h e o r y . T h e i n t e r e s t e d r e a d e r w i l l s e e t h e p a p e r s of L a s c o u x a n d S c h t i t z e n b e r g e r [58] ,[59], [60], [62], [63], [95], [97]

a n d of T h o m a s [103], [104], (105], [lob], [107], t h e e x t e n s i o n to

t h e n i l p l a c t i c monoid [66] a n d t h e s u r v e y p a p e r of C a r t i e r a t t h e " S k m i n a i r e B o u r b a k i [7]. F o r t h e c l a s s i c a l t h e o r y of s y m m e t r i c f u n c t i o n s , s e e t h e book of Macdonald [72]

o r S t a n l e y 6983, F o u l k e s [l8]

,

J e u de t a q u i n . Let

X

contained i n

and

Fx

.

U

be two p a r t i t i o n s s u c h t h a t t h e F e r r e r s d i a g r a m

The diagram

g r a m s , is a n i n t e r v a l of

TI

F

cr

is

Fx\FcI , t h e d i f f e r e n c e of t h e t w o F e r r e r s d i a -

= Z x Z

A s k e w Young t a b l e a u of s h a p e

( o r d e r e d by t h e p r o d u c t o r d e r ( 2 ) ) . X\P

is a l a b e l i n g of

FA\Fcr with i n t e -

g e r s s u c h t h a t they a r e weakly i n c r e a s i n g i n t h e r o w s ( f r o m l e f t t o r i g h t ) a n d s t r i c t l y i n c r e a s i n g i n t h e c o l u m n s (down-up, i n t h e " F r e n c h notation").

G. Viennot

438

p = (5,5,2) = (8,7,5,5,2) 0

comer cell of F

r

F i g u r e 14 - A s k e w Young t a b l e a u .

T h e lljeu de taquin" i s d e s c r i b e d as f o l l o w s . F i r s t we c h o o s e one of t h e c o r n e r c e l l s (defined i n

9 4) of the F e r r e r s d i a g r a m F . We t a k e the m i n i m u m CI

of the t w o v a l u e s that a r e l o c a t e d at t h e North a n d E a s t , a n d move t h i s v a l u e into the c o r n e r c e l l . T h i s c r e a t e s a "hole" i n t h e s k e w Young t a b l e a u . In a sim i l a r way t o t h e c o n s t r u c t i o n of the d u a l i n

5 4, we

hole h a s moved t o the N o r t h - E a s t o u t s i d e of

Fx

r e p e a t t h e p r o c e s s until t h e

. In c a s e the

two v a l u e s a t t h e

N o r t h o r the E a s t of a hole a r e e q u a l , we move i n the hole the value l o c a t e d a t t h e N o r t h (thus p r e s e r v i n g the condition t o have a s k e w Young t a b l e a u ) . A l s o note t h a t if the hole i s a t t h e " b o r d e r " of

Fx , at m o s t one of the N o r t h a n d E a s t

c e l l s i s n o n - e m p t y . We move a l s o t h i s s i n g l e v a l u e i n t h e h o l e . T h e p r o c e s s t e r m i n a t e s when both t h e N o r t h a n d E a s t c e l l of t h e hole a r e e m p t y (that i s i n f a c t when the hole i s a c o r n e r c e l l of

Fx

1.

T h e t e r m i n o l o g y b n d t h e t h e o r y !) lljeu de taquin" is due t o S c h u t z e n b e r g e r . T h e lljeu de taquin" is a g a m e played with a n x n b o a r d . I n s i d e the b o a r d , 2 t h e r e a r e n -1 e l e m e n t a r y c e l l s . On e a c h c e l l i s p r i n t e d a l e t t e r . At e a c h s t e p of the g a m e , one c a n move one of t h e n e a r e s t c e l l s to t h e unique e m p t y p o s i t i o n . At e a c h s t e p t h i s e m p t y position t a k e s the p l a c e of the c e l l which h a s m o v e d . T h e r u l e i s t o obtain some w o r d s ( o r l e t t e r s i n a l p h a b e t i c o r d e r ) .

439

Chain and antichain families

F i g u r e 15

-

T h e lljeu de taquin"

We o b t a i n a n o t h e r s k e w Young t a b l e a u with s h a p e ( r e s p . FuI ) i s obtained f r o m

X'\bi',

Fx

1

( r e s p . F@) by r e m o v i n g a c o r n e r c e l l .

Fx , Now, if we r e p e a t t h e p r o c e s s going f r o m a t a b l e a u

t o a t a b l e a u with s h a p e

where

Y

with s h a p e

I\@

F is e m p t y , we o b t a i n a U Young t a b l e a u , c a l l e d t h e " r e d r e s s d " of t h e s k e w Young t a b l e a u Y u n d e r t h e A'\P'

until t h e d i a g r a m

s u c c e s s i v e c h o i c e s of t h e "jeu de taquin". A r e m a r k a b l e f a c t i s t h e i n v a r i a n c e of the " r e d r e s s C " u n d e r t h e "jeu de taquin",

i . e . the " r e d r e s s d " i s independant

of t h e s u c c e s s i v e c h o i c e s of t h e c o r n e r c e l l s of e a c h d i a g r a m

F

u

. This inva-

r i a n c e h a s b e e n p r o v e d by S c h a t z e n b e r g e r [ 9 7 ] a n d T h o m a s [lo51 ( s e e a n i d e a of t h e proof b e l o w ) . T h e " r e d r e s s C " of the s k e w Young t a b l e a u

Y

is d e n o t e d by

R(Y). We follow a g a i n S c h u t z e n b e r g e r ' s t e r m i n o l o g y ; " r e d r e s s 6 " i s b a d l y t r a n s l a t e d by s a y i n g t h a t t h e South-West b o r d e r of s k e w Young t a b l e a u

Y

is n o

m o r e a zig-zag line. F o r example, the tableau where

a

11

d(Y)

defined i n

8

4 is nothing but

R ( Y \ all)

d e n o t e s t h e ( s k e w ) Young t a b l e a u o b t a i n e d by d e l e t i n g t h e v a l u e

Y \ all (1,l) c e l l f r o m the Young t a b l e a u

i n the

Y

.

In the e x a m p l e d i s p l a y e d i n f i g u r e 16, we h a v e s h o r t e n e d t h e c o n s t r u c t i o n by giving only the t a b l e a u x o b t a i n e d o n c e t h e "hole" i s o u t s i d e of the d i a g r a m

Fx

. We h a v e

c i r c l e d t h e v a l u e s which m o v e f o r e a c h c h o i c e of t h e c o r n e r c e l l s

of t h e s u c c e s s i v e d i a g r a m s

F

U

.

,w

G. Viennot

440

&+

Y =

3 8 9

choice of the

comer cell

0

F i g u r e 16

-

The "redressk"

v a l u e w i n g in the " j e u d e taquin"

R ( Y ) of a s k e w Young t a b l e a u

Y

.

6 . We denote by Y(o) the s k e w Young n t a b l e a u obtained by w r i t t i n g t h e s u c c e s s i v e v a l u e s U(l), ,u(n) f r o m N o r t h Let

u

be a p e r m u t a t i o n of

...

West t o S o u t h - E a s t ( s e e f i g u r e 17). A f u n d a m e n t a l p r o p e r t y of t h e lljeu de taquin", a n d giving a n o t h e r definition of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e is t h e following

PROPOSITION 8 .

(Schiitzenberger)

-

F o r any permutation

" r e d r e s s k " of t h e s k e w Young t a b l e a u P-symbol

P(u)

Y(U)

coding

u

u

of

en ,

the

i s identical to t h e

o b t a i n e d by t h e Robinson-Schensted c o r r e s p o n d e n c e .

Define a s t r i p t o be a d i a g r a m

Fcl a r e disjoint and s u c h t h a t

Fx\Fu

Fx\ Fu

F

s u c h t h a t t h e b o r d e r s of

d o e s not c o n t a i n a n y s q u a r e

x

F

(292)

and of

Chain and antichain families f o u r c e l l s . Such d i a g r a m s with

n

44 1

c e l l s a r e coded by a w o r d of l e n g t h

two l e t t e r s . T h e r e e x i s t s a t r i v i a l b i j e c t i o n between p e r m u t a t i o n s of s k e w Young t a b l e a u x with d i s t i n c t e n t r i e s

1, 2,

. . .,n

6,

n-1

on

and

a n d with a s t r i p s h a p e ,

F r o m s u c h t a b l e a u x , we a s s o c i a t e a p e r m u t a t i o n by r e a d i n g t h e e n t r i e s f r o m N o r t h - W e s t t o S o u t h - E a s t following t h e s t r i p . C o n v e r s e l y , f r o m a p e r m u t a t i o n 0

we define a s k e w Young t a b l e a u

the c e l l containing c e l l containing

iff

having a s t r i p s h a p e a n d s u c h t h a t

i s c o n s e c u t i v e a n d a t t h e E a s t ( r e s p . South) of t h e

a(it1)

O(i)

Str(0)

0(i) < o(it1)

( r e s p . 0 ( i ) 3 o ( i t 1 ) ) . In o t h e r w o r d s ,

t h e s t r i p i s nothing but a coding of t h e up-down s e q u e n c e of t h e p e r m u t a t i o n

8

4

Y h)

-

defined i n

.

k-, 3 9

F i g u r e 17

-

T h e s k e w Young t a b l e a u x Y ( 0 ) a n d Str(ff) coding a p e r m u t a t i o n 0 .

Obviously, the t a b l e a u

Str(0)

c a n be o b t a i n e d f r o m

Y(0)

by applying

t h e "jeu de taquin", a n d t h u s , by S c h f i t z e n b e r g e r ' s t h e o r e m s ( i n v a r i a n c e of t h e " r e d r e s s k " u n d e r t h e "jeu de taquin" a n d p r o p o s i t i o n 8 ) , t h e " r e d r e s s k " R(Str(0))

i s the P-sy mb o l

P(0) of t h e p e r m u t a t i o n

U

.

T h e r e a d e r will

c h e c k t h i s f u n d a m e n t a l fact with f i g u r e 16 f o r t h e g e n e r i c p e r m u t a t i o n a p p e a r i n g i n all the e x a m p l e s of t h i s p a p e r . T h e i n v a r i a n c e of t h e " r e d r e s s k " u n d e r t h e lljeu de taquin" i s p r o v e d by introducing t h e following f u n c t i o n s , b a s e d on G r e e n e ' s i n t e r p r e t a t i o n of the s h a pe d the t a b l e a u x

P(0) a n d

Q(0)

(we s h a l l a s s u m e t h a t t h e e n t r i e s a r e d i s -

t i n c t , t h e g e n e r a l c a s e c a n be d e d u c e d f r o m t h a t ) .

G. Viennot

442 Let

Y

be a s k e w Young t a b l e a u (with p o s s i b l y a "hole" i n s i d e ) . We d e -

f i n e a n i n c r e a s i n g s u b s e q u e n c e of

< xk

such that f o r e v er y

E a s t of t h e c e l l w h e r e we define

i , 1G i

x.

to be a s e q u e n c e of i n t e g e r s k , x.

1t1

. ..

x

1 i s i n a c e l l which i s a t t h e S o u t h -

i s l o c a t e d . T h e n , f o r any i n t e g e r s

k

a2

and

1,

Y

to be the m a x i m u m c a r d i n a l i t y of s u b s e t s of e n t r i e s of

Gk,,(Y)

t h a t a r e unions of

Y

k

.

i n c r e a s i n g s u b s e q u e n c e s with v a l u e s

i s i n v a r i a n t u n d e r the e l e m e n t a r y 'k, sliding of t h e 'ljeu de taquin". T h e f a c t mentioned above i s p r o v e d f r o m the r e It c a n be shown t h a t t h e function

m a r k t h a t a s t a n d a r d Young t a b l e a u is c h a r a c t e r i z e d by the function

G

k,

.

u + (R(0) , R( 0-l))

T h e n one c a n s h o w d i r e c t l y that the c o r r e s p o n d e n c e

i s a b i j e c t i o n . T h e "bumping p r o c e s s " c a n be s e e n as p a r t i c u l a r c a s e of t h e s l i d i n g s of the lljeu de taquin", a n d t h u s we h a v e p r o p o s i t i o n 8 . F r o m t h e r e , G r e e n e ' s i n t e r p r e t a t i o n and p r o p e r t i e s of Knuth's t r a n s p o s i t i o n s c a n be d e d u c e d . Other p r o p e r t i e s mentioned in

5

4 a r e a l s o d e d u c e d . F o r e x a m p l e , the p r o p e r t y

r e l a t i n g t h e up-down s e q u e n c e and t h e "line of r o u t e ' ' i s e a s i l y shown a s a p r o p e r t y i n v a r i a n t u n d e r the "jeu de taquin". A l s o , taking t h e dual of a t a b l e a u c o r r e s p o n d s to r e v e r s i n g the I'jeu de taquin", t h a t is r e v e r s e the o r d e r b e t w e e n i n t e g e r s and move t h e e n t r i e s t o t h e N o r t h - E a s t ( i n s t e a d of t h e S o u t h - W e s t ) .

P r o d u c t s of Young t a b l e a u x . T h e above c o m b i n a t o r i a l c o n s i d e r a t i o n s c a n be e x p r e s s e d in a m o r e a l g e b r a i c way by introducing the plactic m o n o i d . Let

Y

and

Z

be two Y o u n g t a b l e a u x . Denote by

Young t a b l e a u obtained by putting

Z

the skew

Y

a s shown o n

a t t h e S o u t h - E a s t of

f i g u r e 18. T h e p r o d u c t of the two Young t a b l e a u x the " r e d r e s s 6 "

Y. Z

Y

Z

and

i s defined to be

R(Y. Z ) . By t h e i n v a r i a n c e of t h e " r e d r e s s & " u n d e r t h e lljeu de

taquin", t h i s p r o d u c t i s a s s o c i a t i v e . T h e e m p t y t a b l e a u i s t h e identity e l e m e n t . A , and with the

T h e s e t of Young t a b l e a u x with e n t r i e s i n the s e t of i n t e g e r s p r o d u c t j u s t defined, i s t h e p l a c t i c monoid g e n e r a t e d by

A

.

443

Chain and antichain families

=B 1

1

2

Y.2

-@ F i g u r e 18 -

=%

n Young t a b l e a u x r e d u c e d t o one c e l l

insertion process

I(T,x)

with the s i n g l e - c e l l

x

R] 2

2

4

T h e p r o d u c t of two Young t a b l e a u x .

P r o p o s i t i o n 8 s a y s nothing but t h a t t h e P - s y m b o l p r o d u c t of the

m =

R(Y*Z)

5

defined in

P(0)

is the p l a c t i c

O(l), 0 ( 2 ) ,

. . . ,~

( n .) T h e

3 i s in f a c t t h e p l a c t i c p r o d u c t of

T

placed a t the South-East.

A n o t h e r way t o define t h e p l a c t i c monoid c o m e s f r o m K n u t h ' s t r a n s p o s i be a t o t a l l y o r d e r e d a l p h a b e t . L e t

tions . Let

A

n e r a t e d by

A , t h a t i s the s e t of w o r d s

w = wl.. .w

a n d with p r o d u c t the c o n c a t e n a t i o n p r o d u c t : f o r

be the f r e e monoid g e with l e t t e r s

n

of

x,y,z

A

x y z s y x z

(6) f o r any l e t t e r s

x,y

of

A

x y x s y x x

such that and

x c z

in

A


v = vl. . . v q ,

t o be the c o n g r u e n c e

or

y < z c x ,

z x y s z y x ,

such that and

and

= ul'.

%

f o r any letters

w. 1

. We define t h e p l a c t i c c o n g r u e n c e l', vl.. A* g e n e r a t e d by the r e l a t i o n s

uv= u of

A*

x < y ,

y x y - y y x .

In e a c h e q u i v a l e n c e c l a s s , t h e r e i s one and only one "tableau" ; t h a t i s ,

a w o r d t h a t c a n be obtained by r e a d i n g a Young t a b l e a u r o w by r o w , f r o m l e f t t o r i g h t i n e a c h row, t h e r o w s being o r d e r e d up-down ( t h i s c o r r e s p o n d s t o the r o w - c a n o n i c a l p e r m u t a t i o n s defined in monoid above.

A* / s

4). It c a n be shown t h a t t h e quotient

is i s o m o r p h i c t o the p l a c t i c monoid g e n e r a t e d by

A

defined

G. Viennot

444

4

7

-

Grids. T h i s s e c t i o n p r e s e n t s a s u m m a r y of a f o r t h c o m i n g p a p e r of t h e a u t h o r [127],

w h e r e c o m p l e t e p r o o f s of the new r e s u l t s announced will be g i v e n . A word

with

0

pq

Y

e x i s t s a Young t a b l e a u such t h a t the

pq

and c o l u m n s of

distinct letters i s called a with d i s t i n c t e n t r i e s a n d

e n t r i e s of

Y

Y

(p,q)-grid

pxq

rectangular shape

a r e exactly t h e l e t t e r s of

a r e (monotone) s u b s e q u e n c e s of

It is e a s i l y s e e n t h a t s u c h a t a b l e a u

u

iff t h e r e

0

, and the r o w s

.

is unique a n d i s i n f a c t t h e P-

Y

P ( 0 ) . A l s o we have the following l e m m a

symbol

L EMMA 9

-

Let

u

be a w o r d with d i s t i n c t l e t t e r s ( o r p e r m u t a t i o n ) . T h e f o l -

lowing conditions a r e equivalent i s a grid,

(i)

d

(ii)

the s h a p e of

(iii)

f o r any

P(u) i s r e c t a n g u l a r ,

i ' r l , the ith c o l u m n of

y - c o o r d i n a t e s of t h e points of

a^

l i n e of

6

P(0)

i s c o m p o s e d of t h e th lying o n t h e i outstanding

.

P e r m u t a t i o n s with r e c t a n g u l a r s h a p e f o r t h e a s s o c i a t e d G r e e n e d i a g r a m have m a n y o t h e r nice p r o p e r t i e s . T h e n u m b e r of s u c h p e r m u t a t i o n s , with a fixed

px q

r e c t a n g l e a s s o c i a t e d G r e e n e d i a g r a m , h a s b e e n c o n s i d e r e d by

H i l l e r [44], i n t e r m of the "Schubert c a l c u l u s " d e s c r i b i n g t h e "cohomology of the c o m p l e x g r a s s m a n manifold". In t h i s s e c t i o n , we a r e m o s t l y i n t e r e s t e d i n s u b g r i d s of p e r m u t a t i o n s , t h a t i s s u b s e q u e n c e s that a r e g r i d s (see e x a m p l e i n f i g u r e 19). When a p e r m u t a t i o n

u

is viewed a s a p o s e t

POS(U) , the concept of s u b g r i d s i s c l o s e l y r e l a t e d t o

the c o n c e p t of "orthogonal f a m i l i e s " i n t r o d u c e d by F r a n k [23] i n h i s proof of G r e e n e ' s T h e o r e m ( s e e definition

5

8 below).

F i r s t , we h a v e the following

L E M M A 10 - L e t

0

has Greene diagram contained in

be a p e r m u t a t i o n f o r which the a s s o c i a t e d p o s e t G(Pos(0))

(shape of

P(u)).F o r a n y

G ( P o s ( U ) ) , t h e r e e x i s t s a (p, q ) - s u b g r i d of

u

pxq

.

POS(U)

rectangle

Chain and antichain families T h e proof r e l i e s o n G r e e n e ' s i n t e r p r e t a t i o n of a n t i c h a i n s k - f a m i l i e s of T

of

0

diagram

p

G ( P o s ( 0 ) ) with c h a i n s a n d

( p r o p o s i t i o n 6 ) : f i r s t one takes a s u b s e q u e n c e

Pos(0)

t h a t i s a union of

445

chains (increasing subsequences). The Greene

G ( P o s ( 7 ) ) i s not n e c e s s a r i l y the s a m e a s t h e d i a g r a m

obtained f r o m

C ( P o s ( 0 ) ) by deleting all r o w s i n d e x e d by

b e r of e l e m e n t s of t h e

pth

pth

r o w of

Gp(Pos(0)),

i > p , but t h e n u m -

G ( P o s ( T ) ) i s not l e s s t h a n t h e n u m b e r of

G ( P o s ( 0 ) ) . Applying t h e same r e a s o n i n g o n P with a n t i c h a i n s ( d e c r e a s i n g s u b s e q u e n c e s ) l e a d s t o l e m m a 10. e l e m e n t s of t h e

r o w of

T

A s shown i n f i g u r e 19, t h e c o n v e r s e of l e m m a 10 i s not t r u e . A m a j o r p r o b l e m i s to give a c h a r a c t e r i z a t i o n of s u b g r i d s giving e x a c t l y t h e G r e e n e d i a g r a m G(Pos(0))

a n d t h u s o b t a i n a n i n t e r p r e t a t i o n of

G(Pos(0)) symmetric in rows

and c o l u m n s ( i . e . c h a i n s a n d a n t i c h a i n s ) .

Cxeene diagram of C = 6 1 7 2 5 8 3 9 4

T = 6 1 7 2 8 3 9 4 is a (2,3)-subgrid of r.

F i g u r e 19

-

A ( 2 , 4 ) - s u b g r i d of a p e r m u t a t i o n with n o c o r r e s p o n d i n g 2 x 4 - r e c t a n g l e .

G. Viennot

446

Let

u

t h e i n t e r i o r of the g r i d of

T

and T be a s u b g r i d of a . We define n t o be t h e i n t e r s e c t i o n of the f o u r shadows ( s e e 4 5)

be a p e r m u t a t i o n of T

6

r e l a t e d to the f o u r p o s s i b l e d i r e c t i o n s of t h e light : N o r t h - W e s t , South-

W e s t , S o u t h - E a s t a n d N o r t h - E a s t . In o t h e r w o r d s , t h i s i s a l s o the s e t of points (x,y)

of

IRxIR

7

s u c h that t h e r e e x i s t s a t l e a s t one point of

f o u r q u a d r a n t s with o r i g i n a t

in e a c h of t h e

(x, y ) .

T h e i n t e r i o r of a g r i d i s a union of e l e m e n t a r y c e l l s having the p r o p e r t y t h a t f o r e v e r y p a i r of c e l l s on the s a m e v e r t i c a l ( r e s p . h o r i z o n t a l ) , all the c e l l s between t h e m a r e i n t h i s union. Such o b j e c t s a r e known i n c o m b i n a t o r i c s a s convex polyominoes ( s e e [9] a n d the r e f e r e n c e s t h e r e i n ) .

1

2

3

F i g u r e 20

Let

P ( T )= Y

T

be a

-

5

6

7

8

9

T h e i n t e r i o r of a g r i d .

( p , q ) - s u b g r i d of the p e r m u t a t i o n

d

of

en

and

t h e unique c o r r e s p o n d i n g t a b l e a u i n t h e definition of g r i d s . W e s h a l l

s a y t h a t the s u b g r i d

quences

4

al,

. . .,aP

T

of

i s extendable (in u ) iff t h e r e e x i s t s i n c r e a s i n g s u b s e 0

and decreasing subsequences

PI,.

..#Pq

of

0

Chain and antichain families

441

s a t i s f y i n g t h e t h r e e following conditions ( s e e f i g u r e 21). (i)

j ,

for any


4j

s u b s e q u e n c e of (iii)

, the i

th

r o w of

Y

is a (increasing) subse-

a. ,

q u e n c e of (ii)


i , 14 i

f o r any

e v e r y point of

6

t h e jth

c o l u m n of

Y

i s a (decreasing)

J

t h a t i s i n the i n t e r i o r of

al. . . . , n , B,,

the s u b s e q u e n c e s

P

T

b e l o n g s t o one of

. . . ,p . 9

T h e definition of e x t e n d a b l e s u b g r i d s i s not c h a n g e d i f we r e q u i r e t h a t t h e s u b sequences

al, . . . , a

P

a r e pairwise disjoint, and sim ila rly f o r

pl,

. . , ,p . q

T h e ( 2 , 4 ) - s u b g r i d d i s p l a y e d i n f i g u r e s 19 and 20 is not e x t e n d a b l e . T h i s c o m e s f r o m the f a c t t h a t ( 2 , 5, 8 ) a n d ( 7 , 5, 3 ) a r e r e s p e c t i v e l y i n c r e a s i n g a n d

, with t h e point l a b e l e d 5 in t h e i n t e r i o r of

d e c r e a s i n g s u b s e q u e n c e s of

0

M o r e g e n e r a l l y , a point

of the p e r m u t a t i o n

f o r the s u b g r i d

T

x

0

u

.

i s s a i d t o be a c r i t i c a l point

iff t h e r e e x i s t s e n t r i e s of the Young t a b l e a u

P(0) s u c h

) is a n i n c r e a s i n g s u b s e q u e n c e a n d 1 ,x,a. * x~ a i t l , jtl 1, j t l (aitl, j i s a d e c r e a s i n g s u b s e q u e n c e of 0 . Obviously, s u c h c r i t i c a l point x i s i n t h e

that

(a. . 1, j

i n t e r i o r of t h e ( 2 , 2 ) - s u b g r i d (of

7)

is a l s o i n the i n t e r i o r of the subgid

{ a i , j , a i t l , , ai, j + l , ai+l, j t l 3 , a n d t h u s . Now one c a n s e e t h a t x c a n n o t be

T

" i n c o r p o r a t e d " i n one of t h e ( m o n o t o n e ) s u b s e q u e n c e s of r o w s a n d c o l u m n s of t h e Young t a b l e a u

0

f o r m e d with t h e

Y = P ( o ) , ( s e e f i g u r e 22).

T h u s , a s u b g r i d having a c r i t i c a l point i s not e x t e n d a b l e . Note t h a t t h e c o n v e r s e i s not t r u e ( s e e f i g u r e 2 3 ) . Note a l s o t h a t t h e s h a p e ( 4 , 2 ) of t h e s u b g r i d of f i g u r e 23 i s not c o n t a i n e d i n t h e G r e e n e d i a g r a m .

G. Viennot

448

// a" ,a

0

0

0

\ I

\

\

0

0

p i n t s of the subgrid

T

X

pints of u i n the i n t e r i o r of T but not not i n T

0

okher points of the pnmtation u

Figure 21

-

- - - I

border of the i n t e r i o r of the subgrid T increasing subsequences o^) (chains of decreasing subsequences (antichains of o^)

A n extendable (3,4)-subgrid.

449

Chain and antichain families

11 10 9

8

7 6 5 4 3 2 1

u = 10 3 2 7 1 6 5 11 9 8 4 T

=lo 3 2 111 9 8 4

Greene diaqran

G(o) =

F i g u r e 22

-

A c r i t i c a l point of a s u b g r i d .

F i g u r e 23

-

A non-extendable subgrid having no c r i t i c a l p o i n t .

T h e m a i n r e s u l t of t h i s s e c t i o n is

PROPOSITION 11

-

Let

u

be a p e r m u t a t i o n and

by t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e . T h e v a l u e (i, j )

c e l l of

i t s P - s y m b o l obtained

P(U)

P..(o)l o c a t e d i n the 'J

P(a) is e q u a l t o P..(u) = m i n ( m a x 'J 7

w h e r e t h e m i n i m u m i s t a k e n o v e r all e x t e n d a b l e

T)

,

(i, j ) - s u b g r i d s

T

of

d

.

G. Viennot

450 In o t h e r w o r d s , if 1< x


u

i s a p e r m u t a t i o n in

, t h e r e e x i s t s a unique p a i r

of a n extendable

( i ,j )

F u r t h e r m o r e , this

.

and

such that

x

( i , j ) - s u b g r i d a n d s u c h t h a t no extendable

e x t r a c t e d f r o m the s u b s e q u e n c e of

u

(i, j )

en

u

X

an integer

i s t h e m a x i m u m value ( i , j ) - s u b g r i d c a n be

obtained by deleting a l l v a l u e s

i s p r e c i s e l y the position of

x

2 x

,

i n the P - s y m b o l of

T h i s kind of "topological" p r o p e r t y of points d i s p l a y e d i n the plane i s a

"local" definition of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e (and a f o u r t h one a f t e r the "bumping p r o c e s s " of

4

3 , the shadows and s k e l e t o n s of

l'jeu de taquin" and p l a c t i c monoid of

5

5 and t h e

5 6).

P r o p o s i t i o n 11 i s t r i v i a l i n t h e c a s e of row (or c o l u m n ) c a n o n i c a l p e r m u t a tions ( s e e

9 4). T h u s

p r o p o s i t i o n 11 i s p r o v e d by showing the i n v a r i a n c e of t h e

m i n - m a x quantity u n d e r Knuth's t r a n s p o s i t i o n s . In f a c t , m o r e c a n be s a i d . One c a n i n t r o d u c e a n a l o g o u s l y "extendable" s u b g r i d s of skew Young t a b l e a u x (with " h o l e s " ) . T h e r e i s n o m o r e " i n t e r i o r " of a s u b g r i d but a n a n a l a g o u s definition i s p o s s i b l e f r o m the n e x t s e c t i o n . T h e c o r r e s p o n d i n g m i n - m a x f u n c t i o n p l a y s the r o l e of t h e f u n c t i o n in

4

introduce d k, 1 6 . Again one c a n deduce m o s t of t h e p r o p e r t i e s of t h e R o b i n s o n - S c h e n s t e d G

c o r r e s p o n d e n c e f r o m the i n v a r i a n c e of t h i s m i n - m a x function u n d e r the e l e m e n t a r y s l i d i n g s of the 'ljeu de taquin". T h e p r o o f s a r e m o r e c o m p l i c a t e d , but one h a s the s a t i s f a c t i o n of keeping the s y m m e t r y between r o w s a n d c o l u m n s , t h a t i s between c h a i n s and a n t i c h a i n s .

8

8

-

Extensions t o a r b i t r a r y p o s et s . In conclusion, we d i s c u s s ( v e r y b r i e f l y ) p o s s i b l e e x t e n s i o n of the Robinson-

S c h e n s t e d c o r r e s p o n d e n c e a n d of t h e g r i d c o n c e p t t o a r b i t r a r y p o s e t s . T h i s s e c tion i s m a i n l y b a s e d o n w o r k of G r e e n e [ 3 5 ] , F o m i n [lb], G a n s n e r [ 2 7 ] , F r a n k [ 2 3 ] a n d a f o r t h c o m i n g p a p e r of t h e a u t h o r [128].

Let [n] = (1,

P

be a p o s e t of c a r d i n a l i t y

. . . ,n ) . T h e

p e r m u t a t i o n of

8

n

.

cp

Let

be a l i n e a r e x t e n s i o n i n

p r o b l e m is t o a s s o c i a t e a s t a n d a r d Young t a b l e a u

s u c h t h a t t h i s t a b l e a u is

values

n

P ( u ) ( r e s p . Q(0))

, and cp

a(i) ( r e s p . indices

i n the c a s e

Y(P,cp)

P = P o s ( u ) with

is t h e l a b e l i n g of t h e v e r t i c e s of

u

POS(O)by the

i ).

A s shown by F o m i n [16], a solution c o m e s f r o m G r e e n e ' s t h e o r y ( t h e o r e m 1 and p r o p o s i t i o n 6 ) a n d the following l e m m a : if

P'

i s t h e p o s e t obtained by

45 1

Chain and antichain families deleting f r o m

P

G(P') a n d

Here

6

fined in

G(P)

n

a r e t h e G r e e n e ' s d i a g r a m s of

P'

1,

.

,

iG ;n],

let

P.

.,i . The F e r r e r s diagrams

i n t h e u n i q u e c e l l of

G(P ) =

6

but w i t h

P

and

G(P).

as d e -

b e t h e s u b p o s e t o b t a i n e d by t a k i n g t h e v e r t i c e s

( G(Pi) , 1 d i


u n d e r i n c l u s i o n . We c a n d e f i n e a s t a n d a r d Young t a b l e a u i

G(P')

( m a x i m u m value), then

2.

F o r any labeled

the vertex labeled

G(Pi)\ G(Pi-l)

(for any

cp

Y(P,cp)

i E [n],

). F i g u r e 24 gives a n example with the poset

form a chain

with t he convention P =Pos(a)

0

= 6 1 7 2 5 8 3 9 4

V e r y l i t t l e i s known a b o u t t h i s c o r r e s p o n d e n c e

of f i g u r e 1,

and Q-

P

different f r o m the two n a t u r a l l a l e l i n g s giving t h e

s y m b o l of t h e g e n e r i c p e r m u t a t i o n

by p l a c i n g

. .

(P,cp) + Y

Greene's pro-

p e r t i e s a b o u t k - m a t c h i n g of p e r m u t a t i o n s ( s e e § 4 ) h a v e b e e n e x t e n d e d by G a n s n e r [27], w h e r e h e g i v e s a l s o a d u al analogue with k - s c a t t e r s . A l t o u g h w e a r e g o i n g t o g i v e a n e x t e n s i o n of p r o p o s i t i o n 11 t o a r b i t r a r y posets, no simple algorithm, analogous to the Robinson-Schensted algorithm, h a s b e e n f o u n d f o r t h e c o n s t r u c t i o n of

Y(P,cp).

& 1

Figure 24 - The correspondence

(P,cp)-

2

Y

4

7

9

G. Viennot

45 2

P

Let

y

be a p o s e t . A s u b s e t p

c a n b e p a r t i t i o n e d into j , l

f o r any

~

j

of

'f

c

i s s a i d t o be a

( p , q ) - g r i d iff

{ c ~ , ~ ~ . . . < 1c, . 1 9 i 4 p , s u c h t h a t 1,q ) i s a n a n t i c h a i n of P

chains

~ [ cq. , 1.j " " '

P

.

prj

With t h i s e x t e n s i o n of t h e g r i d definition, i t i s e a s y t o s e e that l e m m a 10

P

i s s t i l l valid f o r any p o s e t

,

But f i g u r e 24 g i v e s a n e x a m p l e of a ( 2 , 4 ) - g r i d

in a p o s e t whose G r e e n e ' s d i a g r a m d o e s not c o n t a i n a 2 x 4 r e c t a n g l e . A s in $ 7 , we give a c h a r a c t e r i z a t i o n of g r i d s with s h a p e c o n t a i n e d i n t h e G r e e n e d i a g r a m . F o r t h a t , we u s e t h e following notion i n t r o d u c e d by F r a n k [Z3] Let

(a,,

. . . ,% P ] P ,

f a m i l y of the poset

be a c h a i n f a m i l y a n d

.

[B1, . . . , B )

be a n a n t i c h a i n 9 with t h e c h a i n s ( r e s p . a n t i c h a i n s ) s u p p o s e d p a i r w i s e

d i s j o i n t . T h e s e f a m i l i e s a r e s a i d t o be orthogonal iff they s a t i s f y t h e two conditions :

( U a,)U ( U ")

(i)

P =

(ii)

i f o r any

j i , 1 4i 4 p

, and

j

It i s e a s y t o s e e that the c h a i n f a m i l y

[ p , , . . . ,!3 ) ) 9 , 1d j & q )

mily

1
(a,,

. . . , aP 3

# fi .

( r e s p . antichain fa-

h a s m a x i m u m c a r d i n a l i t y . A l s o , the s u b s e t i s obviously a ( p , q ) - g r i d of

y

We s h a l l s a y t h a t a g r i d

e x i s t orthogonal f a m i l i e s of c h a i n s and s u b s e t s

, 1 6 j G q , ain B j

I s [p] , J 2 [q]

P

of

P

y =

.

[ai n B j '

i s c o m p l e t e l y extendable iff t h e r e

{ a l , . . . ,a ) P

and antichains

[ Pl, . . . , B q

3

such that Y =

{ai n B

j '

~ E I j, C

JI .

T h e concept of orthogonal f a m i l i e s i s e s s e n t i a l i n F r a n k ' s proof of G r e e n e ' s T h e o r e m . R e c a l l t h a t t h i s proof r e l i e s on l i n e a r p r o g r a m m i n g t e c h n i q u e s . F r o m F r a n k ' s proof of G r e e n e ' s T h e o r e m ( s e e t h e o r e m 3 of [23]), we c a n deduce

PROPOSITION 12 -

T h e G r e e n e d i a g r a m of t h e p o s e t

r e c t a n g l e iff t h e r e e x i s t s a

P

contains a

(p, q ) - c o m p l e t e l y extendable g r i d i n

P

px q

.

A l s o , using G r e e n e ' s T h e o r e m a n d r e s u l t s of F o m i n [16], we c a n d e d u c e f r o m p r o p o s i t i o n 12.

Chain and antichain families PROPOSITION 13 -

P

Y

and

T h e value

Let

P

45 3

be a p o s e t , cp : P + [n]

be a n a t u r a l l a b e l i n g of

be t h e s t a n d a r d Young t a b l e a u defined by t h e a b o v e c o r r e s p o n d e n c e

Y..(P) 'J

l o c a t e d in t h e

(i,j)

c e l l of

Y..(P) = min(max 'J Y

Y

i s equal to

V)

w h e r e the m i n i m u m i s t a k e n o v e r a l l c o m p l e t e l y e x t e n d a b l e max(Y)

d e n o t e s t h e g r e a t e s t l a b e l of the v e r t i c e s of

y

(p, q ) - g r i d s , a n d

.

I have not b e e n a b l e t o f i n d a d i r e c t c o m b i n a t o r i a l proof of P r o p o s i t i o n 12 and 13, not using l i n e a r p r o g r a m m i n g t e c h n i q u e s . T h i s p r o p o s i t i o n c a n be a p p l i e d i n t h e c a s e

P =Pos(0)

where

uEBn

.

It i s p o s s i b l e t o define the c o n c e p t of c o m p l e t e l y e x t e n d a b l e g r i d e x t r a c t e d f r o m a s k e w Young t a b l e a u , show i t s i n v a r i a n c e u n d e r t h e " j e u de taquin" a n d after

6 m a k e a n u l t i m a t e s y n t h e s i s of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e

and t h e p l a c t i c monoid [127]. I n s t e a d of extending the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e t o a r b i t r a r y p o s e t s , one c a n a l s o look f o r g e n e r a l i z a t i o n s involving o t h e r kinds of t a b l e a u x t h a n t h e Young t a b l e a u x u s e d a b o v e . In

6

3 we have m e n t i o n e d the s h i f t e d Young

t a b l e a u x . An a n a l o g of the "bumping p r o c e s s " h a s been defined by S a g a n [83], w h e r e he d e s c r i b e s a b i j e c t i o n between p e r m u t a t i o n s a n d p a i r s of " c o l o r e d s h i f t e d Young t a b l e a u x " having t h e s a m e s h a p e . T h i s b i j e c t i o n p r o v e s t h e a n a l o g of r e l a t i o n (1) f o r p r o j e c t i v e r e p r e s e n t a t i o n s of the s y m m e t r i c g r o u p . O t h e r g e n e r a l i z a t i o n s h a v e been m a d e by Z e l e v i n s k y [121] t o c e r t a i n kind of d i a g r a m s c a l l e d p i c t u r e s , i n connection with t h e i n t e r t w i n i n g n u m b e r of r e p r e s e n t a t i o n s of t h e s y m m e t r i c g r o u p , by Stanton a n d White

[loll, [I151

t o r i m hook t a b l e a u x giving a

c o m b i n a t o r i a l proof of t h e o r t h o g o n a l i t y of t h e c h a r a c t e r s of

Gn

a n d " r i m hook"

v e r s i o n of t h e "bumping p r o c e s s " a n d of t h e l'jeu de taquin", a n d by B e r e l e a n d R e g e v [4] t o " s e m i - s t a n d a r d tableaux" i n t w o s e t s of v a r i a b l e s i n c o n n e c t i o n with the r e p r e s e n t a t i o n t h e o r y of t h e g e n e r a l l i n e a r g r o u p

G L ( k , F ) , W e y l ' s "Strip"

t h e o r e m [114] and t h e t h e o r y of P. I . a l g e b r a s . A l s o we m e n t i o n e d a b o v e i n

4 4 the

Grassl-Hillman correspondence, rela-

t e d to t h e t h e o r y of p l a n e p a r t i t i o n s , a s u b j e c t t h a t we do not touch h e r e but c l o s e l y r e l a t e d t o the R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e ( s e e f o r e x a m p l e A n d r e w s [l] c h a p t e r 11, Mac Mahon [73], [74] c o l l e c t e d p a p e r s , c h a p t e r s 11 a n d 12,

G. Viennot

45 4

Stanley [ 9 8 ] ) . G a n s n e r [ 2 6 ] showed t h a t t h i s c o r r e s p o n d e n c e , when viewed through t h e F r o b e n i u s c o r r e s p o n d e n c e (between r e v e r s e plane p a r t i t i o n s and c e r t a i n p a i r s of t a b l e a u x ) , i s a n e x t e n s i o n of c o l u m n i n s e r t i o n , a n d c l o s e l y r e l a t e d t o t h e " B u r g e c o r r e s p o n d e n c e " [ 6 ] . In

[ill]

V o a n d White showed that

(a s l i g h t v a r i a t i o n , equivalent t o the o r i g i n a l , of) the G r a s s l - H i l l m a n c o r r e s p o n -

d e n c e i s j u s t row i n s e r t i o n in d i s g u i s e . T h i s c o r r e s p o n d e n c e h a s been e x t e n d e d to t h e t w o o t h e r f a m i l i e s of p o s e t s with hooklengths quoted i n

8

4 by S a g a n [ 8 5 ] .

We conclude t h i s p a p e r by mentioning b r i e f l y s o m e v e r y r e c e n t a n d i n t e r e s t i n g w o r k of E d e l m a n and G r e e n e [123], [124], In [125], S t a n l e y c o n j e c t u r e d P i s equal n of s t a n d a r d Young t a b l e a u x having t h e " s t a i r c a s e " s h a p e

that t h e n u m b e r of m a x i m a l c h a i n s i n the ( w e a k ) B r u h a t o r d e r of

f

t o the n u m b e r

W

IJ = (n-1, n - 2 ,

...,l).

the r e l a t i o n s

ud

(The weak Bruhat o r d e r on

Q

8

if

i s obtained f r o m

6

i s the o r d e r g e n e r a t e d by n u by a t r a n s p o s i t i o n of a d j a -

cent i n c r e a s i n g e l e m e n t s ) . An a l g e b r a i c proof i s given by Stanley [126]. A b i j e c t i v e proof i s g i v e n by E d e l m a n a n d G r e e n e . In f a c t , t h r e e b i j e c t i o n s a r e n e e d e d , using S c h i i t z e n b e r g e r ' s o p e r a t o r s [ 9 4 ] , [ 9 7 ]

Y + YJ

a p a r t i t i o n of

1

4

described in n

and

i s a labeling

dinct e n t r i e s m e n t of i t s

t.. 'J 1, 2, .

"hook"

( a n a l o g o u s to the dual

4), a n d t h e new c o n c e p t bf b a l a n c e d t a b l e a u x . L e t

X*

be

t h e conjugate p a r t i t i o n . , A b a l a n c e d t a b l e a u of s h a p e

of the c e l l s

. . ,n

A

( i ,j )

of the F e r r e r s d i a g r a m

, such that the value

H . . , where 'J

*

r . = 1. - i t 1 1j J

i s the

r,t.h

and where

t.

H. 1j

Fx w i t h d i s largest ele-

1J

1j

i s the m u l t i s e t

i s a balanced

t a b l e a u of s h a p e

1 = [ 5 ,2 , 2 , 1 ) .

One of t h e m o s t s u r p r i s i n g f a c t is t h a t t h e n u m b e r of b a l a n c e d t a b l e a u x of shape

A

is

f X , t h e n u m b e r of s t a n d a r d Young t a b l e a u x of s h a p e

1

(see

f o r m u l a e ( 3 a ) , (3b) a n d ( 3 c ) ) . Again t h e r e is d e e p c o m b i n a t o r i c s behind t h i s r e l a t i o n between t a b l e a u x a n d c h a i n s i n the B r u h a t o r d e r , r e l a t e d t o s o m e r e c e n t w o r k of L a s c o u x and S c h u t z e n b e r g e r [65], [ 6 6 ] a b o u t what they c a l l e d S c h u b e r t polynomials a n d

Chain and antichain families

45 5

the nilplactic monoid. T h i s m o n o i d i s d e f i n e d by a v a r i a t i o n of t h e r e l a t i o n ( 6 ) , ( m i x i n g p l a c t i c and C o x e t e r c o n g r u e n c e s ) , a n d c o r r e s p o n d s t o a modified Robinson-Schensted i n s e r t i o n a l g o r i t h m . S t a n l e y ' s c o n j e c t u r e c a n a l s o be d e d u c e d u s i n g t h e n i l p l a c t i c monoid. Acknowledgements ,

I t h a n k o n e of t h e r e f e r e e s f o r n u m e r o u s d e t a i l c o r r e c t i o n s of t h e o r i g i n a l manuscript.

-

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G. Viennot

458

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*.

A . J . H o f f m a n a n d D . E . S c h w a r t z - O n p a r t i t i o n s of a p a r t i a l l y o r d e r e d B 2 3 (1977), 3-13. set, J . Combinatorial Theory,

m.

C . G . Jacobi - De functionibus al tern an ti b us 360-371 ( = W e r k e 3, 4 3 9 - 4 5 2 ) .

. ..,

J . C r e l l e , 22 (1841),

G . J a m e s a n d A . K e r b e r - T h e r e p r e s e n t a t i o n T h e o r y of t h e S y m m e t r i c G r o u p , E n c y c l o p e a d i a of M a t h e m a t i c s a n d i t s A p p l i c a t i o n s , v o l . 16, G . C . R o t a e d . , A d d i s o n - W e s l e y , R e a d i n g , 1981. S . V . K e r o v a n d A . M . V e r s i k , S o v i e t M a t h . D o k l . , 18 (1977). D o k l . A c a d . N a u k . 233 (1977), 1024-1028. D . E . Knuth - P e r m u t a t i o n s , m a t r i c e s a n d g e n e r a l i z e d Young t a b l e a u x , P a c i f i c J . M a t h . , 3 4 (1970), 7 0 9 - 7 2 7 . D . E . Knuth - T h e A r t of C o m p u t e r P r o g r a m m i n g , v o 1 . 3 , S o r t i n g a n d Searching, Add is o n -W es l ey , Reading, 1973.

G . K r e w e r a s - S u r une c l a s s e d e p r o b l ' e m e s de d C n o m b r e m e n t l i 6 s a u x t r e i l l e s d e s p a r t i t i o n s d e s e n t i e r s , C a h i e r s d u B . U . R . O . , 6 (1965), 1-105. G . K r e w e r a s - T r a i t e m e n t s i m u l t a n 6 du " p r o b l & m e d e Young" e t d u " p r o b l ' e m e de S i m o n N e w c o m b " , C a h i e r s d u B . U .R . O . , 10 (1967), 23-31 G. K r e w e r a s - Classification d e s permutations suivant certaines propri6tCs o r d i n a l e s d e l e u r r e p r e s e n t a t i o n p l a n e , i n P e r m u t a t i o n s , a c t e s d u C o l l o q u e 1972, A . L e n t i n e d . , G a u t h i e r - V i l l a r s , P a r i s / M o u t o n , P a r i s L a H a y e , 1974, p . 97-115.

G . Kreweras - Description d e certaines f o r & t s p a r des couples de suites d e Young e n d u a l i t 6 , Young D a y , W a t e r l o o , J u n e 1 9 7 8 . G . K r e w e r a s - Sur l e s e x t e n s i o n s l i n 6 a i r e s d ' u n e f a m i l l e p a r t i c u l i & r e d ' o r d r e s p a r t i e l s , D i s c r e t e M a t h . 27 (1979), 2 7 9 - 2 9 5 .

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A . L a s c o u x - T a b l e a u x d e Young e t f o n c t i o n s de S c h u r - L i t t l e w o o d , s 6 m i n a i r e D e l a n g e - P i s o t - P o i t o u , I . H . P . , P a r i s , 4 (1974), 1 - 7 . A . L a s c o u x - T a b l e a u x de Young g a u c h e s , s t m i n a i r e d e t h e o r i e d e s n o m b r e s , expost? n " 4 , 1974/75, P u b l . d e 1 ' U n i v . de B o r d e a u x I , 1975 A . L a s c o u x - P r o d u i t de K r o n e c k e r d e s r e p r e s e n t a t i o n s du g r o u p e s y m e t r i q u e , s C m i n a i r e D u b r e i l - M a l l i a v i n , 1978/79, i n L e c t u r e N o t e s i n Math., n"795, Springer-Verlag, Berlm/Heidelberg/New-York. A . Lascoux - Calcul de S c h u r e t extensions g r a s s m a n i e n n e s d e s h. - a n n e a u x , i n C o m b i n a t o i r e e t R e p r C s e n t a t i o n d u G r o u p e S y m e ' t r i q u e , S t r a s b o u r g 1976, D . F o a t a e d . , L e c t u r e N o t e s i n M a t h s . n " 5 7 9 , S p r i n g e r V e r l a g , Berlin/Heidelberg/New-York, 1977, p.182-216. A . L a s c o u x a n d M. P . S c h u t z e n b e r g e r - L e monoi'de p l a x i q u e , i n Nonc o m m u t a t i v e s t r u c t u r e s i n a l g e b r a a n d g e o m e t r i c , A . de L u c a s e d . , Q u a d e r n i de l a R i c e r c a S c i e n t i f i c a n"109, 1981, p.129-156. A . L a s c o u x a n d M . P . S c h u t z e n b e r g e r - C r o i s s a n c e d e s p o l y n 6 m e s de F o u l k e s - G r e e n , C . R . A c a d . S c i . P a r i s , 288 (1979), 9 5 - 9 8 . A . L a s c o u x a n d M. P . S c h u t z e n b e r g e r - A new s t a t i s t i c s on w o r d s , i n Combinatorial Mathematics, Optimal d e s ig n s and t h e i r applications, J . S r i v a s t a v a e d . , A n n a l s of D i s c r e t e M a t h . 6 (1980), 251-255. A . L a s c o u x and M . P . S c h u t z e n b e r g e r - PolynBmes de Schubert, C . R . A c a d . S c i . P a r i s , 294 (1982), 4 4 7 - 4 5 0 . A . L a s c o u x a n d M . P . S c h u t z e n b e r g e r - S t r u c t u r e d e Hopf de l ' a n n e a u de c o h o m o l o g i e e t de l ' a n n e a u de G r o t h e n d i e c k d e la variiSt6 d e d r a p e a u x , C . R . A c a d . S c i . P a r i s , (1983).

N . L i n i a l - E x t e n d i n g the G r e e n e - K l e i t m a n t h e o r e m t o d i r e c t e d g r a p h s , J. C o m b i n a t o r i a l T h e o r y , S e r . A 30 (1981), 331-334.

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N . L i n i a l - A new d e r i v a t i o n of t h e counting f o r m u l a f o r Young t a b l e a u x , J. C o m b i n a t o r i a l T h e o r y , A 33 (1982), 3 4 0 - 3 4 2 . D . E . L i t t l e w o o d - T h e T h e o r y of G r o u p c h a r a c t e r s a n d M a t r i x r e p r e s e n t a t i o n s of g r o u p s , 2nd e d . , O x f o r d Univ. P r e s s , O x f o r d , 1950. D . E . Littlewood and A . R . R ic h a r d s o n - G r o u p c h a r a c t e r s and a l g e b r a , P h i l . T r a n s . R o y . S O C . London, A 233 (1934), 99-141. B . F . L o g a n a n d L . E . S h e p p - A v a r i a t i o n a l p r o b l e m f o r r a n d o m Young t a b l e a u x , Adv. i n M a t h . , 26 (1977), 2 0 6 - 2 2 2 .

I . G . Mac Donald - S y m m e t r i c f u n c t i o n s a n d H a l l p o l y n o m i a l s , C l a r e n d o n P r e s s , O x f o r d , 1979. P . A . Mac Mahon

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P . A . Mac Mahon - M e m o i r o n the t h e o r y of t h e p a r t i t i o n of n u m b e r s . P a r t IV. On t h e p r o b a b i l i t y t h a t t h e s u c c e s s f u l candidate a t a n e l e c t i o n by b a l l o t m a y n e v e r a t a n y t i m e have f e w e r v o t e s t h a n t h e one who i s u n s u c c e s s f u l ; on a g e n e r a l i z a t i o n of t h i s q u e s t i o n ; a n d o n i t s connexion with o t h e r q u e s t i o n s of p a r t i t i o n , p e r m u t a t i o n a n d c o m b i n a t i o n , P h i l . T r a n s . , 209, (1909), 153-175. S e e a l s o C o l l e c t e d p a p e r s , vol.1, G . A n d r e w s e d . , T h e MIT P r e s s , C a m b r i d g e , 1978, page 1311-1314. A . Regev - A s y m p t o t i c v a l u e s f o r d e g r e e s a s s o c i a t e d with s t r i p s of Young d i a g r a m s , Adv. i n M a t h . , 41 (1981), 115-136.

J.B. R e m m e l - B i j e c t i v e p r o o f s of f o r m u l a e f o r t h e n u m b e r of s t a n d a r d Young tableaux, L i n e a r a n d Multil. A l g . , (1982). G . de B . Robinson - On t h e r e p r e e n t a t i o n of the s y m m e t r i c g r o u p s , A m e r . J. M a t h . , 60 (1938) 745-760 ; 69 (1947) 286-298‘; 70 (1948) 2 7 7 - 2 9 4 .

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H . A . Rothe - In S a m m l u n g Combinatorisch-analytischer A b h a n d l u n d e r , K.F. Hindenburg e d . , 2 , L e i p s i g , 1800, p . 2 6 3 - 3 0 5 . B . S a g a n - P a r t i a l l y o r d e r e d s e t s with hook l e n g t h s , a n a l g o r i t h m i c a p p r o a c h , P h . D . t h e s i s , M . I . T . , 1979. B . S a g a n - An a n a l o g of S c h e n s t e d ’ s a l g o r i t h m f o r s h i f t e d Young t a b l e a u x , J. C o m b i n a t o r i a l T h e o r y , S e r . A 27 (1979), 10-18.

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B . S a -g a n E n u m e r a t i o n of p a r t i t i o n s with hooklengths, E u r o p . J. Combin a t o r i c s , 3 (1982), 8 5 - 9 4 . M . S a k s - Duality p r o p e r t i e s of f i n i t e s e t s y s t e m s , P h . D . T h e s i s , M . I . T . , (1980). M. S a k s - A s h o r t proof of t h e e x i s t e n c e of k - s a t u r a t e d p a r t i t i o n s of p a r t i a l l y o r d e r e d s e t s , A d v . i n M a t h . , 33 (l979), 207-211. M. Saks - Dilworth n u m b e r s , incidence m a p s and product p a r t i a l o r d e r s , SIAM J. A l g . D i s c . M e t h . , 1 (1980), 211-216.

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C , Schensted - Longest i n c r e a s i n g and d e c r e a s i n g subsequences, Canad. J . M a t h . , 13 (1961). 179-191.

I . S c h u r - U b e r d i e D a r s t e l l u n g d e r s y m m e t r i s c h e n und d e r a l t e r n i e r e n d e n G r u p p e d u r c h g e b r o c h e n e l i n e a r e S u b s t i t u t i o n e n , J . C r e l l e , 139 (1911), 155-250 ( = G e s . Abhandlungen 1, 846-441). M. P. S c h u t z e n b e r g e r - Q u e l q u e s r e m a r q u e s s u r une c o n s t r u c t i o n de S c h e n s t e d , Math. S c a n d . , 12 (1963), 117-128. M. P. S c h d t z e n b e r g e r - S u r une c o n s t r u c t i o n de G i l b e r t de B . R o b i n s o n , s e m i n a i r e D u b r e i l (alg'ebre), 25'eme a n n k e , 1971/72, e x p o s k n " 8 , I . H . P . , Paris. M . P . S c h i i t z e n b e r g e r - S u r un thkor'eme de G . de B . R o b i n s o n , C A c a d . S c i . P a r i s , 272 (1971), 420-421.

L

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M . P. S c h u t z e n b e r g e r Promotions des morphismes d'ensembles o r d o m e s , D i s c r e t e M a t h . , 2 (1972), 73-94.

M . P. S c h u t z e n b e r g e r - P r o p r i k t k s n o u v e l l e s d e s t a b l e a u x de Young, skmin a i r e D e l a n g e - P i s o t - P o i t o u , 19kme a n n k e , 1977/78, e x p o s k n " 2 6 , I . H . P . , s e c r k t a r i a t mathe'matique, P a r i s , 1978. M. P. S c h t i t z e n b e r g e r - E v a c u a t i o n s , in colloquio I n t . s u l l e T e o r i e C o m b i n a t o r i e , R o m a 1973, Torno I, A c c a d . N a z . d e i L i n c e i , 1976, R o m a , p.257-264. M. P. S c h t i t z e n b e r g e r - La c o r r e s p o n d a n c e de R o b i n s o n , i n C o m b i n a t o i r e e t R e p r C s e n t a t i o n du G r o u p e s y m k t r i q u e , S t r a s b o u r g 1976, D . F o a t a e d . , L e c t u r e Notes in Maths, n"579, S p r i n g e r - V e r l a g , Berlin/Heidelberg/ New-York, 1977, p. 59-135. R . P. Stanley - T h e o r y a n d a p p l i c a t i o n of p l a n e p a r t i t i o n s , (I) a n d (11), S t u d i e s i n Applied M a t h . , 50 (1971), 167-188, 2 5 9 - 2 7 9 . R . P. S t a n l e y n"119, 1972.

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O r d e r e d s t r u c t u r e s a n d p a r t i t i o n s , M e m o i r s of t h e A . M . S . ,

R . P. S t a n l e y - S o m e c o m b i n a t o r i a l a s p e c t s of t h e S c h u b e r t c a l c u l u s , i n C o m b i n a t o i r e et R e p r C s e n t a t i o n d u G r o u p e S y m k t r i q u e , S t r a s b o u r g 1976, D. F o a t a e d . , L e c t u r e Notes i n Maths. n"579, S p r i n g e r - V e r l a g , B e r l i n / H e i d e l b e r g / N e w - Y o r k , 1977, p.217-251. D . W . Stanton a n d D . E . White b l e a u x , p r e p r i n t , 1982.

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F . StCltzer a n d M. C l a u s e n - J e u de t a q u i n a n d c o n n e c t e d s t a n d a r d s k e w t a b l e a u x , D i s c r e t e M a t h s . , 45 (1983), 133-135.

C .P. T h o m a s - B a x t e r a l g e b r a s a n d S c h u r f u n c t i o n s , T h e s i s , Univ. of W a l e s , S w a n s e a , 1974.

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G . P . T h o m a s - A g e n e r a l i z a t i o n of a c o n s t r u c t i o n due t o Robinson, C a n J . M a t h . , 28 (1976), 665-672. G . P . T h o m a s - On a c o n s t r u c t i o n of S c h u t z e n b e r g e r , D i s c r e t e M a t h . , 17 (1977), 107-118. G . P. T h o m a s - F r a m e s , Young tableaux a n d B a x t e r s e q u e n c e s , A d v . i n M A . , 26 (1977), 275-289. G . P. T h o m a s - On S c h e n s t e d ' s c o n s t r u c t i o n and t h e m u l t i p l i c a t i o n of S c h u r f u n c t i o n s , Adv. i n M a t h . , 30 (1978), 8 - 3 2 .

R . M. T h r a l l - A c o m b i n a t o r i a l p r o b l e m , Michigan M a t h . J . , (1952) 81-88. G . Viennot - Une f o r m e gkomCtrique de la c o r r e s p o n d a n c e de RobinsonS c h e n s t e d , i n C o m b i n a t o i r e e t R e p r C s e n t a t i o n du G r o u p e SymCtrique, S t r a s b o u r g 1976, D . F o a t a e d . , L e c t u r e N o t e s in M a t h s . n " 5 7 9 , S p r i n g e r V e r l a g , Berlin/Heidelberg/New-York, 1977, p. 2 9 - 5 8 . G . Viennot - Q u e l q u e s a l g o r i t h m e s de p e r m u t a t i o n s , i n J o u r n 6 e s a l g o r i t h m i q u e s , A s t C r i s q u e n " 3 8 / 3 9 , S O C . Math. de F r a n c e , 1976, p. 2 7 5 - 2 9 3 .

K . P . Vo - T h e S c h e n s t e d c o r r e s p o n d e n c e a n d l e x i c o g r a p h i c m a t c h i n g on m u l t i s u b s e t l a t t i c e s , SIAM J . A l g . D i s c r e t e M e t h . , 2 (1981) 324-332.

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D . B . W e s t - E x t r e m a l p r o b l e m s i n p a r t i a l l y o r d e r e d s e t s , in O r d e r e d S e t s , I . Rival e d . , Nato Advanced Study I n s t i t u t e S e r i e s , S e r i e s C, v o l . 83, D . R e i d e l , Dordrecht/Boston/London, 1982, p. 473-521. H . Weyl, T h e c l a s s i c a l g r o u p s , t h e i r i n v a r i a n t s a n d r e p r e s e n t a t i o n s , P r i n c e t o n Univ. P r e s s , 1939. D . White - A b i j e c t i o n proving o r t h o g o n a l i t y of t h e c h a r a c t e r s of Adv. i n Maths, 50 (1983), 160-186.

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D . White, S o m e connection between t h e L i t t l e w o o d - R i c h a r d s o n r u l e and t h e c o n s t r u c t i o n of S c h e n s t e d , J . C o m b i n a t o r i a l T h e o r y , &. A 30 (1981), 237-247. S . G . W i l l i a m s o n a n d E . R . Canfield - Hook length p r o d u c t s a n d C a y l e y o p e r a t o r s of c l a s s i c a l i n v a r i a n t t h e o r y , p r e p r i n t , 1980.

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T h e c o l l e c t e d p a p e r s of A l f r e d Young, Univ. of T o r o n t o A . Young P r e s s , T o r o n t o , 1977.

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A . V . Z e l e v i n s k y - A g e n e r a l i z a t i o n of the L i t t l e w o o d - R i c h a r d s o n r u l e a n d t h e R o b i n s o n - S c h e n s t e d Knuth c o r r e s p o n d e n c e , J . A l g e b r a , 6 9 (1981), 8 2 - 9 4 . R . M . B a e r a n d P. B r o c k - N a t u r a l s o r t i n g o v e r p e r m u t a t i o n s p a c e s , Math. C o m p . 2 2 (1968), 385-410. P. E d e l m a n a n d C . C r e e n e - C o m b i n a t o r i a l c o r r e s p o n d e n c e s f o r Young t a b l e a u x , b a l a n c e s t a b l e a u x a n d m a x i m a l c h a i n s in t h e B r u h a t o r d e r of S , p r e p r i n t ( r e s e a r c h a n n o u n c e m e n t ) (1983). n C . C r e e n e - S o m e r e m a r k on S c h i i t z e n b e r g e r ' s t a b l e a u o p e r a t o r s , in preparation.

R . P. Stanley - A combinatorial conjecture concerning the s y m m e t r i c g r o u p , p r e p r i n t (1982). R . P . S t a n l e y - On t h e n u m b e r of r e d u c e d d e c o m p o s i t i o n s of e l e m e n t s of C o x e t e r g r o u p s , p r e p r i n t (1982). C . Viennot - A l o c a l definition of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n dence, i n pr ep ar at i o n . C . Viennot - Young t a b l e a u x a n d g r i d s i n p o s e t s , i n p r e p a r a t i o n .

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PARTVIII ORDER AND EFFECTIVENESS

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- PARTIEVIII ORDRE ET EFFECTIVITE

Joseph G. ROSENSTEIN Recursive linear orderings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p . 465

Irhe GUESSARIAN - Maurice NIVAT About ordered sets in algebraic semantics, . . . . . . . . . . . . . . . . . . .p . 477 J.K. LENSTRA - A.H.G. RINNOOY KAN Two open problems in precedence constrained scheduling. . . . . . . p . 509

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Annals of Discrete Mathematics 23 (1984) 465-476 0 Elsevier Science Publishers B.V. (North-Holland)

465

RECURSIVE LINEAR ORDERINGS Joseph G. ROSENSTEIN Department of Mathematics Rutgers- The State Universityof NewJersey New Brunswick, New Jersey 08903 U.S.A.

RESLJm.Dans ce bref discours sur les chaines rscursives, je voudrais vous pr6senter certaines questions qu'on sc pose et vous donner un 6chantillon des mgthodes et techniques qu'on emploie pour r6pondre h ces questions. Le contexte de ce discours est fourni par la question g6ni.rale suivante: C t a i t tlontzL' 2.m th'ort'rnc

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c m b i > i a t ~ i r ce J n c t m i u n t I e s LThainc.s, sst-ii Si n o i , cmrn-nt peut-on mesui=i;r 1 ' e f - f e c t i o i t P

Un exemple de th6orsme combinatoire vrai effectivement est le r6sultat de Cantor selon lequel toute chaine dsnombrable est isomorphe 2 un sous-ensemble des nombres rationnels. On peut modifier la d6monstration standard de ce r6sultat pour dBcrire un algorithme qui produit le th6orCme suivant:

Touts chainc rdcursivc est d c u r s i v e m e n t isornorplie a' un sousenscmbie rz'zursif des nnmbres rationneis. Remarquez que tout emploi du mot "recursif" doit Ctre prScis6ment dgfini; ainsi par exemple, la phrase "chaine recursive" entraine l'existence d'un algorithme au moyen duquel on peut determiner si a < b ou non, et la phrase "r6cursivement isomorphe" entraine l'existence d'un algorithme au moyen duquel on peut calculer les valeurs de la fonction. Dans l'exemple que j'ai cit6 ci-dessus, la d6monstration traditionnelle peut Ctre effectivisge; il y a des exemples oij cela n'est pas possible. Dansces cas-li on doit ou fabriquer un nouvel algorithme ou demontrer qu'il n'existe pas d'algorithme convenable. Par exemple, la dgmonstration standard quetoutensemble bien ordonn6 a une extension lin6aire bien ordonn6e ne peut pas s'effectiviser. Cependant, un autre algorithme, que j'ai dgcrit avec H. Kierstead, demontre que:

Tout ensemb7,e bier1 o r r h n 8 qui est re'cursif i unp extension line'aire qui est bien ordonne'e ct re'cursive. La question de savoir si tout ensemble ordonne qui est dispers6 et rdcursif a une extension lingaire qui est dispers6e et rgcursive reste ouverte. 11 n'y a pas toujours une manisre unique d'effectiviser une formulation combinatoire. Par exemple, une autre manisre d'approcher l'exemple mentionnd cidessus est de demander si tout ensemble ordonne qui est r6cursif et recursivement bien ordonn6 a une extension lin6aire qui est rgcursive et recursivement bien ordonnke. ("Recursivementbien ordonn6" signifie que l'ensemble ordonn6 n'a aucune suite infinie r6cursive qui soit dgcroissante). Cette approche mPne plutst 1 une solution nggative, due h moi-m&ne et h R. Statman: I 1 existe un ensemble ordonni q u i est re'rursif e t re'cursivement bien ordonne', mais qui n'a pas d'extension 7in6aire qui soit re'cursivc et re'cursivement bien ordonnie. AprSs avoir introduit la higrarchie arithm6tique de Kleene, nous rgpondons B l'autre moitig de la question g6nCrale: Etant donn6 un ensemble ordonn6 qui est recursif et r6cursivement bien ordonn6, quelle complexit6 l'extension lin6aire d6sir6e doit-elle poss6der ?

466

J.G. Rosenstein Tout ensemble ordonnd q u i e s t r e c u r s i f e s t re'cursivement ordonnd a m e extension Lindaire qui e s t rdcursivement bien ordonneeet e s t s i t u d e au niveau h 2 dans La hidrarchie arithmdtique de KZeene.

La dgmonstration de ce thSorPme e s t un exemple de l ' u t i l i s a t i o n des "argumentat i o n s d i a g o n a l e s " dans l a t h g o r i e de l a r g c u r s i v i t g .

Les v e r s i o n s e f f e c t i v e s de quelques a u t r e s r g s u l t a t s combinatoires s o n t d i s c u t g e s dans l ' a r t i c l e y compris l e thLorOme de Dushnik e t M i l l e r s e l o n l e q u e l t o u t e c h a i n e dgnombrable p e u t s ' a b r i t e r dans un sous-ensemble p r o p r e d ' e l l e - m i k e . Ces s u j e t s s o n t d i s c u t g s 3 fond dans mon l i v r e L i n e a r Orderings (Academic P r e s s , 1982) dans l e c h a p i t r e i n t i t u l 6 , comme on p e u t s ' y a t t e n d r e , L i n e a r Orderings and Recursion Theory.

*** The framework f o r t h i s survey a r t i c l e about r e c u r s i v e l i n e a r order i n g s i s provided by t h e f o l l o w i n g Reneral q u e s t i o n s : Given a c o m b i n a t o r i a l theorem about l i n e a r o r d e r i n g s , i s i t t r u e e f f e c t i v e l y ? I f n o t , how can t h e e f f e c t i v e n e s s of t h e r e s u l t be measured ? Among t h e s p e c i f i c c o m b i n a t o r i a l theorems considered i n d e t a i l are ( a ) Every c o u n t a b l e l i n e a r o r d e r i n g i s order-isomorphic t o a s u b s e t of t h e r a t i o n a l s ( C a n t o r ) , ( b ) Every well-founded ( r e s p . , s c a t t e r e d ) p a r t i a l o r d e r i n g h a s a well-founded ( r e s p ; , s c a t t e r e d ) l i n e a r e x t e n s i o n , and ( c ) Every c o u n t a b l e l i n e a r o r d e r i n g can be embedded i n t o a p r o p e r s u b s e t of i t s e l f (Dushnik and M i l l e r ) . I n t h i s b r i e f t a l k about r e c u r s i v e l i n e a r o r d e r i n g s , I would l i k e t o convey t o you some s e n s e of t h e q u e s t i o n s that a r e r a i s e d and some of t h e f l a v o r of t h e I w i l l not methods and t e c h n i q u e s t h a t a r e used i n answering t h e s e q u e s t i o n s . t r y t o be comprehensive. The framework f o r t h i s t a l k i s provided by t h e f o l l o w i n g g e n e r a l q u e s t i o n :

Given a combinatorial theorem about linear ordsrings, i s it true e f f e c t i v e l y ? If not, how can the effectiveness of the result be measured? I u s e t h e phrase " l i n e a r o r d e r i n g s " t h e way o t h e r s u s e t h e p h r a s e s " t o t a l o r d e r i n g s " o r "chains." Information about l i n e a r o r d e r i n g s can be found i n my r e c e n t book [ S ] on t h e s u b j e c t ; t h e c h a p t e r on r e c u r s i v e l i n e a r o r d e r i n g s also c o n t a i n s t h e a p p r o p r i a t e material on r e c u r s i o n t h e o r y . A l l l i n e a r o r d e r i n g s t h a t I d i s c u s s w i l l be c o u n t a b l e , and t h e r e f o r e w e need o n l y look at s u b s e t s of Q ( t h e r a t i o n a l numbers.) This i s c o r r e c t by t h e f o l l o w i n g theorem, due t o Cantor:

E v e v countable linear ordering i s (order) isomorphic t o a subset of Q . Is t h i s c o m b i n a t o r i a l theorem t r u e e f f e c t i v e l y ? Before answering t h i s q u e s t i o n . we must f i r s t f o r m u l a t e and e x p l a i n t h e e f f e c t i v e v e r s i o n of C a n t o r ' s theorem. Loosely speaking, t h e way t o a r r i v e a t a n e f f e c t i v e v e r s i o n of a c o m b i n a t o r i a l r e s u l t is t o add t h e word " r e c u r s i v e " a t a l l a p p r o p r i a t e l o c a t i o n s . Using t h i s h e u r i s t i c , we a r r i v e at t h e f o l l o w i n g statement:

E v e y recursive linear ordering is recureiveZy isomorphic t o a recursive subset of Q.

46 7

Recursive linear orderings

Each of the three usages of the term "recursive" requires an explanation. Let us think of a countable linear ordering as a structure where N is the set of natural numbers and R is a binary relation on N which defines a linear ordering of N. Then we define the linear ordering to be recursive if there is an algorithm which, given a and b, will determine whether aRb or bRa is correct. Similarly, a subset A C- Q is recursive if there is an algorithm which, given a E Q, will determine whether or not a E A. Also, a function f:N + Q is recursive if there is an algorithm which, given a E N, will produce f(a) E Q; thus to say that is recursively isomorphic to a subset A C_ Q means that there is a recursive function f:N + A which defines an order isomorphism between and . With these definitions. it turns out that the effective version of Cantor's theorem stated above is correct, and moreover that the traditional proof can easily be effectivized, since it already has an algorithmic flavor. Thus consider the induction step for n = 6. We assume as the induction hypothesis that if

3 R 2 R 5 R l R 4

then If now 3 R 6 R 2 . the diagram below:

f(3)-

f(3)

<

f(2)

<

f(5)

<

f(1)

then we would choose f(6)

f(4).

between f(3)

f(5)-

f(2)-

<

-

f(1)

and

f(2).

f

as in

C

In order for f to be defined recursively, we must specify f(6) in a way which can be systematically applied to define the other values of f . The natural solution to this problem seems to be to define f(6) to be the midpoint of the interval, but this leads to some difficulty for the following reason. We have specified that the image of f be a recursive subset of Q so that we must be able to provide an answer to questions like "Is (f(5) + f(1))/2 in f [N]?" Now if for some a E N we have 5 R a R 1, then indeed the midpoint of the interval [f(5), f(l)] will be in Q, but otherwise it will not be. But the question of whether there is such an element a E N cannot be answered recursively, for the recursiveness of R only enables us to decide simple issues like the relative position of two elements, but not more complicated questions like the non-emptiness of an interval. Thus another definition of f(6) must be produced. Let us specify in advance a fixed canonical enumeration Iqnln E N) of 0 (which will be referred to elsewhere in this paper) and arrange the construction so that if qn is none of f(l), f(2), f(n), then qn is not in f [N] at all; then to determine whether or not b E f[N] we need only determine b's place in the enumeration and, assuming that b qm, carry out the first m applications of the algorithm, and at that time determine whether or not b E f[N]. To achieve this effect, we define f(6) to be the first rational which is between f(3) and f(2). and number on the list 96, 97, 98, similarly for each other value of n. This algorithm then has all the required properties.

...,

-

...

In the example above the traditional proof can be effectivized; there are examples where it cannot. In such cases, one must either produce a new algorithm or prove that no suitable algorithm exists.

J.G. Rosenstein

468

A s a second example of a c o m b i n a t o r i a l r e s u l t , I w i l l , i n d e f e r e n c e t o t h e o r g a n i z e r s of t h i s c o n f e r e n c e , d i s c u s s e x t e n d a b i l i t y of p a r t i a l o r d e r i n g s . ( I am r e p o r t i n g h e r e on d i s c u s s i o n s w i t h H. K i e r s t e a d and R. Statman.) Consider t h e following statement:

If A

is a well-founded partial ordering, then A a well-founded linear extension.

Is t h i s t r u e r e c u r s i v e l y ?

If A

has

That i s , i s t h e f o l l o w i n g s t a t e m e n t c o r r e c t ?

A is a well-founded recursive partial ordering, then has a well-founded recursive linear extension.

Here of c o u r s e we a r e t h i n k i n g of a r e c u r s i v e p a r t i a l o r d e r i n g a s a r e l a t i o n on N f o r which t h e r e i s a s u i t a b l e a l g o r i t h m .

R

The s t a n d a r d proof of t h i s r e s u l t ( s e e f o r example Bonnet and Pouzet [ l ] ) proceeds by " l e v e l s . " That i s , we d e f i n e A0 to c o n s i s t of a l l minimal e l e m e n t s of A, and proceed i n d u c t i v e l y t o d e f i n e A, t o c o n s i s t of a l l minimal e l e m e n t s of A {Aglg
u

T h i s proof c l e a r l y cannot be e f f e c t i v i z e d . I n f a c t , we cannot even b e g i n t h i s way s i n c e t h e r e i s no way of t e l l i n g r e c u r s i v e l y which elements of A a r e minimal. There i s however a n o t h e r a l g o r i t h m which works. This a l g o r i t h m and t h e proof t h a t i t works a r e due t o myself and H. K i e r s t e a d . A t stage n, f o r a l l a , b < n, which e x t e n d s
..., ...,

we assume t h a t w e have a l r e a d y d e f i n e d whether a
...,

.1 [email protected][email protected]., 11, 2 , ..., n - l l g /

W e could p l a c e n anywhere between t h e l a r g e s t squared element and t h e s m a l l e s t c i r c l e d element. W e choose t o p l a c e n a t t h e l o c a t i o n i n d i c a t e d by t h e arrow; t h a t i s , n i s p l a c e d j u s t below t h e B-smallest m < n f o r which n < ~ m . Thus n is p l a c e d as high as p o s s i b l e among 11, 2 , n-llg.

...,

I t i s c l e a r t h a t w i t h t h i s a l g o r i t h m w e do end up w i t h a r e c u r s i v e l i n e a r e x t e n s i o n of A -- f o r t o determine whether a b, and a
Recursive linear orderings

469

c f o r which c < a and c
a.

Now suppose t h a t B i s not well-founded, s o t h a t t h e r e i s a s u b s e t of B S O t h a t b l > B b2 > B b3 > B We may assume w i t h o u t { b l , b2, b3, . . . I l o s s of g e n e r a l i t y t h a t b l < b2 < b3 < For e a c h i < j , e i t h e r b i >A b j b j , s o by Ramsey's theorem and t h e assumption t h a t A is w e l l or bi founded, we may a l s o assume t h a t bi ( A b j f o r e v e r y i and j . For each i > 1 we t h u s have b i ( A b l , b i > b l , and b i 1 t h e r e i s some c i such t h a t c i < b l and b i 1.

... .

IA

... .

I f we c o n t i n u e t h i s procedure i n d u c t i v e l y , we conclude t h a t f o r t h e o r i g i n a l t h e r e i s a sequence c1, c 2 , c 3 , such t h a t sequence b l > B b2 >B b3 >B f o r a l l i > j we have b i j i > j). i m p l i e s t h a t c i 3 c j , we know t h a t c 1 < c2 < c 3 <

...

...

... .

...,

We now c l a i m t h a t c 1 >A c2 >A c 3 >A c o n t r a d i c t i n g t h e assumption Indeed, s i n c e c j + l c j , c +I j + l . T h i s c o n t r a d i c t s t h e c h o i c e of c j since < c j . Hence c j + l
ii

d

As noted by s e v e r a l p e o p l e , t h i s a l g o r i t h m p r o v i d e s a new proof t h a t i f K i s an i n f i n i t e c a r d i n a l number and K A than there is a l i n e a r extension B of A f o r which K B.

4

It is i n t e r e s t i n g t o see what happens i f t h i s p a r t i c u l a r a l g o r i t h m i s a p p l i e d t o a p a r t i a l Ordering which i s n o t well-founded. L e t A be t h e f u l l b i n a r y t r e e , l a b e l l e d as i n t h e diagram:

J.G. Rosenstein

470

It i s e a s y t o s e e t h a t i f t h i s tree were extended t o a l i n e a r o r d e r i n g u s i n g t h e a l g o r i t h m p r e s e n t e d above, then t h e even numbered p o i n t s would form a set of Indeed f o r each n , t h e Z n - l even numbers o r d e r t y p e rl (of t h e r a t i o n a l s ) . a t l e v e l n of t h e t r e e would, i n t h i s l i n e a r e x t e n s i o n , be l o c a t e d i n t h e 2"-1 i n t e r v a l s formed by t h e Z n - l - l even numbers a t h i g h e r l e v e l s of t h e tree. Of c o u r s e , one can extend t h i s p a r t i a l o r d e r i n g t o a s c a t t e r e d l i n e a r o r d e r i n g by u s i n g t h e o p p o s i t e a l g o r i t h m , namely, by l o c a t i n g each p o i n t a t t h e lowest p o s i t i o n p o s s i b l e . The example above a r i s e s when one a s k s whether every s c a t t e r e d r e c u r s i v e p a r t i a l o r d e r i n g has a s c a t t e r e d r e c u r s i v e l i n e a r e x t e n s i o n . The c o m b i n a t o r i a l v e r s i o n of t h i s q u e s t i o n has a p o s i t i v e answer, due t o Bonnet and Pouzet ( s e e [ l ] ) . and a l s o t o Galvin and MacKenzie. Whether t h e e f f e c t i v e v e r s i o n is t r u e i s an open q u e s t i o n .

Open question: Does every scattered recursive partial ordering have a scattered recursive linear extension? The a s t u t e r e a d e r w i l l perhaps have n o t i c e d t h a t i n g i v i n g t h e e f f e c t i v e v e r s i o n of our second example we d i d not q u i t e f o l l o w t h e p r e s c r i p t i o n l a i d down in our f i r s t example, namely, t o add t h e word " r e c u r s i v e " a t a l l a p p r o p r i a t e l o c a t i o n s . I n t h e theorem w e assumed t h a t A had "0 s u b s e t of o r d e r t y p e w* and showed t h a t then B had "0 s u b s e t of o r d e r t y p e w*. What i f we assume can we t h e n extend A o n l y t h a t A has no r e c u r s i v e s u b s e t of o r d e r t y p e w*; t o a r e c u r s i v e l i n e a r o r d e r i n g which has no r e c u r s i v e s u b s e t of o r d e r type w*?

L e t u s t h e n s a y t h a t A is r e c u r s i v e l y well-founded i f A has no r e c u r s i v e s u b s e t of o r d e r t y p e wR. The c o n c l u s i o n below, due t o myself and R. Statman, answers the q u e s t i o n above:

There i s a recursive partial ordering which i s recursively well-founded but has no recursively well-founded recursive linear extension. The proof of t h i s result--by c o n s t r u c t i n g a s u i t a b l e counterexample--has two s t e p s . The f i r s t s t e p i n v o l v e s c o n s t r u c t i n g a r e c u r s i v e b i n a r y tree--a subo r d e r i n g of t h e t r e e above--which though i n f i n i t e (and t h e r e f o r e c o n t a i n s a n i n f i n i t e p a t h by Konig's Theorem) c o n t a i n s no i n f i n i t e r e c u r s i v e p a t h . T h i s k i n d of c o n s t r u c t i o n is f a m i l i a r t o r e c u r s i o n t h e o r i s t s ; I w i l l s a y more a b o u t how one c o n s t r u c t s such a t r e e l a t e r in t h i s paper. The second p a r t i n v o l v e s showing t h a t such a p a r t i a l o r d e r i n g cannot be extended t o a r e c u r s i v e l y well-founded r e c u r s i v e l i n e a r o r d e r i n g . Assume t h e n t h a t A i s a r e c u r s i v e s u b t r e e of t h e f u l l b i n a r y t r e e which is r e c u r s i v e l y W e will well-founded and assume t h a t B i s a r e c u r s i v e l i n e a r e x t e n s i o n of A. show t h a t B has a r e c u r s i v e s u b s e t of o r d e r t y p e w* by c o n s t r u c t i n g a r e c u r s i v e w*-sequence of elements of B. This w i l l of c o u r s e prove t h e r e s u l t . The method of c o n s t r u c t i n g such a sequence might be c a l l e d a B - f i r s t s e a r c h through A, g e n e r a l i z i n g t h e n o t i o n s of b r e a d t h - f i r s t s e a r c h ( i . e . l e v e l s e a r c h ) and d e p t h - f i r s t s e a r c h (i.e. greedy s e a r c h ) through A which c o r r e s p o n d t o p a r t i c u l a r l i n e a r e x t e n s i o n s B of A. A t t h e end of s t a g e n w e w i l l have d e f i n e d a descending B-sequence { a l , a2, an} and a pool Pn of elements of A from which an+l w i l l be s e l e c t e d . We assume a s part of t h e i n d u c t i o n a,,) forms a t e r m i n a l p a r t of t h e tree A, t h a t hypothesis t h a t ( a l , a2, is, f o r each i, i f b is above a 1 on t h e t r e e , t h e n b = a j f o r some j < i, and t h a t t h e p o o l Pn c o n s i s t s of a l l immediate A-predecessors of an} e x c e p t t h o s e which are a l r e a d y i n e l e m e n t s of { a l , a 2 , {al, a2, an]. Thus a t s t a g e 1 we l e t a1 b e t h e t o p element of t h e tree

...,

..., ...,

...,

Recursive linear orderings

47 1

and l e t P1 be t h e A-predecessor of a l . Proceeding i n d u c t i v e l y we l e t an+l b e t h e B - l a r g e s t element of Pn and we o b t a i n Pn+l from Pn by d e l e t i n g a n + l and adding i t s A-predecessors. T h i s completes t h e c o n s t r u c t i o n of what i s c l e a r l y a r e c u r s i v e s u b s e t of A. To show t h a t t h i s s e t has o r d e r t y p e w* i n B , i t s u f f i c e s t o show t h a t a t + l < B a t f o r each t . But o t h e r w i s e a t + l > B a t , f o r s i n c e a t 4 Pt w e cannot have a t + l = a t . Also a t t l f P t - l f o r o t h e r w i s e we would have chosen i t t o be a t . But a t + l f Pt s o i t must be a n A-predecessor of a t , so t h a t a t + l
Open question: Does every recursively scattered recursive partial ordering have a recursively scattered recursive linear extension? Now t h a t w e have a c o m b i n a t o r i a l s t a t e m e n t about l i n e a r o r d e r i n g s t h a t i s not t r u e e f f e c t i v e l y , we go back and a s k how t h e e f f e c t i v e n e s s of t h e o r i g i n a l r e s u l t can be measured.

To d i s c u s s t h i s q u e s t i o n , we have t o i n t r o d u c e t h e c l a s s i f i c a t i o n of t h e a r i t h m e t i c a l sets. We s a y t h a t a s e t K of numbers i s a En-set i f t h e r e i s a r e c u r s i v e p r e d i c a t e R(x1, x2, x,, y ) s u c h t h a t

...,

where t h e n q u a n t i f i e r s a r e a l t e r n a t e l y 3 and b' , s t a r t i n g w i t h 3. S i m i l a r l y , w e s a y t h a t K i s a IIn-set i f t h e r e i s a r e c u r s i v e p r e d i c a t e R(x1, x2 xn, m) such t h a t

...,

m

E

K

f--+

--.,%,

( V x l ) ( 3 x 2 ) ( v ~ 3 )* * * (OnXn) R(x1, ~ 2 .

m)

d.

where a g a i n t h e q u a n t i f i e r s a l t e r n a t e , t h i s t i m e s t a r t i n g w i t h A set K i s a An-set i f it is both a En-set and a IIn-set. A set i s c a l l e d an a r i t h m e t i c a l set i f i t i s a En-set o r a IIn-set f o r some I . . That t h e s e s e t s form a h i e r a r c h y was shown by S.C. Kleene ( f o r r e f e r e n c e s e e [ 5 ] ) ; t h a t i s , i n t h e diagram below a l l i n c l u s i o n s ( i n d i c a t e d by a r r o w s ) a r e p r o p e r i n c l u s i o n s .

A1

...

For example, a set E is r e c u r s i v e l y enumerable i f t h e r e i s a Turing machine which w i l l produce a l l of i t s members; t h a t i s , i f f o r some t ,

47 2

J.G. Rosenstein m

E

E

+--+

(3s)

T ( t , s , m)

where T ( t , 6 , m) i s t h e r e c u r s i v e p r e d i c a t e which s a y s t h a t "Turing machine number t w i l l produce m w i t h i n s s t e p s . " Now f o r each f i x e d t , t h e p r e d i c a t e T t ( s , m) :T ( t , s , m) i s a r e c u r s i v e p r e d i c a t e R ( s , m), so t h a t m E E +--+ ( 3 s ) R ( s , m). Thus every r e c u r s i v e l y eIIUmeKable s e t i s a E l - s e t ; t h e converse i s a l s o c o r r e c t . A s e t is r e c u r s i v e i f and only i f both i t and i t s complement a r e r e c u r s i v e l y enumerable, or El-sets. Note t h a t a set is a En-set i f and o n l y i f i t s complement is a Hn-set, so t h a t t h e Al-sets a r e p r e c i s e l y t h e r e c u r s i v e sets. We l e t Wt {mi ( 2 s ) T ( t , says t h a t the enumerated,)

d e n o t e t h e t ' t h r e c u r s i v e l y enumerable s e t : Wt = (Hidden in t h e n o t a t i o n i s t h e Enumeration Theorem which r e c u r s i v e l y enumerable s e t s can themselves be r e c u r s i v e l y For which t i s Wt non-empty? T h i s s e t i s C 1 s i n c e s , m)}.

non-empty

Wt

+---+

( 3 m ) ( g s ) T ( t , s , m).

(In t h i s p r e d i c a t e , t h e two e x i s t e n t i a l q u a n t i f i e r s can be combined i n t o one by u s i n g t h e r e c u r s i v e f u n c t i o n f ( n , x ) = (x), = t h e power of t h e n ' t h prime i n X. Thus t h e above i s e q u i v a l e n t t o ( 3 x ) T ( t , ( x ) l , ( x ) ~ ) which can be s i m p l i f i e d i n f i n i t e ? This s e t i s H2 s i n c e t o ( 3 x ) R(t, x).) For which t i s Wt

wt

infinite

+--+

( V n ) ( 3 m) (m

which can be r e w r i t t e n i n t h e a p p r o p r i a t e

II2

>

n

m

E wt)

form.

One a d d i t i o n a l t o p i c t h a t we need t o d i s c u s s b r i e f l y i s t h e use of an o r a c l e . During a c o n s t r u c t i o n we may f i n d i t n e c e s s a r y t o r e f e r t o a set A which i s e x t e r n a l t o t h e c o n s t r u c t i o n , and i n q u i r e whether given numbers a r e i n A. In t h a t c a s e t h e set or f u n c t i o n c o n s t r u c t e d i s s a i d t o be r e c u r s i v e in A. I t t u r n s o u t t h a t i f A i s a '&-set o r a lIn-set, then sets r e c u r s i v e in A a r e a l l An+l-sets. Thus, f o r example, i f d u r i n g a c o n s t r u c t i o n we need t o know f o r i s i n f i n i t e , then whatever we c o n s t r u c t w i l l be a A3-set; each t whether Wt in t h i s case, t h e o r a c l e being c o n s u l t e d i s I t l W , i s i n f i n i t e } . The a r i t h m e t i c a l h i e r a r c h y p r o v i d e s a u s e f u l measure of t h e complexity of The v a l u e judgment i t propounds i s t h a t t h e lower g i v e n sets o r c o n s t r u c t i o n s . i n t h e h i e r a r c h y t h e outcome of a p a r t i c u l a r c o n s t r u c t i o n , t h e b e t t e r t h e construction. We now r e t u r n t o OUK p r e v i o u s q u e s t i o n . Given a r e c u r s i v e l y well-founded r e c u r s i v e p a r t i a l o r d e r i n g A , how complicated need a d e s i r a b l e l i n e a r e x t e n s i o n B of A be? I n Order f o r B t o be r e c u r s i v e l y well-founded, it is s u f f i c i e n t t o g u a r a n t e e t h a t no Wt p r o v i d e s a B-decreasing sequence of elements of A. How do we do t h i s ? Suppose t h a t a f t e r s s t e p s we have enumerated ( i n o r d e r ) t h e ak of Wt. I f a j B a2 >B >B ak and t h a t , even worse, t h i s p a t t e r n w i l l c o n t i n u e u n t i l t h e very end. So we must f i n d some a i and a j w i t h i < j ( e i t h e r from t h o s e elements s o f a r g e n e r a t e d o r from t h o s e n o t y e t g e n e r a t e d ) f o r which a t i s not y e t B-larger t h a n a j , Since a t each s t e p of t h e c o n s t r u c t i o n of B we make and d e f i n e a i
...,

...

.

Recursive linear orderings

473

The t r o u b l e is t h a t w e have t o do t h e same t h i n g f o r each W t and s o w e cannot wait around t o f i n d o u t i f Wt is infinite. I f i t i s , we w i l l e v e n t u a l l y f i n d a s u i t a b l e i and j , but i f i t is n o t , then o u r s e a r c h w i l l go o n f o r e v e r and no I3 w i l l be c o n s t r u c t e d a t a l l . I f a t a given s t a g e of t h e t h e n w e w i l l need t o know whether o r n o t c o n s t r u c t i o n we want t o d i s p o s e of Wt Wt i s i n f i n i t e ; i f i t i s , w e w i l l s e a r c h f o r t h e r i g h t a i and a . , and i f i t is n o t , we w i l l go on t o W t + l . Thus t h e c o n s t r u c t i o n of B he r e c u r s i v e i n a Il2 s e t . The r e a d e r may have observed t h a t we can u s e a s i m p l e r o r a c l e , namely, one t h a t answers whether W t h a s a n o t h e r element. S i n c e t h i s q u e s t i o n i s 11, t h e c o n s t r u c t i o n of B w i l l be r e c u r s i v e i n an 11-set. W e t h u s come t o t h e f o l l o w i n g c o n c l u s i o n :

will

i s a recursively well-founded recursive p a r t i a l ordering, then A has a r e c u r s i v e l y well-founded A2 l i n e a r extension.

If* A

By a s i m i l a r argument i t is p o s s i b l e t o show t h a t

i s a r e c u r s i v e l y scattered recursive p a r t i a l ordering, then A has a r e c u r s i v e l y scattered A2 l i n e a r extension.

If A

The proof w e p r e s e n t e d above is an example of t h e u s e of " d i a g o n a l arguments" i n recursion theory. Before going on t o t h e next example, l e t me pause t o p o i n t o u t how t o c o n s t r u c t an i n f i n i t e r e c u r s i v e b i n a r y t r e e which i s r e c u r s i v e l y well-founded; S t a r t generating the f u l l binary s u c h a tree w a s used i n an e a r l i e r argument. t r e e . A t a c e r t a i n t i m e , we may d e c i d e t o c l o s e o f f a c e r t a i n node; by t h i s we mean t h a t n o t h i n g new w i l l go below t h a t node, so t h a t below t h a t node t h e t r e e w i l l remain f i n i t e T W h e n do we make such a d e c i s i o n ? W e s i m u l t a n e o u s l y begin and i f i t happens a t some p o i n t t h a t we have enumerated t enumerating a l l W t , elements of a p a r t i c u l a r Wt and t h o s e elements form a descending c h a i n s o f a r on t h e t r e e , t h e n we c l o s e o f f t h e lowest of t h e t nodes. The r e a d e r should v e r i f y t h a t t h e t r e e t h a t r e s u l t s has t h e d e s i r e d p r o p e r t i e s . We now c o n s i d e r a t h i r d example of a c o m b i n a t o r i a l theorem about l i n e a r o r d e r i n g , t h i s one due t o Dushnik and Miller [ 2 ] .

Any countable l i n e a r ordering can be embedded i n t o a proper subset of i t s e l f . Can any r e c u r s i v e l i n e a r o r d e r i n g be r e c u r s i v e l y embedded i n t o a p r o p e r r e c u r s i v e s u b s e t of i t s e l f ? Those of you who have s e e n t h e proof of Dushnik and Miller's theorem w i l l r e c a l l t h a t i n t h e s c a t t e r e d c a s e t h e d e s i r e d embedding w a s t h e i d e n t i t y o u t s i d e of a s u b s e t of o r d e r type o ( o r a*) and was t h e s u c c e s s o r ( o r p r e d e c e s s o r ) map on t h a t s u b s e t . The t w i s t came i n t h e n o n - s c a t t e r e d case where a s u b s e t of order type n was s e l e c t e d and A was embedded i n t o t h a t s u b s e t u s i n g C a n t o r ' s theorem. More p r e c i s e l y , t h i s procedure was needed o n l y when A i s a dense sum of f i n i t e l i n e a r o r d e r i n g s . Thus we need t o f i n d a dense s u b s e t of a dense sum of f i n i t e l i n e a r o r d e r i n g s . Can w e make such a s e l e c t i o n r e c u r s i v e l y ? For example, suppose t h a t A i s a r e c u r s i v e l i n e a r o r d e r i n g of o r d e r type 2-11. The right-hand e n d p o i n t s form a s u b s e t of o r d e r type n b u t t h e r e i s no way t o t e l l r e c u r s i v e l y which p o i n t s a r e right-hand e n d p o i n t s . (Show t h a t t h i s i s a A2-set.) Nevertheless, t h e e f f e c t i v e v e r s i o n of Dushnik and Miller's theorem i s c o r r e c t i n t h i s situation.

414

J. G.Rosenstein Any recursive subset A of Q of order type Z'n can be recursively embedded i n t o a proper recursive subset of itself.

...

} in o r d e r of d i s c o v e r y and To prove t h i s we enumerate A = { a l , a 2 , a 3 , begin t h e d e f i n i t i o n of a f u n c t i o n P: A A f o r which a 1 4 P [ A ] by s e t t i n g P(a1) = a2. How do we d e f i n e P ( a 2 ) ? I f f o r example a2 > a1 we can w a i t f o r some am > a2 and then d e f i n e P(a2) = am. But t h a t w i l l g i v e u s problems l a t e r i f a, is t h e s u c c e s s o r of a2 b u t a2 i s not t h e s u c c e s s o r of a l . I n s t e a d w e wait f o r an a, which i s not t h e s u c c e s s o r of a2. HOW t o do t h i s s y s t e m a t i c a l l y we l e a v e f o r t h e r e a d e r . ( A more complete d i s c u s s i o n of t h e p r o o f s of t h i s and subsequent r e s u l t s i n t h i s paper can be found i n my book I S ] . ) ---+

The same argument works whenever A is a dense sum of bounded f i n i t e l i n e a r o r d e r i n g s . The r e a d e r might t r y h i s hand a t t h e a p p a r e n t l y open q u e s t i o n of what happens when A i s an a r b i t r a r y r e c u r s i v e dense sum of f i n i t e l i n e a r orderings. We s t a r t e d t h i s d i s c u s s i o n by mentioning how e a s i l y Dushnik and M i l l e r ' s theorem works in t h e s c a t t e r e d c a s e . This i s f a r from t r u e r e c u r s i v e l y , s i n c e one cannot i d e n t i f y s u c c e s s o r s o r determine t h e s i z e of i n t e r v a l s r e c u r s i v e l y The f o l l o w i n g r e s u l t , due t o L. Hay and even i f A has o r d e r type w. R o s e n s t e i n , is proved i n (51.

There i s a recursive subset A of Q of order type w f o r which there is rw recursive mzp from A t o A other than the identity. There a r e s e v e r a l i n t e r e s t i n g q u e s t i o n s which were r a i s e d in t h e above d i s c u s s i o n which merit f u r t h e r a t t e n t i o n . Does e v e r y non-scattered r e c u r s i v e s u b s e t of Q have a r e c u r s i v e s u b s e t of o r d e r t y p e n? The embedding of 2-II i n t o i t s e l f does y i e l d a r e c u r s i v e s u b s e t of o r d e r type n but t h a t i s t h e b e s t t h a t can be done. That i s , i t i s p o s s i b l e t o c o n s t r u c t a r e c u r s i v e (5 i s t h e o r d e r type of t h e i n t e g e r s ) which s u b s e t of Q of o r d e r type 5.17 h a s no A 2 dense s u b s e t (Lerman and R o s e n s t e i n [ 4 ] ) ; t h e b e s t one can t h u s s a y i s t h a t such a s e t has a II2 dense s u b s e t ( t r y t o prove t h i s ) . I s u s p e c t , b u t I cannot prove, t h a t t h e r e is a r e c u r s i v e n o n - s c a t t e r e d l i n e a r o r d e r i n g which has no a r i t h m e t i c a l dense s u b s e t .

Open question: 16 there a recursive non-scattered l i n e a r ordering which has rw arithmetical dense subset? n a r e c u r s i v e nons c a t t e r e d l i n e a r o r d e r i n g of o r d e r type E n * n which has no A2n r e c u r s i v e dense s u b s e t , and t h e n perhaps one of o r d e r type 5 which has no a r i t h m e t i c a l dense s u b s e t . A p o s s i b l e l i n e of a t t a c k would be t o c o n s t r u c t f o r each

S i n c e t h e r e i s no end t o t h e c o m b i n a t o r i a l r e s u l t s about l i n e a r o r d e r i n g s (and o t h e r s t r u c t u r e s ) , t h e r e i s a l s o no end t o t h e q u e s t i o n s of e f f e c t i v e n e s s of c o m b i n a t o r i a l r e s u l t s t h a t can arise. I w i l l d i s c u s s b r i e f l y two more t y p e s of q u e s t i o n s . The f i r s t i s t h e f o l l o w i n g : Given a r e c u r s i v e l i n e a r o r d e r i n g which has an automorphism, must i t have a r e c u r s i v e automorphism? In my book I c o n s t r u c t examples of r e c u r s i v e l i n e a r o r d e r i n g s of o r d e r type 5 and 2*r, which have no r e c u r s i v e automorphisms. (The l a t t e r c o n s t r u c t i o n u s e s a p r i o r i t y argument, which i s a more s o p h i s t i c a t e d t y p e of d i a g o n a l argument.) A r e c e n t t h e s i s by Steven Schwarz [ 6 ] c o n t a i n s t h e f o l l o w i n g d e f i n i t i v e r e s u l t : L e t a be a r e c u r s i v e o r d e r type ( i . e . t h e o r d e r type of a r e c u r s i v e linear o r d e r i n g ) . Then t h e r e i s a r e c u r s i v e s u b s e t A of Q of o r d e r type a which has no r e c u r s i v e automorphism, u n l e s s a has an i n t e r v a l of o r d e r t y p e n.

Recicrsiue linear orderings

41 5

F i n a l l y , I should d i s c u s s t h e o l d e s t r e s u l t concerning e f f e c t i v e v e r s i o n s of theorems about l i n e a r o r d e r i n g s . Consider t h e c l a s s i c a l f a c t t h a t every i n f i n i t e l i n e a r o r d e r i n g h a s e i t h e r a s u b s e t of o r d e r t y p e w o r a s u b s e t of o r d e r t y p e w*. What i s t h e c o m b i n a t o r i a l v e r s i o n of t h i s f a c t ? Tennenbaum showed a number of y e a r s a g o t h a t t h e r e is a r e c u r s i v e s u b s e t of 0 o f o r d e r t y p e w + a* which h a s no r e c u r s i v e s u b s e t of o r d e r t y p e w o r w*; s u c h a s e t c a n be t h o u g h t of a s e f € e c t i v e l y f i n i t e . Watnick [ 7 ] e x t e n d e d Tennenbaum's c o n s t r u c t i o n t o show t h a t f o r e v e r y r e c u r s i v e o r d e r t y p e a t h e r e i s a r e c u r s i v e s u b s e t of 0 of o r d e r t y p e w + 6.a + w* w h i c h h a s no r e c u r s i v e s u b s e t of o r d e r t y p e a o r w*. Looking i n a d i f f e r e n t d i r e c t i o n , i t is p o s s i b l e t o show t h a t e v e r y r e c u r s i v e l i n e a r o r d e r i n g h a s a r e c u r s i v e s u b s e t of o r d e r t y p e w, w*, w + w*, o r w 5-11 + w*; Lerman [ 3 ] showed t h a t t h i s was b e s t p o s s i b l e by c o n s t r u c t i n g a r e c u r s i v e l i n e a r o r d e r i n g of t h e l a s t o r d e r t y p e which h a s no r e c u r s i v e s u b s e t of any of t h e o t h e r t h r e e o r d e r t y p e s .

+

A s I h a v e m e n t i o n e d s e v e r a l t i m e s a l r e a d y , t h e s e r e s u l t s and c o n s t r u c t i o n s a r e d i s c u s s e d a t some l e n g t h i n my book [ 5 ] , i n t h e c h a p t e r t i t l e d , n o t s u r p r i s i n g l y , L i n e a r O r d e r i n g s and R e c u r s i o n T h e o r y .

I hope t h a t i n t h i s t a l k I have s u c c e e d e d i n c o n v e y i n g some of t h e f l a v o r of t h e s u b j e c t of r e c u r s i v e l i n e a r o r d e r i n g s .

REFERENCES B o n n e t , R. and P o u z e t , M., L i n e a r e x t e n s i o n s of o r d e r e d s e t s , i n : R i v a l , I. ( e d . ) , O r d e r e d S e t s , P r o c . of t h e NATO Advanced S t u d y I n s t i t u t e , B a n f f , Canada (1981). D u s h n i k , B. and M i l l e r , E.W., C o n c e r n i n g s i m i l a r i t y t r a n s f o r m a t i o n s of l i n e a r l y o r d e r e d s e t s , B u l l . h e r . Math.. SOC. 3 (1940). 322-326. [MR 1, p. 3181. Lerman, M., On r e c u r s i v e l i n e a r o r d e r i n g s , i n : Lerman, M., S c h m e r l , J. and S o a r e , R. ( e d s . ) . P r o c e e d i n g s , L o g i c Year 1979-80: The U n i v e r s i t y of o f C o n n e c t i c u t , L e c t u r e Notes i n M a t h e m a t i c s , Vol. 859, pp.132-142. ( S p r i n g e r - V e r l a g , B e r l i n and New York, 1981). Lerman M. and R o s e n s t e i n , J . G . , R e c u r s i v e L i n e a r O r d e r i n g s 11, i n : P r o c e e d i n g s , P a t r a s C o n f e r e n c e 1980, Amsterdam ( N o r t h - H o l l a n d P u b l . ,

1981). R o s e n s t e i n , J.G.,

L i n e a r O r d e r i n g s (Academic P r e s s , New Y o r k , 1982).

S c h w a r z , S . , Q u o t i e n t L a t t i c e s , I n d e x S e t s , and R e c u r s i v e L i n e a r O r d e r i n g s , Ph.D. T h e s i s , U n i v e r s i t y of C h i c a g o (1982). W a t n i c k , R., A g e n e r a l i z a t i o n of Tennenbaum's t h e o r e m o n e f f e c t i v e l y f i n i t e r e c u r s i v e o r d e r i n g s , t o a p p e a r i n J. Symbolic Logic.

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Annals of Discrete Mathematics 23 (1984)477-508 0 Elsevier Science PublishersB.V. (North-Holland)

411

ABOUT ORDERED SETS IN ALGEBRAIC SEMANTICS Irene GUESSARIAN et C.N.R.S.-L.I.T.P.

Maurice NIVAT UnivrrsitQ Paris 7

-

L.I.T.P.

U.E.R. dr MATHEMATIQUES Universitd Paris 7 2 , Place Jussieu 75251 PARIS CEDEX 05

Au Professeur E. COROMINAS, 5 l'occasion de s o n Emdritat.

RQsumd

La sgmantique algdbrique ddcrit les schemas de programmes par des systemes S

d'gquations rdcursives dont, intuitivement, les membres gauches repr.5-

sentent les "noms d e p r o c e d u r e " et les membres droits les " c o r p s d e p r o c e d u r e ' ' les d6finissant ; les dquations d'un tel systeme sont des dgalitds entre termes bien formds d'une me

S

Z-algsbre libre ordonnde. La fonction calculde par un program-

peut alors stre caractdrisde par la solution du systeme d'dquations

S

dans une Z-algebre ordonnde compliite libre : la compldtude de l'algsbre oii l'on rejoint

S

unique

TS

permet d'assurer d'une part que tout eel systiime S

,

admet une solution

dans l'alglbre libre, d'autre part que cette solution est calculable

par approximations successives. La compldtion la plus utilisde est une compldtion par id6aux de la Z-algiibre libre. La fonction calculde par le programme une algebre complete quelconque est alors l'image de

TS

S

dans

par un morphisme cano-

nique. On montre ensuite que la complgtion par iddaux s'avsre insuffisante

pour prendre en compte des varibtds, dventuellement non dquationnelles, d'algsbres et qu'elle contient "trop de bornes supErieures", dont certaines ne sont pas ddfinissables par systemes d'gquations. Ceci nous amiine 1 :

I. Guessarian, M. Nivat

418 (I).

D6finir des completions continues (intermgdiaires entre compl6tions par cou-

pures et par idEaux). (2).

D6finir des algPbres libres ordonn6es recursivement complPtes (consistant

exclusivement de solutions de schdmas de programmes). ( 3 ) . Etudier diverses vari6t6s d'algsbres ordonnges utiles du point de vue de l a

programmation et les alglbres libres complstes correspondantes (ce qui conduit 2 une Etude de cornpl6tions de quotients et P des combinaisons des points ( I ) et ( 2 ) ci-dessus).

Abstract : We show how complete ordered algebras are used in algebraic semantics. We introduce the notions of interpretation (or model, or algebraic structure), and of free and Herbrand interpretations. We define classes of interpretations. We

characterize and construct the free and Herbrand interpretations for a class. We define algebraic, equational and relational classes of interpretations and study the construction of the Herbrand interpretation in each case. We also give some hints on how this study can be applied to derive proof systems for program equivalences. I.

INTRODUCTION We here show how Universal Algebra methods, and more specially the

theory of ordered sets, can be applied to programming language semantics thus yielding an algebraic semantics. Problems in semantics can be expressed in terms of ordered algebras ; the theory of ordered algebras, within the theory of ordered

sets then helps i n providing partial answers for them. This in turn usually leads to new questions or different formulations of problems, hence new problems in the theory of ordered algebras thus resulting in a two way communication channel between the two disciplines. Algebraic semantics main goal has been to provide a clean and sound semantics of programming languages by splitting as much as possible the syntactic and semantic parts of a program. Using tools from Universal Algebra, one can then describe abstractly, i.e. independent of any interpretation, the syntactic properties of a program ; after what, one is well equipped for, given any concrete interpretation, translating the abstract or syntactic properties, via that interpretation, into concrete, or semantic properties, of the real program. Moreover, this can be done stepwise, introducing at each step the exactly needed amount of semantic knowledge. Algebraic semantics thus makes easier and more natural the concepts of modularization and abstraction, essential in software development. I n algebraic semantics, interpretations are nothing but certain ordered

479

Ordered sets in algebraic semantics algebras, or equivalently X-structures or models in the sense of

/GR/.

The main

tool in characterizing the syntax of a program is to have i t compute symbolically in a free interpretation, i.e. i n an absolutely free algebra of terms. The set of all possible computations then is naturally endowed with an ordering corresponding to the notion of successive approximations of a computation, or, equivalently, the relation "to be less defined than" for partial functions. For this ordering, the set of all finite symbolic computations has a lattice structure, hence can be represented by an infinite tree (or term) ; this tree belongs to an ordered algebra, complete with respect to the ordering "to be less defined than" ; moreover, this infinite tree (or term) characterizes the behavior of the program with respect to all possible interpretations : properties of programs are thus described by properties of the associated infinite tree (or term). One of the main problems one has to deal with in proving any program property, is to prove equivalences of programs : this approach shows that two programs are equivalent w.r.t. all possible interpretations iff their associated infinite trees are equal. However, equivalence w.r.t. all possible interpretations is by far too exacting to be of any real use. In practice one has to take into account some of the constraints or properties verified by the interpretations one is interested in ; this extra information should even be modularized according to needs. We thus have to look for alternate syntactic objects and structures for finding and describing the set of all possible symbolic computations under some given constraints. The constraints will be described by a class of interpretations C, and we will look for a generic any given class I)

-

or free, or Herbrand - interpretation H

for

C, together with effective ways of describing :

the free interpretation H

2) The equivalence w.r.t.

H ; this equivalence will in turn characterize

the equivalence w.r.t. the class C. This H

corresponds roughly to what is usual-

ly called a free complete C-algebra, or an algebra free relative to the class of complete ordered algebras. The description of this free interpretation H, its elements and its equivalence can then be fruitfully applied to prove various kinds of program properties, transformations, simplifications, etc... The main concepts of universal algebra used in our approach are :

-

complete ordered F-algebras

- free (and related) algebraic systems in various classes

-

classes of algebraic systems equational, quasi-equational (or inequational, or relational) and other

kinds of classes of algebraic systems. This paper is primarily a survey written with an intuition minded bias : we give numerous examples and informal explanations, but refer the reader to the literature and mainly /GU/ where most of the results here given are proved. It is organized as follows : section 2 introduces the preliminaries on algebraic semantics,

I. Guessarian, M. Nivat

48 0

in relation with ordered set theory ; ordered algebras. complete algebras, program schemes and their interpretations, specially the free and Herbrand ones are there defined. Section 2 introduces the "class of intcrpretations" approach, illustrates problems and questions inherent to i t and relates i t to the literature ; in sections 4 and 5 we study in a more detailed way the equational and first order definable

classes, the most interesting from a practical standpoint. 2.

PROGRAM SCHEMES

2.1.

-

SEMANTICS - FREE COMPLETE ALGEBRAS

Programs and program schemes We will briefly outline in this section the basic results of algebraic

semantics. The fundamental ideas of algebraic semantics are :

1) using universal algebra tools, characterize a program by a mathematical object which is an element of a free complete algebra /GU,GUI, N/. 2) Prove the existence of the free complete algebras needed and construct

them. 3) Use the mathematical representation of the program to obtain sound

proofs of its properties.

To these ends, programs are represented by program schemes, i.e. systems of mutually recursive equations whose right hand sides represent procedure names and left hand sides represent procedure bodies ; the equations of such a system are equalities between well formed terms of a free ordered algebra. Let us illustrate this by an example. Example I : Suppose we wish to compute nk using only addition and multiplication. The simplest idea, in view o f the binary representation, is to write down the definition of nk with 2 as explicit factors in the exponent, i.e. : 2 x [k/2]

nk

=

+ r (k)

n

, where [k/2] and r (k) denote the quotient and

the remainder of the integer division of k by 2 . Whence the recursive procedure : Recursive procedure G(n,k) G(n,k)

=

;f k=O

where p(n,b)

then 1 else

: =

; integer

n,lc ;

G(nx n , [k/2]) x p(n,r(k))

b=O then 1 else n .

(1)

}

p

Notice that the first line of this definition consists of procedure declarations inessential from the computational standpoint. We will forget that first line and concentrate on the last two lines which contain the procedure body. If we now abstract the previous program into a "program scheme", obtained by "forgetting" the nature and meaning of the base functions and data, we we obtain the scheme :

48 1

Ordered sets in algebraic semantics

S : G(n,k)

Scheme

g(a,k,b, m(G(m(n,n), d(k)), p(n,r(k)))) (2) gives program P back if we consider the interpretation Z having domain

=

S

N (i.e. the data belong to N), and s u c h that = 0, bI= A , mI= x , dI (k) = [k/2], gI(x,y,z,t) = if x = y __ then z I else t, p (n,b) = if b = 0 then 1 else n. Now, we consider the interpretation 3 , __ I having the same domain as I and giving the same meaning to the base function symbols

a

except for b and m which are respectively interpreted as bJ this case, GJ(n,k)

nxk.

=

=

0 and m

J

=

+. In

T h u s abstracting a program into a program scheme leads

to a greater generality. We now come to the problem of semantics

-

namely, givenaprogram scheme

S and an interpretation I , how to define the function T (S,G) computed by the scheme I is the least fixpoint of equation TI(S,G)

with respect to the interpretation I ?

(I)(in the algebra of partial functions N 2

defined than

T (S,G)

g).

I

-f

N ordered by: f


iff

f

is less

can be computed by successive approximations as follows : 0

is everywhere undefined which is abbreviated by :

the first approximation tI

to(n,k) = 1. Then, for q in N , the q+l-th approximation is defined by substiI tuting t; for G in the right hand side of equation ( I ) , whence :

and more generally :t

The sequence It:, q

=

!

nk

if

0< k

< Zq-'-l

1 otherwise

q € I N ] is increasing with respect to the ordering "to be less

defined than" ; the function {t:,

(n,k)

(n,k)

I-+

nk

is the least upper bound of the sequence

€IN}. This example suggests that interpretations be ordered algebras where

increasing sequences have least upper bounds, i.e. algebras which are complete in 0

some sense.

2.2 Complete F-algebras

Let us sketch more formally the algebraic semantics approach here. For more details see /GU/. Let

F

=

{f,g,h, . . . I

(resp.

= {G,H,K,

... I )

be a ranked alphabet of

base function symbols (resp. of function variables) ; the rank of a symbol denoted by p ( s )

and

F (resp. 0 )

(resp. function variables) of rank presentingthe"undefined";

algebra. Let

V

=

0

is

n. Let R be a special rank 0 symbol in F, re-

F i s asignature, or similarity type in terms of universal

{u,v,w,x,y,z, ... ) be a set of variable symbols (intended t o re-

present parameters or positions) of rank in

s

denotes the set of base function symbols

which may have arbitrary ranks.

0

, as opposed t o the function variables

48 2

I. Guessarian, M. Niuat An ordered F-algebra /ADJ/, or F-magma /N/, or algebraic structure

/AN, C , G R / , M

is an ordered set

a total and monotone mapping element of

DM. Ordered

(DM,

fM : D i

+

5)

together with for each

DM, and such that

RM

f

in

F

n’

is the least

F-algebras are actually a special kind o f algebraic

systems o f similarity type, or signature, F U

{
The class of a l l ordered F-algebras forms a quasi-variety (i.e. a class axiomatisable by quasi-atomic formulae) in the sense of /MA/. See also /GR/ p. 339, and /AN/(section 4 ) , /GU/, for quasi equational logic. Since quasi-varieties are epireflective in their similarity class, all the nice properties of similarity classes of algebraic systems are inherited by ordered F-algebras too. Recall now that a subset ted iff any pair

E

d,d’ of elements of

An F-algebra

M

of an ordered algebra is said to be direcE

is said to be

has an upper bound in

directed subsets (resp. countable chains) of f’s

are continuous, i.e. preserve l.u.b!s

M chains).

DM

have a 1u.b. in DM

and the

of directed sets (resp. countable

I n the sequel we will consider mainly

exactly similar for

E.

A-complete (resp. w-complete) iff all

A-complete magmas (the theory is

w-complete ones) and we will call them complete to shorten

notations. Whenever we consider a different notion of completeness, this will be mentioned explicitly. Notice that the notions of completeness here considered are slightly weaker than the usual ones in lattice theory : 1 ) n o t all subsets of

M

are required to have l.u.b!s.

2) no greatest lower bounds are required, hence the F-algebras we consider are not even lattices. Example 1 (continued) : here

F = In, a, b, d, r, p, m, g], where

p(Q) = p(a) = p(b) = 0 , p(d) = o(r) = 1 , p(p) = p(m) = 2 , p ( g ) = 4 . The F-algebra I has domain DI =IN u {l],ordered by the discrete (or flat) ordering which can b e represented by : 0

1

2

48 3

Ordered sets in algebraic semantics and the algebra operations o n

I

RI = I ,

aI

I

m (n,n') = I

=

1

if

b

and

n

if

b GIN

and

n 21

1

if

b = l

or

n=l

n

x

=

0

= I,

n,n' E M n = l or n ' = l

if

n'

I

bI

0,

n

if

z

if l # x = y # l

t

if 1 f x # y # I if

DI

are defined by :

x=lor

y=l

together with these operations is obviously a continuous F-algebra. Define the category of

0

(A-)complete F-algebras, or F-magmas by :

objects are (A-)complete F-algebras ; morphisms are the continuous homomorphisms, that is, in addition to being an F U { S } homomorphism they have to preserve lubs of directed sets too. We shall define Mm(F,V) freely generated by

V

in the category of

definition comes later. The category of

as being the algebraic system

A-complete F-algebras. A more precise

w-complete F-algebras can be defined simi-

larly.

It might be of interest to note that i t was proved in category of

/LP, P/ that the

w-complete F-algebras is reducible to a variety of partial algebras

in the sense of

/AN/.

This shows the strong relationship of algebraic semantics

with well investigated concepts of universal algebra. A

complete)

FUV-algebra

I is said to be free over generators V iff

for any (complete) FUV-algebra M, every set map from V extension into a morphism

v

into M

has a unique

: I +M.

I is also called the free (resp. free comp1ete)F-algebra generated by

V.

I. Guessarian, M. Nivol

48 4

The free and free complete F-algebras exist and can be constructed as follows : the free F-algebra

H generated by V

has as domain the set of finite,

well-formed (with respect to ranks) trees (i.e. terms) on the alphabet V +H

the set map that : (a) s2 generated by

V

An ideal of

and

<

is the identity. H is ordered by the least ordering

such

is the least element, (b) the algebra operations are monotone. The

free F-algebra generated by

E

F UV ;

V

is denoted by

M(F,V).

The free complete

F-algebra

is the ideal completion of M(F,V) cf. /B/ and is denoted by Mw(F,V).

M(F,V)

E

is a directed subset

of

M(F,V)

such that for any

d

in

is in E. Note that again the notion of ideal here i s slightly

d' < d , d'

different from the usual lattice theory notion. Recall tion of an algebra

M

/GU/

that the ideal comple-

is obtained by endowing the set of ideals of

by set-theoretical inclusion, with the obvious complete intuitively, Mm(F,V)

M , ordered

F-algebra structure. More

is described as the set of all finite and infinite trees

(terms) on F U V ; the ordering

<

extends the ordering on

M(F,V)

and can be

T <TI for any trees (terms) T,T' iff T' can be deduced from T by replacing s2b by arbitrary trees. This ordering corresponds to inclusion, namely T
ideal associated with

T'

.

2.3. Recursive program schemes A recursive program scheme, in short RPS, on

S

S : Gi(v

],...,

i

vn ) = ti

= I,

i

, G i E 4n ,

= ],...,XI

u

(2)

...,vn. 1 )

U 4 , {vl,

t i E M(F

i tree in M(F

(S,t), where

...,n

i where for

i s a pair

F

n equations :

is a system of

and

is a

t

4 , V).

It is associated with a schematic tree rewriting system (or context free

/BO, EN, GU, R / ) , defined by

tree grammar

i

= I , . ..,n

:

Gi(v,

,..., vn. )

, and which is also denoted by S. Let L(S,t)

=

+

ti + s2

, for

{t'/t'EM(F,V),

t

5

t')

S

be the tree language generated by It can be shown 1)

L(S,t)

2)

lub(L(S,t))

/GU/

S

with axiom

that

is a directed subset of

<)

,...,vP) = t E M(F, Gi(v,, ...,vn ) = t. + s2 Go(vI

T(S,t)

=

M(F,V)

is the least solution (with respect to inclusion, o r

equivalently the ordering

Let

t.

i lub(L(S,t)).

on

F(F,V)

,...,vP 1 ) for i = I , ...,n Iv,

of the system :

Ordered sets in algebraic semantics

485

Example 1 (end) : T(S,G(n,K))

here is the infinite tree defined as the 1.u.b.

of the chain of finite trees :

to

< t l < .. .< t < t < ... 9 q+l

= R

t

t1 =

a

Ag\ /”\ k

b

R

n/p\r

I

k For any

each

t

4

q

in N t h e tree t

q

is defined inductively by :

represents the formal computation of S

times the recursive definition of

tp

=

obtained by “unwinding”

S. For instance :

q

I. Guessarian, M. Nivat

48 6

The least upper bound

T ( S , G(n,k))

of the

t's can be pictured as the infinite q

tree

where

. ..

represents the

infinite branch of the tree. ?

m. I

,

When interpreted by I, t

k

,

I

q

naturally yields the function

.

t; previously defined; the

1.u.b. T(S,G(n,k))

of the

t's is then interpreted as 4

the 1.u.b. TI(S,G(n,k))

-

2 . 4 Interpretations

pretation is an interpretation of

; equivalently it is a pair

in P(F,V)

together with a valuation

induces, on any complete F-algebra

v m : Mm(F,V) + I

under the unique morphism T

of the tq' I S .

(1,~)

v : V +DI.

Mm(F,V), we can deduce the following :

TI : Iv +. I , such that for each valuation v : V Also, since each

nk

is a complete F-algebra ; a valuated inter-

F

F U V

consisting of an interpretation I From the freeness of

T

I+

Herbrand interpretations

An interpretation I of

any tree

: (n,k)

in P(F,V)

+

I , TI(\)) v

extending

I

,a

function

is the image of

is the least upper bound of the

0

in M(F,V!

which are below it, and these form a directed set, then it is clear that least upper bound of the

BI's.

We can thus infer : Theorem 1 :

(i) (ii)

for any

TI
T, T'

in Mm(F,V)

for all I I , TI

=

lub

iff

{e, I

T 0

< T'

< T 1

;

Let now (S,t) and (S',t') be program schemes, applying theorem 1 with T

=

T(S,t)

and

TI

=

T(S',t')

leads to:

T

. TI

is the

187

Ordered sets in algebraic semantics Corollary : Let

(i)

<.

for all

be two RPS6 :

(Sit')

T(S,t)I G I T(S',t')I T(S,t)

(ii)

(S,t) and

for all

I

iff

T(S',t')

I, T(S,t)I

lub

=

{@,/El


(ii) expresses the fact that the function computed by

(S,t)

w.r.t. I

i s defined as a lub of finite computations, by successive approximations ; and (i)

says that the infinite tree

T(S,t)

characterizes the behaviour of

all interpretations, which was goal # 1 Since the free complete F-algebra #'2

(S,t) w.r.t.

stated at the beginning of this section.

Mm(F,V)

has been explicitly constructed, goal

also has been achieved. As for goal #3,

let u s lay down the following refi-

nement of the notion of a free algebra :

Definition I : An interpretation H iff : for any

T,T'

TI Q I Ti

is said to be a Herbrand interpretation

in M~(F,v) : for all

I iff TH G ~ T ~ .

Then Theorem 1 expresses the fact that M"(F,V), identity valuation u(v)

=

v,

for all v

in V

,

together with the

is a Herbrand interpretation.

This fact can be applied to obtain proofs of programs properties. The Herbrand interpretation gives necessary and sufficient conditions for program equivalences to hold, thus leading to valid and complete proof systems (see a 2 . 5 ) . On the other hand, the free interpretation leads only to sufficient conditions for

such equivalences to hold. Here, Herbrand and free interpretations coincide, but this is not always true. I n section 4 . 2 , we shall see an example of a Herbrand interpretation H

which leads to a complete equational proof system for the corresponding

class of interpretations, even though that class is not a variety. Of course H does not belong to the class i t characterizes. From these remarks the reader might infer that Herbrand interpretations are the only ones of interest and might wonder why at all introduce free interpretations ; the reason is that free interpretations are usually much easier to construct, and might still suffice for many purposes. Note finally that Herbrand interpretations, like free interpretations, are not necessarily unique (cf. Example 3 below).

Notations due to typographic restrictions, the sign F the sign t-

represents semantical validity represents logical validity.

I, Guessarian, M . Nivat

488

2.5. Application to proofs of program equivalences

By characterizing the semantic behaviour of a program scheme by a syntactic object, its associated infinite tree, Theorem 1 provides us with the corner-stone of algebraic semantics. Let us illustrate this by two very simple, though interesting, applications.

7

Let

1

denote the class of all interpretations ; then, even though

is clearly not first-order, theorem 1 can be viewed as a Rirkhoff like completeness formalism of logics, TI 5 T i

I

in all models Ax

I , i.e.

for all

1F

is denoted by

Note first that, in the

T
T S T '

is semantically valid

. We

then can introduce a set T
of axioms and deduction rules for inequational logic such that

Ax b- T

< T' .

I F T

Hence theorem I can be restated as

Q

iff

.

iff Ax k T Q T'

T'

of axioms is defined by the axiom scheme :

The set Ax (3)

,

1

in

.

I

theorem for deriving all valid (in)quations in

t-nst

t

for any

in M"(F,v)

Consider then the following set of deduction rules : (4)

t

(5)

< t' ti < t;

(6)

for any

t < t

and

t

i

for

for any (7)

t'

Vi E N

< t" t = I

in M(F,V)

t

t S t"

,...,r

ti, ti

in M(F,V),

3j E W

/ t i S t' j

(reflexivity)

for any

t,t',t"

in M(F,V)

i =I

1-

,..., r,

lub(ti)

and

in F (monotonicity)

f

< lub(t!),J

(transitivity)

,..., ti)

f(tl ,..., tr)
/-

for any

ti,t;

in M(F,V)

(algebraicity and continuity). For

t

and

deducible from axiom iff

I-

t

t'

finite trees in M(F,V),

!-

let

using deduction rules ( 4 ) - ( 6 )

Ax

t

< t'

iff

; then clearly :

t Q t'

is

t
< t'. For T

and

T'

(possibly infinite) trees in Mm(F,V)

, it can be shown

that it is necessary and sufficient to add the induction rule (7) ; let 1 " T S T ' . Theorem 1 can thus be translated into two completeness theorems for finite and infinite trees :

, I F t < t' for T,T' in M~(F,v), T F T S T ' for

t,t'

in M(F,V)

< t'

iff

AX

C

t

iff

AX

I-"

T <TI

.

This makes clear how algebraic semantics can be applied to yield results in more model theoretic or logic minded approaches as in

/BL ,

BT, G /

.

This con-

nection between algebraic semantics and logics will be further investigated in the subsequent sections. Remark finally that, since

1F T

=

T'

iff

1 F T G T'

and

1 F T'


the above also provides us with a complete proof system for equational logic ; however this proof system does not immediately translate into a proof system using the deduction rules of equational logic involved in

/GU2/

.

/B,BT/

. See a

sample of the difficulties

48 9

Ordered sets in algebraic semantics

2.6 Some different standpoints Notice that, since a program scheme (S,t) is characterized by the associated infinite tree

T(S,t),

we may w.1.g. study infinite trees instead of

program schemes, although not every infinite tree is associated with a program scheme. However, the facts that : 1)

there are much more infinite trees than trees associated with program schemes

2 ) i t might then be tuo exacting t o require that all directed sets,

even those which are not associated with any program scheme, have lubs (in order to ensure completeness) led some authors t o introduce weaker notions o f interpretations. To this end, they define algebras which, though they are incomplete in the above sense, contain enough lubs to express the functions computed by the programs one is interested in. Let us briefly outline some of these approaches.

We need first recall some terminology from denotational semantics. Note first that if A

is any

F-algebra (not necessarily ordered or complete), we can

define as above a derived operation

tA(xl, ...,xn) = Wx

(t) for any t in ~ . * * ~ n Now, let A be an F-algebra

in M"(F,V)).

M(F,V)

(but not necessarily any

and

be a recursive program scheme defined by a system of equations ( 2 ) , let n i' -+ D ~ ) X... X(D~' -+D*) be the set 0 : n-tuples of mappings D~ - + D ~, for

u

=

S

nl (D*

t

-+ g

. .,n . Any

in D defines an F U @-algebra A(:) by : = (gl,.. .,gn) in F and G =g.forG. in 0 Hence, we can associate to 1 1 iA(3 each recursive program scheme S a mapping SA : D -+D defined by i = I,.

fA);(

=

fA

for

Now, an proper) iteration

.

f

F-algebra S,

An ordered

SA

A

is said to be iterative iff, for any ideal (or

has a unique fixpoint in

F-algebra

A

-

lub IS:

(aA,.. .,QA) /

GI, NE, T/.

is said to be

- regular iff, for any iteration is defined by

D /E,

S,

SA

has a least fixpoint which

n E N } / G , GPP, PP, T / .

]-rational iff, for any recursive program scheme

fixpoint which is defined by

lub { S i ( Q A , .

. . ,QA)

/ n €IN) /G/

S,

SA

has a least

.

By allowing for higher type schemes, Gallier also defines n-rational algebras which we won't consider here in order t o keep the notations simple. A subset of

DA

of the form

{Si ( G A ,

...,fiA)

/ n €IN}

for some itera-

tion (resp. RPS) S , is called a regular (resp. an algebraic) subset of

A

.

Regular

subsets are also called iterations. The following then holds

:/BG2, G, NE/

Theorem 2 : Any complete F-algebra (and any interpretation) is regular and rational. M"(F,V)

is iterative.

I. Guessarian, M. Nivat

490

Still different approaches consist in embedding metric spaces into cpo's/WE/, or endowing the set of infinite trees with a metric /ARNI, COM/ : one then studies computations of program schemes through their approximations by Cauchy sequences in metric spaces. 3 . CLASSES OF INTERPRETATIONS

We showed in the previous section that the characterization of a program in a

scheme (S,t) by an infinite tree, which is the function computed by (S,t)

Herbrand interpretation, can be rewarding. However, this approach is usually far too general : in practice, one never considers all interpretations I , but rather

I , where, e.g. some operation is associative and commutative, or

subclasses of

"if... then . . .*.

some base function symbol is always interpreted as an

.

.'I

Hence, if one wants to come any closer to real programs, one has to take into account some constraints on interpretations. For instance, equivalence with respect to all interpretations is far too exacting as is illustrated by the following : Example 2 :

Let

S : G(v)

(S,G(v))

be program schemes defined by

S'

f(v, f(a,v))

=

: G'(v)

S'

and

S

= f(f(v,a),v)

(S' , G'(v))

and

are not equivalent with respect to all interpretations

since T(S,G(V))

=

f (v, f (a,v))

T(S' ,C' (v))

=

f (f (v,a) ,v)

are incomparable

However, let us restrict our attention to interpretations where

in M"(F,V).

is associative, and let then

c

and

= {I/ fI(x,

fI(y,z))

For any

I

c ,

(S,

denoted by

(S',

G(v))

Kc

G(v))

and

, or

G'(v))

=

in DI}

TI(S',

G'(v))

(S',

T(S,G(v))

fI

I defined by

for any x.y,z

then : TI(S, G(v))

expressed by saying that (S,

be the subclass of

fI(fI(x,y),z)

=

in

c

G'(v)) are

c'

. , which

will be

C-equivalent, and

T'(S',

G'(v))

.

Formally, let us define : Definition 2 : A

class c

of interpretations is a subclass of

class of interpretations and

T,T'

in Mm(F,V)

iff for any

I

C

TI

%

T GCT'

iff for any

I in C

TI

=

C C < C' C << C ' C C'

and

C

T s+ T' For

M

C'

C'

in

, define

I

:

Ti

Ti.

two classes of interpretations, define :

zc

iff

'c,

5

iff

:ct

_C

-c

iff

'c,

=


iff

E~~

=

=

-C

. For c

a

Ordered sets in algebraic semantics Note that, clearly,

c < c' * c << c'

and

49 1

c c' * c 2 c' , counter %

examples for reverse inclusions will be shown later (see theorem 8 below and / G M / ) . Note also that

c

C C'*

C

< c' .

We will now try to generalize the approach of section 2 to classes of

C of interpretations, we first have to

interpretations : namely, given a class

characterize the behaviour o f a program scheme w.r.t. all interpretations in C

by

the function i t computes in Some Herbrand interpretation. We will show now that this is much less straightforward than in the previous completely free case : we have to accept a trade-off between a better modelling o f reality versus an increased complexity of proofs and results. We need one more definition :

C be a class of interpretations and 1 an interpretation ; I is said to be C-Herbrand iff C 2r {I} . A valuated interpretation (1,~) is Definition 3 : Let said to be

v' : V

+

v(v(v)) with

C-free (over generators V)

iff for any

DI', there exists a unique morphism 'P : I

=

w')

Remark 1

v'(v)

for any

--*

I' in C and valuation I' such that

in V , (i.e. the restriction of

v

~p

to

w(V)

coincides

. a) We are implicity considering classes of non-valuated interpretations (or functional interpretations). Valuated interpretations can be treated similarly /GU/. b) Note that our definitions of

C-free and

C-Herbrand are slightly

different from the classical notions of universal or free object in universal algebra or category theory /AN, C, GR, MA/ : we do not require that the class

C

. As

belongs to

C-free or

C-Herbrand interpretation belong to the

a result, it may indeed be the case that none of them

C

; for instance, Mm(F,V)

C(§

3 . 2 and 4 . 2 ) .

C-free interpretation belongs to C

C-free for any

C

.

We

C-Herbrand interpretations I

will see later various examples of which are not in

is

Note however that, whenever a

,

then (i) it is also

C-Herbrand,

(ii) it is the unique C-free interpretation belonging to C (up to isomorphism). c) We can also ensure uniqueness of the free interpretation by postulating the following stronger definition /AN/

:

Definition 3.1 Let c be the preordering defined on valuated interpretations by : ( 1 , ~ )_C (I',v') iff there exists a unique morphism ~p : DI +DI, such that : for any v

in V

iff (i) it is i.e. (Io,wo)

q(w(v))

=

w'(v)

C-free, (ii) for any is the largest

. Then

(Io,wo)

is said to be strongly

C-free

C-free interpretation (1,~) , ( 1 , ~ )_C (Io,wo)

C-free interpretation w.r.t. the preordering

5 .

d) Note finally that the notion of Herbrand interpretation can be extended for arbitrary atomic formulae of the form P(t arbitrary finite trees and

P

] , . . . , tn)

is a predicate symbol :

, where the t i ' s are

I. Guessarian, M. Nivat

492

Definition 3.2 : An interpretation I is said to be strongly C-Herbrand iff for c F@ 0 any atomic formulae 0 : I F 0 +f

This concept, even though it was never given a name, has been investigated in universal algebra (see e.g. / A N / ) .

3.1 Constructing C-free and

C-Herbrand interpretations.

For completeness sake, let u s state the following /GU page 7 9 / : Proposition I : For any class C a

of interpretations, there exists a

C-free and

C-Herbrand interpretation. and

C

The five element set

F

Example : Let F a=b or

c=d

.

=

{fi,a,b,c,d}

least element n , is both

be the class of algebras such that either

,

together with the discrete ordering with

C-Herbrand ; in general though, Herbrand

C-free and

and free interpretations need not coincide. See below. However, the proof of proposition 1 is highly non-constructive : it consists in taking some suitable subclass of the infinite product of all interpre-

. Hence

tations in C

this result is of no help and we have to find alternate characC-Herbrand interpretation. Let u s give first some

terizations of the C-free and terminology about preorderings.

Definition 4 : A preordering on an ordered F-algebra M sitive relation

T

containing

structure of M , namely : for any d.

TI

1

d!

1

...,dr)

imply

fM(dl,

imply

lub{e/e in E }

TI

f

d'

T

. When

in DM, for

in Fr, di, di

,...,di) .

f (d! M i in Dw, and any

for any directed set E

is a reflexive and tran-

s and which is compatible with the M = V(F,V)

i = I,

...,r

,

is said to be continuous iff

T

in DM , e

d'

F-algebra

T

d'

for all

e

in E ,

with the syntactic ordering


is said to be

-

...,T

substitution-closed if for any

t,t'

in M(F,{vI,

,...,Tn/vn)

...,vn})

and any

t'(T1/vl,...,T /v ) n n Equivalently, one may require that for any endomorphism h : Mm(F,V), t T t'

TI,

implies h(t)

t

< T,

t

,

in V(F,V)

T

T

t'

implies

t(Tl/vl

T

.

.

h(t')

algebraic

in M(F,V),

t

iff for any

T,T'

there exists some

in M"(F,V), t' < T I ,

T n T'

implies : for any

in M(F,V),

t'

such that

tnt'. For a preordering

T

on

Mm(F,V)

let

CT

=

11

/

T

C<{Il}

.

We can

now state the : Proposition 2 : /GU/ : C

-+%

and

'II

+Cs

define a Galois correspondence between

classes of interpretations and continuous, substitution-closed preorderings on p(F,V). One might then expect to use the above Galois correspondence in order to get somehow more tractable characterizations of the

c

-free and Herbrand interpre-

tations. This goal can be only partially fulfilled, as will be shown by the sequel.

49 3

Ordered sets in algebraic semantics For any ordered

M , and preordering

F-algebra

- Mm denote the ideal completion of

n

M , let :

on

M /B,CR,GU page 3 3 /

- M/n

denote the F-algebra obtained by factoring M through the -I equivalence IT = IT fl IT associated with n , and ordering the factor algebra by a / n . The equivalence class of an element t of M modulo

n

will be denoted by

Let

'II

be a continuous and substitution closed preordering on Mm(F,V).

c

We now try to find choice would be

.

[ tIn

I

I

c

-Herbrand interpretations. The most natural

.

fl(F,V)/n

=

However,

-free and/or

c

is usually neither

mediate reason for that i s that

C -Herbrand. The most im-

-free nor

I , being usually not complete, is not large

enough and does not contain enough lub;. However, there are more subtle causes for which

I

c

cannot be

I

-free or Herbrand : even when

is complete, lub; might

be there just by chance and might not be the right lub;. Hence : the operations on

I)

I may be non continuous

I are continuous, I may n o t be

2) even when the operations on

c

p :

-free because the unique mapping

may be not continuous, hence p

I +I'

into an

I'

cn

in

will not be a morphism.

This will be made clearer by the following example : Example 3 : We will show here continuous, substitution-closed and algebraic preorderings

such that

IT

Let Define b n h(b),

n

F =

Mm(F,V)/n

is neither

{n, a, b, c, h} r(n)

bn

c,

c IT h(c),

/ n E N}

f o r any n in N

h(c) =

hI

TI

c. Then

and

I

=

=

r(b) :

r(c)

=

k'n

Mm(F)/n

lub {hn(b)I

c

-Herbrand nor

as being the preordering generated by

lub {hn(a)I

-free.

= 0

r(h)

= 1

.

N , hn(a) nb, a n h(a),

€

is complete:

/ n E N}

=

cI , and

hr(c),c

.

However hI is not continuous since

> l~b{h~+'(a)~] f

c

= r(a)

=

[b],

=

bI

.

hI(bI)

=

[h(b)],

hI(lub{hn(a) n

=

Hence I is not even an interpretation.

n C

Figure 1 might help the reader's intuition. h2(d)

/ h(b)

/

I1 )

494

I. Guessarian, M. Niuat c

/

m m

1

Figurc 1 Now, even when the operations are continuous, I may still be not

c

-free. Let

TI' be the (continuous, algebraic, substitution-closed) preordering generated by II

.

together with the relation h(b) rr'b

I'

Then

=

the hn(a)I,'s

by mere chance, and this results in

v

instance, the unique

c

I' is

,-Herbrand and

:

I'

+

I"

I"

=

(M(F)/n)=

is both

Mm(F)/n'

is clearly a

bI, happens to be the lub of

complete F-magma, with continuous operations. But

I'

being not

CTI,-free.For

is clearly not continuous. However,

C,,-Herbrand and

c

,-free : the freeness

results from proposition 3 below, the rest is left to the reader. This example shows that neither M"(F,V)/n

c -Herbrand

C -free or

nor

(Mm(F,V)/~)m

in general, even for a very smooth choice of

can be

. We

IT

ne-

vertheless can state the : Proposition 3 : Let

IT

be a preordering on

Mm(F,V)

; then

(M(F,V)/IT)~ i s

C -free. 0 Example 4 : Note that, except when is usually not Let

CTI-Herbrand.Let

T . = lub {h:

clearly M(F) / TI

(il)), = M(F)

i =

for and

TI

is algebraic (cf. Theorem 3) (M(F,V)/TI)~

I n , h 1 ' h 21 with r(h,) = r(h2) = 1 . 1,2 , and IT be defined by : T I TI T2 . Then, F

=

is n o t

(M(F)/T)~ = F(F)

c

0

-Herbrand.

The previous example amply illustrate that the standard completion by ideals method cannot give us a

C-Herbrand interpretation. However, for any

continuous and substitution-closed preorder and

c

IT

, one can construct a c -free

-Herbrand interpretation by a more refined completion method : one has to

perform a "continuous" completion preserving lub; of algebraic subsets o f M"(F,V)/r

; by transfinite induction, one than gets the required

interpretation /CR,

C -Herbrand

G U 3 , MI/ thus yielding the :

Proposition 4 : For any continuous and substitution-closed preordering Mm(F,V)

we can construct a

c

-free and

c

IT

on

-Herbrand interpretation.

C -free and CT-Herbrand interpretation still exists by proposition 1, but cannotbe constructed explicitly.

i I n case TI does not satisfy these hypotheses, the

495

Ordered sets in algebraic semantics __ Proof : Let

Let

A

be a continuous and substitution-closed preordering on

II

, ordered by

denote the algebra Mm(F,V)/n

D

it exists, belongs to

.

For any subset D

namely the least closed ideal containing D

of

.

.

T/*

of

to be closed iff for any directed subsets D '

D

An ideal of

the 1.u.b. of

-

M

let

c(E)

The set A

.

Mm(F,V)

is said

A

D', whenever

denote its closure,

of closed ideals of

A

,

ordered by set inclusion, is a complete lattice. It has a complete F-algebra structure defined by :

f(D

,,...,Dn)

=

c({f(t,

,...,tn)

1

=

c({t})

=


{t'/t'

and

t' E A }

=

/ t E

{i(t)

i

=

I ,..., nj)

:

is a continuous and injective morphism.

-

Let u s define the continuous completion Ac

.A

ti E

is a complete F-algebra. Now, i : A + ' A

Henceforth, A i(t)

- / definedDi byfor

of

inductively by :

A

A) C A

a :

and for any ordinal

"

A a + I = Aa

{lubi E / E

is an ideal of

Aa} C A

I

is a limit ordinal. Since all A s ned in A , there exists an ordinal y

such that Ay

are contai-

= :Aa

Av ordered by inclusion and equipped with the restrictions of the - 1

f , for

operations

checks that A

c

-Herbrand. A

Mm(F,V).

Y Y

f

in F , is a complete subalgebra of

A

.

By induction, one

has the required universal property in order to be is thus the continuous completion of

c

-free and

and is also denoted by

A

For more details see / C R , GU3, MI/. This construction by transfinite induction is however only very slightly

more effective than the bare existential result of proposition 1 . I n particular, this construction can lead neither to a nice characterization of the function computed by an RPS w.r.t. a class C , nor to the faintest hope of getting a complete proof system for deducing valid inequations zation of

%.

t

5

t' , nor even to some characteri-

Hence, in order to get more manageable results, we will have to

consider somehow more specific classes of interpretations. Which, as we will see, can nevertheless cover most situations of interest in computer science. This is what we shall do extensively in the next sections. However, in order to get the flavour of oncoming results, let u s define here some classes of interpretations and state some of their properties. 3 . 2 . Algebraic and topological classes of interpretations A

class

C

is said to be algebraic iff

5

-

Applications

is an algebraic preordering.

The intuition behind the notion of algebraic class is the following : the behaviour of any infinite tree with respect to an algebraic class

c

can be characterized by

I. Guessarian, M . Nivat

49 6

c

the behaviour of its finite subtrees w.r.t. ; equivalently, all the computations of a program scheme are completely described by its finite computations : this makes sense since finite computations are the only ones which can be at hand. For any

Cn

=

F

dp

5

...

of elements of

...

of elements of

uI

c, s

0, the class of discrete interpretations. Clearly :

=

s

c2

Moreover, one has

c3

5

...

$ C m $ C w $ '

/CG,GU/

Proposition 5 : (i) (ii)

F

Cm

Q

Cw

%

I.

%

D < C3 < C4 < #

#

... < Cn < #

#

Cn+I <

... <

+

be defined inductively by

. Then

tn+l

%n

for all

and not

t

Corollary I :

, where n €IN

tn+, = f(v,tn)

< t,+,,

n tn

(tn+l

The classes

F

#

the proof of (ii) stems easily from the following : let

in F

is finite)

C, has a single element, the trivial interpretation reduced

c,

QE, and that

Proof :

and

f

to = 0 and

Corollary 2 :

F, Cm

C

Cw

F, Cm

or

and

.

is a rank 2 symbol

but a l s o :

.

Gcn+Itn)

Cn, for

For the proof of (i) see /GU page

in E4 , form a strict hierarchy for the

n

Col are algebraic and M"(F,V)

F

is

C-Herbrand for

for instance is a class of interpretations such that

F , as soon as F contains at least

no F-Herbrand interpretation can belong to

one base function symbol of rank greater than 1 . For, M (F,V) being the chain

c : Q

$

h(Q)

$

h2(Q)

...

$

F-Herbrand interpretation can be finite.

I is said to be algebraic iff its domain

Recall that an interpretation

is an algebraic complete partial order B

=

. Clearly, if

itI / t E M(F)}

F-Herbrand,

is infinite. c must thus be infinite in

F-Herbrand interpretation ; hence no

set

loo/.

0

This shows that

DI

tn

<.

ordering

any

has at

DI

i s finitel

Note that

=

5 .. . 5

d2

distinct elements} {I / any chain d l S 1 ... < d G I I P

= {I / D I

to

5

dl

n

most

cw =

in Dl, define :

n

11 / any chain

/B,GU,S/

with compact elements the

I is algebraic, the class

is alge-

111

braic, but the converse is false. For any preordering

IT''

t

IT

; and for any preordering

M(F,V) on

< T,

Mm(F,V)

by : for T,T'

there exists a

t'

on P'

Mm(F,V), on

in Mm(F,V),

in M(F,V),

t'

IT^

define its restriction

M(F,V)

T

< T' ,

to

define its algebraic closure

IT'^

T'

iff for any

such that

t IT' t'

t

.

in M(F,V),

0

49 I

Ordered sets in algebraic semantics

We now can state Theorem 3 : A class

c

/GU/ :

is algebraic iff i t satisfies the following equivalent

conditions :

(i)

(<;)a

GC

=

(ii) (iii)

<:)-

/

(M(F,V)

C-Herbrand

is

There exists an algebraic

0

C-Herbrand interpretation.

The importance of algebraic classes stems from the fact that the behaviour of infinite trees w.r.t. finite subtrees. Hence, suppose as in section 2.2, tions

t

5

a

i s characterized by the behaviour of their

GC

c

is an algebraic class and we have obtained,

complete deduction system C

holding between finite trees

t'

t,t'

for proving all valid inequain M(F,V) ; then comes

"for free" a complete deduction system for proving all valid inequations T GC T' holding between infinite trees T,T'

too : i t suffices to add to

an induction rule (7) : / G /

the rules of V

in Mm(F,V)

3 j E N

i E N for

ti,t! J

ti

< t' k j

in M(F,V)

< lub(t!)

lub(ti)

J

.

This is exactly what we did in section 2 . 2 . We shall see later on further instances of a similar situation (Proposition 7 , Theorem 6 ) . Note finally that T

sc T'

T,T'

is generally undecidable for infinite, or even only algebraic, trees

in Mm(F,V)

; hence usual deduction systems, such as equational logic

/C0/

/B,BT/, first-order logic, etc ..., will be incomplete in general and one will have to allow for some infinitary inference rule such as (7)

/G,GM/, the other alter-

natives being to consider still more restricted classes of interpretations (eg. first order classes of interpretations, see section 5 ) , or a smaller set of inequations, or both. Algebraic classes of interpretations present the advantages of being :

- general enough to include all usual

classes of interpretations

- well-behaved enough so that any inequation between infinite trees will be provable by computation induction : this is the reason why, by adding the single infinitary induction rule ( 7 ) one gets a complete proof system for deducing all valid inequations. Example : consider the schemes defines as follows /AK/ : S,

s;

:

G(n)

:I

H(n,r)

=

G'(n)

g(c(n),a,m(G(p(n)), =

d(n)))

H(n,a) =

g (c(n) ,r,H(p(n) ,m(r,d(n)))).

These schemes, when suitably interpreted, take

DI =IN

u

{o}

, with the

discrete ordering, aI = I , pI(x) = x - 1 , nI = x, gI(x,y,z) = if x = 0 then y else z , and cI equal to the identity function, give a recursive and iterative

I. Guessarian, M. Nivat

498

version of the factorial : TI(SI,G(n)) TI(S;, G'(n))

computes

C

.

(...(((I.n).(n-l)).(n-2))...1)

can prove that T(SI ,G(n)) conditions :

Let

computes (...(((l.l).2).3)...n)

:c

T ( S ; , G'(n)),

Then using our method, we

assuming only the very weak following

(i)

m(f2,r)

(ii)

m(m(x,y) ,z)

(iii)

m(g(n,x,y) ,z) = g(n,m(x,z) ,m(y,z))

=

and

f2 =

m(m(x,z) .y)

be the class of interpretations defined by the conditions (i), (ii) and

(iii). The proof uses the following lemma : L

m : let

Hq denote the q-th approximation of

T(S;, H(n,r)),

times the recursive definition of

S; , then for any

unwinding q

obtained by

Ec m(Hq(n,x),y). q in IN ; Hq(n, m(x,y)) Proof : for q = 0 the equivalence reduces to

f2

=

which holds by (i).

m(f2,y)

Suppose the equivalence holds for q = 0 , . ..,k-I, then, by definition : k H (n,m(x,y)) = g(c(n), m(x,y), Hk-l (p(n), m(m(x,y), d(n)))) = g(c(n),

m(x,y),

= g(c(n),

m(x,y),

Y)))

Hk-'(p(n), m(m(x,d(n)), k- 1 m(H (p(n), m(x,d(n))),

by (ii)

Y)) by the induc-

tive hypothesis m(g(c(n), x, Hk-'(p(n), m(x,d(n)))), k by definition. = m(H (n,x), Y) An immediate induction on q then shows that : =

G"(n)

Zc

Gq(n) (where Gq(resp. GIq)

(resp. T(S;, G'(n)))

. This is clear

integers, then by the definition af G"(n)

Y)

by (iii)

denote the q-th approximation of

0

T(SI, G(n))

for q = 0 ; suppose it holds for the q-I first GIq ,

=

Hq(n,a) = g(c(n) ,a,Hq-](p(n) ,m(a,d(n))))

=

g(c(n) ,a,m(Hq-' (p(n) ,a) ,d(n)))

=

g(c(n) ,a,m(Gq-' (p(n)) ,d(n)))

=

Gq(n)

by the above lemma by the inductive hypothesis

by the definition of

Gq

Hence, by the induction rule (7) : T(Sl, G(n))

Ec

0

T(S;. G'(n)).

3 . 3 Related approaches

Note that most o f the problems in constructing a C-Herbrand and/or

c-

free interpretation stemmed from completeness and continuity. Hence considering classes of

X-interpretations, where

X

is intended to be replaced by iterative,

regular or I-rational, one would expect to cancel some of these problems, since completeness and continuity are replaced by weaker conditions that only those lubs

499

Ordered sets in algebraic semantics

one effectively wants to compute should exist and be preserved by operations. This i s indeed partly the case.

An

X-morphism of

ving lubs of of

X-algebras is obviously defined as a mapping preser-

X-sgbsets ( o r unique fixpoints of ideal iterations). For a class

X-algebras, one can define as previously

X-C-free and

X-C-Herbrand interpre-

tations or algebras i n the present formalism (in the case when

5

simply has to replace free case, when Theorem 4 :

C

=

I

by

Zc)

.

C

X

=

iterative, one

First note the following for the completely

:

/BG2, G , GPP, GU, N I /

Let Alg(F,V)

(resp. Reg(F,V))

be the subset

m

of algebraic (resp. regular) infinite trees in M (F,V), namely trees of the form T(S,t)

for some RPS(resp. iterative scheme) (S,t)

and Herbrand I-rational algebra and

Then Alg(F,V)

is the free

is the free and Herbrand iterative

0

(and regular) algebra. Reg(F,V)

Reg(F,V)

.

is also denoted by

RTF(V)

in the A D J

terminology.

Notice that we still are going to obtain a Galois connection between classes of

X =

X-algebras and suitably defined preorderings (resp. congruences if

iterative)

on the free

X-algebra. Note now that, in the case where

X

iterative,

=

; these

we are going to run into troubles similar to the ones we had in section 3.1

troubles will no more be due to completeness but to unicity of solutions, which is at least as bad as completeness /CO,GPP/. This is illustrated by the following : Example 4 : we show here that factoring an iterative algebra, even by a very smooth congruence, one usually does not get an iterative algebra. Let with f(a,a)

r(a) 1~

= 2.

r(f)

= 0

f(f(a,a),a)

associated with

T

Then

S : z = f(z,a) t , = lub { S ;

n tion, and t2

and

T

f(f(a,a),a)

m

F

=

In, a,

be the preordering generated on Mm(F) 'TI

is generated by

(5

I = M (F)/:

f(a,a) f(a,a)

and let 5

f]

,

by

: denote the congruence

f(f(a,a),a)).

is not iterative since the ideal iteration

has the two distinct solutions

(a,)] =

Let

= lub In,,

f(a,a),

; and

f(n,a),, tl

2

f(f(C2,a),a)I,...) tp

, which is the least soluB = { w , a l , with

since the F-algebra

f (w,w) = f ( w , a ) = f ( a , w ) = w and f ( a , a ) = a i s a regular (and B B B B =o+t = a . complete) F-algebra, which is compatible with : and such that t

w Q a , aB = a ,

IB

See /GPP/ for more details. I' = Reg(F)

*B

/ : is similarly regular but not iterative. 0

In order to avoid such troubles we shall in the sequel limit X meanings "regular" o r "I-rational". The definition of an

to the

X-continuous preordering

will then clearly paraphrase definition 4 and we shall omit it. Now there clearly is Galois connection between classes C

of

X-algebras and

X-C-free interpretation by factoring the free

X-continuous, substi-

. We moreover can obtain the X-algebra A by % , namely :

tution-closed preorderings on the free X-algebra

A

5 00

I. Guessarian, M . Nivat

Proposition 6 : The X-C-free algebra H

is defined by

H

= A/GC ;

H is an

X-algebra belonging to c , hence is also X-C-Herbrand for any class

c

of

X-algebras. The same result can be achieved by categorical methods /BGI/. Note also that this result also holds for a higher-type notion of that

n-rationality

/G/, and

0-rationality corncides with regularity. Note finally that there is a trade-

off between the simplified construction of the X-C-Herbrand algebra H fact that H

may contain some meaningless lubs, e.g. bI

normal forms are also troublesome in H (cf.

and the

in example 2 where

H

=

I ;

/CO,G/).

4 . EQUATIONAL AND RELATIONAL CLASSES OF INTERPRETATIONS 4 . 1 Relational classes

Definition 5 : A class of interpretations is said to be relational iff it is of the form CR = {I/R

E

for some binary relation R

5 M(F,V)

x

M(F,V).

R

If

is

an equivalence relation, CR. is said to be equational. Relational classes are called algebraic varietal in

/M2/, varieties in

/BL/, and semi-varieties in /G/, who considers classes of rational algebras, defined by some relation R

possibly involving infinite trees. Relational classes are

those classes definable by a set of inequations between finite terms ; they are the most tractable classes of interpretations : for a relational class

%

an easy characterization of

c

restricted to finite trees, show that

, we will

c

get

is alge-

braic, hence obtain a complete deduction system within inequational logic to prove

c

all valid inequations in

(cf. sections 2.2 and 3.2).

Let u s state first a Birkhoff theorem characterizing relational classes in terms of closure operations : / W / . Theorem 5 :

c

is relational iff it is closed under products, continuous subalge-

bras and factor algebras, and ideal completions. A slightly simpler version of this theorem is proved in /G/ for semi-varieties of rational algebras and in /BL/ for varieties of ordered algebras. For a relational class preordering containing R abbreviate

<

by

-5

zR

C,

on M(F,V). and

z

by

define


as the least substitution-closed

Then, clearly, C, =R

=

c< . Finally, let us R

.

CR We can now state the main theorem of this section /GU/ : Theorem 6 : Let

R be a binary relation on M(F,V)

then :

(i)

5

is

(ii)

ZRf

= cR

(iii)

cR

CR-free and and

-R

=

CR -Herbrand
R

is an algebraic class.

0

and

HR = (M(F.V)/
,

Ordered sets in algebraic semantics

501

The importance of this theorem can be illustrated by its consequences. We simply state two of them.

c

Corollary I : A class class

cR

, i.e.

is algebraic iff i t is equivalent to some relational

C

iff

%

cR

(cf. definition 2 ) . 0

Recall that, as noted in section 2 . 2 , - t <-R

t'

is equivalent t o

cR F

- t <

t'

amounts t o s;.ying

that

5

t

t

t'

5 t'

can be deduced from

(R U <)

using the deduction rules ( 4 ) - ( 6 ) of inequational logic, completed by induction rule (7) and rule (8) expressing substitution closure : (8)

I-

t

5

v

: M ~ ( F , v )- + M ~ ( F , v ) .

t'

v(t)

5 v(t')

Hence the statement

zR =

for any

in M(F,V)

and endomorphism

(ii) of the above theorem can be res-

in

<:

t,t'

tated as the following completeness theorem : Corollary 2 (completeness theorem) : Let

1-

t 2 t'

for any

for any

T

t,t'

in M ~ ( F , v ) Then, for any

in M ( F , V )

be the axiom system defined by : AxR such that (t,t') is in R , and I - R 5 T

. t,t'

in Mm(F,V)

,

cR F

t

2 t'

iff AxR

1:

t

2 t'

using the deduction rules ( 4 ) - ( 8 ) . 0 One can obtain similar results when considering varieties (or relational classes) of regular /T/, rational /G/ or recursive /BGI/ algebras. Note however that i n the above cases, even though proposition 6 leads t o a similar complete deduction system, the inference rule for infinite trees i s obviously more complicated than a mere computation induction rule and must explicitly involve infinite lubs

/G/.

In the case of varieties of iterative algebras though, the existence and construction of a free or Herbrand interpretation for a variety of iterative algebras is more problematic /CO,GPP,PP/, because unicity of solutions is not preserved through factoring. Theorem 6 has many other applications which we merely list here for lack of space :

-

proofs of program properties (with or without induction) and program equivalences /G,GU,BK/.

-

program simplifications and transformations w.r.t. classes of interpretations and correctness proofs of such transformations /CO,EG,GU,K/

4 . 2 . Application to "if

...then...else

...I'

We will however detail some more the study of classes of interpretations

where a given base function symbol is interpreted as a test i.e. an "if...then... else..." . This will give some clues about how relational and equational classes can help in stydying non relational ones. Moreover, the tests (or conditionals,

I. Guessarian, M . Nivat

502

case or select functions,...) no matter what name they are given, arewellknownto be fundamental in computer science. Our approach replies on fundaments laid down

/MAC/, and pursued, among others, by

by

/BT,SE/.

B

We will start by recalling a result from /BT/. Let where

r(tt) = r(ff)

and

= 0

r(g)

=

3

.

and

tt

= {Q,

tt, ff, g }

ff are intended to be interpre-

ted as the constants "true" and "false" in any (not necessarily ordered) B-algebra I ; we will thus say that

(9)

for any

x,y.z

gI

is a

in DI

test

iff i t satisfies :

gI(x,y,z)

i

=

y

if

x = tt

z

if

x

nI

K

Let

K ,

=

=

I

ff

I otherwise

is a B-algebra satisfying (9)). It is well-known that

{I/I

being not closed under products, is neither equational nor relational. However,

the following completeness theorem has been shown in /BT/ : Theorem 7 : For any

-

A

t,t'

,K F

in M ( F , V )

t

5

t'

iff A 1 E- t

f

t'

, where

is the following set of equational axioms, in which g(x,y,z)

been abbreviated into

has

[x,y,z] :

A

-

A

I E-

means : t

t E t'

5

t'

is deducible from A

rules of equational logic /B,BT/, or, equivalently, t

5

tible and substitution closed congruence containing A

, which

= -A

*

using the deduction

is in the least compawe also denote by

o Intuitevely, this means that H

where A

K

t'

=

{I/I

= M(B.V)/EA

B-algebra satisfying A )

is a

K

5A

K K-Herbrand algebra which does not belong to is not equational ; hence

and

e=

is both

. Note that

H

"K

and

A-Herbrand",

is not in K

since

A , and this is an example of a

K.

Note that theorem 7 was obtained assuming two restrictive hypotheses, namely that one deals with unordered algebras, and that the unique operation allowed in the signature is the test F-algebras whose signature F

g

. We

can now extend this result to complete

contains base function symbols F'

provided those new symbols in F'

other than g

represent strict operations : this is a not too

stringent restriction since, in practice (see for instance abstract data types), the only base function symbol which is assumed to be non strict, and needs to be such, is the test

g

Let now F

. =

B

,

U F', for some set

F'

of base function symbols, and

Ordered sets in algebraic semantics

let

C

503

be the following set of axioms : for any

I

C

f

in

F t n ,n

,..., x i-1' n.

>

, and

1

i

1

,...,n

)

~[U,f(Xl,...,xi_l,v,xi+l,..., Xn) , f (XI,.* . f(xl

Let

A'

= A U

A' =

{I/I

C

:

xi+ ,,...,x ) E n

,,...,Xi-], [U,V,W1,Xi+] , . . . . x

f(x

=

'Xi-]'w'xi+l I *

and define :

is an

F-algebra satisfying

{I/I F A ' }

A') =

A:

=

A' f l 7

=

{I/I

is a complete F-algebra satisfying

A

=

A' n D

=

{I/I

is a discrete interpretation of

d

o

*.

A'}

F

satisfying

A']

.

And define similarly KS = K

n

{I/I

Ks = K

n

A'

is an F-algebra where, for any

f

FA , n

in

>

1

' fI

is strict) (recall an operation

fI

K:

DI whenever one of its arguments is QI)

=

is strict iff i t yields the result KS n 1

Ki

and

Then, extending in an obvious way the relations

ZC

Kz nD

=

and

RS

.

of definition

2 to classes of algebras which are not necessarily complete, one obtains the following

/GM/ :

A'

= A,'

(ii)

K:

Q

(iii)

K: Z A : ,

Theorem 8 : (i)

%

A:

KS% K:

, A;

K:

; i

K:

A:,

Q

, K' d #< A 'd '

0

This theorem has several consequences. It first yields a nice characterization of the equivalences valences coincide with generated by neral,

E

,

EC

for C

{Ks,

in

Ki, Ki,

A;

A;]

: by (i) those equi-

, which is by Birkoff's theorem, the congruence

ZA

.

E

A '

Note that, here, = ,, also coincides with E A l , whereas in ge-I -A0 -
CR with finite trees only).

3 ER

Hence we can state : s s A ' , Ks , KO , Kd}, zC

C in {A,',

Corollary I : For

closed congruence generated by Corollary 2 : K:

is not

A'

.

D-equational

This stems from :

is the compatible substitution-

d

K:

the only p o s s i b l e congruence is

C

A:

EA,

/GU/.

but

l(K:

, which

A:)

: by the first equivalence,

is excluded by the second inequiva-

lence. U Similar results hold slightly modifying lity test ; let

B'

=

{Q, 9')

satisfying (10)) , where :

, r(g')

=

4 , and let

B

in order to obtain an equa-

K'

=

{I/I is a

B'-algebra

I. Guessarian, M . Nivat

504

(10)

for any

x,y,u,v

in DI

nI PphY,U,V)

=

v

RI

x

if

x = y t n I

if

~ ~ + x + y + a ~

Then there exists an axiom system

=

or

y

=

/BT/ such that for any

.

M(B~,v) : K ' F t : t' iff 1 -E t = t' Letting ; i l = u C , where C = C U {(a),(b)l, with, for f in FA , nzl , and i=l,

A

RI

if

g(u,v,f(xl ,...,xi-l'u'xi+l,"' .Xn),Y)

(b)

g(u,v,g(f(xl

in

A

...,n

(a)

t,t'

:

:

g(u,v,f(x I , . . . ,x i ~ l , ~ , x i + l , . . . , x n ~ , Y ~

,. * .

'xi-I'u'xi+l,

-

f

,Xn),Y,Z,ZI),W)

g(u,v,g(f (x,, . * * ,x,i-l 'v'xi+l,*

* *

f

,xn) ,y,z,z').w)

The proofs and results of the previous case easily go through (see /GM/ for more details). 4 . 3 . Related approaches

As in section 3.3, we can define X-relational classes. I n the case when

X

= regular or I-rational, we will similarly obtain X-C -free and Herbrand interR pretations by factoring the corresponding completely free X-algebra by <;? ; when

X

=

iterative though, example 4 shows that no good solution can be expected. An alternate approach is to use category theory in order to build, without

any reference to either an ordering or a metric, free algebras in which all recursive program schemes have solutions, and which consist exactly of sets of solutions to recursive program schemes /BGI/

: this can be done via pushouts and coequali-

zers. Then, the factoring goes through very smoothly and guarantees the existence of a

C-free and

C-Herbrand algebra, where all RPSs have solutions and consisting

exclusively of solutions to RPSs, for any class

c

of algebras presented by gene-

rators and relations. This approach can also lead to an algebraic semantics of non determinism BO, MI/

/BG2/ and hopefully concurrency different from the ones in

/ARN,

and somehow closer to the one in /W/.

5 . FIRST-ORDER CLASSES OF INTERPRETATIONS

Because of the conditions that any algebraic subset (or iteration) have a lub (or a unique fixpoint), none of the classes of interpretations (or X-algebras) introduced

so

far are first-order definable ; hence, first-order logic will

be incomplete w.r.t. those classes. We introduce in the present section the last well-behaved classes of interpretations, nemely first-order definable classes. Definition 6 : Let A

be a set of closed first-order formulae

/BS/ and

C a

5 05

Ordered sets in algebraic semantics class of interpretations ; definr

. Mod ( A ) the set of . C(A) = c fMod l (A) and belong to

A

A class

A

some set

C'

models of

A

thr class of interpretations which are models of

c .

is said to be a first-order class iff

C'

=

Mod (A)

for

0

of closed first-order formulae.

The main result of this section is the following characterization theo-

rem

/GU/

Theorem 9 : in N

c

a class

and some set

A

is first-order iff i t is of the form

First-order classes are algebraic.

Corollary 2 :

For any

Cn

and

:

in N

n

Cn(A)

and set

A

Cn(A)

for some

n

0

of closed first-order formulae.

Corollary I :

classes

Cn)

(cf. section 3.2 for the definition of

o f closed first-order formulae, the

are algebraic. 2

this result was obtained in /GUI, CG/ together f = D) on with a decision procedure for deciding zu (recall that for n = 2 Note that, for

n

=

c

finite trees. Corollary 3 :

Any finite intersection of first-order and relational classes is

first-order and algebraic. Corollary 4 :

The classes

ucond(of discrete interpretations where a given base

function symbol is interpreted as a test, see

/GU/), K:

, KAs

(of section 4 . 2 )

are first-order and algebraic. Together with the fact that algebraic classes are closed under finite unions and products, Corollary 3 enables one to show that the classes one is interested in are algebraic. Note that this corollary is hard to improve since we can construct an algebraic class of discrete interpretations such that

c

cR

fl

is not algebraic

c

and a relation R

/GU/.

Note finally, that, first-order classes being algebraic, the following completeness result is clear : Proposition 7 : Let

c

=

Mod (A)

be a first-order class of interpretations, (i)

first-order logic is a complete system for deducing all valid inequations with

t,t'

in M(F,V),

from the set of axioms A

.

t

zC t',

(ii) first-order logic together with the induction rule (6) is a complete system for deducing all valid inequations

t <

-C

t'

from the axioms A

.

0

Acknowledgments : It is a pleasure to thank H. Andreka, I. Nemeti and I. Sain for helpful comments on a first draft of this paper. Thanks a l s o to S. Bloom, E. Nelson and A. Na'it Abdallah for valuable comments. Last but not least, thanks to the anonymous referee whose insightful comnents greatly helped in making this paper more readable.

I. Guessarian, M . Nivat

506

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509

TWO OPEN PROBLEMS IN PRECEDENCE CONSTRAINED SCHEDULING J.K. Lenstra Centre for Mathematics and Computer Science P. 0. Box 4079, 1009 AB Amsterdam

A.H.G. R~MOOY Kan Erasmus University

P.O.Box 1738, 3000 DR Rotterdam

The computational complexity of a machine scheduling problem can be affected in various ways if a partial order is imposed on the set of jobs that has to be executed. Some typical complexity results for such problems are discussed in the light of two prominent open problems in this area: the minimization of total tardiness for unit-time jobs on a single machine subject to chainlike precedence constraints, and the minimization of maximum completion time for unit-time jobs on three identical parallel machines subject to arbitrary precedence constraints.

RESUME La thbrie de l'ordonnancement traite de l'affectation temporelle de ressources limitks h des activitds. Dans ce contexte, des ordres partiels apparaissent de fawn naturelle: un ordre partiel sur l'ensemble des activites impose des contraintes sur l'ordre dans lequel les activitks doivent Ctre exkutkes et il rkduit ainsi l'ensemble des affectations admissibles. L'interCt du problkme consiste h incorporer ces contraintes de prdcddence de manibre aussi efficace que possible dans des algorithmes destinks a dkterminer une affectation admissible qui soit optimale par rapport a un certain critere. L'effet des contraintes de prWence peut Ctre double. Si le problkme sans contraintes peut Ctre rksolu par une methode efficace, alors leur adjonction conduira h modifier l'algorithme. Dans certains cas, cette adaptation n'affecte pas l'efficacitk de l'algorithme. Dans d'autres, l'efficacitk est diminuk au point

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J.K. Lenstra, A.H.G. Rinnooy Kan

que parfois la nouvelle methode de resolution revient B humerer entikrement les affectations admissibles. Si le probleme sans contraintes de precedence est dkjh si complique qu’une approche par enumkration semble inkvitable, on peut tabler sur le fait que l’addition de contraintes de prkkdence peut rkduire le nombre d’affectations admissibles. Dans le premier cas, les contraintes rendent le probleme plus difficile; dans le second, le probleme devient un peu plus simple. En termes de complexiti de calcu(, l’addition de contraintes de prkkdence peut transformer un probleme bien soluble en un probleme NP-dur, ou encore rendre un probleme d‘ordonnancement NP-dur plus facile B resoudre en pratique. Nous nous concentrerons sur le premier phknomene et nous l’illustrerons dans le cas ou les ressources limitks correspondent a des machines MI,. . . ,M, dont chacune peut triter au plus l’une des activitks ou tdches J l , . . . ,J , h la fois. De nombreux problemes spkdiques ont pu Ctre formules et ktudies dans ce cadre gknkral. Le crit2re d’optimalitk qui doit Ctre minimis& joue un r61e proeminent dans ce contexte. Pour chaque ordonnancement admissible conduisant a un temps de fin d’exkution Cj pour J, (j = I,...,n), on fait l’hypothkse fondamentale que le critkre est une fonction non dkroissante de chacune des variables C1, . . . ,C,. Nous en donnerons des exemples plus loin. Panni les caractkristiques relatives aux tiches qui definissent des types particuliers de probkmes, on peut avoir des contraintes depricidence de la forme J j + Jk c’est-bdire que J j doit Ctre termink avant que Jk puisse commencer. De telles contraintes ont Ctk abondamment etudiks et certains types de contraintes de prkMence ont ktC mis en kvidence. En termes du graphe de p r W e n c e G dont les sommets sont I, ...,n et dont les arcs sont les paires ( j , k ) telles que Jj Jk, on s’est surtout intkressk au cas ou G est une collection de chaines, m e forit ou un graphe sirie-parallde. Beaucoup d’autres cas particuliers entre un graphe sans arcs et un graphe arbitruire ont aussi Cte examinks. En general, l’effort principal a ktC mis sur la dktermination d’une frontikre aussi bien dklimitee que possible entre les problkmes bien solubles et les problkmes NP-durs. Ceci a etk fait en determinant le cas le plus gkneral de contraintes de prkkdence qui peuvent Ctre traitks en temps polynomial et le cas le plus simple qui conduit h un probleme NP-dur. Dans cette note, nous nous concentrerons sur d ew problkmes ouverts connus: d’une part la minimisation du retard total pour des tiiches de durke unite h exkuter sur une machine avec des contraintes de prkckdence dkfinies par des chaines; d’autre part, la minimisation du maximum des temps d‘exkution sur trois machines paralleles identiques avec des contraintes de p r W e n c e quelconques. Les resultats connus pour des problkmes voisins seront passes en revue et gknkralisks ti l’aide de deux nouvelles preuves de NP -difficulte.

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1. INTRODUCTION The theory of scheduling is concerned with the allocation over time of scarce resources to activities. In this context, partial orders arise in a natural fashion: a partial order on the activity set imposes constraints on the order in which the activities can be executed and as such delimits the set of feasible allocations. The challenge is to incorporate these precedence constraints as efficiently as possible in algorithms designed to determine a feasible allocation that is optimal with respect to some criterion. The effect of precedence constraints can be twofold. If the problem without precedence constraints can be solved efficiently, their addition will generally require the algorithm to be adapted. In some cases, this adaptation does not affect the efficiency of the algorithm; in other cases, it does, possibly to the point that the new solution method amounts to complete enumeration of all feasible allocations. If the unconstrained problem is already so difficult in itself that an enumerative approach seems unavoidable, one may capitalize on the addition of precedence constraints by exploiting the fact that they reduce the number of feasible allocations. In the former case, precedence constraints make the problem harder to solve; in the latter case, it becomes a little easier. The theory of computational complexity of combinatorial problems has served to formalize the preceding informal discussion. We will settle here for a very brief review of the main concepts of this theory and refer the reader for more details to [Cook 1971; Karp 19721 (the first two papers on the subject), [Garey & Johnson 19791 (a comprehensive textbook) and [Lawler & Lenstra 19821 (a survey likely to be readily available to the current readership). The size of a combinatorial problem is defined as the number of bits needed to encode its data, and the running time of an algorithm as the number of elementary operations (such as additions and comparisons) required for its solution. If a problem of size s can be solved by an algorithm with running time 0 (p(3)) where p is a polynomial function, then the problem is said to be well solvable; there are good theoretical and practical justifications for this notion. Many problems have been shown to be well solvable, simply by the construction of a polynomial-time algorithm. Only few problems have been proved to be not well solvable, but there is a large class of problems for which it is strongly suspected that this is indeed the case. These are the NP-hard problems, which share a notorious reputation for computational intractability as well as the property that a polynomial-time algorithm for any one of them would yield polynomial-time algorithms for all problems in an important subclass, the NP-complete problems - a very unlikely event.

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One establishes NP-hardness of a problem P by taking another NP-hard problem Q and showing that Q is reducible to P (Q e P ) , i.e., that for any instance of Q a corresponding instance of P can be constructed in polynomial time such that solving the latter will solve the former as well. This implies that Q is a special case of P , and since Q is NP-hard, P is NP-hard too. (This recipe obviously does not apply to theflrst NP-hardness proof - for this, see [Cook 19711.) Rephrased more formally, then, the addition of precedence constraints may turn a well-solvable problem into an NP-hard one, or may make an NP-hard scheduling problem easier to solve in practice. We shall focus on the former phenomenon, and illustrate it for the case , each of which that the scarce resources correspond to machines M . . . ,M,,, can handle at most one of the activities or jobs J 1, . . . ,J,, at a time. Within this general setting, many specific problem types have been formulated and studied. For a detailed problem classification and a survey of the complexity results in this area, we refer to [Graham et al. 1979; Lawler et al. 19821. A prominent role in this classification is played by the optimality criterion to be minimized. With every feasible schedule leading to a completion time CJ for J j 0 = l , ...,n), the basic assumption is that the criterion is a function of CI, . . . ,C,,, nondecreasing in every variable. We shall encounter various examples below. Among the various job characteristics that further specify a problem type, there may be precedence constraints of the form J j +Jk, sijpfying that JJ has to be completed before Jk can start. Such constraints have long formed a research subject in the area, whereby several types of precedence constraints have been distinpshed. In terms of the precedence graph G with vertex set { 1,...,n } and arc set {(j,k):J j + J k } , separate attention has been paid to the case that G is a collection of chains, aforest, or series-parallel. Many other special cases inbetween an empy and an arbitrary arc set have been investigated as well. In general, the effort has been to draw as sharp a borderline as possible between well-solvable and NP-hard problems, by identification of the most general type of precedence constraints that can be coped with in polynomial time versus the simplest type that leads to NP-hardness. For a review of the results obtained so far, we refer to [Lawler & Lenstra 19821. In this note, we concentrate on two prominent open problems in this area, while surveying known related results. 2. A SINGLE MACHINE PROBLEM

Let us assume that there is a single machine (m = 1) and that each of the n

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jobs JJ (j = 1,...,n ) has to spend an uninterrupted processing time of pJ time units on the machine. Each JJ becomes available for processing at time 0 and incurs, upon its completion at time CJ,a tardiness cost T, = max{O,CJ - d J } , where dJ is a given due date. The criterion to be minimized is the total tardiness E;= I T / .

Thls is perhaps the most notorious open problem in single machine scheduling theory. It can be solved by dynamic programming techniques in O(n4EpJ)time [Lawler 19771; although the running time is obviously exponential in the problem size (which is O(Z(logpJ +log dJ))),the algorithm in question is called pseudopolynomial since the running time is polynomial in the problem data themselves. We will concentrate on the special case of unit-time jobs, i.e., pJ = 1 (j=1, ...,n ) . The cost cIJ of scheduling JJ in the i-th position is now given by qJ.= max(0,i - d J } , and the problem is to find a permutation u of { 1,...,n } rmnimizing X/”= caOh. If there are no precedence constraints, this is an ordinary linear assignment problem, which can be solved in O(n3) time (see, e.g., [Lawler 19761). If arbitrary precedence constraints between the jobs are allowed, the problem becomes NP-hard [Lenstra & Rinnooy Kan 19781. It is not known, however, what the effect of chain-like precedence constraints is, and this is our first open problem:

Given a directed graph G with vertex set { 1,...,n } in which each vertex j has an associated integer dj, indegree at most one and outdegree at most one, find a permutation u of { 1,...,n } satisfiing U(J) < u ( k ) whenever ( j , k ) is an arc of G, such that E,?= 1 max{O,u(j ) -a’, } is minimized. An optimality criterion related to the total tardiness I: T, is the number of late jobs XUJ, where UJ = 0 if C, < dJ, UJ = 1 if CJ > dJ. Since we know of no problem type for which minimizing XUJ is harder than minimizing XT, and since the problem of minimizing XUJ for unit-time jobs on a single machine subject to chain-like precedence constraints is NP-hard [Lenstra & Rinnooy Kan 19801, the most plausible conjecture is that the above problem will eventually turn out to be NP-hard. Three immediate generalizations of our open problem are worth considering: (1) The processing timesp, (j = 1,...,n ) are arbitrary nonnegative integers. The resulting problem is NP-hard (Theorem 1). (2) Each JJ (j = 1,...,n ) has to be completed no later than a given deadline eJ (not to be confused with the due date dJ). This problem is NP-hard as well (Theorem 2). (3) Each JJ (j = 1, ...,n ) becomes available for processing at a given release

J.K. Lenstra, A.H.G. IZinnooy Karl

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date r,. This problem is still open and, of course, also suspected to be NP-hard. As a preparation for the proofs of Theorems 1 and 2, we recall an NPhardness result for the total weighted tardiness criterion Zy= w, 5 , where w, is a given weight of J, (j = 1,...,n ) . Lemma 1 [Lawler 1977; Lenstra et al. 19771. The problem of scheduling jobs with arbitrary processing times on a single machine in the absence of precedence constraints so as to minimize total weighted tardiness Z w, Tj is NP-hard.

Proof [Lenstra & Rinnooy Kan 19801. We have to show that a known N P hard problem is reducible to the Z wjT, problem. Our starting point will be the following NP-hard problem [Garey & Johnson 19791: 3-PARTITION: Given a set S = {1,...,3t} and positive integers a1, . . . , a 3 t , b with lhb < aj < ‘hb for a l l j E S and X j E s a, = t b , does S have a partition into t 3-element subsets Si such that ZjEs, aj = b ( i =0, ...,t - l)? Given any instance of 3-PARTITION, we construct an instance of the Z w, T/ problems as follows: there are 4t - 1 jobs; - for each j ES, there is a job J, with processing time pi = a,, due date d, = Oandweightw, = a,; - for each i E { 1,...,t - l}, there is a job J’i with processing time p’i = 1, due date di = i ( b 1) and weight w’; = 2.

+

We claim that 3-PARTITION has a solution if and only if there exists a schedule with value Z w, T, C y , where y = & c j c k G 3 t ajak 4- %(t - 1)tb. This would imply that a polynomial-time algorithm for the Z w, Ti problem could be used to solve 3-PARTITION in polynomial time as well and therefore prove the theorem. Let us first ignore the jobs J’i ( i = 1,...,t - 1). Since dj = 0 for all j E S , we have X,Es wj T/ = 2,ES w j C j ; moreover, since pi = w, for all j ES,the value of X j E S wjCj is not influenced by the ordering of S. That is, for any schedule of the jobs J j (jES) without machine idle time we have

2 j ES

wjTj =

2

ajak* 1 cj 9 k 9 3 t

Let us now calculate the effect of inserting a job J’i in such a schedule. Suppose that Yi is completed at time Ci and define L’i = Ci -di . Since all jobs Jj (/ E S ) that are processed after J’i are completed one time unit later, the value of Xj ES w, Ti is increased by the total weight of these jobs, and we have

Two open problems in precedence constrained scheduling

2

+

wjT, wli T i =

j ES

=

2 2

+ 1 -d’, -L’i +21nax{O,L’j } a,uk +tb +I-d’i + IL’j I. a,ak +tb

1 < j
I =Zj< k

5 15

=Z31

More generally, insertion of all jobs J l i , resulting in completion times = dfi+L’i ( i = 1,...,t - 1) such that C’
x wj Tj

1-1

=

2

19j 9 k 931

a,uk

+ x ( t b +i

-d’<,)+ IL’<j)I)

i=l

1-1

i=l

irrespective of the permutation T . It follows that a schedule has value Z w, Ti < y if and only if there is no idle time and moreover the jobs J f i are completed at times Ci = d’i = i ( b 1) ( i = 1, ...,t - 1). Such a schedule exists if and only if the jobs J j (jE S ) can be divided into t groups, each containing 3 jobs and requiring b units of processing time, i.e., if and only if 3-PARTITION has a solution. 0

+

The proof of Lemma 1 provides the basis for our proofs of Theorems 1 and 2. We will specify reductions from 3-PARTITION to both Z T, problems in which the number of jobs created is Q ( t b ) and O(tb2)respectively. This may raise some eyebrows, as the size of an instance of 3-PARTITION is only O(t log b ) . However, 3-PARTITION has been shown to be NP-hard even when problem size is measured in a pseudopolynomial fashion as O ( t b ) [Garey & Johnson 19791, and hence the reductions below suffice to establish N P hardness.

Theorem 1. The problem of scheduling jobs with arbitrary processing times on a single machine subject to chain-likeprecedence constraints so as to minimize total tardiness Z Ti is NP-hard. Proof. Given any instance of 3-PARTITION, we first construct an instance of the Z wJ T, problem as in the proof of Lemma 1 and then transform it into an instance of the ZT, problem with chain-like precedence constraints as follows. Each job JJ with processing timepJ, due date dJ and weight wJ (whether it is a ”partition” job JJ (j E S ) or a ”splitting” job J’, ( i = 1,...,t - 1)) is replaced by a chain of wJ unit-weight jobs. The first job in the chain has processing timepJ and due date d J , the next wJ - 1 ones have processing times 0 and due dates dJ

*

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The resulting problem instance has tb +2(t - 1) jobs. Given any feasible schedule in which the jobs of some chain are not scheduled consecutively, one can obtain another schedule by processing all the zero-time jobs of that chain directly after its first job. This schedule is still feasible, and its I: Ti value has not increased. Hence, each chain of length wj can be considered as a single job with weight w,, and we are back at our original construction. The reader who dislikes zero-time jobs could quite easily replace them by unit-time jobs and multiply the lengths of the other jobs by a factor polynomial in t and b such that the equivalence argument still carries through. 0

Theorem 2. The problem of scheduling unit-timejobs on a single machine subject to arbitrary deadlines ej and chain-like precedence constraints so as to minimize total tardiness I: T/ is NP-hard. Proof. Our proof is again related to the proof of Lemma 1, although it is not such a straightforward extension as the proof of Theorem 1. Given any instance of 3-PARTITION, we construct an instance of the Z Ti problem with unit-time jobs, deadlines and chain-like precedence constraints as follows: - there are n = tb2 t - 1 jobs; - for each j E S , there is a chain of ba, unit-time jobs:

+

4

with due dates and deadlines defined by

d j k ) = n ( k = 1,...,(b - l)aj), d,W-‘) = -I

(I=aj

- 1,

...,o),

e r ) = n (k = 1,...,baj);

-

for each i E { 1, ...,z - l}, there is a unit-time job J’i with due date and deadline defined by di = e’i = i(b2+1). We claim that 3-PARTITION has a solution if and only if there exists a feasible schedule with value Z Ti < z , where z = bZlGjike3, ajuk +%(t - 1)tb. Before we prove this claim, we make some introductory remarks about the way in which the job weights occurring in the proof of Lemma 1 have been simulated in the present construction. For each chain 0’ ES), the due dates have been specified such that in any schedule without machine idle time only the last aj jobs in the chain contribute to the criterion; if all these jobs are completed one time unit later, this adds aj units to Z Ti, which corresponds to the original weight wj = a j . For each job J’, ( i = 1, ...,t - l), we previously used a weight w ’ ~= 2 in combination with an upper bound y on I: wj Ti to enforce an implicit deadline di ; we now simply have an explicit deadline e‘i = d i . Consider any feasible schedule with value Z Tj < z . Without loss of gen-

6

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Two open problems in precedence constrained scheduling

<

erality, we assume that the schedule contains no machine idle time, that each (j E S ) job J’, ( i = 1 , ...,t - 1) is completed at time d,,and that the chains do not preempt each other; the latter two statements can be proved by means of simple interchange arguments. The jobs J’, (i = 1 , ...,t - 1) do not contribute to the Z T/ value of the schedule. The contribution of the chains J/ (j E S ) consists of two terms. First, there is the total tardiness of all jobs in the chains when the chains are processed from time 0 onwards without interruption. It is not hard to see that this term is given by b E l , G k <3t a, ak , irrespective of the ordering of S . Secondly, there is the increase in total tardiness due to the insertion of the - l , d , ]= [ d , - , + b 2 , d ’ , ] ( i = 1 , ...,t - l), where jobs J’, in the intervals [d, = 0. Let S, C S denote the index subset of chains that are completed in + b 2 ] ,and let A, = 2, €s, a, ( i = O ,...,t - 1 ) . Note that bA, the interval [d,,d, is equal to the total length of all chains completed in the final interval [ d - I , d ’ , - 1 + b 2 ] , SO that A t - 1 2 b . More generally, we have that 2; -, Ah 2 ib ( I = 1 , ...,t - 1). Since all chain lengths as well as the interval lengths are integer multiples of b , we know that, if j ES, , the last b jobs of J/ and in particular the last a, ones (the only ones that contribute to 2 T,) must be processed in [a,,d’, +b2],so that J/ contributes ia, additional units to Z T/ . Thus, the second term is given by r-1

t-1

t-1

t-1

It follows that 2 T/ < z if and only if Ai = b (i =O ,...,t only if 3-PARTITION has a solution . 0

-l),

i.e., if and

3. A PARALLEL MACHINE PROBLEM We now assume that there are m machines and n jobs J j ( j = l , ...,n ) . The machines are parallel in the sense that each job can be assigned to any one of them, and they are identical in the sense that, when J j is assigned to some machine, it requires an uninterrupted processing time p i , irrespective of the machine. The criterion to be minimized is the maximum completion time Cm, - mal<j
J.K. Lenstra, A.H.G. Rinnooy Kan

518

cal parallel machines subject to precedence constraints, specified in the form of a directed graph G : (1) If m is arbitrary (i.e., specified as part of the problem instance) and G is an inforest (each vertex has outdegree at most one) or an outforest (each vertex has indegree at most one), the problem is solvable in O ( n ) time [Hu 19611. (2) If m = 2 and G is arbitrary, the problem is also well solvable; algorithms that have subse uently been developed require O ( n 3 ) time [Fujii et al. 1969, 19711, O(n ) time [Coffman & Grahman 19721, "almost linear" time [Gabow 19821, and O ( n ) time [Gabow & Tarjan 19831. (3) If m and G are arbitrary, the problem is NP-hard [Ullman19751. These results do not resolve the complexity status of the problem if G is arbitrary and rn is fixed but greater than 2. In particular, the case that m = 3 has withstood all attacks, and this is our second open problem:

P

Given a directed graph G with vertex set { 1,...,n }, Jind the minimum value of C for which there exists a function a:{ 1,...,n } + { 1,...,C } satisaing a(j) < a ( k ) whenever u , k ) is an arc of G and 10' E {1,...,n } : a(j) = t } l < 3 for all t E { 1,...,C}. In the course of research on this problem, progress has been made for several special types of precedence constraints other than forests. We mention the following results. Let the height h of G be defined as the number of arcs in a longest path in G . If m is arbitrary and h = 2, the problem is still NP-hard, and there exists no polynomial-time (approximation) algorithm that guarantees a relative error less than one third of the optimal C, value unless all NPcomplete problems are well solvable [Lenstra & Rinnooy Kan 19781. If both m and h are fixed, the problem is well solvable in O ( n h ( m - l ) 1+ l t h e [Dolev& Warmuth 3982Bl. Suppose G is an interval order: each vertex j corresponds to an interval [aJ. ,b.] on the real line and u , k ) is an arc of G whenever b, < a k . In J t h s case, the problem is solvable in O(n2) time [Papadimitriou & Yannakakis 19791. Suppose G is a level order: any two incomparable vertices with a common predecessor or successor have identical sets of predecessors and successors. If m is fixed, this problem is well solvable in O(n"'-') time [Dolev & Warmuth 1982Cl. Suppose G is an opposing forest, consisting of the disjoint union of an inforest and an outforest. If rn is arbitrary, this problem is NP-hard [Garey et al. 1983, Mayr 19811. If m is fixed, it is well solvable in O(n2m-210gn ) time [Dolev & Warmuth 1982Cl. If m = 3, there is an

Two open problems in precedence constrained scheduling

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O ( n ) algorithm [Garey et al. 1983; Dolev & Warmuth 1982Al. These results give little additional insight into the complexity status of the three-machine problem with arbitrary precedence constraints. Recent rumors on a proof of its well-solvability have not been substantiated so far, but such a proof should be extendable to the case that m is any fixed constant. We believe that the problem stands a good chance to be the seventh one to be removed from the list of twelve open problems in [Garey & Johnson 19791.

4. CONCLUDING REMARKS The above discussion has illustrated that very detailed insights exist on the way in which partial orders on the job set affect the computational complexity of machine scheduling problems. The two problems considered in the preceding two sections figure prominently on the list of open problems that is produced by the computer program MSPCLASS [Lageweg et d. 1981,19821. This program keeps track of the complexity status of 4,536 machine scheduling problems, 390 of which are currently still open. Resolution of many of these problems, in particular of the two above ones, would seem to require the development of new algorithmic approaches or transformation techniques. ACKNOWLEDGEMENTS We are grateful to D. de Werra who translated our extended abstract into a risumt ditailli. Our research was supported by NSF grant MCS78-20054. REFERENCES E.G. COFFMAN, JR. & R.L. GRAHAM (1972) Optimal scheduling for twoprocessor systems. Acta Informal. I, 200-2 13. S.A. COOK ( 1971) The complexity of theorem-proving procedures. Proc. 3rd Annual ACM Symp. Theory of Computing, 151-158. D. DOLEV & M.K. WARMUTH (1982A) Scheduling flat graphs. Research Report RJ3398, IBM, San Jose, CA. D. DOLEV & M.K. WARMUTH (1982B) Sceduling precedence graphs of bounded height. Research Report RJ3399, IBM, San Jose, CA; J. Algorithms, to appear. D. DOLEV & M.K. WARMUTH (1982C) Profile scheduling of opposing forests and level orders. Research Report RJ 3553, IBM, San Jose, CA.

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M. FUJII, T. KASAMI & K. NINOMIYA (1969,1971) Optimal sequencing of two equivalent processors. SIAM J. Appl. Math. 17, 784-789. Erratum. 20, 141. H.N. GABOW (1982) An almost-linear algorithm for two-processor scheduling. J. Assoc. Comput. Mach. 29, 766-780. H.N. GABOW & R.E. TARJAN (1983) A linear-time algorithm for a special case of disjoint set union. Proc. 15th Annual ACM Symp. Theory of Computing, 246-25 1. M.R. GAREY & D.S. JOHNSON (1979) Computers and Intractability: a Guide to the Theory of NP-Completeness, Freeman, San Francisco. M.R. GAREY, D.S. JOHNSON, R.E. TARJAN & M. YANNAKAKIS (1983) Scheduling opposing forests. SIAM J. Algebraic Discrete Methods 4, 72-93. R.L. GRAHAM, E.L. LAWLER, J.K. LENSTRA & A.H.G. RINNOOY KAN (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math. 5, 287-326. T.C. HU (1961) Parallel sequencing and assembly line problems. Oper. Res. 9, 841-848. R.M. KARP (1972) Reducibility among combinatorial problems. In: R.E. MILLER, J.W. THATCHER (eds.) (1972) Complexity of Computer Computations, Plenum, New York, 85- 103. B.J. LAGEWEG, E.L. LAWLER, J.K. LENSTRA & A.H.G. RINNOOY KAN (1981) Computer aided complexity classification of deterministic scheduling problems. Report BW 138, Mathematisch Centrum, Amsterdam. B.J. LAGEWEG, J.K. LENSTRA, E.L. LAWLER & A.H.G. RINNOOY KAN (1982) Computer aided complexity classification of combinatorial problems. Comm. ACM 25, 817-822. E.L. LAWLER (1976) Combinatorial Optimization: Networks and Matroids, Holt, Rinehart & Winston, New York. E.L. LAWLER (1977) A "pseudopolynomial" algorithm for sequencing jobs to minimize total tardiness. Ann. Discrete Math. I , 331-342. E.L. LAWLER & J.K. LENSTRA (1982) Machine scheduling with precedence constraints. In: I. RIVAL (ed.) (1982) Ordered Sets, Reidel, Dordrecht, 655-675. E.L. LAWLER, J.K. LENSTRA & A.H.G. RINNOOY KAN (1982) Recent developments in deterministic sequencing and scheduling: a survey. In: M.A.H. DEMPSTER, J.K. LENSTRA & A.H.G. RINNOOY KAN

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(eds.) ( 1982) Deterministic and Stochastic Scheduling, Reidel, Dordrecht, 35-73. J.K. LENSTRA & A.H.G. RINNOOY KAN (1978) Complexity of scheduling under precedence constraints. Oper. Res. 26, 22-35. J.K. LENSTRA & A.H.G. RINNOOY KAN (1980) Complexity results for scheduling chains on a single machine. European J. Oper. Rex 4, 270275. J.K. LENSTRA, A.H.G. RINNOOY KAN & P. BRUCKER (1977) Complexity of machine scheduling problems. Ann. Discrete Math. I , 343-362. E. MAYR (1981) Well structured parallel programs are not easier to schedule. Report STAN-CS-81-880, Department of Computer Science, Stanford University. C.H. PAPADIMITRIOU & M. YANNAKAKIS (1979) Scheduling intervalordered tasks. SIA M J. Cornput. 8, 405-409. J.D. ULLMAN (1975) NP-complete scheduling problems. J. Cornput. System Sci. 10, 384-393.

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- PART IX ORDER AND SOCIAL SCIENCES

- PARTIEIX ORDRE ET SCIENCES SOCIALES

Jean-Pierre BARTHELEMY - Bruno LECLERC - Bernard MONJARDET

Ensembles ordonnes et taxonomie mathematique.

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Annals of Discrete Mathematics 23 (1984) 523-548 0 Elsevier Science Publishers B.V. (North-Holland)

523

ENSEMBLES ORDONNES ET TAXONOMIE MATHEMATIYUE

J.P.

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BARTHELEMY", B. LECLERC"",

B . MONJARDET**"

ENST, 4 6 r u e B a r r a u l t , 75634 PARIS Cedcsx 13 CAMS, 54 Bd R a s p a i l , 7 5 2 7 0 PAKIS Cedex 06 o*% U n i v e r s i t e P a r i s - V e t ChMS. J

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En hommage a u p r o f e s s e u r C o r o m i n a s , b l ' o c c a s i o n d e s o n Pmtritat. Le t e r m e t a x o n o m i e m a t h 6 m a t i q u e r e c o u v r e un e n s e m b l e d e p r o c t d u r e s d e s t i n b e s i o b t e n i r des c l a s s i f i c a t i o n s hic!rarchiques ou ( e t ) d e s g r o u p e s homoggnes d a n s u n e p o p u l a t i o n d " ' i n d i v i d u s " d e n a t u r e v a r i t e . Ces t e c h n i q u e s f o n t p a r t i e d e s d b v e l o p p e m e n t s r b c e n t s d e 1 " ' a n a l y s e d e s donnbes" e t s o n t l a r g e m e n t u t i l i s t e s d a n s l e s s c i e n c e s s o c i a l e s , Gconomiques e t d e g e s t i o n . L ' i m p o r t a n c e d e c o n c e p t s e t dc t e c h n i q u e s o r d i n a l e s s ' e s t m a n i f e s t e r a u moins d e p u i s l e s t r a v a u x d e J a r d i n e e t S i b s o n (Mathematical Taxonomy, 1 9 7 3 ) e t a donnc! l i e u i p l u s i e u r s c o n t r i b u t i o n s , ti d e s n i v e a u x v a r i t s (nutamment p a r Boorman, O l i v i e r , J a n o w i t z , S c h a d e r , Day, Mc M o r r i s e t l e s a u t e u r s d e c e p a p i e r ) . Nous n o u s p r o p o s o n s i c i d e d o n n e r une s y n t h h s e d e c e s c o n t r i b u t i o n s , e n p r 6 s e n t a n t l a t a x o n o m i e m a t h t m a t i q u e sous l ' a n g l e ( p a r t i e l ) d e s s t r u c t u r e s o r d i n a l e s q u ' e l l e met e n j e u . ABSTRACT A f t e r t h e i n t r o d u c t i o n ( s e c t i o n 1 1 , we b e g i n w i t h a n o v e r v i e w on t h e a i m s a n d t h e m e t h o d s o f m a t h e m a t i c a l taxonomy ( s e c t i o n 2 ) . A t a x o n o m i s t c l a s s i f i e s o b j e c t s d e s c r i b e d by "raw" d a t a ( ~ e n e r a l l yv a l u e s o f d i f f e r e n t a t t r i b u t e s , v a r i a b l e s , . .) t h a t c a n be c o n s i d e r e d as " i n p u t s " . A c h o s e n method of c l a s s i f i c a t i o n t r a n s f o r m s these inputs i n t o "uutputs" t h a t art' c l a s s i f i c a t o r y " m o d e l s " , a n d w e e m p h a s i z e t h e f o u r main s u c h models : p a r t i t i o n ; n - t r e e , c a l l e d a l s o h i e r a r c h y ( ' ; i i t r a r c h i e " , i n french) ; ordered t r e e ( " h i t r a r c h i e s t r a t i f i P e " i n french) ; valued tree ( " h i G i - a r c h i e i n d i c b e " in f r e n c h ) , o r t h e e q u i v a l e n t n o t i o n of u l t r a m e t r i c d i s t a n c e . We r e c a l l t h a t most ( b u t n o t a l l ) m e t h o d s of c l a s s i f i c a t i o n b e g i n w i t h t h e c o m p u t a t i o n of a d i s s i m i l . a r i t y c o e f f i c i e n t (a1 so c a l l e d d i s s i m i l a r i t y i n d e x ) h e t ween t h e o b j e c t s t o be c l a s s i f i e d , f r o m t h e raw d a t a . I n s e c t i o n 3 , we s y s t e m a t i c a l l y s t u d y t h e o r d i n a l s t r u c t u r e s o f t h e d a t a ( 3 . 1 . and 3 . 2 ) a n d of t h e m o d e l s of s e c t i o n 2 . A d i s s i m i l a r i t y c o e f f i c i e n t i s d e f i n e d as a map w i t h domain t h e s e t B , ( X ) o f a l l t h e p a i r s o f e l e m e n t s o f a s e t X ( t h e o b j e c t s t o b e c l a s s i f i e d ) and w i t h codomain a l i n e a r l y o r d e r e d s e t L , w i t h a l e a s t e l e m e n t ( u s u a l l y , L i s a s u b s e t of c o n t a i n i n g 0 ) . The s e t o f a l l L - d i s s i m i l a r i t y i n d i c e s i s o n e t o o n e w i t h t h e s e t of L - c h a i n s o f t h e l a t t i c e 8 ( 8 , ( X ) ) of a l l s y m m e t r i c r e l a t i o n s on X , w h e r e a L - c h a i n i s an i s o t o n e a p p l i c a t i o n F : L -B t ? ( p n ( X ) ) s a t i s f y i n g two a d d i t i o n a l c o n d i t i o n s ( o r , e q u i v a l e n t l y , a s n o t i c e d by J a n o w i t z , a r e s i d u a l map F : L + f ( f ' ? ( X ) ) ) . A f t e r a r e c a l l on t h e l a t t i c e o f p a r t i t i o n s , w e s t u d y l a t t i c e s of u l t r a o i e t r i c s , e m p h a s i z i n g t h a t t h e i r p r o p e r t i e s a r e s i m i l a r t o t h e p r o p e r t i e s o f p a r t i t i o n l a t t i c e s ; f o r example, t h e y a r e semimodular. Then, we p o i n t o u t t h a t t h e s e t of a l l " d i s s i m i l a r i t y r e l a t i o n s " ( i . e . t h e t o t a l p r e o r d e r s on 6 ) 2 ( X ) , "pr6ordonnance"in f r e n c h ) and t h e s e t of a l l u l t r a d i s s i m i l a r i t y r e l a t i o n s a r e " f r a m e " s e m i l a t t i c e s , t h u s m e d i a n s e m i l a t t i c e s ( a n o t i o n d e f i n e d by S c h o l n n d e r , i n 1954).

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I n s e c t i o n 4 , we p r e s e n t , f i r s t o f a l l , t h e a b s t r a c t f r a m e w o r k o f o r d i n a l m e t h o d s o f c o m p a r i s o n , f i t t i n g o r a g g r e g a t i o n . S o , t h e c o m p a r i s o n of two o r

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s e v e r a l e l e m e n t s of a n o r d e r e d s e t i s b a s e d on t h e " n a t u r a l " m e t r i c s d e f i n e d on i t . F i t t i n g o r a g g r e g a t i o n p r o b l e m s le,id t o d e f i n e maps be tw e e n o r d e r e d s e t s s a t i s f y i n g a l g o r i t h m i c , a x i o n i n t i c o r o p t i m i s n t i o n c o n d i t i o n s . Th a r e a p p l i e d t o o r d e r e d s t r u c t u r e s o f s e c t i o n 3. T h u s , we ~ i l i t ~ i i-n and sometirnc.s c h a r a c t e r i z e - many o l d - o r new - d i s t a n c e s u s e f u l t o compare p a r t i t i o n s o r 11t r e e s , by means o f r e s u l t s on m e t r i c s a s s o c i a t e d w i t h l o w e r o r u p p e r va 1tl;ttions on o r d e r e d s e t s . I n s e c t i o n 4 . 3 , we g i v e t h r e e e x a m p l e s of o r d i n a l me thods in o r d e r t o f i t some u l t r a m e t r i c o r u l t r a d i s s i m i l a r i t y r e l a t i o n t o ;i g i v e n c l i s s i m i l a r i t y c o e f f i c i e n t o r r e l a t i o n . The f i r s t e x a m p l e , d e f i n e d h e r e a s t h e " u l t r a m e t r i c a n t i c l o s u r e " ( o n t h e s e t of a l l L - d i s s i m i l a r i t y i n d i c ~ e s ) i s e q u i v a l e n t t o t h e well-known " s i n g l e l i n k a g e " method ; t h e o t h e r e x a m p l e s a r e due t o Zalin and S c h a d e r . Then ( s e c t i o n 4 . 4 . ) , w e i n t r o d u c e t h e J a n o w i t z o r d e r t h e o r e t i c model f o r c l u s t e r a n a l y s i s " ( 1 9 7 8 ) , t h a t , f o r i n s t a n c e , a l l o w s t o c h a r a c t c r i z e " f l a t " me t h o d s ("mGthodes 2 s e u i l " i n f r e n c h ) of c l u s t e r a n n I y s i s . I n s e c t i o n 4 . 5 , we p o i n t o u t t h a t a " G a l o i s " a p p r o a c h t o c l a s s i f y o b j e c t s d e f i n e d by " p r e s e n c e - a b s e n c e " a t t r i h u t e s , l e a d s t o f i x e d p o i n t p r o b l e m s on o r d e r e d s e t s . F i n a l l y ( s e c t i o n 4 . b ) , we t a c k l e t h e s o - c a l l e d " c o n s e n s u s proble nI" i n taxonoiiiy, i . e . t h e problem t o "aggregate" o r "summarize" a t b e s t s e v e r a l c l a s s i f i c a t i o n s i n t o a c l a s s i f i c a t i o n ( t h e similar p r o b l e m f o r p r e f e r e n c e s l e a d s t o t h e w e l l known Arrow t h e o r e m ) . One a l s o o b t a i n s " d i c t a t o r i a l " o r " o l i g a r c h i c " r e s u l t s , l i k e t h e M i r k i n t h e o r e m f o r p a r t i t i o n s , g e n e r a l i z e d t o u l t r a m e t r i c s by L e c l e r c . B u t , i n t h e c a s e of me d i a n s e m i l a t t i c e s , o n e c a n o b t a i n a x i o m a t i c o r o p t i m i s a t i o n c h a r a c t e r i z a t i o n s of more " p o s i t i v e " c o n s t r u c t i v e p r o c e d u r e s , 1 i k e t h e "median" p r o c e d u r e ( a g e n e r a l i z a t i o n of t h e well-known m a j o r i t y r u l e ) .

1 . INTRODUCTION. Les S c i e n c e s Humaines o n t h e s o i n d ' a n a l y s e r d e s "donnbe s " r e c u e i l l i e s e t c o n s t r u i t c s p a r d e s moyens v a r i C s a l l a n t d e l ' o h s e r v a t i o n " p u r e " h l ' e x p 6 r i r n e n t a t i o n en l a b o r a t o i r e . Les t e c h n i q u e s s t a t i s t i q u e s u t i l i s b e s j u s q u ' i c i o n t b t b rtcemment e n r i c h i e s p a r l ' a p p o r t d ' a u t r e s mGthodes p e r m e t t a n t notamment d e t r a i t e r d e s d o n n b e s m u l t i d i m e n s i o n n e l l e s n o m b r e u s e s , d a n s une p e r s p e c t i v e p l u s d e s c r i p t i v e e t e x p l o r a t o i r e q u ' i n f t r e n t i e l l e e t c o n f i r m a t o i r e . L e u r e s s o r e s t dCi t a n t h d e s p r o g r k s t h C o r i q u e s qu'au dbvel oppement c o n s i d b r a b l e d e s nioycns i n f o r m a t i q u e s d e c a l c u l . Comme r e p r 6 s e n t a t i f s d e c e c o u r a n t , on p e u t c i t e r 1"'Rcole F r a n s a i s e d ' h a l y s e d e s Donn6es" ( c f . notamment B e n z b c r i , 1973) OLI l e s d b v e l o p p e m e n t s a n g l o Sax o n s a u t o u r du " M u l t i d i m e n s i o n a l S c a l i n g " ( c f . , p a r e xe mple , C a r o l 1 e t A r a b i e , 1 9 8 1 ) . La ta x o n o mi e mat hPmat i que ( c l u s t e r a n a l y s i s ) f a i t p a r t i e de c e s m6thode s "n o u v ell e s", so n h u t s p G c i f i q u e t t a n t l a r e c h e r c h e d e g r o u p e s homoghnes d a n s une population d"'individus" de nature varibe. Eventuellement, ces groupes pourront Btre o r g a n i s C s d a n s u n s s t r u c t u r e h i t r a r c h i q u e , r e f l t t a n t p a r f o i s un phtnomtne G v o l u t i f , comme d a n s l e s c l a s s i f i c a t i o n s b o t a n i q u e s o u zoologiques. C ' c s t d ' a i l l e u r s 3 p r o p o s d e t e l l e s c l a s s i f i c a t i o n s que d e s n o t u r a l i s t e s comme Adanson f o n t i n t e r v e n i r l e s p r e m i h r e s c o n s i d b r a t i o n s m a t h b m a t i c o - l o g i q u e s a u 186me s i e c l e . Ma is c e n ' e s t que c e s d e r n i t r e s a n n 6 e s q u e l a t a x o n o m i e r n a t h h a t i q u e c onna 'it un d t v e lo p p eme n t c o n s i d t r a b l e ( c f . , p a r exernpl e, S okal e t S n e a t h 1963, Lerman1970, 1981, J a r d i n e e t S i b s o n 1971, BenzC cri 1973, Bock 1974, H a r t i g a n 1975, Jambu 1978, Diday 1980, Gordon 1981, Chandon e t P i n s o n 1 9 8 1 ) . L ' i m p o r t a n c e d e c o n s i d t r a t i o n s o r d i n a l e s e n t a x o n o m i e rnat hbmat i que a p p a r a ' i t au m oins d h s l e s t r a v a u x d e J a r d i n e e t Sibson, p a r f o i s d e m a n i t r e voilcie ( l a f o r m a t i o n h a b i t u e l l e d e s "taxonomistes" n e les p r t d i s p o s a n t pas B r c c o n n a y t r e e t 2 u t i l i s e r d e s s t r u c t u r e s d ' o r d r e s ) . Mais c e s s t r u c t u r e s s o n t e x p l i c i t e e s d a n s d e s t r a v n u x comrne c e u x d e Boorman e t A r a b i e ( 1 9 7 2 ) , Boorman e t O l i v i e r ( 1 9 7 3 ) , Degenne , Fla me nt e t V e r g e s ( 1 9 7 8 ) , Barth b l e my ( 1 9 7 9 ) , B art l i el emy e t M o n j a r d e t ( 1 9 8 1 ) , BerthClerny, L e c l e r c e t M o n j a r d e t ( 1 9 8 3 ) , S c h a d e r ( 1 9 8 1 ) , Day ( 1 9 8 1 ) . Mc M o r r i s e t Neumann ( 1 9 8 3 ) , l e s i m p o r t a n t s d b v e l o p p e m e n t s du m o d t l e de J a r d i n e e t S i b s o n p a r J a n o w i t z ( 1 9 7 8 , 1979, 1981) e t l e s c o n t r i b u t i o n s r c c e n t e s de R . L e c l e r c ( 1 9 7 9 , 1981, 1982, 1 9 8 3 ) .

Nous t e n t o n s d a n s c e t e x t e une p r 6 s e n t a t i o n s y n t h G t i q u e d e s r e l a t i o n s a c t u e l l e s e n t r e ensembles ordonnbs e t taxonomic mathematique e t de l e u r s n p p o r t s m u t u e l s .

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I1 n ' e s t t!vidcmment p a s q u e s t i o n d e c r o i r e q u e l e s s t r u c t u r e s o r d o n n b e s s o i e n t l e s e u l i n s t r u m e n t - ou mGme l ' i n s t r u m e n t p r i v i l t g i b - p o u r r g s o u d r e l e s p r o b l k m e s d e s t a x o n o m i s t e s . On v e r r a d ' a i l l c u r s a p p a r a f t r e d a n s c e t e x t e l ' i n t t r g t d ' a u t r e s d i ~ m a i n e s m a t h & m , i t i q u e s comme l a t h g o r i e d e s g r n p h e s ou p l u s g t n b r a l e m e n t l a combin a t o i r c . , y c o m p r i s l e s p r o h l 6 m c s d ' o p t i m i s a t i o n ( c f . , s u r c e s p o i n t s , H u b e r t 1974, Hubert e t S c l i u l t z 1976, Hubert 1977, Matula 1977, Monjardet 1 9 8 0 ) . Mais l ' u n d e s p r o b l P m e s m a j e i i r s d u t a x o n o m i s t e c o n c e r n e l e c h o i x d e l a m6thode q u ' i l v a u t i l i s e r . A c e t t ! g a r d , l a p r o l i j - a r , i t i o n d ' a l g o r i t h m e s d e c l a s s i f i c a t i o n a u q u e l on a a s s i s t 6 ( e t q u i c o n t i n u e ) n ' e s t pas s a t i s f a i s a n t e . E l l e d o i t s'acconlpagner d ' u n e f f o r t d ' e x p l i c i t a t i o n d e s m s t h o d e s S O L I S - j a c e n t e sp o u r l e s c o m p a r e r e t l e s v a l i d e r . Une a p p r o c l i e " e x p b r i m e n t a l e " ( s t y l e procedures d e Monte C a r l o ) p e u t S t r e un e l b r n e n t d e r t p o n s e . Mais i l f a u t a u s s i r e c h e r c h e r l e s b a s e s t h 6 o r i q u e s d e s methodes u t i l i s G e s . Or l e s o h j c t s (clonn6es ou m o d $ l e s ) d e l a t a x o n o m i e m a t h b m a t i q u e o n t t r k s ( 7 . s o u v e n t u n e n a t u r e o r d i n a l e t a n t nu n i v e a u i n d i v i d u e l " ( u n e " h i b r a r c h i e s t r a t i f i b e l ' e s t un s u p - d e m i - t r e i l l i s arborescent) qu'au niveau " c o l l e c t i f " (l'ensemble des h i 6 r a r c h i e s s t r a t i f i t e s e s t un d e m i - t r e i l l i s h m g d i a n e s ) . D h s l o r s u n e m b t h o d e d e c l a s s i f i c a t i o n p e u t a p p a r a P t r e comme u n e a p p l i c a t i o n e n t r e e n s e m b l e s o r d o n n e s , g t r e GtudiGe d e c e p o i n t de vue e t , d a n s l e m e i l l e u r d e s c a s , S t r e c a r a c t t r i s b e a x i o m a t i q u e m e n t . On c o n s o i t q u e d a n s c e c o n t e x t e o n p u i s s e u t i l i s e r p r o f i t a b l e m e n t d e s r e s u l t a t s gbnbraux - o u s p e c i f i q u e s - s u r d e s a p p l i c a t i o n s e n t r e ensembles o r d o n n 6 s . De m&ne 1 3 t h t i o r i e d e s m k t r i q u e s sur l e s e n s e m b l e s o r d o n n t s v a Ctre u t i l e pour d b f i n i r e t p r G c i s e r l e s p r o p r i t i t t s d e d i s t a n c e s ( c o n d u i s a n t 2 d e s c o e f f i c i c n t s d ' n c c o r d s ) e n t r e c l a s s i f i c a t i o n s . On c o n s t a t e i c i l ' a p p o r t d e s s t r u c t u r e s o r d i n a l e s pour l a v a l i d a t i o n e t l a comparaison "axiomatique" d e mhthodes c l a s s i f i c a t o i r e s . Inversement, l e s a s p e c t s o r d i n a u x de l a taxonomie mathtmatique peuvent c o n d u i r e 3 d ' i n t b r e s s a n t s problgmes pour l e s s p g c i a l i s t e s d e s ensembles o r d o n n t s c o n t r i b u a n t a i n s i Zi l e u r d b v e l o p p e m e n t ( c f . , p a r e x e m p l e , l e s " F l a t problems" d e J a n o w i t z 1 9 7 8 a ) . Donnons m a i n t e n a n t l e p l a n d e ce t e x t e . Le p a r a g r a p h e 2 e s t u n e i n t r o d u c t i o n A p r t s a v o i r 6voque l e s sommaire 5 la p r o b l g m a t i q u e de l a t a x o n o m i e m a t h 6 m a t i q u e . d o n n b e s " b r u t e s " s u r l e s q u e l l e s l e " t a x o n o m i s t e " t r a v a i l l e , on m e t e n h v i d e n c e l e s q u a t r e p r i n c i p a u x modtles c l a s s i f i c a t o i r e s q u ' i l u t i l i s e : p a r t i t i o n , h i h r a r c h i e , h i t r a r c h i e s t r a t i f i6e (chaine etendue de p a r t i t i o n s , ultra-prbordonnance), h i h r a r c h i e i n d i c e e ( k + - c h a P n e d e p a r t i t i o n s , u l t r a m b t r i q u e ) . On r a p p e l l e l a mbthod o l o g i c l a p l u s c o u r a n t e q u i v i s e 2 o b t e n i r de t e l s m o d t l e s 5 p a r t i r d ' i n d i c e s de d i s s i m i l s r i t b s . Le p a r a g r a p h e 3 G t u d i e s y s t b m a t i q u e m e n t l e s s t r u c t u r e s o r d i n a l e s d e s d o n n P e s e t d e s rnodeles d u p a r a g r a p h e 2 . L e s i n d i c e s d e d i s s i m i l a r i t 6 h v a l e u r s d a n s un e n s e m b l e t o t a l e m e n t o r d o n n b ( a v e c p l u s p e t i t b l t m e n t ) L s o n t m i s e n c o r r e s p o n d a n c e b i - j e c t i v e n v e c l e s L-chaPnes d u t r e i l l i s d e s r e l a t i o n s s y m b t r i q u e s q u i s o n t a u s s i l e s a p p l i c a t i o n s r t s i d u e l l e s d e L d a n s c e t r e i l l i s . A p r h s un r a p p e l s u r l e t r e i l l i s d e s p a r t i t i o n s , on b t u d i e e n s u i t e d e s t r e i l l i s d ' u l t r a m b t r i q u e s duaux d e s t r e i l l i s d e s L-cha?nes de p a r t i t i o n s , e n m o n t r a n t q u e l e u r s p r o p r i b t 6 s s o n t v o i s i n e s d e c e l l e s du t r e i l l i s d e s p a r t i t i o n s (en p a r t i c u l i e r , c e s t r e i l l i s s o n t s e m i - m o d u l a i r e s s u p s r i e u r e m e n t ) . On c o n s i d k r e e n s u i t e l e s s t r u c t u r e s o r d i n a l e s d e l'ensemble des prkordonnances, d e s ultra-prtordonnanres et d e s h i e r a r c h i e s q u i s o n t t o u t e s d e s d e m i - t r e i l l i s d e " c h a r p e n t e s " e t done d e s demil a problbmatique genbrale t r e i l l i s 2 m b d i a n e s . 1.e p a r a g r a p h e 4 p r h s e n t e d ' a h o r d d e s m t t h o d e s o r d i n a l e s d e c o m p a r a i s o n , a j u s t e m e n t ou a g r e g a t i o n : a i n s i l a compar a i s o n de d e u x o u p l u s i e u r s t l t m e n t s d ' u n e n s e m b l e o r d o n n 6 v a r e p o s e r s u r l e s r n P t r i q i r e s n a t u r e l l e m c n t a s s o c i 6 e s 5 sa s t r u c t u r e d ' o r d r e ; l e s p r o b l 5 m e s d ' a j u s t e ment ou d ' a g r t g a t i o n c o n s i s t e n t h d 6 f i n i r d e s a p p l i c a t i o n s e n t r e d e u x ensembles ordonneis E e t F B p a r t i r d e c o n t r a i n t e s a l g o r i t h m i q u e s , a x i o m a t i q u e s o u / e t d ' o p t i m i s a t i o n . On p a r t i c u l a r i s e e n s u i t e c e t t e p r o b l k m a t i q u e a u c a s d e s s t r u c t u r e s o r d i n a l e s du p a r a g r a p h e 3 . En 4 . 2 . on r e t r o u v e p l u s i e u r s d i s t a n c e s e n t r e c l a s s i f i c a t i o n s et hventuellement leurs c a r a c t t r i s a t i o n s axiomatiques B p a r t i r de rbsult a t s s u r l e s v a l u a t i o n s - i n f b r i e u r e s oil s u p b r i e u r e s - d ' u n e n s e m b l e o r d o n n C . A p r h s a v o i r d b c r i t t r o i s m 6 t h o d e s o r d i n a l c s , on p r h s e n t e un " m o d t l e o r d i n a l " d e l a c l a s s i f i c n t i o n dG 2 - J a n o w i t z . Ce m o d t l e , a b s t r a c t i o n de c e l u i d e J a r d i n e e t S i b s o n , p e r m e t notamment d e d t f i n i r e t c a r a c t & r i s e r l a c l a s s e d e s p r o c P d u r e s

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p o i n t f i x e d a n s un e n s e m b l e o r d o n n 6 . On t e r m i n e p a r I e s probIi!mes d ' a g r G g a t i o n olii l ' a p p r c ~ c h e o r d i n a l e p e r m e t d ' o b t e n i r ou d e r e t r o u v e r d e s c n r a c t 6 r i s n t i o n s axionrat i q u e s d e p r o c e d u r e s r6sumant p l u s i e u r s c l a s s i f i c a t i o n s . S i g n a l o n s p o u r f i n i r q u e l a l i s t e d e s r c f c r m c c s , s a n s p r 4 t e n d r e :I l ' e x h n u s t i v i t P , c o n t i e n t p r e s q u e t o u t c e q u e n o u s c o n n c r i s s o n s s u r l e tli6me ensembles o r d o n t i c s e t taxonomie math6mntique.

N . B . Nous i n d i q u o n s c i - d e s s o u s q u e l q u e s n o t , i t i o n s o u definitions u t i l i s c e s t l a n s c k , texte. On n o t e X' l ' e n s c m b l e d e s c o u p l e s d ' c l h m e n t s d e l ' e n s e m b l e X , V(X) l ' e n s e m b l c d e s p a r t i e s d e X , 6',(X) l ' e n s e m h l e des p a r t i e s ?I d e u x 6 1 6 m e n t s d e X , nX l ' e n s e m b l e cles p a r t i t i o n s d e X. S o i t E un e n s e m b l e o r d o n n e a v e c p l u s p e t i t e t p l u s g r a n d ~ l 6 1 1 1 e n;t u n e C k ~ . i f n c c e s deux P l b m e n t s .

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Dans l ' e n s e m b l e o r d o n n k ( I ; , < ) line p a r t i e A d e E e s t : - une pcci.t%e finissantc: s s i ( s i e t s e u l t ~ n i e n t s i ) [ x E A A1 ;

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co-rqdseau. Une a p p l i c , n t i o n f e n t r e deux e n s e m b l e s o r d o n n h s e s t 1:::otond ( c z n t i t o n i s ) s s i x 5 y i m p l i q u e f x 5 f y ( f x 2 f y ) . On n o t e Is0 ( E , F ) l ' e n s e m h l e d e s a p p l i c a t i o n s isotiines d e E dans F.

2 . DONNEES, MODELES ET METHODKS. 2.1

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DONNIlES.

Le p r o c e s s u s q u i c o n d u i t B d e t e r m i n e r d e s c l a s s i f i c a t i o n s d ' u n e n s e m b l e d ' o b j e t s c o m p o r t e p l u s i e u r s & t a p e s . Nous s u p p o s o n s i c i r c s o l u s l e s p r o b l t m e s - f o n d a m e n t a u x - d e c h o i x e t d e r e c u e i l d e s d o u n h e s e t q u e celles-ci s e p r 6 s e n t e n t s o u s l a forme d e tableaux du t y p e s u i v a n t :

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Dans l e t a b l e a u d e l a F i g u r e 1 , I e s t l ' e n s e m b l e d e s " i n d i v i d u s " ( d e n a t u r e q u e l r o n q u e ) , J e s t l ' e n s e m b l e des " v a r i a b l e s " ( c a r a c t k r e s , a t t r i b u t s ) I e s d C c r i v a n t , a . . e s t l a " v a l e u r " d e l a v a r i a b l e i p o u r l ' i n d i v i d u i . Ce t y p e d e donneies e s t 1J t r h s f r C q u e n t : t a b l e a u x d e m e s u r e s , d e n o t e s , d e r e p o n s e s c o d e e s B un q u e s t i o n n a i r e , d e p r b s e n c e ou a b s e n c e d e c a r a c t g r e s , d e r e l a t i o n e n t r e d e u x e n s e m b l e s . L a F i g u r e 2 e s t un e x e m p l z 6 1 C m e n t a i r e d e t e l t a b l e a u a v e c d i x i n d i v i d u s ( p l a r t s c c t t e f o i s en l i g n e ) e t d e u x v a r i a b l e s x e t y . On p e u t d o n c i c i r e p r b s e n t e r c e s i n d i v i d u s comme d i x p o i n t s d u p l a n , c e q u i e s t f a i t 2 l a F i g u r e 3 , c i - a p r h s . C e t exemple - a r t i f i c i e l - nous s e r v i r n d ' i l l u s t r a t i o n d a n s l a s u i t e .

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Un a u t r e t y p e d e t a b l e a u d e donnGes f r 6 q u e m m e n t r e n c o n t r 6 e s t c e l u i ob I = J . Un t e l t a b l e a u c o r r e s p o n d 1'' l a d o n n e e d ' u n e r e l a t i o n b i n a i r e v a l u t e , comme on en o b t i e n t p a r exemple d a n s l e s metbodes d e coinparaison p a r p a i r c s . 2 . 2 . MODELES. 1.e b u t d e l ' a n a l y s e d e s t a b l e a u x d e d o n n 6 e s c o n s i d 6 r 6 s c i - d e s s u s e s t i c i d ' o b t e n i r u n e t y p o l o g i e d e l ' e n s v m b l e d e s i n d i v i d u s , 011 d e l ' e n s e m b l e d e s v a r i a b l e s , o u d e s d e u x . Dans l a s u i t e , n o u s a p p e l l e r o n s X l ' e n s e m b l e d e s 6 1 6 m e n t s 5 c l a s s e r , q u ' i l s ' a g i s s e d e s i n d i v i d u s ou d e s v a r i a b l e s . Nous a l l o n s m a i n t e n a n t p r e c i s e r les m o d e l e s m a t h 6 m a t i q u e s d e t y p o l o g i e l e s p l u s couramment u t i l i s 6 s . A c e t e f f e t n o u s r e p r o d u i s o n s a u x F i g u r e s 4 e t 5 c i - d e s s o u s , des s o r t i e s d e programmes u s u e l s d e c l a s s i f i c a t i o n a u t o m a t i q u e , p o u r l e s d o n n 6 e s d e l ' e x e m p l e p r 6 c C d e n t ( F i g u r e s 2 e t 3).

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Dans l a s o r t i e d u programme NUDOK ("Nu6es Dynamiques", C f . D i d a y 19801, on o b t i e n t une p a r t i t i o n d e X en 3 c l a s s e s : t 6 1 , {8,9,lOI, { 1 , 2 , 3 , 4 , 5 1 . C ' e s t un e x e m p l e d u p l u s s i m p l e m o d s l e t y p o l o g i q u e , c e l u i c o n s t i t u 6 p a r une p a r t i t i o n d e X .

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La sortie du programme de classification hierarchique CLIH, Figure 5 , est souvent appelee dendrograrrirne o u arbre de cZassi~ication.Nous lui avons donne une forme un peu differente B la Figure 6. L'information apportee par ces figures comporte differents niveaux qui examines successivement, vont nous fournir trois autres modgles typologiques. 1") I1 est clair que la Figure 6 est une representation du diagramme de Hasse d'un sup-demi treillis arborescent (sup-demi treillis oh tout element different de 1'ClCment universe1 est couvert par un Clement unique). Ce sup-demi treillis a pour elements minimaux l e s dix QlGments B classer ; B tout autre Clement, numerot6 de 1 1 B 19, correspond une partie de X, formee des elements minimaux inferieurs B cet element. Par exemple, B 17 correspond la partie { 1 , 2 , 3 , 4 , 5 1 . Aux 19 "noeuds" de l'arbre de classification correspond donc une famille de parties de X, contenant X, les parties B un element de X et verifiant la condition que deux parties incomparables ont une intersection vide (autre manigre d'ecrire que l'ensemble de ces parties, ordonne par inclusion, est arborescent). Une telle famille de parties sera appelee une h i i m r c h i e sur X et notee HX. Ses Plements (i.e. les parties de X appartenant B H ) seront appeles les classes de la hierarchie. Un X algorithme de classification hierarchique fournit toujours une telle hierarchic. Dans l'exemple traitb, on a H = ~1,2,3,4,5,7,8,9,10,12,78,910,125,678,1245, X 67C91O , X } . 2 " ) Les Figures 5 et 6 comportent une echelle numeriquc graduee de 0 B 100. Cette echelle permet d'attribuer B chaque noeud, (i.e. B chaque classe A de la hierarchic H ) un indice h(A). Par exemple, X(11) = h({1,2)) = 16 ; X X(18) = X({6,7,8,9,101) = 58. On en deduit un prPordre total s u r HX : A 5 B s s i

h(A) 5 A ( B ) . Ce preordre est compatible avec l'ordre d'inclusion sur P(X) : A c B implique A < B . De maniere generale, on appellehie'rarchie stratififie'e sur X, une hierarchie sur X munie d'un prbordre total sur s e s classes, compatible avec l'ordre d'inclusion.

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4 " ) Coupons le dendrogramme p a r une h o r i z o n t a l e d ' o r d o n n b e X . L e s c l a s s e s d e l a h i b r d r c h i e c o r r e s p o n d a n t a u x n o e u d s s i t u C s j u s t e a u - d e s s o u s ou s u r c e t t e h o r i z o n t a l e , forment une p a r t i t i o n 7x d e s 616ments de X . Par exemple, F i g u r e 6 , n50 = i 1 2 4 5 , 3 , 6 7 8 , 9 1 0 ) . TI\

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I n v e r s e m e n t h t o u t e ultra mC-

t r i q u e s u r X , on p e u t f a i r e c o r r e s p o n d r e u n e h i b r a r c h i e i n d i c b e ou u n e R 2 c h a i n e d e p a r t i t i o n s ( c f . 3 . 2 . 3 . ) . L e s t r o i s n o t i o n s d e h i b r a r c h i e s i n d i c b e s H X , RZchaEne de p a r t i t i o n s d e X e t d ' u l t r a m b t r i q u e s u r X , s o n t donc b q u i v a l e n t e s e t c o n s t i t u e n t l e

mzme modhle t a x o n o m i q u e . Dans l e c a s d ' u n e h i b r a r c h i e s t r a t i f i b e , on a b g a l e m e n t l ' h q u i v a l e n c e a v e c d eu x a u t r e s n o t i o n s ; c e l l e d e c h d n e e ' t e n d u e d e p a r t i t i o n s d e X , e t c e l l e d'uZtra-

J.-P. Barthelemy et al.

530

pr6ordonnunce de X, c'est-i-dire d'un prkordre total sur l'ensenible des paires d'kl6ments de X v6rifiant l'analogue de la condition ultrametrique (cf.3.3.2.). En particulier les ultra-preordonnances B deux classes correspondent aux partitions de X. En conclusion, nous considkrons que nous disposons de quatre modhles taxonomiques pour classer des objets : partition, hibrarchie, hihrarchie stratifibe ( o u chafne etendue de partitions ou ultraprbordonnance) , hierarchic indic6e ( o u k? cha'ine de partitions o u ultram6trique). 2.3. METHODES

.

Le prohleme est maintenant de passer des donn6es B l'un - ou plusieurs - des modeles taxonomiques recherch4s. Par exemple, comment est-on pass6 du tableau de la Figure 2 au dendrogramme de la Figure 5 ?

Certaines methodes dites parfois "directes", partent du tableau de donnees lui-mcme ou d'informations issues directement de ce tableau. Par exemple, dans la mbthode des partitions centrales (Regnier 1965), on associe B un tableau I x J de donnees les partitions de relkvement sur l'ensemble des ohjets B classer, puis on agrege ces partitions. Ainsi si l'on veut classer l e s elements de I, on considhre l e s IJI partitions T . de I d6finies par les variables de J : i et i' sont J

dans la mcme classe cle

71

= a i l j ; puis on agrgge ces IJI partitions ij (cf . 4 . 6 . ) .

ssi a

j'

en une partition "centrale"

Toutefois les methodes les plus usuelles passent par l'intermediaire d'une "proxirnite'" d6finie sur l'ensemble des objets b classer. L'id6e est que si l'on veut classer les elements de I b partir du tableau I x J, deux individus i et i ' proches - c'est-5-dire tels que les lignes i et i' le soient - doivent Etre classes ensemble. I1 faudra donc &valuer numbriquement la proximit6 de deux lignes du tableau, c'est-b-dire d6finir un indice de proximite ou ce qui est 6quivalent et plus courant un i n d i c e de dissimilurit6 d(i,i'), Le choix de l'indice utilis6 conditionne evidemment les resultats ohtenus et ce choix pose de nombreux probl5mes que nous n'ahorderons pas ici (cf. l e s ouvrages de taxonomie math6matique cites dans l'introduction). Pour l'exemple Figure 2, on a pris comme dissimilarit6 la distance euclidienne entre les points, ce qui conduit au tableau de la Figure 7 ci-dessous :

1

2

3

4

5

6

7

e

9

1

0.00

2 3

1.00 3.61

2.83

0.00

4

2.24

3-16

5.10

0.00

5

2.00

2.24

3.00

2.24

6

5.00

5.10

4.24

4.47

3.00

0.00

7

7.00

7.07

5.83

6. 32

5.00

2.00

0.00

e

8.06

8.25

7.21

7.07

6.08

3.16

1.41

0.00

9

10.20

111.05

8.06

9.85

8.25

5.39

3.61

3.61

0.00

10

8.54

8.25

6.00

8.60

6.71

4 - 2 4

3.16

4.00

2.24

10

0.00

0.00

0.00

Figure 7 Ggngralement, une dissimilarit.6 est une application de X2 dans IR' verifiant d(x,y) 2 0 , d(x,x) = 0 et d(x,y) = d(y,x). On peut aussi, plus Qconomiquement, l a

Ensembles ordonnes et taxonomie niathematique c c i n s i d c r e r comme m e a p p l i c a t i o n d e f',(X)

dnns

531

k?+.

S i on a c o n s t r u i t une d i s s i m i l a r i t 6 s u r l ' e n s e m b l e X d ' o b j c t s B c l a s s e r , i l e s t n a t u r e 1 d e r a p p r o c h e r cl ' abord l e s o b j e t s d e f a i b l e d i s s i m i l a r i t 6 , p u i s d ' a u g -

E R+,on pose :

mcnter p r o g r e s s i v e m e n t l e s e u i l . A i n s i , pour

R(h)

RO)

est

=

!(x,y)

I n r e l a t i o n au s e u i l

E X2

> ,p o u r

t e l quc

d(x,y)

5 !!

l a d i s s i m i l a r i t 6 d . C ' e s t u n e r e l a t i o n sym6-

t r i q u e e t on a : h

<

),'

implique

R(A)

5

R(A')

Une t e l l e a p p l i c a t i o n d e R+ d a n s l e t r e i l l i s bool bim d e s r e l a t i o n s b i n a i r e s s y m 6 t r i q u e s s u r x s ' a p p e l l e r a line R*c 2 t 2 1' 1 ; s o n image e s t u n e chaTne 6 t e n d u e d e r e l a t i o n s s y m t t r i q u e s . La b i j e c t i o n e n t r e d i s s i m i l a r i t 4 s e t IR*chaTnes s e r a p r k c i s 6 e a u p a r a g r a p h e 3 . 1 . 1 . La d o n n t c d ' u n e d i s s i m i l a r i t 4 s u r l ' e n s e m b l e X pe rme t d ' b v a l u e r s i d e u x 6 1 6 m en ts s o n t p l u s p r o c h e s q u e deux a u t r e s , ( i . e . s i d ( x , y ) 5 d ( z , t ) ) . Dans c e r t a i n s c a s , c ' e s t u n i q u e m e n t c e t o r d r e e n t r e l e s d i s s i m i l a r i t 6 s ( e t non l e u r s v a l e u r s a b s o l u e s ) q u i e s t s i g n i f i c a t i f e t q u i d o i t @ t r eu t i l i s 6 p o u r r e c h e r c h e r u n e c l a s s i f i c a t i o n . O n e s t a i n s i c o n d u i t 2 l a n o t i o n d e yr~S~i2.~ion,zLrnci, sur X, i.e. d ' u n p r 6 o r d r e t o t a l s u r X2 d o n t l a c l a s s e m i n i m a l e e s t form6e d e s c o u p l e s (x,x). On p e u t a u s s i d 6 f i n i r u n e p r 4 o r d n n n a n c e comme un p r 6 o r d r e t o t a l s u r 8 (X). Nous

2

notons P l a pr4ordonnance a s s o c i 6 e canoniquement 2 l a d i s s i m i l a r i t 6 d : ssi d(x,y) 5 d ( z , t ) . (x,y) Pdfz,t) On p e u t m a i n t e n o n t p r b c i s e r l a m 6 t h o d o l o g i e g 4 n 6 r a l e d e l a t a x o n o m i e mathbmat i q u e . A p a r t i r d e s donn4es " r e c u e i l l i e s " s u r l ' e n s e m b l e d e s Clbments B c l a s s e r , o n c o n s t r u i t d e s d o n n 6 e s d e r e s s e m b l a n c e e n t r e c e s 616me nts ( d i s s i m i l a r i t & , prbordonnance, p-uple de p a r t i t i o n s , ) p u i s on u t i l i s e une procbdure l e s t r a n s f o r mant en l ' u n d e s modilles t a x o n o m i q u e s d 6 f i n i s c i - d e s s u s . Ces p r o c 6 d u r e s s o n t e x t r z m e m e n t n o m b r e u s e s e t d e m a n d e r a i e n t elles-msmes h @ t r e c l a s s i f i 6 e s . Nous n o u s c o n t e n t e r o n s d e d o n n e r q u e l q u e s p r i n c i p e s d e c l a s s i f i c a t i o n , notamment c e u x m e t t a n t en l u m i k r e d e s a s p e c t s o r d i n a u x .

...

1 . On p e u t d ' a b o r d d i s t i n g u e r l e s p r o c 6 d u r e s s u i v a n t l a n a t u r e d e s d o n n 6 e s d e ress e m b l a n c e e t d e s mod6les t a x o n o m i q u e s q u ' e l l e s p r e n n e n t e n c o m p t e . En ne r c t e n a n t p o u r l e s d o n n b e s q u ' u n e d i s s i m i l a r i t 6 ou u n e p r b o r d o n n a n c e e t p o u r l c s m o d e l e s q u e les q u a t r e d 6 f i n i s p l u s h a u t ( p a r t i t i o n , h i 4 r a r c h i e , h i 4 r a r c h i e s t r a t i f i b e ou i n d i c 6 e ) on o b t i e n t d C j 2 h u i t t y p e s e t e n p a r t i c u l i e r l e s de ux l e s p l u s frkquemment r e n c o n t r k s : t r a n s f o r m a t i o n d ' u n e d i s s i m i l a r i t 6 en s o i t une p a r t i t i o n , s o i t une hi6rarchie indic6e. 2 . Les e n s e m b l e s d ' o b j e t s - d o n n b e s ou modhl es - d ' u n c e r t a i n t y p e r e n c o n t r b s e n taxonomie math6matique, o n t d e s s t r u c t u r e s o r d i n a l e s q u i s e r o n t dbveloppdes au p a r a g r a p h e s u i v a n t . On p e u t a l o r s d i s t i n g u e r l e s p r o c b d u r e s q u i u t i l i s e n t e x p l i c i t e m e n t ces s t r u c t u r e s ou l e u r s d 4 r i v b e s ( f e r m e t u r e s , m 6 t r i q u e s ... ) a u n i v e a u d e s r & g l e s d e c o n s t r u c t i o n ou d ' e x i g e n c e s a x i o m a t i q u e s ( p a r e x e m p l e , G t r e i s o t o n e ) . On en v e r r a d e s e x e m p l e s e n 4 . 3 . a v e c l ' o u v e r t u r e u l t r a m b t r i q u e ou l a m6thode d'ajustement de Schader. 3 . Une a u t r e d i s t i n c t i o n e s t f o n d b e s u r l e s p r o p r i 6 t i . s d ' " i n v a r i a n c e o r d i n a l e " d e l a p r o c 6 d u r e . Nous n e c o n s i d b r e r o n s i c i q u ' u n e d e c e s p r o p r i 6 t b s d ' i n v a r i a n c e b i e n q u ' o n p u i s s e en d 6 f i n i r p l u s i e u r s ( c f . 5 c e p r o p o s , B e n z 6 c r i 1 9 6 7 , J o h n s o n 1967, Lerman 1 9 7 0 , S i b s o n 1 9 7 0 , 1972, e t l e s t r a v a u x d e J a n o w i t z ) . D i s o n s q u e deux d i s s i m i l a r i t 6 s sont 6 q u i v a l e n t e s s i l e u r s pr6ordonnances associt5es P e t Pd, sont d 6gales : d(x,y) 5 d ( z , t ) s s i d ' ( x , y ) 5 d ' ( z , t ) , ( i l r e v i e n t a u &me d e d i r e q u ' i l e x i s t e u n e i n j e c t i o n i s o t o n e cp d e k+ d a n s k?+ t e l l e que d ' = c p d ) . S o i t

J . 2 Barthelerny et al.

532

une procedure transformant une dissimilarit6 en une autre dissimilaritl (par exemple, une ultrametrique). On dira que @ est globalerient i n v a r i a n t e si 1'4quivalence de d et d' implique celle de @(d) et O(d'). Cette approche ordinale permet de d6finir une classe de procedures comprenant 6videmment toutes celles dont l a donn6e de depart est une preordonnance, plus celles qui ne dependent en fait - d e facon 6vidente ou non - que de la pr6ordonnance associee i la dissimilarit6 utilisee. On en donnera trois exemples ci-dessous. 4 . Donnons enfin une autre distinction valide uniquement pour l e s procldures de classification hibrarchique, i.e. celles conduisant i une kkchafne de partitions. On peut, d'une part, utiliser un critere "global" pour calculer cette k k h a i n e , par exemple en cherchant l'ultram&~rique " B 6loignement minimum" de la dissimilarite initiale. On peut, d'autre part, obtenir cette IR?cha'ine, pas 2 pas, en calculant s e s partitions successives B partir soit de la partition triviale TI '(pro-

c&iure

agregative ou ascendante) soit de la partition grossiere

TI

(proc6dure

divisive ou descendante). Par exemple, i chaque &tape d'une proc6dure a?re.gative, on calcule la partition suivante en optimisant un critere local ou global parmi les partitions couvrant celle qu'on vient d'obtenir. Dans la procedure la plus rbpandue (algorithme agregatif de Johnson, 19671, on se donne un indice de dissimilarite entre parties de X, et la nouvelle partition est celle obtenue en fusionnant les deux classes de dissimilarit6 minimum de la partition precedente (critkre local). En choisissant comme dissimilarit6 entre deux parties respectivement le minimum, le maximum ou la mkdiane des dissimilarites entre leurs elbments, on obtient des procedures appel6es respectivement lien simple, lien complet et lien m6dian. I1 est clair que ces trois procedures sont globalement invariantes au sens defini ci-dessus. On verra d'autres d6finitions de celle du lien simple au paragraphe 4 . 3. LES STRUCTURES ORDINALES DES DONNEES ET DES MODELES. 3.1. INDICES DE DISSIMILARITE ET L-CHAINES DE RELATIONS. 3.1.1. Dissimilarites. La donnee de dCpart est le plus souvent (cf. 2.3.) un indice de dissimilarit6, c'est-i-dire une application d : $ , ( X I --t L, OL L est une partie de I R ' contenant le z6ro. Ici, nous retiendrons seulement que L e s t un ensemble totalement ordonne avec un p l u s petit element note 0 . due essentielleUne transformation des indices de dissimilarit6 ,)d'(E''L ment B Janowitz (1S78a) reprenant la problematique de Jardine et Sibson (1971) met l'accent sur l'aspect ordinal du problkme de la classification. Nous presentons d'abord cette idee, que l'on trouve Bgalement s o u s une forme moins 6laborCe et dans le cas particulier des ultram6triques (8.3.2.2.) dans l'article de Boorman et Olivier (1973) : A d E L8z(x) et 2 un seuil X E L, on associe la relation sym6trique R(d,A) = {e E p,(X)/d(e) 5 X I , avec les proprietes imediates :

Pour d fix6, on dBfinit une application f : L--t8(p2(X)) par fd(X) = R(d,X). f est une indiciation par L de la chafne Im f L ) ={ R(d,X)/A E L l du treillis d fd booleen 6(p,(X)). On dira que fd est une L-chafne de ce treillis. I1 est immediat que les L-chafnes f sont caract6risees parmi les applications isotones de L dans par les conditions (LCI) et (LC2) suivantes :

6'(6',(X))

Ensembles ordonnis et tawonomie mathematique (LC1)

€-l(

533

92(X)) # 0

(1.~2) ( V e E

P,(x))

I X E L/e E f(h)I

a un minimum.

quand L est fini. Des formes differentes de (LC2) ont CtC On a (LCl)*(LC2) donnGes par Boorman et Olivier et par Janowitz. Toutes ces fornies sont bquivalentes pour L = R+. Une application f : L + 6'(6',(X)) v6rifiant (LC1) et (LC2) dPfinit un indice de dissimilarit6 d . f ' df(e) = Min { A E L/e E f(X)l. (V e E f',(X)) Les applications d + f et f + df sont inverses et, d'aprhs la proposition 1, antitones. Elles Ctablissenf une correspondance bijective entre les indices de dissimilarit6 i v,ileurs dans L et les L-chafnes.

3.1.2. Rtsiduation. Soient E, F deux cnsembles ordonnks. On note l s o ( E , F ) l'ensemble des applications isotones de E dans F. On dit que f E Iso(E,F) est i-e'sidue'e -resp. re'sidueZZe- ssi, pour tout y,G F , f-l({y€F/y y o } ) est un ideal princiual de E (resp. E Y/y > yo}) cstun filtre principal de E. Si f est rbsidube-rCsiduelle-, f-'({y il existe une application residuelle - residube - unique g E I s o ( F , E ) telle que le couple (f,g) &tablit une correspondance de Galois entre E et l'ensemble ordonne F* dual de F - entre E" et F - (Blytli et Janowitz 1972). Les indices de dissimilarit6 se ramenent aux fonctions residuelles en prenant E = L, F = 8(82(X)). Proposition 3.2. est rhsiduelle.

Une application f : L + P ( I P ~ ( X ) ) verific (Lc1) et (Lc2) ssi elle

Cette Gquivalence, qui reformule celle Ctablie par Janowitz (1978a), reste vraie si L est un sup-demi-treillis avec un 6ltment nul. L'application rCsidu6e g correspondant 5 f est dkfinie par : (V

R C@?(X))

g(R)

=

max {df(e)/e c1 R ] .

Dans I'etude des structures des ensembles de classifications intervient la question suivante : F Ptant un demi-treillis, l'ensemble Res(E,F) des applications r6siduelles de E dans F est-il stable pour 1'opPration de demi-treillis induite dans FE ? @and E est u n ordre total et F un inf-demi-treillis, la stabilitt pour l'infimum de Kcs(E,F) est imm6diate. Rlyth et Janowitz proposent en exercice de montrer que la stabilitt pour l e supremum est egalement v6rifiPe lorsque F est un ordre total. Un autre cas de stabilitb pour le supremum est celui oh F est un sup-demi-treillis fini (mais elle disparaft si E n'est plus un ordre total). Ce dernier cas est celui des L-cliaTnes de p ( p , ( X ) ) , ce qui entrafne le fait que l a correspondance d + fd est un anti-isomorphishie de treillis :

Proposition 3.3.

f d v d , = f d h f d , , et fd A d l

= fdV

fd'

3.1.3. Applications 5 la taxonomie. Soit F 5 &'(4:2(::)) un ensemble de relations symbtriques sur X ayant de bonnes proprietes classificatoires. En posant Y = y:F y , on vbrifie que l'on a, pour f F ( (P(y))L telle que Imf(L) 5 F :

-

f F Res(L,F) f E Kes(L,p(Y)). Si Res(L,F) n'est pas vide, on a Y E F, donc Y = yu: maximum de F. En gbnbral, aucun Clement de f f ? ( X ) n'est exclu a priori des classifications consid6rhes et l'on a Y = P2(X) Alors aux L-chafnes de F sont associGes des indices

.

J.-P. Barthelemy et al.

534

d c clissiini1;irith

p a r t i c u l i c 3 r s ; on n o t c d F I 'ensemble dc' c e s i n d i c e s .

Deux u s a g e s d e 1 3 c o r r e s p o n d a n c e F

+

d

I?

se

:

prhsentcnt.

- L ' u n d e s b u t s d e l a t axonomi c mnt l i &mat i que e s t tlc c o n s t r u i r e une bunnt3 9,(XI

:ipplication

a

: L

+

d F , c e q u i p e u t G t r e ramen&

Ii

l a rc~ctii.rclii. d ' u n e a p p l i c ; i -

t i o n 0 ' : Res ( L, p ( B , ( X ) ) ) + Kes ( L , F ) . Janowitz ( 1 9 7 8 a ) p o s e a i n s i l e p r o b l t n i c d e l a r e c h e r c h e e t d e 1' ; t i i de d e s iiietliudes d e c l a s s i f i c a t i o n dans d e s t c r m e s t o u t - i - f a i t o r d i n a u x . C e t t e a p p r o c l i c C s t ;ihordCe a u p a r a g r a p t i e 4.4.

- P o u r G t i i d i r r l a s t r u c t u r e d ' o r d r e dt, (IF, u n u t i l i s e l e f . i i t q u ' c ~ ll e <'st d u n l e d e c e l l ? d e R e s ( L , F ) . P a r e x e m e l c , s i F e s t u n t r e i l l i s , R L - s ( I . , F ) e s t un s o u s - t r c i l l i s d e I s o ( L , f ? ) , donc d e F d F e s t d o n c u n t r e i l l i s , q u i p"ut c u n s e r v e r c e r t a i n e s p r o p r i t t g s d u t r e i l l i s F . Au p a r a g r a p h e 3 . 2 . 2 . , c c t t e d&rnarc~tie(1st appliqu&t, h 1 ' & t i l d e d e s t r e i l 1 i s d ' u l t r a n i 6 t r i q u e s .

.

3 . 2 . PARTITIONS e t ULTMMETR1C)UES : TKEIT.I,TS SEMI-MODULAIKES.

.

3.2.1

P a r t it ions

.

s L e s p a r t i e s c ~ e63, (x) I C S p1iis t y p i q u e m e n t c l n s s i f i c n t o i r e s s u n t ~ e Cquivalences stir X. On n o t e r a nX ( o u n) l e t r e i l l i s d e s C q u i v a l e n c e s s u r X ( i d i , n t i f i 6 e s aux p a r t i t i o n s d e X a v e c l ' < i r d r e de f i n c x s s e ) , s o n ciltmeiit n u l , q u i e s t l ' b q u i v a l e n c e

:I

1x1

=

i[

=ff2(X)

s o n niaximuni, T O

n classes.

Le t r e i 1 1 i s g t om; t r i q u e ( rl t omi q u e c' t s emi -niodu 1a i r e i n f C: r i e 11 renie n t ) d c s p a rt i t i o n s e s t b i e n connu ( c f . p a r c'xernple R i r k h o f f 1967, Bnrhut e t M o n j a r d e t 1 9 7 0 , Crawley e t D i l w o r t h 1973, G r z t z e r 1 9 7 8 ) . Sa c o m p l e x i t & a Ct6 h t a b l i e a v e c l a d t m o n s t r a t i o n d r l a c o n j e c t u r e d e Whitman ( 1 9 4 6 ) p a r Pudlrik c t Tcimii ( 1 9 8 0 ) .

3.2.2.

UltramGtriqucs.

I.es 111 t r m i t t r i q u e s o n t p r i s u n e g r a n d e i m p o r t a n c e en ta xonomic d e p u i s 1 ' o h s e r v a t i o n s i m u l t a n b e p a r .Julinson ( 1 9 6 7 ) e t B e n z e c r i ( 1 9 6 7 , r e p r i s dans Benzccri 1973) d e l ' t q u i v a l e n c e s u i v a n t e :

P r o p o s i t i o n 3.5. ( 1 ) ( V h C L) (2)

(V x,y,z

E X)

8, (XI

Les d e u x c o n d i t i o n s s u i v c i n t e s s o n t t q u i v a l e n t e s p o u r d E L R ( d , E ) C rlX

d(x,z)

5

max ( d ( x , y ) , d ( y , z ) )

I:n d ' a u t r e s t e r m c s l ' c n s e m b l e d a n s I. ( L . - u l t r a m t t r i q u e s

$,x

(ouu) d e s u l t r a m t t r i q u e s s u r X

s u r X) e s t i d e n t i q u r 3 dT,

u s u e l , e s t un t r e i l l i s a n t i - i s o m o r p h e d e s t a n t i - i s o m o r p h e :i n .

[ i n t j i a l i t t (iltram6triquel.

h Re

(par.3.1.4.),

(TA,n), f:n

I:

valeurs

e t , avec l ' o r d r e

p a r t i c u l i c r , p o u r I*

=

IO, 1 t ,

n

p o u r 1 ' C . t u c l e dtis p r o p r i c t c s d e s t r e i l i s u e t K c s ( L , n ) , on p e u t d ' a b o r d s u p p o s e r I, f i n i . Dans c e c a s , l e s t r e i l l i s f i n i s I s o ( L , n ) o n t p o u r r e l a t i o n d e r o i ~ v e r t u r el a r e s t r i c t i o n d e c e l l e d e : ( V f , f ' t n'a) f > f ' Q "3 h o E I.) f ( h ) > f ' ( h o ) l A [(V h E L - I h 1) €(A) = f'(h)l. Les t r o i s t r e i l l i s n l ' , I s o ( l , , n ) e t r c s ( l , , n ) o n t l a m e m e g r a d u a t i o n = g ( f ( h ) ) e t s o n t s e m i - m o d u l a i r e s s u p & r i e u r e m e n t (Moniez 1 9 8 2 ) , AEL [,cur lonpueur commune e s t I + ( I L I - 1 ) ( 1 x 1 - 1 ) . Le t r c i l l i s f i n i U Gt:int g(f)

sous-suo-denii-trei

I 1 i s d e L p 2 ( X ) c u n t e n a n t s o n 6lGment n u l . Y i n d u i t

un

L I ~ C ouvc'r-

Ensembles ordonnes et taxonomie mathematique

535

L'application lilt e s t idempotente, isotone, anti-extensive [ult(d) 5 d] image e s t u . L'ultrambtrique ult(d) -0cii.e d a de nombreuses appellations traditionnelles : ultramtitrique inftrieure maximale (ultimax), s o u s dominante, distance du moindrc saut, fernirture transitive floue de d (par dualiti.) ... Elle correspond h I'une des methodes de classification les p l u s connues dont on rappellc quelques propri6tii.sau paragraphe 4 . 3 . ) . e t son

L'ouverture ultram6trique esiste encore lorsque L n'est p a s fini. Pour l a construire, on utilise le fait que L

est un treillis d'applications dbfinies P2 (X) sur un ensemble fini, en posant, pour d t L , L~ = Imd(82(x)) et '2(x)

e, (x)

ult ( d ) , oil u l t d cst I'ouverture ultrambtrique dans I. . De facon d d analogue, on montre que le treillis U vhrifiela condition de semi-modularitb d e Dubreil-Jacotin, Lesieur et Croisot (1953) (Cf. Leclerc 1979). ult(d)

=

Une Ctude plus combinntoire des ultram6triques et de leur treillis a b t b faite par Leclerc (1979, 1981). Reprenant une idbe d e Hubert ( 1 9 7 7 1 , on y gbnbralise la coatomicitb de n par l'hcriture de t o u t e ultrambtrique comme supremum d'au plus n-1 ultrametriquessup-irrbductibles. Si L a un hlbment universel, il y a une ghn6ralisation analogue de 1'atomicitP de n . Finalement, les proprietbs ordinales des treillis d'ultrambtriques apparaissent trks proches de celles d e s treillis des partitions. Pour L fini, le dbnombrement d e s L- ultram6triques de msme graduation serait intbressant pour la statistique des classifications. Cn donne plus loin une facon de dhnombrer toutes les L-ultrambtriques. 3.3.ULTRAPREORDONNANCES ET HIERARCHIES : DEMI-TREILLIS A MEDIANES. 3 . 3 . 1 . Demi-treillis b mhdianes.

Un demi-treillis 5 mhdianes est un inf-demi-treillis vhrifiant les deux conditions suivantes (Sholander 1954 ; cf. Bandelt et Hedlikova 1983) : (Ml) Tout idbal principal de M est un treillis distributif. (M2) Pour tout triplet x,y,z de M tel q u e x v y , y v z , z v x existent, x v y v z existe. On dbfinit dualement un sup-demi-treillis 5 mbdianes. Sholander montre notamment qu'un demi-treillis B medianes M admet un plongement avec de bonnes propribths dans un treillis distributif M'. I1 suffit en fait de prendre pour M' le treillis d e s parties commencantes des irrbductibles de M : M est une partie commencante de M' et tout element de M' est supremum d'une partie finie d e M. I1 en resulte que le plongement est isom6trique. Le fait pour un ensemble d'avoir cette structure peut donc a priori faciliter l'obtention d e solutions aux problkmes d'ajustement et d'agrhgation. Des exemples en seront donnbs aux paragraphes 4 . 3 . et 4.6. 11s utilisent notamment le fait que tout triplet x,y,z E M posshde une mediane unique (x A y) v (y A z ) v ( z A x). Pour M fini et k impair, tout k-uple a egalement une mbdiane. De plus, celle-ci conserve les propribtbs mbtriques &tablies par Barbut (1961) pour l a mbdiane dans les treillis distributifs (Bandelt et Barthblemy 1983).

Les demi-treillis 5 mbdianes qui vont Stre considerhs ici sont des demitreillis de charpentes (Bandelt 1982). Soit E un ensemble ordonnh fini, Min E(Max E ) l'ensemble de s e s Clbments minimaux (maximaux). Une partie C de E une charpente ("frame") de E ssi les deux conditions suivantes sont verifihes

est

J,-P.Barthelemy e t al.

536 par C : (Cl) (Min E) U (Max E ) 5 C ( V x E Min E ) ( V y E Max E ) (C2)

(z

E C/x 5 z 5 y) est totnlement ordonnh.

L'ensemble e ( E ) des charpentes de E, ordonn6 par l'inclusion, est un demitreillis 5 m6dianes (avec pour Cl6ment nu1 Co = (Min E) U (Max E)). On dira de deux charpentes C et C ' qu'elles sont presque disjointes lorsque C n C ' = Co. L'exemple le plus imm6diat de demi-treillis de charpentes est don& par l'ensemble des chafnes Ctendues (c'est-$-dire contenant le maximum et le minimum) d'un treillis fini. Si E est un treillis distributif, et F l'ensemble ordonn6 des bl6ments sup-irrbductibles de E, l'inf-demi-treillis des chaines btendues (charpentes) de E est dualement isomorphe au sup-demi-treillis des pr6ordres totaux compatibles avec F. Celui-ciest donc h medianes. En particulier l e supdemi-treillis des preordres totaux sur un ensemble fini S est h mtdianes. 3.3.2. Ultrapreordonnances. Une prbordonnance (dissimilarity relation, PR) sur X est un pr4ordre total P sur P z ( X ) . Soient S 1 , ...,Sk 5 f'z (XI les classes de P indicCes dans 1 'ordre induit par P, C en posant Ro

=

P

= (Ro,

8, Ri

=

Rl,

..., %)

la chafne Ctendue de

S 1 , Rz = S i U S z , .

..R.1

= S.U 1

associbe

ff(82(X))

Ri-l,...,\

=

6)2(X).

i P

On

reprend la demarche du paragraphe 3.2.2. en s'intbressant aux prbordonnances P pour lesquelles les blbments de la chaine Cp ont de bonnes propribtfs, et plus prtcisbment au cas oij ce sont des equivalences sur X. Proposition 3.6. Les deux conditions suivantes sont equivalentes pour une priordonnance D sur X : (1)

La cha?ne CD est une chaine de

nX'

(2) Pour tous x,y,z E X, on a : ({x,y},{x,z})ED

ou ((x,y),(y,z1)€

D.

Une pr4ordonnance vbrifiant ( 1 ) et (2) est une ~rltruprdordonnance (ultradissimilarity relation, UPR) sur X. Les conditions de la proposition 3.6. correspondent respectivement h celles de la proposition 3.5. : le prbordre D(u) induit sur 6 ' 2 ( X ) par une ultrambtrique u est une ultrapreordonnance sur X tandis que l'image de L par laL-chaine fU est la chaine des partitions C (privbe de D(u) R O si fu(0) # no). L'ensemble

d3X

des UPR sur X est un sup-demi-treillis B mbdianes, dualement

isomorphe au demi-treillis des charpentes (chaines) de T l . De plus, d'aprgs l a condition (2) ci-dessus, c'est une partie finissante du demi-treillis px des prbordonnances sur X et, donc, un sous-sup-demi-treillis absorbant (Schader 1980). On dispose dc formules de recurrences pour le c a l c u l du nombre D des -n,k ultraprbordonnances B k classes (de mgme graduation) et du nombre total D des -n ultraprbordonnances sur X (avec n = 1x1). S dCsignant les nombres de Stirling de seconde espece -9P Proposition 3 . 7 .

Les nombres D -n

et

vCrifient :

n- 1

D -n

=

=

i=l

sn,; D i

(Lerman 1970)

n- 1

D -n,k

=

E S D i=1 -n,i -i,k-1

(Schader 1980)

Ensembles ordonnis et taxonomie mathematique

avcc

O n iin t l e d u i t , p o u r I. f i n i , I c nombre U -11,. " = IL :

531

des L - u l t r a m ~ t r i q u c s s u r

1x1

soit,

U

-n,V

i=l 154 u l t r a m k t r i q u e s sur 1 1 , 2 , 3 , 4 3 h v a l e u r s d a n s

I 1 y a , pCir exemple, {O, 1,2,3 t.

3.3.3.

lli6rnrcliies.

P o u r iine u l t r n p r G o r d o n n a n c e D s u r X e t p o u r l a cii;i?ne & t e n d u e CD = ( T ~ , T T ~ , . . . , ~ a~ s)s o c i g e , on p e u t c o n s i d b r e r I ' e n s e m b l e H = HD d e s c l a s s e s d e s C q u i v a l e n c c s d e CD : i l e s t n a t u r e 1 en c l a s s i f i c a t i o n d e s ' i n t h r e s s e r a u x c l a s s e s .

H vPrifie lcs propribtes : X E H b ?! f l (H3) ( V x E X ) [xi E H (H4) ( V h , h ' E H) h n (H1)

(H2)

Une note

11'

E {h,h',@!

hiirarchie s u r X e s t uii h l g m c n t d e P ( p (X)) v k r i f i a n t ( H l ) 5 (H4). On

VX l ' e n s e r n b l e

des h i e r a r c h i e s s u r X.

Pour H

EgXe t

Y

5 X,

l'ensemble

{ h F H/Y 5 h l est totalement ordonnt p a r l ' i n c l u s i o n , c e q u i a deux consequences : e s t I'ensemble dcs charpentes de

p(X) - i01

d e m i - t r e i l l i s arborescent, c'est-;-dire h l v 11: c 111 v h3 ou h l V ti2 5 hz minimale i n i q u e d e H c o n t e n a n t h n h ' . Soit alors D

H

V

; t o u t e h i t r a r c h i e H e s t un s u p -

v t r i f i a n t : (V h , , h , , h i € H) h ; . Pour h , h ' E H , h v h ' e s t l a c l a s s e

l e preordre p a r t i e l d e f i n i s u r ( { x , y j , { z , t } ) E DH

Q

x v y

5

P?(X) p a r

:

z v t.

I,e p r b o r d r e DH v e r i f i e l a c o n d i t i o n ( 2 ) d e l a p r o p o s i t i o n 3 . 7 . , mais n ' e s t p a s t o t a l . P a r c o n t r e , t o u t e p r k o r d o n n a n c e s u r X c o n t e n a n t DH e s t u n e u l t r a p r t o r d o n n a n c e s u r X . 011 a s s o c i e a i n s i 2 t o u t e hierarchic u n e c l a s s e d'UPR : S o i e n t kI u n e h i t r a r c h i e s u r X e t D un p r t o r d r e t o t a l s u r $?(X) Proposition 3.8. c o n t e n a n t DH e t m i n i m a l avec c e t t e p r o p r i g t i . . A l o r s D e s t u n e u 1 t r a p r ; o r d o n n a n c e s u r X e t HD = H .

Les h i g r a c h i e s s o n t d o n c e x a c t e m e n t l e s e n s e m b l e s d e c l a s s e s d e f i n i s p a r l e s u l t r a p r e o r d o n n a n c e s nu l e s u l t r a m e t r i q u e s s u r X ( c e s o n t les f a m i l l e s d e b o u l e s de c e s d e r n i k r e s ) ; d'oh l e s a p p e l l a t i o n s c l a s s i q u e s , B l a s u i t e de Benzecri, de h i G r a r c h i e s s t r a t i f i g e s pour les u l t r a p r 6 o r d o n n a n c e s , e t d e h i e r a r c h i e s i n d i c h e s p a r les u l t r a m e t r i q u e s ( r e s p e c t i v e m e n t " o r d e r e d t r e e s " e t "valued trees" chez Boorman e t O l i v i e r ) . Une e t u d e d e s d e m i - t r e i l l i s d e h i e r a r c h i e s , o r i e n t g e v e r s l a t y p o l o g i e e t l a c o m p a r a i s o n d e c e l l e s - c i , a e t b f a i t e p a r L e c l e r c ( 1 9 8 2 ) . On y met notamment l e demi-treillis e n r e l a t i o n avec l ' o r d r e rnodulaire d e s p a r t a g e s d e I ' e n t i e r n - 1 e n e x h i b a n t un m o r p h i s m e d ' o r d r e , p r e s e r v a n t l a c o u v e r t u r e , d u p r e m i e r s u r l e s e c o n d . Le p r o b l P m e d u dbnombrement d e s h i e r a r c h i e s r e m o n t e B S c h r i j d e r ( 1 8 7 0 ) . Comtet ( 1 9 7 0 ) a dCnombr6 l e s h i e r a r c h i e s d e m&ne g r a d u a t i o n p a r l e s n o m b r e s d e S t i r l i n g 2 - a s s o c i e s d e s e c o n d e e s p t c e , e t e n a d b d u i t un c a l c u l p a r r b c u r r e n c e e t un G q u i v a l e n t a s y m p t o t i q u e d e s n o m b r e s ( d u q u a t r i k m e p r o b l k m e ) d e S c h r o d e r .

xX

Les h i e r a r c h i e s maximales ( d i t e s gbneralement h i e r a r c h i e s b i n a i r e s ) o n t 2n-1 b l b m e n t s . P o u r l a s t a t i s t i q u e d e s h i e r a r c h i e s , l a r e p o n s e B c e r t a i n e s q u e s -

J.-P. Barthelerny et al.

5 38

tions surl'intersection des hiCrarchies serait intCressantc. Plus gbneralement, les problemes classiques d'intersertion de ch,iPnes peuvent stre posGs pour l e s intersections de chdrpentes. Par exemple, pour un ensemble ordonn6 F fini, on pc'iit chercher l e nombre maximum de ses charpentes maximales presque disjointes deux 1' deux. Si E a un plus petit et un plus grand flfment c'est le nombre maximum d c chaPnes maximales presque disjointes de E. Une rCponse est fournie p a r l e th6ot-Cme de Menger (1927, cf. Berge 1 9 7 0 ) . 4 - COMPARAISONS, AJUSTEMENTS, AGREGATION : METHODES ORDINALLES.

4.1. Situations. Nous nous intkressons, ici, d u s mCthodes de classification qui conduisent construire une fonction 9 : E + F, E et F Ctant deux ensembles ordonnes (dans l a pratique, il s'agira, bien sSr, des ensembles ordonnbs, treillis ou demi-treillis dCcrits au paragraphe 3 ) . Une telle fonction sera appelCe Line prucd&irt di,

classification. Deux cas se rencontrent essentiellement : 2 F. On dira que 9 est un ajustement. Un exemple classique est : E = L "(') (ensemble des indices de dissimilarit4 bur X h valeurs dans I.) et F = L,X (ensemble des ultramktriques sur X, 2 vnleurs dans L). Un autre serait E = 8 ( 8 , ( X ) ) (ensemble des relations reflexives et sym6triques sur X) et F = r'' X (ensemble des kquivalences, ou des partitions, de X). Dans le premier cas on "approche" une dissimilarit6 par une distance ultrametrique de manihre B construire une hi6rarchie indicke ; dans le second on "construit" une partition 5 partir d'une "relation de ressemblance" quelconque. (1) E

u

(2) E = U FV. On dira alors que 9 est une agre'gaiation. Ce c a s se rencontre, par exemple,"dans la situation suivante : l'ensemble X est dkcrit par v variables, chacune d e ces variables induit sur X une partition : deux blkments pour lesquels elle prend la mOme valeur sont dans l a m@me classe. I1 s'agira, des lors de rksumer, en une seule, ces v partitions (on a donc, ici, F = I$,,). Bien scr la fonction 9 ne sera pas n'importe quai. On exigera qu'elle vCrifie certaines "contraintes" (au sens le plus large d e c e terme). On peut regrouper ces contraintes en trois rubriques (d'ailleurs largement empibtantes). a) .Rhgles On indique explicitement l'art et la manihre de . . . . . . .de . . . construction. ........... construire la fonction 9. On ohtient ainsi les proce'dures a2gorithrnique.s (par exemple, la construction d'une hierarchic indicbe, ou stratifiee, en fusionnant, h chaque niveau, des classes, de maniere B optimiser un critgre). On se restreint aux fonctions 0 verifiant des b) ContrainteS-"axjomatiques". conditions jugees indispensables du point de vue oh l'on se place. On dCbouche alors sur des problemes d'existence et le cas &heant, d'unicite o u sur la caract6risation de toutes les "solutions". ___________S__-T_____-------

On s'intkresse aux fonctions 9 qui permettent c) Contraintes_dloetjmjsatjon_. --'-r-------"--T-r--n------Par exemple, lorsque est un ajustement, d 6tant d'optimiser un critere une distance sur E, on demande que 0(r) soit une solution de Min d(r,s).

.

sEF

Cette r6partition en (a), (b), (c) appelle un certain nombre de remarques : Remarque 1. Les ensembles ordonn6s interviennent fort diversement dans ces trois cas. Si les constructions algorithmiques (cas (a)) ne sont qu'exceptionnellement fond6es sur des crithres ordinaux (la methode dite du "lien simple" que nous rencontrerons un peu plus loin constitue une telle exception), les "axiomes" (cas (b)) le plus souvent rencontr6s sont fortement libs aux ordres de E et F. Enfin, pour le cas (c), les criteres qu'utilise le taxonomiste sont parfois lies B ces structures ordonn6es.

Ensembles ordonnes et taxonomie mathematigue

5 39

Remarque 2 . 1.a r u b r i q u e ( c ) ( e t d a n s une c e r t a i n e m e s u r e ( a ) ) i m p o s e l ' e x t e n s i o n de l a notion de procedure de c l a s s i f i c a t i o n . I 1 n ' y a , en e f f e t aucune r a i s o n de p r c s u p p o s e r 1 ' r i n i c i t c d ' u n h l c m e n t d e F q u i o p t i m i s e un c r i t t r e . On e s t a i n s i conclriit 311x < ,< l -L i icati:or;, q u i s o n t d e s a p p l i c a t i o n s d e E d a n s 1 ' e n s e m b l e cles p a r t i e s d c F . I'

Keni;irqur. 3 . L ' c x e n i p l e q u ' o n R d o n n e p o u r ( c ) m o n t r e q u ' i l p e u t 8 t r e n G c e s s a i r e d ' i i t i l i s e r une d i s t a n c e (ou un i n d i c e d e d i s t a n c e ) s u r E . Indgpendamment d c t o u t s o u c i d ' a j u s t e m e n t , d e t e l l e s m c t r i q u e s s o n t f o r t u t i l e s pour comparer d e s c l a s s i f i c a t i o n s . A c e p r n p o s , nous r e t r o u v o n s nos t r o i s r u b r i q u e s . Les r t g l e s d e c o n s t r u c t i o n p c rme t t e t i t d e c a 1c u 1 e r r f f e c t ivemen t un e d i s t a n c e Le s c o n t r a i n t e s a x i o n i a t i q u e s c u n d u i s e n t 5 ne c u n s i d e r e r q u e c e l l e s q u i v 6 r i f i e n t c e r t a i n e s c o n d i t i o n s , j u ~ : G e s " n a t u r e l l e s " clans l e c o n t e x t e oh l'on t r a v a i l l e . E n f i n , on p e u t t'xiger q u ' u n c r i t G r e s o i t o p t i m i s r ( e n t r a v a i l l a n t , p a r exemple, a v e c d e s " d i s t a n c e s d u p l u s c o u r t rhemin")

.

.

Reniarque 4 . Cornme s o t i v e n t , p l u t 8 t q u e 1 ' C t u d c d e ('11, ( b ) , ( c ) p r i s i s u l e m e n t , un p o i n t t o u t 11 f a i t e s s e n t i e l s e r a l a d G t e r r n i n a t i o n d e r a p p o r t s e n t r e c e s t r o i s c a s . P a r e x e m p l e , Line p r o c e d u r e v e r i f i a n t u n e c o n t r a i n t e d ' o p t i m i s a t i o n s e p e u t c l l e c o n s t r u i r e I: l ' , l i d e d ' u n n l g o r i t h m e ? q u e l l e s s o n t s e s p r o p r i G t 6 s ? ( c o n f r o n t a t i o n aux a x i o m a t i q u c s d ' a u t r c s p r o c G d u r e s : q u e c o n s e r v e - t - o n , q u ' a b a n d o n n e t-on ? ) . P e u t - o n n l l e r j u s q u ' 2 lcas c a r a c t 6 r i s e r a x i o m a t i q u e m e n t ? D;ins c c t t e qlJatrii.nie p a r t i e , n o u s G t u d i e r o n s s u c c e s s i v e m e n t l e s c o m p a r a i s o n s , a j u s t e m e n t s e t a,qrGgntions d e c l ~ s s i f i c a t i o n s , : I l ' a i d e d ' e x e m p l e s , sous l ' 6 c l a i rage des t r o i s rubriques ( a ) , ( b ) , ( c ) . 4 . 2 . Compnraisuns de c l , i s s i f i c a t i o n s . 1.e s ~ 1 6 < ~ ~ & s iS~U~I !E< ce o: n s t i t u e n t d e s m G t r i q u e s f o r t l i 6 c . s h sa s t r u c t u r e ordonnee. L k t e l l e s d i s t a n c e s s ~ n td c s " d i s t a n c e s du p l u s c o u r t chemin" d n n s l e g r a p h e ( n o n o r i e n t h ) d e c u u v e r t u r e d e E , u n e a r S t e uv d e c e g r a p h e e t a n t 6valui:e p a r I . b ( u ) - $ ( v ) I , 9 G t a n t une f o n c t i o n , 5 v a l e u r s r e e l l e s , s t r i c t e m e n t c r o i s s i i n t c siir E . O n t r o u v e r a , clans l ' n r t i c l e d e s y n t h s s e d c M o n j a r d e t ( 1 9 8 1 ) , l e s d i v e r s e s p r o p r i 6 t i . s a c t u e l lenient c o n n u e s d e c e s d i s t a n c e s a v e c l e s i n d i c a t i o n s b i b l i o g r a p h i q u e s c o r r e s p o n d a n t e s . En p a r t i c u l i e r , u n e h y p o t h t s e s u r $ ( v a l u a t i o n i n f e r i e u r e ou s u p 6 r i e u r e ) f o u r n i r a une c o n d i t i o n n g c e s s a i r e e t s u f f i s a n t e p o u r q u e l a d i s t a n c e g G o d 6 s i q u e dd, ( d o n c d e f i n i e p a r une c o n t r a i n t e d ' o p t i m i s a t i o n ) p u i s s e S t r e c a l c r i l t e p a r l a "formule" :

(1)

d (r,s)

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cb(r) + g ( s ) - 2 + ( r

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? , b ( r v s ) - , + ( r ) - $ ( s ) , s i E e s t un sup-derni-treillis.,

A

s ) , s i E e s t un i n f - d e m i - t r e i l l i s .

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P a r a i l l e u r s , m @ m e l o r s q u e (1 n ' c s t p a s m e v a l u a t i o n ( i n f b r i e u r e o u s u p e r i e u r e ) , l e s f o r m u l e s ( 1 ) e t ( 1 * ) p e u v e n t C t r e u t i l i s 6 e s p o u r c a l c u l e r un i n d i c e de d i s t a n c e . Celui-ci s e r a p e r t i n e n t en " t h t o r i e de I'Information". Lorsque 9 r e p r t s e n t e une m e s u r e d ' i n f o r m a t i o n (on s e p l a c e d a n s I e c a s E = II ) , ( 1 ) e t ( 1 9 0 X r e p r G s e n t e n t u n c summe d ' i n f o r m n t i o n s c o n d i t i o n n e l l e s ( p o u r q u e l q u e s p r 6 c i s i o n s c f . Kampe d e F e r i e t 1 9 7 4 , L o s f e l d 1 9 7 4 ) . I1 e s t d ' a i l l e u r s r e m a r q u a b l e q u e l e s mesures, c l 3 s s i q u e s , d'informnt ion conduisent des "valuations". I'
partition

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p r t s , , + ( T ) e s l~' e n t r o p i e d e r i ) . De m G m e , on o b t i e n d r a une v a l u a t i o n s u p t r i e u r e e n p o s a n t +(n) = p . C e t t e d e r n i e r e q u a n t i t ; e s t l i G e :' 1 " ' i n f o r m a t i o n h y p c r b o P

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De n o m b r e u s e s d i s t a n c c s (gt!odGsiques e t / o u v t r i f i s n t ( 1 ) u u ( 1 " ) a i n s i q u e q u c , l q u e s a ~ i t r c s ) l i h e s o u n o n i l a t h C o r i e d e l ' i n f o r m a t i o n o n t ttt! G t u d i e e s ,

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e t l e s i i i C r a r c I i i e s (E

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l e p r e m i e r c a s , on p e u t c i t e r : A r n h i e e t Doorman ( 1 9 7 8 ) , Boorman ( 1 9 7 0 ) , I3oorm;in e t A r a b i c ( 1 9 7 2 ) , BarthGlemy ( 1 9 7 9 ) , Day ( 1 9 8 1 ) . D a n s l e s e c o n d c u s : Robisucl e l Hobisud ( 1 9 7 2 ) , H u b e r t o t 13;iltcr ( 1 9 7 7 1 , Lec l e r c ( 1 9 8 P ) , Mnr}:ush ( 1 9 8 2 ) , S o h 1 e t RohlF ( 1 9 6 2 ) , Watermnn e t S m i t h ( 1 9 7 8 1 , Wil1i;irns e t C l i f f o r d ( 1 9 7 1 ) . N o u s v e n o n s d e v o i r i n t e r v e n i r l e s r u b r i q u e s ( 3 ) ('t ( c ) . Q u ' c n e s t - i l de ( h ) ? O n s e r a t t n c h e i i i U I I problGme dia Iri c o r a c t 6 r i s ; i t i o n n x i m a t i q u e de d i s t a n nnns l e r a s d e s r e l a t i o n s b i n r i i r e s c t ' theme reniontc Kcnicny ( 1 9 5 9 ) d o n t l a tiGniarc h t . a ;tt r e p r i s e p o u r l a c l a s s i f i c a t i o n p;ir M i r k i n e t Cliernyi ( 1 9 7 0 ) q u i rardL't 6 r i s ; S r e n t l a d i s t a n c e d e I n d i f f G r e n c e syni;triqrie e n t r e d e s r e l n t i o n s d ' 6 q u i v a l e n c c . N o t o n s q u ' i l s'agit, en f n i t , d e I n d i s t a n c e g & o d t s i q u e s u r !Ix p o u r P 2 TI = {Al, A ] ) . Le ccls d e s I i i G r a r c t i i e s ;i c t c G t u t l i G par , l ( n ) = 1 [ A i l (avoc i =1 P L e c l e r c ( 1 9 8 2 ) e t Plargush ( 1 9 8 2 ) . Remarquons e n f i n ( R a r t h 6 l e n i y 1979h) q u e C E S c n r a c t 6 r i s n t i o n s a x i o m a t i q u r s dependent 6 t r o i L e m e n t d e l a s t r u c t r i r e orc1imni.e de E e t sous d e s l i y p o t h t s e s c o n v e n a b l c s ( I n s e m i - m o d u l a r i t 6 , p a r r x c 9 n i p l e ) a p p a r a i s s e n t s o u v e n t comme des a p p l i c a t i o n s d e rssul t a t s b e a u t o u p p l u s gt!nGr;iux.

...,

4 . 3 . A j u s t e m e n t s I : t r o i s cxxemples. 1 , ' e x e m p l e l e p l u s c l n s s i q u e d ' a j u s t e m e n t e s t s a n s d o u t e l a tr.2 sivipTc ( " s i n g l e l i n k a g e " ) , C e l l e - c i a g t t ! abordcc. d e d i v e r s e s m a n i b r e s e t a r r s u d i v e r s noms ( c f . 3 . 2 . 2 . ) . h n s une p e r s p e c t i v e d c c l u s s i f i c a t i o n a s c e n d a n t e tiit!r a r c h i q u e , on f u s i o n n e deux c l a s s e s d o n t deux 6 l b m e n t s s o n t l e s p l u s p r o c h e s v o i s i n s ( d ' o h son n o m ) .

I-n d ' a u t r e s t e r m e s , i l s ' n g i r a d ' a p p r o c h e r u n e d i s s i m i l a r i t 6 p a r u n e u l trcim t t r i q u e , o u , c e q u i r e v i e n t a u m@rne, ( c f . 3.1.3.) d e t r a n s f o r m e r Line a p p l i c a t i o n r g s i d u e l l e f : T,+ ff(e,(X))e n uiie a p p l i c a t i o n g : L - , f ( I [ x ) . La m t t h o d e d u l i e n simple c o n s i s t e , p o u r uiie d i s s i m i l a r i t 6 d , <' c a l c u l e r I ' u l t r a m C t r i q u e supremum d e s u l t r a m 6 t r i q u e s i n f c r i e u r e s B d . I1 s ' n g i t 1,'i d e I ' o u v e r t u r e u l t r a m b t r i q u e de 3 . 2 . 2 . . C e t t e d e r n i h r e se c o n s t r u i t e f f i c a c e m e n t en e x t r a y n n t du g r a p h e c o m p l e t s u r X , v a l u 6 p a r d , un a r h r e m i n i m a l . C e t t e r e m a r q u e i l l u s t r e l ' a s p e c t ( a ) ( r 6 g l e s d e c o n s t r u c t i o n ) . L e mot " o u v e r t u r e " , quaiit 5 l u i , r e n d c o m p t e d e ( b ) . En e f f e t , c h e r c h o n s B d 6 t e r m i n e r l c s f o n c t i o n s 11,: E + F , o t ~E = F = I, B ' ( x ) d(x,y) < d'(x,y)),

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une d i s t a n c e u l t r a m 6 t r i q u e ( c o n d i t i o n c l n s s i f i c a t o i r e )

S i d e s t une u l t r a m b t r i q u e , cations);

;

0. l a i s s e d i n c h a n g 6 e ( i n v a r i a n c e d e s c l a s s i f i -

n'augmente p a s l e s m e s u r e s d e d i s s i m i l a r i t 6 ; n ' i n v e r s e pas les d i s s i m i l a r i t t s

(i.e.

O(d) a u g m e n t e o v e c d ) .

On r e m a r q u e q u e U *, U j , U s o n t d e s p a r a p h r a s e s d e , r e s p e c t i v e r n e n t : @ e s t 4 i d e m p o t e n t e , t e s t a n t i e x t e n s i v e , @ e s t i s o t o n e . C'est donc e n c o r e l ' o u v s r t u t - e u l t r a m G t r i q u c q u e l ' o n r e n c o n t r e . On t r o u v e r a d a n s J a r d i n e e t S i b s o n ( 1 9 7 1 ) unc c a r a c t 6 r i s a t i o n a x i o m a t i q u e d e c e t t e o u v e r t u r e oh i n t e r v i e n n e n t d e s considerations topologiques.

Kemnrquons a u s s i que s o l u t i o n de :

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e n t r e les indices de d i s s i m i l a r i t 4 d e t d ' .

b(d,d')

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x ,Y

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Li'(x.y)I

N o t e r q u ' o n p e u t s ' i n t ~ ! r r s s c - r ail p r o b l s m e d u a l , b e a u c o u p p l u s d i f f i c i l e , d e m i n i m i s a t i o n de 6 ( d , u ) p o u r l e s u l t r a m k t r i q u e s s u p G r i e u r e s ou & g a l e s h d ( c f . D e f a y s 1 9 7 5 ) . Lc-s s o l u t i o n s , non n & c e s s a i r e r n e n t u n i q u e s , s e t r o u v e n t p a r m i 1 ' e n semble d e s u l t r n m P t r i q u e s s u p c r i c u r e s m i n i m a l e s d e d , d o n t l a c a r a c t G r i s a t i o n e t l ' o b t e n t i o n s o n t d e s p r o b l h m e s o u v e r t s ( l a m c t h o d e du l i e n c o m p l e t , k v o q u b e e n 2 . 3 . 4 . p r o d u i t line t e l l e u l t r a m G t r i q u e ) . Deux n u ~ r e st e c h n i q u e s d ' a j u s t e m e n t s o n t , :I d e s d e g r 6 s d i v e r s , l i k e s a u x s t r u c t u r e s c i ' o r d r e : 1 3 mettiode dc- Zalin ( 1 9 6 4 ) e t c e l l e d e S c h a d e r ( 1 9 8 0 ) .

La p r e m i t r e c o n s i s t e :I a j u s t e r u n e r e l a t i o n s y m g t r i q u e S p a r une r e l a t i o n d ' G q u i v a l e n c e IT. P o u r r e f a i r e , on c h o i s i t u n e G q u i v a l e n c e q u i m i n i m i s f 6 ( S , n ) , 6 P t a n t l a d i s t a n c e d e l a difference s y m g t r i q u e e n t r e d e u x r e l a t i o n s s y m b t r i q u e s . I1 s ' a g i t d o n c d ' u n e m u l t i p r o c G d u r e . C ' e s t p a r I ' i n t e r m G d i a i r e d e 6 , q u i e s t d C f i n i e p a r line v a l u a t i o n s u r f ' ( 6 ' ? ( X ) ) , q u e s e f a i t 1 e l i e n a v e c l e s ensembles o r d o n nGs. I'our le p o i n t ( a ) m e n t i o n n o t i s q u e 1 ' G q u i v a l e n c e d e Zahn e s t s o l u t i o n d ' u n programme l i n t a i r c en n o m b r e s e n t i e r s . Le p o i n t ( b ) e s t , q u a n t 5 l u i , un p r o b l t m e ouvert. C e t t e tprhnique f a i t p a r t i e de la v a s t e c l a s s e des "proctdures mtdianes" (Bartti6lemy-Monjardet 1981). 1.a mGthode d e S c h a d e r e s t t r t s u r d i n a l e d ' e s p r i t , e t d e m i s e e n o e u v r e . I1 s l a g i t d ' a j u s t e r une p r t o r d o n n a n c e p a r une u l t r n p r C o r d o n n a c e . Les p r 4 o r d o n n a n c e s s u r X f o r m e n t un s u p - d e m i - t r e i l l i s s e m i - m o d u l a i r e s u p t r i e u r e m e n t ( 3 . 3 . ) , d o n c g r a d u k p a r u n e v a l u a t i o n s u p C r i e u r e @ ( M o n j a r d e t 1 9 8 1 ) . La d i s t a n c e du p l u s c o u r t c h e m i n y e s t donc donnPe p a r l a t o r m u l e d ( u , v ) = 2 $ ( u v v ) - $ ( u ) - $ ( v ) . L ' e n semble d e s u l t r a l , r ; . o r d o n n a n c e s sur X f o r m e 1111 r G s e a u d a n s l ' e n s e m b l e o r d o n n k d e s prbordonnances. Schnder propose d ' a s s o c i e r 5 l a pr4ordonnance P 1' u l t r a p r t o r donnance (en f i i i t e l l e e s t u n i q u e ) s o l u t i o n d e :

DX

+

I1 s ' a g i t d o n r d'une p r o c P d u r e d P f i n i e p a r u n e c o n t r a i n t e d ' o p t i m i s a t i o n . Une c n r a c t b r i s a t i o n a x i o m a t i q u e d e c o u l e i m m k d i a t e m e n t d e l a r e m a r q u e g k r 4 r n l e c i - d e s s o u s , ob I ' o n c o n s i d t r e une d i s t a n c e " a s s o c i k e " 5 u n e f o n c t i o n $ ( : E + l R , s t r i c t e m e n t c r o i s s a n t e ) p a r l a f o r m u l e 1" du p a r a g r a p h e 4 . 2 . , ou sa g h b r a l i s a t i o n au cas n o n l a t t i c i e l ( c f . M o n j a r d e t 1 9 8 1 ) . Proposition 4.1. S o i t E un e n s e m b l e o r d o n n 4 a v e c un p l u s g r a n d 6 l C m e n t . S o i t A un r e s e a u f i n i s s a n t d e E e t s o i t d u n e d i s t a n c e " n s s o c i k e " h une f o n c t i o n (1 b v a l e u r s r e e l l e s e t s t r i c t e m e n t c r o i s s a n t e s u r E . L'blernent d e A q u i m i n i m i s f d ( a , r ) , p o u r r € E , e s t u n i q u e e t e s t l ' i m a g e d e r p a r une f e r m e t u r e $ s u r E . Le f a i t q u e l a p r o c 6 d u r e d e S c h a d e r e s t u n e f e r m e t u r e p e r m e t , o u t r e u n e c a r a c t 6 r i s a t i o n a x i o m a t i q u e , d ' o b t e n i r un a l g o r i t h m e f o r t s i m p l e . On r e n d compte a i n s i d e s p o i n t s ( a ) , ( b ) , ( c ) . I1 f a u t cepcrndant n o t e r q u e c e t t e rnbthode s e &v&l e peu d i s c r i m i n a n t e .

4 . 4 . A j u s t e m e n t 1 1 . Les p r o c k d u r e s h s e u i l e t l e m o d s l e o r d i n a l d e J a n o w i t z . La mCthode d u l i e n s i m p l e , d G c r i t e a u p a r a g r a p h e p r e c b d e n t a p p a r t i e n t 5 l a c l a s s e d e s p r o c k d u r e s i s e u i l ( " f l a t m e t h o d s " ) . Ces d e r n i g r e s , i n t r o d u i t e s p a r Jardine e t Sibson ( 1 9 7 1 ) o n t kt4 e t u d i k e s ( e t g6nGralisbes) dans l e cadre d'un modsle purement o r d i n a l p a r J a n o w i t z .

5 42

J . 2 Barthelerny et al.

I.eur p r i n r i p e p e u t s e resumer con~me s u i t : u n e m4thode d e c l a s s i f i c . a t i o n t r a n s f o r m e une d i s s i m i l a r i t 6 d en une d i s s i m i 1 ; i r i t b t t ' ( d ) ( o n o u b l i e i c i 1 'casi>:encc s u p p l 6 m e n t a i r e : & ( d ) e s t une u l t r a m G t r i q u e ) . A c h a q u e s r u i l X E L c o r r c s p o n d I n r e l a t i o n s y m C t r i q u e f d ( X ) = t I x , y l t e l q u e c l ( x , y ) 5 A , } . On demandc q u e f ~ ; , ( ~ i ) ( I ) ( i . c . " @ ( d ) a u s e u i l A " ) ne d6pcnde que de f d ( A ) ( i . e . tie " d a11 s e u i l A"). Te c h n iq u e me n t : i c h a q u e d i ssi ni i I n r i t e d c o r r c s p o n d I J I.-clinine f d E R e s( L, 8 ( p,(X)) ( 3 . 1 . 2 ) . On c i i t a l o r s clue I n p r o c 6 d u r e d e c l n s s i i c a t i o n '11 : I, i p 2 ( X ) + I . p 3 2 ( X ) e s t L: scL4i/ s ' i l e x i s t e t e l l e q u e , pour t o u t e L - d i s s i m i l n r i t & d : a l o r s npye1i.e u n e j b n i r t i o n

line

y

o

a p 7 l i c a t i o n y : x'(P,(x))+ fd = 1,a [ u n c t i o n y

P( P , ( x ) ) rst

Dks l o r s s e p o s e n t deux q u e s t i o n s (les " f l a t proble ms " tle J n n o w i t z 1 9 7 8 ~ ) . 1) c a r a c t t r i s e r I e s p r o c P d u r e s h s e u i l , 2) c a r a c t b r i s e r l e s f o n c t i o n s s e u i l . J a n o w i t z l e s t r a n s f o r m e e n p r o h l 6 m e s plus g b n e r a u x e t pure me nt o r d i n n u x e n r e m p l a c a n t L p a r un s u p - d c m i - t r e i l l i s ;aver u n p l u s p e t i t 616nrent e t ff( B~(x)) p a r deiix e n s e m b l e s o r d o n n 6 s q u e l c o n q u e s M e t N , a v e c un p l u s g r a n d 6 l b m e n t . Dans c e c a d r e , u n e f o n c t i o n s ' 1 s e r a une a p p l i c a t i o telle q u c , pour t o u t g F Re s( L,M ) , y o g E R e s ( L , N ) . De rnSnie ;in<' [ I m c ' 7 s e r a line a p p l i c n t i o n @ : Res(L,M) + Res(T,,N) t e l l e qu'il existe y : M + N vi'rifiant ,?(f) = y o f . O n t r o u v e clans J a n o w i t z l a c a r a c t i ' r i s a t i o n c o m p l t t e d e c e s proc-6tiures :I s e u i l e t de c e s fonctions s e u i l (dans l e premier c s , selon l a n a t u r e de L : ch aPn e o u non c h a f n e o n o b t i e n t d e s r b s u l t a t s d i f f r e n t s ) . Une c ons i'que nc c rc ma rquable des r 6 s u l t a t s de Janowitz e s t l ' i s o t o n i e d e s proc6dures L s e u i l e t d e s f o n c t i o n s s e u i l ( p o u r c e s d e r n i i i r e s , l o r s q u e L ri'est p a s une c h a i n e :i d r u x Clements).

Les p r o c e d u r e s 2 s e u i l o n t donnb l i e u i d i v e r s a f f a i b l i s s e m e n t s t e l s clue l e s " s e m i - f l a t me t h o d s" ( J a n o w i t z 1978b) e t l e s "monotone e q u i v a r i a n t m e t h o d s " ( J a n o w i t z 1 9 7 9 a ) . C e s d e r n i k r e s a p p a r a i s s e n t comme une c l a s s e b i e n c a r a c t 6 r i s t ! e , d'un p o i n t d e v u e o r d i n a l , d e s mktlrodes h i 6 r a r c h i q u e s a s c e n d a n t e s .

E n f i n , i l c o n v i e n t d e m e n t i o n n e r i c i q u e c e t y p e d e proc G dure r e l t v e d e c o n s i d t r a t i o n s l i C e s i la t h b o r i e d e s g r a p h e s ( l e s f o n c t i o n s s e u i l 6 t a n t d e s t r a n s f o r m a t i o n s d e g r a p h e s ) . P a r exemple, la mCthode d u l i e n s i m p l e r e v i e n t :i a s s o c i e r , h chnque r e l a t i o n ( g r a p h e ) s e u i l , l ' k q u i v a l e n c e obtenue p a r f e r m e t u r e t r a n s i t i v e ( d o n t l e s c l a s s e s s o n t l e s c l a s s e s c o n n e x e s d e c e g r a p h e ) . L e thPme " g r a p h e s e t ta x o n o mi e " e s t d 6 v e l o p p 6 notamment d a n s H u b e r t (1974 a e t b), Ma tuln ( 1 9 7 7 ) , Day ( 1 9 7 7 , 1 9 7 8 ) , M o n j a r d e t ( 1 9 8 0 ) .

4.5. A j u s t e m e n t 111 - L ' a p p r o c h e g a l o i s i e n n e . Nous a v o n s e u l ' o c c a s i o n d e c o n s t a t e r q u e l e s f e r m e t u r e s ( e t l e u r s d u d e $ l e s o u v e r t u r e s ) i n t e r v i e n n e n t en t a x o n o m i e . L ' a p p r o c h e g a l o i s i e n n e 1 e s u t i l i s e d t : l i b c r g m e n t , a i n s i q u e l ' i m p o r t a n t e n o t i o n d e p o i n t f i x e d a n s un e n s e m b l e ordonnC. L ' i d b e d e s c l a s s i f i c a t i o n s g a l o i s i f n n e s e s t d ' o b t e n i r une p a r t i t i o n d o n t l e s c l a s s e s s o n t formCes d ' o b j e t s " p o s s P d a n t t o u s e t p o s s b d a n t seuls un ou p l u s i e u r s c a r a c t k r e s communs" ( L a l a n d e , V o c a b u l a i r e d e l a p h i l o s o p h i c , c f Degenne, Fla me nt e t V e r g t s 1978).

.

On p a r t d ' u n t a b l e a u en 0 , l c r o i s a n t un e n s e m b l e d ' o b j e t s X e t un e n s e m b l e d e c a r a c t 6 r e s Y . Un t e l t a b l e a u t r a d u i t une r e l a t i o n R e n t r e X e t Y e t on p e u t l u i a s s o c i e r une c o r r e s p o n d a n c e d e Galois ( B i r k h o f f 1 9 4 0 ) , ( n , B ) e n t r e P(X) e t I'(Y) : a ( A ) = {y/y F Y e t (x,y) B ( B ) = {x/x E X e t ( x , y )

t R p o u r t o u t x E A1 E R p o u r t o u t y F B}

Ensembles ordonnes e t taxonomie mathematique

t!~i

est

iin

dit

q u ' r ~ n c - p a r t i t i o n r F [Ix

:ilL)rs

5 43

c'st

f e r m b pour 1 3 f e r m e t ~ i r e f i a l o i s i e n n e

1.i~.

P l u s ::bnCrnlement, c o n s i d C r o i i s Line f e r m e t u r e ip s u r 1 ' e n s e m b l e p(X) e t c h e r clions I c s p a r t i t i o n s I: d o n t c h a q u c c l n s s e est u, - f e r m 6 e ( n 5 i p ( 8 ( X ) ) . C o n s i d & r o n s l a f o n c , t i o n f : ) I x + nx d c f i n i c p a r : f ( n ) = l e s " c c ~ m p o s n n t e s c o n n e x e s " d e s

ip(Ai) des

o t ~ Ics

ip(Ai)

s m t l c s cl.isscs d e

1..

: on p r o r t d c p a r f u s i o n s t i c c e s s i v e

TI

lion d i s j o i n t s .

L ' a p p l i c . i t i o n f esL i s u t o n r ' e t e x t e n s i v e . On s a i t a l o r s ( F l a m e n t 1978) q u ' i l e x i s t c sur tine ftmrnieturc ', t c - l l e qire : y(nX) = 1 ; 1 F 1 ! ~ / f ( ~ r )= II} ( p o i n t s f i x e s c1c7

f ) . c)r,

TI

c -

ip(

P(x))

s i e t seiilenient s i f ( n )

= 71.

I . ' u t i l i s , i t i o n d ' u n "tliGortmc. d e p o i n t f i x e " a donc p e r m i s d ' G L a b l i r l ' e x i s tc'nce d ' u n e c l a s s i t i c a t i o n g a l o i s i e n n e , e t s u n p r i n c i p e i t 6 r a t i f c o n d u i t t o u t n;i t l i r e 1 1 ement ;I un a I g o r i t lime

.

4 . 6 . Agrcgation (vu c o n s e n s u s ) , P o u r line a g r b g a t i o n , i l s ' a g i r n d e c o n s t r u i r e , p o u r chaque e u t i e r v , u n e a p p l i c a t i o n ~1~ = EV E . On r e t r o i i v e 111 un d o m n i n c c l a s s i q u e d e s m a t h 6 m a t i q u e s s o c i a l e s : l a tliCcirie d u c t r o i x r ~ i l l c c t i t . L ' n p p r o c h e a x i o m a t i q u e , i n a u g u r s e e n c e d o m a i n r p a r Arrow ( 1 9 6 2 ) a C t ; r c a p r i s e p a r M i r k i n ( 1 9 7 5 , c f . a u s s i 1 9 7 9 ) p o u r ' r des r e l a t i o n s d ' e ! q u i v n l e n c e . L e s c o n d i t i o n s a x i o m a t i q u e s q u ' o n i m p o s e R U X p r o c b d u r e s d ' a j i r 6 g a t i o n p e u v e n t , e n t a i t , s't3crii-e d a n s n ' i m p o r t e q u e l ensemb l e o r d o n n C . C o n s i d c r o n s d u n c , d ' u n e m a n i h r e j i b n 6 r a l e , un e n s e m h l e o r d o n n t E e t u n e n t i e r p o s i t i t v , unc a p p l i c a t i o n : EV + I: e t n o t o n s V I ' e n s e n i h l e [I, vl --f

...,

+

pour t o u t i , r n t r a i n e s lorsque : ( r l l ,..., r;),

5

'0

(I:).

LW

i I s

(NM) p o u r t o u t

1 ' i nc 1 u s i o n :

ri =

(r

,...,r v ) t i I s S

c t ri'

5 r.i 5 (rI)

=

irl,

5

,..., r '

ti I s'

implique

pour t u u t s E F, 1'bgalitG e n t r a T n e qiie :

r.']

5

rli1

s'

5

),pour tout s , s ' F F, entrai'ne que :

P (7').

T o u t e f o n c t i o n F - n e u t r e m o n o t o n e e s t F - d & c i s i v e . L o r s q u e F = E , on d i r a d t c i s i v e ( r e s p . n e u t r e monotone) a u l i e u d e F - d e c i s i v e ( F - n e u t r e monotone). C o n s i d G r o n s m a i n t e n a n t line f a n i i l l c f i n i s s n n t e F d e p a r t i e s d e V. On d i r a q u e 0 e s t F - r q j ( i l i t a i rc 1o r s q u e : p o u r t o u t r) = ( r ,,...,r v ) , l ' e c r i t u r e : V

IEF

A

r.

i 6 1

a un s e n s d a n s l ' e n s c m b l e o r d o n n e E ( c e s e r a l e c a s , e n p a r t i c i i l i e r , I o r s q u e E e s t 1111 t r e i l l i s ) e t c o i n c i d e RVL'C 0 (n). L o r s q u e 6r c o i ' n c i d e a v e c l e f i l t r c e n g e n d r k p a r H 5 V , on o b t i e n d r a line a p p l i c a t i o n B - m a j o r i t a i r e s i e t seulement s i E est un i n f - d e m i - t r e i l l i s . Cellec i s e r a a l o r s d e l a fornie :

5 44

J.-P.Barthelemy et al.

On dira alors que 0 est une fonction H-olic{urchique. Si P e s t un ultrafiltre 0 est une projection (alias une "procbdure dictatoriale")

.

Les caractkrisations des fonctions parbtiennes et monotones actuellement connues conduisent toutes a des fonctions F-majoritaires qui peuvent @tre de formes tr$s varibes suivant l'ensemble ordonnG considbrc!. Ainsi : Proprosition 4.2. (Monjardet 1982). Si E est un inf-demi-treillis r'i mtdianes ct si F est l'ensemble des sup-irreductibles de E, (D : Ev + E est F-neutre inondtone et parbtienne si et seulement si il existe une f a m i l ~ efinissante F telle que @ soit F-majoritaire. En taxonomie, ce rbsultat s'applique aux hibrarchies, aux hierarchies strntifibes et aux ultraprbordonnances. Dans le cas ob E = IIx, on obtiendra une version (aux hypothtses affaiblies) d'un rbsultat d e Mirkin. Proposition 4.3. (Mirkin 1975) 'D : II;

+

est decisive et par6tienne si et s e u -

IIx

lementsi il existe un sous-ensemble tl de V tel q u e

0

soit 13-oligarchique.

Cette derniPre proposition illustre le fait, que comme en agrbgation des pr6fCrences, des rhgles d'apparcnce "raisonnables" peuvent conduire 5 une procbdure "triviale" (Noter q u e le cas H = 0 , correspond h la fonction constante, quij tout 11 associe la partition grossitre I! 1. On peut maintenant considerer cette proposition comme un cas particulier d'un resultst de B. Leclerc (1983) portant sur l'agr4gation des relations transitivement valubes. On trouvera dans Barth&lemy, Leclerc, Monjardet (19831, l'bnoncb de ce resultat dans le cas particulier des ultramGtriques : sous des hypothtses paretiennes et d e decisivitt, l'agrbgation d'ultrametriques est une "oligarchie gh6ralis6e". Donnons maintenant un exemple de multiprocbdure 0 contrainte d'optimisation : les partitioris c e ntraze s

:

Ili

+

IIx

vbrifiant une

de Regnier (1965). Pour V

E 6(n',ni), i=l 6 btant la distance de la difference symCtrique sur RX. L'aspect ordinal d e cette multiproc4dure vient du fait que, ainsi q u e nous l'avons mentionn4 en 4.2., 6 est definie par unc valuation inferieure sur 0 =

( n 1 ,...,II) ,

O(q)

est l'ensemble des solutions de

Min

II'E

n

".

TWL

comme la methode de Zahn, cette multiprockdure appartientb la classe des

procedures medianes et lelecteur est renvoye B Barthelemy et Monjardet (1981) pour la description de son environnement et 1'4noncG de quelques-unes d e ses propriCtCs ( c f . notamment, Regnier 1965, Regnier et de la Vega 1976, Michnud et Marcotorchino 1979). Par ailleurs, si des propribtes du type "confrontation aux conditions d'Arrow" sont bien Gtablies, le probleme de l a caracterisation axiomatique d e s partitions centrales demeure ouvert. Notons enfin que, dans un inf-demi-treillis 5 medianes E, gradu4 par 9, la mkdiane d u v-uple (r ,,,..,r ) (v impair), v A r ( F d4signant les I C F i E I v+ 1 parties d e V a - 6lbments) est l'uciaue solution de : 2 Min d (s,ri) $ s F E i=l

Ensembles ordonnes et taxonomie mathematique

5 45

(tLirb u t( 1 9 3 0 ) p o u r l e s t r e i l l i s d i s t r i h u t i f s , B a n d e l t e t b a r t h 6 l e m y ( 1 9 8 3 ) p o u r l e s d e m i - t r e i l l i s ?i m t d i a n e s ) . D a n s l e c a s p a r t i c u l i e r d e s h i 4 r a r c h i e s , on r e t r o u v e a i n s i u n r P s u l t a t d e Margush e t Mac M o r r i s ( 1 9 8 1 ) . En o u t r e , e n p a r t i c u l a r i s a n t l e s c o n d i t i o n s d e 13 p r o p o s i t i o n 4 . 2 . on o b t i e n t d e s c a r a c t b r i s a t i o n s a x i o m a t i q u e s d e l a m6dinne ( M o n j a r d e t , 198'). On r e n d a i n s i c o m p t e , d a n s l e c a s d e l a mGdiane, d e s t r o i s p o i n t s a ) , b ) , c ) . C e p e n d m t , 1,) n k d i a n e d ' u n e famil l e d e % sera s o u v e n t l a % h i 6 r a r c h i e t r i v i a l e ( c e ph6noniene f i t a n t e n r o r e a c c e n t u G p o u r l e s h i t r n r c h i e s s t r a t i f i 6 e s ) . C e t t e rernarque t e n d r e s t r e i n d r e , d a n s c e c o n t e x t e , l'usage d e l a r g g l e m a j o r i t a i r e ,I l a r e c h e r c h e cte compromis e n t r e d i v e r s e s h i 6 r a r c h i e s s u f f i s a m m e n t " v o i s i n e s " .

ARABIE, P. e t BOORMAN, S . A . , b etween p a r t i t i o n s , i

M u l t i d i m e n s i o n n a l s c a l i n g of m e a s u r e s of d i s t a n c e s Z., 17, ( 1 9 7 8 ) , 31-63.

iilz'dzcaz valzies, 2nd e d . , J. Wile y & S o n s , NewYork,

1962, ( p r e m i s r e C d i t i o n , 1951).

BANDELT,

Communication p e r s o n n e l l e , 1982.

H.J.,

BANDELT, M.J. e t BAKTHELEMY, J . P . , !'isci?. :"lcrtl!. 1 9 8 3 .

BANDELT, H . J .

Medi ans i n media n g r a p h s , t o a p p e a r i n A p p l .

e t HEDLIKOVA, J., Median a l g e b r a s , Discrete bkztlr. 4 5 , ( 1 9 8 3 )

1-30.

BARBUT, M . , M t d i a n e , d i s t r i b u t i v i t 6 , r l o i g n e m e n t s , P u b l i c a t i o n s du C e n t r e d e M n t h t m a t i q u e S o c i a l e , E . P . H . E . , G m e S e c t i o n , P a r i s 1961, e t Muth. Sci. hum. 7 0 , ( 1 9 8 0 ) , 5-31. BARBUT, M . e t MONJAKDET, B . , Ordre e t c l a s s i f i c a t i o n , alg6bre e t cornbinatoire, Tome I e t 11, H a c h e t t e , P a r i s , 1970.

.

BARTHELEMY, J . P , t'xprie't6s r?fQi,iz,k[ues des erisefiibZes ardonnks. L'orrpamison e t tr(irc5:rutiurz d r r e / a t i o ? z s b-inaires, T h e s e d e D o c t o r a t d ' E t a t d e M a t h b m a t i q u e s , U n i v e r s i t C d e Uesanc;on, 1979a. BARTHELEMY, J . P . , C a r a c t r r i s a t i o n s n x i o m a t i q u e s d e l a d i s t a n c e d e l a d i f f b r e n c e s y m r t r i q u e e n t r e l e s r e l a t i o n s b i n a i r e s , Matl!. S c i . hiin!. 67, ( 1 9 7 9 b ) 85-1 1 3 . BARTHELEMY, J . P . , FLAMENT, C . e t MONJARDET, B . , O r d e r e d s e t s a n d s o c i a l s c i e n c e s , i n I . R I V A L ( e d . ) Ur'lcred s e t s , L ondon, D. REIDEL, 1982, 721-758. BAP,TFEI,EEF!, J . P . , LECLERC, B . e t MONJARDET, B. Q u e l q u e s a s p e c t s du c o n s e n s u s e n c l a s s i f i c a t i o n , i n Dutu Analysis anti Informatics 111,N o r t h - H o l l a n d , 1984. BARTHELEMY, J . P . e t MONJARDET, B . , The medi an p r o c e d u r e i n c l u s t e r a n a l y s i s a nd s o c i a l c h o i c e t h e o r y , i % z t h . Soc. Sci. 1 , ( 1 9 8 1 ) 235-267. BENZECRI, J . P . , D e s c r i p t i o n math6matique d e s c l a s s i f i c a t i o n s , cles donnhes, Tome I , I,a Tazinomie, Dunod, P a r i s , 1973.

1967 i n L ' A n a l y s e

B E N Z E C K I , J . P . , e t C o l l a b o r a t e u r s , L ' a n a l y s e d e s donne'es, Tome I , Tome 11, L'Ancrlyse des correspandances, Dunod, P a r i s , 1973. BERGE, C .

La Tazinornie,

, Crnphes e t h2pergrnphes, Dunod, P a r i s , 1970.

B I R K H O F F , G . , L a t t i c e t h e o r y , 3 r d e d . , h e r . Mat h. S o c . , P r o v i d e n c e 1 9 6 7 , (premiPre 6 d i t i o n 1940). BLYTH, T . S . e t JANOWITZ, M.F.,

BOBISUD, H . M . e t BOBISUD, L . E . ,

RQsiduation t h e o r y , Pergamon P r e s s , O x f o r d , 1972. A metric f o r c l a s s i f i c a t i o n s , ii'axon

21, (1972),

607-6 1 3 . BOCK, H . H . , Autorrntische K Z a s s i f i k a t , k m , t h e o r e t i s c h e und p r a k t i s c h e Methoden z u r G r u p p i e r u n g und S t r u k t u r i e r u n g von D a t e n , Vandenhoech und R u p r e c h t , G o t t i n g e n 1974.

Barthelemv et al.

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BOORMAN,

S.A.

B : ) ~ L ( ? c : ~ oj"

e t ARABIE, 1'.

,

L'o

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oij,jcxti;, s e n i o r h o n o r s t h e s i s , tla rvLird

S t r u c t u r a l m e a s u r e s and t h e method of s o r t i n K , i n 1971.

~ ~ ~ I t i i j i r i i ~ t ~ s i oSr icnaaI /i n ~ ~Vol , . 1 , T h c o r y , S t m i n a r Press, New-York, BOORMAN, S.A. e t O L I V I E R , D . C . ,

M e t r i c on s p a c e s of f i n i t e t r e e s ,

"7

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p y C k

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ordies c t a; y ~ ~i c w t ~ ' c ~ i i s ,

~ ~ ~ J L J ? O ~ / Z

COMTET, L . , S u r l e q u a t r i k m e probl kme e t l e s nombres d e S c h r i i d e r , 2 . h . Actr(l. S c L . , P a r i s , A . 271, ( 1 9 7 0 ) 913-916. CKAWLEY, P e t DILWORTH, R . P . , A/!iebi7aic t i Englewood H i l l s , 1973.

DAY, W . H . E . ,

V a l i d i t y o f c l u s t e r s Eormed by g r a p h - t h e o r e t i c

c l u s t e r methods,

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cnl Diosciences, 4 1 , ( 1 9 7 8 ) 1-19. The c o m n l e x i t y oE c o m p u t i n g m e t r i c d i s t a n c e s b e t w e e n p a r t i t i o n s , 1 , 3 , (1981)269-287.

DAY, W . H . E . ,

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