STUDIES IN LO-GIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 118
Editors J. BARWISE, Stanford D. KAPLAN, Los Angeles H. J . KEISLER, Madison P. SUPPES, Stanford A. S. TROELSTRA. Amsterdam
NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD
THEORY OF RELAmONS
R.FRAISSE Universitk de Provence Marseille France
1986
NORTH-HOLLAND AMSTERDAM 0 NEW YORK OXFORD
ELSEVIER SCIENCE PUBLISHERS B.V., 1986 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.
ISBN: 0 444 878653
Translationof ThLorie des relations Translated by P. Clote Published by: Elsevier Science Publishers B.V. P.O. Box 1991 1000BZ Amsterdam The Netherlands Sole distributors for the U.S.A. and Canada: Elsevier Science PublishingCompany, Inc. 52VanderbiltAvenue NewYork, N.Y. 10017 U.S.A.
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F r a i s d , Roland. Theory of relations.
(Studies i n l o g i c a d the foundations of matheratics ; v. 118) Translation of: Thhrie des relations. Bibliography: p. Includes index. 1. Set theory. I. Title. 11. Series. QA248.FT75 1986 511.3'22 85-20701
ISBN 0-444-87865-3
PRINTED IN THE NETHERLANDS
V
INTRODUCTION Relation theory goes back t o the 1940's ; i t o r i g i n a t e s i n the theory o f order types, due t o HAUSDORFF (Grundzuge der Mengenlehre 1914), S I E R P I N S K I ( L e ~ o n ssur l e s nombres t r a n s f i n i s 1928, taken up again i n Cardinal and o r d i n a l numbers 1958), SZPILRAJN (Sur l ' e x t e n s i o n de l ' o r d r e p a r t i e l 1930), DUSHNIK, MILLER (Concerning s i m i l a r i t y transformations of 1i n e a r l y ordered sets 1940), GLEYZAL (Order types and s t r u c t u r e o f orders 1940), and t o HESSENBERG (Grundbegriffe der Mengenlehre 1906, i n t r o d u c i n g the negative and r a t i o n a l o r d i n a l s ) . A t t h a t t i m e , r e l a t i o n theory j u s t extended t o a r b i t r a r y r e l a t i o n s t h e elementary n o t i o n s o f order type and embeddability. Relation theory i n t e r s e c t s o n l y weakly w i t h graph theory, w i t h which i t i s sometimes s t i l l confused. F i r s t l y , echniques i n r e l a t i o n theory o n l y r a r e l y d i s t i n g u i s h between graphs, i.e. s j m t r i c b i n a r y r e l a t i o n s , and r e l a t i o n s o f a r b i t r a r y a r i t y . A d d i t i o n a l l y , as opposed t o graph theory, i n r e l a t i o n theory one considers e q u a l l y t h e two t r u t h values (+) and ( - ) taken on by a r e l a t i o n w i t h base
E
f o r each element o f
E2
(or o f
En
f o r the a r i t y
n ).
On the o t h e r hand, r e l a t i o n theory uses techniques e s p e c i a l l y from combinatorics, the l a t t e r which can be defined as f i n i t e s e t theory. Anything concerning r e l a t i o n s w i t h f i n i t e bases, o r counting isomorphism types o f f i n i t e r e s t r i c t i o n s o f a given r e l a t i o n , o r again the study o f permutations o f t h e base which preserves a given r e l a t i o n ( i .e. automorphisms o f t h e r e l a t i o n ) , makes use o f combinatorics. From a more t e c h n i c a l viewpoint, see t h e combinatorial lemnas i n ch.3 5 4, and the study o f the incidence m a t r i x i n ch.3 5 5. ' a
As f o r mathematical l o g i c , i t s i n t e r s e c t i o n w i t h r e l a t i o n theory i s r a t h e r import a n t . One can even say t h a t t h e two p r i n c i p a l sources f o r r e l a t i o n theory are the study o f order types, already mentioned, and l i n e a r l o g i c , i . ? . f i r s t - o r d e r o n e - q u a n t i f i e r l o g i c ; t h a t i s the study o f u n i v e r s a l formOlas (prenex formulas only having u n i v e r s a l q u a n t i f i e r s ) , and boolean combinations thereof, w i t h the p a r t i c u l a r case o f q u a n t i f i e r - f r e e formulas. From a semantic, o r model-theoretic viewpoint, t h i s i s the study o f u n i v e r s a l classes o f TARSKI,' VAUGHT 1953, and o f boolean combinations t h e r e o f .
I f one presents mathematical l o g i c from a r e l a t i o n a l t h e o r e t i c viewpoint, the basic n o t i o n i s t h a t o f l o c a l isomorphism, i . e . isomorphism o f a r e s t r i c t i o n o f the f i r s t r e l a t i o n onto a r e s t r i c t i o n o f the second one: see ch.9 5 1.4. For example, t h e f r e e i n t e r p r e t a b i l i t y o f a r e l a t i o n S i n another r e l a t i o n R w i t h the same base, i s a l g e b r i c a l l y defined by t h e c o n d i t i o n t h a t every l o c a l automorphism o f R ( l o c a l isomorphism from R i n t o R ) i s a l s o a l o c a l automorphism o f S . E q u i v a l e n t l y , f r e e i n t e r p r e t a b i l i t y i s l o g i c a l l y defined by the existence o f a q u a n t i f i e r - f r e e formula which defines S i n t h e s t r u c t u r e of R . For example, i f R i s a chain, o r t o t a l ordering, then t h e betweenness r e l a t i o n S(x,y,z) = + i f f z i s between x and y , i s defined by t h e q u a n t i f i e r - f r e e formula (Rxz h Rzy) v (Ryz A Rzx) This equivalence between algebraic and l o g i c a l n o t i o n s e x i s t s even above t h e f r e e - q u a n t i f i e r and t h e o n e - q u a n t i f i e r cases; since l o g i c a l ( o r elementary) equivalence between R and S , saying t h a t R and S s a t i s f y t h e same f i r s t - o r d e r formulas, i s equivalent t o t h e i r being (k,p)-equivalent f o r a l l i n t e g e r s k and p , which i s a p u r e l y algebraic n o t i o n : see my Course o f mathematical l o g i c 1974 vol. 2 Coming back t o t h e l i n e a r case ( o n e - q u a n t i f i e r l o g i c ) , as common notions and techniques i n both mathematical l o g i c and r e l a t i o n theory, we have those o f 1-isomorphism, 1-extension, p r o j e c t i o n f i l t e r ( a v a r i a n t o f u l t r a p r o d u c t ) : see ch.10 5 1 . And f o r each o r d i n a l o( , t h e 4 -morphism (ch.10 5 4), which i s n o t
.
.
vi
Introduction
a one-quantifier notion, b u t i s i n d i s p e n s i b l e in r e l a t i o n theory f o r t h e study of embeddability: see ch. 10 5 4 and 5 5.3. From the 1970's, an important connection appears between r e l a t i o n theory and t h e theory of permutations. See the study of o r b i t s (ch.11 § 2 ) , t h e theorem on the increasing number of o r b i t s (LIVINGSTONE, WAGNER, ch.11 5 2.8) and the theorem on s e t - t r a n s i t i v e , o r homogeneous groups (CAMERON, ch.11 5 5 . 1 0 ) . Let us mention, a l s o from t h e 1970's, some unexpected connections between r e l a t i o n theory and topology ( c h . 1 5 8 and ch.7 5 2 ) ; and even connections w i t h l i n e a r algebra (ch.11 5 2 . 6 ) . We s h a l l now b r i e f l y present the p r i n c i p a l notions s t u d i e d , by mentioning f i r s t t h a t chapters 1 through 8 concern the theory of p a r t i a l and t o t a l orderings ( o r c h a i n s ) , while chapters 9 through 12 concern t h e general study of r e l a t i o n s . In chapter 1, we review b a s i c set t h e o r e t i c a l r e s u l t s , i n general without proofs, which allow t h e reader t o know, f o r i n s t a n c e , i n which p r e c i s e sense we use the notion of f i n i t e set (TARSKI's sense r a t h e r than OEDEKIND's), o r the notion of c a r d i n a l i t y of a s e t . T h i s allows us t o p r e c i s e , throughout t h e r e s t of t h e book, which axioms a r e used f o r each proof: ZF a l o n e , t h e axiom of choice, dependent choice, t h e u l t r a f i l t e r axiom, t h e continuum hypothesis, e t c . Moreover i t seems t h a t even among l o g i c i a n s , t h e r e a r e few who a r e aware t h a t , while O1 > W i s provable in ZF a l o n e , y e t the countable axiom of choice, f o r i n s t a n c e , i s used t o prove t h a t W 1 i s r e g u l a r . Or t h a t KONIG's theorem (ch.1 0 1.8), even i n t h e very p a r t i c u l a r case of two ordered pairs of s e t s , i s not provable in ZF alone. Or t h a t t h e p o s s i b l e equivalence between the axiom o f choice and t h e statement t h a t t h e range of a function i s subpotent w i t h i t s domain, i s s t i l l an open problem, already p u t f o r t h by RUBIN 1963. Thus t h i s chapter could be useful as a memory brush-up f o r the axiomatic s e t t h e o r e t i c i a n . In chapter 2 , in a d d i t i o n t o a review of b a s i c r e l a t i o n t h e o r e t i c a l n o t i o n s , s i m i l a r in s p i r i t t o chapter 1, we introduce some notions which a r e no longer c l a s s i c a l , y e t which extend well-known concepts. For example the coherence lemma ( 5 1 . 3 ) , a not well-known version of the u l t r a f i l t e r axiom. Another example, the c o f i n a l i t y of a p a r t i a l o r d e r i n g , a s well a s t h e r e l a t e d notion of c o f i n a l height ( 5 5.4 and 5 7 ) . C l a s s i c a l l y , t h e notion of c o f i n a l i t y i s r e l e g a t e d t o t h e s i n g l e case of c h a i n s , o r t o t a l o r d e r i n g s , which while i n t e r e s t i n g i s too much r e s t r i c tive. In chapter 3 , we present RAMSEY's theorem and important refinements of i t , due t o GALVIN and t o NASH-WILLIAMS ( 5 2 ) . Furthermore, the " i n i t i a l i n t e r v a l theorem" o r GALVIN's theorem i s presented t w i c e , with very d i f f e r e n t proofs: POUZET's proof i n 5 2 and LOPEZ'S proof using t h e c l a s s i c a l Ramsey s e t s of r e a l s , i n 5 6. Then we a r e led t o the p a r t i t i o n theorems of DUSHNIK, MILLER and of ERDOS, RADO. We a l s o present a combinatorial study of the incidence matrix, w i t h the l i n e a r independence lemma due t o KANTOR. In chapter 4 , we begin t h e study of p a r t i a l o r d e r i n g s , w i t h t h e notions of good and bad sequence, of a f i n i t e l y f r e e p a r t i a l o r d e r i n g , and t h a t of a well p a r t i a l e present HIGMAN's c h a r a c t e r i z a t i o n of a well p a r t i a l ordering (the ordering. W s e t of i n i t i a l i n t e r v a l s is well-founded under i n c l u s i o n ) ; a l s o HIGMAN's theorem on words i n a well p a r t i a l o r d e r i n g , and RADO's well p a r t i a l ordering ( 5 4 ) . Also e present t h e notions of i d e a l , t r e e , dimension, bound of an i n i t i a l i n t e r v a l . W the theorem of t h e maximal reinforced chain f o r a well p a r t i a l ordering, due t o D E JONGH, PARIKH ( 5 9 ) . The chapter ends ( 5 10) w i t h POUZET's theorem on r e g u l a r ( o r f i n i t e ) c o f i n a l i t y of any f i n i t e l y f r e e p a r t i a l ordering.
Introduction
vii
In chapter 5, we consider embeddability between orderings, the well pa rtia l ordering of f i n i t e trees (KRUSKAL), the existence of immediate extensions and of faithful extensions (HAGENDORF), Cantor's theorem for pa rtia l orderings (DILWORTH, GLEASON). Then the existence of s t r i c t l y decreasing i n f i n i t e sequences of chains of reals: the denumerable sequence due t o DUSHNIK, MILLER and the continuum length sequence due t o SIERPINSKI. Finally a brief study of SUSLIN's chain and t r e e , in connection with SUSLIN hypothesis; also ARONSZAJN t r e e , SPECKER chain. In chapter 6 , we introduce the scattered chain, which does not admit any embedding of the chain Q of the r at i o n al s . Also the indecomposable, as well as the right and the l e f t indecomposable chain. We present HAGENDORF's theorem of unique decomposition of an indecomposable chain ( 5 3.3) and some connected re sults (JULLIEN, LARSON). We begin t o study the covering of a chain by r l g h t or l e f t indecomposable in te r v a l s, or by doublets of indecomposable i n t er v als. We present the hereditarily indecomposable chain with L A V E R ' S r e s u l t s , and f i n a l l y the indivisible relation o r chain. I n chapter 7 , we proof supplementary r es u l t s a b o u t f i n i t e l y free pa rtia l orderinas and t h e i r reinforcements by chains. We extend t o the s e t of i n i t i a l intervals the topology already introduced in ch.1 5 8 , and give some applications, namely BONNET'S r e s u l t s . Then we prove the following important theorem of POUZET: every directed well p a r t i a l ordering has a cofinal r es t r ic tion which i s a dire c t product of f i n i t e l y many d i s t i n c t regular alephs. The chapter ends with a short study of Szpilrajn chains (BONNET, JULLIEN); two interesting re sults due t o TUKEY and t o KRASNER are presented as exercises.
I n chapter 8 , we introduce the important notion of ba rrie r due t o NASH-WILLIAMS; the p a r t i t i o n theorem ( 5 1 . 4 ) , the theorem on the minimal bad barrier sequence ( 5 2 . 2 ) ; the forerunner and successor b ar r i er . This i s the main t o o l i n the proof of the very important theorem of LAVER: every s e t of scattered chains forms a well quasi-ordering under embeddability ( 5 4.4). In other words, there e x is t s neither an i n f i n i t e s t r i c t l y decreasing sequence nor an i n f i n i t e s e t of mutually incomparable scattered chains. LAVER proved even more, in extending his result t o chains formed from a countable union of scattered chains. However his proof has n o t yet been s u f f i ci en t l y simplified t o be presented in a textbook of a reasonable s i z e . In t h i s chapter, we also study the be tte r partial ordering, a notion due t o NASH-WILLIAMS, both f o r i t s i n t r i n s i c inte re st and for i t s applications t o chains. In chapter 9 , we begin the general theory of r el at ions, w i t h the notions of local isomorphism, free i nt er p r et ab i l i t y and free operator (which i s the re la tionist version of a logical f r ee formula, and links relation theory t o logic ). We study constant, chainable, monomorphic r el at i o n s . I n the case of a binary relation with cardinality p , we present the deep r es u l t due t o JEAN: (~-2)-monomorphy implies (p-1)-monomorphy ( 5 6 . 7 ) . We present the profile increase theorem (POUZET, 5 7 ) . Finally we extend t o ar b i t r ar y relations the homomorphic image ( 5 8 ) , and in 5 9 we introduce the bivalent t ab l e, which apparently yields d i f f i c u l t problems, one of them being very p ar t i al ly solved by L O P E Z . Most of r e l a t i o n i s t researchers seem t o be discouraged by t h i s branch of relation theory, which i s s t i l l a marginal study inside relation theory, considered i t s e l f as being marginal d u r i n g t o o long a time.
I n chapter 10, we c l as s i f y relations according t o t h e i r age: two representatives of the same age have the same f i n i t e r e s t r i c t i o n s , up t o isomorphism. This i s equivalent t o classifying relations by the s e t of universal formulas which they s a t isf y . Then we study maximalist or ex i s t en t i al l y closed relations ( 3 3.8), rich r e l a t i o n s, inexhaustible relations ( 5 4 and 5 ) , and relations which are rich f o r t h e i r age. This notion, connected t o saturated relations, leads t o the existence c r i t e r i o n of POUZET, VAUGHT ( 5 7 ) . The f i n i t i s t and almost chainable relations are presented in 5 8 and 9.
Introduction
viii
Chapter 11 i s concerned w i t h correspondence between r e l a t i o n theory and permutations, the l i n k between them being the homogeneous r e l a t i o n s and r e l a t i o n a l systems. We already mentioned t h e theorem o f i n c r e a s i n g number o f o r b i t s , due t o LIVINGSTONE, WAGNER. I n 5 3 and f o l l o w i n g , we introduce t h e c o m p a t i b i l i t y modulo a permutation group, which y i e l d s a marginal study i n s i d e permutation group theory, w i t h many open problems. The n o t i o m o f i n d i c a t i v e group and i n d i c a t o r l e a d t o FRASNAY's r e d u c t i o n theorem ( 5 4 ) . The p a r t i c u l a r case o f Q - i n d i c a t i v e groups leading t o the s e t - t r a n s i t i v e group theorem o f CAMERON. F i n a l l y we study t h e pseudo-homogeneous r e l a t i o n s , t h e prehomogeneous r e l a t i o n s w i t h POUZET's existence c r i t e r i o n ( 5 7), t h e set-homogeneous r e l a t i o n s . I n chapter 12, we introduce the bounds o f a r e l a t i o n R : f i n i t e r e l a t i o n s nonembeddable i n R b u t whose proper r e s t r i c t i o n s are embeddable i n R . We present several important theorems due t o FRASNAY: the reassembling theorem ( 5 3); t h e existence o f an i n t e g e r p such t h a t , from t h i s p o i n t on, p-monomorphism i m p l i e s c h a i n a b i l i t y ; and t h e f i n i t e n e s s o f the number o f bounds f o r a chainable r e l a t i o n . This study uses t h e method o f permuted chains, o r c o m p a t i b i l i t y modulo a permutat i o n group, already presented i n chapter 11. Proofs have been s i m p l i f i e d by using, as a powerful t o o l , the p-well r e l a t i o n due t o POUZET. The chapter, and t h e book, are ending w i t h the study o f reduction, reassembling, monomorphism and chainabil i t y thresholds: c a l c u l a t e d f i r s t by FRASNAY, they were improved by HODGES, LACHLAN, SHELAH, then proved again by FRASNAY t o have the smallest p o s s i b l e value. I n 5 6 we added some easy considerations about u n i v e r s a l classes. I n order t o keep t h i s book t o a reasonable s i z e , we suppressed two planned chapters. One about the celebrated problem o f r e c o n s t r u c t i o n , i . e . the problem t o know i n what cases a r e l a t i o n w i t h base E i s completely determined, up t o isomorphism, by the isomorphism types o f i t s r e s t r i c t i o n s t o proper subsets o f E The reader may c o n s u l t BONDY, HEMMINGER 1977, LOPEZ 1978, 1982, 1983, POUZET 1979', STOCKMEYER 1977, ULAM 1960 (see B i b l i o g r a p h y ) . The o t h g r y i s s i n g chapter concerned the n o t i o n o f i n t e r v a l i n r e l a t i o n theory: see FRAISSE 1984 i n Bibliography.
.
I would l i k e t o thank those among my colleagues - professors, researchers, students and ex-students - who solved o r c o n t r i b u t e d t o t h e s o l u t i o n o f a l l problems presented here; and t o those who, by s i m p l i f y i n g t h e i n o r d i n a t e l y long o r d i f f i c u l t p r o o f o f the o r i g i n a l paper, have made these r e s u l t s accessible, hence s u i t a b l e f o r p r e s e n t a t i o n i n t h i s textbook. T h e i r names are mentioned together w i t h t h e i r c o n t r i b u t i o n . As f o r myself, I have t h e f r e e conscience o f having accomplished my work as "chef d ' e c o l e " : namely the presentation, i n a form accessible t o a wide audience, o f r e s u l t s obtained by those who loved my research area and surpassed me.
1
CHAPTER
1
REVIEW OF AXIOMATIC SET THEORY
The purpose o f t h i s chapter i s t o s i t u a t e p r e c i s e l y "theory o f r e l a t i o n s " w i t h i n the framework o f axiomatic s e t theory, which i n i t i a l l y w i l l be t h a t o f ZERMELOFRAENKEL. The axioms f o r ZF are introduced below i n
91
and 92. Our i n i t i a l
n o t a t i o n w i l l be introduced there. I n r e f e r r i n g t o the f i r s t and sometimes second chapter, we w i l l i n d i c a t e throughout the book which statements r e q u i r e o n l y the axioms o f ZF and those which require, t o our knowledge, t h e axiom o f choice, o r r a t h e r the weaker u l t r a f i l t e r axiom (boolean prime i d e a l axiom), o r t h e axiom o f dependent choice, e t c . Most o f the proofs, as w e l l as c l a s s i c a l d e f i n i t i o n s from the f i r s t and second chapter, are l e f t t o t h e reader.
§
1 - FIRST GROUP
OF A X I O M S FOR
ZF, F I N I T E SET, A X I O M OF CHOICE,
KONIG'S THEOREM We begin w i t h the axioms of e x t e n s i o n a l i t y , p a i r , union, power s e t ( s e t o f a l l subsets o f a s e t ) and t h e scheme o f separation, a l l supposedly known t o the reader. We denote t h e empty s e t by We denote the union o f the s e t
a
0
by
, inclusion C , s t r i c t inclusion u a , and the power s e t by p ( a )
I f b s a , we designate the d i f f e r e n c e by (simply c a l l e d p a i r s ) are denoted by a}
a v {a)
of
a
o f the empty set;
,
\ a,b)
, etc.
.
.
unordered p a i r s
The successor s e t
{ 0) i s t h e successor .( 0,l) i s t h e successor o f 1 , e t c . This nota-
i s denoted by 2 = 1+1 =
. So
. Singletons,
a-b
C
a+l
that
1 = 0+1 =
t i o n coincides w i t h the n o t a t i o n f o r o r d i n a l a d d i t i o n , introduced i n fj 3 below.
1.1. FINITE SET Following TARSKI 1924', we d e f i n e a s e t set
b
o f subsets o f
a
a
t o be f i n i t e i f f every non-empty
contains an element which i s minimal w i t h respect t o
i n c l u s i o n , i . e . an element
c E b
such t h a t no
xE b
satisfies
x c c
.
Taking complements, i t i s e q u i v a l e n t t o say t h a t every non-empty s e t o f subsets o f
a
a i s f i n i t e e x a c t l y when contains a maximal element. A n o n - f i n i t e
s e t i s s a i d t o be i n f i n i t e . The empty set, a s i n g l e t o n , a p a i r are f i n i t e sets. Every subset o f a f i n i t e set i s f i n i t e .
If a
i s f i n i t e , then so i s the s e t composed o f
element. I n p a r t i c u l a r , the successor
a+l
of
a a
together w i t h an a d d i t i o n a l i s finite.
2
THEORY OF RELATIONS
f
Scheme of-injuction-for f i n i t e s e t s . If a condition i s t r u e f o r the e m p t y j e t , and i f f o r every s e t a satisfying and every s e t u , the s e t a u \ u j s m s a t i s f i e s f , t+ i s t r u e f o r every f i n i t e s e t . I f a s e t a and a l l i t s elements are f i n i t e , then the union u a i s f i n i t e . This i s often expressed in the following form called pigeonhole principle: i f we p a r t i t i o n an i n f i n i t e s e t into f i n i t e l y many subsets, then a t l e a s t one of these subsets i s i n f i n i t e .
f
1 . 2 . COUPLE OR ORDERED PAIR, CARTESIAN PRODUCT Given two s e t s a , b the couple or ordered pair (a,b) i s the s e t {{a} , {a,b]\ formed of the singleton { a ) and the (unordered) pair i a , b ) . This definition goes back t o KURATOWSKI 1921 (see also AJDUKIEWICZ). The s e t a i s said t o be the f i r s t term and b the second term of the couple. Clearly two couples are equal i f f they have the same f i r s t and same second terms. The Cartesian product a x b i s the s e t of couples (x,y) where x belongs t o a and y belongs t o b . FUNCTION, DOMAIN, RANGE A function or mapping from a onto b i s a subset f of a x b such t h a t every element x of a appears as f i r s t term in exactly one couple (x.y) belonging t o f and every element y of b appears as a second term in a t l e a s t one couple belonging t o f The s e t a = Dom f i s called the domain, the s e t b = Rng f i s the range of f . For each element x of a , the second term y of the unique couple (x,y) having f i r s t term x i s denoted y = f ( x ) or y = fx and is called the value of f on x , or the image of x under f . For every superset c 7 Rng f we say t h a t f i s a function from a c THE TRANSFORMATION f " AN0 ITS INVERSE I f uc_ Dom f , we denote by f " ( u ) the s e t of elements fx where x u The function t h u s denoted f " i s a function on the s e t of subsets of Dom f and i s called the transformation associated with f . This transformation f " ( v ) . However preserves inclusion, in the sense t h a t u c_ v implies f " ( u ) s t r i c t inclusion i s not preserved. I f v C Rng f , then the inverse image of v by f , denoted ( f - 1) " ( v ) , i s the s e t of elements x such t h a t fx belongs t o v . So we define the inverse I t preserves s t r i c t transformation associated w i t h f , denoted (f-')" inclusion as well as inclusion. INJECTION, INVERSE FUNCTION, PERMUTATION, TRANSPOSITION The function f i s said t o be a n injection o r injective function i f f x # x ' implies fx # f x ' f o r a l l x , x ' in Dom f . I f a i s the domain, b the range,
.
-
.
into
.
c
.
3
Chapter 1
t h e n an i n j e c t i o n i s s a i d t o be a b i j e c t i o n f r o m The i n v e r s e o f an i n j e c t i o n
a
i s denoted by
f-l
i n j e c t i v e , the transformation associated w i t h
f-l
f
onto
, so
.
b
t h a t i n t h e case o f
coincides w i t h
(f-l)"
i s f i n i t e , then
Rng f
f
(the
l a t t e r e x i s t s f o r every f u n c t i o n f i n i t e . For
x
f ). i n j e c t i v e o r not, i f
a
i s a b i j e c t i o n from
, the transposition
a
and y
Oom f
is
i n j e c t i v e , t h e converse i s t r u e .
f
A permutation o f
of
,
f
Given a f u n c t i o n
(x,y),
a
onto
a
.
Given two elements
i s the permutation o f
a
and i s t h e i d e n t i t y on e v e r y o t h e r element o f
a
x, y
which interchanges
.
FIXED POINT LEMMA (KNASTER 1928, g e n e r a l i z e d by TARSKI 1955) Let
a
hx
of
be a s e t and
. Suppose
a
h that
a f u n c t i o n which takes each subset h
x
f o r e v e r y x, y 5 a . Then: (1) t h e r e e x i s t s e t s x a m a j o r i z e d by
of
xc
i s i n c r e a s i n g under i n c l u s i o n :
a
t o a subset
implies that
y
hx C- hy example (2)
x
if
x
h
,in
t h e sense t h a t
x
5 hx
;for
can be t a k e n as t h e empty s e t ; i s majorizedA
h
, then
hx
&
i s majorized
u o f a l l majorized subsets-satisfies
(3) the union
h ;
hu = u
.
1.3. RESTRICTION, EXTENSION, COMPOSITION Given a f u n c t i o n & t
of
to
f
f
b
w i t h domain
a
which t h e f i r s t t e r m belongs t o Putting
, we
g = f/b
say t h a t
and a subset
, the
, denoted f!b
b
f
b
of
and
g
, with
Dom(g,f)
, we
c a l l the r e s t r i c of
f
.
i s an e x t e n s i o n o f
g
We l e a v e i t t o t h e r e a d e r t o d e f i n e t h e c o m p o s i t i o n g,f f
a
s e t o f ordered p a i r s belonging t o t o t h e domain
.
a
o f the functions
= Dom g n Rng f
EQUIPOTENCE, SUBPOTENCE
A s e t b i s s a i d t o be e q u i p o t e n t w i t h onto b . A set
b
i s s a i d t o be subpotent w i t h
A set b i s s t r i c t l y w i t h a but a i s n o t subpotent w i t h valent t o saying t h a t b i s subpotent Every s e t e q u i p o t e n t w i t h a f i n i t e s e t s t r i c t l y subpotent w i t h every i n f i n i t e one b e i n g s u b p o t e n t w i t h t h e o t h e r . I f potent w i t h
product
b
a x b
I
i s f i n i t e . If
a
i f f there exists a b i j e c t i o n o f
a
f f t h e r e e x i s t s a subset o f
a
a
a
equi-
subp k t w i t h a i f f b i s subpotent By theorem 1.4 below, t h i s i s e q u i b
.
but not equipotent w i t h
a
.
i s i t s e l f f i n i t e . Every f i n i t e s e t i s s e t . Two f i n i t e s e t s a r e always comparable, a
and
b
are f i n i t e , then the Cartesian
i s f i n i t e , t h e n so i s t h e power s e t p ( a )
.
A f i n i t e s e t i s n o t e q u i p o t e n t w i t h any o f i t s p r o p e r subsets. E q u i v a l e n t l y , i f a i s f i n i t e , then every i n j e c t i o n o f a i n t o a i s a permutation o f a 0 Suppose t h a t
a
f
i s an i n j e c t i o n s a t i s f y i n g
which i s minimal among a l l subsets
x
of
f"(a) a
c a
. Take
satisfying
.
a subset
fo(x)c x
m
. Then
of
THEORY OF RELATIONS
4
c
f"(f"(m))
f"(m)
f : t h i s c o n t r a d i c t s the m i n i m a l i t y . 0
by t h e i n j e c t i v i t y o f
DEDEKIND-FINITE SET A set
a
i s s a i d t o be D e d e k i n d - f i n i t e i f f
a
i s n o t equipotent w i t h any proper
subset o f i t s e l f (DEDEKIND 1888); i t i s D e d e k i n d - i n f i n i t e i n the opposite case. The converse w i l l be proved i n 2.6 by using
Every f i n i t e s e t i s Dedekind-finite.
the denumerable subset axiom (weaker than t h e axiom o f choice). 1.4. BERNSTEIN-SCHRODER THEOREM
Given sets
then a
a
and
b
,if
a
i s equipotent w i t h
i s subpotent w i t h
. The
b
b and
b
subpotent w i t h
a
,
f o l l o w i n g p r o o f i s i n FRAENKEL 1953 and a t t r i -
buted t o WHITAKER. It does n o t use the n o t i o n o f i n t e g e r , which i s used i n the c l a s s i c a l " m i r r o r p r o o f " ; see a1 so SUPPES 1960. 0 Let
into to
be an i n j e c t i o n from
f
a
. It suffices
a-o
a
into
t o f i n d a subset
by t h e f u n c t i o n
,or
go
b , and g be an i n j e c t i o n from b u of
a
equivalently
consider t h e f u n c t i o n which takes each subset
such t h a t u = a
-
of
a
x
b-f"(u)
into
a
i s sent
. To
g"(b-f"(u))
-
do t h i s ,
.
g"(b-f"(x))
This f u n c t i o n i s i n c r e a s i n g under i n c l u s i o n . By t h e f i x e d p o i n t lemma, the union u of all
x
such t h a t
x
5
a
-
g"(b-f"(x))
s a t i s f i e s the above.
1.5. CANTOR'S LEMMA Let
a
be a set. There i s no f u n c t i o n , i n j e c t i v e o r otherwise, w i t h domain
9 (a)
and range
( s e t o f subsets o f
a
a ).
CANTOR'S THEOREM (1) Every s e t
(2) I f a
a
9 (a) .
i s s t r i c t l y subpotent w i t h
i s non-empty, then every s e t o f m u t u a l l y d i s j o i n t subsets o f
s t r i c t l y subpotent w i t h
a
9 (a) .
1.6. EXPONENTIAL Given sets from
a
a
into
and b
However aO = 0 subsets o f
a
b
, the
. Thus
exponential o r power
i s the set o f functions
b . I n p a r t i c u l a r '0 = 1 . . For each s e t a , t h e s e t 7 ( a ) , where 2 = { O , l ) ) .
'b = {O) = 1 f o r each
f o r each non-empty s e t i s equipotent w i t h
a2
We have the f o l l o w i n g equipotences. For potent w i t h t h e Cartesian product w i t h t h e product
ab
(Ca)x(cb)
a
b
and
. Finally
c
. The
(ba)x(ca) C(ba)
disjoint, set
'(a
(b"c)a
x b)
i s equipotent w i t h
of
i s equi-
i s equipotent (bxc)a
.
1.7. CHOICE SET AND CHOICE FUNCTION Let
a
be a s e t o f non-empty m u t u a l l y d i s j o i n t s e t s
a s e t whose i n t e r s e c t i o n w i t h each element
x
of
a
x
.A
choice s e t f o r
i s a singleton.
a
is
5
Chapter 1
If a Let
i s f i n i t e , t h e r e i s always a choice s e t f o r a
be a s e t o f non-empty sets
which t o every element
x
of
a
.
x
a
( p r o o f by i n d u c t i o n ) .
A choice f u n c t i o n f o r
associates an element
f i n i t e , then there i s a choice f u n c t i o n f o r
fx
a
i s a function
of
.
x
a
If
is
.
a
AXIOM OF CHOICE (ZERMELO 1908) Every set, even i n f i n i t e , o f non-empty mutually d i s j o i n t sets admits a choice set. .____ E q u i v a l e n t l y every s e t o f non-empty sets admits a choice f u n c t i o n . An immediate consequence o f the axiom o f choice i s t h e f o l l o w i n g . Given a f u n c t i o n f
, injective
o r otherwise,
given a non-empty s e t subpotent w i t h
a
Rng f
, every
a
i s subpotent w i t h
.
Dom f
I n o t h e r words,
s e t o f mutually d i s j o i n t subsets o f
a
is
.
Problem. Does the preceding statement imply t h e axiom o f choice (problem mentioned i n RUBIN 1963 p. 5 note 1). A seemingly weaker consequence o f the axiom o f choice i s the assertion that
Dom f
i s never s t r i c t l y subpotent w i t h
. This
Rng f
does
5
1, 5 2 and 2.4; see ch.10 exerc. 2, where a FRAENKEL-MOSTOWSKI model i s constructed w i t h Dom f s t r i c t l y subpotent w i t h Rng f , a r e s u l t which i s t r a n s f e r a b l e t o ZF v i a the
n o t f o l l o w from ZF alone
i.e.
from t h e axioms mentioned i n
theorem o f JECH-SOCHOR (observation due t o HODGES).
1.8. GENERALIZED CARTESIAN PRODUCT Let a
a
be a non-empty s e t whose elements a r e non-empty. The Cartesian product o f
i s t h e s e t o f choice f u n c t i o n s which, t o each element
an element o f product
. If
ai
b x c
o f 1.2.
a
reduces t o t h e p a i r \ b , c ) , a
If
ai
of
a
associate
we have again the Cartesian
i s i n f i n i t e , i t f o l l o w s from t h e axiom o f choice
t h a t t h e Cartesian product o f
a
i s non-empty.
KONIG'S THEOREM
Let
I be a non-empty s e t o f elements
associated a p a i r o f s e t s
ai,
bi
with
i ( c a l l e d i n d i c e s ) , t o each o f which i s ai s t r i c t l y subpotent w i t h bi .
Then t h e union o f t h e product o f t h e
bi
ai ( i 6 I ) i s s t r i c t l y subpotent w i t h the Cartesian (axiom o f choice i s used).
h from u ai
0 Suppose there e x i s t s a b i j e c t i o n
the
bi
. For each
take i t s value
i and each x
(hx)(i)
. Thus we
of
ai
, take
onto Tr bi the f u n c t i o n
d e f i n e a f u n c t i o n from
ai
axiom o f choice, t h e range o f t h i s f u n c t i o n i s subpotent w i t h subpotent w i t h value o f associates
hx
bi on
ui
. Hence
t h e r e i s an element
i f o r any i s not i n
x
in
h " ( u ai)
t o see t h a t the union o f t h e
ai
ai
. The
ui
of
bi
, the
product o f
hx E 17bi
and
. By
into
bi
ai
, thus
the
strictly
which i s not the
choice f u n c t i o n which t o each
i
: c o n t r a d i c t i o n . We leave i t t o t h e reader
i s subpotent w i t h t h e product o f t h e
bi
.0
THEORY OF RELATIONS
6
Problem. Can the above theorem be proved from only the axioms of ZF in the case where the s e t I of indices i s f i n i t e w i t h cardinality greater than o r equal t o 2. Note th a t i f , in addition t o I being f i n i t e , we have f o r each index i tha t T ( a i ) i s subpotent with bi , then by CANTOR'S lemma 1 .5 , the axioms o f ZF suffice f o r the proof. For Card I = 2 , KONIG's theorem i s a consequence of ZF plus the axiom which a s s e r t s t h a t Dom f i s never s t r i c t l y subpotent with Rng f , or of ZF plus the apparently weaker axiom which as s er t s t h a t if a (resp. a ' ) i s s t r i c t l y b , b ' d i s j o i n t , then a u a ' i s s t r i c t l y subpotent with b (resp. b ' ) subpotent with b u b ' . § 2 - SECONDGROUP OF A X I O M S FOR ZF: FOUNDATION, SUBSTITUTION; O R D I N A L , INTEGER, COUNTABLE SET
INFINITY,
AXIOM OF FOUNDATION The axiom of foundation i s the statement t h a t every non-empty s e t a admits an element d i sj o i n t from a . I t follows t h at x $ x f o r any x Moreover f o r any XY Y i t i s impossible t h at x E y and Y E x , e t c . The axiom of foundation was introduced by ZERMELO 1930, inspired by a statement of von NEUMANN 1929. As t o i t s consistency, supposing t h a t a l l other axioms of ZF are consistent, see exercise 1. PREDECESSOR Given a s e t a , the successor a+l = a v j a } i s d i s t i n c t from a , since a $! a . Moreover i f a+l = b + l then a = b ; otherwise we would have a € b + l with a b , so a E b and similarly b E a , contradicting the axiom of foundation. Given a s e t c , the s e t whose successor i s c (which i s unique i f i t e x i s t s ) i s called the predecessor of c , denoted by c-1 . Finally, given a s e t a and i t s successor a t 1 , there i s no s e t x such t h a t a E x E a+l TRANSITIVE SET, TOTALLY ORDERED SET A s e t a i s t r a n si t i ve i f f , f o r every x, y , conditions y 6 x E a imply y E a If a i s t r a n si t i v e and non-empty, then every element o f a i s a proper subset of a . Also 0 E. a ( 0 i s the only element o f a which i s disjoint from a ) . Every union and intersection of t r an s i t i v e sets i s tra nsitive . I f a i s t r a n s i t i v e , then so i s a+l , A s e t a i s t o t a l l y ordered (by membership relation) i f f , f o r every x, y of a , e ith e r X E . y or y E x or x = y For example, a l l singletons are t o t a l l y ordered. However the singleton o f 1, i . e. 11) ={{O)): i s not tra nsitive . is The s e t { O , l , { l ) } i s t r an s i t i v e b u t n o t t o t a l l y ordered. The s e t {0,(1)) neither t r a n s i t i v e nor t o t a l l y ordered. Every intersection of t o t a l l y ordered s e t s i s t o t a l l y ordered. A union of such s et s i s not necessarily t o t a l l y ordered;
.
+
.
.
.
I
Chapter 1
however i f t h e s e t o f t o t a l l y o r d e r e d s e t s i s d i r e c t e d under i n c l u s i o n ( i . e . any two such s e t s a r e i n c l u d e d i n a t h i r d such s e t ) , t h e n t h e u n i o n i s t o t a l l y ordered. Finally i f
a
, then
i s t o t a l l y o r d e r e d by E
a + 1
so i s
.
2.1. ORDINAL
. For
An o r d i n a l i s a t r a n s i t i v e s e t which i s t o t a l l y o r d e r e d by E
example
,
0
1 = t o } , 2 = {0,1) . Every element o f an o r d i n a l i s an o r d i n a l . The successor s e t o f an o r d i n a l i s an o r d i n a l . The predecessor ( i f i t e x i s t s ) o f an o r d i n a l i s an o r d i n a l . The i n t e r s e c t i o n o f any s e t o f o r d i n a l s i s an o r d i n a l . An o r d i n a l
a
iff
a€ b
or
i s s a i d t o be l e s s t h a n o r equal t o an o r d i n a l
iff
a€ b
a = b ; an o r d i n a l
. Hence
If
, then
a,c b+l
a s b ac b
t o s t r i c t inclusion 0 By t r a n s i t i v i t y
a Eb
As from let
d
a& b ac b
. Let
a cb
b-a
. So e i t h e r so t h a t
d = a
. As
b
b
a< b )
(or
a
c_ b
i s equivalent
.
.
d E b-a
(yielding
u ca c
a E b
i s equivalent t o
be an element d i s j o i n t f r o m b-a
. Also
d c_ a
a € b ), o r
d ca
e b , t h i s d i s an o r d i n a l and d c b u E a-d
a
between o r d i n a l s . S i m i l a r l y
condition
. Hence
implies
Conversely, suppose t h a t
, denoted a .4( b , , denoted a < b ,
.
a = b+l
, the
b
b
> ( s t r i c t l y g r e a t e r than) a r e d e f i n e d .
or
fi
a
b
i s s t r i c t l y less than
< i s synonymous w i t h 6
( g r e a t e r t h a n o r equal t o ) and Given two o r d i n a l s
a
since
.
i s an o r d i n a l and
u
d
.
i s disjoint
I f t h e l a t t e r occurs, E b
and
d Eb
u E d o r d E u o r u = d . If U E d , t h i s contradicts I f d E u , t h e n s i n c e u E a , we have d E a which c o n t r a d i c t s I f u = d , t h e n d E a-d so d E a , again c o n t r a d i c t i n g d E b-a
,
we have e i t h e r
u E a-d d e b-a
.
.
TRICHOTOMY Given any two o r d i n a l s
a, b,
either
0 As we know, t h e i n t e r s e c t i o n
a n b = b case
or
a n b
so
a s b
a = b
a nb
b Ea
a€ b
i s s t r i c t l y included i n both or
a c b
and t h u s
a
.
a€ b
, so
that
a nb
and
a b
.
r\
. b = a
or
I n the f i r s t
A s i m i l a r conclusion i s
reached i n t h e second case. I n t h e t h i r d case, we have a 0b E b
a = b
i s an o r d i n a l . E i t h e r
.0
a
A
b
a
and
belongs t o i t s e l f , c o n t r a d i c t i n g t h e axiom o f
foundation. 0 Let
a
and
b
be two o r d i n a l s ; i f
b b a
then
b 3 a+l
or
b = a
.
We l e a v e i t t o t h e r e a d e r t o d e f i n e t h e maximum o r minimum o r d i n a l o f a s e t o f o r d i n a l s , denoted
mum:
take
b
Max, Min
belonging t o
.
Every non-empty s e t
a
and d i s j o i n t f r o m
a a
o f o r d i n a l s admits a m i n i -
.
More g e n e r a l l y we have t h e f o l l o w i n g scheme o f statements: g i v e n a c o n d i t i o n
‘8
which i s s a t i s f i e d by a t l e a s t one o r d i n a l , t h e r e i s a minimum o r d i n a l s a t i s f y i n g f Every t r a n s i t i v e s e t o f o r d i n a l s , e v e r y u n i o n o f a s e t o f o r d i n a l s i s an o r d i n a l .
8
THEORY OF RELATIONS
We leave i t t o t h e reader t o d e f i n e upper bound and lower bound o f a s e t o f o r d i nals. Given a s e t u o f o r d i n a l s , we denote the union o f u by Sup u It i s
.
If
i s an o r d i n a l and
o(
then
6 o( .
Sup u in
>
q
Given an o r d i n a l w i t h domain
.
o(
I n t h i s case
oc
i s t h e l e n g t h o f the sequence.
, the elements o r = o f
u
f o r which the f i r s t term called indices o f
EXTRACTED SEQUENCE
o( -SEQUENCE;
, an d - z e q u e x e , o r ordinal-indexed sequence, i s a f u n c t i o n
o(
Given a sequence
, or
u
u
are a l l ordered p a i r s
i i s an o r d i n a l s t r i c t l y l e s s than OC
u
i s indexed by
i
<
. The
o(
or
u(i)
. The
(i,ui) i's
are
second terms o f the
ordered p a i r s (which are a r b i t r a r y s e t s ) are c a l l e d the values o f ui
,
.
ORDINAL-INDEXED SEQUENCE,
2.2.
/3 e u i m p l i e s f i s a < Sup u , then there e x i s t s an o r d i n a l
a s e t o f o r d i n a l s such t h a t
u
I n o t h e r words i f
(3
u with
.
, i . e . t h e l e a s t upper bound o f u
u
the supremum o f
u and denoted
.
I n the p a r t i c u l a r case o f an h -sequence w i t h o r d i n a l values, we leave i t t o the reader t o d e f i n e increasing, decreasing, s t r i c t l y i n c r e a s i n g and s t r i c t l y decrea-
sing sequences. Given an o r d i n a l o(
and an 4 -sequence
u with
h
, where
h
, we
u
1364
u t o be a sequence w i t h l e n g t h
from
define an e x t r a c t e d sequence
, obtained by composition v o f
i s a s t r i c t l y i n c r e a s i n g &-sequence w i t h values i n
O(
;
v = u,h and v. = u f o r each i < f j . The n o t i o n o f e x t r a c t e d sequence i h(i) 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 antisynnnetric. For instance, by t h e axiom o f
so
i n f i n i t y introduced i n 2.4 below, given two d i s t i n c t sets t h e a-sequences 2.3.
a,b,a,b,..
and
, each
b,a,b,a,..
a, b,
we w i l l d e f i n e
e x t r a c t e d from t h e other.
INTEGER, n-ELEMENT SET, WORD, n-TUPLE
By non-negative integer, o r i n t e g e r , o r n a t u r a l number, we mean a f i n i t e o r d i n a l .
Every element o f an i n t e g e r i s an i n t e g e r . Every non-zero ( i . e . non-empty) i n t e g e r has an i n t e g e r predecessor. If
a
i s an i n t e g e r and
As
a
i s finite,
b
bE a
(or
b
, then
i s s t r i c t l y subpotent w i t h
b
i s s t r i c t l y included i n
. Thus
a
a
equipotent i n t e g e r s
are i d e n t i c a l .
f
We thus have t h e scheme o f induction: i f a c o n d i t i o n
each i n t e g e r
a
the condition f ( a )
implies f?(a+l)
,
holds f o r 0
, and
if for
z/fholds f o r every
integer. Every f i n i t e s e t i s equipotent w i t h an i n t e g e r . Given a s e t c a l l e d the c a r d i n a l , o r c a r d i n a l i t y o f
A s e t equipotent w i t h an i n t e g e r A f i n i t e sequence o r word i s an of l e n g t h n i s c a l l e d an n-t=.
n
a
and denoted
i s c a l l e d an
n-sequence, where
a
Card a
, t h i s integer i s
.
n-element s e t . n
i s an i n t e g e r ; such a word
.
9
Chapter 1
When r e s t r i c t e d t o words, the n o t i o n o f e x t r a c t e d sequence becomes antisymmetric; i . e . two words each o f which i s e x t r a c t e d from the o t h e r are i d e n t i c a l . 2.4. A X I O M OF INFINITY, SUCCESSOR AND LIMIT ORDINAL
The axiom o f i n f i n i t y asserts t h e existence o f an i n f i n i t e set. A more useful and stronger version asserts t h e existence o f a D e d e k i n d - i n f i n i t e s e t . More p r e c i s e l y the existence o f a s e t
if
x
belongs t o
a
, then
a
c o n t a i n i n g t h e element
t h e successor
x+l = x u ( x )
0
and such t h a t
belongs t o
a
. With
an appropriate a p p l i c a t i o n o f t h e separation scheme, the axiom o f i n f i n i t y y i e l d s t h e existence o f the s e t o f integers, denoted by 0 . The s e t
>
i n f i n i t e o r d i n a l , the smallest o r d i n a l
0
w i s an
w i t h o u t a predecessor.
A limit o r d i n a l i s an o r d i n a l w i t h o u t a predecessor. A successor o r d i n a l i s an o r d i n a l w i t h a predecessor. SUBSTITUTION SCHEME
A t t h i s p o i n t we replace t h e separation scheme by t h e more general s u b s t i t u t i o n scheme (due t o FRAENKEL 1925), o f which the reader i s assumed t o be f a m i l i a r . With t h i s scheme we can define, f o r example, L3+ W = w . 2 : beginning w i t h the set
w o f integers, associate t o each i n t e g e r i the o r d i n a l w+i defined
below i n s e c t i o n 3.1. Then using the s u b s t i t u t i o n scheme d e f i n e t h e s e t o f ~
+
i runs through . Another example: denote by No the s e t G) o f integers, and f o r each i n t e g e r i l e t Ni+l = y ( N i ) ( t h e s e t o f subsets o f Ni) ; the s u b s t i t u t i o n scheme allows one t o d e f i n e the s e t o f Ni f o r i belonging t o w
.
as
1 t o t h e present 5 2.4 ( n o t i n c l u d i n g the axiom of choice nor i t s weakened versions such as choice among f i n i t e sets),
The axioms p r e v i o u s l y introduced, from
5
are c a l l e d the axioms of ZF. I f no special assumption i s e x p l i c i t e l y mentioned i n a theorem, then t h i s i n d i c a t e s t h a t t h e theorem i s proved i n ZF alone.
If, however, the axiom o f choice o r o t h e r supplementary axioms ( f o r t h e most p a r t weakened versions o f the axiom o f choice, s t a t e d below) are used, then we c l e a r l y i n d i c a t e such. We have already done t h i s f o r KONIG's theorem i n 1.8. Recall t h a t t h e axiom of choice has been proved c o n s i s t e n t w i t h
ZF ( i f ZF i t s e l f
i s c o n s i s t e n t ) by GODEL 1938. The negation o f the axiom o f choice has been proved equiconsistent w i t h ZF by COHEN 1963 (see t h e Bibliography COHEN 1966). 2.5. DENUMERABLE SET, COUNTABLE SET, COUNTABLE AXIOM OF CHOICE A s e t i s s a i d t o be denumerable, resp. countable, i f i t i s equipotent, resp. subpotent w i t h
w , t h e s e t o f i n t e g e r s . ZF alone s u f f i c e s t o show t h a t t h e
union o f two denumerable sets, t h e Cartesian product o f two denumerable sets, and t h e s e t o f a l l f i n i t e subsets o f Following 2.2, we c a l l an &-sequence by t h e s e t o f i n t e g e r s .
w
are a l l denumerable.
a sequence o f l e n g t h
w , hence indexed
i
THEORY OF RELATIONS
10
The c o u n t a b l e axiom o f c h o i c e i s a p a r t i c u l a r case o f t h e axiom o f c h o i c e . I t s t a t e s t h a t f o r e v e r y c o u n t a b l e s e t o f non-empty d i s j o i n t s e t s , t h e r e i s a c h o i c e s e t . T h i s axiom i s s t r i c t l y weaker t h a n t h e axiom o f choice; i . e . i f ZF i s c o n s i s t e n t , t h e n t h e r e i s a model o f ZF and c o u n t a b l e c h o i c e which s a t i s f i e s t h e n e g a t i o n of t h e general axiom o f c h o i c e (JECH 1973). Note t h a t t h e c o u n t a b l e axiom o f c h o i c e i m p l i e s t h a t e v e r y denumerable u n i o n o f denumerable s e t s i s denumerable. Indeed, t h i s axiom a l l o w s one t o choose, f o r each o f t h e denumerable s e t s i n t h e union, a b i j e c t i o n f r o m t h a t s e t o n t o t h e i n t e g e r s . On t h e o t h e r hand, t h e above statement i s n o t p r o v a b l e from ZF alone: t h e r e i s a model o f ZF itn which t h e continuum i s a denumerable u n i o n o f denumerable s e t s ( A z r i e l LEVY, u n p u b l i s h e d ) .
2.6. DENUMERABLE SUBSET A X I O M T h i s axiom s t a t e s t h a t e v e r y i n f i n i t e s e t has a denumerable subset. I t f o l l o w s from t h e c o u n t a b l e axiom o f c h o i c e . 0 Let
a
be an i n f i n i t e s e t . F o r each i n t e g e r
o f elements f r o m one o f these
a
. By
i-tuples.
o f t h e chosen
i
, associate
the s e t o f
i-tuples
i
c o u n t a b l e choice, we can a s s o c i a t e t o each i n t e g e r
I t remains t o t a k e t h e W-sequence formed f r o m t h e terms
1-tuple,
... . 0
2-tuple,
The denumerable subset axiom i s s t r i c t l y weaker t h a n t h e c o u n t a b l e c h o i c e (JECH 1973). Let
a
be an i n f i n i t e s e t ; t h e f o l l o w i n g t h r e e c o n d i t i o n s a r e e q u i v a l e n t :
(1) t h e r e e x i s t s a denumerable subset o f (2) there e x i s t s a b i j e c t i o n o f
a
a ;
a ; i n o t h e r words,
o n t o a p r o p e r subset o f
a
i s D e d e k i n d - i n f i n i t e (see 1.3);
(3) there e x i s t s a choice f u n c t i o n a s s o c i a t e s an element
fx
which t o each f i n i t e subset
f
i n t h e complement
a-x
x
.
of
a
Consequently, t h e d e n u m e r a b l e subset axiom i s e q u i v a l e n t t o s a y i n g t h a t f i n i t e n e s s c o i n c i d e s w i t h D e d e k i n d - f i n i t e n e s s . However w i t h ZF alone, t h e r e can e x i s t an i n f i n i t e s e t having f o r each i n t e g e r
i a subset e q u i p o t e n t w i t h
i
,yet
having
no denumerable subset. 2.7. Having d e f i n e d t h e i n t e g e r s , we can now complete t h e i n i t i a l remarks f r o m
5
2 by adding t h a t , w i t h t h e axiom o f f o u n d a t i o n , t h e r e a r e no O - s e q u e n c e s
with ger
ui+l r
belonging t o
ui
f o r each i n t e g e r
, t h e r e i s no c y c l e u1C u2 E
. .. E
i
. I n particular,
ur E u1
u
f o r every i n t e -
.
2.8. The axiom o f f o u n d a t i o n i s e q u i v a l e n t t o t h e f o l l o w i n g axiom scheme. Let
f
be a c o n d i t i o n which h o l d s f o r
ment o f a g i v e n s e t
a
, then
f
0
holds f o r
- h o l d s f o r each e l e . Under these hypotheses, f holds
and such t h a t , i f a
f o r e v e r y s e t . Note t h a t we can e l i m i n a t e t h e h y p o t h e s i s
"
f
h o l d s f o r 0 "; t h i s
Chapter 1
11
b e i n g a p a r t i c u l a r case o f t h e second h y p o t h e s i s , made p r e c i s e as f o l l o w s : e i t h e r t h e r e e x i s t s an element o f 0
Let
a
a
s a t i s f y i n g "not
condition holding f o r
and f o r e v e r y s e t
0
(e s a t i s f i e s o u r hypotheses, b u t Conversely, l e t a
a2
a
f.
satisfies
'6
be t h e
which does n o t belong t o
a
'if does n o t h o l d f o r e v e r y element o f
a
. Let
al
be t h e s e t o f elements o f
be t h e s e t o f elements o f t h e u n i o n u al
be t h e s e t o f elements o f u a2
a
. .
Then
f".
s a t i s f y i n g "not
s a t i s f y i n g "not
If: . "
Let
a3
s a t i s f y i n g " n o t I$", e t c . Then t h e u n i o n o f t h e
(ii n t e g e r ) f a l s i f i e s t h e axiom o f f o u n d a t i o n .
ai
, or
I'
be a c o n d i t i o n s a t i s f y i n g o u r hypotheses, b u t such t h a t t h e s e t
does n o t s a t i s f y
Let
f
be a s e t which f a l s i f i e s t h e axiom o f f o u n d a t i o n , and l e t
0
A necessary and s u f f i c i e n t c o n d i t i o n f o r a s e t a t o be an o r d i n a l i s t h a t ao, al, ... , an w i t h a. = a and ai+l b e l o n g i n g t o
2.9.
f o r e v e r y f i n i t e sequence ai
for
i
< n , every
ai
i s t r a n s i t i v e . I n o t h e r words, an o r d i n a l i s a s e t
which i s h e r e d i t a r i l y t r a n s i t i v e ; use 2.1: e v e r y t r a n s i t i v e s e t o f o r d i n a l s i s an o r d i n a l . Equivalently,
a
i s an o r d i n a l i f f
a
and
all
elementsof
a
are t r a n s i t i v e
(see f o r i n s t a n c e POWELL 1975 p. 223). Analogously, we l e a v e i t t o t h e r e a d e r t o p r o v e t h a t a s e t iff
a
i s empty o r
a
a
i s an i n t e g e r
i s a successor s e t , and e v e r y element o f
a
i s either
empty o r a successor s e t (communicated by HATCHER 1977). 2.10. A X I O M OF CHOICE FOR FINITE SETS Now t h a t i n f i n i t e s e t s have been i n t r o d u c e d , we i n d i c a t e h e r e an i m p o r t a n t weakening o f t h e axiom o f c h o i c e , which a s s e r t s t h e e x i s t e n c e o f a c h o i c e s e t f o r e v e r y s e t o f non-empty,
f i n i t e , m u t u a l l y d i s j o i n t s e t s . T h i s weakened f o r m
i s n o t i m p l i e d by and does n o t i m p l y t h e c o u n t a b l e axiom o f c h o i c e f r o m n o r t h e denumerable subset axiom o f 2.6.
0
2.5,
I n f a c t t h e axiom o f dependent choice,
which i s s t r o n g e r t h a n c o u n t a b l e c h o i c e , does n o t i m p l y c h o i c e among f i n i t e s e t s : see ch.2
5
1.6.
2.11. We s h a l l c a l l i n d u c t i o n , o r t r a n s f i n i t e i n d u c t i o n , t h e f o l l o w i n g reasoning. Suppose t h a t i f a c o n d i t i o n then
If
holds f o r
'f
holds f o r every o r d i n a l s t r i c t l y less than
o(
4 ; under t h i s h y p o t h e s i s , f h o l d s f o r e v e r y o r d i n a l . T h i s
" i s a f o r m o f t h e scheme s t a t e d i n 5 2.1: i f " n o t i s s a t i s f i e d by a t l e a s t one ordinal, there i s a least ordinal satisfying "not Often, i n d u c t i o n i s broken up i n t o a p r o o f f o r 0, a p r o o f f o r t h e t r a n s i t i o n between an a r b i t r a r y o r d i n a l o(
e'' .
and i t s successor o( + 1 , and a p r o o f f o r o( a l i m i t o r d i n a l . A d e f i n i t i o n by r e c u r s i o n i s made by i n t r o d u c i n g a statement f ( o ( ,a) which u n i q u e l y a s s o c i a t e s a s e t a t o each o r d i n a l o( T h i s s t a t e m e n t w i l l u s u a l l y be
.
THEORY OF RELATIONS
12
o f the f o l l o w i n g form. "There e x i s t s one and o n l y one f u n c t i o n
w i t h domain
f
d + 1 ( t h e successor o f o(), such t h a t t h e i n i t i a l ordered p a i r
belongs t o (0,u) (where u i s a r b i t r a r i l y given), the f i n a l ordered p a i r (o( ,a) belongs t o , and such t h a t f o r each (3 o( t h e p a i r ( (3 ,b) belongs t o f , provided
f f
that
b
has been obtained i n a c e r t a i n ( s u i t a b l y defined) manner from t h e s e t o f f w i t h f i r s t term < 13 Because o f the uniqueness , the f u n c t i o n f ' corresponding t o oc' w i l l be an exten-
ordered p a i r s belonging t o of
, when
f
'I.
t o t h e domain a( '+1
f
sion o f
q'>d,
. Some
examples o f d e f i n i t i o n by recursion:
0
sum, product, exponentiation f o r o r d i n a l s i n t i o n o f fundamental rank i n
0
3; aleph rank i n
0
6.4. The d e f i n i -
5.2 i s a l s o by recursion, i f one begins by associa-
.
t i n g t o each o r d i n a l o( t h e s e t o f a l l sets w i t h fundamental rank o(
Note t h a t d e f i n i t i o n by r e c u r s i o n using the axioms o f ZF i s e a s i e r t o j u s t i f y than d e f i n i t i o n by simple r e c u r s i o n i n f i r s t - o r d e r Peano a r i t h m e t i c , such as i s generally presented today (however, the o r i g i n a l t e x t o f PEANO 1894 i s w r i t t e n i n second-order l o g i c ) . I n order t o j u s t i f y d e f i n i t i o n by r e c u r s i o n i n f i r s t order a r i t h m e t i c , one i s led, i n the manner o f GOOEL 1931, t o use the "Chinese b = a!
remainder theorem". For instance, one defines the f o l l o w i n g :
as an a b b r e v i a t i o n f o r
u, v such t h a t the remainder a f t e r
there e x i s t two i n t e g e r s
"
u by v + l i s 1 , the remainder a f t e r d i v i s i o n o f u by 1 i s b , and f o r each i ( 1 s i d a) one obtains the remainder a f t e r
division o f
+
(a+l)v
u by ( i + l ) v + 1 from the remainder a f t e r d i v i s i o n o f 1 by m u l t i p l y i n g the l a t t e r by i+l " .
u
division o f iv
+
5 3 - REVIEW
CANTOR
ORDINAL ALGEBRA,
OF
by
NORMAL FORM, INDECOMPOSABLE
ORDINAL 3.1. SUM
Y
We say t h a t o( +/3 =
f
f i +1
w i t h domain
f
p a i r belonging t o then
(u+l,v+l)
)
belong t o
belongs t o
ordered p a i r s
(x,y)
then t h e p a i r
(Sup x, Sup y )
Given
and
o(
d,b, 8
w i t h f i r s t term
(/5 ,
and f i n a l p a i r
(where
(hence, f o r each
are o r d i n a l s ) i f f t h e r e e x i s t s a f u n c t i o n
u&
fi
there i s one and only one ordered
u ), such t h a t t h e i n i t i a l p a i r
f ;if
(u,v)
f ; and f i n a l l y such t h a t i f
f o r which the f i r s t terms
/3 , the
belongs t o
belongs t o
f
x
f
(0, m ) u < fs
where
f
contains as elements
4 (5,
admit a supremum Sup x
.
reader can prove by i n d u c t i o n on
/3
the existence and
uniqueness o f the preceding f u n c t i o n , hence the existence and uniqueness o f the ordinal
5
=A
+ fi
. I n the
same manner, one proves f o r every o(
l i t i e s d + O = O+o( = % and oC+( /5+1) = ( * +f3)+1 every s e t o f o r d i n a l s For a l l o(
, 13 P
0
u
, and
,/s
the supremum e q u a l i t y o( +(Sup u) = Sup(&
we have o(+ /3,
> o( .
t h e equa-
f o r every o(
For a1 1 o( , p we have 4+
+ u)
>, (3
and
. where
,
13
Chapter 1
: f o r instance
e q u a l i t y i s p o s s i b l e w i t h non-zero o(
The supremum e q u a l i t y does n o t h o l d on t h e l e f t : i f S u p ( i + w ) = ~d #
then
(Sup i )
+
.
1 + 4) = G, i i s an a r b i t r a r y i n t e g e r
LJ = ~ d t.3. +
Ordinal a d d i t i o n i s a s s o c i a t i v e . Commutativity holds f o r integers, o r f i n i t e 1 + CJ = c d # id+ 1 . f i 2 o( , there e x i s t s one and o n l y one satisfying ; t h i s 2( i s c a l l e d t h e d i f f e r e n c e /3 - o( .
ordinals; however For a l l O(
CA +
8
=
and
0
6
The i n e q u a l i t y
w
+ /3
= O(
II
+
< . b =8 .
implies
The i n e q u a l i t y % $
<
general f o r
o< + &'
i m p l i e s o(+ /3$
same r e s u l t f o r s t r i c t i n e q u a l i t y
+ I f . This
6
i m p l i e s o( +
.
, as
and conversely. Also the
Hence a d d i t i o n i s l e f t cancellable, i.e. does n o t hold i n
0 + W = 1 + W Hence a d d i t i o n i s n o t r i g h t cancellable. 1 and consequently every f i n i t e o r d i n a l i s absorbed by
F i n a l l y the o r d i n a l
1 + o( = o( f o r q
every i n f i n i t e o r d i n a l , i n t h e sense t h a t
infinite.
3.2. PRODUCT We say t h a t o(
. /s
=
?f
i f f there exists a function
f
and t h e f i n a l p a i r
(
w i t h domain
0 ,g )
fi +
,
1
f
,
O i s equivalent t o o ( = O o r /3= , w i t h o( f i x e d and u an a r b i t r a r y o r d i n a l , are
O
Ordinals o f the form o(.u c a l l e d t h e m u l t i p l e s o f o(
.
such t h a t the i n i t i a l p a i r and such t h a t i f that i f Sup x
(x,y) E f
,< f i
For a l l o(
then
(0,O)
f
(u,v)
where
for all (Sup x
x
u
< fs f
= 0.N = 0
For every o( and every s e t o f o r d i n a l s
d .(sup u) = sup(@ .u)
(u+l,v+o( ) 6 f
, and
such
belonging t o a s e t which admits a supremum
, Sup y ) E
, p we have o( .O
then
belong t o
. .
and d .( &+1) = o(. 0 + o(
, we
u
have the supremum e q u a l i t y
. Moreover d.0 = For example
Every m u l t i p l e o f M I augmented by o( o f a set o f multiples o f d
0
.
i s a m u l t i p l e o f every o r d i n a l .
,yields
i s a m u l t i p l e o f o(
a m u l t i p l e o f o(
. Finally
. The
supremum
every m u l t i p l e o f o(
i s obtained from 0 by these two i n d i c a t e d orocesses. More r i g o r o u s l y i f a condit i o n i s t r u e f o r 0 and i s preserved i n t h e passage from an o r d i n a l
u
to
u+w
as w e l l as i n the passage t o supremum, then t h i s c o n d i t i o n i s t r u e f o r every m u l t i p l e o f o(
.
The supremum e q u a l i t y on t h e r i g h t , given above, does n o t h o l d on t h e l e f t : i f
i designates an a r b i t r a r y i n t e g e r , then
Sup(i.2) = W # (Sup i ) . 2 = CJ .2
M u l t i p l i c a t i o n i s a s s o c i a t i v e and d i s t r i b u t i v e on t h e r i g h t : r . ( M + / 3 )
.
=r.q+r,(,p I
D i s t r i b u t i v i t y on the l e f t and commutativity h o l d f o r integers; however ( w + l ) . w = c J . o # w . b + 1 . ~ 3 and 2 . ~ = ~ # ~ 3 I.t c2a n. h a p p e n t h a t c ( . p i s n o t a m u l t i p l e of (3 , e.g. (0+1).2 = w . 2 + 1 : i t i s n o t a m u l t i p l e o f 2 For 4 # 0 the i n e q u a l i t y 1)4 $ i m p l i e s & . (3 6 &.$ and conversely. The same r e s u l t holds f o r s t r i c t i n e q u a l i t y . Thus m u l t i p l i c a t i o n i s cancellable on
.
THEORY OF RELATIONS
14
the l e f t except f o r
0 ; i.e. f o r
The i n e q u a l i t y o( & f s since 1.a = 2 . w = W
non-zero
O(
implies
o(
.
0= d
.y,r 0 . 8 .
. Multiplication i s
Given two o r d i n a l s d and f i # 0 quotient, and a unique o r d i n a l E
g .fs = q . v i m p l i e s
This does n o t s u b s i s t f o r thus n o t cancellable on the r i g h t .
<
, there
i s a unique o r d i n a l called the c a l l e d t h e remainder i n t h e d i v i s i o n o f o(
0 : consequence
by f i , w i t h g = f i $ + and E <' maximum o r d i n a l u such t h a t /s u 4
4
o f t h e existence o f a
.
3.3. POWER OR EXPONENTIATION We say t h a t
&Is
i f f there e x i s t s a function
=
w i t h domain
f
f i +1 , such
t h a t t h e i n i t i a l p a i r ( 0 , l ) and t h e f i n a l p a i r ((3,&) belong t o f , and such t h a t i f ( u , v ) E f where u < (3 then ( u t l . v . 4 ) f , and such t h a t i f (x,y) E f f o r x belonging t o a s e t which admits a supremum Sup x 6 (3, then (SUP X I SUP Y) E f . For a l l cx,p we have oCo = 1 and cc ('+'I = 4'. 4 For a l l o( and every s e t o f o r d i n a l s u , we have the supremum e q u a l i t y o( (sup = Sup ( o i ') . Moreover W p = 0 i s e q u i v a l e n t w i t h o( = 0 and f i # 0 The e q u a l i t y M p = 1
.
.
i s equivalent t o o( = 1 o r /3 = 0 Ordinals o f the form o( , w i t h cx f i x e d and u an a r b i t r a r y o r d i n a l , are c a l l e d powers o f O( For example 1 i s a power o f every %r I f v i s a power o f o(
.
.
.
then so i s v.* The supremum o f a s e t o f powers o f i s a power o f 4 F i n a l l y , every power o f o( i s obtained from 1 by these two processes.
.
The supremum e q u a l i t y on t h e r i g h t given above does n o t s u b s i s t on t h e l e f t : i f i designates an a r b i t r a r y i n t e g e r , then Sup(i 2 ) = W # (Sup i)'= W 2 We have o( ( * + ' = & f i . 4 r f o r a l l O( ,P, I t i s impossible i n general =w.2 f 2 1 . 2 w = w . t o interchange terms i n t h e product; f o r example 2( w+l)
.
r.
&(b'r)
(~&fi)~= f o r a l l o<,p, 8 . I t i s impossible i n general t o = W 2 # 2('*@) = interchange terms o f t h e product i n t h e exponent: (2 ")' The e q u a l i t y (ac).(bc) = (a.b)c which holds f o r integers, does n o t s u b s i s t i n 2 2 general, even f o r f i n i t e exponents: ( Q ~ ) . Z =~ u 2 . 4 # ( w . 2 ) = o .2 . This e q u a l i t y does n o t s u b s i s t f o r a, b f i n i t e and an i n f i n i t e exponent: (2LJ).(ZW)=L32#4W= w . We have
For o(
>/ 2 , the
<.
i n e q u a l i t y f34 21 i m p l i e s NP< c ( and ~ conversely; same
Thus we have c a n c e l l a t i o n : r e s u l t with The i n e q u a l i t y o( 4 -fir since z W = 3 w = w
= o(
.
F i n a l l y f o r o((
f s the o r d i n a l w4 i s absorbed by
3.4. Given two o r d i n a l s o( and /3 maximum exponent among t h e o r d i n a l s
>/
'
implies
fs = 'b
.
; t h i s does n o t s u b s i s t f o r
&*,
i.e.
<
aa+ W f i = k)?
2 , there i s a unique o r d i n a l which i s the u satisfying fi \< 4 .
Chapter 1
Given o( and
(3 2 2 and t h e maximuA exponent
s
b,
82 ...
8
/33-6
such t h a t
o(
, there
( f i r ) . $,< o( . Moreover < (-5 and a s t r i c t l y decreasing, t h u s f i n i t e sequence o f o r d i n a l s
e x i s t s a maximum o r d i n a l Given
15
2
such t h a t
$ ( I ) '> g(2), ... and a c o r r e s p o n d i n g se uence o f o r d i n a l s t ( l ) , $ ( 2 ) , each s t r i c t l y l e s s t h a n /3 , we have b x > f i 9 ( ' ) . $ ( 1 ) t $ ( 2 ) t ...
fig(*).
8
( p r o o f by i n d u c t i o n on
).
3.5. CANTOR NORMAL FORM Given
4 and f3 >, 2 , t h e r e e x i s t s a decomposition o f o( i n t o a f i n i t e sum o f f i r , 6 , w i t h c o e f f i c i e n t s $< /3 and exponents s t r i c t l y decreasing.
terms
Furthermore t h i s decomposition i s unique. I t i s c a l l e d t h e Cantor decomposition o f o( i n t o powers o f that
fi
=
0
o r Cantor normal f o r m o f 4 i n base f ,
6 are
u , the c o e f f i c i e n t s
.
I n t h e case
integers.
3.6. DECOMPOSABLE AND INDECOMPOSABLE ORDINAL An o r d i n a l o( i s c a l l e d decomposable i f f t h e r e e x i s t
o( = f l + 8
r'c.C w i t h
& < q and
; o t h e r w i s e o( i s c a l l e d indecomposable.
I f o( i s indecomposable,
then e v e r y sum o f two non-zero o r d i n a l s which i s equal t o o( has second t e r m equal t o d , and c o n v e r s e l y . A non-zero o r d i n a l o( i s indecomposable i f f
O(
i s a power o f
a .T h i s
follows
from t h e e x i s t e n c e and uniqueness o f t h e Cantor decomposition i n t o powers o f t o g e t h e r w i t h t h e a b s o r p t i o n statement (end o f 3.3).
54
-
4.1.
EQUIPOTENT WITH THE CONTINUUM
LC:
,
EQUIPOTENT W I T H THE CONTINUUM, C O N T I N U U M HYPOTHESIS, R E A L
A s e t i s s a i d t o be e q u i p o t e n t w i t h t h e continuum i f f i t i s e q u i p o t e n t w i t h ?(a), t h e power s e t o f t h e i n t e g e r s , o r e q u i v a l e n t l y w i t h " 2 on w t a k i n g values
0
s t r i c t l y subpotent w i t h Let
a
or
1
. By
9 ( IC) )
, the
s e t o f functions
CANTOR'S theorem 1.5, e v e r y c o u n t a b l e s e t i s
.
, b be two d i s j o i n t denumerable s e t s . By 1.6 we have t h a t a2 x b2 i s
equipotent w i t h
(a
" b ) 2 . Hence
t h e C a r t e s i a n p r o d u c t o f two s e t s each equi-
p o t e n t w i t h t h e continuum i s i t s e l f e q u i p o t e n t w i t h t h e continuum. The same r e s u l t h o l d s f o r t h e C a r t e s i a n p r o d u c t o f a c o u n t a b l e s e t w i t h a s e t which i s e q u i p o t e n t w i t h t h e continuum. Similarly
2)
i s equipotent w i t h
(w
w)2
.
Hence
e q u i p o t e n t w i t h t h e continuum, t h e n t h e s e t o f &-sequences
a
i s a set
w i t h values i n
a
i s a l s o e q u i p o t e n t w i t h t h e continuum.
4.2.
I f we s u b t r a c t an a r b i t r a r y denumerable subset
w i t h t h e continuum, t h e n t h e d i f f e r e n c e
c-a
a from a s e t c equipotent i s e q u i p o t e n t w i t h t h e continuum.
16
THEORY OF RELATIONS
This i s a special case o f t h e f o l l o w i n g p r o p o s i t i o n .
Let
be an i n f i n i t e s e t which i s equipotent w i t h the Cartesian product
a
c = y(a)
and l e t
. Then
t h e d i f f e r e n c e set, obtained by removinq from a
a r b i t r a r y subset which i s e q u i p o t e n t w i g 0
Since
a
i s equipotent w i t h
i s a l s o equipotent w i t h
cxc
which i s equipotent w i t h
a
f
range o f a b i j e c t i o n ordered p a i r
on
f x = (y,z)
2xa
,is
equipotent w i t h
by 1.6. Hence t h e d i f f e r e n c e o f
.
.
c
and a subset
Each element
o f elements
y, z
x of
of c
a
. Let
cxc
i s associated t o an
o f t h i s p a i r . The f u n c t i o n thus obtained has domain
cannot have range
c
?(a)
, by
and the
us associate t o each
t h e f i r s t term y
=
an
, t h e s e t c , which i s equipotent w i t h a2 ,
i s equipotent w i t h t h e d i f f e r e n c e o f a
c
,
2xa c
a
x
and
CANTOR'S lemma 1.5. Thus there e x i s t s an element
f o f an element o f a , f o r any z belonging t o c . Hence t h e d i f f e r e n c e o f cxc and f " ( a ) includes a subset which i s equipotent w i t h c , and so by BERNSTEIN-SCHRODER 1.4 i s equipotent u
of
with
c
c
f o r which
(u,z)
i s n o t the value by
.0
4.3. L e t
a
be a s e t equipotent w i t h the continuum. For every p a r t i t i o n o f
a
i n t o denumerably many subsets, one o f t h e subsets i s equipotent w i t h the continuum (uses t h e axiom of choice). 0
ai
Suppose on the c o n t r a r y t h a t there i s a p a r t i t i o n o f ( i i n t e g e r ) , and t h a t every
a
i n t o d i s j o i n t subsets
i s s t r i c t l y subpotent w i t h
ai theorem 1.8 (axiom o f choice), t h e union
a
o f the
ai
a
. Then
by KONIG's
i s s t r i c t l y subpotent
w i t h the Cartesian product o f an a - s e q u e n c e o f sets, each equipotent w i t h t h e continuum. But t h i s Cartesian product i s equipotent w i t h the continuum: contradiction. 0 4.4. CONTINUUM HYPOTHESIS, GENERALIZED CONTINUUM HYPOTHESIS The axiom c a l l e d continuum hypothesis asserts the non-existence o f a s e t which
o and y( a) ZF, and even o f ZF p l u s t h e axiom o f
i s s t r i c t l y intermediate, w i t h respect t o subpotence, between This axiom i s l o g i c a l l y independent o f
.
choice (COHEN 1963, see B i b l i o g r a p h y 1966). The axiom c a l l e d generalized continuum hypothesis asserts t h e non-existence o f a s e t s t r i c t l y intermediate, w i t h respect t o subpotence, between f o r every i n f i n i t e s e t
a
.
a
and p ( a )
When added t o t h e axioms o f ZF, t h i s i m p l i e s the
axiom o f choice (see ch.2 exerc. 1). 4.5. REAL We leave i t t o t h e reader t o r e d e f i n e p o s i t i v e and negative i n t e g e r , and then r e a l , as an ordered p a i r formed from an i n t e g e r which i s c a l l e d t h e i n t e g e r -
-
p a r t , and an i n f i n i t e s e t o f non-negative i n t e g e r s . The l a t t e r s e t w i l l be i d e n t i f i e d w i t h an W-sequence o f terms
ui
( i non-negative i n t e g e r ) w i t h
,
17
Chapter 1
ui = 0
or
1 according t o whether
i belongs t o t h e i n f i n i t e s e t o f i n t e g e r s
o r not. This sequence i s c a l l e d the b i n a r y expansion o f the r e a l , which always contains i n f i n i t e l y many occurrences o f zero. The notions o f r a t i o n a l r e a l and dyadic r e a l , i . e . r a t i o n a l whose denominator i s a power o f 2
, are assumed t o be f a m i l i a r , as w e l l as t h e denumerability o f the
set o f rationals. The s e t o f r e a l s i s equipotent w i t h the continuum: remove from the s e t o f a l l sets o f i n t e g e r s , the denumerable s e t o f f i n i t e sets o f i n t e g e r s , and use 4.2. We leave i t t o t h e reader t o d e f i n e the o r d e r i n g on t h e r e a l s : l e s s than o r equal
( 3 ) , greater
( 3 ) , and the r e l a t e d s t r i c t i n e q u a l i t i e s . Also the reader can d e f i n e the notions o f dense, c o f i n a l , c o i n i t i a l s e t o f r e a l s to
than o r equal t o
(an example being t h e r a t i o n a l s o r t h e dyadic r e a l s ) . The reader can define a closed, open, half-open i n t e r v a l o f r e a l s , an i n i t i a l , f i n a l i n t e r v a l , an _upper
bound and
lower bound o f a s e t o f r e a l s , t h e maximum, t h e minimum, a r e a l valued
sequence which i s s t r i c t l y ( o r otherwise) increasing, decreasing. Every s e t o f mutually d i s j o i n t i n t e r v a l s o f r e a l s which are n o t reduced t o s i n g l e t o n s i s countable: enumerate the r a t i o n a l s and associate t o each i n t e r v a l the f i r s t r a t i o n a l which belongs t o i t . Consequently, every s t r i c t l y i n c r e a s i n g ( o r s t r i c t l y decreasing) ordinal-indexed sequence o f r e a l s i s countable. 4.6.
DEDEKIND'S THEOREM
I f we p a r t i t i o n the r e a l s i n t o an i n i t i a l i n t e r v a l
final interval b
b
, both
non-empty, then e i t h e r
a
a
and i t s complement the
has a maximum element o r
has a minimum element.
Consequently, f o r any s e t
a
o f r e a l s , i f t h e r e e x i s t s an upper bound, then
there e x i s t s a l e a s t upper bound c a l l e d t h e supremum o f Analogous d e f i n i t i o n o f the infimum which i s denoted f o r every s e t
a
a
Inf a
and denoted
.
Sup a
.
I n o t h e r words,
o f reals, t h e r e e x i s t s a smallest i n t e r v a l ( w i t h respect t o
inclusion) including
a : the i n t e r v a l
(Inf a
, Sup a) which i s closed, open
o r half-open, i n i t i a l , f i n a l o r c o n t a i n i n g a l l t h e r e a l s , depending on the case. When u s e f u l , we w i l l use the
and p r o d u c t o f r e a l s , which the reader i s
presumed t o know.
4.7.
To see some i n i t i a l d i f f i c u l t i e s provided by the axiom o f choice, which
i n d i c a t e t h a t t h i s axiom i s n o t "obvious", note t h a t i t i s impossible i n ZF p l u s the axiom o f choice, t o define and prove uniqueness o f a choice f u n c t i o n which associates t o each non-empty s e t o f r e a l s one o f i t s elements. S i m i l a r l y i t i s impossible t o uniquely d e f i n e a choice s e t p i c k i n g one f u n c t i o n from each p a i r of r e a l f u n c t i o n s
h,-h
, where
f o r each r e a l
x
, t h e value o f -h
THEORY OF RELATIONS
18
on x
i s t h e a d d i t i v e inverse o f
h(x).
To o b t a i n a p r o o f o f uniqueness, com-
p l e t i n g the existence (which i s guaranteed by the axiom o f choice), i t i s necessary f o r example t o add t o ZF the axiom o f c o n s t r u c t i b i l i t y o f GODEL 1940.
§
5
- TRANSITIVECLOSURE,
HEREDITARILY F I N I T E SET, FUNDAMENTAL RANK,
CARD I NAL 5.1. TRANSITIVE CLOSURE For every s e t
a
, there
e x i s t t r a n s i t i v e supersets o f
a
, and
among these t h e r e
e x i s t s one which i s included i n a l l t h e others. T h i s s e t i s formed from t h e values o f a l l f i n i t e sequences xl, ... ,xh ( h i n t e g e r ) such t h a t x1 € a and xi+l E xi f o r each i (1 6 i < h) . We s h a l l c a l l t h i s s e t t h e t r a n s i t i v e closure of a For each non-empty s e t a , t h e t r a n s i t i v e closure o f a i s t h e union of a together w i t h t h e t r a n s i t i v e closures o f t h e elements o f a -
.
.
I f a s b then
(closure o f
a )
5 (closure o f
b )
,
HEREDITARILY FINITE SET A h e r e d i t a r i l y f i n i t e s e t i s a s e t whose t r a n s i t i v e closure i s f i n i t e . For i n s t a n ce, every f i n i t e t r a n s i t i v e s e t i s h e r e d i t a r i l y f i n i t e . I n p a r t i c u l a r every i n t e ger ( i . e . every f i n i t e o r d i n a l ) i s h e r e d i t a r i l y f i n i t e . The s i n g l e t o n o f 1 i s n o n - t r a n s i t i v e y e t h e r e d i t a r i l y f i n i t e . Every h e r e d i t a r i l y f i n i t e s e t i s f i n i t e , as i t i s included i n i t s t r a n s i t i v e closure which i s f i n i t e . Every element and every subset o f a h e r e d i t a r i l y f i n i t e s e t i s h e r e d i t a r i l y f i n i t e . Every f i n i t e s e t o f h e r e d i t a r i l y f i n i t e s e t s i s hered i t a r i l y f i n i t e . S i m i l a r l y f o r f i n i t e unions, f i n i t e Cartesian products, and t h e power s e t o f h e r e d i t a r i l y f i n i t e sets. A necessary and s u f f i c i e n t c o n d i t i o n f o r a s e t a t o be h e r e d i t a r i l y f i n i t e i s t h a t , f o r every f i n i t e sequence xO, ...,xh ( h i n t e g e r ) w i t h xo = a and xi+l
E xi
f o r each
i< h
, the
terms
xi
are f i n i t e .
5.2. FUNDAMENTAL RANK Let
a
be a s e t and
c
be t h e t r a n s i t i v e closure o f the s i n g l e t o n
ia 1 .
We say
t h a t t h e o r d i n a l o( i s t h e fundamental rank o f a , i f there e x i s t s a f u n c t i o n f w i t h domain c , t a k i n g o r d i n a l values 4 o( , such t h a t t h e i n i t i a l ordered p a i r (0,O) and t h e f i n a l ordered p a i r (a, cx ) belong t o f : so t h a t f ( 0 ) = 0 and f ( a ) = d ; and such t h a t i f u E c then the value f ( u ) i s t h e s m a l l e s t o r d i n a l s t r i c t l y g r e a t e r than f ( x ) f o r a l l x belonging t o u I t f o l l o w s from the axiom o f foundation t h a t every s e t has a unique fundamental
.
rank. -
.
Suppose t h a t a i s non-empty and Indeed, t h e empty s e t 0 has rank 0 t h a t every element o f a has a rank. Then by t h e preceding d e f i n i t i o n , a has rank equal t o t h e s m a l l e s t o r d i n a l which i s s t r i c t l y g r e a t e r than t h e ranks of
Chapter 1
19
a l l i t s elements. The e x i s t e n c e of rank r e s u l t s from the axiom of foundation i n the form of scheme 2.8. For every ordinal oc t h e fundamental rank i s g . 5.3. For every ordinal o( , t h e r e i s a s e t Vd o f a l l s e t s of ranks s t r i c t l y l e s s t t . 4 . Moreover , V has fundamental rank o( . 0 Obvious f o r 0 s i n c e Vo i s empty. I f t h i s i s t r u e f o r cl( , then i t i s t r u e Finally for q f o r @ + l with Vq +1 = s e t of elements and s u b s e t s of . ,V a l i m i t o r d i n a l , ,V i s t h e union of t h e Vi f o r i s t r i c t l y l e s s than o( . 0 Note t h a t f o r each ordinal & , t h e s e t Vatl - ,V of s e t s of rank o( i s nonempty, s i n c e , V and 4 belong t o t h i s s e t . For i an i n t e g e r , o r f i n i t e o r d i n a l , the s e t of sets of rank i i s f i n i t e . I t follows t h a t every i n f i n i t e s e t has rank a t l e a s t equal t o w . Note t h a t a s e t i s h e r e d i t a r i l y f i n i t e i f f i t s fundamental rank i s f i n i t e . The set of h e r e d i t a r i l y f i n i t e sets i s t h e i n t e r s e c t i o n of a l l sets which contain 0 and which, i f they contain x and y , a l s o contain x u { y ) a s an element.
5.4. CARDINAL, OR CARDINALITY Given a set a , consider sets equipotent w i t h a and among t h e s e , those of minimum fundamental rank. By t h e preceding, these form a non-empty s e t which we c a l l t h e c a r d i n a l o r c a r d i n a l i t y of a , denoted by Card a : d e f i n i t i o n from SCOTT 1955. T h u s every set has a c a r d i n a l , and two s e t s a r e equipotent i f f they have t h e same c a r d i n a l . Note t h a t every s e t a i s equipotent, not t o Card a , but t o an a r b i t r a r y element o f Card a T h i s i s only a minor inconvenience i n the d e f i n i t i o n .
.
Given two c a r d i n a l s a and b , t h e ordering of l e s s than o r equal t o , o r g r e a t e r than o r equal t o , means t h a t every s e t of cardinal a i s subpotent w i t h every s e t of cardinal b . Obvious d e f i n i t i o n of s t r i c t ordering; notations 6 , < .
5.5. CARDINAL SUM, CARDINAL PRODUCT AND EXPONENTIATION Let a and b be c a r d i n a l s ; the cardinal sum a + b i s defined as t h e cardinal e denote of the union of two d i s j o i n t sets of cardinal a , r e s p e c t i v e l y b W t h e cardinal sum by + (boldface) t o avoid confusion w i t h the ordinal sum + i n 3.1. Thus we can i d e n t i f y , i n 5 6 below, Card w w i t h W i t s e l f , and w r i t e a + 1 = W and y e t W + ~ > L S .To be rigorous, we should a l s o d i s t i n g u i s h between t h e ordering r e l a t i o n f o r c a r d i n a l s and f o r o r d i n a l s . In p r a c t i c e the context will always permit t h e d i s t i n c t i o n . Since cardinal m u l t i p l i c a t i o n and exponentiat i o n a r e denoted by a x b and a b ( n o t a t i o n s from 1 . 2 and 1.6), t h e r e w i l l be no confusion w i t h t h e operations of ordinal m u l t i p l i c a t i o n and exponentiation a.b and ba . In p a r t i c u l a r t h e cardinal notation “ i s not necessary: ~3~ w i l l be s u f f i c i e n t .
.
THEORY OF RELATIONS
20
+
The sum a b does n o t depend upon t h e choice o f d i s j o i n t s e t s o f c a r d i n a l and cardinal b Cardinal a d d i t i o n i s commutative and a s s o c i a t i v e . We have a + O = a . F i n a l l y a b a ' and b & b ' imply a + b , ( a'+b' .
a
.
The cardinal product o f a set o f cardinal
a x b i s defined as t h e c a r d i n a l o f t h e Cartesian product a with a s e t o f c a r d i n a l b . There w i l l be no inconvenience
i n using t h e same symbol f o r c a r d i n a l m u l t i p l i c a t i o n and f o r t h e Cartesian product o f two sets (see 1.2). The c a r d i n a l product does n o t depend upon the choice o f t h e sets o f cardinal a , resp. b Cardinal m u l t i p l i c a t i o n i s commutative, associat i v e , and d i s t r i b u t i v e over c a r d i n a l a d d i t i o n : ( a + b ) % c = ( a x c ) + ( b x c )
.
We have a % 0 = 0 and a x 1 = a
. Finally
.
aGa'
and
b,
imply a x b s a ' x b '
The cardinal power ab i s defined as t h e c a r d i n a l o f t h e power between sets o f cardinal a , resp. b ( n o t a t i o n from 1.6). Cardinal exponentiation does n o t depend upon t h e choice o f t h e s e t s o f c a r d i n a l a , resp. b . We have a' = 1 , 'a = a , aO = 0 f o r a # 0 , and al = 1 . Moreover f o r b # 0 , conditions a,< a ' and b,< b ' imply ab C a ' b ' . F i n a l l y the equipotences i n d i c a t e d i n 1.6 become c a r d i n a l e q u a l i t i e s : (b+c)a = ( b a ) $ ( C a ) ; then
§
6 - ALEPH, HARTOGS,
'(ash)
= ('a)x('b)
; and
C(ba) = ( b * c ) a
.
ALEPH RANK
6.1. ALEPH I n the case o f a s e t
a which has an o r d i n a l equipotent t o it, we take as t h e d e f i n i t i o n o f the c a r d i n a l o f a , denoted s t i l l by Card a , t h e smallest o r d i n a l equipotent t o a I n s p i t e o f t h e very d i f f e r e n t n o t i o n o f c a r d i n a l as defined
.
i n 5.4, t h i s new Card a sets
b i j e c t i v e l y corresponds t o t h e o l d notion, a t l e a s t f o r
a which are equipotent t o an o r d i n a l . Such sets a r e c a l l e d well-orderable
-
i n ch.2 5 2.5 below. The c a r d i n a l o f such a s e t i s c a l l e d an aleph. D e f i n i t i v e l y , we have the f o l l o w i n g a r t i f i c i a l b u t general and r i g o r o u s d e f i n i t i o n : i f a i s equipotent t o an o r d i n a l , then t h e smallest such i s Card a ; otherwise Card a i s the s e t o f a l l sets o f minimum fundamental rank which a r e equipotent w i t h a . Another d e f i n i t i o n o f aleph, which i s e q u i v a l e n t t o t h e preceding one: an aleph i s an o r d i n a l o( which i s equipotent t o no o r d i n a l < o( ( l e s s than w i t h respect t o the o r d e r i n g o f t h e o r d i n a l s ) . I n p a r t i c u l a r , t h e f i n i t e alephs are t h e integers, the f i r s t i n f i n i t e aleph i s 0 . We w i l l see i n ch.2 5 2.5 t h a t , with t h e axiom o f choice, every c a r d i n a l i s an aleph ( e q u i v a l e n t l y every s e t i s well-orderable). a r e two equipotent o r d i n a l s , then every intermediate Notice t h a t i f o( and o r d i n a l i s equipotent t o them. Moreover, f o r every i n f i n i t e o r d i n a l a( , t h e
21
Chapter 1 successor
o(
+1
.
i s e q u i p o t e n t w i t h iA
I t f o l l o w s t h a t e v e r y i n f i n i t e aleph i s
a l i m i t ordinal. 6.2.
HARTOGS SET, OR HARTOGS ALEPH
Let
a
be an i n f i n i t e s e t . We say t h a t an o r d i n a l
t h e r e e x i s t s an i n j e c t i o n o f
u
equipotent w i t h
.
If
u
u
u
every i n t e g e r i s i n j e c t a b l e i n
Given a s e t 0
a
, the
a
For each o r d i n a l
a
, or
, then
a
i s injectable i n a
. However
every ordinal
.
a
Since
a
su
, iff a
is
and
i s infinite, a
,
see 2.6.
a
form a set.
u and each i n j e c t i o n f o f u
u , i s d e f i n e d by t h e s e t o f o r d e r e d p a i r s
quently
a
i n o r d e r t h a t c3 be i n j e c t a b l e i n
be D e d e k i n d - i n f i r i t e :
ordinals injectable i n
i s injectable i n
u
e q u i v a l e n t l y i f a subset o f
i s injectable i n
every o r d i n a l e q u i p o t e n t w i t h i t i s necessary t h a t
into
Such a s e t i s a r e l a t i o n , i n t h e sense o f ch.2
5
into
a
(fx,fy)
, this
, and
f
f o r which
conse-
x< y
1 below; and a l l these r e l a t i o n s
form a s e t by t h e axioms o f ZF. 0 a
i s an aleph, which we
. This
i s also the smallest
Hence, t h e s e t o f a l l o r d i n a l s i n j e c t a b l e i n a g i v e n s e t s h a l l c a l l t h e Hartogs s e t , o r t h e Hartogs a l e p h o f o r d i n a l which i s n o t i n j e c t a b l e i n
a
a
(HARTOGS 1915).
6.3. SUCCESSOR ALEPH, LIMIT ALEPH If
a
i s i t s e l f an aleph, t h e n t h e Hartogs a l e p h o f
d i a t e l y greater than
a
,in
n a l ( w i t h r e s p e c t t o subpotence) between t h e l a t t e r by
a+
a
i s t h e unique aleph imme-
t h e sense t h a t t h e r e i s no s t r i c t l y i n t e r m e d i a t e o r d i a
and i t s Hartogs a l e p h . We s h a l l denote
and c a l l i t t h e successor a l e p h o f
a
.
, and i s t h e s e t o f a l l w 1 = LJ+ countable o r d i n a l s ; o r a g a i n t h e l e a s t uncountable o r d i n a l . L e t t i n g W o = , f o r each i n t e g e r i we l e t i+l= ( (3 i)+ . denotes t h e successor a l e p h o f w
For example
The u n i o n o r supremum o r d i n a l o f an a r b i t r a r y s e t o f alephs i s a g a i n an aleph. For example, f r o m t h e p r e c e d i n g W i ( i i n t e g e r ) , we l e t
W U = Sup( w i )
an a l e p h . We c a l l a non-successor aleph, such as &J o r
Urn
,a
, which
is
l i m i t aleph.
6.4. ALEPH RANK
u , f o r an i n f i -
We g e n e r a l i z e t h e p r e c e d i n g n o t a t i o n . Given an a r b i t r a r y o r d i n a l n i t e aleph
a
we w r i t e
( t h e successor o r d i n a l o f the f i n a l ordered p a i r (x,y)
belongs t o
belongs t o form to
(x,y) f
.
f
Wu
a =
w i t h domain
x
Sup x ,C u
u+l
(0, W ) and
belong t o
f ; and such t h a t , i f t h e o r d e r e d p a i r
an o r d i n a l
<
u and y an aleph, t h e n (x+l,y+)
f
contains a s e t o f ordered p a i r s o f the
f ; and f i n a l l y such t h a t i f
with
f
u ) , such t h a t t h e i n i t i a l o r d e r e d p a i r
(u,a) with
i f there exists a function
then the ordered p a i r
(Sup x, Sup y)
belongs
.
THEORY OF RELATIONS
22
every i n f i n i t e aleph We c a l l
u
, there
u
Thus f o r each o r d i n a l a
5
For every o r d i n a l
.
a
We have
wx
=
u
Conversely f o r
such t h a t
.
a = Ou
; e q u a l i t y i s p o s s i b l e : see
W
6.6. u
, the
we have Wu+l = ( G )u)+
Moreover, f o r e v e r y s e t o f o r d i n a l s sup
,<
u
.
G,,
t h e r e e x i s t s a unique o r d i n a l
t h e a l e p h rank o f
f o r i n s t a n c e ch.2
e x i s t s a unique aleph
x
, we
successor a l e p h o f
.
(3
have t h e supremum e q u a l i t y :
w
(SUP x ) . Hence an i n f i n i t e a l e p h i s a successor o r a l i m i t aleph, a c c o r d i n g t o whether i t s a l e p h rank i s a successor o r d i n a l o r l i m i t o r d i n a l ( i n c l u d i n g 0, s i n c e
a,,=
id).
I t f o l l o w s f r o m t h e correspondence between alephs and a l e p h ranks t h a t , g i v e n a
s e t o f alephs 6.5.
ai
and an a l e p h
,
b
if
ai4
b
for all
ai
then
Sup ai
6
b
.
I n t h e presence o f t h e axiom o f choice, t h e continuum h y p o t h e s i s i s equiva-
&2
lent t o the equality
= O1
.
However, w i t h t h e axioms o f ZF a l o n e i n t h e
absence o f t h e axiom o f choice, t h e e q u a l i t y L32 =
i s , a p r i o r i , a stronger
cd1
a s s e r t i o n t h a n t h e continuum h y p o t h e s i s . Indeed, t h e r e may e x i s t a model o f ZF w i t h o u t choice, where t h e r e i s no s t r i c t l y i n t e r m e d i a t e s e t ( w i t h r e s p e c t t o sub-
,yet
potence) between G) and " 2
where
W 1 i s incomparable w i t h & 2
.
The s i t u a t i o n i s d i f f e r e n t w i t h 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 , which i m p l i e s t h e axiom of c h o i c e (see ch.2 e x e r c . 1 ) . Thus 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 implies the equality
a 2 = a+
f o r each i n f i n i t e a l e p h
.
a
However, i t seems p o s s i b l e t o c o n s t r u c t a model o f ZF s a t i s f y i n g every i n f i n i t e aleph
a
, and
a2 = a+
for
y e t h a v i n g non-aleph c a r d i n a l s which a r e incomparable,
t h u s n e g a t i n g t h e axiom o f c h o i c e . Elementary p r o p e r t i e s c o n c e r n i n g a d d i t i o n , m u l t i p l i c a t i o n and e x p o n e n t i a t i o n of w i l l be o b t a i n e d w i t h t h e h e l p o f r e l a t i o n s and isomorphisms, i n
i n f i n i t e alephs ch.2
§
5
3.8 t o 3.10.
7 - FILTER,
ULTRAFILTER
AXIOM
7.1. FILTER Given a s e t a a , such t h a t (1)if
~
(2) i f x,y E
, recall
€
5
and
t h a t a f i l t e r on x s y s a , t h e n
, then
s e c t i o n o f elements o f
F
a
,--
f
i s a set
Y E T ;
the intersection i s an element o f
xny E
F; t h u s
Let
be a f i l t e r on
a
, and
a
i s a f i l t e r on
b5 a
every f i n i t e i n t e r -
F.
F o r h x a m p l e , t h e s e t o f complements o f f i n i t e subsets o f Every i n t e r s e c t i o n o f f i l t e r s o n
o f non-empty subsets o f
such t h a t
a bnx
k)
i s a f i l t e r on c . 3
. i s non-empty f o r e v e r y
23
Chapter 1
element x of LT. Then the s e t of intersections b n x constitutes a f i l t e r on b , cal'led the f i l t e r induced by on b . Let & be a s e t of subsets of a f o r which every f i n i t e intersection of i t s e l e . ments i s non-empty. Then the s e t of supersets of these intersections constitutes a f i l t e r on a , called the f i l t e r generated by h! i s said t o be finer than 3 , or Let 3 , be two f i l t e r s on the same s e t .
3
9
.
t o extend 3 , i f i t includes ; s t r i c t l y f i n e r i f i t s t r i c t l y includes F . Given a f i l t e r on a and b C_ a , e i t h e r b E or a-b E , or every intersection of any x 6 with b and with a-b i s non-empty. Hence there exists a f i l t e r f i n e r t h a n 7 which contains b , or a f i l t e r f i n e r t h a n 3 which contains a-b . Let a be a s e t . If a s e t of f i l t e r s on a i s t o t a l l y ordered by the comparison relation " f i n e r than", or more generally i f t h i s comparison relation i s a directed partial ordering ( i . e . given two f i l t e r s , there i s a t h i r d f i l t e r which i s finer than b o t h ) , then the union of the f i l t e r s i s a f i l t e r on a . 7 . 2 . ULTRAFILTER, ULTRAFILTER AXIOM Given a s e t a , an u l t r a f i l t e r on a i s a f i l t e r f o r which there i s no s t r i c t l y finer f i l t e r on a . For example i f u E a , then the s e t of subsets of a containing u i s an u l t r a f i l t e r , said t o be t r i v i a l . Already f o r the s e t w , the axioms of ZF alone are n o t s u f f i c i e n t t o prove the existence of a non-trivial u l t r a f i l t e r . I t i s necessary t o add, f o r instance, the u l t r a f i l t e r axiom ( a l s o called boolean prime ideal axiom), which asserts t h a t for every s e t a and every f i l t e r on a , there e x i s t s an u l t r a f i l t e r on a which i s f i n e r t h a n 9 . For example t h i s implies the existence of an u l t r a f i l t e r on 0 which contains as elements a l l complements of f i n i t e subsets of w . W e will see in ch.2 5 2.8 t h a t the axiom of choice implies the u l t r a f i l t e r axiom. For a model of ZF having no u l t r a f i l t e r s other t h a n the t r i v i a l u l t r a f i l t e r s , see BLASS 1977. A necessary and s u f f i c i e n t condition t h a t a f i l t e r 7 on a be an u l t r a f i l t e r , or a-x e 3 . i s t h a t f o r every subset x o f a , e i t h e r x E 7 Let be an u l t r a f i l t e r and-x E Then f o r every p a r t i t i o n of x % 1 f i n i t e number o f d i s j o i n t subsets, one and only one of these subsets belongs-
-
F.
t.3.
Every f i l t e r 7 on a i s the intersection of a l l u l t r a f i l t e r s on a which are f i n e r than 7 (uses the u l t r a f i l t e r axiom). To calculate the number of f i l t e r s on a s e t , see ch.2 exerc. 2 . To see the impossibility of countably generating an u l t r a f i l t e r on CS, see ch.2 5 8.1.
THEORY OF RELATIONS
24
§
8 - TOPOLOGY ON
Start with the set sets F
F, G
of
N
SETS OF INTEGERS
N
, let
o f t h e n a t u r a l i n t e g e r s . F o r each o r d e r e d p a i r o f f i n i t e G UF denote t h e s e t o f those subsets o f N which i n c l u d e
and a r e d i s j o i n t f r o m G . F o r F and G empty, we o b t a i n t h e e n t i r e s e t Y ( N ) G UF i s non-empty i f f F and G a r e d i s j o i n t .
Note t h a t
The i n t e r s e c t i o n
is
U F n U:
any u n i o n o f p r e c e d i n g
.
U~~~~
Consequently, d e f i n i n g an o p e n s e t t o be
s e t s , t h e n t h e i n t e r s e c t i o n o f any two open s e t s i s
U
T (N) . U s e t s , t h u s an open s e t ; so t h a t each
s t i l l an open s e t ; so t h a t we o b t a i n a t o p o l o g y on The complement o f a U
s e t i s a union o f
U
s e t i s b o t h open and closed, i . e .
each
U
t h e complement o f an open s e t ; more b r i e f l y
i s a clopen s e t .
T h i s t o p o l o g y i s H a u s d o r f f : g i v e n two subsets d i s t i n c t , take a f i n i t e s e t f i n i t e set G G t h e n UF and
included i n
and
A
F which i s i n c l u d e d i n B
yet not i n
F separate A from
UG
B
, with
A
.
of
B
A
, supposed t o be B , and a
N
y e t not i n
F and
n o t b o t h empty:
G
8.1. CONVERGENT SEQUENCE, CLOSURE Consider an a - s e q u e n c e o f subsets
Hi
of
f o r every i n t e g e r some i n d e x on ; o r complement o f
x
of
H
, e i t h e r x belongs t o
x
( i n a t u r a l i n t e g e r ) . We say t h a t
N
t h i s sequence converges, and t h a t a subset
i s the
N
and t h e n
H
belongs t o t h e complement o f
H
limit
x
o f the
belongs t o
and t h e n
x
,if
Hi
from
Hi
belongs t o t h e
f r o m some i n d e x on.
Hi
The c l o s u r e o f a s e t
S
o f s e t s o f i n t e g e r s , i s d e f i n e d as b e i n g t h e s e t of a l l
l i m i t elements f o r a l l convergent
ra-sequences o f elements o f
i s also the smallest closed superset o f
S
, with
S
.
This closure
respect t o inclusion.
8.2. THE TOPOLOGY I S COMPACT
I n o t h e r words,
if
T(N)
i s covered by a u n i o n o f open s e t s , t h e n i t i s covered
by a f i n i t e number o f t h e s e s e t s . 0,Suppose
(ii n t e g e r ) of Assume t h a t f o r each i , t h e u n i o n
we have an a - s e q u e n c e o f o r d e r e d p a i r s
f i n i t e subsets o f
N
, and l e t
Ui = UG(!) F(1) i s s t r i c t l y included i n
Uo u U1 u ... u U 1. i s d i s t i n c t from o f a l l t h e s e Ui We see t h a t t h e r e e x i s t s a f u n c t i o n
T(N) . h
.
(F(i),G(i))
Q(N) . We
which, t o each
w i l l show t h a t t h e u n i o n i
, a s s o c i a t e s an element
hi o f F ( i ) u G ( i ) , which can be assumed t o be non-empty. G i v i n g hi t h e s i g n (+) o r ( - ) a c c o r d i n g t o whether i t belongs t o F ( i ) o r G ( i ) , we can choose h t o v e r i f y t h e f o l l o w i n g c o n d i t i o n . F o r each difference hl,...,hi
T(N)
-
(Uo u
... u Ui) , which
i
, there
e x i s t s an element i n t h e
c o n t a i n s as an element a l l t h o s e
o f s i g n ( - ) , and none o f s i g n (+). It f o l l o w s t h a t t h e i d e n t i t y
ho, hi
=
hj
Chapter 1
25
i # j ) i m p l i e s t h a t hi and h . have t h e same s i g n . F i n a l l y t h e s e t o f a l l J o f s i g n ( - ) , does n o t belong t o t h e u n i o n o f t h e Ui f o r a l l integers i 0
(with
.
hi
It f o l l o w s immediately f r o m t h i s compactness, t h a t t h e clopen s e t s a r e e x a c t l y t h e
f i n i t e unions o f t h e p r e c e d i n g U
sets.
8.3. Consider an a - s e q u e n c e o f o r d e r e d p a i r s ( i i n t e g e r ) , and assume t h a t
i
each
. Let
h
F(i)
and
G(i)
(F(i),G(i))
o f f i n i t e subsets o f
N
a r e b o t h non-empty and d i s j o i n t , f o r
be a f u n c t i o n which, t o each
i
, associates
an element
hi
in
F ( i ) u G ( i ) ; w i t h t h e c o n d i t i o n t h a t i f hi = h . (i# j ) , t h e n e i t h e r hi belongs J Then we a s s o c i a t e t h e s i g n both t o F ( i ) and F ( j ) , o r b o t h t o G ( i ) and G ( j )
.
.
( - ) t o hi , a c c o r d i n g t o whether i t belongs t o F ( i ) o r t o G ( i ) L e t Vh be t h e open s e t formed by t h e subsets X o f N f o r which t h e r e e x i s t s an i such t h a t hi has s i g n (+) and hi E X , o r hi has s i g n ( - ) and hi X . (t) o r
Then t h e open s e t , u n i o n o f t h e functions
h
previously defined.
UG(i) F(i)
, i s the intersection o f the
Vh
f o r a l l the
8.4. DENSE SET, BAIRE'S CONDITION
A set i s called
dense, i f
i t s i n t e r s e c t i o n w i t h e v e r y non-empty open s e t i s non-
empty.
i an element hi o f N , Vh p r e v i o u s l y d e f i n e d i s dense. Thus t h e r e e x i s t i n f i n i t e l y many dense open s e t s , a l t h o u g h t h e o n l y dense c l o s e d s e t i s 9 (N) For e v e r y f u n c t i o n
h
a s s o c i a t i n g t o each i n t e g e r
t a k i n g i n f i n i t e l y many v a l u e s
hi
w i t h any s i g n , t h e open s e t
.
N o t i c e t h a t t h e i n t e r s e c t i o n o f two dense open s e t s i s a dense open s e t . Every compact t o p o l o g y s a t i s f i e s BAIRE's c o n d i t i o n : e v e r y c o u n t a b l e i n t e r s e c t i o n o f dense open s e t s i s non-empty,
and even dense. We l e t t h e p r o o f t o t h e reader,
f o r i n s t a n c e by t a k i n g a f i n i t e s e t o f dense ooen s e t s ; and t h e n i n f i n i t e l y many. L e t us s k e t c h a d i r e c t p r o o f f o r t h e p r e s e n t t o p o l o g y . Every open s e t i s a u n i o n G UF , t h u s by 8.3, a c o u n t a b l e i n t e r s e c t i o n o f open s e t s Vh L e t us prove BAIRE's c o n d i t i o n f o r t h e dense Vh , hence those c o r r e s p o n d i n g t o
0
.
o f sets
Vh , by r e d u c i n g h t o o n l y t h o s e values w i t h s i g n (t), o r o n l y those o f s i g n ( - ) , assumed t o be
the
h
t a k i n g i n f i n i t e l y many values. To do t h i s , r e p l a c e each
i n f i n i t e i n number. Denote by W t h e i n f i n i t e s e t o f these values. The open s e t i s reduced t o t h e s e t o f subsets t h e s e t o f subsets
5
X
such t h a t
X
of
N
W n (N-X)
such t h a t
W nX
i s non-empty. We s h a l l see i n ch.2
8.1 ( c o u n t a b l e case), t h a t t h e r e e x i s t s a s u i t a b l e s e t
W nX
and W n (N-X)
Vh i s non-empty, o r t o
i n f i n i t e . Thus t h e r e e x i s t s a s e t
W
f o r a l l the
X
i n c l u d i n g any g i v e n
f i n i t e s e t and e x c l u d i n g any g i v e n f i n i t e s e t ( d i s j o i n t each f r o m t h e o t h e r ) ; hence t h e r e e x i s t s an
X
,with
X
b e l o n g i n g t o any g i v e n open s e t . 0
THEORY OF RELATIONS
26
I n conclusion, an open s e t defined by an a - s e q u e n c e
i with
§
F(i)
d i s j o i n t from
9 - NATURAL
F and
G(i)
) ) i s dense i f f , t h e r e e x i s t s an
(F(i),G(
F, G
f o r every ordered p a i r o f f i n i t e d i s j o i n t subsets
of
d i s j o i n t from
N
.
G
SUM AND PRODUCT FOR ORDINALS
From the unique decomposition o f an o r d i n a l i n t o a sum o f decreasing powers o f 0 , one defines the commutative operations o f n a t u r a l sum and n a t u r a l product f o r o r d i n a l s : t h i s goes back t o HESSENBERG 1906 ; see a l s o BACHMANN 1967 p. 107. For the sum, we begin w i t h O( coefficients
ml,
...,mh
o r d i n a l exponents, and
=
w0( (l).ml +
...
are i n t e g e r s , and where
p=
+
Wo((l).nl
... +
, where the
t W w(h).mh
cx ( 1 ) 7 Ga(h).nh
...
7
o( ( h )
. We
are
can always assume
t h a t t h e o r d i n a l exponents are t h e same f o r both decompositions, i f necessary by i n s e r t i n g terms w i t h c o e f f i c i e n t zero. Then t h e n a t u r a l sum o( @
+
c3 M(l).(ml+nl)
...
fs
i s defined as:
+ ~ c ) ~ ( ~ ) . ( m ~. + n ~ )
For the n a t u r a l product, one f i r s t defines the product o f being
*@
. Then f o r
a sum o f powers o f
G)
' u
and U ' a s
two a r b i t r a r y o r d i n a l s , each w r i t t e n i n the form o f
, one m u l t i p l i e s them as w i t h polynomials.
With respect t o these notions, one defines the " r e l a t i v e " ( i . e . p o s i t i v e o r negat i v e ) o r d i n a l s , by s u b s t i t u t i n g i n t h e c o e f f i c i e n t s
m
, positive
and negative
i n t e g e r s . Then t h e r a t i o n a l o r d i n a l s by t a k i n g the q u o t i e n t f i e l d : see S I K O R S K I 1948, and a l s o BACHMANN 1967. More d i f f i c u l t and l e s s " n a t u r a l " , one can d e f i n e t r a n s f i n i t e r e a l s ; see f o r instance KLAUA 1959 and 1960.
EXERCISE 1
-
CONSISTENCY OF THE AXIOM OF FOUNDATION
To see t h a t the a d d i t i o n o f the axiom o f foundation does n o t imply a c o n t r a d i c t i o n , i f the o t h e r axioms o f ZF are c o n s i s t e n t , we d e f i n e an o r d i n a l t o be a s e t n o t
only t r a n s i t i v e and t o t a l l y ordered by the membership r e l a t i o n , b u t a l s o s a t i s f y i n g the f o l l o w i n g foundation c o n d i t i o n . For every non-empty subset o r d i n a l , there e x i s t s an element
y
o f , x which i s d i s j o i n t from
x
o f an
x : see
BERNAYS 1968 p. 80.
Next, one must r e d e f i n e an i n t e g e r as a f i n i t e o r d i n a l . Then say t h a t a s e t well-founded i f every sequence belonging t o
xi
f o r each
xi
indexed by i n t e g e r s , w i t h
xo = a
and
a
is
xi+l
i , i s f i n i t e . F i n a l l y , one v e r i f i e s t h a t the w e l l -
founded sets w i t h t h e membership r e l a t i o n , s a t i s f i e s a l l t h e axioms o f ZF, i n c l u d i n g the axiom o f foundation. N.B.
I t i s n o t s u f f i c i e n t t o d e f i n e a well-founded s e t ( o t h e r than an o r d i n a l ) by
21
Chapter 1
the f i r s t foundation c o n d i t i o n
-
EXERCISE 2
ALEXANDROFF-FODOR THEOREM
m 1, the
A subset o f
s e t o f c o u n t a b l e o r d i n a l s , i s s a i d t o be c o f i n a l i n
ifi t i s e q u i p o t e n t w i t h element i n t h e subset. We say t h a t a f u n c t i o n f every x i n Dom f .
w 1 , so
t h a t e v e r y element o f
w i t h domain
W1
- {O}
i s regressive, i f
1 - If f
i s r e g r e s s i v e , t h e n t h e r e e x i s t s a c o u n t a b l e o r d i n a l o(
set o f
such t h a t
x
,
W
U 1 admits a g r e a t e r fx
<x
for
f o r which t h e
f x = d i s c o f i n a l (ALEXANDROFF 1935, c o u n t a b l e axiom o f o( (0) , i f f o r e v e r y /s < o< ( 0 ) f i , t h e n t h e r e e x i s t s an m ( l ) 7 a( (0)
choice used). S t a r t i n g w i t h a c o u n t a b l e o r d i n a l there are c o u n t a b l y many
x
where
such t h a t from t h a t p o i n t on, sequence
fx =
>/
fx
( i ) ( i i n t e g e r ) , and so
O(
b l e axiom o f c h o i c e ) , and s a t i s f i e s veness o f
w
2
.
f
O( ( 0 )
.
I t e r a t i n g t h i s , we o b t a i n an GO-
8 = Sup O( ( i ) , thus f 8 >/
i s c o u n t a b l e by 2.5 ( t o u n t a contradicting the regressi-
We say t h a t a s e t o f o r d i n a l s i s c l o s e d i f , f o r e v e r y s t r i c t l y i n c r e a s i n g -sequence o f elements, t h e supremum o r d i n a l belongs t o t h e s e t . C l e a r l y , t h e
i n t e r s e c t i o n o f two c l o s e d c o f i n a l s e t s i s c l o s e d c o f i n a l . A l s o t h e i n t e r s e c t i o n o f a countable s e t o f c l o s e d c o f i n a l s e t s i s c l o s e d c o f i n a l : t h i s reduces t o an 0 -sequence which i s d e c r e a s i n g w i t h r e s p e c t t o i n c l u s i o n .
We say t h a t a s e t i s s t a t i o n a r y , i f i t i n t e r s e c t s e v e r y c l o s e d c o f i n a l s e t . Note t h a t every s t a t i o n a r y s e t i s c o f i n a l . If f
i s r e g r e s s i v e , t h e n t h e r e e x i s t s a c o u n t a b l e o r d i n a l o(
of
such t h a t
x
fx =
o(
,is
Suppose t h e c o n t r a r y . Since f o r e v e r y c o u n t a b l e o r d i n a l /?J
/?I )
(fx =
f o r which t h e s e t
s t a t i o n a r y (FODOR 1966, uses axiom o f c h o i c e ) .
, the set o f x C
i s n o t stationary, there exists a closed cofinal set
,-,
(axiom
/5. S t a r t i n g w i t h an a r b i t r a r y c o u n t a b l e o r d i n a l w.( 0 ) , ( /5 < o ( ( 0 ) ) . I t e r a take d(1)> 4 ( 0 ) where 4 (1) belongs t o n C
of choice) on which
fx #
,-,
(ii n t e g e r ) t i n g t h i s , we o b t a i n an w - s e q u e n c e o( (i) Hence
belongs t o A C
/s < 3
-
‘E( , and so f y
( /?J
21 =
thus
.
) , and so f 8 i s d i s t i n c t f r o m e v e r y , which c o n t r a d i c t s t h e regressiveness o f f . Dom f
i s stationary.
EXERCISE 3
A CLASSICAL INTERPRETATION OF THE ORDINAL EXPONENTIATION
Let o(
be two o r d i n a l s . Consider f u n c t i o n s
. We
Sup o( ( i )
<6
>,
Extend t h i s p r o p o s i t i o n t o t h e case where
, fi
, and
f
w i t h domain
/3 , t a k i n g
f i s almost z e r o i f f ( i ) # 0 f o r a t most i o f /3 . Given two almost z e r o f u n c t i o n s f and g , i f t h e r e e x i s t s an i i n (3 w i t h f ( i ) < g ( i ) i n t h e usual o r d e r i n g
values i n o(
say t h a t such an
f i n i t e l y many elements let
f< g
f o r o r d i n a l s , and
f ( j )= g ( j )
for a l l
j such t h a t
i< j < b
.
Let
f< g
if
28
THEORY OF RELATIONS
f < g o r f = g . I n o t h e r words, t h e s e t o f such almost z e r o f u n c t i o n s by l a s t d i f f e r e n c e . 1 - Show t h a t
6 i s a w e l l - o r d e r i n g on t h e s e t o f almost z e r o f u n c t i o n s t h e
r e a d e r i s assumed t o know t h i s n o t i o n ; o t h e r w i s e see ch.2 2
-
s ordered
5
2.4).
Show t h a t t h i s w e l l - o r d e r i n g i s i s o m o r p h i c w i t h t h e e x p o n e n t i a l
(induc-
o("
t i o n on (3 ) . EXERCISE 4
-
A FAMILY OF SUBSETS I N A DENUMERABLE SET, OR I N THE CONTINUUM
1 - Given a denumerable s e t
E
, show t h a t t h e r e e x i s t continuum many denumerable
A o f E , h a v i n g p a i r w i s e a f i n i t e i n t e r s e c t i o n . Indeed t a k e f o r E t h e s e t o f a l l f i n i t e sequences w i t h p o s s i b l e values 0 and 1 . Then d e f i n e each A by an w - s e q u e n c e o f 0 and 1 ; say t h a t a g i v e n f i n i t e sequence belongs t o A i f f i t i s an i n i t i a l i n t e r v a l o f t h e W-sequence a s s o c i a t e d w i t h A ( t h e n o t i o n o f i n i t i a l i n t e r v a l i s obvious; i f necessary see ch.4 § 2.1). subsets
2
-
Modulo t h e axiom o f choice, complete t h e g i v e n f a m i l y o f s e t s
t h e r e e x i s t s an 3
-
A
Now c o n s i d e r a s e t
h a v i n g c a r d i n a l i t y c.dl E
of
,
E
A n X be i n f i n i t e . E w i t h continuum c a r d i n a l i t y . Show t h a t , modulo t h e A
of
E
, each
, and h a v i n g p a i r w i s e a c o u n t a b l e i n t e r s e c t i o n . Indeed
t h e s e t o f a l l c o u n t a b l e o r d i n a l - i n d e x e d sequences o f 0 and 1
Then d e f i n e each
A
A
by an
W1-sequence
.
o f 0 and 1 ; say t h a t a g i v e n c o u n t a b l e
A
i f f i t i s an i n i t i a l i n t e r v a l o f t h e ul-sequence asso( n o t e t h a t t h e axiom o f c h o i c e i s needed t o see t h a t t h e s e t E
sequence belongs t o ciated with
X
such t h a t t h e i n t e r s e c t i o n
axiom o f choice, t h e r e e x i s t ( 2 t o t h e power al) many subsets take f o r
i n order
A
t o s a t i s f y t h e f o l l o w i n g c o n d i t i o n : f o r e v e r y denumerable subset
o f a l l c o u n t a b l e sequences i s e q u i p o t e n t w i t h t h e continuum). 4
-
Complete t h e g i v e n f a m i l y o f s e t s
E
h a v i n g c a r d i n a l i t y O1
A
X
have c a r d i n a l i t y
, there
cS1
.
A
i n o r d e r t h a t , f o r e v e r y subset
e x i s t s an
A
such t h a t t h e i n t e r s e c t i o n
X
of
29
CHAPTER
2
RELATI ON, PARTIAL ORDER1 NG, CHAI N , ISOMORPH I Sp1, COFI WALITY
§
1 - RELATION,
MULTIRELATION,
EXTENSION,
RESTRICTION,
COHERENCE
LEMMA, AXIOM OF DEPENDENT CHOICE
E be a s e t and n an i n t e g e r . I n c h . 1 5 2.3 we d e f i n e d t h e n o t i o n o f
Let
n - t u p l e w i t h values i n
+
denoted
-
and
relation. w i t h e value
R(xl
,...,xn)
. We
E
s e t a s i d e two elements c a l l e d values, which a r e
( f o r i n s t a n c e , t h e s e can be d e f i n e d by 0 and 1). An n-ary E =
, o r based on E , i s a f u n c t i o n R + o r - t o each n - t u p l e x1 ,..., xn
nience, we o f t e n denote t h e n - t u p l e b y i t s i n d i c e s
1
to
which a s s o c i a t e s t h e in n
E
( f o r conve-
instead o f from 0
.
t o n - 1 ) . The s e t E , t h e base o f R , w i l l be denoted I R I The i n t e g e r n w i l l be c a l l e d t h e * o f R . F o r n = 1,2,3, we w i l l say a unary, b i n a r y , ternary r e l a t i o n . n = 0
For on
E
, we
, which - .
adopt t h e c o n v e n t i o n t h a t t h e r e e x i s t two 0 - 9 r e l a t i o n s based
we denote by
(E,t)
and
(E,-)
: the 0-ary r e l a t i o n s w i t h value
t
and value
We adopt t h e c o n v e n t i o n t h a t , f o r each p o s i t i v e
n
, there
e x i s t s a unique n-ary
r e l a t i o n w i t h empty base. However, t h e r e e x i s t two 0 - a r y r e l a t i o n s w i t h empty base:
(O,+)
and
. These
(0,-)
conventions agree w i t h t h e c a l c u l a t i o n o f t h e
number o f n - t u p l e s w i t h v a l u e s t a k e n f r o m a base o f f i n i t ' e c a r d i n a l p
The number o f n-ary r e l a t i o n s based on
p ; i.e.
elements i s " 2 t o t h e power pn
ordinal exponentiation coincides w i t h cardinal exponentiation
np
, for
'I.
p
Here
n and p
finite. Examples o f r e l a t i o n s . The u s u a l o r d e r i n g o f t h e i n t e g e r s i s t h e r e l a t i o n which s a t i s f i e s
R(x1,x2)
=
+
when
a ternary r e l a t i o n t a k i n g t h e value x1.x2 # x3 , where we s h a l l o f t e n use
x1 t
when
.
x2
and
-
when
x1.x2 = x3
x1
> x2 . A
and t h e v a l u e
i s t h e c o m p o s i t i o n l a w o f t h e group. I n s t e a d o f x,y,z .
R
group i s
- when x1,x2,x3
A multirelation with base E i s a f i n i t e sequence R o f r e l a t i o n s R1, ...,Rh ( h i n t e g e r ) , each w i t h base E Each Ri (i= 1, ...,h ) i s c a l l e d a component o f the multirelation R We c a l l t h e arity o f R t h e sequence (nl, ...,nh) o f a r i t i e s o f t h e components R1, Rh We say t h e n t h a t t h e m u l t i r e l a t i o n R i s (nl, ..., n h ) - x . The l e n g t h h o f t h e sequence o f i n d i c e s can be zero: i n t h i s case, t h e m u l t i r e l a t i o n i s reduced t o i t s base E . I n s t e a d o f t h e n o t a t i o n R1, R2,R3 , o f t e n we s h a l l use R,S,T I n t h e case where h = 2 , we w i l l say
.
.
.
...,
.
n
.
THEORY OF RELATIONS
30
a b i r e l a t i o n ; f o r h = 3 a t r i r e l a t i o n , e t c . F i n a l l y , t h e base o f a m u l t i r e l a t i o n R s h a l l be denoted I R I
.
Example. An ordered group i s a (3,2)-ary
b i r e l a t i o n which i s formed o f t h e t e r n a r y
group r e l a t i o n and t h e b i n a r y o r d e r i n g r e l a t i o n . denumeraaccording t o whether i t s base i s f i n i t e , i n f i n i t e , countable, denumerable or continuum-equipotent. The c a r d i n a l o f t h e mu t i re1 a t i o n R i s the c a r d i n a l o f i t s base J R I
A r e l a t i o n o r m u l t i r e l a t i o n w i l l be c a l l e d f i n i t e , i n f i n i t e , countable o r continuum-equipotent,
.
Given two m u l t i r e l a t i o n s
E
w i t h common base
R, S
, we
c a l l t h e concatenation
o f R and S , denoted (R,S) , the sequence o f components o f R f o l l o w e d by the components o f S , i n which case f o r the l a t t e r t h e i n d i c e s a r e increased by the number of terms i n R . 1.1. n-ARY RESTRICTION, n-ARY EXTENSION R be an n-ary r e l a t i o n w i t h base
Let
E
, and
let
F be a subset o f
E
. We
c a l l t h e n - a 3 r e s t r i c t i o n o f R t o F , denoted by R/F , t h e n-ary r e l a t i o n t a k i n g t h e same value f o r each n - t u p l e w i t h values i n F . The n o t i o n o f r e s t r i c t i o n o f a f u n c t i o n i n ch.1 5 1.3, i s more general than t h a t o f n-ary r e s t r i c t i o n : t h e former would a l l o w one t o r e s t r i c t R t o an a r b i t r a r y subset o f t h e s e t 'E o f n-tuples w i t h values i n E , and n o t n e c e s s a r i l y t o a subset o f t h e form 'F w i t h F S E However i n p r a c t i c e , t h e context w i l l make t h e meaning o f t h e ad-
.
j e c t i v e "n-ary" obvious: we t a c i t l y assume t h i s . For t h e a r i t y 0, t h e r e s t r i c t i o n t o F o f t h e 0-ary r e l a t i o n
(F,+) ; s i m i l a r l y w i t h
-
Given a r e l a t i o n R w i t h base E
sion o f Let X
R
R, R '
3
to
E+
be two
and a superset
any r e l a t i o n w i t h base E+
6
n
, we
have
Given a m u l t i r e l a t i o n R = (R1,-..,Rh) we d e f i n e t h e r e s t r i c t i o n o f R t o
.
F
R/X = R ' / X
R, R '
.
of
E
-
of .
, we
E
. If
, then
9-
c a l l an
E
is
R
.
f o r every subset
R = R'
.
w i t h base E and a subset F o f E , by R/F , t o be t h e m u l t i r e l a R w i t h base E and a superset any m u l t i r e l a t i o n w i t h base E+ + where any sequence (R;,. ,Rh)
1, ... ,h)
be two m u l t i r e l a t i o n s o f common a r i t y
I f f o r each subset X base E have R/X = R ' / X , then R = R '
-
E+
w i l l be
(E,+)
, denoted
Given a m u l t i r e l a t i o n t i o n (R1/F, ...,Rh/F) E+ o f E , we c a l l an extension o f R t o Ef whose r e s t r i c t i o n t o E i s R . E q u i v a l e n t l y , + each Ri i s an extension o f Ri t o E+ ( i = Let
.
whose r e s t r i c t i o n t o
n-ary r e l a t i o n s w i t h common base
E with cardinal
F
; t h i s remains v a l i d f o r empty
(nl,
. ...,nh)
E w i t h c a r d i n a l 6 Max(nl,
.
and w i t h common
...,nh) ,
Chapter 2
31
1.2. COMPATIBLE RELATIONS Two r e l a t i o n s ( o r m u l t i r e l a t i o n s ) w i t h t h e same a r i t y a r e s a i d t o be compatible i f f t h e y have t h e same r e s t r i c t i o n t o t h e i n t e r s e c t i o n o f t h e i r bases.
Let
6%
be a s e t o f m u t u a l l y c o m p a t i b l e r e l a t i o n s ( o r m u l t i r e l a t i o n s ) :
&, , based
( 1 ) t h e r e e x i s t s a common e x t e n s i o n o f t h e r e l a t i o n s i n
on t h e u n i o n
o f t h e i r bases; ( 2 ) l e t us denote by
t h e u n i o n o f t h e bases and by
E
n
t h e common a r i t y , o r
t h e maximum o f t h e common a r i t y ( f o r m u l t i r e l a t i o n s ) ; i f each of
E
n-element subset
i s covered by one o f t h e bases, t h e n t h e common e x t e n s i o n i s unique.
1.3. COHERENCE LEMMA Consider a s e t 9 o f s e t s F f o r each o f which we have a f i n i t e non-empty s e t UF o f m u l t i r e l a t i o n s based on F ( a l l o f t h e same a r i t y ) w i t h t h e f o l l o w i n g hypotheses: (1) 3 i s a d i r e c t e d system: i f F, F ' belong t o )3 , t h e n t h e r e e x i s t s an F" i n 3 with F"? F u F' ; ( 2 ) i f F, F ' belong t o 3 and F ' C F , t h e n e v e r y m u l t i r e l a t i o n b e l o n g i n g t o
UF , when r e s t r i c t e d t o F ' , y i e l d s an element o f U F , ; i n t h i s case, t h e r e e x i s t s a m u l t i r e l a t i o n R based on t h e u n i o n o f t h e s e t s F i n 3 , such t h a t f o r each F t h e r e s t r i c t i o n R/F belongs t o UF (uses t h e u l t r a f i l t e r axiom; ZF s u f f i c e s ift h e F a r e f i n i t e and t h e i r u n i o n c o u n t a b l e ) . 0 Denote by E t h e u n i o n o f t h e F i n 3 To each F a s s o c i a t e t h e s e t VF o f e x t e n s i o n s t o E o f m u l t i r e l a t i o n s b e l o n g i n g t o UF . The supersets o f t h e VF c o n s t i t u t e a f i l t e r on t h e s e t o f m u l t i r e l a t i o n s based on E w i t h t h e g i v e n a r i t y . Indeed i f F, F ' belong t o 3 , t h e n t h e r e e x i s t s i n 3 an F " ? FuF' ; hence VFn V F 8 i s a s u p e r s e t o f VF,, . Take an u l t r a f i l t e r e x t e n d i n g t h i s f i l t e r .
.
For each F o f '3 , p a r t i t i o n t h e m u l t i r e l a t i o n s i n VF
i n t o a f i n i t e number
F o f these m u l t i r e l a t i o n s .
o f classes, each c l a s s d e f i n e d by t h e r e s t r i c t i o n t o
One and o n l y one o f t h e s e c l a s s e s i s an element o f o u r u l t r a f i l t e r : denote by
RF
t h e c o r r e s p o n d i n g r e s t r i c t i o n , so t h a t
RF
belongs t o
UF
. Hence
the
RF
a r e m u t u a l l y c o m p a t i b l e i n t h e sense o f 1.2 above: t h e e x i s t e n c e o f t h e m u l t i relation
R
stated i n our proposition follows. If
a r e f i n i t e subsets o f
E
, then
i s c o u n t a b l e and t h e
E
F
t h e u l t r a f i l t e r becomes s u p e r f l u o u s , so t h a t t h e
axioms o f ZF a r e s u f f i c i e n t . 0
1.4. The coherence lemma i m p l i e s , and hence i s e q u i v a l e n t t o t h e u l t r a f i l t e r axiom. 0
Let
e
be a s e t ,
p(e)
be t h e s e t o f subsets o f
e
, and 'bQ
a f i l t e r on
e
F be a f i n i t e s e t o f subsets o f e which i s c l o s e d w i t h r e s p e c t t o union, i n t e r s e c t i o n and t a k i n g complements ( i n e ) . To each F a s s o c i a t e t h e s e t UF o f unary r e l a t i o n s X w i t h base F which s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : Let
.
THEORY OF RELATIONS
32
, i f a E 8 t h e n t h e v a l u e X(a) = + ; i f e-a F , we have o p p o s i t e values X(e-a) # X(a) ;
f o r each
a& F
f o r each
a E
i f a, b E F so a n b e F and X(a) = X(b) = + , t h e n i f a, b E F and a c b and X(a) = + , t h e n X(b) = The s e t
. The
F
i s non-empty f o r each
UF
set o f the
2 then
E
+
X(anb) =
.
+
F
X(a) = -;
;
forms a d i r e c t e d system,
so we can a p p l y t h e coherence lemma. Consequently t h e r e e x i s t s a unary r e l a t i o n
based on T ( e )
whose r e s t r i c t i o n t o each
F
belongs t o
. The
UF
subsets o f
which g i v e t h e value (+) t o t h i s unary r e l a t i o n c o n s t i t u t e an u l t r a f i l t e r on which i s f i n e r t h a n
w.0
e e
1.5. A v a r i a n t o f t h e coherence lemma i s g i v e n by RADO 1949. Consider a s e t o f
f i n i t e mutually d i s j o i n t sets consider a choice f u n c t i o n fI(a)
of
the J
a
, and
a
of
I
. Then
, and
a
fI
there e x i s t s a choice f u n c t i o n
I
f o r each f i n i t e s e t
with
f/I
f o r each f i n i t e s e t
which a s s o c i a t e s t o each o f the
equal t o t h e r e s t r i c t i o n
of
a
,
I an element
whose domain i s t h e s e t o f
f
, there
a
I o f sets
a
e x i s t s a f i n i t e superset
.
f,/I
The preceding RADO's lemma p l u s t h e axiom o f c h o i c e f o r f i n i t e s e t s i s e q u i v a l e n t t o t h e coherence lemma (BENEJAM 1970). 1.6. AXIOM OF DEPENDENT CHOICE Let
E
, such
satisfying
R(x,y)
be a b i n a r y r e l a t i o n w i t h base
R
e x i s t s a t l e a s t one
y
of
E
c h o i c e a s s e r t s t h a t , g i v e n such an ai
of
satisfying
E
R(ai,ai+l)
R =
+
, there
t h a t f o r each =
+ . The
x
of
E
there
axiom o f dependent
e x i s t s an w - s e q u e n c e o f elements
f o r each i n t e g e r
i (MOSTOWSKI 1948).
The axiom o f dependent c h o i c e o b v i o u s l y f o l l o w s f r o m t h e axiom o f c h o i c e . I t i s proved t h a t t h e dependent c h o i c e i s s t r i c t l y weaker t h a n t h e axiom o f
c h o i c e : see f o r i n s t a n c e JECH 1973 p. 122 and f o l l o w i n g . The c o u n t a b l e axiom o f choice, s t a t e d i n ch.1
5
2.5,
f o l l o w s from t h e axiom o f
dependent choice. 0 S t a r t f r o m an
take
R
R(x,y) =
w -sequence o f non-empty m u t u a l l y d i s j o i n t s e t s ai
t o be t h e b i n a r y r e l a t i o n based on t h e u n i o n o f t h e
+
i f f t h e r e e x i s t s an
i with
x
E
ai
ai
and y e ai+l
(ii n t e g e r ) by
, defined
.0
The c o u n t a b l e axiom o f c h o i c e i s s t r i c t l y weaker t h a n t h e axiom o f dependent c h o i c e : see JECH 1973 p. 119 and f o l l o w i n g . F i n a l l y , from t h e axiom o f dependent choice, assumed t o be c o n s i s t e n t , one cannot deduce t h e axiom o f c h o i c e f o r f i n i t e s e t s , s t a t e d i n ch.1
5
2.10.
The p r o o f i s
due t o MOSTOWSKI 1948 w i t h o u t t h e axiom o f f o u n d a t i o n , and t o FEFERMAN 1965 w i t h foundation.
,
Chapter 2
33
1.7. NEGATION, CONJUNCTION, DISJUNCTION Given a r e l a t i o n
, i t s negation
R
i s t h e r e l a t i o n w i t h same base and a r i t y ,
R
1
always t a k i n g t h e o p p o s i t e v a l u e . Given conjunction
R
disjunction
R v S
A
takes t h e v a l u e (+) i f f
S
w i t h t h e same base and a r i t y , t h e
R, S
and
R
t a k e t h e value ( + ) . The
S
t a k e s t h e v a l u e (+) i f f e i t h e r
or
S
, denoted
by
R
t a k e s t h e v a l u e (+).
OF A BINARY RELATION, RETRO-ORDINAL
CONVERSE
Given a b i n a r y r e l a t i o n
R
, the
converse o f
R
,i s
R-
the r e l a t i o n
f o r e v e r y x, y . I n p a r t i c u l a r w i t h t h e same base, such t h a t R-(x,y) = R(y,x) we c o n s i d e r an o r d i n a l o( as t h e b i n a r y r e l a t i o n based on t h e s e t a l r e a d y denot e d by
o(
(the set o f ordinals
QUASI-ORDERING,
P A R T I A L ORDERING,
CHAIN,
WELL-ORDERABLE SET,
WELL-ORDERING,
ORDERING,
) , and t a k i n g t h e value (+) i f f
o(
x E y
or
w i l l be c a l l e d
x ay ) . The converse r e l a t i o n o(
x = y (denoted a l r e a d y b y a retro-ordinal.
2 -
<
WELL-FOUNDED PARTIAL
HAUSDORFF-ZORNAXIOM
A q u a s i - o r d e r i n g ( o r p r e - o r d e r i n g ) i s a b i n a r y r e f l e x i v e and t r a n s i t i v e r e l a t i o n ( t h e s e n o t i o n s assumed t o be known). I f A x 4 y (mod A)
elements i n t h e base, t h e n equal t o y
, means
that
A(x,y)
o r i s g r e a t e r t h a n o r equal t o than
y
,if
o r again y
x 6 y x
. We
+
. We
x
x
, denoted b y x
y
two
y
>, x (mod A) o r
x < y (mod A) A(x,y)
=
or
x
y
follows
s t r i c t l y less
+ and A(y,x)
=
y
if
x
6y
l y (mod A)
or y
6x
. x
t h e base, t h e e q u i v a l e n c e c l a s s of
such
that
A(x,y)
= A(y,x)
=
+
;
; otherwise
An e q u i v a l e n c e r e l a t i o n i s a symmetric q u a s i - o r d e r i n g . Given a element x
-
; a l s o " s m a l l e r " i s synonymous w i t h " l e s s
i s comparable w i t h
i s incomparable w i t h
also write
write
x, y
precedes or i s l e s s t h a n o r
x
; i n o t h e r words
s t r i c t l y greater than
than". We say t h a t x
and y$
=
x
i s a q u a s i - o r d e r i n g and or
(mod A)
i s the s e t o f those y
of
.
A p a r t i a l o r d e r i n g i s an a n t i s y m m e t r i c q u a s i - o r d e r i n g ( n o t i o n assumed t o be
known). We a l r e a d y have t h e example o f i n c l u s i o n .
A
Given a q u a s i - o r d e r i n g
, the
e q u i v a l e n c e r e l a t i o n generated by
A , i s the
r e l a t i o n w i t h t h e same base, t a k i n g t h e v a l u e (+) i f f x < y and y 4 x (mod A ) . F o r each element x , t h e e q u i v a l e n c e c l a i s o f x (mod A) i s t h e c l a s s o f x modulo t h e e q u i v a l e n c e r e l a t i o n generated by
A
.
Take as a new base t h e s e t o f e q u i v a l e n c e c l a s s e s , and w r i t e
.
( e q u i v . c l a s s o f x)
We t h u s o b t a i n a p a r t i a l o r d e r i n g ( e q u i v . c l a s s o f y ) , i f x \ < y (mod A) c a l l e d t h e p a r t i a l o r d e r i n g generated by t h e q u a s i - o r d e r i n g A
4
Let
A
be a p a r t i a l o r d e r i n g ,
n o t i o n of maximum o f minimum, denoted
D (mod A)
Min D
. Recall
D
a subset o f t h e base
IA I
. . We
assume t h a t t h e
, denoted Max D , i s known. S i m i l a r l y f o r t h e t h a t an element i s maximal i n
D (mod A ) ,
i f it
34
THEORY OF RELATIONS
belongs t o D and there i s no element of D which s t r i c t l y follows i t . Analogous notion of a minimal element. The maximum, i f i t e x i s t s , i s maximal, b u t the converse i s f a l se . Similarly f o r the minimum. These notions extend in an obvious manner t o a quasi-ordering. Here there can e x i s t several maximums and several minimums, which are equivalent t o each other in the sense of the qenerated equivalence relation. The reader i s assumed t o know the notion of uoper bound of a s e t D (mod A ) as well as t h a t of lower bound. The supremum of D , denoted by Sup D , i f i t e x i s t s , i s the minimum in the s e t of upper bounds. Hence x > r Sup D i s equivalent t o x greater than or equal t o every element in D . If Sup D belongs t o D , then i t i s the maximum. Analogous definition of the infimum, denoted by Inf D These notions appeared already in ch.1 5 2.1 f o r ordinals, in ch.1 5 4.5 and 4.6 f o r reals.
-
.
INTERVAL, INITIAL AND FINAL INTERVAL The reader i s assumed t o be familiar with the notion of an element z between x and y (mod A) , or z intermediate between x and y , as well as t h a t of an element s t r i c t l y intermediate. An interval of A i s a subset of the base which i s closed with respect t o the notion of intermediate (mod A ) . An i n i t i a l interval o r i n i t i a l segment of A i s a subset closed with respect t o " l e s s than" . A final interval i s a subset closed with respect t o "greater t h a n "
.
2 . 1 . Let A be a p a r t i al ordering. Then every subset of the base I A 1 without a minimal element i s i n f i n i t e . Similarly for a subset without a maximal element. 0 To each element x of the subset 0 under consideration, associate the s e t Dx of elements of D which are less t h a n or equal t o x (mod A ) . None of the Dx i s minimal under inclusion (see ch.1 5 1.1, definition of a f i n i t e s e t ) . 0 2 . 2 . AMALGAMATION LEMMA
Let A , B be two p a r t i al orderings having the same re stric tion t o the intersection of the bases. Then there e x i s t s a partial ordering which i s an extension of b o t h A .-and B , based on the union of the bases. OWrite x G y when x , y G l A l and x s y ( m o d A ) , o r w h e n w e h a v e t h e s a m e condition for B , or when x belongs t o I A l , y belongs t o I B I and there (mod A) and t $ y (mod B ) , e x is t s an element t i n the intersection with x or when we have the same condition when interchanging A and B . Finally write x\y in the other cases. 0
-
st
2.3. CHAIN, ORDERABLE SET A chain, or total orderinq, i s a p ar t i al ordering whose elements are mutually
Chapter 2
35
comparable. For example, we shall denote by Z the chain of the positive and negative integers, and by Q the chain of the rationals. The previous amalgamation lemma 2 . 2 extends t o the case of two chains, the common extension i t s e l f being a chain. However, t h i s lemma does not extend t o t r e e s , defined i n ch.4 5 6. 0 Take a t r e e on a,b,c,d with a,b,c mutually incomparable, d < a , d c b and d I c ; and another t r e e on a,b,c,e with e < b , e < c and e l a . Then e i t h e r d < e < c or e Q d < a : contradiction. 0 W e say t h a t a s e t E i s orderable i f f there e x i s t s a chain based on E . Using only the axioms o f ZF, every f i n i t e s e t i s orderable (induction: see ch.1 5 1.1). ORDERING AXIOM The ordering axiom a s s e r t s t h a t every s e t i s orderable. I t follows from the ultraf i l t e r axiom, or equivalently from the coherence lemma 1.3. 0 Let E be a s e t ; t o each f i n i t e subset F of E , associate the s e t U F of chains based on F . By 1.3 there e x i s t s a relation R based on E every of whose f i n i t e r e s t r i c t i o n i s a chain; thus R i s a chain. 0 The ordering axiom i s s t r i c t l y weaker t h a n the u f t r a f i l t e r axiom (JECH 1973 p.100). The ordering axiom implies the axiom of choice f o r f i n i t e s e t s (see ch.1 5 2.10). 0 Given a s e t of mutually d i s j o i n t f i n i t e s e t s , i t suffices t o take a chain A based on the union: t o each f i n i t e s e t we associate i t s minimum (mod A ) . 0 The axiom of choice f o r f i n i t e s e t s i s s t r i c t l y weaker than the ordering axiom: see LAUCHLI 1964 f o r ZF without foundation, completed f o r ZF by PINCUS 1972. The axiom of choice f o r f i n i t e s e t s does not follow from the axiom of deoendent choice: see 5 1 . 6 . Hence the ordering axiom does n o t follow from dependent choice. 2.4. WELL-FOUNDED PARTIAL ORDERING OR QUASI-ORDERING; WELL-ORDERING
W e say t h a t a partial ordering o r quasi-ordering i s well-founded i f f every non-empty subset of i t s base has a t l e a s t one minimal element. A well-founded chain, o r t o t a l ordering, i s called a well-ordering. Every f i n i t e p a r t i a l ordering i s well-founded. Every r e s t r i c t i o n of a well-founded p a r t i a l ordering i s well-founded. Given a p a r t i a l ordering A , the reader i s assumed t o know the notion of a sequence with values in A which i s incr?asing, decreasing, s t r i c t 1 2 or otherwise.
A s a t i s f i e s the following conditions: (1) there i s no s t r i c t l y decreasing (mod A) W-sequence; ( 2 ) every t o t a l l y ordered r e s t r i c t i o n of A i s well-founded, hence a well-ordering; equivalently every non-empty t o t a l l y ordered r e s t r i c t i o n of A has a minimum. Every well-founded p a r t i a l ordering
THEORY OF RELATIONS
36
Conversely, each of the conditions ( l ) , ( 2 ) implies, hence i s equivalent t o saying t h a t A i s well-founded. This uses the axiom of dependent choice, y e t ZF suffices i f A i s countable, or i f the base I A l i s well-orderable, in the sense below. In the general case, apply dependent choice t o the relation y < x (mod A ) .
2 . 5 . WELL-ORDERABLE SET We say t h a t E i s well-orderable i f f there e x i s t s a well-ordering based on E . For example any f i n i t e or denumerable s e t i s well-orderable. A s e t E i s well-orderable i f f there e x i s t s a choice function on the s e t < non-empty subsets of E 0 Let f be a choice function on non-empty subsets. Let a. = f(E) . Let u be a non-zero ordinal, Du the s e t of a l l a i ( i i u ) Let a u = f(E-DU) , as long as possible, thus reaching a s e t DU = E . 0 WELL-ORDERING AXIOM, TRICHOTOMY AXIOM I t follows t h a t the axiom of choice i s equivalent t o saying t h a t every s e t i s well-orderable. Or again t h a t every cardinal i s an aleph, or t h a t every i n f i n i t e cardinal has the form c . ) ~ ( o( ordinal index: see ch.1 5 6.1 t o 6 . 4 ) . The axiom of choice i s equivalent t o the trichotomy axiom which says t h a t , given any two cardinals a , b , e i t h e r a < b or a = b or a > b . 0 If every cardinal i s an aleph, then trichotomy holds. Conversely, given a s e t a and the Hartogs u of a (see ch.1 5 6.2), i f trichotomy holds then necessar i l y a i s subpotent t o o( , hence a i s well-orderable. 0
-
.
.
2.6. MAXIMAL CHAIN Let A be a partial ordering and C be a t o t a l l y ordered r e s t r i c t i o n of A . The chain C i s said t o be maximal (under inclusion, mod A) i f f every t o t a l l y ordered r e s t r i c t i o n of A extending C i s identical t o C . Let E be a s e t ; denote by X any well-ordering based on a subset of E Write
.
X6 X '
.
i f f X i s an i n i t i a l interval of X ' The well-founded p a r t i a l ordering thus defined on the s e t of X will be called the interval-orderinqon E . (1) A s e t E i s well-orderable i f f there e x i s t s a maximal chain which i s a r e s t r i c t i o n of the interval-ordering on E ( 2 ) Let A be a partial ordering and C be a t o t a l l y ordered r e s t r i c t i o n of A Let U be any chain which i s both a r e s t r i c t i o n of A and an extension of C Every function f which t o each U associates f(U) , a t o t a l l y ordered rest r i c t i o n of A and extension of U , has a fixed point V such t h a t f(V) = V 0 Index by ordinals a sequence of chains U i s t a r t i n g with Uo = C ; s e t Ui+l = f ( U i ) and, f o r i a l i m i t ordinal, l e t Ui be the common extension of U j ( j < i ) t o the union of t h e i r bases. 0
.
.
.
.
37
Chapter 2
2.7. MAXIMAL CHAIN AXIOM, OR HAUSDORFF-ZORN AXIOM T h i s axiom, g o i n g back t o HAUSDORFF 1914, t h e n t a k e n up by KURATOWSKI, MOORE and
A
t h e n ZORN, i s s t a t e d as f o l l o w s . Given a p a r t i a l o r d e r i n g which i s a r e s t r i c t i o n o f
and a c h a i n
C
e x i s t s a c h a i n which i s an e x t e n s i o n o f
C
.
and maximal (mod A) By 2.6.(2)
, there
A
above, t h e axiom o f c h o i c e i m p l i e s t h e maximal c h a i n axiom. By 2.6.(1),
the
By 2.5,
t h e maximal c h a i n axiom i m p l i e s t h a t every s e t i s w e l l - o r d e r a b l e . maximal c h a i n axiom i s t h e n e q u i v a l e n t t o t h e axiom o f c h o i c e . 2.8.
The u l t r a f i l t e r axiom f o l l o w s f r o m t h e axiom o f choice.
0 Consider t h e s e t o f f i l t e r s on a g i v e n s e t , w i t h t h e comparison o r d e r i n g
" f i n e r f i l t e r t h a n " . Take a maximal c h a i n e x t e n d i n g t h e c h a i n reduced t o a g i v e n
. The
filter
u l t r a f i l t e r g i v e n by t h e u n i o n o f t h e f i l t e r s b e l o n g i n g t o t h e
F.0
maximal c h a i n i s f i n e r t h a n
The u l t r a f i l t e r axiom i s s t r i c t l y weaker t h a n t h e axiom o f choice: HALPERN, LEVY 1971 p. 83-134. 2.9. FREE SUBSET, ANTICHAIN, MAXIMAL FREE SUBSET, MAXIMAL ANTICHAIN
,a
A
Given a p a r t i a l o r d e r i n g
subset o f i t s base i s c a l l e d
i t s elements a r e m u t u a l l y incomparable (mod A) f r e e subset
D
i s c a l l e d an a n t i c h a i n (mod A )
r e l a t i o n based on
D
. The
free (mod A)
restriction
. It reduces
A/D
iff
t o such a
t o the i d e n t i t y
.
A f r e e subset, and t h e c o r r e s p o n d i n g a n t i c h a i n , a r e c a l l e d maximal (under i n c l u -
A
s i o n ) i f f t h e r e i s no p r o p e r s u p e r s e t which i s f r e e . Given a p a r t i a l o r d e r i n g and a f r e e subset
D
, there
e x i s t s a maximal f r e e subset i n c l u d i n g
axiom o f choice; ZF s u f f i c e s i f
A
0 (uses
i s c o u n t a b l e ) : a p p l y t h e maximal c h a i n axiom
t o t h e i n c l u s i o n among f r e e subsets.
3 - ISOMORPHISM^
AUTOMORPHISM, H E I G H T OF A WELL-FOUNDED P A R T I A L ORDERING, SUM AND PRODUCT OF CHAINS, HOMOMORPHIC IMAGE §
Let
n
an
be a non-negative i n t e g e r ,
n - a r y r e l a t i o n w i t h base
We say t h a t
f
transforms o r
f
i s an isomorphism o f
(=
+
o r -)
and we s e t
f"(R)
= (El,+)
n-ary r e l a t i o n w i t h base
E' ; l e t
f
takes
into
onto
f o r a l l elements
A relation R' from
R
an
R
x1 or
R'
R
be a b i j e c t i o n f r o m
iff R ' ( f x l
,...,xn R = (E,-)
in
. This
E
, denoted
,...,fx,)
. For
and t h e n
i s s a i d t o be isomorphic w i t h
R onto R '
R'
R
R'
= R(xl
n = 0
E'
onto
E 5
and
E
f"(R)
R'
.
, or that
,..., xn)
, either
f"(R) = (El,-)
R = (E,+)
.
i f f t h e r e e x i s t s an isomorphism
c o n d i t i o n i s r e f l e x i v e , symmetric and t r a n s i t i v e ,
y i e l d i n g an e q u i v a l e n c e r e l a t i o n o n e v e r y s e t o f r e l a t i o n s o f a g i v e n a r i t y .
38
THEORY OF RELATIONS
ISOMORPHISM TYPE, ORDER TYPE Modeled after the definition of "cardinal" in ch.1 0 5.4, we consider the relations isomorphic with R , and among such, those whose base has minimum fundamental rank. These form a set, called the isomorphism type of R (the order type if R is a chain, or total ordering). Thus two relations are isomorphic iff they have the same type. AUTOMORPHISM, EMPTY FUNCTION Given a relation R with base E , a permutation f of E i s called an automorphism of R iff f i s an isomorphism from R onto R . The automorphisms of R form a group of permutations of E . We adopt the convention that the empty function, which is a bijection of the empty set onto itself, is also an automorphism of each relation with empty base. In particular it is an automorphism of the 0-ary relation with empty base and value ( + ) , denoted (O,+) , and also of (0,-). However (O,+) and (0,-) are not isomorphic. These definitions and conventions extend to mu tirelations. Given a multirelation R = (R1 ,..., Rh) with base E and R' = (Ri,. .,Rb) with base E ' , a bijection f from E onto E' transforms R into R' or is an isomorphism from R onto R' , denoted R' = f"(R) , iff for each i = 1, ...,h , the function f is an isomorphism of the component Ri onto the component R; . In other words fo(R1 ,...,Rh) = (fo(R1) ... .,fo(Rh)) . 3.1. (1) Let A be a well-ordering and f be an isomorphism from A onto a restriction of A . Then fx & x (mod A ) for each element x of A . ( 2 ) Given a well-ordering, its unique automorphism is the identity. Given two well-orderings A, B , there exists at most one isomorphism from A onto B . ( 3 ) Given a well-ordering A , no proper initial interval of A is isomorphic with A . In particular two isomorphic ordinals are identical.
-
3.2. HEIGHT IN A WELL-FOUNDED PARTIAL ORDERING
Let A be a well-founded partial ordering. To each element x of IAI , associate as follows an ordinal called the height of x (mod A) and denoted Ht x If x is a minimal element, let Ht x = 0 . Let o( be a non-zero ordinal; assume that each ordinal (o( has been associated to at least one element, but that there still remain elements in the base to which no height (q has been associated. Then associate the height o( to minimal elements among these. Given a well-founded partial ordering A , there i s a unique height associated to each element of the base I A l . Moreover, for each element x of height cr( and every ordinal /3 < d , there exists at least one element < x (mod A) with height /3.
.
Chapter 2 However, g i v e n
>
elements (mod A) 0
Let
x
x
(d
> /s
o f heights and
x (y
(mod A)
, then
, it
i s p o s s i b l e t h a t no element
x
fs. a
< e <.d
and i t s o n l y s t r i c t upper bound i s
If
f5 2 4 , even i f t h e r e e x i s t
of h e i g h t o( and an o r d i n a l
(mod A)
exists with height a
39
<
Ht x
with
elb
, with
d
and
height 3
elc
.
Then
e
has h e i g h t 1
.0
.
Ht y
Two d i s t i n c t elements of t h e same h e i g h t a r e incomparable. But two incomparable elements may have d i f f e r e n t h e i g h t s . 3.3. HEIGHT OF A WELL-FOUNDED PARTIAL ORDERING The h e i g h t s of t h e elements o f a w e l l - f o u n d e d p a r t i a l o r d e r i n g an o r d i n a l , c a l l e d t h e h e i q h t o f H t A = Sup((Ht x ) + l )
for all
A
x (mod A )
The h e i g h t o f an element
and denoted
IAI
X E
.
Ht A
.
A
constitute
E q u i v a l e n t l y we have
i s also the heiqht o f the r e s t r i c t i o n o f
I n p a r t i c u l a r , t h e fundamental r a n k o f a s e t
a
(ch.1
5
5.2) i s t h e h e i g h t o f
t h e w e l l - f o u n d e d p a r t i a l o r d e r i n g based on t h e t r a n s i t i v e c l o s u r e o f d e f i n e d by h
i f f t h e r e e x i s t s a f i n i t e sequence
x
i n t e g e r and
ti
E ti+l
f o r each
3.4. Every w e l l - o r d e r i n g A
i< h
x
into
o f t h e o t h e r , by c h . 1
with
Ht A
t h e isomorphism
.
one i s i s o m o r p h i c t o an
n i t i a l interval
A
2 . 1 ( t r i c h o t o m y ) . Given a w e l l - o r d e r i n g
i s isomorphic w i t h an i n i t i a l i n t e r v a l o f
A
restriction o f
5
, and
.
Ht x
Consequently, g i v e n two w e l l - o r d e r i n g s ,
a
... th = y
to = x,
i s isomorphic t o t h e o r d i n a l
phism t r a n s f o r m s each element
A
.
, each I f two w e l l -
o r d e r i n g s a r e each isomorphic t o a r e s t r i c t i o n o f t h e o t h e r , t h e n t h e y a r e isomorphic. 3.5.
of
(1) L e t
A
A
be a well-founded p a r t i a l o r d e r i n g . Then each r e s t r i c t i o n
i s a well-founded p a r t i a l ordering w i t h
Ht B
6
Ht A
.
.
Given a w e l l - f o u n d e d p a r t i a l o r d e r i n g ordered r e s t r i c t i o n o f
A
A
does n o t r e a c h
B
Indeed f o r each
x o f 1 6 1 we have H t x (mod B ) 4 H t x (mod A) ( 2 ) I f B i s an i n i t i a l i n t e r v a l o f A c o n t a i n i n g x , then H t x (mod H t x (mod A) . Consequence o f 3.2 ( i n d u c t i o n on H t x ) . element
B)
=
, it
can happen t h a t a maximal t o t a l l y H t A . Even t h e supremum o f t h e
.
Ht A i , t a k e a c h a i n isomorphic t o i ; t h e s e c h a i n s a r e assumed t o be m u t u a l l y incomparable. We o b t a i n a well-founded p a r t i a l o r d e r i n g w i t h h e i g h t s o f t h e maximal c h a i n s can be s t r i c t l y l e s s t h a n
0 F o r each i n t e g e r
height
A
<x .
t o elements
a ,b u t
i n which e v e r y maximal c h a i n i s f i n i t e .
40
THEORY OF RELATIONS
Another example. Take denumerably many c o p i e s isomorphic t o t h e p r e v i o u s o r d e r i n g Above t h e
jth copy ( j i n t e g e r ) , p u t a c h a i n isomorphic t o
j
. Let
these
d i f f e r e n t o r d e r i n g s w i t h d i s j o i n t bases be m u t u a l l y incomparable. We o b t a i n a
, in
well-founded p a r t i a l o r d e r i n g w i t h h e i g h t w.2
which e v e r y maximal c h a i n
i s finite. 0 3.6. ORDINAL SUM, HOMOMORPHIC IMAGE OF A CHAIN
A, B be two c h a i n s w i t h d i s j o i n t bases. We c a l l t h e o r d i n a l sum, o r s i m p l y t h e sum A+B t h e c h a i n based on t h e u n i o n o f t h e bases, which i s t h e common e x t e n s i o n o f A and B f o r which each element i n I A I precedes each element Let
in
IBI
.
T h i s g e n e r a l i z e s t h e n o t i o n o f sum among o r d i n a l s (ch.1
T h i s sum agrees w i t h isomorphism, i n t h e sense t h a t i f and
6'
to
, then
B
i s isomorphic t o
A'+B'
commutative, even up t o isomorphism:
A
More g e n e r a l l y , l e t ly disjoint intervals
\Ail
t&of
. The
A
chain
, again
Ai
denoted
= rAi
A A
. The
i
.
Take t h e new base t o be t h e s e t
I by p u t t i n g
i s called the
A
i< j
I A J. 1 . T h i s
set o f the intervals
i f f each element
chain
I i s called
i s c a l l e d a decomposi-
Ai
I-=o r t h e sum a l o n g
I
o f the
Ai
:
.
(iE I ) Each homomorphic image o f a c h a i n A notation
A
It i s associative but not
w + 1 and l + w a r e n o t i s o m o r p h i c .
i s s t r i c t l y l e s s t h a n each element o f
a homomorphic image o f
.
3.1).
be a c h a i n . Consider a p a r t i t i o n o f i t s base i n t o mutual-
o f t h e s e i n t e r v a l s , and d e f i n e t h e c h a i n of
A+B
5
i s isomorphic t o
A'
i s isomorphic t o a r e s t r i c t i o n o f
A
(uses axiom o f c h o i c e ) : t a k e one element i n each i n t e r v a l o f t h e decomposition. The converse i s f a l s e . F o r example, t h e c h a i n w o f t h e i n t e g e r s i s a r e s t r i c t i o n o f W+1
, but
i n t e r v a l . Hence
each decomposition o f ~
+ i n1t o i n t e r v a l s has a l a s t
w i s n o t a homomorphic image o f w + l Q
Another counterexample. The c h a i n
.
o f the rationals i s a restriction o f the
c h a i n o f t h e r e a l s , b u t n o t a homomorphic image. Indeed DEDEKIND's theorem
5
4.6) would be v i o l a t e d by a decomposition a l o n g Q o f t h e c h a i n R of t h e r e a l s , and a p a r t i t i o n o f Q , t h u s a p a r t i t i o n o f R i n t o an i n i t i a l i n t e r v a l w i t h o u t a maximum and t h e complementary i n t e r v a l w i t h o u t a minimum. (ch.1
Note t h a t a homomorphic image o f a homomorphic image o f 3.7.
A, B t h e o r d i n a l p r o d u c t
A
.
i s t h e c h a i n based on t h e
A.B
C a r t e s i a n p r o d u c t o f t h e bases, d e f i n e d by p u t t i n g
<
i s an image o f
ORDINAL PRODUCT OF CHAINS
Given two c h a i n s y
A
y ' (mod B )
t o each y
of
t h e ordered p a i r
or
y = y'
and
x
1B1 the chain A (y,x)
Y
6 x'
(mod A)
.
iff (y,x),( (y',x') I n o t h e r words, by a s s o c i a t i n g
o b t a i n e d by r e p l a c i n g each
x
of
A
5
3.2).
, t h e n t a k i n g t h e 8-sum o f t h e A
Y ' T h i s p r o d u c t g e n e r a l i z e s t h e n o t i o n o f p r o d u c t among o r d i n a l s (ch.1
by
41
Chapter 2
A'
T h i s p r o d u c t agrees w i t h isomorphism: i f then
A'.B'
i s isomorphic t o
up t o isomorphism:
~3
and
.2
A.6
.
i s isomorphic t o
A
and
B'
to
B,
I t i s a s s o c i a t i v e b u t n o t commutative, even
a r e n o t isomorphic.
2. W
3.8. CARDINAL SUM, PRODUCT AND EXPONENTIATION REVISITED L e t us r e t u r n t o t h e o p e r a t i o n s between alephs and between c a r d i n a l s (ch.1 § 5.5) w i t h t h e means now a f f o r d e d by t h e n o t i o n s o f isomorphism and w e l l - o r d e r i n g . a
(1)
be an i n f i n i t e aleph; t h e n
( 2 ) k t a, b b = a x b a
a
+a
= a x a = a
.
be two alephs, a t l e a s t one o f which i s i n f i n i t e ; t h e n = Max(a,b) ( f o r t h e p r o d u c t we assume t h a t a, b # 0 ) .
+
0 It s u f f i c e s t o e s t a b l i s h
a x a = a : indeed
+a
a
= a
f o l l o w s by BERNSTEIN-
SCHRODER; by t h e same argument ( 2 ) f o l l o w s from (1). L e t E be t h e f o l l o w i n g t o t a l o r d e r i n g r e l a t i o n between two couples o f o r d i n a l s ( d , f s ) and ( ~ ' , f i ': )e i t h e r M a x ( & , p ) < M a x ( o ( ' , / j ' ) , o r the maximums are equal and fi <& or A = 0 ' anb 4 F o r coup1 es o f o r d i n a l s , t h i s comparison i s a w e l l - o r d e r i n g . Thus t h e r e e x i s t s a c o n d i t i o n ( & , f i , r ) which t o each c o u p l e ( d , ( 3 ) which i s isomorphic w i t h t h e s e t o f associates one and o n l y one o r d i n a l
so('.
couples
) . We have ?f>/ M a x ( d , p ) . f ("i, ~ 0 , ' 6 )a s s o c i a t e s t o each o(
< (6.0)(mod
take /3 = 0 : t h e n This
8
product
increases as o( increases, and
.
o<
o( x
, t h e n Sup gi
3
an o r d i n a l
o(
i s equipotent w i t h the Cartesian
Moreover, g i v e n a s e t o f o r d i n a l s
are associated t o them by
I n particular,
4
xi
and t h e
which
.
i s associated t o
Everything comes down t o p r o v i n g t h a t , f o r o( i n f i n i t e , t h e a s s o c i a t e d 8 i s the smallest = w Denote by F o r o( = w we have
.
.
equipotent w i t h o(
i s s t r i c t l y subpotent t o , t h u s t o o( x o ( . T h i s o( does n o t have any o r d i n a l o( ' <' P( which i s e q u i p o t e n t w i t h o( Indeed we
o r d i n a l f o r which o(
.
would have o( Thus q
8
'
e q u i p o t e n t w i t h o(
i s n e c e s s a r i l y an aleph. Denote by o( t h e o r d i n a l a s s o c i a t e d t o ct
Fi 6 4
.
o(
a x 4.
equipotent w i t h
the ordinals
.
<
o(
ri
and by
by By h y p o t h e s i s each is so s t r i c t l y subpotent t o o( , so Ti < 4 ; t h u s By t h e p r e c e d i n g d i s c u s s i o n = Sup x i * % : contradiction. 0
, and
equipotent w i t h o( Sup
' x 6 ', hence
Modulo t h e axiom o f choice, o u r p r o p o s i t i o n extends t o a r b i t r a r y c a r d i n a l s . Note t h a t t h e e q u a l i t y
a x a = a
f o r every i n f i n i t e cardinal
a
i s equiva-
l e n t t o t h e axiom o f c h o i c e (TARSKI 1924; see a l s o RUBIN 1963). 3.9. k t a
be an i n f i n i t e a l e p h and l e t
Indeed a = a x b
~
so
a2 = (axb)2 = a(b2)
2 6 b
>,
6a
ab
.
; then ab = a2
.
Modulo t h e axiom o f choice, t h i s i d e n t i t y extends t o a r b i t r a r y c a r d i n a l s .
THEORY OF RELATIONS
42
3.10. Given an i n f i n i t e c a r d i n a l
a
,let
o(
be the Hartogs aleph f o r sets o f
c a r d i n a l a (ch.1 5 6.2). E i t h e r N = w ; then a i s i n f i n i t e b u t n o t D e d e k i n d - i n f i n i t e (ch.1
5
2 . 6 ) ; then
+
1 i s immediately g r e a t e r than a , i n t h e sense t h a t i t i s t h e c a r d i n a l sum a s t r i c t l y g r e a t e r than a (as a c a r d i n a l ) y e t t h e r e i s no s e t s t r i c t l y intermediat e w i t h respect t o subpotence. O r o( i s a t l e a s t equal t o w 1 ; then the c a r d i n a l sum a g r e a t e r than a i n the preceding sense (see TARSKI 1954).
+
0 Consequence o f t h e f a c t t h a t a r e s t r i c t i o n o f t h e o r d i n a l o(
d
i s immediately
e i t h e r i s isomor-
p h i c w i t h o( o r w i t h an o r d i n a l /3 s t r i c t l y l e s s than o( ; then a + CardfS= a.D Modulo t h e axiom o f choice, every c a r d i n a l i s an aleph. Hence f o r each c a r d i n a l a there e x i s t s a c a r d i n a l b which i s immediately g r e a t e r than a i n t h e stronger f o l l o w i n g sense: every c a r d i n a l > a i s b b = a+ , the successor c a r d i n a l o f a , I n the mentioned paper, i t i s proved Put t h a t the preceding statement i s e q u i v a l e n t t o the axiom o f choice. I n the absence o f t h e axiom o f choice, t h e r e can e x i s t several c a r d i n a l s immedia-
.
t e l y g r e a t e r than a given c a r d i n a l ( i n the weak sense). For example, w i t h t h e continuum hypothesis w i t h o u t t h e axiom o f choice, w 1 and a l s o t h e continuum
w
a r e immediately g r e a t e r than
§
4
-
and p o s s i b l y incomparable: see ch.1
5
6.5.
REINFORCED RELATION, REINFORCED PARTIAL ORDERING
4.1. REINFORCEMENT, WEAKENING
.
L e t A be an n-ary r e l a t i o n w i t h base E An n-ary r e l a t i o n B w i t h t h e same base i s c a l l e d a reinforcement o f A , and A i s a weakening o f B i f f every n - t u p l e i n E having t h e value (+) i n A s t i l l has t h e value (+) i n B . Reinforcement i s a p a r t i a l o r d e r i n g on t h e s e t o f n-ary r e l a t i o n s w i t h base A s t r i c t o r proper reinforcement o f
A
i s a reinforcement d i s t i n c t from
A
E
.
.
S i m i l a r l y we speak o f a s t r i c t o r p r o p e r weakenin&. L e t A be a p a r t i a l o r d e r i n g which i s n o t a chain. We o b t a i n a s t r i c t r e i n f o r c e ment o f A by t a k i n g two elements u, v which are incomparable (mod A) , then d e f i n i n g , f o r a l l elements x, y o f t h e base, x d y i f f e i t h e r x d y (mod A) o r x,< u and v g y (mod A ) Generalize t h i s procedure as f o l l o w s .
.
L e t A be a p a r t i a l o r d e r i n g w i t h base E . L e t D g : E and B be a p a r t i a l order i n g w i t h base D which i s a reinforcement o f t h e r e s t r i c t i o n A/D Then there e x i s t s a p a r t i a l o r d e r i n g C which i s a reinforcement o f A and an
.
extension o f
6
2
E
. Moreover,
minimal among these reinforcements ding i s a reinforcement o f
Co
.
t h e r e e x i s t s a p a r t i a l o r d e r i n g Co C
which i s
, i.e. every C which v e r i f i e s t h e prece-
Chapter 2 0 Let
x, y
be two elements o f
e x i s t two elements
XI,
of
y'
E D
.
.
Co ; t h e
are reinforcements o f
x+ y
iff
x
x 4 x ' (mod A)
with
y ' 6 y (mod A )
C's
Put
43
6 y (mod A ) and
o r i f there
x ' d y ' (mod
B) and
The p a r t i a l o r d e r i n g t h u s o b t a i n e d i s t h e minimal r e i n f o r c e m e n t Co
.0
4.2. REINFORCEMENT AXIOM
-
This axiom a s s e r t s t h a t each p a r t i a l o r d e r i n g has a c h a i n among i t s r e i n f o r c e m e n t s . This f o l l o w s from t h e u l t r a f i l t e r axiom (SZPILRAJN 1930); ZF s u f f i c e s i n t h e case o f a countable p a r t i a l o r d e r i n g .
A
0 Let
we l e t
be a p a r t i a l o r d e r i n g w i t h base
UF
cement o f
E
. F o r each
denote t h e s e t o f a l l c h a i n s w i t h base A/F
. The
f i n i t e subset
F of
E
,
F each o f which i s a r e i n f o r -
a r e non-empty and v e r i f y t h e hypotheses o f t h e coherence
UF
E which s a t i s f i e s C i s a chain, s i n c e i t s f i n i t e r e s t r i c t i o n s a r e chains; i t i s a r e i n f o r c e m e n t o f A 0 I n t h e l i g h t o f t h e p r e c e d i n g 4.1, t h e r e i n f o r c e m e n t axiom can be s t a t e d equival e n t l y : g i v e n a p a r t i a l o r d e r i n g A and a c h a i n C which i s a r e i n f o r c e m e n t o f lemma 1.3. Hence t h e r e e x i s t s a r e l a t i o n C/F E UF
f o r each
F
C
based on
(uses u l t r a f i l t e r axiom). T h i s
.
a restriction o f
A
ment o f
A
,there
e x i s t s a c h a i n which i s s i m u l t a n e o u s l y a r e i n f o r c e -
and an e x t e n s i o n o f
C ,
5 2.3. It s u f f i c e s , g i v e n , t o r e i n f o r c e i n t o a c h a i n t h e p a r t i a l o r d e r i n g which reduces t o t h e
The r e i n f o r c e m e n t axiom i m p l i e s t h e o r d e r i n g axiom o f a set
E
i d e n t i t y on
E
.
Note t h a t t h e r e i n f o r c e m e n t axiom i s s t r i c t l y weaker t h a n t h e u l t r a f i l t e r axiom: see FELGNER 1974 p. 375, making r e f e r e n c e t o t h e unpublished t h e s i s o f t h e a u t h o r . Also t h e o r d e r i n g axiom i s s t r i c t l y weaker t h a n t h e r e i n f o r c e m e n t axiom: see MATHIAS 1974. 4.3. Each weakened p a r t i a l o r d e r i n g o f a well-founded p a r t i a l o r d e r i n g i s w e l l founded. On t h e o t h e r hand, a r e i n f o r c e m e n t o f a w e l l - f o u n d e d p a r t i a l o r d e r i n g i s n o t n e c e s s a r i l y well-founded: i n t o an w 4.4.
r e i n f o r c e t h e i d e n t i t y w i t h a denumerable base
c h a i n (converse o f w )
.
For each w e l l - f o u n d e d p a r t i a l o r d e r i n g
reinforcement o f
A
A
, there
e x i s t s a well-ordered
(uses axiom o f choice; ZF s u f f i c e s i f A
i s countable o r
has w e l l - o r d e r a b l e base).
i s t r i c t l y l e s s t h a n H t A , t a k e a w e l l - o r d e r i n g Ci based i (axiom o f c h o i c e ) . Then t a k e t h e sum according t o increasing i 0
For each o r d i n a l
on t h e f r e e s e t o f elements w i t h h e i g h t
of the Ci
.
THEORY OF RELATIONS
44 4.5.
If
A, B
Let
be two well-founded p a r t i a l orderings w i t h t h e same base.
. Consequently
H t x (mod A)
5 5 - COFINAL SUBSET,
, then f o r
A
i s a reinforcement o f
B
each
.
Ht B 3 Ht A
x
H t x (mod B) >/
i n t h e base,
Proof by i n d u c t i o n ,
C O - I N I T I A L SUBSET, C O F I N A L I T Y OF A C H A I N
COFINAL. CO-INITIAL SUBSET AND RESTRICTION
5.1.
A be a D a r t i a l orderino. A subset D o f IAI i s s a i d t o be c o f i n a l (mod A ) . and A / D i s s a i d t o be a c o f i n a l r e s t r i c t i o n o f A i f f f o r each x i n I A l Let
there e x i s t s a y
in
D with
. Analogous
y + x (mod A)
definition for a
c o - i n i t i a l subset and a c o - i n i t i a l r e s t r i c t i o n . I f t h e base i s non-empty, then every c o f i n a l o r c o - i n i t i a l subset i s non-empty. Each superset o f a c o f i n a l s e t i s c o f i n a l ; s i m i l a r l y w i t h c o - i n i t i a l . Each c o f i n a l r e s t r i c t i o n K a c o f i n a l re_z_tt--c_t_ion_ i s a qozinal r e s t r i c t i o n ; similarly with co-initial.
A
Let
be a p a r t i a l ordering,
B
. For each
in
E
base
p a i r \x,y)
. Then
then remove t h i s
y
c o f i n a l (mod A)
, and
B
i s a well-ordering,
there e x i s t s an and
x
x
(mod B)
would e x i s t an
x
x
>y
,let x
are never ordered i n
D
i s a well-founded p a r t i a l o r d e r i n g L
.
Indeed f o r each
y
which i s removed,
among those s a t i s f y i n g
x
>y
(mod A)
. i s a well-ordering.
B
B/X : then
be the minimum o f
d i c t i n g the minimality o f
,
(mod B)
D : i f i t were removed, then there (mod A) and x ' < x < y (mod B) , c o n t r a d i c t i n g the
, we do n o t have
D
>x
(POUZET 1979, unpublished).
For otherwise there would e x i s t a y belong t o
and
belongs t o
x
Now consider t h e case t h a t D
A/D
then A
o f l e a s t h e i g h t (mod B)
. This
minimality o f the height o f of
x (mod A)
.
B
i s c o f i n a l (mod A)
D
x'>
<
y
D obtained a f t e r a l l suc~-removals i s
and modulo
which i s a c o f i n a l r e s t r i c t i o n o f 0 F i r s t we see t h a t
, if
E
the set
any two d i s t i n c t elements o f
opposite senses modulo A Moreover i f
a well-founded p a r t i a l o r d e r i n g w i t h t h e same
x
y
>
in
For each non-empty subset
.
.
X
i s a minimal element (mod A/X)
with y
X
x (mod 8)
(mod B/X)
x
<
Hence y
.
.
x (mod A) Since x, y < x (mod B) , contra-
B i s a well-ordering, i f we i (mod B) , then D i s t h e s e t o f
From an i n t u i t i v e p o i n t o f view, note t h a t when denote by elements = the
Then
bi
bi c
the element w i t h h e i g h t
defined as f o l l o w s among t h e o f least height
c2 = bi(*)
elements that the
.&
co
= the
and
cu = b .
1(u)
bi
4 c1 (u
i(1)
O(
)
.
L e t , ,c
= bo
. Then
# 0 (mod B) , among t h e elements
o f l e a s t height (mod A)
<
b
.
i(2)
>
i(1)
(mod B)
I n general f o r each o r d i n a l
are defined. Then
c o( - bi(%)
c1 = bi(l)
$
co (mod A)
, among t h e o( , assume = the
bi
of
.
Chapter 2
45
l e a s t height i ( % ) 7 i ( u ) f o r a l l u
. ..
9.
founded). Take a w e l l - o r d e r i n g o f
+
IA1
.
and use t h e preceding p r o p o s i t i o n .
5.2. STRATIFIED PARTIAL ORDERING A p a r t i a l o r d e r i n g A i s s a i d t o b e . s t r a t i f i e d i f f t h e union o f t h e r e l a t i o n s o f incomparability (mod A) and i d e n t i t y forms a t r a n s i t i v e r e l a t i o n , hence an equivalence r e l a t i o n . Consequently x 4 y and y I z i m p l i e s t h a t x < z ; s i m i l a r l y with > i n the place o f < . The equivalence classes o f i n c o m p a r a b i l i t y - i d e n t i t y form a t o t a l ordering, by p u t t i n g "class o f x " < "class o f y I' i f f x < y (mod A) . This t o t a l order i n g o f t h e equivalence classes w i l l be c a l l e d t h e p r i n c i p a l t o t a l o r d e r i n g of t h e s t r a t i f i e d p a r t i a l ordering A Every t o t a l l y ordered r e s t r i c t i o n o f A which i s maximal under i n c l u s i o n i s isomorphic w i t h t h e p r i n c i p a l t o t a l ordering.
.
5.3. L e t A be a well-founded p a r t i a l ordering. Then t h e r e e x i s t s a c o f i n a l r e s t r i c t i o n C o f A s a t i s f y i n g H t C 6 Card H t A (POUZET 1979, unpublished). 0 To each o r d i n a l i < H t A associate the c l a s s Bi o f elements o f h e i g h t i (mod A) Order t h e s e t o f t h e Bi by a w e l l - o r d e r i n g isomorphic w i t h i t s cardinal, which i s Card H t A Denote by B the well-founded s t r a t i f i e d p a r t i a l ordering w i t h base IA I , defined by p u t t i n g x < y (mod B) i f f "class o f x I' "class o f y I' according t o t h e preceding well-ordering. Then H t B = Card H t A By 5.1, t h e r e e x i s t s a c o f i n a l r e s t r i c t i o n C o f A such t h a t any two elements o f I C I are never ordered i n opposite senses by A and B
-
.
<
.
.
.
Hence i f x, y belong t o I C l and x < y (mod A) , then x ( y (mod B) , as incomparability (mod 8) i s impossible since x and y belong t o two d i s t i n c t classes Bi Thus B / K I i s a reinforcement o f C = A/)CI . By 4.5, we have
.
Ht C
4
Ht(B/ICI)
6
H t B = Card H t A
.
5.4. COFINALITY, CO-INITIALITY Let A be a p a r t i a l ordering. If, among t h e c o f i n a l sets (mod A) , there e x i s t s one o f l e a s t c a r d i n a l , then t h i s c a r d i n a l i s c a l l e d t h e c o f i n a l i t y o f A , denoted by Cof A Analogous d e f i n i t i o n o f c o - i n i t i a l i t y . With the axiom o f choice, every c a r d i n a l i s an aleph, hence t h e c o f i n a l i t y and coi n i t i a l i t y e x i s t f o r each p a r t i a l ordering. With o n l y t h e axioms o f ZF, these
.
46
THEORY OF RELATIONS
o n l y e x i s t i n p a r t i c u l a r cases, f o r example when t h e base i s w e l l - o r d e r a b l e . T h e i r s t u d y i s v e r y d i f f e r e n t i n t h e case o f a t o t a l o r d e r i n g , t h e c l a s s i c a l case c o n s i d e r e d i n t h e p r e s e n t and n e x t s e c t i o n s , f r o m t h e general case o f a p a r t i a l
5
o r d e r i n g , such a case i n t r o d u c e d and s t u d i e d by POUZET: see
be a t o t a l o r d e r i n g w i t h w e l l - o r d e r a b l e base. Then t h e r e e x i s t s a c o f i n a l
A
Let
7 below.
subset
o f t h e base, w i t h
U
A/U
a w e l l - o r d e r i n g isomorphic t o
Cof A ; same
result with co-initiality. 0 Take a c o f i n a l subset
Totally order and
A
by
D
A/D
D
w i t h l e a s t c a r d i n a l , hence o f c a r d i n a l equal t o
a c c o r d i n g t o i t s c a r d i n a l . Then a p p l y 5.1,
, and
C o r o l l a r y . L e t o( exists a
D ) , which i s
be an o r d i n a l ;
~~t
be
4 . To
Cof
Then we have
N i ( i E I ) = (Sup d i )
o r d i n a l sum o f t h e
Mi
Obviously
a
cofinal
subset
of
i of
each element
f o r which t h e r e
4 ).
o (
-
I
with
w / I
we asso-
zcxi
= o(, where
designates the : see 5 3.6.
i
along the order o f increasing
>, (Sup
oCi
D
c( ( e q u i v a l e n t l y :
di , t h e i n i t i a l i n t e r v a l o f d formed by t h e elements
ciate the ordinal
0
I
indecomposable ordinal.
an
u
i s the least ordinal
u-sequence o f successor o r d i n a l s whose u n i o n i s
o f l e a s t o r d e r t y p e , hence equal t o
.
Cof A by
.0
Cof A
u-sequence of successor o r d i n a l s whose u n i o n i s
a s t r i c t l y increasing 5.5
Cofd
E
D isomorphic t o t h e c a r d i n a l o f D
by a w e l l - o r d e r i n g o f
B
( i . e . t h e l e a s t o r d i n a l based on
replacing
o( . It remains t o e l i m i n a t e t h e case o f 2 Hi > o( and l e t u be t h e l e a s t e l e z o( i(i< u ) >/ o( . E i t h e r u has a predecessor v
Ni)
=
s t r i c t i n e q u a l i t y . Suppose t h a t
I
ment o f (u = v+l)
f o r which
,
<
i n which case o( i s equal t o
< d V hence
which i s
o( i(i v ) p l u s a non-zero o r d i n a l I n t h i s case o( i s n o t indecomposable: con-
tradiction. Or
u
i s a l i m i t o r d i n a l , hence
o f the disjoint intervals i s cofinal f o r greater than
d
di(i<
Ni(i<
. Hence
u
>, Cof
u)
u ) = o(
.
Then t h e s e t o f minimums
y i e l d s an o r d i n a l isomorphic t o
, and
o(
so t h e o r d i n a l o f
d/I
u
, and
is strictly
C o f M : contradiction. 0
N o t i c e t h a t i t i s necessary t o assume t h a t o( i s indecomposable: t a k e t h e counterexample o( = W . 2
with
i i n t e g e r and di = 6~+ i
t o t a k e OC / I
isomorphic t o 2 t h e counterexample LX= Cr) and
o(
=
LJ.2
+
~
(
i W- )
, and
Cof d with
i
for J ,(
<
d . 2
6
.
It i s a l s o necessary
not only o f cardinal and o( i < (2.2
.
=
Cof o( : t a k e
c3 + i f o r f i n i t e i
,
41
Chapter 2
§
6 - REGULAR
AND SINGULAR
ACCESSIBILITY
ALEPH,
6.1. REGULAR AND SINGULAR ALEPH An a l e p h ( i . e . t h e c a r d i n a l of a w e l l - o r d e r a b l e s e t ) i s s a i d t o be r e g u l a r i f f , considered as an o r d i n a l , e v e r y c o f i n a l subset i s e q u i p o t e n t w i t h i t . I n o t h e r words, i t s c o f i n a l i t y i s equal t o i t . An a l e p h i s s a i d t o be s i n g u l a r i n t h e o p p o s i t e case where i t s c o f i n a l i t y i s s t r i c t l y s m a l l e r . For example
3
1 and w a r e r e g u l a r . Each i n t e g e r
s i n g u l a r . The c a r d i n a l ww o r Modulo t h e c o u n t a b l e z i ' l m
2
has c o f i n a l i t y 1, hence i s
has c o f i n a l i t y r i j
o f choice,
, hence
i s singular.
i s r e g u l a r . However t h e i n e q u a l i t y
U1
w 1 > W i s o b t a i n e d i n c h . 1 5 6.3 u s i n g o n l y ZF. 0 Associate t o each c o u n t a b l e subset D o f ul t h e c o u n t a b l e u n i o n o f those
.
D
countable o r d i n a l s which a r e elements o f
5 2.5, a1 . 0
By c h . 1
countable o r d i n a l , hence cannot be t h e e n t i r e s e t L e t A be a c h a i n w i t h w e l l - o r d e r a b l e base; t h e n
Cof A
t h i s union i s a
i s a r e g u l a r aleph.
Consequence o f t h e f a c t t h a t a c o f i n a l r e s t r i c t i o n o f a c o f i n a l r e s t r i c t i o n i s i t s e l f c o f i n a l ( 5 . 1 above). 6.2. Every successor a l e p h i s - y e g - u x (uses axiom o f c h o i c e ) . 0 Our a l e p h i s o f t h e f o r m
c o f i n a l i t y . Take a see c o r o l l a r y 5.4. Suppose
u
ONtl
where
o(
i s an o r d i n a l . L e t
u be i t s
u-sequence o f successor o r d i n a l s whose u n i o n i s
c3d+1 : From some p o i n t on, t h e s e o r d i n a l s a r e e q u i p o t e n t w i t h wq
, hence
s t r i c t l y less than
Card u
6 ad . Then
.
the union
o f the ordinals i n our
u-sequence has c a r d i n a l a t most equal t o fdH x Ld4 hence a t most equal t o fAd (axiom o f c h o i c e g i v i n g a b i j e c t i o n o f each o r d i n a l
6.3. L e t
a
be an i n f i n i t e a l e p h .
sufficient that there exists a set s t r i c t l y subpotent w i t h 0
If a
a
, whose
For
.0
u = Wdtl
Contradiction proving that onto do(). a
t o be s i n g u l a r , i t i s necessary and
u , s t r i c t l y subpotent w i t h a , o f elements union y i e l d s
a
(uses axiom o f c h o i c e ) .
i s s i n g u l a r , t h e n o u r c o n c l u s i o n i s obvious. Conversely i f
regular, then l e t
u be a s e t o f subsets o f a whose u n i o n i s a
o f t h e subsets i s c o f i n a l , hence o f c a r d i n a l
a
and t h e s e t o f l e a s t upper bounds i s c o f i n a l i n
. Or a
a
.
is
E i t h e r one
each subset i s bounded above,
, hence
o f cardinal
a
.
Replace each upper bound by one o f t h e c o r r e s p o n d i n g subsets (axiom o f c h o i c e ) : the s e t
u has a t l e a s t c a r d i n a l
a
.0
6.4. The p r e c e d i n g p r o p o s i t i o n suggests t h e f o l l o w i n g g e n e r a l i z a t i o n . A c a r d i n a l a
( n o t n e c e s s a r i l y an a l e p h ) i s s a i d t o be s i n g u l a r i f f i t i s t h e u n i o n o f a s e t
THEORY OF RELATIONS
48
a , whose elements a r e s t r i c t l y subpotent w i t h a ; i t i s s a i d t o be r e g u l a r o t h e r w i s e . I n t h e presence o f t h e axiom o f choice, e v e r y c a r d i s t r i c t l y subpotent w i t h
n a l i s an aleph, and we have t h e c l a s s i c a l d e f i n i t i o n i n 6.1. I n t h e absence o f t h e axiom o f choice, we do n o t know whether t h i s g e n e r a l i z e d d e f i n i t i o n o f r e g u l a r and singular cardinal yields interesting results. With t h e axiom o f c h o i c e and t h e continuum h y p o t h e s i s , we know t h a t t h e c a r d i n a l o f t h e continuum equals
CJ
and so i s r e g u l a r . With o n l y t h e axiom o f c h o i c e , t h e r e
e x i s t models where t h e continuum i s a r e g u l a r aleph, and o t h e r s where t h e continuum i s a s i n g u l a r aleph. I t can have any i n f i n i t e c o f i n a l i t y , except i t cannot equal
. This
ww
w
: f o r example
r e s t r i c t i o n on t h e c o f i n a l i t y r e s u l t s f r o m t h e f a c t
t h a t , f o r any p a r t i t i o n o f t h e continuum i n t o a c o u n t a b l e number o f subsets, t h e r e i s a t l e a s t one which i s e q u i p o t e n t w i t h t h e continuum: see c h . 1
6.5.
(1) Let
a
b (14 b
be a r e g u l a r aleph; f o r e v e r y
< a)
5
4.3.
we have
ba = a
(TARSKI 1938; uses g e n e r a l i z e d continuum h y p o t h e s i s ; ZF s u f f i c e s f o r
a = W ;
a = cJ1 , ZF p l u s c h o i c e p l u s continuum h y p o t h e s i s ) . ( 2 ) L e t a be a l i m i t aleph; f o r e v e r y c, d c a we have prop. 9 ; g e n e r a l i z e d continuum h y p o t h e s i s i s used).
(Ibid.
for
0 (1) The s t a t e m e n t i s t r u e f o r
a = W . Suppose t h a t
a = ‘2
s o r aleph, hence o f t h e f o r m
i s an i n f i n i t e succes-
( g e n e r a l i z e d continuum h y p o t h e s i s ) , and b ba = b(C2) = ( b x c ) 2 = ‘2 = a s i n c e b x c
1 b $ c . Then we have by 3.8. Now suppose t h a t a i s a l i m i t a l e p h which i s s t i l l r e g u l a r and
satisfies = c
a
‘d
a .There e x i s t s an i n c r e a s i n g a-sequence o f successor
s t r i c t l y greater than ai
<
a (iruns t h r o u g h a)
ai7
b
, hence b(ai)
cardinals we have
r e g u l a r , no values i n i n each
ai
= ai
b-sequence i s c o f i n a l i n a
a = Sup ai
with
a
. Hence
the set o f a l l
i varies, o f the sets o f b ba = Sup( (ai)) = Sup ai = a 0
i s t h e union, as
.
. From a
c e r t a i n p o i n t on,
by t h e p r e v i o u s d i s c u s s i o n . Since
is
b-sequences w i t h values
.
Hence
a
b-sequences w i t h
( 2 ) L e t u be a successor a l e p h s a t i s f y i n g c < u < a and d < u < a u i s r e g u l a r , hence ‘d \< JI‘ = u by (1); hence ‘d < a . 0
0
. Then
Statement (1) does n o t h o l d f o r s i n g u l a r alephs. Let
a = OU and
b = U. Decompose
and a s s o c i a t e t o each W i t h e union o f the
cc, i+l
. Hence we
a
i n t o t h e u n i o n o f t h e W i (ii n t e g e r )
. By
i t s successor
K O N I G ‘ S theorem ( c h . 1
5
i s s t r i c t l y subpotent t o t h e Cartesian product o f the
Oi
have
<
w (
W
)
.
Even i n ZF (KONIG‘s theorem u s i n g t h e axiom of c h o i c e ) , we have t h e f o l l o w i n g counterexample. L e t
a.
= (d , and f o r each i n t e g e r
i l e t us d e f i n e :
1.8)
49
Chapter 2 = ( 2 t o t h e power ai) , and then a = Sup ai . Then we have a2 5 @ a , ai+l hence a G t a : t o get the f i r s t i n e q u a l i t y , associate t o each subset u o f
<
ui = a,.n
the sequence o f the i n t e r s e c t i o n s Let
6.6.
a
be a r e g u l a r l i m i t aleph; then t h e aleph rank o f
i n o t h e r words 0
a = W
( b < a)
a
.
wa = a
Assume t h e contrary, t h a t
so
u , and n o t i c e t h a t ui 6 ai+l
. Then
a
< ma , and
either
b
a
.
itself;
be t h e aleph rank o f
i s a successor o r d i n a l , so ( i < b) i s c o f i n a l i n a
b
wi
r t h e sequence o f the sor aleph. O
let
& a
a
a
,
i s a succes-
, hence
a
is
not r e g u l a r . 0 On t h e o t h e r hand, we can have t h e l i m i t aleph o f t h e sequence
ba= a
where
a(0) =
u ,a(1)
a
i s a s i n g u l a r c a r d i n a l . Take = Wu
,...,
i : t h e aleph obtained has c o f i n a l i t y w
f o r each i n t e g e r
a(i+l)
, hence
= W a(i) i s singular.
6.7. ACCESSIBLE CARDINAL, A X I O M OF ACCESSIBILITY
An i n f i n i t e cardinal a set with
, strictly
b
, which
a
subpotent w i t h
a
i s s a i d t o be accessible i f
subpotent w i t h
a
a = CJ
, or
i f there e x i s t s
and whose elements a r e s t r i c t l y subpotent
u b = a ; o r f i n a l l y i f there exists a set c s t r i c t l y (c) A cardinal i s said a , and where a i s subpotent w i t h
satisfies
.
t o be inaccessible otherwise. The axiom o f a c c e s s i b i l i t y asserts t h a t every s e t has accessible c a r d i n a l i t y .
ZF i s consistent, t h e theory
Under the assumption t h a t
ZF plus the axiom o f acces-
s i b i l i t y i s consistent: see SHEPHERDSON 1952. Every accessible l i m i t aleph d i f f e r e n t from W i s s i n g u l a r (uses generalized continuum hypothesis). Consequently i n view o f 6.2, this 0 Let
a
i s regular i f f
, hence of
f o r an accessible aleph
a # k~,
i s a successor aleph.
be our aleph, where
c3,
smaller c a r d i n a l i s an
+c
a
Ui ( i
4 i s a non-zero l i m i t o r d i n a l . Every s t r i c t l y
<
c a r d i n a l i t y lrJi
&)
. There
i s no s e t s t r i c t l y subpotent w i t h
, w i t h wd
subpotent w i t h
?(wi)
: indeed
(generalized continuum hypothesis) and i+l< d . F i n a l l y as ?(mi) = o u r aleph i s assumed t o be accessible and d i s t i n c t from .LI , t h e r e e x i s t s a s e t strictly LJ4
subpotent w i t h a d
, and
whose union i s
1J,
, whose
elements are as w e l l s t r i c t l y subpotent w i t h
: our aleph i s thus s i n g u l a r (see 6 . 3 ) . 0
An aleph i s s a i d t o be weakly i n a c c e s s i b l e i f i t i s d i f f e r e n t from 0 , r e g u l a r and a l i m i t aleph. The preceding p r o p o s i t i o n i s e q u i v a l e n t t o saying t h a t , w i t h t h e axiom o f choice and t h e generalized continuum hypothesis, every weakly inaccess i b l e aleph i s inaccessible. Using o n l y t h e axiom o f choice, every inaccessible aleph i s weakly inaccessible.
THEORY OF RELATIONS
50
Indeed, i t i s d i f f e r e n t from w , r e g u l a r , and t h e r e i s no s t r i c t l y smaller c f o r which i t i s subpotent w i t h T ( c ) : hence i t i s not a successor aleph. Problem. Assume t h a t ZF p l u s choice p l u s t h e existence o f a weakly i n a c c e s s i b l e aleph l e s s than the continuum i s consistent. Then does t h i s theory remain c o n s i s t e n t i f we r e q u i r e t h a t t h e continuum i t s e l f be weakly inaccessible. C l a s s i c a l l y we have an a f f i r m a t i v e answer i f we assume the consistency o f ZF p l u s choice p l u s t h e existence o f a weakly i n a c c e s s i b l e aleph g r e a t e r than t h e continuum. 6.8. Some unpublished r e s u l t s of BLASS, i n a l e t t e r t o HODGES, 1982.
Consider t h e f o l l o w i n g statements: (1) every countable union o f countable sets i s countable (see ch.1 5 2 . 5 ) ; ( 2 ) i f a s e t a and i t s elements have c a r d i n a l s cdl , then t h e union o f a has c a r d i n a l $ u 1 ; (3) given a f u n c t i o n f , if Rng f = d l , then U1 i s subpotent t o Dom f
.
Then i t i s proved t h a t n e i t h e r o f (1) and (2) i m p l i e s the other, and t h a t (2) i s equivalent t o (3) p l u s t h e r e g u l a r i t y o f w1 .
§
7 - COFINALITYOF
A PARTIAL ORDERING, COFINAL HEIGHT
The c o f i n a l i t y of a p a r t i a l o r d e r i n g A , denoted Cof A and introduced i n 5.4, was s t u d i e d above o n l y i n t h e case o f a t o t a l ordering. I f A i s n o n - t o t a l l y ordered, then Cof A can A by taking, and p u t t i n g them together Given a p a r t i a l o r d e r i n g
0 Construct
c o f i n a l subset Consequently
U
of
Cof(A/D)
D
be a s i n g u l a r aleph. f o r each i n t e g e r i , a c h a i n of order type wi mutually incomparably: then Cof A = W, 0 A and a c o f i n a l s e t D (mod A) , t h e r e e x i s t s a
.
with
= Cof
A
Card U = Cof A
,
.
f o r each c o f i n a l s e t
D
(uses axiom o f choice;
ZF s u f f i c i e n t i f A i s countable o r has well-orderable base). Take a s e t V c o f i n a l (mod A ) and w i t h l e a s t c a r d i n a l , hence Card V = Cof A , Then replace each element x o f V by an element i n D which i s g r e a t e r
0
than
x
.0
.
7.1. L e t A be a p a r t i a l o r d e r i n g and u = Cof A There e x i s t s a c o f i n a l subset U o f c a r d i n a l u , t h e r e s t r i c t i o n A/U being a well-founded p a r t i a l
.
o r d e r i n g w i t h h e i g h t Ht(A/U) 6 u A refinement o f c o r o l l a r y 5.1, due t o PDUZET 1979, unpublished. Uses t h e axiom o f choice; ZF s u f f i c e s i f A I s countable o r has we1 1-orderable base. 0
Let
D be a c o f i n a l subset w i t h l e a s t c a r d i n a l u
= Cof A
. Order
D by
Chapter 2
51
bi
(i< u)
ai
i t s c a r d i n a l i t y , which we assume t o be an aleph, and c a l l quence (axiom o f c h o i c e ) . E x t r a c t a sequence
t h i s se-
by removing, f o r each
i
, the
.
a such t h a t j > i and a . < a i (mod A) The sequence o f t h e bi has l e n g t h j J a t most equal t o u . By 5.1, t h e i n e q u a l i t y i < i ' i m p l i e s bi < o r I bin
.
(mod A)
Moreover, t h e s e t
U
o f values
b
i s cofinal i n
A
and
is a
A/U
well-founded p a r t i a l o r d e r i n g . For each s u b s c r i p t (mod A/U)
is
& i
i< u
, show
. First
that there exists a l e a s t
i with
bi
,
i
by i n d u c t i o n on
o f a l l , a l l the
bi
o f height
t h a t the height o f
2
are
>
bk
o f height
7.2. For e v e r y w e l l - f o u n d e d p a r t i a l o r d e r i n g A (uses axiom o f choice; ZF s u f f i c e s i f
A
, we
. Then
i (mod A/U)
i , w i t h b k < bi (mod A) ; and hence which c o n t r a d i c t s t h e m i n i m a l i t y o f i . F i n a l l y t h e h e i g h t o f exists a
have
k
bi
. Assume
o r \ b o (mod A)
A/U
Cof H t A
there
by 5.1,
is
,<
6 u .
0
Cof A
i s countable o r w i t h well-orderable
base).
D o f l e a s t c a r d i n a l i t y , so Card D = Cof A . The h e i g h t s D f o r m a s e t H , which i s c o f i n a l i n t h e o r d i n a l H t A . By t h e axiom o f c h o i c e , H i s subpotent w i t h D , hence we have Cof H t A ,< Card H 4 Card D = Cof A . 0
0
Take a c o f i n a l s e t
(mod A) o f t h e elements o f
If
A
i s isomorphic w i t h
A
On t h e o t h e r hand, t a k e t h e w e l l - f o u n d e d p a r t i a l o r d e r i n g
A
h
Cof A
i s a well-ordering,
then
Ht A
, hence
Cof H t A =
.
nent c3
,a
component
formed o f a compo-
w+1 , t h e elements o f one c h a i n p u t incomparable w i t h
those o f t h e o t h e r . Then
Cof A = W
,
H t A = W+1 so
Cof H t A
=
1
.
7.3. COFINAL HEIGHT Given a p a r t i a l o r d e r i n g
, then
of
A
of
A , and denoted
A
,
i f there e x i s t s a well-founded c o f i n a l r e s t r i c t i o n
t h e l e a s t h e i g h t o f such r e s t r i c t i o n s i s c a l l e d t h e c o f i n a l h e i g h t Cofh A
f o r every p a r t i a l ordering
.
With t h e axiom o f choice, t h e c o f i n a l h e i g h t e x i s t s
A : see c o r o l l a r y 5.1. Using o n l y t h e axioms o f ZF,
the c o f i n a l height e x i s t s a t l e a s t f o r a l l well-founded p a r t i a l orderings. (1) F o r e v e r y p a r t i a l o r d e r i n g
A ,.we have
Cofh A
6
.
Cof A
quence o f 7 . 1 and uses t h e axiom o f choice; ZF s u f f i c e s i f
A
T h i s i s a consei s countable o r
has w e l l - o r d e r a b l e base. S t r i c t i n e q u a l i t y can happen: f o r example when i t s base
E , t h e n Cofh A
= 1 and
A
Cof A = Card E
i s reduced t o t h e i d e n t i t y on
.
( 2 ) I,f A i s a t o t a l o r d e r i n g , t h e n Cofh A = Cof A . Consequence o f (1) and o f t h e f a c t t h a t t h e c a r d i n a l o f a w e l l - o r d e r e d r e s t r i c t i o n o f A i s a t most equal t o i t s h e i g h t ; same c o n d i t i o n s as i n (1).
52
(3) If
THEORY OF RELATIONS
A
i s a well-founded p a r t i a l ordering, then
Cofh A
4
Card H t A : another
form of 5.3. 7.4. Given a p a r t i a l ordering A , t h e c o f i n a l height i s t h e l e a s t height of wellfounded cofinal r e s t-r.i c t i o n s of A ,whose cardinal i s equal t o Cof A (uses axiom of choice; ZF s u f f i c e s i f A i s countable o r has well-orderable base). S t a r t w i t h a cofinal subset D such t h a t A/D i s a well-founded p a r t i a l ordering of l e a s t height H t ( A / D ) = Cofh A Take a subset D ' of D which is cofinal and of l e a s t cardinal Cof A (see beginning of present 5 7, using t h e axiom of c h o i c e ) . Then Ht(A/D') $ H t ( A / D ) = Cofh A by 3.5. 0
.
-
7.5. I f B i s a cofinal r e s t r i c t i o n of t h e p a r t i a l ordering A , then Cofh B i Cofh A (assuming t h a t these c o f i n a l h e i g h t s e x i s t ) . Indeed each cofinal r e s t r i c t i o n of B i s a cofinal r e s t r i c t i o n of A The following example, due t o POUZET 1979, shows t h a t s t r i c t i n e q u a l i t y i s p o s s i b l e , contrary t o t h e s i t u a t i o n f o r c o f i n a l i t y (beginning of present 5 7 ) . . For each ordered p a i r ( i , j ) 0 Take a s base t h e Cartesian product w x with i i n t e g e r , j countable o r d i n a l , p u t ( i , j ) 6 ( i ' , j ) f o r every i n t e g e r i For each even i n t e g e r i , p u t ( i , j ) s (i,j') for all j . For each odd i n t e g e r i , we l e t t h e ordered p a i r s w i t h f i r s t term i be mutually incomparable. Then we complete by t r a n s i t i v i t y . The p a r t i a l ordering A thus obtained has a cofinal s u b s e t of a l l ordered p a i r s w i t h odd f i r s t term: thus we have Cofh A = a.The r e s t r i c t i o n t o ordered p a i r s with even f i r s t term has cofinal height w1 . 0
.
i'a
.
j'a
For each p a r t i a l ordering A , t h e cofinal height of A i s a cardinal (uses axiom of choice; ZF s u f f i c e s i f A i s countable, o r has well-orderable base, o r i s well-founded; POUZET 1979, unpublished). 0 Take a well-founded cofinal r e s t r i c t i o n B of A w i t h l e a s t h e i g h t , so H t B = Cofh A : see c o r o l l a r y 5.1. By 7.5 we have Cofh A $ Cofh B . By 7 . 3 . ( 3 ) we have Cofh B 6 Card H t B = Card Cofh A . 0 7.6.
Let A be a well-founded p a r t i a l ordering: any of t h e possible comparisons " s t r i c t l y l e s s than", " s t r i c t l y g r e a t e r " , "equal" can be obtained f o r the cardin a l s Cofh A and Cof H t A . (1) S t a r t from t h e chain of i n t e g e r s . To each i n t e g e r i a s s o c i a t e an element i ' . P u t i ' 7 i ; p u t i ' incomparable with i n t e g e r s b i t 1 ; f i n a l l y f o r a l l i n t e g e r s i and j # i , p u t i ' incomparable with j ' . Denote by A t h i s p a r t i a l ordering: we have H t A = Cof H t A = c3 and Cofh A = 1
7.7.
.
(1') In t h e above example 7.5 we have
Cofh A = W and
Cof H t A
= W1
.
Chapter 2
( 2 ) For each ordinal q
, we
have
Cofh o(
=
53
Cof o( = Cof H t o(
.
( 3 ) Construct A by s t a r t i n g with the ordinal c3,; f o r each integer i , add by a bifurcation another chain w i a f t e r the i n i t i a l interval c d i . Then we have H t A = awso Cof H t A = W and Cofh A = w w . W e can also have Cof H t A 4 Cofh A where Cofh A i s a regular aleph. See below
in ch.7
§
5
3.10 where Cof A = Cofh A
8 - SET
= W1
and H t A =
OF SUBSETS, SET OF INJECTIVE
ON THE REALS,
INTERMEDIACY,
dl.w
FUNCTIONS,
CONSECUTIVITY,
CYCLE,
thus
INCREASING
Cof H t A = W . FUNCTIONS
CYCLIC RELATION
8.1. Let E be an i n f i n i t e s e t , I a s e t of subsets Ei ( i 6 I ) of E where I i s subpotent with E and each Ei i s equipotent with E . (1) There e x i s t two d i s j o i n t subsets C , D E such t h a t f o r each i & t intersections CnEi and DnEi are non-empty. ( 2 ) More strongly, there e x i s t d i s j o i n t C , D such t h a t CnEi and DnEi are equipotent with E (uses axiom of choice; ZF suffices when E i s countable; see BERNSTEIN 1908 or KURATOWSKI 1966 p. 514. 0 (1) Well-order E according t o i t s cardinal a , and I according t o i t s cardi nal b ( l e s s t h a n or equal t o a ) . P u t the f i r s t element xo of Eo into C , the second yo into 0 . I n general f o r each ordinal i < b , p u t the f i r s t xi of Ei which i s d i s t i n c t from a l l x j and y j ( j < i ) into C , and the second yi d i s t i n c t from a l l x and y into D . 0 j j 0 ( 2 ) Replace the s e t I by a sequence of the terms Ei with repetitions, as follows. If b = a so t h a t I i s equipotent with E , then take the f i r s t term Eo , then Eo followed by E l , and in general f o r each i , take the sequence of
of
.
E j ( 0 6 j ,c i ) I f b < a , then keep the above sequence of the Ei , b u t keep repeating so as t o obtain a sequence with length a = Card E In e i t h e r case, repeat the preceding proof: f o r each s e t Ei , the intersections CnEi and DnEi have cardinal a 0 Consequently, i f E i s denumerable and 3 i s a non-trivial u l t r a f i l t e r on E , then there i s no countable s e t of subsets Ei f E which generates g ,in the sense t h a t the elements of are obtained by taking supersets of f i n i t e intersections of the E i . 0 Denote by Ei the Ei from the statement and also t h e i r f i n i t e intersections. Then the s e t s C a n d D , which are d i s j o i n t , would b o t h belong t o the ultraf i l t e r : contradiction. 0 More generally, l e t E be i n f i n i t e and 3 be an u l t r a f i l t e r on E whose Then elements are equipotent with E cannot be generated by a s e t , equipotent with E , of elements of g .
.
.
.
THEORY OF RELATIONS
54
8.2. (1) Let E be an i n f i n i t e s e t , I a s e t of injective functions f i ( i € I ) from E i n t o E , where I i s subpotent w i t h E , and each f i s a t i s f i e s f i ( x ) # x on a subset of E equipotent w i t h E Then there -e x i s t s a subset C of - E , equipotent w i t h E , w i t h the non-inclusion f i o ( C ) q C f o r each i (notation f " from ch.1 5 1 . 2 ) . ( 2 ) Under the same hypotheses, _______ there e x i s t two subsets C , D Df E which are d i s j o i n t and equipotent w i t h E , w i t h the-_?on-inclusion f i o ( C ) F C W D and similarly by interchanging C and D (uses axiom of choice; SIERPINSKI 1950; see also ROSENSTEIN 1982). 0 (1) Well-order E by i t s c a r d i n a l i t y a ; well-order I f i r s t by i t s cardinalit y , then w i t h repetitions according t o the ordinal a , in the case where Card I < Card E . P u t the f i r s t xo of E s a t i s f y i n g f o ( x O ) # xo into C , and p u t
.
xb = f o ( x O ) into E - C . I n general, f o r each i < a , l e t xi be the f i r s t e l e ment of E such t h a t f i ( x i ) # xi , and where xi and x(i = f i ( x i ) are d i s t i n c t from a l l x and x! ( j < i ) . Such an x i e x i s t s , because the number of j < i j J i s s t r i c t l y l e s s t h a n the cardinal a , and there are ( c a r d i n a l i t y a)-many x which are d i f f e r e n t from f i ( x ) , as f i i s i n j e c t i v e . Finally p u t xi i n t o C and x i into E - C . 0 ( 2 ) The preceding proof modified as follows. F i r s t l y , p u t the f i r s t xo # fo(xO) i n t o C and xb = fo(xO) i n t o E - ( C u 0 ) . Secondly, p u t the f i r s t element
0
yo # f o ( y o ) , where yo and
fo(yo)ar%othd i s t i n c t from xo
and
xb , into D ,
and yb = fo(yo) i n t o E - ( C u D) . I n general, f o r each i < a , f i r s t l y find the f i r s t element xi # f i ( x i ) where both x i and x; = f i ( x i ) a r e d i s t i n c t from ( j< i )
x j , x!, y . , y j
all
J
J
. Then
put
xi
into
C
Secondly consider the f i r s t yi # f i ( y i ) with yi xj,
xi,
yj, y j ( j < i )
y! = f . ( y . ) 1
1
1
into
E
-
as well as from x i , x i ( C u D)
.
and and
xi
E
into
-
( C uD)
.
f i ( y i ) d i s t i n c t from a l l
Then p u t yi
into
D
, and
put
.0
8.3. The reader i s familiar w i t h the notion o f increasing o r decreasing ( s t r i c t l y or otherwise) function on the r e a l s . Let E be the s e t of r e a l s and R t h e i r t o t a l ordering, F a subset of E and Ft the smallest interval including F ( w i t h endpoints Inf F and Sup F modulo R ) For each increasing (mod R ) function f from F into E , there e x i s t s a t l e a s t one increasing function f + which i s an extension o f f t o the domain Ft If F i s dense f o r the t o t a l ordering R/F+ , then the extension f' of f is unique. Moreover, i f f i s s t r i c t l y increasing and F *for R/F' , then f' i s s t r i c t l y increasing and hence i s an isomorphism from R/F+ i n t o R . 0 For each element u of F+-F , l e t f t ( u ) be the infimum of the f ( x ) f o r a l l x in F such t h a t x > u . 0
.
.
-
Chapter 2
55
8.4. Let E be t h e s e t of r e a l s , R t h e i r t o t a l ordering, F a non-empty subset of E The s e t of increasing (mod R ) functions from F into E i s equipotent with the continuum. 0 Assume f i r s t of a l l t h a t F i s an i n t e r v a l . An increasing function from F i n t o E has countably many points of d i s c o n t i n u i t y , s i n c e t o t h e s e points correspond non-singleton mutually d i s j o i n t i n t e r v a l s ( s e e ch.1 5 4 . 5 ) . Hence an increasing function i s defined by i t s values on the r a t i o n a l s f o r example, plus i t s values on the points of d i s c o n t i n u i t y : a l t o g e t h e r making a countable sequence of real values. I n the case of an a r b i t r a r y F , l e t Ft be t h e s m a l l e s t i n t e r v a l including F . To each increasing function f from F i n t o E , a s s o c i a t e the function f + from F+ i n t o E which i s increasing and an extension of f , obtained by t h e previous proposition. The set of f f i s equipotent w i t h the continuum, hence so i s the s e t of r e s t r i c t i o n s f of f t t o t h e domain F 0
.
.
8.5. Let R be the t o t a l ordering of t h e r e a l s , and F a s e t of r e a l s . I f f o r each automorphism h of R which i s d i f f e r e n t from t h e i d e n t i t y , t h e r e exists an element x f F with h ( x ) # x , then F is dense f o r R
.
Let a < b be r e a l s . There e x i s t s a t l e a s t one automorphism, o r s t r i c t l y increasing function f from R onto R , s a t i s f y i n g f ( x ) = x f o r a l l x 6 a and a l l x ) b , and f ( x ) # x f o r a l l x ( a 4 x G b ) . Hence F must have an e l e ment between a and b . 0
0
8.6. INTERMEDIACY, CYCLIC RELATION, CONSECUTIVITY, BINARY CYCLE We f i n i s h t h i s chapter by introducing several r e l a t i o n s frequently associated with p a r t i a l and t o t a l o r d e r i n g s , and used i n l a t e r c h a p t e r s .
Given a p a r t i a l ordering A , we c a l l intermediacy associated with A , the ternary r e l a t i o n “z i s between x and y (mod A ) ” ; in o t h e r words x & z 6 y o r y x and t h e r e does not e x i s t t s a t i s f y i n g x < t < y . Given a t o t a l ordering A , t h e binary c y c l e , o r cycle o f consecutivity associated with A , i s t h e r e l a t i o n of c o n s e c u t i v i t y w i t h the following possible modification: i f t h e r e e x i s t s a minimum u and a maximum v of A , then we p u t C ( v , u ) = + .
THEORY OF RELATIONS
56
EXERCISE 1 - THE GENERALIZED CONTINUUM HYPOTHESIS IMPLIES THE AXIOM OF CHOICE (SIERPINSKI 1947; see also COHEN 1966) 1 - Let A , B be two s e t s . If A u B i s equipotent with the s e t ? ( 2 x A ) , then T(A) i s subpotent with B . 0 Let f be a bijection from AuB onto y(2 x A ) = T(A) x ? ( A ) . Each element x of A i s transformed into the ordered pair f ( x ) = (y,z ) of elements of T(A) . To each x associate the f i r s t term y of t h i s ordered pair. By CANTOR'S lemma ch.1 5 1.5, there ex i s t s an element u of T(A ) f o r which none of the pairs (u,z) i s the image of any x of A Hence there e xists a subset of B which i s bijectively transformed by f into the s e t ? ( A ) of the second terms z of the ordered pairs (u,z) . 0 2 - For each s e t A , l e t To(A)= A ; Tl(A) = T(A) , the s e t of subsets of A ; f o r each integer i , l e t Ti+l(A)= ( 9i(A)) If A has a denumerable subset, then Pi(A)= 2 x T i ( A ) f o r each positive integer i 0 By hypothesis A i s equipotent with A augmented by an element. Hence we have
.
.
.
P(A)= A2 equipotent with 2 x A2 . I t follows tha t ? ( A ) i s equipotent with i t s e l f augmented by an element: then we i t e r a t e . 0 3 - Let A be an i n f i n i t e s e t , assumed t o be equipotent with 2 x A . If f o r i = 0,1,2,3, the sets T i ( A ) and (j)i+l(A) do not have any s e t which i s s t r i c t ly intermediate with respect t o subpotence, then there e xists a well-ordering of the s e t A . 0 The s e t A i s equipotent with a proper subset of i t s e l f , hence has a denumerable subset: ch.1 5 2 . 6 . Hence Ti(A) i s equipotent with 2 x Ti(A) for each integer i : see ( 2 ) above. Consider the well-orderings based on subsets of A , and the isomorphism cl.asses of these well-orderings. From t h i s point on, denote Ti(A) by Pi . The pairs of elements of A belong t o P1 The ordered pairs belong t o P2 : see ch.1 5 1.2. The well-orderings based on subsets of A belong t o P3 Finally the isomorphism classes belong t o P4 The s e t H of isomorphism classes i s equipotent with the Hartogs aleph o f A : see ch.1 5 6 . 2 ; and i t i s included i n P4 . The union HuP3 is intermediate, under subpotence, between P3 and P3 u P4 , the l a t t e r equipotent with P4 , since P3 i s equiBy hypothesis, H u P3 potent with a subset of P4 as well as with 2 x P3 i s equipotent e i t h e r with P4 or with P3 I n the f i r s t case, i t i s equipote n t with ( 2 x P3) , and by the previous (1) P4 i s subpotent w i t h H , hence well-orderable, so A i s well-orderable as well. Then H and P2 are Consider the second case: H v P3 i s equipotent with P3 both subpotent with P3 , hence H u P2 subpotent with 2 x P3 and hence with P3 By hypothesis H u P 2 i s equipotent ei t h er t o PJ or t o P2 I n the f i r s t case, by the previous ( l ) , the s e t P3 i s subpotent with H ,
.
.
.
.
.
9
.
.
.
Chapter 2
hence well-orderable,
so t h a t
i s subpotent w i t h
. By
hence
H
I J
A
P2
A
51
i s well-orderable.
I n t h e second case,
i t e r a t i n g , we o b t a i n t h a t
.
H u P1 P1 ,
i s equipotent w i t h A o r w i t h P1 The f i r s t case i s excluded, The previous (1) shows A i s n o t subpotent w i t h A
.
since the Hartogs aleph o f that
i s subpotent w i t h
H
i s subpotent w i t h
, hence
well-orderable, and consequently A
itself i s well-orderable. 0 4 - L e t A be an i n f i n i t e s e t . Take the union o f A w i t h JL. and l e t B = F ( A u w ) . Then by ( 2 ) , we have 2 x B equipotent w i t h B . Using the gener a l i z e d continuum hypothesis, the statement (3) shows t h a t B i s well-orderable, P1
so A u W and hence EXERCISE 2
-
A
H
as w e l l .
THE CARDINAL OF THE SET OF FILTERS
be an i n f i n i t e s e t w i t h c a r d i n a l a . We s h a l l prove t h a t the cardinal o f the s e t o f f i l t e r s , o r as w e l l t h e c a r d i n a l o f t h e s e t o f u l t r a f i l t e r s on E Let
E
equals
2
t o t h e power
(a2)
(TARSKI; see BELL, SLOMSON 1969 p . 108; uses the
axiom o f choice). Since each f i l t e r i s a s e t of subsets o f
E
, the
s e t o f f i l t e r s has a t most the
above s t a t e d c a r d i n a l i t y . Hence i t s u f f i c e s t o c o n s t r u c t a s e t o f u l t r a f i l t e r s
E having t h i s c a r d i n a l . For each X we denote by (X) the s e t o f f i n i t e X . By the axiom o f choice, F ( X ) is equipotent w i t h X f o r each i n f i n i t e s e t X . I n the f o l l o w i n g , we o b t a i n u l t r a f i l t e r s on F ( ( E ) ) which i s equipotent w i t h E 1 Divide E i n t o two complementary subsets A, B o f t h e same c a r d i n a l a , and l e t f be a b i j e c t i v e mapping from A onto B To each subset X o f A associate X+ = the union o f X and (B-fo(X)) (notation, O from ch.1 5 1.2).
on
subsets o f
.
-
.
X, Y
Show t h a t f o r two d i s t i n c t subsets
X+, Y+ i s included i n the other. 2 To the s e t E associate t h e s e t
-
E'
, neither
A
of
= F(E)
-
o f the two images
o f a l l f i n i t e subsets o f
E
.
.
A associate X ' = E ' F(X+) Note t h a t the t r a n s f o r mation from X i n t o X ' i s i n j e c t i v e , hence t h e c a r d i n a l o f t h e s e t o f X ' i s a2 Moreover, f o r X, Y d i s t i n c t subsets o f A , we have X ' , Y ' n e i t h e r included i n t h e other. To each subset
X
of
.
3
-
Let
X
be a subset o f
A
Y1,
and
...,Yn .
( n integer)
be a f i n i t e s e t o f
Y o f A , a l l d i s t i n c t from X Then t h e i n t e r s e c t i o n o._ fI-It h e images " Y' = E' -3(Y') o f t h e Y i s n o t included i n the image X ' f X
subsets
Indeed t o each index
i = 1, ...,n g(i)
The f i n i t e s e t o f t h e Hence i t does n o t belong t o
Y; = E '
- F(Yf)
associate an element
i s included i n X ' = E'
-
3(X+)
X+
g(i)
of
X+
. -
y e t n o t included i n any y e t does belong t o each
, hence belongs t o t h e i n t e r s e c t i o n o f t h e
Y'
.
(YtnX')
Yt .
.
THEORY OF RELATIONS
58
t h a t E " has c a r d i n a l a . Denote by X ' o f subsets X o f A . Hence has c a r d i n a l a2 i s the set o f a l l F ( E ' - F ( X + ) ) L e t f denote t h e s e t o f a l l F ( X ' ) Hence where X i s an a r b i t r a r y subset o f A . Thus i s a l s o o f c a r d i n a l a2 . Moreover, each element o f i s a subset o f E" L e t H, K be two f i n i t e non-empty d i s j o i n t subsets o f . Then t h e i n t e r s e c t i o n 4
-
Let
E" = F ( E ' ) =
T(3 ( E ) ) , so
t h e s e t o f p r e c e d i n g images
.
.
K i s not included i n the union
A
Indeed
i s a set o f
H
F(X;)
U H
(i= 1,
.
...,m)
and
T(Y!) J
i s a set of
K
...,n )
w i t h m, n i n t e g e r s . Each X; and Y ! i s t h e image o f a subset J Y . o f A , a l l d i s t i n c t . F i x an i n d e x i 6 m : by ( 3 ) above, t h e i n t e r s e c J t i o n o f t h e Y ! ( j = 1, ..., n ) i s n o t i n c l u d e d i n X; . Hence t h e r e e x i s t s an J element h ( i ) which belongs t o t h i s i n t e r s e c t i o n and n o t t o X i . The s e t of h ( i ) (i= 1, ...,m) i s a f i n i t e subset o f each Y ' , hence an element o f each j ( Y j ) , hence an element o f t h e i n t e r s e c t i o n n K . However t h i s element i s n o t i n c l u d e d i n any o f t h e X; , hence belongs t o no F ( X ; ) , and so i s n o t an e l e ( j = 1,
or
Xi
uH
ment o f t h e u n i o n
. i s non-
By t h e p r e v i o u s statement, each f i n i t e i n t e r s e c t i o n o f elements of
-
empty. I n p a r t i c u l a r , two complementary subsets o f
5 - Let
, where
E"-S
. Associate
S
e.
U+ =, U p l u s a l l complements belongs t o t h e d i f f e r e n c e s e t 't'- U F o r two d i s t i n c t U ,
be a subset o f
U
cannot b o t h belong t o
E"
to it
.
V , t h e r e e x i s t s f o r example an element S o f which belongs t o U , hence t o U+ , and whose complement E"-S belongs t o V+ . I t f o l l o w s t h a t U+ and V+ a r e d i s t i n c t . F o r o t h e r w i s e , two complementary elements would belong t o U+ , these elements b e i n g o b t a i n e d f r o m two complementary elements o f f . say
U
and
Thus t h e s e t o f a l l power
(a2)
U+
, as
the set o f a l l
Beginning w i t h an a r b i t r a r y non-empty s u b s e t p o s i t i v e f i n i t e number o f elements o f
S1,. .. ,Sm
of
U and t h e elements
the difference s e t
if - U .
E"-T
cardinality
U+
E"-T1,.
of
U
, by
$ , c o n s i d e r an a r b i t r a r y
. . ,E"-Tn
T
, hence
where t h e s e
U
T
belong t o
the intersection o f the E"
S S and
extending
U+
.
correspond two d i s t i n c t u l t r a f i l t e r s , because o f t h e e x i s -
tence of complementary elements ( w i t h r e s p e c t t o
E" ) b e l o n g i n g t o t h e c o r r e s -
.
, j u s t as t h e s e t o f
ponding s e t s U+ Thus t h e s e t of u l t r a f i l t e r s on E" subsets o f f , has c a r d i n a l i t y 2 t o t h e power (a2)
-
t o the
d i s t i n g u i s h i n g t h e elements
i s non-empty. Thus t h e r e e x i s t s an u l t r a f i l t e r on
TO two d i s t i n c t
6
2
By t h e p r e v i o u s ( 4 ) , t h e i n t e r s e c t i o n o f t h e
i s n o t included i n t h e union o f the the
, has
U
.
.
Modulo 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 , we have a much s i m p l e r p r o o f
of t h e preceding, y i e l d i n g , f o r a s e t f i l t e r s on
E has c a r d i n a l i t y
2
E o f cardinal
t o t h e power
(a2)
a
.
, t h a t the set o f u l t r a -
.
Chapter 2
59
To see t h i s , consider a l l p a r t i t i o n s of E into two d i s j o i n t a , and t o t a l l y order t h i s s e t of p a r t i t i o n s by i t s cardinal Let uo be the f i r s t p a r t i t i o n , whose two associated subsets Eo(+) (associated t o the 1-sequence t ) and E o ( - ) . For each
subsets of cardinal b = a2 . shall be denoted of these subsets, Eo ( + ) f o r example, take the f i r s t p a r t i t i o n ul(+) such t h a t the intersection of Eo ( + ) with each of the subsets of E given by u l ( + ) has cardinal a Denote by El(++) and E l ( + - ) these two intersections. Do the same thing f o r the I n general, f o r each ordinal i s t r i c t l y l e s s t h a n 2-sequences (-+) and (--) b , note t h a t Card i 6 a by the generalized continuum hypothesis. We have t o Fix i and f i x an a r b i t r a r y i-sequence x with values (+) and ( - ) define the p a r t i t i o n u i ( x ) . Assume t h a t we have obtained, f o r each j < i , the sequence of the E . ( x . ) where xj+l i s the i n i t i a l interval of x with J J+1 length j+l By 8.1, the s e t of the E . ( x ) generates a f i l t e r which i s n o t J . j+l an u l t r a f i l t e r . So there e x i s t s a p a r t i t i o n u i ( x ) yielding f o r every intersection of f i n i t e l y many of the E . ( x . ) a s e t of cardinal a J J+1 Finally, for each b-sequence of values (+) and ( - ) , we obtain a f i l t e r such t h a t two d i s t i n c t b-sequences give two d i s t i n c t f i l t e r s . Hence there are b2 many such f i l t e r s .
.
.
.
.
.
EXERCISE 3 - A CLASSICAL PROOF OF HAUSDORFF-ZORN WITHOUT ORDINALS With the notions of partial and t o t a l ordering, b u t without using ordinals, we outline as follows the classical proof, using the axiom of choice, of the existence of a maximal chain: HAUSDORFF-ZORN axiom. Let A be a partial ordering, C a t o t a l l y ordered r e s t r i c t i o n of A We denote by X any chain which i s simultaneously a r e s t r i c t i o n of A and an extension of C Suppose t h a t no X i s maximal under inclusion, and l e t f be a choice function which associates t o each X a s t r i c t extension f(X) of X , again a r e s t r i c t i o n of A and an extension of C . We shall obtain a contradiction as follows. Denote by & every s e t of chains X satisfying the following conditions: (1) ‘6 i s t o t a l l y ordered under r e s t r i c t i o n (or extension); admits (2) i s closed with respect t o taking the union of the bases; hence a maximum element, which i s the common extension of a l l the chains in t o the union of t h e i r bases; (3) f o r each X in 6 other than the maximum chain, f(X) belongs t o and no s t r i c t intermediate chain between X and f(X) belongs t o There e x i s t such : f o r instance the singleton of C and the pair {C,f(C)} . Given two such E , say and E ’ , we shall prove t h a t one i s an i n i t i a l interval of the other ( f o r the ordering of extension defined between the chains X ) .
.
.
e
6
.
8,
THEORY OF RELATIONS
60 Let
1
be t h e s e t o f
&
elements o f
X
belonging t o the i n t e r s e c t i o n
which a r e r e s t r i c t i o n s o f
restrictions o f
, are
X
t h e same.
3
X
E
fi
E'
and t h e elements o f
, f o r which t h e E' which a r e
i s non-empty s i n c e t h e element
C
belongs
t o it. Define the chain
U
3 .
as t h e common e x t e n s i o n o f a l l elements o f
By ( 2 ) , t h e
U belongs t o t h e i n t e r s e c t i o n $ , A I ' . Moreover, e v e r y element X o f E which i s a r e s t r i c t i o n o f U , e i t h e r i s i d e n t i c a l t o U , o r i s a r e s t r i c t i o n o f a c h a i n b e l o n g i n g t o 9 : i n t h i s case X belongs t o 3 . Thus U belongs chain
to 9 . Either U
i s t h e maximum o f
f',
i s t h e maximum o f chain
f(U)
, so
&'
that
&
?,
U
and
i s an i n i t i a l segment o f
i s an i n i t i a l segment o f
n
belongs t o t h e i n t e r s e c t i o n
mediate c h a i n between longs t o
E
so t h a t
f(U)
belongs t o
which c o n t r a d i c t s t h e m a x i m a l i t y
We s h a l l prove t h a t t h e r e e x i s t s a maximum
.
& , and
( w i t h r e s p e c t t o i n c l u s i o n ) ; we
&' V
ment t h e c h a i n
V
f(V) denote t h e u n i o n o f a l l t h e
&
,
is t h a t no t o which we add as l a s t e l e -
.
e k i s indeed an
I n o t h e r words, t h a t
t i s f i e s ( l ) , ( Z ) , ( 3 ) . C o n d i t i o n (1) i s obvious, as w e l l as ( 2 ) s i n c e as l e a s t element. To o b t a i n ( 3 ) : i f
X
l y intermediate
f(X)
Y
between
X
and
c o n t a i n i n g as elements
f(X)
and
belong t o
belongs t o
X, f ( X )
and
one i s an i n i t i a l segment o f t h e o t h e r ) . Then t h i s contradiction.
E.
which i s t h e common e x t e n s i o n o f a l l t h e elements o f a l l t h e
It i s s u f f i c i e n t t o prove t h a t
l e a s t one
o f t h e maxi-
. So suppose
then adding the s t r i c t extension
maximum and l e t
U
or
E' . Moreover no s t r i c t l y i n t e r € , n o r t o & ' ; t h u s f ( U ) beo f U . Hence o u r c l a i m i s proved.
s h a l l t h e n o b t a i n a c o n t r a d i c t i o n by c o n s i d e r i n g t h e maximum c h a i n mum
%';
O r by ( 3 ) , t h e
Y
€*.
& * , and
&'
E"
sa-
has
V
if a strict-
then there e x i s t s a t
( s i n c e , f o r any two
E
,
i t s e l f does n o t s a t i s f y ( 3 ) :
61
CHAPTER
3
RAMSEY THEOREMS, PART I T I ONS , COMB1NATORIAL P R I NC IPLES
5 1 - RAMSEY'S
THEOREM
1.1. We s t a t e f i r s t t h e i n f i n i t a r y f o r m o f t h i s theorem (1926). P a r t i t i o n t h e
k classes ( k
(unordered) p a i r s o f n a t u r a l numbers i n t o
c a l l c o l o r s . Then t h e r e e x i s t s an i n f i n i t e s e t pairs included i n
, which
set E 0
m-element s e t s , o r s e t s w i t h f i n i t e c a r d i n a l
a r e assumed t o be p a r t i t i o n e d i n t o
c o l o r s . There e x i s t s an i n f i n i t e
k
of i n t e g e r s such t h a t a l l m-element subsets o f
The s e t
which we
have t h e same c o l o r .
E
T h i s g e n e r a l i z e s t o t h e case o f m
finite)
o f i n t e g e r s such t h a t
E
E have t h e same c o l o r .
i s c a l l e d monochromatic.
E
Case o f p a i r s . P a r t i t i o n t h e non-zero i n t e g e r s
t o t h e c o l o r of t h e p a i r { O,x{
0 = 0 and l e t
0 ul,
UO
t i t i o n the integers . 1 t h e p a i r {ul,x}
0 u2,
..
x
k
into
classes according
: a t l e a s t one o f t h e s e c l a s s e s i s i n f i n i t e . L e t
1 0 u1 = u1 and p a r -
be t h e elements o f t h i s c l a s s . L e t (i2 2)
x = up
k
into
classes according t o the c o l o r o f
1 u3, 1 u2,
: a t l e a s t one o f t h e s e c l a s s e s i s i n f i n i t e . L e t
t h e elements o f t h i s c l a s s . L e t
A t t h e end we o b t a i n t h e i n f i n i t e s e t w i t h elements
1 vo =.O, v1 = ul,
( i i n t e g e r ) , s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n . For e v e r y i n t e g e r \vi,vi+l\ integer
..
, {vi,vi+?J, i
. Then
-
--
vi
have t h e same c o l o r . 0
0,1,2,
...,m-2 , and
p a r t i t i o n the integers
x
uo 0 = 0, u11 = 1,
... , Um-2 m-2
l e a s t one o f these c l a s s e s i s i n f i n i t e . L e t
ulIf, u I - ~..,
t i t i o n the integers x
element
and y m- 1
x =
be t h e elements o f t h i s c l a s s . L e t t~:-~
(i>/ m)
m-1
. into
: at = m-2
,
and p a r -
=
i n t o a f i n i t e number o f classes, by p u t -
i n t o t h e same c l a s s i f f f o r each (m-1)-element
set
, t h e s e t I augmented w i t h x and I augmented w i t h
two m-element s e t s w i t h t h e same c o l o r (which depends on f i n i t e l y many
m-1
2
,..., m-2,x)
ting
u?1
have t h e same c o l o r , which we say i s a s s o c i a t e $ t o t h e
c l a s s e s , a c c o r d i n g t o t h e c o l o r o f t h e m-element s e t { O , l
and l e t
=
, the pairs
i
Case o f m-element s e t s (m 2 2 ) . Assume t h a t t h e statement i s t r u e f o r
Put a s i d e t h e i n t e g e r s k
... , v 1.
a t l e a s t one c o l o r i s a s s o c i a t e d w i t h i n f i n i t e l y many i n t e g e r s
i : a l l p a i r s subsets o f t h e s e t o f 0
.. be
2 1 u 2 = u 2 and i t e r a t e .
I , one o f t h e s e c l a s s e s i s i n f i n i t e : l e t
I
with last y
yield
I ) . As t h e r e a r e o n l y ul-',
ul;:,
..
be t h e
THEORY OF RELATIONS
62
u;
elements o f t h i s c l a s s . L e t
and i t e r a t e t h i s w i t h t h e (m-1)-element
.
ui
s e t s w i t h l a s t element
= ui-'
vo = uo 0 = 0, v1 = u11 = 1, .. i vi = ui f o r each i n t e g e r i , s a t i s f y i n g t h e f o l -
A t t h e end we o b t a i n t h e i n f i n i t e s e t w i t h elements
~ = - m-l, ~1
v
..
and i n general
l o w i n g c o n d i t i o n . For each (in-1)-element l a s t element i s
,
vi
set
I formed o f elements
t h e m-element s e t s o b t a i n e d by a d d i n g a
v
and whose
v j ( j > i )t o
I
,
a l l have t h e same c o l o r , which we c a l l t h e c o l o r a s s o c i a t e d w i t h t h e (m-1)-element
I
set
.
By t h e i n d u c t i o n hypothesis, a p p l y t h e theorem f o r
i n f i n i t e s e t o f elements
,all
v
m - 1 : we o b t a i n an
o f whose (in-1)-element subsets have t h e same as-
s o c i a t e d c o l o r . Hence an i n f i n i t e s e t , a l l o f whose m-element subsets have t h e same c o l o r . 0 1.2. Given a r e l a t i o n
A
,a
sequence whose values a r e elements o f t h e base
s h a l l be c a l l e d a sequence i n
.
A
If
A
and d e c r e a s i n g sequences were i n t r o d u c e d i n ch.2 introduced i n ch.1
u
Let
5
5
2.4;
e x t r a c t e d sequence was
2.2.
be an a - s e q u e n c e i n a p a r t i a l o r d e r i n g
ce e x t r a c t e d f r o m
IAI
i s a p a r t i a l ordering, then increasing
u
, which
A . Then t h e r e e x i s t s an w-sequen-
i s constant, o r s t r i c t l y . i n c r e a s i n q ,
o r s t r i c t l y de-
c r e a s i n g , o r c o n s i s t i n g o f elements which a r e p a i r w i s e incomparable (mod A ) . 0 P a r t i t i o n the set o f pairs
{i,j) o f integers i n t o f o u r classes ( p u t
f i x t h i n g s f o r d i s c u s s i o n ) , by d e f i n i n g a f i r s t c l a s s by t h e e q u a l i t y a second by
I
ui
< uj
,a
ui > u . (mod A) J u j (mod A) ; t h e n a p p l y RAMSEY's theorem. 0 ui
(mod A )
t h i r d by
, and
i< j ui = u
to
j '
a f o u r t h by
1.3. FINITARY FORM OF RAMSEY'S THEOREM; MONOCHROMATIC SET m, k, p 5 m
Given t h r e e i n t e g e r s
f o r every s e t o f cardinal k
>/
t
, there
, whose
e x i s t s an i n t e g e r p+>/ p such t h a t , m-element subsets a r e p a r t i t i o n e d i n t o
p p-element subset, a l l of whose
colors, there e x i s t s a
t h e same c o l o r . I t i s c a l l e d a monochromatic Consider t h e case
m
classes i s i n f i n i t e " , a t least
l/k
=
2
.
m-element subsets have
p-element subset.
Repeat t h e p r o o f o f 1.1, b u t i n s t e a d o f "one of t h e
say "one o f t h e c l a s s e s i s l a r g e " , meaning t h a t i t c o n t a i n s
o f t h e o r i g i n a l elements. 'It s u f f i c e s t o t a k e
p+ = (kp).k(kp-l)
= p.kkp i n o r d e r t o o b t a i n , a f t e r k p - 1 o p e r a t i o n s , a sequence o f l e n g t h >/ kp o f elements v , analogous t o t h o s e i n 1.1. Thus we have a l a r g e c l a s s o f v , o f
cardinality
>/
p
.0
63
Chapter 3
1.4. RAMSEY NUMBERS p+
The l e a s t such
i n t h e p r e c e d i n g p r o p o s i t i o n i s c a l l e d a Ramsey number, denoted
(p): . T h i s l o o k s i k e t h e usual Erdos-Rado n o t a t i o n , where t h e arrow w i l l be r e placed by = o r < o r > , e t c . We g i v e s e v e r a l v a l u e s .
-
Case m = 1 have
.
I f each o f t h e
k.(p-1)
a t l e a s t one c l a s s w i t h principle" : i f
k
c l a s s e s had
k(p-1)
p
+
p (p);
k
pigeonholes, t h e n
= p
.
p = m : a p-element s e t i s necessary monochromatic, t h u s
.
(3); = 6 Consider t h e elements 1,2, (b,c)
or
+ 1 t o obtain
= k(p-1)
objects.
...,6
; p a r t i t i o n t h e edges
two c o l o r s . A t l e a s t one c o n t a i n s t h r e e edges or
(p):
1 objects are partitioned i n t o
k = 1 : a s i n g l e class, thus
Calculation o f 0
elements, t h e e n t i r e s e t would
elements. T h i s argument i s c a l l e d t h e "pigeonhole
a t l e a s t one o f t h e pigeonholes has
Case Case
p-1
elements. Hence i t s u f f i c e s t o t a k e
(l,a),
(1,2)
(l,b),
.
= m
(m):
to
(1,6)
.
(1,c)
into
Either
(a,b)
(c,a)
has t h e same c o l o r , o r t h e s e t h r e e edges have t h e o p p o s i t e 2 c o l o r : t h i s shows t h a t ( 3 ) 2 i 6 . 2 To see t h a t ( 3 ) 2 > 5 , t a k e t h e usual pentagon w i t h one c o l o r , and t h e s t a r r e d pentagon w i t h t h e o p p o s i t e c o l o r . 0 Calculation o f
0
(3);
= 17
Consider t h e elements
(GLEASON, GREENWOOD 1955). 1,2,.
..,17
and p a r t i t i o n t h e 16 edges
(1,2)
to
(1,17)
(l,al), ... , ( l , a 6 ) . I t remains t o p a r t i t i o n t h e edges (ai,a.) ( i , j = 1 t o 6) i n t o two c o l o r s : hence we J 2 2 f a l l back t o t h e case ( 3 ) 2 = 6 ; t h i s shows t h a t ( 3 ) 3 6 17 . i n t o t h r e e c o l o r s . A t l e a s t one c o n t a i n s 6 edges, say
The f o l l o w i n g counterexample shows t h a t
1+1 = 0
o f t h e i n t e g e r s 0 and 1 w i t h
2 (3)3
>
16
. Consider t h e
t h e r i n g o f p o l y n o m i a l s on t h i s f i e l d w i t h t h e i d e n t i t y (0 o r 1)
composed o f 16 elements
These elements a r e e x a c t l y element i s a power
xi
+
(0 o r 1).x
x4 = x + l
.
, and
This r i n g i s
+ ( 0 o r 1).x 2 + (0 o r l ) . x 3
.
2 ,. . .,x14 (we have x15 = 1). Every non-zero
O,l,x,x
(i= O , l ,
f i e l d composed
( t h e f i e l d o f t h e i n t e g e r s modulo 2)
..., 14) , and
has i n v e r s e
x15-i
. Hence
this ring
i s a f i e l d . P a r t i t i o n the p a i r s o f polynomials i n t o three colors, according t o whether t h e d i f f e r e n c e o f these two p o l y n o m i a l s i s a cube i s o f the form
x3'+'
or
x3'+'
.
x3" ( u = O , l ,
...,4)
or
It s u f f i c e s t o see t h a t t h e sum o f two non-zero
cubes i s n o t a cube. 0 1.5. L e t
E be a f i n i t e s e t ; p a r t i t i o n i t s m-element subsets i n t o k c o l o r s i n t e g e r s p1 ,..., pk 3 m , by (pl ,..., pk)m we denote
u1 ,..., uk . Given k the l e a s t cardinal o f
E
f o r which t h e r e e x i s t s e i t h e r a pl-element
subset
THEORY OF RELATIONS
64 with color tion
(p
t h e ramsey number Calculation o f 0
.
u l , ... , o r a pk-element subset w i t h c o l o r uk
,,...,pk)m
i s symmetric. Moreover, t a k i n g (p)!
(p,.
=
(3,412 = 9
. . ,p) m .
1
=
..*
The f u n c -
pk = p, we o b t a i n
=
. .
9
We show t h a t t h i s number i s
E i t h e r among t h e 8 edges
p
(1,2)
to
J o i n up t h e i n t e g e r s 1 t h r o u g h 9 by edges. (1,9)
t h e r e e x i s t 4 edges o f c o l o r ( + ) . T h i s
t h e n y i e l d s e i t h e r a 3-element s e t w i t h c o l o r (+) o r a 4-element s e t w i t h c o l o r ( - ) , O r t h e r e e x i s t 6 edges w i t h c o l o r ( - ) , which t h e n y i e l d s e i t h e r a 3-element subset
(+) o r a 4-element subset ( - ) . Or f i n a l l y none o f t h e p r e c e d i n g cases i s r e a l i z e d f o r any o f t h e p o i n t s
1 through
9
. Then
f r o m each p o i n t t h e r e emanate e x a c t l y
3 edges (+) and 5 edges ( - ) . B u t t h i s i s i m p o s s i b l e , s i n c e we would t h e n have 3.(9/2) = 27/2 edges ( + ) . 2
>
2 C a l c u l a t i o n o f ( 4 ) 2 = 18
.
We now show t h a t t o t h e edge
.
8 Take t h e i n t e g e r s 0 t o 7, and g i v e t h e c o l o r (+) i f f t h e a b s o l u t e v a l u e o f y - x i s 3, 4, or 5. 0
(3,4)
(x,y)
Take t h e i n t e g e r s 1 t o 18. Among t h e edges emanating f r o m 1, t h e r e a r e a t l e a s t
0
9 o f t h e same c o l o r which we d e s i g n a t e ( + ) . They j o i n 1 t o t h e i n t e g e r s d e s i g n a t e d
...,
al,. . . , ag . By t h e preceding, i n t h e s e t o f al, ag t h e r e e x i s t s e i t h e r a 3-element s e t w i t h c o l o r ( + ) , o r a 4-element s e t o f t h e o p p o s i t e c o l o r ( - ) . Hence t h i s Ramsey number i s a t most 18.
\<
The f o l l o w i n g ewample w i l l prove t h a t t h e Ramsey number i s n o t
17. Take t h e
i n t e g e r s modulo 17, so 0 t o 16. For any two d i s t i n c t x, y i n t h i s s e t , we g i v e t o t h e edge (x,y) t h e c o l o r (+) i f f x i s congruent t o y modulo a q u a d r a t i c r e s i d u e , so mod
1, +, 2,
f 4,
o r 2 8 ; t h e c o l o r ( - ) i n t h e o p p o s i t e cases.
Suppose t h a t t h e r e e x i s t 4 i n t e g e r s
a,b,c,d
same c o l o r . We can r e p l a c e these i n t e g e r s by
such t h a t a l l t h e i r edges have t h e and 0 , and t h u s
a-d, b-d, c-d
.
can c o n s i d e r o n l y t h e case o f O,a,b,c We can r e q u i r e t h a t t h e 6 i n t e g e r s a, b, c, b-a, c-b, a-c be non-zero and e i t h e r a l l r e s i d u e s o r a l l non-residues. M u l t i p l y i n g by t h e i n v e r s e o f the 5 integers
a
, we
can reduce t h i s t o t h e case o f
b, c, b-1, c-1, c-b
r e s i d u e s . Then t h e p o s s i b i l i t i e s f o r b = -1 c
-
, we
b # -7
we o b t a i n
have
c
b
and
c
# -1 , c # 2 s i n c e c-b # 3
(3,5)L = 14
-1, +2, -8 c # -8 s i n c e
a r e reduced t o
. Moreover
. The same argument f o r c = -1 : i m p o s s i b l e . F o r b-c = -7 : i m p o s s i b l e . 0
Calculation o f
0, 1, b, c
with
which a r e a l l non-zero and a l l q u a d r a t i c
b = 2
and
. For
c = -8
.
Take a s e t w i t h 14 p o i n t s and l e t a be i n t h i s s e t . E i t h e r f r o m a t h e r e eman a t e a t l e a s t 5 edges w i t h c o l o r (+), which y i . e l d s a 3-element s e t monochromatic
0
Chapter 3
65
w i t h c o l o r ( + ) , o r a 5-element monochromatic s e t w i t h c o l o r ( - ) . O r f r o m
a
t h e r e emanate a t most 4 edges (+), hence a t l e a s t 9 edges ( - ) . Then s i n c e (3,4) 2 = 9 , t h i s y i e l d s e i t h e r a (+)-monochromatic 3-element s e t , o r a (-)-monochromat i c 5-element s e t . Thus
(3,5)2
i s bounded above by 14.
To see t h a t i t equals 14, t a k e t h e 13 i n t e g e r s 0 t h r o u g h 12. Give t h e c o l o r (t) t o the p a i r
{x,y)
(where
the absolute value o f
x
and y
a r e d i s t i n c t elements among O , l ,
..., 12)
iff
equals 2, 3, 10 o r 11; c o l o r ( - ) i n o t h e r cases. 0
y-x
Other known ( o r almost known) v a l u e s o f b i n a r y Ramsey numbers: (3,6)2 = 18 : KALBFLEISCH 1964; i n d e p e n d e n t l y KERY 1964; n
(3,7)L = 23 : GRAVER, YACKEL 1968; (3,8)
2
= 28 o r 29 ; a l s o
(3,9)
2
= 36 : GRINSTEAD, ROBERTS 1982.
1.6. Below we l i s t some i n e q u a l i t i e s f o r t h e s m a l l e s t Ramsey numbers whose e x a c t value i s n o t known. B i n a r y numbers.
,<
25
(4,5)[
4
28 : l o w e r bound o f 25 by KALBFLEISCH 1964; upper
bound o f 28 by WALKER 1971. 42
6 (5); ,< 55
: l o w e r bound o f 42 by I R V I N G 1973; a l s o GARCIA 1975; a l s o HANSON
1976; upper bound o f 55 by WALKER 1971. 2 128 4 ( 4 ) 3 d 254 : l o w e r bound o f 128 by HILL, I R V I N G 1982; upper bound o f 254 e a s i l y o b t a i n e d v i a (2,4,4) 2 = (4,4) 2 = 1 8 and (2,3,4) 2 = (3,4) 2 = 9 ; t h e n (3,3,4)26 51
6
(3);
34
and
6 65
(3,4,4)2
6
85
.
: l o w e r bound o f 51 by CHUNG 1973; upper bound of 65 by FOLKMAN
1974, a l r e a d y announced by WHITEHEAD 1973. 6 ( 3 ) 52 322 : l o w e r bound 159 by FREDRIKSON 1979; t h e upper bound i s easy.
<
159
S t i l l c o n s i d e r i n g b i n a r y Ramsey numbers, we have t h e easy i n e q u a l i t y : 2 (3)k
1) + 2 , which y i e l d s
2 (3)k
4
k ! e + 1 ( c l a s s i c a l number e ) .
I n t h e o t h e r d i r e c t i o n , we e a s i l y o b t a i n
2 (3)k
3
(3
5
2 k((3)k-1
-
on Schur’s numbers. Improved by 2 (3)4
>/
Moreover ratio:
2 (3)k+13
2 3.(3)k
+
k
+ 3)/2
2 (3)k-2
-
: see e x e r c i s e 1
3
( f o r example
2 3.(3)3 = 3.17 = 51 ); see CHUNG 1973. 2 ( ~ + 1 i)s~bounded above, up t o a c o n s t a n t f a c t o r , by t h e f o l l o w i n g (2p)!(Log Log p ) / (p!)[Log p
Ternary number.
13
4
(4);
bound o f 15 by GIRAUD 1969.
6
: see YACKEL 1972.
15 : l o w e r bound o f 13 by ISBELL 1969; upper
66
THEORY OF RELATIONS
2 - LEXICOGRAPHICALLYORDERED THEOREM, NASH-W ILLI AMS ' THEOREM §
SET,
GALVIN'S
INITIAL
INTERVAL
2 . 1 . LEXICOGRAPHICALLY ORDERED SET, LEXICOGRAPHIC RANK Totally order the s e t of f i n i t e s e t s of integers lexicographically, by f i r s t difference: ( s e t a ) ,< ( s e t b ) i f f the l e a s t integer in a i s s t r i c t l y l e s s than the l e a s t integer in b ; or in the case of equality, compare the second l e a s t integer of a with the second l e a s t integer of b , e t c . The empty s e t i s defined
t o precede a l l other s e t s in t h i s ordering. Finally i f a i s a proper i n i t i a l interval o f b , we p u t a < b . of f i n i t e s e t s of integers i s said t o be lexicographically well-ordered, A set i f the lexicographic ordering of elements o f i s a well-ordering. The corresponding order type i s called the lexicographic rank of F . I t i s a countable ordinal. For example, the s e t of singletons of integers i s lexicographically well-ordered with rank w, the s e t of pairs with rank U 2. The s e t o f a l l f i n i t e s e t s of integers i s n o t lexicographically well-ordered. Indeed we obtain a lexicographically decreasing W-sequence, by taking the singleton i l } , then the p a i r { 0 , 2 ) , , etc. then {0,1,3} , then {0,1,2,4} be a s e t of f i n i t e s e t s of integers, such t h a t : Let are mutually incomparable under inclusion; (1) the elements of ( 2 ) every i n f i n i t e s e t of integers includes a subset which belongs t o 3 ; i s lexicographically well-ordered (communicated by POUZET in 1980). then t o be lexicographically well-ordeNotice t h a t ( 2 ) alone i s not s u f f i c i e n t f o r red: take the s e t of a l l the f i n i t e s e t s of integers; (1) alone i s not s u f f i c i e n t : take the above decreasing W-sequence. Consider a non-empty subset of and show t h a t there e x i s t s a minimum f o r the lexicographic ordering. Let a. be the l e a s t integer such element in t h a t there e x i s t s an element of beginning with a. I f the singleton ao) belongs t o , then i t i s the minimum of Otherwise, take the elements of which begin with a. , and l e t al be the l e a s t integer such t h a t there e x i s t s an element of beginning with ao, a l . If the pair {ao,al} belongs t o , then i t i s the minimum of Otherwise, i f t h i s procedure never terminates, then we obtain an i n f i n i t e increasing sequence a. < al < . . < ai < . . ( i inte-
0
5
5 3
5
5
9
F,
5
5.
.
.
ger). By our hypothesis ( 2 ) , there e x i s t s a f i n i t e : denote by ah the the ai and belonging t o , ah also a f i n i t e set beginning with ao, a l , to B u t t h i s contradicts our hypothesis ( I ) of
F.
4
.
...
.
s e t composed of certain of l a s t among these. There e x i s t s and belonging t o , hence incomparability. 0
5
67
Chapter 3
2 . 2 . INITIAL INTERVAL THEOREM Let p be a s e t of f i n i t e s e t s of integers, such t ha t every i n f i n i t e s e t of Then there e xists an integers includes as a subset a t l e a s t one element of in f in i t e s e t E of integers such t h at every i n f i n i t e subset of E has an elea s a n i n i t i a l interval (GALVIN 1968; the following proof, using only -.ment of the axioms of ZF, i s due t o POUZET 1980, unpublished). Note f i r s t t h a t RAMSEY's theorem eas i l y follows from the preceding statement. Indeed, p a r t i t i o n the pairs of integers into two colors (+) and ( - ) . Then either there e x i s t s an i n f i n i t e s e t of integers a l l of whose pairs belong t o (+), or every i n f i n i t e s e t includes an element of color ( - ) . Then by the above statement, there e x i st s an i n f i n i t e s e t E such t ha t every i n f i n i t e subset of E begins with a pair belonging t o ( - ) : in other words every pair belongs t o ( - ) . To simplify the proof, note t h at i t i s always possible t o assume tha t the elements of are mutually incomparable with respect t o inclusion. Indeed, starting with an a r b i t r ar y we obtain the subset T o by taking those which are minimal with respect t o inclusion. Every i n f i n i t e s e t elements of The i n i t i a l interval theorem, of integers includes a t l e a s t one element of F' when r e s t r i c t e d by the preceding condition, says t ha t there e xists an infinite set E , such t h a t every i n f i n i t e subset of E has an element of 3", hence of F , as an i n i t i a l interval. By the preceding 2 . 1 , we see t h a t i t suffices t o prove the i n i t i a l interval theorem f o r an arbitrary lexicographically well-ordered s e t (whose elements will no longer necessarily be incomparable under inclusion). Thus we are led t o prove the following statement.
F.
3,
.
2.3. Let be a lexicographically well-ordered s e t of f i n i t e subsets of integers could contain the empty s e t as an element); then: (1) e i t h e r there e x i s t s an i n f i n i t e s e t of integers which includes no element
(3'
of F-; ( 2 ) or there e x i s t s an i n f i n i t e s e t of integers, each of whose i n f i n i t e subset has an element of 3- as i n i t i a l interval (the empty s e t i s considered as an i n i t i a l interval of every s e t ) . 0 W e argue by induction on the lexicographic rank of F . Suppose f i r s t tha t the rank i s equal t o 1 , so t h at 3 i s the singleton of a f i n i t e s e t F of integers. Then the i n f i n i t e s e t of a l l integers not belonging t o F s a t i s f i e s our conclusion (1) i f F i s non-empty, our conclusion ( 2 ) i f F i s empty. Let o( be a countable ordinal. Suppose the statement i s true f o r every s e t of lexicographic rank < o(. We shall prove i t f o r every s e t of rank n( .
THEORY OF RELATIONS
68
More strongly, in order to avoid use of the axiom of choice, or even a weakened form of choice, suppose t h a t there exists a function h which, to each ordered p a i r (E,F) where E i s an infinite set of integers, If a set of f i n i t e subsets of E of lexicographic ranks < I% , associates an infinite set h ( E , F ) E satisfying one of the conclusions (1) or ( 2 ) . More precisely, either h ( E , y ) includes as a subset no element of , or every infinite subset of h ( E , begins with an initial interval which belongs t o W e will prove that there has exists an analogous function for the ordered pairs whose second term lexicographic rank d . Thus h will be progressively extended t o all countable lexicographic ranks, so t o a l l ordered pairs (E,F) S t a r t with an infinite s e t E of integers and a set of f i n i t e subsets of E with lexicographic rank O C . For each integer i of E , denote by Fi the i s the union of subset of those elements of which begin w i t h i Hence of 3;i s the sum along w of the the and the lexicographic ordering o( lexicographic orderings of the Fi . If there exist infinitely many integers i of E with empty, then the set of these i does not include as a subset any element of F . This will be, by definition, h ( E , F ) which then verifies our conclusion (1). Consider the other case, and l e t m(0) be the least integer of E after which Fi i s never empty. Denote by Mo the set of integers of E which are >,m(O) and by the set Mo with i t s minimum m(0) removed. The Ti all have lexicographic ranks t d , hence the function h i s already defined for these. and then = the set of the elements of , each of W e put $ = whose m i n i m u m m(0) has been removed: the lexicographic r a n k of this is s t i l l < e( P u t El = h(Mg, 2 Mi . Either there exist infinitely many integers i of El for which El includes no element of Ti . Then the set o f these i i s by definition h ( E , g ) , which satisfies our conclusion (1). Or i n the opposite case, l e t m(1) be the least integer of El , from which point on every Ti restricted t o i t s elements which are subsets of El i s never empty. Denote by M1 the set of integers of El which are >/ m(1) , and Mi the set M1 with i t s minimum m(1) removed. Let = res-
s
5
F)
F.
.
.
Fi
Fi
M i
Fm(o)
.
go
50 5 0)
.
5i
5;
zl Fm(l)
tricted to elements which are subsets of M1 Then = the set of the elemnts of each w i t h i t s minimum removed: the lexicographic rank of
5
.
$1
and of i s s t r i c t l y less than o( Let E2 = h(M;, 5 Mi , and iterate this procedure. Then either, a t the end of a f i n i t e number of steps, we obtain an Er ( r integer) with infinitely many integers i of E r f o r which no element of Fi i s a subset of E r . Then by t o be the set of these i which satisfies (1). definition, we p u t h ( E , $ ) Or this described process continues indefinitely: we must consider two subcases.
5 i)
Chapter 3
69
First subcase. There e x i s t i n f i n i t e l y many integers k f o r which the s e t Take the i n f i n i t e Ek+l = h(Mi , contains as a subset no element of corresponding s e t of minimums m ( k ) . Then m ( k ' ) t Z Ek+l Mk f o r a l l k , k ' > k . , hence no So the s e t of minimums m(k) contains as a subset no element of element of F m ( k ) f o r any k Finally our s e t of m ( k ) contains as a subset no element of 8 : we take i t as our definition of h ( E , 7 ), which s a t i s f i e s ( 1 ) .
9
9 k)
.
%K
.
In the second subcase, because of the definition of the function h , there e x i s t s an integer ko from which point on, every i n f i n i t e subset o f Ek+l begins by a
% .
A f o r t i o r i , i f K denotes the s e t of minimums possibly empty element of m ( k ) f o r k >/ kg , then every i n f i n i t e subset of K begins by an element of a
k , hence by an element of s a t i s f i e s our conclusion ( 2 ) . 0
F.
Thus the s e t
K
, which
we take f o r
h(E,
3
)
2.4. NASH-WILLIAMS' THEOREM (1965)
5k
Consider two d i s j o i n t s e t s of f i n i t e s e t s o f integers. Suppose t h a t no , and vice-versa. Then element of F i s an i n i t i a l interval o f an element of there e x i s t s an i n f i n i t e s e t E of integers, which con ains a s a subset no element . NASH-WILLIAMS only assumes of F , orwhich contains as a s u b T t no element of , one i s never an i n i t i a l interval t h a t , f o r two d i s t i n c t elements of
9
p l _ l _ _ _
9 5 of the other. The present stronger statement r e s u l t s from a remark by HODGES.
-
F w
Motice t h a t RAMSEY's theorem follows. Indeed, given two d i s t i n c t pairs of integers,
or in general, f o r p a fixed integer, given two d i s t i n c t p-element s e t s of integers, one i s never an i n i t i a l interval of the other. 0 Either there e x i s t s an i n f i n i t e s e t o f integers having
no subset which i s an
element of 3 . Or every i n f i n i t e s e t of integers has a sirbset which i s an element of F . I n the l a t t e r case, by GALVIN's theorem 2 . 2 , there e x i s t s an i n f i n i t e s e t E of integers, such t h a t every i n f i n i t e subset of E has an element of T as i n i t i a l i n t e r v a l . Then E has no subset which i s an element of . Indeed, i f i t , then take an i n f i n i t e subset X of E contained as a subset an element G of with i n i t i a l interval G There e x i s t s an element F of which i s an i n i t i a l interval of X : thus F i s an i n i t i a l interval of G , or G an i n i t i a l interval of F : contradiction. 0
.
5 3 - UNCOUNTABLE E R D ~ S , RADO
CASE,
9
5
PARTITION
THEOREMS:
DUSHNIK,
MILLER,
3.1. SIERPINSKI 'S COUNTEREXAMPLE (1933) There e x i s t s a partition of the pairs of reals into two colors, such t h a t every monochromatic s e t i s countable (uses axiom of choice).
,
70
THEORY OF RELATIONS
0 Take a w e l l - o r d e r i n g
with
o f t h e s e t o f r e a l s . Then t o each p a i r o f r e a l s
A
i n t h e usual o r d e r i n g , g i v e t h e c o l o r (+) i f
x c y
x < y (mod A)
x,
y
and
t h e c o l o r ( - ) i f x > y (mod A) , The p r o p o s i t i o n f o l l o w s f r o m t h e f a c t t h a t e v e r y s t r i c t l y i n c r e a s i n g ( o r s t r i c t l y decreasing) sequence o f r e a l s i s c o u n t a b l e (see ch.1
5
4.5). 0
As a p a r t i c u l a r case of 3.4 below, we mention h e r e t h a t , f o r e v e r y p a r t i t i o n o f t h e p a i r s o f elements of a s e t o f c a r d i n a l
i n t o two c o l o r s , t h e r e i s a
a 1(uses
monochromatic subset o f c a r d i n a l
axiom o f c h o i c e p l u s t h e continuum
hypothesis). We can summarize t h i s s i t u a t i o n by u s i n g t h e n o t a t i o n f o r Ramsey numbers w i t h f i n i t e o r i n f i n i t e c a r d i n a l values. Then t h e usual Ramsey theorem i s w r i t t e n
( w ); = W f o r a l l i n t e g e r s m, k 2 w i t h 3.4 y i e l d s ( w 1 ) 2 = Ld2 . 3.2.
.
The p r e c e d i n g p r o p o s i t i o n complemented
PARTITION LEMMA (DUSHNIK, MILLER 1941)
L e t A = C d N be an i n f i n i t e r e g u l a r aleph. P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s which we d e s i g n a t e by (+) and ( - ) . Then e i t h e r , f o r e v e r y subset B which i s e q u i p o t e n t w i t h A , t h e r e e x i s t s an element
a
of
B
and a s e t o f elements
equipotent w i t h
A
.
of
B
A
>, ,a
al
in
B
a.
A ,
.
equipotent w i t h
f i r s t c o n c l u s i o n . Consider t h e elements o f the ordinals. Let
which i s e q u i p o t e n t w i t h
O r t h e r e e x i s t s a subset o f - A
, a l l o f whose p a i r s have c o l o r (+)
Assume t h e r e e x i s t s a subset
0
f B
x
have c o l o r (-)
where a l l t h e p a i r s {a,x)
B
A
, which
negates t h e
o r d e r e d by t h e usual o r d e r i n g on
B ; by h y p o t h e s i s t h e r e e x i s t s an
be t h e minimum o f
1
, such t h a t f o r e v e r y x >, al t h e p a i r { aO,x has c o l o r (+). i C w* , assume t h a t f o r t h e j < i we have a s t r i c t l y
By i n d u c t i o n , g i v e n
i n c r e a s i n g sequence o f elements
<
a
j
of
B
,
such t h a t a l l p a i r s o f t h e
a.
J
have
i , t h e r e a r e l e s s than wd many x i n B such t h a t the p a i r {aj,x) has c o l o r ( - ) . Since u4 i s r e g u l a r , t h e s e t o f a l l such x f o r a l l j < i has c a r d i n a l i t y < ud . Thus t h e r e e x i s t s an ai of B , which i s s t r i c t l y above a l l t h e a , and such t h a t f o r a l l x >, ai j and a l l j c i , t h e p a i r ( a j , x ) has t h e c o l o r ( + ) . F i n a l l y we o b t a i n an
t h e c o l o r (+). For e v e r y
j
C30(-sequence o f elements, a l l o f whose.pairs have c o l o r ( + ) . 0 Notice t h a t the proposition i s f a l s e f o r every singular cardinal be t h e c o f i n a l i t y of . For e v e r y i < l e t T < W, Ai
of
A = W4
, such
s t r i c t l y subpotent w i t h
A
r~~
.
Indeed
, t a k e a subset
t h a t t h e u n i o n o f t h e Ai i s A , b u t e v e r y Ai is . F o r e v e r y p a i r c o n t a i n e d i n an Ai , g i v e t h e c o l o r
( - ) , and f o r p a i r s o f elements b e l o n g i n g t o d i s t i n c t
Ai
's, the color (+).
71
Chapter 3
3.3. PARTITION THEOREM (DUSHNIK, MILLER 1941) Let A be an a r b i t r ar y i n f i n i t e s e t ; partition the pairs of elements of two colors (+) and ( - ) . Then ei t h er there ex i s t s a denumerable subset of
A into A A which i s equipotent
which i s (-)-monochromatic, or there ex i s t s a subset of w i t h A fi (+)-monochromatic (uses axiom o f choice). DUSHNIK and MILLER mention the influence of ERDOS. A different proof of the theorem i s given by ERDOS, RADO 1956. Using Ramsey numbers notation and replacing A by an aleph w, , we have ( ~ 3 w , W ) '= L&, Replace A by an aleph which we designate by &>,;,(axiom of choice), and assume f i r s t t h a t t h i s aleph i s regular. By the preceding lemna, i f our second conclusion i s f a l s e , then there ex i s t s an a. in A for which the s e t A. of x 7 a. (mod A ) such t h at {ao,x] has color ( - ) i s equipotent with A . Take t h i s a. minimum (mod A) Then replace A by A. , t h u s yielding an element al of A. satisfying the same condition and taken minimum. By i t e r a t i o n , we obtain an w-sequence o f elements ai ( i integer) , a l l of whose pairs of elements have color ( - ) . Assume now t h a t A = ud i s singular. Then c4 i s a lim it ordinal (ch.2 9 6.2, axiom of choice). Let UUc and o(. be the co f i n al i t y of un , SO Thus M, i s the ordinal l i mi t of the r-sequence w , where i < 2( m(i) and g ( i ) < o( Moreover, we can choose the LX ( i ) t o be s t r i c t l y increasing with i , and every o \ ( i ) > 8 Finally every w g ( i ) can be assumed t o be regular, replacing i f need be d ( i ) by i t s successor. Suppose t h a t the f i r s t conclusion f a i l s : there i s no denumerable subset o f A Then by the preceding, there e x i s t s a subset a l l of whose pairs have color ( - ) B of A , equipotent with A , such t h at f o r every x in B , there are many y in B with { x,y) having color ( - ) s t r i c t l y l e ss t h a n For every subset X of B , denote by M ( X ) the s e t of elements of B-X which, together w i t h a t l e a s t one element of X , have color ( - ) . Let U be any subset of B equipotent with A , and l e t i be an ordinal s t r i c t l y less than 8 . We shall show t h at there e x i s t s a subset W of U with cardinal IAM ( i ) ' satisfying the two following properties: every pair of elements of W has color (+) ; the s e t M ( W ) has cardinal s t r i c t l y l ess t h a n ma. Indeed, by our f i r s t paragraph and because f o r every i < 8 the cardinal L3 %(i) i s regular, there e x i s t s a subset V of U with cardinal w N c (i) , a l l of whose pairs have color (+) . For every j < d , denote by V the s e t of j elements x of V , such t h a t there e x i s t a t most woc(j) elements in B - i x ] which together with x have color ( - ) . Then V = \I V . ( j < b') since, J by our third paragraph, no x together with wd many elements of B , has color ( - ) , and i s the limit of the (j C I f ) Recall t h a t the
.
.
r<
.
x+
.
.
Q,
.
.
THEORY OF RELATIONS
72
.
c a r d i n a l wec(i) o f V i s r e g u l a r and s t r i c t l y g r e a t e r than It f o l l o w s t h a t there e x i s t s a t l e a s t one o r d i n a l k 6 8 w i t h Vk equipotent w i t h V : p u t
<
. Then Card M(W) ,< W,c(i). L3 o ( ( k ) do(. Thus the two p r o p e r t i e s s t a t e d above f o r W are obtained. It remains t o c o n s t r u c t a subset o f B which i s equipotent w i t h B and thus w i t h A , a l l o f whose p a i r s have c o l o r (+) . L e t W1 be a subset o f B w i t h cardinal w , a l l o f whose p a i r s have c o l o r (+) , w i t h Card M(W1) s t r i c t l y o((1) l e s s than ~3~ I t e r a t e by t a k i n g W2 a subset o f B (W1 u M(W1)) w i t h card i n a l G)o((2) , a l l o f whose p a i r s have c o l o r (+) , w i t h Card M(W2) s t r i c t l y
W = Vk
.
-
.
l e s s than ucANote t h a t , i n t h e union W1u W2 , a l l t h e p a i r s have c o l o r (+). L e t i < 8 and assume t h a t t h e Wi are defined f o r j < i Then the union
.
For (Wj u M(W.)) ( f o r a l l j < i ) has c a r d i n a l s t r i c t l y l e s s than a d . J otherwise t h e i-sequence o f c a r d i n a l s Max( W ,Card M(W.)) would y i e l d a d(j) J and Card M(Wj) < c 3 ~ f o r each j < i sum >, Wd , w i t h Cdo((j) J
.
<
Hence t h e c o f i n a l i t y o f wOc would be 6 i < 8 , c o n t r a d i c t i n g t h e f a c t t h a t F o r U take the difference B u ( W . u M(Wj) f o r 8 i s the c o f i n a l i t y o f J a l l j < i) : t h i s has c a r d i n a l i t y ad o f t h i s d i f f e r e n c e which s a t i s f i e s the two proHence t h e r e e x i s t s a subset Wi w i t h a l l p a i r s o f Wi having c o l o r (+) , and p e r t i e s : Card Wi = CJ M(i) Card M(Wi) I t remains t o note t h a t the union o f t h e s t r i c t l y l e s s than L,J~ 0 Wi ( i < 8 ) has c a r d i n a l W, and t h a t a l l i t s p a i r s have c o l o r ( t ) Example. For u 1 we o b t a i n e i t h e r a denumerable (-)-monochromatic subset, o r With t h e n o t a t i o n o f Ramsey a (+)-monochromatic subset having c a r d i n a l W l numbers: ( a, =
.
-
.
.
.
.
a1
-
3.4. PARTITION LEMMA (ERDOS 1942) Let
oa be
an i n f i n i t e aleph. Set
A
=
WM+2
and assume t h e generalized
.
i ) 2 = c3 i+l f o r every i.4< d Then e i t h e r P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s (+) and (-) 3 , o~r t h e+r e ~ there e x i s t s a (+)-monochromatic subset o f A w i t h c a r d i n a l ~ See E R D b 1942 o r e x i s t s a (-)-monochromatic subset w i t h c a r d i n a l w 0(+2 2 ERDOS, RADO 1956. With Ramsey numbers: ( w o(+l, %+2) = c30(+2 . 0 We prove f i r s t t h a t t h e r e e x i s t s a monochromatic subset o f A w i t h c a r d i n a l o(+l Take ‘4 = 0 , so t h a t A = W 2 : we s h a l l o b t a i n a monochromatic s e t w i t h c a r d i n a l CC, The p r o o f w i l l e a s i l y extend t o t h e general case. We say t h a t a sequence of terms ai i n A (io r d i n a l ) i s pre-monochromatic i f f for every index i , t h e c o l o r o f t h e p a i r s {ai,aj) remains t h e same f o r a l l j > i : we say t h a t t h e c o l o r i s associated t o t h e index i Construct as f o l l o w s continuum hypothesis i n t h e form
(
.
.
.
.
.
Chapter 3
73
a pre-monochromatic u l - s e q u e n c e . W e then immediately extract a monochromaGJ l-sequence, by taking a l l those indices i with the same color, provided t h i s s e t i s cofinal in CJ . a. denote the minimum of A ( i n the usual well-ordering of A = "2 ; thus a. = 0 , the value i s unimportant). Partition the elements x # a. of A into two classes: the class of those x such that { a o , x j has color (+), and the class 1 similarly defined with ( - ) . Let a; be the minimum in the class into two subclasses: the 8 , and then partition the elements x # a; of such tic that Let
df
i
1
ri
t'
class
i-
of those x for which { a l , x ) has color (+), and the class similarly defined with ( - ) . Similarly, l e t a; be the minimum of the class , and then p a r t i t i o n the elements of d i s t i n c t from t h i s minimum, into
8 ;+ and
two subclasses I n general, l e t
u
xi
r i-
, defined as previously. be a n ordinal s t r i c t l y less than w
.
If
u
has a prede-
d
cessor u-1 , assume t h a t the classes :-1 are already defined, each characterized by a sequence s of + and - , with length u-1 , hence defined on the indithe minimum of , provided t h i s ces s t r i c t l y l e s s t h a n u . Denote by a:-1 class i s non-empty. Partition the elements in t h i s class which are d i s t i n c t from the minimum (assuming of course t h a t there are such), into two subclasses: the
r:-l
class
$
:+,
characterized by the sequence
u , of
hence a sequence of length
gous definition f o r the class
1(
s completed by the
x f o r which {a:-l,x)
:-.
( ~ - 1 ) term '~ + ,
has color ( + ) . Analo-
Suppose now t h a t u i s a l i m i t ordinal. Given a sequence s of length u , hence of indices s t r i c t l y l e s s t h a n u , consider f o r each i < u the r e s t r i c t e d sequence s / i taking the same values as s , b u t defined only f o r indices < i Then we define the class as the intersection of the classes 8 :Ii f o r a l l
.
r:
ordinals
i
<
u
. Finally,
whether
u
i s a l i m i t ordinal o r n o t , we define the
element a: as the minimum of f: , provided t h a t t h i s c l a s s i s non-empty. Since the ordinals considered a r e a t most countable, there are continuum many sequences s , hence u 1 many, since we assume the continuum hypothesis. Hence there are cjl many classes 4 a n d t h e i r minimums a , f o r a l l indices u and a l l sequences s . Since A has cardinality &J2 , there e x i s t other elements besides the minimums a
.
Let r be one such. Beginning with a. , pick e i t h e r a; o r a; , depending on whether the pair \ a o , r ) has color (+) or ( - ) . I f we have chosen a: , then choose either u
<W
a?
or
, if
a;by the same consideration. Continue thusly: given an ordinal we have already chosen the sequence s with length u-1 and values
THEORY OF RELATIONS
14 (+) and ( - ) , choose
a:'
, depending on whether the pair {a:-l,r}
or a:-
has
the color (+) o r ( - ) . For every l i m i t ordinal u , take the sequence s with length u which i s the l i m i t of the sequences already obtained, and take the corresponding a s . U By the definition of the classes , whenever we reach a term a: in the sequence defined by r , a l l following terms belong t o the same class 8 , i.e. t h a t which contains r . Hence f o r every v u a l l pairs { a u , a v } have the same color as { a " , r ) . Moreover, the class y through which we pass i s never empty, since r belongs t o i t . W e thus obtain a pre-monochromatic W1-sequence from which we e x t r a c t , as already explained, a monochromatic wl-sequence. 0
>
Let us now take u p the proof of our stated lemma. Assume t h a t the second conclusion does not hold, so t h a t every (-)-monochromatic subset of A has cardinality a t most W 1 . Take a subset Do of A , which i s maximal with respect t o inclusion among the . For every e l e (-)-monochromatic subsets; hence Do has cardinality ,< W ment x of A-Do , there e x i s t s a t l e a s t one element y of Do such t h a t the pair ( x , y ) has color ( t ) . Associate t o each x of A-Do the minimum such y . Thus we p a r t i t i o n the elements of A-Do into classes, each defined by an element a. of Do . W e denote by 'd (a,) the class thus associated with a, , with the color (+) f o r {ag,x) f o r a l l x in t h i s c l a s s . If If(a,) i s non-empty, then take a subset Dl(ao) of t h i s class which i s maximal with respect t o inclusion among (-)-monochromatic subsets; hence D1 has cardinality U l . As previousl y , p a r t i t i o n the elements of (a,) - Dl(ao) into classes which are defined by an element a l of Dl(aO) . W e denote by (a,,al) the class thus associated with the sequence (a,,al) , with the color (+) f o r { ao,x) and f o r { al,x) for every x in t h i s c l a s s . The i t e r a t i p n can be continued in an obvious manner f o r a l l successor ordinals u s t r i c t l y less t h a n cJ1 For a l i m i t ordinal u and a sequence s with length u , having values a i ( i < u) , we define the class 8 ( s ) as the intersection of the g ( s / i ) f o r a l l i < u ( r e c a l l t h a t s / i i s the r e s t r i c t i o n o f s t o indices s t r i c t l y l e s s than i ) . Of course, each such sequence s = ao,al, ...,a i ' " must s a t i s f y the following conditions. For every successor ordinal i < u , the term a i belongs t o Di(ao,al,.. . , a i - l ) , which i s a maximal (-)-monochromatic subset of r i ( a D , a l ,...,a i - l ) For every l i m i t ordinal i < u , the term ai belongs t o Di(ao,a l , . . . , a j , . . ) ( j < i ) , which i s a maximal (-)-monochromatic
0
.
.
subset of
r i ( a o , al , . . . , a j , . . )
, the l a t t e r being the intersection of the
.
8 j ( a o , a l , . . . , a j - l ) (j successor ordinal < i ) Notice t h a t the axiom of choice D : we do not have i n i t i a l l y a well-ordering of the s e t o f i s used t o define subsets of A .
75
Chapter 3
<w
u
For e v e r y o r d i n a l
, there
a r e a t most & ( U 1)
many sequences
...
w i t h l e n g t h u , t h u s a t most W 1 many, s i n c e we assume t h e continuum ao,al, h y p o t h e s i s . Hence t h e s e t o f a l l t h e a has c a r d i n a l i t y W 1 and l i k e w i s e f o r the union o f the belong t o no Let
.
D
n o t belong t o f o r every
quence o f these
a.
, pick
Do
u
A
such t h a t
al
pick
au
a l l contain
A
r
. Then
belongs t o
r
as
does
; and i n general,
. The
Z f ( a ,,...,au)
,
se-
s i n c e t h e above
RADO 1953)
<
an a l e p h
B
.
A
Partition the pairs o f
i n t o two c o l o r s (+) and ( - ) . Then e i t h e r t h e r e e x i s t s a subset of and ( + ) - m o n o c h r o m a t i c , z t h e r e -
A
potent w i t h
(-)-monochromatic
a
r(ao)
as an element, and so a r e a l l non-empty. 0
r
A equipotent w i t h (2) Let
e x i s t elements which
8 (ao,al)
belongs t o
such t h a t
be a r e g u l a r l i m i t aleph,
elements o f
belongs t o
r
r
such t h a t
PARTITION THEOREM (ERDOS, A
, there
W2
i s (+)-monochromatic and has l e n g t h (,d1
au
considered c l a s s e s
(1) L e t
has c a r d i n a l i t y
.
D
be one such. P i c k
r
3.5.
Since
B
be an i n f i n i t e aleph,
b
e x i s t s a subset o f
w-
A
(uses g e n e r a l i z e d continuum h y p o t h e s i s ) .
t h e l e a s t aleph s a t i s f y i n g
ba
>
a
. Let
A be t h e a l e p h a+ , t h e successor o f a . P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s (+) and ( - ) . Then e i t h e r t h e r e e x i s t s a subset o f A e q u i p o t e n t with A (+)-monochromatic, t h e r e exl’sts a subset of A e q u i p o t e n t w i t h
- and b
and (-)-monochromatic
(uses axiom o f c h o i c e ) .
0 (1) Assume t h a t e v e r y (+)-monochromatic
s h a l l c o n s t r u c t a subset o f
A
A
subset o f
equipotent w i t h
<
has c a r d i n a l i t y
A
. We
and (-)-monochromatic.
B
X o f A , a s s o c i a t e a subset T(X) o f X which i s (+)-monochromatic and maximal w i t h r e s p e c t t o i n c l u s i o n , among (+)-monochromatic
To e v e r y non-empty subset subsets o f worst
X
T(X)
(uses axiom o f c h o i c e ) . N o t i c e t h a t
For e v e r y element ment set
fi(x) fo(x)
Now l e t j
x
A
of
of
A
. Fix
< i
i
and e v e r y o r d i n a l
x ; take
fo(x) = x
t o be t h e l e a s t element i n
{
A ) , f o r which t h e p a i r
of
x,fo(x))
T(A)
if
Either there already e x i s t s a
case, s e t
i s non-empty: a t t h e
fi(x)
,Or
= x
a l l the
, we x
d e f i n e as f o l l o w s t h e e l e belongs t o
T(A)
.
has c o l o r ( - ) (uses m a x i m a l i t y o f j
J
f.(x)
J i f o r which
<
f.(x)
Otherwise
( w i t h respect t o the well-ordering
i be a non-zero o r d i n a l , and assume t h a t
.
T(X)
c o u l d be a s i n g l e t o n .
T(A) ) .
i s d e f i n e d f o r each f . ( x ) = x and i n t h i s J x and d i s t i n c t among
are d i s t i n c t from
themselves, and t h e p a i r s t h e y f o r m among themselves o r w i t h c o l o r ( - ) . Then l e t color (-) f o r a l l
j
, then
fi(x)
T(U)
set
maximality o f
T(U)
U
be t h e s e t o f t h e
<
i
.
= x
, there
I n particular
.
x a l l have t h e A f o r which ( y , f j ( x ) ] has belongs t o U I f x belongs t o in
y
.
x
Otherwise, i f
x
belongs t o
e x i s t elements
z
of
T(U)
U-T(U)
,
w i t h {x,z}
t h e n by t h e having
THEORY OF RELATIONS
16 c o l o r ( - ) . Take
fi(x)
z
t o be t h e l e a s t such
.
A
i n the well-ordering o f
x o f A , t h e r e e x i s t s an o r d i n a l ix ix, t h e f i ( x ) a r e d i s t i n c t f r o m x and d i s t i n c t f r o m each o t h e r , and t h e p a i r s t h e y f o r m among themselves o r w i t h x have c o l o r ( - ) ; and f i ( x ) = x f o r i 3 i x ' For e v e r y o r d i n a l i , l e t Mi be t h e s e t o f these f i ( x ) , w i t h i f i x e d and x r u n n i n g through A . We s h a l l p r o v e t h a t Mi has c a r d i n a l i t y < A f o r e v e r y By t h e p r e c e d i n g c o n s t r u c t i o n , f o r e v e r y
such t h a t , f o r a l l
i
<
i
. First of
A
<
all,
Mo
i s contained i n
<
hence by h y p o t h e s i s has c a r d i n a l i t y
me t h a t f o r e v e r y i
-= C
we have
.
A
Card Mi
T(A)
which i s (+)-monochromatic,
Consider an o r d i n a l
<
.
A
Since
A
C
and assu-
i s r e g u l a r by hypo-
Mi ( i < C ) i s A . Otherwise A would be t h e u n i o n o f s t r i c t l y < A many s e t s , each s t r i c t l y subpotent w i t h A L e t D denote t h e maximum o r supremum c a r d i n a l o f t h e Mi . To e v e r y element x o f A , a s s o c i a t e t h e sequence o f t h e
t h e s i s , t h e maximum c a r d i n a l o r t h e supremum c a r d i n a l o f t h e s t r i c t l y less than
.
fi(x)
. The
( i 4 C)
. Now by
(Card
number o f d i s t i n c t such sequences i s l e s s t h a n o r equal t o
the f a c t that
A
5
i s a l i m i t aleph, and by ch.2
r a l i z e d continuum h y p o t h e s i s ) , t h i s c a r d i n a l i s s t r i c t l y l e s s t h a n
6.5 ( 2 ) (gene-
.
A
Two i d e n t i -
U , hence t h e same T(U) o f c a r d i n a l i t y < A . For any x , t h e new element f C ( x ) belongs t o T(U) . Hence as x v a r i e s , t h e
c a l sequences g i v e t h e same s e t p o s s i b l e sequences regularity o f
A
By h y p o t h e s i s
B
. For
ir >/ B
fi(x)
, the
( i d C)
set
<
MB
A
.
A
A
o f cardinality
<
has c a r d i n a l i t y
with cardinality
b
.
a
many s e t s fC(x)
would be i d e n t i c a l w i t h
fi(r) a r e d i s t i n c t f o r c h r o m a t i c and o f c a r d i n a l i t y B . 0 (2) For
A
a+
, suppose
i
<
B
, and
. Because o f
T(U)
r
in
A
such t h a t
, contradicting
A
the
.
has c a r d i n a l i t y C A
Thus t h e r e e x i s t s an
ceding. Hence t h e
0
<
MC o f a l l p o s s i b l e
i s a cardinal
otherwise,
give
t h e pre-
t h e i r s e t i s (-)-monochro-
t h a t e v e r y (+)-monochromatic subset of
We s h a l l c o n s t r u c t a (-)-monochromatic subset o f
A
.
x o f A , c o n s t r u c t as p r e v i o u s l y t h e f i ( x ) and t h e Mi f o r a l l o r i (axiom o f c h o i c e ) . We s h a l l p r o v e t h a t , f o r e v e r y i o f c a r d i n a l i t y b , hence f o r e v e r y i < b , t h e s e t Mi has c a r d i n a l i t y 6 a Notice
For e v e r y dinals
<
.
f i r s t that
b
6
a
i s (+)-monochromatic hence
c
<
a
, and
>
since
.
a
The
set
and hence has c a r d i n a l i t y assume t h a t
o r t h e supremum o f t h e
Card Mi
Card Mi
(i
i s contained i n
6
a
. Take
c)
is
a a
. The
since
Card c
x
running through
Since
b
,< a , t h e
A
,
is
,<
union o f the
. (i<
c
. The
<
< b . As Mc
,
b
maximum
fi ( x ) (i < c)
sequences
p r o o f o f (1) above, i t f o l l o w s t h a t t h e c a r d i n a l i t y o f t h e s e t
, which
T(A)
an o r d i n a l
6 a f o r every i < c
, hence 6
(Card ')a
have c a r d i n a l i t y a t most
<
Mo
of
i n the
fc(x)
for
a.a = a Mi
b)
has c a r d i n a l i t y
6
b.a = a
.
Let
Chapter 3
77
r .be an element of A not in this union. Then r # fi(x) for every x of A and every i < b . In particular r # fi(r) for every i < b : thus i, 3 b . It follows that the set of the fi(r) (i < b) is (-)-monochromatic and has cardinality b . 0 Note that, in the proof of ( 2 ) , the cardinal A = a, is not a limit aleph. This condition, and also the use of generalized continuum hypothesis, is only necessary for (1) in order to apply ch.2 5 6.5.(2). The example of w1 , given at the end of 5 3.3, can be expressed here by taking a = d , b = w and A = b J 1 . Recall that with generalized continuum hypothesis, the only regular limit alephs are CJ and the inaccessible alephs (ch.2 5 6.7). The statement (1) holds only for these. On the other hand, with only the axiom of choice, the continuum can be regular or not, a limit or a successor aleph, with cofinality >/a(ch.2 § 6.4). §
4 - COMBINATORIAL LEMMAS,
COLOR AND INCLUSION
4.1. (1) Let E be a set, p, q two integers such that ptq 4 Card E . Take a set of p-element subsets included in E and call this the color u . If every (p+q)-element subset includes the same number k f p-element subsets with color & , then every (ptqt1)-element subset includes the same number with color k(p+qtl)/(qtl) of p-e&%n_t_Aybsets (2) Given two not necessarily disjoint sets of p-element subsets o f E , call these the colors ?d, and 2'. If every (p+q)-element subset includes as many p-element subsets with color p-element subsets with color ?f, then the same is true fmeve_ry (p+qtl)element subset of E (1) Let F be a (p+q+l)-element set. The cardinality of the set o f (p+q)element subsets of F is ptq+l Each includes k many p-element subsets with color k , which yields k(p+qtl) ordered pairs, each formed with a p-element set having color ?,(, and with a (ptq)-element set which includes it. For every p-element subset with color % in F , there are qtl many (ptq)as the number element subsets which include it. This yields k(p+q+l)/(q+l) of p-element subsets included in F and having color & . 0 (2) Let F be a (p+q+l)-element set. Every (p+q)-element subset o f F Thus we includes as many p-element subsets with color % as with color have that in F , there are the same number of ordered pairs, each with first term a p-element set with color 21 and second term a (p+q)-element set including the first term, as of ordered pairs, each with first term a p-element
u.
u
~
as
.
.
r.
THEORY OF RELATIONS
78
set having color 'It and second term a (p+q)-element set including the first term. We obtain the number of p-element subsets with color % by dividing the preceding number by q+l . Similarly for the color 'v . 0 4.2. (1) Let E be a finite set, p. q two integers such that p+q 6 Card E . Let 'LL be a color of certain p-element subsets of E . Let ss p and s 6 (Card E) - p - q . If every (ptq)-element subset includes the same number of p-element subsets , then every s-element subset is included in the same number of_ with color p-element subsets with color /d~. (2) Let %, be two colors of p-element sets, not necessarily disjoint. If every (p+q)-element set includes as many p-element subsets with color % as p-element subsets with color 1/, then every s-element subset is included in as many p-element sets with color 2 p-element sets with color (communicated by POUZET in 1975). 0 First we prove (2). For s = 0 , this follows from 4.1.(2) iterated from p+q to Card E . Assume that s 31 and assume that the statement is true for s - 1 and p+q . In other words, for every E with finite cardinality >/ s+p+q-1 and every (s-1)-element subset of E . We shall prove this for s and p+q , hence for E with finite cardinality 3 s+p+q and a s-element subset H G E . Let u be an element of H . By the induction hyoothesis, there exists a same number k of p-element subsets with color % as with color 2 / , included in E and including H - {u) . Similarly, there exists a same number 1 of p-element subsets with color as with color included in E - { u) and including H - {u} (the cardinality of these sets being respectively s+p+q-1 and equal to s-1 ). By subtraction, there existsthesame number k-1 of p-element subsets with color ?A,'! as with color , included in E and including H .O 0 Statement (1) follows from (2). Indeed, let H and H' be two s-element subsets of E . Take a permutation f of E which transforms H into H ' . Take the p-element subsets with color , and let 21' be the color of their images via f . Then every (p+q)-element subset X includes as many p-eleas one can see by taking the ment subsets with color as with color (p+q)element subset (f-l)"(X) (notation from ch.1 !j 1.2). By ( 2 ) , the s-element subset H ' is included in as many p-element subsets with color %& as with color . But the latter are the images via f of p-element subsets with and including H . Thus H and H ' are included in the same number color of p-element subsets with color u . 0
u
u
u
r,
u
u
u
u'
a',
Chapter 3
79
4.3. (1) Let E be a set, p an integer less than or equal to Card E , and let be a non-empty set, called color, of p-element subsets.
u
such that 2p + q Card E , and if every (p+q)element set includes the same number of p-cgment subsets with color u , then every p-element subset has color ld . (2) Given E and p less than or equal to Card E , let I d , 3‘ be two sets, called colors, of p-element subsets. If there exists an integer q such that 2p + q 4 Card E , and for which every (p+q)-element subset includes as many p-element subsets with color p-element subsets with color 1;’ , then the colors and 2/ are identical. I f there exists an integer q
_
_
~
_
^
_
_
can assume that E is finite, by replacing E if necessary by a finite subset of cardinality at least equal to 2p + q . Now take the preceding statements with s = p . By statement (l), every p-element subset is included in the same number of p-element subsets with color 16. In other words, every p-element subset has color 1 1 (since it is assumed that 2 is non-empty). By statement (2), every p-element subset is included in as many p-element subsets with color 2 as p-element subsets with color 17. In other words, the are identical. 0 colors ‘lG and
0 We
If Card E < 2p + q , then by taking s 6 (Card E)-p-q , it is easy to give an example in which the color does not extend to the entire set of p-element subsets. Thus with E = {a,b,c,dj hence of cardinality 4 , with p = 2 , q = 1 , and only the edges ab and cd with color % , every 3-element subset contains such an edge, and every element belongs to such an edge. Adding ac and bd with color every 3-element subset contains an edge with each color, and all elements belong to an edge of each color.
v,
Calculation. If e designates the cardinality of E , and k the number of p-element subsets of color u? contained in every (p+q)-element subset, then by 4.1.(1 , the number of all p-element subsefis with color a is k.e!q!/(p+q)!(e-p)! . The number of those which contain a given element u , hence are not contained in the (e-1)-element subset E - (u) , is obtained by subtraction. It equals / (p+q)!(e-p)! , by assuming e >/ p+q+l . The number of those which k.(e-l)!q!p , hence which contain u and are not contained in contain a given pair (u,v) the (e-1)-element subset E - { v ) , i s obtained by subtraction. It equals , in assuming e >/ p+q+2 . In general, the numk.(e-2)!q!p.(p-1) / (p+q)!(e-p)! ber of those containing a given s-element subset is: , in assuming s & e-p-q . k.(e-s)!q!p! / (p+q)!(e-p)!(p-s)!
THEORY OF RELATIONS
80
§
5 - INCIDENCEMATRIX, MULTICOLOR THEOREM
KANTOR'S
L I N E A R I N D E P E N D E N C E LEMMA,
5.1. INCIDENCE MATRIX Let p. q be two integers, E a s e t of f i n i t e cardinal h Represent as "ordinate values" the s e t of p-element subsets of E , of cardinality h!/p!(h-p)!, and as "abscissa values" the s e t of (p+q)-element subsets, of cardinality To each couple ( x , y ) where x i s a (p+q)-element h!/(p+q)!(h-p-q)! s e t and y i s a p-element s e t , a t t r i b u t e the value 1 i f f y s x and the value 0 otherwise. The rectangular t ab l e thus obtained shall be called the incidence matrix of E f o r p and q Note t h a t i f h = Card E 2 2p + q , then each row of the matrix, corresponding t o a p-element s e t , i s a t l e a s t as long as each column, corresponding t o a (p+q)-element s e t : indeed p!(h-p)! 3 ( p + q ) ! ( h - p - q ) ! The reader i s assumed t o be familiar with the elementary theory of determinants and with the notion of l i n ear dependence. If Card E < 2p + q , then i t i s possible t h a t a row of the incidence matrix depends linearly on one or several other rows. For example, f o r p = q = 1 and Card E = 2 , the matrix reduces t o two rows and one column, with value 1 . B u t f o r Card E 3 2 p + q , we have the following r e s u l t . LINEAR INDEPENDENCE LEMMA I f Card E 3 2p + q , the rows of the incidence matrix a re linearly independent: no row i s a l i n e a r combination of other rows. Equivalently, every non-zero determinant extracted from the matrix and depending on a f i n i t e number r of rows can be extended t o a non-zero determinant based on the previous rows together with an a r b i t r ar y (r+l)st row. I n the case t h a t E i s f i n i t e , i t follows t h a t there e xists a non-zero determin a n t depending on a l l the rows. Hence there ex i s t s an~injection _ _ which _ t o each p-element s e t y associates a (p+q)-element s e t including y as a subset (KANTOR 1972). 0 To each permutation f of E associate the corresponding permutation f " of Hence f " permutes the s e t of rows. There corresponds p-element subsets of E t o f as well a permutation of the s e t of (p+q)-element subsets, hence of the s e t of columns, b u t i t i s unnecessary t o consider t h i s , since we are working with l i n e a r combinations of rows and reasoning by the coefficients a ttribute d t o each row in a given l i n ear combination. Assume t h a t E has f i n i t e cardinal h ; we argue ad absurdum. Assume t h a t there e x is t s a p-element subset, hence a row which i s a line a r combination of a l l the other rows, with positive, negative or zero rational coefficients, since these are quotients of determinants with values 0 or 1
.
.
.
-
.
~
.
.
Chapter 3
L e t us c a l l
b this
81
p-element s e t and t h e corresponding row. Given an a r b i t r a r y
permutation f o f E which preserves the s e t b ( b u t not n e c e s s a r i l y each element o f b ), then f o preserves t h e row b and permutes t h e s e t o f t h e other rows; two rows which are transformed one i n t o t h e o t h e r represent two p-element sets y, y ' such t h a t b n y and b A y ' are equipotent. Transform the given l i n e a r combination by a l l p o s s i b l e fa
, the
number o f such
being (h-p)!p! , then take t h e combination which i s t h e a r i t h m e t i c average o f the combinations thus transformed. By s y n e t r y , a l l t h e rows which represent p-element sets d i s j o i n t from b w i l l have t h e same c o e f f i c i e n t . S i m i l a r l y f o r a l l rows which represent p-element s e t s having a s i n g l e element i n common w i t h b , and i n general f o r a l l rows which represent p-element sets having equipotent intersections with b Consider a column a representing a (p+q)-element s e t d i s j o i n t from b : t h i s
.
.
I n the column a , t h e p-element sets included a e x i s t s since h >/ 2p + q i n a a r e a l l d i s j o i n t from b , and so a l l have t h e same c o e f f i c i e n t i n o u r combination. Moreover, i f we denote these p-element sets by y , these are the i n t h e incidence matrix, o n l y ones y i e l d i n g t h e value 1 i n p o s i t i o n (a,y) w h i l e t h e m a t r i x has t h e value 0 i n p o s i t i o n (a,b) It f o l l o w s t h a t t h e i r
.
c o e f f i c i e n t i s zero, hence each row which represents a p-element s e t d i s j o i n t from b has c o e f f i c i e n t zero. The problem i s thus answered n e g a t i v e l y f o r p = 1 , since i n t h i s case the p-element sets d i s t i n c t from b are d i s j o i n t w i t h b , hence t h e above assumed l i n e a r combination does n o t e x i s t . 2 , and consider a column al representing a (ptq)-element Assume t h a t p s e t which i n t e r s e c t s b i n a unique element. Then t h e rows y f o r which t h e m a t r i x has value 1 i n (a,,y) a r e those which represen! e i t h e r a p-element s e t d i s j o i n t from b , hence w i t h c o e f f i c i e n t zero, o r a p-element s e t i n t e r s e c t i n g b i n a s i n g l e p o i n t . By t h e preceding discussion, t h e l a t t e r have t h e same c o e f f i c i e n t i n the combination. Since the m a t r i x has the value 0 i n (al,b) , t h i s c o e f f i c i e n t i s zero. The problem i s thus answered n e g a t i v e l y f o r p = 2 , s i n c e i n t h i s case t h e p-element s e t s d i s t i n c t from b have a t most one element i n common w i t h
b
. I n the
general case, by i t e r a t i n g t h e preceding
argument, we prove t h a t a l l t h e c o e f f i c i e n t s are zero, .hence t h a t t h e above assumed l i n e a r combination does n o t e x i s t . The r e s u l t f o l l o w s imnediately i n t h e case o f E i n f i n i t e . F i n a l l y , f o r t h e cnnclusion concerning t h e e x t e n d i b i l i t y o f a non-zero determinant, assume on t h e c o n t r a r y t h a t t h e r e e x i s t s a non-zero determinant which i s n o t extendible, and deduce t h a t an a r b i t r a r y row o f t h e m a t r i x i s a l i n e a r combination o f rows o f the submatrix which corresponds t o t h i s determinant,
82
THEORY OF RELATIONS
5.2. In t h e "degenerate case" where h = Card E < 2p+q , t h e number of columns i s s t r i c t l y l e s s than t h e number of rows. In t h i s case t h e columns of t h e incidence matrix a r e l i n e a r l y independent; i n o t h e r words, t h e r e e x i s t s a non-zero determinant based on t h e col umns. 0 Interchange each p-element s e t y w i t h t h e (h-p)-element s e t E-y , and each (p+q)-element s e t x with t h e (h-p-q)-element s e t E-x . Then the inclusion E-y . The r o l e of p i s played by p ' = h-p-q ; y c x i s equivalent t o t h e r o l e of p+q i s played by p ' + q ' = h-p , so t h a t q ' = q . We have 2p' + q ' = 2 h - 2 p - q < h : hence we can apply t h e l i n e a r independence lemna with rows and columns interchanged. 0
E-xc
5.3. MULTICOLOR Let E be a f i n i t e s e t , h i t s cardinal and p , q two i n t e g e r s . P a r t i t i o n the p-element subsets of E i n t o a f i n i t e number k of c l a s s e s which a r e c a l l e d colors uo, ul, ..., u ~ .- For ~ each (p+q)-element subset a of E , we c a l l t h e multicolor of a t h e function which t o each c o l o r u i ( i k ) associates t h e number of p-element s e t s of c o l o r u i which a r e included i n a When t h i s number i s non-zero, we say t h a t t h e c o l o r u i f i g u r e s in the multicolor. MULTICOLOR THEOREM I f Card E 2, 2p+q , then t h e number of m u l t i c o l o r s ' o f (p+q)-element subsets
.
of
E i s a t l e a s t equal t o t h e number of c o l o r s of p-element subsets. More p r e c i s e l y , t h e r e e x i s t s an i n j e c t i o n which t o each c o l o r u ( t o which a t l e a s t one p-element s e t belongs) a s s o c i a t e s a multicolor i n which u f i g u r e s , and t o which a t l e a s t one (p+q)-element s e t belongs (POUZET 1976). Assume f i r s t t h a t E has f i n i t e cardinal h > , 2p+q . Hence t h e number of (p+q)-element s e t s i s a t l e a s t equal t o t h a t of t h e p-element s e t s , and the rows of t h e incidence matrix a r e l i n e a r l y independent. To each c o l o r t h e r e corresponds a f i n i t e s e t of rows of t h a t c o l o r . Replace these by a unique row which i s t h e i r sum, obtained by adding the values 0 o r 1 i n each column. T h u s each new row represents a c o l o r u Each column continues t o represent a (p+q)-element s e t , and i n d i c a t e s the number of p-element sets of c o l o r u which a r e included i n t h i s (p+q)-element set. Note t h a t , i n t h e new matrix t h u s obtained, t h e rows a r e l i n e a r l y independent. I t s u f f i c e s t o see t h a t , given a matrix w i t h k independent rows ( k 3 2 ) , the replacement of two rows b and b ' by t h e i r sum y i e l d s a matrix w i t h k-I independent rows. Indeed, t h e r e e x i s t s a non-zero determinant based on the k-2 i n t a c t rows. So t h a t the only o t h e r p o s s i b i l i t y would be t h a t t h e row sum o f b and b' i s a l i n e a r combination of the k-2 i n t a c t rows. B u t then the row b , f o r example, would be a l i n e a r combination o f t h e k-2 i n t a c t rows plus
.
83
Chapter 3
the row b' , c o n t r a d i c t i n g t h e hypothesis. T h u s , i f k i s now t h e number of c o l o r s , hence of rows, we have a non-zero determinant of order k . Take i n this determinant a sequence of k ordered p a i r s ( x , y ) where x i s a column and y a row, w i t h non-zero value of the new matrix i n each considered ordered p a i r . We thus o b t a i n the i n j e c t i v e function i n the theorem. This i n j e c t i o n a s s o c i a t e s , t o two d i s t i n c t c o l o r s y. y ' two ( p + q ) element s e t s x , x ' whose m u l t i c o l o r s a r e d i s t i n c t . Otherwise we would have two i d e n t i c a l columns i n t h e determinant. Thus t h i s is an i n j e c t i o n from t h e s e t of colors i n t o the s e t of m u l t i c o l o r s . I t remains t o consider t h e case when E i s countably i n f i n i t e . I f we only have a f i n i t e number of c o l o r s , then we r e s t r i c t E t o a s e t of f i n i t e c a r d i n a l i t y a t l e a s t equal t o 2p+q and including p-element s u b s e t s of each c o l o r . The rows, which r e p r e s e n t t h e c o l o r s , a r e l i n e a r l y independent, and remain so when one takes up the e n t i r e i n f i n i t e set E . I f t h e r e a r e i n f i n i t e l y many c o l o r s , then we s t i l l have l i n e a r independence. Then a s mentioned f o r t h e l i n e a r independence lennna, every non-zero determinant i s extendible t o a non-zero determinant over one more row, hence one more c o l o r . The e x i s t e n c e of t h e i n j e c t i v e function i n the theorem follows. 0
5 6 - RAMSEY
SEQUENCE: ANOTHER PROOF OF
GALVIN'S
THEOREM
The following notion of Ramsey sequence of conditions is a form of the c l a s s i c a l Ramsey s e t : see ERDOS, RADO 1952. The connected proof of GALVIN's i n i t i a l i n t e r v a l theorem i s due t o LOPEZ 1983'. As opposed w i t h POUZET's proof i n 5 2 , here we need n e i t h e r lexicographic rank nor t r a n s f i n i t e induction. As well a s i n 5 2 , t h e axioms of ZF w i l l be s u f f i c i e n t : see 5 6.5 below.
6.1. Given two s e t s A, B of i n t e g e r s , p u t A < B o r B > A i f f every element of B i s s t r i c t l y g r e a t e r than every element of A . We adopt the convention t h a t the empty s e t i s < and > any s e t ; so t h a t < i s i r r e f l e x i v e and t r a n s i t i v e only f o r non-empty s e t s . L e t H be a f i n i t e s e t , Z an i n f i n i t e set of i n t e g e r s . A f i n i t e sequence of conditions g i ( H , Z ) ( i = 1, ..., r ) i s s a i d t o be a Ramsey sequence i f f we have the following: V H fin i n f X > H 93, i n f YCX A ( b z i n f Z c y 3rl(H,Z))V ...
VX
[
... v ( d zi n f
Z s Y
+Vr(H,Z))]
( n o t a t i o n s : f i n = f i n i t e , i n f = i n f i n i t e set of i n t e g e r s ; obvious l o g i c a l symbols).
THEORY OF RELATIONS
84
Example. P a r t i t i o n t h e p a i r s o f i n t e g e r s i n t o two c o l o r s (+) and (-). Take f o r
i n H , a l l p a i r s 1h.z) h ) " . Then alone
f(H,Z) t h e f o l l o w i n g statement: " f o r each i n t e g e r h where z belongs t o Z , have same c o l o r (depending on
,
c o n s t i t u t e s a Ramsey sequence. Another example. Take a c o n d i t i o n f and d e f i n e i g as t h e negation o f d i s o f t e n a Ramsey sequence. I t i s t h e case, for Then t h e sequence ( ff, -I instance, i f yf (H,Z) means t h a t the preceding p a i r s { h,z) have c o l o r (t). I n the case o f two such opposite conditions, the above formula means t h a t , given
.
e)
H , the s e t o f a l l i n f i n i t e Z s a t i s f y i n g C(H,Z) i s Ramsey i n the sense o f ERDOS, RADO 1952. I n other words, there e x i s t s an i n f i n i t e s e t Y o f i n t e g e r s such t h a t e i t h e r each i n f i n i t e Z C_ Y belongs t o t h e s e t defined by and H o r each i n f i n i t e Z c Y belongs t o t h e complement. Among sets o f i n f i n i t e sets o f integers, i . e . among sets o f r e a l s , i t i s known t h a t t h e f o l l o w i n g a r e Ramsey: a l l open sets (see t h e topology defined i n ch.1 exerc. 4 ) ; Bore1 sets (see GALVIN, PRIKRY 1973); a n a l y t i c s e t s (SILVER 1970). See a l s o ELLENTUCK 1974, who characterizes t h e "completely Ramsey sets" by t h e Bai r e property.
el ,..., er , we have t h e f o l l o w i n g
6.2. Given a Ramsey sequence
statement,
modulo t h e axiom o f dependent choice:
J ~ i ~ f f l ~ f i(vzinf ,, (
...
(Vz
inf
H C A A Z S A ~ Z > H ) V~1 ( H J ) ) V - - *
( H c A h Z E A A Z > H ) 3 f,(H,Z))
0 The p r o o f generalizes the f i r s t p a r t o f RAMSEY's p r o o f 1 . 1 , i n o b t a i n i n g e l e -
.
.
S t a r t from uo = 0 , Ho = { D l and Xo = s e t o f i n t e g e r s # 0 We ments vi get an i n f i n i t e Ys Xo , c a l l e d Yo and s a t i s f y i n g t h e above c o n d i t i o n i n
.
and A = {O)uYa Then l e t u1 be t h e f i r s t element { uo,ul) and XI = Yo -{ul) We g e t an i n f i n i t e
brackets, where H = Ho of Yo S t a r t from HI
.
.
=
Y1s X1
which s a t i s f i e s our above c o n d i t i o n i n brackets, where H = and A = \uo,uI) u Y1 Then s t a r t from H i = {ul) and = Y1 infinite Y1 which s a t i s f i e s our condition, where H = Ho o r and A = {uo,ul)u Yi Then l e t u2 be t h e f i r s t element o f Y i
Yi
.
.
Xi
.
Yi
and X2 = - {u2). from H2 = {uo,u1,u2) s a t i s f i e s our condition, where H = Ho o r
.
Ho
or
H1 an or H i
. We g e t HI Start
We g e t an i n f i n i t e Y2 5 Y i HI o r H i o r H2 and where
which
A = {uo,u1,u2\ u Y2 I t e r a t e , t a k i n g f o r H i , H;,.. a l l the sets w i t h l a s t element u2 , and so g e t t i n g Y h y Y i , before d e f i n i n g u3, Hg and Y3 ; and so on. F i n a l l y take f o r A t h e s e t o f ui ( i i n t e g e r ) The axiom o f dependent choice i s used f o r choosing s e t s Y . 0
...
.
6.3. L e t H, F be f i n i t e sets o f i n t e g e r s and Then t h e p a i r o f t h e f o l l o w i n g c o n d i t i o n e(H,Z) (HJ) c ( H , Z ) : 3 F fin F c Z A
a
.
b (H,F)
be an a r b i t r a r y c o n d i t i o n .
w i t h i t s negation i s Ramsec
Chapter 3
85
0 Suppose the contrary. There e x i s t a f i n i t e s e t
H
and an i n f i n i t e s e t
X 7 H
such t h a t , f o r every i n f i n i t e s e t Y 5 X , there e x i s t two i n f i n i t e subsets Z1 and Z2 with (H,Z1) and the negation iff (H,Z2) Then each f i n i t e subset F of Z2 s a t i s f i e s (H,F) Now replace Y by Z2 : there e x i s t s an i n f i n i t e subset Z i of Z2 such t h a t 'f: ( H , Z i ) . Thus there e x i s t s a f i n i t e subset F of (H,F) : contradiction. 0 5 Z2 which s a t i s f i e s
.
.
a
Zi
6 . 4 . A PROOF OF GALVIN'S INITIAL INTERVAL THEOREM be a s e t of f i n i t e s e t s of integers, assumed t o be mutually incomparable Let under inclusion, and t o s a t i s f y GALVIN's hypothesis: every i n f i n i t e s e t of integers includes a t l e a s t one element of 3 as a subset. Take 3 (H,F) t o be the following condition: "the union H u F admits an i n i t i a l interval belonging t o " ; more b r i e f l y " H,J F has i . i . " . Then by 6 . 2 there e x i s t s an i n f i n i t e s e t A such t h a t , f o r every f i n i t e subset H of A : either (1) inf ( Z g A A Z > H ) =) 3, f i n ( F c Z A H u F has F i . i . )
v,
or ( 2 )
tJz
inf ( Z 5 A
A
Z 7 H) =)
vF f i n
( F c Z
+
HuF
has no
'3 i . i . ) .
F i r s t l y we prove t h a t , assuming GALVIN's hypothesis, there e x i s t s an i n f i n i t e s e t A such the above ( 2 ) i s f a l s e : so only (1) i s true. 0 For H empty, the above conclusion ( 2 ) i s f a l s e . Indeed ( 2 ) reduces t o saying t h a t , f o r every i n f i n i t e s e t Z , there does n o t e x i s t any f i n i t e subset of Z which belongs t o Now l e t G be a f i n i t e s e t of integers. Assume t h a t the above ( 2 ) i s f a l s e f o r every subset H of G . Then i t suffices t o prove t h a t there e x i s t s an integer g 7 M a x G such t h a t every H 5 G u{g> f a l s i f i e s ( 2 ) . For t h i s purpose, i t suffices t o prove t h a t , f o r each subset H of G , there e x i s t only f i n i t e l y many integers h Z Max G such t h a t H u { h ) s a t i s f i e s ( 2 ) : indeed i t will suffice t o choose g s t r i c t l y greater t h a n a l l such h Arguing ad absurdurn, assume the existence of an i n f i n i t e sequence h l , h 2 , ..., h i , ... with Hu{hi} satisfying ( 2 ) . Take Z t o be the i n f i n i t e s e t of these h i : by hypothesis H f a l s i f i e s ( 2 ) , so H v e r i f i e s (1). Thus there e x i s t s a f i n i t e subset F of Z such t h a t H u F has an i n i t i a l interval which belongs t o 3-. Let h P (p integer) be the f i r s t element of F ; then H w{h ) f a l s i f i e s (2): contraP diction. 0
-
.
.
Secondly we obtain GALVIN's conclusion. A of 6.2, now assumed t o s a t i s f y only the above ( 1 ) . Let B be an a r b i t r a r y i n f i n i t e subset of A Let K be a f i n i t e subset of B belonging t o , and denote by H the i n i t i a l interval of B which ends with Max K . Then by ( 1 ) above, there e x i s t s a f i n i t e subset F of 6-H such
0 Consider the i n f i n i t e s e t
.
THEORY OF RELATIONS
86
.
t h a t HuF has an i n i t i a l interval which belongs t o 3 This i n i t i a l interval are mutually incomparable under cannot surpass Max K , since elements of inclusion. Consequently our i n i t i a l interval of H u F reduces t o an i n i t i a l interval of H , thus of B . 0 6.5. To end t h i s section, l e t us prove t h a t the axiom of dependent choice, used in 6.2, i s avoidable in view of obtaining GALVIN's theorem.
Come back t o our hypotheses in 6.1. Given a f i n i t e s e t H and a n i n f i n i t e set X > H : e i t h e r (1) 3 i n f x A z' i n f ( Z C y 3~ f i n (FcZ A H u F has $-i.i.))
0
ys
or ( 2 )
3,
inf Y c _ X
A
vz inf(ZcY=> 'dF f i n ( F c Z =>
HuF
has n 0 F i . i . ) ) .
Either ( 2 ) i s f a l s e ; in other words:
vy inf
Y c _ X =>
2 z inf
(ZSY
3, f i n
(FcZAHUF
has 5 i . i . ) ) .
I n such a case, take Y = X ; then take any Y c X b u t change the notation, writing Z instead of Y : we get the following 3, inf Y = X A \dz inf ( Z S X => 3, f i n (FcZ A HuF has F i . i . ) )
So we obtain (1) strengthened by the unambiguous definition
Y = X
.
.
Or ( 2 ) i s t r u e , with (1) true or f a l s e , which i s immaterial. In t h i s case, take a l l the i n f i n i t e s e t s Y which s a t i s f y ( 2 ) , and note t h a t each i n f i n i t e subset of a Y i s s t i l l a Y satisfying ( 2 ) . Let us proceed lexicographically: we take the l e a s t integer uo f o r which there e x i s t s a Y beginning with uo . Then the l e a s t u l , uo f o r which there e x i s t s a Y beginning with uo, u1 ; and so on. Finally we adopt the unambiguous definition Y = i u o , u l , . . . ) .0 EXERCISE 1 - SCHUR NUMBERS 1 - Given a p a r t i t i o n of the s t r i c t l y positive integers into a f i n i t e number of classes called columns, show t h a t a t l e a s t one of the columns contains three d i s t i n c t integers a,b,c with c = a+b (SCHUR 1916). Hint. To each column U associate the class of pairs of integers x , y such t h a t the absolute value I x-yl belongs t o U ; then apply RAMSEY's theorem. 2 - A s e t U of integers i s said t o be additively f r e e i f f the sum of any two integers belonging t o U does not belong t o U Given an integer k , show that there e x i s t s an integer k+, k such t h a t , f o r each p a r t i t i o n of the integers 1,2, ...,k+ i n t o k classes called columns, there is a t l e a s t one nonadditively f r e e column. The smallest k+ will be denoted by s ( k ) and called the Schur number r e l a t i v e t o k Show t h a t s ( 1 ) = 2 , s ( 2 ) = 5 , s(3) = 14 ( s t a r t with the column 5,6,7,8,9).
.
.
Chapter 3
3
-
87
I n 1961, Leonard D. BAUMERT (see p u b l i c a t i o n 1965) e s t a b l i s h e d , w i t h t h e a i d
o f a computer, t h a t s ( 4 ) = 45. Here i s t h e example he gave o f a p a r t i t i o n o f t h e f i r s t 44 s t r i c t l y p o s i t i v e i n t e g e r s i n t o 4 a d d i t i v e l y f r e e columns: 1 2 4 9 3 7 6 1 0 11 5 8 13 15 18 20 12 17 21 22 14 19 24 23 16 26 27 25 29 28 33 30 31 40 37 32 34 42 38 39 35 44 43 41 36 2 Show t h a t t h e Ramsey number ( 3 ) k >/ s ( k ) + l : a s s o c i a t e t o each column U t h e s e t o f p a i r s o f integers from
-
Show t h a t
s(k+l)
gers
1 to
p = s(k)-1
gers
p+l
4
2p + 1
to
.
>/
ned
& 158 , thus
I n general
s(5) 2 (3)5
k
3 ( 3 +1)/2
s(k)
k
>/
.
1
46
.
Begin
with the p a r t i t i o n o f the inte-
columns. Add a
(k+l)st
column o f t h e i n t e -
Then complete each column formed o f i n t e g e r s
Hence
2p+l+u
s ( k ) f o r which t h e a b s o l u t e v a l u e o f t h e
2 (3)4
3.s(k) into
.
integers s(5)
1 to
U ; thus
d i f f e r e n c e belongs t o
+ 159
u by t h e
>, 135 . FREDRICKSON 1979
3 134 and (3);
obtai-
. 3 (3k-4.89/2) + 1/2 f o r k >/ 4 . T h i s
and even
i n e q u a l i t y i s improved by ABBOTT, HANSON 1972 who o b t a i n , i f one r e c t i f i e s t h e i r numerical e r r o r ,
-
EXERCISE 2
1
-
Let
s(k)
3
89(k-7)/4.1201
+
1 for
k
>/
4
.
RAMSEY'S THEOREM WITH m-TUPLES
m, k
be two i n t e g e r s ; t a k e an
m-sequence o f i n t e g e r s
ai (i= 1, ..., m).
P a t t e r n e d a f t e r t h e f i n i t e Ramsey theorem, p r o v e t h a t t h e r e e x i s t s an of i n t e g e r s Fi Xm)
bi
+ ai
o f cardinality
where
subsets
xi
, for
belongs t o Fi
Gi
bi
s a t i s f y i n g t h e f o l l o w i n g . Given an
with -
Gi
i n the Cartesian product
m = 2 with
(kp).kkp
and
p
x
... x
k
Gm
m-tuples
(x
,.... ,
c o l o r s , t h e r e e x i s t s a sequence o f
, such t h a t a l l t h e m-tuples
have t h e same c o l o r . Note t h a t n o t h i n g
p a i r w i s e d i s j o i n t , which i s o f t e n convenient.
Fi al = a2 = p
. Beginning w i t h
a set
F1
o f cardinality
F2 o f c a r d i n a l i t y (p).kkp , we o b t a i n G2 5 F2 o f c a r d i n a l i t y H1 C, F1 o f c a r d i n a l kp , such t h a t , f o r e v e r y o f H1 , a l l t h e o r d e r e d p a i r s w i t h f i r s t t e r m x and second term
and an " i n t e r m e d i a t e " s e t
element
, into
o f c a r d i n a l i t y ai -
G1 x
i s changed i n assuming t h e Example:
every p a r t i t i o n o f the s e t o f Fi
m-sequence
m-sequence of s e t s
THEORY OF RELATIONS
88
c
an element of G2 have the same c o l o r . F i n a l l y we obtain G1 H1 p , w i t h t h e monochromatic Cartesian product G1 x G2 . 2 - For m = 2 and a l = a 2 = 2 , we can take bl = 3 and b2 = 9 sely. W e search f o r a symmetric s o l u t i o n bl = b2 ; show t h a t bl = i n s u f f i c i e n t . On t h e o t h e r hand, f o r the common value 5 , we always
of cardinal
, o r converb2 = 4 i s have a mono-
.
chromatic Cartesian product with Card G1 = Card G2 = 2 One could assume f i r s t t h a t t h e r e e x i s t s an element x of F1 joined t o a t l e a s t 4 elements of F2 by the same c o l o r , denoted ( + ) . Call t h i s subset of F2 of cardinal 4. Then e i t h e r t h e r e e x i s t s another element of F1 y i e l d i n g t h e c o l o r (+) w i t h two elements of , o r two elements of F1 d i s t i n c t from x y i e l d the opposite c o l o r ( - ) w i t h two elements of F; Next, we assume t h a t each element of F1 y i e l d s a p a r t i t i o n of 2 as opposed t o 3 elements of F2 f o r t h e c o l o r s . Such an element x i s c a l l e d (+)-major i t y o r (-)-majority, according t o whether t h e r e a r e 3 edges ( o r p a i r s ) emanat i n g from x with c o l o r (+), o r 3 w i t h c o l o r ( - ) . Then t h e r e e x i s t 3 elements of F1 of t h e same majority: this y i e l d s our conclusion. 3 - Attempt t o extend t h e statement in s e c t i o n 1 above t o t h e case of i n f i n i t e Note t h a t for m = 2 and al = a? = W the c a r d i n a l s a ( i = 1,. ,ml
Fi
Fi
.
,
.. . i = b2 = ~3 do not hold:
values bl we do not have a simple i n f i n i t a r y theorem analogous t o RAMSEY's theorem. Indeed, taking F1 = F2 = is , t h e s e t of i n t e g e r s , i t s u f f i c e s t o c o l o r (+) t h e ordered p a i r s ( x , , ~ , ) i f f x l \ < x2 . However, with t h e axiom of choice, ERDOS, RADO 1956 proved t h a t one can take bl countable and b2 equipotent w i t h t h e continuum, o r conversely: s e e e x e r c i s e 3 below. EXERCISE 3
-
A PARTITION THEOREM WITH POINTS, i . e . ORDERED PAIRS
Take a denumerable s e t D , a s e t C equipotent w i t h t h e continuum, and t h e Cartesian product D x C whose elements ( x , y ) s h a l l be c a l l e d points with t h e abscissa x in D and the o r d i n a t e y i n C P a r t i t i o n the p o i n t s i n t o two c l a s s e s c a l l e d t h e c o l o r s (+) and ( - ) . We s h a l l show t h a t t h e r e e x i s t s either a denumerable s u b s e t Do fo D subset Co of C equipotent w i t h t h e continuum, w i t h Do x Co e n t i r e l y of c o l o r ( - ) , o r two denumerable s u b s e t s D1 0-f D and C1 C with D1 x C1 e n t i r e l y of c o l o r (+) (uses axiom of choice; ERDOS, RADO 1956 p. 482).
.
-
1 Assume t h a t t h e r e does not e x i s t any (-)-monochromatic s e t which i s t h e Cartesian product of a denumerable s e t w i t h a continuum-equipotent set. Take an
a r b i t r a r y denumerable subset X of D and an a r b i t r a r y s u b s e t Y of C which is continuum-equipotent: by hypothesis X x Y i s never (-)-monochromatic. Call Y o t h e projection onto Y of t h e s e t of points of X x Y w i t h c o l o r (+), and note t h a t , by our hypothesis, Card(Y-Yo) i s s t r i c t l y l e s s than t h e
Chapter 3
89
continuum. Every p a r t i t i o n of a continuum-equipotent s e t i n t o countably many c l a s s e s y i e l d s a t l e a s t one c l a s s which i s continuum-equipotent (ch.1 5 4 . 3 , axiom of c h o i c e ) . T h u s there e x i s t s an element xo i n X y i e l d i n g continuum many points w i t h abscissa x o and o r d i n a t e s i n Y with t h e c o l o r (+). 2 - W i t h the s e t s X, Y a s above, f o r each point y in Y , denote by f ( y ) the s e t of a b s c i s s a s x i n X such t h a t the point ( x , y ) has c o l o r (t). Note t h a t f ( y ) i s not always f i n i t e , f o r otherwise t h e r e would e x i s t a continuumequipotent s e t of o r d i n a t e s y w i t h t h e same f ( y ) , hence the same complement X - f ( y ) and f i n a l l y a (-)-monochromatic s e t which i s t h e Cartesian product of a denumerable and continuum-equipotent s e t . Thus t h e r e exists an element y o of Y y i e l d i n g continuum many p o i n t s w i t h o r d i n a t e yo and a b s c i s s a s i n X , having t h e c o l o r (t). 3 - Beginning w i t h Xo = D and Yo = C , take x1 i n Xo y i e l d i n g continuum many points w i t h a b s c i s s a x1 and c o l o r ( + ) . Denote by Y1 t h e continuumequipotent set of o r d i n a t e s . Take y1 i n Y1 y i e l d i n g a denumerable s e t of points with o r d i n a t e y1 and c o l o r ( + ) . Denote by XI t h e denumerable set of {xl} and Y1 - { y l \ , thus a b s c i s s a s . I t e r a t e t h i s , beginning v i t h X1 obtaining an element x2 of t h e f i r s t s e t , and y2 an element of t h e second, w i t h continuum many points with a b s c i s s a x2 and c o l o r (+), and denumerably many points w i t h o r d i n a t e y2 and c o l o r (+). The f o u r points w i t h a b s c i s s a s x l , x2 and o r d i n a t e s y l , y2 have c o l o r ( + ) . By i t e r a t i o n , obtain a denumerable s e t of a b s c i s s a s x i and a denumerable s e t o f o r d i n a t e s y i y i e l d i n g t h e color (t) f o r a l l p o i n t s ( x i , y . ) ( i , j p o s i t i v e i n t e g e r s ) . J 4 - I t follows t h a t i f the plane, which i s the Cartesian product of two sets both equipotent w i t h t h e continuum, i s p a r t i t i o n e d i n t o two c o l o r s , then e i t h e r t h e r e exists a denumerable " g r i d " of t h e f i r s t c o l o r , or t h e r e e x i s t both a g r i d obtained a s t h e product of denumerably many a b s c i s s a s w i t h continuum many o r d i n a t e s , and a g r i d obtained as t h e product of continuum many abscissas with denumerably many o r d i n a t e s , both of t h e second c o l o r . On t h e o t h e r hand, using SIERPINSKI's counterexample i n 3.1, we can p a r t i t i o n t h e plane i n t o f o u r c o l o r s so t h a t every monochromatic g r i d i s a t most the product of a denumerable s e t of a b s c i s s a s w i t h a denumerable s e t of o r d i n a t e s .
-
90
THEORY OF RELATIONS
Problem. Does there e x i s t a p a r t i t i o n of the plane into two colors, or into three colors, such t h a t every monochromatic grid i s a t most the product o f a denumerable s e t with a denumerable set. EXERCISE 4 - SPERNER'S LEMMA Let E be a s e t with f i n i t e cardinality 2h or 2 h + l . Then every s e t o-f sub. s e t s of E which are mutually incomparable with respect t o inclusion has 2 (even case) ( 2 h + l ) ! / h ! ( h + l ) ! (odd case) cardinality a t most ( 2 h ) ! / ( h ! ) In other words, the largest possible cardinality i s obtained by taking the s e t of a l l h-element subsets. Beginning with a s e t Jf o f subsets of E , none of which i s included in another, by replacing i f necessary each s e t by i t s complement, we can always assume t h a t We shall prove t h a t the smallest cardinality of the elements of 4 i s p,< h i f p < h (even case) or p 4 h (odd case) , then we can injectively substitute for every subset A of minimum cardinality p a superset B of A of cardinality p + l . Indeed, i f t h i s i s possible, then the B will be d i s t i n c t
.
or
.
and of the same cardinal p + l , hence will be mutually incomparable with respect t o inclusion. Moreover, no B can be included in an element of of larger cardinality, since then the p-element s e t A from which B was obtained would i t s e l f be included in t h a t s e t . Now note t h a t SPERNER's lemma e a s i l y follows from the l i n e a r independence lemma 5.1 in the case of E f i n i t e and q = 1 For further developments on t h i s subject, see POUZET, ROSENBERG 1982.
4
.
91
CHAPTER
4
GOOD AND BAD SEQUENCE, FINITELY FREE PARTIAL ORDERING, WELL PARTIAL ORDERING, IDEAL, TREE, DIMENSION
§
1 - PARTIAL
ORDERING
OF THE INITIAL
INTERVALS,
PARTITION
NI
SLICES
1.1. INITIAL INTERVALS Let A be a p a r t i a l ordering, k a s e t of i n i t i a l intervals of A ; by an abuse of language, we confuse the notion of interval as subset of the base with t h a t o f r e s t r i c t i o n of A , and speak of the union and intersection of &! The union of i s an i n i t i a l interval o f A and i s the supremum or l e a s t upper bound of with respect t o inclusion; similarly the intersection of 4 i s an i n i t i a l interval and i s the infimum of
.
u
u.
I n the case of a chain, o r t o t a l 01 jering A , we get a total ordering of the i n i t i a l intervals of A . To each i n i t i a l interval X of A , associate the final i s said t o be interval X ' , the complement of X ; the ordered pair ( X , X ' ) a of A . The comparison ordering between cuts i s defined by inclusion of t h e i r f i r s t terms, which are i n i t i a l intervals.
cut
Separate the s e t of cuts of A into two complementary subsets '& and v , such i s l e s s t h a n every cut in v ; we have DEDEKIND's r e s u l t : that every cut in has a maximum c u t , or ?f has a minimum cut. 0 Take the union C of the i n i t i a l intervals in the cuts in , and the final interval C ' , the complement of C . Then (C,C') i s the supremum of If t h i s cut belongs t o then i t i s the maximum; i f i t belongs t o t ' then i t i s the minimum. 0
u
u
u
.
A . If A',A") i s a cut of A , then of B and A ' t o the intersection of the bases, and the common r e s t r i c t i o n B" of B and A " constitutes a cut ( B ' , B " ) of B , called the restriction, of the cut (A'"") t o B We also say t h a t (A'"") is a cut induced by ( B ' , B " ) on A Obviously there may e x i s t many such cuts.
l e t A be a chain, B the common r e s t r i c t i o n
a r e s t r i c t i o n of B'
.
. .
1 . 2 . Let E be a s e t , f a s e t of subsets of E Two d i s t i n c t elements of E are said t o be separable by f , i f there e x i s t s an element of f which contains one and n o t the other.
be a s e t of subsets of E , which we assume t o be t o t a l l y ordered under Let t o be a maximal chain with respect t o inclusion, i t i s inclusion. necessary and s u f f i c i e n t t h a t f? be closed under unions and intersections
For
92
THEORY OF RELATIONS
( f i n i t e or otherwise), and t h at any two d i s t i n c t elements of
E
be separable
txv. I n general, l e t L& be a s e t of subsets of E , closed under union and intersection. Let f? be a subset of & which i s t o t a l l y ordered by inclusion. t o be maximal among t o t a l l y ordered r es t r i ctions of LR , i t i s necessary For and intersection, and tha t any t w q and s u f f i c i e n t t h a t f be closed under union.-___ d is t i n c t elements of E which are separable by ‘4 are also separable by f? .
Assume t h a t i s a chain which i s maximal among the re stric tions of &? . Then i s closed under union and intersection, f o r otherwise we could add the unions and intersections of elements of ‘f . Moreover, i f x , y are two d i s t i n c t elements of E which are separated by an element X o f ‘4 with x belonging t o X yet n o t y , then denote by U the union of the elements of which contain neither x nor y , and by V the intersection of the elements of which contain both x and y . By the preceding discussion, U and V are . elements of Since (e i s t o t a l l y ordered by inclusion, then U i s s t r i c t l y included in V Either ‘Cr contains an element separating x and y , in which case our conclusion holds. Or every element of f i s a subset o f U or a superset of V . Hence U and V are consecutive with respect t o inclusion in the chain So ( U u X ) n V i s an element of (R , situated between U and V with respect t o inclusion, and d i s t i n c t from U and V as i t contains x b u t n o t y . This . contradicts the maximality o f Conversely, assume t h at the chain i s not maximal among the t o t a l l y ordered r e s tr i c t i o n s of & . We can thus add t o f? a subset W of E which i s an element of L & , and e i t h e r including Or included in every s e t which belongs t o which are included in W , . Denote by U the union of those elements of and by V the intersection of those elements o f which include W Then e ith e r i t i s the case t h at U or V does not belong t o : so tha t is e ith e r n o t closed under union or under intersection. Or i t i s the case that U and V belong t o ‘t? : so t h a t W i s d i s t i n c t from U and V , and hence i s properly situated between U and V . Let u be an element of W-U and v an element of V-W : the elements u and v are separated by the element W .0 of dz , and y e t are n o t separated by any element o f
0
.
.
.
1.3. For any p a r t i a l ordering A , denote by 3 ( A ) the partial ordering o f the i n i t i a l intervals o f A (with respect t o inclusion). We know tha t 9 ( A ) i s closed under union and intersection. Moreover any two elements x , y of I A I are separable by 3 ( A ) : indeed we can assume t h a t x < y o r x l y (mod A) , and i t suffices t o take the interval 5 x (mod A) t o separate x and y
.
Chapter 4
93
To every t o t a l l y ordered reinforcement B of A (see ch.2 § 4 ) there corresponds the t o t a l ordering 3 ( 6 ) of the i n i t i a l intervals of B . These intervals are chains (mod B ) , and although n o t necessarily chains (mod A ) , they remain i n i t i a l i s an element of 3(A) . intervals (mod A ) . I n other words, every element of J ( B ) Let A be a p a r t i a l ordering. If B i s a t o t a l l y ordered reinforcement of A , then 3 ( B ) i s a total ordering which i s maximal among those tota lly ordered r e s tr i c t i o n s of 'j ( A ) . Conversely, every maximal t o t a l l y ordered re stric tion of 3 ( A ) i s of the form g(B) , where B i s a t o t a l l y ordered reinforcement of A ; moreover this B i s unique (BONNET, POUZET 1969). 0 Starting with a t o t a l l y ordered reinforcement B of A , we already know t h a t
3(B) i s closed under union and intersection. Moreover, any two d i s t i n c t elements x , y of IA I are separable by J (B) : suppose for example tha t x c y x (mod 6 ) . By the preceding 1 . 2 , the total (mod B ) and take the interval ordering 3(B) i s maximal among those t o t a l l y ordered re stric tions of 3 ( A ) . Conversely, l e t f be a t o t al ordering which i s maximal among those tota lly ordered r e s t r i c t i o n s of 3 ( A ) . By 1 . 2 , t h i s kf i s closed under union and intersection, and any two d i s t i n c t elements in the base I A I are separable by an element of To obtain = j ( B ) , define the t o t a l l y ordered reinforcement B of A by the condition t h a t , when given two elements x , y of A , we p u t xd y (mod 6 ) i f f every element of which contains y as an element also contains x as an element. The antisymmetry of B follows from the fa c t t h a t , f o r d i s t i n c t x, y , there ex i s t s an element of which separates them. Moreover, since f i s a t o t al ordering, B i s also. Finally, the uniqueness of B follows from the f a c t t h at two d i s t i n c t t o t a l orderings B and B ' yield two d i s t i n c t 3 ( B ) and Y ( B ' ) . 0
.
e
1 . 4 . I n p a r t i c u l a r , l e t E be an ar b i t r ar y s e t . For every chain B based on E , the i n i t i a l intervals of B form a maximal t o t a l l y ordered r e s t ri ct i o n 3 ( 6 ) of the p ar t i al ordering of inclusion. Conversely, every chain which i s a maximal r es t r i ct ion of the pa rtia l ordering of inclusion of subsets of E i s the s e t of i n i t i a l intervals 3(B) for a certain chain B on E .
AXIOM OF MAXIMAL CHAIN OF INCLUSION This axiom asserts t h a t , f o r every s e t E , the p artia l ordering of inclusion among subsets o f E admits a maximal t o t al ordering among i t s re stric tions. I t i s considerably weaker than the general maximal chain axiom, as stated in ch.2 5 2.7: the HAUSDORFF-ZORN axiom, equivalent t o the axiom of choice. By the preceding proposition, the axiom of maximal chain o f inclusion i s equivalent t o the ordering axiom, ch.2 5 2.3.
94
THEORY OF RELATIONS
AXIOM OF MAXIMAL C H A I N ON THE SET OF INITIAL INTERVALS This axiom a s s e r t s t h a t , given a partial ordering A , the partial ordering of inclusion on the i n i t i a l intervals of A admits a maximal chain among i t s r e s t r i c .-t I t i s equivalent t o the reinforcement axiom, ch.2 5 4.2. 1.5. Starting with an a r b i t r a r y partial ordering A , there e x i s t s t r a t i f i e d partial ordered reinforcements of A (see ch.2 5 5 . 2 ) : f o r examole the chains which are reinforcements of A . We obtain as follows a minimal s t r a t i f i e d reinforcement, in the sense t h a t i f x ( y (modulo the reinforcement), then e i t h e r x c y (mod A) , or there e x i s t x ' , y ' w i t h x x' I y ' + y or with x & X I I y ' < y (mod A ) (however the converse can be f a l s e ) .
-=
LEMMA FOR PARTITION IN SLICES Let A be a p a r t i a l ordering. There e x i s t s an equivalence relation R for which the equivalence classes are free subsets (mod A ) , and a & c
on H
IAI
on the s e t of equivalence classes, such t h a t f o r any two elements x , y of I A I : (equivalence class of x ) 6 (equivalence class of y ) modulo H i f f there e x i s t two elements X I , y ' which are equivalent (mod R ) and s a t i s f y and y ' d y (mod A ) (uses axiom o f choice; BONNET, POUZET 1969).
x + x'
F i r s t of a l l , denote by R any equivalence relation on the base 1 A l , and by H any chain on the s e t of the equivalence classes of R , which s a t i s f y the following condition: (equivalence class of x ) 6 (equivalence class of y ) (mod H) i f f e i t h e r x i s equivalent t o y (mod R ) , or there e x i s t s an equivalence class U of R which i s a f r e e subset of I A l and two elements x ' , y ' of U with x & x ' and y ' b y (mod A) . Such ordered pairs ( R , H ) e x i s t : i t suffices t o take R t o be the t r i v i a l equivalence relation with one equivalence class I A I , and H the chain on the singleton of I A I . An ordered pair ( R ' , H ' ) i s said t o be finer t h a n ( R , H ) i f every equivalence class (mod R ) which i s f r e e (mod A) remains an equivalence class (mod R ' ) , and moreover every equivalence class (mod R ) i s a union of equivalence classes (mod R ' ) constituting an interval of H ' , and f i n a l l y f o r any two of these i n t e r v a l s , the t o t a l ordering induced by H ' i s identical t o the t o t a l ordering (mod H ) of the corresponding equivalence classes. The comparison " f i n e r than" defines a p a r t i a l ordering o n the ordered pairs ( R , H ) We shall prove t h a t an ordered pair ( R , H ) f o r which the equivalence classes are n o t a l l free (mod A) admits an ordered pair (R' ,HI) which i s s t r i c t l y f i n e r . For thisn take an equivalence class U of R which i s not free. Take a subset V of IJ which i s maximal free (mod A) (ch.2 5 2 . 9 ) , and p a r t i t i o n U into three d i s j o i n t subsets: V , the s e t of elements having a greater element (mod A ) in V , and the s e t of elements having a smaller element (mod A ) in V
0
.
.
Chapter 4
95
Define the equivalence classes (mod R ' ) as those (mod R ) p a r t i t i o n e d i n t o three equivalence classes (mod R ' ) as
H
, except
ordering
f o r the s u b s t i t u t i o n o f
U
.
, excepting U which i s
The chain
i s t h e same
H'
by t h e t h r e e classes w i t h t h e obvious
.
Using HAUSDORFF-ZORN axiom, consider a maximal ordered p a i r obtained from a maximal chain o f ordered p a i r s than". For t h i s maximal a l l f r e e (mod A)
(R,H)
, totally
(R,H)
, the
ordered by the comparison " f i n e r
equivalence classes defined by
R
are
.0
1.6. There e x i s t s a p a r t i a l o r d e r i n g w i t h c a r d i n a l i t y W 1 i n which every chain and every a n t i c h a i n i s countable (uses axiom o f choice; compare w i t h ch.7 exerc. 4 ) . 0
To every countable o r d i n a l
u i n j e c t i v e l y associate a r e a l r ( u )
choice). Define a p a r t i a l o r d e r i n g based on the countable o r d i n a l s setting
u
o r d i n a l s and
i f f simultaneously r(u)
<
(axiom o f
...
u, v
by
u < v modulo t h e usual ordering o f the
modulo the usual o r d e r i n g o f t h e r e a l s . To o b t a i n a
r(v)
t o t a l l y ordered r e s t r i c t i o n o f t h i s p a r t i a l ordering, one must take a s t r i c t l y i n c r e a s i n g sequence o f r e a l s : such a sequence i s countable: ch.1 fj 4.5. To obtain a f r e e set, one must take a s t r i c t l y decreasing sequence o f r e a l s . 0 1.7. The existence o f a
tree (as
defined i n fj 6 below) o f c a r d i n a l
in
cJ1
which every chain and every a n t i c h a i n i s countable, i s n o t provable i n
ZF and
even n o t provable using t h e axiom o f choice and the continuum hypothesis. This problem i s r e l a t e d t o SUSLIN's hypothesis: see ch.5 fj 5 below. 1.8. GENERATED INITIAL INTERVAL, FINITELY GENERATED INTERVAL Given a p a r t i a l o r d e r i n g A generated by
D
(mod A)
A
An i n i t i a l i n t e r v a l o f
and a subset
0 o f i t s base, t h e i n i t i a l i n t e r v a l
i s t h e s e t o f elements having an upper bound i n
D
.
i s s a i d t o be f i n i t e l y generated i f i t i s generated
by a f i n i t e subset o f the base. We denote by r ( A ) t h e p a r t i a l o r d e r i n g (under i n c l u s i o n ) o f f i n i t e l y generated i n i t i a l i n t e r v a l s o f A .
If A
i s a well-founded p a r t i a l ordering, then so i s I f ( A )
(uses axiom o f
dependent choice; see BIRKHOFF 1948).. 0
To each f i n i t e l y generated i n i t i a l i n t e r v a l
mal elements (mod A)
, which
X
associate t h e s e t o f i t s maxi-
i s f i n i t e and non-empty (unless
X
i s itself
empty), and which i s a f r e e set, i . e . a s e t o f mutually incomparable elements. L e t US say t h a t a s e t
Y
i s smaller than
Y'
i f f f o r each element y
in Y
THEORY OF RELATIONS
96
t h e r e e x i s t s a t l e a s t one upper bound o f
Y ' (mod A)
in
y
. Then
i t suffices t o
prove t h a t t h e r e does n o t e x i s t any s t r i c t l y d e c r e a s i n g u - s e q u e n c e o f f i n i t e f r e e s e t s , w i t h r e s p e c t t o t h e p r e c e d i n g comparison (dependent c h o i c e ) . Given such a sequence
Yi
, either
( i integer)
each element o f
belongs t o f i n i t e l y many
Yo
Yo i s d i s j o i n t f r o m a l l o t h e r s e t s Y O r t h e r e e x i s t elements o f Yo which b e l o n g t o i n f i n i t e l y many s e t s Yi . Then t o which i t belongs. By i t e r a remove one o f these elements and keep o n l y t h e Yi t i n g t h i s procedure, we e l i m i n a t e a l l such elements f r o m Yo b u t n o t a l l elements o f Yo . F i n a l l y we o b t a i n a new, non-empty Yo which i s d i s j o i n t f r o m a l l o t h e r s e t s Yi . I t e r a t i n g t h i s , we o b t a i n a s t r i c t l y d e c r e a s i n g w-sequence o f f i n i t e
sets
Yi
: t h e n remove these
so t h a t
Yi
.
free, mutually d i s j o i n t sets. Zi
be these d i s j o i n t s e t s . There e x i s t s a t l e a s t one element
zo
i n Zo zo i n Z1 w i t h s m a l l e r elements i n a l l s e t s Zi (i> 1) ; and so on. Thus we have a s t r i c t l y d e c r e a s i n g w - s e q u e n c e o f elements (mod A): c o n t r a d i c t i o n . 0 Let
w i t h s m a l l e r elements i n a l l o t h e r s e t s
Zi
. Then
a t l e a s t one
z1
<
5 2 - LESS THAN
R E L A T I O N AMONG SEQUENCES, EMBEDDING BETWEEN SEQUENCES, GOOD, BAD, MINIMAL BAD, STRONGLY MINIMAL BAD SEQUENCE u i n t h e p a r t i a l o r d e r i n g A , t h e i n i t i a l i n t e r v a l generated u i s t h a t generated by t h e s e t o f values o f u : see 1.8 above.
Given a sequence by
LESS THAN RELATION, EMBEDDING BETWEEN SEQUENCES Let
A
be a p a r t i a l o r d e r i n g and
w i t h t h e same l e n g t h g smaller than
.
u , or u i < o(
f o r each i n d e x
u, v be two sequences w i t h values i n A , b o t h v i s s a i d t o be l e s s t h a n u (mod A ) o r
The sequence
i s s a i d t o be q r e a t e r t h a n
. This
thus d e f i n e s a p a r t i a l o r d e r i n g on e v e r y s e t o f sequences i n Given a p a r t i a l o r d e r i n g
Consequently ( l e n g t h o f
if
vi<
ui (mod A)
u, v
in
A
A
o f a given length.
, we say t h a t v i s
u admits an embedding o f v (mod A) , i f t h e r e e x i s t s
embeddable i n u o r t h a t an e x t r a c t e d sequence o f
and two sequences
A
,
v (mod A)
comparison i s r e f l e x i v e , a n t i s y m m e t r i c , t r a n s i t i v e ,
u
(ch.1
5
2.2) which i s g r e a t e r t h a n
v)s ( l e n g t h o f
v (mod A)
.
u). Embeddability i s r e f l e x i v e , t r a n s i t i v e
b u t n o t a n t i s y m m e t r i c , s i n c e t h e n o t i o n o f e x t r a c t e d sequence i s n o t i t s e l f a n t i symmetric: i t d e f i n e s a q u a s i - o r d e r i n g on e v e r y s e t o f sequences i n A
.
Two sequences which a r e each embeddable i n t h e o t h e r , a r e n o t n e c e s s a r i l y e x t r a c t e d one f r o m t h e o t h e r : i n t h e t o t a l o r d e r i n g o f t h e p o s i t i v e i n t e g e r s , t a k e t h e i n c r e a s i n g sequences o f even i n t e g e r s , r e s p e c t i v e l y odd i n t e g e r s .
I f a sequence
v
i s embeddable i n
u
, then
t h e i n i t i a l i n t e r v a l generated b y v
i s i n c l u d e d i n t h e i n i t i a l i n t e r v a l generated by axiom o f choice, s t a r t i n g w i t h an i n i t i a l i n t e r v a l suffices t o well-order
IA I
u
. Note A
a l s o t h a t , modulo t h e of a p a r t i a l o r d e r i n g , i t
i n o r d e r t o o b t a i n a sequence which generates
A
.
97
Chapter 4
PARTIAL ORDERING OF WORDS Starting with a p a r t i a l ordering o f elements o f t h e base I A l
.
A
, take
t h e s e t o f words, o r f i n i t e sequences
E m b e d d a b i l i t y between words i s a n t i s y m m e t r i c and
so d e f i n e s a p a r t i a l o r d e r i n g o f words. If A
i s a w e l l - f o u n d e d p a r t i a l o r d e r i n g , t h e n t h e p a r t i a l o r d e r i n g o f words i n
A
i s well-founded.
A , o b t a i n a minimal word i n t h i s
0 S t a r t i n g w i t h a non-empty s e t o f words i n
n
set. F i r s t l y t a k e t h e subset o f words o f minimum l e n g t h u, uo, u1 where
words beginning w i t h
2.1. INITIAL INTERVAL OF A SEQUENCE Given a sequence u w i t h l e n g t h o( u
tion o f
, and among such, those
o f minimal value. Then among t h e l a t t e r , those
words beginning w i t h a t e r m
u1 has minimal value; and so on. 0
, an
i n i t i a l interval o f
t o an o r d i n a l domain s m a l l e r t h a n o r equal t o o(
u
i s any r e s t r i c -
. The
val i s c a l l e d s t r i c t o r p r o p e r i f f i t s l e n g t h i s s t r i c t l y l e s s t h a n Let A in A
be a p a r t i a l o r d e r i n g ,
.
If the length o f
interval o f 0 Let
h(0)
u
u
u, v
o(
.
be two o r d i n a l - i n d e x e d sequences w i t h values and i f each p r o p e r i n i t i a l
i s a l i m i t ordinal
i s embeddable i n v
i n i t i a l inter-
(mod A )
,then
u
vh(o)
be t h e l e a s t o r d i n a l such t h a t
i s embeddable i n v
uo (mod A)
. By
.
induction,
i s t r i c t l y l e s s than , assume t h a t t h e s t r i c t l y i n c r e a s i n g o f t h e h ( j ) (j < i) i s d e f i n e d , and l e t h ( i ) be t h e l e a s t o r d i n a l Clearly t h i s g r e a t e r t h a n t h e h ( j ) and such t h a t v ~ ( ~ ) >ui' (mod A) t h e s m a l l e s t p o s s i b l e , and by h y p o t h e s i s f o r each i s t r i c t l y l e s s than h ( i ) i s s t r i c t l y less than t h e length o f v : our proposition follows. 0
given an o r d i n a l sequence strictly h(i)
is
7 , our
.
2.2. GOOD SEQUENCE, BAD SEQUENCE
Let A
be a p a r t i a l o r d e r i n g . A sequence
u
in
A
w i t h l e n g t h ( o r domain)
i, j w i t h i < j .C a and ui Q u. (mod A ) . Otherwise a sequence i s c a l l e d bad, i . e . J i f f every t e r m i s < o r I (mod A) w i t h t h e terms o f s t r i c t l y s m a l l e r i n d e x . Every sequence e x t r a c t e d f r o m a bad sequence i s bad. Given a bad c s - s e q u e n c e u , f o r e v e r y uo o b t a i n e d f r o m u by a p e r m u t a t i o n o( i s c a l l e d &(mod
A) i f f t h e r e e x i s t a t l e a s t two i n d i c e s
o f the s e t o f i n d i c e s , t h e r e e x i s t s a bad
w
-sequence e x t r a c t e d f r o m
uo
.
We can even r e q u i r e t h a t t h e bad e x t r a c t e d sequence b e g i n w i t h , f o r i n s t a n c e , the f i r s t t e r m
uo
of
u
.
2.3. MINIMAL BAD SEQUENCE Let A
be a p a r t i a l o r d e r i n g . A bad a - s e q u e n c e
u
in
A
i s s a i d t o be
THEORY OF RELATIONS
98
minimal bad i f , f o r any &-sequence
v
e x t r a c t e d from
u , every bad sequence
.
v (mod A) i s equal t o v e x t r a c t e d from a minimal bad sequence i s minimal bad. I n a minimal bad sequence, a l l t h e terms are m u t u a l l y incomparable. 0 L e t u be a minimal bad sequence i n t h e p a r t i a l o r d e r i n g A , and l e t p C q Define t h e e x t r a c t e d sequence v by be two i n t e g e r s w i t h u > u (mod A) P 9 vo = uo ,..., v = up-l , v = u q < u , and vpti = uq+i f o r every i 3 1 P P P: 1 Then v i s bad since i t i s e x t r a c t e d from u Moreover v i s a sequence l e s s u p-1 ' up ' Uq+l ' uq+2 ,... which i s e x t r a c t e d from u , and than uo as v i s d i s t i n c t from t h i s e x t r a c t e d sequence, t h i s c o n t r a d i c t s t h e m i n i m a l i t y of u . 0 l e s s than
Every a-sequence -
.
.
,...,
L e t A be a p a r t i a l ordering, u a minimal an integer, and x < u (mod A) Then x P p o s s i b l y f i n i t e l y mani. 0 Assume on the c o n t r a r y t h a t there e x i s t s a integers p < f ( 1 ) < f ( 2 ) < Cf(i)<
bad sequence i n A , and l e t p be terms o f u (mod A) , except
. - < all
...
the sequence x, uf(l)
,..., uf(i) ,...
s t r i c t l y i n c r e a s i n g sequence o f with u < o r 1 x (mod A) . Then f(i) i s a bad sequence l e s s than up, uf(l),...
...
.
u ~ ( ~ ) , . . which c o n t r a d i c t s t h e m i n i m a l i t y o f u 0 Notice t h a t a sequence w i t h incomparable terms which s a t i s f i e s t h e c o n d i t i o n o f t h e preceding p r o p o s i t i o n i s n o t n e c e s s a r i l y minimal. A t o be a well-founded p a r t i a l o r d e r i n g formed from denumerably many minimal elements which we denote by t h e i n t e g e r s 0, 1, 2 , , . and w i t h O'> 0
0 Take
,
...,
1'> 0 and 1 ' > 1 , and f o r every i n t e g e r i an element i ' 7 0, 1, i and f i n a l l y make t h e i' m u t u a l l y incomparable. Then t h e sequence 0 ' , 1' , 2 ' ,.. . has incomparable terms and s a t i s f i e s t h e c o n d i t i o n o f t h e preceding p r o p o s i t i o n , but t h e sequence 0, 1, 2, considered. 0 2.4.
...
i s a bad sequence which i s l e s s than t h e sequence
Let A be a p a r t i a l ordering. I f u i s a minimal bad w-sequence
(mod A), then the image of i n d i c e s i s minimal bad. 0 Denote by
the terms of
u by an a r b i t r a r y permutation o f t h e s e t o f
uo t h e image o f u under a permutation o f t h e i n d i c e s . By 2.3, u
, hence
of
uo
, are
m u t u a l l y incomparable, so t h a t
uo i s
bad. Suppose t h a t uo i s n o t minimal bad. Then t h e r e e x i s t s an W -sequence v e x t r a c t e d from u0 and a s m a l l e r w -sequence w , which i s bad and d i s t i n c t from v We can assume t h a t the f i r s t terms o f v and w are d i s t i n c t , so
.
.
wo < vo (mod A) Retransform uo, v and w by t h e i n v e r s e o f our i n i t i a l permutation. Then we have again u and we o b t a i n t h e images o f v and w L e t w' denote t h e l a t t e r image. By 2.2 t h e r e e x i s t s a bad m-sequence
.
Chapter 4
99
extracted from w ' , which begins by wo (more exactly, by the term of w ' whose value i s wo ) . This contradicts the minimality of u ( t h i s proof using only ZF i s communicated by H O D G E S ) . 0 2.5. Given a partial ordering A and a minimal bad sequence u in A , the i n i t i a l interval of A generated by u i s a well-founded partial ordering (uses dependent choice; ZF suffices i f A i s countable, or obviously i f A i s well-founded). 0 Assume on the contrary t h at there ex i s t s a term u of u with a s t r i c t l y P > ai > ... (mod A) (dependent -sequence up decreasing a. > a l 7 choice). Then a. 6 a l l except f i n i t e l y many of the terms of u : see 2.3.
...
>
Hence there exists an integer q such that a. 6 uqti Finally we have ai < uq+i f o r every integer minimality of the sequence u 0
.
i
for every integer i .
, thus contradicting the
2.6. Let A be a p a r t i al ordering, u an (.J -sequence in A with mutually incomparable terms. Then u i s minimal bad i f f every bad a-sequence embeddable in u i s extracted from u 0 If u i s minimal bad, our conclusion follows immediately from the definition. Conversely, suppose t h at u has incomparable terms, so i s bad, and yet not minimal bad. Then there e x i s t s a s t r i c t l y increasing a-sequence of integers < f ( i ) < ... with a bad sequence x = (xo,xl ,..., xi ,...) which f(0) < f ( l ) < i s a d i st i n c t sequence less than u f ( o ) , u f ( l ) ,..., u f t i ) ,... . T h u s there e xists an integer p such t h a t x q u (mod A) . Consider the sequence o f the P f(P) x (j = 0,1,2, ...) This sequence i s embeddable in u and moreover i s bad p+j since x i s bad. By hypothesis, t h i s sequence i s extracted from u , hence (mod A) , which contradicts the hypothesis there e x i st s a term u = x 4 u q P f(p) t h a t the terms o f u are mutually incomparable. 0
.
...
.
The hypothesis o f incomparability of the terms of u i s necessary. Take A t o be a well-founded partial ordering with O < 0 ' and denumerably many integers 1,2,3, ... which are s e t t o be mutually incomparable, and also Then the sequence 0',0,1,2,3, ... i s bad yet incomparable with 0 and 0 ' not minimal bad (since 0 ' > 0 ) , and every bad w-sequence embeddable in i t i s extracted from i t . 0
.
Moreover, a bad non-minimal bad w -sequence u can sa tisfy the condition t h a t every bad sequence l es s than u i s equal t o u nTake an element a s e t t o be minimal, and the even integers s e t t o be mutually Take the odd integers 1,3,.. a l l to incomparable, with 0 > a , 2 > a , . be minimal and incomparable w i t h the preceding elements. Then the sequence of a l l
.
.. .
.
THEORY OF RELATIONS
100
integers
..
0,1,2,.
i s n o t minimal s i n c e
a
i s l e s s t h a n e v e r y even i n t e g e r . Yet
e v e r y d i s t i n c t sequence l e s s t h a n i t must t a k e t h e v a l u e
a
i n the position o f
an even i n t e g e r , and hence i s good. 0
A
Problem. L e t
be a p a r t i a l o r d e r i n g ,
r a b l e terms. F o r
u
u
an w - s e q u e n c e i n A
t o be minimal bad, does i t s u f f i c e t h a t e v e r y
w i t h incompa-
w -sequence
u be e x t r a c t e d f r o m u
w i t h incomparable terms which i s embeddable i n
.
2.7. THEOREM ON THE MINIMAL BAD SEQUENCE Let
A
be a well-founded p a r t i a l o r d e r i n g . F o r e v e r y bad w - s e q u e n c e
u
t h e r e e x i s t s a minimal bad sequence which i s embeddable i n choice; ZF s u f f i c e s i f
elements
x & uo
uo
of
u
by
v o c < uo w i t h
vo
.
u
,
minimal among those
f o r which t h e r e e x i s t s a bad w - s e q u e n c e which begins by
and i s embeddable i n
A
i s denumerable).
A
Replace t h e f i r s t t e r m
0
&
u
(uses dependent
Denote by
m - s e q u e n c e embeddable i n
u
.
vo, w1
w1
Replace
x
t h e f i r s t two terms i n such a bad by
w1
vl&
with
v1
minimal
among those
f o r which t h e r e e x i s t s a bad a - s e q u e n c e which begins by xBwl and i s embeddable i n u . Denote by vo, vl, w2 t h e f i r s t t h r e e terms i n
vo, x
such a bad w - s e q u e n c e embeddable i n v = vo, vl,
... which
v2,
over, f o r e v e r y i n t e g e r by
vo, vl,
...,
To see t h a t
v
vi-l,
i and e v e r y
x
vf(l)
,...,
,... .
vf(i)
, no w - s e q u e n c e b e g i n n i n g
.
u
v
... < f ( i ) < ...
vo, vl,
and a bad
which i s d i s t i n c t and l e s s t h a n
Then t h e r e e x i s t s an i n t e g e r
p
with
x
< v f(P) P
v
f(O) ’ (mod A)
v ~ ( ~ ) -xp, ~ , x ~ + ~ , . . .. T h i s sequence i s
...,
x
.
u
and so embeddable i n
sequence i s good. As t h e sequences
C f(p)
(dependent c h o i c e ) . More-
u
(mod A )
f(l)<
f(O)<
..., xi,...)
x = (xo, xl,
Consider t h e sequence
(mod A )
x < vi
i s minimal, suppose on t h e c o n t r a r y t h a t t h e r e e x i s t s a s t r i c t l y
embeddable i n j
By i t e r a t i o n , we o b t a i n t h e sequence
i s s i m u l t a n e o u s l y bad and embeddable i n
increasing u-sequence o f integers W -sequence
.
u
i s bad and embeddable i n
and
By t h e p r e c e d i n g i n e q u a l i t y , t h i s
v
a r e bad, t h e r e e x i s t s an i n t e g e r
and an i n t e g e r k >/ p , hence j
f(k)
with
vj
6 xk 6 v
f(k)
2.8. STRONGLY MINIMAL BAD SEQUENCE Given a p a r t i a l o r d e r i n g
A
,a
bad
minimal bad i f , f o r each i n t e g e r
w -sequence
u
in
i and each element
w -sequence i n A b e g i n n i n g by uo, ul,
....
ui-l,
x
A
x
i s called strongly
< ui
(mod A)
, every
i s good.
Every s t r o n g l y minimal bad sequence i s minimal bad. 0
Suppose t h a t
u i s s t r o n g l y minimal bad b u t n o t minimal bad. There e x i s t s a
s t r i c t l y increasing W-sequence o f integers a d i s t i n c t sequence e x i s t s an i n t e g e r
x p
l e s s than with
f(O)<
f(l)<
... < f ( i ) < ...
and
u ~ ( ~ u) ~, ( ~ ) , . . . , u ~ ( ~ ) , . . . . Hence t h e r e
x < u ~ ( ~(mod ) A) P
. The
w -sequence
uo, ul,
...
Chapter 4
u ~ ( ~ ) -xp,~ , x ~ + ~ , . .i .s good, since u
101
i s strongly minimal. As the sequences
ka
u and x are bad, there e x i s t s an integer j < f ( p ) and also p with and u . d x k ,< u f ( k ) (mod A) , hence u i s good: contradiction.
j 4 f(p)5 f(k)
J
0
There e x i s t minimal, non-strongly minimal sequences. More precisely, there e x i s t s a n w-sequence extracted from a strongly minimal bad sequence, which i s not strongly minimal; however by 2.3 i t i s minimal bad. In addition, there e x i s t s a strongly minimal bad sequence and a sequence obtained from i t by permutation of the s e t of indices, which i s not strongly minimal bad; however by 2.4 i t i s minimal bad. Take two minimal elements a , b and p u t a l l even numbers 0 , 2 , 4 , . . . t o be mutually incomparable and incomparable with b , b u t a l l 7 a . P u t a l l odd numbers 1, 3 , 5 , ... t o be mutually incomparable, a l s o incomparable with the even numbers a n d with a , y e t a l l 7 b . Then the sequence b , 0 , 2, 4, ... i s strongl y minimal; b u t 0 , 2, 4 , ... i s n o t , since, f o r each integer i , the sequence 0 , 2, 4 ,..., 2 i , a , 1, 3 , 5 ,... i s bad. 0
0
.
Take a minimum element a , then b , c mutually incomparable and > a Put the even numbers 0 , 2 , 4 , ... t o be mutually incomparable and incomparable with c , yet a l l 7 b P u t the odd numbers t o be mutually incomparable, also incomparable with the even numbers and with b , y e t a l l 7 c . Then the sequence c , 0 , 2 , 4 , ... i s strongly minimal bad; b u t some of i t s permutations are not: indeed every strongly minimal bad sequence begins with b or c . 0
0
.
2.9. There e x i s t s a minimal bad sequence which i s n o t extracted from any strongly minimal bad sequence. 0 Let A be the well-founded partial ordering constructed in the following manner. Begin with the sequence of integers i in increasing order. To each i associate an element i ' 7 i , the i ' being mutually incomparable, each i ' incomparable with a l l integers > i . Finally add i n f i n i t e l y many minimal elements ao, a l , ... which a r e incomparable with the i and i ' . Let u be the sequence of the i ' This sequence i s minimal bad, since f o r each extracted W -sequence, every d i s t i n c t smaller sequence i s good. Suppose t h a t u i s extracted from a strongly minimal bad GJ -sequence v . Then v i s obtained from u by inserting elements a i Let p be an integer such
.
.
t h a t the pth term of v i s a term of u , in other words an i ' ( i integer .\< p ) . Hence v begins by O', l ' , Z', ..., i ' between which can be inserted elements a j I t suffices t o replace i ' by i , which i s incomparable with 0 ' , 1' , , ( i - 1 ) ' and which i s < i' , and t o add a f t e r i an a-sequence of elements a . which are not already inserted, in order t o obtain a bad sequence. J
...
.
THEORY OF RELATIONS
102
This contradicts the strong minimality o f
v
.0
2.10. Theorem 2.7 as s t a t e d does n o t e x t e n d t o s t r o n g l y minimal bad sequences. 0
Take t h e w e l l - f o u n d e d p a r t i a l o r d e r i n g o f t h e second counterexample 2.6, w i t h
t h e minimal element
a
f o l l o w e d by t h e even i n t e g e r s
0,2,
...
together w i t h
t h e odd i n t e g e r s t a k e n as minimal and incomparable w i t h t h e even i n t e g e r s . Then
0,2,... i s bad and admits o n l y i t s e l f and good
t h e sequence o f even i n t e g e r s
sequences as embedded sequences. I t i s n o t s t r o n g l y minimal bad, s i n c e f o r each integer
i
, the
sequence
,..., 2i,a,1,3 ,...
0,2,4
However, g i v e n a well-founded p a r t i a l o r d e r i n g w-sequence i n A
, then
uo, x
i f t h e r e e x i s t s a bad
A
i s countable).
uo minimal among t h e f i r s t terms o f bad
an element minimal among t h o s e g i n n i n g by
,
t h e r e e x i s t s as w e l l a s t r o n g l y minimal bad sequence
(uses dependent choice; ZF s u f f i c e s i f 0 Take an element
A
i s bad. 0
x
a-sequences, then
f o r which t h e r e e x i s t s a bad
w-sequence be-
etc. 0
F i n a l l y , by t h e method i n t h e f i r s t paragraph o f t h e p r o o f o f 2.7, we see t h a t : given a well-founded p a r t i a l ordering e x i s t s a bad and each
w -sequence
x< v
P
(mod A)
v
, every w
and which i s embeddable i n We can c a l l t h i s sequence t h e sequence
§
u (as i n 2.7,
3 - FINITELY
A
and a bad w - s e q u e n c e
embeddable i n
u
,
u
in
A
, there
such t h a t f o r each i n t e g e r
-sequence b e g i n n i n g by
vo, v1 ,... , vp-l,
p x
i s good.
u
s t r o n g l y minimal w i t h r e s p e c t t o e m b e d d a b i l i t y z
v
we use dependent choice; ZF s u f f i c e s i f
FREE PARTIAL
ORDERING,
WELL PARTIAL
A
countable).
ORDERING
FINITELY FREE PARTIAL ORDERING T h i s i s a p a r t i a l o r d e r i n g such t h a t any f r e e subset o f i t s base, o r any a n t i chain, i s f i n i t e (see ch.2 § 2.9). A p a r t i a l o r d e r i n g can be w e l l - f o u n d e d w i t h o u t b e i n g f i n i t e l y f r e e : t a k e an i n f i n i t e base and t h e p a r t i a l o r d e r i n g o f i d e n t i t y , t h e n a l l elements a r e m u t u a l l y incomparable. A p a r t i a l o r d e r i n g can be f i n i t e l y f r e e w i t h o u t b e i n g well-founded: t a k e t h e n a t u r a l o r d e r i n g on t h e n e g a t i v e i n t e g e r s . Every f i n i t e p a r t i a l o r d e r i n g i s f i n i t e l y f r e e . Every r e s t r i c t i o n o f a f i n i t e l y f r e e p a r t i a l o r d e r i n g i s f i n i t e l y f r e e . Every p a r t i a l o r d e r i n g which i s a r e i n f o r c e m e n t o f a f i n i t e l y f r e e p a r t i a l ordering
i s f i n i t e l y f r e e (see ch.2
5
4.1).
3.1. Given a f i n i t e l y f r e e p a r t i a l o r d e r i n g a t o t a l l y ordered r e s t r i c t i o n o f
A
A w i t h i n f i n i t e base, t h e r e e x i s t s A (uses axiom o f
which i s e q u i p o t e n t w i t h
Chapter 4
103
choice; ZF s u f f i c e s i f A i s countable o r i f Card A i s a r e g u l a r aleph!. P a r t i t i o n t h e p a i r s o f elements x, y o f the base I A l i n t o two c o l o r s , according t o whether x and y are comparable o r incomparable (mod A). Assuming the
0
axiom o f choice, take an aleph equipotent t o t h e base, and apply the p a r t i t i o n theorem ch.3 5 3.3 (DUSHNIK, M I L L E R ) . Then e i t h e r t h e r e e x i s t s a denumerable subs e t w i t h a l l i t s elements incomparable: c o n t r a d i c t i o n . O r t h e r e e x i s t s a subset w i t h same c a r d i n a l i t y as t h e base, a l l o f whose elements a r e comparable. 0 3.2. WELL PARTIAL ORDERING This i s a p a r t i a l o r d e r i n g which i s well-founded and f i n i t e l y f r e e . For example, every f i n i t e p a r t i a l o r d e r i n g i s a w e l l p a r t i a l ordering. S i m i l a r l y f o r any well-ordering, o r even a w e l l - o r d e r i n g where each element i s replaced by a f r e e f i n i t e set. E j g , A quasi-ordering ( r e f l e x i v e and t r a n s i t i v e b i n a r y r e l a t i o n ) i s s a i d t o be a w e l l q u a s i - o r d e r i n g iff t h e p a r t i a l o r d e r i n g o f t h e equivalence classes i s a w e l l
p a r t i a l o r d e r i n g (each c l a s s i s formed by elements each o f which i s both g r e a t e r and s m a l l e r than other elements). i s a w e l l p a r t i a l o r d e r i n g i f f f o r any non-empty subset
(1) A
X
o f the base,
the s e t o f minimal elements (mod A/X) i s f i n i t e and non-empty. 0 Let
A Since A
be a w e l l p a r t i a l o r d e r i n g and X a non-empty subset o f t h e base. i s well-founded, t h e r e e x i s t minimal elements (mod A/X): ch.2 5 2.4.
These elements are mutually incomparable (mod A), hence there are o n l y f i n i t e l y many such, since A i s f i n i t e l y f r e e . Conversely i f A i s n o t a w e l l p a r t i a l ordering, then e i t h e r there e x i s t s a subset X o f the base w i t h o u t any minimal element, o r t h e r e e x i s t s an i n f i n i t e f r e e subset X I n t h e l a t t e r case a l l t h e elements o f X are minimal (mod A/X). 0
.
( 2 ) A necessary and s u f f i c i e n t c o n d i t i o n f o r
A t o be a w e l l p a r t i a l ordering, Cd-sequence i n A be good ( t h e s u f f i c i e n c y uses dependent choice; ZF s u f f i c e s i f A i s countable o r has well-orderable base). 0 L e t A be a w e l l p a r t i a l ordering. I f t h e r e e x i s t s a bad w-sequence i n A ,
i s that
every
then t h e r e i s an e x t r a c t e d
w -sequence which i s e i t h e r s t r i c t l y decreasing o r
with a l l i t s terms m u t u a l l y incomparable: see ch.3 5 1.2 (RAMSEY). Conversely, i f A i s n o t a w e l l p a r t i a l ordering, then e i t h e r A i s n o t w e l l founded, so there e x i s t s a s t r i c t l y decreasing w-sequence (ch.2 § 2.4, dependent choice). O r t h e r e e x i s t s an i n f i n i t e f r e e set, hence an w-sequence o f m u t u a l l y incomparable elements: i n both cases, a bad sequence. 0 (3) A necessary and s u f f i c i e n t c o n d i t i o n f o r A
t o be a w e l l p a r t i a l o r d e r i n g i s
t h a t , f o r every w-sequence i n A w i t h d i s t i n c t values, there e x i s t s an
104
THEORY OF RELATIONS
extracted w -sequence which i s s t r i c t l y increasing (mod A ) . Same conditions and same proof as f o r ( 2 ) . 3.3. A necessary and s u f f i c i e n t condition f o r A
t o be a well partial ordering
i s t h a t , for each element u of the base I A I , the r e s t r i c t i o n of A elements < 1 u i s a well partial orderinq. 0 I f A i s a well partial ordering, then every r e s t r i c t i o n of A i s a well partial ordering: hence the necessity of our condition. Conversely, assume t h a t the condition holds. Let X be a non-empty f r e e subset of the base. Take an
.
-
By hypothesis X { u) i s f i n i t e , so X i s f i n i t e : thus A element u of X i s f i n i t e l y f r e e . Suppose now t h a t X i s an a r b i t r a r y non-empty subset of the base. Take an element u of X and l e t Y be the s e t of those elements n o t greater than o r equal t o u (mod A) Then e i t h e r Y i s empty, so t h a t u i s minimal in X Or Y i s non-empty and so by hypothesis has a minimal element which i s also minimal in X Thus A i s well-founded (proof of POIZAT). 0
.
.
.
3.4. THE EXTRACTION THEOREM FOR WELL PARTIAL ORDERINGS Let A be an i n f i n i t e well partial ordering and C J a ~ given aleph. Let u be a sequence in A with d i s t i n c t values a n d length ad . Then there e x i s t s a sequence extracted from u which i s s t r i c t l y i n c r e a g x (mod A ) and has same length as u (consequence of DUSHNIK, MILLER 1941; uses axiom of choice i f wd i s singular; ZF suffices i f ad i s r e g u l a r ) . 0
Partition the pairs of indices
i,j
(i
<j <
ad )
into two colors: the
color (t) i f u i < u . (mod A ) , and ( - ) i f ui > or I u . (mod A ) . Using ch.3 J J 5 3.3 (OUSHNIK, M I L L E R ) , e i t h e r there e x i s t s a s e t of indices equipotent t o U 4 , a l l of whose pairs have color ( + ) , and hence an C+-sequence extracted from u and s t r i c t l y increasing. Or there e x i s t s a denumerable s e t of indices a l l of whose pairs have color ( - ) , and hence a bad 0 -sequence (mod A ) , thus contradicting 3 . 2 . ( 2 ) . 0 Conversely by 3 . 2 . ( 3 ) , only well partial ordering5 s a t i s f y our conclusion. However, the ordinal product 0 l.( W - ) ; and the partial ordering with denumerably many mutually incomparable components, each isomorphic with W , s a t i s f y our conclusion, when r e s t r i c t e d t o sequences u of length Wac( o( # 0) (communicated by POUZET; see also 9.3 below). 3.5.(1) Every p a r t i a l l y ordered reinforcement of a well partial ordering i s a well partial ordering. Let A be a well partial ordering, and B a p a r t i a l l y ordered reinforcement of A If X i s a free s e t (mod B ) , i . e . a s e t whose elements are incomparable
.
105
Chapter 4
(mod B), then X i s f r e e (mod A ) and hence f i n i t e ; thus B i s f i n i t e l y f r e e . Now l e t X be a non-empty subset o f t h e c o m n base IAI = I B I L e t Y be t h e s e t o f those elements o f X which are minimal (mod A) . Since these elements are incomparable (mod A), t h e r e a r e o n l y f i n i t e l y many. Hence t h e r e e x i s t s i n Y an
.
element y which i s minimal (mod B) . We s h a l l show t h a t y f o r t h e s e t X , which then i m p l i e s t h a t B i s well-founded.
i s minimal (mod B) I f t h e r e e x i s t s an
element x of X such t h a t x < y (mod B) , then the non-empty s e t o f elements of X which are 6 x (mod A) ha5 a minimal element z ; so t h a t 2 6 x < y (mod B) and z i s an element of Y . But t h i s c o n t r a d i c t s the m i n i m a l i t y o f y (mod B) i n Y ( p r o o f communicated by POIZAT 1976, using o n l y ZF). 0 (2) A necessary and s u f f i c i e n t c o n d i t i o n f o r A t o be a w e l l p a r t i a l ordering, i s t h a t every t o t a l l y ordered reinforcement o f A be a well-ordering(sufficiency uses the reinforcement axiom p l u s dependent choice; ZF s u f f i c e s i f A i s countable). 0 If A
i s a w e l l p a r t i a l ordering, then by t h e previous p r o p o s i t i o n every t o t a l l y ordered reinforcement o f A i s a w e l l - o r d e r i n g . Conversely, suppose t h a t A
i s n o t a w e l l p a r t i a l ordering. Then e i t h e r A
is
not well-founded, so t h a t t h e r e e x i s t s a s t r i c t l y decreasing w-sequence (mod A ) : see ch.2 5 2.4, dependent choice. O r A has an i n f i n i t e f r e e subset, hence a denumerable f r e e subset. I n the l a t t e r case, take a t o t a l o r d e r i n g 0 - (converse o f I% , see ch.2 5 1.7) on t h i s f r e e subset. By the reinforcement axiom, and i t s immediate consequence i n ch.2 0 4.2, there e x i s t s a t o t a l o r d e r i n g which extends fd and i s a reinforcement o f A 0
.
( 3 ) I f every t o t a l l y ordered reinforcement there -- o f A i s a well-orderina,and.jf e x i s t s such a t o t a l ordering, then A i s a w e l l p a r t i a l o r d e r i n g ( p r o o f using ZF alone, POUZET 1979, unpublished). 0 Let
C
A
be a w e l l - o r d e r i n g which r e i n f o r c e s
i f n o t then t h e r e would e x i s t a subset
D
. Then
A
o f t h e base w i t h
i s well-founded, f o r A/D
having no m i n i -
mal element, so C/D w i t h o u t any minimal element. Suppose now t h a t t h e r e e x i s t s an i n f i n i t e f r e e (mod A) subset H L e t I denote t h e i n i t i a l i n t e r v a l o f those
.
.
elements x f o r which t h e r e e x i s t s an element y > x (mod A) w i t h y d H S i m i l a r l y denote by F t h e f i n a l i n t e r v a l o f those x f o r which t h e r e e x i s t s an e l e ment Y < x w i t h y 6 H F i n a l l y l e t L be the s e t o f those elements which are
.
.
incomparable (mod A) w i t h a l l elements o f H The f o u r s e t s H, I, F, L a r e d i s j o i n t and form a p a r t i t i o n o f t h e base, Then t h e t o t a l o r d e r i n g C/I t C/L .t (C/H)- + C/F i s a reinforcement of A (where (C/H)- i s t h e converse chain o f C/H ) . However, as H i s i n f i n i t e , t h e w e l l - o r d e r i n g C/H i s isomorphic t o an o r d i n a l >/ r+, , and hence i t s converse i s n o t a well-ordering. 0
THEORY OF RELATIONS
106
3.6. (1) -For every i n f i n i t e well partial ordering A , we have Card H t A = Card A (uses axiom of choice f o r f i n i t e s e t s ; ZF suffices i f A i s countable or has
well-orderable base). To each ordinal i
H t A associate the f i n i t e s e t Ai of elements with height i , then the f i n i t e s e t of t o t a l orderings based on Ai . Finally associate t o each i one of these t o t a l orderings called Ci (choice f o r f i n i t e s e t s ) . Denote by C the well-ordering which i s the sum of the C i according t o increasing values of i To each component w of the ordinal H t A , bijectively associate the component w of C obtained by the substitution of Ci f o r Thus the order type of C d i f f e r s from the ordinal H t A by a t most each i a f i n i t e ordinal: t h e i r cardinals are t h u s equal. 0 (2) If A i s a f i n i t e l y f r e e p a r t i a l ordering, then e i t h e r Cof A i s f i n i t e Cofh A = Cof A (uses axiom of choice; ZF suffices i f A and - Cofh A = 1 i s countable or has well-orderable base). 0 Recall t h a t , f o r any p a r t i a l ordering A , we have Cofh A 6 Cof A (see ch.2 5 7 . 3 . ( 1 ) , axiom of choice). Take a cofinal s e t F of l e a s t i n f i n i t e cardinal, so t h a t Card F = Cof A , with A/F a well-founded ordering, hence a well partial ordering of minimum height. Hence Ht(A/F) = Cofh A : see ch.2 5 7.4. 0
.
.
For each i < Ht(A/F) , there e x i s t only a f i n i t e number of elements with height i (mod A/F) Yet F i s assumed t o be i n f i n i t e . Hence Card H t ( A / F ) = Card F = = Cof A , SO Cof Ad Ht(A/F) = Cofh A 0 (3) For every well p a r t i a l ordering A , we have Cof H t A Cofh A (uses axiom of choice; ZF suffices i f A i s countable). 0 Take a cofinal r e s t r i c t i o n B of A with l e a s t cardinal and l e a s t height. To each i < H t B associate the maximum height (mod A) of elements ( i n f i n i t e number) with height i (mod B ) . The s e t of those maximum heights i s cofinal H t B = Cofh A 0 in the ordinal H t A . Hence Cof H t A I t i s necessary t o assume t h a t A i s a well partial ordering: recall t h a t we defined in ch.2 5 7 . 7 . ( 1 ) , a well-founded p a r t i a l ordering with Cofh A = 1 and H t A = ul , so Cof H t A = w . For a well p a r t i a l ordering, s t r i c t inequality may occur: see ch.7 5 3.13 below where Cofh A = W1 and H t A = Ldl. W thus Cof H t A = Cd .
.
.
<
<
.
4 - I N I T I A L INTERVALS OF A WELL P A R T I A L O R D E R I N G ; RADO'S WELL P A R T I A L ORDERING; HIGMAN'S THEOREM ON THE WELL PARTIAL O R D E R I N G OF WORDS
§
4.1. THEOREM ON INITIAL INTERVALS OF A WELL PARTIAL ORDERING For a partial ordering A t o be a well partial ordering, i t i s necessary
Chapter 4
107
and s u f f i c i e n t t h a t t h e p a r t i a l o r d e r i n s of inclusion of i n i t i a l i n t e r v a l s of A be well-founded (HIGMAN 1952; the n e c e s s i t y uses dependent choice; ZF s u f f i c e s if A i s countable). 0 Let A be a well p a r t i a l o r d e r i n g , and suppose t h e statement of t h e proposit i o n i s f a l s e . By ch.2 5 2.4 (dependent c h o i c e ) , t h e r e e x i s t s an w -sequence of i n i t i a l i n t e r v a l s Ai ( i i n t e g e r ) which i s s t r i c t l y decreasing w i t h respect t o inclusion. For edch i , choose an element u i in Ai-Ai+l . Each u i i s < o r I (mod A ) t o t h e u w i t h indices -6 i Hence our k, -sequence i s bad, cont r a d i c t i n g 3.2.(2). Conversely, suppose t h a t A i s not a well p a r t i a l ordering. Then e i t h e r A is not well-founded o r A i s not f i n i t e l y f r e e . In t h e f i r s t case, t h e r e e x i s t s a subset D of t h e base I A I such t h a t A/S has no minimal element. To each e l e ment x of D , a s s o c i a t e t h e i n i t i a l i n t e r v a l Dx of eiements Gx (mod A ) The s e t of t h e Dx has no element minimal with r e s p e c t t o inclusion; so t h e p a r t i a l ordering of t h e i n i t i a l i n t e r v a l s i s not well-founded. I f A i s not f i n i t e l y f r e e , then t h e r e e x i s t s an i n f i n i t e s e t H o f incomparable elements (mod A ) . By t h e d e f i n i r i o n of f i n i t e n e s s ( c h . 1 5 l . i ) , t h e r e e x i s r s a s e t of subsets X of H , no one of which i s minimal w i t h respect t o inclusion. Complete each X in the i n i t i a l i n t e r v a l X+ generated by X . Then X+n H = X
.
.
f o r every X , hence X # Y implies X+ p Y+ , and even X c Y implies Xt c Y+ f o r every X, Y . The X+ form a set of i n i t i a l i n t e r v a l s of A , no one of which i s minimal w i t h r e s p e c t t o i n c l u s i o n . Hence t h e p a r t i a l ordering of t h e i n i t i a l i n t e r v a l s i s n o t well-founded (proof of s u f f i c i e n c y using only ZF and communicated by POUZET). 0 4.2. RADO'S WELL PARTIAL ORDERING If A i s a well p a r t i a l ordering, then t h e p a r t i a l ordering of inclusion f o r the i n i t i a l i n t e r v a l s of A i s not n e c e s s a r i l y a well p a r t i a l ordering. In f a c t i t i s not n e c e s s a r i l y f i n i t e l y f r e e . The following example i s c a l l e d Rado's well p a r t i a l o r d e r i n g (RADO 1954). I t will be used i n p a r t i c u l a r i n ch.8 5 5.2 f o r the theory of b e t t e r p a r t i a l orderings. Consider t h e couples of natural numbers x , y and set (x,y) 6 ( x ' . y ' ) i f f x = x ' and y d y ' in t h e usual ordering, o r i f X' xty and y i s a r b i t r a r y . The comparison r e l a t i o n thus defined i s a well p a r t i a l ordering. Yet t h e r e a r e i n f i n i t e l y many i n i t i a l i n t e r v a l s which a r e mutually incomparable w i t h respect t o inclusion. The reader couple ( x , y ) , since e i t h e r sum of t h e i r
can v e r i f y r e f l e x i v i t y , antisymmetry and t r a n s i t i v i t y . Given a t h e r e a r e only f i n i t e l y many couples l e s s than i t , t h e i r f i r s t term i s x and t h e i r second term i s .\< y , o r t h e terms i s x . T h u s the p a r t i a l ordering i s well-founded.
THEORY OF RELATIONS
108
Given
(x,y)
,
any c o u p l e which i s incomparable w i t h i t , has f i r s t t e r m
s t r i c t l y less than
x+y
. Hence
t h e r e e x i s t f i n i t e l y many p o s s i b l e f i r s t terms.
Moreover, two incomparable couples n e c e s s a r i l y have d i s t i n c t f i r s t terms. Hence t h e p a r t i a l o r d e r i n g i s f i n i t e l y f r e e , and t h u s a w e l l p a r t i a l o r d e r i n g . Consider now, f o r each i n t e g e r couples (x.y)
with
(i,y)
with A
and
or
x : > Max(x,y)
i
, the
i n i t i a l interval
Ai
formed o f t h e
an a r b i t r a r y n a t u r a l i n t e g e r , and t h e couples
: f o r an a r b i t r a r y i n t e g e r
x+y,(i
Ai
y
j
+
i
, the
i n i t i a i intervals
a r e incomparabie w i t h r e s p e c t t o i n c l u s i o n . 0
j A second example o f a w e l l p a r t i a l o r d e r i n g w i t h t h e same incomparabie i n i t i a l i n t e r v a l s : t a k e a g a i n t h e couples o f i n t e g e r s w i t h e i t h e r x = x i and y & y ' ,
dnd y '
arbitrary.
A t h i r d example, wnich w i l l be g e n e r a l i z e d i n ch.8 e x e r c . 2: tegers with e i t h e r
4.3. L e t A
x = x'
and
y,< y '
, or
X I >
Max(x,y)
couples o f i n and y < y ' .
be a p a r t i a l o r d e r i n g and
u
a minimal bad w-sequence .in
t h e i n i t i a l i n t e r v a l o f t h o s e elements
x
of
i fin
g p
x
< ui
(mod A )
A
.
Then
I A I such t h a t t h e r e e x i s t s an i n t e -
i s a w e l l p a r t i a l o r d e r i n g (uses dependent choice;
ZF s u f f i c e s i f A i s c o u n t a b l e j .
Compare w i t h 2.5 above.
i4e prove f i r s t o f a i l t h a t , g i v e n a t e r m u ( p i n t e g e r ) o f t h e minimal bad P sequence, t h e i n i t i a l i n t e r v a l o f t h o s e elements u (mod A j i s a w e l l p a r t i a l P o r d e r i n g . By 2.5 t h i s i n t e r v a l i s well-founded (dependent c h o i c e ) . Suppose t h a t 0
<
t h e r e e x i s t i n f i n i t e l y many m u t u a l l y incomparable elements, hence an (J -sequence of such elements , , ,a al,. .., a.,. a l l < u . Then each a . < up , so each J P J a./ail except a f i n i t e riumber o f terms of u (see 2.3). Thus t h e w -sequence J o f t h e a . i s embeddable i n u , and hence by 2.6 i s e x t r a c t e d f r o m u , so t h a t J t h e r e e x i s t terms u ( q > p ) which a r e 6 u . Yet a l l t h e terms o f u a r e q P m u t u a l l y incomparable by 2.3: c o n t r a d i c t i o n . Suppose now t h a t t h e p r o p o s i t i o n i s f a l s e . There e x i s t s a bad w -sequence v
..
such t h a t , f o r each i n t e g e r i , t h e r e e x i s t s a j w i t h v i < uj (mod A ) : see 3.2.(2) u s i n g dependent c h o i c e . A t most a f i n i t e number o f terms o f v a r e < uo, or
< ul,
e t c . Hence t h e r e e x i s t s an (,d -sequence
e i t h e r s t r i c t i y less than
4.4. If
.
u
from
u (mod A)
w
extracted from
v
which i s
o r s t r i c t l y l e s s t h a n a sequence e x t r a c t e d
This contradicts the minimality o f
u
.0
THEOREM ON "ELL PARTIAL ORDERING OF WORDS A
i s a well p a r t i a l ordering, then t h e p a r t i a l orderinq o f embeddabilitv f o r
words i n
A
suffices i f
i s a w e l l p a r t i a l o r d e r i n g (HiGMAN 1952; uses dependent choice; A
i s countable).
ZF
Chapter 4 By 3.2.(2)
109
(dependent c h o i c e ) , i t s u f f i c e s t o show t h a t t h e r e does n o t e x i s t any
bad &J -sequence o f words w i t h r e s p e c t t o e m b e d d a b i l i t y ( d e f i n e d i n 5 2 ) . I f such a bad sequence e x i s t s , t h e n t h e r e e x i s t s a l s o a s t r o n g l y minimal bad sequence U (see 2.10).
We s h a l l show t h a t t h i s i s i m p o s s i b l e . Denote by
ith term o f
.
i s such t h a t no bad w-sequence o f words beby removing a t l e a s t one Uo element. No bad sequence b e g i n n i n g by Uo c o n t i n u e s w i t h a word s h o r t e r t h a n U1 , U
By h y p o t h e s i s ,
the
( i integer)
Ui
Uo
g i n s by a s h o r t e r word, i . e . a word o b t a i n e d f r o m
e t c . N o t i c e t h a t no
Ui
of t h e ui
U . ( j> i ) a r e
i s empty, s i n c e t h e
respect t o embeddability: t h e f i r s t term
ui
J
of
Ui
<
I Ui
or
with
always e x i s t s . The sequence
t a k e s i t s values i n t h e w e l l p a r t i a l o r d e r i n g
A
and so by 3.2.(3)
has an e x t r a c t e d 0 - s e q u e n c e which i s e i t h e r c o n s t a n t o r s t r i c t l y i n c r e a s i n g . Denote by
n ( i j ( i integer)
t h e indices corresponding t o t h i s constant o r increa-
V be t h e word u h(i) h(i) removed. Consider t h e sequence Uo, U1,..., Uh(o)-l,
s i n g sequence o f f i r s t terms, and l e t
u
term
h(i) This sequence begins as
U
, but
placed by t h e s h o r t e r word
Vh(o)
a t the position
.
good. Now, two words i n p o s i t i o n C h ( 0 ) V
with
2
h(0) uh(i).
Uh(i)'<
that a
Ui
i s embeddable i n a
ble i n
Uh(k)
-= h ( 0 ) ) .
che h o r d
Uh ( 0 )
*
i s re-
cannot a d m i t an embedding o f one i n t h e
i <j : g i v e n t h e f a c t t h a t
h(j) embedding would y i e l d (i
,
V h ( 0 ) ' 'h( 1 )
So by h y p o t h e s i s , t h i s l a t t e r sequence i s
o t h e r . S i m i l a r l y f o r two words i n p o s i t i o n and
h(0)
with i t s f i r s t
. Finally,
Uh(j)
, which <
V
are o f t h e form
uh(j)
, such
(mod A)
h(i) an
t h e r e remains t h e p o s s i b i l i t y
Vh(k) ( k i n t e g e r )
However t h i s c o n t r a d i c t s t h e assumption t h a t
and hence embeddaU
i s bad.
A be a w e l l - f o u n d e d p a r t i a l o r d e r i n g and o( i t s h e i g h t . I f f o r e v e r y i < < , t h e r e a r e o n l y f i n i t e l y many-elements o f h e i g h t i , then A i s not necessarily f i n i t e l y free.
4.5.
Let
ordinal
i , add an i ' i' m u t u a l l y incom-
Take t h e t o t a l o r d e r i n g o f t h e i n t e g e r s , and f o r each i n t e g e r s e t t o be
>i
and incomparable w i t h i n t e g e r s
> i , the
parable. LEMMA OF THE CHAIN MEETING EVERY HEIGHT
A
Let
be a w e l l - f o u n d e d p a r t i a l o r d e r i n g h a v i n g a t most f i n i t e l y many elements
o f each h e i g h t , and l e t c4 = H t A C
of
A
.
Then t h e r e e x i s t s a w e l l o r d e r e d r e s t r i c t i o n
which i s isomorphic w i t h o(
. More
p r e c i s e l y , t h e base
.
IC
I
contains
i < 4 a unique element o f h e i g h t i (mod A) Uses u l t r a f i l t e r axiom; ZF s u f f i c e s i f A i s countable; communicated by POUZET 1979. f o r each o r d i n a l For
o(
= a , t h i s i s j u s t a f o r m o f KONIG's lemma: see below ch.5 5 2.5.
Compare t h e p r e s e n t p r o p o s i t i o n w i t h 3.1, which uses t h e axiom o f choice; h e r e c h o i c e would be u s e f u l i f we had added t h e equipotence o f
I A l and H t A
.
110
THEORY OF RELATIONS
0 For e v e r y f i n i t e s e t
h
F o f ordinals
which a s s o c i a t e s t o each
h i ( i E F) of
i of
F
< 4( , t h e r e an element
i (mod A ) . The
o f height
a r e m u t u a l l y comparable, so f o r m a f i n i t e t o t a l l y o r d e r e d r e s t r i c t i o n
A : it suffices t o start with the largest
each
e x i s t s a t l e a s t one f u n c t i o n
hi
j < i t h e r e e x i s t s an element
i in
F
and t o r e c a l l t h a t f o r
h j < h i (mod A) ; see ch.2
5
3.2.
Consider t h e o r d i n a l o( and t h e base I A [ as d i s j o i n t s e t s . To each ciate
F+ = F p l u s those elements i n
I A l whose h e i g h t belongs t o
F
F
asso-
. The
F+
Ft
F , formed by t h e
, together
with the bina-
t h u s f o r m a d i r e c t system w i t h r e s p e c t t o o r d e r i n g by i n c l u s i o n . Given an t o each
h
w i t h domain
F a s s o c i a t e t h e b i r e l a t i o n w i t h base
+
unary r e l a t i o n t a k i n g t h e v a l u e
+
ry r e l a t i o n t a k i n g the value
on
F and
-
on
F+-F
f o r ordered p a i r s
(i,hi)
where
t o each
F t h e non-empty s e t
The
s a t i s f y t h e hypotheses i n t h e coherence lemma ch.2
UF
UF
o f these b i r e l a t i o n s f o r a l l
5
h
c(
Associate
w i t h domain
IAIu
a(
,
A. 0
onto a t o t a l l y ordered r e s t r i c t i o n o f
A c l o s e l y connected p r o p o s i t i o n i s g i v e n i n e x e r c i s e 4 below.
-
5
NET,
OR DIRECTED P A R T I A L ORDERING:
IDEAL,
DECOMPOSITION OF A
F I N I T E L Y FREE P A R T I A L ORDERING INTO I D E A L S
A p a r t i a l o r d e r i n g i s s a i d t o be a
net, o r
directed p a r t i a l orderinq, i f given
any two elements i n t h e base, t h e r e i s a t h i r d element g r e a t e r t h a n b o t h . F o r example, a c h a i n ( t o t a l o r d e r i n g ) , o r a p a r t i a l o r d e r i n g w i t h a maximum. Given a p a r t i a l o r d e r i n g
A
, an
ideal i n
A
i s any i n i t i a l i n t e r v a l which i s
d i r e c t e d . F o r example, e v e r y i n i t i a l i n t e r v a l o f a c h a i n i s an i d e a l ; o r again, f o r an a r b i t r a r y p a r t i a l o r d e r i n g an element
a
A
, the
and a l l elements l e s s t h a n
i n i t i a l i n t e r v a l o b t a i n e d by t a k i n g a (mod A )
o r d e r i n g o f i n c l u s i o n among subsets o f a s e t
E
, an
of
E
of
E : hence an i d e a l i s t h e complement o f a f i l t e r .
.
I n t h e case o f t h e p a r t i a l
i d e a l i s a s e t o f subsets
which i s c l o s e d under t a k i n g subsets and t a k i n g t h e u n i o n o f two subsets
Given a s e t o f i d e a l s
U
which i s t o t a l l y o r d e r e d by i n c l u s i o n , o r s i m p l y i f
i n c l u s i o n i s a net, then the union o f t h e
U
i s an i d e a l .
Consequently t h e MAXIMAL IDEAL AXIOM: every ideal i n a p a r t i a l ordering
A
i s i n c l u d e d i n a maximal i d e a l o f
A
(with
respect t o inclusion), i s e q u i v a l e n t t o t h e axiom o f c h o i c e . 0 It
U of
f o l l o w s f r o m t h e maximal c h a i n axiom (ch.2 A
, take
5
F
1.3 ( e q u i v a l e n t t o
t h e u l t r a f i l t e r axiom). Hence t h e r e e x i s t s a b i r e l a t i o n on t h e u n i o n which d e f i n e s an isomorphism f r o m
.
iE F
2.7): s t a r t i n g w i t h an i d e a l
a maximal c h a i n i n t h e p a r t i a l o r d e r i n g o f i n c l u s i o n o f i d e a l s
Chapter 4
of
A
111
and t h e n t a k e t h e u n i o n o f t h e i d e a l s i n t h i s c h a i n , which i s a maximal
i d e a l .. Conversely, t h e maximal i d e a l axiom i m p l i e s t h e w e l l - o r d e r i n g axiom: t a k e
5
a maximal i d e a l i n t h e i n t e r v a l - o r d e r i n g , d e f i n e d i n ch.2 5.1. I n a p a r t i a l o r d e r i n g , an i n i t i a l i n t e r v a l Initial intervals 0 Suppose t h a t
ez
.
, if
X u Y = I
then
i s an i d e a l i f f f o r any two
I X = I
.
Y = I
i s an i d e a l which i s t h e u n i o n o f two i n i t i a l i n t e r v a l s
I
X # I
and t h a t
X, Y
2.6. Cl
and
Y # I
. Thus
t h e r e e x i s t s an element
x E I-X
X, Y
and an e l e -
z such t h a t z E I and z 3 x and z 5,y . Hence z 4 X and z + Y : c o n t r a d i c t i o n . Conversely, suppose t h a t I i s an i n i t i a l i n t e r v a l which i s n o t an i d e a l . So
ment
y
I-Y
Consequently, t h e r e e x i s t s an element
Define
U
I
u, v o f
t h e r e e x i s t two elements
w i t h o u t any common upper bound i n
t o be t h e i n i t i a l i n t e r v a l o f those elements
a common upper bound o f
x
u
and
in
t o be t h e i n i t i a l i n t e r v a l o f t h o s e
I : hence u E U
.
F i n a l l y we have
U u V
=
I
.
I
f o r which t h e r e e x i s t s and
v+U
.
Define
V
z ay i n I , I : hence u I+ V and
f o r which t h e r e e x i s t s a
y
such t h a t t h e r e i s no common upper bound o f v g V
x
z
and
u
in
.0
5.2. Every w e l l p a r t i a l K d e r i n q i s a - f i n i t e u n i o n o f i d e a l s (uses dependent
ZF s u f f i c e s f o r a c o u n t a b l e o r d e r i n g ) .
choice; 0
Let
A
be a w e l l p a r t i a l o r d e r i n g ; r e c a l l t h a t t h e p a r t i a l o r d e r i n g o f i n i t i a l
intervals o f
A
i s w e l l - f o u n d e d (see 4.1,
dependent c h o i c e ) . Suppose t h a t
n o t a f i n i t e u n i o n o f i d e a l s . Among t h e i n i t i a l i n t e r v a l s o f
A
f i n i t e unions o f i d e a l s , t h e r e e x i s t s a minimal such i n i t i a l i n t e r v a l r e s p e c t t o i n c l u s i o n . As
M
M
, and
Uu V = M
.
each a f i n i t e u n i o n o f i d e a l s ; so 5.3.
is
M
, with
i s n o t an i d e a l , i t i s non-empty. By t h e preceding
p r o p o s i t i o n 5.1, t h e r e e x i s t two i n i t i a l i n t e r v a l s t i n c t from
A
which a r e n o t
By m i n i m a l i t y o f M
U, V o f M , which a r e d i s M , t h e i n t e r v a l s U, V a r e
as w e l l : c o n t r a d i c t i o n . 0
(1) F o r a p a r t i a l o r d e r i n g t o be f i n i t e l y f r e e , i t i s necessary and s u f f i -
c i e n t t h a t e v e r y i n i t i a l i n t e r v a l be a f i n i t e u n i o n o f i d e a l s (BONNET 1975; t h e n e c e s s i t y uses axiom o f c h o i c e ) .
u which a r e m u t u a l l y incomu t h e i d e a l o f t h o s e elements < u : t h e u n i o n o f
0 Suppose t h a t t h e r e e x i s t i n f i n i t e l y many elements
p a r a b l e . A s s o c i a t e t o each
these i d e a l s i s an i n i t i a l i n t e r v a l which i s n o t decomposable i n t o a f i n i t e u n i o n
of i d e a l s. Conversely, l e t val o f
A
A
. By ch.2
be a f i n i t e l y f r e e p a r t i a l o r d e r i n g , and § 5.1,
I
an i n i t i a l i n t e r -
c o r o l l a r y (axiom o f c h o i c e ) , t a k e a w e l l o r d e r i n g
which i s a c o f i n a l r e s t r i c t i o n o f
I
.
By t h e p r e c e d i n g 5.2,
J
J
i s a f i n i t e union
112
THEORY OF RELATIONS
of ideals. Complete each ideal X in J into an ideal of elements in III which are bounded above by an element of f i n i t e union of ideals. 0
I
by adding those 1x1 . T h u s I i s a
(2) Every f i n i t e l y f r e e p a r t i a l ordering i s a f i n i t e union of ideals-. ( 3 ) Every i n f i n i t e f i n i t e l y free partial ordering has an i n f i n i t e ideal; consequences of ( 1 ) . The converse i s f a l s e : take a non-finitely f r e e p a r t i a l ordering which has a maximum. For further informations, see MILNER, 1982. 5 . 4 . For every net A , there e x i s t s a cofinal r e s t r i c t i o n f A which i s a well-founded net (uses axiom of choice; ZF suffices i f A i s countable). 0 By ch.2 5 5.1, corollary (axiom of choice), there e x i s t s a cofinal F with A/F well-founded. For any two elements x , y o f F , by hypothesis there e x i s t s an element z satisfying z ax and z a y (mod A ) Since F i s cofinal, there x and t > / y : hence A/F e x i s t s an element t i n F with t 3 z and so t i s directed; in other words, a net. 0
.
5.5. For every denumerable net A , there e x i s t s 2 t o t a l l y p-de-r-d-restriction ofA , which i s cofinal and isomorphic t o LJ 1 . 0 Take an a-sequence of the elements a i ( i integer) of the base; then take bo = a. bo
and
, bl = the element with l e a s t index which i s greater t h a n (mod A ) b o t h a l , then b2 greater than b l and a 2 , e t c . 0
On the other hand, the d i r e c t product w X (*1 defined in $ 7 below, has no t o t a l l y ordered cofinal r e s t r i c t i o n . 0 T h a t would require a total ordering of order-type w 1 formed o f ordered pairs ( i , j ) with i , j increasing and i running through cu and j running through a 1 , which i s impossible. 0
§
6
- TREE
tree i f , f o r each element x in the base, the s e t of a l l predecessors of x i s t o t a l l y ordered. For example, every t o t a l ordering i s a t r e e . Every f r e e partial ordering, reduced t o the identity, i s a t r e e . Another example: beginning with a s e t a and a s e t A o f subsets of a , where any two elements of A are e i t h e r d i s j o i n t or one i s included in the other, and no element of A i s empty. Then reverse inclusion constitutes a t r e e based on A . A partial ordering i s called a
Conversely, l e t A be a t r e e with base E . To each element x in E , associate the s e t Ax of those elements >/ x (mod A ) : the Ax ordered by reverse inclusion form a t r e e , isomorphic with
A
.
Chapter 4
6.1.
If
A
113
i s a f i n i t e t r e e , t h e n e v e r y base o f a maximal c h a i n o f
A
and e v e r 1
maximal f r e e s e t , have one and o n l y one element i n common. (KUREPA 1952). Let
F
be t h e base o f a maximal t o t a l l y o r d e r e d r e s t r i c t i o n o f
i d e n t i c a l o r incomparable (mod A) t o each element o f to
, since
G
which i s d i s t i n c t f r o m
and
v
.
u ; f o r o t h e r w i s e v c o u l d be added t o belongs t o
u
.
Then
v
v
since
A/F
belongs
of
v
G
,
is a strict
F and so
i s maximal. Thus
a
G
u is
Then e i t h e r
A/F
would
u ,
i s comparable w i t h e v e r y predecessor o f
v
F
and
I n t h a t case
u (mod A)
and comparable w i t h
u
n o t be a maximal c h a i n . Thus and hence
G
i s a maximal f r e e s e t . O r t h e r e e x i s t s an element
G
predecessor o f
.
u be t h e maximum o f t h e c h a i n A/F
maximal f r e e s e t ; l e t
,
A
i s common t o
F
.
G
T h i s r e s u l t does n o t h o l d f o r an i n f i n i t e t r e e . i , a s s o c i a t e an i ' incomparable w i t h i n t e g e r s > i F i n a l l y t h e i ' a r e
Take t h e t o t a l o r d e r i n g o f t h e i n t e g e r s and t o each i n t e g e r
.
i ' >i b u t
element
s e t t o be m u t u a l l y incomparable. Then t h e s e t o f i n t e g e r s d e f i n e s a maximal chain, and t h e s e t o f "primed" i n t e g e r s i s a maximal f r e e s e t (or d e f i n e s a maximal a n t i c h a i n ) ; example due t o KUREPA. The r e s u l t no l o n g e r h o l d s f o r an a r b i t r a r y f i n i t e p a r t i a l o r d e r i n g . 0
Take f o u r elements and l e t a
Then t h e c h a i n
6.2. L e t
A
a
< b,
a
-= c,
b ' c c, w i t h
i s maximal and t h e a n t i c h a i n
b
b I c, a
I b'
be a t r e e and
E
i t s base. There e x i s t s a c h a i n
such t h a t f o r each element
x
fo
t e r v a l (mod C) w i t h minimum
x
E
, the
interval
I
b',
is maximal. C
3 x (mod A )
b
Ib' .
0
based on
E
,
becomes an i n -
(uses u l t r a f i l t e r axiom: ZF s u f f i c e s i f
A
is
countable). Suppose f i r s t t h a t
E
i s f i n i t e . Let
ul, ..., uh be t h e minimal elements (mod A) u1 < ...
and i n an a r b i t r a r y manner o r d e r these elements, o b t a i n i n g f o r example
< uh .
i = 1,.
Then f o r each
. .,h
replace
ui
by a sequence b e g i n n i n g w i t h
and t o t a l l y o r d e r i n an a r b i t r a r y manner t h e immediate successors o f Continue i n t h i s f a s h i o n u n t i l a l l t h e elements o f t h u s we o b t a i n t h e c h a i n NOW
E
UF
t i o n for the r e s t r i c t e d t r e e
.
are contained i n t h i s chain:
F o f E , as F which s a t i s f y t h e p r o p o s i -
i s i n f i n i t e . To each f i n i t e subset o f c h a i n s w i t h base
A/F
. Then
the
UF
s a t i s f y t h e hypotheses o f t h e
coherence lemma ch.2
5
1.3 ( e q u i v a l e n t t o u l t r a f i l t e r axiom). Then we o b t a i n a
chain
C
w i t h base
E
,all
Hence
A
and
C
ui
(mod A)
.
C
c o n s i d e r t h e case where
s o c i a t e t h e non-empty s e t
E
ui
o f whose f i n i t e r e s t r i c t i o n s s a t i s f y t h e p r o p o s i t i o n .
s a t i s f y the proposition. 0
THEORY OF RELATIONS
114
Consequently, given a tree A , there exists a chain w i t h t h e s a m e b a s e , and a set___.__. a r e mutually i n c l u s i v e o r d i s j o i n t , such U of i n t e r v a l s (modulo t h e chain) which t h a t A i s isomorphic t o reverse inclusion on U . ~
-
6.3. Given a s e t E and -a s e.t U of subsets of E , which a r e mutually i n c l u s i v e o r d i s j o i n t , t h e r e e x i s t s a chain with base E , f o r which t h e e l e m e n t s f U i n t e r v a l s ( u l t r a f i l t e r axiom; ZF s u f f i c e s i f E i s countable). 0 Add t o U the s i n g l e t o n s of elements of E , and apply t h e preceding proposit i o n t o t h e s e t thus obtained, which i s ordered by reverse i n c l u s i o n . Replace the chain by i t s r e s t r i c t i o n t o t h e s e t of s i n g l e t o n s ; each singleton being i d e n t i f i e d with i t s unique element. 0
are
6.4. REDUCED TREE Let A be a t r e e . We introduce an equivalence r e l a t i o n c a l l e d reduction (mod A ) . Two elements x, y of t h e base a r e s a i d t o be equivalent i f they a r e comparable, f o r example i f x < y (mod A ) and a l s o every element a x i s comparable w i t h y Equivalently, every t o t a l l y ordered r e s t r i c t i o n of A , which i s maximal (mod A ) with respect t o inclusion and which contains x , a l s o contains y . The r e l a t i o n thus defined i s r e f l e x i v e , symmetric and t r a n s i t i v e . An equivalence c l a s s is a t o t a l l y ordered r e s t r i c t i o n of A . We say t h a t an equivalence c l a s s precedes another equivalence c l a s s , i f t h e e l e ments of t h e f i r s t precede (mod A ) t h e elements of t h e second. The p a r t i a l order i n g on t h e equivalence c l a s s e s thus defined, i s a t r e e , and c a l l e d t h e reduced t r e e o r reduct of A . Given a t r e e A , a s s o c i a t e t o each element x of t h e base the s e t Ax of maximal cha ns ( w i t h r e s p e c t t o i n c l u s i o n ) which contain x in t h e i r base. For x and y incomparable (mod A ) , t h e s e t s Ax and A a r e d i s j o i n t . For x b y Y (mod A ) , we have Ax 2 Ay . Hence these s e t s , ordered by r e v e r s e i n c l u s i o n , form a t r e e somorphic t o the reduct of A .
.
-
6.5. INITIALLY MAXIMAL CHAIN e say t h a t a chain in A i s i n i t i a l l y maximal Let A be a p a r t i a l ordering. W i f i t i s an i n i t i a l i n t e r v a l of a maximal chain ( w i t h r e s p e c t t o i n c l u s i o n ) . In o t h e r words, i f x belongs t o t h e base of an i n i t i a l l y maximal chain U , then every predecessor of x (mod A ) e i t h e r belongs t o t h e base U o r i s incompar a b l e (mod A ) t o some element i n t h i s base. The s e t of i n i t i a l l y maximal c h a i n s , ordered by i n c l u s i o n , forms a t r e e . F i n a l l y r e c a l l t h a t t h e amalgamation lemma ch.2 5 2 . 2 and 2.3, which holds f o r p a r t i a l orderings and f o r t o t a l o r d e r i n g s , no longer holds f o r t r e e s : a counterexample has been given i n ch.2 § 2.3.
115
Chapter 4
7 - DIRECT PRODUCT OF
§
P A R T I A L O R TOTAL O R D E R I N G S ; D I M E N S I O N
Given a s e t o f p a r t i a l o r d e r i n g s
(i EI)
Ai
, the
d i r e c t product o f t h e
is
Ai
d e f i n e d t o be t h e p a r t i a l o r d e r i n g w i t h base t h e C a r t e s i a n p r o d u c t o f t h e bases, hence w i t h base t h e s e t o f f u n c t i o n s
c:
f
w i t h domain
I t a k i n g a value
i ; t h e comparison r e l a t i o n b e i n g d e f i n e d by f Q g iff f o r e v e r y i E I . T h i s comparison r e l a t i o n i s r e f l e x i v e , f(i),( g ( i ) (mod Ai) a n t i s y m m e t r i c and t r a n s i t i v e . f(i)
Ai
f o r each
I n p a r t i c u l a r , f o r two p a r t i a l o r d e r i n g s A
5
.
B
A, B t h e d i r e c t p r o d u c t i s denoted by Up t o isomorphism, t h e o p e r a t i o n x i s commutative and a s s o c i a t i v e .
Another p a r t i c u l a r case: i f a l l t h e p a r t i a l o r d e r i n g s
Ai a r e i d e n t i c a l t o a i r u n s t h r o u g h an o r d i n a l , t h e n we o b t a i n t h e " l e s s than" comparison r e l a t i o n between sequences i n A o f t h e same o r d i n a l fixed p a r t i a l ordering
l e n g t h (see
5
A
, where
2).
7.1. (1) L e t A, B be two w e l l - f o u n d e d p a r t i a l o r d e r i n g s ; t h e n t h e d i r e c t p r o -
duct
A
B i s we1 1 -founded.
%
0 S t a r t w i t h a non-empty subset
D
Then t a k e an element
D
then y E (2) Let
(x,y)
of
o f t h e C a r t e s i a n p r o d u c t o f t h e two bases.
, where
x 6 1Al
w i t h minimal h e i g h t (mod A),
lBl w i t h minimal h e i g h t (mod B) : hence (x,y) A, B
be two w e l l p a r t i a l o r d e r i n g s ; t h e n t h e d i r e c t p r o d u c t
5
a w e l l p a r t i a l o r d e r i n g (uses denumerable subset axiom: ch.1 if 0
A
and
B
D
i s minimal i n
2.6;
.0
Ax B
1s.
ZF s u f f i c e s
are countable).
By t h e p r e c e d i n g ( l ) , t h e d i r e c t p r o d u c t i s w e l l - f o u n d e d . Suppose t h a t t h e r e
e x i s t i n f i n i t e l y many incomparable elements, and hence an w - s e q u e n c e of incomp a r a b l e elements. Then we have an e x t r a c t e d cd -sequence o f elements where t h e f i r s t terms
x
a r e i n c r e a s i n g (mod A)
(x,y) f r o m t h i s sequence, ano-
, and
t h e r e x t r a c t e d a - s e q u e n c e where t h e second terms
a r e i n c r e a s i n g (mod B ) :
y
t h i s contradicts the incomparability. 0 7.2.
(1) L e t
A,
B
be two well-founded p a r t i a l o r d e r i n g s ,
I A I and y an element of
base
lB(
.
Then
Ht(x,y)
x
(mod A x
an element o f t h e
B) i s equal t o
H t x (mod A) Q H t y (mod B) : see c h . 1 5 9 a r e non-empty w e l l - f o u n d e d p a r t i a l o r d e r i n g s , then:
t h e commutative sum A, B
(2) If
Sup ( H t A, H t B) 0
If
x
& H t ( A x B) < H t A @ H t B
i s minimal (mod A) and
y
.
minimal (mod
B) t h e n t h e c o u p l e (x,y)
i s minimal (mod A x B ) : so t h e above e q u a l i t y h o l d s f o r
8 with
Ht x = Ht y = 0
be an o r d i n a l # 0 ; assume t h a t t h e e q u a l i t y h o l d s f o r each H t x'
0 Ht
y'
< y .
.
x'
. Let
and each y '
Assume moreover t h a t t h e e q u a l i t y h o l d s t o o i f
THEORY OF RELATIONS
116
Ht(x',y') (mod A x B) < . Let I be the initial interval of those (x',y') with height 4 f , and J the complement of I . So (x,y) belongs to J iff 8 . By definition of height, a couple (x,y) is minimal Ht x 8 Ht y in J iff Ht(x,y) (mod A x B) = li . So it suffices to prove that (x,y) is minimal in J iff Ht x @ Ht y = . For (x,y) in J , either Ht x @ Ht y > $ . Then there exists for instance an ordinal j < Ht y with Ht x EI j = $ ; and so an element y'< y (mod B) with Ht y' = j (see ch.2 5 3.2). Hence (x,y') < (x,y) and (x,y') belongs to J , so that (x,y) is not minimal in J . Or Ht x Q Ht y = 8 . Then for any (x',y') < (x,y) (mod A K B) we have either x ' < x (mod A ) with y'+ y (mod B) , or conversely x'< x with y ' < y . By the properties of commutative ordinal addition, this always gives Ht x'@ Ht y' < y , so that (x',y') belongs to I , and (x,y) is minimal in J . 0 (2) The first inequality is obvious. For the second inequality, notice that the height of a well-founded partial ordering A equals Sup(i+l) where i denotes the height of any element in A . Similarly for B we obtain Sup(j+l) ; and finally for A s B we obtain Sup(i @ j + l ) , taking in account the preceding (1). Then the second inequality immediately results from the following, which is a consequence of the definition of commutative sum , easily provable by the reader: Sup(i 0 j + 1) < (Sup i+l ) @ (Sup jtl ) . 0
0
7 . 3 . CONJUNCTION OF A SET OF PARTIAL ORDERINGS.
Given a partial ordering A and a set of partial orderings Bi which are all reinforcements of A with common base IAI , we say that the partial ordering A is the conjunction of the Bi if, for any x, y in I A I , we have x \< y (mod A ) iff x, y (mod B.) . J DIMENSION The dimension of a partial ordering A is the least cardinal of a set of chains, each with base IAI and whose conjunction is A . Modulo the axiom of choice, every partial ordering has a dimension; in ZF at least every finite partial ordering has a dimension. The notion of dimension goes back to DUSHNIK, MILLER 1941. We shall denote by Dim A the dimension of a partial ordering A .
Let A A
be a partial ordering, B a restriction of A ; if dimensions exist for and B , then Dim B 4 Dim A .
Every chain has dimension 1
. Given a set
E of cardinality
2 , the free
Chapter 4
117
o r d e r i n g , o r i d e n t i t y r e l a t i o n , has dimension 2 : t a k e a c h a i n based on t h e converse c h a i n ( s o we use t h e o r d e r i n g axiom; ZF s u f f i c e s i f
(1) L e t
7.4.
A
be a p a r t i a l o r d e r i n g w i t h dimension
h
E
E
and
i s countable).
( p o s i t i v e i n t e g e r ) . Add
u which w i l l be t h e minimum o f a p a r t i a l o r d e r i n g , e x t e n s i o n o f A
a new element
\Alu{u}
t o t h e new base
.
h ; similar-
Then we o b t a i n an o r d e r i n g w i t h dimension
l y i f we add a maximum. 0 Take t h e c h a i n s
element
u
(2) Let
h
whose c o n j u n c t i o n i s
Ci
be an i n t e g e r 3 2
, and
A, B
let
j o i n t bases, each one w i t h dimension & h u n i o n o f t h e bases, common e x t e n s i o n o f 0 Assume
h = 2
C, C '
conjunction i s
B
A
t h e p a r t i a l o r d e r i n g based on t h e
a d B
and i n which e v e r y element
l B l , has dimension \< h
.
of
.
whose c o n j u n c t i o n i s
, and
A
t h e two c h a i n s
D, D '
Then, on t h e u n i o n o f t h e two bases, t a k e t h e chains
whose CtD
.0
D'+C'
.
7.5. Every f i n i t e t r e e , e i t h e r i s a c h a i n , o r has dimension 2 0
t h e minimum
Ci
( t h e p r o o f immediately extends t o any g r e a t e r i n t e g e r ) . Consider
t h e two c h a i n s and
add t o each
be two p a r t i a l o r d e r i n g s w i t h d i s -
. Then
i s i n c o m p a r a e w i t h e v e r y element o f
IAl
, then
A
.0
We c o n s t r u c t t h e t r e e f r o m i t s maximal elements, by a f i n i t e sequence o f t h e
two f o l l o w i n g o p e r a t i o n s . (1) u n i o n o f two t r e e s w i t h d i s j o i n t bases, each e l e ment o f one base b e i n g incomparable w i t h each element o f t h e o t h e r ; ( 2 ) a d d i t i o n o f a minimum; f i n a l l y use t h e p r e c e d i n g 7.4. 0 7.6. L e t
ty
a
a
, is
be a c a r d i n a l ; e v e r y d i r e c t p r o d u c t o f c h a i n s whose s e t has c a r d i n a l i a p a r t i a l o r d e r i n g w i t h dimension
cement axiom s u f f i c e s i f
a
4
a
(uses axiom o f choice; r e i n f o r -
i s f i n i t e : see ch.2 Cj 4 . 2 ) . T h i s statement w i l l be
completed i n 7.9 below. 0 Denote t h e c h a i n s by
product o f t h e
Ai
.
( i € I with cardinality
Ai
, consider
i
For each i n d e x
a)
, and
let
A
be t h e d i r e c t
t h e d i r e c t product
Di
o f the
A j ( j # i) . By t h e r e i n f o r c e m e n t axiom, t h e r e e x i s t s a t l e a s t one t o t a l l y o r d e r e d Di
.
then a s s o c i a t e t h e c h a i n
Bi
reinforcement f(i)
< g(i)
Ci
(mod Ai)
of
or
To each
i a s s o c i a t e a unique
f ( i ) = g(i)
A
be a p a r t i a l o r d e r i n g and
restriction o f
A
has dimension
&
p
(axiom o f c h o i c e ) ,
. Then p
iff
(sequence o f t h e f ( j ) f o r j # i)4
with
(sequence o f t h e g ( j ) f o r j # i) (mod Ci) j u n c t i o n o f t h e c h a i n s Bi . 0 7.7. L e t
Ci
I A I , such t h a t f Q g (mod Bi)
w i t h base
t h e d i r e c t product
A
i s t h e con-
a p o s i t i v e integer. I f every f i n i t e
, then
A
has dimension
4p
(uses
THEORY OF RELATIONS
118
u l t r a f i l t e r axiom; ZF suffices i f
A
i s countable).
To each f i n i t e subset F of the base I A l , associate the s e t UF of multirelations ( C , ,..., C p ) , a sequence of p chains with cornon base F , such t h a t , f o r every x , y in F , we have X Q y (mod A ) i f f x G y (mod C1) a n d ... a n d x g y (mod C ) By hypothesis U F i s non-empty f o r every F . Moreover i f F'S F P then every multirelation which i s an element of UF , when r e s t r i c t e d t o F ' , yields an element of U F , . By the coherence lemma ch.2 5 1.3 (equivalent t o the u l t r a f i l t e r axiom), there e x i s t s a multirelation with base I A l whose r e s t r i c t i o n t o each F belongs t o UF . Hence t h i s multirelation i s a sequence of p chains, each of whose base i s I A 1 , and whose conjunction i s the partial ordering A . 0 Ll
.
I n p a r t i c u l a r , i t follows from 7 . 5 t h a t every t r e e ( f i n i t e or i n f i n i t e ) i s e i t h e r a chain or has dimension 2 . be a s e t ; t o each element a of E , associate the singleton a ' of = E-a' of t h i s singleton. Then f o r any given s e t of subs e t s of E which contains a l l the a ' and a " as elements, the partial ordering of inclusion has dimension equal t o Card E (DUSHNIK, MILLER 1941).
7.8. Let
E
a and the complement a"
Let U S denote by < any chain which i s a reinforcement of the partial ordering of inclusion. Then there e x i s t s a t most one element a of E f o r which a " < a ' . Indeed, i f we have two d i s t i n c t elements a , b with a " < a ' and b " < b ' , hence we obtain a " < a ' s b " < b ' s a " thus a = b . For each element a of E , the two s e t s a ' and a" are incomparable with respect t o inclusion. Hence among the chains whose conjunction i s the partial ordering of inclusion, one a t l e a s t s a t i s f i e s the inequality a " < a ' , with x ' < x" f o r a l l elements x # a . Associate t h i s chain t o a : then the s e t of those chains corresponding t o a l l elements of E gives the desired ordering of inclusion. To see t h i s , i t remains t o consider two subsets X, Y of E which are incomparable with respect t o inclusion. So there e x i s t an element x of E with x c X and x I# Y , and an element y w i t h y € Y and y f X . Then the chain associated with x gives Y 5 x " < x ' t, X so Y < X ; similarly the chain associated with y gives X < Y 0
0
.
The preceding statement extends as follows (POUZET, 1969'). Let A be a partial ordering with base E . To each element a of E , associate the i n i t i a l interval a ' of elements x s a (mod A ) , and the i n i t i a l interval a" of elements x < or I a (mod A ) Then f o r any given s e t of subsets o f E which contains a l l the a ' fi a " as elements, the partial ordering of inclusion has dimension equal t o the l e a s t cardinality of those s e t s of chains which a r e r e s t r i c tions of A and whose bases cover the whole s e t E (uses axiom of choice).
.
0
For each a , b
in
E
, if
a"< a'
and b " < b '
(where
c
denotes a chain
119
Chapter 4
which r e i n f o r c e s t h e i n c l u s i o n ) , then the preceding argument proves t h a t a and b a r e comparable: a d b o r b,< a (mod A ) . Hence t o each chain which reinforces the i n c l u s i o n , t h e r e corresponds a t o t a l l y ordered r e s t r i c t i o n I of A , such t h a t the elements u of 111 s a t i s f y u " < u' , w i t h however x ' < x" f o r those elements x in E - I11 . Conversely, given an a r b i t r a r y t o t a l l y ordered r e s t r i c tion I of A , t h e i n e q u a l i t i e s u"( u ' f o r each u i n 1 1 1 , a r e mutually compatible, and a r e compatible with i n c l u s i o n . We can always t a k e each I t o be maximal with r e s p e c t t o inclusion (axiom of c h o i c e ) , and then these conditions have t o be completed by x ' ( x" f o r every x in E - 111 . We end a s in the preceding proof. 0 7.9. Given a d i r e c t product of a - many chains (where a is finite or infinite), each chain being reduced t o t h e elements 0 and 1, we obtain a p a r t i a l ordering isomorphic t o inclusion f o r subsets of a , by replacing each subset b of a by
i t s c h a r a c t e r i s t i c f u n c t i o n , taking t h e value 1 f o r each chain belonging t o and 0 f o r each chain belonging t o a-b . Hence, by t h e preceding, t h e d i r e c t product just considered has dimension Consequently, t h e d i r e c t product of elements, has dimension a . I t has dimension by 7.6. 0
0
3
a
a
-
b
a
,
.
many chains, each havinq a t l e a s t two
by the preceding argument and 7 . 3 , and dimension
4
a
For several developments, e s p e c i a l l y concerning p a r t i a l orderings w i t h dimension 2 , see e x e r c i s e 5 below. Let us f i n a l l y c i t e a r e s u l t due t o HIRAGUCHI 1951 and 1955: f o r every p a r t i a l ordering w i t h f i n i t e c a r d i n a l i t y 4 , we have dimension most equal t o p/2 .
pa
§
8
at
- BOUND
Let A be a p a r t i a l ordering and B an i n i t i a l i n t e r v a l of A . An element u in the d i f f e r e n c e s e t I A l - I B I i s c a l l e d a -(more p r e c i s e l y a minimal s t r i c t upper bound) o f B (mod A ) i f every element s t r i c t l y l e s s than u (mod A ) i s an element of l B l
.
8.1. The bounds of an i n i t i a l i n t e r v a l a r e pairwise incomparable. Consequently i f A i s f i n i t e l y f r e e , then there a r e only f i n i t e l y many bounds f o r each i n i t i a l i n t e r v a l of A . 8.2. Let B be an i n i t i a l i n t e r v a l of a p a r t i a l ordering A u (mod A ) f o r every bound u of B .
x
3
. If
x 6 IBI , then
THEORY OF RELATIONS
120
if
Moreover
A
i s well-founded,
x 2 u
condition
0 Assume t h a t
A
f o r e v e r y bound
IAI
If
A
A
if
x
u
. l B l , there exists
does n o t belong t o
i s a bound. 0
then t h e previous proposition i s f a l s e .
+
t o be t h e t o t a l o r d e r i n g L3
converse o f w ) , and
B
i s e q u i v a l e n t Lo-the
which i s minimal among those elements i n t h e d i f f e -
u s x IBI : this
-
i s n o t well-founded,
0 Take
u f
i s well-founded;
a t l e a s t one element rence s e t
x e IBI
then t h e c o n d i t i o n
(where C J -
W-
i s the retro-ordinal
B t o be t h e i n i t i a l i n t e r v a l u : t h e n t h e r e does n o t
e x i s t any bound. 0 8.3. L e t quence .h)
A
o f elements w i t h
vi
. For
B an i d e a l i n A
be a p a r t i a l o r d e r i n g and ( i = 1, ..., h )
ui
, and
B
< ui
f o r each
(mod A )
e v e r y f i n i t e sevi
f o r e v e r y sequence
o f bounds o f
( i = 1, ...
i , t h e r e e x i s t s an element
t
o f t h e base I B I , s a t i s f y i n g t h e c o n d i t i o n s t 3 vi and t 2 ui f o r each i . 0 Consequence o f t h e d e f i n i t i o n s o f i d e a l and bound, t a k i n g i n t o account that
<
vi
u.
implies
1
The p r o p o s i t i o n i s no l o n g e r t r u e when 0 Take
Let
A
w i t h t h r e e elements
B
B
a,b,c
= u2 = c
with
v1 = a
i s an a r b i t r a r y i n i t i a l i n t e r v a l o f
where
a
. Then .
be t h e r e s t r i c t i o n t o (a,b)
u1
take
i .0
vi E I B I f o r each
and
and
b c c
and
A.
a I b (mod A ) .
i s a bound and i t s u f f i c e s t o
v2 = b
N o t i c e a l s o t h a t t h e converse o f t h e p r o p o s i t i o n i s f a l s e .
A e
0 Take
d
w i t h f i v e elements and
a,b,c B
so t h a t
a,b,c,d,e
where
d< a
m u t u a l l y incomparable. L e t
<e
and
d
e
and
be t h e r e s t r i c t i o n t o
B
i s n o t an i d e a l . However t h e c o n c l u s i o n o f t h e p r o p o s i t i o n
h o l d s . Indeed t h e o n l y bound i s
c
and t h e o n l y element
t = b , o r even
t = d
is
d : it suffi-
.0
ces t o t a k e
t = a
8.4.
be a w e l l - f o u n d e d p a r t i a l o r d e r i n g such t h a t e v e r y element has o n l y
Let
A
or
f i n i t e l y many predecessors, and l e t sequence o f bounds such t h a t t h e
ui
ui
and t o enumerate them, r e p e a t i n g ui
. Then
be an i d e a l i n B
, there
.
Then f o r e v e r y f i n i t e
6
t
t
of
IBI
.
t h e f i n i t e s e t o f t h o s e elements
ui ui
A
e x i s t s an element
a r e a l s o some bounds of t h e i n t e r v a l
0 I t s u f f i c e s t o a s s o c i a t e t o each
than
B
( i = 1, ..., h )
as many t i m e s as t h e r e a r e elements
v < ui v less
apply t h e preceding proposition. 0
One s h o u l d n o t t h i n k t h a t t h e i d e a l
B
i s n e c e s s a r i l y reduced t o an element and
i t s predecessors; f o r i n t h i s case t h e base o f
B
0 To have an i d e a l w i t h i n f i n i t e base, t a k e
t o be t h e s e t o f words formed o f
t h e two l e t t e r s
A
would be f i n i t e .
a, b, p a r t i a l l y o r d e r e d by e m b e d d a b i l i t y (see
5
2, t h e elements
Chapter 4
121
a , b assumed t o be incomparable). Then the s e t of words formed with the single l e t t e r a , repeated a f i n i t e number of times, constitutes an i n f i n i t e ideal. The only bound of t h i s ideal i s the word reduced t o the singleton of b . For t one can take any f i n i t e sequence of
a
.0
The previous proposition no longer holds f o r an a r b i t r a r y well-founded partial ordering. 0 Take A t o be the t o t a l ordering o +1 and’ B t o be the i n i t i a l interval w of A . The only bound i s the maximum element of A . 0 The converse of our proposition i s f a l s e , as shown by the counterexample in the preceding 8.3. 8.5. Let A be a well partial ordering, and u l , ...,u h a f i n i t e sequence of elements. Suppose t h a t there e x i s t s an W -sequence of d i s t i n c t elements t . (j inJ teger) such t h a t , f o r each j , the elements ul, ...,u h are some bounds of the i n i t i a l interval \< t . Then there e x i s t s an i n f i n i t e ideal i n A , which has j ul, ..., u h among i t s bounds (uses dependent choice; ZF suffices i f A countable).
A i s a well partial ordering, there e x i s t s a s t r i c t l y increasing w -se(see 3 . 2 . ( 3 ) , dependent quence (mod A ) extracted from the sequence of the tj choice). Denote t h i s sequence again by the terms t . , and l e t B be the s e t of J those elements x f o r which there e x i s t s a t l e a s t one j with t . > x . Then J f i r s t l y , B i s an i n f i n i t e ideal since i t s base contains the t . Secondly, f o r j each element u = u.1 (i= 1 or ... or h ) , we have t h a t u i s a bound of each inHence f o r each j a n d so u 4 I B 1 . Moreover, terval of the form 6 t j j f o r every element x ( u , we have x c to , hence x E: I B I . I t follows that u i s a bound of B . 0
0 Since
.
ugt
8.6. Let A be a well partial ordering such t h a t every element has a t most f i n i t e l y many predecessors. Let u l , ..., u h be a f i n i t e sequence of elements. Assume t h a t there e x i s t s an &I -sequence of d i s t i n c t elements t . ( j integer) such t h a t , J f o r each j , the bounds of the i n i t i a l i n t e r v a l s t are exactly ul,..., uh j plus elements having each a t l e a s t j predecessors (mod A) Then there e x i s t s an i n f i n i t e ideal in A which has exactly the bounds ul, ....u h (same conditions as in preceding 8 . 5 ) .
.
As in the preceding proof, take a s t r i c t l y increasing &-sequence extracted from the t , and denoted again by the t . Let B be the ideal generated by j j Supthese t . . By the preceding proposition, each u l , ..., u h i s a bound of B J pose t h a t there e x i s t s another bound v of B . Then v I+ 1 B I , so v & t for j each j Moreover, every element w e v belongs t o I B I and hence s a t i s f i e s
0
.
.
122
THEORY OF RELATIONS
w4 t j
f o r a l l j s u f f i c i e n t l y l a r g e . By hypothesis, t h e r e a r e o n l y f i n i t e l y many elements smaller than v , so t h e r e e x i s t s an i n t e g e r j a f t e r which a l l t h e w are ,< t , hence a f t e r which v i s a bound of the i n t e r v a l 4 t . . This conj J t r a d i c t s the hypothesis t h a t there are 3 j many predecessors o f v 0
.
We s h a l l see i n ch.12
0
2.2 and 2.3, t h a t the two preceding p r o p o s i t i o n s are s t i l l
t r u e f o r embeddability between f i n i t e r e l a t i o n s o f a given a r i t y . Obviously embedd a b i l i t y i s a well-founded p a r t i a l o r d e r i n g f o r such r e l a t i o n s , and s a t i s f i e s t h e c o n d i t i o n t h a t each element has o n l y a f i n i t e n o t a w e l l p a r t i a l ordering. The proofs a r e a over, t h e p r o p o s i t i o n 8.5 becomes f a l s e i f A O r d e r i i g ; taRe A t o be t h e i d e n t i t y relatl'on
number little i s an on an
§ 9 - MAXIMAL REINFORCED CHAIN THEOREM DE JONGH.PARIKH: EXTRACTION PROPERTY
o f predecessors; b u t i t i s l e s s simple than here. Morea r b i t r a r y well-founded p a r t i a l infi'nl'te base.
FOR A WELL PARTIAL
ORDERING:
L e t A be a w e l l p a r t i a l ordering, B an i n i t i a l i n t e r v a l o f A , thus a w e l l p a r t i a l ordering. Denote by A-B t h e w e l l p a r t i a l o r d e r i n g r e s t r i c t i o n o f
9.1.
A t o t h e d i f f e r e n c e s e t I A I - I B I . Recall t h a t 3 (A) denotes the p a r t i a l o r d e r i n g o f i n i t i a l i n t e r v a l s o f A , which i s well-founded by 4.1 (dependent choice). L e t us denote by Ht"(A) t h e h e i g h t o f t h e maximum A o f g(A) , i . e . t h e h e i g h t o f 3 ( A ) minus one; analogous n o t a t i o n s w i t h B and A-B Then we have t h e f o l l o w i n g i n e q u a l i t i e s : Ht"(B) + Hto(A-B) 4 Ht"(A) & Ht"(B) @ Ht"(A-B) , where t i s t h e usual o r d i n a l sum and @ i s t h e n a t u r a l , commutative sum: see ch.1 5 9. 0 Consider the isomorphism which, t o each i n i t i a l i n t e r v a l X o f A-B , associat e s t h e i n i t i a l i n t e r v a l BuX o f A The p a r t i a l o r d e r i n g 3 (A) i s an extens i o n of 3 (B) followed by 3(A-B) : t h i s y i e l d s t h e f i r s t i n e q u a l i t y . On another hand, t o each i n i t i a l i n t e r v a l X o f A , associate t h e couple whose f i r s t term i s t h e r e s t r i c t i o n X / ( \ X ( n l B ( ) and whose second term i s t h e rest r i c t i o n X / ( l X \ A IA-BI) T h i s i s an isomorphism which transforms 3 ( A ) i n t o a r e s t r i c t i o n of the d i r e c t product 3 ( B ) x 'J(A-B) : t h i s y i e l d s t h e second i n e q u a l i t y , by using 7.2. 0
.
.
.
9.2. L e t us s t a t e as follows t h e MAXIMAL REINFORCED CHAIN THEOREM.
Given a w e l l p a r t i a l o r d e r i n g A
and t h e associated well-founded p a r t i a l o r d e r i n g
3
( A ) which is isomorphic t o J(A) , t h e r e e x i s t s a well-ordered r e s t r i c t i o n o f the h e i g h t of 3(A) , thus which has t h e maximum p o s s i b l e height.
Equivalently, t h e r e e x i s t s a w e l l - o r d e r i n g which i s a reinforcement o f A , isomorphic t o t h e h e i g h t o f 3 ( A ) minus one, thus isomorphic t o t h e maximum
123
Chapter 4
p o s s i b l e o r d i n a l (DE JONGH, PARIKH 1977; uses axiom o f c h o i c e ) . 0
F i r s t o f a l l , t h e e q u i v a l e n c e o f b o t h p r e c e d i n g statements i s an immediate conse-
, the
quence o f 1.3 (BONNET, POUZET 1969); n o t e t h a t f o r any o r d i n a l
well-
.
o r d e r i n g o f i n i t i a l i n t e r v a l s i s isomorphic w i t h o ( t 1
We s h a l l prove t h e f i r s t statement, c o n c e r n i n g t h e maximal c h a i n (up t o isomor-
.
phism) among r e s t r i c t i o n s o f J ( A ) Note t h a t t h e r e s u l t i s obvious f o r also i f
A
We procede by i n d u c t i o n on t h e h e i g h t Ht"(A). empty, t h u s H t o ( A ) = 0 . I t i s obvious
assume t h a t
Ht"(A)
i s a decomposable o r d i n a l
i s t h e l a s t component o f
previous decomposition
/s + 'd
.
Ht"(A) = 1
i s based on a s i n g l e t o n , t h u s
A
Assume f i r s t t h a t
oc
o( =
. Moreover we
/3+
i n t h e Cantor normal form. Then t h e
h o l d s as w e l l w i t h t h e n a t u r a l , o r commutative
sum as w i t h t h e usual o r d i n a l sum. Denote by
B an i n i t i a l i n t e r v a l o f
hypothesis, t h e r e e x i s t s a c h a i n o f t y p e S i m i l a r l y w i t h J(A-B)
and
. Hence
8
A
elements a r e i n i t i a l i n t e r v a l s o f
form w ktl
, where
h e i g h t CJ
modulo
k) k+l
4
w
Hto(A-B) =
/3
modulo
including B
.
. Then
the
induction
which i s a r e s t r i c t i o n o f
8
3 (B) . whose
F i n a l l y t h e sum o f these
.
i s an indecomposable o r d i n a l o f t h e p a r t i c u l a r
Ht"(A)
k i s an o r d i n a l . Take an i n i t i a l i n t e r v a l B o f A w i t h g ( A ) . Then by 9 . 1 we have t h e f o l l o w i n g i n e q u a l i t y :
8 Ht"(A-B)
, thus
Ht"(A-B)
.
= L3 k+l
Bi
By i t e r a t i o n , we o b t a i n an c ~ - s e q u e n c e o f i n i t i a l i n t e r v a l s
A which i s s t r i c t l y i n c r e a s i n g under i n c l u s i o n , w i t h Hto(Bitl-Bi) each
3 (A)
. By o u r
8
there e x i s t s a chain o f type
two c h a i n s has t h e isomorphism t y p e o( Secondly we assume t h a t
13
with height
A
two i n e q u a l i t i e s i n t h e p r e v i o u s 9.1 y i e l d
i (obvious n o t a t i o n Bitl-Bi
f o r the r e s t r i c t i o n o f
( i integer)
of for
= W
t o the diffe-
A
rence s e t o f t h e bases). By o u r i n d u c t i o n h y p o t h e s i s , we a s s o c i a t e t o each
o f i n i t i a l i n t e r v a l s , each one s i t u a t e d between
w
i a chain o f order type 0
Bi
and
. Hence
Bitl
-sum o f t h e s e c h a i n s y i e l d s a c h a i n o f i n i t i a l i n t e r v a l s w i t h t y p e
T h i r d l y we assume t h a t Denote by
u = Cof d
Ht"(A)
. For
the
w k+l .
i s an a r b i t r a r y indecomposable o r d i n a l o(
and t a k e a
u-sequence o f i n c r e a s i n g o r d i n a l s o(
.
(i< u)
i < u , t a k e an i n i t i a l i n t e r v a l Ai o f A w i t h h e i g h t C t i modulo (axiom o f c h o i c e ) . Hence H t o ( A i ) = o( for each i < u L e t bi be an element i n t h e d i f f e r e n c e s e t I A ( - lAil . Denote with
Sup
o(i
=
K
each
.
by
hi
t h e h e i g h t modulo
r(A)
(A)
o f the i n i t i a l interval
s t r i c t l y l e s s t h a n o r incomparable w i t h
Sup
/si
= o(
.
From t h e
bi
. Then
u-sequence o f i n d i c e s
u-sequence f o r which a l l elements
bi
o(
o f elements
Bi
4 fi
< 4
thus
i K u , we e x t r a c t another
and a l l o r d i n a l s
bi
k
are distinct.
THEORY OF RELATIONS
124
Since A i s a well p a r t i a l ordering, by 3.4 we may require t h a t the u-sequence of the bi be s t r i c t l y increasing (mod A ) . Thus the sequence of corresponding intervals Bi i s s t r i c t l y increasing with respect t o inclusion. By a new extrac0 p i ( n a t u r a l , or commutative sum) t i o n , we may require t h a t fs i+l >,
pi
f o r each index i < u . Then we have Ht"(Bi+l-Bi) >/ pi f o r each i , using 9 . 1 above. By the induction hypothesis, we can take in each difference Bi+l-Bi a chain of i n i t i a l intervals of A , situated between B i and B i t l and whose length i s a t l e a s t h i , The sum of these chains yields a d e f i n i t i v e chain whose Oi = 4 by ch.2 5 5.5. CI length i s 9.3. THE EXTRACTION PROPERTY Given an aleph , we shall say t h a t a p a r t i a l ordering A has the extraction u of d i s t i n c t elements in the property f o r w,, i f f o r every &,-sequence base A , there e x i s t s a s t r i c t l y increasing (mod A) sequence extracted from u and with the same length as u (notion communicated by ZAGUIA 1983). By 3 . 2 . ( 3 ) , an i n f i n i t e p a r t i a l ordering A i s a well p a r t i a l ordering i f f A has the extraction property f o r 0 . By 3.4, any well partial ordering has the extraction property f o r any i n f i n i t e aleph. Let A be a well p a r t i a l ordering, J ( A ) the p a r t i a l ordering of i n i t i a l intervals of A (with respect t o inclusion). Then 'j ( A ) has the extraction property f o r every regular aleph s t r i c t l y greater than w ( Z A G U I A 1983). 0 For any i < U* (regular aleph), l e t B i be an i n i t i a l interval of A . Denote by Mi the f i n i t e s e t of minimal elements (mod A ) in the complement s e t I A l - l B i l . For each integer p , denote by C the class of those Mi whose P , i s regular, then there e x i s t s an integer p such cardinal i s p . Since a that C has cardinality Reindexing the sequence of Mi , we consider i t P as an G)&-sequence in the d i r e c t Droduct Ax ...% A (p times) (choice f o r f i n i t e s e t s ) . This d i r e c t product being a well p a r t i a l ordering by 7.1.(2) (denumerable subset axiom), the proposition follows by 3.4. 0 I n the cited t h e s i s , i t i s proved t h a t any d i s t r i b u t i v e l a t t i c e which has the extraction property f o r every regular aleph s t r i c t l y greater than w , admits a
.
chain isomorphic t o i t s height: t h i s i s another proof f o r theorem 9 . 2 .
§
10 - COFINALITY
OF A FINITELY
FREE PARTIAL
ORDERING
10.1. INCREASING SEQUENCE LEMMA Let A be a p a r t i a l ordering, and l e t k = Cof A Then a necessary and s u f f i c i e n t condition t h a t A be generated by a s t r i c t l y increasing sequence ( i . e . t h a t every element have an upper bound in the sequence) i s t h a t every subset of the base with
.
Chapter 4
.
-.c a r d i n a l 4 k have an upper bound i n A
125
Moreover
0 If
i s necessarily
regular
i s generated by a s t r i c t l y i n c r e a s i n g o r d i n a l - i n d e x e d sequence
A
k = Cof C
thus
k
, has
k
<
subset w i t h c a r d i n a l Card H
(iC k )
.
k
=
k
b
in
an upper bound i n
H
. Conversely
C
suppose t h a t e v e r y
has an upper bound. Take a c o f i n a l subset
Well-order
t h e elements o f
o f elements
, then
C
i s r e g u l a r . Moreover e v e r y subset o f t h e base, w i t h c a r d i n a l
s t r i c t l y less than with
k
.
(uses axiom o f choice; communicated by POUZET)
H
. Let
H ; consider
. Then
k
of
A
as an o r d i n a l and denote by
ai
H
d e f i n e as f o l l o w s t h e i n c r e a s i n g
.
bo = a.
16 i < k
Let
k-sequence
; by h y p o t h e s i s t h e r e e x i s t s
Then d e f i n e b . ( j < i). J
an upper bound o f a l l a and a l l a l r e a d y d e f i n e d j as b e i n g such an upper bound, b e l o n g i n g t o H 0
.
bi
10.2. REGULAR INCREASING SEQUENCE THEOREM
A
Let A
be a p a r t i a l o r d e r i n g and
<
has c o f i n a l i t y
k
.
k = Cof A
I f every proper i n i t i a l i n t e r v a l o f
, t h e n ( 1 ) e v e r y subset o f t h e base,
has an upper bound; ( 2 )
k
i s a r e g u l a r aleph and
,
with c a r d i n a l i t y < k
i s generated by an i n c r e a -
A
sing k-sequence .(uses axiom o f choice; communicated by POUZET). A
0 By t h e p r e c e d i n g lemma, ( 2 ) f o l l o w s f r o m (1). Moreover we can always reduce
A
t o a c o f i n a l r e s t r i c t i o n , s t i l l denoted by To each element or Hu
I
i n t h e base, a s s o c i a t e
t h e s e t o f elements
every subset in
x
x (mod A) ; t h e n l e t
F
<
k x = Cof Ax x
<
k
cardinality
. F i r s t we
, has
. uck ,
see t h a t , g i v e n
an upper bound. Indeed f o r a l l
has a t most c o f i n a l i t y
Ax
k
Ax = i n i t i a l i n t e r v a l o f elements < . F o r each c a r d i n a l u < k , denote by
kxG u
such t h a t
F o f Hu w i t h Card F
, the union o f corresponding
k
, having
uxCard F
<
x
.
k
So t h a t t h i s u n i o n does n o t c o v e r t h e e n t i r e base: hence t h e i n t e r s e c t i o n o f c o r responding f i n a l i n t e r v a l s 3 x
, i s non-empty.
Now l e t us prove t h e r e g u l a r i t y o f consider
Hu
each o f which h a v i n g c a r d i n a l i t y < k a set
.
k
u
F o r each
as t h e u n i o n o f an a r b i t r a r y i n c r e a s i n g
, since
Card Hu<
: so t h a t each has an upper bound. We o b t a i n
Ku o f these upper bounds, w i t h Card KU 6 Cof k
. Now when
(Cof k ) many values, we o b t a i n a c o f i n a l s e t modulo A 2 n a l i t y l e s s t h a n o r equal t o ( C o f k ) = Cof k : t h i s proves t h a t
by t a k i n g
We a r e f i n i s h e d p r o v i d e d we n o t e t h a t each s e t
r i l y i n c l u d e d i n some = k
Hu
(u < k )
,
k
( C o f k)-sequence o f subsets,
F with
Card F
<
u varies , having cardik = Cof k k
,is
.
necessa-
F would be t h e u n i o n o f (Cof k )
; f o r otherwise
many d i s j o i n t non-empty subsets: c o n t r a d i c t i o n . 0
10.3. COFINALITY THEOREM FOR FINITELY FREE PARTIAL ORDERINGS
Every f i n i t e l y f r e e p a r t i a l o r d e r i n g , e i t h e r has f i n i t e c o f i n a l i t y , o r i n f i n i t e r e g u l a r c o f i n a l i t y (POUZET 1979; uses axiom o f c h o i c e ) . 0 With previous notations,
suppose t h a t
We can always suppose t h a t
Card A = k
k = Cof A and t h a t
i s an i n f i n i t e s i n g u l a r aleph.
A
i s w e l l - f o u n d e d (ch.2
5
5.1,
THEORY OF RELATIONS
126
c o r o l l a r y ) . Then by t h e p r e c e d i n g 10.2, t h e r e e x i s t s a p r o p e r i n i t i a l i n t e r v a l with cofinality
.
k
Thus t h e r e e x i s t s a s t r i c t l y d e c r e a s i n g a - s e q u e n c e o f i n i -
. Take
an
element i n each d i f f e r e n c e s e t : we o b t a i n a bad w - s e q u e n c e (mod A). Since
A
t i a l i n t e r v a l s (under i n c l u s i o n ) , each o f which h a v i n g c o f i n a l i t y
k
i s well-founded, we have i n f i n i t e l y many m u t u a l l y incomparable elements. 0 EXERCISE 1
Let
A
the
set
-
MAC NEILLE COMPLETION
be a p a r t i a l o r d e r i n g w i t h base
then t h e s e t val o f
o f l o w e r bounds o f
X+
X'
and t h a t t h e a s s o c i a t i o n o f
A
.
E
To each subset X'
is
>/
. Notice
that
X+
o f upper bounds (each element o f
X'
X+zX , idempotence: t h e n X+C Y + .
we have t h e i n c l u s i o n X EY
increasing: i f
to
X+
1 - Give examples t o show t h a t ( X u Y)+ ( X n Y ) + can be a p r o p e r subset o f X + A
of
X
, associate
E
X ),
e v e r y element o f
i s an i n i t i a l i n t e r -
i s a closure operation; i . e .
X
X++ = X+ ; and t h e o p e r a t i o n i s
can p r o p e r l y i n c l u d e
.
Y+
X+
u
Y+
, and t h a t
I n any case, e v e r y i n t e r s e c t i o n o f
closed sets i s closed (closed s e t = i d e n t i c a l t o i t s closure). 2 - Let
x
be an element o f
4x
t i a l interval
.
(mod A )
.
E
The c l o s u r e o f t h e s i n g l e t o n o f
I d e n t i f y each
x
x
i s the i n i -
with this i n i t i a l interval: the
s e t o f c l o s u r e s , p a r t i a l l y o r d e r e d by i n c l u s i o n , becomes a p a r t i a l o r d e r i n g e x t e n ding
, called
A
F o r each s e t
X
t h e MAC-NEILLE c o m p l e t i o n o f
, the closure
A (see MAC-NEILLE 1937).
i n c l u d e s t h e i n i t i a l i n t e r v a l generated by
X+
I n g e n e r a l , i t i s d i s t i n c t f r o m i t . Indeed, t a k e and
a Ib
and c o n s i d e r t h e i n i t i a l i n t e r v a l 4a.b)
o r d i n a l W +1 and i t s i n i t i a l i n t e r v a l G,
u
-
Let
be c h a i n s w i t h base
Ci
Ci
c a l l t h e component
Ti
The components o f a l l the chain Ci
.
and
X
.
b< c
example: t a k e t h e
take the d i r e c t product
w i t h index T
t
i
T
A . Suppose t h a t
i the closure o f
r e l a t i v e t o a given index
. For
A
i s the i
,
T (mod Ci) : t h e n Ti 3 T i are i n i t i a l intervals o f
.
be c l o s e d (mod A)
each i n d e x
, hence t h e i r s e t i s a c h a i n ( w i t h r e s p e c t t o i n c l u s i o n ) a s s o c i a t e d
Ci
T (mod Ci)
f o r each
which r e i n f o r c e
E
: see 7.3. L e t
Moreover, e v e r y c l o s e d s e t
Indeed, i f of
. Another
. Finally
a
.
J ( ,
conjunction o f the
to
with
and n o t i c e t h a t t h e c l o s u r e s a r e t h o s e r e c t a n g l e s d e f i n e d by two i n i -
w
t i a l intervals o f 3
a,b,c
.
T
i s an element o f each f o r each Hence
i
.
Hence A
, then
t$(mod
t ,< (mod A) e v e r y
Consequently t h e c o m p l e t i o n o f
i s t h e i n t e r s e c t i o n o f i t s components
Ti
Ci)
t &(mod Ci)
e v e r y upper bound o f
u. b. o f T (mod A)
Ti.
e v e r y upper bound T (mod A)
.
has t h e same dimension as
A
.
-
EXERCISE 2 BOUNDEDLY FINITELY FREE PARTIAL ORDERING L e t A be a p a r t i a l o r d e r i n g w i t h base E , whme f r e e subsets a r e assumed t o
D be one of t h e f r e e subsets w i t h maximum c a r d i n a l . Then t o each element i f D , be f i n i t e and a l l o f c a r d i n a l i t i e s l e s s t h a n a maximum f i n i t e i n t e g e r . L e t
Chapter 4
127
of
Ai
t h e r e corresponds a t o t a l l y o r d e r e d r e s t r i c t i o n
A
whose base i n t e r s e c t s
i , such ______ t h a t t h e u n i o n o f t h e bases l A i l 12 E (see DILWORTH 1950; t h e p r o o f q i v e n h e r e i s due t o PERLES 1963). We assume t h a t t h e D
with the singleton o f
bases o f these chains, o r t o t a l l y o r d e r e d r e s t r i c t i o n s , a r e d i s j o i n t , which i s obviously possible.
.
1 - F i r s t assume t h a t E i s f i n i t e , and argue by i n d u c t i o n on Card E Fix A D , and examine t h e f i r s t case where D i s d i f f e r e n t f r o m t h e s e t o f m i n i mal elements (mod A) and f r o m t h e s e t o f maximal elements. Then c o n s i d e r t h e r e s and
triction
A+
of
A
t o those
analogous r e s t r i c t i o n tion i s true for
A+
x
f o r which t h e r e e x i s t s i n
D
i6 x ;
an element
A- w i t h i x . By t h e i n d u c t i o n h y p o t h e s i s , t h e p r o p o s i and A- , which a r e p r o p e r r e s t r i c t i o n s o f A . Then t o each
i i n D , there i s associated a chain
At
and a c h a i n
AT : d e f i n e
Ai
as t h e i r
common e x t e n s i o n . I n t h i s f i r s t case, n o t e t h a t t h e s e t o f minimal elements
moduA i s always a subset o f t h e s e t o f minimums f o r a l l c h a i n s Ai . Examine t h e second case where D i s t h e s e t o f minimal elements (mod A), and moreover t h e r e e x i s t s a n o t h e r f r e e s e t D ' w i t h same c a r d i n a l i t y , which i s n e i t h e r t h e s e t o f minimal, n o r t h e s e t o f maximal elements. Then by o u r l a s t r e lo
D
mark i n t h e f i r s t case, we can a s s o c i a t e w i t h with
D'
. Analogous
t h e same s e t o f chains
Ai
as
argument w i t h maximal elements.
Now examine t h e t h i r d case where
D
i s , f o r i n s t a n c e , t h e s e t o f minimal elements,
b u t t h e o n l y o t h e r p o s s i b l e f r e e s e t w i t h maximum c a r d i n a l i t y i s t h e s e t o f maximal elements. Take
i in
D
, then
a maximal c h a i n ( u n d e r i n c l u s i o n ) whose m i n i -
i . Then remove f r o m E t h e elements o f t h i s maximal chain. Then use t h e i n d u c t i o n h y p o t h e s i s on t h i s p r o p e r r e s t r i c t i o n o f A and t h e subset D - ti} .
mum i s
2
-
Suppose t h a t
E
i s i n f i n i t e and assume t h e u l t r a f i l t e r axiom. L e t
f r e e s e t w i t h maximum ( f i n i t e ) c a r d i n a l i t y . To each f i n i t e subset cluding each
D
i in
, associate D
as p r e v i o u s l y a l l p o s s i b l e p a r t i t i o n s o f
a t o t a l l y ordered r e s t r i c t i o n o f
A/F
D
F of
be a E
in-
F yielding for
containing
i as an e l e -
ment. Consider each p a r t i t i o n as an e q u i v a l e n c e r e l a t i o n , and c a l l
UF t h e s e t of t h e s e e q u i v a l e n c e r e l a t i o n s v e r i f y i n g o u r p r o p o s i t i o n : i t s u f f i c e s t o a p p l y t h e coherence lemma ch.2
5
1.3.
N o t i c e t h a t t h e p r o p o s i t i o n does n o t e x t e n d t o t h e case o f an a r b i t r a r y f i n i t e l y f r e e p a r t i a l o r d e r i n g . F o r example, t a k e
A
t o be t h e d i r e c t p r o d u c t o f
w i t h i t s e l f , which i s a w e l l p a r t i a l o r d e r i n g by 7 . 1 . ( 2 ) .
w1
Yet e v e r y decomposi-
t i o n i n t o a sum o f c h a i n s c o n s i s t s o f W1-many c h a i n s (PERLES). 3 - L e t us t a k e up a g a i n t h e case o f a p a r t i a l o r d e r i n g E ; c o n s i d e r a g a i n t o t a l l y o r d e r e d r e s t r i c t i o n s Ai of me t h a t t h e i r bases a r e m u t u a l l y d i s j o i n t . L e t that
y
covers
x
if y
x, y
A w i t h a f i n i t e base A , b u t no l o n g e r assu-
be elements o f
i s an immediate successor (mod A)
of
x
E : we say
.
We say t h a t
128
A
THEORY OF RELATIONS i s t r a n s i t i v e l y generated by t h e
Ai
e x i s t s a f i n i t e sequence g o i n g f r o m
x
if for to
, each
y
,
and y > x (mod A)
x
there
t e r m o f which covers t h e
p r e c e d i n g term, any two c o n s e c u t i v e terms of t h e sequence b e l o n g i n g t o the base o f a same Ai F o r t h i s i t s u f f i c e s t h a t , f o r e v e r y x and el'ery y c o v e r i n g
.
x
, there
Ai
e x i s t s an
x
containing
and y
as elements. I n o r d e r t o a v o i d i s i s o l a t e d i f i t i s incom-
E
useless c o m p l i c a t i o n s , we say t h a t an element o f
p a r a b l e (mod A) w i t h e v e r y o t h e r element: l e t us remove a l l i s o l a t e d elements. Consider t h e o r d e r e d p a i r s l e s s t h a n o r equal t o or
(x',y')
by
A"
.
.
= (x,y)
(x,y)
(x',y')
y
where
where
covers covers
y'
. We
x x'
, if
say t h a t either
(x,y)
is
y,< x ' (mod A)
N o t i c e t h a t t h i s r e l a t i o n forms a p a r t i a l o r d e r i n g denoted
The c o m p a r a b i l i t y (mod A " )
o f two o r d e r e d p a i r s i s e q u i v a l e n t t o t h e
o f a l l t h e f o u r elements which c o n s t i t u t e t h e s e p a i r s .
c o m p a r a b i l i t y (mod A)
Do be a f r e e s e t o f o r d e r e d p a i r s , which i s maximal w i t h r e s p e c t t o i n c l u -
Let
A"
sion, f o r t h e p a r t i a l o r d e r i n g
.
Then t o each element
i
of
ef
A
, such
j e c t i v e l y associate a t o t a l l y ordered r e s t r i c t i o n l e r cardinality
A
generates
. Moreover no
Ai
t r a n s i t i v e l y qenerated by t h e
Ai
Do
we can b i -
is.
A
that
s e t o f c h a i n s w i t h s t r i c t l y smal-
(see BOGART 1970). T h i s l a t t e r p o i n t i s o b v i o u s . Ai
Thus we o n l y have t o prove t h e e x i s t e n c e o f t h e c h a i n s
.
F o r t h a t purpose,
a p p l y t h e statement i n paragraph 1 t o t h e p a r t i a l o r d e r i n g o f o r d e r e d p a i r s w i t h one element c o v e r i n g t h e o t h e r , s t a r t i n g w i t h t h e f r e e s e t
.
Do
For each o f t h e
c h a i n s obtained, i t s u f f i c e s t o t a k e t h e elements i n t h e o r d e r e d p a i r s , and t o complete
transitively.
-
EXERCISE 3
CONJUGATE PARTIAL ORDERINGS
Let
E
and
B a r e conjugates i f , f o r each p a i r o f elements
y
be a s e t and
A, B two p a r t i a l o r d e r i n g s w i t h base
, then
a r e comparable (mod A)
Example:
A
the d i r e c t product r i n g d i r e c t product Notice t h a t i f
A
1 - Let
A, B
x g y
j u n c t i o n of
E
. Another
then
C
z< x
, and 6 i s
example: l e t
C
x
D-
where
D
, and
x
A
and
conversely.
be two chains;
i s a conjugate o f B-
, then
B
A
D
(see ch.2
y (mod A o r mod B)
A u B and
A
,is
. Deduce
u B-
1.7).
i s also a conjugate o f the
a chain. Notice t h a t
that
with the following condition: i f
or z 7 y (mod C)
5
.
be conjugates; show t h a t t h e d i s j u n c t i o n r e l a t i o n
iff x (
if
the free p a r t i a l
C, D
i s t h e converse o f
s i o n i s 1 o r 2 (see 7.3). O r e q u i v a l e n t l y i f f Cement
E
say t h a t
,
E
C x D d e f i n e d i n 5 7 has as a c o n j u g a t e t h e p a r t i a l o r d e -
converse p a r t i a l o r d e r i n g
by
. We
E of
t h e y a r e incomparable (mod B)
i s a t o t a l o r d e r i n g , o r c h a i n on
o r d e r i n g ( i d e n t i t y r e l a t i o n ) on
x, y
.
Au B d e f i n e d A
i s t h e con-
A
has a c o n j u g a t e i f f i t s dimen-
A
has a t o t a l l y o r d e r e d r e i n f o r -
x< y
and
x Iz
and
y I z (mod A)
,
Chapter 4
-
2
S t a r t w i t h a chain
C
and a s e t
U
129
o f intervals o f
U
i n c l u s i o n . Show t h a t one o b t a i n a c o n j u g a t e o f
C
,
p a r t i a l l y ordered by
by o r d e r i n g i n t e r v a l s which C
a r e incomparable w i t h r e s p e c t t o i n c l u s i o n , by means o f t h e p o s i t i o n i n
C
t h e i r l e f t c u t (i.e. the i n i t i a l interval o f
of
s i t u a t e d on t h e l e f t ) . One can as
w e l l o r d e r t h e s e incomparable i n t e r v a l s by t h e p o s i t i o n o f t h e i r r i g h t c u t . Deduce t h a t a p a r t i a l o r d e r i n g has dimension 1 o r 2 i f i t i s isomorphic t o t h e p a r t i a l o r d e r i n g o f i n c l u s i o n between i n t e r v a l s o f a g i v e n c h a i n . I n p a r t i c u l a r u s i n g 6.2 ( u l t r a f i l t e r axiom), e v e r y t r e e has dimension 1 o r 2, which was a l r e a d y mentioned i n 7.7.
-
3
Conversely, l e t
, followed
A
and
B
be c o n j u g a t e s . Consider t h e c h a i n
A- u B
with
A u B w i t h base E ' (an isomorphic copy o f E ) . Associate t o each element x o f t h e p a r t i a l o r d e r i n g A , hence x b e l o n g i n g t o E , t h e interval (x,x') where x ' i s t h e r e p l i c a o f x i n E ' (communicated by POUZET i n 1968, u n p u b l i s h e d ) .
base
E
EXERCISE 4 A
Let
-
by
WELL-ORDERED RESTRICTION OF MAXIMAL LENGTH (LEMMA 4.5 REVISITED)
be a w e l l - f o u n d e d p a r t i a l o r d e r i n g . Then e i t h e r t h e r e e x i s t i n f i n i t e l y
many elements, a l l o f d i f f e r e n t h e i g h t s (mod A) which a r e m u t u a l l y incomparable, o r there e x i s t s a well-ordered r e s t r i c t i o n o f
w i t h t h e same h e i g h t ( u s e s u l t r a -
A
f i l t e r axiom; POUZET 1979).
-
1
Suppose t h e f i r s t c o n c l u s i o n does n o t h o l d , and argue by i n d u c t i o n on t h e
height o f
/3 +
8
A
.
Ht A
Assume f i r s t t h a t
with
/3
non-zero and
i s decomposable, hence o f t h e form
I < /3 + J' .
Let
denote t h e r e s t r i c t i o n o f
C
A t o elements whose h e i g h t (mod A ) i s g r e a t e r t h a n o r equal t o A . For each x i n ICI , we have H t x (mod A ) = 0 + H t x (mod C) ( p r o o f by i n d u c t i o n ) . Hence
. By t h e
Ht C = C"
C
of
induction hypothesis, there e x i s t s a well-ordered r e s t r i c t i o n
with height
. Let
h e i g h t >/
/3 (see
ch.2
ordered r e s t r i c t i o n of
A
with height
2 - Assume now t h a t
5 B"
3.5).
be t h e minimum o f
u
u
t o elements s t r i c t l y l e s s t h a n
. The
C"
A
restriction o f
i s a well-founded p a r t i a l ordering
B
with
By t h e i n d u c t i o n h y p o t h e s i s , t h e r e e x i s t s a w e l l -
of
B
w i t h h e i g h t ('3.
B" +
Then
C"
i s a restriction
fi + 8 . Ht A =
c(
, an
i n d e c m p o s a b l e o r d i n a l ; and suppose t h a t t h e
f i r s t c o n c l u s i o n does n o t h o l d . By 4.5 ( u l t r a f i l t e r axiom), i t s u f f i c e s t o prove that there exists a r e s t r i c t i o n o f
A
w i t h t h e same h e i g h t d,
f r e e and hence a w e l l p a r t i a l o r d e r i n g . L e t pose d i n t o a sum s t r i c t l y l e s s t h a n o( For each
i
c. k , c o n s i d e r
Sup
o(
=
o(
the restriction
i s b o t h g r e a t e r t h a n o r equal t o
2 OC
, which
i s finitely
be t h e c o f i n a l i t y o f o( ; decorn-
( i c k ) with the
o< , and
k
(ch.2
Ai
5 of
increasing, a l l
o(
5.5). A
t o elements whose h e i g h t
( j < i) and s t r i c t l y l e s s t h a n
o(
j
130
THEORY OF RELATIONS
i) . This Ai has height H . By the induction hypothesis, and since there do not exist infinitely many mutually incomparable (mod Ai) elements with different heights, there exists a well-ordered restriction Ci of Ai with the same height M i . Let D denote the restriction of A to the union of the bases hence greater than or lCil . Firstly Ht D is strictly greater than each o( equal to o( = Sup M i ; so that Ht D = o( . Secondly, there do not exist infinitely many mutually incomparable elements in D , since they would all have different heights (mod A), which contradicts our hypothesis. Hence D is finitely free, s o that we can apply 4.5. ( j6
3 - As opposed to what happens in 4.5, we cannot require that the well-ordering having the same height as A , have one and only one element of each height (mod A). Consider the following counterexample due to POUZET. Take the points (XJ) of coordinates natural numbers with y~ x . Let (x,y) < (x',y') iff either x = x' and y < y' , or x d x ' and y' 3 y+2 . Then the height of each point (x,y) is y , hence Ht A = w . However, a totally ordered restriction of A isomorphic with w can have at most a finite sequence of points with heights 0,1,2, ... ; for example the points (u,O), ( u , l ) , ... , (u,u) (where u is an integer). After which one must pass to a point with ordinate greater than or equal to u t 2 , hence with height greater than or equal to u+2 .
EXERCISE 5
- REINFORCEMENT OF A WELL PARTIAL ORDERING (REVISITED)
1 - Given a well partial ordering A , there exists a well-ordered reinforcement of A . This is an immediate consequence of the reinforcement axiom (ch.2 5 4.2) by using proposition 3.5.(1) above. We propose to obtain this result in a more economic manner, using only the axiom of choice for finite sets (ch.1 5 2.10). For this, to each ordinal u strictly less than Ht A , associate the finite set Fu of those elements with height u in A ; then the finite set of chains with base Fu . Then the axiom of choice for finite sets associates to each u one of these chains. Finally it suffices to take the sum along the u . 2 - Notice that with the considered axiom, a partial ordering A is a well partial ordering iff: (i) every totally ordered reinforcement of A is a well-ordering, and (ii) there exists a totally ordered reinforcement of A (use 3.5.(3): ZF suffices). More precisely, the preceding proposition is equivalent to the following weakening of the axiom of choice for finite sets: "for every well-orderable set of finite mutually disjoint sets, there exists a choice set"(communicated by POUZET in 1979).
131
5
CHAPTER
EMBEDDABILITY BETWEEN PARTIAL OR TOTAL ORDERING
§
1 - EMBEDDABILITY,
IMMEDIATE
FAITHFUL
EXTENSION,
EXTENSION
EMBEDDABILITY, EQUIMORPHISM Let
be two r e l a t i o n s o f t h e same a r i t y . We say t h a t
R, S
R
i s embeddable
S o r i s s m a l l e r t h a n S under e m b e d d a b i l i t y , o r t h a t S admits an embedding o f R o r i s g r e a t e r t h a n R , i f f t h e r e e x i s t s a r e s t r i c t i o n o f S in
R ; we w r i t e R + S o r S i s s t r i c t l y embeddable i n S
isomorphic w i t h We say t h a t
.
o r s t r i c t l y smaller than
R
, or
S admits a s t r i c t embedding o f R o r i s s t r i c t l y g r e a t e r t h a n R , denoted
that by
R
R
R < S
or
We say t h a t
S > R , i f f
R$S
but
i s equimorphic w i t h
R
The comparison r e l a t i o n
4
S $ R .
S , denoted R
5 S
, iff
R 6'S
and
S
6
R
i s r e f l e x i v e and t r a n s i t i v e , hence d e f i n e s a quasi-
o r d e r i n g o n each s e t o f r e l a t i o n s . Moreover equimorphism i s symmetric and hence d e f i n e s an e q u i v a l e n c e r e l a t i o n . E m b e d d a b i l i t y i s n o t a n t i s y m m e t r i c , even up t o isomorphism: see t h e f o l l o w i n g examples. Let
be t h e c h a i n o f t h e r a t i o n a l s , and
Q
a l a s t element: t h e n
Q+l t h e e x t e n s i o n o b t a i n e d by adding
.
Q I Q+l
L e t LJ be t h e c h a i n o f t h e n a t u r a l numbers, and ch.2
5 5
1.7). Then
u-.G)E
a-
t h e converse c h a i n ( s e e
1 + ( W - . w ) (the ordinal product i s defined i n
3.7). I n t h e c h a i n o f n a t u r a l numbers, r e p l a c e each even number by Z ( t h e c h a i n of p o s i t i v e and n e g a t i v e i n t e g e r s ) and each odd number by a f i n i t e chain. We o b t a i n continuum many m u t u a l l y non-isomorphic chains, a l l o f which a r e equimorphic.
ch.2
1.1. L e t
R, S
be two equimorphic r e l a t i o n s . Then t h e r e e x i s t s a p a r t i t i o n o f
I R I i n t o two d i s j o i n t subsets D, D ' , and a p a r t i t i o n o f 1st i n t o two d i s j o i n t subsets E, E ' w i t h R/D isomorphic t o S/E and R / D ' isomort h e base phic t o where
S/E'
f
and
.
Repeat t h e p r o o f o f BERNSTEIN-SCHRODER's theorem (ch.1 g
5 1.4),
become isomorphisms from one r e l a t i o n o n t o a r e s t r i c t i o n o f
the other. The converse i s f a l s e , even f o r c h a i n s . Indeed, t h e c h a i n w o f t h e n a t u r a l numbers and t h e c h a i n
w+1
g i v e r i s e t o p a r t i t i o n s s a t i s f y i n g t h e above condi-
t i o n s . S i m i l a r l y f o r t h e incomparable c h a i n s U+c.4-
and
Z = W-+
.
.
132 1.2.
THEORY OF RELATIONS IMMEDIATE EXTENSION
Given a r e l a t i o n R , we say t h a t S i s an immediate extension o f R i f S i s an extension, and furthermore i f there does n o t e x i s t any s t r i c t l y intermediate r e l a t i o n T such t h a t R < T < S w i t h respect t o embeddability. We say a l s o t h a t S , o r any r e l a t i o n equimorphic w i t h S , i s an immediate successor o f R w i t h respect t o embeddability.
.
Moreover .if R For each r e l a t i o n R , t h e r e e x i s t s an immediate extension o f R has a r i t y >/ 1 , then t h e r e e x i s t a t l e a s t two immediate extensions (HAGENDORF 1977, p r o p o s i t i o n VI.5.6). 0 Suppose f i r s t t h a t
i s a 0-ary r e l a t i o n , say
R
R = (E,+)
: then i t s u f f i c e s t o
replace t h e base E by a s e t w i t h immediately g r e a t e r c a r d i n a l i t y : see ch.2 5 3.10. Suppose t h a t R i s a unary r e l a t i o n . L e t a be t h e c a r d i n a l i t y o f the s e t o f e l e -
R
ments g i v i n g t h e value (+) t o i t s u f f i c e s t o replace e i t h e r using ch.2 5 3.10. Suppose now t h a t R
a
has a r i t y
, and or
b t h e analogous c a r d i n a l i t y f o r (-). Then b by an immediately g r e a t e r c a r d i n a l , again
n >/ 2
. Add
t o t h e base E o f
R
a set
D dis-
t o have base E u D w i t h R+/E = R and R+/D j o i n t w i t h E , and d e f i n e R' always (t), and f i n a l l y w i t h R+ t a k i n g the value (+) f o r those n-tuples contain i n g a t l e a s t one term i n D F i n a l l y choose f o r d = Card D t h e l e a s t aleph f o r which Rt i s n o t embeddable i n R , hence R+> R L e t us prove t h a t R' i s an
.
.
immediate extension o f R ; t h e r e l a t i o n R- s i m i l a r l y defined by exchanging (t) and ( - ) , being another immediate extension, obviously incomparable w i t h R' with respect t o embeddabi 1ity. Suppose f i r s t t h a t d i s an i n f i n i t e aleph, and t h a t t h e r e e x i s t s a s t r i c t l y i n t e r mediate r e l a t i o n T w i t h R < T < R' . Consider T as a r e s t r i c t i o n o f Rt Then
.
.
D n IT1 has c a r d i n a l i t y d Indeed i f i t had c a r d i n a l i t y c d , R (more p r e c i s e l y , i n a r e s t r i c t i o n o f Rt t o E increased w i t h (c:d) many elements o f D ). Now p a r t i t i o n DnITI i n t o two the i n t e r s e c t i o n
then T would be embeddable i n
d i s j o i n t subsets, each w i t h c a r d i n a l i t y d
, say
-
D'
and D"
. Then
T
i s isomor-
p h i c w i t h i t s r e s t r i c t i o n t o I T 1 D" , so t h a t R i s embeddable i n t h i s r e s t r i c t i o n , and f i n a l l y R+ i s embeddable i n T : c o n t r a d i c t i o n . Now i t remains t o consider t h e case where d = Card D i s f i n i t e . We f i r s t see t h a t d = 1 . Indeed assume t h a t d f i n i t e and >, 2 Then by hypothesis, the extension of R obtained by adding o n l y one element u t o t h e base I R 1 , w i t h value (t)
.
f o r a l l n-tuples c o n t a i n i n g u , i s embeddable i n R , thus equimorphic w i t h R I t e r a t i n g t h i s , t h e s i m i l a r extension obtained by adding 2 elements, i s s t i l l equimorphic w i t h
R , and so on u n t i l we add d elements: c o n t r a d i c t i o n .
Now examine t h e case where singleton constitutes
D
d = 1
. Call
. Consider
u the supplementary element, whose
again the intermediate r e l a t i o n T as a
.
Chapter 5
133
.
r e s t r i c t i o n o f R+ ; obviously u belongs t o t h e base I T We say t h a t an e l e ment x i n the base i s a (+)-element (mod R ) i f f every n - t u p l e which contains x gives value (+) t o R . Analogous d e f i n i t i o n f o r a (+)-element (mod T) ; i n p a r t i c u l a r , u i s a (+)-element (mod T) . Every (+)-element (mod R) belongs t o the base I T 1 and i s a (+)-element (mod T). Indeed otherwise, i f x i s a (+)-element (mod R ) and does n o t belong t o I T I , t h e n by r e p l a c i n g u by x we could embed T i n R : c o n t r a d i c t i o n . E i t h e r t h e r e e x i s t D e d e k i n d - i n f i n i t e l y many (+)-elements (mod R ) . Then R+ isomorphic w i t h R : c o n t r a d i c t i o n . O r t h e s e t o f (+)-elements (mod R ) has D e d e k i n d - f i n i t e c a r d i n a l i t y , say
h
. Then t h e r e e x i s t a t l e a s t
is
h + l many
(+)-elements (mod T) . Consider a r e s t r i c t i o n R ' o f T which i s isomorphic with R Since we have e x a c t l y h many (+)-elements (mod R ' ) and a t l e a s t h t l many (+)-elements (mod T) , t h e r e e x i s t s a t l e a s t one (+)-element (mod
.
T) ,
.
say v , which does n o t belong t o t h e base I R ' I Then the r e s t r i c t i o n o f T t o t h e base I R'I p l u s t h e element v i s isomorphic w i t h R' : c o n t r a d i c t i o n . 0
1.3. FAITHFUL EXTENSION
.
(1) L e t R , S be two n-ary r e l a t i o n s ( n >r 1) Assume t h a t S does not Then t h e r e e x i s t s a s t r i c t l y g r e a t e r extension T admit an embedding o f R
.
.
of S which does n o t _admit an embedding o f R We c a l l i t a f a i t h f u l extensio? o f S modulo R . Moreover we can choose T t o be an immediate extension o f S : see HAGENDORF 1977. The statement i s obviously f a l s e f o r a r i t y zero.
R aqd S are unary. L e t at be 'the c a r d i n a l i t y o f t h e s e t o f elements g i v i n g the value (+) t o R , and a- the analogous c a r d i n a l i t y f o r (-); s i m i l a r l y l e t b+ and b- be t h e analogous c a r d i n a l i t i e s f o r S . Since R $ S ,
0 Suppose f i r s t t h a t
.
e i t h e r b + < a+ o r b-< aSuppose t h e f i r s t case holds, t h e argument being analogous f o r t h e second case. It s u f f i c e s t o take an extension of S i n which bt i s preserved and b- i s replaced by an immediately l a r g e r c a r d i n a l . Suppose t h a t R and S have a r i t y n 3 2 Add t o t h e base E o f S a s e t '0
.
which i s d i s j o i n t from
E
, and
d e f i n e the extension T+
of
S w i t h base
EvD',
THEORY OF RELATIONS
134
t a k i n g t h e v a l u e (+) f o r those choose
w i t h c a r d i n a l ( a l e p h ) s u f f i c i e n t l y l a r g e t o have
D+
+
w i t h t h e value ( - ) , t h u s o b t a i n i n g
and
0-
T->
.
S
.
Do t h e same
R
4:'
T+> S
We c l a i m t h a t
. Also
D
n - t u p l e s c o n t a i n i n g a t l e a s t one t e r m o f
or
, which y i e l d s o u r c o n c l u s i o n . Indeed suopose t h e c o n t r a r y , and c o n s i d e r R as a r e s t r i c t i o n o f T+ , The base o f R i s n o t a subset o f E , s i n c e R $ R $T-
hence t h e r e e x i s t s an element
u+
R
such t h a t
u+
n - t u p l e c o n t a i n i n g a t l e a s t one t e r m equal t o
S,
takes t h e v a l u e (+) f o r each
. There
e x i s t s an analogous e l e -
ment f o r t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0 be a r e l a t i o n w i t h a r i t y 5 2
R
(2) Let
e x i s t s a common e x t e n s i o n o f The c o n t r a p o s i t i v e i s : i f X
3 S2 , t h e n
or
6 S1
R
and
S1
and
S1&
R
R,< X
f o r every
R 6 S2
.
X
and
El
.
E2
p a r t i t i o n o f t h e base o f e v e r y element =
R
have d i s j o i n t
S2
taking
E2
and t h e o t h e r i n
El
R.
and . ,Sh.
.
E2
or
S+
contrary; then there e x i s t s a
i n t o two non-empty d i s j o i n t subsets such t h a t , f o r
i n one subset and
u
. Suopose t h e
R
.
S1,.
f o r t h e value ( - ) . It s u f f i c e s t o see t h a t e i t h e r
S-
does n o t a d m i t an embedding o f
S-
and
S1
St be t h e common e x t e n s i o n w i t h base El
Let
there
X >,S1
Extend t h i s t o any f i n i t e sequence
t h e value (+) f o r those o r d e r e d p a i r s w i t h one t e r m i n Analogously d e f i n e
. Then
S2 & R
which s a t i s f i e s b o t h
0 Take t h e case o f a b i n a r y r e l a t i o n , and suppose t h a t
bases
and
which does n o t a d m i t an embedding o f
S2
v
i n t h e o t h e r , we have
R(u,v) = R(v,u)
=
+ ; same c o n c l u s i o n w i t h t h e v a l u e ( - ) . Note t h a t , g i v e n two p a r t i t i o n s o f t h e u, v
base, each w i t h two non-empty d i s j o i n t s e t s , t h e r e e x i s t two elements
in
t h e base which a r e separated b o t h by t h e f i r s t p a r t i t i o n and by t h e second. Thus R(u,v)
=
+ and - :
2 and non-empty base. Suppose t h a t
S
& R1
u, v
t h e r e e x i s t two elements
g i v i n g simultaneously
contradiction. 0 The p r o p o s i t i o n i s o b v i o u s l y f a l s e f o r unary r e l a t i o n s . (3) Let
S
. Then
S%R3
>/
have a r i t y
there e x i s t s a proper extension
embeddabilities
R1 ,
S+*
S+$
R2
and
S+ f S
S+$ R3
.
element Let
S2
a
and s e t t i n g
S,(a,x)
of
= S (x,a)
. Let
S =
+
S1
, say
r o l e of
. Hence
i n t h e base o f t h i s
t o f i x t h e ideas, p l a y i n g t h e r o l e o f a
and
R2
x
in
S3
. Suppose f i r s t l y
a
t h a t t h e base
in
.
Si
(i
IS1
i n t h e base S3
=
.
be o b t a i n e d w i t h
S (x,a) = - f o r e v e r y x i n I S ( , and moreover 3 w i t h t h e same c o n d i t i o n s , except t h a t S4(a,a) = - .
R
and
be o b t a i n e d by a d d i n g a new
f o r every
Suppose o u r c o n c l u s i o n i s f a l s e . Then t h e r e e x i s t two al
R1
and
an embedding o f a same
$ R2
which r e s p e c t s t h e non-
1 be s i m i l a r l y o b t a i n e d w i t h ( - ) ' i n s t e a d o f ( + ) . L e t
S (a,x) = + 3 F i n a l l y S4
S
F o r t h e a r i t y 1 o r f o r empty
base, t h e p r o p o s i t i o n i s o b v i o u s l y f a l s e , even w i t h o n l y 0 Consider t h e 4 f o l l o w i n g e x t e n s i o n s
,
S3(a,a)
1,2,3,4)
= +
.
which a d m i t
R , t h e r e e x i s t s an element
S1 and an a3 p l a y i n g t h e
I R I has c a r d i n a l i t y
32 .
Chapter 5 Then
al
and
and
R(x,a3)
a3 =
-
are d i s t i n c t , since x # a3
f o r every
.
135
R(x,al)
+
=
R(al,a3)
takes s i m u l t a n e o u s l y t h e
f o r every
value (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . Analogous argument f o r and
S1
,
S4
S2
and
S3
,
and
S2
Suppose now t h a t t h e base I R I takes t h e v a l u e ( - ) . Since S >
,
S4
x # al
= R(al,x)
Moreover
and
S3
and
S1
,
S2
.
S4
has c a r d i n a l i t y 1, and t o f i x t h e ideas, t h a t R R and by h y p o t h e s i s S non-empty, n e c e s s a r i l y S
i s r e f l e x i v e . As p r e v i o u s l y d e f i n e e x t e n s i o n s S1, S2, S3 which now a r e a l l t h r e e r e f l e x i v e . E i t h e r R1 = R 2 = R3 = R and t h e n o u r c o n c l u s i o n h o l d s . O r R1 and possibly
R , t h u s have c a r d i n a l i t i e s
are d i s t i n c t from
R2
our conclusion i s false: then
, and
( i = 1,2,3)
R1
1.4. Consider an 13 -sequence o f r e l a t i o n s
Ria A
i
f o r every
, then
R+
suppose Si
A
Ri
( i integer)
3
o f common a r i t y
A
2
be a r e l a t i o n o f t h e same a r i t y .
t h e r e e x i s t s a comnon e x t e n s i o n o f a l l t h e
which does n o t a d m i t an embedding o f 0 Let
. Again
t h e argument t e r m i n a t e s as p r e v i o u s l y . 0
and w i t h m u t u a l l y d i s j o i n t bases. L e t
.-have
2
f o r i n s t a n c e i s embeddable i n a t l e a s t two
Ri
.
denote t h e common e x t e n s i o n o f t h e
Ri
on t h e u n i o n o f t h e bases, which
takes t h e v a l u e (+) f o r a l l t h o s e n - t u p l e s ( n = a r i t y ) c o n t a i n i n g a t l e a s t two terms t a k e n f r o m two d i s t i n c t bases. Analogously d e f i n e t h e e x t e n s i o n that
A
i s embeddable b o t h i n
, there
Ri
R+
and
R-
n e c e s s a r i i y e x i s t two elements
. Since x, y
t r a n s f o r m e d i n t o two elements i n two d i s t i n c t
A
, simultaneously
and second embedding. tience f o r an n - t u p l e c o n t a i n i n g b o t h for
I A I , which a r e
i n t h e base
lRil
. Suppose
R-
i s n o t embeddable i n any
x
and
f o r the f i r s t
, we
y
have
t h e v a l u e (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0
A
5 2 - EMBEDDABILITY
BETWEEN PARTIAL
(KRUSKAL); CANTOR'S GLEASON); TOURNAMENT
OF FINITE
TREES
(DILWORTH,
ORDERINGS;
WELL PARTIAL ORDERING ORDERINGS
THEOREM FOR PARTIAL
2.1. There e x i s t i n f i n i t e l y many f i n i t e p a r t i a l o r d e r i n g s which a r e m u t u a l l y incom-parable w i t h r e s p e c t t o e m b e d d a b i l i t y . 0 Let
a'< i
be t h e p a r t i a l o r d e r i n g on 5 elements
A1 b'<
,let
v Ai
until
ui-l
with
be t h e p a r t i a l o r d e r i n g based on 2i+3 elements
U ~ ~ ~ , V ~ , ~ , V ~ , ~ , . w. i .t h, Va ~<~b ~ , ~
bilities
a,b,a',b',v
a(
b
,
and i n c o m p a r a b i l i t y elsewhere. Now more g e n e r a l l y f o r each i n t e g e r
u1 < v ~ , and ~
and
and
a'<
u1 < v ~ , and ~ t h e n u2 < v1,2 ui-1
< vi-l,i
a,b,a',b',u
b;< and
vi-l,i
ui
l,..., and
comparaand so
; and i n c o m p a r a b i l i t y elsewhere.
The p a r t i a l o r d e r i n g s t h u s d e f i n e d a r e m u t u a l l y incomparable w i t h r e s p e c t t o emb e d d a b i l i t y (example comnunicated i n 1969 by JULLIEN). 0
I36
THEORY OF RELATIONS
2.2.
( 1 ) There e x i s t s a s t r i c t l y decreasing (under e m b e d d a b i l i t y ) w - s e q u e n c e
o f denumerable p a r t i a l o r d e r i n g s . 0
To each s e t
I o f integers, associate the p a r t i a l ordering
extension o f the f i n i t e p a r t i a l orderings
Ai
(i
e
AI
o b t a i n e d as an
I) , taken t o be m u t u a l l y
incomparable. Then t a k e an i n f i n i t e s t r i c t l y d e c r e a s i n g sequence o f s e t s
I
.0
( 2 ) There e x i s t continuum many denumerable p a r t i a l o r d e r i n g s which a r e m u t u a l l y incomparable under e m b e d d a b i l i t y . 0
Take t h e p r e c e d i n g
AI
I, J
and n o t e t h a t , f o r two s e t s
o f which i n c l u d e s t h e o t h e r , t h e n
AI
and
AJ
o f integers neither
a r e incomparable. 0
Problem posed by HAGENDORF. E x i s t e n c e o f a s t r i c t l y d e c r e a s i n g G1-sequence o f denumerable p a r t i a l o r d e r i n g s . Same problem f o r r e l a t i o n s , posed by POUZET. 2.3. THEOREM OF THE WELL PARTIAL ORDERING
OF FINITE TREES
Embeddability between f i n i t e t r e e s i s a w e l l p a r t i a l o r d e r i n g (KRUSKAL 1960). 0
We can always assume t h a t each o f t h e c o n s i d e r e d f i n i t e t r e e s has a minimum
element: i t s u f f i c e s t o add a minimum, and even a new minimum i f t h e r e a l r e a d y e x i s t s a minimum, t o each f i n i t e t r e e ; t h e n e m b e d d a b i l i t y o r non-embeddability i s preserved. Suppose t h a t e m b e d d a b i l i t y between f i n i t e t r e e s w i t h a minimum, i s n o t a w e l l p a r -
w -sequence o f such t r e e s , which i s bad w i t h
t i a l o r d e r i n g . Then t h e r e e x i s t s an r e s p e c t t o e m b e d d a b i l i t y : see ch.4
5
s t r o n g l y minimal bad w-sequence:
ch.4
3 . 2 . ( 2 ) c o u n t a b l e case. Hence t h e r e e x i s t s a
5
2.iO.
I n t h i s sequence, t h e terms a r e
m u t u a l l y incomparable w i t h r e s p e c t t o e m b e d d a b i l i t y : ch.4 5 2.3 and 2.8. L e t U denote t h i s sequence, and Ui ( i i n t e g e r ) each term. The s e t o f f i n i t e t r e e s w i t h a minimum, which a r e s t r i c t l y embeddable i n a
5
see ch.4 Let
H
Ui
,
i s a well p a r t i a l ordering:
4.3 and 2.8.
denote t h i s c o u n t a b l e s e t (up t o isomorphism) o f f i n i t e t r e e s w i t h a m i n i -
mum. Embeddability between words, o r f i n i t e sequences o f elements o f w e l l p a r t i a l o r d e r i n g : ch.4
5
4.4 (HIGMAN). To each t r e e
H
,
is a
a s s o c i a t e one o f t h e
Ui f i n i t e sequences o b t a i n e d by t o t a l l y o r d e r i n g i n an a r b i t r a r y manner t h e immediate successors o f t h e minimum, t h e n by r e p l a c i n g each o f them by t h e s u b - t r e e o f t h o s e g r e a t e r o r equal elements. Then t h e p r e c e d i n g sequence
words i n Ui Ui
H
, hence
U
becomes a sequence o f
i s good. Moreover, i f t h e word t h u s s u b s t i t u t e d f o r t h e t r e e
i s embeddable i n t h e word s u b s t i t u t e d f o r UJ. (i,j i n t e g e r s ) , t h e n t h e t r e e i t s e l f i s embeddable i n U . . Hence t h e sequence U i s good: c o n t r a d i c t i o n . 0 3
2.4. A b i n a r y r e l a t i o n
A
i s a p a r t i a l ordering i f f
A
does n o t a d m i t an embed-
ding o f the following f i n i t e relations: t h e b i n a r y r e l a t i o n w i t h c a r d i n a l i t y 1 and v a l u e ( - ) ( t h i s e n s u r e s r e f l e x i v i t y ) ; t h e r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 ( a n t i s y m n e t r y ) ; t h e r e f l e x i v e b i n a r y cycle with cardinality 3
, and
f i n a l l y t h e c o n s e c u t i v i t y r e l a t i o n on 3 elements
Chapter 5
137
associated w i t h t h e c h a i n o f c a r d i n a l i t y 3 : see ch.2
5
8.6 ( t h e s e two non-embed-
d a b i l i t i e s ensure t r a n s i t i v i t y ) .
2.5.
or
Every denumerable p a r t i a l o r d e r i n g admits an embedding o f t h e o r d i n a l w
i t s converse
(J-
o r t h e denumerable f r e e p a r t i a l o r d e r i n g .
I n p a r t i c u l a r , e v e r y denumerable f i n i t e l y f r e e p a r t i a l o r d e r i n g , hence e v e r y denu-
.
merable chain, admits an embedding o f w o r L,I-
5
WiXh ch.4
0 Enumerate t h e elements
integers ai < a ch.3
5
j
T h i s i s KONIG's lemma: compare
4.5 i n t h e case o f a w e l l - f o u n d e d p a r t i a l o r d e r i n g .
i, j
ai
o f t h e base, and p a r t i t i o n t h e p a i r s o f
( i integer)
where we assume
i< j
,
i n t o 3 c o l o r s , a c c o r d i n g t o whether
o r l a . (modulo t h e g i v e n p a r t i a l o r d e r i n g ) . By RAMSEY's theorem or 7 a j J 1.1, t h e r e e x i s t s a denumerable s e t o f i n t e g e r i n d i c e s i n which a l l p a i r s
have t h e same c o l o r . According t o whether i t i s t h e f i r s t , second o r t h i r d c o l o r , the given p a r t i a l o r d e r i n g admits an embedding o f
cc)
or
w - or
a denumerable
relation o f identity. 0 Modulo t h e denumerable subset axiom (ch.1
w
admits an embedding o f
A
or w -
5
2.6),
every i n f i n i t e p a r t i a l ordering
o r t h e denumerable f r e e p a r t i a l o r d e r i n g .
well-founded p a r t i a l o r d e r i n g does n o t a d m i t an embedding o f
w- .
Conversely,
a denumerable p a r t i a l o r d e r i n g ( o r more g e n e r a l l y a p a r t i a l o r d e r i n g w i t h w e l l orderable base) which does n o t a d m i t an embedding o f
w - , i s well-founded. With
the axiom o f dependent choice, e v e r y p a r t i a l o r d e r i n g which does n o t admit an embedding o f
w - i s well-founded: see ch.2 5 2.4.
2.6. CANTOR'S THEOREM FOR PARTIAL ORDERINGS
, the
Given a p a r t i a l o r d e r i n g A
, admits
A
vals o f
I n particular, i f embedding o f 0 Let
A g B
A
p a r t i a l o r d e r i n g o f i n c l u s i o n among i n i t i a l i n t e r -
a s t r i c t embedding o f A
A
(DILWORTH, GLEASON 1964).
i s a chain, t h e n t h e c h a i n o f c u t s o f
A
B denote t h e p a r t i a l o r d e r i n g o f i n i t i a l i n t e r v a l s o f
, it
admits a s t r i c t
.
s u f f i c e s t o a s s o c i a t e t o each element
x
of
A
. To
see t h a t
I A I the i n i t i a l interval
.
6 x (mod A ) NOW suppose t h a t BG A and l e t f be an isomorphism o f B i n t o A Some i n i t i a l i n t e r v a l s X o f A s a t i s f y t h e r e l a t i o n f ( X ) E X : f o r example t h e e n t i r e
o f elements
x
f(U)
< f(U)
Phhm f ( V ) < Thus
. L e t U be t h e i n t e r s e c t i o n o f a l l t h e s e i n t e r v a l s . We s h a l l 4 U . Indeed i f f ( U ) E U , t h e n t h e i n t e r v a l V o f those e l e -
X = \A[
interval prove t h a t ments
.
f(U)
+U
there e x i s t s
(mod A)
satisfies
f ( U ) (mod A ) and s i n c e
one o f t h e s e
, hence U
VcU
( s t r i c t i n c l u s i o n ) . Hence by t h e isomor-
f(V) E V
and so
Vz
i s t h e i n t e r s e c t i o n o f those X
, say
Xo
, which
U : contradiction. X
such t h a t
f ( X ) i5 X,
does n o t c o n t a i n t h e element
f(U).
THEORY OF RELATIONS
138
Thus we have Moreover 2.7.
Xoz U
f(U)
+
Xo
and
# U , hence by isomorphism f ( U ) < f(Xo) (mod A)
and so
f(Xo)
$
Xo : c o n t r a d i c t i o n .
TOURNAMENT
A binary r e l a t i o n A
i s c a l l e d a tournament i f f i t i s r e f l e x i v e , antisymmetric by a n t i and comparable: f o r a l l x, y e i t h e r A(x,y) = + (and thus A(y,x) =
-
-
symnetry) o r A(y,x) = t (and A(x,y) = ) . I n the f i r s t case we say t h a t f o l l o w s x (mod A) ; i n t h e second case y precedes x (mod A)
y
. E - u
For each element u o f t h e base E o f A , t h e elements i n are divided i n t o two complementary sets, formed r e s p e c t i v e l y o f those elements which f o l l o w
u and those which precede u
.
Every r e s t r i c t i o n o f a tournament i s a tournament.
A tournament i s a chain i f f i t i s t r a n s i t i v e . The b i n a r y c y c l e on 3 elements (see ch.2 5 8.6), obviously m o d i f i e d t o be r e f l e x i ve, i s a tournament. Yet a b i n a r y r e f l e x i v e c y c l e w i t h c a r d i n a l i t y 3 4 i s not antisymmetric, thus i s n o t a tournament.
A tournament i s a chain i f f i t does n o t admit an embedding o f the b i n a r y r e f l e x i v e c y c l e w i t h c a r d i n a l i t y 3.
A b i n a r y r e l a t i o n i s a tournament i f f i t admits no embedding o f t h e b i n a r y r e l a t i o n w i t h c a r d i n a l i t y 1 and value ( - ) (ensures r e f l e x i v i t y ) ; no embedding o f t h e r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 (ensures antisymmetry); f i n a l l y no embedding o f the i d e n t i t y r e l a t i o n w i t h c a r d i n a l i t y 2 (ensures c o m p a r a b i l i t y ) .
§
3
- DENSE CHAIN:
A-DENSE CHAIN, FOR AN I N F I N I T E CARDINAL A
3.1. DENSE CHAIN A chain i s s a i d t o be dense i f f i t s base i s i n f i n i t e , and i f between any two d i s t i n c t elements x < y t h e r e e x i s t s an element z : x c z < y Hence between any two elements x and y 7 x t h e r e e x i s t i n f i n i t e l y many e l e ments.
.
To each z ( x < z c y ) associate the i n t e r v a l (x,z) : t h e r e i s no minimal interval
(x,z)
w i t h respect t o i n c l u s i o n (use ch.1 tj 1.1). 0
Every denumerable dense chain w i t h o u t any minimum o r maximum, i s isomorphic to_ t h e chain
Q o f the ratj.1..
Every countable chain i s embeddable - -- i n__ Q
.
Every dense chain admits an embedding o f Q (uses dependent choice; i f t h e base i s denumerable, o r o n l y w e l l - o r d e r a b l e ) .
ZF s u f f i c e s
Chapter 5
U
139
Consider t h e s e t o f f i n i t e s t r i c t l y i n c r e a s i n g sequences o f elements o f t h e chain,
say u o c u1 < ... < uh ( h i n t e g e r ) sequences w i t h a r b i t r a r y l e n g t h h t h e form
<
vo
the r e l a t i o n 3.2. L e t
,u, R
A
, and t h e r e l a t i o n R which t o each , a s s o c i a t e s any sequence w i t h l e n g t h
2h
+ 1of
c v1 c u 1 < . . . < vh < uh< v ~ .+ Then ~ a p p l y dependent c h o i c e t o . The o b t a i n e d w - s e q u e n c e y i e l d s a r e s t r i c t i o n isomorphic w i t h Q.
be a denumerable c h a i n . I f
ordinal, then
o f these
A
A
0
admits an embedding o f e v e r y c o u n t a b l e
Q
admits an embedding o f t h e c h a i n
(uses c o u n t a b l e axiom o f
choice). 0
To each c o u n t a b l e o r d i n a l
embedded i n
A
@
, a s s o c i a t e 4 +1+a
. Hence a s s o c i a t e
a(& )
an element
i n t h e l o w e r i n t e r v a l and i n t h e i n t e r v a l above ble, y e t there are element
a
wl-many
which by h y p o t h e s i s can be such t h a t 4
i s embeddable
) . The c h a i n
a(*
A
i s denumera-
c o u n t a b l e o r d i n a l s 6 .So t h e r e e x i s t s a t l e a s t one
such t h a t e v e r y c o u n t a b l e o r d i n a l i s embeddable b o t h below and above
a
( c o u n t a b l e axiom o f choice: we use t h e f a c t t h a t e v e r y denumerable u n i o n o f denume-
5
r a b l e o r d i n a l s i s a denumerable o r d i n a l : see c h . 1 a restriction o f
3.3. L e t
A
A
be a
which i s isomorphic w i t h
2 . 5 ) . By i t e r a t i o n , we o b t a i l i
.0
Q
non-empty c h a i n w i t h an i n i t i a l i n t e r v a l and a f i n a l i n t e r v a l ,
b o t h d i s j o i n t and i n each of which
A
i s embeddable. Then
(uses dependent choice; ZF s u f f i c e s i f
A
Q
i s embeddable i n A
i s denumerable o r w i t h w e l l - o r d e r a b l e
base). 0 By i t e r a t i o n , t h e h y p o t h e s i s a l l o w s one t o d i v i d e
v a l s , i n each o f which that
A
A
i n t o three d i s j o i n t i n t e r -
A
i s embeddable. Hence t h e r e e x i s t s an element
i s embeddable b o t h b e f o r e and a f t e r A
choice, we o b t a i n a r e s t r i c t i o n o f
a
.
a
such
By i t e r a t i o n u s i n g dependent
which i s isomorphic w i t h
Q
.
0
3 . 4 . a-DENSE CHAIN Given an i n f i n i t e c a r d i n a l
a
, we
say t h a t a c h a i n i s
n i t e and between any two d i s t i n c t elements equal t o
a
Sense o f 3.1,
. Analogous is
(
d e f i n i t i o n f o r a chain
y
a--
, the
( 3 a)--.
i f i t s base i s i n f i i n t e r v a l has c a r d i n a l i t y A dense chain, i n t h e
a
, there
exists a chain with cardinal
a
which
a-dense (uses axiom o f c h o i c e ) .
c3 S t a r t w i t h t h e o r d i n a l p r o d u c t
a
and
w )-dense (uses denumerable subset axiom).
Given an i n f i n i t e c a r d i n a l
is
x
Q.a
where
Q
i s t h e c h a i n o f t h e r a t i o n a l s , and
i s an o r d i n a l , more p r e c i s e l y an a l e p h ( r e c a l l t h a t , w i t h choice axiom, e v e r y
c a r d i n a l i s an aleph: c h . 1
0
6 . 1 ) . The c h a i n
Q.a
has c a r d i n a l i t y
a
and i s ob-
v i o u s l y dense w i t h o u t any minimum or maximum. A l s o t h e s e t o f f i n i t e sequences o f elements i n
Q.a
has c a r d i n a l i t y
a
. We
o r d e r t h i s s e t l e x i c o g r a p h i c a l l y . Now i t
THEORY OF RELATIONS
140
a-dense o r d e r i n g . Indeed, c o n s i d e r two f i n i t e se-
s u f f i c e s t o see t h a t i t i s an
u
quences
and
Denote by
i with
index ui
ui
and
strictly after
u
and
v
vi
the
ui
< vi
, and
vi
( i integer)
w i t h r e s p e c t t o l e x i c o g r a p h i c a l comparison. i - t h terms. E i t h e r t h e r e e x i s t s a l e a s t
wi
(mod Q . a ) ; then we t a k e
we have
which i s s t r i c t l y between
a-many f i n i t e sequences b e g i n n i n g w i t h
t h e y a l l a r e s i t u a t e d between
u
and
.
v
u
Or
U ~ , . . . , U ~ - ~ , W ~:
i s an i n i t i a l i n t e r v a l o f
3.5.
< vh
wh
an element
(h = length o f
v
,
f o l l o w e d by
u
and we have t h e same c o n c l u s i o n by t a k i n g sequences b e g i n n i n g w i t h
u ). 0
L e t w , be an i n f i n i t e r e g u l a r a l e p h . Every c h a i n w i t h c a r d i n a l i t y
admits an embedding e i t h e r o f t h e o r d i n a l
o r i t s converse
a , ,
w,
,
,
CJ,
W , -dense c h a i n .
Let
0
A
be a c h a i n w i t h c a r d i n a l i t y w 4 , i n which n e i t h e r ad n o r i t s conver-
se i s embeddable. L e t
x, y
I A l ; we p u t
be elements o f
<
i f t h e i n t e r v a l between them has c a r d i n a l i t y
.
LJ&
x
equivalent w i t h
y
The e q u i v a l e n c e c l a s s e s a r e
.
Every e q u i v a l e n c e c l a s s has c a r d i n a l i t y < L d M . Indeed l e t T be an e q u i v a l e n c e c l a s s . Take an o r d i n a l indexed sequence o f elements o f T which i s
intervals o f
A
s t r i c t l y i n c r e a s i n g (mod A) and c o f i n a l i n less than
~3~
,
f i r s t t e r m o f t h e sequence, i s a u n i o n o f nality than
<
ui
. This
T
s i n c e w , i s n o t embeddable i n
.
By r e g u l a r i t y o f w ,
a d .Same
<
sequence has l e n g t h s t r i c t l y
.
The subset o f
w,-many
, this
argument f o r t h e subset o f
A
T
above t h e
i n t e r v a l s , each w i t h c a r d i -
u n i o n has c a r d i n a l i t y s t r i c t l y l e s s T
b e f o r e t h e f i r s t t e r m o f t h e se-
quence, by c o n s t r u c t i n g a d e c r e a s i n g o r d i n a l - i n d e x e d sequence. Since t h e e q u i v a l e n c e c l a s s e s a l l have c a r d i n a l i t y s t r i c t l y l e s s t h a n
ad
i s regular, there e x i s t
wd
and
C d d -many e q u i v a l e n c e c l a s s e s . Take a r e p r e s e n t a t i -
ye element f r o m each c l a s s . Again by r e g u l a r i t y , between two r e p r e s e n t a t i v e s t h e r e necessarily e x i s t
LJ, -many e q u i v a l e n c e c l a s s e s , hence as many r e p r e s e n t a t i v e s .
Finally, the r e s t r i c t i o n o f
A
t o t h e s e t o f r e p r e s e n t a t i v e s i s an W, -dense
chain. 0 T h i s p r o p o s i t i o n i s no l o n g e r t r u e i f a d i s a s i n g u l a r a l e p h . 0
Take t h e sum
cc,- + LJ
+ ... +
i~ .1
+ ...
where
i i s an a r b i t r a r y i n t e g e r
and (Ji designates t h e r e t r o - o r d i n a l converse o f o i . The sum i s t h u s a c h a i n with cardinality W . F i r s t l y , t h e o n l y w e l l - o r d e r e d r e s t r i c t i o n s of t h i s sum a r e t h e f i n i t e c h a i n s and u
. No dense
c h a i n i s embeddable i n t h i s sum.
Secondly, a r e t r o - o r d i n a l i s embeddable i n t h i s sum o n l y i f i t i s embeddable i n one o f the
wi
3.6. L e t aleph
,
A
w,
i s n o t embeddable. 0
be a p a r t i a l o r d e r i n g whose base has c a r d i n a l i t y an i n f i n i t e r e q u l a r
. Suppose t h a t ,
(dw
elements
so
f o r each element u o f t h e base I A l , t h e s e t o f I u has c a r d i n a l i t y s t r i c t l y l e s s t h a n c d K . Then t h e r e e x i s t s
141
Chapter 5
a t o t a l l y ordered r e s t r i c t i o n of
A
which i s isomorphic with the ordinal L J .~
a Well-order the base I A I i n order-type and l e t C denote the chain thus obtained. P a r t i t i o n the p a i r s of elements x, y of the base i n t o two c o l o r s . I f x < y (mod C ) , we say t h a t the p a i r i s ( + ) i f f x < y (mod A ) and ( - ) i f f x > y or xly (mod A ) . For a given element a i n the base, t h e r e a r e s t r i c t l y l e s s than wA-many elements x f o r which the p a i r { a , x ) has c o l o r ( - ) : so t h a t e i t h e r x < a (mod C ) o r x < a o r l a (mod A ) . Now apply ch.3 5 3.2: there e x i s t s a subset equipotent t o the base, i n which a l l p a i r s have color (+) . This y i e l d s a r e s t r i c t i o n isomorphic t o the ordinal d d 0 The proposition i s f a l s e f o r a s i n g u l a r aleph: For each i n t e g e r i , take a s e t Ei with c a r d i n a l i t y oi , the Ei being mutually d i s j o i n t . Define a p a r t i a l ordering on the union, which has c a r d i n a l i t y a W, by taking each Ei a s a f r e e s e t , each element i n Ei being l e s s than each element of E f o r any two i n t e g e r s j > i Then every t o t a l l y ordered j r e s t r i c t i o n has order-type a t most G)
.
.
.
3.7. (1) Let W , be a r e g u l a r l i m i t aleph. Then each ad-dense chain admits e i t h e r an embedding of wd o r i t s converse, o r an embedding of a l l o r d i n a l s s t r i c t l y l e s s than ad and t h e i r converses. ( 2 ) Let Ud be a r e g u l a r aleph. Then L3 d + l -dense chain admits e i t h e r an embeddinq of o r i t s converse, o r an embedding of a l l o r d i n a l s and their converses,(ERDOS, RADO 1953; (1) and ( 2 ) use equipotent w i t h the generalized continuum hypothesis; f o r o( = 0 , ZF p l u s choice s u f f i c e s ; f o r d = 1 , ZF plus choice plus continuum hypothesis s u f f i c e s ) . (1) Let A be an -dense chain. By r e s t r i c t i n g t o one o f i t s i n t e r v a l s , we can assume A t o have c a r d i n a l i t y a&. Let C denote a well-ordering with base 1 A I and order-type a0(. P a r t i t i o n the p a i r s { x,y} i n the base i n t o two c o l o r s : assuming t h a t x < y (mod C ) , the p a i r will have c o l o r (+) i f f x ( y (mod A ) and ( - ) i f f x > y (mod A ) . By ch.3 5 3 . 5 . ( 1 ) (gen. continuum
each
hypothesis), e i t h e r t h e r e e x i s t s a (-)-monochromatic subset equipotent t o the Or f o r each base: thus A admits an embedding of t h e retro-ordinal cardinal b < W, , t h e r e e x i s t s a (+)-monochromatic subset w i t h c a r d i n a l i t y b : thus A admits an embedding of every ordinal s t r i c t l y l e s s than C C ) ~. The
.
proof ends by interchanging the c o l o r s . ( 2 ) Let A be an Wbc+l -dense chain. Since u d i s regular by hypothesis, the l e a s t cardinal b s a t i s f y i n g b ( G), ) md i s b = &IM : see ch.2 5 6.5(1) u s i n g gen. continuum hypothesis. In o r d e r t o prove ( 2 ) , we now denote by A those oc+1 -dense chains which embed n e i t h e r the ordinal W o(+l nor i t s converse. P a r t i t i o n the p a i r s of elements of I A I i n t o two c o l o r s , e x a c t l y a s in (1) above.
>
142
THEORY OF RELATIONS
By ch.3 § 3 . 5 . ( 2 ) , e i t h e r t h e r e e x i s t s a
(-)-monochromatic subset equipotent w i t h
.
O r a (+)-monochromatic subthe base: thus A admits an embedding of @ N + l s e t w i t h c a r d i n a l i t y Cdd : thus A admits an embedding o f C d M Hence i n every case, A admits an embedding o f t h e o r d i n a l ae and i t s converse. ai ( i < wo( ) be a s t r i c t l y i n c r e a s i n g (mod A ) a,-sequence. L e t ao,al
.
,..., ,...
w , + ~-dense chain which 3 w O c . Assume, i n order
L e t C be t h e l e a s t o r d i n a l f o r which there e x i s t s an admits no embedding o f c , By the preceding we have c
.
Then c i s the l i m i t t o o b t a i n a c o n t r a d i c t i o n , t h a t c has c a r d i n a l i t y u d o f a s t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s ci < c , indexed by i running through a t most
(30(
. Each
ci
aH+l -dense
i s embeddable i n every
chain. For
each i , the o r d i n a l , o r t h e aleph equal t o ci , i s embeddable i n the i n t e r v a l Hence the sum o f t h e ci i s embeddable i n A and so c i s embeddable (ai,ai+l) i n A . This c o n t r a d i c t i o n shows t h a t A admits an embedding o f every o r d i n a l equipotent w i t h W M . The same argument proves t h a t A admits as w e l l an embedding of t h e corresponding converse o r d i n a l s . 0
.
5 4 - IMMEDIATE An o r d i n a l
U
EXTENSION OF A CHAIN: FAITHFUL EXTENSION OF A CHAIN
i s indecomposable i f f ever.y o r d i n a l
(indecomposable o r d i n a l i s defined i n ch.1
5
V< U
satisfies
V.2
3.6).
.
.
U be an indecomposable o r d i n a l and \I < U Then V i U = U so V . 2 < U Conversely, suppose t h a t U i s decomposable: t h e r e e x i s t V < U and W < U
0 Let
with
V+W = U
. Let
T = Max(V,W)
. Then
T
and
T.Z&V+W
I n next chapter, indecomposability w i l l be extended t o chains (ch.6 preceding p r o p o s i t i o n must t h e r e be m o d i f i e d (ch.6 4.1.
Then
5
.0
= U
5
3 ) . The
3.4).
L e t A, A ' be two chains and U t h e l e a s t ordi-nal such t h a t U i s indecomposable (HAGENDORF 1971).
A+U+A'4
A+A'
L e t Y < U , so t h a t A+V+A' I A+A'(equimorphism L defined i n 5 1 above). Take an isomorphism from A+V+A' i n t o A+A' L e t V = W+W' w i t h W embeddable i n A and W ' embeddable i n A ' I n o r d e r t o f i x our ideas, suppose t h a t W $ W ' .
0
.
.
Then
W'+A',(
A'
(W' .2)+A',< A '
so
4
and so
(W' .4)+A'
< A'
, hence
(V.2)+A1
A ' and so A+(V.2)+A',< A+A' and f i n a l l y V.2< U . Argue analogously t o o b t a i n t h e same conclusion i f W>, W' Thus U i s indecomposable, by t h e prece-
.
ding proposition. 0 4.2.
IMMEDIATE EXTENSION OF A CHAIN
We say t h a t a chain
B
i s an immediate extension o f
i s immediately q r e a t e r than A
A
(up t o isomorphism)
( w i t h respect t o embeddability) i f
B >A
or
and
.
Chapter 5 t h e r e does n o t e x i s t any c h a i n
X
satisfying
be a c h a i n and
U
the
immediate e x t e n s i o n o f
A
(HAGENDORF 1972).
Let
A
143
<X<
A
.
B
l e a s t o r d i n a l such t h a t
N o t i c e t h a t t h e p r o p o s i t i o n n o l o n g e r h o l d s i f we r e p l a c e F o r example, l e t and 0
A =
A+U = IJ - . 2
The o r d i n a l
U
B
tion of
AtU
we have
converse o f w
,
1+ W -
2+ W
.
Then
U
satisfying
, which
an o r d i n a l . I f
A
B
< A+d .
t h u s decomposes
,
8" < U
the d e f i n i t i o n o f
by a r e t r o - o r d i n a l .
U = W - ; y e t between
B"
Consider an isomorphism o f B
B
i n t o an i n i t i a l i n t e r v a l
such t h a t
B ' G A
and
B'
. Hence
, which
and a complementary f i n a l i n t e r v a l
U
.
decomposable we have
A'
The c h a i n
A"+U = U
A"
5
A
U
B = B'+U
i s an o r d i n a l . I f
is
B
.
Hence
hence
A" = U
and
the existence o f a f i n a l i n t e r v a l o f
A+1
A
such t h a t U
i s in-
B'+U = B
A+U,<
,
.
A = A'+U
i s an i n f i n i t e indecomposable o r d i n a l , i n which case
(equimorphy). The preceding argument, w i t h
decomposes
A"
< U , since
A"
A+U = A ' t A " + U = A ' + U
so
thus c o n t r a d i c t i n g t h e d e f i n i t i o n o f Then e i t h e r
B"
B = B ' + B " , C A+B" 6 A t h u s c o n t r a d i c t i n g B" = U and B = B'+U (by = we mean "isomorphic t o " ) . onto a r e s t r i c t i o n o f
A",(
and a com-
chain
i t follows that
i n t o an i n i t i a l i n t e r v a l and
onto a r e s t r i c -
< U . The
B"
A
A+1
A
- , etc.
Consider an isomorphism f r o m A
6 B'
iLaL
o f t h e p r o p o s i t i o n i s non-zero and by t h e p r e c e d i n g 4.1 i t i s i n -
plementary f i n a l i n t e r v a l
A'
A+U
I n o r d e r t o o b t a i n a c o n t r a d i c t i o n , l e t us suppose t h a t t h e r e e x i s t s
decomposable. a chain
, the
U-
. Then
A+U $ A
so
1
A+l , proves
r e p l a c e d by
which i s isomorphic w i t h
U : contra-
diction. U = 1 and so
Or
A'
.
A'+1
A = A'+1
B = B ' + 1 , hence A ' + 1 (
and
I t e r a t e t h e p r e c e d i n g argument: we see t h a t
c e r t a i n i n i t i a l i n t e r v a l and o f t h e f i n a l i n t e r v a l (.d phic with
A
, which
contradicts the hypothesis t h a t
- .
, so
B'+l
Hence
that
i s t h e sum o f a A+1
i s isomor-
.0
U = 1
4.3. Every i n f i n i t e c h a i n admits a t l e a s t two immediately g r e a t e r c h a i n s ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) , which a r e m u t u a l l y incomparable (HAGENDORF 1972). 0
Let
A
be an i n f i n i t e c h a i n and
embeddable i n
A ; similarly l e t
i s n o t embeddable i n than
A
,
.
A
U U'
t h e l e a s t o r d i n a l such t h a t
A+U
By t h e p r e c e d i n g 4.2,
It remains t o prove t h a t t h e y a r e m u t u a l l y incomparable w i t h r e s p e c t t o
Take an isomorphism f r o m interval
U'+A
t h e s e sums a r e immediately g r e a t e r
e m b e d d a b i l i t y . We s h a l l argue ad absurdum: t o f i x t h e ideas, suppose n o t embed
i s not
be t h e l e a s t r e t r o - o r d i n a l such t h a t
U'+A of
morphic w i t h
A
in
A
U'+A
, nor
in
onto a r e s t r i c t i o n o f A+V
f o r any
which i s c o f i n a l l y embeddable
U : o t h e r w i s e t h e r e would e x i s t
which c o n t r a d i c t s t h e d e f i n i t i o n o f
U'
.
A+U
< U . Thus i n U . This
.
V
V< U
A+U
.
there exists a f i n a l final interval i s iso-
such t h a t
F i n a l l y by 4.1,
U'+A,(
T h i s isomorphism can-
U
U'+A$
A+V$
A
i s indecomposable.
,
144
THEORY OF RELATIONS
If U is an infinite indecomposable ordinal, then 1 < U so A+1 I A . The preceding argument, where A is replaced by A+l , proves the existence of a final interval of A+1 which is isomorphic with U : contradiction. Hence U = 1 and U'+A d A + 1 . Thus every isomorphism of U'+A into A+l has in its range the maximum element of A+1 ; for otherwise U'+A would be embeddable in A . Thus A has a maximum. By iteration, we see that A has a final interval which is isomorphic with the retro-ordinal w - . Thus A+1 is isomorphic with A , contradicting the inequality A+l = A+U > A . 0 An infinite chain can have 3, 4, ... immediately greater chains. has the 3 immediately greater chains For example the product A = b -. B = A+l , C = W + A , D = ( w 2 ) - + A , where ( W 2 ) - is the converse of the
4.4.
ordinal
W
.
These are the only immediate extensions of A
.
The reader easily see that there does not exist any chain strictly intermediate between A and B , A and C , A and 0 . Let us see that every chain strictly greater than A admits an embedding of B or C or 0 . Indeed, every chain which is an extension of A can be obtained as follows. Either by adding new elements at the end, so admitting an embedding of B . Or by adding a chain at the beginning, in which w is embeddable (embedding of C ) , or in which the converse of an ordinal is embeddable (from the ordinal w 2 we get an embedding of D ) . Or by partitioning the new elements into a finite number of components W - of A , which is the same as adding them at the beginning. Or by modifying infinitely many components, by adding an ordinal CC, (embedding of C ) , or by replacing these components by a retro-ordinal ( s o getting an embedding of D from the converse of u 2) . 0
0
Another example: the chain W1 + Q , where Q is the chain of the rationals, has the 5 following immediate extensions: the addition of &J1 or its converse at the end or between k, and Q , the addition of W1- at the beginning. For each integer p 3 2 , the chain sions (among chains).
G,
.(p-1)
has exactly p immediate exten-
There exist chains having infinitely many immediately greater chains. For example, every chain A which has no strict restriction isomorphic with A : see the method of SIERPINSKI 1950, used in 5.3 below. FAITHFUL EXTENSION BETWEEN CHAINS Let A be an infinite chain, B a chain in which A is not embeddable. lhen there exists a chain, immediate extension of B , in which A is not embeddable (uses axiom of choice; HAGENDORF 1972; the case for scattered chains .e. chains
4.5.
145
Chapter 5
without any embedding of 0 If
, was
Q
a l r e a d y proved by JULLIEN 1969).
B i s a f i n i t e chain, t h e n i t s u f f i c e s t o t a k e
i n f i n i t e . We d i s t i n g u i s h two cases:
B
B IA
and
.
B+l
Suppose t h a t
B
is
( w i t h r e s p e c t t o embeddabi-
lity).
I n the case
A ,let
B
diately greater than o r A,(
.
B"
B'
and
be two m u t u a l l y incomparable chains, each imme-
B"
(see 4 . 3 ) . Suppose t h e p r o p o s i t i o n i s f a l s e . Then
B
since B ' l B" . Hence A < B ' f o r i n s t a n c e , and so between B and B ' : c o n t r a d i c t i o n .
It remains t o examine t h e case where
A
dum by supposing t h a t
B
B
into
t h e complementary f i n a l i n t e r v a l . L e t B ; similarly
.
UA
6U
,
V
and t h e converse o f
A
i n t o three intervals
.
B
If
UA<
U
D
then A A
morphic t o t h e r e t r o - o r d i n a l
V
i s included i n A
, then
E
\< B , c o n t r a d i c t i n g
U i n E'
, then
U+D',<
,
.
D'
with
Thus
UA = U
>
U
B
.
By 4.1,
\< E gC
Hence
U = 1
D = E
and
, and
0' =
E'4
C'
so
,
V so
preceding, we have
,
and
. Similar-
and an i n t e r v a l i s o E,<
C
,
E'Q
C'
.
A . I f the interC+C' = B : i n o t h e r
intervals o f
D+U+D',(
A = D+U+D'&
B , contradicting the , isomorphic t o an indecom-
C+C' =
U
V , isomorphic t o a r e t r o - o r d i n a l .
V = 1 : the intervals
U = V = 1 a r e t h e same, t h u s
.
be an a r b i t r a r y element o f t h e base
of those elements
0 4 C
the hypothesis. I f there e x i s t s a f i n a l i n t e r v a l o f
similarly
E'
and
A = D+U+D'
E , E'
A = E+V+E' and
A = D+UA+D'
and t h e n
h y p o t h e s i s . Thus we must suppose t h a t t h e i n t e r v a l
x
C+V+C'
B , c o n t r a d i c t i n g t h e hypothe-
into intervals
posable o r d i n a l , i s i n c l u d e d i n t h e i n t e r v a l
Let
.
B
i s an i n i t i a l i n t e r v a l
C
be t h e l e a s t o r d i n a l such t h a t
C+UA+C',<
such t h a t D+U
,
UA
A<
Envisage a l l t h e r e l a t i v e p o s i t i o n s o f U
We argue ad absur-
a r e indecomposable o r d i n a l s . By h y p o t h e s i s Consider an isomorphism o f k o n t o a r e s t r i c t i o n o f t h i s sum: t h i s
i s incomparable w i t h
val
U
B'
,
V
l y we d e f i n e t h e decomposition o f
words
IA .
t h e l e a s t r e t r o - o r d i n a l such t h a t
U
0 ' 4 C'
sis that
B
, where
C+C'
C+U+C'>
<
A,<
A z B"
i s a s t r i c t l y intermediate
i s i n f i n i t e and
and C '
A C+U+C' decomposes
A
and
i s embeddable i n e v e r y c h a i n s t r i c t l y g r e a t e r t h a n
I n an a r b i t r a r y manner decompose
the o r d i n a l
A E B'
I t i s i m p o s s i b l e t o have b o t h equimorphisms
6X
(mod 6 )
Bx+l+B;
>B
and
IS1
.
Let
Bx
be t h e i n i t i a l i n t e r v a l
>
the final interval
B; hence > / A
. Moreover,
a s s o c i a t e t o each x an element f x o f t h e base d e s i g n a t e t h e i n i t i a l i n t e r v a l of t h o s e elements
x (mod B)
. By
the
by t h e axiom of choice,
I A I , such t h a t i f we l e t Ax (mod A ) , t h e n we have
< fx
A = Ax+l+A;( w i t h A x < Bx and A<,; B; . We s h a l l prove t h a t t h e f u n c t i o n f which t o each e:ement x o f t h e base I B I a s s o c i a t e s f x , an element o f A i s s t r i c t l y i n c r e a s i n g . From t h i s we w i l l deduce t h a t t h e h y p o t h e s i s o f i n c o m p a r a b i l i t y o f A and B . TO do t h i s , we argue ad absurdum by supposing t h a t t h u s t h e r e e x i s t two elements x, y o f I S ( w i t h
B,(
A
, which
,
contradicts
f i s not s t r i c t l y increasing: x < y (mod B) , and so w i t h
THEORY OF RELATIONS
146
<
. Yet
B
Bxtl&
Y
Bx t 1
\
4.6.
+
fy
4 BY
B' Y
A
and
,
B
A
and
Obvious f o r
A 4B
.
than
B
A i d A ' 6 B;I
and so
,
B' = B Y
A
thus
Y
+
A = Ax
1 + A;
c o n t r a d i c t i n g the hypothesis. 0
have t h e same s t r i c t l y g r e a t e r c h a i n s ( w i t h r e s p e c t t o B a r e equimorphic.
a r e i n f i n i t e , t h e n i t even s u f f i c e s t h a t e v e r y c h a i n s t r i c t l y qrea-
be
B
t e r than 0
B
+
(mod A)
then
I f two c h a i n s
embeddability), If
(only) greater than
A
finite or
B
A
, and
conversely.
f i n i t e . Suppose t h a t
,
A
B a r e i n f i n i t e and t h a t
Then by t h e p r e c e d i n g p r o p o s i t i o n , t h e r e e x i s t s a c h a i n s t r i c t l y g r e a t e r and n o t g r e a t e r t h a n
A
.0
4.7. N o t i c e (by JULLIEN 1969) t h a t t h e f a i t h f u l e x t e n s i o n no l o n g e r h o l d s i f we
A
replace 0
by two c h a i n s
For example n e i t h e r
A1
w
A1 =
and
A2
+1 n o r
. A2
=
However, e v e r y c h a i n which i s s t r i c t l y g r e a t e r t h a n
Al
or of
A2
.
-+ W i s embeddable i n W w admits an embedding o f
Z = W
.0
The e x i s t e n c e o f a f a i t h f u l e x t e n s i o n between chains, no l o n g e r h o l d s i f we r e p l a c e B 0
by two c h a i n s
B1
and
For example A = W + B2 = w . W -
converse and
B2
c3
.
B2
.
-
i s n e i t h e r embeddable i n
Yet
A
B1 = U - .
nor i n i t s
i s embeddable i n e v e r y c h a i n i n which b o t h
a r e embeddable: t h i s w i l l be proved i n ch.6
B1
3.7. 0
We w i l l have an o p p o s i t e r e s u l t i f we a l l o w e x t e n s i o n s by a r b i t r a r y b i n a r y r e l a t i o n s , i n s t e a d o f r e s t r i c t i n g o u r s e l v e s t o c h a i n s : see e x e r c . 2 below. Problem posed by SABBAGH i n 1975. E x i s t e n c e o f two chains
A, B which a r e incom-
p a r a b l e w i t h r e s p e c t t o e m b e d d a b i l i t y , and have a supremum c h a i n
C
.
More p r e c i s e -
X > / A and X B, . I t has been proved by HAGENDORF 1977 t h a t t h i s supremum c h a i n does n o t e x i s t f o r
ly, f o r every chain
A
or
B
X
we would have
X >/C
iff
s c a t t e r e d ; t h e general case remains unsolved.
For t h e dual o f t h e above statement, i . e . t h e e x i s t e n c e o f t h e infimum, we have t h e easy example
A = w + 1 and
5 5 - DECREASING SEQUENCES DUSHN I K, MILLER, S IERP INSKI 5.1.
Let
A, B
B = Z
w i t h t h e infimum
C =
AND SETS OF INCOMPARABLE
w . CHAINS
OF REALS:
be two chains, each o f which i s embeddable i n t h e r e a l s . Then
t h e r e a r e a t most continuum many r e s t r i c t i o n s o f
B
isomorphic w i t h
0 Consider t h e base
E
o f r e a l s , and l e t
lBl
as a subset o f t h e s e t
A
. F be
Chapter 5
IBI
a subset o f
B/F
such t h a t
For e v e r y subset
X
i s isomorphic w i t h
1 B l such t h a t
of
147
A
( i f t h e r e e x i s t s such).
i s isomorphic w i t h
B/X
f X f r o m F o n t o X , hence f r o m F t h e r e a r e continuum many s t r i c t l y i n c r e a s i n g maps f r o m F i n t o
5
8.4;
and f i n a l l y
5.2. L e t
A
X = fXo(F)
i s determined by
f X( n o t a t i o n
A
, there
into
a s t r i c t l y i n c r e a s i n g map
E
exists
. Now
E : see c h . 2 O
i n ch.1
5
1.2). U
embeddable i n t h e
be a c h a i n o f continuum c a r d i n a l i t y , which i s
chain o f the reals. (1) There e x i s t s a s t r i c t l y s m a l l e r ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) r e s t r i c t i o n A which has continuum c a r d i n a l i t y (DUSHNIK, MILLER 1940; uses t h e axiom o f
of
choice). ( 2 ) Even s t r o n g e r , t h e r e e x i s t s a r e s t r i c t i o n l i t y , such t h a t no i n t e r v a l o f
,of
A
B
of
A
w i t h continuum c a r d i n a -
continuum c a r d i n a l i t y , i s ernbeddable i n
B
(HAGENDORF 1977, u n p u b l i s h e d ) . ( 3 ) F o r e v e r y denumerable c h a i n preceding ( 2 ) ) . 0 (2) Let
X
U
we have
be any subset o f t h e base
an i n t e r v a l o f
A
A $ B.U
(where
I A 1 such t h a t A/X
B
satisfies the
i s isomorphic w i t h
o f continuum c a r d i n a l i t y . For each such i n t e r v a l , by t h e p r e -
c e d i n g p r o p o s i t i o n , t h e r e a r e a t most continuum many corresponding s e t s
X
. More-
R o f r e a l s , each i n t e r v a l o f A i s t h e r e s t r i c t i o n t o I A l o f an i n t e r v a l o f R , which i s i t s e l f
over, i f we c o n s i d e r
A
as a r e s t r i c t i o n o f t h e c h a i n
d e f i n e d by i t s e n d p o i n t s . Consequently t h e r e a r e continuum many such i n t e r v a l s . X
Finally, the set o f a l l the
5
Apply ch.2 where
C
has a t most continuum c a r d i n a l i t y .
8.1 (axiom o f c h o i c e ) . There e x i s t s a s e t
and
D
=
IAl
a l l the intersections thus the r e s t r i c t i o n
-C
C
included i n
IAI ,
b o t h a r e e q u i p o t e n t w i t h t h e continuum, as w e l l as
.
C n X and D n X Consequently no X i s i n c l u d e d i n C B = A/C a d m i t s no embedding o f any i n t e r v a l o f A which
has continuum c a r d i n a l i t y . 0 0
( 3 ) Suppose
A 6B.U
.
Then t h e base
I A I i s p a r t i t i o n e d i n t o c o u n t a b l y many
i n t e r v a l s , each c o r r e s p o n d i n g w i t h an element o f
U
. At
t e r v a l s i s e q u i p o t e n t w i t h t h e continuum: see c h . 1 § 4.3, o v e r i t i s embeddable i n
B
, contradicting
l e a s t one o f t h e s e i n axiom o f c h o i c e . More-
the preceding (2). 0
N o t i c e t h a t (1) i m p l i e s t h e e x i s t e n c e , s t a r t i n g f r o m t h e c h a i n o f t h e r e a l s , o f a s t r i c t l y decreasing w -sequence o f c h a i n s . We s h a l l see t h a t such a sequence does n o t e x i s t f o r s c a t t e r e d c h a i n s , i . e . those i n which t h e c h a i n
Q
o f the
r a t i o n a l s i s n o t embeddable: see ch.8
5
5.3.
R t h e c h a i n o f t h e r e a l s . There e x i s t two
Let
subsets
E C,
be t h e s e t o f r e a l s and
D
of
E
4.4.
which a r e d i s j o i n t , e q u i p o t e n t w i t h t h e continuum, dense
,
148
THEORY OF RELATIONS
in
R and such t h a t , f o r each subset X f D , every r e s t r i c t i o n of R isomorphic w i t h R / ( C u X ) , e i t h e r has base C u X , o r i t s base contains a t l e a s t one
element o f E
-
(C u D)
.
Consequently f o r Y C X cD , we have the s t r i c t embeddability R / ( C uY)< R/(CuX). For X and Y s u b s e t s of D which a r e incomparable with r e s p e c t t o i n c l u s i o n ,
the preceding two restrictions are incomparable with respect to embeddability (SIERPINSKI 1950; see a l s o ROSENSTEIN 1982; uses axiom of c h o i c e ) . Take the s e t s C , D in ch.2 5 8.2.(2) (axiom of c h o i c e ) , where t h e f i designate a l l isomorphisms from R i n t o R , d i s t i n c t from t h e i d e n t i t y . For such an isomorphism f , i f a real x i s mapped t o f x # x , f o r example i f f x > x (mod R ) , then every real i n t h e i n t e r v a l ( x , f x ) i s mapped t o a s t r i c t l y g r e a t e r r e a l ,
0
hence f y # y f o r continuum many r e a l s y . Notice t h a t C and 0 a r e d i s j o i n t and each equipotent with t h e continuum. Moreover by t h e same proposition we have f " ( C ) # C f o r each considered isomorphism f ; s i m i l a r l y with D T h u s by ch.2 5 8 . 5 , t h e s e t s C and D a r e both dense (mod R ) . Take an a r b i t r a r y subset X of D and an isomorphism g of R / ( C u X ) i n t o R , which i s d i s t i n c t from t h e i d e n t i t y . Since C , hence C u X , i s dense, t h e r e
.
e x i s t s an isomorphism gt of R i n t o R , which extends g t o t h e domain E of a l l r e a l s : see ch.2 5 8.3. Hence g+ i s one of t h e previously considered i s o morphisms f . By ch.2 5 8.2, t h e s e t of images g"(C) = ( g + ) " ( C ) i s not i n c l u ded in C u D . 0 This immediately implies t h e e x i s t e n c e of a s t r i c t l y decreasing (with r e s p e c t t o embeddability) sequence, indexed by t h e continuum, of chains of r e a l s . Also t h e e x i s t e n c e of a s e t , equipotent w i t h t h e continuum, of mutually incompar a b l e chains of r e a l s (SIERPINSKI 1950).
§
6 - SUSLINC H A I N
AND
SUSLINTREE
in A : Given a chain A , t h e reader i s acquainted w i t h t h e notion of a s e t a subset D of t h e base f o r which, given any two elements x < y (mod A ) , t h e r e e x i s t s an element t of D w i t h x st d y (mod A ) The chain of r e a l s , and more g e n e r a l l y any chain A which i s embeddable in the chain of r e a l s , s a t i s f i e s t h e two following conditions: (1) t h e r e e x i s t s a countable s e t which i s dense in A ; (2) every s e t of mutually d i s j o i n t i n t e r v a l s of A , none o f which a r e s i n g l e t o n s , i s countable.
The condition (2) follows from ( 1 ) . I f D i s countable and dense i n A , then every non-singleton i n t e r v a l contains a t l e a s t one element of D . Two d i s j o i n t i n t e r v a l s cannot contain a same element
0
Chapter 5
of
D , so there are countably many i n t e r v a l s .
149 0
SUSLIN'S HYPOTHESIS (see SUSLIN 1920) The axiom called S u s l i n ' s hypothesis, a s s e r t s that the preceding condition (2) implies ( l ) , hence t h a t ( 1 ) & (2) are equivalent. This axiom i s neither provable nor refutable in ZF, even with the axiom of choice and even with the generalized continuum hypothesis. More precisely JECH and TENNENBAUM have proved the consistency of the existence o f a Suslin t r e e ( i . e . the negation of the axiom) with ZF (modulo the consistency of Z F ) . Whereas SOLOVAY and TENNENBAUM have proved the relative consistency of the axiom: see JECH 1978. For a detailed discussion of Suslin chains and Suslin t r e e s , as well as f o r the advanced r e s u l t s o f JENSEN, see for example DEVLIN, JOHNSBRiTEN 1974.
6.1. SUSLIN CHAIN I t i s more convenient t o work with the negation of Suslin's hypothesis, rather than the hypothesis i t s e l f . We say t h a t a chain i s a Suslin chain i f i t s a t i s f i e s ( 2 ) and n o t ( l ) , i . e . i f every s e t of non-singleton mutually d i s j o i n t intervals i s countable, y e t there e x i s t s no countable s e t which i s dense in the chain. A Suslin chain i s uncountable; moreover i t admits an embedding of t h e c h a i n Q rationals (uses axiom of choice). 0 The inexistence of any countable dense s e t implies t h a t the chain i t s e l f be
of
uncountable: i t s cardinality i s a t l e a s t W1 (axiom of choice). If Q i s n o t embeddable in i t , then e i t h e r the ordinal W 1 or i t s converse i s embeddable in i t : see 3.5. Hence there e x i s t uncountably many non-singleton mutually d i s j o i n t intervals. 0
6 . 2 . Every Suslin chain has cardinality exactly w 1 (uses axiom of choice and the continuum hypothesis). 0 Let A be a Suslin chain; we already know t h a t A i s uncountably i n f i n i t e , so has cardinality a t l e a s t w 1 (axiom of choice). Suppose t h a t A has cardinality a t least w . Replace A by a r e s t r i c t i o n o f cardinality O 2 and l e t B be a well-ordering of type o 2 on the same base. Partition the pairs of elements x, y of the base into two colors: l e t t i n g x < y (mod A ) , we say t h a t the pair has color (+) i f x < y (mod B ) , and color (-) i f x ) y (mod B ) By the ERDOS partition lemma (ch.3 5 3.4) f o r d = 0 (hence using only the continuum hypothesis), there e x i s t s a subset of the base of cardinality G, , a l l of whose pairs have a same color. Hence there e x i s t s a s t r i c t l y increasing or a s t r i c t l y decreasing 0 l-sequence, and hence wl-many non-singleton mutually d i s j o i n t intervals: contradiction. 0
.
THEORY OF RELATIONS
150
6.3. SUSLIN TREE We say t h a t a t r e e i s a S u s l i n t r e e i f i t has c a r d i n a l i t y
w1
and i f every chain
( o r t o t a l l y ordered r e s t r i c t i o n ) and every a n t i c h a i n i s countable. The existence o f a S u s l i n chain o f c a r d i n a l i t y
W1 S u s l i n t r e e ( t h e a d d i t i o n a l assumption o f c a r d i n a l i t y
i m p l i e s t h e existence o f a
w 1 allows us t o avoid
using the axiom o f choice: ZF s u f f i c e s ) . 0
Let
be a
A
S u s l i n chain o f c a r d i n a l i t y
w1 .
i ,
To each countable o r d i n a l
defined by i t s two endpoints ui < vi (mod A) , where Ai an a r b i t r a r y are d i s t i n c t . To do t h i s , begin w i t h A. = (uo,vo)
associate an i n t e r v a l all
ui
and
vi
interval. Let
i be a non-zero countable o r d i n a l , and suppose t h a t t h e
A
for
j j < i have already been defined so t h a t they are m u t u a l l y e i t h e r d i s j o i n t o r one
contained i n the other. The s e t o f endpoints uj, v j ( j < i ) i s countable: by hypothesis i t i s n o t dense i n A , hence there e x i s t two elements u, v i n t h e base o f
, between
A
.
v. ( j < i ) L e t Ai = J vi = v : t h i s i n t e r v a l must be e i t h e r d i s j o i n t o r included
which there i s no endpoint
uj
or
ui = u and A. ( j < i ) J thus obtained has c a r d i n a l i t y w 1 Reverse i n c l u s i o n The s e t o f i n t e r v a l s Ai defines a t r e e on the s e t o f these Ai . Every antichain, i . e . every s e t o f i n t e r which are m u t u a l l y d i s j o i n t , i s countable. F i n a l l y , a chain, o r s e t o f v a l s Ai (u,v)
so
.
i n each
Ai
intervals
.
which are mutually comparable w i t h respect t o i n c l u s i o n , i s w e l l f o r every p a i r o f countable o r d i n a l s A . c Ai J Such a chain i s countable; f o r i f i t had c a r d i n a l i t y w 1 , then
ordered by the o r d i n a l i n d i c e s w i t h
i, j ( i
<j ) .
using the endpoints o f preceding i n t e r v a l s , we could o b t a i n W l-many
mutually
d i s j o i n t intervals. 0 6.4. The existence o f a S u s l i n t r e e i m p l i e s , and hence i s e q u i v a l e n t w i t h t h e existence o f a S u s l i n chain (here we use the r e g u l a r i t y o f
W1
, thus
f o r example
the countable axiom o f choice). 0 Let
be a S u s l i n t r e e ; we can assume t h a t
A
A
i s a well-founded p a r t i a l orde-
A by a c o f i n a l well-founded p a r t i a l ordering. To 5.1 w h i l e n o t i n g t h a t , by hypothesis, t h e base o f A i s
r i n g , i f necessary by r e p l a c i n g see t h i s , apply ch.2
5
.
well-orderable w i t h order type w 1 More p r e c i s e l y l e t ui ( i < td 1) be an indexation o f the base; then remove each u f o r which t h e r e e x i s t s an i<j j w i t h ui u j (mod A) To see t h a t what remains i s s t i l l a S u s l i n t r e e , note t h a t
>
.
i , the removed elements u are a l l < ui (mod A) j there are countably many such; thus t h e r e remain w l-many elements.
f o r each index
We can f u r t h e r assume t h a t every non-empty f i n a l i n t e r v a l o f W
. To
see t h i s , l e t
many successors (mod A )
x
A
, and
hence
has c a r d i n a l i t y
be an element o f the base which has o n l y countably
. Those
x
o f minimal h e i g h t are m u t u a l l y incomparable,
Chapter 5
151
x and the ir successors. Neither A nor any f in al interval of A i s f i n i t e l y fre e . For otherwise, by ch.4 5 3.1 (using the regularity of LJ there would e x i s t a t o t a l l y ordered re stric contradicting our hypotheses. tion of A with cardinality w W e can assume t h a t A has a minimum, whose singleton will be denoted by Eo For each countable non-zero ordinal i , l e t Ei be the denumerable s e t of elements of height i : we require t h at t h i s s e t be i n f i n i t e . For each element x of Ei we require that there e x i s t denumerably many elements which are immediate successops of x (mod A ) . This s e t i s denoted Eitl ,X and the union of these se ts must be E i t l . For each countable l i mi t ordinal i and each x of height < i , we require t h a t there e x i s t i n f i n i t e l y many elements in Ei which are 7 x (mod A ) . Finally, for each countable l i mi t ordinal i and each t o t a l l y ordered re stric tion X of A containing elements of a l l heights < i , we require tha t there e xist either a unique element of Ei which i s a successor of a l l elements of X , or none such. These requirements are easy t o s a t i s f y . For example f o r the minimum, take an arbitrary element having W l-many successors. Then having obtained Ei , the s e t of elements x with height i , note t h at f o r each x in Ei , the final interval > x has cardinality w 1 and i s not f i n i t e l y f r e e . Hence take a denumerable free subset as E i + l , x and among the successors of x , retain only those which Finally, f o r each counare identical t o or successors of an element of Ei+l ,X table limit ordinal i and each chain X containing elements of a l l heights 4 i , i f there e x i s t elements above X , then decide t o retain one such plus the wl-many successors of t h i s element. For each element x of the base IAl with height i , t o t a l l y order the denumeNow consider the with the order type of a dense chain C i + l , x rable s e t Ei+l ,X set of a l l maximal t o t a l l y ordered r e s t r i c t i o n s , or maximal chains of A . This set i s t o t a l l y ordered by the preceding dense chains. Indeed, given two distinc t maximal chains U and V : none of the two bases i s included in the other. Moreover there e x i st s a l e a s t element u among those elements of I U 1 which do not belong t o I V I , and a l e a s t element v among those elements of I V I which do not belong t o I U I By the preceding, there ex i s t s a l a s t element x whose height will be denoted i , common t o b o t h bases of U and V , and having u and v as immediate successors. Let U < V i f u < v (mod C i + l , x ) : t h i s tota lly orders the s e t of maximal chains. Let H be the chain thus obtained. We shall prove tha t H i s a Suslin chain. =i-r+ -11 a rnt n irrhirh i c h n c o i n H rannot be countable. For i f i t were, and so there are countably many such: i t suffices t o remove these
.
.
.
.
THEORY OF RELATIONS
152
the i n t e r v a l o f maximal chains passing through of D
z
does n o t c o n t a i n any element
.
Now suppose t h a t t h e r e e x i s t o l-many non-singleton m u t u a l l y d i s j o i n t i n t e r v a l s I n each i n t e r v a l take two elements, o r maximal chains U and V . As of H before, take an element x whose h e i g h t i s denoted i and t h e elements u, v
.
immediate successors o f x (mod A) ; and take w between u and v modulo the chain Ci+,lx Then these w thus associated w i t h our d i s j o i n t i n t e r v a l s o f H ,
.
are m u t u a l l y incomparable (mod A): they must be countably many; c o n t r a d i c t i o n . 0
§
7 - ARONSZAJN
TREE,
SPECKER CHAIN
7.1. ARONSZAJN TREE
This i s a well-founded t r e e o f c a r d i n a l i t y ~3~ whose chains and h e i g h t l e v e l s are countable. I t i s n o t r e q u i r e d t h a t every a n t i c h a i n be countable. Hence every well-founded S u s l i n t r e e i s an Aronszajn tree; b u t the converse p o s s i b l y depends on s e t - t h e o r e t i c axioms: see the problem a t the end o f 7.4. The f o l l o w i n g c o n s t r u c t i o n o f an Aronszajn t r e e , using ZF p l u s choice, goes back t o KUREPA 1935 p . 96, c i t i n g a l e t t e r from ARONSZAJN i n 1934. The elements o f the t r e e w i l l be o r d i n a l sequences o f i n t e g e r s ai ( i < ) y w i t h o u t r e p e t i t i o n , where o( v a r i e s over a l l countable o r d i n a l s . We say t h a t such a sequence u precedes v o r t h a t v f o l l o w s u , i f u i s an i n i t i a l i n t e r v a l o f v . Moreover we r e q u i r e t h e f o l l o w i n g c o n d i t i o n s o f convergence and denumerability. Convergence. For each sequence ai (i< o ( ) , the sum o f the inverses l/ai is f i n i t e . Furthermore for each sequence u w i t h l e n g t h o( , each countable o r d i n a l and each p o s i t i v e r e a l number r , t h e r e must e x i s t , i n our set, a sequence w i t h l e n g t h o i + /s , f o l l o w i n g u , and such t h a t the sum of the inverses l/ai f o r o( i
.
denumerably many o( -sequences (uses axiom o f choice). Using t h e preceding convergence c o n d i t i o n , we see t h a t , given a l i m i t countable o r d i n a l 4
we can r e t a i n
only o( -sequences u such t h a t , f o r each i ( 4 , t h e i n i t i a l i n t e r v a l o f u w i t h l e n g t h i a l r e a d y belongs t o our set; more e x a c t l y , we r e t a i n denumerably many such o( -sequences. F i n a l l y f o r each sequence u i n our set, every i n i t i a l i n t e r v a l o f u w i l l belong t o our set. These c o n d i t i o n s a l l o w us, f o r each o( , t o c o n s t r u c t a " d e f i n i t i v e " denumerable s e t o f o( -sequences, which w i l l n o t be unduly increased by the u l t e r i o r construct i o n o f longer sequences. Because t h e n o n - r e p e t i t i o n of values
ai
, every
t o t a l l y ordered r e s t r i c t i o n , o r
Chapter 5
153
chain, of the preceding Aronszajn t r e e , i s countable. In the preceding t r e e , f o r each countable ordinal o( and each o( -sequence u , there e x i s t u1-many sequences which are successors of u However t h i s i s not true for a l l Aronszajn t r e e s ; f o r instance we can add t o the preceding tree new ordinal sequences without successors, or with countably many successors.
.
7.2. A chain A i s embeddable in the chain of r e a l s , i f f there e x i s t s a countable subset which i s dense in A (uses countable axiom of chsice). 0
If
A
i s a r e s t r i c t i o n o f the chain of r e a l s , then f o r each integer
n
take a
partition o f the reals into intervals of length l / n , and choose an element of I A l in each interval (countable axiom o f choice). Conversely, i f there i s a r e s t r i c t i o n I o f A , which i s denumerable and dense in A , then i f necessary complete I i n t o J , the l a t t e r which i s isomorphic with the chain of rationals. Then i n each cut o f J , we take the unique element o f I A I ( i f i t exists) defined by i t s position with respect t o the elements of I I I . Thus we embed A into the reals. 0
7.3. SPECKER CHAIN This i s an uncountable chain A such t h a t neither w nor i t s converse are embeddable in A ; moreover any chain which i s embeddable both in A and in the chain of reals, i s countable. A Specker chain has no countable dense subset. For i f s o , then by the preceding, i t would be embeddable in the chain of r e a l s as well as in i t s e l f , and consequently i t would be countable. Every Specker chain has cardinality W (uses choice plus continuum hypothesis). Suppose on the contrary t h a t there e x i s t s a Specker chain of cardinality 3 w 2 By 3.5, t h i s chain admits an embedding e i t h e r of w 2 , or i t s converse: contradiction with the definition o f Specker chains. Or our chain admits an embedding of an W *-dense chain. By 3.7 ( 2 ) (choice and continuum hypothesis), the l a t t e r admits an embedding o f w 1 : contradiction. 0
0
.
7.4. Construct as follows a Specker chain, s t a r t i n g with the Aronszajn t r e e of 7.1, hence on a s e t of ordinal sequences of integers, without repetition. Apply ch.4 5 6.2 (using u l t r a f i l t e r axiom) which associates t o the given t r e e A , a chain C with the same base. For each element u (ordinal sequence of integers without repet i t i o n ) , the final interval 3 u (mod A ) becomes an interval (mod C ) with minimum u .
THEORY OF RELATIONS
154
We s h a l l show f i r s t o f a l l t h a t t h e c h a i n C embedding o f
t h u s defined does n o t admit an
u1, n o r o f i t s converse,
C can be decomposed i n a denumerable sum o f d i s j o i n t i n t e r v a l s , By ch.4 5 6.2, each corresponding t o an i n t e g e r a , and formed from some sequences which begin
with
a
.
Consequently, given a s t r i c t l y i n c r e a s i n g (mod C)
W1-sequence whose
terms are denoted ui ( i countable o r d i n a l ) , t h e r e e x i s t s an i n t e g e r al such t h a t , from a c e r t a i n o r d i n a l on, t h e sequence ui begins w i t h al By i t e r a t i o n ,
.
f o r every countable o r d i n a l k , t h e r e e x i s t s a k-sequence o f i n t e g e r s such t h a t , from a c e r t a i n o r d i n a l index on, t h e sequence a j' * * ( j < k ) begins w i t h the above k-sequence. F i n a l l y , these al, a2, ... , aj, a s t r i c t l y i n c r e a s i n g wl-sequence o f elements o f the t r e e A , thus an
..
al,
a*,
...
ui form o l-se-
quence o f i n t e g e r s w i t h o u t r e p e t i t i o n : c o n t r a d i c t i o n . 0 L e t us now show t h a t if t r i c t i o n D dense i n H
H jA_an uncountable chai_n-_which has a denumerable res,then H i s n o t embeddable i n C .
D i s denumerable, i t i s formed o f ord'nal sequences o f i n t e g e r s w i t h o u t r e p e t i t i o n , a l l o f l e n g t h l e s s than a c e r t a i n countable o r d i n a l . L e t o( be a countable o r d i n a l s t r i c t i y g r e a t e r than a l l these lengths. Since H i s uncounta-
0 Since
ble, t h e r e e x i s t s a t l e a s t one sequence u w i t h l e n g t h o( having ul-many sequences i n I H I which extend u . By ch.4 4 6.2, t h e r e e x i s t s an i n t e r v a l (mod C) formed o f a l l those elements, o r sequences extending u Thus any two o f them are n o t separated (mod C) by any sequence belonging t o I D l : t h i s contra-
.
d i c t s the density o f
D
in H
.0
Problem. I n 7.1 we constructed a well-founded Aronszajn t r e e , based on c e r t a i n o r d i n a l sequences o f i n t e g e r s ( w i t h o u t r e p e t i t i o n ) , u s i n g ZF p l u s choice. Whereas a S u s l i n t r e e can o n l y be constructed by u s i n g t h e negation o f S u s l i n ' s hypothes i s . Consequently, w i t h ZF p l u s choice p l u s t h e axiom c a l l e d S u s l i n ' s hypothesis, we are insured t h a t t h e r e e x i s t s an Aronszajn t r e e which i s n o t a S u s l i n t r e e . However i f we admit ZF p l u s choice p l u s t h e negation o f S u s l i n ' s hypothesis, then we do n o t know whether the Aronszajn t r e e i n 7.1 i s o r n o t a S u s l i n t r e e . And more g e n e r a l l y w i t h these s e t - t h e o r e t i c axioms, we do n o t know whether t h e r e e x i s t Aronszajn t r e e s which are n o t S u s l i n t r e e s .
Chapter 5
5 8 - FAITHFUL
INFINITE
155
MALITZ'
EXTENSION:
AND
LOPEZ' COUNTEREXAMPLES
8.1. FAITHFUL INFINITE EXTENSION Let A1,
...,Ah
be a f i n i t e s e t o f f i n i t e r e l a t i o n s w i t h common a r i t y , and
be a f i n i t e r e l a t i o n w i t h
R
I f there e x i s t e x t e n s i o n s o f
2A1
and ... and 3 Ah - 3 A1 are
and
...
R
.
and $ A h
w i t h a r b i t r a r y l a r g e f i n i t e c a r d i n a l i t i e s , which
R
, t h e n t h e r e e x i s t s a denumerable e x t e n s i o n o f
R
which respects t h e same c o n d i t i o n s .
0 We can assume t h a t
1, ... ,p
i s d e f i n e d on t h e i n t e g e r s
R
and t h a t , f o r each
i , t h e r e e x i s t s an e x t e n s i o n Ri o f R based on t h e i n t e g e r s 1 t o p+i and r e s p e c t i n g t h e c o n d i t i o n s . F o r i n f i n i t e l y many i n t e g e r s i , t h e Ri have a same r e s t r i c t i o n S1 t o 1,. . . ,p+1 . Among these, t h e r e a r e i n f i n i t e l y many i n t e g e r s i f o r which t h e Ri have a same r e s t r i c t i o n S2 t o 1, ...,p+ 2 integer
I t e r a t i n g t h i s , we t h u s d e f i n e
S
j
t a k e t h e common e x t e n s i o n o f t h e
...,Ah
A1,
8.2. L e t
and
common a r i t y , and l e t and
...
and
>/
Bk
R
B1,
u
...,Bk
satisfy
f o r each i n t e g e r j . I t now s u f f i c e s t o , based on t h e s e t o f a l l i n t e g e r s . 0 j be two f i n i t e s e t s o f f i n i t e r e l a t i o n s o f
R3AA1
and
... and
#Ah
R >/El
as w e l l as
.
Then t h e r e e x i s t s an i n t e g e r equal t o
S
u
such t h a t
every
R
with cardinality _ a t_ l e_ a s-t .
s a t i s f y i n g t h e p r e c e d i n g c o n d i t i o n s has a r e s t r i c t i o n
t i n g t h e same c o n d i t i o n s , and such t h a t
R'
R'
respec-
has a denumerable e x t e n s i o n s t i l l
respecting the conditions. 0
Let
v
F o r each
be t h e sum o f t h e c a r d i n a l i t i e s of t h e r e l a t i o n s
B1
through
s a t i s f y i n g the conditions, there e x i s t s a r e s t r i c t i o n
R
R'
.
Bk
of
R
v , which s a t i s f i e s t h e c o n d i t i o n s . Consider , which a r e o n l y f i n i t e l y many, up t o isomorphism. F o r each,
w i t h c a r d i n a l i t y a t most equal t o a l l these
R'
.
e i t h e r t h e r e e x i s t s a denumerable e x t e n s i o n s a t i s f y i n g t h e c o n d i t i o n s . O r t h e r e e x i s t s an i n t e g e r
u(R')
a l l extensions o f
R'
which i s s t r i c t l y g r e a t e r t h a n t h e c a r d i n a l i t i e s o f
r e s p e c t i n g t h e c o n d i t i o n s . Then i t s u f f i c e s t o s e t
t o be t h e maximum o f t h e s e
u(R')
.0
u
THEORY OF RELATIONS
156
8.3. MALITZ' COUNTEREXAMPLE Can we require t h a t R ' = R ; in other words, does there e x i s t an integer u such t h a t , i f R has cardinality greater than or equal t o u and s a t i s f i e s the conditions, then there e x i s t s a denumerable extension of R satisfying them. A negative answer i s due t o MALITZ 1967. 0 Take the base of integers from 0 t o n-1 Let I n be the usual chain of these integers; l e t Cn be the consecutivity r e l a t i o n ( y = x + l ) ; l e t O n be the unary relation called the singleton of zero, i . e . the relation taking (+) f o r 0 and ( - ) elsewhere; and l e t U n be the relation singleton of n-1 . Finally l e t Rn be the quadrirelation (In,Cn,On,Un) . From n = 7 on, a l l the R have the n same r e s t r i c t i o n s of c a r d i n a l i t i e s 1, 2 and 3 , up t o isomorphism. Let A1, ..., Ah be those quadrirelations of the same a r i t y and c a r d i n a l i t i e s 1, 2 , 3 which are n o t embeddable in R, , and hence in Rn ( n 3 7 ) . We see t h a t every extension of an Rn t o a new element added t o i t s base admits an embedding of one of the A1,...,Ah . An analogous b u t rather complicated counterexample i s obtained f o r binary relations by LOPEZ 1973. 0
.
8.4. LOPEZ ' COUNTEREXAMPLE Given the f i n i t e relations A1, ...,Ah and B1,...,Bk , one can ask whether there e x i s t two integers u , v such t h a t , f o r every R with cardinality greater t h a n or equal t o u , there e x i s t v elements of the base, such t h a t the r e s t r i c t i o n of R t o i t s base with these v elements removed respects the embedding inequal i t i e s in the 6 ' s and has an extension of a r b i t r a r y large cardinality respecting the non-embedding inequalities in the A's . Negative answer by LOPEZ 1973. 0 For the base, take the s e t of points, or ordered pairs of integers called the abscissa and ordinate, and which vary from 0 t o n-1 Let Rn be the multirelation on t h i s base, which i s composed of the following 4 unary relations and 6 binary relations. The unary relation O n takes the value (+) f o r the points with abscissa 0 The relation U n takes (+) f o r the points with abscissa n-1 Similarly 01; and U,', are defined by interchanging the abscissas and ordinates. The s t r a t i f i e d partial ordering I n takes the value (+) f o r each ordered p a i r of points ( i , x ) , ( j , y ) whose abscissas s a t i s f y i -c j < n , with a r b i t r a r y ordinates x , y ; moreover I n i s reflexive. The equivalence relation E n takes (+) for any two points with a same abscissa and a r b i t r a r y ordinates. The equivalence classes of t h i s relation are thus the classes of elements which a r e pairwise incomparable modulo I n . The binary relation Cn , which by abuse of notation we shall c a l l a consecutivity, takes the value (t) f o r each ordered pair of points
.
.
.
157
Chapter 5
( i , x ) , ( i + l , y ) whose abscissas are consecutive. Finally, the s t r a t i f i e d partial ordering I,', , the equivalence relation E,', and the consecutivity C,', are obtained from the preceding by interchanging abscissas and ordinates. From n = 7 on, every Rn has the same r es t r i ct i o ns B1, ...,B k with cardinalities 1, 2 , 3 ( u p t o isomorphism). Let A1, ...,Ah be the other multirelations of the same a r i t y and car d i n al i t i es 1, 2 , 3 . We see t h a t every proper extension of Rn ( n 3 7 ) admits an embedding of a t l e a s t one of the A's Indeed, add a new element t t o the base of Rn . Consider the case where e i t h e r O n or U n or 0,', or U,', takes the value (+) f o r t , and reduce t h i s t o the preceding 5 8.3. Now consider the case where a l l the preceding unary relations take the value ( - ) f o r t Then e i t h e r there ex i s t s an equivalence class of En t o which t belongs: again reduce t o 5 8.3. Or t occurs between two consecutive equivalence classes of En I n t h i s case, use the consecutivity Cn t o see t h a t the extension of Rn t h u s obtained admits an embedding of one o f the A's . Now suppose the existence of u and v satisfying our hypothesis; take n > u and > v . Let Sn be a r es t r i ct i o n of Rn in which the B's are embeddable, and which i s obtained by removing v points. Then in each equivalence class of Add a En , there remains a t l e a s t one element of I S n l ; similarly f o r E,', new element t t o the base o f Sn , and attempt t o require t h a t the extension of Sn t o i t s base with t added admit only embeddings of the B's and not of the A's . This leads us t o s i t u a t e t in the chain of the equivalence classes By using Cn , one sees t h at t necessarily belongs t o one o f the equiof En valence classes: t cannot be situated between two consecutive classes. Thus we obtain an element in the base of Sn , which i s equivalent with t (mod E n ) , and another element equivalent with t (mod E,',) From t h i s , we deduce tha t t is the unique element common t o both equivalence classes. Thus,we have again a restriction of Rn obtained by removing v-1 points: t h i s i s our extension of Sn Iterating t h i s , we obtain Rn i t s e l f , and a t the following step we obtain a proper extension o f Rn , in which necessarily one of the A's i s embeddable. 0
.
.
.
.
.
.
.
EXERCISE 1 - RESTRICTIONS OBTAINED BY REMOVING ONE OR TWO ELEMENTS 1 Existence of a relation R with base E and two elements a , b o f E such t h a t R z R / ( E - j a } ) I R / ( E - {b}) y et R > R / ( E - { a , b ) ) Let R be the binary relation based on the s e t formed with negative integers -1, -2, and ordered pairs of natural numbers i , j = 0 , 1, 2 , .. , taking the value (+) in only following cases. R(u-1,u) = + f o r each negative u ; and R ( - l , ( O , O ) ) = t ; and R ( ( i , j ) , ( i + l , j ) ) = R ( ( i , j ) , ( i , j + l ) ) = + f o r a l l natural numbers i , j Show t h a t R s a t i s f i e s the stated conditions with a = ( 0 , l ) and b = ( 1 , O ) (communicated by POUZET in 1975).
-
.
...
.
2
- Similarly with
R
> R/(E -
\aj )
>
R/(E
-
{a,b})
ye t
R
L
R/(E
- { b)) .
158
Let
THEORY OF RELATIONS R
be t h e b i n a r y r e l a t i o n based on t h e s e t o f p o s i t i v e and n e g a t i v e i n t e g e r s ,
which we denote by
O,l,-1,2,-2,..
and n a t u r a l numbers, which a r e assumed t o be 0 ' ,1' ,2'
d i s t i n c t from t h e i n t e g e r s and denoted by o n l y i n f o l l o w i n g cases:
+
R ( i ,i')=
i3 0
f o r every i n t e g e r
value; f i n a l l y
R(i',(i+l)')
. Show
= R(i,i+2)
R(i,i+l)
that
(communicated by LOPEZ i n 1977).
For
Similarly with R
R
R
> R/(E
i ' w i t h equal
i ' and i t s immediate suc-
satisfies the stated conditions with
(i+l)'
b = 0'
-
v a l u e (+) i s t a k e n on
and f o r t h e n a t u r a l number
f o r e v e r y n a t u r a l number
cessor 3
,. . . The
+ f o r a l l i n t e g e r s i ; and
=
- { a } ) :R/(E - { b ) )
- {a,b}
:R/(E
a = 0
and
.
)
t a k e t h e c o n s e c u t i v i t y on p o s i t i v e and n e g a t i v e i n t e g e r s , w i t h
a, b
two
c o n s e c u t i v e i n t e g e r s (communicated by HAGENDORF i n 1977).
-
EXERCISE 2
Val o f
Let
A, B, C
B satisfying
Defining
by
D
be t h r e e c h a i n s such t h a t
Let
he an isomorphism f r o m
C+D 5 C
A+B
t h e non-empty f i n a l i n t e r v a l o f f"(Do)
i s a t t h e l e f t of
which a r e a t l e f t o f
Do I t e r a t i n g t h i s , we o b t a i n
Di
Either a l l o f the elements o f
B into
B
and
B
C<
i n i t i a l inter-
C
(see HAGENDORF 1977 lemma 1.10).
A
if
A+C+D
,then
A+C
3
in
Do
, and l e t
A+C
onto a r e s t r i c t i o n o f
B which i s t h e complement o f
. Let
A+B
. The
C
be t h e i n t e r v a l o f those elements
D1
and more g e n e r a l l y
D2
Di
.
B , i n which case t h e i d e n t i t y on those , when completed by f , g i v e s an isomorphism o f
Di
h y p o t h e s i s . O r a t l e a s t one o f t h e
hence a l l t h e
times i s an isomorphism o f
included i n
A+B
A
.
I n t h i s case,
onto a r e s t r i c t i o n o f
iterated
, hence A+B
A
. Or
take
, even
B
if
C<
B
. For
instance take
A =
B
=
.
A = U - + W1
-+
Prove t h a t
Q +
";+
0
= Q
i s an
C
l + w - and
compare w i t h
has 3 immediate e x t e n s i o n s : 1+A
, A+1
5
+
; and
has 5 immediate e x t e n s i o n s :
Q
+ W - . W1
, not
B+1,
wl+B
4.4).
and a l s o
which i s n o t o b t a i n e d by s i m p l y a d d i n g an i n t e r v a l t o
B
.
2 A
.
EXERCISE 3 - IMMEDIATE EXTENSIONS (HAGENDORF 1983, unp.; 0 W
(i+l)
A = B = Q+O1+Q and C = W1+Q , where Q i s t h e c h a i n o f A = converse o f W2 and B = Z and C = 0
the r a t i o n a l s . O r again
Prove t h a t
,
A
hits
Di
f
Note t h a t o u r statement no l o n g e r h o l d s i f we remove t h e h y p o t h e s i s t h a t C = CJ-
.
f"(Do) i
f o r each i n t e g e r
are included i n
preceding the
i n i t i a l interval o f
be
Do
image s e t
y e t g r e a t e r t h a n o r equal t o some element of
, contradicting the D . (j > i) a r e J
C
1
.
A+C I A
f
A+B
, we have e q u i v a l e n t l y t h a t
B = C+D
either
.o- r
then
A+B 5 A+C ;
,w
o b t a i n e d by a d d i n g an i n t e r v a l .
.
A
+
B
,
159
CHAPTER
6
SCATTERED CHAI 11, NEIGHBORHOOD, INDECOMPOSAB I LITY
§
1 - SCATTERED CHAIN
A chain i s said t o be s cat t er ed i f the chain Q of rationals i s n o t embeddable in i t . For example, every well-ordering or converse well-ordering, the chain Z of positive and negative integers, the product G J - . ~ or i t s converse. I f a chain has no dense r e s t r i c t i o n , then i t i s scattered. Conversely by ch.5 5 3.1 (using dependent choice), a scattered chain has no dense re stric tion. 1.1. Every sum of scattered chains, ordered alona a scattered chain, i s scattered. I n p a r t i c u l a r , the ordinal sum and the ordinal product of two scattered chains i s scattered.
Let Ai be the chains in consideration, and A the sum of the Ai along If Q i s embeddable in A , then e i t h e r there e x i s t two elements chain I in the same interval Ai , in which case Q i s embeddable in Ai , or there a t most one element of Q in each Ai , in which case Q i s embeddable in
0
.
1 . 2 . Let
be a chain; i f every i n i t i a l interval of i s scattered. Same statement with "final interval". A
A
the of Q e xists I 0
.
i s scattered, then A
1.3. (1) Given a s c a t t e r e d chain A , the chain A . 2 = A+A i s scattered and s t r i c t l y greater than A with respect t o embeddability (uses dependent choice; ZF suffices i f A i s denumerable or with well-orderable base). This follows from the preceding 1.1 and ch.5 5 3.3. ( 2 ) For every countable s e t of denumerable scattered chains, there e xists a denume-
rable scattered chain in which a l l are embeddable. Particular case of 1.1. 1.4. Let A be a denumerable chain. If. A i s scattered, then there are only countably many i n i t i a l intervals (uses countable axiom of choice). If A i s not scattered, then there are exactly continuum many i n i t i a l intervals o f A
.
For the second conclusion, i t suffices t o note t h a t Q i s embeddable in A and there are exactly continuum many i n i t i a l intervals of Q Conversely, suppose t h a t A has uncountably many i n i t i a l intervals: we shall prove t h a t Q i s embeddable i n A . For t h i s , we shall prove tha t there e xists an element u of the base 1 A I , such t h a t uncountably many i n i t i a l intervals do n o t
0
.
THEORY OF RELATIONS
160
contain
, and
u
uncountably many i n i t i a l
i n t e r v a l s do contain
.
u
Suppose the contrary. We say t h a t an element o f the base i s "on t h e r i g h t " i f there are uncountably many i n i t i a l i n t e r v a l s which do n o t c o n t a i n i t . Analogously define an element t o be "on t h e l e f t " . The elements on t h e l e f t form an i n i t i a l i n t e r v a l
B , and the elements on t h e r i g h t form a f i n a l i n t e r v a l
C
. Every
element i s
e i t h e r on t h e l e f t o r r i o h t ; moreover by our assumption o f the non-existence o f u
, no
element i s both on the l e f t and r i g h t . Thus
l e a s t one o f the two, say
and
B
are complements. A t
C
, has uncountably many i n i t i a l i n t e r v a l s . Yet B i s
B
countable: enumerate the elements o f i t s base. Now B
has no maximum element, f o r
otherwise i t would be preceded by uncountably many i n i t i a l i n t e r v a l s . Hence t h e r e exists a s t r i c t l y increasing a-sequence cofinal i n
. Partition
B
B according
t o t h i s sequence. We then have denumerably many i n t e r v a l s , each o f which has count a b l y many i n i t i a l i n t e r v a l s , and y e t t h e i r union has uncountably many i n i t i a l i n t e r v a l s . This c o n t r a d i c t s t h e countable axiom o f choice: see ch.1 The existence o f
i s then proved: c a l l i t
u
uo
.
5
2.5.
I t e r a t i n g t h i s , we o b t a i n an
analogous element
ul< uo (mod A) w i t h uncountably many i n i t i a l i n t e r v , a l s which u1 and uncountably many ones which c o n t a i n u1 b u t n o t uo And
.
do n o t contain
an analogous element
, and
u2 7 uo
which i s isomorphic w i t h
Q
so f o r t h : t h i s y i e l d s a r e s t r i c t i o n o f
A
.0
A , the chain o f i n c l u s i o n o f the i n i t i a l i s s c a t t e r e d (uses dependent choice; ZF s u f f i c e s i f the base i s
1.5. Given an i n f i n i t e s c a t t e r e d chain intervals of
A
well-orderable).
B be the chain o f the i n i t i a l i n t e r v a l s o f
Let
0
chain
Q
o f r a t i o n a l s i s embeddable i n B
e x i s t s an element u
, both
I A I , for
u o f the base
intervals containing
u
, and
. We
A
, and
suppose t h a t t h e
s h a l l prove f i r s t t h a t t h e r e
which the chain o f those i n i t i a l
the chain o f those i n i t i a l i n t e r v a l s n o t c o n t a i n i n g
admit an embedding o f
Q
.
It suffices t o s t a r t with a r e s t r i c t i o n o f
B which i s isomorphic w i t h Q , and t o p a r t i t i o n i t i n t o a sum o f three terms, each isomorphic w i t h of
A
,in
Q
, then
t o take two elements, i . e . two i n i t i a l i n t e r v a l s
the middle term; and f i n a l l y t o take an element
Having proved the existence o f analogous element
ul<
we use dependent choice;
u
, call
uo (mod A)
1.6. L e t 0
A
A
and
it
A
uo
i n the difference
. Iterating this,
we o b t a i n an so f o r t h (here i s countable o r has well-orderable
u 2 7 uo (mod A)
ZF s u f f i c i e n t i f A
base). This y i e l d s a r e s t r i c t i o n o f
of
u
A
o f these two i n i t i a l i n t e r v a l s o f
isomorphic w i t h
, and
Q
.0
be an i n f i n i t e s c a t t e r e d chain. Then t h e s e t o f i n i t i a l i n t e r v a l s
i s equipotent w i t h
L e t t h e aleph
A
(uses axiom o f choice).
be t h e c a r d i n a l of the base
I A I , and suppose t h a t t h e
Chapter 6
set o f i n i t i a l intervals o f
Q
p o s i t i o n 1.5, t h e c h a i n
A
5
i s embeddable i n &+l-sequence
w
t h e p r e c e d i n g proB
of
l i k e e v e r y successor B
admits an em-
Hi
Hi
. We
(i
of
A
.
Take an element
then obtain a s t r i c t l y increasing A
, contradicting t h e hypothesis
.0
5 2 - DECOMPOSITION NEIGHBORHOOD;
. By
n the chain
o r i t s converse. To f i x ideas, suppose t h a t
o f elements i n t h e base of
Card A = fd,
+1
, hence t h a t t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g
o f i n i t i a l intervals
d+l-sequence
o(
and s i n c e
0 o(+l B
i n each d i f f e r e n c e o f successive that
W
6.2 u s i n g axiom o f c h o i c e ) , t h e c a i n
bedding o f e i t h e r t h e o r d i n a l
w
>/
has c a r d i n a l i t y
o f r a t i o n a l s i s n o t embeddable
these i n i t i a l i n t e r v a l s . By ch.5 § 3.5, aleph, i s r e g u l a r (ch.2
161
OF A CHAIN
INTO
A SUM OF SCATTERED CHAINS: NEIGHBORHOOD RANK
STRONGLY SCATTERED CHAIN,
Every n o n - s c a t t e r e d c h a i n can be u n i q u e l y decomposed i n t o a dense sum o f
2.1.
s c a t t e r e d c h a i n s (HAUSDORFF 1914; uniqueness uses t h e axiom o f dependent c h o i c e ) . 0
L e t two elements
x, y
o f t h e base be e q u i v a l e n t i f t h e i n t e r v a l
(x,y)
is
s c a t t e r e d . T h i s c o n d i t i o n y i e l d s e q u i v a l e n c e c l a s s e s which a r e themselves s c a t t e r e d i n t e r v a l s . No two o f these i n t e r v a l s can e v e r be consecutive; t h u s t h e y c o n s t i t u t e a dense chain. I f t h e r e were two d i s t i n c t decompositions o f a c h a i n i n t o a dense sum o f s c a t t e r e d
chains, t h e n one o f these s c a t t e r e d chains would a d m i t an embedding o f a dense
Q : see ch.5
sum, hence an embedding o f
2.2.
x, y
o f t h e base o f a c h a i n
i f f t h e i n t e r v a l w i t h endpoints x
and y
t h i s r e l a t i o n a r e c a l l e d 0-neiqhborhoods f i n i t e , o r isomorphic w i t h L e t o( o(
3.1,
u s i n g dependent c h o i c e . 0
NEIGHBORHOOD
Two elements
x
5
w
be an o r d i n a l . I f o(
A
a r e s a i d t o be
0-neighbors (mod A)
i s f i n i t e . The e q u i v a l e n c e c l a s s e s of
. These
a r e i n t e r v a l s which a r e e i t h e r
o r i t s converse, o r isomorphic w i t h i s a successor, t h e n
x
and y
Z
.
a r e s a i d t o be
-neighbors i f f t h e r e e x i s t o n l y f i n i t e l y many ( W -1)-neighborhoods between and y . I f o( i s a l i m i t o r d i n a l , t h e n x and y a r e s a i d t o be o( -neigh-
b o r s i f t h e r e e x i s t s an o r d i n a l
fi <
d
f o r which t h e y a r e
/3-neighbors.
The e q u i v a l e n c e c l a s s e s f o r t h i s e q u i v a l e n c e r e l a t i o n , a r e i n t e r v a l s and a r e called
o( -neighborhoods.
2.3. STRONGLY SCATTERED CHAIN A i s s t r o n g l y s c a t t e r e d i f t h e r e e x i s t s no dense s e t o f
We say t h a t a c h a i n
THEORY OF RELATIONS
162
.
non-empty mutually d i sj o i n t intervals of A Here, i f I and J are two disjoint in te r v a l s, then we write I < J i f every element of I i s le ss t h a n (mod A ) every element of J ; t h i s yields the notion of a dense chain of inte rva ls. I f A i s strongly scattered, then no dense chain i s embeddable in A Hence every strongly scattered chain i s scattered. Every scattered chain i s strongly s cat t er ed , then b o t h notions are equivalent, modul o the axiom of dependent choice (ZF suffices f o r a denumerable chain).
.
0 Suppose t h a t
A i s not strongly scattered, and take a dense s e t of mutually d is j o in t intervals. By ch.5 5 3.1 (using dependent choice), there e x i s t s a subset formed of intervals which constitutes a chain isomorphic with Q Then take an element i n each o f these intervals (countable axiom of choice): t h i s yields a restr ic tio n of A isomorphic with Q 0
.
.
2 . 4 . NEIGHBORHOOD RANK Given a strongly scattered chain, there ex i s t s an ordinal o( such tha t from t h a t ordinal on, a l l elements of the base are <-neighbors. Moreover for a l l i 4 j ( j < 4 ) , there e x i s t j-neighbors which ar e n o t i-neighbors. 0 The i-neighborhoods form a non-dense chain; hence there e x i s t two consecutive i-neighborhoods. The ordinal LX thus defined i s called the neiqhborhood rank of the chain. I n particular t h i s rank i s 0 i f f the chain i s f i n i t e or isomorphic with e ithe r w or i t s converse or Z
.
If A i s a strongly scattered chain and i f B i s embeddable in A , then B j s strongly scattered and the neighborhood rank of B is-at__ most e q u a l t o the neighborhood rank of A If B i s equimorphic with A , then the neighborhood ranks a re equal. The converse of A has the same neighborhood rank as A . 2.5. Let A be a strongly scattered chain having a minimum element, and l e t o( be the neighborhood rank of A (1) If c% = 0 , then A i s f i n i t e or isomorphic with W . If o( 3 1 , then e i t h e r o( i s a successor ordinal and A i s a f i n i t e sum or the sum of an o -sequence o f chains w i t h neighborhood ranks < o( Or cd i s a l i m i t ordinal; l e t t i n g $ = Cof o( , then A i s the sum of a quence of chains with neighborhood ranks < o( . ( 2 ) If 4 = 0 , then every proper i n i t i a l interval o f A i s f i n i t e ; otherwise, of chains h -neighborhood ranks every proper i n i t i a l interval i s a-f--i n i t e sum -- -_ - _ __ w i t - - __ _ ._ . < d _____g
.
.
.-
.
f-x-
( 3 ) Tf. Cof o( i s i n f i n i t e , then Cof A = Cofw ( t h i s conclusion ( 3 ) uses axiom of choice) . We have the analogous statement i f A has a maximum element: replace i n i t i a l
163
Chapter 6
i n t e r v a l by " f i n a l i n t e r v a l o f (1) Let
a
A
"
and c o f i n a l i t y by " c o - i n i t i a l i t y
be t h e minimum element o f
A
. I f IX
of
A
"
.
i s a successor o r d i n a l , then
A i s a f i n i t e sum o r t h e sum o f an cd-sequence o f ( & -1)-neighborhoods. Ifo( i s a l i m i t o r d i n a l , l e t (3 (i < = Cof a( ) be a ?f-sequence of strictly i n c r e a s i n g o r d i n a l s whose supremum i s OC The base 1 A l i s t h e n t h e u n i o n o f t h e these neighborhoods: t h e c h a i n A i s t h e sum fSi-neighborhoods o f a . C a l l Ai o f t h e d i f f e r e n c e i n t e r v a l s Ai+l-Ai .
.
( 2 ) Immediate consequence o f ( 1 ) . ( 3 ) Using t h e axiom o f choice, t a k e an element i n each d i f f e r e n c e
.
Ai+l-Ai
2.6. Every s t r o n g l y s c a t t e r e d c h a i n i s e i t h e r t h e sum o f two s t r i c t l y l e s s e r chains ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) ; or t h e sum o f s t r i c t l y l e s s e r c h a i n s a l o n g an which i s a r e g u l a r aleph, o r t h e sum o f s t r i c t l y l e s s e r c h a i n s a l o n g
ordinal
t h e converse o f a r e g u l a r a l e p h . Either
A
t h e base,
i s t h e sum o f two s t r i c t l y l e s s e r c h a i n s . O r , t a k i n g an element
A
i s equimorphic w i t h t h e f i n a l i n t e r v a l w i t h minimum
i n i t i a l i n t e r v a l w i t h maximum a
. Now a p p l y
a
a
in
, o r with the
t h e p r e c e d i n g p r o p o s i t i o n , and n o t e
t h a t , i n t h e case o f a f i n i t e sum o f i n t e r v a l s w i t h neighborhood ranks s t r i c t l y less than t h a t o f
A , we can s t i l l decompose A . 0
A
i n t o two i n t e r v a l s , n e i t h e r o f
which admits an embedding o f
We s h a l l g i v e a n o t h e r p r o o f , which does n o t use neighborhood ranks and assumes t h a t
A i s s c a t t e r e d , y e t n o t n e c e s s a r i l y s t r o n g l y s c a t t e r e d . T h i s p r o o f uses t h e axiom o f choice, which however i s t o o much: i n d e e d dependent c h o i c e s u f f i c e s t o i d e n t i f y scattered w i t h s t r o n g l y scattered chains. 0
Suppose t h a t
i s n o t decomposable i n t o a sum o f two s t r i c t l y l e s s e r chains. We
A
t h e n say t h a t a c u t i n a right cut i f
A
i s a l e f t cut i f
A
i s embeddable above t h a t c u t , and
i s embeddable b e f o r e i t . Every c u t i s e i t h e r l e f t o r r i g h t . No
A
c u t i s b o t h a l e f t and r i g h t c u t , f o r o t h e r w i s e and t h u s t h e c h a i n
Q
A+A
would be embeddable i n
5
o f r a t i o n a l s would be embeddable i n A : see ch.5
A
3.3.
Every l e f t c u t i s s i t u a t e d b e f o r e e v e r y r i g h t c u t . There e x i s t s e i t h e r a maximum
5
l e f t c u t o r a minimum r i g h t c u t : see Dedekind ch.4 decomposition o f A =
B and
C
A
1.1. L e t
A = B+C
d e f i n e d by t h i s c u t . F i x i d e a s by assuming
i s empty; f o r o t h e r w i s e , embedding
B+C
in
A 6
B
be t h e
.
Then
B , we would o b t a i n
a r i g h t c u t b e f o r e o u r Dedekind c u t : c o n t r a d i c t i o n . F i n a l l y every c u t other than
(A, empty s e t )
i s a l e f t c u t . Take a s t r i c t l y i n -
c r e a s i n g o r d i n a l sequence which i s c o f i n a l i n A Cof A
(axiom o f choice, ch.2
5
5.4). Then
and has minimum l e n g t h equal t o
A becomes t h e sum o f an o r d i n a l se-
quence o f i n t e r v a l s , each o f which i s s t r i c t l y l e s s
than
A
.0
164
2.7.
THEORY OF RELATIONS INDUCTION SCHEME FOR STRONGLY SCATTERED CHAINS (HAUSDDRFF 1914)
Ip
Suppose t h a t a c o n d i t i o n
holds f o r the empty chain and f o r every s i n g l e t o n holds f o r each chain
chain. I f , supposing t h a t
Ai
(i
<8
= an i n t e g e r o r an
6'
along and along i n f i n i t e r e g u l a r aleph), then f holds f o r the sum o f the Ai holds f o r every s t r o n g l y s c a t t e r e d chain. the converse o f t h i s o r d i n a l . Then Consequence o f 2.5,
by i n d u c t i o n on the neighborhood rank.
5 3 - INDECOMPOSABLE CHAIN, RIGHT OR LEFT INDECOMPOSABLE We say t h a t a chain
A
CHAIN
i s indecomposable i f f f o r every decomposition
A = A'+A"
A" a f i n a l i n t e r v a l , A i s embeddable e i t h e r i n A ' o r A" ; so t h a t A i s equimorphic w i t h e i t h e r A ' o r A" A chain i s , , s a i d t o be decom osable o t h e p i i s e . Some authors say " a d d i t i v e l y i n & composable or "additi&sable where
A ' i s an i n i t i a l i n t e r v a l and
.
.
3.1. RIGHT OR LEFT INDECOMPOSABLE CHAIN, STRICTLY OR NOT We say t h a t
A
i s r i g h t indecomposable i f
f i n a l i n t e r v a l . We say t h a t decomposition of
A
A
i s embeddable i n every non-empty
i s s t r i c t l y r i g h t indecomposable i f , f o r every
A
i n t o an i n i t i a l and a non-empty f i n a l i n t e r v a l ,
A
i s embed-
dable i n the f i n a l i n t e r v a l b u t n o t i n the i n i t i a l i n t e r v a l . Analogously, we define l e f t indecomposable and s t r i c t l y l e f t i n d e c o m p o s a B chain. Every r i g h t indecomposable o r l e f t indecomposable chain i s indecomposable. However, s t a r t i n g w i t h the chain
Q
o f r a t i o n a l s , the sum
1 + Q
t
1 i s indecom-
posable y e t n e i t h e r r i g h t n o r l e f t indecomposable. There e x i s t s , though, an i n t e r val of t h i s sum, namely Q , which i s equimorphic w i t h the e n t i r e chain and which i s both r i g h t and l e f t indecomposable: see 3.3 below.
(1) If A
i s an i n f i n i t e chain, both r i g h t and l e f t indecomposable, then
equimorphic t o (2) If A
A.2 = A+A
A
&
.
i s i n f i n i t e and equimorphic w i t h
r a t i o n a l s i s embeddable i n A
A.2
, then the chain Q o f the ZF s u f f i c e s i f A i s
(uses dependent choice,
5 3.3. i s s t r i c t l y ____-r i g h t inde(3) Every non-empty, scattered, r i g h t indecomposable composable; s i m i l a r l y w i t h " l e f t " (uses dependent choice; consequence o f (1), ( 2 ) ) . denumerable). This i s another form o f ch.5
However
Q.0
i s s t r i c t l y r i g h t indecomposable y e t n o t scattered.
3.2. THEOREM OF THE MAXIMUM RIGHT INDECOMPOSABLE INITIAL INTERVAL (1) I f A
i s a s t r i c t l y r i g h t indecomposable chain, then every chain which
i s equimorphic w i t h
A
i s s t r i c t l y r i g h t indecomposable; s i m i l a r l y w i t h " l e f t " .
B be equimorphic w i t h A . Decompose B = B'tB" w i t h 6" non-empty. Then A = A ' t A " w i t h A',< B' and A " 6 B" and A " non-empty, f o r otherwise A would be embeddable i n a proper i n i t i a l i n t e r v a l . Then B 6 A,< At',< B" and B 4 B'. Let
165
Chapter 6
( 2 ) Given a chain A
which i s n o t r i g h t indecomposable, t h e r e e x i s t s a decomposi-
and A'* A" (axiom o f choice). t i o n A = A'+A" w i t h A" non-empty Ai (i< 4) and note t h a t , i f 0 Set H = Cof A ; s t a r t w i t h a decomposition A i s non-cofinally embeddable i n any f i n a l our conclusion i s f a l s e , then each Ai interval of A . 0 ( 3 ) Every chain A admits a maximum ( w i t h respect t o i n c l u s i o n ) r i g h t indecompo-
-
-
sable i n i t i a l i n t e r v a l
=z
I
, possibly
empty. Moreover i f
I# 0
and A = I+B
,
.
then B i s t h e maximum f i n a l i n t e r v a l s t r i c t l y l e s s than A BY DEDEKIND (ch.4 tj 1.11, there e x i s t s a maximum c u t ( 1 8 ) of A such t h a t A i s embeddable i n each f i n a l i n t e r v a l which i s s t r k t l y l a r g e r than B I f I # 0 I L e t X < I : then X + B C A , since an embedding from then n e c e s s a r i l y B < A
.
.
i n t o X+B would map I i n t o X . F i n a l l y l e t J be a f i n a l i n t e r v a l o f I : < I and J non-empty, then J+B 5 A and J+B < A : c o n t r a d i c t i o n proving t h a t I i s r i g h t indecomposable. Moreover I i s maximum since B < A 0 I+B
if J
.
UNIQUE DECOMPOSITION THEOREM FOR AN INDECOMPOSABLE CHAIN (HAGENDORF 1977) L e t A be an indecomposable chain. Then e i t h e r A i s s t r i c t l y right,indecompo-sable, o r s t r i c t l y l e f t indecomposable, o r A i s uniquely decomposable i n t o 3.3.
a sum A = B+H+C -
where
B < A and C < ' A
and H i s equimorphic w i t h
A and
both r i g h t and l e f t indecomposable. These t h r e e cases a r e m u t u a l l y exclusive; i n t h e t h i r d case only, we have A equimorphic w i t h A.2
.
A i s a r i g h t c u t if A i s embeddable before i t b u t n o t i s embeddable a f t e r i t b u t n o t before; b i l a t e r a l i f A i s e w
0 We say t h a t a c u t i n
after; l e f t i f A
beddable both before and a f t e r . Suppose f i r s t t h a t there o n l y e x i s t r i g h t and l e f t cuts. By DEDEKIND (ch.4 cut. I n t h e f i r s t case, cuts, so A case.
5
l . l ) , t h e r e e x i s t s a maximum l e f t c u t o r a minimum r i g h t
i s embeddable i n a f i n a l i n t e r v a l which has o n l y r i g h t i s s t r i c t l y l e f t indecomposable. Analogous conclusion i n t h e second A
Now consider the case where t h e r e e x i s t b i l a t e r a l cuts. Then each b i l a t e r a l c u t has o t h e r b i l a t e r a l c u t s both before and a f t e r it. Hence t h e r e can be n e i t h e r a minimum n o r a maximum b i l a t e r a l c u t . Thus t h e r e e x i s t s a maximum l e f t c u t ( o r no l e f t c u t ) and a minimum r i g h t c u t ( o r none o f them). So t h a t A i s decomposable i n t o a sum B + H + C , w i t h B < A and C C A and H equimorphic w i t h A ; t h e i n t e r v a l s B
and C may be empty. Every c u t i n H i s b i l a t e r a l (mod A),
.
so i s again b i l a -
t e r a l (mod H) Hence H i s both r i g h t and l e f t indecomposable. F i n a l l y t h e e x i s tence of b i l a t e r a l cuts gives A equimorphic w i t h A.2
.
The uniqueness of the decomposition f o l l o w s from t h e f a c t t h a t a lengthening o f B t o t h e r i g h t would g i v e B a A ; s i m i l a r l y f o r a lengthening o f C t o t h e l e f t ; a lengthening o f H t o t h e l e f t o r r i g h t would destroy i t s b i l a t e r a l indecomposability. 0 C o r o l l a r i e s . (1) For every indecomposable chain A
, there
e x i s t s an i n t e r v a l
which i s l e f t o r r i g h t indecomposable ( o r both), and equimorphic w i t h
A
.
THEORY OF RELATIONS
166 ( 2 ) If A
i s i n f i n i t e and equimorphic w i t h
A.2
, then
A
i s equimorphic w i t h
a chain both r i g h t and l e f t indecomposable.
(3) Every non-empty, s c a t t e r e d and indecomposable- chain i s s t r i c t l y r i g h t o r s t r i c t l y l e f t indecomposable: reinforcement o f 3.1.(3), uses dep. choice (JULLIEN 1969). 3.4. A chain A i s indecomposable i f f , f o r every X < 0 1 1 A , w i t h respect t o embeddability, we have X.2 < Lr I A ; HAGENDORF 1977 proves the s t r o n g e r r e s u l t : if A # 2
i t suffices t h a t
0 Suppose t h a t
A
X < or 1A y i e l d s
i s indecomposable and t h a t
Conversely, suppose t h a t
A
X.2
6 or
)A
X.2 > / A : then
i s decomposable, so
( t h . 1.12). X>,
A
.
A = B + C
3
w i t h B < A and A = B+C so C & B ; s i m i l a r -
C C A , y e t A s a t i s f i e s our conclusion. Then 8.2 ly B$C Moreover (C+B) .2 = C+B+C+B 2 A so C+B >, A = B+C and thus B ,< C o r C & B : contradiction. 0 Notice t h a t the chain Z o f the i n t e g e r s i s decomposable, y e t v e r i f i e s the condi-
.
.
t i o n t h a t every X < Z y i e l d s X.2 < o r L 2 The p r o p o s i t i o n i n ch.5 5 4 about o r d i n a l s , does n o t extend t o chains. Indeed
A = w-. i s indecomposable and w < A b u t W.2 I A , and n o t c A . I f a chain A i s indecomposable, then every i n i t i a l i n t e r v a l X < A s a t i s f i e s 2 However, the r e t r o - o r d i n a l ( 0 + W ) - i s decomposable and every i n i t i a l X.2 < A S i m i l a r l y A = Q+ cJ1 i n t e r v a l X < A i s isomorphic w i t h 0 - hence X.2 < A i s decomposable and y e t v e r i f i e s our c o n d i t i o n . S i m i l a r l y A = ( w + 1 ) -
.
.
.
-
Problem. L e t A be a chain. I f every chain X < A y i e l d s X.2
-
5
4.6.
(1) .-I f A < A.2 and A+B> B + B 6 A+B ; then A.3 or AtBG B ; c o n t r a d i c t i o n . 0 3.5.
0 Suppose A.2
then
t
+
+
A.2 B S A.2
B >AtB B,(A+B
.
. Thus e i t h e r
A.Z,(A
(2) L e t A be a chain. I f every chain X < A s a t i s f i e s X + A S A ,.-then A is indecomposable. Moreover, there e x i s t s a r i g h t indecomposable chain equimorphic w i t h A (HAGENDORF 1976). 2 The converse i s f a l s e : the chain A = Z. (J i s r i g h t indecomposable w i t h w < A but
W2tA
3A.
0 We can suppose t h a t
A
i s i n f i n i t e . It s u f f i c e s t o prove t h a t
A
sable; t h e r e s t o f t h e conclusion f o l l o w s from 3.3,
since the case o f
l e f t indecomposable i s excluded: indeed 1 < A b l e i n one o f i t s proper f i n a l i n t e r v a l s .
1+A<
so
A
i s indecompo-
hence A
A
strictly i s embedda-
.
We argue ad absurdum, supposing t h a t A = B+C w i t h B < A and C < A Then by hypothesis BtAG A so 8.2 + C 6 B+C = A > C . By the preceding ( l ) ,we have 8.2 equimorphic w i t h B , so B i s indecomposable.
Chapter 6 Furthermore
C
<
A
167
i m p l i e s by hypothesis t h a t
.
C+A4 A
so
. Let
C+B+Cd B+C
.
us
embed C+B+C i n B+C Then e i t h e r C+B i s embedded i n B O r , since B+C i s not embeddable i n C , the o n l y remaining p o s s i b i l i t y i s t h a t C admits an em-
.
bedding o f C i t s e l f preceded by a f i n a l i n t e r v a l o f B which i s < B B u t B i s indecomposable, and so i s embeddable i n t h e complementary i n i t i a l i n t e r v a l . Hence i n a l l cases we have C+B* B and so A = B+C&B+C+B,( B . 2 4 B : contrad i c t i o n p r o v i n g t h a t A i s indecomposable. 0 (3) L e t A be a denumerable chain. I f every X < A which i s an i n i t i a l i n t e r v a l o , then A i s equimorphic w i t h Q (and indecomposable). -f A s a t i s f i e s X+A,(A 0
Repeat the preceding p r o o f t o o b t a i n
B.2
equimorphic w i t h
B
. Moreover
B
is
not empty since i t s complement C < A : so B i s n e c e s s a r i l y denumerable. Then apply 3.1.(2) (denumerable case where ZF s u f f i c e s ) . 0 This p r o p o s i t i o n no longer holds i n t h e uncountable case: take A = Q +
q.
3.6. L e t A
be an i n f i n i t e r i g h t indecomposable chain,
If C > A composable chain, and C an a r b i t r a r y chain. C 2 A+B C >/ B+A (JULLIEN 1969).
or
B
an i n f i n i t e l e f t inde--
-
and C 3 B
, then
either
onto a r e s t r i c t i o n o f C , and l e t CA be t h e C formed by those elements l e s s than (mod C) o r equal t o t h e images f x as x runs through I A I Then A i s embeddable i n every non-empty f i n a l i n t e r v a l o f CA S i m i l a r l y l e t CB be a f i n a l i n t e r v a l o f C such t h a t B i s embeddable i n every non-empty i n i t i a l i n t e r v a l o f CB Then e i t h e r CA and CB are d i s j o i n t , and hence C b A + B O r t h e r e e x i s t s an element u common t o the bases o f CA and CB Then B i s embeddable before u i n t h e i n t e r s e c t i o n CAnCB , and A i s embeddable a f t e r u : hence C>,B+A
0 Let
f
be an isomorphism o f A
i n i t i a l interval of
.
.
.
.
.
.
w + W , which i s embeddable n e i t h e r i n A = W - . W , i s however embeddable i n every common extension o f A a
3.7. The chain B = W
.
nor i n d B
(JULLIEN 1969). The chain A i s r i g h t indecomposable and B l e f t indecomposable. Thus i f a chain X 2 A and B , then e i t h e r X 3 A+B >/ k) + c3- o r X >/ B+A >/ &J +
0
3.8. Problem posed by HAGENDORF 1977.. L e t
A
W-.O
be a s t r i c t l y r i g h t indecomposable
chain. I f f o r every chain X < A , we have X + l d A , then i s A i t s e l f a w e l l ordering. A p o s i t i v e response f o r s c a t t e r e d chains i s due t o LARSON 1978. 3.9. L e t A be a s t r o n g l y s c a t t e r e d r i g h t indecomposable chain, and borhood rank.
o(
i t s neigh-
If OC. = 0 , then every proper i n i t i a l i n t e r v a l o f A i s f i n i t e . I f o( >, 1 , then every proper i n i t i a l i n t e r v a l i s a f i n i t e sum o f chains w i t h neighborhood ranks s t r i c t l y l e s s than g
.
168
THEORY OF RELATIONS
Consequently we find again 3 . 1 . ( 3 ) : A i s s t r i c t l y right indecomposable. Moreover in the case where o( & 1 : i f a proper i n i t i a l interval of A i s indecomposable, then i t has rank s t r i c t l y less than- cx . 0 If A has a minimum, then we are in the case of 2.5. Otherwise, take an element U of the base. Let B be the i n i t i a l interval of elements s t r i c t l y l e s s than u (mod A ) and C the final interval beginning with u . By hypothesis A i s equimorphic with C , so the neighborhood rank of C i s o( The i n i t i a l interval B i s embeddable in a proper i n i t i a l interval of C , hence by 2.5 i f o( = 0 then B i s f i n i t e , and i f o( >, 1 then B i s embeddable in a f i n i t e sum of chains with neighborhood ranks s t r i c t l y less t h a n o( . Thus B i t s e l f i s such a f i n i t e sum. 0 _ _ l l l _
.
5 4 - UNION
A N D I N T E R S E C T I O NOF INDECOMPOSABLE C H A I N S , C O V E R I N G BY INDECOMPOSABLE CHAINS OR BY DOUBLETS 4.1. Let A be a chain which i s the u_nion of an initicallngrval and a final interval, both having a---t l e a s t one common element and b o t h of which are right indecompoSame statement f o r " l e f t " . sable. Then A i s right indecomposable. -
-
~
Let B be the i n i t i a l i n t e r v a l , C the final interval and 0 t h e i r intersection. Then B i s embeddable in D . Either 0 = C so t h a t A = B i s right indecomposable. Or C has the form O+E and so A = B+E i s embeddable in C = D+E , hence A i s again right indecomposable. 0
0
4.2. Consider a chain which i s the union of a r i g h t indecomposable i n i t i a l intervalB and a l e f t indecomposable final interval C , b o t h i n f i n i t e and having a t l e a s t one common element. Then the intersection BnC i s both l e f t and r i g h t indecompo-.sable and admits an embedding of the chain Q of rationals (uses dependent choice; t h i s i s a strengthening of 3.1.( 1) and ( 2 ) ) . 4.3. COVERING BY RIGHT OR LEFT INDECOMPOSABLE CHAINS Let A be a chain. W e say t h a t two elements u , v of the base are equivalent with respect t o right indecomposable chains i f there e x i s t s an interval o f A which i s r i g h t indecomposable and contains the elements u and v . The condition thus defined i s reflexive and symmetric. Moreover by 4.1 i t i s t r a n s i t i v e . Analogously we define the equivalence relation with respect t o l e f t indecomposable chains; we c a l l these covering by r i g h t or l e f t indecomposable chains. There can be i n f i n i t e l y many equivalence classes of t h i s covering relation. For instance, take the converse 0 - of w : the equivalence classes f o r covering by right indecomposable chains are singletons. An equivalence class f o r covering by r i g h t indecomposable chains i s not necessar i l y a right indecomposable chain: take the chain Z of the rational integers.
Chapter 6
169
4.4. DOUBLET OF INDECOMPOSABLE CHAINS
A d o u b l e t i s a c h a i n which i s t h e u n i o n o f a l e f t indecomposable i n t e r v a l and a r i g h t indecomposable i n t e r v a l , b o t h h a v i n g a t l e a s t one common element. For example, t h e c h a i n
Z
o f the integers i s a doublet, being the union o f the
f i n a l i n t e r v a l w and t h e i n i t i a l i n t e r v a l d d - , which can be choosen t o have one o r s e v e r a l common elements. On t h e o t h e r hand, a l t h o u g h t h e p r o d u c t & - . W i s r i g h t indecomposable and i t s converse
c3. (.d -
t
W - . bJ
W. W -
i s l e f t indecomposable, t h e sum
i s n o t a d o u b l e t ; n o r even i s t h e sum
The l a t t e r example i s as w e l l a c h a i n which i s t h e u n i o n o f verse: i f we decompose we have
Z
i n t o CJ-
a sum i s o m o r p h i c w i t h
W-.
W.
Lc)-
and o and t h e n a t t a c h w -
i3
t
Z
t
Csr-. W
and i t s con-
W-. W
to
&-. W ,
, and s i m i l a r l y f o r t h e converse chains;
y e t t h e c o n s i d e r e d indecomposable chains a r e no l o n g e r i n t e r v a l s o f t h e f i n a l c o n s t r u c t e d sum. Every r i g h t o r l e f t indecomposable c h a i n i s a p a r t i c u l a r k i n d o f d o u b l e t , i n which one o f t h e i n t e r v a l s reduces t o a s i n g l e t o n . Note t h a t i t i s n o t r e q u i r e d t h a t o u r indecomoosable c h a i n s be i n i t i a l o r f i n a l i n t e r v a l s : one o f them may be a m i d d l e i n t e r v a l . I n t h e case o f a s c a t t e r e d c h a i n which i s a d o u b l e t , e i t h e r one o f t h e indecomposable c h a i n s i s a r e s t r i c t i o n o f t h e o t h e r , o r t h e l e f t indecomposable c h a i n i s an i n i t i a l i n t e r v a l and t h e r i g h t indecomposable c h a i n i s__ a f i n- a l i n t e r v a l o f t h e
_ _ _ I _ _
doublet. T h i s f o l l o w s f r o m 4.2 (uses dependent c h o i c e ) . On t h e o t h e r hand
1tQt1, where
Q
i s the chain o f r a t i o n a l s , i s a non-scattered
1tQ , a r i g h t indecomposable i n i t i a l i n t e r v a l , Qt1, a l e f t indecomposable f i n a l i n t e r v a l .
d o u b l e t h a v i n g a decomposition i n t o and
4.5. EQUIVALENCE RELATION FOR COVERING BY DOUBLETS Consider a s c a t t e r e d c h a i n and l e t u, v be two elements i n t h e base. We say t h a t u and v a r e e q u i v a l e n t w i t h r e s p e c t t o d o u b l e t s i f t h e r e e x i s t s a d o u b l e t o f indecomposable c h a i n s c o v e r i n g
u and v
. This
c o n d i t i o n i s r e f l e x i v e and sym-
m e t r i c . We s h a l l show t h a t i t i s t r a n s i t i v e (uses dependent choice); t h e conside r e d r e l a t i o n w i l l be c a l l e d c o v e r i n q by d o u b l e t s . 0 Take t h r e e elements
hand,
v
and w
u
<w
(modulo t h e c h a i n ) . I f
u and v , on t h e one
on t h e o t h e r hand, a r e covered by two r i g h t o r two l e f t inde-
composable chains, t h e n a p p l y 4.1.
I f u, v
a r e covered by a r i g h t indecomposable chain, n e c e s s a r i l y i n f i n i t e and
h a v i n g no maximum, and i f
v,
w a r e covered by a l e f t indecomposable chain,
i n f i n i t e and h a v i n g no minimum, t h e n e i t h e r one o f these indecomposable c h a i n s
u and w , o r t h e i r i n t e r s e c t i o n i s i n f i n i t e and b o t h r i g h t and l e f t indecomposable, hence admits an embedding o f Q (see 4.2, dependent c h o i c e ) :
covers
contradiction.
THEORY OF RELATIONS
170
v are covered by a doublet and v , w by a r ight indecomposable chain, then by 4.1 we are in the case of a doublet covering u and w . If u, v are covered by a doublet and v , w by a l e f t indecomposable chain, necessarily i n f i n i t e , and If
U,
i f the second term of the doublet i s i n f i n i t e , then we obtain a contradiction again using 4.2. If the second term reduces t o a singleton, then by 4 . 1 we have a l e f t indecomposable chain covering u and w . Now assume t h a t u, v are covered by a doublet formed of A , a l e f t indecomposable, and B , a right indecomposable chain; and similarly v , w are covered by a doublet formed of C , a l e f t indecomposable, and D , a right indecomposable chain. Then e i t h e r the cut situated t o the r i g h t of B f a l l s in or a f t e r 0 , and so by 4.1 we have a doublet covering u and w Or the cut situated t o the l e f t of C f a l l s in or before A , and we have the same conclusion. Or f i n a l l y Q i s embeddable i n the intersection of B and C : contradiction.
.
Transitivity no longer holds f o r a chain in which Q i s embeddable. 0 Take A = 1 + Q + W1. W and l e t u be the minimum of A ; l e t v be the minimum of the final interval W1. G ) ; l e t w be separated from v by the f i r s t component O 1 . Then u and v are covered by the doublet 1+Qt1 formed of a r i g h t indecomposable 1tQ and a l e f t indecomposable chain Qt1 , and v and w are covered by the indecomposable chain al. W . However no common doublet covers u and w , since 1tQt1 i s neither l e f t nor right indecomposable. 0 Even i f we would reinforce the definition of the doublet, by requiring t h a t the l e f t indecomposable chain always be an i n i t i a l i n t er v al , and the right indecomposable chain be a final interval of the doublet, both with a common element, then consider A = 1+Q+1and l e t u be the minimum, w the maximum of A , and v an element o f the middle interval Q Then u i s equivalent t o v by a right indecomposable chain, v equivalent t o w by a l e f t indecomposable chain, ye t u and w are no longer equivalent elements since 1+Q+1 i s no longer a doublet.
.
4.6. For a scattered chain, the equivalence relation of covering by doublets, i s the union of the equivalence relation f o r ,right indecomposable chains and the equivalence r e l a t i o n fo r l e f t indecomoosable chains. In other words, x and y are equivalent with respect t o doublets i f f there e xists a f i n i t e sequence of elements, from x until y , where two consecutive terms are equivalent with respect t o right o r l e f t indecomposable chains. I n ch.8 5 4 . 5 we shall see t h a t , f o r a scattered chain, there are only f i n i t e l y many equivalence classes with respect t o covering by doublets; moreover these classes are themselves doublets, possibly being right or l e f t indecomposable chains.
171
Chapter 6
s
5
-
HEREDITARILY
INDECOMPOSABLE CHAIN
Here i s f i r s t an i n t u i t i v e d e f i n i t i o n o f h e r e d i t a r i l y indecomposable chains, o r h-indecomposable c h a i n s . The empty chain, e v e r y s i n g l e t o n chain, e v e r y c h a i n isomorphic w i t h a r e g u l a r aleph, as w e l l as t h e converse o f such, i s h-indecomposable. Ai ( i < o( ) i , t h e s e t of i n d i c e s j (ic j C. c o f i n a l i n o( , t h e n t h e sum o f t h e
i n f i n i t e a l e p h and
If
o(
i s a regular
a r e h-indecomposable c h a i n s such t h a t , f o r each %
)
f o r which along
Ai
o(
Ai i s embeddable i n A is j o r a l o n g i t s converse, a r e
h-indecomposable. Moreover t h e o n l y h-indecomposable chains, a r e those chains which can be so c o n s t r u c t e d . The p r e c e d i n g " d e f i n i t i o n " i s easy t o use, and i n f a c t we s h a l l u s u a l l y t a k e i t . But i t i s i n c o r r e c t f r o m a l o g i c a l p o i n t o f view. I t i s a d e f i n i t i o n scheme, i n t h e same sense as t h e axiom schemes o f s e p a r a t i o n o r s u b s t i t u t i o n . Indeed t h e l a s t sentence o f t h e p r e c e d i n g paragraph ( t h e o n l y h-indecomposable chains.. .) c o u l d be e x p l i c i t e d t h u s l y : i f an a r b i t r a r y c o n d i t i o n \p h o l d s f o r e v e r y r e g u l a r aleph i s p r e s e r v e d i n t a k i n g any sum o f c h a i n s Ai along a and i t s converse, i f Ai be embeddable i n A for j h o l d s f o r e v e r y h-indecomposable chain.
r e g u l a r a l e p h o r i t s converse, assuming t h a t each a cofinal s e t o f indices
j ; then
f
T h i s t y p e o f procedure i s a d m i s s i b l e as an axiom scheme, which r e p r e s e n t s i n f i n i t e l y many axioms, each f o r a g i v e n c o n d i t i o n
. Obviously t h i s
i s no more admissi-
b l e as a d e f i n i t i o n , which must be of f i n i t e l e-. ngth. ____--.__ We propose t h e f o l l o w i n g f i n i t e (and a d m i s s i b l e ) d e f i n i t i o n , which i s e q u i v a l e n t t o t h e p r e c e d i n g scheme and so w i l l p e r m i t us t o use t h e scheme, i n p r a c t i c e .
. Suppose t h a t I c o n t a i n s A . Suppose t h a t , g i v e n a r e g u l a r i n f i n i t e a l e p h o( , i f I c o n t a i n s t h e i n t e r v a l s Ai (i < W ) which a r e m u t u a l l y d i s j o i n t and s i t u a t e d i n A w i t h o u t i n t e r m e d i a t e i n t e r v a l s , i n t h e o r d e r o f i n c r e a s i n g i o r o f decreasing i , and such t h a t f o r each i t h e s e t o f i n d i c e s j f o r which Ai i s embeddable i n A . i s c o f i n a l i n o( , J t h e n I c o n t a i n s t h e u n i o n i n t e r v a l o f t h e Ails (which i s an i n t e r v a l of A ) . Let
be a c h a i n and
A
I a set o f intervals o f
A
t h e empty i n t e r v a l and e v e r y s i n g l e t o n i n t e r v a l i n
Now we say t h a t
A
i s h-indecomposable,
o r h e r e d i t a r i l y indecomposable, i f t h e
i t s e l f belongs t o any s e t I s a t i s f y i n g t h e p r e c e d i n g . 2 F o r example &, & , 0 . W - , Z . id and t h e i r converses, a r e h-indecompochain
A
sable. F o r e v e r y h-indecomposable c h a i n , i t s converse i s h-indecomposable.
, such t h a t A i s t h e sum o f t h e Ai a l o n g d i n t h e above c o n s t r u c t i o n , i s t h e c o f i n a l i t y o f A ; t h e c o - i n i t i a l i t y i n t h e case of a sum a l o n g t h e converse o f o(
Note t h a t t h e r e g u l a r a l e p h o(
.
THEORY OF RELATIONS
172
5.1. Every h-indecomposable chain i s r i g h t o r l e f t indecomposable, according t o whether i t i s a sum along a r e g u l a r aleph o r along i t s converse. 5.2. An o r d i n a l i s h-indecomposable i f f i t i s indecomposable, i n the sense o f ch.1 _______..~ 3.6. Proof by i n d u c t i o n on the o r d i n a l i t s e l f ( s t a r t w i t h a c o f i n a l r e s t r i c t i o n ~
5
o f minimum length, i . e . isomorphic w i t h the c o f i n a l i t y ) . 5.3. Every h-indecomposable chain i s s t r o n g l y scattered ____ (see d e f i n i t i o n i n 2.3). A i n t o the i n t e r v a l s
Moreover, i n the decomposition o f anh-indecomposable chain
,each
has a neighborhood rank s t r i c t l y l e s s than the rank < Cof A) Ai the c o n d i t i o n A Use f o r instance the d e f i n i t i o n scheme, t a k i n g f o r " i s s t r o n g l y scattered" . F i n a l l y use 3.9. (i
Ai
of
.
We s h a l l see l a t e r t h a t the h-indecomposable chains are e x a c t l y the s c a t t e r e d indecomposable chains: see below, ch.8 5 4.3 using the axiom o f choice. Recall t h a t , modulo the axiom o f dependent choice, t h e s t r o n g l y s c a t t e r e d chains are e x a c t l y the s c a t t e r e d ones ( 2.3 above). 5.4. THEOREMS ON HEREDITARILY INDECOMPOSABLE CHAINS L e t A be a s c a t t e r e d chain. Suppose t h a t t h e h-indecomposable r e s t r i c t i o n s f
A form a w e l l quasi -ordering w i t h respect t o embeddability. Then A i s a f i n i t e sum o f h-indecomposable chains (LAVER 1968, uses dependent choice). For the w e l l pre-ordering o r w e l l quasi-ordering, 0 The chain
A
see ch.4
5
3.2
i s s t r o n g l y s c a t t e r e d (modulo the axiom o f dependent choice), hence
has a neighborhood rank. Suppose t h e statement i s f a l s e , and l e t o( be the l e a s t o r d i n a l f o r which t h e r e e x i s t s an
A
o f neighborhood rank o(
, whose h-indecompo-
sable r e s t r i c t i o n s f o r m a w e l l q a s i - o r d e r i n g w i t h respect t o embeddability, y e t i s n o t a f i n i t e sum o f h-indecomposable chains. We can always assume t h a t a minimum o r a maximum element. Indeed, take an element place
A
by the i n i t i a l o r f i n a l i n t e r v a l w i t h endpoint
rank remains a t most equal t o a well
A
has
o f t h e base, and rea : the neighborhood
H and the h-indecomposable r e s t r i c t i o n s s t i l l form
quasi-ordering.
To f i x ideas, suppose t h a t A
a
A
A
has a minimum element, and apply 2.5. Then e i t h e r
.
i s a f i n i t e sum o f chains w i t h neighborhood ranks s t r i c t l y l e s s than o(
In
which case, by hypothesis, each o f these chains i s a f i n i t e sum o f h-indecomposa-
A i s as w e l l . O r there e x i s t s a r e g u l a r i n f i n i t e aleph 8 such t h a t A i s t h e sum o f a 8-sequence of chains Ai ( i < )' ) w i t h neighi s a f i n i t e sum o f borhood ranks < o( . I n which case, by hypothesis, each Ai h-indecomposable chains: from t h i s p o i n t on, l e t the designate these chains Ai themselves We s h a l l prove t h a t t h e r e e x i s t s a f i n a l i n t e r v a l o f which y i e l d s an h-indecomb l e chains, so t h a t
.
posable f i n a l i n t e r v a l on
A : the induced i n t e r v a l being the sum o f the
Ai
173
Chapter 6
i belong t o t h e f i n a l i n t e r v a l o f
whose i n d i c e s 2.5.(2)
8 . This w i l l
s u f f i c e , s i n c e by
t h e complementary i n i t i a l i n t e r v a l i s a f i n i t e sum o f c h a i n s w i t h neighbor-
hood ranks s t r i c t l y l e s s t h a n d ; hence by h y p o t h e s i s , a f i n i t e sum o f h-indecom-
i , the s e t ) f o r which Ai i s embeddable i n A i s cofinal i n g j Suppose t h e c o n t r a r y . There e x i s t s an i n d e x i ( 0 ) and an i ' ( 0 ) 7 i ( 0 ) such t h a t f o r a l l i 3 i ' ( 0 ) , t h e c h a i n A i(o) i s n o t embeddable i n Ai . Then t h e r e e x i s t s
posable c h a i n s . Thus we s h a l l prove t h a t f r o m some i n d e x on, f o r each j ( i
of
<
j
%
r
.
an i(1) 7 i ' ( 0 ) and an i ' ( l )i(1) > such t h a t f o r a l l i >/ i ' ( 1 ) , n e i t h e r Ai(l) n o r , by t h e preceding, A , i s embeddable i n Ai I t e r a t i n g t h i s , we i( 0 ) have an w -sequence o f c h a i n s which i s bad w i t h r e s p e c t t o e m b e d d a b i l i t y . Hence t h e chains Ai do n o t c o n s t i t u t e a w e l f i u a s i - o r d e r i n g : see ch.4 5 3.2; c o n t r a -
.
diction. 0 5.5. L e t
A
be a s c a t t e r e d indecomposable c h a i n . Suppose t h a t t h e h-indecomposable
restrictions o f
0
A
f o r m a w e l l q u a s i q r d e r i n g w i t h r e s p e c t t o e m b e d d a b i l i t y . Then
i s h-indecomposable (LAVER 1968, uses axiom o f c h o i c e ) .
A
A
By 3.3 c o r o l l a r y ( 3 ) , t h e c h a i n
i s s t r i c t l y r i g h t o r l e f t indecomposable; t o
f i x ideas, suppose i t i s s t r i c t l y r i g h t indecomposable. Take a c o f i n a l r e s t r i c t i o n
of
A which i s isomorphic t o t h e r e g u l a r i n f i n i t e a l e p h
Cof A
(axiom o f choice).
L e t o( denote t h i s r e s t r i c t i o n , which we i d e n t i f y w i t h t h e o r d i n a l
i CIA l e t Ai be t h e i n t e r v a l o f and t h e element i+l( e x c l u s i v e ) . Thus A each
Bi ( i
D e f i n e as f o l l o w s t h e i n t e r v a l s o f the
, and
Bi
r e q u i r e t h a t no
Bi
Bi i s
a final interval o f =
. Let
A.
i s d i f f e r e n t from
A'
i s t h e sum o f t h e
B
i s embeddable i n e v e r y
suppose t h a t we have d e f i n e d t h e Then
between t h e element
A
Ai
Cof A . F o r i (inclusive) along & .
< 4 ) , so t h a t A i s t h e sum a l o n g u
every
e m b e d d a b i l i t y . L e t Bo
A
A
j that
, so
with Bi<
j> i
A
. Moreover we
w i t h respect t o
u be an o r d i n a l s t r i c t l y l e s s t h a n 6( , and < u) . L e t A ' be t h e sum o f these Bi
.
Bi ( i
, s i n c e o t h e r w i s e e i t h e r u would be c o f i n a l i n
, o r u would have an immediate predecessor u-1 w i t h Bu-l a final interv a l o f A , t h u s equimorphic w i t h A , c o n t r a d i c t i n g o u r hypotheses. o(
Let
A"
be t h e f i n a l i n t e r v a l o f
A
, complement
of
A"
.
of
A'
.
Since
A
i s r i g h t inde-
A" i s equimorphic w i t h A S e t Bu t o be a p r o p e r i n i t i a l i n t e r v a l ( s o Bu< A) , i n which Buml i s embeddable ( i n t h e case t h a t u i s a
composable,
successor o r d i n a l ) , o r i n which t h e sum o f t h e case t h a t
Bi
( i < u) i s
embeddable ( i n t h e
u i s a l i m i t o r d i n a l ) . F i n a l l y , e n l a r g e t h i s i n t e r v a l Bu , i f necessa-
ry, so t h a t t h e sum o f t h e
Bi
t h u s we a r e e n s u r e d t o e x h a u s t By t h e p r e c e d i n g 5.4, each we a g a i n d e s i g n a t e by
Bi
.
(i u) A
i n c l u d e s t h e corresponding sum o f t h e
by t h e sum o f a l l t h e
Bi (i< 4 )
Ai
.
Bi i s a f i n i t e sum o f h-indecomposable chains, which F o r each o r d e r e d p a i r ( i , j ) w i t h i< j < 4 , the
:
THEORY OF RELATIONS
174
chain
i s indecomposable and i s embeddable i n a f i n i t e sum o f t h e f o r m
Bi
B . + Bj+l + ... , hence i n a c h a i n B o f i n d e x g r e a t e r t h a n o r equal t o J Thus t h e c h a i n A i s h-indecomposable. 0
- INDIVISIBLE
6
§
R E L A T I O N OR C H A I N
We say t h a t a r e l a t i o n partition of or
R
6 R/D .
E
.
j
R w i t h i n f i n i t e base
E
i n t o two complementary subsets
i s indivisible i f C
and
f o r every
, either
D = E-C
R Q R/C
Some a u t h o r s say "indecomposable" i n s t e a d o f " i n d i v i s i b l e " .
F o r example, t h e c h a i n G, o f n a t u r a l i n t e g e r s i s i n d i v i s i b l e . Every i n d i v i s i b l e c h a i n i s a f o r t i o r i indecomposable. However t h e c h a i n where
,
2. W
i s t h e c h a i n o f i n t e g e r s , i s indecomposable y e t n o t i n d i v i s i b l e : i f we
Z
partition
2
and W - . Lc:
i n t o LL: and i t s converse
, neither
w - , t h e n we have t h e r e s t r i c t i o n s
o f which admits an embeddinq o f
Lc)
2
.
Z.
6.1. Every r e g u l a r a l e p h i s i n d i v i s i b l e . If
0
I
i s a r e g u l a r a l e p h and we p a r t i t i o n
t h e n one o f these subsets must be c o f i n a l i n
.0
I
isomorphic w i t h
I i n t o two complementary subsets, I , and t h u s y i e l d s a r e s t r i c t i o n
6 . 2 . F o r an o r d i n a l , i n d i v i s i b i l i t y i s i d e n t i c a l w i t h i n d e c o m p o s a b i l i t y . 0 I t s u f f i c e s t o show t h a t e v e r y indecomposable o r d i n a l i s i n d i v i s i b l e , hence
w , say W U , i s i n d i v i s i b l e ( i n view o f ch.1 9 3 . 6 ) .
t h a t e v e r y power o f
Proceed by i n d u c t i o n on t h e exponent ordinal:
+
u = v
1
.
Then
wU
u
.
Suppose f i r s t l y t h a t
u
i s a successor
i s a sum a l o n g dc:, o f i n d i v i s i b l e o r d i n a l s equal
t o civ . Hence a p a r t i t i o n o f t h e subsets as a sum a l o n g
uu i n t o two complementary subsets, g i v e s one o f w o f components w v .
Now suppose t h a t t h e o r d i n a l a sequence o f o r d i n a l s regular aleph
I
=
Cof u
u
i s a l i m i t ordinal; then
.
w' , t h e
t i o n o f t h e base y i e l d s , f o r each : c a l l t h i s the
, which
i s cofinal i n
i
,at
u n i o n o f which i s
w' .
~
~ i(s ~
The p a r t i -
l e a s t one r e s t r i c t i o n i s o m o r p h i c w i t h
I
. Thus
i-majority r e s t r i c t i o n f o r a set o f indices
we o b t a i n a t l e a s t a r e s t r i c t i o n o f
i s isomorphic w i t h t h e union, o r supremum o f t h e the given l i m i t ordinal
i runs t h r o u g h t h e
i - m a j o r i t y r e s t r i c t i o n . Then a t l e a s t one o f t h e two
subsets i n o u r p a r t i t i o n y i e l d s an i
i s t h e supremum o f
By t h e i n d u c t i o n h y p o t h e s i s , each power f
an i n d i v i s i b l e i n i t i a l i n t e r v a l o f
0 v(i)
u
v ( i ) ; we can suppose t h a t t h e i n d e x
Wu
.0
, hence
wu
which
isomorphic w i t h
)
Chapter 6
6.3. Given an i n d i v i s i b l e c h a i n tion o f A/D
A 0
w i t h i n f i n i t e base
A
i n t o two complementary subsets
E
175
and
C
E
, there
D = E-C
exists a parti-
ytJ i
and
A E A/C
(HAGENDORF 1975; uses axiom o f c h o i c e ) .
The i n d i v i s i b l e c h a i n
equimorphic w i t h
A
, and
A.2
i s a f o r t i o r i indecomposable. By 3.3, e i t h e r so t h e d e s i r e d p a r t i t i o n e x i s t s . O r
is
A
i s strictly
A
r i g h t o r l e f t indecomposable: t o f i x ideas, suppose t h a t i t i s r i g h t indecomposable. Call along
I t h e r e g u l a r aleph, c o f i n a l i t y o f
I o f intervals
With t h e i n t e r v a l
Ai
, associate
A.
A ; and decompose
( i running through
i n t o a sum
A
I ; axiom o f c h o i c e ) . A
t h e complementary f i n a l i n t e r v a l , i n which
i s embeddable, and i n t h i s f i n a l i n t e r v a l t a k e a p r o p e r i n i t i a l i n t e r v a l
A6 = A.
which
A.
i s embeddable; t h e n c a l l
A;
After
A!
i t remains a f i n a l i n t e r v a l equimorphic w i t h
and
t h e two c o p i e s o f
A6
in
.
A.
A : i t e r a t e t h i s by embedA i and A: o f A1 , t h e b e i n g embeddable i n each copy and t h e r e m a i n i n g f i n a l i n t e r v a l b e i n g
ding, i n t h i s f i n a l i n t e r v a l , two c o n s e c u t i v e c o p i e s i n t e r v a l A1 equimorphic w i t h to
I
.
.
A
T h i s procedure can be i t e r a t e d f o r e v e r y i n d e x
Indeed f o r each
ordinal index
i
, the
i belonging
i-sequence o f t h e sums
( j< i) o f c o p i e s cannot exhaust A ; s i n c e o t h e r w i s e e i t h e r i would J J be c o f i n a l i n I ( c o n t r a d i c t i n g t h e r e g u l a r i t y o f I ) , o r i would have an imme-
A! + A'!
d i a t e predecessor
i-1 with
A;-1
AS-1
or
a final interval o f
A
, which
is
excluded. F i n a l l y , the union o f the
A;
and t h e u n i o n o f t h e
A;
for all
i in
I
,yield
the desired p a r t i t i o n . 0 It i s proved by POUZET 1977 ( u n p u b l i s h e d ) t h a t , g i v e n an i n d i v i s i b l e r e l a t i o n
R
E , t h e r e e x i s t s a p a r t i t i o n o f E i n t o two complementary D w i t h R equimorphic t o R/C and t o R/D .
w i t h denumerable base subsets
C
and
Problem: t h e case o f an uncountable
indivisible relation.
EXERCISE 1 - Given a denumerable c h a i n
A
onto a proper r e s t r i c t i o n o f Either the chain in
Q
, the
Q
, there
e x i s t s an isomorphism o f
A
A
o f r a t i o n a l s i s embeddable i n
statement i s obvious. O r
A
A , and as
A
i s embeddable
i s s c a t t e r e d , and even s t r o n g l y s c a t t e r e d
(see 2.3, denumerable case), so t h a t t h e r e e x i s t two c o n s e c u t i v e 0-neighborhoods
A . The f i r s t o f these two c o n s e c u t i v e 0-neighborhoods admits an embedding o f o r t h e second admits an embedding o f w - . P o s s i b l y A reduces t o a unique 0-neighborhood, i s o m o r p h i c w i t h k, o r i t s converse o r Z . I n a l l cases t h e s t a t e of
w
176
THEORY OF RELATIONS
-
EXERCISE 2 1 - Let
IMMEDIATE EXTENSIONS OF A SCATTERED C H A I N be a s c a t t e r e d , r i g h t indecomposable c h a i n . Then
A
< A t i < ...
proper i n i t i a l i n t e r v a l , thus dent c h o i c e i n o r d e r t h a t
.
v
of
A+1
, then
A
each
A+i
has neighborhood
A+l
% = /j+l i s a successor o r d i n a l ; t h e n t a k e t h e l a s t / 3 - n e i g h b o r h o o d o f
which c o n t a i n s
.
v
I f t h i s neighborhood reduces t o t h e s i n g l e t o n o f
n e x t t o l a s t 0 - n e i g h b o r h o o d , i n which
?(I
rank a t most equal t o
c o(
exists in a
A
v
o(
,
A+l
, take
i s embeddable. I n b o t h cases
: contradiction. O r
A
the
has
i s a l i m i t ordinal; then there
u and v a r e & - n e i g h b o r ,
such t h a t
1/ -neighborhood:
< ...
u be an element o f A : i t s u f f i c e s t o see t h a t t h e maxii s n o t an q - n e i g h b o r o f u . Assume t h e c o n t r a r y . E i t h e r
Indeed l e t
mum element
A+2
be s t r o n g l y s c a t t e r e d ) .
cx i s t h e neighborhood rank o f
Moreover i f rank o(+l
A
< A+1<
A
.
i Indeed A i s n o t embeddable i n any i s n o t embeddable i n A ( 3 . 1 . ( 3 ) u s i n g depen-
f o r each p o s i t i v e i n t e g e r
and
A
i s embeddable
contradiction.
A . A n t i c i p a t i n g on ch.8 g 4.5, decompose A i n t o a f i n i t e sum o f d i s j o i n t , indecomposable i n t e r v a l s . Each i n t e r v a l i s s t r i c t l y r i g h t o r s t r i c t l y l e f t indecomposable; moreover we can assume 2
-
More g e n e r a l l y t a k e an i n f i n i t e s c a t t e r e d c h a i n
t h a t no i n t e r v a l i s absorbed by any o f i t s two a d j a c e n t i n t e r v a l s . Then e i t h e r we have a t l e a s t one l e f t i n t e r v a l . t h e n d e f i n e
Ai
i elements j u s t
by adding
Ai by adding i elements a t 1, e a s i l y see t h a t A < A1 < A 2 < ...
b e f o r e t h e f i r s t l e f t i n t e r v a l . Otherwise d e f i n e t h e end o f EXERCISE 3
A
-
. Then
using the preceding
0
THEOREM OF THE INTERMEDIATE CHAIN BETWEEN A
Given an i n f i n i t e c h a i n
A
,e i t h e r
A.2
a t l e a s t one c h a i n which i s s t r i c t l y i n t e r m e d i a t e between
A.2
AND
i s equimorphic w i t h A
A
o r there exists
a&
A.2
with
r e s p e c t t o e m b e d d a b i l i t y (HAGENDORF 1984 ). Decompose
A
into
B
+
L
, where
indecomposable (see 3 . 2 . ( 3 ) ) .
If
L
i s t h e maximum f i n a l i n t e r v a l which i s l e f t
L = A
then e i t h e r
A.2
equimorphic w i t h
A
or
.
l + A i s s t r i c t l y i n t e r m e d i a t e between A and A.2 If L i s empty, t h e n each i n i t i a l i n t e r v a l o f A i s s t r i c t l y l e s s t h a n A , so t h a t A+1 i s s t r i c t l y i n t e r mediate. I n t h e f o l l o w i n g we assume t h a t L # A and L non-empty. Consider t h e sum A1 + A2 where A1 and A2 a r e b o t h isomorphic w i t h A , and decompose t h i s sum, o b t a i n i n g C + L' where L ' i s t h e maximum l e f t indecompoL ' i s s t r i c t l y g r e a t e r t h a n A2 : t h e n A.2 i s equimorA . O r L ' = L and C = A+B K ' A . 2 by 3.2.(3). E i t h e r C i s s t r i c t l y i n t e r m e d i a t e between A and A.2 . O r C i s equimorphic w i t h A Then e i t h e r L.2 i s equimorphic w i t h L : t h e n A = B+L E B + L.2 = A+L E C+L = A.2 Or f i n a l l y L i s s t r i c t l y l e f t indecomposable. Since A = B+L :B+L+B and B < A by 3.2.(3), t h e n L+B i s embeddable i n L , so t h a t 1 + B B L and B < L Thus L+B+L i s s t r i c t l y i n t e r m e d i a t e between A = B+L and A.2 = B+L+B+L . sable i n t e r v a l . E i t h e r phic with
.
.
.
CHAPTER
7
USE OF SCATTERED CHAINS FOR THE STUDY OF FINITELY FREE AND WELL PARTIAL ORDERINGS
§
1 - REINFORCEMENT
OF A PARTIAL
ORDERING
BY SCATTERED CHAINS
1.1. Let A be a p a r t i a l ordering; then t h e following four conditions a r e equivalent (BONNET, POUZET 1969; uses axiom of c h o i c e ) : (1) A i s f i n i t e l y f r e e and does not admit an embedding of t h e chain Q of rationals: ( 2 ) every t o t a l l y ordered reinforcement of A i s s c a t t e r e d ; ( 3 ) the chain Q i s not embeddable i n t h e p a r t i a l ordering of inclusion f o r t h e i n i t i a l i n t e r v a l s of A ; ( 4 ) f o r every function f mapping t h e r a t i o n a l s i n t o t h e base I A l , t h e r e e x i s t two r a t i o n a l s x , y with x < y (mod Q ) f x 2 f y (mod A) .
and
7
Condition ( 2 ) implies (1). Indeed i f Q i s embeddable in A , then Q i s embeddable i n every reinforcement of A . On t h e o t h e r hand, i f A has an i n f i n i t e , hence a denumerable f r e e s u b s e t , then take a chain B isomorphic w i t h Q on t h i s s u b s e t , and then by ch.2 5 4 . 1 take a p a r t i a l l y ordered reinforcement of A which extends B . By t h e reinforcement axiom i n ch.2 5 4.2, there exists a t o t a l l y ordered reinforcement of A i n which Q i s embeddable. 0
0
Conditions ( 2 ) and ( 3 ) a r e e q u i v a l e n t . First suppose t h a t t h e r e e x i s t s a t o t a l l y ordered reinforcement C of A i n which Q i s embeddable. Then t h e chain of inclusion f o r i n i t i a l i n t e r v a l s of C i s a r e s t r i c t i o n of t h e p a r t i a l ordering of i n c l u s i o n f o r i n i t i a l i n t e r v a l s of A , and admits an embedding of Q . Conversely, suppose t h a t Q i s embeddable i n t h e p a r t i a l ordering of inclusion f o r i n i t i a l i n t e r v a l s of A Take a chain U , i n which Q i s embeddable, and which i s maximal w i t h respect t o i n c l u s i o n , and formed of i n i t i a l i n t e r v a l s of A (see ch.2 5 2 . 7 , axiom of c h o i c e ) . Then U i s t h e chain of i n i t i a l i n t e r v a l s of a c e r t a i n t o t a l l y ordered reinforcement V of A : see ch.4 5 1 .3 . Moreover by
0
.
ch.6
5
1 . 5 , t h e chain
Q
i s embeddable i n V ( t h i s uses dependent c h o i c e ) .
0
Conditions ( 2 ) and ( 4 ) a r e e q u i v a l e n t . Suppose t h a t t h e r e e x i s t s a t o t a l l y ordered reinforcement of A in which Q i s embeddable. Then t h e r e e x i s t s a b i j e c t i o n f of Q onto a r e s t r i c t i o n of A , which maps any two r a t i o n a l s x < y (mod Q ) i n t o f x < o r I f y (mod A ) . Conversely, i f t h e r e e x i s t s such a ( n e c e s s a r i l y i n j e c t i v e ) f u n c t i o n , then t h e image f " ( Q ) i s a reinforcement of t h e r e s t r i c t i o n A / f " ( l Q l ) ( r e s t r i c t i o n of A
0
178
THEORY OF RELATIONS
t o the s e t of images of r a t i o n a l s ) . By ch.2 5 4.1 and the reinforcement axiom, there e x i s t s a t o t a l l y ordered reinforcement of A in which Q i s embeddable. 0 Condition (1) implies ( 4 ) . Suppose t h a t A i s f i n i t e l y free and t h a t Q i s not embeddable in A , y e t there e x i s t s a function f from Q into A , such t h a t f o r x < y (mod Q ) we have fx < or I fy (mod A ) : hence f i s i n j e c t i v e . Then the r e s t r i c t i o n A / f " ( \ Q I ) i s a denumerable f i n i t e l y f r e e p a r t i a l ordering in which Q i s n o t embeddable. By applying RAMSEY (ch.3 5 1 . 2 ) , there e x i s t s a denumerable, maximal, t o t a l l y ordered and scattered r e s t r i c t i o n of A/fo(lQl) . Thus there e x i s t two rationals xo and yo > x o (mod Q) with f ( y o ) an immediate successor of
0
f ( x o ) (mod A / f " ( I Q ( ) . I t e r a t e t h i s , by replacing Q by i t s r e s t r i c t i o n t o the open interval (xo,yo) . We obtain, f o r each integer i , an interval ( x i , y i ) with yi > x i (mod Q ) and each interval ( X ~ + ~ , Y ~ + properly ~ ) included in ( x i , y i ) (mod Q) . For each i we have f ( x i ) < f ( y i ) (mod A ) , without any element of f " ( l Q I ) between f ( x i ) and f ( y i ) (mod A ) : i t suffices t o consider a rational x and a l l possible positions of fx , e i t h e r f o r x between xi-l and yi-l (mod Q ) , o r f o r x elsewhere in the chain Q
.
Again using RAMSEY, we see t h a t there e x i s t denumerably many integers i with a l l values f ( x i ) and f ( y i ) mutually comparable (mod A) . Hence there e x i s t i and j 7 i with, f o r instance, f ( x . ) situated between f ( x i ) and f ( y i ) (mod A ) : J contradiction. 0
1 . 2 . Recall t h a t a partial ordering A i s said t o be s t r a t i f i e d (ch.2 5 5.2) i f incomparability (mod A ) together with i d e n t i t y , forms an equivalence relation. The equivalence classes form a chain, called the princioal chain. A s t r a t i f i e d p a r t i a l ordering i s said t o be scattered i f i t s principal chain i s scattered. Let A be a partial ordering. There e x i s t s an i n i t i a l interval of A which i s maximal with respect t o inclusion, among those i n i t i a l intervals having a scattered s t r a t i f i e d reinforcement. Analogous statement f o r final intervals (uses axiom of choice). Consider those i n i t i a l intervals of A having a scattered s t r a t i f i e d reinforcement. There e x i s t such, f o r example the empty interval. The s e t of these intervals i s ordered by inclusion; so l e t C be a maximal chain of inclusion (see ch.2 5 2 . 7 , axiom of choice). Let X be the i n i t i a l intervals of A which are elements of C , and l e t I be the i n i t i a l interval of A ' which i s the union of the X No i n i t i a l interval of A which i s a proper extension of I has a scattered s t r a t i f i e d rein-
0
.
Chapter 7
reinforcement ment
of
Y+
of
X+
i s s a i d t o be f i n e r t h a n a s c a t t e r e d s t r a t i f i e d r e i n f o r c e -
Y
, i f t h e base o f X i s i n c l u d e d i n t h e base o f
X
p a r a b i l i t y class o f principal chain o f
I f X, Y
179
i s an i n c o m p a r a b i l i t y c l a s s o f
X+
C
Y
, and each incom-
Y
f i n a l l y i f the
of
X+
which i s f i n e r t h a n of
.
, there
X X+
.
e x i s t s a scattered s t r a t i f i e d
Keep t h e i n c o m p a r a b i l i t y c l a s s e s o f
s t r a t i f i e d reinforcement
Z
by i t s i n t e r s e c t i o n w i t h
Y
.
Indeed, t a k e an a r b i t r a r y s c a t t e r e d
r e p l a c e each c l a s s o f
2
Y+
1x1 i s i n c l u d e d i n I Y J , t h e n f o r each
and i f t h e base
scattered s t r a t i f i e d reinforcement reinforcement o f
+ , and
i s an i n i t i a l i n t e r v a l o f t h e p r i n c i p a l c h a i n o f
X+
belong t o
Y
(IYl - 1x1)
X+
and
(provided t h i s i n -
t e r s e c t i o n i s non-empty). Then w h i l e p r e s e r v i n g t h e i r o r d e r , p l a c e these c l a s s e s coming f r o m ment o f
Z
IYI
t h e base
X
a f t e r a l l the classes o f
- 1x1 .
X+
.
This i s possible, since every ele-
i s g r e a t e r t h a n o r incomparable (mod A) w i t h e v e r y element o f
The s e t o f s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t s of t h e elements o f ordered by t h e comparison " f i n e r t h a n "
.
i s partially
C
Take a maximal c h a i n , a g a i n u s i n g ch.2
5 2.7, hence axiom o f c h o i c e . It s u f f i c e s t o p r o v e t h a t t h i s maximal c h a i n t e r m i nates w i t h a s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t o f I F o r t h i s , n o t e t h a t t h i s
.
maximal c h a i n cannot t e r m i n a t e w i t h a r e i n f o r c e m e n t from
I
.
Indeed, t a k e a
Y
in
C
,a
X+
o f an
proper extension o f
X
in
X
. The
graph shows t h e e x i s t e n c e o f a s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t o f
C
, distinct
p r e c e d i n g para-
Y
which i s
X+ : c o n t r a d i c t i o n . Furthermore, t h i s c h a i n n e c e s s a r i l y has a maximum
f i n e r than
element, which i s t h e r e i n f o r c e m e n t o f an element o f
C
. Indeed,
given a set
X o f C , w i t h o u t any X maximum, and t o t a l l y o r d e r e d by t h e comparison " f i n e r than", t a k e t h e i n i t i a l i n t e r v a l J formed 'by t h e u n i o n o f t h e X b e i n g considered, and t h e n t h e r e i n f o r c e m e n t o f J which i s t h e common e x t e n s i o n o f t h e X+ T h i s r e i n f o r c e m e n t i s f i n e r t h a n each X+ , and i s s c a t t e r e d , s i n c e e v e r y c h a i n which i s t h e u n i o n o f s c a t t e r e d p r o p e r composed of reinforcements
X+
o f elements
.
i n i t i a l i n t e r v a l s i s s c a t t e r e d : see ch.6 1.3. L e t
A
5
1.2.
be a p a r t i a l o r d e r i n g ; t h e n t h e f o l l o w i n g t h r e e c o n d i t i o n s a r e e q u i -
v a l e n t (BONNET, POUZET 1969; uses axiom o f c h o i c e ) : (1) t h e c h a i n
9
i s n o t embeddable i n
A ;
(2) there e x i s t s a scattered t o t a l l y ordered reinforcement o f A ; (3) t h e r e e x i s t s a s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t o f A . _ I _ .
C o n d i t i o n ( 2 ) o b v i o u s l y i m p l i e s (1) and ( 3 ) . C o n d i t i o n ( 3 ) i m p l i e s ( 2 ) . Indeed, s t a r t i n g w i t h a s t r a t i f i e d reinforcement o f
A
w i t h a s c a t t e r e d p r i n c i p a l chain, i t s u f f i c e s t o choose a w e l l - o r d e r i n g (hence a s c a t t e r e d c h a i n ) based on each i n c o m p a r a b i l i t y c l a s s o f t h e s t r a t i f i e d p a r t i a l o r d e r i n g (axiom o f choice); and t h e n t o t a k e t h e o r d i n a l sum o f these w e l l - o r d e r i n g s along t h e p r i n c i p a l c h a i n .
THEORY OF RELATIONS
180
We now show t h a t (1) i m p l i e s ( 3 ) . F o r t h i s , suppose t h a t e v e r y s t r a t i f i e d r e i n f o r c e ment o f
admits an embedding o f t h e c h a i n
A
dable i n A
. Applying
.
Q
t h e p r e c e d i n g 1.2, l e t
I
We s h a l l show t h a t
Q
i s embed-
A
be an i n i t i a l i n t e r v a l o f
which i s maximal w i t h r e s p e c t t o i n c l u s i o n , among those i n i t i a l i n t e r v a l s h a v i n g a scattered s t r a t i f i e d reinforcement. L e t ned. L e t
E be t h e base o f
J
be a f i n a l i n t e r v a l , s i m i l a r l y d e f i -
E - ( I I l u l J I ) i s non-empty. If o f I and J'
A ; the difference
Since i f n o t , t h e n t a k e s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t s of
J
. Take
I+
as new i n c o m p a r a b i l i t y c l a s s e s t h e c l a s s e s o f
f o l l o w e d by, i n
o r d e r , t h e c l a s s o f non-empty i n t e r s e c t i o n s o f t h e c l a s s e s o f This then y i e l d s a s t r a t i f i e d reinforcement o f sum o f t h e p r i n c i p a l c h a i n o f
I'
with
J+
E -
III .
whose p r i n c i p a l c h a i n i s t h e
A
and a r e s t r i c t i o n o f t h e p r i n c i p a l c h a i n o f
J',
hence a s c a t t e r e d chain: t h i s c o n t r a d i c t s o u r assumDtion.
u be an element i n t h e d i f f e r e n c e E - (111 u I J I )
Let
restriction of
t o those elements
A
> u , on
t o those elements
< u , on
of
We s h a l l p r o v e t h a t t h e
Q
t h e o t h e r hand, have o n l y r e i n f o r c e m e n t s i n which
i s embeddable. N o t i c e f i r s t t h a t t h e r e e x i s t elements to
.
t h e one hand, and t h e r e s t r i c t i o n
which do n o t belong
I11 . F o r i f n o t , t h e n I augmented by u o n l y , would be an i n i t i a l i n t e r v a l A and a p r o p e r e x t e n s i o n o f I , y e t i t has a s c a t t e r e d s t r a t i f i e d r e i n f o r c e -
I
ment, o b t a i n e d by t a k i n g a s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t o f by a l a s t c l a s s formed by t h e s i n g l e t o n o f
u
Suppose t h a t t h e i n i t i a l i n t e r v a l o f those elements f i e d r e i n f o r c e m e n t , say of
I
. Then
U
. Take
, completed
has a s c a t t e r e d s t r a t i -
again a scattered s t r a t i f i e d reinforcement
IIlulUl , take the incomparability classes o f
on t h e u n i o n
.
I
: t h i s contradicts maximality o f
I
I+
+ ,
i n t h e i r o r d e r , f o l l o w e d by, a g a i n i n t h e i r o r d e r , t h e non-empty i n t e r s e c t i o n s o f classes o f forcement o f
U
f o r the f i n a l interval Now l e t
uo
- ( l U l n 111) . We , contradicting
lUl
with
A/( IIIu I U l ) be
cements i n which
J
. Since
u Q
I
. Analogous
and t h e f i n a l i n t e r v a l o f those elements the i n i t i a l interval
< u
< u1
>u
proof
(mod A)
.
has o n l y s t r a t i f i e d r e i n f o r -
u l < uo such t h a t
i s embeddable, t h e r e e x i s t s an element
each o f t h e two i n t e r v a l s , t h o s e elements ween
thus obtain a scattered s t r a t i f i e d r e i n the maximality o f
and those elements s t r i c t l y b e t -
u1 and uo , has o n l y s t r a t i f i e d r e i n f o r c e m e n t s i n which Q i s embeddable.
S i m i l a r l y , t h e r e e x i s t s an element 'u2 > uo w i t h t h e same p r o p e r t y f o r t h e f i n a l u2 and t h e i n t e r v a l o f those elements s t r i c t l y between uo and u2 interval
>
.
I t e r a t i n g t h i s and u s i n g dependent c h o i c e , we o b t a i n a denumerable s e t o f elements ui (ii n t e g e r ) g i v i n g a r e s t r i c t i o n o f
A
isomorphic w i t h
Q
.0
Problem, suggested by t h e lemma f o r p a r t i t i o n i n s l i c e s , o r t h e minimal s t r a t i f i e d r e i n f o r c e m e n t i n t h e sense o f ch.4 5 1 . 5 . Does t h e c o n d i t i o n ( 3 ) remain e q u i v a l e n t w i t h (1) and ( 2 ) i f we s t r e n g t h e n i t by r e q u i r i n g t h a t , i n t h e s c a t t e r e d s t r a t i -
A , f o r e v e r y o r d e r e d p a i r o f elements a, b ( a s b modulo t h e r e e x i s t a ' and b ' such t h a t a 6 a ' (mod A) and
f i e d reinforcement o f t h e reinforcement),
Chapter 7
b ' g b (mod A)
and e i t h e r
a' = b '
or
181
a ' I b'
modulo t h e reinforcement.
52
- SUPPLEMENTARY RESULTS ABOUT FINITELY FREE PARTIAL ORDERINGS; TOPOLOGY ON I N I T I A L INTERVALS 2.1. L e t A be a f i n i t e l y f r e e p a r t i a l ordering. Then every s e t o f mutually i n comparable i d e a l s o f A ( w i t h respect t o i n c l u s i o n ) , i s countable (BONNET 1975, uses axiom o f choice).
0
Suppose the existence o f an uncountable s e t o f mutually incomparable i d e a l s of
A
, hence
a1
a set o f c a r d i n a l i t y
o f such i d e a l s
Hi
( i countable o r d i n a l ) .
Hi we associate a p a r t i a l o r d e r i n g Ci which i s a well-founded c o f i n a l d i r e c t e d r e s t r i c t i o n o f Hi : see ch.4 5 5.4 (axiom o f choice). Since A i s f i n i t e i s a w e l l p a r t i a l ordering. l y f r e e , so i s each Ci , and hence each Ci Given two d i s t i n c t countable o r d i n a l s i,j, by using the i n c o m p a r a b i l i t y w i t h To each
respect t o i n c l u s i o n o f the bases o f belonging t o
J
, let
Ai
element o f
H j ' For each countable o r d i n a l
values
. More
j
Card Ai
have
<
i
Ai
precisely, l e t (J
.
and H , d e f i n e an element ui,j j ui ,j > o r I (mod A) w i t h every
Hi
.
but not t o I H . I
I CiI
Thus
be t h e s e t o f the
ui,;
f o r a l l countable
A : we
be the corresponding r e s t r i c t i o n o f
I n general, f o r each s e t
o f countable o r d i n a l s and
U
i , l e t Ai(U) be the s e t o f t h e ui,j for all j # i Ai(U) w i l l be the corresponding r e s t r i c t i o n o f A Ai(U) i s a well p a r t i a l ordering, as i t i s a r e s t r i c t i o n o f
each countable o r d i n a l
.
and j E U Each Ai
More p r e c i s e l y
o r each
.
the w e l l p a r t i a l o r d e r i n g
Ci . Suppose f i r s t the existence o f a s e t
i n O 1 and such t h a t f o r each Let
Uo
U
designate t h i s
elements
j # i(0)
class i f
ui(o),j
of
U o f countable o r d i n a l s , which i s c o f i n a l
i of
and l e t
U
i(0)
, the
restriction
i n t o classes, by p u t t i n g
U
.
= ui(o),j,
in
, and note j e U1 .
U1
that
ko 7 o r
I t e r a t e t h i s by l e t t i n g P a r t i t i o n the elements
i(1) j
ui ( l ) , j = ' i ( l ) , j l
equality
of
i s countable.
j
and
j'
i n the same
There are countably many such classes, since
, a t l e a s t one o f k,, = ui(o) f o r the j J w i t h a l l elements o f every
i s countable. Since the c a r d i n a l i t y o f U Ai(o)(U) these classes, say U1 , has c a r d i n a l i t y w 1 . L e t for
Ai(U)
be i t s l e a s t element. P a r t i t i o n the
I
(mod A)
is W1
"j
be the l e a s t o r d i n a l index belonging t o U1
( j # i(1))
. There
U1
.
i n t o classes defined by the
are countably many such classes, since
i s included i n A (U) and the l a t t e r i s countable. A t l e a s t one i(1) o f these classes, say U2 , has c a r d i n a l i t y . Temporarily l e t kl = ui(l)
Ai(l)(U1)
f o r the
j
in
U2
. By
the preceding,
ko
>
or I kl (mod A)
.
,j
182
THEORY OF RELATIONS that
ko
and
kl
be incomparable (mod A)
i s an element o f
(0) hence
1(1)
> or I ko
1
.
(mod A)
i f necessary by a common upper bound o f
C i(l)
kl
, belonging t o Ci(l) . F i n a l value o f kl , i s > o r ! (mod A)
and
A
U
i in U countable is A io) triction
.
L e t U1
of cardinality W 1
o r d i n a l s , take
i(0)
such t h a t
a w e l l p a r t i a l ordering, by ch.4 of
Bo
be t h e s e t o f countable
. This
t o the base I B O l
cofinal i n
indices
j
. > i ( 0 ) , such ,just
that
of
as
has c a r d i n a l i t y U1 . Then take a well-ordered r e s t r i c t i o n
Ai(l)(U1)
, isomorphic w i t h
Ai(l)(U1)
, such
j 7 i(1)
that
u
r
For each i n t e g e r
r
cJ1
,a
Ur
, consider
U.
1(
>
r ) ,j
I (mod
A)
or
I
(mod A)
vr
>
or
wr
an element
in
H
I wo
.
which i s
>
or
v1
xo
. Then
\ or of
and
vr
i n the well-ordering
Br
o f t h i s subset. Since
H , i t f o l l o w s t h a t vr > o r j and o f and so f o r t h . I n p a r t i -
~ ,+ v~ ~ ,+ . .~. (mod A)
Iv vr
by an element
which i s
i(1) Replace
take i n H
w i t h a l l elements o f
s o f o r t h . We obtain,
and a w e l l - o r d e r i n g
u where j takes t h e i(r),j i s a countable subset o f the base I Br\ .
are mutually incomparable (mod A)
xb
, and
i(r)
w i t h each element o f
I t remains t o replace each that the
be the s e t o f countable i n d i c e s
countable o r d i n a l
w i t h each element o f
c u l a r , we have
U2
v1
i( 2 )
Hi(o)
I w i t h a l l elements o f Hi(ll
> or
wr
of
. H
i n such a manner
. For t h iis(,r ) l e t
by a common upper bound w1
in
an element, again c a l l e d
, which
, and take an element x1 in
xo
. Take
wo = vo
I w i t h a l l elements o f Hi(o)
, hence
Hi(l!
of
is
>
H i ( 2 ) which i s
or
>
, and replace v2 by a common upper bound w2
v2, xo, x1 ; and so f o r t h . This now c o n t r a d i c t s our hypothesis t h a t
f i n i t e l y free.
B1
the s e t o f those
.. . . This
values i ( r + l ) , i( r t 2 ) , Take the l e a s t upper bound
. Let
belongs t o the base [ B 1 l
i(l),j
, a set isomorphic w i t h w 1 .
f o r each i n t e g e r Br
u
be1ongs i(0) ,j Bo . I n t h e case now i ( 1 ) i n U1 , such
considered, we can i t e r a t e t h i s , as f o l l o w s . Take an element that
of all
1(0) 3 . 1 there e x i s t s a well-ordered res-
has c a r d i n a l i t y cJ1
U1
e x i s t s an
Uo
has c a r d i n a l i t y U1 . Since
A.
5
w 1 , there
Starting w i t h the set
, isomorphic w i t h c3
Ai(o)
kl
i s f i n i t e l y free.
Now consider the case where, f o r each s e t Ai(U)
i s d i r e c t e d , replace
1
H f o r j e Up . I t e r a t i n g t h i s y i e l d s an i n f i n i t e j ( r i n t e g e r ) , which are m u t u a l l y incomparable (mod A) :
kr
t h i s c o n t r a d i c t s our hypothesis t h a t with
For t h i s , n o t i c e t h a t
Since
l y , note t h a t the o l d value, hence a l s o the new
w i t h a l l elements o f every sequence o f elements
.
, and 1 7 o r I (mod A) w i t h a l l elements
A.
A
is
0
A p a r t i a l o r d e r i n g can be denumerable and f i n i t e l y f r e e , and can have continuum many mutually incomparable i n i t i a l i n t e r v a l s .
Chapter 7
183
0 Take two c h a i n s , each i s o m o r p h i c w i t h t h e c h a i n o f r a t i o n a l s , e v e r y element o f
t h e f i r s t c h a i n b e i n g incomparable w i t h e v e r y element o f t h e second. Then t o each
x , a s s o c i a t e i n t h e f i r s t c h a i n t h e i n i t i a l i n t e r v a l I ( x ) which as a c u t . S i m i l a r l y i n t h e second chain, a s s o c i a t e t o x t h e i n i t i a l i n t e r v a l which r e p r e s e n t s t h e o p p o s i t e number -x as a c u t , say J ( - x ) . F i n a l l y r e a l number
represents
x
i n the given p a r t i a l ordering, the union
I ( x ) u J(-x)
y i e l d s , when
x
runs
through t h e r e a l s , continuum many incomparable i n i t i a l i n t e r v a l s (example due t o BONNET 1975). 0
2.2. TOPOLOGY ON I N I T I A L INTERVALS; BASIC CLOPEN SET; OPEN SET AND CLOSURE Let
A
3 (A)
be a p a r t i a l o r d e r i n g ; we s h a l l e x t e n d t o t h e s e t
vals o f
A
, t h e t o p o l o g y a l r e a d y i n t r o d u c e d i n ch.1
o f natural integers, i . e . f o r
g(N)
where
o f i n i t i a l inter-
§ 8 f o r the set o f a l l sets
reduces t o t h e i d e n t i t y , o r f r e e
N
p a r t i a l o r d e r i n g on n a t u r a l i n t e g e r s . Let
F, G
IAI
be two f i n i t e subsets o f t h e base
i n i t i a l intervals o f
A
which i n c l u d e
5
G empty, we o b t a i n t h e e n t i r e s e t element i n
F
The i n t e r s e c t i o n any u n i o n o f
<
is
U
or
I (mod A)
G G' UF n UF,
Note t h a t
G
.
For
F
and
G i s non-empty i f f each
UF
. Consequently,
Gu G ' UFuF,
is
UFG be t h e s e t o f those
and a r e d i s j o i n t from
.
w i t h each element i n
G
.
d e f i n i n g an open s e t t o be
s e t s , t h e n t h e i n t e r s e c t i o n o f any two open s e t s i s s t i l l an open
.
'j ( A )
s e t : we have a t o p o l o g y on
s e t s , t h u s an open s e t ; so t h a t o u r We c a l l t h e s e
F (A)
. Let
The complement o f a U
U s e t i s a union o f U
s e t s a r e c l o p e n s e t s : b o t h open and closed.
U t h e b a s i c c l o p e n s e t s . T h i s t o p o l o g y i s H a u s d o r f f : see ch.1 8. Hi , where i < d and d i s
Consider an o r d i n a l sequence o f i n i t i a l i n t e r v a l s
an o r d i n a l , which we can assume t o be a r e g u l a r aleph. We say t h a t an i n i t i a l i n t e r val
H
either x
i s t h e l i m i t o f t h e sequence Hi , i f f o r e v e r y element x i n t h e base, x belongs t o H , and t h e n x belongs t o Hi f r o m some p o i n t on ; o r
belongs t o t h e complement o f
H
, and
then
x
belongs t o t h e complement o f
Hi
from some p o i n t on. Then t h e c l o s u r e o f a s e t
S
o f i n i t i a l i n t e r v a l s , i s d e f i n e d as t h e s e t o f a l l
l i m i t elements f o r a l l convergent o r d i n a l sequences o f elements o f We l e t i t t o t h e r e a d e r t o v e r i f y , as i n ch.1
5
8
S
.
, t h a t t h i s t o p o l o g y i s compact.
So t h a t t h e c l o p e n s e t s a r e e x a c t l y t h e f i n i t e unions o f b a s i c c l o p e n s e t s . 2.3.
ISOLATED ELEMENT, SPARSE SET OF INITIAL INTERVALS
Let
S
be a s e t o f i n i t i a l i n t e r v a l s o f
A
.
An element
X
of
S
i s s a i d t o be
i s o l a t e d (mod S ) , i f t h e r e e x i s t s an open s e t , and consequently a b a s i c clopen s e t U such t h a t X i s t h e o n l y element i n t h e i n t e r s e c t i o n SnU . A set
S
o f i n i t i a l i n t e r v a l s i s s a i d t o be sparse, i f e v e r y subset
c o n t a i n s a t l e a s t an i s o l a t e d element (mod S ' )
.
S'
of
S
THEORY OF RELATIONS
184
Now we are able t o complete by t o p o l o g i c a l considerations, our p r o p o s i t i o n 1.1.
A
Let
be a p a r t i a l ordering; then the f o l l o w i n g c o n d i t i o n i s e q u i v a l e n t t o any o f
o f conditions
sets 0
(1)
to
3 (A)
(T) t h e s e t
i n 1.1 :
(4)
o f i n i t i a l intervals o f
, i s sparse, w i t h respect t o the
A
(POUZET, unpublished; uses axiom o f choice).
Condition (T) i m p l i e s 1 . 1 . ( 3 ) .
Q
t h a t the chain intervals o f
A
Indeed suppose t h a t t h i s ( 3 ) i s n o t v e r i f i e d , i . e .
o f r a t i o n a l s i s embeddable i n
. Call
J(A)
which form a chain isomorphic w i t h
those i n i t i a l
H
under embeddability; then
Q
K any i n i t i a l i n t e r v a l o f A which i s the union o f a s t r i c t l y i n c r e a s i n g W-sequence (under embeddability) o f i n t e r v a l s H . The s e t o f these K i s nonempty (obvious) and has no i s o l a t e d element. Indeed each K i s the t o p o l o g i c a l call
l i m i t o f an o - s e q u e n c e o f
H
, and
an o -sequence o f o t h e r i n t e r v a l s
between any two o f these H
, whose
K
union i s a
H
. So
, we
can take
(T)
t h a t condition
i s falsified. i m p l i e s ( T ) . Indeed suppose t h a t (T) i s f a l s e , i . e . t h a t t h e r e
Conversely 1.1.(4)
e x i s t s a s e t o f i n i t i a l i n t e r v a l s which does n o t contains any i s o l a t e d element. Call
S
t h i s s e t , which i s obviously i n f i n i t e . We c l a i m t h a t there e x i s t s an e l e -
ment
x
i n t h e base
S
which contain
I A I , such t h a t t h e s e t
, and
x
Sx,
Sx
formed o f those elements i n
,
x
formed o f those elements which do n o t c o n t a i n
are both i n f i n i t e and w i t h o u t any i s o l a t e d element. Indeed take i n
S
two d i s t i n c t elements and
x
belonging t o one o f them and n o t
t o the other. Then consider the basic clopen sets o f a l l i n i t i a l i n t e r v a l s which contain
x
and a l l those which do n o t contain
element i n
Sx
or i n
Now define as f o l l o w s the f u n c t i o n take f o r Sx an element
z
element
y
f
. The existence
. Take
f(x) = 0
p l a y i n g the same r o l e as
p l a y i n g the same r o l e f o r
= -1 ; and so f o r t h ,
x
o f an i s o l a t e d
would imply t h e existence o f an i s o l a t e d element i n
Sx,
Sx,
f o r the preceding x
in
; and then l e t
u s i n g successively the f o u r i n t e r v a l s
x
S
. Then
S ; and s i m i l a r l y an
f ( y ) = 1 and
f(z) =
before -1, between -1
and 0, between 0 and 1, a f t e r 1 ; and so exhausting a l l t h e s e t o f r a t i o n a l s . Then we o b t a i n our f u n c t i o n
f-l
(mod Q)
(image of r)
2.4.
always g i v i n g
Let
A
mapping the r a t i o n a l s i n t o the base
<
o r I (image o f s) (mod A)
be the base o f
A
. By
t h e preceding 2.3,
Q
i s not
A
BONNET 1975; uses axiom o f choice; generalizes ch.6
E
r< s
.0
be an i n f i n i t e , f i n i t e l y f r e e p a r t i a l o r d e r i n g i n which
embeddable. Then the s e t o f i n i t i a l i n t e r v a l s o f
0 Let
I A l ,with
5
1.6.We g i v e POUZET's proof.
the s e t
5 (A)
of initial
Chapter 7
185
i n t e r v a l s i s sparse w i t h respect t o t h e given toDology. Use as f o l l o w s t h e CantorBendixson procedure ( u s i n g choice axiom). F i r s t take an ordinal-indexed sequence of a l l i s o l a t e d elements
Hi
of
I) (A)
. To
each
Hi
l e t us associate a basic
clopen s e t UG(!) , where F ( i ) and G ( i ) are two f i n i t e subsets o f E , such F(1) t h a t Hi i s t h e o n l y element o f 3 ( A ) which belongs t o t h i s basic clopen s e t .
3
Removing from
(A)
a l l these i s o l a t e d elements, i t then remains a sparse subset
( 1 ( A ) ) ' , the f i r s t d e r i v a t i v e o f 'j (A) . I t e r a t i n g the procedure, l e t us denote by H a l l the i s o l a t e d elements i n t h i s f i r s t d e r i v a t i v e ; j i s j an o r d i n a l index, s t r i c t l y g r e a t e r than a l l the preceding i n d i c e s i Again we
denoted by
.
o b t a i n f o r each
j a basic clopen s e t U G ( j ) such t h a t H i s the o n l y element F(J) j i n t h e f i r s t d e r i v a t i v e , belonging t o t h i s clopen s e t . I t e r a t i n g t h i s t o successive
d e r i v a t i v e s , which are defined e i t h e r by removing a l l i s o l a t e d elements, o r i n the case o f a l i m i t o r d i n a l rank, by t a k i n g the i n t e r s e c t i o n o f preceding d e r i v a t i v e s , we f i n a l l y reach t h e empty s e t . Now n o t i c e t h a t d u r i n g the procedure, a given basic clopen s e t
G UF
can occur
only once. Indeed t h e f i r s t time t h a t t h i s clopen s e t i s used i n order t o remove an element
Hk
of
'j (A)
, then
G UF
this
can c o n t a i n some
i < k , but
with
Hi
.
H with j > k Consequently j the c a r d i n a l i t y o f 3 (A) cannot exceed the c a r d i n a l i t y o f the s e t o f a l l ordered p a i r s (F,G) o f f i n i t e subsets o f E ; hence cannot exceed Card E . 0 cannot contain as an element any o f the f o l l o w i n g
2.5.
Let A
be a denumerable p a r t i a l ordering; then t h e f o l l o w i n g two conditions
are equivalent: (1) A
(2)
i s f i n i t e l y f r e e and does n o t admit an embedding o f
the set o f i n i t i a l intervals o f
A
Q ;
i s denumerable.
We use axiom o f choice t o deduce ( 2 ) from (1).
A
0 If
has a denumerable f r e e ' s u b s e t o r i f
Q
i s embeddable i n A
, then there
are continuum many i n i t i a l i n t e r v a l s : hence ( 2 ) i m p l i e s (1). The converse i s a p a r t i c u l a r case o f the preceding p r o p o s i t i o n 2.4. 0 2.6.
Let
A
be an i n f i n i t e p a r t i a l ordering; then the f o l l o w i n g two conditions
are e q u i v a l e n t (uses axiom o f choice): (1) A
i s f i n i t e l y f r e e and does n o t admit an- e. m b--e d d m
Q ;
( 2 ) there e x i s t s a s c a t t e r e d chain o f the same c a r d i n a l i t y as t o t a l l y ordered reinforcement o f A i s embeddable.
A
,in
which every
0 The c o n d i t i o n ( 2 ) i m p l i e s t h a t every t o t a l l y ordered reinforcement o f
A
is
scattered, hence ( 2 ) i m p l i e s (1) by 1.1 above ( c o n d i t i o n s (1) and ( 2 ) ) . Conversely, i f
A
s a t i s f i e s our ( l ) , then again by 1.1, c o n d i t i o n s (1) and ( 3 ) ,
the p a r t i a l ordering o f i n i t i a l intervals o f
A
does n o t admit an embedding o f
Q
.
THEORY OF RELATIONS
186
B of this partial
Hence by 1.3, there e x i s t s a t o t a l l y ordered reinforcement o r d e r i n g o f i n c l u s i o n , i n which B
has the same c a r d i n a l as
reinforcement
of
A'
A
.
A
, the
i s n o t embeddable. On the one hand, by 2.4,
Q
On the o t h e r hand, f o r every t o t a l l y ordered
chain o f i n i t i a l i n t e r v a l s o f
.
B
triction of
Finally
5 3 - EVERY DIRECTED WITH
ISOMORPHIC 3.1. L e t
A
WELL PARTIAL
A DIRECT
A
, hence
a res-
B .
i s embeddable i n
A'
i s a restriction
A'
o f the p a r t i a l o r d e r i n g o f i n c l u s i o n f o r i n i t i a l i n t e r v a l s o f
ORDERING
HAS A COFINAL
be a d i r e c t e d w e l l p a r t i a l ordering. L e t
8
RESTRICTION
(POUZET)
PRODUCT OF REGULAR ALEPHS
= Cof A
and l e t
. Then
a c o f i n a l subset o f l e a s t c a r d i n a l i t y , hence o f c a r d i n a l i t y
F
be
under these
conditions and m3dulo the axiom of choice: (1) there e x i s t s a
-sequence
forming a c o f i n a l subset o f
bi
(i
< E()
converse o f A ; i . e . i f i<j < 1 (2) f o r each element c i n F there e x i
-
quence w i t h values i n an upper bound;
F
w i t h o u t r e p e t i t i o n , o f elements
F ;moreover t h i s sequence i s bad w i t h respect t o the
then
bj
>
or I bi
(mod A ) ;
, beginning w i t h c , and f o r which n o e l e m e n t o f A
~
we say t h a t t h i s sequence i s unbounded; 8-sequences i n
( 3 ) given two unbounded s t r i c t l y i n c r e a s i n g
F , there exists a
t h i r d s t r i c t l y i n c r e a s i n g r - s e q u e n c e i n which t h e f i r s t two are embeddable, i n
5 2; (4) the -.i d e a l s generated by t h e unbounded s t r i c t l y i n c r e a s i n g r-sequences i n
the sense o f ch.4
F
t. c o n s t i t u t e a d i r e c t e d p a r t i a l o r d e r i n g w i t h respect t o i n c l u s i o n ; moreover ?
0 (1) T o t a l l y order
quence o f the j
>i
G
of
F
ai
(i
< ai J
a.
and
. Thus
G
.
F
union o f these i d e a l s includes
F according t o i t s c a r d i n a l i t y . In t h e thus obtained 8 - s e -
< 7/ ) , f o r (mod A )
.
each index
i
, remove those a. f o r which J
This e x t r a c t e d sequence y i e l d s a c o f i n a l subset
has c a r d i n a l i t y
and our e x t r a c t e d sequence i s a i$-se-
quence s a t i s f y i n g (1). (2) Let
bi
be the preceding
&-sequence
s a t i s f y i n g (1). By ch.4
0
3.4 ( e x t r a c -
t i o n theorem), t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g (mod A) e x t r a c t e d Suppose t h a t t h e r e e x i s t s an upper bound u we can assume t h a t o f the
bi's
-sequence.
o f the elements o f t h i s sequence:
u belongs t o the c o f i n a l s e t
G
o f (1); hence
u
i s one
: c o n t r a d i c t i o n proving t h a t our sequence i s unbounded.
Now l e t c be an element o f
F
. Since
, and ui ( i A
<
r ) an unbounded s t r i c t l y increa-
i s d i r e c t e d , t o each
i we associate an e l e c and ui Moreover, we can r e q u i r e t h a t the vi be d i s t i n c t . For t h i s , l e t vo be a ) common upper bound o f c and uo . I n general, given an index i ( 1 4 i < sing
'6-sequence i n F
ment
vi
i n the c o f i n a l s e t
F
, which
i s a comnon upper bound o f
.
Chapter 7
187
and v . ( j < i ) , note t h a t t h e s e t o f these v has c a r d i n a l i t y s t r i c t l y l e s s J j than , and so i s n o t c o f i n a l i n A . Thus there e x i s t s an x i n F w i t h v . . Take vi t o be a common upper bound o f c, ui and x . J d i s t i n c t from t h e v . ( j < i ) . F i n a l l y , again using the extracJ t i o n theorem ch.4 0 3.4, we e x t r a c t a s t r i c t l y i n c r e a s i n g Y-sequence. Since the
x 7 o r I w i t h each This makes
yi
sequence o f the
i s embeddable i n i t , t h i s s t r i c t l y i n c r e a s i n g
ui
unbounded. Furthermore, i t i s formed o f elements g r e a t e r than (3) Let
ui
and
vi
be two unbounded s t r i c t l y i n c r e a s i n g
&--sequences.
i associate a common upper bound wi
i s d i r e c t e d , t o each
the procedure i n the preceding ( 2 ) , we can ensure t h a t the ly, e x t r a c t a s t r i c t l y increasing
r-sequence i s
c (mod A)
of
ui
and
. Since
A
. By
vi
are d i s t i n c t . F i n a l -
wi
r - s e q u e n c e from the sequence o f the
.
wi
(4) Follows immediately from ( 2 ) and ( 3 ) . 0 3.2. The f o l l o w i n g p r o p o s i t i o n s u n t i l 3.9
are due t o POUZET 1979, unpublished.
LJ, = Cof A . There e x i s t i d e a l s Ai ( i < W , ) o f A w i t h Ai included i n A f o r i< j < a d , and such ji s A ; f i n a l l y Cof Ai Cof A f o r each i (uses t h a t the union o f the Ai Let
A
be a d i r e c t e d p a r t i a l o r d e r i n g and l e t
<
axiom o f choice). 0
X o f the base o f A , X , which we designate by u(X) In the i s the s i n g l e t o n o f the element x , we r e q u i r e t h a t u(X) = x
By t h e axiom o f choice, associate t o each f i n i t e subset
.
a common upper bound o f the elements o f case t h a t
X
For each subset f i n i t e subsets u(X) D(h)
X
D
of
. Then
f o r a l l f i n i t e subsets
, which
Note t h a t
D+
o f the base, we d e f i n e
D
D+
X
D ( l ) = the s e t o f t h e
f o r each i n t e g e r of
D(h)
we s h a l l c a l l the closure o f includes
i s equipotent w i t h
D
and t h a t
,let let
.
for all
O(h+l) = t h e s e t o f the D+
be the union o f the
D w i t h respect t o the f u n c t i o n u .
Dc D '
implies
D+cD"
; and f i n a l l y t h a t
D .
S t a r t i n g now w i t h a c o f i n a l (mod A) subset
<
h
. Finally
u(X)
C
o f minimum c a r d i n a l i t y cd& = Cof A
,
. To each
i < W4 , i t s u f f i c e s t o associate Di = the s e t o f the c . ( j < i ) , then the closure s e t Di+ , and J t o take f o r Ai the i d e a l , o r d i r e c t e d i n i t i a l i n t e r v a l generated by Di+ let
ci
(i
C.dM
)
be a w e l l - o r d e r i n g o f
C
.
3.3.
Let
A
be a d i r e c t e d p a r t i a l o r d e r i n g f o r which t h e r e e x i s t s a c o f i n a l (mod A)
r e s t r i c t i o n isomorphic w i t h the d i r e c t product (see ch.4
5
7) o f a f i n i t e number
o f well-orderings. Then we can always suppose: ( i ) t h a t each o f these well-orderings i s isomorphic w i t h a r e g u l a r aleph; ( i i ) t h a t t h e c a r d i n a l i t i e s o f two d i s t i n c t well-orderings are d i s t i n c t ; (iii) t h a t the l a r g e s t o f these alephs i s t h e c o f i n a l i t y o f A (ZF s u f f i c e s , provided t h a t Cof A e x i s t ) .
THEORY OF RELATIONS
188
0
representable by a sequence o f t $ t ' (mod A)
iff
tid ti
o f t h e c o f i n a l subset under consideration i s
t
By hypothesis, each element
n
1 to
o r chains, numbered from
coordinates:
...,t n )
t = (tl,
i = 1, ..., n
f o r each
.
(n integer), with
Note t h a t the well-orderings,
. We
, are n o t themselves r e s t r i c t i o n s o f A
n
o b t a i n a well-ordered r e s t r i c t i o n o f
only by t a k i n g a l l t h e coordinates cons-
A
t a n t , except f o r one which v a r i e s . S t a r t w i t h a c o f i n a l (mod A) s e t equal t o
Cof A
.
Replace each
o f minimum c a r d i n a l i t y , hence o f c a r d i n a l i t y
C
x
of
by an upper bound
C
, where
t(x)
a choice f u n c t i o n whose values are some o f the preceding sequences o f nates. We o b t a i n again a s e t which i s c o f i n a l and has c a r d i n a l i t y Moreover, we can replace each o f the
n
n
Cof A
is
t
coordi-
.
chains by a c o f i n a l r e s t r i c t i o n , hence
by a w e l l - o r d e r i n g whose order type i s a r e g u l a r aleph. And i f two chains, say the f i r s t and the second, have t h e same c a r d i n a l i t y , hence are isomorphic, then we again o b t a i n a c o f i n a l s e t by only t a k i n g those tl : t h i s amounts t o decrease the number
n
t
f o r which
t2 i s equal t o
.
o f chains by 1
Cof A
F i n a l l y , note t h a t one o f t h e chains n e c e s s a r i l y has c a r d i n a l i t y
, while
a l l the o t h e r chains have s t r i c t l y smaller c a r d i n a l i t y . The c a r d i n a l i t y o f t h e d i r e c t product i s t h e l a r g e s t c a r d i n a l i t y of these various chains. 0 Let
3.4.
A
be a d i r e c t e d p a r t i a l ordering, having a c o f i n a l r e s t r i c t i o n which i s
the d i r e c t product o f a f i n i t e p o s i t i v e number
n
o f i n f i n i t e well-orderings.
Let
A w i t h Cof B < Cof A . Then there e x i s t s an intermediate i d e a l B 5 C c A w i t h Cof C Cof B . Moreover C has a c o f i n a l r e s t r i c r
-
B
be an i d e a l i n
C
such t h a t
<
t i o n which e i t h e r i s the d i r e c t product o f
n-1
well-orderings ( i f
n & 2)
,
n = 1) (uses axiom o f choice, i n order t h a t c o f i -
which has a maximum element ( i f n a l i t i e s always e x i s t ) . 0
Cof A = W thus
The p r o o f i s obvious i f
Cof B = 1
element. I n t h e f o l l o w i n g , we s h a l l assume t h a t Let
3
W1
.
be a c o f i n a l (mod B ) r e s t r i c t i o n o f l e a s t c a r d i n a l i t y , hence o f cardina-
B'
Cof B
lity
Cof A
, s o t h a t B has a maximum
. To
each
x
in B'
associate a unique upper bound t ( x )
belon-
g i n g t o the d i r e c t product mentioned i n our hypotheses. The s e t o f these t ( x ) has c a r d i n a l i t y a t most Cof B . One o f t h e well-orderings o f our d i r e c t product, say the f i r s t chain, has c a r d i n a l i t y a t l e a s t product i s c o f i n a l i n A
.
Each well-ordering,
Cof A > C o f B
, since
our d i r e c t
o r chain, can be reolaced by a
c o f i n a l r e s t r i c t i o n whose order type i s a r e g u l a r aleph. Since
Cof B
< Cof
A , there e x i s t s an element
g r e a t e r than a l l values in
B'
nates o f
t h e element t(x)
tl(x)
(u,t2
u i n the f i r s t chain, which i s
(modulo t h i s f i r s t chain). Associate t o each
,..., t n )
where the
ti ( i = 2 ,...,n)
o t h e r than the f i r s t one. I t now s u f f i c e s t o take
i n i t i a l i n t e r v a l generated by these new elements, t h e r e being
,<
x
are the coordiC
t o be the
Cof B many such.
189
Chapter 7
More p r e c i s e l y , i n order f o r
t o be d i r e c t e d , we complete the previous by
C
associating t o each f i n i t e subset of
of
X
C
a common upper bound o f t h e elements
, and by i t e r a t i n g t h i s . To achieve the i t e r a t i o n , note t h a t our f i r s t
X
chain has l e n g t h a t l e a s t equal t o
>, Lu
Cof A
, so
an upper bound f o r denumerably many successive
8
Consider a r e g u l a r aleph
3.5.
(i
and Ai
<
Ai
is
t h a t each Ai
w1
.
with
u's
Let
t h a t there always e x i s t s
i n the f i r s t chain. 0 be a d i r e c t e d p a r t i a l ordering
A
f o r i c j ; moreover the union j f o r each index i < 8 , and has a c o f i n a l r e s t r i c t i o n which i s the d i r e c t product o f a f i n i t e
)
be i d e a l s i n
.
A
A
Suppose t h a t
number o f well-orderings.
Ai c A
< Cof
Cof Ai
A
Under these conditions,
Cof A =
8
Cof A
and hence
i s a r e g u l a r aleph (uses axiom o f choice). 0 Since
3
w 1 i s r e g u l a r , we can assume t h a t t h e r e i s a constant integer,
say n , f o r the f i n i t e number o f well-ordering5 whose d i r e c t product c o n s t i t u t e s a c o f i n a l r e s t r i c t i o n o f Ai ( i < 3 ) . More p r e c i s e l y we choose n t o be the minimum p o s s i b l e i n t e g e r corresponding t o any c o f i n a l sequence of i d e a l s
sa-
Ai
t i s f y i n g our hypotheses. We have
>/
Cof A
; indeed since
r i l y has elements n o t belonging t o Cof A
Cof Ai
set o f indices I n t h i s case
i s constant and equal t o a c e r t a i n aleph
i constituting a
Cof A
.4< 8 , and
Suppose now t h a t
if-sequence,
2 / r d = Max(r,d) d
by hypothesis, thus
< 2(
. But
Ci
A
by
Ai+l
d
i s s t r i c t l y l e s s than d
and
B
by
Ai
proposition, including
by
Cof Ci
Ci
Cof A =
having c a r d i n a l
mi
have
i , t h i s being always possiNow apply the preceding 3.4,
Ai
. Thus
i
.
there e x i s t s an intermediate i d e a l
, we n
well-orde-
Ai < Cof A , b u t w i t h n-1 i n s t e a d o f n . Hence we have n = 0 , so each Ai has a
, and then the s e t o f t h e mi
8 .0
n-1
again s a t i s f y a l l t h e hypotheses o f our
6 Cof
c o n t r a d i c t i n g the m i n i m a l i t y o f maximum element
a
Cof A
< Cof A 4 d . Hence we 8 .
having a c o f i n a l r e s t r i c t i o n which i s the d i r e c t product o f
r i n g s . Replacing the
, for
i s s t r i c t l y increasing i n
b l e by e x t r a c t i n g a c o f i n a l sequence o f such replacing
d
c o f i n a l i n the o r i g i n a l sequence.
: i f not, we would have
by the previous discussion Cof Ai
.
Cof Ai
i s l e s s than o r equal t o t h e sum o f t h e
Suppose f i r s t t h a t
Cof A
Cof Ai< Cof A , a c o f i n a l s e t i n A necessaAi , f o r each i 4 8 . On the o t h e r hand,
i s a c o f i n a l subset o f
A
,
THEORY OF RELATIONS
190
A
3.6. L e t
be a d i r e c t e d p a r t i a l o r d e r i n q and o(
a non-zero o r d i n a l w i t h d6
r e g u l a r a1 eph. Then t h e q u a s i - o r d e r i n g o f e m b e d d a b i l i t y (see ch.4 sing wd-sequences
in
A
,is
5
2) f o r t h e s t r i c t l y increa-
f i n i t e l y f r e e (uses denumerable s u b s e t axiom).
Suppose on t h e c o n t r a r y t h a t t h e r e e x i s t denumerably many s t r i c t l y i n c r e a s i n g
0
c3& -sequences
.. , ui ,. .
ul,.
(ii n t e g e r ) which a r e m u t u a l l y incomparable under j # i , a s s o c i a t e an element
and t o each i n t e g e r
ui
e m b e d d a b i l i t y . To each
a. o f ui such t h a t no t e r m o f u . i s g r e a t e r t h a n o r equal t o ai,j 1 ,j J Since wd i s r e g u l a r and s t r i c t l y g r e a t e r t h a n 0 , t a k e an element bi
(mod A)
in
.
ui
a . . (if i x e d , j a r b i t r a r y i n t e g e r # i). 1 ,J a r e m u t u a l l y incomparable, hence A i s n o t f i n i t e l y f r e e : c o n t r a d i c t i o n . 0
which i s g r e a t e r t h a n (mod A) a l l t h e The
bi
3.7. L e t
A
be a p a r t i a l o r d e r i n g and
a non-zero o r d i n a l such t h a t
ad
is
a r e g u l a r aleph. Then f o r e v e r y i n c r e a s i n g ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) -sequence o f s t r i c t l y i n c r e a s i n g W&-sequences
W,
-
s t r i c t l y i n c r e a s i n g W, Let Let of
ui
,
bo
A
in
A
, there
exists a
i n which each i s embeddable.
ai be t h e jth t e r m i n ui (i,j < Wa). ,j be t h e f i r s t t e r m i n u1 which i s a common upper bound
be t h e g i v e n sequences and
bo = ao,O ; l e t
term i n
-sequence i n
aOl, ui
and
bl
.
al,l
i < Wo( , l e t bi be t h e f i r s t b . ( j< i) and t h e a J i,j s a t i s f i e s the conclusion. 0
I n aeneral f o r each
which i s a common upper bound o f t h e
4 i). Then
(j
the
wd
-sequence o f t h e
bi
A be a d i r e c t e d w e l l p a r t i a l o r d e r i n e w h o s e c o f i n a l i t y i s a r e g u l a r a l e p h w 4 w i t h o< # 0 . L e t F be a c o f i n a l s u b s e t o f l e a s t c a r d i n a l i t y , hence o f c a r d i n a l i t y L+ . Consider t h e i d e a l s which a r e generated, each o f 3.8.
Let
them by an unbounded s t r i c t l y i n c r e a s i n g L3* ch.4
5
-sequence w i t h values i n
1.8). Then t h e r e e x i s t s a s e t o f a t most
c30(
F
(see
many i d e a l s which, under
inclusion, constitute a directed well p a r t i a l ordering o f c o f i n a l i t y s t r i c t l y less than
+.c
; moreover t h e u n i o n o f t h e bases o f t h e s e i d e a l s i n c l u d e s
(uses axiom o f c h o i c e )
F
.
By 3.6, t h e i d e a l s c o n s i d e r e d i n t h i s p r o p o s i t i o n f o r m a f i n i t e l y f r e e p a r t i a l o r d e r i n g under i n c l u s i o n , which f u r t h e r m o r e i s d i r e c t e d by 3 . 1 ( c o n d i t i o n ( 3 ) ) .
I n t h i s p a r t i a l o r d e r i n g , t a k e a w e l l - f o u n d e d c o f i n a l r e s t r i c t i o n , hence a 5 5.1. There a r e a t most wOcmany i d e a l s
d i r e c t e d w e l l p a r t i a l o r d e r i n g : see ch.2
191
Chapter 7
F belongs t o an unbounded mM -sequence, by 3.1, condition ( 2 ) . Thus the union of our s t r i c t l y increasing ideals includes F , and t h i s holds as well when taking the s e t of ideals in a cofinal s e t . The c o f i n a l i t y of the partial ordering under inclusion, f o r our ideals, i s a t most the cardinal of t h i s s e t , hence a t most ad. . Yet t h i s cofinality cannot be equal t o W d , since by 3.1, condition ( 2 ) , there would
under consideration: see 2.4 above. Every element of
then e x i s t unbounded s t r i c t l y increasing W & -sequences of i d e a l s , contradicting 3.7. Thus the c o f i n a l i t y i s s t r i c t l y less t h a n w+
.0
3.9. THEOREM O N THE COFINAL RESTRICTION OF A DIRECTED WELL PARTIAL ORDERING (POUZET 1979, unpubl ished) Let A be a directed well partial ordering. Recall t h a t Cof A i s a regular aleph (ch.4 5 1 0 ) . There e x i s t s a cofinal r e s t r i c t i o n of A which i s isomorphic with the d i r e c t product of f i n i t e l y many d i s t i n c t regular alephs, the largest of which i s Cof A (uses axiom of choice)
.
We already know t h a t a directed well p a r t i a l ordering of countable c o f i n a l i t y , e i t h e r has a maximum element and so i t s cofinality i s 1 ; o r has a cofinal r e s t r i c t i o n isomorphic with W : see ch.4 5 5.5. Proceed by induction. Let wd ( o( non-zero) be an i n f i n i t e regular aleph, and assume t h a t , f o r each regular aleph k s t r i c t l y l e s s t h a n W * and each directed well partial ordering X with c o f i n a l i t y k , there e x i s t s a cofinal rest r i c t i o n of X which i s a d i r e c t product of f i n i t e l y many d i s t i n c t regular alephs, the largest of which i s k . Apply the preceding 3.8. Let B be the directed well p a r t i a l ordering formed by the ideals in 3.8. By the induction hypothesis, the cofinality of B , which
0
i s s t r i c t l y l e s s than wd and regular, i s e i t h e r equal t o 1 or t o w a , with an ordinal f i s t r i c t l y l e s s t h a n q If Cof B = 1 , then there e x i s t s a s t r i c t l y increasing ud -sequence which generates a cofinal subset of A , and we are finished. Then by our induction hypothesis, C d f i i s Suppose now t h a t Cof B = Wn regular and there e x i s t s an integer n and a cofinal r e s t r i c t i o n C o f B which i s isomorphic with a d i r e c t product of n regular alephs, the largest of which i s w f i . Consequently, t o each ideal in A which i s an element of the
.
.
base \ C l , bijectively associate an n-tuple of coordinates ( tl,...,tn), each of which runs through a d i s t i n c t regular aleph, the f i r s t coordinate tl running through W R . For each n-tuple, l e t I ( t l . . . . , t n ) designate an unbounded, s t r i c t l y increasing
192
THEORY OF RELATIONS
A which generates the i d e a l whose coordinates are
W , -sequence i n
..., tn
tl,
I i s a choice f u n c t i o n , several sequences y i e l d i n g a same i d e a l ) . modulo t h e d i r e c t Given two n-tuples, the i n e q u a l i t y (tl, ...,t n ) \< ( t i , ..., ), ' , t product, i . e . the s e t o f i n e q u a l i t i e s tl 6 t i ,. . . , tn6 t; , i s e q u i v a l e n t t o
(obviously
the c o n d i t i o n t h a t the sequence
I ( t l,...,tn)
i s embeddable i n
I ( t) i ,,..., ! , t
(mod A) , i n t h e sense o f ch.4 5 2 . We are now going t o replace each U&-sequence
under consideration by an e x t r a c t e d
sequence o f the same length, w i t h the aim i n mind o f b i j e c t i v e l y a s s o c i a t i n g t o each
(nt1)-tuple
( i ,tl,.
an element o f the base
\A
.. ,tn) I
ic a , and the t are as p r e v i o u s l y , a ( i ,tl,. . ,tn) , which more p r e c i s e l y I ( t l....,tn) . We s h a l l do t h i s i n a manner t h a t where
.
designated by
w i l l be a term i n the sequence
,... ,tn)Q( i ' , t i ,... ,tA) modulo t h e new d i r e c t i . e . the s e t o f i n e q u a l i t i e s i s i ' , tlb t i , ... , tndt,', , i s t o the i n e q u a l i t y a ( i , t l ,...,t n ) 6 a ( i ' , t i ,...,t,!, ) modulo A .
the i n e q u a l i t y
(i,tl
U,, many
n-tuples
( tl , . . . ,tn)
and
wR <
. We
Q~
f i r s t take the u
t o be a l l d i s t i n c t . I n general, given an o r d i n a l
a(0,t l,...,tn)
a(u,t l,...,tn)
ensure t h a t t h e cho sen values
being d i s t i n c t from t h e already defined
equivalent
a a r e d i s t i n c t , since there
F i r s t o f a l l , i t i s easy t o ensure t h a t our elements are
product,
< 0,
we
are a l l d i s t i n c t , as w e l l as
values
... ,tn) ...,t n )
a(i,tl,
i< u . belongs t o t h e for
We do t h i s w h i l e respecting the c o n d i t i o n t h a t a(u,tl, sequence I(tl, ..., t n ) and i s s t r i c t l y g r e a t e r (mod A) than t h e already defined values
...,t n ) f o r i < u . A l l t h a t i s easy because cdd i s r e g u l a r , u < wd , no u-sequence e x t r a c t e d from the sequence I i s
a(i,tl,
hence f o r each cofinal.
Secondly, we s h a l l ensure t h a t , f o r two incomparable (ul
,...,un) , every
n-tuples
I ( t l ,...,t n ) . . ,un) . To do t h i s ,
value o f a term i n
w i t h every value o f a term i n
I(ul,.
( t l,...,tn)
, designate by ( u l,...,un)
rable with
(tl
,..., t n ) . Thus
any and a l l
t h e sequence
respect t o embeddability (mod A)
, with
. ,tn)
(tl,..
and
i s incomparable (mod A ) f o r each
n-tuples which are incompa-
,..., t n )
I(tl
n-tuple
each sequence
I(ul,
i s incomparable, w i t h
...,un) . Take
a term
o f the f i r s t , f o r which no term o f the second i s g r e a t e r (mod A) than i t , and designate i t by (ul,
..., un)
b(tl
....,tn;ul
,...,u,) . Then,
vary, and replace t h e sequence
fixing
(tl
...,t n )
I(tl,
,...,t n )
, let
by an e x t r a c t e d se-
quence o f t h e same l e n g t h od , b u t beginning w i t h an upper bound o f a l l the many b preceding b . This i s p o s s i b l e because t h e r e i s a t most
.
T h i r d l y , we s h a l l ensure t h a t , f o r two d i s t i n c t
<
(ul
,...,un)
I(ul, ...,un)
, there
do n o t e x i s t values
a
s a t i s f y i n g the opposite i n e q u a l i t y
n-tuples s a t i s f y i n g in
I(tl
,...,t n )
...
(tl, ,tn) and b i n
a 7 b (mod A ) : the o n l y
Chapter 7
193
p o s s ib i l i t i e s being incomparability a I b (mod A) or the inequality a < b (mod A). To do t h i s , f o r each n-tuple ( t l,..., t n ) l e t (ul ,...,un) designate any and a l l d i s t i n c t l e ss e r n-tuples: t h u s the sequence I ( t l , . . . , t n ) i s not embeddable (mod A) Take a term of the f i r s t sequence, for which no term of in any I(ul, ...,u n ) the second i s greater, and continue as in the preceding paragraph. Fourthly and f i n a l l y , we shall define by induction on i the elements a ( i , t l , . . . , t n ) as previously announced. To the (n+l)-tuple ( O , O , . . ,0) associat e the beginning term of the sequence I ( 0 ,...,0 ) . Let ( t l,..., t n ) be an n-tuple and suppose t hat a(O,ul,. . . , u n ) has already been defined for a l l n-tuples
.
.
(ul ,..., un) < ( t l,..., t n ) I(tl,
..., t n )
.
Then s e t a ( O , t l
,...,t n )
t o be the f i r s t term of
which i s s t r i c t l y greater (mod A) than a l l
a(O,ul,
...,u n )
for
(ul ,..., u n ) < ( t l,..., t n ) . These values of a f o r f i r s t coordinate 0 shall be kept in the end. To the (n+l)-tuple ( l , O , ...,0 ) associate the f i r s t term of I ( 0 , ...,0 ) which i s s t r i c t l y greater t h a n a(O,O,. . . ,0) and which i s not s t r i c t l y less (mod A) than any value a(O,ul ,..., u n ) f o r any n-tuple (ul ,...,u n ) . This i s possible, since there i s no element in the base I A I which i s greater (mod A) t h a n every Note t h at by the paragraph "thirdly", the element of the sequence I ( 0 , . .. ,O) value a ( l , O , ...,0) will be incomparable (mod A) with a l l those a with f i r s t coordinate 0 followed by a non-zero n-tuple ( u1 ,. . . , u n ) . Indeed by construction a(l,O, ...,0 ) i s n o t s t r i c t l y less (mod A) than these values; nor i s i t
.
strictly greater, since otherwise the inequality o f two terms would be i n t h e opposite sense o f the inequality o f the n-tuples (0 ,...,0) < (ul ,..., u n ) . Now l e t
( t l,..., t n ) be an n-tuple, and suppose t h at a (l,ul
dy been defined f o r a l l n-tuples a(l,tl
(ul
,..., t n ) take the f i r s t term
(mod A ) than
a(O,tl
f o r a l l n-tuples
,...,t n )
,...,un)
,..., u n ) < ( t l,... ,tn). Then
of
I(tl
,..., t,)
has alrea-
for
which i s s t r i c t l y greater
and which i s s t r i c t l y greater t h a n
a(l,ul
,..., u n )
(ul, ...,u,) < ( t l ,...,t n ) ; and f i n a l l y which i s n o t s t r i c t l y
.
lesser (mod A J t h a n a(O,vl, ..., v n ) f o r any n-tuple ( v l , ...,v n ) This l a s t condition i s possible by the f a c t t h a t no element in I A I i s an upper bound of the sequence I ( t l , ... , t n ) Note t h a t , by the paragraph "secondly", t h i s value
.
a ( l , t l , ...,t n ) will be incomparable (mod A) with a l l the values a w i t h f i r s t coordinate 0 or 1 followed by an n-tuple incomparable with ( t l ,...,t,) .
Our value a ( 1 , t l , . . . , t n ) will also be incomparable with the a with f i r s t coordinate 0 followed by an n-tuple ( v l ,...,v,) 7 (tl,...,t,) . Indeed i t i s n o t s t r i c t l y l e s s e r (mod A) by construction, nor i s i t s t r i c t l y greater, since otherwise the inequality of the terms would be in the opposite sense o f the
THEORY OF RELATIONS
194
inequality of the n-tuples. I n general, l e t k be an ordinal s t r i c t l y l e s s e r t h a n , and suppose t h a t a l l values of a with f i r s t coordinate < k have been defined. To the ( n + l ) tuple ( k , O , ..., 0 ) associate the f i r s t term of I ( 0 , 0 ) which i s s t r i c t l y greater than a(i,O, ...,0) f o r a l l i < k , and which i s n o t s t r i c t l y l e s s e r (mod A) than any a ( i , u l ,..., u n ) . The value a ( k , O ,..., 0 ) will be incomparable (mod A) with a l l values a having f i r s t coordinate i < k followed by a nonzero n-tuple. Now l e t ( t l,..., t n ) be an n-tuple and suppose t h a t the values a(k,ul ,..., u n ) have been defined f o r a l l n-tuples (ul ,...,tin) C (tl,..., t n ) . Then take
...,
a(k,tl
,..., t n )
than
a(i,tl,
I ( t l ,..., t n ) which i s s t r i c t l y greater
t o be the f i r s t term of
...,t n )
f o r a l l n-tuples
(ul
for all
and s t r i c t l y greater than
i < k
,...,u n ) <
a(k,u l , . . . . u n )
( t l,...,t n ) , and f i n a l l y which i s n o t s t r i c t l y
l e s s (mod A) than a ( i , v l ,..., v n ) f o r any i < k and any n-tuple The reader can verify the desired incomparabilities. 0
3.10. Recall t h a t f o r every well-founded p a r t i a l ordering A Cof H t AS Cof A : see ch.2 0 7 . 2 . S t r i c t inequality Cof H t A c Cof A directed well p a r t i a l ordering A . 0 Consider the ordinal product
A = W
Yet
1
x
Cof A =
1.
.
Then
,..., v n )
have
can be obtained, even in the case of a
W 1. 4
Ht A
, we
(vl
=
and then the d i r e c t product
ol. (h,
so t h a t Cof H t A
=
c3
.
a, . 0
Note t h a t , in agreement with the preceding theorem, we have the d i r e c t product W l x k J as a cofinal r e s t r i c t i o n of A
.
.
195
Chapter 7
5 4 - SZPILRAJN We in ly If as
CHAIN,
DENUMERABLY
SZPILRAJN
CHAIN
say t h a t a chain C i s a Szpilrajn chain i f , f o r every p a r t i a l ordering A which C i s not embeddable, there e x i s t s , modulo the axiom of choice, a t o t a l ordered reinforcement of A in which C i s n o t embeddable. C i s a Szpilrajn chain, then so i s every chain equimorphic with C , as Well with the converse C- .
Among the f i n i t e chains, only the empty chain and the singleton chain are Szpi7r a j n chains. Indeed, f o r each integer p z / 2 , the f r e e partial ordering on p elements, when reinforced, necessarily yields a chain of cardinality
p
4.1. I f a chain C does n o t admit an embedding of C + l , then f o r each the chain C+p i s n o t a Szpilrajn chain. 3 are n o t Szpilrajn chains. For example W+2 , ~ + ,... 0 S t a r t with the p a r t i a l ordering formed by the chain C followed by p l y incomparable elements. 0
.
pa 2
,
mutual-
4.2. The chain W of the natural integers i s a Szpilrajn chain. We shall show t h i s , equivalently, f o r the converse a-. A partial ordering A , in which 12-i s n o t embeddable, i s well-founded: see ch.2 5 2 . 4 , using dependent choice. Then there e x i s t s a well-ordered reinforcement of A : see ch.2 5 4 . 4 (axiom of choice).
0
4.3. The chain Q of the rationals i s a Szpilrajn chain. This i s another s t a t e ment of the equivalence of conditions (1) and ( 2 ) of 1 .3 (modulo the axiom of choice). 4.4. We say t h a t a chain
C
i s denumerably Szpilrajn i f
C
i s denumerable and
i f , f o r every denumerable p a r t i a l ordering A in which C i s n o t embeddable, there e x i s t s a t o t a l l y ordered reinforcement of A , in which C i s not embeddable. If a chain i s denumerable and Szpilrajn, then i t i s denumerably Szpilrajn. However, we shall see below t h a t + 1 i s denumerably Szpilrajn without being Szpi 1 rajn. We see immediately that o + p , with an integer pa 2 , i s n o t denumerably Szpil rajn.
196
4 . 5 . The chain
THEORY OF RELATIONS
C =
CAJ
+ a - i s not denumerably Szpilrajn.
S t a r t with A = w -. LJ and B = A- = U . W - . Take the p a r t i a l ordering formed by A and B with each element of one incomparable with each element of the other. Then C i s not embeddable in t h i s partial ordering; y e t C i s embeddable in every t o t a l l y ordered reinforcement, by ch.6 § 3.7. 0
0
4.6. Let C , D be two chains. rf C + l &a. l + D are Szpilrajn, then so i s C+l+D ; same statement f o r "denumerably Szpilrajn" (JULLIEN 1969).
Let A be a p a r t i a l ordering in which C + l + D i s not embeddable. Call H the i n i t i a l interval of those elements x above which there i s a chain in A isomorThen phic w i t h 1+D ; the element x being the minimum of this chain 1 + D C + l i s n o t embeddable in H . Moreover, the chain l + D i s n o t embeddable in the final interval H ' complementary t o H . Take a t o t a l l y ordered reinforcement of A formed by an i n i t i a l i n t e r v a l , which i s a reinforcement of H y e t in which C + l i s n o t embeddable; followed by a final i n t e r v a l , which i s a reinforcement of H ' y e t in which l + D i s n o t embeddable. Then C+l+D i s n o t embeddable in such a chain. 0
0
.
I n particular, the chain Z of the positive and negative integers i s Szpilrajn, since (13- = d - + 1 and LJ = 1 + W are Szpilrajn. 4 . 7 . The ordinal w+1 i s denumerably Szpilrajn, b u t i s n o t Szpilrajn: see JULLIEN 1969.
We shall prove t h a t w + 1 i s denumerably Szpilrajn. S t a r t with a denumerable partial ordering A in which c3+1 i s n o t embeddable. I f A has a maximum element, then even LJ i s n o t embeddable in A , and so by the above 4.2, there e x i s t s a t o t a l l y ordered reinforcement of A , i n which w i s not embeddable. If A has no maximum element, then take an w-sequence of elements ai ( i integ e r ) , forming a cofinal s e t in A with the condition a . > o r I ai (mod A ) f o r J a l l i and j 7 i Possibly we can have only a f i n i t e cofinal s e t of maximal elements, thus a f i n i t e sequence.
0
.
Associate t o each i the r e s t r i c t i o n Ai of A t o those elements which are 6 ai b u t .$ ao, ... , ai-l . The bases of the Ai are mutually d i s j o i n t and t h e i r union i s the base of A None of the Ai admit an embedding of w , since otherwise, taking in account the maximum ai , the ordinal w+1 would be embeddable. By 4 . 2 , there e x i s t s a t o t a l l y ordered reinforcement Bi of Ai , i n which W i s not embeddable. Then the sum of the Bi according t o increasing i i s a t o t a l l y ordered reinforcement of A , in which 0+1 i s not embeddable. 0
6
.
Chapter 7
197
Now we prove t h a t w +1 i s n o t Szpilrajn. Take an uncountably i n f i n i t e s e t U To each enumeration f , without r e p e t i t i o n , of a denumerable subset of U and to each natural integer i , associate the ordered p a i r ( f , i ) . Take as base E
0
.
the union of U a n d the s e t of these ordered pairs. Set ( f , i ) < ( f , j ) , f o r the same enumeration f , i f i < j in the usual orderina of integers. Moreover f o r each ordered p a i r , s e t ( f , i ) < f ( i ) : t h i s second term being an element of U . Hence by t r a n s i t i v i t y we have ( f , i ) < f ( j ) f o r a l l j > i . Apart from these cases, two d i s t i n c t elements of E shall be incomparable. Call A the thus defined partial ordering on E : we e a s i l y see t h a t w i s embeddable in A , b u t not 0+1 . Let B be a t o t a l l y ordered reinforcement of A , based on E There e x i s t s a t least one element u of U having a t l e a s t denumerably many elements x < u (mod B ) with x E U . Take an enumeration f of such elements. The ordering A , hence B as well, admits as a r e s t r i o t i o n the chain of the ( f , i ) where f i s which has the element u as fixed and i = 0,1,2, ... : a chain of order type an upper bound (mod B ) . Thus w+1 i s embeddable in B 0
.
.
COROLLARIES due t o JULLIEN 1969 (1) If a denumerable chain C s a t i s f i e s the s t r i c t inequality C + l > C respect t o embeddability), then C+1 i s n o t Szpilrajn.
(with
-
has no maximum element. Then use the preceding argument: replace W by C ; replace each f by an injection of the base I C l onto a denumerable subset of U ; the inequality i < j being made modulo C. Or C has a maximum element. Denote by C-1 the chain C a f t e r removing i t s maximum: C-1 does n o t admit an embedding of C . Then i t suffices t o consider the partial ordering obtained from C-1 by adding two maximal and mutually incomparable elements. 0
0 Either
C
( 2 ) The ordinal
0.2
, and
integer >/ 2 )
more generally 0 . p (with p
&
n o t Szpilrajn. Take an uncountably i n f i n i t e s e t U which i s the union of uncountably many On each U k take a chain isomorphic with w disjoint denumerable subsets U k Furthermore any two elements of U which belong t o two d i s t i n c t U k , will be 0 hcomparable. Then i t suffices t o terminate as f o r 0+1
0
.
.
4.8. I t i s proved by GALVIN and MAC KENZIE in 1969 (unpublished) t h a t w i s the only denumerable Szpilrajn ordinal. I t i s proved by BONNET in 1971 (see BONNET, POUZET 1982) that: The only denumerable Szpilrajn chains ( u p t o equimorphism) a r e the following: the chain Q of rationals; the scattered chains defined as follows by ordinal products and sums: PI = 0 ; P2 = w - . CJ ; P 3 -- CJ. W - . W ; in general
.
THEORY OF RELATIONS
198
p.,+1. = p i-. (,d f o r each i n t e g e r s o r o r d i n a l . Furthermore P, =
i , and more generally f o r each countable succesP i ( i i n t e g e r ) ; more g e n e r a l l y , given any denu-
merable limit ordinal u , take any set i n u : then we s e t p U = Z. p i
-sequence o f i n d i c e s
i
forming a cofinal
.
F i n a l l y , f o r each preceding P i ( i countable o r d i n a l ) , the converse
Pi
-
ii
S z p i l r a j n ; furthermore each sum Pi- + Pi i s S z p i l r a j n . Note t h a t l + P i = P i , so the l a t t e r sum i s obviously S z p i l r a j n by 4.6 above. For any two countable o r d i n a l s i and j > i , we obviously have P i + P j S z p i l r a j n ; however t h i s i s already mentioned, s i n c e the l a t t e r sum i s equimorphic w i t h P. . J
4.9. Let C be a denumerable chain, I t h e denumerable i d e n t i t y r e l a t i o n ; in o t h e r words, the denumerable a n t i c h a i n . We say t h a t t h e ordered p a i r (C,I) i s u n i v e r s a l l y S z p i l r a j n i f , f o r every p a r t i a l ordering A in which n e i t h e r C nor I i s embeddable, t h e r e does not e x i s t any t o t a l l y ordered reinforcement of A in which C is embeddable (even when u s i n g the axiom of c h o i c e ) . For example ( Q , I ) , where Q i s the chain of r a t i o n a l s , i s universally Szpilrajn by 2.6.
If C i s a denumerable chain, and C' equimorphic with universally S z p i l r a j n , t h e n so i s (C',I) .
C , and
Given a denumerable s c a t t e r e d chain C , t h e ordered pair: (C-+C,I u n i v e r s a l l y S z p i l r a j n . Same r e s u l t w i t h C+C- . 0 First n o t i c e t h a t C-+C i s not embeddable i n C : use the sam
(C,I)
is
i s not
argument as a s i n ch.5 0 3 . 3 . Then t a k e a p a r t i a l ordering, composed of two chains C and C- w i t h d i s j o i n t bases, every element of t h e f i r s t being incomparable w i t h every element of the second. I t s u f f i c e s t o see t h a t C-+C i s a t o t a l l y ordered reinforcement, y e t not embeddable in our p a r t i a l ordering. 0
It is rable sally Pi ( i
proved by BONNET in 1971 ( s e e BONNET, POUZET 1982) t h a t ;he only denumechains (up t o equimorphism) which y i e l d with t h e i d e n t i t y r e l a t i o n a univerS z p i l r a j n p a i r , a r e t h e chain countable o r d i n a l ) ,
Q o f r a t i o n a l s and the preceding products
4.10. The reinforcement of R being defined f o r an a r b i t r a r y r e l a t i o n R , l e t us mention possible g e n e r a l i z a t i o n s of t h e notion of S z p i l r a j n chain. Using ch.5 0 2.4 and 2.7, r e c a l l t h a t R is a p a r t i a l ordering i f f R does n o t admit an embedding of: A1 = the r e l a t i o n w i t h c a r d i n a l i t y 1 and value (-) ; A2 = r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 ; A3 = r e f l e x i v e binary cycle w i t h c a r d i n a l i t y 3 ; A4 = c o n s e c u t i v i t y w i t h c a r d i n a l i t y 3 Moreover R i s a chain
.
199
Chapter 7
i f f R does n o t admit any embedding of A1 or A2 or Ag o r A5 = identity relation with cardinality 2 . Now we can t r a n s l a t e as follows the definition of a Szpilrajn chain. An arbitrary binary relation R i s Szpilrajn i f f : ( i ) R does not admit any embedding of A1 or A2 o r A3 or A5 (tra nsla te : R i s a chain); ( i i ) for every relation A , i f A $ R and A does n o t admit any embedding of A1 or A2 or A3 or A4 ( t r an s l at e: A i s a partial ordering), then there exists a reinforcement of A which does n o t admit any embedding of R or A1 or A2 or A3 o r A5 . I n the so "translated" d ef i n i t i o n , there only occur general notions as embedding and reinforcement, which are defined f o r ar b i t r ar y relations. Hence we have many possible generalizations, f o r instance by replacing A1 to A5 by an arbitrary f in it e sequence of f i n i t e r el at i o n s . We do n o t know i f t h i s yields interesting problems.
EXERCISE 1
-
TUKEY'S THEOREM O N PAIRS OF DIRECTED PARTIAL ORDERINGS
Let us consider two directed p ar t i al orderings A and B . A function f from A into B i s said t o be convergent, i f every cofinal s e t (mod A ) , say X , i s transformed i n t o a cofinal s e t f"(X) , modulo B . Following TUKEY 1940, we shall prove, modulo the axiom of choice, the equivalence between (1) and ( 2 ) : (1) there e x i st s a convergent function from A i n t o B , and similarly a converpent function from B into A ; ( 2 ) there e x i st s a directed p ar t i al ordering C which i s a common extension of A and B (more exactly A and an isomorphic image of B ) , such t h a t both A and B are cofinal r es t r i ct i o n s modulo C .
-
1 - To see t h a t ( 2 ) implies ( l ) , i t suffices t o associate, t o each element of A , an element f ( x ) of B with f ( x ) >/ x (mod C )
x
.
-
Conversely, suppose t h at A and B s a t i s f y ( 1 ) . Denote by f a convergent function from A into B Consider the directed pa rtia l ordering of final intervals of A , ordered by reverse'inclusion, then similarly the directed partia l ordering of f i nal intervals of B ; then denote by C the dire c t product of these two partial orderings, which i s obviously a directed partial ordering. Now define as follows the embedding of A i n t o C To each element x in A , associate the ordered pair (Ax,Bx) where Ax i s the final interval of elements 3 x (mod A) and Bx i s the final interval of the upper bounds (mod B ) of a l l the images f ( t ) where t 3 x (mod A ) . 2
.
.
THEORY OF RELATIONS
200
Note t h e f o l l o w i n g easy lemma. Given (mod A)
such t h a t
x
f ( t ) < o r I f ( x ) (mod B )
I n o t h e r words, t h e r e e x i s t s an element So t h a t i f
v(x)
3
-
B)
such t h a t e v e r y x
and
From t h e p r e c e d i n g lemma, e a s i l y deduce t h a t
same argument by exchanging A into
, then
.
the
i n t o a subset o f t h e
f"
. A
, when mapped i n t o
previously indicated, y i e l d s a cofinal r e s t r i c t i o n i n B
i s < o r 1 u(x)
t
u ( x ) (mod A)
i s t r a n s f o r m e d under
A)
tax
s e t o f those elements
do n o t c o n s t i t u t e a c o f i n a l s e t i n A .
u(x)
i s a common upper bound o f
>/ v ( x ) (mod >, f ( x ) (mod
final interval final interval
, the
in A
and
B
. Finally
C
as
C
repeat the
and u s i n g t h e convergent f u n c t i o n from
.
A
As a c o r o l l a r y , n o t e t h a t c o n d i t i o n (1) (and o b v i o u s l y ( 2 ) ) i m p l i e s t h a t Cof A = Cof B EXERCISE 2
A
Let
-
. KRASNER'S LEMMA ON DIRECTED PARTIAL OROERINGS
be a d i r e c t e d p a r t i a l o r d e r i n g , and l e t
u = Cof A
. F o l l o w i n g KRASNER
1939, l e t us prove, modulo t h e axiom o f choice, t h a t , g i v e n a c o f i n a l r e s t r i c tion
w i t h minimum c a r d i n a l i t y
A'
Card A ' = u
, there
e x i s t s an i n j e c t i v e ,
i n c r e a s i n g f u n c t i o n whose domain i s t h e s e t o f a l l f i n i t e subsets i n under i n c l u s i o n , and whose range i s We can always suppose t h a t
A' = A
A'
, so
u
, ordered
. that
u
= Card A = Cof A
.
1 - As a p r e l i m i n a r y e x e r c i s e , c o n s i d e r t h e case where A i s denumerable. We know by ch.4 5 5.5 t h a t t h e r e e x i s t s a c o f i n a l r e s t r i c t i o n o f A which i s isomorp h i c t o W ( o r t o 1 , i n which case a l l i s o b v i o u s ) .
u
0 = s e t o f i n t e g e r s . Enumerate t h e base o f A as an u - s e q u e n c e (ii n t e g e r ) . Then d e f i n e as f o l l o w s t h e d e s i r e d f u n c t i o n f . and f(1) = al , and n o t e t h a t Take t h e i n t e g e r s 0 and 1 o n t o f ( 0 ) = a. f ( i ) = ai, w i t h i ' g r e a t e r t h a n o r equal t o i . Take t h e p a i r ( 0 , l ) onto t h e ai w i t h t h e l e a s t i n d e x i , p r o v i d e d t h a t ai i s an upper bound (mod A)
Then ao,
=
..., ai,..
a. = f ( 0 ) and al = f ( 1 ) . Then t a k e t h e i n t e g e r 2 o n t o f ( 2 ) = ai where i i s t h e l e a s t i n d e x n o t y e t used: s o t h a t t h i s i i s g r e a t e r t h a n o r equal and {1,2) Then t o 2 . D e f i n e as p r e v i o u s l y t h e images o f t h e p a i r s {0,2) t a k e t h e 3-element s e t {0,1,2} o n t o t h e element ai w i t h t h e l e a s t index i , p r o v i d e d t h a t t h i s ai i s n o t y e t used and i s a common upper bound (mod A) o f
of
.
t h e images under elements i n A
f
o f the three pairs
{O,l}
, {0,2}
,
{1,2).
. Note
that a l l
a r e f i n a l l y used; moreover t h e procedure always works, s i n c e f o r
any f i n i t e s e t o f elements
a
in A
, t h e r e e x i s t denumerably many common upper
bounds.
-
2 Now e x t e n d t h e same p r o c e d u r e t o t h e g e n e r a l case. Consider u = Cof A = Card A as an ordinal-indexed sequence, o r u-sequence o f terms i < u .
Chapter I
20 1
.
Well-order the base of A as a u-sequence ai Then repeat the previous procedure for ordinals 0 and 1 . I n general, l e t i be an ordinal index s t r i c t l y less t h a n u ; suppose t h a t f i s already defined f o r a l l f i n i t e se ts of indices j < i . Moreover suppose t h a t f ( j ) = a . with j' greater t h a n or equal t o j , J' for every such j . Take i onto f ( i ) = a i , where i ' a i i s the l e a s t index not yet used. Then take every pair i i , j j where j -z i , onto the element a with the l e a st possible index, provided t h a t t h i s a i s n o t ye t used, and i s a common upper bound (mod A) of the images f ( i ) and f ( j ) Then take every 3-element s e t \ i , j , k } where j , k < i , onto the element a with the l e a s t possible index, provided t h at a i s n o t y et used, and i s a common upper bound of the images under f of the three pairs included in { i , j , k ) ; and so on. The procedure always works, since f o r any f i n i t e s e t of elements a in A ,
.
on one hand there e x i s t u many comnon upper bounds; on the other hand the number of f in i t e s e t s of indices less than or equal t o i < u i s i t s e l f < u .
-
I n connection with the two preceding exercises, say tha t two partial orderings
A, B have same convergence type i f f condition ( l ) ,or equivalently ( 2 ) in exer-
cise 1 holds. Then the reader may be interested t o know t h a t , in the case of cofinality w , then i t i s consistent with the axioms of ZF t o suppose e ithe r that there e x i s t ( 2 t o the power ol) many d i f f er ent convergence types; or tha t there exist only three possible convergence types, namely: ( i ) the case of an increasing cofinal wl-sequence; ( i i ) the type of the dire c t product W x W1 ; ( i i i ) the type of the ordered s e t (under inclusion) of a l l f i n i t e subsets of W 1 ; t h i s r e s u l t i s due t o TODORCEVIC 1984.
EXERCISE 3
-
THE POSSIBLE COFINAL RESTRICTIONS OF A DENUMERABLE PARTIAL ORDERING
I n the particular case of a denumerable, directed p artia l ordering, we know by ch.4 5 5.5 t h a t there ex i s t s e i t h e r a maximum, or a cofinal re stric tion which i s isomorphic with I n the general denumerable case, show t h at the possible cofinal re stric tions are the following: (1) an antichain (with f i n i t e or denumerable c a rdina lity); (2) the union of components each of which i s isomorphic with w , with mutual incomparability for elements belonging t o different components; ( 3 ) the W-tomic t r e e , i . e . the t r e e with denumerably many edges from each vertex; ( 4 ) the union of 2 or 3 among the preceding components, with mutual incomparability . EXERCISE 4
-
ON SIERPINSKI'S PARTIAL ORDERING
In connection with Sierpinski's counterexample (ch.3 5 3.1), a pa rtia l ordering is called a Sierpinski p ar t i al ordering, i f we are i n the framework of ZF plus choice plus continuum hypothesis, so t h at the chain of real numbers i s equipotent
202
THEORY OF RELATIONS
. Take
w
to
a chain
o f r e a l s , such t h a t each f i n a l i n t e r v a l o f
R
nuum many elements; a l s o take an a r b i t r a r y w e l l - o r d e r i n g
. Then
morphic w i t h the o r d i n a l w t e d w i t h the b i c h a i n x
(mod A)
(mod A)
x
if
x, y
if
(R,U)
has conti-
R
on the base
U
A
the S i e r p i n s k i ( p a r t i a l ordering)
, iso-
IRI
associa-
i s defined on the same base by t h e conditions that
i s s t r i c t l y l e s s than
y
both modulo
are ordered i n opposite senses modulo
1 - We see t h a t every S i e r p i n s k i
A
and
R
R
and
U
U
, and
xly
.
i s f r e e i n t e r p r e t a b l e i n the associated b i -
chain; y e t the converse i s f a l s e : take the t r a n s p o s i t i o n exchanging t w o incompar a b l e elements (mod A)
5
compare w i t h ch.4 Show t h a t set
D
.
R
, there
1.6. Cof A = w
Show t h a t
i s c o f i n a l (mod A)
C
each f i n a l i n t e r v a l o f
-
i s countable:
x, y
there e x i s t s
z
>
modulo
x,y
= continuum; indeed given a countable
. More
are o n l y countably many elements having an upper bound i n D
precisely a set 2
see t h a t every chain o r a n t i c h a i n i n A
i s d i r e c t e d : given two elements
A
and modulo
U
. We
Show t h a t i f
R
iff
C
.
i s a l o c a l automorphism o f
f
has continuum many elements i n A
(ch.9
5
1.7) and i f Dom f
has continuum c a r d i n a l i t y , then t h e r e e x i s t s a r e s t r i c t i o n o f
f
t o continuum
many elements which i s a l o c a l automorphism o f the associated b i c h a i n
.
(R,U)
To see it, show t h a t ( 1 ) given a countable subset H o f Dom f , there always e x i s t s an element
u
U , and f ( u )
modulo
in
-
(Dom f )
H
, such t h a t u i s an upper bound o f
i s an upper bound o f the image s e t
and ( 2 ) every common l o c a l automorphism o f phism o f
U
and A
f"(H)
H
modulo U ;
i s s t i l l a l o c a l automor-
( i t s u f f i c e s t o prove t h i s f o r an automorphism on 2 elements).
R
3 - By an easy refinement o f the argument i n ch.1 exerc. 4, show t h a t there e x i s t ( 2 t o the continuum power) many c o f i n a l sets C (mod A) having pairwise only a countable i n t e r s e c t i o n . Now take f o r
R
a r i g i d chain o f r e a l s (BONNET 1978 p. 7 ) , i n t h i s sense that
every l o c a l automorphism ( f ( x ) # x)
of
f
. Assume w i t h o u t
R
moves o n l y countably many elements
proof that
R
x
e x i s t s and can have continuum many
elements i n each r e a l i n t e r v a l (non-singleton). Consequently, given two cofinal sets
C, C '
A/C'
has o n l y a countable domain.
i n the previous f a m i l y , every l o c a l isomorphism from
A/C
into
By completion, o b t a i n a family of ( 2 t o t h e continuum power) c o f i n a l sets
such t h a t , f o r any two such sets Fc,cl
C, C '
y i e l d i n g a countable i n t e r s e c t i o n
isomorphism from
A/C
into
every c o f i n a l s e t
X
(mod A)
that the intersection
CnX
A/C'
, there e x i s t s i n CnC'n FC,!,
R
C
a f i n a l interval
. Consequently,
a local
cannot have a c o f i n a l domain. F i n a l l y f o r
, t h e r e e x i s t s a t l e a s t one be c o f i n a l
C
i n the family,
(mod A) (comnunicated by POUZET).
such
203
CHAPTER
8
BARRIER, BARRIER SEQUENCE, FORERUNNING, EMBEDDABILITY THEOREM FOR SCATTERED CHAINS, BETTER PARTIAL ORDERING
5 1-
BARRIER, B A R R I E R
P A R T I T I O N THEOREM, S U C C E S S I V E ELEMENTS,
SQUARE OF A BARRIER Let E be a denuneyable set of natural numbers empty subsets o f
E
, whose union i s
E
. The
, and
set
U a set of finite non-
i s a barrier i f f :
U
(1) t h e elements o f U are mutually non-inclusive; ( 2 ) f o r every i n f i n i t e subset X o f E , there e x i s t s a f i n i t e i n i t i a l i n t e r v a l o f X ( i n i t i a l w i t h respect t o the usual o r d e r i n g o f the i n t e g e r s ) which belongs to
u .
Examples. For
E take the s e t o f a l l n a t u r a l sumbers , t h i s case being the most
f r e q u e n t l y encountered. Then the s e t o f s i n g l e t o n s o f t h e i n t e g e r s
i s a barrier.
S i m i l a r l y , the s e t o f unordered p a i r s o f the i n t e g e r s i s a b a r r i e r , as w e l l as the s e t o f p-element subsets o f t h e integers, f o r any given p o s i t i v e i n t e g e r Another example o f a b a r r i e r : the union o f the s e t o f p a i r s whose minimum i s and t h e s e t o f 3-element subsets w i t h minimum 1, ... , and f o r each i n t e g e r
i a 2 , the
s e t o f i-element subsets w i t h minimum
i-2
the s e t o f those elements o f
U which are subsets o f
every b a r r i e r included i n
i s thus obtained, i . e .
U
o f i n t e g e r s and r e s t r i c t i n g
U
.
0
,
.
P an i n f i n i t e subset o f t h e union U U
1.1. L e t U be a b a r r i e r and
p
P
b
is
. Then
b a r r i e r . Moreover,
by t a k i n g an i n f i n i t e s e t
t o those elements which are subsets o f t h i s i n f i -
n i t e set. The elements o f
U
P are m u t u a l l y incomparable w i t h
which are subsets o f
respect t o i n c l u s i o n . Moreover, f o r every i n f i n i t e subset exists a f i n i t e i n i t i a l interval o f For the second assertion, l e t union
u V .
For each element
X
which i s an element o f
V be a b a r r i e r included i n r of
U which i s a subset
P , there
of
X
U
.
, and P be the o f P , take an i n U
P which begins w i t h r : t h e r e e x i s t s an element o f V which i s an i n i t i a l i n t e r v a l o f R , and t h i s can o n l y be r . 0 f i n i t e subset
R
of
Note t h a t , i f
U
i s a b a r r i e r and
P an i n f i n i t e subset o f u U , then t h e s e t
U does n o t n e c e s s a r i l y form a b a r r i e r . For instance, s t a r t i n g w i t h the b a r r i e r U o f p a i r s o f integers, l e t
o f intersections o f
P
w i t h t h e elements o f
P be the s e t o f s t r i c t l y p o s i t i v e i n t e g e r s (we remove zero). Then we o b t a i n
204
THEORY OF RELATIONS
b o t h the singleton of 1 , coming from the pair {1,2} in which the singleton i s included.
{ 0 , l ) , and
for example the pair
1.2. Let U be a barrier and r a f i n i t e s e t of natural integers which i s aper i n i t i a l interval of an element of U . Then the s e t of elementsof the form x - r , where x i s any element of U beginning with the i n i t i a l interval r , i s a barrier. Let V be the s e t of our difference s e t s x - r . The union of the elements of beginning with the i n i t i a l interval r i s i n f i n i t e ; so u V i s i n f i n i t e . Any two of these elements are incomparable with respect t o inclusion: t h i s subsists when removing the i n i t i a l interval r Finally, f or every i n f i n i t e subset P of the union u V , which necessarily begins with an integer s t r i c t l y greater t h a n Max r , there e x i s t s a f i n i t e i n i t i a l interval y of P , such tha t the union r L J belongs ~ t o U , so y belongs t o V 0
0
U
.
.
1.3. Every b a r r i e r i s lexicographically well-ordered: t h i s i s a particular case 0 2 . 1 . In other words, the s e t of the elements of a ba rrie r, when ordered lexicographically s t a r t i n g with the usual ordering of the integers, forms a denumerable well-ordering.
o f ch.3
Every barrier U t h u s has a lexicographic rank, in the sense of ch.3 5 2 . 1 : the order type o f the well-ordering of the elements of U when ordered lexicographical ly. f o r example, the b a r r i er o f the singletons has rank W . The ba rrie r of the p-element subsets ( p = positive integer) has rank u p The ba rrie r formed of the pairs w i t h minimum 0 , and the 3-element subsets w i t h minimum 1 , and f o r each w integer i 3 2 , the i-element subsets with minimum i - 2 , has rank 0 .
.
f o r each denumerable indecomposable ordinal, i .e. each power of a ,say , there e x i s t s a barrier of rank 8 (POUZET 1972"). More generally: The lexicographic ranks of b ar r i er s are exactly a l l the ordinals u p ( p = positiv e integer) and ( w'( ) . p , where o( i s a denumerable ordinal and p a positiv e integer (ASSOUS 1974). 1 . 4 . BARRIER PARTITION THEOREM Let U be a b a r r i e r and E be the union of U Partition the elements of U into two complementary s e t s U ' and U" Then there e x i s t s an i n f i n i t e subset H f E such t h a t the elements of U which are subsets of H , a l l belong to U ' or a l l belong t o U" (NASH-WILLIAMS 1968). Note t h a t these elements form a b ar r i er by 1.1. I n pa rtic ula r, we obtain RAMSEY's theorem by considering a positive integer p and taking f o r U the s e t of a l l
.
.
205
Chapter 8
p-element subsets o f
E
.
0 Given two d i s t i n c t elements o f
, one i s never i n c l u d e d i n t h e o t h e r , hence
U
one i s never an i n i t i a l i n t e r v a l o f t h e o t h e r : t h e theorem now f o l l o w s f r o m NASH-WILLIAM'S theorem i n ch.3 0
5
2.4.
0
We g i v e a n o t h e r more d i r e c t p r o o f , due t o COROMINAS i n 1 9 7 0 , unpublished.
A s s o c i a t e t o each b a r r i e r
> o . Assume
U be a b a r r i e r o f r a n k M ( U )
Now l e t
every b a r r i e r o f rank s t r i c t l y lesser than for
U
b a r r i e r has rank
Let
uo
where
o(
t h a t t h e theorem h o l d s f o r
(U) : we s h a l l prove t h a t i t h o l d s
. be t h e l e a s t i n t e g e r i n U . Let
n o t belong t o set
.A
i t s l e x i c o g r a p h i c r a n k c( ( U )
U
i f f i t i s a s e t o f s i n g l e t o n s ; i n t h i s case t h e theorem i s obvious.
o
x U1
i s an element o f i s a barrier.
U f o r which t h e s i n g l e t o n o f uo does
E = u
be t h e s e t o f those elements o f t h e f o r m
U1
U
beginning w i t h the i n t e g e r
We have t h e s t r i c t i n e q u a l i t y
i n t e g e r . The p a r t i t i o n o f o f elements o f
U
U
U'
and
, hence
uo
beginning w i t h
{uo}
o((U1)
< M(U)
: indeed
uo
,
which b e g i n w i t h a s t r i c t l y g r e a t e r
U
into
-
U which b e g i n w i t h
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 g , a f t e r t h e elements o f we s t i l l have p o s t e r i o r elements o f
x
U"
gives a p a r t i t i o n o f the set
also a partition o f
. By t h e
U1
i s s t r i c t l y less than
i n d u c t i o n h y p o t h e s i s , s i n c e t h e l e x i c o g r a p h i c rank o f
U1 H ( U ) , t h e r e e x i s t s an i n f i n i t e subset H1 o f t h e u n i o n w U1 , such t h a t , l e t t i n g V1 be U1 r e s t r i c t e d t o t h e subsets o f HI , t h e n t h i s b a r r i e r V1
e i t h e r i s included i n
U'
or in
U"
(whose elements have o b v i o u s l y been modiuo ) . According t o t h e case, we say t h a t
f i e d by removing t h e i r minimum i n t e g e r uo Let
i s linked t o u1 Up
o r linked t o
be t h e l e a s t i n t e g e r i n
belong t o Let
U'
U
, since
U" H1 =
. i~
V1 ; t h e s i n g l e t o n o f
t h e r e e x i s t s an element o f
be t h e s e t o f t h o s e elements o f t h e f o r m
element o f
u1
does n o t
uo, u1
which begins w i t h
U
x
- {ul) , where
U which begins w i t h u1 and i s a subset o f H1
. The
. Then
x
i s an
U2
is a
.
U i n t o U' and U" g i v e s a p a r t i t i o n o f t h e s e t o f elements o f U b e g i n n i n g w i t h u1 , b a r r i e r whose r a n k i s s t r i c t l y l e s s t h a n hence a l s o a p a r t i t i o n o f
U2
. By
N(U)
partition of
t h e i n d u c t i o n h y p o t h e s i s , t h e r e e x i s t s an i n -
u U 2 , such t h a t , l e t t i n g V 2 be U2 H~ -, {,+\ = r e s t r i c t e d t o t h e subsets o f H2 , t h e n t h i s b a r r i e r V2 e i t h e r i s i n c l u d e d i n f i n i t e subset
U'
or in
integer linked t o
H2
of
(whose elements have been m o d i f i e d by removing t h e i r minimum
U"
u1 ). According t o t h e case, we say t h a t u1 i s l i n k e d t o U ' U"
or
.
I t e r a t i n g t h i s , we have a s t r i c t l y i n c r e a s i n g &-sequence each o f which i s l i n k e d w i t h e i t h e r of i n f i n i t e s e t s
Hi
U'
or
,
uo : by 1 . 2 above, t h e
o f integers ui , U" ; and t h e c o r r e s p o n d i n g sequence
, which i s d e c r e a s i n g w i t h r e s p e c t t o i n c l u s i o n . L e t
H
be
206
THEORY OF RELATIONS
f o r example t o uo, ul,
u2,
..
.
U'
,all
ui
an i n f i n i t e s e t o f these Then
o f which are l i n k e d t o t h e same subset, say
s a t i s f i e s our conclusion. Indeed, enumerate
H
here forming an e x t r a c t e d sequence o f the o l d
: the
ui renumber t h e corresponding b a r r i e r s
H
.
Vi
The elements o f
as
H
ui
, and
U which are subsets o f
remain p a i r w i s e incomparable w i t h respect t o i n c l u s i o n . Moreover, given an i n f i -
n i t e subset
of
P
, this P
H
ponding w i t h the b a r r i e r of
begins w i t h a c e r t a i n u (p i n t e g e r ) , corresP .And t h e i n i t i a l i n t e r v a l o f P which i s an element
V
P+i i s the union o f the s i n g l e t o n { u
U
P
1
and an element o f
Vp+,,hence
it
U' , t o which u i s l i n k e d . P
belongs t o the subset
,
The p r o o f i s f i n i s h e d ; however i n order t o see t h a t i t o n l y needs the axioms ZF avoiding f o r instance the axiom o f dependent choice when we d e f i n e t h e i n f i n i t e sequence o f sets me t h a t
, we
Hi
make p r e c i s e the c o n s t r u c t i o n o f each
y i e l d f o r instance a b a r r i e r included i n
U'
, choose
. Assu-
Hi
i s already defined. Among the i n f i n i t e subsets o f
Hi-l
Hi-l
which
those i n f i n i t e subsets which
begin w i t h t h e l e a s t p o s s i b l e i n t e g e r , say h t . Then among the i n f i n i t e subsets 1 hi and y i e l d a b a r r i e r included i n U ' , choose those i n f i n i t e 1 2 subsets which begin w i t h hi, hi where h: i s the l e a s t p o s s i b l e i n t e g e r ; and
which begin w i t h
so on. F i n a l l y we take f o r
t h e s e t o f a l l these
Hi
hr
( i fixed, k varies). 0
1.5. SUCCESSIVE ELEMENT
, we say t h a t r precedes i s a successive element o f r , denoted r
Given two f i n i t e sets o f n a t u r a l numbers or that
s
s
succeeds
r
, or that
s
and
s
ra s , i f s i s obtained from r by adding on i n t e g e r s which are a l l s t r i c t -
by
l y g r e a t e r than
Max r
and then by removing
For example, given two i n t e g e r s t o n of
b
iff
a
cedes the p a i r \b,c) (1) Let
of
U
.
a, b,
Min r
.
the s i n g l e t o n o f
Given t h r e e i n t e g e r s
a< b t c
a
precedes t h e s i n g l e -
, the p a i r
{a,b)
pre-
.
be a b a r r i e r and
r, s
two elements of
U ; p u t m = Card r
.
< Min s , then t h e r e e x i s t s a sequence o f m + l successive elements goingfrom r s ,say r = r o a r l Q r 2 Q ...ar , = s .
I f Max r
0
r
U
Let
I be an i n f i n i t e s e t o f i n t e g e r s which begins w i t h t h e i n i t i a l i n t e r v a l
f o l l o w e d by
t o the union o f
and f i n a l l y by i n t e g e r s l a r g e r than
s
U
. Start with
i t s minimum, then l e t
ro = r
and
I,,= I
r1 be t h e i n i t i a l i n t e r v a l o f
.
Max s
,all
. Then d e f i n e
belonging I1 = .I
I1which belongs t o
minus
U
,
.
so t h a t r o qrl I t e r a t e u n t i l reaching rm= s 0 ( 2 ) L e t U be a b a r r i e r and r, s be two a r b i t r a r y elements o f U . Then t h e r e e x i s t s an element t o f U and two f i n i t e sequences o f successive elements o f U , t h e f i r s t sequence going from r t o t and t h e second going from s 2 t 0 Take an a r b i t r a r y element t i n U w i t h Min t s t r i c t l y l a r g e r than both
-
-
.
207
Chapter 8
Max r and 1.6. L e t
, then
Max s
a p p l y p r o p o s i t i o n ( 1 ) above. 0
be a b a r r i e r ,
U
Given an i n t e g e r
k
be an element o f
l a r g e r t h a n o r equal t o
successive elements o f a sequence o f
r
beginning w i t h
U
successive elements of
k+l
0 F i r s t c o n s t r u c t a sequence o f
... Q sk , u s i n g
r = s o d sl(l
r
i n i t i a l interval
U
... a
so = r QslQ
and same l a s t terms, say
m
k+l
, there
e x i s t s a sequence o f
, say ro =
r
.
and m = Card r
U
, both
U
r 4 rl
m+l
. . . <1 rm and
sequences h a v i n g same f i r s t
sk = rm.
successive terms b e g i n n i n g w i t h
r
, say
an i n f i n i t e s e t o f i n t e g e r s which begins w i t h t h e
t h e i n t e g e r s b e l o n g i n g t o t h e u n i o n o f U . Since k m = Card r , t h e minimum o f sk i s an i n t e g e r s t r i c t Using t h e p r e v i o u s 1.5.(1), we c o n s t r u c t t h e sequence of
,all
i s l a r g e r t h a n o r equal t o l y l a r g e r than
m+l
1.7. L e t Let
Max r
.
U
successive elements o f
U
h
going from
a f u n c t i o n from
f
couple of successive elements
x Qy
(POUZET 1977) .
f (.t ).
Suppose on t h e c o n t r a r y t h a t
wing d i s c u s s i o n , and l e t teger
k
beginning w i t h k
choices f o r
s
kl
x
in
s t r i c t l y less than U
-
with
m = Card s
of
.
h(f(s))
f(y)
.
less than
f(t),< h ( f ( s ) ) - 1
.
U , which s h a l l be f i x e d f o r t h e f o l l o We s h a l l prove t h a t t h e r e e x i s t s an i n -
m+l
successive elements o f
U
so = s d s l c \ s2 d... (1sm , t h e v a l u e f(s,) is f(s,)< k ; i n s p i t e o f t h e i n f i n i t e l y many p o s s i b l e
...,s, .
. The
f(sl) value
by k2 = M a x ( h ( x ) ) - 1 continue t h u s l y u n t i l where
.
t h a t f o r each
, say
, say sl,
Indeed, t h e v a l u e note by
.0
, such
into W
at always i m p l i e s t h a t s
such t h a t , f o r e v e r y sequence o f
bounded by
rm= s k
f(x) t
S Q
. s
S t a r t w i t h an a r b i t r a r y element
U
we have
Then t h e r e e x i s t two successive elements
0
to
be a f u n c t i o n f r o m t h e s e t G, o f non-negative i n t e g e r s i n t o G,
be a b a r r i e r and
o r equal t o
r
i s bounded by t h e maximum f(s2)<
, where f(sm)<
x
h(f(sl))-1
h(f(s))-1
, which we de-
i s bounded by t h e maximum, denoted
...,
r u n s t h r o u g h t h e i n t e r v a l 0,1,2, kl . We , bounded by k, = M a x ( h ( x ) ) - 1 , h(f(sm-l))-l
runs through the i n t e r v a l
. We
0,1,2,...,km-1
then take
k = km + 1
,
ka m . Then by t h e p r e c e d i n g p r o p o s i t i o n , so = s 4 s l a sm and to = s Q tld . . .
We can always assume t h a t t h e bound t h e r e e x i s t two sequences, say Q
t k = sm
. The
o u r hypotheses, r e q u i r e s t h a t diction. 0
...(I
f i r s t sequence r e q u i r e s t h a t f(s,)
-
f(s)
f(s,)
< k , and
t h e second, g i v e n
3 k , hence t h a t f(s,)
2
k : contra-
THEORY OF RELATIONS
208
1.8. SQUARE AND POWER OF A BARRIER Let
U
be a b a r r i e r . To each ordered p a i r o f elements
sa t , associate the union s u t as t h a t element o f
U
. Note
s, t
of
such t h a t
U
t h a t t h i s union uniquely determines
which i s an i n i t i a l i n t e r v a l o f the union. S i m i l a r l y
s t
i s uniquely determined, as being t h e union w i t h i t s minimum element removed. The unions i n consideration are m u t u a l l y non-inclusive; moreover, every i n f i n i t e subset o f the union s e t o f
has such a union
U
s y t
Thus these unions form a b a r r i e r , c a l l e d t h e square o f For each p o s i t i v e i n t e g e r
1
U = U and Uh+'
h
as an i n i t i a l i n t e r v a l . 2 and denoted by U
.
U
, the power Uh i s d e f i n e d by i n d u c t i o n .
.
= (Uh)2
These notions w i l l be used already i n
2 - BARRIER SEQUENCE, (NASH-W I LLIAMS)
§
0
2 o f t h i s chapter, f o r b a r r i e r sequences.
THEOREM ON THE MINIMAL
BAD BARRIER
SEQUENCE
GOOD OR BAD BARRIER SEQUENCE
Given a b a r r i e r
A , we d e f i n e a b a r r i e r sequence,
and a p a r t i a l o r d e r i n g
U
U - b a r r i e r sequence i n A
more p r e c i s e l y a
f u n c t i o n w i t h domain
U
o r w i t h values i n
A
, to
such t h a t the range i s included i n t h e base
We a l s o more simply speak o f a
be a \A1
.
U-sequence.
A b a r r i e r sequence f o f U i n A i s c a l l e d goodif there e x i s t two elements s , t i n U w i t h s q t and f s g f t (mod A) ; t h e sequence f i s c a l l e d bad otherwise. I n the t r i v i a l case o f a b a r r i e r o f singletons, r e c a l l t h a t f o r u, v non-negative integers, we have {u) 4 i v ) i f f u < v i n t h e usual o r d e r i n g o f integers. We have again t h e n o t i o n o f good and bad a-sequence,
5
2.2. Consequently,
if
A
i n the sense o f ch.4
i s a w e l l p a r t i a l ordering, then t h e b a r r i e r sequen-
ces whose domain i s the b a r r i e r o f s i n g l e t o n s
are good: see ch.4
This r e s u l t does n o t extend t o a l l b a r r i e r s . Indeed, l e t o r d e r i n g o f RADO (see ch.4
5
4.2) and
U
0
, our
.
be t h e w e l l p a r t i a l
A
U
i s a bad b a r r i e r sequence:
f u n c t i o n takes t h e p a i r {x,y)
Two successive p a i r s are o f t h e form {x,y)
x (y
3.2.(2)
be the b a r r i e r o f a l l p a i r s o f non-negative
i n t e g e r s . Then the f o l l o w i n g f u n c t i o n from f o r a l l i n t e g e r s x, y ( x < y ) couple o f i n t e g e r s (x,y-x+l)
A
5
and i y , z \
i n t o the
w i t h three i n t e g e r s
. They are taken r e s p e c t i v e l y i n t o the couples (x,y-x+l) . These two couples have d i s t i n c t f i r s t terms. Therefore
and
t h e second (y,z-y+l) couple can be g r e a t e r (modulo the w e l l p a r t i a l o r d e r i n g o f RADO) than t h e f i r s t only i f y
>/
x
+
(y-x+l) = y+l : contradiction. 0
209
Chapter 8
2 . 1 . RESTRICTION OF A BARRIER SEQUENCE, PERFECT BARRIER SEQUENCE Let A be a partial ordering and U , V be barriers with V included in U As with any function, we say t h a t , given a U-sequence f , i t s re stric tion t o V denoted by f/V , i s the V-sequence taking f o r each element of V the same value as f . Let U be a b a r r i e r and A a p ar t i al ordering. A ba rrie r sequence f which i s a function from U into the base of A i s called perfect if( for each ordered pair of elements s , t in U , the condition s at implies f s 6 f t (mod A ) .
.
Let U be a b a r r i e r and f a function from U i nto a partial ordering. Then there e x i s t s a barrier V included in U , such t ha t the re stric tion f/V e ith e r perfect or bad (NASH-WILLIAMS 1965' p . 705). Consider the square U2 of the b ar r i er U , which i s formed of unions = s u t where s , t are successive elements in U . Recall tha t each element r in U 2 determines s as the i n i t i a l interval of r which belongs t o U , as well as t , which i s r with i t s minimum removed. Partition the elements r of U2 into two classes, according whether f s $ f t (modulo the part i a l ordering) or not. By the b ar r i er p ar t i t i o n theorem 1 . 4 , there e xists an i n f i n i t e s e t E of integers in which the elements of U2 are a l l in the same class; and by 1.1 they constitute a b ar r i er V . According t o whether the class in consideration corresponds t o the condition f s 6 f t or to i t s negation, the r e s t r i c t i o n f/V i s ei t h er perfect or bad. 0
0
r
2 . 2 . INF-RESTRICTION, MINIMAL BAD SEQUENCE
Let U, V be two barriers with V included in U , and l e t A be a partial ordering. A V-sequence g with values in A i s called an inf-restriction (mod A) of the U-sequence f (with values in A ) , i f for each element x of V we have g x d fx (mod A ) . We say t h a t a barrier sequence f with domain U and with values in A i s minimal bad (mod A ) i f f f i s bad and i f every bad inf-re stric tion of f i s a r e s t r i c t i o n . I n other words, f o r each barrier V included in U , each bad V-sequence g such th at g x g fx s a t i s f i e s gx = fx (and not gx < fx ) for every x in V This generalizes the notion of a minimal bad u-sequence: see ch.4 5 2 . 3 .
.
,
210
THEORY OF RELATIONS
I n t h e case where quence and we
f
i s the t r i v i a l b a r r i e r o f singletons,
U
becomes an a - s e -
f i n d t h e minimal bad O - s e q u e n c e o f ch.4 § 2.3.
THEOREM ON THE MINIMAL BAD BARRIER SEQUENCE Let
be a w e l l - f o u n d e d p a r t i a l o r d e r i n q and
A
values i n
a bad b a r r i e r sequence w i t h
f
.
A
f
and i s minimal bad (mod A) (NASH-WILLIAMS 1965', lemma 24 p . 707; uses dependent choice). 0
Let
denote t h e b a r r i e r domain o f
U
maximum i n t e g e r
, and l e t
f
ro
. Let
which i s an i n f - r e s t r i c t i o n o f
ro be an element i n
Uo
i s the l e a s t possible. Let
a
and which c o n t a i n s t h e element A
5
T h i s g e n e r a l i z e s t h e theorem on t h e minimal bad sequence i n ch.4
i n t h e sense t h a t no o t h e r c h o i c e o f
Uo
fo be a bad f u n c t i o n f r o m
and
Uo
whose
U
be a b a r r i e r i n c l u d e d i n U
and f o r which
f
2.7.
fo(ro)
into
i s minimal (mod A)
fo g i v e s a v a l u e
,
fo(ro) s t r i c t l y
l e s s t h a n t h e above value. Let
be an element o f
rl
which i s d i s t i n c t from
Uo
ger i s the l e a s t possible. Let t a i n s t h e elements
ro, r1
i s an i n f - r e s t r i c t i o n o f we o b t a i n , a barrier
U1
. Let
Uo
and which con-
be a bad f u n c t i o n f r o m
U1
into
fo and f o r which
and a f u n c t i o n
fi
i s t h e s e t o f those elements o f
Ui
be a b a r r i e r i n c l u d e d i n fl
f o r each n a t u r a l number Ui
ro and whose maximum i n t e -
it from
a f i n i t e set
Ui
into
A
ri
o f n a t u r a l numbers,
(dependent c h o i c e ) . Each
which a r e subsets o f t h e u n i o n
U
A , which
i s minimal. I t e r a t i n g t h i s ,
fl(rl)
u
,
Ui
and each o f these unions i s i n c l u d e d i n t h e p r e c e d i n g one. Note t h a t numbers
= fo(ro), and i n general
fl(rO) i and
strictly)
in
j
>
ri
. The
maximum o f
f.(r.) = fi(ri) J
ri
f o r a l l natural
1
i n c r e a s e s (however n o t always
i and tends t o i n f i n i t y , s i n c e t h e r e a r e o n l y f i n i t e l y many s e t s
of n a t u r a l numbers the
i
h a v i n g a g i v e n maximum. We s h a l l p r o v e t h a t t h e s e t
of
V
i s a barrier. ri
Indeed, two d i s t i n c t
a r e m u t u a l l y n o n - i n c l u s i v e , s i n c e t h e y belong t o
Moreover, e v e r y element
r
in
U
which i s i n c l u d e d i n t h e u n i o n o f t h e
.
U
ri
, i t s u f f i c e s t o see t h a t t h e r e o n l y e x i s t f i n i t e l y many s e t s o f n a t u r a l numbers w i t h maximum a , hence t h a t i s an
ri : l e t t i n g
r w i l l be one of t h e and u U1 and u U 2 i n t h e u n i o n of t h e
a
denote t h e maximum o f
ri
o r o v i d e d t h a t i t i s s i m u l t a n e o u s l y i n c l u d e d i n u Uo i s indeed t h e case, s i n c e r i s i n c l u d e d
... . Here t h i s ri . F i n a l l y t h e
as t h e i n t e r s e c t i o n o f t h e
Ui
Let
g
g(ri) f
. We
ri
, as
barrier
i s the set o f the
V
ri
. Moreover t h e
for
i = 0,1,2,
s h a l l prove t h a t
g
union
w e l l as t h e i n t e r s e c t i o n o f t h e
denote t h e b a r r i e r sequence w i t h domain = fi(ri)
, as
ri
, as w e l l as t h e s e t o f those elements o f
which a r e i n c l u d e d i n t h e u n i o n o f t h e t o t h e u n i o n of t h e
r
...
. Note
that
V
g
,
u
u V
well
U
i s equal
.
Ui and which i s d e f i n e d by
i s a bad i n f - r e s t r i c t i o n o f
i s minimal bad. Suppose t h e c o n t r a r y : t h e r e e x i s t s
21 1
Chapter 8
i n c l u d e d i n V , and a bad W-sequence h which i s an i n f - r e s t r i c , and an i n t e g e r i f o r which ri E W w i t h h ( r i ) < g(ri) (mod A) .
a barrier tion of
W
g
be t h e l e a s t i f o r which t h i s h o l d s . C a l l W+ t h e s u b - b a r r i e r o f V d e f i n e d by u W+ = t h e u n i o n o f ro, o f rl, .. . , o f rn-l and o f a l l t h e e l e -
Let
n
ments o f
W : hence
+ h (ri)
with
particular r 6 W+
:
W
= h(ri)
h+(rn)
c W+s V . for
rig W
h(rn)<
=
Let
h+
, and
be t h e b a r r i e r sequence w i t h domain W+
+ h (ri) .
g ( r n ) (mod A)
hence o u r b a r r i e r sequence
h+
= g(ri)
Note t h a t
i = O,l,
...,n - 1
we have
h(ri)
+
1
. Since
a contradiction with the minimality o f element
h+(r)$
h+
, and e i t h e r
= g(ri)
h ( r . ) = g(ri)
+
W -W ; so i n
ri E
h+(rn)<
fn
f o r every
g(r)
.
g
i s an i n f - r e s t r i c t i o n o f
To o b t a i n a c o n t r a d i c t i o n , i t s u f f i c e s t o show t h a t b o t h cases we have
for
i s bad. Indeed f o r or
ri E W+-W
g ( r n ) (mod A)
ri E W : i n
, we
obtain
, more p r e c i s e l y t h e m i n i m a l i t y o f t h e
f n ( r n ) = g ( r n ) : see t h e i t e r a t i v e d e f i n i t i o n o f
fo, fl,
... , fn
in
o u r second paragraph. Thus suppose t h a t
h+
i s good, and o b t a i n a c o n t r a d i c t i o n as f o l l o w s . By o u r
assumption, t h e r e e x i s t two successive elements i n t h e b a r r i e r W+ , say s q t w i t h h + ( s ) s h + ( t ) . Then s CZW , f o r o t h e r w i s e h+(s) = g(s),< h + ( t ) d g ( t ) (mod A )
, so
g
would be good; c o n t r a d i c t i o n .
From t h e f a c t t h a t t i n c t f r o m each
s EW
, we deduce t h a t h+(s)
ri (i= O , l ,
...,n-1)
= h(s)
.
Moreover
s
i s dis-
; f o r o t h e r w i s e we would have t h e f o l l o w i n g
c o n d i t i o n s : h+(s) = h ( s ) = g(s),( h + ( t ) $ g ( t ) , t h u s g would be good. S i n c e t h e maximum o f ri i n c r e a s e s i n i , t h e maximum o f s i s g r e a t e r t h a n o r equal t o t h e maximum o f
u
W
ri ( i = O , l ,
t h e elements o f t h e
§
3
3.1.
rn . Hence e v e r y element o f
, e i t h e r because i t belongs t o
s
, o r because
..., n ) .
Finally
, so h i s good: c o n t r a d i c t i o n .
h+(t) = h ( t )
belongs t o t h e union
t
i t i s s t r i c t l y greater than
t6W
h ( s ) = h+(s)
so
<
0
- FORERUNNINGBETWEEN B A R R I E R S A N D
B A R R I E R SEQUENCES
FORERUNNER AND SUCCESSOR BARRIER
Given a b a r r i e r
U
, recall
that the m i o n u U
i s t h e i n f i n i t e s e t o f those
.
fore-
We say t h a t a b a r r i e r U i s a i n t e g e r s which belong t o elements o f U r u n n e r o f V o r t h a t U f o r e r u n s V , o r a g a i n t h a t V i s a successor o f w i t h respect t o forerunning, i f the union U V element i n
V
i s included i n
UU
has a (unique) i n i t i a l i n t e r v a l b e l o n g i n g t o
U
.
i s a successor o f
U
. The
F o r example, each b a r r i e r i n c l u d e d i n U a l l pairs of
,
natural
U
and each barrier o f
numbers i s a successor o f t h e b a r r i e r o f a l l s i n g l e t o n s .
The b a r r i e r o f t h o s e p a i r s which a r e subsets o f an a r b i t r a r y i n f i n i t e s e t o f
212
THEORY OF RELATIONS
integers i s a successor of the b a r r i e r of a l l singletons. The relation of forerunning i s reflexive, antisymmetric and t r a n s i t i v e : i t i s a partial ordering on the s e t of the b a r r i e r s . The notion of forerunning, f o r barriers and b a r r i e r sequences, i s introduced by NASH-WILLIAMS 1965' p . 714; the following r e s u l t s a b o u t forerunning are a t l e a s t implicitly obtaines by him, b u t some of them a r e improved and c l a r i f i e d by LAVER 1978 and ROSENSTEIN 1982. 3.2. COMPLETION OF A SUCCESSOR BARRIER Let U be a b a r r i e r and V a successor of U , with respect t o forerunning; assume t h a t V i s n o t simply a sub-barrier of U Let p be the l e a s t maximum integer of those elements of U which are subsets of u V y e t do not belong t o V Call W the s e t of those elements of U which are n o t subsets of u V y e t which are formed of natural numbers 6 p or of elements of u V . Then V u W i s a b a r r i e r , and furthermore a successor of U Note t h a t p can be replaced by any smaller n a t u r a l number. I n the particular case where V i s a sub-barrier of U , take p t o be an arbi-
.
.
.
t r a r y natural number. Then the preceding definition o f conclusion s t i l l holds with the same following proof.
W
i s preserved and our
The s e t s V and W are obviously d i s j o i n t . Prove f i r s t t h a t any elements s in V a n d t in W are incomparable with respect t o inclusion. Indeed, there e x i s t s an i n i t i a l interval r of s with r.s U . If s c_t then r s s ~ t _ and, since r and t belong t o U , we have r = s = t . Yet r i s a subset of u V and t i s not: contradiction. Now i f t s s , then t i s a subset of of u V : contradiction. Now l e t X be an i n f i n i t e s e t of integers in u ( V u W ) , thus a s e t of integers p o r which belong t o u V . We shall prove t h a t there e x i s t s an i n i t i a l interval of X , which belongs e i t h e r t o V or t o W . Since U i s a b a r r i e r , there e x i s t s an i n i t i a l interval s of X , which belongs t o U Then e i t h e r s i s n o t a subset of yV , and hence s E W and we are finished. Or s i s a subset of u V . I n t h i s case, e i t h e r s E V and we are finished. Or s V and then Max s 3 p by the definition of the integer p . Then the elements of X which are s t r i c t l y greater t h a n Max s a l l belong t o W V , as well as the elements of s . Thus there e x i s t s an i n i t i a l interval of X which belongs
0
.
4
to
v . 0
3.3. FORERUNNER AND SUCCESSOR BARRIER SEQUENCE Let A be a p a r t i a l ordering and 6 be any function, which t o each element in the base I A I associates an ordinal. Let U be a b a r r i e r and l e t f be a U-sequence w i t h values in A Similarly l e t V be a successor b a r r i e r of
.
U
213
Chapter 8
and
g
be a
g (mod A,
s)
V-sequence w i t h v a l u e s i n or that
g
r u n n i n g , i f f o r each element which belongs t o
s
C t
and t h e n
gt
< fs
in
t
, either
U
A
.
We say t h a t
i s a successor o f
f (mod A,
i s a forerunner o f
f
s)
w i t h respect t o fore-
and f o r t h e i n i t i a l i n t e r v a l
V
s
of
t
s = t and t h e n g t = f s ; o r we have and f u r t h e r m o r e S g t < J f s ( i n t h e usual
we have t h a t
(mod A)
ordering o f ordinals). F o r e r u n n i n g f o r b a r r i e r sequences i s r e f l e x i v e , a n t i s y m m e t r i c and t r a n s i t i v e :
A ,
i t d e f i n e s a p a r t i a l o r d e r i n g on t h e s e t o f b a r r i e r sequences w i t h values i n
S.
f o r a given function
I n following applications, specially i n next
$
r e l a t i o n on a s e t o f c h a i n s , and which t o each c h a i n
9 4 , we
take
t o be t h e embedding
A
the (non-strictly) increasing function
a s s o c i a t e s t h e neighborhood rank o f
x
s
F o r t h i s reason we w i l l c a l l
$
the present theory, t h a t
(see ch.6 § 2.4).
x
a r a n k i n g f u n c t i o n ; however we o n l y need, i n
A
a s s o c i a t e s t o each element o f
any o r d i n a l value.
3.4. COMPLETION OF A SUCCESSOR BARRIER SEQUENCE (1) L e t
Let in
be a
f
8
be a p a r t i a l o r d e r i n g and
A
function w i t h o r d i n a l values). L e t
U-sequence w i t h values i n
A ; assume t h a t
g
belong t o
V
.
Call W
s ) . Let
(mod A,
f
t h e s e t o f those elements i n
U
V
equal t o
u f
( 2 ) Moreover
i s a b a r r i e r by
W
3.2, t h e b a r r i e r sequence
be t h e l e a s t
. , defined
h
, i s a successor o f f
(mod A,
t o be
$) .
W
and t o
i f
f
! Ig a r e bad, t h e n h i s bad (ROSENSTEIN 1982 ch.10). p
on
p
on
As i n 3.2 above,
g
V
can be r e p l a c e d by any s m a l l e r n a t u r a l
t h e p a r t i c u l a r case where
i s a sub-barrier o f
V
.
which a r e n o t subsets of
v V , y e t a r e formed o f i n t e g e r s 6 p o r b e l o n g i n g t o u V Then as
U
u V y e t do n o t
which a r e subsets o f
U
( i . e . any
a successor o f
V
, and g be a V-sequence w i t h values
A
i s a successor o f
maximum i n t e g e r o f those elements i n
A
be a r a n k i n g f u n c t i o n on
be a b a r r i e r and
U
U
number. Moreover i n
, take
t o be an a r b i t r a -
p
r y n a t u r a l number. 0
(1) F o r e v e r y
interval s = t
s E W
of
s
we have
t
gt = fs
by h y p o t h e s i s we have and 0
, we have hs
$ ht < Sfs .
= fs
U
which belongs t o gt
f
and
g
s'
and
s"
(mod A)
c o n s i d e r two elements
and by h y p o t h e s i s s'
Suppose t h a t
and s'E W
and
Sgt
and f o r t h e i n i t i a l
t E V
ht =
. Then fs . In
t h e case
sc t
< Sfs
; hence
ht (fs
(mod A)
have
ht = gt
i n t h e case
0
a r e bad, t h e n p r o v e t h a t of
We d i s t i n g u i s h f o u r cases. I f s ' and argue i f
For every
by h y p o t h e s i s , hence a g a i n
( 2 ) Now suppose t h a t
hs" = gs"
.
, we
s"
gs'+
s"
, with
belong t o
gs" (mod A)
S'Q
s"
V , then
i s bad. F o r t h i s
. hs' = gs'
, hence h s ' $ hs"
and
. Similarly
W , by r e p l a c i n g g by f . E V , and l e t s be t h e i n i t i a l i n t e r v a l o f s "
belong t o and
V v W s"
h
THEORY OF RELATIONS
214
which belongs t o U , hence s i s n o t included in s ' (since S ' E U ) , hence S ' Q s . Then hs' = f s ' and hs" = g s " 6 f s (mod A) . Since f i s bad, we have f s ' + f s (mod A) : thus h s ' + hs" (mod A) . Finally suppose t h a t s ' 6 V and s " e W , and l e t s be the i n i t i a l interval of s ' which belongs t o U ; hence s 4 s " . Then e i t h e r s = s ' , in which case we have hs' = gs' = f s and hs" = f s " ; and by hypothesis f s p f s " (mod A) , so t h a t h s ' 4 hs" (mod A) Or s c s ' , in which case s $ V , y e t s i s a subset of u V , hence Max s > / p . I n t h i s case i t follows t h a t s , s ' and s " are subsets of uV , hence s " Q W : contradiction. 0
.
3.5. MINIMAL BAD BARRIER SEQUENCE WITH RESPECT TO FORERUNNING Let A be a Dartial ordering, and cf be a ranking function with domain I A I . Say t h a t a b a r r i e r sequence f with values in A i s minimal bad (mod A, with respect t o forerunning, i f f i s b a d , and i f every bad successor of f reduces t o a r e s t r i c t i o n of f . I n other words, denoting by U the b a r r i e r Dom f , there does n o t e x i s t any barrier V successor of U , non-included in U , with a V-sequence which i s a bad successor of f (mod A, ).
s)
s
s
.
Let A be a p a r t i a l ordering, and a rankinq function with Dom $ = 1 A I Let U be a b a r r i e r and f a bad U-sequence with values in A Then t& e x i s t s a minimal bad barrier sequence (mod A, $ ) , wh&hLs a successor of f ( LAVER 1978 ; uses deoendent choice). Note the analogy with the theorem on the minimal bad b a r r i e r sequence in 2 . 2 above. However here A i s an a r b i t r a r y p a r t i a l ordering, n o t necessarily wellfounded; indeed we shall use the present proposition in order t o prove t h a t certain partial orderings are well-founded, and even are well p a r t i a l orderings.
.
I n compensation, the present statement uses the ranking function associated with the partial ordering A
.
.
Set Vo = U a n d f o = f Suppose t h a t there e x i s t s a b a r r i e r V1 , a successor of U which does n o t reduce t o a sub-barrier of U , a n d a bad b a r r i e r sequence f l with domain V1 , a successor of f (mod A, ) Choose V1 and f l such t h a t the l e a s t maximum integer p,, of those elements of U which are subsets of u V1 y e t n o t elements of V1 i s the l e a s t possible. Using the preceding completion statement, we can assume t h a t u V1 contains a l l those integers 6 po which belong t o u U . I t e r a t e t h i s (axiom of dependent choice). For each natural integer i , we have a barrier V i and a bad Vi-sequence f i , with f i + l a successor of f i modu10 A, 6 If one of these f i i s minimal, we are finished, by using the t r a n s i t i v i t y of forerunning.
0
6 .
.
Chapter 8
215
Suppose the contrary: f o r each integer index i , the ba rrie r Vi+l i s not a sub-barrier of V i Then t o each i i s associated an integer pi , the l e a st possible maximum of those elements of V i which are subsets of the y et n o t elements of V i t l By the t r a n s i t i v i t y of forerunning, union s e t v V i t l we see t h a t pi increases in i , t h o u g h not necessarily s t r i c t l y . Indeed i f we had p i + l < pi , then the b ar r i er sequence f i + * , or another giving pi+l or giving an integer 6 pi+l , would appear before f i + l . However pi tends t o i n f i n i t y , since there are only f i n i t e l y many s e t s of integers with maximum pi , and a given f i n i t e s e t can a t most once be an element of Vi without being an By the preceding 3.4, we can assum t h a t each p i belongs t o element of Vi+l for a l l j i , hence f o r a l l integers j . Same remark f o r the union s e t u V j pi . any element of u V i which i s The union s e t s uVi form a decreasing 0 -sequence with respect t o inclusion. Let H be t h e i r intersection, which i s i n f i n i t e since every pi belongs t o H . For any i n f i n i t e subset X of H and any integer i , there e xists a (unique) This si f i n i t e i n i t i a l interval s i of X which belongs t o the ba rrie r V i increases in i , although not necessarily s t r i c t l y . Because of the decreasing values $ f i ( s i ) , f o r X fixed, there ex i s t s an i from which point on s i remains constant. Call s t h i s constant s e t , and note t h a t , when X varies, these s form a b a r r i er V with u V = H . Indeed by construction each i n f i n i t e subset of H admits a certain s as an i n i t i a l i nte rva l; moreover given any two such f i n i t e i e t s s and t , there ex i s t s an i from which point on s and t are b o t h elements of the b ar r i er V i : so t h a t they are incomparable with respect t o inclusion. Finally the b ar r i er V i s a successor of each V i . Define the V-sequence g as follows. For each s E V , take an i from which point on s E V i : then we s e t gs = f i ( s ) , t h i s value being independent from the chosen i . The b ar r i er sequence g so defined, i s bad, since the f i are bad. Also g i s a successor of each f i , in p ar tic ula r a successor of f o = f (mod A, 6 ) . I t remains t o see t hat g i s minimal. Take an ar bitra ry ba rrie r W successor of V , which i s not simply a sub-barrier of V Take a W-sequence h which i s a bad successor of g (mod A, ) . Then there e x i s t s an element t of W with an i n i t i a l interval s belonging t o V and d i s t i n c t from t , so s c t Consider the l e a s t i f o r which we have simultaneously pi > Max s and s an By t r a n s i t i v i t y h i s a bad successor of f i The s e t s element o f V i belongs t o V i y e t not t o W , hence h should have been taken instead of f i + l , because i t leads t o Max s ( o r t o a l es s er integer) instead o f pi : contradiction. 0
.
.
.
.
s
.
.
.
.
216
THEORY OF RELATIONS
5 4
-
EMBEDDABILITY
(LAVER)
THEOREM FOR SCATTERED CHAINS
be a s e t o f h-indecomposable c h a i n s (see ch.6 § 5 ) , which i s q u a s i 4.1. L e t o r d e r e d under e m b e d d a b i l i t y . Assume t h a t & i s c l o s e d w i t h r e s p e c t t o t a k i n g
s
any h-indecomposable i n t e r v a l o f a c h a i n . L e t
be t h e r a n k i n g f u n c t i o n which
a s s o c i a t e s t o each c h a i n i t s neighborhood rank (ch.6 Let
U
be a b a r r i e r and
f
be a bad
V
a bad V-sequence -
which i s a successor o f
successor o f
64 which
The c h a i n s i n
2.4).
44 . Then there
U which i s n o t a s u b - b a r r i e r o f U , and
exists a barrier
0
5
U-sequence w i t h values i n
& , 6 ) (uses
(mod
f
belong t o t h e range o f
f
axiom o f c h o i c e ) .
are a l l i n f i n i t e . For other-
wise, e i t h e r t h e empty c h a i n o r a c h a i n c o n s i s t i n g o f a s i n g l e t o n would b e l o n g Rng f , and so t h e b a r r i e r sequence f would be good. P a r t i t i o n t h e elements o f U i n t o two s e t s , a c c o r d i n g t o whether t h e image under f i s r i g h t o r l e f t indecomposable. A t l e a s t one o f these s e t s i n c l u d e s a b a r r i e r ,
to
which we a g a i n denote by U : see p a r t i t i o n theorem 1.4. To f i x i d e a s , suppose t h a t a l l t h e c h a i n s a r e r i g h t indecomposable. Compose t h e f u n c t i o n
f
8
with the function
which t o each c h a i n a s s o c i a t e s i t s
8
c o f i n a l i t y : we o b t a i n a b a r r i e r sequence
.f t a k i n g o r d i n a l values, and more p r e c i s e l y values which a r e r e g u l a r a l e p h s . I t f o l l o w s t h a t 8 .f has no bad r e s t r i c t i o n : t a k e an
w -sequence of successive elements so 4 s1 4 . . . Q siQ..
.in
u
cannot be s t r i c t l y ( i n a t u r a l number); t h e n t h e values y ,,f(si) decreasing. By 2.1, t h e r e e x i s t s a b a r r i e r i s p e r f e c t . L e t us denote t h i s t h a t t h e r e s t r i c t i o n of b a r r i e r a g a i n by U .
2(
.f t o t h i s b a r r i e r i s p e r f e c t . L e t us denote t h i s
To each h-indecomposable c h a i n
8 (A)
along the ordinal
w i t h t h e c o n d i t i o n t h a t each rank of
, and
A
i n which
Ai
A
,of
, associate
in
a decomposition i n t o a sum,
Ai ( i < x ( A ) ); has neighborhood rank s t r i c t l y l e s s t h a n t h e
h-indecomposable i n t e r v a l s Ai
t h a t f o r each
i
a r e Z ( A ) many A . ( i C j < J 5. T h i s decomposition o f each A
, there
i s embeddable: see ch.6
5
(A) ) into
t h e Ai i s chosen once f o r a l l and s h a l l be c a l l e d t h e s t a n d a r d decomposition (axiom o f c h o i c e ) . Since
f
i s bad, f o r e v e r y
embeddability fs' = j
f s d fs'
A j . Since
8
.
s
and
s'
in
U
with
s d s '
, we
Consider t h e s t a n d a r d decompositions i s p e r f e c t , we have
r f s
<
Yfs' runs through a r e g u l a r a l e p h a t l e a s t as l a r g e as t h a t which of
have t h e non-
fs =
E
Ai
, hence
and
the index
i r u n s through.
I t f o l l o w s t h a t t h e r e e x i s t s a t l e a s t one i n t e r v a l
i n any
A' j
t = s v s'
. The . lhe
f i r s t o f these
Ai
Ai which i s n o t embeddable s h a l l be a s s o c i a t e d w i t h t h e u n i o n
s e t o f these unions i s a b a r r i e r V square o f a s u b - b a r r i e r o f U , hence a successor o f
. More U
.
precisely
V
i s the
Chapter 8
Now we define the b a r r i e r sequence g w i t h domain a successor of f (mod L/z , 6 ) : indeed f o r each i n t e r v a l s of t belonging t o U , we have s c b i l i t y . Furthermore by ch.6 5 5.3, t h e i n t e r v a l Ai s t r i c t l y l e s s than t h e rank of A : i n o t h e r words I t remains t o see t h a t g i s bad. Let t and t '
t at'
217
g t = Ai . This g i s in V and f o r the i n i t i a l t and g t < f s under embeddahas a neighborhood rank sgt 4 Sfs . be elements of V such t h a t V
, by
t
.
Then t i s a union s u s ' w i t h sas'; s i m i l a r l y t ' i s of the form with s ' q s" and s , s ' , s " elements of U . The chain g t i s an i n t e r v a l Ai of the standard decomposition of A = f s ; s i m i l a r l y g t ' i s an i n t e r v a l A' of the standard decomposition of A' = f s ' ; and f i n a l l y Ai has j been defined t o be non-embeddable i n any term of t h e standard decomposition of A ' : so AiCA: , i . e . g t b g t ' . 0 s'
u
s"
3
4.2. Every s e t of h-indecomoosable chains forms a well quasi-ordering under embeddab i l i t y (uses axiom of choice). I t i s even a b e t t e r quasi-ordering, i n t h e sense of 5 5 and 6 below. 0 Suppose t h e c o n t r a r y , t h a t t h e r e ? x i s t s an w-sequence taking h-indecomposable
chains a s values, which i s bad w i t h respect t o embeddability (ch.4 5 3 . 2 . ( 2 ) ) . Consider this w -sequence a s a b a r r i e r sequence on s i n g l e t o n s . By 3 . 5 , there e x i s t s a minimal bad b a r r i e r sequence which i s a successor of the above, modulo the p a r t i a l ordering of embeddability and the ranking function which, t o each chain, a s s o c i a t e s i t s neighborhood rank. B u t t h e preceding proposition 4 . 1 (axiom of choice) proves, on the contrary, t h a t every bad b a r r i e r sequence of t h a t kind, has a bad successor b a r r i e r sequence which i s not simply a r e s t r i c t i o n t o a sub-barrier: c o n t r a d i c t i o n . 0 4.3. Every s c a t t e r e d chain i s a f i n i t e sum of h-indecomposable chains (uses axiom of c h o i c e ) . In p a r t i c u l a r , every indecomposable s c a t t e r e d chain i s h-indecomposabie. In other words, f o r a s c a t t e r e d chain, t h e notion of indecomposability coincides with t h a t of h-indecomposability. By t h e preceding p r o p o s i t i o n , every set of h-indecomposable chains i s well quasi-ordered under embeddability. Our proposition now follows from ch.6 5 5.4 and 5 . 5 . 0
0
4.4. EMBEDDABILITY THEOREM FOR SCATTERED CHAINS Every s e t of s c a t t e r e d chains forms a well quasi-ordering under embaddability (LAVER 1968; s e e a l s o 1971; uses axiom of c h o i c e ) . 0
The r e l a t i o n of embeddability f o r h-indecomposable c h a i n s , forms a well quasi-
218
THEORY OF RELATIONS
ordering, by the preceding 4.2 (axiom of choice). Now use HIGMAN's theorem on words, ch.4 5 4.4. The relation of embeddability between words, or f i n i t e sequences of h-indecomposable chains, i s also a well quasi-ordering. The same holds f o r the embeddability among f i n i t e sums of h-indecomposable chains: indeed embeddabil i t y on sums i s a reinforcement of embeddability on sequences. By 4.3, every scattered chain i s a f i n i t e sum of h-indecomposable chains: our theorem i s proved.
I n LAVER 1971, t h i s theorem i s extended t o countable unions of scattered chains. Unfortunately, the proof i s t o o complicated t o be presented here. I n COROMINAS 1984, the theorem i s extended t o countable t r e e s . COVERING BY A FINITE NUMBER OF DOUBLETS Given a scattered chain, the equivalence relation of covering by doublets of indecomposable chains (see ch.6 5 4.5 and 4.6) has only a f i n i t e number of equivalence classes. Each class i s an indecomposable interval o r a doublet (uses axiom of choice). 4.5.
Using the preceding 4.3, decompose our chain - n t o a f i n i t e number of r i g h t or l e f t indecomposable i n t e r v a l s . Replace any two contiguous such intervals by t h e i r union, provided t h i s union i s indecomposable. When i t becomes impossible t o e f f e c t these replacements, then the intervals thus obtained, or the unions of two contiguous i n t e r v a l s , constitute the covering by doublets. The uniqueness of t h i s decomposition follows from ch.6 propositions 4.1 and 4.2. 0
0
For a non-scattered chain, the relation of covering by right or l e f t indecomposable chains, i s s t i l l an equivalence relation: see ch.6 5 4.3. Hence the union of both equivalence r e l a t i o n s , i s again an equivalence r e l a t i o n . However there can be i n f i n i t e l y many equivalence classes f o r t h i s equivalence relation (BONNET 1972, unpublished; uses axiom of choice). S t a r t with A. = the chain of the r e a l s . By ch.5 5 5.2 (DUSHNIK, M I L L E R ) , we have a s t r i c t l y decreasing (with respect t o embeddability) a-sequence of chains Ai ( i integer) , where each Ai has cardinality of the continuum; moreover we can require t h a t Ai $ w f o r each i (same paragraph, propo-
0
.
for s i t i o n ( 3 ) ; uses axiom of choice). On the other hand, we have Q,< Ai i ; indeed Ai has a t l e a s t W1 many elements, and neither the ordinal nor i t s converse i s embeddable in the r e a l s , hence in Ai : use ch.5 5 3.5 the particular case where o( = 1 T h u s Ai Q + Ai+l + Q +...+ Q + Ai+h for any two natural numbers i and h . Let U = U1 + W1 and consider the sum of the O-sequence A. + U t A1
each W1
in
.
... +
U + A.1 + U +
... . We
+ U +
shall prove t h a t each interval isomorphic with
i s one of the desired equivalence classes; hence t h a t there e x i s t i n f i n i t e l y
U
0
219
Chapter 8
many equivalence c l a s s e s . Indeed, t a k e two elements x
U and
belongs t o
We must j o i n
x
to
x
and y
i n two c o n s e c u t i v e components: f o r example
belongs t o
y
U
f o l l o w i n g t h e considered component
Ai
.
by f i n i t e l y many i n t e r m e d i a t e elements, such t h a t any two
y
consecutive elements be e i t h e r r i g h t e q u i v a l e n t ( i . e .
covered by a same r i g h t i n -
decomposable i n t e r v a l ) o r l e f t e q u i v a l e n t . We can assume t h a t
x
a r e them-
and y
selves c o n s e c u t i v e elements; t h e n i t s u f f i c e s t o see t h a t t h e y a r e n e i t h e r r i g h t nor l e f t equivalent. First,
I
a non-final i n t e r v a l
which c o n t a i n s
x
and y
i s o b v i o u s l y decompo-
I
sable i n t o a f i n i t e sequence o f d i s j o i n t s u b - i n t e r v a l s i n which embedded.
cannot be
Secondly, a f i n a l i n t e r v a l i s o b v i o u s l y n o t l e f t indecomposable; n o r
, for
i s i t r i g h t indecomposable; f o r o t h e r w i s e , i t would be necessary t h a t Ai U + Ai+l + U + +U+
...
example, be embeddable i n a sum o f t h e f o r m an i n t e r v a l o f
U
which i s a r e s t r i c t i o n o f
Ai
. But
i s countable, since i t i s iso-
morphic w i t h t h e u n i o n o f a w e l l - o r d e r e d s e t o f r e a l s and t h e converse o f such a w e l l - o r d e r e d s e t . So i t must be t h a t
i s embeddable i n
Ai
Q
+ Q + ...
+
, contradicting
the prerious discussion. 0 + Q + The p r e c e d i n g c h a i n w i l l be c o n s i d e r e d a g a i n i n e x e r c i s e 3. 4.6. L e t
A
be an i n f i n i t e s c a t t e r e d c h a i n . I f e v e r y c h a i n
then
X.2 6 A , unpublished).
A
X< A
satisfies
, i s indecomposable (uses axiom o f choice; HAGENDORF 1982,
5
As a l s o n o t i c e d by HAGENDORF, i t t h e n r e s u l t s f r o m LARSON's theorem (ch.6 that 0
A
U s i n g 4.3 (axiom o f c h o i c e ) , decompose
i n t o a f i n i t e sum o f e i t h e r r i g h t
indecomposable o r l e f t indecomposable i n t e r v a l s . L e t v a l s , which i s
>/
I
or
I
be one o f t h e s e i n t e r -
a l l o t h e r s under e m b e d d a b i l i t y . We can assume t h a t
i s i n f i n i t e , and r i g h t indecomposable, t o f i x i d e a s . L e t i n t e r v a l s equimorphic w i t h i s equimorphic w i t h X = I(k-1) + 1
, this
X = 1.k
X
Let
of
A
A
I
,in
A ; then i f
, we have X <
our hypothesis. O r
4.7.
3.8)
i s e i t h e r a w e l l - o r d e r i n q o r t h e converse o f a well-ordering.
A
I.k
; then
t h e c o n s i d e r e d decomposition. E i t h e r
5
3.3) y e t
I(ktl)4 A
I
be t h e number o f
k = 1 we a r e f i n i s h e d . I f k (use ch.5
A
k
X.24A
(use again ch.5
1.k
3 2 , setting , contradicting
5
3.3);
setting
contradicts our hypothesis. 0
be an i n f i n i t e s c a t t e r e d c h a i n . I f e v e r y p r o p e r i n i t i a l i n t e r v a l
satisfies
X.24 A
, then A
i s r i g h t indecomposable (uses axiom o f
choice; HAGENDORF 1982, u n p u b l i s h e d ) . For
A
non-scattered,
we have t h e counterexample
Q + W1
.
I f the hypothesis
be an i n i t i a l i n t e r v a l s t r i c t l y l e s s t h a n A i s weakened by r e q u i r i n g t h a t 2 under e m b e d d a b i l i t y , t h e n we have t h e counterexample ( W + W)- a l r e a d y mentioned i n ch.6 5 3.4. X
220
THEORY OF RELATIONS
Decompose A i n t o a f i n i t e sum o f e i t h e r r i g h t o r l e f t indecomposable i n t e r v a l s . We can assume t h a t there are a t l e a s t two such i n t e r v a l s i n the decomposition; denote by D the l a s t such i n t e r v a l . I f D i s i n f i n i t e and l e f t indecomposable, then take an element d i n D and l e t X be t h e i n i t i a l i n t e r v a l o f A w i t h l a s t element d Then X i s equi-
.
morphic w i t h by ch.5 element
5
A
and so
A.2
equimorphic w i t h
A , thus
A
i s n o t scattered,
3.3. I f D i s i n f i n i t e and r i g h t indecomposable, then again take an d i n D and l e t X be the i n i t i a l i n t e r v a l o f A w i t h l a s t element
.
Then X . 2 6 A d generated by X.2
and X.2 has a l a s t element. Thus the i n i t i a l i n t e r v a l Y s a t i s f i e s Y . 2 5 A , and so X.4 < A Hence e i t h e r X . 2 4 X
.
and then n e i t h e r X nor A i s scattered. O r D ; so A equimorphic w i t h X + D y i e l d s A,<
XB
D yet
D and thus
i s not cofinal i n A itself i s right
indecomposable. F i n a l l y i f D reduces t o a s i n g l e t o n , then l e t t i n g X = A-D : we have X . 2 6 A and even X.2 + 1 6 A ( d i s t i n g u i s h t h e case where X has a maximum); hence X . 2 6 X so t h a t X i s n o t scattered. We o b t a i n t h e contradict i o n i n t h e f i r s t and i n t h e t h i r d cases: so t h a t o n l y t h e second case occurs, and A i s r i g h t indecomposable. 0
§
5 - INDECOMPOSABLE
SEQUENCE, BETTER PARTIAL ORDERING
TAIL OF AN ORDINAL SEQUENCE; INDECOMPOSABLE SEQUENCE Given a p a r t i a l o r d e r i n g A and an o r d i n a l sequence u o f l e n g t h o( which takes values i n A , we c a l l a o r f i n a l i n t e r v a l of u , any sequence obtained by t a k i n g a . f i n a l i n t e r v a l (3 o f the l e n g t h a( , then the r e s t r i c t i o n
tail,
u / p , then r e i n d e x i n g t h i s r e s t r i c t i o n , s u b s t i t u t i n g f o r (3 t h e o r d i n a l i s o morphic w i t h p . Recall t h a t t h e connected n o t i o n o f i n i t i a l i n t e r v a l o f a sequence, has already been introduced i n ch.4 5 2.1. An o r d i n a l sequence
u w i t h values i n A i s s a i d t o be indecomposable (mod A)
u i s embeddable (mod A) i n every non-empty t a i l o f u (embeddability among sequences has been introduced i n ch.4 9 2 ) . This requires t h a t the o r d i n a l l e n g t h of t h e sequence u be i t s e l f an indecomposable o r d i n a l , hence a power o f w : see ch.1 5 3.6. if
I n t h e c o n t r a r y case, t h e o r d i n a l sequence i s s a i d t o be decomposable (mod A ) . Note t h a t , w i t h an indecomposable o r d i n a l , u f o r example, one can c o n s t r u c t decomposable sequences o f l e n g t h U : s t a r t w i t h t h e f r e e o a r t i a l o r d e r i n g based on two elements a, b taken t o be incomparable, then take the w-sequen-
.
... We can even c o n s t r u c t W-sequences none o f whose t a i l s are indecomposable: s t a r t w i t h the i d e n t i t y p a r t i a l o r d e r i n g on a denumerable base
ce b,a,a,a, E
and take an &-sequence
in
E , without repetition.
Chapter 8
22 1
5.1. o( -BETTER PARTIAL ORDERING Let o( be an indecomposable ordinal. A p ar t i al ordering A i s said t o be an m -better partial orderinq i f every cx -sequence in A has a non-empty indecomposable t a i l . Obvious definition of a ( < H ) or ( 6 a ) - b e t t e r partial ordering.
Every well partial ordering i s an w - b et t er p ar t i a l orderinq, and conversely. The "converse" part uses dependent choice; however ZF suffices i f the given part i a l ordering i s denumerable. Let A be a well p ar t i al ordering, and u be an w-sequence in A , with terms ui ( i natural number ) . Consider those integers h such tha t there e xist only f i n i t e l y many integers i with ui 2 uh (mod A ) . Then there are only f i n i tely many such h . For otherwise, there would e x i s t an w -sequence h ( i ) ( i natural integer) with u h ( j ) < o r I u (mod A) for a l l i and j 7 i , hence h(i 1 a bad d-sequence. The t a i l obtained by beginning with an index s t r i c t l y greater t h a n these values h i s an indecomposable sequence. Conversely, suppose t h at A i s not a well p ar t i al ordering, and l e t u be a bad W -sequence: see ch.4 5 3 . 2 . ( 2 ) , dependent choice. For such a sequence, no t a i l i s embeddable in a proper t a i l of i t s e l f . 0
0
BETTER PARTIAL O R D E R I N G A Dartial ordering i s said t o be a b et t er p ar t i al ordering, i f i t i s an o( -better partial ordering f o r every ordinal o( (more precisely, f o r every indecomoosable ordinal o( ) . By the preceding, every b et t er p ar t i a l ordering i s a well partial ordering. -The only obvious example of a b et t er p ar t i al ordering i s the ordering defined on a unique element a In t h i s case, every ordinal sequence repeats the element a ; i f q i s the length of the sequence, then the indecomposable t a i l i s obtained by taking the l a s t term in the Cantor normal form of o( with respect t o base w .
.
Every r e st r i c t i o n , and every p ar t i al l y ordered reinforcement of an o( -better p a r ti a l ordering, i s an 6 - b et t er p ar t i al ordering. Same statement 'with " b e tt e r p a r t ia1 orderi ng," A quasi-ordering i s said t o be an o( -better quasi-ordering, i f the partial ordering of the equivalence classes (each formed o f elements simultaneously greater than and l e ss e r t h a n each o t h er ) , i s an o( - b et t er pa rtia l ordering. Equivalently , i f every o( -sequence has a non-empty indecomposable t a i l . A quasi-ordering i s said t o be a b et t er quasi-ordering, i f i t i s an o( -better quasi-ordering f o r every ordinal o(
.
.
5.2. The well partial ordering of RADO, defined in ch.4 5 4.2, i s an example of a well p a r t i a l ordering which i s n o t a b et t er p ar t ia l ordering.
THEORY OF RELATIONS
222
2 More exactly, Rado's well-partial ordering i s not an W -be tte r partial ordering. 0 Recall t h a t t h i s partial ordering i s defined on ordered pairs of non-negative 2 integers. Take the lexicographical LJ -sequence (O,O), ( O , l ) , (0,2), ... , ( l , O ) , ( l , l ) , ( 1 , 2 ) , ... A t a i l U i s necessarily an U2-sequence which contains, for a s u f f i c i e n t l y large integer p , the ordered pairs ( p , O ) , ( p , l ) , ( p , 2 ) , ... followed by (p+l,O) . If we attempt t o embed U in i t s t a i l which begins with (pf1,O) , then f o r each integer i the ordered pair ( p , i ) i s not embeddable before (p+i,O) Hence t o embed a l l these ordered pairs ( p , i ) where p i s fixed and i varies, we exceed a l l the terms of U : i t i s then impossible t o embed the term (p+l,O) . 0
.
.
5.3. (1) Let d be an indecomposable ordinal. If A ( 6 4 )-be tte r A with lengths partial ordering, then the s e t of ordinal indexed sequences in .--.__I_-----s t r i c t l y l e ss than o( constitutes a well quasi-ordering with respect t o embeddab i l i t y . Uses dependent choice; ZF suffices i f A i s countable. In the case where o( = w , t h i s i s j u s t HIGMAN's theorem on the well pa rtia l ordering o f words (ch.4 4.4) in view of the equivalence between well pa rtia l ordering and W - b e t t e r p ar t i al ordering, 0 5.1 above. Since A i s a ( 6 %)-better p ar t i al ordering, every sequence in A with length less than or equal t o o( i s a f i n i t e sum of indecomposable (mod A) sequences. Indeed, i t suffices t o operate regressively, s t a r t i n g with the indecomposable t a i l of the given sequence. Now consider a s e t of indecomposable (mod A) sequences of lengths s t r i c t l y le ss t h a n o( . Let B be the quasi-ordering between these sequences, with respect t o embeddability. We shall prove t h at B i s a well quasi-ordering. Take an w -sequence of these indecomposable sequences s i ( i non-negative inte ge r). Then take t h e i r sum u , whose length i s a t most equal t o o( , since a i s a power of w By hypothesis, there ex i s t s a t a i l of u which i s indecomposable; l e t s ( p integer) be i t s f i r s t term. I n p ar t i cu l ar, t h i s t a i l i s embeddable i n P i t s proper t a i l which begins with s . Then sP i s embeddable in a f i n i t e sum P+ 1 of si ( i p t l ) ; hence by indecomposability, t h i s s i s embeddable in some P si ( i 7 , p+1) . T h u s the sequence of the si i s good, and so B i s a well quasiordering (dependent choice, see ch.4 § 3 . 2 . ( 2 ) ) . By HIGMAN's theorem already mentioned, every s e t of words formed of indecomposable (mod A ) sequences of lengths s t r i c t l y l es s t h a n o< constitutes a well quasiordering. The same i s true f o r the s e t of f i n i t e sums corresponding t o these words, t h e i r quasi-ordering with respect t o embeddability being a reinforcement o f the quasi-ordering of the words: use ch.4 5 3 . 5 .(1). Since every sequence in A with length s t r i c t l y l es s than O( i s such a sum, our proposition i s proved. 0
0
.
223
Chapter 8
( 2 ) Let o( be an indecomposable ordinal and A a partial ordering. If the s e t of 4 -sequences in A constitutes a well-foundedquasi-ordering w i t h - , , r r g c t t o o( -better orderi nq. embeddabi 1 i ty , then A
an o( -sequence in A , and then the s e t of a l l non-empty t a i l s of t h i s sequence. By hypothesis one of these t a i l s i s minimal, thus minimum with respect t o embeddability, i . e . indecomposable. 0
0 Take
( 3 ) Let OC be a denumerable indecomposable ordinal and A a partial ordering. If the s e t of sequences in A with lengths s t r i c t l y less than o< constitutes a
well quasi-ordering under embeddability, then A
4 -better Ordering.
Suppose t h a t A i s not an w -better ordering. Then there e xists an o( -sequence in A which, by countability of o( , has a s t r i c t l y decreasing W-sequence of t a i l s si ( i natural number ); where i t i s understood t h a t , f o r each i , we can choose si + l t o begin with a given term of s i situated as f a r in s i as we desire. Then there e x i s t s an i n i t i a l interval to of so and a natural number i from For otherwise, every proper i n i t i a l interval of so which point on to$ s i would be embeddable in some proper i n i t i a l interval of s i f o r every i ; then by the preceding discussion we could embed a l l of so in s1 f o r example: contradiction. Let h(0) = 0 and l e t h ( 1 ) be the l e a st integer such t h a t for a l l s i . There e x i s t s an i n i t i a l interval tl of larger integers i we have and an i , from which point on we have tl.$ s i . Iterating t h i s , we sh(l) obtain a bad W-sequence, with respect t o embeddability, of intervals ti of Finally the s e t of sequences in A of lengths lengths s t r i c t l y l e ss than o< s t r i c t l y less than q , does n o t constitute a well quasi-ordering. 0
0
.
to4 .
From propositions (1) and (Z), i t follows t h at a necessary and suffic ie nt condition f o r A t o be a b et t er p ar t i al ordering, i s t h a t every s e t of ordinal sequences in A forms a well quasi-ordering under embeddability (necessity uses dependent choice). 5.4. Let o( be a denumerable indecomposable ordinal, l e t A be a (,
0 By 5.1,
A
i s a well p ar t i al ordering; so t h a t i f
U
reduces t o the barrier
of the singletons, then any U-sequence in A i s good by ch.4 5 3.2.(2). I n the following, we remove from U a l l singletons and suppose o< a t l e a s t equal t o a * For each natural number i , l e t U i be the i n f i n i t e s e t of those elements of U which begin with i These elements, when ordered lexicographically,
.
.
THEORY OF RELATIONS
224
form a sequence w i t h l e n g t h s t r i c t l y l e s s t h a n o( ; i n d e e d t h e y a r e l e s s t h a n any element of U which begins w i t h i+l . L e t f be a U-sequence i n A , and l e t denote t h e r e s t r i c t i o n of
fi
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 g . Thus
Ui
. Consider
Ui
, by r e p l a c i n g each element o f
sequence w i t h values i n A lo
t o t h e domain
f
.
l e n g t h s t r i c t l y l e s s t h a n o(
fi as an o r d i n a l by i t s h e i g h t modu-
Ui becomes a sequence i n
fi
with
A
i vary, i t follows from the preceding (dependent c h o i c e ) , t h a t t h e r e e x i s t two n a t u r a l numbers
p r o p o s i t i o n 5.3.(1)
Letting
.
i, j > i such t h a t fi i s embeddable i n f I n o t h e r words, t h e r e e x i s t s a j f u n c t i o n h from Ui i n t o U , which i s s t r i c t l y i n c r e a s i n g w i t h r e s p e c t t o t h e j l e x i c o g r a p h i c o r d e r i n g , such t h a t f o r each element
.
modulo A
s
of
, we have f s Q f h s
Ui
.
s = {i,j) i s an element o f U Then i t s and f s Q f t (mod A) , so f i s good. j ) , whose we have elements i n U b e g i n n i n g w i t h i , j , u ( u elements b e g i n n i n g w i t h j , hence w i t h j , v ( v 7 j ) . As u Suppose f i r s t t h a t
with
, hence
j
s 4t
a t most f i n i t e l y many values h
v
=
begins
images under
h
are
varies, there are
used. Indeed we can never reach t h e images under
.
v
f o r which a l l those elements b e g i n n i n g w i t h
have as images, elements b e g i n n i n g w i t h k
t = hs
I n t h e o t h e r case,
i,j+l . Hence t h e r e e x i s t s a maximum
o f elements b e g i n n i n g w i t h
t h e r e e x i s t s a maximum k
image
>/
f o l l o y e d by an i n t e g e r
j
Moreover, i,j,k
. Indeed
k
j+l s a t i s f i e s t h i s c o n d i t i o n . And on t h e o t h e r hand, t h e r e a r e o n l y f i n i t e l y
many
k
which v e r i f y t h i s c o n d i t i o n , s i n c e t h o s e
o r equal t o t h e maximum p o s s i b l e The elements o f
beginning w i t h
U
Indeed, i f an element o f j,k'&
k+l
ginning w i t h
v
U
which do so, a r e l e s s t h a n
k
. i,j,k
have images which b e g i n w i t h
beginning w i t h
i,j,k
, t h e n t h e elements b e g i n n i n g w i t h i , j , k + l
, contradicting
j,k
.
had an image b e g i n n i n g w i t h would have images be-
.
k s = {i,j,k) i s an element o f U . Then i t s image t = hs begins w i t h j , k , hence s -=It and f s Qft, (mod A ) , so f i s good. I n t h e o p p o s i t e case, we have elements o f U b e g i n n i n g w i t h i,j,k,u > k , whose images j , k ' a k+l
the maximality o f
Suppose now t h a t
under
h
begin w i t h
j,k,v7
k
.
I t e r a t i n g t h i s , we o b t a i n a s t r i c t l y i n c r e a s i n g i(2)
=
k
, ...
begin w i t h
...
i(l), s = i(O),
and
i(O),
i(l), ...
,i(r) .
..., i ( r )
, i ( r ) , ..
r
,
i(1) = j
U which
have images under
h
i(0) = i
which b e g i n w i t h
By t h e d e f i n i t i o n ' o f b a r r i e r , t h e r e e x i s t s an i s an element o f
f s ,
, hence
f
U ; i t s image
t = hs
r
satisfies
f o r which
s 4 t
i s good. 0
From t h e p r e c e d i n g p r o p o s i t i o n , i t f o l l o w s t h a t i f o r d e r i n g f o r e v e r y denumerable o r d i n a l values i n A
,
t h e elements o f
w -sequence
such t h a t , f o r each n a t u r a l number
i s good. I n o t h e r words,
cg , t h e n A
A
i s an o( - b e t t e r p a r t i a l
e v e r y b a r r i e r sequence w i t h
i s a better p a r t i a l ordering with
r e s p e c t t o b a r r i e r s , o r a - b e t t e r p a r t i a l o r d e r i n g i n t h e sense o f
5
6 below.
,
Chapter 8
225
Thus A i s a b e t t e r partial ordering, in view of the equivalence of b o t h notions, below (modulo the axiom of choice). which will be proved in 9 7 5 . 5 . Let us denote the b a r r i e r of a l l pairs of natural numbers by i t s lexicogra2 . phic rank &J Given a p a r t i a l ordering A , a necessary and s u f f i c i e n t condition f o r the partial ordering of inclusion among i n i t i a l intervals of A t . b e a well partial orderinq, i s t h a t every barrier sequence of W L into A i s good (sufficiency uses dependent choice). Recall t h a t the p a r t i a l orderinq of inclusion among i n i t i a l intervals of A i s well-founded i f f A i s a well p a r t i a l ordering, hence i f f every a - s e quence in A i s good: see ch.4 5 3.2.(2) and 4.1.
Suppose f i r s t t h a t the i n i t i a l intervals of A form a well o a r t i a l ordering under inclusion. Consider a b a r r i e r sequence from U 2 into A ; f o r each pair of natural numbers i , j > i , l e t ai . denote the value taken by t h i s barrier ,J sequence a Since A i s a well p a r t i a l ordering, extract an increasing sequence from the sequence of the a 0 , j , which we renumber by aD,l ,< aD,? 4 .... ,<. a o , j d... (mod A) : see ch.4 5 3.2.(3). Simultaneously, preserve only those
0
.
values a f o r which b o t h indices occur in the preceding extracted sequence. By a new i n f i successive i t e r a t i o n s , e x t r a c t from the double sequence of the ai ,j nite double sequence, which upon renumbering, s a t i s f i e s f o r each natural number i , the inequalities a i , i + l & a i , i + 2 4 ... ,< a i , j ,<... (mod A) f o r a l l j > i .
For each natural number i , l e t Ai be the i n i t i a l interval of A generated by the ai ( i fixed and j varies and > i ) . By hypothesis, the Ai form ,j a well p a r t i a l ordering under inclusion: t h e i r sequence i s good (see ch.4 5 3 . 2 . ( 2 ) ) . I n other words, there e x i s t two integers i and j > i with Ai included in A . Since the element ai belongs t o Ai thus t o A , then j ,j j j . < a j , k (mod A ) . B u t we have the conthere e x i s t s an integer k > j with ai , secutivity { i , j } a { j , k \ , so t h a t the barrier sequence a i s good. Conversely, suppose t h a t the i n i t i a l intervals of A do n o t form a well partial ordering under inclusion. Then e i t h e r t h i s p a r t i a l ordering o f inclusion i s n o t well-founded: then A i s n o t a well p a r t i a l ordering (ch.4 5 4.1, dependent choice), and there e x i s t s a bad a-sequence ui ( i integer) in A (ch.4 3 3 . 2 . ( 2 ) , dependent choice). I n t h i s case, the barrier sequence of W 2 i n t o A , defined by ai - ui f o r a l l j > i , i s bad. ,j Or the p a r t i a l ordering of inclusion among i n i t i a l i n t e r v a l s o f A i s wellfounded b u t not a well partial ordering: hence there e x i s t i n f i n i t e l y many mutually non-inclusive i n i t i a l i n t e r v a l s . Choose an w-sequence of such intervals AD, A1, . . . , Ai , . . ( i natural number ; uses denumerable subset axiom).
226
THEORY OF RELATIONS
which belongs t o i, j 7 i , a s s o c i a t e an ai ,j ( c o u n t a b l e axiom o f c h o i c e ) . I t s u f f i c e s t o p r o v e t h a t t h e
To each p a i r o f n a t u r a l numbers Ai
yet not t o
A
b a r r i e r sequence gers
i< j
j
a
t h u s d e f i n e d , i s bad (mod A)
we have
t h e o p p o s i t e case t h a t o f the i n i t i a l i n t e r v a l
{i,jjQ i j , k } and ai,j< aj,k , s i n c e A j , we would have
yet
.
Indeed, g i v e n any t h r e e i n t e -
aiij$
aj,k
(mod A)
a. belongs t o J ,k a. element o f 1 ,j
tion. 0
. For
in
A . by d e f i n i t i o n J A . : contradicJ
I n e x e r c i s e 2, we g i v e an example o f a w e l l p a r t i a l o r d e r i n g , a r e f i n e m e n t o f Rado's p a r t i a l o r d e r i n g , which i s n o t a b e t t e r p a r t i a l o r d e r i n g , b u t f o r which
w 2 i s good, t h u s f o r which t h e i n i t i a l
every b a r r i e r sequence d e f i n e d on
i n t e r v a l s form a w e l l p a r t i a l ordering. 5.6. Given a p a r t i a l o r d e r i n g
A ,let
JO(A) = A ; l e t
o r d e r i n g o f i n c l u s i o n among t h e i n i t i a l i n t e r v a l s o f number p
, l e t 'J p+l(A)
'J l(A)
be t h e p a r t i a l
A ; f o r every natural
be t h e p a r t i a l o r d e r i n g o f i n c l u s i o n among t h e i n i t i a l
i n t e r v a l s o f 3 (A) . P We t h e n o b t a i n t h e f o l l o w i n g g e n e r a l i z a t i o n o f t h e p r e c e d i n g p r o p o s i t i o n 5.5;
the
r e a d e r can e a s i l y adapt t h e p r e c e d i n g p r o o f t o o b t a i n t h e i n d u c t i o n s t e p f r o m p
to
p+l
.
L e t us denote by n a t u r a l numbers
),-I
t h e b a r r i e r formed o f a l l p-element subsets of t h e s e t o f
. Given
a p a r t i a l ordering
t i o n f o r the p a r t i a l ordering
3 P (A)
e v e r y b a r r i e r sequence f r o m
ptl
A
, a necessary and s u f f i c i e n t c o n d i -
t o be a w e l l p a r t i a l O r d e r i n g
into
i s that
A i s good ( s u f f i c i e n c y uses depen-
dent choice). I n t h e a l r e a d y mentioned e x e r c i s e 2, we g i v e , f o r each p o s i t i v e i n t e g e r p , an example o f a w e l l p a r t i a l o r d e r i n g which i s n o t a b e t t e r p a r t i a l o r d e r i n g , f o r which e v e r y b a r r i e r sequence w i t h domain e x i s t s a bad b a r r i e r sequence w i t h domain p a r t i a l ordering y e t not
5 6 - BETTER PARTIAL
3 P (A) ORDERING
w p i s good, y e t f o r which t h e r e c3 ptl
; thus
3
p-l(A)
, f o r the given p a r t i a l ordering WITH
RESPECT TO BARRIERS,
i s a well A
.
OR -BETTER
P A R T I A L ORDERING
We say t h a t simply a
A
i s a b e t t e r p a r t i a l o r d e r i n g w i t h r e s p e c t t o b a r r i e r s , o r more
b e t t e r p a r t i a l o r d e r i n g , i f e v e r y b a r r i e r sequence w i t h values i n A
i s good (see t h e d e f i n i t i o n i n
5
2). This notion i s h i s t o r i c a l l y anterior t o the
b e t t e r p a r t i a l o r d e r i n g , and goes back t o NASH-WILLIAMS 1965'. I t i s proved i n NASH-WILLIAMS 1968 t h a t e v e r y - b e t t e r p a r t i a l o r d e r i n g i s a b e t t e r p a r t i a l o r d e r i n g ,
221
Chapter 8
which we obtain in 5 7 below. Thus b o t h notions are equivalent, as already mentioned a t the end of 5 5 . 4 above. A quasi-ordering i s said t o be a better quasi-ordering, i f the partial ordering of i t s equivalence classes, each formed by elements simultaneously greater and lesser, i s a -better partial ordering.
6.1. Every restriction, and every partially ordered reinforcement of a partial ordering, i s a better partial ordering.
-
better
Every -better partial ordering i s a well partial ordering (uses dependent choice; ZF suffices for a countable partial ordering). Since every barrier sequence i s good, in particular every W-sequence (which can be identified with a barrier on singletons) i s good. Then our statement f o l lows by ch.4 g 3 . 2 . ( 2 ) , using dependent choice. 0
0
6 . 2 . The partial ordering formed by two incomparable elements, i s a -better par-
tial ordering. U be a barrier and s an element of U , By 1.6 above, there exists a sequence of f i n i t e length h of elements ti of U ( i = l , . . . , h ) ; and another sequence of length h + l of elements ti ( i = 1 , . . . , h + l ) , with s a tl 4 t2 4 . . . Q th and s U t i 4 ... at,!+, , b o t h having the same last term
0 Let
ti4
t h = tA+l . Then there necessarily exist two successive elements with the same image in the partial ordering, since there are only two elements in i t . 0 6 . 3 . Every well-ordering, and in particular every finite chain, i s a -better partial ordering. U be a barrier. Take an w-sequence of elements si of U ( i natural number) , such t h a t s i for each i Then for each function f from U into the well-ordering A , there exists an i satisfying f ( s i ) s f ( s i + l j (mod A ) ; so t h a t f i s good. 0
0 Let
.
6 . 4 . Let A be a partial ordering. Part'tion i t s base into two disjoint subsets, B a n d C are -better thus obtaining the two restrictions B and C partial orderings, then A i s a -better partial ordering.
.
Suppose,that our conclusion i s false. Let U be a barrier and f be a bad U-sequence in A . Partition the elements o f U into two classes, according t o whether the image under f belongs t o the base l B l or t o I C l By the barrier partition theorem 1.4, there exists a sub-barrier V of U with f/V a bad barrier sequence from V into B or into C Hence either B or C i s n o t a -better partial ordering. 0
0
.
.
THEORY OF RELATIONS
228
Every f i n i t e p a r t i a l ordering i s a If A of - A
b e t t e r p a r t i a l ordering.
i s a - b e t t e r partial ordering, then so i s every p a r t i a l l y ordered extension t o i t s base augmented by a f i n i t e number of elements.
These statements follow from the preceding proposition, since the ordering on a singleton i s i t s e l f a - b e t t e r p a r t i a l ordering, and even a b e t t e r p a r t i a l orde-
0
ring, as noticed in
5
5.1 above.
6.5. .If A B are both - b e t t e r p a r t i a l orderings, then the d i r e c t product A % B i s a -better p a r t i a l ordering ( d i r e c t product i s defined in ch.4 5 7 ) ; see NASH-WILLIAMS 1965' p . 706. a function from U i n t o A % B . Call p the projection which, t o each ordered pair ( a , b ) in the product of the bases, associat e s the f i r s t term a , and c a l l q the projection which associates the second term b . Since A i s a - b e t t e r p a r t i a l ordering, by 2 . 1 there e x i s t s a barrier V included in U , with (p,f)/V perfect. Since B i s a -better partial ordering, there e x i s t two elements s , t in V with sa t and qfs ,< q f t (mod B ) Moreover p f s s p f t (mod A ) , since p,f i s perfect. Thus f s 4 f t modulo the product A x B : hence f i s good.
0
Let
U
be a b a r r i e r and
f
.
6.6. To each natural number i , associate a f i n i t e s e t Fi of elements; so t h a t , f o r fixed i , the elements in Fi are mutually incomparable, and so t h a t each element in Fi+l i s s t r i c t l y greater than a l l the elements of Fi . We thus obtain a s t r a t i f i e d p a r t i a l ordering, in the sense of ch.2 5 5.2; moreover t h i s ordering i s well-founded and Fi i s the s e t of elements of height i (ch.2 5 3 . 2 ) .
This s t r a t i f i e d partial ordering i s a b e t t e r p a r t i a l ordering. Particular case of the proposition 6.7 below. Assuming t h a t the base i s denumerable (which in general requires the countable axiom of choice on f i n i t e s e t s ) , we have the following easy proof. S t a r t with the s e t of ordered pairs o f natural numbers, hence the d i r e c t pro duct of w with i t s e l f , which by the previous propositions 6.3 and 6 . 5 , i s a better partial ordering. For Fo take a s e t of ordered pairs of natural numbers Then f o r F1 take a s e t of ordeof the form (x,y) with x + y = a. = Card Fo and y > a o and x + y = a. + al where al = red pairs (x,y) with x 2 a.
0
.
Card F1 ; and so f o r t h . The Given s t r a t i f i e d p a r t i a l ordering i s isomorphic with a r e s t r i c t i o n of the d i r e c t Droduct W x W ; by 6.1, t h i s i s a - b e t t e r p a r t i a l ordering. 0 6.7. Let A be a p a r t i a l ordering. If every proper i n i t i a l interval of A f i n i t e , then A i s a - b e t t e r p a r t i a l ordering (POUZET 1977, unpublished).
i2
229
Chapter 8
As in the preceding statement, t h i s proposition gives well partial o r d e r i n g whose elements have f i n i t e heights; y e t they are more varied: i t i s no longer required
t h a t each element of height
i + l be greater t h a n every element of height
i
.
Note f i r s t t h a t every non-empty subset of the base has a minimal element; f o r otherwise, t h i s would yield an i n f i n i t e proper i n i t i a l i n t e r v a l . Moreover every
0
free s e t , or every antichain, i s f i n i t e ; thus A i s a well partial ordering. By hypothesis, each element of the base has a height which i s a natural number. Moreover, i f A i s i n f i n i t e , then by our hypothesis A i s directed, thus A i s an i d e a l . For otherwise, i f a and b are two elements without any common upper bound, then e i t h e r the s e t of non-upper bounds of a , o r the s e t of non-upper bounds of b , i s an i n f i n i t e i n i t i a l interval which i s d i s t i n c t from A : contradiction. Let U be a barrier and g a function from U i n t o A . We shall prove that g i s good. Let h be the function which, t o each element in the base \ A 1 , associates i t s height (mod A ) , which i s a natural number. Let f be the compos i t i o n h o g . Since the chain (A) of the heights i s a b e t t e r p a r t i a l ordering, there e x i s t s a barrier V included in U , such t h a t the r e s t r i c t i o n f/V i s 2 perfect (see 2 . 1 above). Let V ' be the subset of the squared barrier V , formed of the unions s u t of elements s , t in V , such t h a t s 4 t and f s = f t . By the b a r r i e r p a r t i t i o n theorem 1.4, there e x i s t s an i n f i n i t e s e t H of natural numbers which e i t h e r only contains elements of V ' , or only contains 2 elements of V 2 - V ' Let W be the barrier V r e s t r i c t e d t o H ; then W i s e i t h e r included in V ' or included in V2 - V ' . I n the f i r s t case, there e x i s t s a natural number p such t h a t f s = p f o r Indeed, given two elements s , t of W , by 1.5.(2) every element s in W there e x i s t s a t h i r d element u of W with two f i n i t e sequences of successive elements of W , say s Q S ~ . Q U and t q t l 4 . . u Then we have
.
.
..
.a .
f s = f ( s 1) = ... = fu and f t = f ( t l ) = ... = fu . For each element x of the base I A I whose height (mod A) i s p , l e t W x be the subset of W formed of those s such t h a t gs = x . The elements x are mutually incomparable (mod A ) , hence there are only f i n i t e l y many such. Thus there e x i s t s an x with a barrier X included in W x . For two elements s , t in X , which we can take as success i v e , we have gs = g t ; hence g i s good. In the second case, recall t h a t f i s perfect. T h u s f o r every s , t in W , the condition s d t implies f s , < f t B u t here f s # f t and so f s < f t Let k be the function from the s e t w of the natural numbers, into (13, which t o each natural number i , associates the l e a s t j f o r which each element of height 6 i i s less (mod A ) t h a n every element of height j , thus also less than every element of height >, j . This value j = ki e x i s t s , since
.
.
230
THEORY OF RELATIONS
f o r each element a with height i , the s e t of non-upper bounds of a i s a proper i n i t i a l interval of A ; hence i t i s f i n i t e and there are only f i n i t e l y many heights of i t s elements. By 1 . 7 above, there e x i s t two elements s , t in W satisfying s a t and k f s < f t . Thus each element of the same height as g t i s greater (mod A) t h a n every element of the same height as gs ; in p a r t i c u l a r gs 6 g t (mod A): hence g i s good. 0 6.8. Let A be a well partial ordering which has f i n i t e l y many i n f i n i t e ideals. Then A i s a - b e t t e r partial ordering.
Let A be a p a r t i a l ordering which has only f i n i t e l y many i n f i n i t e i n i t i a l intervals. Then A i s a - b e t t e r partial ordering (POUZET 1977, unpublished; uses dependent choice; ZF suffices i f A i s countable). The second assertion follows from the f i r s t , since the partial ordering A under consideration i s necessarily well-founded and f i n i t e l y f r e e . Suppose f i r s t t h a t A i s a directed well partial ordering, with no other i n f i n i t e ideal t h a n i t s e l f . Then A has no i n f i n i t e , proper i n i t i a l i n t e r v a l . Indeed, every i n f i n i t e well partial ordering has as a r e s t r i c t i o n , a t l e a s t one i n f i n i t e ideal (see ch.4 5 5 . 2 , dependent choice); so by the preceding proposition 6 . 7 , A i s a -better partial ordering. I n the general case, we argue by induction. Given a positive integer p , suppose the proposition holds f o r any well partial ordering with a t most p i n f i n i t e ideals, and l e t A be a well p a r t i a l ordering with p + l i n f i n i t e ideals. Let I be an ideal of A , which i s maximal with respect t o inclusion. Partition the base I A I into the union C of those ideals d i s t i n c t from I , and the complement D of C . The r e s t r i c t i o n A/C i s a well p a r t i a l ordering having only p i n f i n i t e ideals, and the r e s t r i c t i o n A/D i s s t i l l a directed well p a r t i a l ordering, hence an i d e a l , having no other i n f i n i t e ideal than i t s e l f (provided D iS i n f i n i t e ) . Each i s thus a -better partial ordering; so by the previous 6.4, A i s a -better partial ordering. 0
0
6 . 9 . -BETTER PARTIAL O R D E R I N G OF WORDS Let A be a -better partial ordering. Then the s e t of a l l words ( i . e . f i n i t e sequences) A forms a - b e t t e r p a r t i a l ordering under embeddability- (uses dependent choice).
Suppose on the contrary t h a t there e x i s t s a b a r r i e r U and a bad U-sequence f taking as values words. The p a r t i a l ordering of words i s a well-founded p a r t i a l ordering: see ch.4 5 2 . So we can assume t h a t f i s minimal bad: see theorem 2 . 2 above, dependent choice. Since f i s bad, f o r every s in U , the word f s i s non-empty.
23 1
Chapter 8
Let s
g
and
in
h
U
, the
hs
is
value barrier
fs
of
V
gs
f s ; the
w i t h i t s f i r s t t e r m removed. By 2.1 above, t h e r e e x i s t s a subsuch t h a t
U
minimal bad, i n g o i n g f r o m hs
as f o l l o w s . F o r each
i s t h e word composed o f t h e f i r s t t e r m o f i s perfect (since
g/V
d e r i n g ) . On t h e o t h e r hand, t h e r e s t r i c t i o n p l a c e d by
U , defined
be b a r r i e r sequences w i t h domain value
to
f
h
, for
h/V
Thus t h e r e e x i s t two elements
s, t
p e c t t o e m b e d d a b i l i t y . Now, as t o e m b e d d a b i l i t y ; so
in
V
fs
b e t t e r p a r t i a l or-
in
s
f
is
, t h e word f s i s r e -
V
w i t h respect t o embeddability.
s Qt
with
and
hs,< h t
, we have f s , ( f t
gs,< g t (mod A)
i s good: c o n t r a d i c t i o n .
f
-
i s good; indeed s i n c e
each
which i s s t r i c t l y l e s s t h a n
is a
A
w i t h res-
w i t h respect
0
6.10. -BETTER QUASI-ORDERING OF ORDINAL-INDEXED SEQUENCES G e n e r a l i z e as f o l l o w s t h e p r e v i o u s p r o p o s i t i o n . Let
A
be a - b e t t e r p a r t i a l o r d e r i n g . Then any s e t o f o r d i n a l - i n d e x e d sequences
w i t h values i n A
forms a - b e t t e r q u a s i - o r d e r i n g under e m b e d d a b i l i t y (NASH-WILLIAMS
1968; t h e f o l l o w i n g p r o o f i s due t o MILNER 1984, unpublished; uses dep. c h o i c e ) . Cl L e t
B
A : this
be a s e t o f o r d i n a l - i n d e x e d sequences i n
under e m b e d d a b i l i t y (ch.4
5
2). We can assume t h a t
B
B
i s quasi-ordered
reduces t o a p a r t i a l orde-
r i n g , by r e p l a c i n g sequences by t h e i r e q u i v a l e n c e c l a s s e s under embeddability. Take t h e r a n k i n g f u n c t i o n which t o each sequence u a s s o c i a t e s t h e l e n g t h o f u.
s
Suppose t h a t and a bad
i s not a -better p a r t i a l ordering: there exists a b a r r i e r
B
U-sequence w i t h v a l u e s i n
e x i s t s a minimal bad b a r r i e r sequence the barrier
B
. By
(mod B,
.
Dom f
P a r t i t i o n t h e elements
s
of
U
theorem 3.5 (dependent c h o i c e ) , t h e r e
J' )
, say f
; we c a l l again
U
i n t o three d i s j o i n t classes, according t o the
U
t h r e e f o l l o w i n g p o s s i b i l i t i e s : e i t h e r t h e sequence
f(s)
has l e n g t h 1
, or
its
l e n g t h i s a l i m i t o r d i n a l , o r i t s l e n g t h i s a successor o r d i n a l s t r i c t l y g r e a t e r t h a n 1. Using t h e b a r r i e r p a r t i t i o n theorem 1.4, we can assume t h a t t h e e n t i r e barrier
U
reduces t o one o f t h e t h r e e c o n s i d e r e d c l a s s e s .
I n t h e f i r s t case, f o r each of
A
. Since
A
s
i n U t h e sequence
i s a -better p a r t i a l ordering, the
f(s)
reduces t o an element
U-sequence
f
i s necessa-
r i l y good: c o n t r a d i c t i o n .
Examine t h e second case where a l l l e n g t h s a r e l i m i t o r d i n a l s . Take any two successive elements
s 4 t
Since t h e l e n g t h o f
f(s)
i n t e r v a l of
f(s)
t h e square b a r r i e r
in U
, so
that
f(s)
$.
f ( t ) under e m b e d d a b i l i t y .
i s a l i m i t ordinal, t h e r e e x i s t s a proper i n i t i a l
which i s non-embeddable i n f ( t ) : see ch.4 g 2.1. Consider 2 V = U and t o each eiement v o f V a s s o c i a t e i t s i n i t i a l
s which belongs t o U and t h e f i n a l i n t e r v a l t = v minus i t s m i n i mum i n t e g e r , so t h a t sa t and v = s u t Then t o t h i s v a s s o c i a t e g ( v ) , t h e minimum p r o p e r i n i t i a l i n t e r v a l o f f ( s ) which i s non-embeddable i n f ( t ) . interval
.
232
THEORY OF RELATIONS
s
Note t h a t $ ( g ( v ) ) G ( f ( s ) ) s i n c e g ( v ) has length s t r i c t l y smaller than f ( s ) . Therefore f foreruns r~ (mod B , $ ) y e t g does not reduce t o a r e s t r i c t i o n of f F i n a l l y g i s bad: take any two successive elements v c l w i n V and t h e corresponding i n i t i a l i n t e r v a l s S Q t i n U ; then g ( v ) i s non-embeddable in f ( t ) thus non-embeddable in g(w) which i s an i n i t i a l i n t e r v a l of f ( t ) . This c o n t r a d i c t s our hypothesis t h a t f i s minimal bad. Examine t h e t h i r d case where a l l lengths a r e successor o r d i n a l s d i f f e r e n t from 1.
.
To each element s of U a s s o c i a t e the l a s t term l ( s ) of the sequence f ( s ) and t h e sequence g ( s ) = f ( s ) minus i t s l a s t term. By 2 . 1 we can replace U by a sub-barrier again c a l l e d U , such t h a t t h e U-sequence 1 i s p e r f e c t . Therefore t h e U-sequence g must be bad, s i n c e f i s bad. Take t h e square b a r r i e r V = U 2 , and t o each element v of V a s s o c i a t e t h e i n i t i a l i n t e r v a l s of v which belongs t o U . Then p u t h ( v ) = g ( s ) . Note t h a t $ ( h ( v ) ) = $ ( g ( s ) ) < S ( f ( s ) ) , t h e r e f o r e f foreruns h (mod B , I? ) y e t h i s not a r e s t r i c t i o n of f . F i n a l l y h is bad; indeed with t h e same notations than i n t h e second c a s e , h ( v ) = g ( s ) i s non-embeddable i n h ( w ) = g ( t ) . This c o n t r a d i c t s our hypothesis t h a t f i s minimal bad. 0
5 7 - EQUIVALENCEOF BOTH N O T I O N S OF BETTER P A R T I A L O R D E R I N G : CHAIN SEQUENCE I N A PARTIAL ORDERING Every - b e t t e r p a r t i a l ordering i s a b e t t e r p a r t i a l ordering ( u s e s dependent c h o t c e ) ; t h e r e f o r e both notions coincide, by 5.4. 0 Let A be a - b e t t e r p a r t i a l ordering. By the preceding 6.10, every s e t of ordinal-indexed sequences i n A forms a - b e t t e r quasi-ordering under embeddabil i t y , thus a well quasi-ordering by 6 . 1 . Using 5 . 3 . ( 2 ) , we see t h a t A is an * - b e t t e r ordering f o r each o( : i n o t h e r words a b e t t e r p a r t i a l ordering. 0 Consequently i n 6.10 we can replace b e t t e r p a r t i a l ( o r q u a s i ) ordering by b e t t e r p a r t i a l ( o r q u a s i ) ordering.
7 . 1 . CHAIN SEQUENCE, DOMAIN CHAIN Let C be a chain and A be a p a r t i a l ordering. A chain sequence i n A , o r C-sequence i n A i s a couple ( C , f ) where f i s a function whose domain i s t h e base I C I and whose range i s included i n 1 A I The chain C i s c a l l e d t h e domain chain of ( C , f ) . I f C is an o r d i n a l , we f i n d again an ordinal-indexed sequence w i t h length C , a s defined in ch.1 5 2 . 2 . In t h i s case i t i s unnecessary t o d i s t i n g u i s h between ( C , f ) and f , s i n c e Dom f i s well-ordered by the membership r e l a t i o n . RESTRICTION, ISOMORPHIC SEQUENCE, EXTRACTED SEQUENCE Return t o t h e general case of a chain sequence ( C , f ) . I f U i s a subset of the base I C l , then t h e sequence (C/U,f/U) obtained by r e s t r i c t i n g both the chain
.
Chapter 8
233
.
and the function t o U i s called the r e s t r i c t i o n of (C,f) t o U If h i s a n isomorphism taking C onto the chain D , then the chain sequence (D,f,h-l) i s said t o be isomorphic with (C,f) , and more precisely t o be the image of (C,f) under the isomorphism h . A chain sequence ( D , g ) i s said to be extracted from (C,f) i f f there e x i s t s a r e s t r i c t i o n of (C,f) which i s isomorphic with ( D , g ) I f C and D are b o t h ordinals, we find again the definition of an extracted sequence (see ch.1 0 2 . 2 ) . LESSER SEQUENCE, INF-RESTRICTION, EMBEDDABILITY BETWEEN CHAIN SEQUENCES Given the chain sequence (C,f) in A , a chain sequence (C,g) with the same domain chain C i s l e s s t h a n (C,f) i f f gx 4 fx (mod A) f o r each x in the domain I C l . Analogously f o r "greater than" For ordinal-indexed sequences, we find again the notion as defined in ch.4 0 2. A chain sequence (D,g) i s said t o be an i n f - r e s t r i c t i o n of (C,f) i f f the chain D i s a r e s t r i c t i o n of C and i f (D,g) i s l e s s t h a n the r e s t r i c t i o n (D,f/ D ) . Note the analogy with barrier sequences, see 2 . 2 above. We say t h a t a chain sequence (D,g) is embeddable in (C,f) or t h a t (C,f) admits an embedding of (D,g) i f f there e x i s t s an i n f - r e s t r i c t i o n o f (C,f) which i s isomorphic with (D,g) Equivalently, i f f there e x i s t s a chain sequence l e s s than ( C , f ) from which (D,g) i s extracted. I n t h i s case D i s embeddable in C However D can be embeddable in C without (D,g) being embeddable in (C,f) : consider two w -sequences, one having the constant value a and the other having the value b incomparable with a Every chain sequence extracted from a chain sequence (C,f) i s embeddable in (C,f) However the converse i s f a l s e : assume t h a t a < b and consider the sequences which reduce t o the singleton of a (resp. the singleton of b ). The notions of r e s t r i c t i o n , isomorphism, extracted from, l e s s than, inf-restriction and embeddability are reflexive and t r a n s i t i v e . The only ones which are antisymmetric are the notions of r e s t r i c t i o n , l e s s than and i n f - r e s t r i c t i o n .
.
.
.
.
.
.
7.2. RIGHT AND LEFT INDECOMPOSABLE CHAIN SEQUENCE Let C be a chain and f a function from I C I into a partial ordering A We say t h a t the chain sequence (C,f) i s right indecomposable i f f , f o r each nonempty final interval D of C , the chain sequence (C,f) i s embeddable in i t s Analogously define a l e f t indecomposable chain sequence. r e s t r i c t i o n (D,f/ D ) I f a chain sequence (C,f) i s right indecomposable, then the domain chain C i s i t s e l f r i g h t indecomposable. However the converse i s f a l s e : see 5 5 above. SUM OF CHAIN SEQUENCES Analogous t o the definition of the sum of chains along a chain I (which i s called a homomorphic image of the sum: ch.2 0 3.6), we define the %of chain sequences f i , where the index i runs through the base of a chain I which shall again be called the image chain.
.
.
234
THEORY OF RELATIONS
For each fi , let Ci be its domain chain, and let C be the sum of the Ci along the image chain I , the bases lCil being taken to be mutually disjoint. Then the sum shall be the couple formed of C and the union of the functions fi in other words, the common extension f of the fi with domain ICl . HEREDITARILY INDECOMPDSABLE OR H-INDECOMPOSABLE CHAIN SEQUENCE A chain sequence in a partial ordering A is said to be hereditarily indecomposable, or h-indecomposable, iff it is obtained by induction from the following procedure. The chain sequence which reduces to the singleton of an element of the base I A I is h-indecomposable. If a is either 0 or 1 or a regular infinite aleph, and fi (i < a ) are h-indecomposable chain sequences such that for each i , the set of indices j (i < j < d ) for which f i is embeddable in f is cofinal j or along its converse are h-indecomin ~, then the sum of the fi along posable. Moreover, the only h-indecomposable chain sequences in A are those which can be so constructed. Every h-indecomposable chain sequence is right or left indecomposable, according to whether it is a sum along an infinite regular aleph or along its converse. Moreover, the considered infinite regular aleph is the cofinality (in the case of a right indecomposable chain sequence) or the co-initiality (in the left case). The empty chain and the chain sequences reduced to a singleton are the only h-indecomposable chain sequences which are both right and left indecomposable.
;
7.3. We immediately extend LAVER'S theorems (ch.6 5 5.4 and 5.5, using axiom of choice) to the case of chain sequences. Let A be a partial ordering, and f be a chain sequence in A whose domain chain i s scattered. Suppose that the h-indecomposable restrictions of f form a well quasi-ordering under embeddability. Then f is a finite sum of h-indecomposable chain sequences. Let f be a chain sequence in A whose domain chain is scattered. Suppose that f is indecomposable and that the h-indecomposable restrictions of f form a well quasi-ordering under embeddability. Then f is h-indecomposable.
7.4. GOOD, BAD BARRIER SEQUENCE AND FORERUNNING REVISITED Consider a partial ordering or quasi-ordering A . Let U be a barrier and f be a barrier sequence which to each element of U associates a chain sequence iff there with values in A . Then such a barrier sequence f is said to be fs under embeddability; exist two elements r, s of U with r q s and fr,( f is said to be bad otherwise, following 5 2 above. To each chain sequence in A whose domain chain is scattered, associate the neighborhood rank of its domain chain: let $ be the ranking function thus defined.
235
Chapter 8
Then the notion of a barrier sequence which foreruns another, or which i s a successor of another (mod A, 6 ) with respect t o forerunning i s defined a s in 5 3 . 3 above. However our present notation (mod A, $ ) replaces the notation ) where J.7 designates the s e t of the considered chain sequences which (mod .-$, take t h e i r values in A and are quasi-ordered under embeddability (mod A) . This forerunning remains reflexive, antisymmetric and t r a n s i t i v e , so tha t i t defines a partial ordering among barrier sequences which themselves take as values chain sequences in A As in 3 . 5 , we define the notion of a minimal bad b arrie r sequence under our new forerunning (mod A, ) . As before, given an ar b i t r ary bad barrier sequence f , there e x i s t s a minimal bad barrier sequence (mod A, which i s a successor of f under forerunning. Now we generalize the proposition 4.1 by replacing the s e t of h-indecomposable , of h-indecomposable chain sequences chains by a s e t , s t i l l denoted by taking t h e i r values in a given b et t er partial ordering denoted by A Our s e t i s quasi-ordered under embeddability. Moreover, we assume tha t (R i s closed under taking any interval of one of the considered chain domains, provided t h i s interval yields an h-indecomposable chain sequence. NOW our proofs 4 . 1 and 4 . 2 extend t o the present case of chain sequences in A , except the f i r s t paragraph in 4.1. Indeed given two successive elements r U s in the b a r r i e r , i f the chain sequence f r reduces to the singleton of an element in A , and i f the chain sequence f s does n o t admit an embedding of f r (since f i s assumed t o be b a d ) , then f s i s not necessarily empty nor necessarily a singleton: i t i s possibly i n f i n i t e , and formed of terms none of which i s greater (mod A) than the unique element of the singleton f r . Moreover, our proof 4.1 corresponds t o the case where A reduces to a singleton. Now i t becomes necessary t o use our hypothesis t h a t A i s a b et t er partial ordering, with possibly infinitely many elements. So t h at we must replace the f i r s t paragraph in 4.1 by the following argument. 0 Let U be a barrier, the domain of f Partition U into two disjoint subsets U ' and U" , where U' i s formed of those elements whose image under f i s a singleton, and U " i s formed of those elements whose image i s an infinite h-indecomposable chain sequence. By the b ar r i er p ar tition theorem 1.4, a t l e a s t one of these two subsets includes a barrier. I f U ' includes a ba rrie r, then t h i s would yield a r e s t r i ct i o n of f which i s a bad barrier sequence with values in the b e t t e r p a r t i a l ordering A : contradiction. Hence there e xists a ba rrie r included i n U" . S t i l l denote by U t h i s b ar r i er , and take up again the proof 4.1 beginning in the second paragraph, now knowing tha t a l l the values taken by f are i n f i n i t e , h-indecomposable chain sequences.
.
s
s)
.
d
.
236
THEORY OF RELATIONS
Now we can take up again the proof 4.2 with "chain sequence in A " instead of "chain" . Moreover in this proof, instead of assuming the existence of a bad w -sequence, we can assume only the existence of a bad barrier sequence. So that we obtain the following generalized statement. Let A be a better partial ordering. Then every set of h-indecomposable chain sequences in A forms a better qug-orderjny under-embeddability (uses axiom of choice).
In view of the preceding 7.3, we obtain the following generalization of 4.3. Let A be a better partial ordering. Then every chain sequence in A with a scattered domain chain is a finite sum of h-indecomposable chain sequences. In particular, every indecomposable chain sequencea scattered domain chain is h-indecomposable (uses axiom of choice). 7.5.
7.6. Let A be a better partial ordering. Then each set o f chain sequences with domain chains is better quasi-ordered under values in A and with scattered____ embeddability (LAVER 1968, uses axiom of choice). In particular, each set of scattered chains is better quasi-ordered under embeddability: take the preceding statement where A reduces to a singleton. 0 We know by 7.5 that each set of h-indecomposable chain sequences forms a better quasi-ordering under embeddability. By 6.9 each set of words, or finite sequences composed o f such h-indecomposable chain sequences forms a better quasi-ordering. This subsists for each set of finite sums of h-indecomposable chain sequences, which yields a reinforcement of the previous quasi-ordering: see 5.1. Now by 7.5, every chain sequence with a scattered domain chain is such a finite sum of h-indecomposable chain sequences. 0 _ l _ _ l
7.7. If. A i s a better partial ordering, so i s the partial ordering 3 ( A ) of initial intervals of A (with respect to inclusion); uses axiom of choice. 0 To each initial interval X associate any ordinal-indexed sequence of elements of X , such that every element of X has an upper bound in the corresponding sequence. Then by 6.10 our sequences constitute a better quasiordering under embeddability. Therefore the corresponding initial intervals constitute, under inclusion, a partial ordering which reinforces the quasiordering o f sequences, hence a better partial ordering by 5.1. 0
231
Chapter 8
EXERCISE 1 - Every f i n i t e l y f r e e p a r t i a l ordering has a c o f i n a l r e s t r i c t i o n which i s a b e t t e r p a r t i a l ordering (POUZET 1979, unpublished, answering a conjecture due t o GALVIN; uses axiom of c h o i c e ) .
1 - Let A be a f i n i t e l y f r e e p a r t i a l ordering. P a r t i t i o n A i n t o a f i n i t e union of i d e a l s : ch.4 5 5 . 3 , axiom of choice. For each i d e a l , take a cofinal r e s t r i c t i o n which i s a well-founded p a r t i a l ordering, hence a d i r e c t e d well p a r t i a l ordering: ch.4 5 5.4. Then in view of 6.4 above, i t would s u f f i c e t o prove our a s s e r t i o n f o r each d i r e c t e d well p a r t i a l ordering. A be a d i r e c t e d well p a r t i a l ordering. By ch.7 5 3.11, t h e r e e x i s t s a cofinal r e s t r i c t i o n of A which i s isomorohic with t h e d i r e c t product of a f i n i t e number o f r e g u l a r alephs. T h i s c o f i n a l r e s t r i c t i o n is a b e t t e r p a r t i a l ordering by 6 . 3 and 6 . 5 above. 2 - Let
EXERCISE 2 - A well p a r t i a l ordering f o r which every b a r r i e r sequence on W good, b u t which i s not a b e t t e r p a r t i a l ordering.
is
Let A be t h e p a r t i a l ordering defined on a l l ordered t r i p l e s of natural nunbers x , y , z by the following condition: ( x , y , z ) 4 ( x ' . y ' , z ' ) i f xb x ' and y,( y ' and z 4 z ' and a d d i t i o n a l l y e i t h e r x = x ' , o r x K x ' and y < x ' , o r f i n a l l y x < x ' and z C y ' . 1 - Prove t h e t r a n s i t i v i t y of the above. Note t h a t A i s well-founded, t h e r e being only f i n i t e l y many predecessors of ( x , y , z ) Prove t h a t A i s f i n i t e l y f r e e , hence a well p a r t i a l ordering. For t h i s , note t h a t t h e d i r e c t nroduct ( x,< x ' and y,< y ' and z ,< z ' ) i s a well p a r t i a l ordering. Suppose t h a t t h e r e e x i s t s an w-sequence of t r i p l e s ( x i , y i , z i ) where i i s a natural number , which a r e mutually incomparable (mod A) . Then e x t r a c t an a - s e q u e n c e w i t h xi i n c r e a s i n g , yi i n c r e a s i n g , z i increasing. Since incomparability r e q u i r e s t h a t xi be s t r i c t l y increasing i n i , t h e r e e x i s t s an i f o r which xi > yo and xi > xo : c o n t r a d i c t i o n .
.
2 - Prove t h a t A i s not a b e t t e r p a r t i a l ordering; indeed we have the bad b a r r i e r sequence with domain i,s3 ( i . e . the set of a l l 3-element subsets of t h e s e t of natural numbers), defined a s follows: each 3-element subset { x , y , z ) ( w i t h x < y < z ) i s taken i n t o the ordered t r i p l e ( x , y , z ) . To see t h i s , l e t
, t h i s being and take an i n t e g e r r > z , and l e t t = { y , z , r ) the only possible manner t o g e t sa t Then ( x , y , z ) and ( y , z , r ) a r e incomparable (mod A) . s = {x,y,z}
.
3 - Now l e t f be a b a r r i e r sequence w i t h domain w ( t h e s e t of unordered p a i r s of i n t e g e r s ) . For each p a i r s of i n t e g e r s , l e t x ( s ) , y ( s ) , z ( s ) be t h e e s h a l l prove t h a t f i s good. coordinates of f ( s ) . W
THEORY OF RELATIONS
238
Denote by B the d i r e c t product ( x x ' and y , c y ' and z sz' ) , which i s a b e t t e r p a r t i a l ordering by 6.3 and 6.5 above. Since A and B a r e both based on the same s e t of ordered t r i p l e s , consider f a s a b a r r i e r sequence i n B and then replace f by a r e s t r i c t i o n which be p e r f e c t (mod B ) : see 2 . 1 above. A f t e r renumbering, this r e s t r i c t i o n of f s t i l l has domain u2 , and now we again consider f a s taking values in A . T h u s f o r any two p a i r s of i n t e g e r s , say s and t w i t h s Q t , we have x(s),( x ( t ) , and s i m i l a r l y w i t h y and with z . Then e i t h e r t h e r e e x i s t s , t w i t h s d t and x ( s ) = x ( t ) , in which case f i s good and we a r e f i n i s h e d .
.
Or s 4 t implies n e c e s s a r i l y t h a t x ( s ) < x ( t ) Then by RAMSEY's theorem, we can require e i t h e r t h a t f o r a l l i n t e g e r s i < j k we have y(J,i,j)) = y({i,k}) , o r . In t h e f i r s t c a s e , take t h a t we have t h e s t r i c t i n e q u a l i t y y ( j i , j j ) ( y ( { i , k ] ) a s t r i c t l y increasing Cr)-sequence of i n t e g e r s i o < i l < , so t h a t successive p a i r s give s t r i c t l y i n c r e a s i n g values f o r x . Then f o r h s u f f i c i e n t l y l a r g e
-=
...
Y(\iO'ih\)< x ( ( i h ' i h + l ) ) and obviously x ( { i O , i h ) ) < x ( i i h , i h + l j ) : so t h a t f i s good. In t h e second c a s e , take again a s t r i c t l y increasing d - s e q u e n c e of i n t e gers g i v i n g , f o r h s u f f i c i e n t l y l a r g e , z({io,il}) < y ( { i l , i h ) ) and obviously x({i,-,,il})<
, and so f
x({il,ih))
i s good.
4 - Generalize t h e preceding f o r the p a r t i a l ordering on the s e t of k-tuples of xk , and addii n t e g e r s , defined by t h e d i r e c t product x l $ and ... and ,or xl< with e i t h e r x2 < or x3< o r ... t i o n a l l y e i t h e r x1 =
xi
xi
or
Xk<
xi
xke
xi
xi
xi-l .
EXERCISE 3 - Taking again BONNET'S chain R = AO+U+...+U+Ai+U+... in 4.5 above, show t h a t every non-empty f i n a l i n t e r v a l of R i s decomposable. Now using B O N N E T ' S r i g i d chain R such t h a t each l o c a l automorphism moves only countably many e l e ments ( s e e ch.7 exerc. 4 ) , obtain t h a t every i n f i n i t e i n t e r v a l of R i s decompos a b l e (communicated by POUZET). EXERCISE 4 - Show t h a t each countable p a r t i a l ordering i s an d l - b e t t e r p a r t i a l ordering. More g e n e r a l l y take a p a r t i a l ordering A such t h a t every bad sequence in A i s countable. EXERCISE 5 - Given two b a r r i e r s U, V having t h e same union s e t , c a l l U+V t h e s e t of unions u u v where u belongs t o U and v belongs t o V and Max u s t r i c t l y l e s s than Min v Show t h a t U+V i s a b a r r i e r . Examine t h e case where V i s t h e s e t of s i n g l e t o n s i n t h e union of U . Given a b a r r i e r U which does not contain any s i n g l e t o n , c a l l U- t h e s e t of elements s - 2 Max s) where s belongs t o U : then U- i s a b a r r i e r .
.
239
CHAPTER 9
ISOMORPHISM AND EMBEDDABILITY BETWEEN RELATIONS, LOCAL ISOMORPHISM, FREE INTERPRETABILITY, CONSTANT RELATION, CHAINABLE AND MONOMORPHIC RELAT ION
§
1 PERMUTATION,TRANSPOSI T ION, '-
LOCAL I S O M O R P HI S M A N D LOCAL
AUTOMORPHISM The notions of isomorphism, automorphism and embeddability a r e def ned in ch.2 5 3 and ch.5 5 1, and previously used f o r p a r t i a l ordering5 and chains ( o r t o t a l ordee now apply these notions t o a r b i t r a r y r e l a t i o n s and mult r e l a t i o n s . For rings). W convenience, t h e d e f i n i t i o n s and statements w i l l be given f o r the case of r e l a t i o n s . Unless otherwise i n d i c a t e d , they can be extended t o the case of m u l t i r e l a t i o n s : t h e r o l e played by t h e a r i t y of t h e r e l a t i o n , t h e r e being played by t h e maximum a r i t y of the component r e l a t i o n s i n the m u l t i r e l a t i o n . 1.1. Let E be a s e t , f a permutation of E , and F a f i n i t e subset of E . There e x i s t s a sequence a l , ...,a h of elements of F , without r e p e t i t i o n , such t h a t f o r each x , i i F , t h e image f x i s equal t o t h e image of x obtained by composition of the successive t r a n s p o s i t i o n s ( a l , f a l ) , ... , ( a h , f a h )
.
P a r t i t i o n t h e elements of F i n t o maximal p a r t i a l o r b i t s of the form (u1,u2 ,...,uk) ( k = p o s i t i v e i n t e g e r ) where u2 = ful , ... , uk = fuk-l Associate t o each maximal p a r t i a l o r b i t the sequence of t r a n s p o s i t i o n s ( u k , f u k )
0
.
(ukm1,uk)
, ... , ( u l , u 2 )
if
f u k # u1 ; and the sequence of t r a n s p o s i t i o n s
i f f u k = u1 ( t h e case of a c y c l e , o r t o t a l o r b i t ) . ( U ~ - ~ , U, ~ )... , (u1,u2) Then each element x of our maximal p a r t i a l o r b i t ( o r t o t a l o r b i t ) has image f x . I t s u f f i c e s t o o r d e r i n an a r b i t r a r y manner the s e t of our o r b i t s , hence t h e s e t of corresponding sequences of t r a n s p o s i t i o n s .
.
1 . 2 . Let R be a r e l a t i o n w i t h base E and f a permutation of E I f f o r each element x iE E , the t r a n s p o s i t i o n ( x , f x ) i s an automorphism of R , -.then f is an automorphism of R (LOPEZ 1969).
n be the a r i t y of R . I f f modifies R , then t h e r e e x i s t s a sequence ,xn) # R(fxl ,... , f x n ) . By the preof n terms xi ( i = 1,... , n ) w i t h R ( x l , . ceding proposition, t h e r e e x i s t s a sequence y l , . . . ,yh formed of elements xi such t h a t the composition of t h e t r a n s p o s i t i o n s ( y l , f y l ) , ... , ( y h , f y h ) takes
4 Let
..
,
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THEORY OF RELATIONS
each x . i n t o f x i and hence modifies t r a n s p o s i t i o n s modifies R . 0
R
. Then
a t l e a s t one of t h e considered
1.3. Let R be an n-ary r e l a t i o n . I f a t r a n s p o s i t i o n ( a , b ) modifies R , then t h e r e e x i s t s a subset F of the base I R I w i t h c a r d i n a l i t y & n + l , which cont a i n s the elements a , b and such t h a t ( a , b ) modifies R / F
.
Let a l ,...,a n be such t h a t R(al ,..., a n ) # R(fal ,..., fa,) Then n e c e s s a r i l y a and b occur the t r a n s p o s i t i o n ( a , b ) t h e f ( a i ) ( i = 1 , ...,n ) . Thus i t s u f f i c e s t o take F t o be and t h e f ( a i ) , which has c a r d i n a l i t y l e s s than o r equal t o
0
.
except where one of t h e s e values i s
ai = f ( a i )
a
, where f denotes among t h e ai and the set of t h e ai n + l : indeed we have
and t h e o t h e r i s
b
.0
1.4. LOCAL ISOMORPHISM Let R, R ' be two r e l a t i o n s of t h e same a r i t y . A local isomorphism from R i n t o R ' i s an isomorphism from a r e s t r i c t i o n of R onto a r e s t r i c t i o n of R' For example, i f R , R ' a r e two p a r t i a l o r d e r i n g s , then a local isomorphism from R i n t o R' i s a b i j e c t i v e mapping f from a subset of t h e base I R I onto a subset of I R ' l , w i t h f x 4 f y (mod R ' ) i f f x < y (mod R ) , f o r every x , y i n Dom f In o t h e r words, f i s order preserving, a s well a s i t s converse f - l
.
.
.
The d e f i n i t i o n of local isomorphism extends immediately t o t h e case of two multir e l a t i o n s of the same a r i t y . I t follows from ch.2 5 3 , t h a t given two m u l t i r e l a ,Rh) : t h e concatenation of t h e tions R , R ' of t h e same a r i t y , where R = (R1,.. . ,R,!,) : t h e concatenation of component r e l a t i o n s R1, .. , Rh ; and R ' =
. (Ri,..
.
R i , . . . , RA
; then a b i j e c t i v e mapping
f
from a subset of
IR (
onto a subset
of I R ' I , i s a local isomorphism from R i n t o R ' i f f f i s simultaneously a local isomorphism from R1 i n t o and and from Rh i n t o R,!, .
Ri
...
For example, i f R , R' a r e chains and S , S ' a r e groups (thus t e r n a r y r e l a t i o n s ) , then the concatenations RS and R ' S ' a r e ordered groups provided t h a t t h e known axioms f o r ordered groups a m s a t i s f i e d ; then a b i j e c t i v e mapping f from a subset o f the base of RS onto a subset of t h e base of R'S' i s a local isomorphism i f , i n a d d i t i o n t o being order preserving between R and R ' , we have ( f x ) . ( f y ) = f z modulo the group S ' i f f x.y = z modulo S , f o r a l l x , y , z i n Dom f
.
Consider the empty function introduced in ch.2 5 3 . Then extending t h e conventions 1 , the empty function i s of th i s paragraph, we say t h a t , f o r every i n t e g e r n a local isomorphism from every n-ary r e l a t i o n i n t o every o t h e r n-ary r e l a t i o n . Moreover f o r a l l s e t s E, E ' t h e empty function i s a local isomorphism from t h e 0-ary r e l a t i o n (E,+) i n t o (El,+) and from (E,-) i n t o (El,-) , but n o t from ( E , + ) into (El,-) o r conversely by exchanging (+) and ( - ) .
Chapter 9
241
Finally given two multirelations R, R ' of the same a r i t y , the empty function i s a local isomorphism from R into R ' e i t h e r when a l l the component relations have positive a r i t i e s , or when, f o r each index i corresponding t o the 0-ary components R i , R; , these l a t t e r have the same value (+) or ( - ) . 1.5. I f f i s a local isomorphism from R into R ' , then f restricted t o an arbitrary subset of i t s domain i s s t i l l a local isomorphism from R into R ' . If f is a local isomorphism from R into R ' , then the inverse function f - 1 i s a local isomorphism from R ' into R . If additionally g i s a local isomorphism from R ' into R" , then the composition g,f i s a local isomorphism from R i n t o R" . In particular, i f f i s an isomorphism from R o n t o R ' , then every restriction of f i s a local isomorphism. More particularly, the identity function w i t h domain a subset of the base o f R i s a local isomorphism from R into R i t s e l f . R ' i s not i n qeneral ex.tendihl-Mto. However, a local isomorphism from R an isomorphism from R onto R ' , even i f R and R ' are isomorphic or even identical. 0 If R i s the chain of natural numbers and R ' the chain of negative , , integers, and f the function taking 0,1,...,p into f0 = - p - l f l = -p fp = -1; then f i s not extendible since R and R' are not isomorphic. Another example. I f R = R ' = the chain of natural numbers , then the function taking 0 into 1 i s a local isomorphism; yet i t i s n o t even extendible t o a range which, apart from 1 , contains 0 .
...
1.6. Let
R, R '
be two n-ary relations. A sufficient condition for a bijection
f with domain F , t o be a local isomorphism from R into R ' , i s that f o r every subset X of F with cardinality 4 n , the function f restricted t o X .is a local isomorphism. If R , R ' are two multirelations w i t h the same a r i t y , then the preceding proposition s t i l l holds, by setting n t o be the maximum of the a r i t y . 0 Let x1 ,... ,xn be a sequence o f n elements in F . The s e t X = {xl ,... ,xn ) i s a subset of F and has cardinality 4 n . Hence by hypothesis we have R ' ( f x l ,..., fx,) = R ( x l ,..., xn) and thus f i s a local isomorphism. Taking R ' = R and taking for f the identity on the base, we find the statement in ch.2 $ 1.1. 1.7. LOCAL AUTOMORPHISM A jocal automorphism of a relation R i s a local isomorphism from R into R . An automorphism f of R , hence also the restriction of f t o an arbitrary subs e t of the base, i s a local automorphism of R
.
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THEORY OF RELATIONS
However i n general, a l o c a l automorphism i s n o t e x t e n d i b l e t o an automorphism: see our example i n 1.5 with the chain o f n a t u r a l numbers and t h e l o c a l automorphism
.
which takes 0 i n t o 1 The f o l l o w i n g statement i s a p a r t i c u l a r case o f 1.6: L e t R be an n-ary r e l a t i o n and l e t f be a b i j e c t i o n whose domain and range are subsets o f t h e base I R I . I f every r e s t r i c t i o n o f f t o 6 n elements i s a l o c a l automorphism o f R , then f i t s e l f i s a l o c a l automorphism o f R
.
i s a permutation o f t h e base, then i t i s an automorphism of
Moreover i f f 1.8. (1) L e t
R
be o f a r i t y
m and S o f a r i t y
R , defined phism o f S , then every l o c a l automorphism I n p a r t i c u l a r , every automorphism o f R i s ( 2 ) With t h e same n o t a t i o n s , i f R and S I f every l o c a l automorphism o f
R
.
n , both having the same base.
on 6 n elements, i s a l o c a l automoro f R i s a l o c a l automorphism o f S an automorphism o f S . have t h e same l o c a l automorphisms on
.
4 Max(m,n) elements, then they have t h e same l o c a l automorphisms; and i n p a r t i c u l a r they have t h e same automorphisms. However, two d i s t i n c t chains isomorphic w i t h W
, both
having t h e same base, have
t h e same automorphism ( t h e i d e n t i t y being t h e i r o n l y automorphism), y e t do n o t have t h e same l o c a l automorphisms.
§
2 - FREE
INTERPRETABILITY,DIMENSIONAL ARITY
L e t R, S be two m u l t i r e l a t i o n s w i t h t h e same base. We say t h a t S i s f r e e l y i n t e r p r e t a b l e i n R i f every l o c a l automorphism o f R i s a l o c a l automorphism
.
of s For example, l e t R be a p a r t i a l ordering. Then t h e r e l a t i o n o f s t r i c t p a r t i a l o r d e r i n g S(x,y) = + i f x t y (mod R) ( i n o t h e r words i f R(x,y) = t i s f r e e l y i n t e r p r e t a b l e i n R ; and R i s f r e e l y i n t e r p r e t a b l e i n S
and x # y)
.
The r e l a t i o n o f intermediacy o r betweenness T(x,y,z) = + i f z i s between x and y (mod R) , is f r e e l y i n t e r p r e t a b l e i n R . I n general, R i s n o t f r e e l y i n t e r p r e t a b l e i n T : f o r example take t h e usual o r d e r i n g o f t h e i n t e g e r s 1, 2, 3; then the permutation which reverses t h i s o r d e r i n g i n t o 3, 2, 1 i s a l o c a l automorphism f o r t h e betweenness r e l a t i o n T , b u t n o t so f o r t h e o r d e r i n g R Another example. L e t R be a chain; t h e c y c l i c r e l a t i o n S associated with R
.
(see ch.2 z I ( x ,
s a t i s f i e s S(x,y,z) = + i f x & y & z o r y < z g x o r (mod R). We see t h a t S i s f r e e l y i n t e r p r e t a b l e i n R , b u t i n general
5 8.6)
R i s n o t f r e e l y i n t e r p r e t a b l e i n S : take t h e chain 1 < 2 < 3 and t h e c i r c u l a r permutation. Free i n t e r p r e t a b i l i t y i s r e f l e x i v e and t r a n s i t i v e , so t h a t i t defines a preo r d e r i n g on every s e t of r e l a t i o n s w i t h common base. It i s n o t antisymmetric;
243
Chapter 9
for example, take the case of an ordering 6 and the s t r i c t ordering < which are freely interpretable each by the other; or take the case of an a r b i t r a r y relation R and i t s negation i R (see ch.2 5 1 . 7 ) . A multirelation freely interpretable in R i s s t i l l freely interpretable in any concatenation RS where S has same base as R . A necessary and s u f f i c i e n t condition f o r S t o be freely intrepretable in R i s t h a t every component relation of S i s f r e e l y intrepretable in R. Let R be a relation with base E If any bijection whose domain and range are subsets of E , i s a local automorphism of R , then R i s freely interpretable in every relation with base E . Take f o r example the identity relation on E , or i t s negation, or the n-ary r e l a t i o n taking always the value (+), e t c .
.
2 . 1 . Let R, S be two relations with common base, and l e t n be the a r i t y of S (the maximum of t h i s a r i t y in the case of a multirelation). Then a s u f f i c i e n t ( a n d obviously necessary) condition for s t o be freely interpretable in R i s t h a t every local automorphism o f R whose domain has cardinality 4 n i s a local automorphism of S This i s 1.8.(1) in other words.
.
2 . 2 . DIMENSIONAL ARITY Let R be a relation. The dimensional a r i t y of R shall be the l e a s t natural number n f o r which there e x i s t s a multirelation M whose a r i t y has n f o r maximum value, and in which R i s freely interpretable. For example, take an equivalence relation R having a f i n i t e number of equivalence classes, say three classes so as t o f i x the ideas. Then R i s freely interpretable in a unary b i r e l a t i o n , formed of the relation U(x) = + i f f x belongs t o the f i r s t equivalence c l a s s , and of V(x) = + i f f x belongs t o the second equivalence class ( t h e third equivalence class being defined by U(x) = V(x) = - ) . Then the dimensional a r i t y o f R i s 1 . For each positive integer
n , there e x i s t s a denumerable rsl.atioj
R
whose a r i t y
.
and dimensional a r i t y are b o t h equal t o n In other words, our n-ary relation i s freely interpretable in no multirelation of a r i t y s t r i c t l y l e s s than n (PABION 1970, unpublished). Take a denumerable base E and an n-ary relation R on E , in which every f i n i t e n-ary relation be embeddable. This i s easy, since there are only f i n i t e l y
0
many n-ary relations on a s e t of p elements ( p integer): in f a c t , 2 t o the power pn such relations. Now suppose t h a t R i s freely interpretable i n a multirelation M formed of h components of a r i t y n-1 . Let (A denote the a r i t y of M , i.e. the sequence of h values equal t o n-1 The number of e - a r y multirelations on a base of p elements, i s equal t o : 2 t o the power (h.pn-’) , T h u s when taking a base of
.
R
244
THEORY OF RELATIONS
c a r d i n a l i t y p > h , the number of n-ary r e l a t i o n s on t h i s base i s s t r i c t l y g r e a t e r than t h e number of p -ary m u l t i r e l a t i o n s . I n our denumerable base E , take a set of p-tuples ( a l , ..., a ) ( p i n t e g e r > h ) P of c a r d i n a l i t y : 2 t o t h e power pn ; so t h a t , t h e b i j e c t i o n taking a l , ...,a into P 1,. . . , p , transforming the r e s t r i c t i o n R/ial ,. . . ,a i n t o an n-ary r e l a t i o n w i t h P base $1,. .. , p ) , these l a t t e r r e l a t i o n s a r e mutually d i s t i n c t . Then t h e r e necessar i l y exist two of our p-tuples, say ( a l ,..., a p ) and (bl ,..., b p ) , f o r which the
1
r e s t r i c t i o n s M/{al, ...,a p \ and M/{bl, ....b 1 a r e transformed i n t o the same P m u l t i r e l a t i o n w i t h base l, 1,..., p ) . I t follows t h a t the function taking a l ,..., a i n t o b l , ..., bp i s a l o c a l automorphism of M , y e t not a local automorphism P of R . In o t h e r words R i s not f r e e l y i n t e r p r e t a b l e i n M : c o n t r a d i c t i o n . 0 2.3. INTERPRETABILITY ARITY To obtain a common g e n e r a l i z a t i o n of t h e notions of dimensional a r i t y of a r e l a t i o n , as defined i n t h e above 2 . 2 , and t h a t of t h e dimension of a p a r t i a l o r d e r i n g , a s defined i n ch.4 5 7.3 and going back t o DUSHNIK, MILLER 1941, we say t h a t an o r d i nal sequence I of length a , taking natural number values, i s an i n t e r p r e t a b i l i t y a r i t y f o r the r e l a t i o n R , i f t h e r e e x i s t s an o( -sequence obtained from I by replacing each term ui i n I ( t h u s i < d ) by a r e l a t i o n Ri o f a r i t y u i and having t h e same base a s R , so t h a t R i s f r e e l y i n t e r p r e t a b l e i n t h e sequence of the R i , in t h e sense t h a t every local automorphism common t o the Ri i s a local automorphism of R Then f o r an a r b i t r a r y r e l a t i o n R , i f we r e q u i r e t h a t I have f i n i t e l e n g t h , then t h e dimensional a r i t y i s the l e a s t possible maximum of t h e sequences I which a r e i n t e r p r e t a b i l i t y a r i t i e s of R . I f R i s a p a r t i a l ordering and i f we r e q u i r e t h a t the r e l a t i o n s Ri be c h a i n s , then the l e a s t p o s s i b l e length of t h e i n t e r p r e t a b i l i t y a r i t i e s I i s obviously l e s s than o r equal t o the dimension of R i n the sense of ch.4 5 7.3. Problem. I f R i s the i d e n t i t y r e l a t i o n , whose dimension i s 2 , and i f we again
.
require t h a t t h e r e l a t i o n s Ri be c h a i n s , t h e n t h e l e a s t possible length under consideration i s 0 (the i d e n t i t y r e l a t i o n being f r e e l y i n t e r p r e t a b l e i n t h e multirel a t i o n reduced t o i t s b a s e ) . Apart t h i s c a s e , t h e dimension of a p a r t i a l ordering is equal t o t h e l e a s t p o s s i b l e length of i t s i n t e r p r e t a b i l i t y a r i t i e s by means of chains. 2.4. Let R , R' be two m-ary r e l a t i o n s and S , S ' be two n-ary r e l a t i o n s ; l e t E be the common base of R and S , and E ' t h e common base of R ' and S ' . I f every r e s t r i c t i o n of t h e concatenation R'S' 4 n elements i s embeddable i n RS , and i f S i s f r e e l y i n t e r p r e t a b l e i n R , then S ' i s f r e e l y i n t e r p r e t a b l e in R' .
245
Chapter 9
f'
Let
be a l o c a l automorphism o f
less than o r equal t o restriction from
.
n
(R'S')/F'
onto a r e s t r i c t i o n o f
RS
(R'S')/(f')'(F')
onto a r e s t r i c t i o n o f
RS
a l o c a l automorphism o f
. Thus
in
R
of
R'
i s f r e e l y interpretable i n
S'
(ch.2
<3
be a chain and
R
8.6). L e t
dinality
and
R'
S/X
n
be the a r i t y o f
f = h,f',g-l
: then
.
S'
Hence every l o c a l automorphism S'
.
By 2.1
( p r o o f communicated by PAILLET). 0
the associated t e r n a r y c y c l i c r e l a t i o n
S
. Then
RS
is
f
l a t t e r being f r e e l y i n t e r p r e t a b l e
i s a chain and
R'
w i t h car-
R'S'
S'
, then f o r every subset
R
i s f r e e l y interpretable i n
there e x i s t s an extension o f 0 Let
. Let
from the
t h e r e e x i s t s an isomorphism
i s the c y c l i c
.
R'
i s freely interpretable i n
S
has c a r d i n a l i t y
i s a l o c a l automorphism o f R'
i s freely interpretable i n
S
the r e s t r i c t i o n
= Dom f '
be such t h a t every r e s t r i c t i o n o f
S'
i s embeddable i n
r e l a t i o n associated w i t h If
n elements
, and
, the
S
i s a l o c a l automorphism o f
f'
For example, l e t
If
and hence o f
R
defined on a t most
above,
2.5.
, such t h a t F '
R'
By hypothesis, there e x i s t s an isomorphism g
S
i s an extension o f
R'
. . ,xn)
(xl,.
f
I f there e x i s t s a l o c a l isomorphism
from
\
R
, then
.
be an n-tuple i n the base into
R'
{xl ,..., xn , then p u t S'(xl ,..., xn) = S(fxl n o t depend on t h e chosen isomorphism, because
o f the base,
R'
which i s f r e e l y i n t e r p r e t a b l e i n
, and
S
and i f
R
X
.
R/X
IR ' I .
having domain
R
. This l a s t value does i s freely interpretable i n R
,..., fx,) S
.
Now i f there does not e x i s t such a l o c a l isomorphism from
R'
into
. Then
S'
i s an extension o f
,.... x n \ , then
domain { x l
R, S
2.6. L e t
-
i s f r e e l y interpretable i n
R/X
,&
S
.
I f , f o r each P
X
the s e t S/X
=
i s f r e e l y interpretable i n
S
of
R
which modifies
.
S
o f elements i n the base, w i t h
( a l,...,an)
R
be t h e a r i t y o f
n
i s not f r e e l y interpretable i n
S
f
automorphism
If
+
.0
R'
have a common base, and
Suppose t h a t
and
=
having
X o f the base w i t h c a r d i n a l i t y l e s s than o r equal t o 2n, the r e s t r i c t i o n
subset S/X
,..., xn)
S'(xl
and i s f r e e l y i n t e r p r e t a b l e i n
S
0
put
R
{ al,. . . .an,fal,.
. . ,fan }
i s a chain, then t h e i n t e g e r
for
n = 2
, the
2n
. Then
there e x i s t s a l o c a l
Thus t h e r e e x i s t s an S(al,
...,an)
# S(fal,
n-tuple
...,fa,) . Then
has c a r d i n a l i t y a t most equal t o
i s not freely interpretable i n
However i n general,
R
R/X
2n ;
.0 n + l : see
can be replaced by
value
.
R
4
5 5.1
below.
cannot be replaced by a smaller
integer. 0 Take a p a r t i a l o r d e r i n g
dinality
>/
2)
, by
R
constructed from two chains
s e t t i n g each element o f
R1
R1,
R2
(each w i t h car-
t o be incomparable w i t h each
246
THEORY OF RELATIONS
element o f =
R1
Take
S
and
S2 =
,yet
table i n
R
cardinal
1, 2
§
.
R2
Sl
chains
3
Ri
= converse o f
. Then
R2
i s freely interpretable i n
S/X
or
t o be t h e p a r t i a l o r d e r i n g s i m i l a r l y o b t a i n e d f r o m t h e i s not freely interpref o r each subset
R/X
CONNECTION WITH FREE INTERPRETABILITY
Given two f i n i t e sequences o f n a t u r a l numbers, say a s s o c i a t e s t o each
m-ary m u l t i r e l a t i o n
m and
an
R
, a f r e e operator
n
n-ary m u l t i r e l a t i o n
h a v i n g t h e same base. We have t h e a d d i t i o n a l c o n d i t i o n t h a t f o r any two R, R '
relations
each l o c a l isomorphism f r o m
T(R)
morphism f r o m
phism. The sequences i s an
(m,n)-ary
with
X
.0
3 - FREE OPERATOR,
9
S
into
R
into
T(R').
m, n
are called the a r i t i e s o f
o p e r a t o r . We say t h a t any
i s also a local iso-
R'
9
I n o t h e r words,
(R) m-ary
p r e s e r v e s l o c a l isomor-
9 , and we
say t h a t
7
9 .
m-ary r e l a t i o n i s a s s i g n a b l e t o
Note t h a t each f r e e o p e r a t o r i s c o m p l e t e l y determined by i t s values on m u l t i r e l a t i o n s whose base i s a f i n i t e subset o f t h e s e t ~3 o f n a t u r a l numbers: t h e v a l u e i n t h e general case f o l l o w s immediately by u s i n g l o c a l isomorphisms. As a l l t h e m u l t i r e l a t i o n s based on subsets o f an
(m,n)-ary
w c o n s t i t u t e a s e t , we can d e f i n e
o p e r a t o r as a f u n c t i o n which, t o each
f i n i t e subset o f CJ
m-ary r e l a t i o n based on a
, a s s o c i a t e s an n-ary r e l a t i o n on t h e same base, w i t h t h e
preceding c o n d i t i o n about l o c a l isomorphisms. A f t e r d e f i n i n g we complete by d e f i n i n g t h e v a l u e t a k e n by
9
f o r each
as i n d i c a t e d ,
m-ary r e l a t i o n : a l l t h i s
w i t h i n t h e framework o f t h e axioms o f ZF. Example. F o r a g i v e n n a t u r a l number each
, negation
n
i s a f r e e o p e r a t o r which takes
n-ary r e l a t i o n i n t o t h e r e l a t i o n h a v i n g t h e same base and always t a k i n g
t h e o p p o s i t e value. Another example. The symmetrizing o p e r a t o r which takes each b i n a r y r e l a t i o n i n t o the binary r e l a t i o n
S
satisfying
S(x,y)
= R(y,x)
for all
R
base. A l s o t h e o p e r a t o r which t a k e s each b i n a r y r e l a t i o n relation An
S ( x ) = R(x,x)
(m,n)-ary
f o r those Max n 3.1.
S
The m u l t i r e l a t i o n
9
i s determined by t h e ordered p a i r s R
(R,
T(R))
having a base o f c a r d i n a l a t most equal t o
R
i s freely interpretable i n
into
S
freely interpretable i n
R
i f f there exists a
R
.
Consider t h e case o f two r e l a t i o n s :
. . ,xn)
i n t o t h e unary
t h e r e a r e o n l y f i n i t e l y many f r e e o p e r a t o r s o f g i v e n a r i t i e s .
f r e e operator taking
(xl,.
R
i n the
.
f r e e operator
m-ary m u l t i r e l a t i o n s
. Hence
x, y
. F o r each
o f elements i n t h e base
R
is
m-ary and
m-ary r e l a t i o n
I X I , either
S
X
is
n-ary and
and each
S
n-tuple
t h e r e e x i s t s an isomorphism
is
2 41
Chapter 9
f
from X/\xl
,..., xn) onto
( 3X ) ( x l , ..., x n )
a r es t r i ct i o n
R/{ fxl ,... ,fxn) : then we define
S(fxl.... , f x n ) ; t h i s l a s t value does n o t depend on the chosen isomorphism. Or no such isomorphism ex i s t s : then we s e t ((?X)(x l , . . . , x n ) = t . 0 =
3.2. INJECTIVE OPERATOR # ?(R') A free operator .j-> i s said t o be injective i f R # R ' implies tha t for a l l R , R ' assignable t o 9 ; or equivalently i f for a l l R , R ' every local isomorphism from q ( R ) into T(R') i s a local isomorphism from R into R ' .
Y(R)
Every injective operator has an inverse. More precisely, i f i s inje c tive , then there e x i s t s a f r e e operator 2 such t h a t $, T(R) = R for each R assignable to .
9
9.
Then Consider the case of r el at i o n s , and l e t m , n be the a r i t i e s of given an n-ary relation Y and an m-tuple ( x l , ..., x,) of elements in the base I Y I , we define ( d , Y)(xl ,..., xm) = X(xl ,..., xm) i f the re stric tion
0
x l , ...,xm] i s the image under 9 of an m-ary relation X having the same Y ) ( x l , ...,xm) = + i f there i s no such X . 0 base; o r
Y/{
(a
Hence, if i s i n j ect i v e, then R and F(R) are each freely interpretable in the other. A converse of t h i s r es u l t shall be proved in 3.5 below.
9
COMPARISON BETWEEN ARITIES We say t h a t the a r i t y n (of a multirelation) i s greater t h a n the a r i t y m , i f each term mi of m can be associated with a term n o f n , with m i $ n j , j in an injective manner: i . e . two d i s t i n c t indices i , i ' in m correspond t o two d i s t i n c t indices j , j ' in n .
If the a r i t y n i s greater t h a n the a r i t y m , then f o r any natural number P, there are more n-ary multirelations with base having cardinality p , than there are of m-ary multirelations with the same base. his However, even i f the above condition i s t r u e f o r every natural number p , tdoes n o t necessarily imply t h at the a r i t y n i s greater than m . 0 For example, take m = (1,l) and n = (0,2) Then f o r a base of cardinality p , there are ( 2 t o the power 2 p ) many m-ary b i r ela tions, and ( 2 t o the power ( l + p 2 ) ) many n-ary b i r el at i o n s , with l + p 2 >/ 2p ; ye t n i s not greater than m.
.
Problem. If the a r i t y n i s greater t h a n m , then there obviously e xists an (m,n)-ary injective operator. Indeed t o each mi-ary component Ri of the multirelation R , i t suffices t o associate the n.-ary component S j ( m i Q n j ) J whose value only depends on the mi f i r s t terms, tha t value being equal to tha t o f Ri . Conversely i f n i s n o t greater t h a n m , then we conjecture t h a t there e xists no injective free operator with a r i t i e s ( m , n ) For example i f m = (1,l) and
.
0
THEORY OF RELATIONS
248
, then
R = (0,2)
there e x i s t s no i n j e c t i v e f r e e
w i t h a base o f three elements the value (+) o n l y f o r
a, b, c
, and
a
R2
(m,n)-ary
operator. Indeed s t a r t
and the b i r e l a t i o n
only f o r
b
. Then
(R1R2)
where
R1
takes
t h e o n l y l o c a l automorphisms
o f t h i s b i r e l a t i o n are the i d e n t i t y on each subset o f the base. On the o t h e r hand, any a r b i t r a r y b i n a r y r e l a t i o n must take the same value, f o r instance f o r and
(a,a)
, and thus admits a l o c a l automorphism o t h e r than the i d e n t i t y on a
(c,c)
subset o f t h e base. Obviously the concatenation o f our b i n a r y r e l a t i o n w i t h a 0-ary r e l a t i o n , changes nothing i n the previous discussion. F i n a l l y , the reader who i s tempted by t h e pseudo-solution which associates t o S(x,y) = R1(x) A R2(y) , w i l l note t h a t i n j e c t i v i t y takes always t h e value ( - ) , f o r example. R1
(R1R2) t h e b i n a r y r e l a t i o n i s no longer s a t i s f i e d when 3.3.
PARTIAL OPERATOR
Let
m
, and
be a f i n i t e sequence o f n a t u r a l numbers
18
a set o f
m-ary m u l t i r e l a -
t i o n s w i t h f i n i t e bases, which i s closed under r e s t r i c t i o n and isomorphism ( t o be rigorous i n the frame o f the axioms o f ZF, we assume t h a t the bases o f our m u l t i r e o f n a t u r a l numbers ) .
l a t i o n s are f i n i t e subsets o f the s e t (m,n)-ary
A
4 i s a function , associates an n-ary
p a r t i a l operator w i t h domain
m-ary m u l t i r e l a t i o n R
belonging t o (R
9
w i t h the same base, such t h a t
preserves l o c a l isomorphisms.
9 to
A necessary 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 a r t i a l operator
i s again t h a t every l o c a l isomorphism from ?(R) phism from
R
into
R'
which t o each multirelation T(R)
, f o r every
R, R '
into
belonging t o
$(R') Dom
be i n j e c t i v e ,
be a l o c a l isomor-
9.
CANONICAL EXTENSION Let
9
arity
be a
,in
m
(m,n)-ary p a r t i a l operator, where t h e a r i t y
9 ,and whose
extending
domain contains a l l
F i r s t , t o each term mi
, such t h a t
i s g r e a t e r than the
of
m
, associate i n an i n j e c t i v e manner a term n . o f
mi$
mi
by
m
and
.,
e i t h e r the r e s t r i c t i o n we s e t
(
The operator If
9
9=
=
i n the base R i s an element o f Dom
. . ,xn) ( T R')(xl,. . . ,xn)
R ' = R/{ xl,.
9 =R)(xl,. . .,xn)
injective.
.
IR I ,
y, and then
thus defined i s c a l l e d t h e canonical extension o f
.
6 n (replace
f o r a m u l t i r e l a t i o n ) , then the canonical extension i s t h e unique f r e e
operator extending I n general,
J by n
n j
; o r not, and then we s e t
i s already defined f o r a l l m-ary r e l a t i o n s o f c a r d i n a l i t y
by Max n
9=
m-ary m u l t i r e l a t i o n s .
n ; i n t h e f o l l o w i n g we denote j Given an m-ary r e l a t i o n R and elements xl,.. x n
n
t h e sense o f 3.2 above. We s h a l l d e f i n e as f o l l o w s an operator
9 .
can be i n j e c t i v e w i t h o u t i t s canonical extension
9=
being
2 49
Chapter 9
m
Let
9
= n = 1 and l e t
associate t o each unary r e l a t i o n t a k i n g always the
9 i s undefined f o r 9 = takes every unary
value (+),theunary r e l a t i o n always ( - ) ; y e t
unary r e l a t i o n s
taking a t l e a s t once, t h e value ( - ) . Then
relation into
the unary r e l a t i o n w i t h same base, always ( - ) ; so t h a t p = i s n o t i n j e c t i v e . 0 3.4. L e t
k
be a n a t u r a l number
. If a partial
7
operator
i s defined f o r every
m-ary r e l a t i o n o f c a r d i n a l i t y
4 k and o n l y f o r such, and i f
then the canonical extension
9=
Moreover i n t h e case where
(
9
i s injective,
i s injective.
i s an
(m,m)-ary
p a r t i a l operator, then we have
9 = ) - I - (9- I ) =
3.5.
n
greater than
i n j e c t i v e p a r t i a l operator w i t h domain
L e t m, n
be two a r i t i e s w i t h
and range
i n t e g e r f o r which
&
. Let 9
m
@ .
Let
be an (m,n)-ary
k
be the l a r g e s t
contains every m-ary m u l t i r e l a t i o n o f c a r d i n a l i t y
Then t h e r e e x i s t s a p a r t i a l operator
+
7 ,which
extending
defined f o r every m-ary m u l t i r e l a t i o n o f c a r d i n a l i t y Consequently every i n j e c t i v e p a r t i a l (m,n)-ary
6 k+l
.
& k
i s i n j e c t i v e and
(POUZET 1973).
operator, where
n
i s g r e a t e r than
m , i s e x t e n d i b l e t o an i n j e c t i v e f r e e operator: go from k t o k + l , e t c . u n t i l reaching
.
Max n
Another consequence i s the f o l l o w i n g converse o f 3.2: Let
If
R R
be an m-ary r e l a t i o n ,
and
S
an n-ary r e l a t i o n , w i t h
i n j e c t i v e f r e e (m,n)-ary
operator which takes
R
w i
t i v e p a r t i a l operator which takes every r e s t r i c t i o n o f of 0
S
n
g r e a t e r than
.
m
are each f r e e l y i n t e r p r e t a b l e i n t h e other, then there e x i s t s an
S
S : s t a r t w i t h the i n j e c i n t o the r e s t r i c t i o n
R
having the same base.
Proof o f the f i r s t assertion. I t s u f f i c e s t o consider the case of an m-ary
relation o f cardinality Let
pk be the
, which
k+l
tions with c a r d i n a l i t i e s
,<
k
9
i)(R)
t o the range the range (Sh-l)
63
; then
@I
9 to
; then
S1 = ( S2 =
R
.
the s e t o f a l l r e l a -
.
Consider t h e sequence o f r e l a t i o n s So = (
td , say
does n o t belong t o t h e domain
r e s t r i c t i o n o f our p a r t i a l operator
9
So, S1,
... , Sh
provided t h a t
-')(So)
;)(
(9 ;)( 7 -l)(S,)
provided t h a t
; and so on, u n t i l t h e f i r s t index
does n o t belong t o the range
@ . Then we
( h integer)
h
(
So S1
belongs belongs t o
( 9 E)(T- l ) T+)(R)t o be Sh .
f o r which
define
, where
Sh =
To see t h a t t h i s procedure always terminates by g i v i n g a r e l a t i o n which does n o t belong t o the range
a,note t h a t t h e
phic, provided t h a t they belong t o
@.
Si
( i = O , l , ...)
Indeed, i f
f
are mutually non-isomor-
were an isomorphism from
250
THEORY OF RELATIONS S . ( 1 6 i < j , < h) , then f would be an isomorphism from J (because the i n j e c t i v i t y of 57 and 7 ; ) , and so f o r t h ; thus
onto
Si Sj-l
an isomorphism from
9
Note t h a t
onto
So
, hence
Sj-i
hence
belongs t o
Sj-i-l
R
R
t i o n as does
.
,
R'
with cardinality
R = R'
+
. First
k+l
y+(R)
=
.
T(R) and
O r neither
nor
R
and
R'
R'
, so
that
&.
belongs t o
R'
by
h
h'
the number s i m i l a r l y associated w i t h
the number o f successive transformations
. Then we
h < h'
, it
to
@i
Si
= S;
obtain t h a t
follows that
f o r each
R
7 '(R)
belongs t o J?i f f
R
similarly with
R
i f we take a proper
R'
belongs t o
on t h i s r e s t r i c -
F+(R'): we
must show
P+(R) belongs
8, and
a,
to
then we have
because t h e i n j e c t i v i t y
R = R'
Then we c l a i m t h a t , i f we denote Si
, then
associated w i t h h = h'
.
R
, and
by
Indeed suppose t h a t
; and since t h i s
'4
R = R'
=
belong t o
R = T-l(S,l,,-h-l)
i and hence
. But
-l(Sj-i-l)
4 , then y + acts
and suppose t h a t
notice t h a t
would be
i s i n j e c t i v e . Indeed, take two m-ary r e l a t i o n s
by the preceding c o n s t r u c t i o n . E i t h e r of
519
onto
R
f
: contradiction.
which belongs t o the domain
F i n a l l y i t remains t o see t h a t R
,&
obviously preserves l o c a l isomorphisms. And
+
r e s t r i c t i o n of
that
from
belongs t o
onto
Si-l
S,l,,-h-l
belongs
: c o n t r a d i c t i o n . F i n a l l y we o b t a i n
.0
Problem due t o POUZET 1973. Can the above be generalized t o t h e case o f i n t e r p r e t a b i l i t y by l o g i c a l formulas ( f i r s t order p r e d i c a t e c a l c u l u s w i t h i d e n t i t y ) . That is, i f
R
,
a r e r e l a t i o n s o f the same a r i t y , each o f which i s i n t e r p r e t a b l e i n
S
the o t h e r v i a a l o g i c a l formula, then does there e x i s t a l o g i c a l formula which operates i n j e c t i v e l y and takes
§
A
4
-
CONSTANT
relation R
R
into
S
.
RELATION
w i t h base
E
i s s a i d t o be constant i f every permutation o f
E
is
an automorphism o f R . For example the n-ary r e l a t i o n t a k i n g always t h e value (+) the n-ary r e l a t i o n always ( - ) , t h e b i n a r y r e l a t i o n o f i d e n t i t y , t a k i n g t h e value
(+) when
x = y
and ( - ) when
x # y
.
The d e f i n i t i o n extends t o m u l t i r e l a t i o n s ; we see t h a t a m u l t i r e l a t i o n i s constant
i f f i t s component r e l a t i o n s are a l l constant. I f R i s constant, then every r e s t r i c t i o n o f 4.1. An n-ary r e l a t i o n sequences i n t o y1
x1 ,. . . ,xn , ... , and
R
i s constant i f f
and yl,. xn
into
. . ,y, yn
R
R(xl
i s constant.
,..., xn)
,...,y n )
= R(yl
for all
such t h a t t h e transformation which takes
,is
x1
an i n j e c t i v e f u n c t i o n .
I n o t h e r words, i f f f o r each p a i r o f i n d i c e s
i, j
(1 5 i < j
6
n)
, we
have t h a t
,
25 1
Chapter 9
xi = x . i f f yi = y j . J It f o l l o w s t h a t R i s constant i f f : (1) a l l r e s t r i c t i o n s o f R .having a same c a r d i n a l i t y , less than o r equal t o the a r i t y , are isomorphic, and ( 2 ) these r e s t r i c t i o n s are constant.
Neither o f the above c o n d i t i o n s (1) and ( 2 ) i s alone s u f f i c i e n t . 0 For every unary r e l a t i o n , any r e s t r i c t i o n o f c a r d i n a l i t y
1 i s constant.
For every r e f l e x i v e and symnetric b i n a r y r e l a t i o n , any r e s t r i c t i o n o f c a r d i n a l i t y 1 o r 2 i s constant. For every tournament (see ch.5
5
2.7),
a l l restrictions with
c a r d i n a l i t y 1 are isomorphic; s i m i l a r l y a l l r e s t r i c t i o n s w i t h c a r d i n a l i t y 2. 0 Given an n-ary constant r e l a t i o n tion of
with cardinality
S
Moreover i f
S
6
n
E
, there
I f Card E
n
, then
, then
4.3.
0
E'
f o r every superset
i s constant, then our conclusion i s obvious. Conversely i f
R
E'
.
a
in
iL
E
,
.
R
1.2 above, there e x i s t s an element
R
i s not
f o f E which modifies R . Using E such t h a t the t r a n s p o s i t i o n (a,fa)
.0
A relation R
base i s
i s constant.
i s constant i f f , f o r a l l elements a, b
constant, then there e x i s t s a permutation modifies
to
S
.
, then
E R
, then
R
S = R
R w i t h base
S : i f every r e s t r i c -
t h i s extension i s unique.
-E
R
R
e x i s t s a constant extension o f
4.2. A r e l a t i o n R , w i t h base the t r a n s p o s i t i o n (a,b) 0 If
i s embeddable i n
has the same base as
Given an n-ary constant r e l a t i o n of
and an n-ary r e l a t i o n
R
i s constant i f f every b i j e c t i o n between two subsets o f the
a l o c a l automorphism o f
R
.
I f our c o n d i t i o n holds, then i n p a r t i c u l a r every permutation o f the base i s an
automorphism, hence trary bijection
f
i s constant. Conversely i f
R
i s constant, then an a r b i -
R
between two f i n i t e subsets o f the base
i s extendible t o a
permutation o f the base, which by hypothesis i s an automorphism o f
R
. Finally
we o b t a i n the case f o r a b i j e c t i o n w i t h i n f i n i t e domain ( i n c l u d e d i n the base), by applying 1.7 above. 0
4.4. A necessary and s u f f i c i e n t c o n d i t i o n f o r an n-ary r e l a t i o n R t o be const a n t , i s t h a t f o r each sequence xl, ..., xn i n R and each index i ( 1 6 i 6 n) , and f o r each element ve f u n c t i o n
f
u
i n the base, d i s t i n c t from
w i t h domain
4 xl,...,xn)
x . # xi (16 j J (communicated by HODGES).
preserves each
which takes
6 n) , s a t i s f y
R(xl,.
, ... , xn , the
x1 xi
.. ,xn)
% 1
u
= R(fxl,.
bijecti-
and which
. . ,fxn)
THEORY OF RELATIONS
252 If
i s constant, then our conclusion f o l l o w s from the preceding statement 4.3.
R
Conversely i f our c o n d i t i o n holds, then by t r a n s i t i v i t y , f o r each
m,<
n
, every
b i j e c t i o n o f an m-element s e t onto another i s a l o c a l automorphism o f R , Using 1.7 above, every b i j e c t i o n between any two subsets o f the base i s a l o c a l automorphism o f
, and
R
so by the preceding 4.3,
4.5. An n-ary r e l a t i o n
nality If
n+l
the r e l a t i o n
R
i s constant. 0
i s constant i f f every r e s t r i c t i o n o f
R
R having c a r d i -
i s constant.
i s constant, we already n o t i c e d t h a t each r e s t r i c t i o n i s constant. Conver-
R
sely, i f each r e s t r i c t i o n t o a t most
n+l
elements i s constant, then the condi-
t i o n i n our preceding statement 4.4 i s s a t i s f i e d , since the s e t
6
has c a r d i n a l i t y
ntl
{ X
l,...,~n,~
)
.0 by
n+l
The p r o p o s i t i o n i s f a l s e i f we replace
n : see i n 4.1 above, the exam-
ples o f any unary r e l a t i o n , o r any r e f l e x i v e and symmetric b i n a r y r e l a t i o n . 4.6. For each n a t u r a l
-
s e t {1,2
,...,n )
on t h i s s e t .
number
, l e t H(n) be t h e number o f p a r t i t i o n s of the
n
; i n o t h e r words, the number o f b i n a r y equivalence r e l a t i o n s
n = 0
For
, we
adopt t h e convention t h a t
H(0) = 1
, by
considering
t h a t the empty b i n a r y r e l a t i o n i s an equivalence r e l a t i o n . Then
H(0)
=
H(n+l) = C:
H ( l ) = 1 and we have the f o l l o w i n g recurrence r e l a t i o n :
H(0)
+
C;
H(l)
... +
+
C:
H(n)
, where
the
Cf
are the
( i s n)
usual binomial c o e f f i c i e n t s . J, l , Z , . . . , n t l ) , l e t A be the equivalence , and l e t k be the c a r d i n a l i t y o f the complementary s e t 21,... ,n+l} - A : so 0 4 k 6 n . For each k , the number o f p o s s i b l e choices
Given an equivalence r e l a t i o n on
class o f
A
for
n+l
, or
e q u i v a l e n t l y f o r t h e complement o f
A
,is
p o s s i b l e p a r t i t i o n s o f t h i s complement, i s t h e product
CL
. Hence
C:.H(k)
the number o f
.O
E be a s e t w i t h c a r d i n a l i t y g r e a t e r than o r equal t o n . Then t h e r e are ( 2 t o the power H(n) ) many constant n-ary r e l a t i o n s w i t h base E , where H(n) i s t h e p r e v i o u s l y defined f u n c t i o n g i v i n g the number o f p a r t i t i o n s o f the s e t
Let
,..., n ) .
{I
Let n-tuple (1
U
be an equivalence r e l a t i o n w i t h base 41 (xl,
...,xn)
6 i < j 6 n)
o f elements i n
we have
xi
= x
4.1 above, an n-ary r e l a t i o n equivalence r e l a t i o n t o (+) o r ( - ) )
, such
U
R
iff
j
R
say t h a t an
i s a (U,n)-tuple i f f o r a l l i,j i and j a r e equivalent (mod U) . By
w i t h base
based on
that
E
....,n} . We
E
{ 1,. .. ,n}
takes t h e value
i s constant i f f t o each i s associated a value v(U)
on a l l
v(U) (equal of
(U,n)-tuples
25 3
Chapter 9
elements o f
E ; the proposition follows. 0
Some i n i t i a l values o f
H : we have
H(2) = 2 ; t h i s y i e l d s the 4 constant b i n a r y
+ , always - , i d e n t i t y and i t s negation. We have H(3)
r e l a t i o n s : always H(6)
=
.
203
5 5 - CHAINABLE Let A
RELATION
be a chain w i t h base
E
. Then a
relation
A
automorphism o f A relation
i s a l o c a l automorphism o f
w i t h base
R
.
A-chainable i f i t i s f r e e l y i n t e r p r e t a b l e i n A
is
= 5 :
H(4) = 15 ; H(5) = 52 ; and
t h i s y i e l d s 32 constant t e r n a r y r e l a t i o n s ; then
.
R
i s s a i d t o be chainable i f there e x i s t s a chain
R
i s s a i d t o be
E
I n o t h e r words, i f every l o c a l f o r which
A
R
A-chainable.
For example the s t r i c t t o t a l o r d e r i n g ring
<
<
, obtained from the chain o r t o t a l orde-
by changing the value (t) i n t o ( - ) along the diagonal, i s
(mod A)
A-chainabl e. The t e r n a r y c y c l i c r e l a t i o n defined i n ch.2 (x,y,z)
iff
x sy,< z
or y
6 z ,< x
5
or
8.6, by g i v i n g (+) t o the t r i p l e
,is
z & x , c y (mod A)
.
S i m i l a r l y f o r the r e l a t i o n o f intermediacy o r betweenness (mod A) 5.1. L e t
R
and
A-chainable.
IAI
be two n-ary, A-chainable r e l a t i o n s (on the same base
R'
I f there e x i s t s an n-element s e t I n o t h e r words, an n-ary
,
F f o r which
, then R
R/F = R'/F
= R'
).
.
A-chainable r e l a t i o n i s determined by i t s r e s t r i c -
t i o n t o an n-element set. This f o l l o w s from ch.2
5
1.1, and t h e uniqueness o f the
isomorphism from a f i n i t e chain onto another equipotent chain. i s A-chainable,
If R R/X
is
I f f o r each subset the r e s t r i c t i o n 0
then f o r every subset
X
o f t h e base, t h e r e s t r i c t i o n
(A/X)-chainable.
X
o f the base o f R w i t h c a r d i n a l i t y 6 n + l ( n = a r i t y o f R), (A/X)-chainable, then R & A-chainable.
is
R/X
Using 1.8.(1) above, i t s u f f i c e s t o see t h a t f o r any two sequences o f n elements
x1
< x2 < .. . < xn
i n t o yi
for
zi = Min(xi,yi)
and y1
< y2 < ... < yn
then t o t h e sequence Z1,...Jn-1,Yn
, the
function taking
.
xi
i = 1, ...,n i s a l o c a l automorphism o f R For t h i s , f i r s t define n : thus zl< z2 < zn (mod A) . (mod A) f o r i = 1
,...,
Now s t a r t w i t h the sequence
to
(mod A)
z1,z2,x3
x1,x2
,..., xn , then
,...,xn , and
and so f o r t h u n t i l
yl,.
... <
go t o t h e sequence
so f o r t h u n t i l
z1
z1,x2
,..., xn ,
,...,zn , and
then
.. ,yn . 0
Given two m u l t i r e l a t i o n s R, S , then the concatenation R and S are both A-chainable.
RS
is
A-chainable i f f
THEORY OF RELATIONS
254
An
n-ary r e l a t i o n
R
is
A-chainable i f f , f o r each n a t u r a l number
i,< n , the
i are a l l isomorphic. These two observations f o l l o w from the uniqueness o f the isomorphism between two f i n i t e chains o f a same c a r d i n a l i t y . r e s t r i c t i o n s o f the concatenation
5.2. A constant r e l a t i o n w i t h base
base
0
E
with cardinality
AR
i i A-chainable f o r every chain
E
having
A
.
Indeed every b i j e c t i o n between two subsets o f
automorphism o f an a r b i t r a r y chain on
,is
E
, and i n p a r t i c u l a r every l o c a l
E
a l o c a l automorphism o f the given
constant r e l a t i o n : see 4.3 above. 0 However, i f a r e l a t i o n then
R
w i t h base
is
E
A
A-chainable f o r every chain
on
E
,
i s n o t n e c e s s a r i l y constant.
R
0 For example, a chain
A on two elements i s both A-chainable and (A-)-chainable A- i s the converse o f A ) . Another example: t h e t e r n a r y c y c l i c r e l a t i o n on three elements (see ch.2 9 8.6) i s
(where
A-chainable f o r any one o f the 6 chains
A
Let
E
R
either
If -
be an is -
R
n-ary r e l a t i o n w i t h base
i s f i n i t e , or that
E
0 It s u f f i c e s t o see t h a t
a sequence xi
( i = 1,
A
A-chainable f o r every chain
F , the r e s t r i c t i o n R/F xl,
...,xn
... ,n) . Now
in fix
R E
. Assume
that
Card E
>/ n + l and t h a t
i s i n f i n i t e and orderable: see ch.2 § 2.3.
E
I t even s u f f i c e s t h a t , f o r every
A ')"
based on these- 3 elements. 0
,then R
on - E
(n+l)-element subset
i s constant.
F .of
E
and every chain
A-chainable. Compare w i t h exercise 2 below.
s i-
s a t i s f i e s the c o n d i t i o n o f 4.4 above. Indeed, take
, and
, d i s t i n c t from the
let
u
be an element o f
E
i and l e t
A
be a chain on
f o r which
E
xi
and
u
are consecutive, and such t h a t those x . # xi (14j 4 n) are e i t h e r s t r i c t l y J l e s s than xi o r s t r i c t l y g r e a t e r than u (mod A) . Then the b i j e c t i o n w i t h domain
{ xl,.
.. ,xn) , which
takes
xi
i s a l o c a l automorphism o f the chain
A
x . # xi , J R : thus the c o n d i t i o n i n 4.4
u and preserves each
into
, hence
of
i s s a t i s f i e d ( p r o o f communicated by HODGES). 0
If R i s a chainable r e l a t i o n w i t h base E , then f o r every superset E'
5.3.
of
E
, there
e x i s t s a chainable extension o f
n i t e , t h i s uses the o r d e r i n g axiom, ch.2
5
, and
R
0 Let
A
be a chain w i t h base
E
R
w i t h base
E' ; for
El-E
infi-
2.3.
be an
A-chainable r e l a t i o n . It s u f f i -
A ' o f A t o E ' , and then t o apply R which be A'-chainable. 0
ces t o take a t o t a l l y ordered extension 2.5 above, t o o b t a i n an extension o f
5.4. A necessary and s u f f i c i e n t c o n d i t i o n f o r a r e l a t i o n
R
t o be chainable
is
t h a t every f i n i t e r e s t r i c t i o n o f R be chainable ( s u f f i c i e n c y uses the u l t r a f i l t e r axiom; ZF s u f f i c e s i f R i s countable).
Chapter 9
A-chainable, then we s a i d i n 5.1 t h a t f o r every subset
is
0 If R
255
base, the r e s t r i c t i o n
R/F
is
Conversely suppose t h a t every f i n i t e r e s t r i c t i o n o f
R
ate, t o each f i n i t e subset
UF
which set
R/F
is
F o f the base, t h e s e t
X-chainable. By hypothesis
UF
i s chainable. Then associ-
X
o f chains
on
, for
F
i s non-empty f o r every f i n i t e sub-
F f i n i t e and a subset G o f F , then each chain , when r e s t r i c t e d t o G , gives an element o f UG .
F o f the base. Given
which i s an element o f
F o f the
(A/F)-chainable.
UF
By the coherence lemma i n ch.2
0
1.3 ( e q u i v a l e n t w i t h the u l t r a f i l t e r axiom, y e t
based on I R ( , ( s o t h a t A/F i s a chain). This r e l a t i o n A i s thus a chain based on I R I Each l o c a l automorphism o f A having f i n i t e domain, i s a l o c a l automorphism o f R : hence R i s A-chainable by 2.1. 0
ZF s u f f i c i e n t f o r a countable base), there e x i s t s a r e l a t i o n
such t h a t f o r each
, t h e r e s t r i c t i o n A/F belongs t o
F
A
UF
.
The preceding p r o p o s i t i o n w i l l be strengthened i n ch.12
-
arity
, there e x i s t s a n a t u r a l number p(n)
n
5
3.4 and 3.5: f o r each
such t h a t every
i s chainable, provided t h a t a l l i t s r e s t r i c t i o n s o f c a r d i n a l i t i e s chainable. Given a s e t o f chainable r e l a t i o n s the common extension o f these
R
R
,on
restriction 5.5.
. Then
R
S/F
has an
.are
be the common exten-
S
F o f the base
as an extension: thus
( 1 ) Given an a r b i t r a r y denumerable r e l a t i o n
subset
p(n)
, which i s d i r e c t e d under extension, then
f o r each f i n i t e subset
R
4
t h e union o f t h e i r bases, i s chainable.
0 This f o l l o w s from t h e preceding p r o p o s i t i o n . Indeed l e t
sion of r e l a t i o n s
n-ary r e l a t i o n
S/F R
, the
IS1
i s chainable. 0
, there
e x i s t s a denumerable
D o f the base and a chain A w i t h base D , which i s isomorphic w i t h W ,
and such t h a t the r e s t r i c t i o n (2) Given an a r i t y
n
R/D
ix
Ac-.-
and a n a t u r a l number
such t h a t e v e r y n-ary
p
, there e x i s t s an i n t e g e r q>/ p >/ q has a chainable r e s t r_i c _ tion ~
.
of cardinality p 0
(1) L e t
n
be t h e a r i t y o f
IRI
chain w i t h base multirelation
RC
.
, which
R
, which
We say t h a t two
n-element subsets
equivalent iffthe r e s t r i c t i o n s
(RC)/X
( i Q n)
. There
for
i-element subsets
we can assume i s >/ 2
. Let
C
be a
i s isomorphic w i t h W ; consider the concatenated and
(RC)/X'
X, X '
o f the base, are
are isomorphic; s i m i l a r l y
are o n l y f i n i t e l y many equivalence classes
f o r t h e equivalence r e l a t i o n thus defined. Using RAMSEY's theorem (ch.3 base, i n which a l l subsets, f o r each
5 l.l),
t h e r e e x i s t s a denumerable subset
n-element subsets are equivalent, as w e l l as a l l
i6 n
.
L e t A = C/D
,a
D of the i-element
chain isomorphic w i t h W . Then every
l o c a l automorphism o f A having domain o f c a r d i n a l i t y \< n i s a l o c a l automorBy 2.1, t h e r e s t r i c t i o n R/D i s f r e e l y i n t e r p r e t a b l e i n A 0 phism o f R/D
.
.
256
THEORY OF RELATIONS
fj 1.3.0
0 ( 2 ) Analogous proof, using t h e f i n i t a r y version o f RAMSEY's theorem: ch.3
5.6.
Given the o r d i n a l
w ,
, each G)-chainable r e l a -
the chain o f n a t u r a l numbers
t i o n i s minimal w i t h respect t o embeddability, among the denumerable r e l a t i o n s ;
i.e.,
every denumerable r e l a t i o n which i s embeddable i n an W-chainable r e l a t i o n i s isomorphic w i t h
R
Let
a
.
R
be an aleph, i . e . an o r d i n a l which i s equipotent w i t h no s t r i c t l y smaller
o r d i n a l . Then each
a-chainable r e l a t i o n i s minimal w i t h respect t o embeddability,
among the r e l a t i o n s o f c a r d i n a l i t y
.
a
The proposition 5 . 5 . ( 1 ) above asserts t h a t , f o r every denumerable r e l a t i o n R , R there e x i s t s a denumerable minimal r e l a t i o n which i s embeddable-i! This r e s u l t does n o t extend t o t h e c a r d i n a l i t y o f t h e continuum. Indeed by DUSHNIK,
.
MILLER, ch.5 fj 5.2.(1), dable i n t h e r e a l s
every chain w i t h continuum c a r d i n a l i t y and which i s embed-
has a r e s t r i c t i o n o f continuum c a r d i n a l i t y which i s s t r i c t l y
l e s s e r , w i t h respect t o embeddability. Problem. Does there e x i s t a r e l a t i o n w i t h continuum c a r d i n a l i t y , which i s minimal among r e l a t i o n s w i t h continuum c a r d i n a l i t y , y e t which i s n o t chainable by mean o f the continuum aleph ( i . e . the smallest o r d i n a l w i t h continuum c a r d i n a l i t y ; the axiom o f choice being used). 5.7.
For each ordered p a i r of n a t u r a l
numbers
n
the S t i r l i n g number, o r number o f p a r t i t i o n s o f an
and
r< n
, l e t S:
n-element s e t i n t o
denote r
non-
empty classes. Zn o t h e r words, the number o f equivalence r e l a t i o n s w i t h cardinality
n
We have
and having e x a c t l y r non-empty equivalence classes. 0 So = 1 : the equivalence r e l a t i o n w i t h empty base i s supposed t o e x i s t ;
0
obviously i t has e x a c t l y each
n 31
, since
non-empty equivalence classes. We have
an equivalence r e l a t i o n on
$=0
for
n elements has a t l e a s t one
non-empty equivalence class. We have the
S;
= S i = 1 f o r each s t r i c t l y p o s i t i v e i n t e g e r
S:
reCUrSiOn e q u a l i t y
Suppose t h a t the s e t
= Sn-'
r-1
{ 1,2,
and p u t aside the element
n
+ r.S:--l
... ,n} . Then
f o r every
n ; moreover we have r
i s partitioned i n t o
( 1 6 r
.
r non-empty classes,
an equivalence r e l a t i o n having
r
non-empty
classes can be obtained, e i t h e r by s t a r t i n g w i t h an equivalence r e l a t i o n on {1,2
,...,n-1)
having
rth class: t h i s y i e l d s r e l a t i o n on {1,2
r-1 classes, t o which we add the s i n g l e t o n o f
:S
,..., n-1)
n
as an
p o s s i b i l i t i e s ; o r by s t a r t i n g w i t h an equivalence having
r classes, and then adding t h e i n t e g e r
possibilities. 0 t o one o f these r classes: t h i s y i e l d s r.S;-' 3 For example S2 = 3 , corresponding t o t h e 3 p o s s i b l e p a r t i t i o n s o f
{ 1,2,3)
n
into
25 7
Chapter 9
4 S2 = 7
a s i n g l e t o n and i t s complement. Another example, 4 p a r t i t i o n s o f 21,2,3,4}
, corresponding
t o the
i n t o a s i n g l e t o n and i t s complement, plus t h e 3 p a r t i -
t i o n s i n t o 2 classes each having 2 elements. Let
n
number and A be a chain w i t h a base o f c a r d i n a l i t y 3 n . ( 2 t o the power V(n) ) many A-chainable n-ary r e l a t i o n s , where
be a n a t u r a l
Then there are
+
V(n) = O! S:
+
l!S!
remove the f i r s t term, since 0
Let
F
... +
+
2! S;
be a subset o f t h e base, w i t h c a r d i n a l i t y
an A-chainable
, we can
n
.
= 0
S:
. For p o s i t i v e
n! Sl
n
. We
have already seen t h a t F : see 5.1
n-ary r e l a t i o n i s determined by i t s r e s t r i c t i o n t o
above. Thus i t suffices t o c a l c u l a t e t h e number o f
n-ary r e l a t i o n s based on
and
xl,
(A/F)-chainable.
L e t us say t h a t two n-tuples
...,xn
F
and yl, ...,yn
in
F are e q u i v a l e n t if the f u n c t i o n mapping xi i n t o yi ( i = 1, ..., n) i s a l o c a l automorphism o f the chain A , thus if t h i s f u n c t i o n i s b i j e c t i v e and order preserving (mod A) R(xl
,..., xn)
.
Then an
= R(yl
n-ary r e l a t i o n
,..., yn)
Now there are e x a c t l y
V(n)
...,n
1,2,
empty classes, by l e t t i n g
xi = x
ordering the s e t o f these
r
i s l e s s than the class o f
j
(1
j same equivalence c l a s s : t h i s y i e l d i n g
r ! = 1.2.
F i n a l l y each
...
.r
n-tuples.
equivalence classes. Indeed, each c l a s s i s defined by
p a r t i t i o n i n g the s e t o f i n d i c e s
yields
(A/F)-chainable i f f we have
is
R
f o r a l l equivalent
Gi
i n t o a f i n i t e number
< jb n )
r
i and j
iff
o f non-
are i n the
SF p o s s i b i l i t i e s ; and f i n a l l y by t o t a l l y
classes and l e t t i n g
, modulo
xi
< x J.
i f fthe class o f
i
t h i s t o t a l o r d e r i n g o f the classes: t h i s
possibilities.
A-chainable r e l a t i o n i s defined by g i v i n g the value (+) o r t h e value
( - ) t o each o f the equivalence classes o f the n-tuples i n F : t h i s y i e l d s (2 t o the power V(n)) p o s s i b i l i t i e s . Note t h a t f o r n = 0 , we have V(0) = 1 0 since O! = 1 and So = 1 ; and we o b t a i n t h e two 0-ary r e l a t i o n s (F,+), (F,-). C a l c u l a t i o n o f the f i r s t values o f r e l a t i o n s , always (+) and always ( - ) t i o n s : always ( + )
, the
chain
A
. We
V
.
, the
have
We have
V(l) = 1 V(2) = 3
, giving
, giving
,
the two unary 8 binary rela-
converse chain, the i d e n t i t y r e l a t i o n , plus
the 4 o t h e r r e l a t i o n s obtained by interchanging (+) and ( - ) ( o r e q u i v a l e n t l y by r e p l a c i n g (+) by ( - ) on t h e diagonal). Then we have 5.8.
V(3) = 13,
V(4) = 75
.
CONJUNCTION OF A SET OF RELATIONS
I n ch.2 i n ch.2
5 5
1.7, we defined the c o n j u n c t i o n o f two r e l a t i o n s having t h e same base; 4.1, we defined the reinforcement o f a r e l a t i o n , and i n ch.4
5
7.3, the
generalized conjunction o f p a r t i a l orderings. Now we extend t h i s n o t i o n t o a r b i t r a r y relations. An n-ary r e l a t i o n
R
i s s a i d t o be the conjunction o f the r e l a t i o n s
Ri
which are
258
THEORY OF RELATIONS
reinforcements o f which each
,if
R
takes t h e value (t) f o r e x a c t l y t h o s e
R
t a k e s t h e v a l u e (+)
Ri
. Recall
n-tuples f o r
t h a t by d e f i n i t i o n , a r e i n f o r c e m e n t
o f R has t h e same base and a r i t y as those o f R . Recall t h a t every p a r t i a l o r d e r i n g A i s the conjunction o f the t o t a l l y ordered reinforcements o f
5
i n ch.2
4.2
A : t h i s i s an immediate consequence o f t h e r e i n f o r c e m e n t axiom
(ZF b e i n g s u f f i c i e n t i f A i s c o u n t a b l e ) .
I n o t h e r words, e v e r y p a r t i a l o r d e r i n g i s t h e c o n j u n c t i o n o f i t s c h a i n a b l e r e i n f o r -
cements, which a r e t h e chains, o r t o t a l l y ordered r e i n f o r c e m e n t s , t o g e t h e r w i t h t h e r e l a t i o n always (+)
.
However t h e r e e x i s t s a b i n a r y r e l a t i o n which i s n o t t h e c o n j u n c t i o n o f i t s chainab l e reinforcements. 0
Take t h e b i n a r y c y c l e , o r c y c l e o f c o n s e c u t i v i t y a s s o c i a t e d w i t h t h e c h a i n o f
i n t e g e r s f r o m 1 t o p ; so t h a t
C(i,i+l)
+ f o r each i
=
; v a l u e ( - ) o t h e r w i s e (see ch.2
C(p,l) = t r e l a t i o n always (+)
, and
the r e l a t i o n C
c h a i n a b l e r e i n f o r c e m e n t s . Hence
#
5
= 1, ...,p-
8.6). T h i s c y c l e
C
1 and f i n a l l y o n l y has t h e
( n e g a t i o n o f t h e i d e n t i t y ) , as i t s
i s n o t the conjunction o f i t s chainable rein-
forcements. 0 If a relation
i s f r e e l y i n t e r p r e t a b l e i n a p a r t i a l ordering, then
R
i s not
R
n e c e s s a r i l y t h e c o n j u n c t i o n o f i t s c h a i n a b l e r e i n f o r c e m e n t s (communicated by HODGES). 0 Let
and
a,b,c b< c
be t h r e e elements and and
a
I
b
. Let
be
R
t h e p a r t i a l o r d e r i n g d e f i n e d by
A
c o m p a r a b i l i t y modulo A
a
, t h u s R(a,b)
=
- , value
(+) o t h e r w i s e . Then a c h a i n a b l e r e i n f o r c e m e n t o f R i s necess a r i l y a b i n a r y r e l a t i o n always (t): t h e c o n j u n c t i o n o f a l l t h e s e c h a i n a b l e r e i n forcements cannot be R . 0 R(b,a)
=
Problem. I n t h e o p p o s i t e d i r e c t i o n , does t h e r e e x i s t a r e l a t i o n which i s n o t f r e e l y i n t e r p r e t a b l e i n any p a r t i a l o r d e r i n g , y e t which i s t h e c o n j u n c t i o n o f i t s chainable reinforcements.
§
6 -
Let
p
MONOMORPHIC
RELATION
be a n a t u r a l number; a r e l a t i o n
restrictions o f Every r e l a t i o n i s
R
t o any two
R
i s s a i d t o be
p-monomorphic i f t h e
p-element subsets o f t h e base, a r e isomorphic.
0-monomorphic.
Every r e l a t i o n based on a t most Any r e f l e x i v e b i n a r y r e l a t i o n i s
p
elements, i s
p-monomorphic.
1-monomorphic. Any b i n a r y tournament i s
2-mono-
morphic; r e c a l l t h a t a tournament i s a r e f l e x i v e , a n t i s y m m e t r i c and comparable b i n a r y r e l a t i o n : see ch.5
5
A r e l a t i o n R i s s a i d t o be n a t u r a l number q,
2.7.
( 6 p)-monomorphic
if
R
is
q-monomorphic f o r each
259
Chapter 9 s s a i d t o be monomorphic i f i t i s
A relation number. p
p-monomorphic f o r e v e r y n a t u r a l
For example a chain, and more g e n e r a l l y a c h a i n a b l e r e l a t i o n , i s monomorphic. There e x i s t monomorphic non-chainable r e l a t i o n s : f o r i n s t a n c e t h e b i n a r y c y c l e , o r c y c l e o f c o n s e c u t i v i t y , on 3 elements. If
R
is
R i s p-monomorphic.
p-monomorphic, t h e n e v e r y r e s t r i c t i o n o f
( 6 p)-monomorphic
Similarly f o r
and f o r monomorphic.
Every denumerable r e l a t i o n has a denumerable, monomorphic r e s t r i c t i o n ; s i n c e , by 5.5.(1) above, i t has a denumerable, c h a i n a b l e r e s t r i c t i o n . F o r each n a t u r a l number .phic y e t not
, t h e r e e x i s t s a unary r e l a t i o n which i s p-monomor-
p
(p-1)-monomorphic.
Take a base o f p elements, and a unary r e l a t i o n t a k i n g t h e value (+) on a s i n g l e element, and t h e v a l u e ( - ) on t h e o t h e r s . 0
0
For each p morphi c . 0
Take a
, t h e w e x i s t s a r e l a t i o n which i s p-ponomorphic y e t n o t (p+l)-=-
(pt1)-ary r e l a t i o n
R
satisfying
R(xl,.
. . , x ~ + ~= ) +
whenever a t l e a s t
a r e e q u a l . Thus a l l r e s t r i c t i o n s o f c a r d i n a two o f t h e elements x1 , ... , xp+l l i t y p a r e isomorphic. To a v o i d t h a t R be (p+l)-monomorphic, t a k e two d i s tinct
(p+l)-element sets
F
and
G ; and d e f i n e
R
t o t a k e t h e value (+) f o r
every (p+l)-sequence o f elements i n F w i t h o u t r e p e t i t i o n , and ( - ) f o r e v e r y (p+l)-sequence i n G w i t h o u t r e p e t i t i o n . 0
-
6.1. A r e l a t i o n R i s p-monomorphic i f f each r e s t r i c t i o n o f p + l & p-monomorphic. 0
Indeed we can go f r o m one
R with cardinality
p-element subset o f t h e base, t o another, by adding
an element and removing a n o t h e r element; o r by a f i n i t e sequence o f these two operations. 0
r & p , a r e l a t i o n R i s r-monomorphic i f f each r e s t r i c R w i t h c a r d i n a l i t y p + l i s r-monomorphic ( f o r r < p , we must assume t h a t Card R >/ p + l ) .
More g e n e r a l l y , g i v e n tion o f
A relation i s is
( & p)-monomorphic i f f each r e s t r i c t i o n w i t h c a r d i n a l i t y
6
p+l
( 6 p)-monomorphic.
6.2. Every monomorphic r e l a t i o n h a v i n g i n f i n i t e base i s c h a i n a b l e . I n o t h e r words,for c o i n c i d e s w i t h c h a i n a b i l i t y ; uses t h e u l t r a an i n f i n i t e r e l a t i o n , mononorphism f i l t e r axiom and t h e denumerable subset axiom; ZF s u f f i c e s f o r a denumerable r e l a tion. 0 Let
subset a chain
be o u r monomorphic r e l a t i o n w i t h i n f i n i t e base
R
D A
of on
. Take
E
(denumerable subset axiom); t h e n u s i n g 5.5.(1),
E D
such t h a t
R/D
is
A-chainable.
Since
R
a denumerable there exists
i s monomorphic,
THEORY OF RELATIONS
260 t o each f i n i t e subset
, associate
E
F of
r e s t r i c t i o n R/f"(F)
such t h a t
f"(F)
an isomorphism
, associate the image o f t h e chain A / f " ( F )
phism f
chainable by t h i s image. For each f i n i t e subset chains w i t h base
F
non-empty f o r each included i n
F
element o f
UG
and by which
. Moreover,
F
, the
set
F
R/F
is
.
R
&
R r
6 Min(p,(Card E)-p) E R
is
E
5
G
, gives an
. Then
t h a t f o r every f i n i t e sub-
f o r every
, the
F
restriction
C-chainable by 5.1. 0
, and p a n a t u r a l number 4 Card
i s also
.
E
r-monornorphic f o r each n a t u r a l number
r-element subset o f
. We
R/F
p-monomorphic, every
>/
2p-1
, then
say t h a t an R
. Let
R
is
(+) denote the isomorphism
E has t h e
t o t h i s s e t i s isomorphic w i t h
. On
the o t h e r hand, we have
we see t h a t a l l the
R
that
6.4. I f a unary r e l a t i o n
E
r-element subset o f
R/F
p-element subset includes the same number o f
ment subsets having the c o l o r (+)
5 4.3.(1), , and hence
G
1.3 ( e q u i v a l e n t t o t h e u l t r a f i l -
, such
E
is
UF
and every
E
restricted t o
is
(POUZET 1976).
i f f the r e s t r i c t i o n of
using ch.3
R/F
be t h e s e t o f
UF
F of
onto a
( 6p)-monomorphic.
is
F be an a r b i t r a r y
c o l o r (t)
R/F
To t h i s isomor-
f-' ; then
, let
E
i s i n f i n i t e , o r even w i t h c a r d i n a l i t y
type of the r e s t r i c t i o n c o l o r (+)
R
is
R
be a r e l a t i o n w i t h base
p-monomorphic, then
p-monomorphic i f f
R
C/F belongs t o UF
restriction
I n particular i f
Let
, when
UF
based on
C
.
D
i s chainable: by the preceding,
By the coherence lemma i n ch.2
(C/F)-chainable, and hence
6.3. L e t
If
R/F
under
F of
f o r every f i n i t e subset
, each chain belonging t o
t e r axiom), there e x i s t s a chain
taking
f
i s a f i n i t e subset o f
is is
R
r-element subsets o f
r+p E
6
,
Since
r-ele-
Card E ;
have t h e
r-monomorphic. 0 1-monomorphic, then
R
i s constant and hence
chainable. If
i s a b i n a r y r e l a t i o n , then
R
R
can be
( 4 2)-monomorphic w i t h i n f i n i t e
base, y e t n o t be chainable.
R t o be r e f l e x i v e
Take
with a t least a cycle If 0
i s a binary,
R
then
R
,
antisymmetric and comparable, i . e . a tournament
R(a,b) = R(b,c)
(63)-monomorphic
= R(c,a)
=
- .0
r e l a t i o n and based on a t l e a s t 4 elements,
i s chainable.
To f i x ideas, suppose t h a t
R
i s r e f l e x i v e . E i t h e r the r e s t r i c t i o n s o f
R
to
two elements are s y n e t r i c ; then since, by hypothesis, these r e s t r i c t i o n s are a l l isomorphic, i t f o l l o w s t h a t O r the r e s t r i c t i o n s o f
R
R
i s a constant, thus a chainable r e l a t i o n .
t o two elements are chains, o r o r i e n t e d r e s t r i c t i o n s .
Then t h e r e s t r i c t i o n s o f R t o 3 elements cannot be c y c l i c : since otherwise, i f we consider 4 elements, the f o u r c y c l i c r e s t r i c t i o n s t o 3 elements would be mutua l l y incompatible. Thus
a l l r e s t r i c t i o n s t o 3 elements are chains, and then
R
261
Chapter 9
i s i t s e l f a chain. 0 The two p r e c e d i n g cases, t h a t o f unary and b i n a r y r e l a t i o n s , a r e b u t t h e f i r s t terms i n a general process. As a l r e a d y s a i d a b o u t c h a i n a b i l i t y ( i n 5.4 above), we s h a l l see i n ch.12
, then R
p
3.4 t h a t f o r each i n t e g e r
least integer
, there
n
( 6 p)-monomorphic
R i s an n-ary,
such t h a t , i f least
5
p a n and f o r t h i s
, there
p
p>, n
relation with cardinality a t
i s c h a i n a b l e . More s p e c i f i c a l l y , f o r each
R i s a (6p)-monomorphic
n
exists a least
n-ary r e l a t i o n w i t h c a r d i n a l i t y
chainable. By t h e above 6.3, we can suppose s i m p l y t h a t
R
there exists a such t h a t i f
q>/ p
>, q , t h e n R is
is
p-monomorphic,
( Lp)-monomorphic.
instead o f
On t h e o t h e r hand, f o r each i n t e g e r w i t h i n f i n i t e base, such t h a t 0 To each
e x i s t s an i n t e g e r
R
, there
n >, 2
e x i s t s an
& (4n)-monomorphic
n-element s e t , a s s o c i a t e a unique
n-tuple,
n-ary r e l a t i o n
R
.vet n o t c h a i n a b l e .
formed o f t h e elements o f
t h i s s e t w i t h o u t r e p e t i t i o n , i n such a way t h a t t h e s e p a r t i c u l a r
n - t u p l e s do n o t
form a c h a i n , f o r which t h e y would be t h e r e s t r i c t i o n s o f c a r d i n a l i t y
n
. Then
R t a k e t h e v a l u e (+) f o r o u r p a r t i c u l a r n - t u p l e s , and t h e v a l u e ( - ) f o r
let
a l l other
n-tuples. 0
-
Problem. Does t h e r e e x i s t a p o s i t i v e i n t e g e r each n a t u r a l number
yet not
p
, there
e x i s t s an
no
such t h a t f o r each
n-ary r e l a t i o n which i s
n >,no
and
p-monomorphic
( f r o m t h e p r e c e d i n g i t f o l l o w s t h a t such r e l a t i o n s
(p+l)-monomorphic
would necessary be f i n i t e ) . 6.5. The n o t i o n of a tournament r e l a t i o n was d e f i n e d i n ch.5
5
2.7.
Every r e s t r i c -
t i o n o f a tournament t o a 3-element s e t , i s e i t h e r t h e c h a i n o f c a r d i n a l i t y 3
, or
the b i n a r y c y c l e o f c a r d i n a l i t y 3 : more e x a c t l y t h e r e f l e x i v e b i n a r y c y c l e o f card i n a l i t y 3 : f o r s h o r t we s h a l l c a l l i t t h e
u
Given an element
3-cycle.
i n t h e base o f a b i n a r y r e l a t i o n , we s h a l l say t h a t a
which i s a r e s t r i c t i o n o f o u r r e l a t i o n , passes t h r o u g h
Let
If
be a tournament and
A
A
.
u
the element
&
(p-2)-monomorphic,
(1) f o r each element
u
independent o f passinq throuqh (2) l e t
(x,y)
u of
i t s base, o f f i n i t e c a r d i n a l i t y
E
u and
and
v
(x',y')
E
, t h e number o f 3-cycles p a s s i n q t h r o u q h u
i s independent o f
1 1
u, v o f E , t h e number o f 3-cvcles
u, v
;
be any two a r b i t r a r y o r d e r e d p a i r s o f elements i m x # y
f
be an isomorphism f r o m A/(E-{x,y))
z
in
E-\x,y)
pa 5 .
then
; f o r each p a i r o f elements
which s a t i s f y t h e c o n d i t i o n s
A/{T,y' , z ' )
3-cycle
u i f i t s base c o n t a i n s
,
x'
# y ' , A(x,y)
onto
A/(E-{x',y'))
= A(x',y')
=
+
E
; and l e t
; t h e n f o r each element
and f o r i t s image z' = f z , t h e r e s t r i c t i o n s A/-(x,y,z)= a r e e i t h e r b o t h 3 - c y c l e s , o r b o t h c h a i n s (POUZET 1977, unpublished).
262
THEORY OF RELATIONS
(1) L e t
0
.
u, v be two d i s t i n c t elements o f E
Since
u
dent o f
and
v ;let
k2
denote t h i s number.
u
number o f 3 - c y c l e s which do n o t pass t h r o u g h
5
4.1.(1)
with
Now l e t
k
p = 3
is
(o-2)-monomorphic,
Given an element
p t q = (Card E)-2 ; l e t
kl
of
u
u , there e x i s t
hl
= k-kl
, the
E
u : a p p l y ch.3
i s independent o f
denote t h i s number.
denote t h e number o f a l l t h e 3 - c y c l e s which a r e r e s t r i c t i o n s o f
F o r each element
.
A
u
many 3 - c y c l e s which pass t h r o u g h
.
u and v d i s t i n c t , t h e r e e x i s t kl many 3 - c y c l e s which do n o t pass t h r o u g h
Given v
and
A
u n o r t h r o u g h v , i s indepen-
t h e number o f 3-cycles which pass n e i t h e r t h r o u g h
, and
u e i t h e r , hence (kl-k2) u without passing through v . F i n a l l y there e x i s t many 3 - c y c l e s which pass t h r o u g h u and t h r o u g h v . 0
among these
k2 many which do n o t pass t h r o u g h
many which pass t h r o u g h h = k-2kl+k2
( 2 ) F o r elements
in E , let
x,y,z
o f t h e 3-cycles p a s s i n g t h r o u g h 3-cycles which pass through
and
y
We r e t u r n t o t h e two o r d e r e d p a i r s fz
z
Let
h(-x,y,z)
denote t h e number o f
without passing through
(x,y)
and
(x',y')
, etc.
x
and t h e elements
i n o u r statement. The number o f 3 - c y c l e s which pass t h r o u g h
z
z
and
i s equal t o
by t h e above (1). On t h e o t h e r hand, t h i s number i s equal t o t h e f o l l o w i n g sum:
hl
h(x,y,z)
+
through
x
+ h(x,-y,z) + h(-x,-y,z) . The number o f 3-cycles which pass z i s h2 by t h e above ( l ) , and i s a l s o equal t o h(x,y,z) + S i m i l a r l y t h e number o f 3 - c y c l e s p a s s i n g t h r o u g h y and z i s h2
h(-x,y,z) and
.
h(x,-y,z) and equals
.
+ h(-x,y,z)
h(x,y,z)
s u b t r a c t i n g t h e f i r s t , we o b t a i n
Adding t h e second and t h i r d e q u a l i t i e s and t h e n h(x,y,z)
same r e s u l t when s u b s t i t u t i n g x ' , y ' , z ' phism f r o m A/(E-{x,y})
6.6. p 2 5
Let
A
A
A
that
&
hl
+
, the
numbers
= h(x',y',z')
y'
due t o POUZET 1977)
,is
. We
h(-x,-y,z)
. Now,
since
f
have t h e
i s an isomor-
h(-x,-y,z)
and
.0
E i t s base, which i s f i n i t e and o f c a r d i n a l
(p-2)-monomorohic. x # y
onto
morphism f r o m A/(E-{x,y)) y
h(x,y,z)
-
x,y,z
i s a chain; o r e l s e , f o r any two o r d e r e d p a i r s
satisfying the conditions x'
= 2h2
for
A/(E- { x ' , y ' } )
be a tournament and
. Suppose
Then e i t h e r
onto
a r e e q u a l : so t h a t
h(-x',-y',z')
0
denote t h e number, e i t h e r 0 o r 1,
h(x,y,z)
.
x,y,z
,
x' # y'
,
A(x,y)
= A(x',y')
, when
A/(E-{x',y'j)
(x,y)
and
= +
, every
extended by t a k i n g
( x SY') iso-
x
to
(JEAN 1969; t h e f o l l o w i n g p r o o f i s
an automorphism o. f A
.
E i t h e r a l l r e s t r i c t i o n s of
A
w i t h c a r d i n a l i t y 3 a r e chains: t h e n
c h a i n . O r t h e r e e x i s t s a t l e a s t one 3 - c y c l e . By t h e p r e c e d i n g 6.5, o f d i s t i n c t elements t h e r e passes a t l e a s t one 3-cycle.
A
is a
f o r each p a i r
Suppose f r o m t h i s p o i n t
on, t h a t we a r e i n t h i s case. Let
(x,y)
element i n
and
(x',y')
E-{x,y}
, and
be t h e two o r d e r e d p a i r s i n o u r statement, and z'
t h e image o f
z
z
under t h e isomorphism i n o u r
an
263
Chapter 9
statement. Suppose t h a t
A(x,z)
=
+
: we s h a l l prove t h a t
towards a c o n t r a d i c t i o n i n supposing t h e c o n t r a r y , t h a t that
A(z',x')
= t
A(x',z') A(x',z')
+ by a r g u i n g
= =
-
and hence
.
i s n o t a c y c l e : by t h e p r e c e d i n g 6.5.(2), t h e r e s t r i c The r e s t r i c t i o n A/jx,y,z) t i o n A/\x',y',z') i s n o t a c y c l e ; hence A ( z ' , y ' ) = + . L e t u be an element such t h a t A/(x,y,u} A(y,u) = A(u,x) A(z',u')
i s a c y c l e : we know t h a t such an element e x i s t s . Then we have
+ . We c l a i m t h a t A(u,z) = + . I f n o t , t h e n A(z,u) = + , so + as w e l l , where u' i s t h e image o f u under t h e isomorphism i n o u r
=
=
z'
h y p o t h e s i s . Then no 3 - c y c l e p a s s i n g t h r o u g h (recall that
nor through y ' Yet
z
A/{x',y',u'J
u'
and
would pass through
x'
i s a c y c l e , by t h e p r e c e d i n g 6 . 5 . ( 2 ) ) .
would be a c y c l e ; so t h a t t h e number o f 3-cycles p a s s i n g through
A/{x,z,u)
u would be s t r i c t l y g r e a t e r t h a n t h e number o f 3 - c y c l e s p a s s i n g through z ' and u' , c o n t r a d i c t i n g 6.5.(1). Thus A(u,z) = A ( u ' , z ' ) = + . It now f o l l o w s t h a t A(z,y) = + ; f o r i f n o t , t h e n no 3 - c y c l e p a s s i n g t h r o u g h z and u would pass t h r o u g h x n o r through y , and and
yet
A/{y',z',u'j
would s t i l l be a c y c l e , c o n t r a d i c t i n g 6.5.(1). Thus o u r h y p o t h e s i s A(x,z) = A ( z ' , x ' ) = + a l l o w e d us t o determine t h e values o f t h e tournament
A
f o r a l l pairs included i n
We t e r m i n a t e t h e p r o o f by t a k i n g an element we know t h a t such an element e x i s t s . Thus Moreover, no 3 - c y c l e p a s s i n g t h r o u g h
1 x,y,z,u]
z, t
Problem communicated by POUZET, 1978. L e t
E with f i n i t e cardinality
x'
phic, then are
A
A'
and
o r by y '
A
A'
and
.
p a 5
u
.
(under
t
. Hence z', t'
0
be two tournaments h a v i n g
I f f o r each subset
E , t h e r e s t r i c t i o n s A/X
by removing two elements f r o m
and from
on t h e one hand, and t h r o u g h
on t h e o t h e r hand, i s d i f f e r e n t , c o n t r a d i c t i n g 6.5.(1).
t h e same base
i s a cycle:
z' and t h r o u g h t h e image t ' o f
t h e isomorphism i n o u r h y p o t h e s i s ) , can be completed by t h e number o f c y c l e s p a s s i n g t h r o u g h
A/{x,z,tj
i s d i s t i n c t from y
t
.
and i n { x ' , y ' , z ' , u ' l
f o r which
t
and
A'/X
X
obtained
a r e isomorphic,
isomorphic.
2
p>/ 5. If A A & (p-1)-monomorphic. Moreover, i f A i s a n o n - t o t a l l y o r d e r e d tournament, t h e n f o r any two elements x & a x ' i n t h e base, t h e r e e x i s t s an automorphism o f A which takes x i n t o x ' (JEAN 1969).
6.7.
Let
A
be a b i n a r y r e l a t i o n w i t h f i n i t e c a r d i n a l i t y
(p-2)-monomorphic,
then
~
-
0 Since t h e r e l a t i o n
by 6.3 ( i n d e e d p ideas, suppose t h a t
A
5
A
is
, so
A
(p-2)-monomorphic, that
2
< p-2
) . Since
i s r e f l e x i v e . Since
A
i s also A
( 4 2)-monomorphic
i s 1-monomorphic, t o f i x o u r
i s 2-monomorphic,
t r i c t i o n s t o 2-element s e t s a r e symmetric, i n which case
A
e i t h e r i t s res-
i s .a c o n s t a n t r e l a -
t i o n , hence monomorphic. O r t h e r e s t r i c t i o n s t o 2-element s e t s a r e o r i e n t e d , i n which case
A
i s a tournament. By t h e p r e c e d i n g 6.6, e i t h e r
A
i s a chain, and
THEORY OF RELATIONS
264
A
hence monomorphic, o r
i s a n o n - t o t a l l y o r d e r e d tournament.
I n t h e l a t t e r case, l e t of
.
E
, and
A
be t h e base o f
E
E i t h e r t h e r e e x i s t two o t h e r elements
let
x, x '
with
y, y '
be two elements
A(x,y)
into
x'
. Thus
into y'
and y
- {XI) and A/(E
A/(E
O r t h e r e do n o t e x i s t two such elements
x
e v e r y ordered p a i r w i t h f i r s t t e r m
,let
l a r g e r than 3 A(x,t)
x'
which takes takes
, while
x
x
into
into
t
A(x',x)
. By
p
x ' : thus
elements i s
= A(x',t)
is
A
(p-1)
Another example: f o r nality
has c a r d i n a l i t y s t r i c t l y
E
x
and
x'
-
=
2
, taken
x
A(x,x')
=
and takes
; s o we have an automorphism o f
(p-1)-monomorphic.
but not
p = 2q
. Then
composition, we have an automorphism o f
I n o p p o s i t i o n t o t h e preceding: f o r each i n t e g e r on
t a k e s t h e v a l u e (+) f o r
A
t a k i n g t h e v a l u e ( - ) f o r e v e r y orde-
x ' ( t o f i x i d e a s ) . Since
. Similarly
+ ; x
(p-1)-monomorphic.
. Then
+ ; t h u s we have an automorphism o f A which preserves
=
into
t
is
A
y, y '
be an element d i s t i n c t f r o m
t
=
taking
A
we have an isomorphism o f t h e r e s t r i c t i o n s
- i x ' ) ) ; so t h a t
r e d p a i r w i t h f i r s t term
A(x',y')
=
t h e n by t h e p r e c e d i n g p r o p o s i t i o n , t h e r e e x i s t s an automorphism o f
A
A
which
0
p a4
, t h e b i n a r y c y c l e based
(p-2)-monomorphic.
, t h e p a r t i a l o r d e r i n g formed o f q c h a i n s o f c a r d i -
t o be m u t u a l l y incomparable.
p a 6 , i f i t i s (p-3)-mono( < 3)-monomorphic, hence c h a i n a b l e : see 6.3 and 6.4. I n g e n e r a l , f o r r a 3 , e v e r y b i n a r y r e l a t i o n based on p + r+3 elements, which i s (p-r)-monomorphic, i s ( 43)-monomorphic, and t h u s c h a i n a b l e .
Note t h a t , f o r a b i n a r y r e l a t i o n w i t h c a r d i n a l i t y morphic, t h e n i t i s
Problem (JEAN 1976, u n p u b l i s h e d ) . L e t based on
p>, 2n+l
elements, which i s
n 3 2
and
k,< n
(p-k)-monomorphic,
.
Is e v e r y
n-ary r e l a t i o n
necessarily
(p-k+l)(,< k)-monomorphic by 6.3. F o r n > / 6 , i s e v e r y n - a r y r e l a t i o n based on p>/ 2n+l elements, which i s (p-6)-monomorphic, n e c e s s a r i l y c h a i n a b l e ( t h i s i s connected t o Jordan h y p o t h e s i s about p e r m u t a t i o n groups ; see ch.11 5 2.2).
monomorphic. Note t h a t i t i s
A
6.8. L e t
If 0
be a tournament w i t h c a r d i n a l i t y
A i s n o t a chain, t h e n
Let
Let
be t h e base of
E
q
p = 3
, modulo
4
p
and which i s
(p-2)-monomorphic.
.
A , o f c a r d i n a l i t y p ; and l e t u be an element i n E u (mod A) ; i n o t h e r words, t h e num-
denote t h e number of successors o f
ber o f
x
such t h a t
A(u,x)
= +
.
u' , A taking u o f successors i s preserved. The p r o d u c t p.q i s
I f we r e p l a c e
u
by a n o t h e r element
t h e n by t h e p r e c e d i n g d i s c u s s i o n t h e r e e x i s t s an automorphism o f into
u'
, hence
t h e number
q
t h u s equal t o t h e t o t a l number o f o r d e r e d p a i r s o f elements g i v i n g t h e v a l u e (+) to
A ; hence equal t o t h e number
q = (p-1)/2
p.(p-1)/2
o f unordered p a i r s o f elements. Thus
, which a l r e a d y shows t h a t p i s odd.
.
Chapter 9
Given an element the s e t
F of F
u u
phism preserves
F and
v
, partition x
(elements
. By the
which preserves cessors o f
E
, and
u
such t h a t
A(x,u)
the s e t =
G
.
v
into
v'
.
p-3
r
(p-l)(p-3)/8
i s a multiple o f
2-set-transitive
be two
A
.0 5
. The
v'
, this
automor-
F which are sucproduct
r.(p-1)/2
F g i v i n g value (+); hence
o f unordered p a i r s i n
4
u
o f elements i n
remains the same number when passing t o
6.9. A n t i c i p a t i n g the notions o f ch.11 is
(p-1)/2 many
+ ) . L e t v and v '
Since i t preserves
Thus the number
i s thus equal t o the t o t a l number o f ordered p a i r s i n and thus
of
G
preceding discussion, there e x i s t s an automorphism o f
and takes
equal t o t h e number
u into
the s e t o f elements d i s t i n c t from
(p-1)/2 many successors o f
predecessors o f elements i n
in
u
265
F . Hence
r = (p-3)/4
,
2.2, we say t h a t a group o f permutation
i f ff o r any two unordered p a i r s o f elements i n t h e base, there
e x i s t s a permutation o f the group, which takes the f i r s t p a i r i n t o the second. The above p r o p o s i t i o n 6.6 then takes the f o l l o w i n g form: i f morphic, n o n - t o t a l l y ordered tournament w i t h c a r d i n a l i t y p automorphisms o f A i s 2 - s e t - t r a n s i t i v e (p i n t e g e r 3 5 ) . Groups which are
A
is a
, then
(p-2)-mono-
the group o f
2 - s e t - t r a n s i t i v e have been s t u d i e d i n p a r t i c u l a r by DEMBOWSKI
1968, p. 96 note 2, under the name o f
2-homogeneous groups. The term " s e t - t r a n s i -
t i v e " i s e q u a l l y used and w i l l be employed i n t h i s book t o avoid confusion w i t h homogeneous r e l a t i o n s (see ch.11
5
1).
From t h e above c i t e d work, i t f o l l o w s t h a t f o r every l y ordered tournament based on
prime congruent t o 3 (mod 4)
p a5
elements,
this
(p-2)-monomorphic, p
non-total-
i s an odd power o f a
.
Thus, a f t e r t h e b i n a r y c y c l e on 3 elements, we have a tournament on 7 elements which i s 5-monomorphic, and hence 6, a z 2 , and 1-monomorphic, y e t n e i t h e r 3
nor
4-monomorphic; t h i s tournament has chains o f c a r d i n a l i t y 3 and cycles o f cardina-
l i t y 3 as r e s t r i c t i o n s . To c o n s t r u c t i t , s t a r t w i t h a heptagon, o r polygon w i t h 7 v e r t i c e s C y c l i c a l l y o r i e n t the edges heptagon
acegbdf
i n the d i r e c t i o n
second s t a r r e d heptagon ae
.
ab, bc,
adgcfbe
ac, ce,
a,b,c,d,e,f,g
. C y c l i c a l l y o r i e n t the s t a r r e d ... , d f , f a . C y c l i c a l l y o r i e n t the
... , f g ,
ga
i n the "opposite" d i r e c t i o n
da, gd,
... , eb,
I n view o f the r o t a t i o n a l symmetry, i t s u f f i c e s t o v e r i f y the isomorphism
between the t h r e e sub-tournaments o f c a r d i n a l i t y 5 : one which is obtained by removing two consecutive v e r t i c e s , a second obtained by removing two v e r t i c e s which are separated by one intermediate vertex, and a t h i r d obtained by removing two v e r t i c e s which are separated by two intermediate v e r t i c e s .
.
THEORY OF RELATIONS
266
§
7 - PROFILE
Let
OF A RELATION,
be a r e l a t i o n w i t h base
R
f i n i t e number
f(p)
f
for all
R
Examples. I f
. To
p
each n a t u r a l number
f ( 0 ) = 1 and i f
.
R
( n o t i o n due
R
f(h) = 1
>h .
i s a chain, o r i n general a monomorphic r e l a t i o n , then the p r o f i l e p & Card E ) .
i s a unary r e l a t i o n t a k i n g the value (+) f o r a f i n i t e number
R
and ( - ) on a l l o t h e r elements, then the p r o f i l e increases from
a
o f elements
f(0) = 1 to
, and then remains s t a t i o n a r y a t t h i s l a t t e r value.
f ( a ) = a+l If
, associate the
p
p-element r e s t r i c t i o n s o f
Card E = h ( f i n i t e ) , then
f u n c t i o n has constant value equal t o 1 (when If
(POUZET)
THEOREM
thus defined i s c a l l e d the p r o f i l e o f
t o POUZET 1972). Note t h a t f(p) = 0
E
INCREASE
o f isomorphism types o f the
The numerical f u n c t i o n and
PROFILE
i s unary and takes t h e value (+) and t h e value ( - ) , each on an i n f i n i t e
R
set, then the p r o f i l e i s
f ( p ) = p+l
.
By taking, f o r instance, the c o n s e c u t i v i t y r e l a t i o n on the n a t u r a l numbers
, we
o b t a i n a p r o f i l e f u n c t i o n w i t h a f a s t e r growth r a t e . I n the case where every f i n i t e r e l a t i o n w i t h the same a r i t y the p r o f i l e o f
R
i s maximum, hence f o r each
phism types o f r e l a t i o n s o f t h e given a r i t y
p
i s embeddable i n
R
,
i s equal t o the number o f isomor-
with cardinality
p
.
7.1. PROFILE INCREASE THEOREM Let
p, q
equal t o
be two n a t u r a l numbers and 2p+q
. Then
to -
R
a relation with cardinality a t least
the number o f isomorphism types o f the r e s t r i c t i o n s o f
R
p+q elements i s a t l e a s t as g r e a t as the number o f isomorphism types o f the p elements. restrictions o f R
2
More p r e c i s e l y , there e x i s t s an i n j e c t i v e f u n c t i o n which, t o each isomorphism type u -
of a restriction of
triction to 0
p+q
R
5
p
elements, which i s an extension o f
This f o l l o w s from the m u l t i c o l o r theorem, ch.3
types on
p
5
u
(POUZET 1976).
5.3, where the isomorphism
elements p l a y the r o l e o f the c o l o r s o f the p-element sets, and (p+q)-element r e s t r i c t i o n s have the same m u l t i c o l o r . 0
two isomorphic
Consequently, i f a r e l a t i o n has an i n f i n i t e base, then i t s p r o f i l e i s increasing. 2h
I f a r e l a t i o n has even c a r d i n a l i t y
integers l e s s than o r equal t o
h
.
, then
i t s p r o f i l e i s increasing f o r
I f a r e l a t i o n has odd c a r d i n a l i t y
i t s p r o f i l e i s i n c r e a s i n g f o r i n t e g e r s l e s s than o r equal t o
h+l
.
2h+l
, then
For t h e case o f an i n f i n i t e base, an a l t e r n a t i v e p r o o f w i l l be given i n ch.10
5
9.9,
about "almost chainable" r e l a t i o n s .
Note t h a t t h e f i r s t o f t h e preceding p r o p o s i t i o n s
i s stronger than the second.
For example, consider a r e l a t i o n on 7 elements. Not only does i t s p r o f i l e increase
267
Chapter 9
f o r integers value f o r 7.2. L e t
Card E
0
p, q
>/
tion of
to
, but
4
i t s value f o r
2 ; and i t s value f o r
, and l e t
t o p+q -
i s g r e a t e r than o r equal t o the
5
i s g r e a t e r than o r equal t o the value f o r
be two n a t u r a l numbers and
2p+q R
6
a r e l a t i o n w i t h base
R
be a permutation o f
f
.
E
E
, where
I f the image o f every r e s t r i c -
f ) i s an isomorphic r e s t r i c t i o n o f
elements (under
R
5
to
p
then t h e image of every r e s t r i c t i o n o f
-
p
1
,
R
elements i s an isomorphic r e s t r i c -
tion.
0
Take an a r b i t r a r y r e s t r i c t i o n o f
have the c o l o r
.
U
i f i t s image under
Now a
of
R/f"(a)
U
U
.
a
5
denote i t s
i s s a i d t o have the c o l o r
E
a
of
Thus
a
4.3.(2).
, the
E
and i t s image
restrictions
f"(a)
V
V
Hence
. It f o l l o w s f
R/a
have the same number
includes t h e same number o f
as w e l l as o f c o l o r
are i d e n t i c a l , by ch.3
U
.
(p+q)-element subset
p-element subsets o f c o l o r
and V
U
are isomorphic, hence
ment subsets o f c o l o r
elements, and l e t
E w i t h t h i s isomorphism type s h a l l
p-element subset o f
has the c o l o r
f
By hypothesis, f o r each and
R
p-element subsets o f
isomorphism type. A l l
p-ele-
t h a t the c o l o r s
takes each
U
p-element subset
i n t o another o f the same c o l o r . 0
§
8 - HOMOMORPHIC
I M A G E OF A N A R B I T R A R Y RELATION
I n t h i s paragraph, we attempt t o generalize t o a r b i t r a r y r e l a t i o n s the c l a s s i c a l n o t i o n o f homomorphic image
among chains (see ch.2
Starting with a relation
w i t h base
let
U1
,
... , Un
R
with the variables
element relations
Ui
x
3 of Ui
and ( - ) i f
...
x,y,z,
meaning E
3.6) o r among groups.
"
(and),
A
o f subsets o f
E"
x
rJ
v (or),
representing elements o f
" f o r every element
of
E
E "
where
belongs t o
E-Ui
Ui(x)
(if
, with
and
E
,
...then),
etc.;
the q u a n t i f i e r s
"there e x i s t s an
; and f i n a l l y w i t h predicates the r e l a t i o n
( i = 1, ...,n) x
5
and w i t h a s e t
be a f i n i t e sequence o f these subsets. Construct a l o g i c a l
formula w i t h the connections* 3 ( n o t ) ,
v x and
E
R
takes t h e value (+) i f
and the unary x
belongs t o
.
The usual semantic t r u t h value o f such a formula being obvious, we say t h a t an n-ary r e l a t i o n
R"
w i t h base
E"
i s a homomorphic image o f
a l o g i c a l formula whose t r u t h value i s through ED . 8.1.
F i r s t example. L e t
R
Ro(U1,
...,Un)
when
be a p a r t i a l o r d e r i n g w i t h base
R i f f there e x i s t s U1
E
, ... , Un , and l e t
run
E"
be
the s e t o f a l l subsets o f E . Then the unary r e l a t i o n R" such t h a t R"(U) = + i f f U i s an i n i t i a l i n t e r v a l o f R i s a homomorphic image o f R , v i a the
268
THEORY OF RELATIONS
vx,y (U(x)
formula
.
3 U(Y)
R(y,x))
A
S i m i l a r l y , we express the r e l a t i o n t a k i n g (+) i f f
vx 3 y R(x,y)
v i a t h e formula
E"
Now t a k i n g f o r
, whose arguments are denoted by
E"
vx U(x) =7
v i a t h e formula
R
i s a c o f i n a l subset (mod R)
t h e s e t o f i n i t i a l i n t e r v a l s (mod R), t h e p a r t i a l o r d e r i n g o f
i n c l u s i o n w i t h base p h i c image o f
U
.
U(y)
A
Second example. Suppose t h a t
R
,
U
V
,
i s a homomor-
.
V(x)
i s a chain w i t h base
E
, and
that
i s a set
E"
o f pairwise d i s j o i n t i n t e r v a l s . Then the corresponding homomorphic image o f i n the usual sense o f ch.2
vx,y(u(X)
A v(Y))
5
3.6,
3 R(x,Y)
.
Problem. Returning t o t h e p a r t i a l o r d e r i n g
w i t h base
R
, and l e t t i n g
E
be
E"
E ; then we conjecture t h a t t h e r e does n o t e x i s t any
the s e t o f a l l subsets o f
l o g i c a l formula o f the preceding kind, which would express t h a t the subset
of
U
i s a c o f i n a l subset (mod R) w i t h minimum c a r d i n a l i t y ( s o d e f i n i n g the c o f i n a l i -
E
ty of
R ).
8.2. Take
R
t o be a t e r n a r y r e l a t i o n o f a group based on
s e t of a l l subsets o f "
,
R
i s expressed by t h e formula:
the r e s t r i c t i o n
R/U
E
. We
i s a subgroup
are homomorphic images o f
R
. Also
"
and
R/U
"
, and
E"
for
i s a normal subgroup
the b i n a r y r e l a t i o n
i s an equivalence class modulo U
group and V
E
the
l e a w i t t o the reader t o see t h a t the unary r e l a t i o n s
"
,
i s a normal sub-
R/U
"
"
.
0 , and f o r E" take the s e t o f equivalence 0 : the reader w i l l see t h a t the usual homomorphic image, o r
Now take a f i x e d normal subgroup classes modulo
q u o t i e n t group, defined on the s e t o f these equivalence classes, i s represented by a l o g i c a l formula o f the preceding kind. Problem o f t r a n s i t-i v i t y o f c e r t a i n homomorphic images. S t a r t w i t h a r e l a t i o n
.--
R
E , and w i t h a R , w i t h base E" . I t e r a t e , by t a k i n g a s e t E"" o f ( n o t n e c e s s a r i l y mutually d i s j o i n t ) subsets o f E " , and a homomorphic image R o o o f R" , w i t h base En" . Associate t o each element o f
w i t h base
E
and a s e t
E"
of m u t u a l l y d i s j o i n t subsets o f
l o g i c a l formula d e f i n i n g a homomorphic image
E"" E""
i t s union, which i s a subset o f
. Note
of
t h a t , t o two d i s t i n c t elements o f
t h e r e correspond two d i s t i n c t unions, because o f t h e above d i s j u n c t i o n of
elements o f E
E
R"
, and
E"
. Hence we
have a b i j e c t i o n from
t h i s b i j e c t i o n transforms
Ron
E""
onto a s e t o f subsets o f
i n t o a r e l a t i o n on subsets o f
r e l a t i o n thus transformed, n e c e s s a r i l y a homomorphic image o f
R
.
E
.
I s the
Chapter 9
§
9
- BIVALENT
269
TABLE
A b i v a l e n t t a b l e i s t h e system formed by two d i s j o i n t sets: t h e s e t
E
o f columns
and t h e s e t
F o f rows; and a f u n c t i o n which, t o each element i n the Cartesian
product
F
E
Y
, associates t h e value (+) o r the value ( - ) . T on E x F and T' on E'x F ' , we say t h a t
Given t h e t a b l e s
, or
i n T'
f
injection all
x
either X
there e x i s t s an i n j e c t i o n e
into
F
and y
in
F'
F
.
, preserving
, or
.
Xt
there e x i s t s an
Otherwise, i f
X+
We l e a v e i t t o t h e reader t o see t h a t the t a b l e X
E'
and an
T ' ( e x , f y ) = T(x,y)
X
for
t o rows, i f
by adding a row, such
f o r every
X+
obtained from
T i s inextensive by X ( r e l a t i v e l y t o rows). X
l e f t ) , which i s n o t embeddable i n the t a b l e r i g h t ) , i s i n e x t e n s i v e by
X+
T,<
i s embeddable
T
into
E
, relatively
X
obtained from
and y e t
T 4 X
by adding a row, then we say t h a t
from
the values:
i s s a i d t o be extensive by t h e t a b l e
T
TQ X
T+
, if
T'
from
E
in
The t a b l e that
T,<
, relatively
T
having two columns (below on the
having f o u r columns (below on the
t o rows:
+ - + + + - + -
t
+
t
-
+
-
t
-
-
+
- -
-
-
-
t
Hint. F i r s t note t h a t you cannot add d i f f e r e n t values t o the t h i r d and f o u r t h columns i n
X
and then you add
, w i t h o u t embedding
-
you must add
T ; i f you add
+ + t o these columns, then
t o t h e second column (because o f t h e second and t h i r d columns);
i s embedded i n the second and f o u r t h co1umns:contradiction.
T
- -
t o the t h i r d and f o u r t h column, then you must add +
T
column (because o f the f i r s t and f o u r t h columns); and then
Now i f
t o the f i r s t
i s embedded i n the
f i r s t and t h i r d columns: c o n t r a d i c t i o n . p-extensive ( p n a t u r a l number )
We say t h a t a t a b l e T
is
f o r every t a b l e having
p rows. I n o t h e r words, f o r every t a b l e X having p X , there e x i s t s X+ (obtained from X by adding a row)
rows and such t h a t
T#
which respects the non-embeddability Finally
T
T$
Xt
iff T
i s extensive
.
i s s a i d t o be extensive ( r e l a t i v e l y t o rows) i f
T
i s extensive by
every t a b l e ; i t i s s a i d t o be i n e x t e n s i v e otherwise. Ev&ry inextensive t a b l e must have a row o f (t), a row o f ( - ) , and a t l e a s t two i d e n t i c a l rows. This i s the case f o r the above example
T
w i t h two columns and
f o u r rows. I t i s proved by LOPEZ 1977, t h a t the above i n e x t e n s i v e t a b l e
T
i s however
p-ex-
.
tensive f o r every i n t e g e r p 3 5 Problem. For each b i v a l e n t t a b l e which i s f i n i t e , i . e . which has a f i n i t e s e t o f rows and columns, do there e x i s t i n f i n i t e l y many i n t e g e r s p-extensive; and even does there e x i s t s an i n t e g e r f o r every i n t e g e r q g r e a t e r than p .
p
p
f o r which
such t h a t
T
is
T i s q-extensive
THEORY OF RELATIONS
270
We terminate t h i s paragraph by g i v i n g a sketch o f the proof, by LOPEZ, o f the above statement. Arguing towards a c o n t r a d i c t i o n , we suppose t h a t there e x i s t s a t a b l e
, w i t h Card F 3
ExF
Cartesian product
remove any column from a row, i n which
E
, with
on the
X
inextensive by
T
X
.
o f columns be minimal, i n the sense t h a t , i f we
E
We can r e q u i r e t h a t the s e t
5
, then
there e x i s t s an
X+
, obtained from
X
by adding
i s no more embeddable. Consequently a l l the columns are d i s t i n c t
T
( s i n c e t h e two columns o f
T
are themselves d i s t i n c t ) ; moreover, every column con-
t a i n s a t l e a s t one (+) and one ( - ) : indeed otherwise, i f f o r example one column contains only (+), then by adding another (+) t o t h i s column, i t could n o t c o n t r i bute t o any embedding o f Given an ordered p a i r
T
(a,b)
. o f columns and two values
v(a)
and
v(b)
, each
equal t o (+) o r t o ( - ) , we say t h a t the ordered p a i r ( v ( a ) , v ( b ) ) i s bad f o r (a,b) , i f T i s embeddable i n t h e t a b l e o f two columns a, b completed by v ( a ) and
v(b)
.
Given two columns
a, b,
only bad p a i r f o r
(a,b)
(-,-)
yet
and t w i c e
the only bad p a i r ; o r ly
(+,-)
(-,+)
and
t h e r e are o n l y f i v e p o s s i b i l i t i e s . E i t h e r
(+,-)
or twice
(+,+)
i s the
(+.+) i t s e l f i s n o t a row i n (a,b) arc rows i n (a,b) . O r (-,-) is
: i n t h e case where
(-,+)
,
(+,-) i s t h e only bad (ordered) p a i r ; o r (-,+) ; o r f i n a l are both bad p a i r s . As an immediate consequence, we see t h a t
a t a b l e X which reduces t o two columns cannot y i e l d t o t h e i n e x t e n s i v i t y o f T . Now given three columns a, b, c, we have t h e f o l l o w i n g lemma: i t i s impossible t h a t (+,+) be bad f o r (a,b) and t h a t (-,-) be bad f o r (b,c) . Indeed by the preceding, assume t h a t t h e t h r e e columns are d i s t i n c t and t h a t every column contains a t l e a s t one (+) and one ( - ) . There n e c e s s a r i l y e x i s t s a f i r s t row f o r (a,b) ( s i n c e (+,+) i s bad). Then the value f o r c i n w i t h values (-,-) would already be pret h i s row i s n e c e s s a r i l y (+); f o r i f i t were ( - ) , then (-,-) sent i n (b,c) , and thus could n o t be a bad p a i r f o r (b,c) ; so our f i r s t row i s (-,-,+) . S i m i l a r l y there e x i s t s a second row w i t h (+,+) f o r (b,c) , and then ( - ) f o r a : so our second row i s (-,+,+) Since the column a must c o n t a i n a t l e a s t one ( + ) , we have a t h i r d row w i t h (+) f o r a , and then ( - ) f o r b , and then (+) f o r c : so t h a t our t h i r d row i s (+,-,+) . S i m i l a r l y c must c o n t a i n a t l e a s t one ( - ) , so t h a t our f o u r t h row i s (-,+,-) F i n a l l y T i s embedded i n t h e columns ( a ,c) : c o n t r a d i c t i o n . From the preceding lemma, i t f o l l o w s t h a t , i f we r e q u i r e t h a t T be i n e x t e n s i v e
.
.
by
X
, then
X
cannot reduce t o t h r e e columns (assumed t o be d i s t i n c t and t o con-
X . the row (+,+,+) (-,-,-) ; so t h a t i f T i s i n e x t e n s i v e by X , then there e x i s t one bad p a i r (+,+) and one bad p a i r (-,-) , c o n t r a d i c t i n g the lemma.
t a i n (+) and ( - ) ) : t o see t h a t , i t s u f f i c e s t o add t o the row
and then a t least
27 1
Chapter 9
EXERCISE 1 - EXTENDIBLE LOCAL AUTOMORPHISM
Given a r e l a t i o n
, we
R
e x i s t s an automorphism o f
extending
R
Consider t h e b i n a r y r e l a t i o n
.
a,b,c,d,r,s,t,u,i,j R(d,t) = R(d,u)
f
say t h a t a l o c a l automorphism
Let
= R(i,r)
f
i s e x t e n d i b l e i f ft h e r e
.
d e f i n e d as f o l l o w s on a base o f t e n elements
R
R(a,r)
R(a,u)
= R(b,r)
=
R(b,s)
= R(j,s)
= R(j,u)
=
+ , w h i l e R takes t h e
=
= R(i,t)
=
R(c,s) = R(c,t
value ( - ) f o r a l l o t h e r o r d e r e d p a i r s . The i d e n t i t y mapping on
{ a,b)
i s o b v i o u s l y a l o c a l automorphism o f
R
, and is
o b v i o u s l y e x t e n d i b l e t o t h e i d e n t i t y automorphism on t h e e n t i r e base. The mapping which preserves
a
and which t a k e s
which p r e s e r v e s
a
and preserves
. Similarly,
s and t , i and j
,is
i into j
e x t e n d i b l e t o t h e automorphism
, and i n t e r c h a n g e s b and d , r and u ,
c
t h e mapping which p r e s e r v e s
b
and which takes
i i n t o j , i s e x t e n d i b l e t o t h e automorphism which p r e s e r v e s b and d , and i n t e r c h a n g e s a and c , r and s , t and u , i and j . L e t f denote t h e mapping which preserves a , and p r e s e r v e s b , and takes i into
j
. Then
f
i s a l o c a l automorphism, s i n c e t h e r e l a t i o n
, t o (a,')
R
i s b i n a r y and
the three r e s t r i c t i o n s o f
f
t o {a,b'J
automorphisms. Prove t h a t
f
i s n o t e x t e n d i b l e t o t h e e n t i r e base. N o t i c e t h a t
r
such t h a t
i s t h e o n l y element
extending
f
x
must p r e s e r v e
r
and t o { b , i }
R(a,x) = R(b,x)
. Taking
=
+ ; hence any automorphism
, we
i into j
are local
must have
= R(j,r),
R(i,r)
which i s f a l s e .
f
Thus a l t h o u g h a l l t h r e e r e s t r i c t i o n s o f e n t i r e base, EXERCISE 2
1
-
Let
-
A
t o 2 elements, a r e e x t e n d i b l e t o t h e
i t s e l f i s n o t extendible.
f
CONDITIONS FOR A RELATION TO BE CONSTANT be t h e usual c h a i n o f t h e n a t u r a l numbers
Q
same base, b u t which i s isomorphic w i t h t h e c h a i n
, and B be a c h a i n on t h e o f t h e r a t i o n a l s , hence
dense and h a v i n g n e i t h e r a maximum n o r a minimum element. Prove t h a t , f o r each n and each p e r m u t a t i o n f o f { 1,2, ... ,nJ , t h e r e e x i s t s a F o f n i n t e g e r s ul, ...,un such t h a t , assuming t h a t these i n t e g e r s a r e o r d e r e d by u1 < u2 < . . . 4 un (mod A) , t h e n t h e y a r e reordered, modulo B , positive integer set
* * . <'f(n)
U f ( l ) < 'f(2)< 2
-
Let
n
numbers. Prove t h a t i f permutation
*
be a p o s i t i v e i n t e g e r , and f
R
i s both
o f t h e s e t {1,2,
an n - a r y r e l a t i o n based on t h e n a t u r a l
R
A-chainable and
...,n )
, there
such t h a t , assuming t h a t these elements of then t h e permutation n
into
un
,is
f
are
ul<
A/G
and
B/G
R/F
. Hint:
t a k e an
F
... < un
transposed by t h e b i j e c t i o n t a k i n g
an automorphism o f
which t h e r e s t r i c t i o n s
F
B-chainable, t h e n f o r e v e r y
exists a set
of
n
integers
(mod A/F)
1 into
u1
n-element s e t
,
,
... ,
G
for
a r e i d e n t i c a l , and t h e n use t h e isomorphisms
THEORY OF RELATIONS
212 of
A/F
onto
n-element s e t
A/G
and o f
B/F
onto
B/G
. From t h i s ,
deduce t h a t f o r each
F , t h e r e s t r i c t i o n R/F i s preserved by every permutation o f F ,
thus i s a constant r e l a t i o n . Using 4.1,
deduce t h a t
R
i t s e l f i s a constant
re1a t i o n . 3
-
Suppose t h a t
R
i s a denumerable n-ary r e l a t i o n s a t i s f y i n g the f o l l o w i n g :
f o r every denumerable r e l a t i o n i s embeddable i n
R
, then
t i o n . H i n t : f i r s t prove t h a t
X
X
, if
every r e s t r i c t i o n t o a t most
i s isomorphic t o R
4
is
-
.
Then
R
elements
i s a constant r e l a -
i s chainable; more s t r o n g l y t h a t , f o r every denu-
merable chain, t h e r e e x i s t s an isomorphic chain
R
R
n
C
w i t h base
IRI
, such
that
C-chainable.
Using the axiom o f choice, extend the above r e s u l t i n ( 3 ) t o the case o f an
arbitrary i n f i n i t e relation.
273
10
CHAPTER
AGE, RICH RELATION, INEXHAUSTIBLE RELAT ON, SATURA ED RELATION, EXISTENCE CRITERION FOR A RICH RELATION OF A GIVEN AGE
1 - PROJECTION FILTER, AND (1,PIISOMORPHISM §
Consider two sets
and
E
EXTENSION, (~,P)-MORPHISM
OLDER RELATION,
E'
and a f u n c t i o n
from
f
a n a t u r a l number ; consider two m-ary r e l a t i o n s , base
.
E'
every
Then
R
i s s a i d t o be t h e inverse p r o j e c t i o n o f
x1 ,..., xm i n
f
, we have R ' ( f x l ,..., fx,)
E
1
R = (f- )"(R')
I n t h i s case we w r i t e If
i s a b i j e c t i o n from
an isomorphism from Given sets
and
E
into
E
w i t h base
R
R E'
E
onto
and under
R'
f
be
m
with
R'
,if
for
,..., xm) .
= R(xl
. E'
onto
. Let
E' E
, then we 5 3.
f i n d t h e usual d e f i n i t i o n of
R ' : see ch.2
, consider
t h e s e t o f a l l f u n c t i o n s from
E
into
.
E'
A f i l t e r on t h i s s e t s h a l l be c a l l e d a p r o j e c t i o n f i l t e r from E onto E ' . The r e l a t i o n R i s c a l l e d the inverse p r o j e c t i o n o f R ' under the f i l t e r F, which we w r i t e by R = - 1 ( R ' ) , i f f o r every xl,. . ,xm i n E , the s e t o f
.
functions
f
filter
.
satisfying
...,fx,)
R'(fxl,
= R(xl,
...,xm)
t i o n (mod 3- ) . -
If R'
and
unique. I f
i s an element o f the
I n other words, t h e preceding e q u a l i t y holds f o r almost every funcf
R'
1 ( f - )'(R')
are given, then t h e inverse p r o j e c t i o n and t h e f i l t e r
3
e x i s t s and i s
are given, then there e x i s t s a t most one inverse
projection. I f the inverse p r o j e c t i o n
';r , then
If
'$.-l(R')
3
e x i s t s and i f
-'(R')
= T-'(R')
.
9
i s a f i l t e r f i n e r than
i s an u l t r a f i l t e r , then t h e inverse p r o j e c t i o n e x i s t s and i s unique.
0 Indeed e i t h e r the s e t o f f u n c t i o n s
f
such t h a t
R'(fxl,
... ,fx,)
= t
, or
the
analogous s e t w i t h t h e value (-), i s ' a n element o f the u l t r a f i l t e r . 0
I f the u l t r a f i l t e r i s t r i v i a l , more p r e c i s e l y i f i t i s formed o f those sets o f f u n c t i o n s which contain as an element the f u n c t i o n
f
, then
the inverse projec-
f
t i o n under the u l t r a f i l t e r coincides w i t h the inverse p r o j e c t i o n under
.
1.1. INJECTIVE FILTER Let
be a p r o j e c t i o n f i l t e r from the s e t
E
onto
E'
. Two
elements
E are s a i d t o be e q u i v a l e n t (mod 3 ) i f f a = f b f o r almost every This c o n d i t i o n i s r e f l e x i v e , symmetric and t r a n s i t i v e .
f
a, b (mod
of
9).
274
THEORY OF RELATIONS
A p r o j e c t i o n f i l t e r from
d i s t i n c t elements
a, b
To each f i n i t e subset into
E
When
onto
E
of F
, we
E
of
E
fa # fb
have
f o r almost every function
, associate the s e t
which a r e i n j e c t i v e on
E'
F varies, the sets
i s s a i d t o be i n j e c t i v e , i f f o r any two
E'
UF
.
f
o f those f u n c t i o n s f r o m
.
F
and t h e i r s u p e r s e t s , c o n s t i t u t e an i n j e c t i v e f i l t e r ,
UF
which i s minimum i n t h i s sense t h a t any g i v e n p r o j e c t i o n f i l t e r f r o m
onto
E
E'
i s i n j e c t i v e i f f i t i s f i n e r than our i n j e c t i v e f i l t e r . e c t i v e f i l t e r from
E %0
E'
.
Every f i l t e r which i s f i n e r t h a n an i n j e c t i v e f i l t e r , i s i n j e c t i v e . I n t h e case o f a t r i v i a l u l t r a f i l t e r , formed o f t h o s e s e t s o f f u n c t i o n s which cont a i n the function
f
, the
u l t r a f i l t e r i s i n j e c t i v e i f f the function
t i v e . I n t h i s case, n e c e s s a r i l y
Card E '
i s injec-
f
.
3 Card E
1.2. OLDER RELATION, YOUNGER RELATION Given two r e l a t i o n s or that
.
S
S
is
,
R
o f t h e same a r i t y , we say t h a t
S
older t h a n
i s younger t h a n
R
, i f every f i n i t e r e s t r i c t i o n o f
R
S
i s embeddable i n
R
T h i s comparison 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 a n t i s y m m e t r i c , even up t o
, and i t s converse,
isomorphism: f o r example t h e c h a i n w o f t h e n a t u r a l numbers a r e each o l d e r t h a n t h e o t h e r . Let
. Then
E'
R =
7 -'(R')
Let
0
iff
F
i s o l d e r than
R'
be a f i n i t e subset o f
, i s qn element o f
F
R'
be a r e l a t i o n o f t h e same a r i t y , w i t h
t h e r e e x i s t s an i n j e c t i v e p r o j e c t i o n f i l t e r
denote t h e common a r i t y o f
m
and l e t on
E , and
be a r e l a t i o n w i t h base
R
base
'F.
E
and
R
,...,fa,,,)
f
satisfying
Since
F
i s f i n i t e , t h e s e t of a l l
.
R'
Suppose t h a t
f
The s e t o f f u n c t i o n s
F o r each m - t u p l e
tions
R'(fal
.
R
.
satisfying
=
R(al
al,..
,..., a,)
m-tuples i n
F
. ,am
in
F
which belongs t o t h i s s e t
Conversely suppose t h a t
R'
there exists a function
f
R/F
onto
E'
, whose
supersets
R'/f"(F)
.
Let
restriction t o
from UF
F
F
into
R E'
s e t o f func-
. And
. F o r each
R/F
every f u n c t i o n
onto
R'/f"(F)
f i n i t e subset
F
of
d e s i g n a t e t h e s e t o f those f u n c t i o n s f r o m UF
E
into
and t h e i r
c o n s t i t u t e a p r o j e c t i o n f i l t e r , o b v i o u s l y i n j e c t i v e , under which R'
. E
, which i s an isomorphism f r o m
i s such an isomorphism. Then these
i s t h e i n v e r s e image ( o r p r o j e c t i o n ) of
.
i s again an element o f
g i v e s an isomorphism f r o m
i s o l d e r than
, the
i s f i n i t e , so t h a t t h e i n t e r -
s e c t i o n o f t h e p r e c e d i n g s e t s o f f u n c t i o n s i s an element o f f
R = T-'(R')
which a r e i n j e c t i v e
R
.0
1.3. The p r e c e d i n g d e f i n i t i o n s and p r o p o s i t i o n s immediately e x t e n d t o t h e case of
m u l t i r e l a t i o n s . Thus we have t h e f o l l o w i n g statement. Let
R
,
R'
be two m u l t i r e l a t i o n s w i t h r e s p e c t i v e bases
E
,
E'
.
,
,
215
Chapter 10
.-I f
i s o l d e r than
R'
, then f o r every S ' w i t h base E ' , there e x i s t s an S E , such t h a t t h e concatenation R ' S ' i s o l d e r
R
o f the same a r i t y and w i t h base
than
RS
(uses t h e u l t r a f i l t e r axiom; ZF s u f f i c e s i f
7
0 Take an i n j e c t i v e u l t r a f i l t e r 7
9
-1
. Another
(S')
w i t h base
s e t o f these
SF
R =
7 -'(R') F of
proof: f o r each f i n i t e subset
, such t h a t
F
such t h a t
i s embeddable i n
(R/F,SF)
i s denumerable).
E
R'S'
, then E
.
set
S =
, there e x i s t s an SF U F designate the
Let
F , and apply t h e coherence lemma o f ch.2 5 1.3. I n
f o r a given
the denumerable case, the u l t r a f i l t e r i s unnecessary. 0
1.4. FILTER IDENTICAL ON A SET Let
,
E
element o f to
be two sets and
E'
7 from
tion f i l t e r
i s s a i d t o be
E'
.A
E n E'
F , formed o f functions from E i n t o D
If R = 'T-'(R')
, is
E'
each o f whose r e s t r i c t i o n
I
) , when
the i d e n t i t y .
, where
same r e s t r i c t i o n t o E'
onto
is a
0 For each f i n i t e subset
into
i s a superset o f
E
which i s i n j e c t i v e and
E'
D - i d e n t i c a l f i l t e r , then
and
R
R'
have the
.
D
i s i n f i n i t e and
If E
projec-
0 - i d e n t i c a l , i f there e x i s t s an
i s t h e i d e n t i t y . I n other'words, almost every f u n c t i o n (mod
D
restricted to
F
a subset of the i n t e r s e c t i o n
D
onto
E
F
of
E'
whose r e s t r i c t i o n t o
. Then
t h e supersets o f these
E
E' UF
E'
, then
there e x i s t s a f i l t e r from
El-identical.
, take
the s e t
UF
o f those functions from
E
i s the i d e n t i t y and which are i n j e c t i v e on c o n s t i t u t e the desired f i l t e r . 0
1-EXTENSION
R
Let
be a r e l a t i o n w i t h i n f i n i t e base
An extension
Rt
of
to
R
E+
E
, and l e t
i s s a i d t o be a
E+
be a superset of
1-extension o f
R
,if
E
.
f o r every
F o f E+ , t h e r e e x i s t s an isomorphism from R+/F onto a r e s t r i c , which reduces t o t h e i d e n t i t y on F n E . For example, i f we add a l a s t element t o t h e chain w o f the n a t u r a l numbers ,
f i n i t e subset tion of
R
then we have a
, then
ment 0
1-extension. However i f we add an element before the minimum ele-
we o b t a i n an extension isomorphic w i t h ~1
, y e t which i s n o t a
1-extension.
Every 1-extension o f
has t h e same f i n i t e r e s t r i c t i o n s as
R
R
, up
t o isomor-
phism. The converse i s f a l s e : take again t h e chain o f n a t u r a l numbers
, and
its
.
extension by adding an element before 0 The n o t i o n o f 1-extension i s t r a n s i t i v e . Moreover, i f
R+
w i t h base
every intermediate s e t of -
R
.
H
E+ (E 5 H
is a E+)
1-extension o f
, the
R
restriction
w i t h base R+/H
E
, then fpr 1-extension
THEORY OF RELATIONS
276
E , and R+ be a r e l a t i o n w i t h t h e same a r i t y , which i s basel on a s u p e r s e t E+ o f E . Then R+ 1-extension o f R i f f t h e r e e x i s t s a p r o j e c t i o n f i l t e r T from E+ E , which i s i n j e c t i v e and E - i d e n t i c a l , and s a t i s f i e s R+ = f - l ( R ) Let
R
be a r e l a t i o n w i t h base
.
onto
0 Take up t h e p r o o f o f 1.2 w i t h t h e f u n c t i o n s f r o m
t h e i d e n t i t y when r e s t r i c t e d t o
R+
, where
R
, based on a s u p e r s e t
t h e r e e x i s t s an e x t e n s i o n
i s a 1-extension o f 0
S+
of
E
o-f S
to
Et
, and l e t S
R+
be a
l-exten-
w i t h base
E
,
RfSt
I n p a r t i c u l a r , f o r each i n f i n i t e s e t Thus f o r e v e r y r e l a t i o n
E - i d e n t i c a l and which v e r i f i e s
.0 .
S+ = ;F -'(S)
, every superset o f
E
a m u l t i r e l a t i o n reduced t o i t s base, i s a E
E
f o r every
, such t h a t t h e c o n c a t e n a t i o n
1
which i s i n j e c t i v e ,
, and t h e n s e t
-l(R)
. Then
E+
(uses t h e u l t r a f i l t e r axiom).
RS
Take an u l t r a f i l t e r
R+ =
i f f there exists
f o r example CHANG, KEISLER 1973, p r o p . 5.2.2.
be a m u l t i r e l a t i o n w i t h i n f i n i t e base
R R
R
the f r e e variables i n the formulas are replaced
. See
by elements o f t h e base I R I 1.5. L e t
of
(called, very improperly,
R
R ) ; i . e . an e x t e n s i o n which s a t i s f i e s t h e same
an elementary e x t e n s i o n o f
sion o f
fi 1-extension
R+
which i s a l o g i c a l e x t e n s i o n o f
l o g i c a l f o r m u l a s as
which reduce t o
E
.0
E
I n connection w i t h l o g i c , note t h a t an e x t e n s i o n o f
into
E+
1-extension o f
w i t h i n f i n i t e base
R
, there e x i s t s a 1-extension o f
E
, c o n s i d e r e d as
.
and f o r e v e r y s u p e r s e t
which i s based on
R
E E
. This
E+
E+
of
i s a weak f o r m
of t h e upward Lowenheim-Skolem theorem.
R
1.6. L e t subset o f D
50 ' s
be a r e l a t i o n w i t h i n f i n i t e base E
E
. Then
, such
there e x i s t s a s e t
that
is a
R
E
, and
let
1-extension o f
be an i n f i n i t e
D
equipotent w i t h
D+
D
, satisfying
R/D+ (uses axiom o f choice;
weak f o r m o f t h e downward Lowenheim-Skolem theorem). 0 L e t us say t h a t two f i n i t e subsets
an isomorphism f r o m
R/F
F
G = F n D =
have t h e same i n t e r s e c t i o n
onto
R/F'
,
F'
of
E+
are equivalent, i f they
F' n D , and i f a d d i t i o n a l l y t h e r e e x i s t s
, which
reduces t o t h e i d e n t i t y on
G
. For
, t h e r e e x i s t c o u n t a b l y many e q u i v a l e n c e c l a s s e s . Indeed i t s u f f i c e s t o choose a r e p r e s e n t a t i v e F f r o m each c l a s s (axiom o f c h o i c e ) , and t h e n t o d e f i n e D+ as t h e u n i o n o f t h e F corresponding t o a l l t h e G . 0
a g i v e n f i n i t e subset
G
of
1.7. Given t h r e e r e l a t i o n s
than S
, then
t h i s copy and ces i f
R
and
D
R
, R' , S
such t h a t
t h e r e e x i s t s an i s o m o r p h i c coey o f R
, which R'
i s younger t h a n
are countable).
S
R
R'
and
R'
a r e b o t h younger
and a common e x t e n s i o n o f
(uses t h e u l t r a f i l t e r axiom;
ZF s u f f i -
271
Chapter 10
be t h e bases o f R , R ' , which we assume t o be d i s j o i n t ; and l e t To each ordered p a i r (F,F') where F i s a f i n i t e subset o f E and F ' a f i n i t e subset o f E' , associate t h e s e t LJF,F, o f a l l f u n c t i o n s h from E u E ' i n t o D , whose r e s t r i c t i o n s t o F and t o F' are r e s p e c t i v e l y an isomorphism from R / F onto S/h"(F) and an isomorphism from R'/F' onto
0 Let
E
,
E'
0 be the base o f S
.
.
By hypothesis, f o r a l l F and F ' , t h e s e t UF,Fl S/ho(F') Moreover, f o r any f i n i t e supersets G o f F and 6 ' o f F '
i s non-empty. have t h e i n c l u -
, we
.
Thus the supersets o f t h e UF,Fl c o n s t i t u t e a f i l t e r on sion UG,Gl c_ UF,Fl the s e t o f f u n c t i o n s from E u E ' i n t o D . Take a f i n e r u l t r a f i l t e r ( o r use the coherence lemma, ch.2
5
1.3).
Match an element a i n E , w i t h an element a' i n E' , i f ha = ha' f o r almost every f u n c t i o n h , modulo the u l t r a f i l t e r . Note t h a t i t i s impossible t o match up an element i n E t o more than one element i n E ' , as w e l l as t o match up an element i n E t o another element i n E , and s i m i l a r l y when exchanging E and E ' . Indeed, the matching i s an equivalence r e l a t i o n , and i f a , b are d i s t i n c t elements o f S/{ha,hb)
E
, then
. Thus
almost every f u n c t i o n h takes ha # hb f o r almost a l l h
R/{a,b)
isomorphically onto
.
L e t n be the common a r i t y o f the t h r e e r e l a t i o n s . Take a sequence o f al,
...,an
i n E u E'
according t o whether
and d e f i n e t h e r e l a t i o n Rt S(hal,
...,han)
=
+
by
t
R (al,
f o r almost every
...,an)
h , or
=
n elements
= + or
-
-
f o r almost
every h , modulo the u l t r a f i l t e r . This r e l a t i o n Rt i s a common extension o f R and R ' . Take an isomorphic copy R" o f R ' , by keeping each element o f E ' which i s matched t o no element, and f o r each element o f E ' which i s matched, by t a k i n g t h a t element matched t o i t ; then R" i s isomorphic w i t h R ' F i n a l l y , the
.
desired common extension o f R and R" , i s t h e r e s t r i c t i o n o f Rt t o t e d by those elements o f E ' which a r e matched t o no o t h e r element. 0
1.8. (1,p)-MORPHISM AND
E augmen-
(1,p)-ISOMORPHISM
.
L e t p be a n a t u r a l number A l o c a l isomorphism f w i t h domain F , from a r e l a t i o n R i n t o a r e l a t i o n S , i s s a i d t o be a (1,p)-morphism from R i n t o S , i f f o r every s e t F ' = F augmented by a t most p elements from the base IRI there e x i s t s a l o c a l isomorphism from R i n t o S which i s an extension o f f t o t h e domain F ' . The l o c a l isomorphism f i s s a i d t o be a (1,p)-isomorphism from R i n t o S , if f i s a (1,p)-morphism and t h e inverse f u n c t i o n f-' S into R Every l o c a l isomorphism i s a (1,O)-isomor~hism. Every (1,p)-morphism i s a (1,q)-morphism f o r each (1,p)-isomorphism.
is a
(1,p)-morphism
from
.
qs p
; similarly for a
,
THEORY OF RELATIONS
218
1-MORPHISM, 1- ISOMORPHISM We say t h a t a l o c a l isomorphism a
(1,p)-morphism
f
b l e t o every f i n i t e superset o f We say t h a t p
.
is a
f
Rng f
i s an e x t e n s i o n o f
is a
into
R
i f it i s a
. Or
, then
R
R
1-morphism f r o m
F o r example, l e t
into
again i f
(1,l)-isomorphism,
o f I R I , the -
F
also a
1-isomorphism.
§
2 - CLOSED &
A set
f o r every integer
and
f-l
0
is a
1-morphism f r o m
and t o
1-morphisms.
IRI
, and
be t h e c h a i n o f
S
R
R
.
Then t h e
. Yet
S
into
it i s not
which i s s t r i c t l y l e s s t h a n 0 (mod R).
(RI
. Then
S
l y e x t e n s i o n i f f , f o r each f i n i t e
i d e n t i t y on
F
1-morphism from
R
S
into
, hence
R
UNDER EMBEDDABILITY,DIRECTED UNDER EMBEDDABILITY, AGE
o f r e l a t i o n s o f t h e same a r i t y , i s s a i d t o be c l o s e d under e m b e d d a b i l i t y
(up t o isomorphism), i f f o r a l l r e l a t i o n s
6;1
X, Y,
Y
if
X
&
belongs t o
which belongs t o
and
i s s a i d t o be d i r e c t e d under e m b e d d a b i l i t y , i f f o r a l l r e l a t i o n s exists a relation Z i n s a t i s f y i n g Z >,X and Z>/ Y
.
R
and an i n t e g e r
n a l i t y l e s s t h a n o r equal t o For c e r t a i n r e l a t i o n s
R
p
, this
p
, the
R
p = 0
p = 1
and
set o f restrictions o f
X,Y
o f cardi-
R
s e t i s also directed: f o r instance i f
5
R
is a
4 and 6.
i s t h e c o n s e c u t i v i t y r e l a t i o n on t h e i n t e g e r s , t h e n e x c e p t f o r
Given a r e l a t i o n
,this R
s e t i s not directed.
and a p o s i t i v e i n t e g e r
o f c a r d i n a l i t y g r e a t e r t h a n o r equal t o
embeddabi 1 it y
4X
i s c l o s e d under e m b e d d a b i l i t y .
c o n s t a n t r e l a t i o n , o r a c h a i n , o r a monomorphic r e l a t i o n : see ch.9 However i f
Y
(J&.
, there
Given a r e l a t i o n
R
Dom f
are both
i s an e x t e n s i o n o f
S
then there e x i s t s a r e l a t i o n isomorphic w i t h
A set i n (R
f
i f it i s i s extendi-
s i n c e t h e n e g a t i v e i n t e g e r (-1) i s s t r i c t l y l e s s t h a n 0
be an e x t e n s i o n o f
subset
f
.
S
(mod S ) , and t h e r e e x i s t s no element i n
S
1-morphism,
(1,p)-isomorphism
be t h e c h a i n o f t h e n a t u r a l numbers
i d e n t i t y on t h e s i n g l e t o n o f
Let
is a
S
I n o t h e r words i f
t h e i d e n t i t y on any subset o f t h e base
t h e p o s i t i v e and n e g a t i v e i n t e g e r s , so t h a t even a
.
p
i s extendible t o every f i n i t e superset o f
f
every f i n i t e superset o f S
R
.
Dom f
1-isomorphism,
I n o t h e r words i f
If
from
f o r e v e r y n a t u r a l number
p p
, the
set o f f i n i t e restrictions o f
i s d i r e c t e d b u t n o t c l o s e d under
.
AGE, REPRESENTATIVE RELATION
Let
R
be a r e l a t i o n . Then t h e s e t o f f i n i t e r e s t r i c t i o n s o f
R
i s c l o s e d and
d i r e c t e d under e m b e d d a b i l i t y . Considered up t o isomorphism, t h i s s e t s h a l l be called the R
of
R ; we s h a l l say t h a t
i s a r e p r e s e n t a t i v e o f t h e age.
R
r e p r e s e n t s t h i s aqe, o r a g a i n t h a t
279
Chapter 10
R i s the s e t o f isomor-
For the reader d e s i r i n g a rigorous d e f i n i t i o n , the age o f
R , which are based on f i n i t e subsets
phic copies o f the f i n i t e r e s t r i c t i o n s o f o f the s e t o o f the i n t e g e r s .
For a l o g i c i a n , an age i s a u n i v e r s a l theory o f f i r s t order predicate calculus w i t h i d e n t i t y ; see f o r example KRAUSS 1971. Since ages are countable sets, they can be compared under i n c l u s i o n . For example the age o f a l l f i n i t e chains, which i s represented by an i n f i n i t e chain, i s included i n the age o f a l l f i n i t e p a r t i a l orderings, which i t s e l f i s represented by those i n f i n i t e p a r t i a l orderings, i n which every f i n i t e p a r t i a l o r d e r i n g i s embeddable.
R, S o f the same a r i t y , R i s younger than S ,or S & R (see 1.2 above) i f f t h e age o f R i s included i n the aqe o f S . The r e l a t i o n s R and S have t h e same age i f f each i s o l d e r (and younger) than Given two r e l a t i o n s o l d e r than the o t h e r . To each age
@
there corresponds t h e negation age o f
each element o f
@,
value: see ch.2
5
, obtained by r e p l a c i n g
by i t s negation ( i . e . t h e r e l a t i o n t a k i n g always the opposite 1.7); o r e q u i v a l e n t l y by r e p l a c i n g a representative by i t s
negation. For example, the negation o f the age o f a l l f i n i t e chains, o r t o t a l orderings, i s the age o f a l l s t r i c t f i n i t e t o t a l orderings ( w i t h
<
instead o f
,<
).
An age may be i d e n t i c a l t o i t s negation: f o r example t h e s e t o f a l l f i n i t e binary r e l a t i o n s , o r t h e s e t o f a l l f i n i t e symmetric b i n a r y r e l a t i o n s .
2.1. Given a s e t
&
o f f i n i t e r e l a t i o n s o f the same a r i t y , which i s closed and
d i r e c t e d under embeddability, t h e r e e x i s t s a countable r e l a t i o n whose age i s
&
(up t o isomorphism). I n o t h e r words, an age i s an i d e a l f o r the p a r t i a l ordering o f embeddability among f i n i t e r e l a t i o n s . 0
Since the s e t
indices then to
i
,?fI
, thus
i s countable (up t o isomorphism), enumerate i t by using i n t e g e r o b t a i n i n g a sequence o f f i n i t e r e l a t i o n s
B1 = the common extension, up t o isomorphism, o f
& , hence
Bo
Ai and
. Let A1
Bo = A.
, which
; belongs
t o our sequence, w i t h the minimum p o s s i b l e index; and so on. Then
take t h e common extension o f the
Bi
, on
the union o f t h e i r bases.
Given a s e t o f ages, t o t a l l y ordered under i n c l u s i o n , t h e union and the i n t e r s e c t i o n are both an age. Given a s e t of ages f o r which i n c l u s i o n i s a d i r e c t e d part i a l ordering, then t h e union i s an age. However i n general, the union o r the i n t e r s e c t i o n o f two ages i s n o t an age.
o f a l l f i n i t e unary r e l a t i o n s always (+), and t h e age o f f i n i t e unary r e l a t i o n s always ( - ) . For t h e case o f an i n t e r s e c t i o n , s t a r t w i t h two d i s j o i n t , non-empty sets E, F Take t h e b i n a r y r e l a t i o n A based on t h e union E v F w i t h A(x,y) = + f o r
0 For t h e case o f a union, take the age formed
.
THEORY OF RELATIONS
280
x with
y belonging t o E B(x,y) = + f o r x
, and y
-
A(x,y) = otherwise. Now take the r e l a t i o n B belonging t o E , and value (-) otherwise. Then
t h e o n l y common f i n i t e r e s t r i c t i o n s o f A and B are t h e r e l a t i o n s always (+) and the r e l a t i o n s always ( - ) : they do n o t c o n s t i t u t e a d i r e c t e d s e t . 0 2.2.
A set
6%o f
f i n i t e r e l a t i o n s o f common a r i t y , which i s closed under embed-
d a b i l i t y , i s an age i f f
&, is n o t t h e-_-_union o f
two subsets d i s t i n c t from
closed under- embeddability. This i s a c l a s s i c a l p r o p e r t y o f i d e a l s : ch.4
5
&@ 5.1.
2.3. L e t A be a r e l a t i o n o f a r i t y 3 2 , with f i n i t e base. Then -t.h e set, up t o isomorphism, o f those f i n i t e r e l a t i o n s which do n o t admit an embedding o f A an i n f i n i t e age. This f o l l o w s from ch.5 5 1.3.(1) and ( 2 ) ( f a i t h f u l common extension) 2.4. Given a r e l a t i o n R and an age i n c l u d i n g t h e age o f R , t h e r e e x i s t s an extension o f R which i s a r e p r e s e n t a t i v e o f ,f ' (uses the u l t r a f i l t e r axiom; ZF
R
suffices i f
i s countable).
Take a r e l a t i o n S representing the age s" , then a common extension o f R and o f an isomorphic copy o f S , t h i s extension being chosen t o be younger t h a n S : see 1.7 above. 0
0
On t h e o t h e r hand, s t a r t i n g w i t h a r e l a t i o n R younger than S , t h e r e does n o t necessarily e x i s t a r e s t r i c t i o n o f S having the same age as. R 0 Take R t o be t h e binary, denumerable r e l a t i o n always (t). For each p o s i t i v e
.
i n t e g e r i , d e f i n e Si t o be t h e b i n a r y r e l a t i o n o f c a r d i n a l i t y i , always t a k i n g t h e value (+) , where t h e bases Ei o f t h e Si are taken t o be d i s j o i n t . L e t S be t h e common extension t o t h e Si , based on t h e union of t h e elements x, y from d i s t i n c t Ei , g i v i n g t h e value (-) , 0
Ei
; two
2.5. L e t us give, w i t h o u t p r o o f , t h e f o l l o w i n g refinement o f 1.7, which concerns
.
the existence o f a common 1-extension, i n s t e a d o f a common extension o f two r e l a .. t i o n s having t h e same age; see FRAISSE 1975. (1) L e t
R and
R'
two r e l a t i o n s having t h e same age. There e x i s t s an isomorphic
c m R" o f R' and a common extension o f bases, which i s a 1-extension o f R
.
( 2 ) Given
R
, R'
R
and
R"
on t h e union o f t h e i r
having the same age. A necessary and s u f f i c i e n t c o n d i t i o n f o r
t h e existence of an isomorphic copy R" o f and R" i s t h a t , f o r every f i n i t e subsets
R' F
w i t h a common 1-extension o f R IRI F' L f IR'I and
of
and
every i n t e g e r p , there e x i s t s a r e l a t i o n T w i t h a (1,p)-isomorphism from R i n t o T w i t h domain F , and a (1,p)-isomorphism from R ' i K T w i t h domain F' ( f o r d e f i n i t i o n s o f 1-extension and (1,p)-isomorphism, see 1.4 and 1.8).
28 1
Chapter 10
2.6. L e t us a p p l y t h e above statement 2 . 5 . ( 2 ) t o t h e c o n c r e t e example o f chains. Consider t h e age o f a l l f i n i t e chains, whose r e p r e s e n t a t i v e s a r e a l l i n f i n i t e
c h a i n s . Given two i n f i n i t e c h a i n s into
T
w i t h a f i n i t e domain
and
R
, and a l o c a l isomorphism
T
, we
F = Dom f
see t h a t
is a
f
( p p o s i t i v e i n t e g e r ) i f f , f o r each o r d e r e d p a i r o f c o n s e c u t i v e elements
, either
modulo
R/F
(fa,fb)
(mod T)
t h e two c o r r e s p o n d i n g open i n t e r v a l s
have t h e same c a r d i n a l i t y
(if a (if a
a
; and f o r t h e open f i n a l i n t e r v a l s
fa
It f o l l o w s e a s i l y t h a t , g i v e n two i n f i n i t e c h a i n s
I R I and F '
of
R
f'
f
F
, which a r e (1,p)-isomorphisms:
and
F'
interval i n
R
a
(mod R)
and
b
and
R
,
R'
i s t h e minimum o f
R/F)
i s t h e maximum o f
R/F).
and f i n i t e subsets
F
I R ' I , t h e r e e x i s t s an i n f i n i t e c h a i n T and two l o c a l i s o -
of
morphisms
from
R
< p , o r b o t h have c a r d i n a l i t i e s 3 p .
S i m i l a r l y f o r t h e open i n i t i a l i n t e r v a l s b e f o r e and b e f o r e
(a,b)
from
f
(1,p)-isomorphism
into
or i n
T
and
, either
R'
from
into
R'
, w i t h r e s p e c t i v e domains
T
i t s u f f i c e s t o a s s o c i a t e , t o each open
an i n t e r v a l w i t h t h e same f i n i t e c a r d i n a l i t y ,
o r another i n f i n i t e i n t e r v a l . Consequently, g i v e n two i n f i n i t e c h a i n s
gJ
R"
and a common
R'
1-extension o f
R R
, R' , there and R" .
e x i s t s an isomorohic CODY
2.7. On t h e o t h e r hand, t h e r e e x i s t ages where t h e common
1-extension does n o t
always e x i s t . Take f o r setting
R
R(x,x)
nition for
R'
t h e usual c h a i n o f n a t u r a l numbers =
+
-
or
a c c o r d i n g t o whether
, whose x
d i a g o n a l i s changed by
i s even o r odd. S i m i l a r d e f i -
by exchanging values (+) and ( - ) on t h e d i a g o n a l . Then
have t h e same f i n i t e r e s t r i c t i o n s , up t o isomorphism. Yet a
R
and
R'
R
is
1-extension o f
o b t a i n e d , as f o r t h e usual c h a i n o f i n t e g e r s , by adding new elements a f t e r a l l t h e i n t e g e r s : so t h a t t h i s
, w i t h value + f o r the
1 - e x t e n s i o n s t i l l has minimum 0
(0,O) . S i m i l a r l y f o r a 1 - e x t e n s i o n o f
ordered p a i r
R'
: t h e y cannot be iden-
t i c a l . T h i s example i s due t o PAILLET. Another example. Take f o r
R
a p a r t i a l o r d e r i n g w i t h a minimum element, f o r
R'
a n o t h e r p a r t i a l o r d e r i n g w i t h two incomparable minimal elements. It i s easy t o ensure t h e same f i n i t e r e s t r i c t i o n s , up t o isomorphism. Moreover t h e minimum i s p r e s e r v e d when p a s s i n g t o a
1-extension; s i m i l a r l y f o r t h e incomparable minimal
elements: we cannot o b t a i n a common
5 3 - RELATIVE
RESTRICTION,
1-extension.
RELATIVE
Example due t o BONNET. 0
ISOMORPHISM,
REL-AGE,
MAXIMALIST
R E L A T I ON 3.1.
A-RELATION, A-RESTRICTION
A
Let of
A
be a r e l a t i o n w i t h f i n i t e base. An
. An
A - r e s t r i c t i o n o f an
A-relation
A-relation i s a r e l a t i o n extension R
i s a restriction o f
R
to a
THEORY OF RELATIONS
282
.
IAJ
superset o f the base
If A
i s n o t mentioned, then we s h a l l say r e l a t i v e
restriction. A-EMBEDDABILITY
A-ISOMORPHISM,
We s h a l l c a l l an tion
A-isomorphism,
onto another
R
A
t r i c t i o n t o the base o f Two extensions o f
, if
there e x i s t s an
c a l l e d an
IAl-jsomorphism from an
, any isomorphism from
S
onto
R
A-rela-
whose res-
S
i s the i d e n t i t y .
are s a i d t o be
A
We say t h a t an extension A
o r more e x a c t l y an
A-relation
A
of
R
A-isomorphic i f such an isomorphism e x i s t s . is
A-embeddable i n another extension
A-isomorphism from
A-embedding o f
into
R
.
S
R
onto a r e s t r i c t i o n o f
S
S
of
, again
Here again we s h a l l speak o f r e l a t i v e i s o -
morphism and o f r e l a t i v e embeddability. &AGE OR REL-AGE Given an extension
, the
A - s o f R w i l l be t h e s e t o f a l l f i n i t e A - r e s t r i c t i o n s o f R , considered up t o A-isomorphism. For t h e reader d e s i r i n g a rigorous d e f i a i t i o n , the A-age o f R i s the s e t o f those A-relations whose bases are f i n i t e sets i n c l u d i n g I A 1 and i n c l u d e d i n the union I A J u w , where W i s the s e t o f i n t e g e r s . I t may be convenient t o assume t h a t the base I A l i s
w
d i s j o i n t from We say t h a t If A
A
of
R
.
R .represents, o r i s a r e p r e s e n t a t i v e , o f the considered
A-age.
i s n o t mentioned, we s h a l l speak o f rel-age.
Each A-age i s a s e t o f f i n i t e
A-relations,
which i s closed and d i r e c t e d under
A-embeddability.
ft
Conversely given a s e t under
tative of
4.
We say t h a t an R
in
is S
A-relations,
A-YOUNGER RELATION
A-relation
A-younqer than
.
which i s - c l o s e d and d i r e c t e d A - r e l a t i o n which i s a represen:
Same p r o o f as i n 2.1.
A-OLDER RELATION,
3.2.
of finite
A-embeddability, t h e r e e x i s t s a countable
,
S
S
is
O r e q u i v a l e n t l y , i f the
A-age represented by
S
A-older than another
i f every f i n i t e
A-relation
A-restriction o f
A-age represented by
R
A - r e l a t i o n s R , R ' and S where , t h e r e e x i s t s an A-isomorphic copy R"
R
and
than - S
of
R'
R
and R"
fices i f
R
, which
and
R'
Consequently, given an R
is
A-younger than
. Same
R
, or that
A-embeddable
i s included i n t h e
S
R'
are
both
A-younqer
and a common extension
(uses the u l t r a f i l t e r axiom; ZF suf-
are countable; same p r o o f as i n 1.7 above). A-relation
, there e x i s t s an extension o f
sentative o f
is
.
Given t h r e e
0-f
R
R R
and an A-age
f
including the
A-age o f
w i t h t h e same c a r d i n a l i t y , which i s a repre-
c o n d i t i o n s and same p r o o f as 2.4.
283
Chapter 10 3.3.
INDUCED REL-AGE, SPECIFICATION OF A REL-AGE
Let
A
be a f i n i t e r e l a t i o n and
B
a f i n i t e extension o f
.
A
Starting with a
A - r e s t r i c t i o n s o f t h e elements o f ( I? ,,considered up t o
@ , the
B-age
phism, c o n s t i t u t e an
A-age which i s s a i d t o be induced by
A-isomor-
@.
A i s empty, t h e n t h e A-age reduces t o t h e a l r e a d y known n o t i o n o f an age: we s h a l l speak o f t h e age induced by a g i v e n B-age. In particular if
The n o t i o n o f induced r e l - a g e i s r e f l e x i v e , t r a n s i t i v e and a n t i s y m m e t r i c , and t h u s d e f i n e s a p a r t i a l o r d e r i n g among r e l - a g e s . For every
one
A-age
A
For
. We
and
B
there exists a t least
B-age s p e c i f i e s
given, i n general t h e r e e x i s t many
B
b4,
belonging t o
say t h a t such a
4
or i s a
4.
specification o f 0
and e v e r y r e l a t i o n
U$
B-age which induces c&
A
S t a r t w i t h t h e age o f f i n i t e chains, so t h a t
B-agerspecifications o f
B
i s empty, and l e t
.
c,+
be t h e
u < v . There e x i s t s a s p e c i f i c a t i o n , a l l o f whose elements u , v . There e x i s t s a n o t h e r s p e c i f i c a t i o n f o r which
c h a i n on two elements
are chains beginning w i t h we have a t most
p
elements between
u and
v
(p given integer). F i n a l l y there
e x i s t s a n o t h e r s p e c i f i c a t i o n w i t h as many elements as one m i g h t w i s h between
u
and
v
v ; o r w i t h as many elements as one w i s h b e f o r e
and a f t e r
, etc.
v
B-age
6
Then t o say t h a t representative o f If
@
and
A
B extension o f
A , w i t h an
,
64 i s e q u i v a l e n t '4 .
i s a specification o f
@
A-age
'4
and
, then
A
empty and
I t may happen t h a t
every
a r e p r e s e n t a t i v e o f &? which i s an exten-
i s not, i n general, a representative o f
o f chains w i t h
t o saying t h a t
i s a representative o f
i s a specification o f B
u and
.
(13
s i o n of
and between
0
Consider two f i n i t e r e l a t i o n s a
u
B
R
: c o n s i d e r a g a i n t h e case
d e f i n e d on two elements.
be a r e p r e s e n t a t i v e o f
R
be isomorphic w i t h
@
&
and t h a t no r e p r e s e n t a t i v e o f
.
R t h e c o n s e c u t i v i t y r e l a t i o n on t h e p o s i t i v e a n d n e g a t i v e i n t e g e r s . A empty; l e t S be a c o n s e c u t i v i t y r e l a t i o n formed o f two components each isomorphic w i t h R ; and f o r B , t a k e t h e r e s t r i c t i o n o f S t o two elements,
0
Take f o r
Take
each i n one o f t h e components. Then e v e r y of
3.4.
S
B-relation representing the
, has a t l e a s t two components, hence cannot be isomorphic w i t h Let
A
Consider two
be a f i n i t e r e l a t i o n and A-ages
tx@ &
d
of , t h e r e A' .
and
B
be a f i n i t e e x t e n s i o n o f
h?'i n c l u d i n g
exists a
B-a2
A . Then
f o r every
including
63 , and which
B-age R
A
.0
.
B-age s p e c i f i c a i s a specifi-
cation o f
0 Take a c o u n t a b l e r e p r e s e n t a t i v e
R
o f L+'and a c o u n t a b l e r e p r e s e n t a t i v e
S
of
THEORY OF RELATIONS
284
@) . This r e l a t i o n extension
A'
of
T
R
i s s t i l l a representative o f
S
and
, up
S
to
T
: see 3.2 above. This r e l a t i o n
o f a certain
B-age i n c l u d i n g
@.
'4 . There
e x i s t s a common
A-isomorphism, which i s a r e p r e s e n t a t i v e o f is a
B - r e l a t i o n , and i s a r e p r e s e n t a t i v e
0
3.5. MAXIMAL REL-AGE
@,
Given an age
&
tion of
and a f i n i t e r e l a t i o n
a;? , an
belonging t o
A
i s s a i d t o be maximal (mod &,),
i f no o t h e r
A-age s p e c i f i c a -
A-age s p e c i f i c a t i o n o f
s t r i c t l y includes i t . Example. Take the age o f a l l f i n i t e chains, and l e t A be the chain on two e l e u < v Then the o n l y maximal A-age i s obtained by a u t h o r i z i n g an a r b i t r a r y
.
ments
.
, a f t e r v , and between u and v
u
f i n i t e number o f elements before
Another example. S t a r t from the age represented by t h e c o n s e c u t i v i t y r e l a t i o n on
A
the n a t u r a l numbers ; and l e t
be the c o n s e c u t i v i t y r e l a t i o n o f a f i x e d f i n i t e
chain. Then t h e r e e x i s t s o n l y one maximal namely the
A-age s p e c i f i c a t i o n o f the given age:
A-age o f a l l t h e c o n s e c u t i v i t y r e l a t i o n s o f f i n i t e chains i n c l u d i n g
as an i n t e r v a l , and t h e i r
A
Now t a k i n g the preceding age and t a k i n g f o r and
v
with
A(u,v)
= A(v,u)
- , we
=
u
element which i s the successor o f ments as one wishes before have another maximal
Given an age
&
. Given
v
and a f t e r
and a f i n i t e r e l a t i o n
@, , there
A
p
v
v
, and
as many e l e -
a positive integer elements between
, we
p
u
and
belonging t o
&, , f o r
each
A-aqe speci-
A-ase which includes i L (and
&, 1.
We could apply the maximal i d e a l axiom: see ch.4
l e n t t o the axiom o f choice, i s unnecessary:
5
5. However t h i s axiom, equiva-
ZF s u f f i c e s , since t h e considered e l e -
ments are f i n i t e r e l a t i o n s which can be based on the s e t o f n a t u r a l numbers.
A
3.6. Given two f i n i t e r e l a t i o n s
@ , then 0 Call
the
4
mal. L e t
be an
A-age i n c l u d i n g tion of
4'
4
B
extension o f
and supPose t h a t
y'z
@ C
A
, and
a maximal
i s maximal y e t C
.
Let
& , and
B-as
@'
be a
4 i s n o t maxi-
l e t ,/$'be
A - r e s t r i c t i o n o f an element o f
: contradiction. 0
an
B-age s p e c i f i c a -
: see 3.4 above. Since ($ i s maximal, then
i s an
0
i s maximal.
and c o n t a i n i n g the element
and hence
i s an element o f
and
@
A - r e l a t i o n which does n o t belong t o
and i n c l u d i n g
i d e n t i c a l t o ($ C
A-aqe induced by
the induced A-age, C
,
v
.
e x i s t s a t l e a s t one maximal
i s s t i l l a specification o f 0
u
A-age by p u t t i n g a unique
and t h e predecessor o f
and a f t e r
u
u
t h e r e l a t i o n on two elements
have a maximal
A-age, by p u t t i n g e x a c t l y
and as many as one wishes before
fication o f
A
A-restrictions.
63'
is
($ ; hence
285
Chapter 10
On the o t h e r hand, a s p e c i f i c a t i o n o f a maximal rel-aqe i s not n e c e s s a r i l v maximal. 0
Take two elements
B
t h e chain
which
a
and
b
, b and l e t
a
.
a< b
The
be the chain o f the s i n g l e t o n o f
A
remain consecutive, i s a s p e c i f i c a t i o n o f the
chains c o n t a i n i n g
. The
a
However, given two f i n i t e r e l a t i o n s A-age which contains
and
A
B-age under consi-
a
extension o f
B
has a t l e a s t one maximal
B
in
B
A-age o f a l l f i n i t e
l a t t e r i s obviously maximal; y e t the
d e r a t i o n i s n o t maximal, since one can add elements between
, and
a
B-age formed o f the f i n i t e chains, extensions o f
and
.0
b
, then every maximal
A
B-age as a s p e c i f i c a t i o n . This
f o l l o w s from 3.3 and 3.5 above. 3.7.
INCOMPATIBILITY LEMMA FOR REL-AGES
6?. be an &, and
Let
a relation
No
A
a f i n i t e r e l a t i o n belonging t o & . Given a maximal A-% A - r e l a t i o n B which does n o t belong t o & , t h e r e e x i s t s
belonging t o
B'
t a i n s both B 0
age and a finite
and
B'
.
lfz,
such t h a t no
&, contains
A-age, s p e c i f i c a t i o n o f
both
B
and a l l t h e elements o f
Suppose t h a t our conclusion i s f a l s e . Take a sequence o f r e l a t i o n s belonging t o Ci
&,
%, , con-
A-age, s p e c i f i c a t i o n o f
i n which every element o f (Ze be embeddable. For each
be a common extension o f
obviously an extension o f
A
B
.
and
B;
, which
belongs t o the age
We can always suppose t h a t
. Moreover,
.
L.f
( i integer)
B;
i
,let
fi , and
is
i s an extension
B;+l
i f i x e d , we can suppose t h a t f o r a l l j > i , by t a k i n g a s u i t a b l e i n f i n i t e sequence e x t r a c t e d from the j . Then the Di c o n s t i t u t e an A-age, s p e c i f i c a t i o n o f & , which contains B and includes '4 : c o n t r a d i c t i o n . 0
of
B;
f o r each
i
we have the same r e s t r i c t i o n
for
Di = C j / ( l B ( v l B ; l )
3.8. MAXIMALIST SUBSET, MAXIMALIST RELATION Given a r e l a t i o n (mod R), i f the relation domain S
of
S
F
R
,a
f i n i t e subset
F o f i t s base i s s a i d t o be maximalist
(R/F)-age represented by
o f the same age as
R
, every
R
i s maximal. Equivalently, f o r each 1-morphism from
R
into
S
with
, i s a 1-isomorphism (see 1.8 above). Equivalently, f o r each extension
R w i t h the same age, and f o r every r e s t r i c t i o n B
of
S
t o a f i n i t e super-
, there e x i s t s an isomorphism from B onto a r e s t r i c t i o n o f R , which F (see 3.2, g e n e r a l i z i n g 1.7 above). Example. Consider the chain Q+ c3 , where Q i s t h e chain o f r a t i o n a l s ; then any
set o f
F
i s the i d e n t i t y on f i n i t e subset o f
Q
i s maximalist, modulo t h e age o f a l l f i n i t e chains; t h i s i s
n o t the case f o r the p a i r o f i n t e g e r s {0,1)
i n the f i n a l i n t e r v a l
.
Every subset o f a maximalist f i n i t e s e t , i s maximalist. This f o l l o w s from 3.6.
A r e l a t i o n R i s s a i d t o be maximalist, i f every f i n i t e subset o f i t s base i s maximalist (mod R).
286
THEORY OF RELATIONS
For example, the chain
Q
A second example. L e t t i n g
o f t h e r a t i o n a l s i s maximalist. C
denote t h e c o n s e c u t i v i t y r e l a t i o n on p o s i t i v e and nega-
C ,
t i v e integers, consider each r e l a t i o n formed o f components isomorphic w i t h w i t h d i s j o i n t bases, and w i t h value ( - )
f o r every ordered p a i r o f elements taken
i n two d i s t i n c t components. For each p o s i t i v e i n t e g e r
p
, the r e l a t i o n formed o f
components i s a maximalist denumerable r e l a t i o n . S i m i l a r l y the c o n s e c u t i v i t y re-
p l a t i o n formed o f countably many such components; a l l are o f the same age.
A t h i r d example. S t a r t w i t h t h e c o n s e c u t i v i t y r e l a t i o n on the n a t u r a l numbers,
say
C
.
To each unary r e l a t i o n
which d i f f e r s from
C
on these i n t e g e r s , associate t h e r e l a t i o n
U
on t h e diagonal, w i t h
Take a denumerable s e t o f
Cu(x,x) = U(x)
Cu
f o r each i n t e g e r
x
.
w i t h d i s j o i n t bases, which are m u t u a l l y l i n k e d by
Cu the value ( - ) , and which s a t i s f y the f o l l o w i n g d e n s i t y c o n d i t i o n . For each f i n i t e
s o f values (+) and ( - ) , take a component Cu w i t h U beginning by s We now d e f i n e the b i r e l a t i o n formed o f the preceding b i n a r y r e l a t i o n , and a unary r e l a t i o n 0 , t a k i n g the value (t) f o r the minimum o f each comoonent, and ( - ) otherwise. Then our denumerable b i r e l a t i o n s are a l l o f t h e same age. There are continuum many such. Moreover, they are a l l maximalist. Indeed, every 1-morphism from such a b i r e l a t i o n R i n t o another, say R ' , associates t o each element o f a component sequence
Cu
of
R
, an
element o f a component , 'C,
corresponding t o t h e same unary r e l a t i o n a same rank i n
Cu
and i n
C,)
. Thus
3.9. MAXIMALIST EXTENSION THEOREM For every denumerable r e l a t i o n
of
, isomorphic w i t h Cu , thus
R'
U ; the corresponding elements having
this i s a R
.
, there
I-isomorphism.
e x i s t s an extension o f
which i s
R
denumerable, maximalist and o f the same age. R , there R w i t h t h e same c a r d i n a l i t y and t h e same age.
More s t r o n g l y , modulo t h e axiom o f choice: f o r every i n f i n i t e r e l a t i o n e x i s t s a maximalist extension o f 0 Let
F be a f i n i t e subset o f t h e base o f
R
.
It s u f f i c e s t o prove t h a t , by
adding countably many elements, then we can o b t a i n an extension t h a t the
(R/F)-age represented by
S
S
of
R
such
i s maximal: the r e s t o f the p r o o f w i l l
f o l l o w by i t e r a t i n g t h i s . Let
&,
designate the
say
@
including
L+$
(R/F)-age represented by : see 3.5.
which i s a representative o f 0
@
Then take
R
. Take
t o be an
S
a maximal
(R/F)-age,
(R/F)-extension o f
R
: see 3.2. 0
I n the case o f an uncountable base, t h e preceding p r o o f i s modified on two p o i n t s .
F i r s t by an o r d i n a l indexing o f a l l f i n i t e subsets ( u s i n g the axiom o f choice); secondly by the c o n s t r u c t i o n o f a common extension which preserves t h e age, o r t h e rel-age ( u s i n g the coherence lemma, ch.2
0
1.3). 0
3.10. A necessary and s u f f i c i e n t c o n d i t i o n f o r a r e l a t i o n
R
t o be maximalist, i s
287
Chapter 10
t h a t every extension o f
w i t h t h e same age, be a
R
uses t h e axiom o f choice; ZF s u f f i c e s i f
R
1-extension ( t h e s u f f i c i e n c y
i s denumerable).
This p r o p o s i t i o n i d e n t i f i e s t h e maximalist r e l a t i o n s as those r e l a t i o n s which are e x i s t e n t i a l l y closed: a n o t i o n used i n t h e model-completeness c r i t e r i o n f o r l o g i c a l theories; see ROBINSON 1963.
R
0 Let
and
be a maximalist r e l a t i o n ,
. Since
the
i s maximal, i t i s i d e n t i c a l t o t h a t represented by is a
.
R w i t h the same age,
an extension o f
R'
F a f i n i t e subset o f the base \ R I
(R/F)-age represented by R'
. Thus
the i d e n t i t y on
R
F
1-isomorphism from
R
into
Conversely, suppose t h a t
R
i s not maximalist. Thus t h e r e e x i s t s a f i n i t e subset
R'
I n other'words
is a
R'
1-extension o f R.
F which i s n o t maximalist (mod R). By t h e preceding p r o p o s i t i o n (axiom o f choice), there e x i s t s an extension Thus t h e i d e n t i t y on
F
R'
of
, having the same age and which i s maximalist. R ' i n t o R , hence R ' i s
R
i s not a
1-morphism from
.0
not a
1-extension o f
s4
R I C H RELATION, d - M O R P H I S M
-
R
A denumerable n-ary r e l a t i o n i s s a i d t o be @, i f every denumerable n-ary r e l a t i o n i s embeddable i n i t . 4.1. For every p o s i t i v e i n t e g e r
n
, there
e x i s t s a denumerable r i c h
n-a2
relation. 0 This i s obvious f o r
n = 1 : take a r e l a t i o n w i t h t h e value (+) i n f i n i t e l y many
times, and t h e value ( - ) i n f i n i t e l y many times. We s h a l l prove the p r o p o s i t i o n f o r
n = 2 ; t h e p r o o f extends immediately t o the
case o f
.
n
s t r i c t l y g r e a t e r than 2
As base, take t h e s e t o f n a t u r a l numbers
. We
s h a l l d e f i n e our b i n a r y r i c h r e l a -
be t h e s e t o f the i n t e g e r s 0, 1, El I n stage 2, l e t E2 be t h e s e t o f integers
t i o n i n several "stages". I n stage 1, l e t and l e t
R(0,O) =
+ and R(1,l)
=
-
.
from 2 through 17, and d e f i n e R as f o l l o w s . From x = 2 t o x = 9 , a t t r i b u t e 3 t o R a l l the 2 = 8 p o s s i b l e systems o f values (+) and ( - ) f o r R(0,x) and S i m i l a r l y , from
x = 10 t o
and
the
possible systems o f values (+) and (-) f o r
8
R(x,x)
.
R(x,O)
x = 17
, attribute t o
R(1,x)
and
.
R(x,x) I n general, l e t
i be a p o s i t i v e i n t e g e r , and suppose t h a t
El
R(x,l)
all
R
and
, ... , Ei
are
R i s already defined on t h e i r union, so t h a t f o r any binary based on {1,2 ,...,i } , t h e r e e x i s t s a sequence u1 ui where
defined, and t h a t relation U
,...,
u belongs t o E. ( j = 1,..., i ) ; so t h a t t h e transformation o f 1,..., i i n t o j 3 u1 ,...,ui i s an isomorphism from U onto the r e s t r i c t i o n R/{ul ,..., ui
288
THEORY OF RELATIONS
Then we take a f i n i t e s e t union
u ... U Ei
El
o f integers
Ei+l
u . .. u Ei . We u . . . u Ei+l , by l e t t i n g u
El
new union
in
Now we e a s i l y see t h a t
R
i an element
integer
and
i+l , by , for
i n stage
R
R(x,u)
and
R(x,x)
r e s t r i c t e d t o the
R
R take a r b i t r a r y values on ordered p a i r s ( 0 , l ) i n stage 1 , o r (2,3) i n stage 2 i s r i c h . Indeed, s t a r t w i t h an a r b i t r a r y denumerable
.
w i t h base the p o s i t i v e i n t e g e r s . We s u b s t i t u t e t o each p o s i t i v e
X
isomorphism from
R(u,x)
achieve t o d e f i n e
El
n o t considered: f o r example binary r e l a t i o n
l a r g e r than those o f the
; and we p a r t i a l l y d e f i n e our r e l a t i o n
a t t r i b u t i n g a l l p o s s i b l e systems o f values a l l elements
,all
x
ui
in
, so
Ei
t h a t the considered transformation be an
i n t o a denumerable r e s t r i c t i o n o f
X
R
.0
4 . 2 . E n t 2 , n o t a l l r i c h denumerable n-ary r e l a t i o n s are isomorphic. 0 Start with a r i c h relation
.
the value ( - ) i n s t e a d o f (+) element
4.3.
a =
Let
would be taken i n t o an
Suppose t h a t
E
F
. Then
a'
and having base
R
and
R' E
R
, so
Then whether
a
S
of
R
b with
by the isomorphism, and we would have
the r e s t r i c t i o n
f o r n-tuples w i t h some terms i n
common extension
0
R/(E-F)
w i t h base
R
E
of
a
F be
E
E'
. Then
, we
R'
arbitrarily i s equimor-
S
isomorphic w i t h
R E'
. Take
E ; then take a
, completed
E u E'
or of
let
i s rich.
and some terms i n
belongs t o the image o f
, and
E
d i s j o i n t from
E'
t h a t there exists a r e s t r i c t i o n o f
embeddable i n the r e s t r i c t i o n o f
have t h a t
t o i t s base minus the element
a
S
R
.
is
.0
CONNECTION WITH INDIVISIBLE RELATIONS
Recall t h a t partition of R
add a new element
reduces t o the s i n g l e t o n o f an element
t o be isomorphic w i t h
4.4.
. Similarly
t o i t s base, and l e t
be a r i c h n-ary r e l a t i o n w i t h denumerable base
R
phic with
a
I f t h e two extensions were isomorphic, then the
+ and - simultaneously: c o n t r a d i c t i o n .
a f i n i t e subset o f 0
a new element
+ f o r a l l x i n the base I R I
R(a,x) =
R(a',b)
. Add
R
w i t h i n f i n i t e base
R E
E
i s s a i d t o be i n d i v i s i b l e , i f f o r every
i n t o two complementary subsets
i s embeddable i n
R/C
, or
in
R/D :.see ch.6
C
5
and
D = E-C
, then
either
6.
A r i c h b i n a r y r e l a t i o n R i s n o t i n d i v i s i b l e , since one can p a r t i t i o n i t s base i n t o t h e s e t o f those x f o r which R(x,x) = + and i t s complement: we o b t a i n two r e s t r i c t i o n s i n t o n e i t h e r o f which However, given a b i n a r y r e l a t i o n
R
R
, give
i s embeddable. the value (+) t o a l l ordered p a i r s
on the diagonal: we o b t a i n the r e f l e x i f i e d r e l a t i o n R (x,x) Then every r e f l e x i f i e d r i c h b i n a r y r e l a t i o n i s i n d i v i s i b l e . 0 Let
R
.
be such a r e f l e x i f i e d r i c h b i n a r y r e l a t i o n on t h e i n t e g e r s .
Take
Chapter 10
w -sequence o f isomorphic copies Ri
an
of
( i i n t e g e r ) . Take the common extension ging t o
and y
Ei
to
289
, we have S(x,y)
Ej
, w i t h m u t u a l l y d i s j o i n t bases
R
o f a l l the
S
, such
Ri
= R(i,j)
(i, j
that f o r
x
Ei
belon-
d i s t i n c t integers).
Then given a p a r t i t i o n o f the base I S ( i n t o two complementary subsets, e i t h e r some Ei i s included i n one o f t h e subsets; o r t h e o t h e r subset contains a t l e a s t one element i n each
.
S
tions o f
Since
4.5.
c( -MORPHISM,
Let
R
from
,
.
i s equimorphic.with
We say t h a t
o(-OLDER RELATION, w i t h domain
S
F
i s an o( -morphism from
f
fi
cx
(3) i f
f
-morphism from
into
S
f
R
R =
f 3 + 1 , then
4 -morphism I R I , there
i s an
f
, extending
t o the domain
f
i s an o( -morphism i f i t i s a
G ;
13 -morphism
.
f o r every /s s t r i c t l y l e s s than o(
, we
be a l o c a l isomorphism
d be an o r d i n a l . i n t o S , i n the f o l l o w i n g cases:
augmented by a f i n i t e number o f elements i n
G = F
i s a l i m i t o r d i n a l , then
For o( = 1
i s indivisible. 0
R
0-morphism;
i s a successor o r d i n a l , say o(
i f f o r every s e t
exists a
, then
R
-YOUNGER RELATION
; and l e t
(1) every l o c a l isomorphism i s a ( 2 ) i f o(
i s embeddable i n one o f the two r e s t r i c -
R
I n both cases,
be two r e l a t i o n s o f the same a r i t y ; l e t
S
into
R
Ei S
f i n d the notion o f
1-morphism: see 1.8.
Every g -morphism i s a (3-morphism f o r each
/s
.
l e s s than o(
An o( -morphism, when r e s t r i c t e d t o any subset o f i t s domain, i s s t i l l an s' -morphism. The composition o f two d -morphisms i s an o( -morphism. An isomorphism from
R
onto
R
i s an O(-morphism from
S
. Similarly
S., f o r every o r d i n a l 6
i n t o any extension o f
f o r a l o c a l isomorphism from
R
into
which
S
i s e x t e n d i b l e t o an isomorphism. I n p a r t i c u l a r , the i d e n t i t y on a subset o f the base o f i n t o every extension o f R , f o r every o r d i n a l o(
We say t h a t a r e l a t i o n S
, if
o( - o l d e r
i s embeddable i n
The n o t i o n o f ly for
then R
S
"
-older than
@
, or R
that
i s &-younger
R
into
S
than
.
i s r e f l e x i v e and t r a n s i t i v e . An a( - o l d e r r e l a t i o n
l e s s than o(
, then
S
.
i s &-older
than
R
f o r every o r d i n a l o(
1-older coincides w i t h t h e n o t i o n o f o l d e r , defined i n 1.2;
R
,
S
be two denumerable r e l a t i o n s .
i s embeddable i n
S
.
similar-
If S &
ul-older
than
R
,
(uses countable axiom o f choice).
f be an W1-morphism from R be an a r b i t r a r y f i n i t e superset o f
0 Let
R
-morphism from
an o( -morphism from
1-younger.
4.6. L e t
G
"
f o r every /3
i s fi-older R
i s o(
S
the empty f u n c t i o n i s an
The comparison If
,is
R
.
R
i n t o S , w i t h f i n i t e domain F , and l e t F ( i n c l u d e d i n t h e base I R I ). It s u f f i c e s
290
THEORY OF RELATIONS
t o prove t h a t
domain
.
G
w l-morphism
i s e x t e n d i b l e t o an
f
g
Indeed, s t a r t i n g w i t h t h e empty s e t as
F
up w i t h an isomorphism from the denumerable r e l a t i o n
G
Since ordinal
, the f u n c t i o n
o(
into
R
, with
S
and i t e r a t i n g t h i s , we end R
onto a r e s t r i c t i o n o f
S
.
augmented by a f i n i t e number o f elements, and since f o r each countable
= F
extending
from
i s an
f
t o the domain
f
(o< +l)-morphism, there e x i s t s a b i j e c t i o n g ,
, which
G
i s an 4 -morphism from
There are only countably many b i j e c t i o n s
IS1
the denumerable base a t l e a s t one
.
g w i t h domain
u l many
While
R
into
countable o r d i n a l s c( : so there e x i s t s
, hence
g which i s an o( -morphism f o r every countable o r d i n a l %
which i s an Cdl-morphism
from
into
R
S
.
S
and range included i n
G
.
Note t h a t t h i s argument uses the f a c t
t h a t a countable union o f countable o r d i n a l s i s a countable o r d i n a l ; i n o t h e r words, we use the countable axiom o f choice, as already i n ch.1 4.7.
NON-EMBEDDABILITY RANK
For
R
,
5
2.5 and i n ch.2
denumerable r e l a t i o n s w i t h same a r i t y , e i t h e r
S
i s o( - o l d e r than
f o r every o r d i n a l o( ); o r
R
R
o l d e r than
R
+
We c a l l o(
, and so a l e a s t o r d i n a l
R
6.1. 0
(and hence
R,<S
i s n o t embeddable i n
S
, and
S
f o r which
by the preceding, there e x i s t s a greatest countable o r d i n a l o(
CX - o l d e r than
5
is
S
+ 1 from which p o i n t on S i s n o t
o(
. 1 the non-embeddability rank o f
ged i f we replace e i t h e r
R
or
S
R
S
. This
rank remains unchan-
by an equimorphic r e l a t i o n .
For example, s t a r t w i t h the chain Q o f r a t i o n a l s , and w i t h the o r d i n a l wOi By i n d u c t i o n , we see t h a t d i s @ - o l d e r than Q . I n o t h e r words, t h e nonembeddability rank o f
Q
in
L&
i s a t l e a s t equal t o o(
. f
More p r e c i s e l y , we see by i n d u c t i o n on o( t h a t a l o c a l isomorphism Dom f = {al
i n t o an o r d i n a l , w i t h
,..., ap)
(al<
o( -morphism, provided t h a t each o f the i n t e r v a l s
, (fap, +
(fap-l,fap)
00 )
.
... < aP (O,fal)
from
Q
mod Q ) , i s an
,
i n the o r d i n a l , i s a t l e a s t equal t o
w
These c o n d i t i o n s can obviously be s a t i s f i e d w i t h t h e o r d i n a l
,
(fal,fa2) w o(
o( +1
...
.
.
Consequently, our preceding p r o p o s i t i o n 4.6 generalizes the p r o p o s i t i o n i n ch.5
5
3.2, which s t a t e s t h a t Q i s embeddable i n a chain, i f every countable o r d i n a l i s i t s e l f embeddable. I n our new terminology, Q i s embeddable i n a r e l a t i o n R
if
R
i s % - o l d e r than
a fortiori,
i f CJ
Q
f o r every countable o r d i n a l o(
d .
i s embeddable i n
R
.
This i s t h e case,
f o r every countable o r d i n a l exponent o(.
NON- R I CHNESS RANK
I n particular, l e t
R
be a denumerable non-rich r e l a t i o n .
We d e f i n e the non-richness rank o f which
R
R
t o be the l e a s t countable o r d i n a l o(
for
i s n o t o( - o l d e r than a denumerable r i c h r e l a t i o n . This rank does n o t
depend on the chosen r i c h r e l a t i o n , s i n c e a l l r i c h r e l a t i o n s are equimorphic.
29 1
Chapter 10
4.8. The non-embeddability rank leads t o two kinds o f problems, according t o whether
R and consider o( - o l d e r r e l a t i o n s than
we f i x
and consider
S
First f i x o(
o(
-younger r e l a t i o n s than
For example, i f
on the c o n t r a r y i f we f i x
S
i s embeddable i n
S
.
i s the denumerable b i n a r y r e l a t i o n always (+), o r again the
S
o f r a t i o n a l s , then i t s u f f i c e s t h a t a denumerable r e l a t i o n X
Q
ger than
If S
, or
and note t h a t i n the usual cases, t h e r e e x i s t s a countable o r d i n a l
S
such t h a t every 4 - y o u n g e r r e l a t i o n than
chain
R
.
S
S
, for
X
t o be embeddable i n
be
l-youn-
.
S
i s r i c h , then every countable r e l a t i o n , hence already every denumerable,
, i s embeddable i n S . i s t h e chain o f n a t u r a l numbers , then a I-younger r e l a t i o n i s necessarily
0-younger r e l a t i o n than If S
a chain. A
S
2-younger r e l a t i o n can have a t most f i n i t e l y many elements between
any two given elements, o r before any given element. So t h a t i t can only be e i t h e r a f i n i t e chain, o r a chain isomorphic w i t h Problem 1
d
. For
, such
each countable r e l a t i o n
S =
Gd , hence embeddable i n
, does t h e r e e x i s t a countable o r d i n a l
S
t h a t every countable @-younger r e l a t i o n than
I n o t h e r words, i f
i s n o t embeddable i n
X
S
, then
rank of
X
in
Now f i x
R
and consider the o( - o l d e r r e l a t i o n s than
o( , then & a
.
R
= chain
i s embeddable i n S
i s the non-embeddability
Q
R
. We
saw i n the preceding
o f r a t i o n a l s and f o r every countable o r d i n a l
i s o( - o l d e r than
.
Q
R o f a r i t y a t l e a s t equal t o 2 , and , does t h e r e e x i s t an Xb R , which i s denumerable
For each countable r e l a t i o n
f o r each countable o r d i n a l O( and o( - o l d e r than rank o f
S
u n i f o r m l y bounded by a countable o r d i n a l .
S
paragraph, t h a t f o r Problem 2
.
i n an
R
.
I n o t h e r words, i s i t p o s s i b l e t h a t t h e non-embeddability
#R
takes a r b i t r a r i l y l a r g e countable o r d i n a l values. A p o s i t i v e answer f o r r i c h r e l a t i o n s , and more g e n e r a l l y f o r i n e x h a u s t i b l e r e l a t i o n s ( d e f i n e d below i n 5 5 ) , w i l l soon be obtained. This i s t h e case, f o r i n s R
X
tance, o f the chain o f n a t u r a l numbers always (+)
5 5 - INEXHAUSTIBLE A relation
, and
o f t h e denumerable b i n a r y r e l a t i o n
.
R
f i n i t e subset
R E L A T I O N A N D INEXHAUSTIBLEA G E
w i t h i n f i n i t e base F of
E
, the
i s s a i d t o be i n e x h a u s t i b l e i f , f o r every
E
relation
R i s embeddable i n i t s r e s t r i c t i o n
R/(E-F) , hence i s equimorphic t o i t . By 4.3, every denumerable r i c h r e l a t i o n i s inexhaustible. Every i n d i v i s i b l e r e l a t i o n (see ch.6
5
6) i s i n e x h a u s t i b l e . For example, the chain
w , the chain Q , the denumerable b i n a r y r e l a t i o n always (+). However, t h e r e e x i s t o t h e r denumerable i n e x h a u s t i b l e r e l a t i o n s . For example, the c o n s e c u t i v i t y r e l a t i o n on n a t u r a l numbers
.
.
292
THEORY OF RELATIONS
5.1. L e t F be a f i n i t e s e t . Consider an m-sequence o f denumerable r e l a t i o n s Ri ( i i n t e g e r ) , a l l o f the same a r i t y
>,
t h a t f o r any two d i s t i n c t i n t e g e r s
i
and
lRjl
Let
H
, each o f whose base includes F
2
,
i s F ; and f i n a l l y w i t h the c o m p a t i b i l i t y c o n d i t i o n
the
i n which
Ri
H
F i r s t t h e r e e x i s t s a common extension o f the r e s t r i c t i o n s
0
i s not embeddable: use ch.5
H
adding
lRil
= R./F . J Ri f o r each
Ri/F
be an i n e x h a u s t i b l e r e l a t i o n o f the same a r i t y , w i t h H $
:
; such
the i n t e r s e c t i o n o f t h e bases
j
5 1.4. Complete
F t o i t s base and preserving each
i
.
i s n o t embeddable.
-
Ri/(lRi(
F)
, i n which
t h i s common extension a r b i t r a r i l y ,
Ri
. Then since
i s inexhaustible,
H
the non-embeddability c o n d i t i o n subsists. 0 5.2. L e t
be a denumerable i n e x h a u s t i b l e r e l a t i o n o f a r i t y
H
For each countable o r d i n a l o( e x i s t s a countable r e l a t i o n
F
j d e n t i t v on
i n which
R
o( -morphism from
, take
For oC = 0
0
and each f i n i t e subset
fi .
Fi
is a
I H I , there
.
R = H/F
Then t o each f i n i t e subset
te a relation
.
R (uses countable ax. o f choice).
-1
Suppose t h a t o( i s a successor o r d i n a l q = for
2
i s non-embeddable, and f o r which t h e
H
H
>/
F o f the base
Fi
A+l
and t h a t our p r o p o s i t i o n holds
o f the base
I H 1 , including
, associa-
F
i n which
H
i s non-embeddable, and f o r which t h e i d e n t i t y on
(5-morphism from
H
into
Ri
change those i n
lRil-
Ri
.
i, j
so t h a t f o r
F
Keep the elements o f
F
, but eventually lRil - F
d i s t i n c t , the s e t d i f f e r e n c e s
I R . 1 - F are always d i s j o i n t . By t h e preceding p r o p o s i t i o n , t h e r e e x i s t s a J common extension R o f the Ri , i n which H i s non-embeddable, and where the and
i d e n t i t y on
F i s ana-morphism from H i n t o R ( t a k e one R i f o r each F i by c.a.c.). /ji ( i i n t e g e r ) be an
Now suppose t h a t o( i s a non-zero l i m i t o r d i n a l , and l e t increasing a-sequence o f o r d i n a l s w i t h supremum o(
H
be a r e l a t i o n i n which
Ri
is a
fii-morphism
from
into
H
.
For each i n t e g e r
i
, let
i s non-embeddable, and such t h a t the i d e n t i t y on
F
. Suppose t h a t
Ri
the d i f f e r e n c e s e t s l R i l - F R o f t h e Ri , i n which H i s i s an 4 -morphism from H i n t o R . 0
a r e mutually d i s j o i n t , and take a common extension non-embeddable. Then the i d e n t i t y on
F
Then our problem 2 i n 4.8 above, has a p o s i t i v e s o l u t i o n i n the p a r t i c u l a r case o f an i n e x h a u s t i b l e r e l a t i o n : i t s u f f i c e s t o take The non-embeddability o f in
H
, or
that
R
and
in
H
H
R
F empty.
means e i t h e r t h a t
R
i s p r o o e r l y embeddable
are incomparable ( w i t h respect t o embeddability).
H = Q (chain o f r a t i o n a l s ) , t h e preceding propoR
I n c e r t a i n cases, f o r instance i f s i t i o n can be strengthened t o 0
We have seen i n 4.7 t h a t U'(
.
, which
generally t h a t , f o r every f i n i t e subset o( -morphism from
Q
i n t o cdat',
is
<
Q
, is
o( - o l d e r than
F o f t h e base
w i t h domain
F
.0
Q
. And more
191 , t h e r e e x i s t s an
293
Chapter 10
However, t h e preceding p r o p o s i t i o n cannot always be so strengthened.
, take
H
For
condition
R
tion
t h e denumerable i n e x h a u s t i b l e b i n a r y r e l a t i o n always ( + ) . Then the would imply t h a t
R< H
i s n o t even
i s f i n i t e and always (+); and such a r e l a -
R
1-older than
H
.0
5.3. FAITHFUL EXTENSION WITH RESPECT TO AN INEXHAUSTIBLE RELATION Let
H
be a denumerable i n e x h a u s t i b l e r e l a t i o n w i t h a r i t y >/ 2
a denumerable
5
1.3.(1)
R+>
i n which
R
.
H i s non-embeddable, there e x i s t s i s s t i l l non-embeddable. Compare w i t h ch.5
R
For every denumerable r e l a t i o n
i n which H
f o r a r b i t r a r y r e l a t i o n s , and ch.5
5
4.5 f o r chains; note t h a t here we
have t h e a d d i t i o n a l requirement t h a t R+ i s denumerable, as i s R . Our proof uses countable axiom o f choice. 0 L e t o< be the non-embeddability rank o f H i n R . By t h e preceding 5.2, there e x i s t s a denumerable r e l a t i o n
S
i n which
ble
Rt
which i s a common extension o f
respects t h e non-embeddability o f
H
R
i s non-embeddable, and which i s
H
o( - o l d e r than H ( t a k e F empty). By ch.5
5
1.3.(2),
t h e r e e x i s t s a denumera-
, up
t o isomorphism, and which
and
S
(see 4.7 f o r the non-embeddability rank). 0
I n p a r t i c u l a r , f o r every denumerable non-rich r e l a t i o n
R
r a b l e non-rich r e l a t i o n which i s s t r i c t l y g r e a t e r than
R
, there
e x i s t s a denume-
( w i t h respect t o embed-
dabi 1it y ) . Consequently, a denumerable r e l a t i o n which admits an embedding o f every denumerab l e non-rich r e l a t i o n , i s n e c e s s a r i l y r i c h . Compare w i t h the case of denumerable chains a d m i t t i n g an embedding o f every coun-
5
t a b l e o r d i n a l : ch.5 5.4.
INEXHAUSTIBLE AGE
We say t h a t an age of
3.2.
d% , which
I f an age
6L
&
i s i n e x h a u s t i b l e , i f there e x i s t s a r e l a t i o n representative
5
i s i n e x h a u s t i b l e i n t h e sense o f
6%
i s i n e x h a u s t i b l e , then
5 above.
v e r i f i e s the f o l l o w i n g equivalent condi-
tions:
(1) For any two f i n i t e r e l a t i o n s t h e r e e x i s t s a common extension o f
A
( 2 ) For every f i n i t e r e l a t i o n
there e x i s t s an element
of
B
t o i t s base minus
of
B
p
d%
&,,ntoijsediaB , which belongs t o & ; belonging t o &, and every p o s i t i v e i n t e g e r p , such t h a t A i s embeddable i n the r e s t r i c t i o n
A, B
A
belonging t o
and
a r b i t r a r y elements;
R be a r e p r e s e n t a t i v e o f the age 62 , and E i t s base; then f o r every E , t h e r e s t r i c t i o n R/(E-F) i s again a r e p r e s e n t a t i v e f i n i t e subset F of ; ( 4 ) For R , S representatives o f 0% w i t h d i s j o i n t bases, there e x i s t s a comnon (3) Let
L
a
of
and
extension o f R s u f f i c e s i f R and
S S
, which represents are denumerable).
0%
(uses t h e u l t r a f i l t e r axiom; ZF
294
THEORY OF RELATIONS
0 S t a r t i n g w i t h ( l ) , take
p+l
bases, and a common extension
A
copies isomorphic w i t h B
and having d i s j o i n t
belonging t o the age: we o b t a i n ( 2 ) .
The c o n d i t i o n ( 2 ) immediately i m p l i e s (3); and ( 3 ) immediately i m p l i e s ( 1 ) . Note t h a t i t s u f f i c e s t h a t t h e r e e x i s t s a r e p r e s e n t a t i v e o f
&,
which s a t i s f i e s ( 3 )
i n order t h a t every r e p r e s e n t a t i v e s a t i s f y ( 3 ) . F i n a l l y ( 1 ) i m p l i e s ( 4 ) by the coherence lemma (ch.2 f i l t e r axiom): apply t h i s lemma t o the unions o f the base 5.5.
(RI
and G
5
1.3, e q u i v a l e n t t o the u l t r a -
F u G where
i s a f i n i t e subset o f ( S l
F
i s a f i n i t e subset
. Obviously
(4) implies (1). 0
INEXHAUSTIBLE EXTENSION THEOREM Every age which s a t i s f i e s the c o n d i t i o n ( 1 ) i n the preceding 5.4, i s inexhaus-
to ( 4 )
t i b l e . Hence each o f the precedinq c o n d i t i o n s (1)
i s e q u i v a l e n t t o inexhaus-
ti b i 1it y .
More s t r o n g l y i f _senting
dL
s a t i s f i e s (l), then f o r every denumerable r e l a t i o n
a , there exists
and represents
6&
a denumerable extension o f
repre-
R
R which i s i n e x h a u s t i b l e
(POUZET 1979 p. 343). w i t h mutually Ri o r o f an age included i n 4$, , w i t h base the union o f the Ei , which
0 Suppose t h a t the c o n d i t i o n ( 1 ) holds. Given a s e t o f r e l a t i o n s
d i s j o i n t bases
Ei
, each
a representative o f
there e x i s t s a common extension o f the represents
&
o r an age i n c l u d e d i n
Ri
6?. . To
@
seethis,use the coherence lemma, as
precedently i n 5.4 f o r proving ( 4 ) from ( 1 ) . Now take an ~1 -sequence o f isomorphic copies
of
Ri
(a representative o f
R
w i t h d i s j o i n t bases. By the preceding, there e x i s t s a common extension
S
Ri , which represents 6$. L e t ai denote the jth element o f j where i t i s understood t h a t t h e r e e x i s t s an isomorphism from Ri
Ri,
i' takes a'. into a ( i and J j Using RAMSEY's theorem (ch.3
i' fixed,
5
l.l),
j
Ei
fi
) y
o f the
( i ,j i n t e g e r s ) ,
onto
which
varying).
given an i n t e g e r
, we o b t a i n an w-sequen-
h
ce e x t r a c t e d from the sequence o f the i n t e g e r s ; such t h a t a f t e r renumbering, t h e b i j e c t i o n which, f o r each i n t e g e r i s a l o c a l automorphism o f
S
i
takes
a6
into
a.
i+l
, ... , a:
into
i+l ah ,
.
By passing t o the l i m i t which respects each Ri , we o b t a i n a r e l a t i o n S , s t i l l , f o r which t h e preceding holds f o r each i n t e g e r h . Then S i s representing
a
isomorphic w i t h i t s r e s t r i c t i o n obtained by removing t h e s e t Hence a l s o by removing any f i n i t e union o f the
Ei
; thus
S
Eo from i t s base. i s inexhaustible. 0
5.6. Following POUZET 1979 p . 326, we s h a l l prove t h a t every age i n c l u d i n g an inexh a u s t i b l e aqe, a l s o includes a maximum i n e x h a u s t i b l e age. More p r e c i s e l y l e t
6%
be an age. The maximum i n e x h a u s t i b l e age i n c l u d e d i n
i s the s e t (denumerable up t o isomorphism) o f f i n i t e r e l a t i o n s a r b i t r a r y f i n i t e s e t o f isomorphic copies o f extension i n ba, .
A
A
d%
f o r which any
w i t h d i s j o i n t bases, has a common
295
Chapter 10
0
It s u f f i c e s t o see t h a t t h i s s e t i s d i r e c t e d . L e t
representing b;L trictions of
.
For each o f the
A
which are isomorphic w i t h
R
R
be a denumerable r e l a t i o n
considered, there e x i s t i n f i n i t e l y many r e s -
, having
A
d i s j o i n t bases. Thus given
and an analogous f i n i t e r e l a t i o n B , there e x i s t s a common extension o f A and B , taken w i t h d i s j o i n t bases, which has i n f i n i t e l y many isomorphic copies which are
A
restrictions of
§
6 - A
.0
R
RELATION
RICH
FOR ITS
AGE,
We say t h a t a denumerable r e l a t i o n
R
r e l a t i o n younger than
R
A RELATION
MINIMAL
FOR ITS AGE
i s r i c h f o r i t s age, i f every countable
i s embeddable i n
R
.
By 2.4 above, i t s u f f i c e s t h a t every denumerable r e p r e s e n t a t i v e o f the same age as
R
, be embeddable i n
tical to
R
. I n o t h e r words,
the n o t i o n o f " r i c h f o r i t s age"
i s iden-
"maximal, w i t h respect t o embeddabili ty, among the denumerable representa-
t i v e s o f i t s age
I'
.
For example, t h e r i c h denumerable b i n a r y r e l a t i o n (see 4.1) i s r i c h f o r i t s age. The chain
Q
o f r a t i o n a l s , o r any denumerable chain i n which
Q
i s embeddable,
o r e q u i v a l e n t l y any chain equimorphic w i t h Q , i s r i c h f o r i t s age. Another example. S t a r t w i t h t h e c o n s e c u t i v i t y r e l a t i o n on p o s i t i v e a n d negative i n t e gers, i . e . the b i n a r y r e l a t i o n t a k i n g the value (+) f o r those ordered p a i r s where
y = x
+
1
. Take
(x,y) denumerably many such c o n s e c u t i v i t y r e l a t i o n s w i t h mutually
d i s j o i n t bases, and c a l l these the components; then complete by g i v i n g value ( - ) f o r p a i r s o f elements belonging t o two d i s t i n c t components. Then every denumerable younger r e l a t i o n i s formed o f analogous components which are e i t h e r f i n i t e , o r the c o n s e c u t i v i t y on n a t u r a l numbers
, or
the c o n s e c u t i v i t y on negative integers,
o r isomorphic w i t h preceding components: such a r e l a t i o n i s embeddable i n our consec u t i v i ty r e l a t i o n . 6.1. There e x i s t s a denumerable p a r t i a l o r d e r i n g which i s r i c h f o r i t s age, and i n which a l l f i n i t e p a r t i a l orderings, thus a l l countable p a r t i a l orderings, are embeddable. 0 Recall t h e amalgamation lemma i n ch,2
5
2.2. S t a r t w i t h the o r d e r i n g reduced t o
a s i n g l e t o n . Take the t h r e e p o s s i b l e p a r t i a l orderings o f c a r d i n a l i t y 2
, extending
our s i n g l e t o n ordering. These are obtained by adding a new element a t the end, o r a t the beginning, o r making i t incomparable w i t h the unique element o f our given s i n g l e t o n . By the amalgamation lemma, there e x i s t s a common extension s i n g l e t o n o r d e r i n g A1 and f o r the t h r e e preceding extensions o f i t s e l f a p a r t i a l ordering.
A1
A2
f o r our
, which i s
I n general, suppose t h a t we have a p a r t i a l o r d e r i n g Ai ( i i n t e g e r ) i n which every i n c r e a s i n g sequence of p a r t i a l orderings w i t h successive c a r d i n a l i t i e s 1,. . ,i
.
THEORY OF RELATIONS
296
i s embeddable. Take a l l p o s s i b l e p a r t i a l l y ordered extensions o f
Ai
, whose
base
i s augmented by a new element. There are o n l y f i n i t e l y many such extensions, and by the amalgamation l e m a , there e x i s t s a common extension p a r t i a l ordering. F i n a l l y the common extension o f a l l the
Ai+l which i s a f i n i t e Ai ( i i n t e g e r ) t o the
union o f the bases, i s a denumerable p a r t i a l o r d e r i n g i n which every countable p a r t i a l o r d e r i n g i s embeddable. 0 6.2. There e x i s t s a denumerable t r e e which i s r i c h f o r i t s age, and i n which a l l f i n i t e t r e e s , thus a l l countable t r e e s are embeddable.
...,
Our base w i l l be the s e t o f a l l f i n i t e non-empty sequences (rl, r ) (p p o s i t i P ve i n t e g e r ) o f r a t i o n a l s . We say t h a t such a sequence i s l e s s than another such sequence
,
( s l,...,sq)
if
q i s g r e a t e r than o r equal t o
p
, and rl
, ... ,
= s1
rp-l = sp-l , and f i n a l l y rP. c s P . The reader e a s i l y shows t h a t t h i s comparison i s r e f l e x i v e , t r a n s i t i v e , antisymmetric, and t h a t any two sequences which are l e s s than a t h i r d , are comparable: thus we have a denumerable t r e e . L e t us show t h a t every countable t r e e i s embeddable i n i t . L e t denumerable base
E
. We
can assume t h a t
E
, and
E which contains the element 0
of
a subset
E'
I = R/E'
i s a maximal chain: each element o f
l e a s t one element o f
E'
. Take
such t h a t the r e s t r i c t i o n
E-E'is incomparable (mod R) t o a t
.
Construct a t o t a l l y ordered extension
.
be a t r e e w i t h
R
i s the s e t o f n a t u r a l numbers
I+o f
I , by adding countably many c u t s
x o f E-E' , add the c u t , o r i n i t i a l i n t e r v a l o f I , formed o f those elements o f 111 which are l e s s than x (mod R) Then embed It i n the chain o f the r a t i o n a l s . Each element o f E ' ,
t o the base
III
More p r e c i s e l y , f o r each element
.
and i n p a r t i c u l a r the i n t e g e r
0
, i s taken i n t o a r a t i o n a l ; o r i n o t h e r words,
i n t o a sequence formed o f a unique r a t i o n a l . As f o r each element
x
in
E-E'
,
i t i s taken i n t o a f i n i t e sequence o f r a t i o n a l s , the f i r s t o f whose terms i s the rational Let
Ur
r which i s the image o f the c u t o f
I p r e v i o u s l y associated w i t h
denote the equivalence c l a s s o f the elements
x
in
E-E'
t o which
x
.
r
i s thus associated. I n each equivalence class a subset tion
of
Ir= R / U i
. As
R/Ur
U,'.
Ur
Ur
, take
-
than Ub
U,'.
x
, and
, add
u(r)
, and
u ( r ) , then take f o r which t h e r e s t r i c -
i s a maximal t o t a l l y ordered r e s t r i c t i o n o f the p a r t i a l o r d e r i n g
before, construct an extension
1:
I IrI . More
cuts t o the elements o f the base
Ur
t h e l e a s t n a t u r a l number
which contains t h e element
of
Ir , by adding countably many
p r e c i s e l y , f o r each element
the c u t formed o f those elements i n the base
(mod R )
. Then embed
1 :
i n p a r t i c u l a r the i n t e g e r
lUhl
x
of
which are l e s s
i n the chain o f the r a t i o n a l s . Each element i n u(r)
,is
taken i n t o a sequence of two r a t i o n a l s
Chapter 10
the f i r s t o f which i s x
in
Ur
- LJ; , i t
f i r s t o f which i s
I:
r
291
and the second i s i t s r a t i o n a l image. As f o r each element
i s taken i n t o a f i n i t e sequence o f a t l e a s t t h r e e r a t i o n a l s , the and the second i s the image o f the c u t o f
r
, p r e v i o u s l y associated w i t h x
Ir , belonging t o
.
I t e r a t e the preceding c o n s t r u c t i o n by p a r t i t i o n i n g the elements o f each class
-
U;
r, s
.
Ur
i n t o subclasses
U
each associated w i t h the ordered p a i r o f r a t i o n a l s r,s Note t h a t every element o f the base E i s taken i n t o a f i n i t e sequence o f
r a t i o n a l s , since we successively take the l e a s t i n t e g e r i n Ur
, then i n each class
etc.,
U
E
, then i n each class
and then we choose a maximal chain going
r,s through t h i s l e a s t i n t e g e r . A l s o note t h a t t h e r e l a t i o n o f being l e s s than, o r g r e a t e r than, o r incomparable (mod R), i s preserved when r e p l a c i n g each element o f the base
of
E
R
by t h e f i n i t e sequence o f r a t i o n a l s which i s associated
with it. 0 6.3. There e x i s t s an age n o t having any denumerable r i c h r e p r e s e n t a t i v e (SPECKER 1957, unpublished). 0
S t a r t w i t h a l l f i n i t e binary r e l a t i o n s , each having i t s base p a r t i t i o n e d i n t o
equivalence classes; on each class, c o n s t r u c t a c o n s e c u t i v i t y r e l a t i o n by t o t a l l y o r d e r i n g the elements, and then g i v i n g the value (+) f o r ordered p a i r s o f consecut i v e elements and t h e value (-) otherwise; hence i n p a r t i c u l a r f o r p a i r s whose terms belong t o d i s t i n c t classes. Now on the diagonal, hence f o r the ordered p a i r s (x,x)
where
x
belongs t o t h e base, assign an a r b i t r a r y value (+) o r (-).
A r e l a t i o n thus constructed s h a l l be c a l l e d a modulated c o n s e c u t i v i t y r e l a t i o n ;
these r e l a t i o n s obviously c o n s t i t u t e an age (closed and d i r e c t e d under embeddab i 1 ity )
.
Consider the f o l l o w i n g denumerable representatives o f t h i s age. S t a r t w i t h the c o n s e c u t i v i t y r e l a t i o n on t h e n a t u r a l numbers \
, modulated
on the diagonal, so
t h a t every f i n i t e sequence o f (+) and ( - ) i s r e a l i z e d by a t l e a s t one sequence o f consecutive i n t e g e r s . To see t h a t t h e r e e x i s t continuum many such representat i v e s , i t s u f f i c e s t o b i j e c t i v e l y associate each r e p r e s e n t a t i v e w i t h an i n f i n i t e s e t o f integers. For example, i f we assign a r b i t r a r y values (+) o r (-) t o those pairs
(x,x)
f o r which t h e i n t e g e r
x
i s a square, then t h e r e remain a r b i t r a r i l y
l a r g e f i n i t e i n t e r v a l s between two consecutive squares, i n order t o ensure the existence o f every f i n i t e sequence o f (+) and ( - ) . Suppose t h a t t h e r e e x i s t s a denumerable r e l a t i o n
R which i s r i c h f o r t h i s age. R Thus R must
Then each o f the preceding r e l a t i o n s must be embeddable i n
.
have components which are obtained by modulation from ei'ther the c o n s e c u t i v i t y r e l a t i o n o f t h e n a t u r a l numbers negative i n t e g e r s . Since
R
, or
the c o n s e c u t i v i t y r e l a t i o n o f the p o s i t i v e and
i s denumerable, there are countably many components;
THEORY OF RELATIONS
298
and i n each component, t h e r e a r e c o u n t a b l y many modulated c o n s e c u t i v i t y r e l a t i o n s c o n s t r u c t e d on t h e n a t u r a l numbers ding, t h e images o f
1, 2, 3,
(up t o isomorphism). Indeed, f o r any such embed-
...
a r e determined as soon as t h e image o f
is
0
chosen. I t f o l l o w s t h a t t h e r e a r e continuum many modulated c o n s e c u t i v i t y r e l a t i o n s c o n s t r u c t e d f r o m t h e n a t u r a l numbers, which a r e n o t embeddable i n
R : contra-
diction. 0 6.4.
Problem 1
.
Let
f o r i t s age, and same age as
R
F
R
be a r e l a t i o n w i t h denumerable base
be a f i n i t e subset o f
,is
i t equimorphic w i t h
E
. Then
(i.e.
R
if
E
, which
R/(E-F)
i s rich
represents the
each one i s i t embeddable i n t h e
other). Problem 2 : f a i t h f u l e x t e n s i o n . I f
i s n o t r i c h f o r i t s age, t h e n does t h e r e
R
e x i s t an e x t e n s i o n s t r i c t l y g r e a t e r t h a n t h e same age as
R
, and
R
under e m b e d d a b i l i t y , which r e p r e s e n t s
which i s n o t r i c h f o r i t s age.
6.5. A RELATION MINIMAL FOR ITS AGE
A denumerable r e l a t i o n R i s s a i d t o be minimal f o r i t s age, i f t h e r e e x i s t s no r e l a t i o n r e p r e s e n t i n g t h e same age, and s t r i c t l y l e s s t h a n R under e m b e d d a b i l i t y . F o r example, t h e c h a i n W o f n a t u r a l numbers , and i t s converse, a r e t h e o n l y minimal r e l a t i o n s f o r t h e age o f c h a i n s .
, are
the relations freely interpretable i n
The W - c h a i n a b l e r e l a t i o n s , i . e .
5
minimal f o r t h e i r age: see ch.9
5.6.
There e x i s t s an age h a v i n g no minimal denumerable r e p r e s e n t a t i v e (POUZET 1979, p. 318). 0
S t a r t w i t h t h e s e t o f a l l f i n i t e sequences and O - s e q u e n c e s o f
0
and
1
,
which i s e q u i p o t e n t w i t h t h e continuum. F i r s t t a k e t h e unary r e l a t i o n
w i t h v a l u e (t) f o r t h e empty sequence, and t h e
v a l u e (-) f o r e v e r y non-empty sequence. Then t a k e t h e unary r e l a t i o n v a l u e (+) f o r each non-empty f i n i t e sequence whose l a s t t e r m i s
0
U
, and
with with
value (-) otherwise. F i n a l l y , take t h e binary r e l a t i o n
R
w i t h v a l u e (+) f o r each o r d e r e d p a i r formed
o f a f i n i t e sequence and o f a c o n s e c u t i v e sequence, t h e l a t t e r o b t a i n e d by adding
0 o r 1 as a l a s t term; a d d i t i o n a l l y R takes t h e v a l u e (+) f o r each o r d e r e d p a i r formed o f a f i n i t e sequence s and an U - s e q u e n c e b e g i n n i n g by t h e i n i t i a l interval
s
. Let
R
t a k e t h e v a l u e (-) o t h e r w i s e .
We see t h a t t h e f o l l o w i n g a r e r e p r e s e n t a t i v e s o f t h e same age. Each o f the m u l t i r e l a t i o n
(R,U,B)
restriction
whose base c o n t a i n s a l l t h e f i n i t e sequences, and
which s a t i s f i e s t h e d e n s i t y c o n d i t i o n , which a s s e r t s t h a t f o r each f i n i t e sequence s
b e l o n g i n g t o t h e base, t h e r e e x i s t s a t l e a s t one element i n t h e base which i s
an Cd-sequence b e g i n n i n g by
s
.
299
Chapter 10
which r e p r e s e n t t h e same Note t h a t any two denumerable r e s t r i c t i o n s o f (R,U,H) age as ( R , U , 0 ) , hence whose bases c o n t a i n a l l f i n i t e sequences and which s a t i s f y t h e d e n s i t y c o n d i t i o n , a r e i d e n t i c a l as soon as t h e y a r e isomorphic. Indeed f o r each c3 -sequence i t s u f f i c e s t o c o n s i d e r i t s successive f i n i t e i n i t i a l i n t e r v a l s . F i n a l l y , i f one o f these denumerable dense r e s t r i c t i o n s o f
(R,U,0)
were minimal
f o r i t s age, t h e n i t would s u f f i c e t o t a k e a p r o p e r dense r e s t r i c t i o n , t o o b t a i n a m u l t i r e l a t i o n o f t h e same age, b u t s t r i c t l y l e s s under e m b e d d a b i l i t y : c o n t r a diction. 0
§
7 - SATURATED RELATION, EXISTENCE AGE (POUZET, VAUGHT)
CRITERION
FOR A RICH
RELATION
FOR ITS
SATURATED SUBSET Let
R
be a r e l a t i o n and
F
a f i n i t e subset o f t h e base. We say t h a t
r a t e d (mod R), i f f o r any r e l a t i o n 1-morphism
, which extends
into
R
f"(F)c G c IS1
ing G
from
f
f-l
S
, there
.
r e p r e s e n t i n g t h e same age as
S
w i t h domain
F
, and
any f i n i t e
R
set
F
i s s&-
, and G
any
satisfy-
R w i t h domain 1-morphism i s d e f i n e d i n 1.8.
exists a
1-morphism f r o m
Recall t h a t the
S
into
For example, g i v e n t h e c h a i n Q o f t h e r a t i o n a l s and t h e c h a i n W o f t h e n a t u r a l numbers, t a k e R = Q + c3 ; t h e n e v e r y f i n i t e subset o f t h e i n i t i a l i n t e r v a l Q i s s a t u r a t e d (mod R ) , b u t no subset o f t h e f i n a l i n t e r v a l W i s s a t u r a t e d . T h i s example has a l r e a d y been g i v e n i n 3.8 ( m a x i m a l i s t subsets). Another example. F o r t h e o r d i n a l W 2 0 F o r any c h a i n
al<
... < ap
A
and any f i n i t e subset
(mod A) ; f o r a f u n c t i o n
it suffices t o put
g
empty s e t i s s a t u r a t e d . G =
al,.
from
A
i n t o the
into Cd2
i t s base, w i t h t o be a
1-morphism,
( i + l ) s t component, l e a v i n g i n f i n i t e l y many elements
i n each o f t h e i n t e r v a l s d e f i n e d by t h e images 7.1.
. . ,a P 4 o f
i n t o t h e second component W o f U 2 , and i n general
g(al)
(i= 1, ...,p)
g(ai)
, the
g(ai)
.0
Every s a t u r a t e d f i n i t e subset i s m a x i m a l i s t , i n t h e sense o f 3.8.
I n o t h e r words, i f
F
1-morphism f r o m
R
, then
i s s a t u r a t e d (mod R)
i s maximal; o r e q u i v a l e n t l y , f o r e v e r y into
S
w i t h domain
S
the
(R/F)-age r e p r e s e n t e d by
r e p r e s e n t i n g t h e same age as
F
,is
also a
R
R
, every
1-isomorphism.
The converse i s f a l s e . 0 Take
and t a k e
R
t o be t h e c o n s e c u t i v i t y r e l a t i o n o n t h e p o s i t i v e a n d n e g a t i v e i n t e g e r s , F
t o be a f i n i t e i n t e r v a l o f t h i s c o n s e c u t i v i t y r e l a t i o n . Then t h e
(R/F)-age r e p r e s e n t e d by
S t o be t h e r e l a t i o n o b t a i R , with t h e value (-) f o r p a i r s whose elements belong t o d i s t i n c t components. Then a 1-morphism f f r o m R i n t o S w i t h domain F , i s a l s o a 1-isomorphism. B u t by t a k i n g G t o be a f i n i t e
R
i s maximal. However, t a k e
ned f r o m two components, each i s o m o r p h i c w i t h
THEORY OF RELATIONS
300
superset of f "( F ) with elements in b o t h of the two components, we see t h a t no local isomorphism from S into R with domain G and which extends f-' , i s a 1-morphism. 0 7 . 2 . Let R be a r e l at i o n , F be a saturated f i n i t e subset of the base I R ) ; l e t G be a subset of F and g be a 1-morphism from R into a relation S representing the same age, with Dom g = G . Then there exists an extension S+ of S S+ , extending g to the of the same age, and a 1-morphism f from R domain F ( t h i s proposition and i t s following consequence are communicated by HODGES; uses the u l t r a f i l t e r axiom).
Let H = Rng g , hence g takes R/G into S/H . Since g i s a 1-morphism, the (S/H)-age represented by S i s included in the image under g of the ( R / G ) age represented by R . By 3.2 above ( u l t r a f i l t e r axiom), there e x i s t s an extension S+ of S such t h a t the (S/H)-age represented by S+ i s exactly the image under g of the (R/G)-age represented by R We can even require t h a t there e x i s t s an isomorphism from R onto a certain r es t r i ct i o n of S+ , which on G i s identical t o g So t h a t t h i s isomorphism, when r es t r i ct ed t o F , i s a 1-isomorphism from R into S+ .
0
.
.
Consequently, every subset of a saturated f i n i t e subset i s saturated (uses ultraf i l t e r axiom). Take up the preceding notations, where F i s a saturated f i n i t e subset of the base I R 1 , and G i s a subset of F . Let g be a 1-morphism from R into S with Dom g = G Using the preceding extension S+ of S and the 1-morphism f we see t h a t , f o r every f i n i t e superset K of f"(F) , included in the base I S 1 , there e x i s t s a 1-morphism from S+ into R with domain K , hence a f o r t i o r i a 1-morphism from S i n t o R , which extends f - l , hence extends g-' . T h u s G i s saturated. 0 0
.
7.3. SATURATED RELATION A relation R i s said t o be saturated, i f every f i n i t e subset of i t s base i s saturated. By the preceding, i t suffices t h at each f i n i t e subset of the base be included in a f i n i t e saturated subset. For example, the chain of the rationals i s saturated. The rich relation defined in 4 . 1 above, i s saturated. The consecutivity relation with denumerably many components constructed from the chain Z o f positive and negative integers ( 5 6 above); as well as the pa rtia l ordering in 6.1 and the t r e e i n 6.2, a l l rich f o r t h e i r ages, are saturated. Every saturated relation i s maximalist, by the preceding 7.1. A non-saturated maximalist relation i s obtained from 3 . 8 with the consecutivity
,
30 1
Chapter 10
r e l a t i o n on
( p o s i t i v e and n e g a t i v e i n t e g e r s ) ; o n l y t h e analogous r e l a t i o n w i t h
Z
denumerably many such components, i s s a t u r a t e d . W i t h t h e t h i r d example o f 3.8, we have an age w i t h o u t any r i c h r e p r e s e n t a t i v e : t o see t h i s , t a k e up SPECKER's argument i n 6.3.
Now by 7.4 below, e v e r y r e l a t i o n which
i s n o t r i c h f o r i t s age, i s n o t s a t u r a t e d . So t h a t w i t h o u r t h i r d example o f 3.8, we o b t a i n an age w i t h o u t any s a t u r a t e d r e p r e s e n t a t i v e , b u t w i t h continuum many maxim a l i s t r e p r e s e n t a t i v e s ; remark due t o HODGES 1979, unpublished. I n t h e t e r m i n o l o g y o f Abraham ROBINSON, a s a t u r a t e d r e l a t i o n i s c a l l e d an e x i s t e n t i a l l y u n i v e r s a l model o f a u n i v e r s a l t h e o r y . See f o r example HIRSCHFELD, WHEELER 1975 p. 31; o r a g a i n SIMMONS 1976 p. 384. Any two denumerable s a t u r a t e d r e l a t i o n s o f t h e same age, a r e isomorphic.
R, R '
More p r e c i s e l y , l e t Then e v e r y
isomorphism f r o m 0 Let
be a
f
onto
R
R
into
w i t h f i n i t e domain, i s e x t e n d i b l e t o an
R'
.
R'
1-morphism f r o m
into
R
, the
inverse function
w i t h domain
. By
G'
, w i t h f i n i t e domain F and range
R'
. For every f i n i t e superset G '
F ' = f"(F)
IR'I
be denumerable s a t u r a t e d r e l a t i o n s o f t h e same age.
1-morphism f r o m
of
F'
which i s i n c l u d e d i n t h e base
i s extendible t o a
f-'
1-morphism f r o m
i t e r a t i n g t h i s a l t e r n a t i v e l y from
R
into
R'
into
R'
R
and back, we
o b t a i n an a - s e q u e n c e o f l o c a l isomorphisms, one e x t e n d i n g t h e o t h e r . Taking c a r e
R
t o i n c l u d e each element o f t h e bases o f
and
morphisms, we end up w i t h an isomorphism f r o m 7.4.
i n t h e domains o f l o c a l i s o -
R'
onto
R
R'
.0
Ever.y denumerable s a t u r a t e d r e l a t i o n i s r i c h f o r i t s age.
0 Let
R
be a denumerable s a t u r a t e d r e l a t i o n , and
S
be a denumerable r e l a t i o n
o f t h e same age. Thus t h e empty f u n c t i o n i s a
1-isomorphism f r o m
Since
G
a
i s s a t u r a t e d , f o r any f i n i t e subset
1-morphism
Let
g
F = g"(G)
7.3), the
R
the
from and
(R/F)-age
(S/G)-age
S
f = g
4
R w i t h domain
into -1
. Since
R
By 3 . 2 above, t h e r e e x i s t s an e x t e n s i o n p r e s e n t s t h e same age as
(R under f
.
I t e r a t i n g t h i s , and s i n c e superset
G1
and
R
I n o t h e r words,
of
, hence
G
R
S
cd
f
under
.
which i s denumerable, which r e -
, and
which r e p r e s e n t s t h e r e l - a g e image of
is a
1-isomorphism f r o m
R
into
T
.
I S I , we o b t a i n a 1-morphism g1 from R , which i s an e x t e n s i o n o f g t o t h e domain
i n c l u d e d i n t h e base
into
GI
; t h e n we o b t a i n an e x t e n s i o n
age, and f o r which
of
T
i s s a t u r a t e d , we have t h a t g i v e n an a r b i t r a r y f i n i t e
T
R
S
f
.
exists
R i s m a x i m a l i s t (see 7.1 and i s maximal. Since g i s a 1-morphism,
i s i n c l u d e d i n t h e r e l - a g e image o f
S
S
into
, there
the r e l a t i o n
r e p r e s e n t e d by
r e p r e s e n t e d by
R
o f t h e base I S 1 G .
from
S
fl = (gl)
into -1
TI is a
By i n c l u d i n g e v e r y element o f t h e base
of
S
which i s denumerable and o f t h e same
R i n t o T1 . I S 1 i n t h e domains o f t h e successive
1-isomorphism f r o m
gi
,
302
THEORY OF RELATIONS
we have t h a t t h e u n i o n o f a l l a restriction o f 7.5.
Let
.
R
( i integer)
F
G
which i s a s p e c i f i c a t i o n o f
which i s t h e image o f
, there
@L
under an
G
s a t u r a t e d subset: see
5
F
R
i s saturated i f f , f o r
onto
R/G
t h e image of
includes
o f t h e base,
G'
e x i s t s a subset
, such
R/G'
.
@
i n terms o f r e l - a g e , o f t h e d e f i n i t i o n o f a
7. We s h a l l use i t i n t h e p r o o f o f 7.6 below. 0
EXISTENCE CRITERION OF A R I C H RELATION OR OF A SATURATED RELATION
Given an age
69 , t h e r e e x i s t s
f o r each f i n i t e r e l a t i o n
A
A-ages s p e c i f i c a t i o n s o f
&.
a denumerable r i c h r e l a t i o n f o r t h e age
@, , t h e r e
belonging t o
e x i s t s a denumerable s a t u r a t e d e x t e n s i o n o f
R
r a t e d e x t e n s i o n of o f choice i f
R
,of
R
&,
of
f-R (uses
axiom
i s uncountable).
R
f o r each f i n i t e r e s t r i c t i o n A
i s a specification (see 3.1).
there e x i s t s a satu-
t h e same c a r d i n a l i t y , which r e p r e s e n t s
Suppose t h a t t h e r e e x i s t s a denumerable r i c h r e l a t i o n
&, . Then
a , there ba. (POUZET 1972,
representing
R
which r e p r e s e n t s
g e n e r a l i z i n g VAUGHT 1961). More s t r o n g l y , f o r e v e r y i n f i n i t e r e p r e s e n t a t i v e
d?, iff
a r e c o u n t a b l y many maximal
Moreover i n t h i s case, f o r each denumerable r e l a t i o n
0
onto
, and f o r any (R/G)-age
F
F-isomorphism f r o m
r e p r e s e n t e d by
0 This i s simply the t r a n s l a t i o n ,
7.6.
. Then
R
o f t h e base, which i n c l u d e s
(R/G')-age
S
a f i n i t e m a x i m a l i s t subset o f t h e base.
(R/F)-age r e p r e s e n t e d by
any f i n i t e subset
that the
i s an isomorphism f r o m
0
be a r e l a t i o n and
R
L e t .t%denote t h e
@
gi
Since
6% , t h e r e
of
of
e x i s t s an
R
r e p r e s e n t i n g t h e age
and each maximal
R
A-relation
U
A-age which
representing
i s r i c h , t h e r e e x i s t s a r e s t r i c t i o n isomorphic w i t h
R
hence a f i n i t e r e s t r i c t i o n
A'
the
R
A'-age r e p r e s e n t e d by
of
R
which i s isomorphic w i t h
d , hence
includes
R
. Thus
~4 ,
, such t h a t
A
i s an isomorphic image o f
F i n a l l y each maximal r e l - a g e which i s a s p e c i f i c a t i o n o f isomorphism, by a f i n i t e r e s t r i c t i o n o f
U
&, , i s
&.
d e f i n a b l e , up t o
t h e r e a r e c o u n t a b l y many such
rel-ages. Conversely, suppose t h a t f o r each element many maximal representative
R
of
&,
G
into
from t h e base
G'
v i a an
IRI
.
Let
F
and
G
F - i d e n t i c a l isomorphism
~4 denote
a maximal r e l - a g e ) , and l e t cifies
6?, , t h e r e a r e c o u n t a b l y R . S t a r t w i t h a denumerable
o f t h e age
which we can always assume t o be m a x i m a l i s t , by 3.9;
and w i t h two a r b i t r a r y f i n i t e subsets Take
A
A-ages which a r e s p e c i f i c a t i o n s o f
@
: see 3.5. Then l e t
the
including h
, o f t h e base.
(R/F)-age r e p r e s e n t e d by
be an a r b i t r a r y maximal T
F
making G '
-
F R
disjoint (which i s
(h"(R/G))-age which spe-
be a denumerable r e p r e s e n t a t i v e o f
@ , hence
303
Chapter 10
a representative of the (R/F)-age . By 3 . 2 there e x i s t s an A-isomorphic copy T ' of T and a common extension R1 of R and T ' , such t h at i f we denote by GI the imaqe of G ' when passing from T t o T ' , then the (R1/G1)-age represented by R1 includes the image of Furthermore we can choose R1 t o be maximalist. Pass from R t o R1 , then i t e r a t e t h i s , by using a l l possible quadruples (F,G, A ,?, , 63 ) . In view of the countability of the s e t of maximal rel-ages, in the lim it, we obtain a denumerable extension of R , each of whose f i n i t e subsets i s saturated. 0 0 In the case where R i s uncountable, the preceding proof requires the u l t r a f i l t e r axiom t o take a common extension by 3 . 2 , and more strongly the axiom of choice t o well-order the base, and then t o well-order the s e t of f i n i t e subsets. 0
a.
7.7. Let R be a relation. If there ex i s t s a saturated f i n i t e subset of the base, then the age of R s a t i s f i e s the preceding c r i t e r i o n , and there e x i s t s a saturated relation of the same age as R (uses the u l t r a f i l t e r axiom). 0 The empty s e t i s saturated by 7 . 2 ( u l t r a f i l t e r axiom). Let
F be empty in 7.5. Then f o r any f i n i t e subset G of the base, and any maximal (R/G)-age 6 , which specifies the age of R , there ex i s t s a subset G ' which i s the image of G under a local automorphism of R , such t h at the (R/G')-age represented by R i s isomorphic t o I t follows t h at there are countably many maximal rel-ages arising from an a r b i t r ar y f i n i t e r es t r i ct i o n of R : the c rite rion i s s a t i s f i e d . 0
a.
7.8. A CLASSIFICATION OF AGES Given an age d? : i f there e x i s t s no denumerable rich relation f o r & , then there e x i s t s a s t r i c t l y increasing w1-sequence (under embeddability) of denumerable relations having age &, such t h a t these relations cannot be a l l embedded in any single denumerable relation of age @, (uses the axiom of choice and the continuum hypothesis).
-
an U1-sequence of a l l relations
Ri ( i countable ordina l), having age 62 and base the s e t o f natural numbers , f o r example. Let So = Ro ; l e t S1 be a common extension of So and R1 ; then S2 be a common extension of S1 and R2 , e t c . ; where the S ar e denumerable and represent the age bt . For each countable l i m i t ordinal o( , l e t Sd be a denumerable common extension of the Si ( i < o( ) . The Si ( i countable ordinal) can be chosen t o be s t r i c t l y increasing, since there does not e x i s t any rich r el at i o n . 0
0 F i r s t take
Problem. Can the conclusion be strengthened t o "these relations cannot be a l l embedded in any single denumerable relation (possibly having age larger t h a n 0%)". Now suppose t h a t there e x i s t s a denumerable rich relation with age
@
; then:
304
THEORY OF RELATIONS
(1) Either a l l denumerable relations which represent 6% are equimorphic. This i s the case f o r denumerable binary relations always (t), which are a l l isomorphic. This i s the case f o r equivalence relations with denumerably many classes of cardinality 2 , and possibly classes of cardinality 1 . ( 2 ) Or there e x i s t s a s t r i c t l y increasing W -sequence (under embeddability) of denumerable non-rich relations having age ($& , such t h a t every denumerable relation of age & , i n which a l l these relations are embeddable, i s rich f o r @ . This i s the case f o r relations formed of f i n i t e l y many components, each isomorphic w i t h the consecutivity relation on Z . ( 3 ) Or there e x i s t s an W1-sequence satisfying the preceding conditions. This i s the case f o r chains, with the sequence of the denumerable ordinals ( 4 ) Or there e x i s t s a relation which represents 6% and which i s immediately l e s s t h a n the denumerable rich relations f o r & .
Problem 1 in 6 . 4 . Problem 2
. The .
impossibility of case ( 4 ) ; t h i s i s , in other words, the problem 2
The incompatibility of cases ( 2 ) and (3) in the same age.
- FINITIST RELATION
8
F of E , a relation R with base E i s said t o be F - f i n i t i s t , i f f a r every p a i r of elements a , b i n E-F , the transposition ( a , b ) i s an autonorphism of R . A relation i s said t o be f i n i t i s t , i f there e x i s t s an F f o r which i t i s F - f i n i t i s t .
Given an i n f i n i t e s e t
E
and a f i n i t e subset
To say t h a t R i s F - f i n i t i s t i s equivalent t o saying t h a t every oermutation of the base E which i s the identity on F , i s an automorphism of R This follows immediately from ch.9 5 1 . 2 . Consequently, t o say t h a t R i s F - f i n i t i s t i s equivalent t o saying t h a t R i s freely interpretable (ch.9 5 2 ) in the sequence of unary r e l a t i o n s , each of which takes the value (+) on one element of F (singleton r e l a t i o n s ) . For example, a unary relation i s f i n i t i s t i f f i t takes the value (+) on a f i n i t e subset of the base, or on the complement of a f i n i t e subset.
.
8.1. KERNEL OF A FINITIST RELATION
If
is 0
R
&
F - f i n i t i s t and
G-finitist
(F, G
f i n i t e subsets of the base),
then
R
(Fn G)-finitist. Take any two elements a , b i n E - (FnG) , then an element c in E - ( F u G ) ,
and replace the transposition
(a,b)
by the composition
(a,c).(b,c).(a,c). 0
Consequently, i f R i s f i n i t i s t , then there e x i s t s a minimum f i n i t e s e t F (with respect t o inclusion) f o r which R i s F - f i n i t i s t . We c a l l t h i s F the kernel of R ,
Chapter 10
A relation
5
i s constant (see ch.9
R
305
4) i f f
R
i s f i n i t i s t w i t h empty k e r n e l .
.
I f u belongs t o the kernel o f R , y e t n o t v , then (u,v) modifies R I f two f i n i t i s t n-ary r e l a t i o n s w i t h t h e same base, have t h e same kernel and t h e same r e s t r i c t i o n t o the kernel p l u s
n
elements, then they are i d e n t i c a l . Indeed n-element subset.
they have the same r e s t r i c t i o n t o each
8.2. L e t If
R
R
be a r e l a t i o n w i t h denumerable base
.
E
i s f i n i t i s t w i t h a non-empty kernel, then t h e r e e x i s t denumerably many
isomorphic copies of
R
w i t h base
then a l l isomorphic copies of
R
.
E
If
has empty kernel, hence i s constant,
R
w i t h base
E
are i d e n t i c a l t o
. If
R
f i n i t i s t , then there e x i s t continuum many isomorphic copies w i t h base
R -1
E .
0 The two f i r s t cases immediately f o l l o w from precedinq remarks.
Suppose t h a t
R
i s n o t f i n i t i s t . Then f o r each f i n i t e subset
e x i s t s a p a i r o f elements fies n
xo, yo
restriction
(xo,yo)
.
ments
xl,
in
R/F1
and nor
y1
Fo
F1-Fo
, such
. Yet
(xl,yl)
belong t o
y1
image o f
R R
Fo
(n+l)-element s e t transposition R
preserves Ri1}
= R
F1
(xo,yo)
and
R,o)
of
Po
and
and a p a i r o f elements
, since
Rto)/Fo
(xl,yl)
(xl,yl)
xi,
yi
in
-
Fi
Fi-l
i
.
,a
and
R40,1)
4
t i o n o f a l l transpositions
I ) ; the
I
,
(xi,yi)
Fi
.
F1
including
Fi-l
F i n a l l y f o r each countable s e t RI
f o r d i f f e r e n t sets
R
neither
; then these
f i n i t e set
I o f i n t e g e r i n d i c e s , we o b t a i n an isomorphic copy
8.3.
= image o f
and a p a i r o f e l e -
a r e d i s t i n c t , and even they y i e l d 4 d i s t i n c t r e s t r i c t i o n s t o
I t e r a t i n g t h i s , we obtain, f o r each i n t e g e r
modi-
modifies both
under
R
(x,y)
modifies t h e
(xl,yl)
R/FO and
= image o f
under the composition o f
, there
E
Fo and a p a i r o f
(xo,yo)
t h a t the transposition
. Call
F of
t h a t the t r a n s p o s i t i o n
(empty 1 Now there e x i s t s a f i n i t e superset
Rlo)/F1
images o f
, such t h a t t h e see ch.9 5 1.3. C a l l
in
R/FO :
under
=
, such
E-F
be the a r i t y ; then there e x i s t s a
elements
x1
in
.
R
Let
x, y
( i belonging t o
of
t h i s gives continuum many copies o f
R
under t h e composiRI
R
being d i s t i n c t
.0
CHARACTERIZATION OF THE FINITIST RELATIONS
If R
i s a r e l a t i o n w i t h i n f i n i t e base
E
, then
r elements, i f f t h e r e e x i s t a t most
o f a t most
o f which has a t l e a s t r+l elements modify R
y
R
r
i s f i n i t i s t w i t h a kernel elements
x
ik
E
such t h a t t h e t r a n s p o s i t i o n s
, each (x,y)
.
13 I f
R
i s f i n i t i s t w i t h a kernel of c a r d i n a l i t y l e s s than o r equal t o
our c o n d i t i o n i s obvious. Converse1y;suopose i s infinite, l e t those
v
u be an element which i s d i s t i n c t from the x
such t h a t
(u,v)
modifies
r
, then
t h a t our c o n d i t i o n holds. Since
R : t h e r e are a t most
.
Associate t o
r many such
E
u
v ;let
306
THEORY OF RELATIONS
be the s e t of such v . For any two elements x , y in E-F , the t r a n s p o s i t i o n s and ( u , y ) , hence a l s o ( x , y ) preserves R . Thus R i s f i n i t i s t and i t s (UJ) kerne i s included i n F . 0 F
8.4. Let R be a f i n i t i s t n-ary r e l a t i o n w i t h a kernel of c a r d i n a l i t y r , and l e t S be another n-ary r e l a t i o n . I f every r e s t r i c t i o n of S w i t h c a r d i n a l i t y l e s s than o r equal t o ( n + l ) ( r + 1 ) 2i s embeddable i n R , then S i s f i n i t i s t and i t s kernel has c a r d i n a l i t y l e s s than o r equal t o r
.
0 Suppose t h a t
S does not s a t i s f y our conclusion. Then by the preceding proposit i o n , t h e r e exist a t l e a s t r + l elements u , each of which has a t l e a s t r t l elements v such t h a t ( u , v ) modifies S There a r e a t most ( r t l ) ’ many such t r a n s p o s i t i o n s ; and f o r each t r a n s p o s i t i o n , t h e r e e x i s t s a s e t H of n t l e l e ments, including u and v , such t h a t ( u , v ) modifies t h e r e s t r i c t i o n S/H ( s e e ch.9 5 1 . 3 ) . This property i s preserved when taking an isomorphism on a r e s t r i c t i o n of R T h u s by t h e preceding p r o p o s i t i o n , R i s not f i n i t i s t , o r i f so, has a kernel w i t h c a r d i n a l i t y s t r i c t l y g r e a t e r than r . 0
.
.
8 . 5 . CHARACTERIZATION BY (1,p)-EQUIVALENCE
We say t h a t two r e l a t i o n s a r e ( 1 , p ) - e q u i v a l e n t , where p i s a natural number, i f f t h e y have the same r e s t r i c t i o n s t o p elements, up t o isomorphism. We have already defined (1,p)-isomorphism in 1.8 above. Then two r e l a t i o n s a r e (1,p)-equivalent i f f the empty function i s a (1,o)-isomorphism from one r e l a t i o n i n t o the o t h e r .
<
If a r e l a t i o n R I s f i n i t i s t , then t h e r e e x i s t s an i n t e g e r p such t h a t every r e l a t i o n w i t h t h e same base and which i s (1,p)-equivalent t o R , i s isomorphic to R . Note t h a t t h e minimum value of p can be s t r i c t l y g r e a t e r than t h e c a r d i n a l i t y ~
~
-
of the kernel. For example i f R i s c o n s t a n t , hence w i t h empty k e r n e l , t h e m i n i p i s the a r i t y of R , by ch.9 § 4.1. The converse of our proposition w i l l be proved i n 9.10 below.
mum value of
.
Let n be t h e a r i t y and r t h e c a r d i n a l i t y of t h e kernel of R Consider a r e l a t i o n S w i t h t h e same base E as R and which i s (1,p)-equivalent w i t h R , where p = ( n + l ) ( r + l ) ’ By t h e preceding proposition, S i s f i n i t i s t and i t s kernel has c a r d i n a l i t y l e s s than o r equal t o r Now by interchanging R and S , we see t h a t the kernel of S has c a r d i n a l i t y e x a c t l y r . There e x i s t s an isomorphism f which embeds i n t o S t h e r e s t r i c t i o n of R t o
0
.
.
an a r b i t r a r y p-element subset including the kernel. For a c e r t a i n choice of t h i s p-element subset, f takes t h e kernel of R i n t o the kernel o f S : take each element x of t h e kernel of R , then f o r each x take r + l elements y such
Chapter 10
that
(x,y)
tion of
modifies to
R
n+l
transposition phism
kernel o f
, t h e n f o r each such o r d e r e d p a i r (x,y)
R
elements, i n c l u d i n g
.
(x,y)
, since
f
p
307
x
and
take a r e s t r i c -
i s m o d i f i e d by t h e
Then t h e s e c o n d i t i o n s h o l d as w e l l f o r
by t h e isomor-
S
i s s u f f i c i e n t l y l a r g e . So t h a t t h e images
fx
constitute the
.
S
f
F i n a l l y extend
t o a permutation o f
, our
E
,still
a t l e a s t equal t o
rtn
same k e r n e l as
and t h e same r e s t r i c t i o n s as
S
by 8.1 above, t h i s image o f
5 9 - ALMOST
permutation under
R
f
is
f
denoted by
takes S
f
.
Since
p
is
i n t o a r e l a t i o n having the
R
t o i t s kernel plus
n
elements:
itself. 0
S
CHAINABLE RELATION
E , a f i n i t e subset F and a c h a i n A w i t h base E-F , a r e l a t i o n
Given a s e t w i t h base
R
, which
y
i s s a i d t o be
E
(F,A)-chainable,
u n i o n o f t h e i d e n t i t y on a subset o f l o c a l automorphism o f We say t h a t
R
is
i f each b i j e c t i o n which i s t h e
and a l o c a l automorphism o f
F
A
, is
a
5
5.
.
R
F-chainable, o r i s almost c h a i n a b l e .
I n t h e case where
F
i s empty, we
9.1. The c o n s i d e r e d r e l a t i o n
R
f i n d t h e c h a i n a b l e r e l a t i o n : see ch.9
is
F-chainable i f f
R
i s freely interpretable
i n t h e m u l t i r e l a t i o n formed o f t h e s i n g l e t o n unary r e l a t i o n s o f each element i n F
, and
o f any t o t a l l y o r d e r e d e x t e n s i o n o f
A
f o r which
F
i s e i t h e r an i n i t i a l
i n t e r v a l , o r a f i n a l i n t e r v a l , o r the union o f both.
-
Consequently, i f
in R
&
R
F-chainable, t h e n e v e r y r e l a t i o n f r e e l y i n t e r p r e t a b l e
F-chainable.
R
is
F-chainable i f f t h e r e e x i s t s a c h a i n
t h a t the b i r e l a t i o n
&
9.2. _ I f R
RA
is
A
w i t h t h e same base as
R
, such
F-chainable.
F-chainable, t h e n f o r each subset
D
o f t h e base, t h e r e s t r i c t i o n
( F n D)-chainable.
R/D
I n particular, i f
R
w i t h base
E
is
F-chainable, t h e n t h e r e s t r i c t i o n
R/(E-F)
i s c h a i n a b l e , which f o l l o w s i m m e d i a t e l y f r o m t h e d e f i n i t i o n . The converse i s f a l s e . 0 Take t h e c h a i n
Z(0,O)
=
-
Z of P o s i t i v e and n e g a t i v e i n t e g e r s , m o d i f i e d by t h e c o n d i t i o n
(instead o f +). This modified Z
i s n o t almost c h a i n a b l e , s i n c e i t s
p r o f i l e i s n o t bounded: see 9.8 below. 9.3.
Let
subset
be a r e l a t i o n and
R X
F
a f i n i t e subset o f i t s base. I f f o r each f i n i t e
o f t h e base which i n c l u d e s
then R &
F
, the
restriction
F-chainable (uses t h e u l t r a f i l t e r axiom).
R/X
F-chainable,
308
THEORY OF RELATIONS
X including F , associate the s e t Ux of chains Y with base X-F such t h a t R/X i s (F,Y)-chainable. By hypothesis Ux i s nonempty f o r each X ; and i f X ' i s included i n X , then every chain belonging t o Ux , when r e s t r i c t e d t o X ' , yields a chain belonging t o U x , . By the coherence lemma (ch.2 0 1.3, equivalent t o the u l t r a f i l t e r axiom), there e x i s t s a chain A with base R -F , f o r which R i s (F,A)-chainable. 0
0 To each f i n i t e subset
9 . 4 . Let R be an i n f i n i t e relation which i s F-chainable, and R ' the r e s t r i c . tion of R t o an i n f i n i t e subset including F , and G be a subset of F . Now if R' i s G-chainable, then R i s G-chainable (uses the u l t r a f i l t e r axiom).
E denote the X of E including X into a subset X ' G-chainable. Letting nable. 0
0 Let
.
base of R and E ' the base of R ' For each f i n i t e subset F , there e x i s t s a bijection which i s F-identical and takes of E ' and R/X into R / X ' Now R / X ' , thus R/X , is
.
X
vary, the preceding proposition shows t h a t
R
is
G-chai-
9.5. KERNEL OF AN ALMOST CHAINABLE RELATION Let R be an i n f i n i t e relation and F , G be two f i n i t e subsets of the base I R I . If R F-chainable and G-chainable, then R (FnG)-chainable (uses the u l t r a f i l t e r axiom). 0 Let E denote the base of R The r e s t r i c t i o n R/(E-G)uF i s (FnG)-chainable by 9.2. Then R i s (FAG)-chainable by the preceding proposition. 0
-
&
.
Consequently, i f R i s almost chainable, then there e x i s t s a minimum f i n i t e s e t (with respect t o inclusion) f o r which R i s F-chainable. We call t h i s F the kernel of R . 9.6. Let
R
be an almost chainable relation. Then
every R '
F
which i s younger than
R i s almost chainable, and the cardinality of the kernel of R ' i s less than or equal t o the cardinality of the kernel of R (uses the u l t r a f i l t e r axiom). E, E ' denote respectively the bases of R , R ' . Let F be the kernel of 9.1, there e x i s t s a chain A with base E , such t h a t i f f o r each element a of F , we denote by Ua the singleton unary relation of a , then R i s freely interpretable in (A.U ,,... ) . By 1.3'above ( u l t r a f i l t e r axiom), there e x i s t s a multirelation formed of a binary relation A ' and of unary relations U i such t h a t ,...) i s younger t h a n ( R , A , U a , ...) . the concatenation The relation A ' i s a chain, since i t i s younger t h a n A Each i s either a unary relation always ( - ) , or a singleton unary relation of an element a ' of E ' . Let F ' denote the s e t of these a ' ; hence F ' i s a subset of E ' and Card F ' ,< Card F . Moreover we can assume t h a t F i s an i n i t i a l interval of A , and so
0 Let
R
. By
(R',A',Ui
.
Ui
309
Chapter 10
F'
A'
i s an i n i t i a l i n t e r v a l o f
Finally
.
i s freely interpretable i n
R'
9.7. L e t
.
E
of
that 0
R
be a r e l a t i o n w i t h denumerable base
&
F-chainable.
Take a c h a i n
w i t h base
A
E
, which 5
n
of
X
0
is
, including
E
F
and such
5.5.(1).
begins by t h e i n i t i a l i n t e r v a l F
and
F
, and
(R,A)/X
and
(R,A)/X'
X, X '
are
, assumed t o be > 2 , t h e n i t o f c a r d i n a l i t y (Card F) t 2n , i n o r d e r t o i n s u r e
designates the a r i t y o f
s u f f i c e s t o consider sets
R'
1.1); by c o n s i d e r i n g two f i n i t e subsets
o f t h e base as e q u i v a l e n t , i f b o t h i n c l u d e F-isomorphic. I f
2.4. Hence
, and F be a f i n i t e subset
E D
G e n e r a l i z a t i o n o f ch.9
t h e n a p p l y RAMSEY's theorem (ch.3
0
ch.9
.0
F'
Then t h e r e e x i s t s a denumerable subset
R/D
, by
...)
(A',U;,
almost c h a i n a b l e w i t h a k e r n e l i n c l u d e d i n
R
F-chainability. 0 9.8. A necessary and s u f f i c i e n t c o n d i t i o n f o r a denumerable r e l a t i o n almost c h a i n a b l e , i s t h a t t h e p r o f i l e o f ch.9 § 7 ; s u f f i c i e n c y 0
Let
uses t h e u l t r a f i l t e r axiom).
be almost c h a i n a b l e , w i t h base
R
and k e r n e l
E
R when r e s t r i c t e d t o p
t h e isomorphism t y p e o f intersection o f
R
be bounded ( f o r t h e p r o f i l e , see
R
F w i t h t h e s e t o f these
p
. F o r each
F
integer p
,
elements, o n l y depends on t h e
elements. Since t h e k e r n e l i s f i n i t e ,
t h e number o f these isomorphism types i s bounded by t h e number o f r e s t r i c t i o n s of
.
R/F
Conversely, l e t
be a r e l a t i o n w i t h denumerable base
R
c h a i n a b l e . To prove t h a t t h e p r o f i l e i s unbounded, l e t we s h a l l c o n s t r u c t
h
restrictions o f
, which i s n o t almost
E
h
be an a r b i t r a r y i n t e g e r ;
R , a l l o f t h e same f i n i t e c a r d i n a l i t y and
m u t u a l l y non-isomorphic. S t a r t w i t h a denumerable, c h a i n a b l e r e s t r i c t i o n The r e l a t i o n
i s n o t younger t h a n
R
Hence t h e r e e x i s t s a f i n i t e subset in
Ro
.
Take a denumerable,
, no
p 3 Card Fo
ger
any r e s t r i c t i o n o f The r e l a t i o n R1
.
to a
F1
of
to a h
to a
Ro
Ro
, are
of
nor than
p
. For every
Ro
R2
of
i s isomorphic w i t h
. Hence R
there exists a
Ro
. For every
with cardinality
Fo
, and
inte-
.
Fo R1
i s n o t embeddable R
i s n e i t h e r embeddable i n
, a restriction o f F1
R/FO
R1
with cardinality
p-element subset i n c l u d i n g h
that
p-element subset i n c l u d i n g
p-element subset i n c l u d i n g
times, we o b t a i n
, such
F1-chainable r e s t r i c t i o n
p 3 Max(Card Fo,Card F1) R1
E
, such t h a t R/F1
E
Take a denumerable,
triction of
of
i s n e i t h e r younger t h a n
R
f i n i t e subset
Fo
(see 9.7 above ) .
R
u s i n g t h e u l t r a f i l t e r axiom.
Fo-chainable r e s t r i c t i o n
restriction o f
R1
of
Ro
Ro : see 9.6,
p
nor i n integer
, and
a restriction o f
m u t u a l l y non-isomorphic.
a resR2
Iterating this
r e s t r i c t i o n s which a r e m u t u a l l y non-isomorphic. 0
THEORY OF RELATIONS
310
9.9.
The f o l l o w i n g remark due t o POUZET, proves t h a t t h e p r o f i l e o f an i n f i n i t e
r e l a t i o n i s i n c r e a s i n g , w i t h o u t u s i n g t h e i n c i d e n c e m a t r i x o r t h e m u l t i c o l o r theorem ( p r o f i l e i n c r e a s e theorem, ch.9 Given a denumerable r e l a t i o n
R
5
t h e base, such t h a t e v e r y r e s t r i c t i o n o f in
R/F
.
Then t a k e an
R
, at
R
5.3).
, take
p
a f i n i t e subset
with cardinality
F-chainable r e s t r i c t i o n o f
t i o n i s t h e same as t h e p r o f i l e o f
5
7.1, u s i n g ch.3
and an i n t e g e r
p
F
of
i s embeddable
R : the p r o f i l e o f t h i s r e s t r i c -
l e a s t up t o t h e v a l u e
p
.
Hence i t
s u f f i c e s t o prove t h a t t h e p r o f i l e i n c r e a s e s i n t h e p a r t i c u l a r case where
R
almost c h a i n a b l e . Now assume to
p
that
R
is
F-chainable. To each isomorphism t y p e o f a r e s t r i c t i o n
elements, a s s o c i a t e a
and t h e i n t e r s e c t i o n
G
associate t o t h i s
G
taking
G with
p-element subset
F nG
H =
t h e isomorphism t y p e o f c a r d i n a l i t y
p l u s an element n o t b e l o n g i n g t o
. Thus
F
t i o n which, t o each isomorphism t y p e o f c a r d i n a l i t y type o f c a r d i n a l i t y
R/G
p+l
having t h i s type,
h a v i n g t h e l e a s t p o s s i b l e c a r d i n a l i t y . Then p+l
, obtained
p
, associates
an isomorphism
.
To see t h e i n j e c t i v i t y : i f t h e same isomorphism t y p e o f c a r d i n a l i t y ned from two e q u i p o t e n t subsets belonging t o
by
we d e f i n e an i n j e c t i v e func-
H, H '
of
F
, each
p+l
i s obtai-
augmented by elements n o t
F , t h e n e v e r y isomorphism o f t h e f i r s t r e s t r i c t i o n o n t o t h e second H i n t o H ' . F o r o t h e r w i s e H ' would n o t be t h e i n t e r s e c t i o n
r e s t r i c t i o n , takes of
F with least possible cardinality.
9.10.
Let
with
We a r e now i n a p o s i t i o n t o prove t h e converse o f 8.5 R
E ; i f e v e r y r e l a t i o n w i t h base E , .-which is . R , R i s finitist.
have base
R
,is
isomorphic w i t h
(1,p)-equivalent
then
~
0 We g i v e an argument f o r t h e case t h a t
E
i s denumerable. Take a r e p r e s e n t a t i v e
R
o f each isomorphism t y p e o f a r e s t r i c t i o n o f
to
6
p
elements, and l e t
F
F i s f i n i t e , by 9.7 which extends R/F and i s
be t h e u n i o n o f t h e bases o f these r e p r e s e n t a t i v e s . Since above, t h e r e e x i s t s a denumerable r e s t r i c t i o n o f
R
F-chainable. By h y p o t h e s i s , t h i s r e s t r i c t i o n i s isomorphic w i t h suppose t h a t Assume t h a t
R
R
R : hence we can
i s almost c h a i n a b l e . is
(F,A)-chainable,
where
A
e i t h e r isomorphic w i t h G, or i s o m o r p h i c w i t h
i s a c h a i n which can a r b i t r a r i l y be Q
. From t h i s
p o i n t on, t h e argu-
ment i n ch.9 e x e r c i s e 2, s l i g h t l y m o d i f i e d t o t a k e account o f t h e k e r n e l included i n transposition
F ) , proves t h a t f o r e v e r y p a i r o f elements (x,y)
preserves
R
. Hence
R
is
x, y
in
E-F
F
(or
, the
F-finitist. 0
EXERCISE 1 - THE EXISTENCE OF A R I C H RELATION I N THE UNCOUNTABLE CASE
Modulo t h e axiom o f c h o i c e and t h e continuum h y p o t h e s i s , we s h a l l p r o v e as f o l l o w s t h e e x i s t e n c e , f o r each i n t e g e r
n
, of
an n-ary r e l a t i o n o f c a r d i n a l i t y W 1
,
31 1
Chapter 10
o r e q u i v a l e n t l y continuum c a r d i n a l i t y , in which every n-ary r e l a t i o n of same card i n a l i t y i s embeddable. Let E be t h e base s e t , which we i d e n t i f y w i t h the ordinal GJ1 . There exist t d l many countable subsets of E , denoted by D i ( i < W 1 ) . To each Di , we a s s o c i a t e a subset Ci of E w i t h c a r d i n a l i t y W 1 , so t h a t a l l these Ci a r e mutually d i s j o i n t . This is p o s s i b l e , s i n c e t h e Cartesian product of W 1 by i t s e l f i s equipotent with w1 . S t a r t w i t h an a r b i t r a r y n-ary r e l a t i o n Ro w i t h base Do . Consider a l l possible manners of extending Ro t o a new element, t h e r e b e i n g a t most continuum many, thus cdl many such manners. To each of these p o s s i b i l i t i e s , a s s o c i a t e an element x of Co , and c o n s t r u c t the corresponding extension on t h e base Do u { x] . This i s p o s s i b l e , s i n c e t h e r e a r e u1many elements i n Co , hence a l s o W 1 many such elements which do not belong t o t h e countable set Do . More g e n e r a l l y l e t u be a countable ordinal index. Suppose t h a t f o r each i < u , we have already defined an n-ary r e l a t i o n Ri cn D i , these Ri being mutually compatible ( i . e . w i t h same r e s t r i c t i o n t o any i n t e r s e c t i o n of t h e i r b a s e s ) . Moreover suppose t h a t by using t h e Ci , we have already ensured a l l possible extensions f o r each Ri t o an additional element. Then define RU on Du , i n an a r b i t r a r y manner with t h e only requirement of c o m p a t i b i l i t y w i t h preceding Ri and t h e i r extensions. Now consider a l l possible extensions of R u t o an additional element. For each p o s s i b i l i t y , use an element x i n Cu . This i s p o s s i b l e ; indeed a l l the n-tuples t o which a value (+) o r ( - ) has already been assigned, have a l l their terms belonging e i t h e r t o a Di o r t o t h e corresponding C i ( i < u) Yet Cu i s d i s j o i n t from these Ci , and t h e union of t h e s e D i i s countable. So t h a t t h e r e remain G) many elements x i n Cu , such t h a t no value has been assigned t o any n-tuple having x among i t s terms. F i n a l l y we obtain on t h e e n t i r e base E , an n-ary r e l a t i o n R such t h a t each countable r e s t r i c t i o n of R i s a r b i t r a r i l y e x t e n d i b l e t o i t s base plus an a d d i t i o nal element: thus every n-ary r e l a t i o n w i t h c a r d i n a l i t y W1 i s embeddable in i t .
.
EXERCISE 2 - FINITIST STRUCTURE AND FRAENKEL-MOSTOWSKI MODEL, IN CONNECTION WITH A FUNCTION WHOSE DOMAIN IS STRICTLY SUBPOTENT WITH ITS RANGE 1 - S t a r t w i t h the set of natural numbers, which we consider a s urelements, i . e . as copies of t h e empty s e t , each having no element. On t h e set N o f these i n t e g e r s , consider t h e s e t of f i n i t i s t r e l a t i o n s , i n t h e sense of 5 8 above. More g e n e r a l l y , t o each ordinal o( , a s s o c i a t e the set of f i n i t i s t s t r u c t u r e s with rank 4 on N , defined a s follows by induction. Each element o f ti , or urelement, i s a s t r u c t u r e w i t h rank 0 . A s t r u c t u r e of rank o( >/ 1 i s a s e t of f i n i t i s t s t r u c t u r e s of ranks s t r i c t l y l e s s than o( , such t h a t t h e r e e x i s t s a f i n i t e s u b s e t F of N , w i t h the condition t h a t every
312
THEORY OF RELATIONS
.
permutation of N-F preserves the s e t or structure of rank o( The reader can verify t h a t these f i n i t i s t s;ructures constitute a mcdel of FRAENKEL-MOSTOWSKI's s e t theory: see FRAiSSE 1958. More precisely t h i s model s a t i s f i e s the axioms of ZF excepting extensionality, which must be weakened as follows: any two non-empty s e t s which have the same elements are identical. One can easily transform the model so as t o s a t i s f y the f u l l extensionality axiom, b u t while abandonning the foundation axiom. Indeed i t suffices t o consider each urelement a as equal t o the singleton of a i t s e l f . In t h i s case, we add the empty s e t t o the elements of N , and we give t o each of them the rank 0 , the rank 1 being given only t o those s e t s which contain several elements of N o r one element of N plus the empty s e t . 2 - Construct the s e t o f words, or f i n i t e sequences without repetition on the s e t N of the urelements; note t h a t t h i s s e t i s f i n i t i s t with F empty. Consider the function f which t o each non-empty word u without repetition associates the word obtained from u by removing i t s l a s t term. This function i s a f i n i t i s t s t r u c t u r e , with F empty. Dom f i s the s e t of non-empty words, Rng f i s the s e t of a l l words (without r e p e t i t i o n ) , including the empty sequence. W e shall prove t h a t in the considered model, Dom f i s s t r i c t l y subpotent with Rng f (example communicated by H O D G E S ) . 0 Indeed a bijection from Rng f o n t o Dom f cannot be f i n i t i s t . For such a bijection h , l e t t i n g 0 denote the empty sequence, consider the a - s e q u e n c e 2 of successive i t e r a t e s 0 , h ( O ) , h ( 0 ) = h ( h ( O ) ) , ... . If t h i s w-sequence were f i n i t i s t , with a f i n i t e subset F of N , then a l l i t s terms would be sequences of elements of F . Since h i s a bijection, these terms are a l l d i s t i n c t , so t h a t F should be i n f i n i t e : contradiction. 0 Problem. Is i t possible t o generalize t o structures the lemmas in ch.9 0 1 . 2 and 1.3, so t h a t we could define a f i n i t i s t structure by using only transpositions instead of general permutations of N-F . Consequently the intersection of two f i n i t e subsets F should be an F , so t h a t each f i n i t i s t structure should have a minimum F called i t s kernel.
313
11
CHAPTER
HOMOGENEOUS RELATION, RELATIONAL SYSTEM, CONNECTION WITH PERMUTATION GROUPS, ORBIT
§
1 - HOMOGENEOUS
Let
p
AMALGAMABLE SET AND AMALGAMABLE AGE
RELATION;
be a n a t u r a l number ; a r e l a t i o n
e v e r y l o c a l automorphism o f
R
morphism o f
.
Every r e l a t i o n i s
i s s a i d t o be
R
d e f i n e d on
R
p
o-homogeneous, if
elements i s e x t e n d i b l e t o an auto-
0-homogeneous, s i n c e t h e empty f u n c t i o n i s
e x t e n d i b l e t o t h e i d e n t i t y mapping on t h e e n t i r e base.
( S p)-homogeneous r e l a t i o n . i s s a i d t o be homogeneous, i f R i s p-homogeneous f o r e v e r y natu-
Obvious d e f i n i t i o n o f a A relation
R
.
r a l number p Example. A b i n a r y c y c l e on a t l e a s t 5 elements
is
1-homogeneous b u t n o t
2-homo-
f o u r c o n s e c u t i v e elements; t h e n t h e mapping which t a k e s a,b,c,d a,c i n t o a,d i s a l o c a l automorphism and i s n o t e x t e n d i b l e . The c h a i n on 2 elements i s n o t 1-homogeneous, b u t i t i s 2-homogeneous, s i n c e geneous: c a l l
t h e o n l y l o c a l automorphism on 2 elements i s t h e i d e n t i t y t r a n s f o r m a t i o n . The f o l l o w i n g b i n a r y m u l t i r e l a t i o n ( w i t h b i n a r y and unary component r e l a t i o n s ) 2-homogeneous y e t n o t
3-homogeneous.
0 S t a r t with the binary r e l a t i o n
.
&
R
g i v e n i n ch.9 exerc.1, on t h e 10 elements
f o r each ordered A such t h a t A(x,y) = + e x a c t l y f o r those o r d e r e d p a i r s (x,y) which a r e an image o f (a,s) under one o f t h e t h r e e f o l l o w i n g p e r m u t a t i o n s : ( 1 ) t h e mapDing which preserves a and c
a,b,c,d,r,s,t,u,i p a i r , say
,j
(a,s)
d
2-homogeneity:
f o r i n s t a n c e , add a b i n a r y r e l a t i o n
and i n t e r c h a n g e s and
I n o r d e r t o ensure t h e
(b,d),
and i n t e r c h a n g e s
(r,u), (s,t), (a,c),
(r,s),
mappings (1) and ( 2 ) , which p r e s e r v e s
( i , j ) ; ( 2 ) t h e mapping which preserves
(t,u),
b
( i , j ); ( 3 ) t h e c o m p o s i t i o n o f
i and j
and i n t e r c h a n g e s
(a,c),
(b,d),
(r,t), ( s , ~ ) . Here t h e images o f (a,s) a r e ( a , t ) , ( c , r ) , (c,u) and o b v i o u s l y i t s e l f , t h e i d e n t i t y p e r m u t a t i o n b e i n g added t o ( 1 ) , ( 2 ) , ( 3 ) . (a,s) The o b t a i n e d m u l t i r e l a t i o n i s n o t 3-homogeneous: as a l r e a d y mentioned i n t h e exerc i s e , t h e l o c a l automorphism which p r e s e r v e s
a
and
b
and i n t e r c h a n g e s
(i,j)
i s n o t e x t e n d i b l e t o an automorphism on t h e e n t i r e base. 0 The p r e c e d i n g example i s n o t
1-homogeneous: f o r examule map
a
into
b ;y e t
i t becomes
1-homogeneous i f we add, f o r i n s t a n c e , a unary r e l a t i o n t a k i n g value
(+) f o r
and
a
c
, and
a n o t h e r t a k i n g (+) f o r
i and j
.
THEORY OF RELATIONS
314
Example of homogeneous r e l a t i o n s . The r e l a t i o n always (+); another example, the chain Q of the r a t i o n a l s . Another homogeneous r e l a t i o n . The equivalence r e l a t i o n w i t h f i n i t e l y many, o r w i t h In c o n t r a s t with t h e two preceding denumerably many c l a s s e s having c a r d i n a l i t y 2 examples, here homogeneity does not s u b s i s t when we remove an element from the base. Problem. For each a r i t y n , does t h e r e e x i s t a threshold s ( n ) above which every n-ary r e l a t i o n which i s ( 4 s(n))-homogeneous, i s homogeneous.
.
1.1. Let E be a denumerable s e t . A r e l a t i o n R with base E i s homogeneous i f f , E , f o r any l o c a l automorphism f R w i t h domain f o r any f i n i t e subset F F , and f o r any element u iJ E-F , t h e r e e x i s t s a local automorphism of R which extends f t o the domain F U {u 1 .
0-f
of
R i s homogeneous, then i t obviously s a t i s f i e s our condition. Conversely, suppose t h a t t h e condition holds. Enumerate the elements of E as a i ( i i n t e g e r ) . S t a r t w i t h an a r b i t r a r y local automorphism f o of R , with f i n i te domain. Add a. t o t h e domain, i f i t does not y e t belong t o this domain, thus obtaining a local automorphism f i extending f o . S i m i l a r l y add a. t o the range of f b , i f i t does not y e t belong t o t h i s range, thus obtaining a local automorphism f l extending f; , t h u s extending f o I t e r a t e t h i s , going from f i t o f i + l ( i i n t e g e r ) by adding a i t o t h e domain and then t o t h e range. F i n a l l y the common extension of these f i i s an automorphism of R 0
0 If
.
.
This proposition does not extend t o t h e case of any uncountable r e l a t i o n . Consider t h e chain Q+R , where Q i s t h e chain of r a t i o n a l s and R i s isomorphic t o t h e chain of r e a l s . Then t h e condition i n our preceding statement i s sat i s f i e d ; y e t mapping an element of the i n i t i a l i n t e r v a l Q i n t o an element of the f i n a l i n t e r v a l R , we cannot extend t h i s l o c a l automorphism t o the e n t i r e base. 0
0
1 . 2 . Note t h a t the r i c h denumerable r e l a t i o n R defined in ch.10 5 4 . 1 s a t i s f i e s t h e following condition. Each f i n i t e r e s t r i c t i o n of R i s a r b i t r a r i l y extendible t o i t s base plus an additional element, which we can f i n d i n t h e base of R We immediately see t h a t R s a t i s f i e s t h e condition i n our preceding statement. Hence f o r each i n t e g e r n , t h e r e e x i s t s a denumerable n-ary r e l a t i o n which i s r i c h and homogeneous.
.
1.3.(1) Any two denumerable homogeneous r e l a t i o n s , with same a r i t y and same age, a r e isomorphic. ( 2 ) Let R be a denumerable homogeneous r e l a t i o n . Then every denumerable r e l a t i o n which i s younger than R , i s embeddable i n R . 0 (1) Let
R and
R'
be denumerable homogeneous r e l a t i o n s w i t h r e s p e c t i v e bases
315
Chapter 1 1
E and E ' , and the same age. Enumerate the elements of E as ai , the elements of E ' as a; ( i i n teg er ) . Embed the r es t r i ct i o n R/{ao) into R ' , thus obtaining a local isomorphism f o from R i n t o R ' with domain { ao\ . Augment the range of f o by adding ah . By the preceding proposition, f o i s extendible t o a local isomorphism go from R i n t o R ' , whose domain contains a. and whose range contains ah . I t er at i n g t h i s f o r each i , we obtain a local isomorohism f i whose domain contains the elements ao, ...,ai and which i s extendible t o a local The union of the isomorphism gi whose range contains the elements a h , . . . ,a! f i (or equivalently the gi ) i s an isomorphism from R onto R ' . 0
.
( 2 ) Same argument, b u t only in one direction, using the ocal isomorphisms from the younger relation i n t o the homogeneous relation R 0
0
.
Consequently, f o r the age of chains, the chain Q of the denumerable homogeneous chain, u p t o isomorphism.
ationals i s the only
1.4. If we remove an element from the base of a denumerable relation which i s rich and homogeneous, then we obtain an isomorphic relation. Similarly i f we change the value of an a r b i t r a r y n - t u p l e ( n = a r i t y ) 0 Indeed, the f i r s t and second operations both preserve age, since t h i s age i s constituted by a l l f i n i t e n-ary relations ( u p t o isomorphism). Moreover the condition t h a t each f i n i t e r es t r i ct i o n be a r b i t r a r i l y extendible, i s preserved. 0
.
Many other relations are isomorphic t o t h e i r r es t r i ction a f t e r removing an arbitrary element. For example the chain Q (which i s homogeneous), the chain W (which i s not homogeneous). We have already seen t h at the equivalence relation with classes of cardinality 2 , i s homogeneous b u t n o t isomorphic with such a re stric tion. Problem. Let R be a denumerable n-ary r el at i o n , which i s isomorphic with any relation obtained by changing the value of an ar b i t ra ry n-tuple; then i s R an homogeneous r e l a t i o n . Note t h at necessarily every n-ary f i n i t e relation i s embeddable in R : i f R i s homogeneous, then i t i s rich. 1.5. AMALGAMABLE SET, AMALGAMABLE AGE A s e t 6;L of f i n i t e relations of the same a r i t y i s said t o be amalgamable i f , for any relations A , B, C belonging to'R, f o r any isomorphism f from A onto a r e s t r i c t i o n of B and any isomorphism g from A onto a re stric tion of C , there e x i s t s a relation D belonging t o 61. , an isomorphism f ' from B onto a r e s t r i c t i o n of D and an isomorphism g ' from C onto a re stric tion of D , such t h a t f o r each element x of the base I A I , we have ( f ' , f ) x = ( g ' o g ) x Obviously the relations in the considered s e t are defined up t o isomorphism. A s e t (R of f i n i t e relations i s said t o be strongly amalgamable i f , f o r any relations B, C belonging t o 6!,and C ' isomorphic w i t h C , with a common
.
316
THEORY OF RELATIONS
r e s t r i c t i o n t o the intersection of the bases l B l and I C ' l , then there e x i s t s a relation D in &, ( u p t o isomorphism), which i s a common extension of B and C ' . For example, the age of a l l f i n i t e chains, and also the age of a l l f i n i t e p a r t i a l orderings, are strongly amalgamable: see ch.2 fj 2 . 2 and 2.3. The age of those f i n i t e unary relations which take the value (+) f o r a t most one element, i s amalgamable y e t n o t strongly amalgamable. The age of a l l f i n i t e trees i s not amalgamable: the example given in ch.2 fj 2.3 contradicts ordinary, as well as strong amalgamability. 1.6. AMALGAMATION THEOREM Given an age 61 , there e x i s t s a countable homogeneous representative of d?, i s amalgamable.
&
iff
Let R be a countable homogeneous relation. Let A , B be two f i n i t e r e s t r i c tions of R , where B i s an extension of A , and l e t g be an isomorphism from A into a t h i r d f i n i t e r e s t r i c t i o n C of R . Then by hypothesis g i s extendible t o an automorphism of R . Let f ' denote the r e s t r i c t i o n of t h i s automorphism t o the base I B I . Let f be the identity on I A l , l e t g ' be the identity on I C l and f i n a l l y l e t 0 be the r e s t r i c t i o n R / ( [ C I y ( f ' ) ' ( I B l ) ) We thus have the condition of amalgamation. Conversely, l e t @, be an amalgamable age. Let B be a f i n i t e relation belonging t o & , and f a local automorphism of B . Let A denote the r e s t r i c t i o n of B t o Dom f . W e shall construct an extension D of B , s t i l l belonging t o bt , which has a local automorphism extending f t o the domain B . For t h i s , l e t C = B , and denote by g the identity on I A I = Dom f . Now using the amalgamabil i t y condition, obtain an extension 0 of B with an isomorphism g ' from B onto a r e s t r i c t i o n of D , such t h a t f o r every x in Dom f , we have g ' x = ( g ' , g ) x = ( f ' , f ) x = fx : so t h a t g ' extends f . S t a r t with an w -sequence of f i n i t e relations Ai ( i i n t e g e r ) , the s e t of whose r e s t r i c t i o n s gives the amalgamable age &. Let Bo = A. Using the preceding, replace Bo by i t s extension B1 belonging t o & , such t h a t every local automorphism of Bo (there are only f i n i t e l y many) has an extension which i s a local automorphism of B1 and whose domain i s Bo Moreover we require t h a t A1 be embeddable in B1 , which i s possible since every age i s directed. , such t h a t Iterating t h i s , we obtain an 0 -sequence of elements Bi of each Bi+l extends B i , and Ai i s embeddable in Bi f o r each integer i Take the common extension R of the Bi t o the union of bases. Then every local automorphism of R with f i n i t e domain i s extendible, with alternatively a domain and a range containing every element in every base I S i ( : the homogeneity i s proved. 0
0
.
.
.
61.
.
Chapter 11
317
5
1.7. Given an amalqamable aqe, t h e c r i t e r i o n o f ch.10
7.6 i s s a t i s f i e d , and
unique denumerable homogeneou? r e p r e s e n t a t i v e o f o u r a q t (up t o isomorphism) i s u e s a t u r a t e d r e l a t i o n o f ch.10
5
7.3 and 7.6.
R be a denumerable homogeneous r e l a t i o n , S a r e l a t i o n r e o r e s e n t i n g t h e same age, and f a l o c a l isomorphism f r o m R i n t o S , w i t h f i n i t e domain F
0
Let
.
Take an a r b i t r a r y f i n i t e s u p e r s e t from
onto a r e s t r i c t i o n o f
S/G
phism
h,f
kh,-'
i s an isomorphism f r o m
G R
of
.
Since
There e x i s t s an isomorphism h
i s homogeneous, t h e l o c a l automor-
R
i s e x t e n d i b l e t o an automorphism
Replacing G
.
fa(F) k
of
R
. Then
onto a r e s t r i c t i o n o f
S/G
k-l0h
is a
(see t h e d e f i n i t i o n s i n ch.10
1-morphism f r o m
5
S
into
R
extends
.
f-l
IS1 ) , t h e same argu-
b y any f i n i t e s u o e r s e t ( i n c l u d e d i n t h e base
ment shows t h a t
t h e composition
, and
R
. Hence
R
i s saturated
7 and 7.3). 0
1.8. I t i s proved by HENSON 1972, t h a t t h e r e e x i s t continuum many m u t u a l l y nonisomorphic denumerable homogeneous r e l a t i o n s . Hence continuum many amalgamable ages. F o r o t h e r r e s u l t s on homogeneous r e l a t i o n s , see JONSSON 1965 and CALAIS 1967.
2 - RELATIONAL SYSTEM: ORBIT: ADHERENCE OF A PERMUTATION GROUP: THEOREM OF THE I N C R E A S I N G NUMBER OF O R B I T S (LIVINGSTONE,WAGNER) 2.1. RELATIONAL SYSTEM Given a s e t
E
, we
say t h a t a r e l a t i o n a l system w i t h base
i s any o r d i n a l sequence whose terms a r e r e l a t i o n s Letting
ni
denote t h e a r i t y o f
s a i d t o be t h e
Ri
, the
Ri
E
, or
based on
( i o r d i n a l ) based on
o r d i n a l sequence o f i n t e g e r s
arity o f t h e system, and each
Ri
ni
,
E
.
E
is
i s s a i d t o be a component o f t h e
system. By t a k i n g a f i n i t e sequence o f r e l a t i o n s based on based on
E
, we
f i n d the m u l t i r e l a t i o n
E (see ch.2 § 1).
The n o t i o n s o f r e s t r i c t i o n o f a system t o a subset o f t h e base, e x t e n s i o n t o a s u p e r s e t of t h e base, isomorphism, automorphism, l o c a l isomorphism o r automorphism, a l l e x t e n d immediately t o t h e case o f
r e l a t i o n a l systems.
However, we have an i m p o r t a n t d i f f e r e n c e which p r o h i b i t s c e r t a i n g e n e r a l i z a t i o n s . Indeed t h e number o f r e l a t i o n a l systems o f a g i v e n a r i t y and o f a g i v e n f i n i t e base, i s i n general i n f i n i t e . I n p a r t i c u l a r RAMSEY's theorem, used when p a r t i t i o ning the
p-element subsets o f t h e base i n t o a f i n i t e number o f classes, c o r r e s -
ponding t o d i f f e r e n t isomorphism types, can no l o n g e r be used s y s t e m a t i c a l l y . The same remark h o l d s f o r t h e coherence lemma. p-HOMOGENEOUS SYSTEM, HOMOGENEOUS SYSTEM The n o t i o n o f homogeneity i n t r o d u c e d i n 5 1 above, can be extended t o r e l a t i o n a l systems. A system R i s s a i d t o be p-homogeneous, i f e v e r y l o c a l automorphism
THEORY OF RELATIONS
318
of
, d e f i n e d on p elements, i s e x t e n d i b l e t o an automorphism o f
R
.
R
A system i s s a i d t o be homogeneous, i f i t i s p-homogeneous f o r e v e r y i n t e g e r
2.2. ORBIT OF AN Let
n-TUPLE, OR OF AN
be a s e t , and
E
Given an i n t e g e r o r b i t o f the
n
and an
Given an If
n - t u p l e o f elements
n - t u p l e (mod H )
permutation belonging t o n-element s e t
s e t images o f
n-ELEMENT SET
a s e t o f permutations o f
H
, n o t n e c e s s a r i l y a group.
...,an
in
n - t u p l e images o f
H
.
n - t u p l e i n t o another, i s an e q u i v a l e n c e r e l a t i o n between
for
n-element s e t s .
n-TRANSITIVE GROUP,
under any
, we c a l l t h e o r b i t o f F (mod H) t h e s e t o f n-element
i s a group, then t h e e x i s t e n c e o f a p e r m u t a t i o n b e l o n g i n g t o
H
c a l l the
... ,an
al,
one
H
, which
takes
n-tuples. S i m i l a r l y
n-SET-TRANSITIVE GROUP
A group o f p e r m u t a t i o n s o f
E
i s s a i d t o be
there e x i s t s a permutation belonging
n - t r a n s i t i v e , i f f o r any two
t o t h e group and t a k i n g one
o t h e r . I n o t h e r words, i f t h e r e e x i s t s o n l y one o r b i t o f
E
.
E has f i n i t e c a r d i n a l i t y a t l e a s t equal t o n+2 ,
Another example, i f t h e base
then t h e a l t e r n a t i n g group formed o f a l l even p e r m u t a t i o n s , i s n - t r a n s i t i v e group i s
n-tuples
n-tuple i n t o the
n-tuples.
F o r example, t h e symmetric group formed o f a l l p e r m u t a t i o n s o f
An
, we
E
.
H
F
the set o f
E
al,
under any p e r m u t a t i o n b e l o n g i n g t o
F
.
p
m - t r a n s i t i v e f o r any
R e c a l l JORDAN'S h y p o t h e s i s , 1893: f o r
n
m
3 6 ,
6
n
every
n-transitive.
. n - t r a n s i t i v e group i s
symmetric o r a l t e r n a t i n g . A f f i r m a t i v e s o l u t i o n ; see f o r example CAMERON 1981 p. 9. A group o f p e r m u t a t i o n s on
i s s a i d t o be
E
n-set-transitive,
i f any two
n-ele-
ment s e t s a r e t r a n s f o r m a b l e one i n t o t h e o t h e r by a c e r t a i n p e r m u t a t i o n o f t h e group. I n o t h e r words, i f t h e r e e x i s t s o n l y one o r b i t o f
n-element s e t s .
2.3. ADHERENT PERMUTATION, GROUP CLOSED UNDER ADHERENCE Let in
be a s e t ,
E E
into
n
an i n t e g e r and
fan
.A
E
permuta-
.
H
n-adherent t o
o f permutations o f
permutation o f under
a s e t o f permutations o f
of
Every p e r m u t a t i o n which i s A set
H
E i s s a i d t o be n-adherent t o H , i f f o r any elements al,. .. ,a n , t h e r e e x i s t s a p e r m u t a t i o n o f H t a k i n g al i n t o fal , ... , and an
f
tion
E which i s
n-adherence,
then
H
E
,is
H
m-adherent f o r any
i s s a i d t o be c l o s e d under
n-adherent t o
H
, belongs
to
m gn
.
i f every i s closed
n-adherence,
H
.
If H
i s c l o s e d under m-adherence f o r each
m>,
n
.
A p e r m u t a t i o n i s s a i d t o be adherent t o H , i f i t i s n-adherent f o r e v e r y n . A s e t H o f p e r m u t a t i o n s i s s a i d t o be c l o s e d under adherence, i f e v e r y adherent p e r m u t a t i o n belongs t o
H
.
Chapter 11
2.4.(1)
Let
a-4
be a r e l a t i o n a l system w i t h base
R
. Then
n
319
, formed
E
t h e group o f automorphisms o f
o f components a
i s c l o s e d under
R
x
n-adherence.
( 2 ) F o r e v e r y r e l a t i o n a l system, t h e group of automorphisms i s c l o s e d under adhe-
rence. Consequences 2.5.(1)
Let
rence.
5
o f ch.9
1.7.
be a group o f p e r m u t a t i o n s o f
G
, which i s c l o s e d under n - m -
E
(6n)-homogeneous
Then t h e r e e x i s t s a
r e l a t i o n a l system, formed o f G (uses axiom o f c h o i c e when
components, whose group o f automorphisms i s
n - u
E is
uncountable). (2) Let
G
E , which i s c l o s e d under adherence.
be a group o f p e r m u t a t i o n s o f
Then t h e r e e x i s t s a homogeneous r e l a t i o n a l system whose group o f automorphisms i s G
(same remark).
( 3 ) Assume t h a t
tions o f
E
-
has f i n i t e c a r d i n a l i t y
E
group o f automorphisms i s 0
h
. Then
f o r e v e r y group
G
o f permuta-
, t h e r e e x i s t s a homogeneous m u l t i r e l a t i o n of maximum a r i t y h , whose
(1) To each
.
G
n - t u p l e o f elements o f
, associate the o r b i t , i.e. the class o f
E
n - t u p l e s which can be o b t a i n e d f r o m i t , by t a k i n g i t s image under any p e r m u t a t i o n in
G
. Then
t o each o r b i t , a s s o c i a t e t h e
t h e v a l u e (+) f o r those
o f choice, t a k e an o r d i n a l sequence Every p e r m u t a t i o n b e l o n g i n g t o automorphism
g
of
t h e same o r b i t . Thus
G
t a k e s each
R
E , which takes
n - a r y r e l a t i o n based on
n - t u p l e s i n t h e o r b i t , and ( - ) o t h e r w i s e . U s i n g t h e axiom
g
belongs t o
R
of t h e s e r e l a t i o n s , which f o r m a system.
i s an automorphism o f
R
, into
an
n-tuple i n
E
, since
G
G
. Conversely,
every
n-tuple belonging t o
i s c l o s e d under
n-adherence.
(6n)-homogeneous. Given a l o c a l automorphism f o f R , d e f i n e d on a domain of c a r d i n a l i t y 6 n , t a k e an a r b i t r a r y n - t u p l e c o n t a i n i n g a l l t h e elements al, ..., an o f F , w i t h p o s s i b l e r e p e t i t i o n s . T h i s n - t u p l e It remains t o see t h a t
and i t s image
fal,
permutation o f phism o f
R
is
R
..., f a n
belong t o t h e same o r b i t of
E which extends
and belongs t o
.0
( 2 ) Analogous p r o o f , b u t where
under adherence, i n s t e a d o f 0
f
G
G
. Thus
, hence
n t a k e s a l l i n t e g e r values,
there exists a
which i s an automor-
G
being closed
n-adherence. 0
( 3 ) P a r t i c u l a r case, where
G
i s c l o s e d under
h-adherence, and w i t h f i n i t e l y
many o r b i t s . 0 By t h e p r e c e d i n g p r o p o s i t i o n s , t o e v e r y r e l a t i o n a l system
a homogeneous system S
R
, there
corresponds
on t h e same base, h a v i n g t h e same automorphisms, and
w i t h o u t augmenting t h e maximum a r i t y o f t h e components. Mote however t h a t , s t a r t i n g from a simple r e l a t i o n
R
, we
can end up a t a homogeneous r e l a t i o n a l system S
w i t h i n f i n i t e l y many components. For example, s t a r t w i t h
R = W (chain o f natural
numbers), whose o n l y automorphism i s t h e i d e n t i t y . Then we end u p w i t h t h e
THEORY OF RELATIONS
320
sequence o f s i n g l e t o n unary r e l a t i o n s o f a l l the i n t e g e r s . 2.6. Here are two examples o f a group o f permutations which i s closed under
0
Let
be a s e t o f c a r d i n a l i t y
E
, and
n+2
an a r b i t r a r y t r a n s p o s i t i o n interchanging two elements o f (n+l)-adherent t o
position
such t h a t
t
G
.
Then
and
g
g
belongs t o
.
Let
Let
G
, where t i s
f,t
g
be a permuta-
G : indeed, there e x i s t s a t r a n s -
are i d e n t i c a l on
f,t
E
.
E
an odd permutation o f
f
be the group o f even permutations generated by the permutations tion
(n+l)-
n-adherence ( n n a t u r a l number ); POUZET 1979, unpublished.
adherence b u t n o t under
n+l
elements o f
E
, and
. Since f i s odd, i t does not belong t o G : it suffices t o see that f is n-adher e n t t o G . Indeed, f o r any x1 ,..., xn belonging t o E , l e t y, z be two d i s -
hence are s t i l l i d e n t i c a l on the
(n+2)nd
and l a s t element o f
t i n c t elements which are a l s o d i s t i n c t from the
.
(y,z)
... , (f,t)xn
= fxl,
(f,t)xl
Then
x
. Let
, which
= fxn
t
E
be the t r a n s p o s i t i o n
proves t h e
n-adherence
f . n
of
This example extends t o the case where in
i s i n f i n i t e , by t a k i n g
E
Another example. L e t
be t h e s e t o f a l l
E
The group
G
i s n o t closed under
tation of
E
w i t h negative determinant, hence
arbitrary points
.. ,an
al,.
in
n-adherence.
respect t o a hyperplane passing through into
On t h e o t h e r hand, mutation
Indeed, l e t
f
al,
i s closed under
G
G
(n+l)-adherent t o
. Take
f
be a l i n e a r permu-
does n o t belong t o
...,an . We G
.
fan
G
G
. Take
, then compose f w i t h the symmetry w i t h
E
w i t h p o s i t i v e determinant, hence belonging t o an
( n i n t e g e r >/ 2). L e t
n+l
E w i t h p o s i t i v e determinants.
be the group o f l i n e a r permutations o f
and
elements
( n + l ) - t u p l e s o f r a t i o n a l s , which we
s h a l l c a l l the r a t i o n a l vector space o f dimension
n
n+2
and repeating the preceding c o n s t r u c t i o n .
E
o b t a i n a l i n e a r permutation
, which
takes
(n+l)-adherence. n+l
points
xl,
al
into
Indeed, l e t
... ,
g
, ...
fal
be a per-
E ,
x ~ + belonging ~ to
which are l i n e a r l y independent. The images gxl,. . . ,g~,+~ are by hypothesis t h e images o f the x under a l i n e a r permutation w i t h p o s i t i v e determinant; hence they are l i n e a r l y independent. Let
u
be any p o i n t i n
t o the p o i n t s v = xl, sis
u 1. x 1 +
...,xn,v of
x
... +
.
Let
u ~ , . . . , u ~ + ~be the coordinates o f 1 1+
.= u .x
... +
are l i n e a r l y dependent. Since t h e i r images under
(n+l)-adherence G
. Similarly
u
relative
~ ~ + ~ .. Now x ~d e f+i n e~
. The
and w = ~ ~ + ~ ,. sox t~h a +t ~u = v + w
un.xn
t a t i o n belonging t o
+ un.(gxn)
E
, so t h a t we have u
g
points
are by hypothe-
i d e n t i c a l t o t h e i r images under a c e r t a i n l i n e a r permu-
, the w
+
dependence r e l a t i o n i s preserved:
and
QW = ~ ~ + l . ( g x , + ~ ); s i m i l a r l y
gv = ul.(gxl) x ~ +are~ l i n e a r l y dependent, s o t h a t we have
gu = gv + gw
=
ul.(gxl)
+
... +
~,.,+~.(gx,,+I)
.
...
321
Chapter 11
Letting
u
vary, we see t h a t
g
. Hence
i n t o gxi ( i = l,.,.,n+l) longs t o G [7
.
i s a l i n e a r permutation which takes each
xi
g has p o s i t i v e determinant, so t h a t
g be-
I n t h e preceding p r o o f , t h e hypothesis n >/ 2 i s used t o go from u = v + w t o gu = gv + gw ; f o r t h i s , the 3-adherence o f g i s required. On the o t h e r hand, take n = 1 thus n + l = 2 , so t h a t E i s the s e t o f the r a t i o n a l p o i n t s , o r vectors i n the plane, and G i s the group o f l i n e a r permutations o f E w i t h p o s i t i v e determinants. Then the permutation g which takes each vector (0,v) i n t o (0,2v) and which preserves each (u,v) when u # 0 ( w i t h r a t i o n a l s ) , i s 2-adherent t o G ; y e t t h i s g does n o t belong t o G
u, v
.
Example, due t o t h e same author, o f a group o f permutations closed under adherence, y e t n o t under n-adherence f o r any n 0 For each i n t e g e r n , take a s e t En w i t h c a r d i n a l i t y n , where the En are mutually d i s j o i n t , and l e t E be t h e i r union. Take G t o be the group o f those
.
permutations o f
E whose r e s t r i c t i o n t o
En
, for
each
n
,is
an even permutation
i s necessarily an'eleo f En . Then a permutation o f E which i s adherent t o G ment o f G . However, given an a r b i t r a r y i n t e g e r n , a permutation f i s n-adherent t o f o r each
G provided t h a t i t s r e s t r i c t i o n t o Ei i s an even permutation o f Ei i& n t l , and an a r b i t r a r y permutation o f Ei f o r each i >/ n t 2 . 0
2.7.(1) L e t p be an i n t e g e r , and R be a p-monomorphic, p-homogeneous r e l a t i o p - s e t - t r a n s i t i v e . Indeed, n a l system. Then t h e group o f automorphisms o f R f o r any two p-element subsets a, b o f t h e base, t h e r e e x i s t s an isomorphism from R/a onto R/b , which i s e x t e n d i b l e t o an automorphism o f R
2
.
.
( 2 ) L e t E be a set, and G be a p - s e t - t r a n s i t i v e group o f permutations o f E Then the closure 'G o f G under adherence i s again p - s e t - t r a n s i t i v e , and conversely. Moreover, there e x i s t s a homogeneous r e l a t i o n a l system whose automorphism group i s
-
-
, and such a r e l a t i o n a l system i s always p-monomorphic. T h i s f o l l o w s from 2.5 above; i f E i s uncountable, then t h i s uses t h e axiom o f choice. G+
2.8. THEOREM ON THE INCREASING NUMBER OF ORBITS
( 1 ) L e t p, q be i n t e g e r s , E a f i n i t e s e t w i t h c a r d i n a l i t y a t l e a s t equal tc , and G a group o f permutations o f E Then t h e number o f o r b i t s (mod G) of t h e (p+q)-element subsets o f E , i s g r e a t e r than or equal t o t h e number o f o r b i t s o f the p-element subsets (LIVINGSTONE, WAGNER 1965).
.
2p + q
-
0 Associate t o t h e group
group i s
G
a homogeneous m u l t i r e l a t i o n R whose automorphism p-element s e t s a, b belong t o the
G : see 2 . 5 . ( 3 ) above. Then two
same o r b i t (mod G) i f f the r e s t r i c t i o n s R/a and R/b are isomorphic: indeed by homogeneity, every isomorphism o f one r e s t r i c t i o n onto another i s e x t e n d i b l e t o an
322
THEORY OF RELATIONS
.
automorphism of R Same r e s u l t f o r the (p+q)-element s e t s . T h u s our proposition follows from t h e f a c t t h a t t h e number of isomorphism types of r e s t r i c t i o n s t o p+q elements i s g r e a t e r than o r equal t o the number of isomorphism types of the restrictions t o
p elements: see ch.9
5
7.1. 0
( 2 ) Let E be a denumerable s e t , and G be a group of permutations of E . To each i n t e g e r p a s s o c i a t e the countable number of o r b i t s of p-element subsets of E . Then t h i s number increases w i t h p (POUZET 1976). G+ of G under adherence, and note t h a t f o r each p , t h e o r b i t s of t h e p-element s e t s (mod G ) a r e the same a s t h e o r b i t s of the p-element s e t s (mod G'). Take a r e l a t i o n a l system R whose automorphism group i s G+ :
0 Consider t h e c l o s u r e
see 2 . 5 . ( 2 ) above. The proof terminates a s before, using t h e p r o f i l e i n c r e a s e theorem, ch.9 5 7 . 1 . However we must note t h a t t h i s theorem extends t o t h e case of a r e l a t i o n a l system with denumerably many components. Indeed, the multicolor theorem i n ch.3 5 5.3 includes the case of i n f i n i t e l y many c o l o r s , hence here of i n f i n i t e l y many isomorphism types f o r c e r t a i n values of
p
.0
( 3 ) In p a r t i c u l a r , l e t E have countable c a r d i n a l i t y a t l e a s t equal t o 2 p + q , and l e t G be a group of permutations of E . I f G i s ( p + q ) - s e t - t r a n s i t i v e , then G i s p - s e t - t r a n s i t i v e . 0 Indeed p - s e t - t r a n s i t i v i t y means t h a t a l l the p-element subsets of E belong t o t h e same o r b i t . Another proof i s obtained from ch.9 5 6 . 3 , i n view of t h e i n e q u a l i t y : p l e s s than o r equal t o Min(p+q,(Card E)-p-q) ; by using a l s o 2 . 7 . ( 2 ) above. 0 2.9. Let p , q be two i n t e g e r s , and E be a s e t of c a r d i n a l i t y g r e a t e r than o r equal t o 2 p + q ; l e t G be a group of permutations of E Then every oermutat i o n of E which preserves the o r b i t s o f (p+q)-element s e t s (mod G ) , a l s o preserves t h e o r b i t s of p-element s e t s .
.
Let G+ be t h e c l o s u r e of G under adherence, and l e t R be a homogeneous r e l a t i o n a l system whose automorphism group i s G+ : see 2 . 5 . ( 2 ) above. Let f be a permutation of E which t a k e s each (p+q)-element s e t i n t o another i n t h e same o r b i t (mod G ) , hence again i n the same o r b i t (mod G'). Then f t a k e s each r e s t r i c t i o n of R t o p+q elements i n t o an isomorphic r e s t r i c t i o n . Hence f t a k e s each r e s t r i c t i o n of R t o p elements i n t o an isomorphic r e s t r i c t i o n : see ch.9 5 7.2. Since R i s homogeneous, f o r any p-element s e t a , t h e r e e x i s t s an automorphism of R , hence an element of G+ , hence an element of G , which t a k e s
0
R/a
i n t o the isomorphic r e s t r i c t i o n
R/f"(a)
.0
2.10. Let E be a s e t of c a r d i n a l i t y g r e a t e r than o r equal t o 2p + q , and l e t G , H be two groups of permutations of E . I f every o r b i t of (p+q)-element sets
Chapter 11
p-element s e t s (mod G)
, then
(p+q)-element s e t s (mod H)
i s i n c l u d e d i n an o r b i t o f
(mod G)
of
323
i s i n c l u d e d i n an o r b i t o f
every o r b i t
.
p-element s e t s (mod H)
T h i s r e s u l t was c o n j e c t u r e d by BERCOV, HOBBY 1970, and a weaker v e r s i o n was proved by them; t h e p r e s e n t r e s u l t i s due t o POUZET 1976. 0
Let
a, b
be two
p-element s e t s b e l o n g i n g t o t h e same o r b i t (mod G)
e x i s t s a permutation tion
g
g
belonging t o
p r e s e r v e s a l l t h e o r b i t s (mod
, such G) , and
that
G
.
g"(a) = b
. There
T h i s permuta-
i n particular the o r b i t s o f the
(p+q)-element s e t s (mod G)
, hence by h y p o t h e s i s g preserves t h e o r b i t s o f t h e
(p+q)-element s e t s (mod H)
.
o r b i t s o f the o r b i t (mod H )
§
p-element s e t s (mod H)
.
3 - CHAINS
.
Hence
a
and
b
g
preserves t h e
belong t o t h e same
0
MODULO A PERMUTATION
COMPATIBLE
BY CHAINS;
By t h e p r e c e d i n g p r o p o s i t i o n ,
GROUP GENERATED
GROUP;
DILATED GROUP, CONTRACTED GROUP
COMPATIBILITY MODULO A GROUP m
Let
be a n a t u r a l number
1,..., m
A, 6
Two c h a i n s
.
A group
G
o f p e r m u t a t i o n s on t h e s e t o f i n t e g e r s
m-ary group, o r a group w i t h
i s s a i d t o be an
arity m
. , i f for < a2 < ...
G-compatible, o r c o m p a t i b l e modulo G
a r e s a i d t o be
every s e t o f
m elements i n t h e i n t e r s e c t i o n o f t h e bases, say
< am
, t h e p e r m u t a t i o n u which r e o r d e r s these elements a c c o r d i n g t o
(mod A)
c
au(l)
au(2)
We see t h a t
< . . . < au(,,,)
(mod B)
, belongs t o t h e group
G
al
.
G - c o m p a t i b i l i t y i s r e f l e x i v e and symmetric. I f t h e chains
t h e same base, then
A, B
have
G - c o m p a t i b i l i t y i s t r a n s i t i v e , hence i s an equivalence r e l a -
tion.
I n t h e general case, n o t e t h a t two c h a i n s a r e
G-comoatible when t h e i n t e r s e c t i o n
o f t h e bases has c a r d i n a l i t y s t r i c t l y l e s s t h a n i s no t r a n s i t i v i t y , s i n c e f o r example i f has base d i s j o i n t f r o m t h e base o f
B
and
on t h e o t h e r hand, a r e
A
n o t t h e case f o r If
m
= 2
and
G
and
A
A
, then
and
A
m ( a r i t y o f G ) . Hence t h e r e A ' have t h e same base and B and
G-compatible f o r e v e r y
on t h e one hand, and
B
G
A'
, which i s o b v i o u s l y
A ' , assumed t o be d i s t i n c t .
reduces t o t h e i d e n t i t y on t h e s e t { 1 , 2 )
, then
G-compati-
b i l i t y means t h a t t h e r e s t r i c t i o n s o f b o t h c h a i n s t o t h e i n t e r s e c t i o n o f t h e i r bases ch.2
i s t h e same. Then we
5 1.2;
f i n d t h e n o t i o n o f c o m p a t i b i l i t y i n t h e sense o f
and t h e r e e x i s t s a common e x t e n s i o n which i s a c h a i n based on t h e
u n i o n o f t h e two bases.
I f m i s an a r b i t r a r y i n t e g e r , and G i s t h e m-ary symmetric group, o r group o f 1,. . ,m , t h e n a l l c h a i n s a r e G-compatible. Two c h a i n s a r e G-compatible i f f t h e i r r e s t r i c t i o n s t o t h e i n t e r s e c t i o n o f t h e a l l permutations o f
bases a r e
G-compatible.
.
1
THEORY OF RELATIONS
324
Let A, B be two chains with the same base E , and l e t G be an m-ary group; l e t n be such t h a t m 6 n Card E Then A and B are G-compatible i f f , f o r each n-element subset X of E , the r es t r i ct i o n s A/X and B/X are G-compatible.
<
.
3.1. Let '7%' be a s e t of chains with the same f i n i t e base E , and l e t p = Card E Then the following three conditions are equivalent. (1) Given three chains A, B , C in A , the image of C under the permutation of E which takes A into B , belongs t o & ( 2 ) There e x i st s a p-ary group G such t h a t i s formed of a chain and i t s images under the permutations belonging t o G . ( 3 ) There e x i s t s an integer qd p and a q-ary group H such tha t i s formed of a chain and a l l chains on E which are H-compatible with i t . 0 (1) and ( 2 ) are obviously equivalent; (2) i s a particular case of ( 3 ) ; f i n a l l y ( 3 ) e a si l y implies ( 1 ) . 0
.
.
4
4
3.2. DILATED GROUP Let G be an m-ary g r o u p , and E be an m'-element s e t , with m ' 3 ni . Let A be a chain based on E ; consider the s e t of a l l chains on E which are G-compatible with A By the preceding proposition, these chains are the images of A under a certain m'-ary group G ' : t h i s G ' i s called the m'-ary dilated group of G and denoted Gm' . Let G be an m-ary group, H a subgroup of G , and l e t m ' be an integer 3 m
.
.
Then the dilated group Given three integers
Hm' m<
i s a subgroup of the dilated group G m '
m',<
m"
and an m-ary group G , then
.
(Gm')m"
= Gm"
.
3.3. Two d i s t i n c t qrouDs of the same a r i t v can have the same dilated qrow of a larger a r i t y .
.
.
Let m = 4 , and l e t G be the group formed of the identity on 1 , . . ,4 Let 5 G ' be formed o f the identity plus the transposition ( 2 , 4 ) . Then G I 5 = G = the group formed o f the identity on 1, ...,5 Indeed, s t a r t with the chain 1 2 3 4 5 . A chain which i s GI-compatible with i t , orders the integers 1,2,4,5 according t o 1 2 4 5 or according t o 1 5 4 2 . Similarly, we have one of the chains 1 3 4 5 or 1 5 4 3 , and one of the chains 2 3 4 5 or 2 5 4 3 . The chain 1 5 4 2 requires t h a t 5 be before 4 , hence implies 1 5 4 3 and 2 5 4 3 . Thus 5 must be b o t h before and a f t e r 2 , a contradiction. Thus we have 1 2 4 5 , which implies 1 3 4 5 and 2 3 4 5 , 0
0
.
3.4. Let G , H
be two groups of the same a r i t y
( G ~ I H ) =~ Gn n Hn identi ty
Then -
.
and -
( G u H ) ~2
m
, and l e t n 2 m
.
Gn w H n ; t h i s inclusion i s n o t always an
325
Chapter 11
The group (GnH)n i s included i n G n and in Hn , by 3.2 above. On t h e o t h e r hand, i f two chains a r e simultaneously G-compatible and
0
H-compatible,
then they a r e (GnH)-compatible. The group ( G u H ) ~ includes G n and H n , by 3 . 2 . F i n a l l y l e t G be the group of c y c l i c permutations on 1,2,3 and H be t h e group generated by t h e t r a n s p o s i t i o n ( 1 , 3 ) . Then G u H i s the symmetric group on 1,2,3 ; hence ( G u H ) 4 i s the symmetric group on 1,2,3,4 . On t h e o t h e r hand,
G4 i s t h e group generated by the cycle ( 1 , 2 , 3 , 4 ) ; and H4 i s t h e group generated by ( 1 , 4 ) ( 2 , 3 ) . So t h a t t h e group generated by t h e i r union i s the dihedral group on 1,2,3,4 formed of 8 permutations generated by t h e considered cycle and t h e r e f l e c t i o n ( o r inversion ( 1 , 4 ) ( 2 , 3 ) ) . 0 3 . 5 . Let G , H be two m-ary groups. If. Hm+' i s s t r i c t l y included i n then ( G n H ) # G -, and H i s not n e c e s s a r i l y included i n G .
Gml
,
-
Suppose t h a t G A H = G . Then G i s included i n H , so Gm+' included in Hmtl. To have H non-included i n G , take G t o be t h e group generated by t h e cycle (1,2,3,4) , and take H t o be t h e group on 1,2,3,4 formed of the i d e n t i t y plus the t r a n s p o s i t i o n ( 2 , 4 ) Then H5 reduces t o t h e i d e n t i t y on 1,2,3,4,5 ( s e e 0 3.3 above), and G5 i s generated by the cycle (1,2,3,4,5)
0
.
3.6.
If
G
has a symmetric d i l a t e d group, then
.
G
i s i t s e l f synnnetric.
Let m be t h e a r i t y of G , and l e t E be a s e t w i t h c a r d i n a l i t y n am . By hypothesis Gn i s symmetric, hence a l l chains on E a r e mutually G-compatible. Taking t h e i r r e s t r i c t i o n s t o a m-element subset D of E , we obtain a l l chains based on D : hence G i s symmetric. 0
0
3 . 7 . GROUP GENERATED BY A SET OF CHAINS; CONTRACTED GROUP Let & be a s e t of chains w i t h t h e same base E , and l e t m be an i n t e g e r l e s s than o r equal t o Card E . The l e a s t m-ary group G such t h a t t h e chains i n a r e a l l G-compatible i s c a l l e d t h e group generated by d This i s a l s o t h e group generated, i n t h e usual s e n s e , by a l l permutations u of { 1 , . . ,m] obtained by taking any two chains i n & and then any m elements i n E , say modulo a l < a 2 < .. . < a m modulo one of the chains, and a u ( l ) a,,(,,,) t h e o t h e r chain. I f A i s just a s e t of two c h a i n s , then we c a l l i t a bichain, and G i s s a i d t o be the w a r y group generated by this bichain. S t a r t w i t h an n-ary group G . Take a chain A of c a r d i n a l i t y n ; then take t o be t h e set formed of A and a l l i t s images under the permutations belonging t o G . Then f o r m 4 n , the m-ary aroup generated by & i s s a i d t o be
.
< ... <
A
4
.
326
THEORY OF RELATIONS
the contracted group o f If
H
i s a subgroup of
S t a r t w i t h an
to arity m
G
, then
G
denoted by
Gm
.
the contracted group Hm i s a subgroup of
G , take
m-ary group
, and
n >/ m
and t h e d i l a t e d group
Gm
.
Gn , and then
the contracted group (G')), . Then (G')), i s a subgroup of G , which can be d i s t i n c t from G . 0 S t a r t w i t h G of a r i t y 4 , formed of the i d e n t i t y and the t r a n s p o s i t i o n ( 2 , 4 ) . 5 Then G5 reduces t o the i d e n t i t y ( s e e 3.3 above), hence ( G ) 4 a s w e l l . 0 Two d i s t i n c t groups can have t h e same contracted group. Indeed, t h e r e e x i s t s only one p o s s i b l e unary contracted group, and two p o s s i b l e binary ( o r 2-ary) contracted groups: t h e group formed of the i d e n t i t y , and the symmetric group formed of the i d e n t i t y and t h e t r a n s p o s i t i o n (1,Z) More gener a l l y , the proposition follows from the increase i n t h e number of m-ary groups of a r i t y m . 0
0
.
S t a r t w i t h an
G , take
n-ary group
then t h e d i l a t e d group
(Gm)n
. Then
mg n
and t h e contracted group Gm , and
(G,Jn
i s an extension of
G , and can be
a DroDer extension. S t a r t w i t h t h e 5-ary group G which preserves 4 and 5 and c i r c u l a r l y permut e s 1 , 2 , 3 Then G4 preserves 4 and symmetrically permutes 1,2,3 . Then
0
.
(G4)5
preserves
4 and
5
and symmetrically permutes
Let G be an m-ary group, and ( G m , ) m , , i s an extension of G,,,
m"g m ' 6 rn
.
1,2,3
.0
Then t h e twice contracted aroup
.
Problem. Are t h e s e two groups always i d e n t i c a l . More s t r o n g l y , given a s e t & of chains a l l with the same base, and another s e t 6.3 of chains w i t h the same base, i f c+ and @ generate the same w a r y group, then do they generate t h e same m'-ary group f o r m ' < m . 3.8. Let
&T
be two m-ary groups and n < m
G, H
(GuH),
=
.
Gn u H n @ (GnH), c_ Gn n Hn ; t h i s inclusion can be proper.
By the preceding, t h e f i r s t qroup includes G n and Hn , hence includes their union. S i m i l a r l y t h e t h i r d group i s included i n t h e i n t e r s e c t i o n . Moreover, i f a chain A i s taken i n t o a chain B by a permutation belonging t o G u H , then t h e r e e x i s t s a f i n i t e sequence of chains going from A t o B , and such t h a t t h e passage from each one t o the next i s e f f e c t e d by a permutation i n G o r i n H . Hence the r e s t r i c t i o n of A t o a given n-element s e t 0 , i s taken i n t o t h e r e s t r i c t i o n of B t o 0 , by a f i n i t e sequence o f permutations i n Gn and i n Hn . I t follows t h a t the f i r s t group i s included i n the second, hence i s equal t o i t .
0
Chapter 1 1
321
To see t h a t t h e i n c l u s i o n o f t h e t h i r d group i n t h e f o u r t h
{
t o be t h e group o f c y c l i c p e r m u t a t i o n s on
G
o f t h e i d e n t i t y and t h e t r a n s p o s i t i o n intersection
, and
t o be formed
H
. Then t h e
{ 1,2,3}
on t h e same s e t
G2
and
.0
3.9. L e t
so
m-ary group i s i n c l u d e d i n
Problem. Is t h e d i l a t e d group
Gn
m-ary c o n t r a c t e d group i n c l u d e s be an
G
G
Problem. L e t t i n g
n< m
3.10. There e x i s t which a r e A
,is
t h e c o n t r a c t e d group
5-ary group
be t h e c h a i n
12 3 4
5-ary group which p r e s e r v e s 4 Let
A ' , B'
t o be
B'
G4
and and
be two e x t e n s i o n s of
5-ary group reduces t o
Gn
t h e i n t e r s e c t i o n o f those
. 1, ...,4
G-compatible e x t e n s i o n s . the chain
B
5
A, B
. Let
3 2 1 4
be t h e
G
. The .
, and c i r c u l a r l y permutes 1,2,3 1,2,3
hence takes
t o the integers
5
G-compatible, i t i s necessary t h a t
4 - INDICATIVE
c o n s i s t s o f t h e iden-
G4
and two c h a i n s on t h e i n t e g e r s
s y m m e t r i c a l l y permutes
mod B ' ) ; b u t t h e n we do n o t have
§
G
G
i s n o t neces-
Gn
m-ary d i l a t e d group
: see 3.3 above. 0
(2,4)
m-ary group i n c l u d e s
a
. Thus
4 - a r y groups whose d i l a t e d
G4-compatible, y e t have no
t r a c t e d group
c o n t r a c t e d group
,..., 5 )
contains the transposition
groups, whose d i l a t e d
i s genera-
Gn
n-ary groups
n-ary groups whose
n-ary groups, whose
.
t i t y . Yet t h e u n i o n o f those
,
.
G
. The
n<m
c o n s i s t o f t h e i d e n t i t y on {l
the identity
Hp
.
w a r y group and l e t
i s included i n G
0 Let
d i l a t e d group
the intersection o f the G
s a r i l y generated by t h e u n i o n o f those 0 Let
. The
n+ m
t h e usual sense, by c o m p o s i t i o n ) f r o m t h e u n i o n o f those
whose c o n t r a c t e d
Let
reduces t o t h e i d e n t i t y . Yet
m-ary group, and l e t
be an
G
(GnH)2
can be p r o p e r , t a k e
and t a k e
hence t h e i r i n t e r s e c t i o n , a r e i d e n t i c a l t o t h e symmetric group on {1,2}
ted ( i n
G n H
(1,2)
1,2,3}
be a f t e r
A
.
1, ..., 5 1,2,3
con-
B
into For
A'
and
(mod A ' and
G-compatibility. 0
GROUP, REDUCTION THEOREM
(FRASNAY)
A group o f p e r m u t a t i o n s i s s a i d t o .be i n d i c a t i v e i f f i t i s generated by a
b i c h a i n w i t h i n f i n i t e base ( w h i c h we can assume t o be denumerable). 4.1.
F o r each i n t e g e r m
, there
e x i s t s an
a base w i t h c a r d i n a l a t l e a s t equal t o
mt
+
m
m
such t h a t e v e r y b i c h a i n on
g e n e r a t e s an
m-ary i n d i c a t i v e group.
0 Suppose t h e c o n t r a r y , and c o n s i d e r an w - s e q u e n c e o f b i c h a i n s
w i t h f i n i t e bases
Ei
where
Card Ei
i s s t r i c t l y increasing w i t h
Bi
( i integer)
i
, such
that
THEORY OF RELATIONS
328
each B i generates the same non-indicative group G . There are only f i n i t e l y many, say h permutations in G . So f o r each integer i , there e x i s t s a subset Fi of Ei with cardinality p 4 h.m , such that the r e s t r i c t i o n Bi/Fi generates G . We can suppose t h a t the Ei are f i n i t e s e t s of integers, and t h a t Fi = 41 ,. . . , p ) Extract from the sequence of the Bi a f i r s t i n f i n i t e sequence giving the same r e s t r i c t i o n of each B i t o the integers 1 , . . . , p Then keep d e f i n i t i v e l y Bo and extract from the sequence of the B i ( i 31) a second i n f i n i t e sequence giving the Then keep definitively same r e s t r i c t i o n of each bichain t o the integers l,..,,p+l and B1 , and i t e r a t e t h i s t o obtain the same r e s t r i c t i o n of each bichain t o Bo the integers 1, ...,p+ 2. e t c . In the sequence definitively obtained, replace each Bi by i t s r e s t r i c t i o n B i t o the integers 1,..., pti . Since each B i , and i t s r e s t r i c t i o n t o 4 1,..., p ) , generates G , then each B(i generates G as well. Now the common extension of the B; t o the s e t of a l l integers i s an i n f i n i t e bichain which generates G ; hence G i s indicative: contradiction. 0
.
.
.
4 . 2 . DESCRIPTION OF INDICATIVE GROUPS; INDICATOR Given an i n f i n i t e bichain, f o r each integer m i t generates an m-ary indicative group G, The function G which t o each m associates Gm i s said t o be the indicator of the bichain (called "fiche" in FRASNAY's terminology). Let us describe as follows a l l the indicators, hence a l l indicative groups. The indicator S gives by d e f i n i t i o n , f o r each m , the m-ary symmetric group denoted by Sm . Take f o r instance the chain of n a t u r a l numbers , and another chain which f o r each integer m realizes a l l possible permutations of m consecutive integers. The indicator I gives f o r each m , the group I,,, consisting only of the identity on m elements. Take a bichain formed of two identical i n f i n i t e chains. The indicator J gives f o r each m , the group J, formed of the identity and of : take an i n f i n i t e chain and i t s converse. the reflection (l,m)(Z,m-l)(3,m-2). The indicator T gives f o r each m , the group Tm of t r a n s l a t i o n s , generated ( i n the usual sense) by the cyclic permutation (1,2,3, . . . . , m ) Take f o r instance two i n f i n i t e chains A , B with d i s j o i n t bases, then the two sums At6 and B+A The indicator D gives the dihedral group Dm , which i s generated (usual sense) by the union of Tm and J, Take f o r instance two i n f i n i t e chains A, B with d i s j o i n t bases, then the sums A+B and A- + B- (sum of converses). Given two integers p , q , the indicator I p y q gives f o r each m , the group I:'q formed of those permutations which preserve the i n i t i a l interval 1, . . . , p and the final interval m-q+l, ...,m and which reduce t o the identity on the integers p + l , p+2, , m-q , assuming t h a t m &p+q In the contrary case where m < p+q , then denotes the w a r y symmetric group. Take f o r instance the chain d 2 ... p followed by an i n f i n i t e chain A followed by the chain 1' 2' ... q ' ;
.
..
.
.
. ..
,Iyq
.
.
329
Chapter 11
and on the o t h e r hand, the chain
..
. Note
q ’ 1’
that
Given an i n t e g e r
1’”
m
r+l,...,m-r
< 2 r , then
...
12
r
r 1
.
JL
m-r+l,
denotes the
...
A, B
J1
followed by
A
..
2’ 3’
.
I
gives the group
Jk
the i n i t i a l i n t e r v a l
which i s generated 1, ..., r i s taken
and conversely; moreover the median i n t e r -
.
m32r
I n t h e opposite case where
m-ary symmetric group. Take f o r instance the chain
followed by an i n f i n i t e chain
Note t h a t
4.3. L e t
p 1 f o l l o w e d by
i s reversed, assuming t h a t
the o t h e r hand 2 ’ 3 ’
..
. Then ...,m
Iiyr and Jm
i n t o the f i n a l i n t e r v a l val
...
, the i n d i c a t o r Jr
r
by the union o f
2 3
i s identical t o
r ’ 1’
followed by
A
1’ 2 ’
followed by t h e converse
i s identical with
A-
... r ’
; and on
followed by
2 3
..
.
J
be two chains w i t h common base o f c a r d i n a l i t y l a r g e r than o r
are
.
or
and
A B D,-compatible, then e i t h e r A 6 , A B- x e Tm-compatible. and B ( w i t h common base) L e t m, r be two i n t e g e r s , w i t h 2 r 6 m If. A -
equal t o
m
and the converse
are
.
Jk-compatible,
then e i t h e r
4.4. The d i l a t e d groups o f The d i l a t e d groups o f If
ptq 6 m
If
m 2,3
, then
and
A
T3
and
B
, op
B-
are
Ik’r-compatible.
.
>/ 4) .
ILyq are the
the d i l a t e d groups o f
2rs m
and
Tm (ma3)
are t h e
are the. Dm (m
D4
A
n >/ m
I:yq for
, then t h e d i l a t e d groups o f J k are the J:
for m3 3
Consequently, two chains whose i n t e r s e c t i o n o f bases has c a r d i n a l i t y T,-compatible
For
m3 4
For
m
32
i f f they are
, they and
are
then t h e chains are For
i f f they are
, are
D4-compatible.
, i f the i n t e r s e c t i o n o f bases has c a r d i n a l i t y n + m ,
I:’q-compatible
i f f they are
I~’qycompatible.
ma3 and 2 r Q m , i f the i n t e r s e c t i o n o f bases has c a r d i n a l i t y
then t h e chains are
+m
T3-compatible.
D,-compatible
p+q 6 m
n
JL-compatible i f f they are
n am
,
J:-compatible.
I n general, the d i l a t e d o r t h e contracted group o f one o f the p r e v i o u s l y described i n d i c a t i v e groups
i s simply obtained by keeping the i n d i c a t o r
and by modifying the a r i t y understood t h a t , i f J; 4.5.
is
p+q >
m
m
, keeping or i f
the i n d i c e s
, then
2r> m
i d e n t i c a l w i t h the symmetric group
S,
p, q
or
the group
r
I , J, T, D, S
, where
it is
o r the group
.
It i s proved by FRASNAY 1965, t h a t t h e o n l y i n d i c a t i v e groups are those o f
.
I , J, T, 0, S For a sketch o f the p r o o f , which i s l i t t l e more complicated than t h e p r o o f concerning the Q - i n d i c a t i v e groups i n 5 5
the f i v e preceding f a m i l i e s
.
330
THEORY OF RELATIONS
below, see propositions 5 . 1 through 5.8 and exercise 2 below. By examining each case described in 4 . 2 above, we see t h a t the group generated by the union of any two indjsativegroups i s indicative. Since there are f i n i t e l y many m-ary groups f o r each integer m , we have the following statements. A group i s indicative i f f i t i s generated by an a r b i t r a r y s e t of chaiFsmcn the same denumerable base. If G i s an m-ary indicative group and n 3 m , then G n i s indicative and (G')), = G . t Given an integer m , there e x i s t s an m m such t h a t every s e t of chains on a ~
~
~
base of cardinality a t l e a s t equal t o m+ generates an indicative m-ary qroup (generalization of 4 . 1 ) . Given an m-ary group G , the dilated group Gn i s indicative f o r n >, m' . 4.6. REDUCTION THEOREM (FRASNAY 1965) Given an m-ary group G , there e x i s t s a maximumnndicative m-ary group H ( G ) included in G . Moreover f o r n 3 m we have H ( G n ) = ( H ( G ) ) n . Taking n m such t h a t G n i s indicative, then we have H ( G ) n = Gn and H ( G ) = (G')),
.
§
5 - Q-BICHAIN,
SET-TRANSITIVE
Q-INDICATIVE GROUP THEOREM
GROUP;
FIVE
Q-INDICATIVE
GROUP THEOREM;
(CAMERON)
Q-BICHAIN, Q-INDICATIVE GROUP We shall c a l l a Q-bichain, any birelation both o f whose components are chains each isomorphic with Q . We say t h a t a group i s Q-indicative, i f i t i s generated by a Q-bichain. We see t h a t the f i v e following groups are Q-indicative, f o r each a r i t y m : the identity I,,, ; the group J m ( i d e n t i t y and r e f l e c t i o n ) ; the group Tm of translations; the dihedral group Dm generated by the union of Jm and Tm ;
and f i n a l l y the symmetric group Sm . We call these temporarily the canonical groups. The group generated by the union of two canonical groups i s canonical. 0 Indeed, the only case where two of these groups are non-inclusive, i s the case of Jm and Tm , whose union generates Dm . 0 Consequently, f o r each group G , there e x i s t s a maximum canonical qroup included in G -
-
.
THE FIVE Q-INDICATIVE GROUP THEOREM The five canonical groups are the only Q-indicative groups. To prove t h i s , we shall show t h a t i f G i s Q-indicative, then i t i s equal t o the maximum canonical group included in G This i s obvious f o r the case of the symmetric group S, ; so we shall consider the four cases I,, Tm, J m , Dm in the following propositions 5 . 1 t o 5.8.
.
33 1
Chapter 11
.
5.1. Consider a Q-bichain with components A , B We say that a pair of elements x , y of the base, i s preserved or inverted, accordin9 t o whether x and y are in the same order modulo A and modulo B , o r in the opposite order. Let AB be a Q-bichain having -a t l e a s t one preserved pair and one inverted pair. AB , f o r every Then e i t h e r the group Jm i s included in the group generated bym , Tm i s included in the group, f o r every m .
or
Let u and v denote two elements such t h a t u < v (mod A ) and v < u (mod B ) . Since A i s isomorphic with Q , there e x i s t i n f i n i t e l y many elements x between u and v (mod A ) . For each such x , we have v < x or x < u (mod B ) , hence there e x i s t i n f i n i t e l y many x s a t i s f y i n g , f o r example, x < u (mod B ) . Using RAMSEY's theorem (ch.3 g l . l ) , e i t h e r there e x i s t i n f i n i t e l y many of these x which form mutually inverted p a i r s , in which case the aroup Jm i s included in the m-ary group generated by AB , f o r each m . Or there e x i s t i n f i n i t e l y many of these x forming preserved pairs. Then since u i s less t h a n (mod A) and greater t h a n (mod B ) these elements, the group Tm of translations i s included in the group generated by AB , f o r each m 0
0
.
5 . 2 . Let G be the m-ary g r o u p generated by a given Q-bichain. If the maximum canonical group included in G I,,, , & t G i s identical t o I,,,
.
Either there only e x i s t preserved pairs in the given bichain, in which case G = I . Or there only e x i s t inverted p a i r s , in which case G = Jm . Or f i n a l l y m there e x i s t s a t l e a s t one pair of each kind, in which case Jm or Tm i s included in G . Only the f i r s t case i s possible under our assumption t h a t I,,, i s the maximum canonical group included in G . 0
0
.
Suppose t h a t 5.3. Let AB be a Q-bichain w i t h base E ~subset U Lf E a l l of whose pairs are preserved, hence Let m be a positive integer; suppose t h a t the group Tm m-ary group generated by the bichain. Then e i t h e r G = Tm symnetri c g r o u p .
there e x i s t s an i n f i n i t e A/U = B / U . i s included in the or G = Sm , the
Suppose t h a t mg 2 , since the case where m = 1 i s obvious. The s e t U i s d i f f e r e n t from E , since Tm i s included in G . For each element x of E-U , l e t xA denote the cut defined on A/U by the i n i t i a l interval of those elements of U l e s s than x (mod A) , and the complementary f i n a l i n t e r v a l . Let xB denote the cut analogously defined with B F i r s t suppose t h a t f o r each x in E-U , we have xA = xB By hypothesis, there e x i s t s an inverted p a i r , say { x,y} which i s thus included in E-U , which then implies t h a t xA = xB = yA = yB I f , before t h i s c u t , we have i n f i n i t e l y many elements of U , then G contains the transposition (m-1,m) , which together with T, generates S, Similarly, i f there e x i s t i n f i n i t e l y many elements
.
.
.
.
THEORY OF RELATIONS
332 greater than t h i s cut, then Tm generates
with
From t h i s p o i n t on, we are i n t h e case where t h e r e e x i s t s an xA # xB
. Suppose t h a t
before
A/U
on U
, and
l i e s before i t , then
G
defined by t h e c y c l e
(2,3
.
E-U
with
a t l e a s t one o f these i n t e r v a l s i s i n f i n i t e . xA and
xB
, and
i f the o t h e r non-empty i n t e r v a l
contains the permutation which preserves
. By composition
,..., m)
S i m i l a r l y i f the i n t e r v a l between
(1,Z
with
1 and which i s
,...,m) , we
generate
xA and xB i s i n f i n i t e and the o t h e r
non-empty i n t e r v a l l i e s a f t e r i t . I f the i n t e r v a l between and t h e i n t e r v a l l y i n g before i s i n f i n i t e , then which, together w i t h t h e t r a n s l a t i o n
(m-1,m)
in
, nor the f i n a l c u t . Then t h e r e e x i s t a t l e a s t two i n t e r v a l s bounded
I f i t i s t h e i n t e r v a l between
Sm
x
a t l e a s t one o f these cuts i s n e i t h e r the i n i t i a l c u t l y i n a
xA and xB
by
, which together
(1,2)
contains t h e t r a n s p o s i t i o n
G
.
Sm
G
xA and
xB
is finite
contains the t r a n s p o s i t i o n
(1,2,
...,m)
, generates
Sm
. Same
r e s u l t i n t h e case o f an i n f i n i t e i n t e r v a l l y i n g a f t e r . xA and
Suppose now t h a t , i f
xB
are d i s t i n c t , then they are extremal ( i . e . one
o f them i s the i n i t i a l cut, and the o t h e r i s the f i n a l c u t ) . We can r e q u i r e t h a t any two elements
x, y
which give t h e same non-extremal c u t
xA = xB = yA = yB
,
form a preserved p a i r . Indeed otherwise, we o b t a i n again t h e t r a n s p o s i t i o n (1,2) o r (m-1,m) Thus augment U by a l l these x corresponding t o non-extremal
.
cuts. From t h i s p o i n t on, every E i t h e r there e x i s t
x, y
. Then
yA f i n a l c u t
identical with
Sm
G
in
x
in with
E-U
E-U
xA and yB i n i t i a l c u t
contains the t r a n s p o s i t i o n
. Then
x
in
G
contains t h e permutation which transforms
are preserved, i n which case
+ A/(E-U)
5.4. L e t
in
G
G
2
, hence G
xA
i s i n i t i a l and
12 3 Sm
. Or
i s t h e sum A/(E-U)
A
= Tm
be generated by a
Tm,=
and
xB and
and hence again i s xB
f i n a l f o r each
e i t h e r there e x i s t s an i n v e r t e d p a i r among these; i n which case
and t h i s together the t r a n s l a t i o n generates A/U
(1,m)
.
O r we are i n the case where, f o r instance,
E-U
y i e l d s two extremal cuts.
... m-1
m into 3 4
...
m 2 1
f i n a l l y a l l the pairs i n
+
A/U
and
B
E-U
i s the sum
.0 Q-bichain. I f the maximum canonical group i n c l u d e d
G=Tm.
Because o f RAMSEY's theorem, there e x i s t s an i n f i n i t e subset
U of E , all of
U , a l l o f whose p a i r s are i n v e r J m i s included i n G , and since by hypothesis Tm i s included i n G as w e l l , we have t h e dihedral group Dm included i n G ,
whose p a i r s are preserved, o r an i n f i n i t e subset t e d . I n t h e second case, t h e group
i s the maximum canonica.1 group included i n Tm by the preceding statement, G = Tm o r G = S, , t h i s l a s t case contra-
c o n t r a d i c t i n g the assumption t h a t G
. Thus
d i c t i n g our assumptions. 0
333
Chapter 1 1
5.5. L e t
be a
AB
of
U
t -s
Q - b i c h a i n w i t h base
.
E
Suppose t h a t t h e r e e x i s t s an i n f i n i t e
, a l l o f whose p a i r s
E
m
a l l o f whose p a i r s a r e i n v e r t e d . Then f o r each AB
i s t h e symmetric group
0 Note f i r s t t h a t
existence o f
and
U
m-ary group generated by ~
.
Sm
have a t most one element i n common. Moreover, t h e
V
shows t h a t t h e group
V
, the
Jm o f t h e r e f l e c t i o n , i s generated. Using
5.3 above, i t s u f f i c e s t o prove t h a t t h e group
Tm o f t r a n s l a t i o n s i s generated,
s i n c e t h e n t h e e n t i r e generated group cannot be reduced t o t h e s i n g l e group Tm and i s t h e n i d e n t i c a l w i t h Sm . F i r s t suppose t h a t f o r each i n t e g e r h , t h e r e e x i s t s an x i n V f o r which t h e cuts
xA
least
h
and
xB
d e f i n e d by
elements o f
.
U
x
, are
A/U = B/U
on t h e c h a i n
I n t h i s case, t h e t r a n s l a t i o n
(l,Z,
separated by a t
...,in)
i s obtained,
hence o u r p r o p o s i t i o n h o l d s . Suppose now t h a t t h e r e e x i s t s an i n t e g e r a t most
h
elements i n
o f elements i n
U
, which
V
h
such t h a t f o r each
between t h e c u t s
xA
and
. Take
xB
i s f o r example d e c r e a s i n g (mod A ) ,
x
in
V
, we have
an w-sequence
hence decreasing
. Then f r o m some p o i n t on, t h e c u t s xA become i d e n t i c a l , as w e l l as t h e become i d e n t i c a l . Thus t h e r e e x i s t i n f i n i t e l y many elements i n U which a r e
(mod B ) xB
e i t h e r a l l g r e a t e r t h a n these c u t s , o r a l l l e s s t h a n these c u t s . T h i s y i e l d s f o r i n s t a n c e , f o r each ..,p
t h e symmetric group 5.6. L e t
pair.
m
and each
and which i n t e r c h a n g e s
AB
be a
Then f o r each
Sm
, t h e p e r m u t a t i o n which p r e s e r v e s 1 2 , ...
p d m
(p+l,m),
(p+Z,m-l),
etc.;
t h i s s u f f i c e s t o generate
.
(I-bichain h a v i n g a t l e a s t one p r e s e r v e d p a i r and one i n v e r t e d , m
,t h e group Tm
i s i n c l u d e d i n t h e group generated by
AB
.
u < v (mod A) w i t h v < u (mod B) . Take i n f i n i t e l y many u and v (mod A ) . Then e i t h e r t h e r e e x i s t i n f i n i t e l y many o f them which a r e > u (mod B ) . I n t h i s case, by RAMSEY's theorem, e x t r a c t an i n f i n i -
0 Consider two elements
elements between
t e subset o f t h e s e elements, a l l o f whose p a i r s a r e preserved, o r a l l o f whose p a i r s a r e i n v e r t e d . I n t h e case where t h e p a i r s a r e preserved, by o u r c h o i c e o f t h e element obtain again
, we
v
t e d , t h e group
Jm Tm .
o b t a i n t h e group
T,
O r t h e r e e x i s t i n f i n i t e l y many elements
(mod A)
. Then
.
I n t h e case where t h e p a i r s a r e i n v e r -
i s obtained; by t h i s r e f l e c t i o n and by o u r c h o i c e o f
<
v (mod B )
, and
v
and
u and v
t h e p r e c e d i n g argument s t i l l works.
O r f i n a l l y t h e r e e x i s t i n f i n i t e l y many elements between between
s t i l l between
u , we
u (mod B)
. Then
u and v (mod A)
and
i n t h e case o f p r e s e r v e d p a i r s , we o b t a i n t h e
group Tm . There remains t h e case o f an i n f i n i t e s e t o f i n v e r t e d p a i r s . We s h a l l prove t h a t i f Tm i s n o t a l r e a d y obtained, t h e n we o b t a i n a l s o an i n f i n i t e s e t o f p r e s e r v e d p a i r s : hence by t h e p r e c e d i n g statement, we o b t a i n t h e symmetric group .S,
THEORY OF RELATIONS
334
To t h i s end, c o n s i d e r two new elements s t i l l denoted by u u
< v (mod A and mod
. The
(mod A)
and
, such
v
B ) . Take a g a i n i n f i n i t e l y many elements between
u
that and
v
same argument as b e f o r e proves t h a t , i f t h e r e e x i s t i n f i n i t e l y many
u (mod B ) , o r i n f i n i t e l y many which a r e , t h e n we again o b t a i n t h e group Tm .
o f these elements which a r e l e s s t h a n greater than Finally
(mod B)
u and
suppose t h a t t h e r e e x i s t i n f i n i t e l y many o f these elements between
, modulo
v
v
and modulo
A
p a i r s , we o b t a i n t h e group reflection, o f
as w e l l . I n t h e case o f an i n f i n i t e s e t o f i n v e r t e d
B
o f t h e r e f l e c t i o n . T a k i n g t h e images, under t h i s
Jm
u, v and o u r i n f i n i t e i n v e r t e d s e t , we a g a i n o b t a i n Tm
. There
remains t h e case o f i n f i n i t e l y many p r e s e r v e d p a i r s . Together w i t h t h e p r e v i o u s l y o b t a i n e d i n f i n i t e s e t o f i n v e r t e d p a i r s , t h e y g i v e t h e symmetric group
S,
,in
view o f 5.5 above. 0 Let
5.7.
be an
G
group i n c l u d e d i n
m-ary group generated by a
Q - b i c h a i n . I f t h e maximum c a n o n i c a l
m. G
i_i Jm ( r e f l e c t i o n ) ,
G
=
Jm
.
0 By h y p o t h e s i s , t h e r e e x i s t s an i n v e r t e d p a i r . I f t h e r e e x i s t s as w e l l a p r e s e r v e d
, hence t h e d i h e d r a l group
G
maximum c a n o n i c a l group i n c l u d e d i n G
i s a t least
5.8. L e t
be an
G
group i n c l u d e d i n 0
Let
w a r y group generated by a G
be t h e c o n s i d e r e d b i c h a i n , and
AB
t h e r e e x i s t s an i n f i n i t e s u b s e t
U
of
E
then
.
G = Dm
be i t s base. By RAMSEY's theorem,
E
, a l l o f whose p a i r s a r e preserved, o r Tm i s i n c l u d e d
a l l o f whose p a i r s a r e i n v e r t e d . I f t h e y a r e preserved, t h e n s i n c e
, we
in
G
Dm
i s t h e maximum c a n o n i c a l group i n c l u d e d i n
have t h a t
G = Tm o r
Hence we have an i n f i n i t e s e t
, and
let
chain o f
B
that
i s included i n
G'
each o b t a i n e d by t a k i n g A
G
.
c o n t r a d i c t i n g o u r assumption t h a t
G
.
. The
group
G'
i n the ordering
rm o f t h e i n t e g e r s
AB-
G
more in
G' G
Since
#
Sm
B-
.
l,Z,
1,2,..
. Hence
by h y p o t h e s i s
by 5.3 we have
G ' = T,
Dm
.,m
F o r such an
s
...,m
s
rm belongs t c
, since
. We
s h a l l prove
i s generated by some p e r m u t a t i o n s
with
G , t h e n s belongs t o G as By t h e preceding, t h e b i c h a i n AB- has p r e s e r v e d p a i r s . Then a r e preserved, and s o A = B- , hence G = Jm , c o n t r a d i c t i n g AB- has a l s o an i n v e r t e d p a i r , and t h e n Tm i s i n c l u d e d i n of
denote t h e converse
B-
m-ary group generated by
m elements which we denote by
s ( l ) , s ( 2 ) ,...,s(m)
and
rmos o f t h e r e f l e c t i o n
by 5.3,
U w i t h inverted pairs. Let
be t h e
G'
G = S,
G ; the
Dm : c o n t r a d i c t i o n which proves
Q - b i c h a i n . I f t h e maximum c a n o n i c a l
( d i h e d r a l group),
Dm
1 2
hypothesis
Dm i s i n c l u d e d i n
.0
G = Jm
t h a t a l l the p a i r s are inverted, so t h a t
. By
G
Tm i s i n c l u d e d i n
t h e group
p a i r , t h e n by t h e p r e c e d i n g 5.6, Jm i s a l s o i n c l u d e d i n
s ,
i n the ordering
, the composition , i s an element
well. either a l l i t s pairs o u r assumptions. O r G'
by 5.6. F u r t h e r -
i s t h e maximum c a n o n i c a l group i n c l u d e d
. Then
G
, which
i s generated by t h e
335
Chapter 1 1
s belongs t o G ' ,
compositions rmos where rm i s t h e m-ary r e f l e c t i o n and s a t i s f i e s t h e i n c l u s i o n G 5 Dm , and t h u s G = Dm . 0 Now the f i v e Q - i n d i c a t i v e group theorem i s proved.
5.9. Given a chain A , we have introduced the c y c l i c r e l a t i o n associated w i t h A in ch.2 8.6. Here we a d d i t i o n a l l y need t h e notion of dihedral r e l a t i o n of A , which i s defined as t h e quaternary r e l a t i o n taking value (+) f o r ( x , y , z , t ) i f x < y < z < t (mod A) , and f o r a l l 4-tuoles obtained from the preceding by any permutation belonging t o t h e dihedral group D4 ( t r a n s l a t i o n s and r e f l e c t i o n s ) . Note t h a t t h e dihedral r e l a t i o n of A expresses t h a t x and z a r e s i t u a t e d in the two opposite i n t e r v a l s defined by y and t on the c y c l i c r e l a t i o n associat e d w i t h A . Thus we can say t h a t the dihedral r e l a t i o n i s a betweenness, o r an intermediacy, on t h e c y c l i c r e l a t i o n .
,
Let R be a r e l a t i o n a l system which i s f r e e l y i n t e r p r e t a b l e i n the chain Q of the r a t i o n a l s . Then t h e group of automorphisms of R i s e i t h e r the symmetric group, o r the group of i n c r e a s i n g b i j e c t i o n s , i . e . of automorphisms of Q , o r the group of i n c r e a s i n g b i j e c t i o n s and decreasing b i j e c t i o n s , i .e. automorphisms of the betweenness r e l a t i o n , o r the group of automorphisms of the c y c l i c r e l a t i o n associated with Q , o r f i n a l l y t h e group of b i j e c t i o n s which preserve o r which inverse the c y c l i c r e l a t i o n , i . e . automorphisms of the dihedral r e l a t i o n of Q (POUZET 1979). 0
Consider a p o s i t i v e i n t e g e r m and a s t r i c t l y increasing Q ) . Transform t h i s by an automorphism f of
... < xm (mod the
m-ary permutation such t h a t
f(xs(l))
< f(xs(2)) <
m-sequence x l < x 2 < R , and l e t s denote
... < f(xs(,,,))
(mod Q )
Let Gm be t h e m-ary group generated by these s , when f v a r i e s . This group xm . Moreover, does not depend on the choice of the increasing sequence x l , we can suppose t h a t t h e images f x a r e the x themselves, permuted by s . Thus
...,
Gm i s a subgroup of t h e group of automorphisms of the r e s t r i c t i o n R/{xl, ... ,xm). Notice t h a t i t can be a proper subgroup: f o r i n s t a n c e , take the r e l a t i o n R such t h a t R(x,y,z) = + i f f x < z and y < z and x # y .
To each automorphism f of R , a s s o c i a t e the bichain (Q,f"(Q)) and notice t h a t t h e m-ary group generated by this bichain i s included i n Gm Indeed, t h e inverse s-l i s nothing e l s e but the reordering of the x l , ,xm by f o ( Q ) . Moreover, as f v a r i e s , the entire group G, i s generated, s i n c e each s i s obtained from an automorphism f . From t h e f i v e Q - i n d i c a t i v e group theorem, i t then follows t h a t the sequence of the groups G, , a s m v a r i e s , i s one of t h e sequences, o r i n d i c a t o r s I , J , T , 0 , S . Each of t h e s e f i v e cases gives one o f the f i v e conclusions of our statement.
...
.
.
THEORY OF RELATIONS
336
5.10. SET-TRANSITIVE GROUP THEOREM
Let
i f f the closure o f
n
t h e f i v e groups o b t a i n e d as f o l l o w s , s t a r t i n g f r o m
is -
G
n-set-
under adherence i s one o f -
G a
. Then
E
be a group o f p e r m u t a t i o n s o f a denumerable s e t
G
t r a n s i t i v e f o r every i n t e g e r
chain
on
Q
, isomorphic
E
Q ;
w i t h t h e c h a i n o f t h e r a t i o n a l s : symmetric group; group o f automorphisms o f
group o f automorphisms o f betwenness (mod Q ) ; group o f automorphisms o f t h e c y c l i c Q ; group o f automorphisms o f t h e d i h e d r a l r e l a t i o n
r e l a t i o n associated w i t h of 0
(CAMERON 1976).
Q
Each o f t h e f i v e mentioned groups i s o b v i o u s l y
n-set-transitive
f o r each
n
,
and s i m i l a r l y so f o r any group y i e l d i n g one o f these f i v e groups under adherence. be a group o f p e r m u t a t i o n s o f a denumerable s e t E , which i s n . By 2.5.(2) above, t h e r e e x i s t s a homogeneous r e l a -
Conversely, l e t
G
n-set-transitive
f o r every
t i o n a l system rence. Since system
is
R
see 2.7.(2)
R G
whose automorphism group i s t h e c l o s u r e
, and so
G+
as w e l l , i s
n-monomorphic f o r e v e r y
above. I t f o l l o w s t h a t
n ; i n o t h e r words,
G
under adhe-
, the
n
i s monomorphic:
R
i s c h a i n a b l e : see ch.9
R
of
G+
n - s e t - t r a n s i t i v e f o r every
5
6.2, which can be
extended t o r e l a t i o n a l systems which a r e O - s e q u e n c e s o f r e l a t i o n s w i t h base
,
E
by s i m p l y t a k i n g t h e l i m i t . T h i s h o l d s i n s p i t e o f t h e use o f RAMSEY's theorem i n types o f m u l t i r e l a t i o n s o f c a r d i n a l i t y Let
A
be a c h a i n i n which
freely interpretable i n
f o l l o w s : t a k e each l o c a l isomorphism
geneous. Indeed by 1.3.(1)
Q
Q
f
from A
into
o f the
Q
, which i s o b t a i n e d as
Q
w i t h a f i n i t e domain
F
. Thus
above, which e a s i l y extends t o r e l a t i o n a l systems, S
, hence
R
w i l l be f r e e l y i n t e r p r e t a b l e i n an isomor-
R
o u r p r o p o s i t i o n w i l l f o l l o w f r o m t h e p r e c e d i n g 5.9.
. Let
S
, and l e t u be an a r b i t r a r y element i n t h e
F
and range
base. Take an isomorphism
h
F'
f
be a l o c a l automorphism o f
from the f i n i t e chain
Q/(Fu{u))
S
,
Now l e t us p r o v e t h e homogeneity o f w i t h f i n i t e domain
onto a r e s t r i c t i o n
.
A , and an isomorphism h ' f r o m Q / F ' o n t o a n o t h e r r e s t r i c t i o n o f A Let v = hu Then t h e b i j e c t i v e c o m p o s i t i o n hl,f,(h-l) i s a l o c a l automorphism o f R Since R i s homogeneous, by 1.1 t h e r e e x i s t s a l o c a l automorphism o f R which extends t h e p r e c e d i n g l o c a l automorphism t o t h e domain h"(F) augmented by v . L e t g denote t h i s e x t e n s i o n . Due t o t h e f a c t t h a t Q i s dense and w i t h o u t end-
of
.
p o i n t s , t h e r e e x i s t s a l o c a l isomorphism f r o m A of
h'-'
t o t h e domain
h"'(F')
F i n a l l y t h e b i j e c t i v e composition extends
,
f " ( R / F ) , and f i n a l l y t h e common e x t e n s i o n o f these images. S have t h e same age. I t s u f f i c e s t o prove t h a t S i s homo-
w i l l be isomorphic w i t h p h i c copy of
f i n i t e number o f isomorphism
.
i s f r e e l y i n t e r p r e t a b l e . Take t h e c h a i n
R
r a t i o n a l s , t h e n t h e system S t h e n t a k e t h e image The systems R and
n
,a
n
t h e beginning, which supposes f o r each i n t e g e r
f
t o i t s domain
F
augmented by h",g,h
augmented by
Q
into gv
. Let
, which h"
i s an e x t e n s i o n
denote t h i s e x t e n s i o n .
i s a l o c a l automorphism o f u
. Then
S
.
S
and
i s homogeneous by 1.1. 0
337
Chapter 11
§
6 - EXTENSIVE
SUBSET, PSEUDO-HOMOGENEOUS
RELATION
Given a relation R with base E , we say t h a t a subset F of E i s extensive (mod R) , i f each local automorphism of R with domain F i s extendible t o an automorphism of R . If F i s extensive and G i s a subset of E with R/G isomorphic t o R/F , then G i s extensive. i s extenNote t h a t R i s homogeneous i f f every f i n i t e subset of the base of R sive. A relation R with base E i s said t o be pseudo-homogeneous, i f each f i n i t e subset of E i s included in an extensive (mod R ) f i n i t e subset. For example, the relation of consecutivity on the positiveand negative integers, i . e . on Z , i s pseudo-homogeneous y e t n o t homogeneous. 0 Complete each f i n i t e s e t F of these integers by the l e a s t interval including F : we obtain a f i n i t e extensive superset of F . I n general F i t s e l f i s n o t extensive: take F = {0,25 and the local automorphism which preserves 0 and takes 2 i n t o 3 . This consecutivity relation associated with Z i s n o t rich f o r i t s age: the saturated, and thus rich denumerable relation representing t h i s age, i s obtained by taking denumerably many components each isomorphic with our consecutivity relation: see ch.10 5 6 and 7.3. Another example: the saturated t r e e , hence rich f o r i t s age, which i s described in ch.10 5 6.2, i s pseudo-homogeneous y e t n o t homogeneous. 0 Repeating our argument of ch.2 5 2.3, take three incomparable elements a , b , c with an element d < a and < b , y e t d I c , and three other incomparable eleThen the ments a ' , b ' , c ' with an element e ' < b ' and < c ' , y e t e ' I a ' local automorphism which takes a , b , c into a ' , b ' , c ' i s n o t extendible t o an automorphism, since the image of d would be < b ' , thus comparable with e '
.
and f i n a l l y less t h a n c ' , unless t h a t e ' be less than a ' To see t h a t our relation i s pseudo-homogeneous, complete each f i n i t e s e t F t o G in a manner t h a t any two elements of F have a common predecessor in G . 0 6.1. Every extensive subset relation i s maximalist (see The converse i s f a l s e : take components, each isomorphic
i s maximalist. Consequently, every pseudo-homogeneous ch.10 5 3 . 8 ) . a consecutivity relation formed of two, or several with the consecutivity relation on Z ; t h i s i s a
maximalist, non-pseudo-homogeneous r e l a t i o n . 6.2. In the case of a pseudo-homogeneous r e l a t i o n , the proposition 1.1 becomes: Let E be a denumerable s e t . Then a relation R based on E i s pseudo-homogeneous i f f there e x i s t s a s e t of f i n i t e subsets F of E such t h a t every f i n i t e
338
THEORY OF RELATIONS
subset i s included in an F ; and furthermore, f o r any local automorphism f of R having an F a s domain and f o r any f i n i t e subset G of E including t h i s F , t h e r e e x i s t s a local automorphism extending f t o G . 6.3. The proposition 1 . 3 . ( 1 ) can be extended: Any two denumerable pseudo-homogeneous r e l a t i o n s , w i t h same a r i t y and same age, a r e isomorphic. Let R and R ' be two denumerable pseudo-homogeneous r e l a t i o n s of the same age. S t a r t with a local isomorphism f from R i n t o R ' , whose domain F i s an extens i v e f i n i t e s u b s e t (mod R) . Let F ' = f " ( F ) be t h e range, and n o t i c e t h a t f o r any f i n i t e subset G ' of the base I R ' ( , including F ' , t h e inverse function f - l Indeed, i s extendible t o a local isomorphism from R ' i n t o R w i t h domain G ' by hypothesis t h e r e e x i s t s a local isomorphism g from R ' i n t o R w i t h domain G ' . Then the composition g,f i s a local automorphism of R with e x t e n s i v e domain F , hence g,f i s extendible t o an automorphism h of R ; and f i n a l l y h-',, g i s t h e desired local isomorphism from R ' i n t o R with domain G ' . Now i t s u f f i c e s t o choose G ' t o be extensive (mod R ' ) ; then by i t e r a t i n g t h i s , t o extend a l t e r n a t i v e l y the l o c a l isomorphism from R i n t o R ' , then the local isomorphism from R ' i n t o R , by taking successively each element of I R ( i n t h e domain of one, and each element of I R ' I in t h e domain of t h e o t h e r . 0
0
.
6.4. The proposition 1 . 3 . ( 2 ) cannot be extended t o pseudo-homogeneous r e l a t i o n s : see t h e example already given, of t h e c o n s e c u t i v i t y r e l a t i o n on Z . There even e x i s t s an age which i s represented by a pseudo-homogeneous r e l a t i o n , y e t by no homogeneous r e l a t i o n , and by no r i c h r e l a t i o n . Let C be the consecutivity r e l a t i o n on the natural numbers . Denote by 0 t h e s i n g l e t o n unary r e l a t i o n taking the value (+) f o r t h e i n t e g e r 0 only. Let A be
0
a unary r e l a t i o n such t h a t , f o r every f i n i t e sequence of (t) and ( - ) , t h e r e e x i s t s a sequence of consecutive i n t e g e r s giving t o A t h i s sequence of values. Then t h e t r i r e l a t i o n ( C , O , A ) i s pseudo-homogeneous, s i n c e each i n t e r v a l beginning w i t h zero and going up t o an a r b i t r a r y i n t e g e r , i s extensive with t h e i d e n t i t y a s the unique local automorphism. However, no denumerable r e l a t i o n i s r i c h f o r t h i s age, because of t h e e x i s t e n c e of continuum many possible components ( c o n s t r u c t e d f o r i n s t a n c e , by s t a r t i n g from t h e c o n s e c u t i v i t y r e l a t i o n on 2 ) which can be added, without strengthening the represented age (example communicated by POUZET; n o t i c e t h e analogy w i t h SPECKER's argument i n ch.10 5 6 . 3 ) . 0 6.5. Let
of 0
R
Let
R be a pseudo-homogeneous r e l a t i o n . Then t h e s e t of the r e s t r i c t i o n s t o extensive f i n i t e s u b s e t s (mod R ) i s amalaamable ( s e e 1 . 5 ) . A , B, C be t h r e e r e s t r i c t i o n s of T t o extensive f i n i t e subsets of the
Chapter 11
base. L e t
A
be an isomorphism f r o m
f
339
onto a r e s t r i c t i o n o f
B
, and g be an
.
A onto a r e s t r i c t i o n o f C The images o f t h e base I A I under f and under g a r e e x t e n s i v e s e t s . Thus t h e r e e x i s t s an automorphism f ' o f R , e x t e n d i n g f - l ; and s i m i l a r l y an automorphism g ' o f R , e x t e n d i n g g-' . Resisomorphism f r o m
trict
t o t h e domain
f'
finite restriction of
lBl
, and g '
p r o p e r t y i s s a t i s f i e d , s i n c e f o r each and
= x
(g',g)x
f'O(IB1) x
be t h e
D
The amalgamation
I A l , we have
(f',f)x
= x
t o prove t h a t e v e r y amalgamable s e t o f f i n i t e r e s t r i c -
, which generates t h e age o f
R
i n t h e base
; then l e t
.
. [3
We l e a v e i t t o t h e r e a d e r tions o f
ICl u g'"(lC1)
t o t h e domain
t o the union
R
under e m b e d d a b i l i t y , i s (up t o isomor-
R
phism) a subset o f t h e s e t o f r e s t r i c t i o n s o f
R
t o extensive f i n i t e sets.
6.6. PSEUDO-AMALGAMABLE AGE An age
d?,
i s s a i d t o be pseudo-amalgamable,
i s amalgamable, and f u r t h e r m o r e generates
i f t h e r e e x i s t s a subset o f
6a. under e m b e d d a b i l i t y ;
&
which
hence t h i s sub-
set i s directed. PSEUDO-AMALGAMATION THEOREM
, there
Given an age ,@ , iff -
6?,
e x i s t s a c o u n t a b l e pseudo-homogeneous r e p r e s e n t a t i v e o f
F (,!
i s pseudo-amalgamable (CALAIS 1967, t o whom t h e n o t i o n o f pseudo-amalgamable
age i s due). 0
Suppose t h a t t h e r e e x i s t s a oseudo-homogeneous r e l a t i o n
Then t h e s e t o f r e s t r i c t i o n s o f
which r e p r e s e n t s
R
6& .
t o e x t e n s i v e f i n i t e subsets o f t h e base, i s
R
amalgamable, by t h e p r e c e d i n g 6.5. And by d e f i n i t i o n o f t h e pseudo-homogeneous r e l a t i o n s , t h i s s e t covers e v e r y f i n i t e subset (under i n c l u s i o n ) ; t h u s o u r age i s pseudo-amalgamable. Conversely, suppose t h a t t h e age
6%
i s pseudo-amalgamable.
Ai
Let
(ii n t e g e r ) be
a f i n i t e , o r an W - s e q u e n c e o f f i n i t e r e l a t i o n s , which under e m b e d d a b i l i t y g i v e s
& , and which
f u r t h e r m o r e forms an amalgamable s e t .
We s h a l l c o n s t r u c t two sequences o f f i n i t e r e l a t i o n s of which i s isomorphic t o an embeddability tion of
Bi+l
t o a same
A
Aid
.
Bi
, and
Bi
Moreover, f o r any two r e s t r i c t i o n s o f
, we
and
Ci
( i i n t e g e r ) , each
A . ( j i n t e g e r ) ; such t h a t f o r each i we have t h e J Bi i s a r e s t r i c t i o n o f Ci , and Ci i s a restricBi
which a r e isomorphic
r e q u i r e t h a t e v e r y isomorphism f r o m one o n t o t h e o t h e r , be exten-
j d i b l e t o an isomorphism f r o m
Bi
onto another r e s t r i c t i o n o f
Ci
. With
t h i s cons-
t r u c t i o n , t h e common e x t e n s i o n o f t h e
Bi ( o r t h e Ci ) t o t h e u n i o n o f t h e i r bases, s a t i s f i e s t h e c o n d i t i o n o f 6.2 above, hence i s pseudo-homogeneous and r e p r e s e n t s t h e g i v e n age. F o r t h i s sketched c o n s t r u c t i o n , s t a r t w i t h
Bo = A.
. Suppose
that
Bi
i s already
obtained, and l e t fi,O , ... , fi ,p be a l l t h e l o c a l automorphisms o f betBi ( j v a r i a b l e ) . By amalgamation, by ween r e s t r i c t i o n s isomorphic t o a same Aj
3 40
THEORY OF RELATIONS
s t a r t i n g from t h e b i j e c t i o n
and f r o m t h e i d e n t i t y on t h e base
f.
130
I A J. 1 , which
A . i n t o a r e s t r i c t i o n o f Bi , we o b t a i n an e x t e n s i o n Di,o of J , which belongs t o o u r age , and such t h a t fi,O i s e x t e n d i b l e t o an isomor-
both transform
&,
Bi
phism f r o m Bi o n t o a n o t h e r r e s t r i c t i o n o f Di,o . I t e r a t e t h i s t o obtain the sequence, w i t h f i n i t e l e n g t h p , o f successive e x t e n s i o n s D i ,I 9 . . . "Ji ,I! c o r r e s ponding t o
fi,l
isomorphic t o an for
Bitl
a common e x t e n s i o n of
-
PREHOMOGENEOUSRELATION
7
§
,..., fi,p . Then t a k e f o r A . F i n a l l y t o be sure j
and
Ci
Ci
Di,p
an e x t e n s i o n o f
which i s
t h a t t h e e n t i r e age i s represented, t a k e Ai+l
, which f u r t h e r m o r e i s an
Aj
.0
G- EXTENSIVE SUBSET
Given a r e l a t i o n F
, we
of
R
say t h a t
F
F
G
and
G
including
G-extensive (mod R ) , i f f o r e v e r y l o c a l automorphism f , t h e r e s t r i c t i o n f / F i s e x t e n d i b l e t o an automorphism o f R . = F , we f i n d t h e e x t e n d i b l e f i n i t e s e t , i n t h e sense o f 5 6.
G
12G-extensive
and i f
F', G'
R , then F '
l o c a l automorphism o f If
F
is
w i t h domain
I n t h e case where
If
and two f i n i t e subsets o f i t s base, say
R
a r e t h e images o f
F, G
under t h e same
i? G I - e x t e n s i v e .
F i s G-extensive, e v e r y subset o f F i s G-extensive; f u r t h e r m o r e F o r F i s GI-extensive f o r every f i n i t e superset G' o f G .
each subset o f
A r e l a t i o n R i s s a i d t o be Prehomoqeneous, i f each f i n i t e subset admits a f i n i t e s u p e r s e t G f o r which F i s G-extensive. Every homogeneous o r pseudo-homogeneous r e l a t i o n i s orehomogeneous. 7.1.
If
F
admits a f i n i t e s u p e r s e t
G
f o r which
F
F o f i t s base
G-extensive,
then
&
F
maximal i s t . 0 Take any e x t e n s i o n
S
R which r e p r e s e n t s t h e same age. Given an a r b i t r a r y
of
H o f t h e base
f i n i t e subset
IS1 , i n c l u d i n g G , t a k e an isomorphism h f r o m
S/H o n t o t h e r e s t r i c t i o n R/ho(H) . By h y p o t h e s i s h/F i s e x t e n d i b l e t o an automorphism o f R . By composition, t h e r e e x i s t s an F-isomorphism f r o m S/H o n t o a restriction of subset, ch.10
R : we see t h a t
5
F
s a t i s f i e s the t h i r d d e f i n i t i o n o f a maximalist
3.8. 0
Consequently, e v e r y prehomogeneous r e l a t i o n i s m a x i m a l i s t . The converse i s f a l s e : t a k e up t h e counterexample i n 6.1;
i.e.
a consecutivity rela-
t i o n formed o f s e v e r a l components, each i s o m o r p h i c w i t h t h e c o n s e c u t i v i t y on 7.2. Any two denumerable prehomogeneous r e l a t i o n s w i t h t h e same age
( g e n e r a l i z a t i o n o f 1.3.(1)
and 6 . 3 ) .
2
.
are isomorphic
341
Chapter 11
Let R , R ' be two denumerable prehomogeneous relations with bases E , E ' , representing the same age. Let F be a G-extensive f i n i t e subset (mod R ) and l e t f be a local isomorphism from R into R ' , with domain G . Let G ' be the range of f . Take a f i n i t e superset H ' of G ' f o r which G ' i s HI-extensive (mod R ' ) . By the same argument as i n , 6 . 3 , we see t h a t the inverse function (f/F)-' i s extendible t o a local isomorphism g ' from R ' into R with domain i s extendible t o a local isomorH ' . Similarly the inverse function ( g ' / G ' ) - ' phism from R into R ' , and so f o r t h , going alternatively from R i n t o R ' and from R ' i n t o R . To terminate the proof and obtain a common extension of our local isomorphisms which i s an isomorphism from R onto R ' , i t suffices t o note t h a t F can contain an a r b i t r a r i l y given element of E , then t h a t G ' can cont a i n an a r b i t r a r i l y given element of E ' , and so f o r t h . 0
'3
7.3. RELATION
RA
Given a chain A , l e t RA be the ternary relation freely interpretable in A which i s defined by RA(x,y,z) = + i f f x < z and y < z (mod A) and x # y (1) Let
Q
be the chain of the rationals; then
RQ
,
.
i s prehomogeneous b u t n o t
pseudo-homogeneous. 0 Given a f i n i t e subset F of the base, l e t G be the s e t F augmented by one Every element which i s s t r i c t l y less (mod Q ) than the minimum element of F local automorphism f of R with domain G , preserves the ordering of the Q elements (mod Q ) , except when concerning the f i r s t two elements of G , where the ordering can be inverted. Hence the ordering of the elements of F i s preserved; so t h a t f/F i s extendible t o an automorohism of Q , hence of RQ Thus F i s G-extensive (mod R ) ; y e t F i s n o t extensive. 0
.
.
Q
( 2 ) Every denumerable younger relation t h a n R i s of the form RA , where A Q i s a denumerable chain. 0 To each f i n i t e subset F of the base, associate the s e t of those chains C based on F and such t h a t R C be the r e s t r i c t i o n t o F of the given relation. Then apply the coherence lemma ch.2 5 1.3. 0
Consequently RQ i s rich f o r i t s age. Hence there e x i s t s an age having a rich representative, y e t having no denumerable pseudo-homogeneous representative. 0 Indeed a denumerable pseudo-homogeneous representative of the age under consideration, would a f o r t i o r i be prehomogeneous, hence isomorphic with R Q . O Compare with the counterexamples in 6.4 above. 7.4. ISOLATING PAIR, ISOLATED REL-AGE Let b;L be an age, and A , B be two f i n i t e relations belonging t o an extension of A
.
& , with
B
THEORY OF RELATIONS
342 The o r d e r e d p a i r
(A,B)
and o n l y one maximal
i s s a i d t o be i s o l a t i n g
We say as w e l l t h a t t h i s maximal F o r example, t a k e
&
) iff
t h e r e e x i s t s one
which c o n t a i n s t h e element
A-age i s i s o l a t e d by
B
.
B
.
& t o be t h e age r e p r e s e n t e d by t h e c o n s e c u t i v i t y r e l a t i o n on
.
t h e n a t u r a l numbers the ordered p a i r
(mod
bE
A-age s p e c i f i c a t i o n o f
If
i s t h e consecutivity r e l a t i o n o f a f i n i t e chain, then
A
i s i s o l a t i n g , and t h e c o r r e s p o n d i n g i s o l a t e d maximal
(A,A)
A-age i s formed o f a l l t h e c o n s e c u t i v i t y r e l a t i o n s o f f i n i t e c h a i n s i n c l u d i n g A as an i n t e r v a l , and t h e i r
A
Now denote by
A-restrictions.
t h e common e x t e n s i o n o f t h e c o n s e c u t i v i t y r e l a t i o n s o f two f i n i t e
chains, which we s h a l l c a l l t h e two components, w i t h t h e v a l u e ( - ) f o r those order e d p a i r s formed o f one element i n each component. A-ages, o b t a i n e d by d e c i d i n g t o p u t t h e
Then t h e r e e x i s t i n f i n i t e l y many maximal
f i r s t component b e f o r e o r a f t e r t h e second, w i t h a non-zero f i n i t e number o f i n t e r mediate elements. T h i s can be done w i t h t h e a i d o f a f i n i t e c o n s e c u t i v i t y r e l a t i o n
B , which i s an e x t e n s i o n o f
denoted by
A
.
Thus such an o r d e r e d p a i r
(A,B)
is
isolating. Now w i t h t h e same
, define
A
the
A-age o b t a i n e d by never t a k i n g t h e u n i o n o f t h e
two components w i t h f i n i t e l y many i n t e r m e d i a t e elements. Then t h e maximal
A-age
t h u s d e f i n e d i s i s o l a t e d by no o r d e r e d p a i r o f r e l a t i o n s . (A,B)
If -
i s i s o l a t i n g (mod
f i n i t e extension o f
A-age s p e c i f i c a t i o n o f (A,B)
If
be t h e
be t h e unique maximal
&' and of
5
3.6.
f o r every
; hence
5
Suppose t h a t t h e r e e x i s t s a maximal B
, and
f' ( i n d u c e d
by
. Then
. d'
B belongs. L e t B and i s i t s e l f
f'
A'-age
t h e r e e x i s t s an
distinct
A-age s p e c i f i c a t i o n
i s maximal. By h y p o t h e s i s
f
which i s a
A ' Lf A
f o r each r e s t r i c t i o n
3 . 3 ) . So A ' c o n t a i n s
B
B'
a l s o belongs t o t h e
B'
A-age t o which t h e r e l a t i o n
c o n t a i n i n g t h e element
, and hence
(A,B')
.
B
(A',B)
(ch.10
r' , which c o n t a i n s
with
so i s
and i s o l a t e d by
A'-age induced by
maximal, by ch.10 from
&
i s i s o l a t i n g , t h e n so i s
A
0 Let
6 L ) , then
and belongs t o t h e age 6%
B
f
i s identical
) i s i d e n t i c a l w i t h c d ' ( i n d u c e d by
&
):
contradiction. 0 7.5. L e t F
be a r e l a t i o n ,
R
&
t h e age r e p r e s e n t e d by
, two f i n i t e subsets o f t h e base.
e x i s t s a unique maximal ment -
R/G
F
j s -
R , and
-
F
G-extensive (mod R)
(R/F)-age s p e c i f i c a t i o n o f
bt , which
,
including
G
, then
there
contains the ele-
.
Moreover, t h e 0 Suppose t h a t
(R/F)-age t h u s i s o l a t e d by F
specifications o f extension
If
U'
of
is
i s t h a t which i s r e p r e s e n t e d by
R/G
G-extensive, y e t t h a t t h e r e e x i s t two maximal
(R/F)-age R/G I n t h e f i r s t , t a k e an R/G , and i n t h e second, t a k e an e x t e n s i o n U " , such t h a t no
fi , which
(R/F)-age s p e c i f i c a t i o n o f
&
.
c o n t a i n t h e element contains both
U'
and
U " : see ch.10
5
3.7.
R
.
Chapter 11
343
H ' , H " o f t h e base, w i t h R / H ' t h e image o f U ' f ' , and R/H" t h e image o f U " under f " . Set F' = f ' ' ( F ) and G ' = f ' " ( G ) , and s i m i l a r l y F" = f " " ( F ) and G" = f " " ( G ) By t h e h y p o t h e s i s o f G-extensiveness, t h e r e e x i s t s an automorphism h ' o f R Now t a k e two f i n i t e subsets
under an isomorphism denoted by
which t a k e s
F' i n t o F w i t h h ' / F '
which takes
F"
t e d by
into
F with
c o n t a i n s t h e element
R
F-isomorphism;
= (f'/F)-'
h"/F" = ( f " / F ) - '
, which
R/h'"(H')
as w e l l as t h e element
and a n o t h e r automorphism
.
R/h""(H")
Then t h e
R/G
i s t h e image o f
age which c o n t a i n s t h e element is
d! , t h e r e
&
. F o r each e x t e n s i o n
R/G
e x i s t s an isomorphism f r o m
o u r p r e v i o u s isomorphism be an age r e p r e s e n t e d by R
:
(R/F)-age con-
denote t h e unique maximal
. And
R
of
U
F-isomorphism. Thus
R
d! i s
, we
(R/F)-
, belonging
R/G
onto a r e s t r i c t i o n o f
U
G-extensive, by c o m p o s i t i o n w i t h an automorphism o f
age o f
under an
U'
, which i s t h e image o f U"
.
We s h a l l now prove o u r second c o n c l u s i o n . L e t to
h"
(R/F)-age represen-
c o n t r a d i c t i o n . Thus we have Droved t h e uniqueness o f t h e maximal taining
.
R
.
Since
F
can r e q u i r e t h a t
included i n the
s i n c e i t i s maximal, i t i s i d e n t i c a l w i t h t h e
(R/F)(R/F)-
.
7.6. L e t R be a r e l a t i o n and d?, t h e age r e p r e s e n t e d by R . Suppose t h a t R i s m a x i m a l i s t , and l e t F and G i n c l u d i n g F
, be
two f i n i t e
i s i s o l a t i n g (mod @, ) . (R/F,R/G) w i t h domain G , t h e r e s t r i c t i o n h/F
subsets o f t h e base, such t h a t t h e o r d e r e d p a i r Then f o r e v e r y l o c a l automorphism
h
iz 1-isomorphism
of -
R
itself.
0
Let
F ' = h"(F)
from and
into
R
G ' = h"(G)
. Since
(R/F')-age r e p r e s e n t e d by
R
I n t a k i n g i t s image under
h-l/F'
t h e element
, and R
t a k e t h e image under i s maximalist, t h i s
, we o b t a i n a maximal
hence a l s o t h e i n v e r s e f u n c t i o n
R
o f the
(R/F')-age
i s maximal.
(R/F)-age which c o n t a i n s
(R/F)-age r e p r e s e n t e d by
,is
h/F
a
R
1-isomorphism f r o m
R be a r e l a t i o n w i t h denumerable base E
p r e s e n t e d by
h-l/F'
( u p t o F-isomorphism). Since t h e o r d e r e d p a i r (R/F,R/G)
R/G
i s o l a t i n g , we o b t a i n p r e c i s e l y t h e
7.7. L e t
R
, and
let
. Thus
is
h-'/F'
,
into itself. 0
R
be t h e age r e -
. R i s m a x i m a l i s t , and
Suppose t h a t ( 1 )
( 2 ) f o r each f i n i t e subset
F
F , such t h a t t h e o r d e r e d p a i r
of
E
, t h e r e e x i s t s a f i n i t e subset G i n c l u d i n g
(R/F,R/G)
i s i s o l a t i n g (mod
R).
Then
R
i s prehomogeneous. More p r e c i s e l y , f o r each of t h e p r e c e d i n g ordered p a i r s
(F,G)
,
t h e subset
0 Let
f
F
&
G-extensive (mod R)
be a l o c a l automorphism o f
automorphism of
R
extending
f/F
R
.
w i t h domain
as f o l l o w s . Set
G
. We
s h a l l c o n s t r u c t an
F' = f"(F)
and
G'
= f"(G)
,
344 and l e t
THEORY OF RELATIONS
F1
be an arbitrary f i n i t e superset of
F1 , such t h a t
(R/F1,R/G1)
F , and
G1
f i n i t e s e t including
i s i s o l a t i n g . By the preceding proposition,
1-isomorphism from R into R t o the domain G1 , which i s a Iterating t h i s , by taking each and in the range, we obtain an union i s an automorphism of R
f/F
is a
. Thus there e x i s t s a function f l extending f/F local automorphism of R . element of the base, alternatively in the domain a-sequence of extensive local automorphisms, whose extending f . 0
7.8. Let & be an age. Then e i t h e r , f o r each f i n i t e relation A belonging t o & , there e x i s t s a f i n i t e extension B of A belonging t o , such t h a t ( A , B ) isolating (mod ) ; or there e x i s t continuum many maximal rel-ages specifications -
of m. . -
Let A be an element of & , having no extension which together with A forms an isolating p a i r . Take two d i s t i n c t maximal A-ages soecifications of fl , and by ch.10 5 3.7, take a relation A. belonging t o the f i r s t and a relation A1 belonging t o the second, such t h a t there e x i s t s no A-age specification of (R , which contains b o t h A. and A1 . By hypothesis, among the relations belonging t o & and extending A. , there e x i s t s none which together with A forms an isolating o a i r . Thus there e x i s t two d i s t i n c t maximal A-ages specifications of R , each of which contains the element A. . Using again ch.10 5 3.7, take an element Ao,O belonging t o the f i r s t and
0
AO,l belonging t o the second, such t h a t there e x i s t s no A-age specification of & , which contains b o t h A0 3 0 and AO,l . Moreover, since each A-age i s directed, we can always require t h a t A and A be extensions o f A. . 0 ,O
091
Dichotomously, in a similar fashion, obtain A and A s t a r t i n g with A1 . 1,o 1 9 1 Finally we obtain continuum many A-ages which we can always take t o be maximal. 0 7 . 9 . EXISTENCE CRITERION FOR A PREHOMOGENEOUS RELATION (1) Given an age , there e x i s t s a denumerable prehomogeneous representative i f f f o r each f i n i t e relation A belonging t o & , there e x i s t s a f i n i t e "f (R extension B A belonging t o , such t h a t ( A , B ) i s i s o l a t i n g (mod ) ( 2 ) Consequently, i f there e x i s t s a denumerable representative o f which i s rich f o r &. , then there e x i s t s a denumerable prehomogeneous representative of & (POUZET 1972). -
fi
of
By the exists a rel-ages ment A
0
existence c r i t e r i o n f o r a rich relation (see ch.10 § 7 . 6 ) , i f there relation rich f o r i t s age bt , then there are countably many maximal specifications of & . Hence by the preceding proposition, f o r each eleof & , there e x i s t s an extension B of A belonging t o such t h a t
8,
.
345
Chapter 1 1
the p a i r
i s i s o l a t i n g . Thus our conclusion ( 2 ) f o l l o w s from ( 1 ) .
(A,B)
To prove (l), note f i r s t t h a t , i f there e x i s t s a prehomogeneous representative o f our age, hence i f f o r every f i n i t e subset G
including
F
F i s G-extensive, then by 7.5 our c r i t e r i o n i s
and f o r which
s a t i s f i e d , since t h e p a i r
R
F o f t h e base, there e x i s t s a f i n i t e
(R/F,R/G)
i s isolating.
Conversely, assume the c r i t e r i o n i n statement ( 1 ) . S t a r t w i t h an a-sequence o f f i n i t e relations
( i integer)
Ai
belonging t o our age
& , each
element o f t h i s
age being embeddable i n an
B1
.
Bo = A.
Set
Ai . be an element o f
Co
61
be an element i n
be such t h a t Bi
Let
(B1,C1)
fi
such t h a t
(Bo,Co)
which i s a common extension o f
Co
i s isolating. Let
and A1 ; then l e t
i s i s o l a t i n g , and so f o r t h . The common extension
( o r e q u i v a l e n t l y the
C1
o f the
R
Ci
) i s a r e p r e s e n t a t i v e o f our age, and s a t i s f i e s the
R
i s maximalist, which i s the f i r s t hypothesis o f 7.7.
second hypothesis o f 7.7. It remains t o show t h a t
To t h a t end, i t s u f f i c e s t o see t h a t each f i n i t e subset o f the base
I R I i s inclu-
ded i n a maximalist f i n i t e subset. Indeed modify t h e preceding c o n s t r u c t i o n as follows. When we o b t a i n
Co
, take
a f i n i t e extension
which i s chosen so as t o make the
Do
Bo-age represented by
t o admit an embedding o f every extension o f
which belongs t o the unique maximal take a f i n i t e extension every extension o f maximal
Bo
R
maximal. Thus choose
t o one a d d i t i o n a l element
Bo-age i s o l a t e d by
Co
. When
we o b t a i n
C1
,
D1 belonging t o our age, which admits an embedding o f t o two a d d i t i o n a l elements, which belongs t o the unique
Bo-age i s o l a t e d by
i s o l a t e d by
Bo
belonging t o our age,
Do
, and
Co
which belongs t o the unique maximal
B1-age
C1 ; and so f o r t h . 0
Recall t h a t t h e r e e x i s t s an age having a prehomogeneous, and even a pseudo-homogeneous representative, y e t having no r i c h representative: see 6.4 above.
5 8 - SET-HOMOGENEOUS RELATION, We say t h a t a f i n i t e subset (mod R)
,
i f f o r every subset
an automorphism o f R/G
onto
If
F
R
taking
F o f t h e base o f a r e l a t i o n R
R/G
G
with
F
into G
isomorphic t o
. Yet
i s set-extensive R/F
, there e x i s t s
n o t each isomorphism from R/F
i s so e x t e n d i b l e .
i s set-extensive and
A relation
RELATION
SET-PSEUDO-HOMOGENEOUS
R/G
i s isomorphic t o
R/F
,then
G
i s set-extensive.
i s s a i d t o be set-homogeneous, i f each f i n i t e subset o f i t s base
i s set-extensive.
A r e l a t i o n i s set-pseudo-homogeneous, i f each f i n i t e subset o f i t s base i s i n c l u ded i n a f i n i t e set-extensive subset.
346
8.1. Given the chain
Q
THEORY OF RELATIONS
Q of the r a t i o n a l s , every relation freely interpretable in
i s set-homogeneous.
For every f i n i t e subset F of the base and every G of the same c a r d i n a l i t y , the isomorphism from Q/F onto Q/G i s extendible t o an automorphism of Q , hence of the given r e l a t i o n , since i t i s freely interpretable in Q . 0
0
For example the relation RQ defined in 7 . 2 , taking the value (+) i f f x < z y < z (mod Q ) and x f y , i s set-homogeneous, y e t n o t pseudo-homogeneous.
and
The consecutivity relation on Z i s pseudo-homogeneous (by associating t o each f i n i t e s e t F of positiveor negative integers, the smallest interval including Hence i t i s set-pseudo-homogeneous, y e t not set-homogeneous.
F ).
Problem. Does there e x i s t a set-pseudo-homogeneous relation which i s neither s e t homogeneous nor pseudo-homogeneous. 8.2. Every f i n i t e set-extensive subset i s maximalist. Consequently, every set-homogeneous or set-pseudo-homogeneous relation i s maximal i s t .
R be a relation and F be a f i n i t e set-extensive subset of the base. Let be a maximal (R/F)-age specification of the age represented by R This rel-age i s generated by an a-sequence of f i n i t e relations Ai ( i integer) , a l l extending R/F , with each A i + l an extension of Ai . For each i , there e x i s t s a r e s t r i c t i o n of R which i s the isomorphic image of Ai under a function denoted by f i . Since F i s set-extensive, there e x i s t s an automorphism g i of R which takes f j o ( F ) i n t o F Since F i s f i n i t e , there are only f i n i t e l y many permutations of F . Hence there e x i s t s an o-sequence of indices i f o r which the composition giofi i s the same permutation of F , say h I t follows t h a t the (R/F)-age represented by R i s a superset of the image of J% under h . Since i s maximal, i t s image under h i s maximal. Finally the (R/F)-age represented by R i s maximal. 0
0
Let
.
.
.
R be a relation and F a f i n i t e subset of i t s base. of F , If F i s set-extensive (mod R ) , then there e x i s t s a f i n i t e superset G such t h a t the pair ( R / F , R / G ) i s isolating modulo the age of R Moreover, the unique maximal (R/F)-age which contains R/G i s the rel-age represented by R . Consequently, every set-homogeneous or set-pseudo-homogeneous relation i s prehomogeneous, by 7.7 (communicated by POUZET). 8.3. Let
.
We know already t h a t F i s maximalist; hence the (R/F)-age A0 represented by R i s maximal. The same holds f o r each o f the f i n i t e l y many (R/F)-ages which are images of under an a r b i t r a r y automorphism of R/F . We shall denote
Chapter 11
phisms o f
, .. . , 4
4
these r e l - a g e s by
p-l
d !
( i = 0,1,.
Ai
We s h a l l f i r s t p r o v e t h a t t h e
.
o f t h e age o f
R
t i n c t from the
Ai .
a restriction of
A,
. . ,p-1) .
a r e t h e o n l y maximal
R
63
Take i n
"4 .
(R/F)-ages s p e c i f i c a t i o n s
Since
B
.
C
'4
there e x i s t s a f i n i t e superset (R/F)-ages a r e t h e
of
G
A
. Since
F
i s s e t - e x t e n s i v e , by
F
LAo
which belongs t o
i s the
of
,
B
: contradiction.
R/F
td0
Since
dis-
, we can t r a n s f o r m B i n t o an e x t e n s i o n
R
which belongs t o one o f t h e
belongs t o no o t h e r dfi
@
(R/F)-age
B which extends R/F and belongs t o t h e age o f R , t h e r e e x i s t s
which i s isomorphic w i t h
Now c o n s i d e r a f i n i t e e x t e n s i o n
and
a finite relation
c o m p o s i t i o n w i t h an automorphism o f
maximal
N o t i c e t h a t c e r t a i n automor-
Indeed, suppose n o t and c o n s i d e r a maximal
does n o t belong t o any
R/F
.
(p integer)
, which a r e d i f f e r e n t f r o m t h e i d e n t i t y , can p r e s e r v e
R/F
consequently p r e s e r v e each
of
347
b u t which
(R/F)-age r e p r e s e n t e d by
such t h a t
. But
C = R/G
i t follows t h a t the p a i r
R
,
since the only
(R/F,R/G)
i s isola-
ting: our conclusion follows. 0 Consequently, any two denumerable set-pseudo-homogeneous age
r e l a t i o n s o f t h e same
a r e isomorphic.
8.4. RELATION
CA
Given a c h a i n
A
that -
CA(x,y,z)
Letting
, we let =
+
iff
CA x # y
denote t h e t e r n a r y r e l a t i o n on t h e same base, and
z
i s consecutive t o
Max(x,y)
denote t h e c h a i n o f t h e p o s i t i v e a n d n e g a t i v e i n t e g e r s , t h e n
Z
such
.
(mod A) Cz
5
prehomogeneous b u t n o t set-pseudo-homogeneous. 0
i s a g i v e n i n t e g e r , and a3, a4, p-2 c o n s e c u t i v e ( p o s i t i w o r n e g a t i v e ) i n t e g e r s , and i f
To see t h e prehomogeneity, n o t e t h a t i f p 2 3
...
, ap
al,
a2
domain
i s a sequence o f satisfy
{
al
Max(al,a2)
,..., aP 5
= a3
-
1
, then
a l o c a l automorphism o f
w i l l necessary t a k e each
ai
( i = 1,..., p )
b3, b49 , bp c o n s e c u t i v e i n t e g e r s and Max(bl,b2) l o c a l automorphism, when r e s t r i c t e d t o { a3, ... ,ap) . - a
phism o f
Cz
.
Now we see t h a t
Cz
i s n o t set-pseudo-homogeneous.
= b3
,is
-
1
with
Cz
into
. So
bi
with
that this
e x t e n d i b l e t o an automor-
Indeed, e i t h e r t h e domain o f
a g i v e n l o c a l automorphism, i s formed o f c o n s e c u t i v e i n t e g e r s , i n which case t h e f i r s t two elements can be t r a n s f o r m e d i n t o two non-consecutive elements; o r t h e r e e x i s t i n t h e domain, a t l e a s t two non-consecutive i n t e g e r s , which can be t a k e n i n t o two elements such t h a t t h e l e n g t h o f t h e i n t e r v a l s e p a r a t i n g
i s n o t preserved.
0
them
THEORY OF RELATIONS
3 48 EXERCISE 1
-
PERMUTATION GROUP GENERATED BY THE FINITE LOCAL AUTOMORPHISMS OF A
RELATION Let
be a r e l a t i o n , o r a r e l a t i o n a l system, w i t h base
R
o f permutations o f
GR
l o c a l automorphism o f
, which
E R
E
.
Consider t h e groups
a r e c l o s e d under adherence, and such t h a t e v e r y
w i t h f i n i t e domain i s e x t e n d i b l e t o an element o f GR . GR f o r e v e r y
Note t h a t t h e symmetric group ( g r o u p o f a l l t h e p e r m u t a t i o n s ) i s a system
.
R
i s a group, and s h a l l be c a l l e d t h e group generaGR t e d bv t h e f i n i t e l o c a l automorohisms o f R
The i n t e r s e c t i o n o f a l l these
.
1 - Note t h a t t h e group t h u s generated, i s c l o s e d under adherence, and t h a t e v e r y
automorphism o f 2
-
belongs t o i t .
R
I n t h e case o f a homogeneous r e l a t i o n a l system
i d e n t i c a l t o t h e group o f automorphisms o f Problem 1
.
. I n o t h e r words,
Is t h e group t h u s generated always a GR
f i n i t e l o c a l automorphism o f the
, t h e group generated i s
R
.
R
extendible t o a permutation belonging t o a l l
R
.
GR
.
Problem 2
Take
t o be t h e c h a i n o f t h e n a t u r a l numbers
R
i s every
. What
i s t h e group
generated. Is i t i d e n t i c a l t o t h e symmetric group, o r i s i t reduced t o t h e i d e n t i t y . EXERCISE 2 - THE ONLY INDICATIVE GROUPS ARE THE I , J , T, D, S F o l l o w i n g § 5 above, we c a l l c a n o n i c a l groups those groups I , J , T, D, S
bed i n
9
4.2.
1 - Let
G
be an i n d i c a t i v e
bichain
AB
.
included
m-ary group, i . e .
a group generated by an i n f i n i t e
Suppose t h a t t h e r e e x i s t s an i n f i n i t e subset
U
o f t h e base
. Show t h a t , i f G does n o t c o n t a i n any p e r m u t a t i o n w i t h (1,2, ...,m) , generates t h e symmetric group Sm , t h e n G i n Tm (group o f t r a n s l a t i o n s , generated by (1,2, ...,m)).
f o r which together
descri
A/U = B/U
F o r each p a i r o f
IABl
which, is
m-ary c a n o n i c a l groups, v e r i f y t h a t t h e u n i o n group, more p r e c i -
s e l y t h e group generated by t h e i r union, i s c a n o n i c a l ( i f i t seems t o o l o n g , t h e reader w i l l assume t h i s ) . Consequently, f o r each group c a n o n i c a l group i n c l u d e d i n
G
Conclude f r o m t h e preceding, t h a t i f
G
maximum c a n o n i c a l group i n c l u d e d i n G 2
-
in
Let G
G is
, there
e x i s t s a maximum
i s an i n d i c a t i v e m-ary group, and i f t h e is
Tm
, then
G = Tm
.
be an i n d i c a t i v e m-ary group. I f t h e maximum c a n o n i c a l group i n c l u d e d Dm
, then G
then e v e r y subset
-
G
.
U
= Dm
. Furthermore,
o f t h e base f o r which
if
AB
i s a bichain generating
A/U = B/U
i s finite.
Let
G
be an i n d i c a t i v e m-ary group generated by an i n f i n i t e b i c h a i n
and l e t
U
be an i n f i n i t e s e t w i t h
3
the permutation
(1,2, ...,m)
A/U = B/U
. Suppose
and t h a t t h e r e e x i s t
,
D,
that
p, q (ptq
G
6 m)
AB
,
does n o t c o n t a i n such t h a t
c o n t a i n s none o f t h e f o l l o w i n g types o f p e r m u t a t i o n s : those which t a k e
ptl
G into
349
Chapter 11
an integer ,< p and preserve p t 2 , p+3, ... , m ; and those which take m-q an integer < m-q and preserve 1, 2 , . . . , m-q-1 . Then G i s included in
into
Iiyq
4 - Let G be an indicative m-ary group. If the maximum canonical group included . in G i s I i y q (with p+q sm) , then G =
Iiyq
5 - If the maximum canonical group included in
G
=
JI
. Moreover,
if
AB
G
i s a bichain generating
is
JI
(with 2 r d m ) , then
JL , then every
U
f o r which
A/U = B/U , i s f i n i t e . Recall t h a t these r e s u l t s are due t o FRASNAY 1965; see also FRAISSi 1974’. EXERCISE 3 - THE NUMBER OF m-ARY INDICATIVE GROUPS For m = 0 , 1, 2 , 3, a l l m-ary permutation groups are indicative. For instance the ternary groups are S3 (symmetric group), 1;” ( i d e n t i t y g r o u p ) , 1;’’ (transposition ( 1 , 2 ) ) , 1;” (transposition ( 2 , 3 ) ) , J i (transposition ( 1 , 3 ) ) , and T3 (cyclic group). Show t h a t there e x i s t exactly eleven indicative quaternary indicative groups, namely S4 (symmetric grouo), 1;” ( i d e n t i t y group), I i y 2 (transposition (3,4)), 1;’’
(transposition (1,2)),
1;”
(union of the two preceding groups),
I i y 3 (symmetric group on the s e t 2,3,4 ) , 1 2 J4 (reflection), J4
1;’’
(symmetric group on
1,2,3 ) ,
(generated by the reflection and the two transpositions
( 1 , 2 ) and ( 3 , 4 ) ) , T4 (cyclic g r o u p ) ,
D4 (dihedral group).
Show t h a t , for m a t l e a s t equal t o 4 , the number of indicative m-ary groups 2 i s 3 plus the maximum integer 4 m / 2 (communicated by FRASNAY).
35 1
CHAPTER 12
BOUND OF A RELATION; WELL RELATION; REASSEMBLING THEOREM
5 1 - BOUND OF A
RELATION
Given a relation R , a bound of R i s any f i n i t e relation A with same a r i t y , such t h a t A is not embeddable i n R , b u t every proper restriction of A i s embeddable i n R . If we consider the partial ordering of embeddability between f i n i t e relations (considered up t o isomorphism), and the i n i t i a l interval formed of a l l f i n i t e restrictions of R , then we find the bound of t h i s i n i t i a l interval, in the sense of ch.4 5 8. If R has f i n i t e cardinality p , then any bound o f R has cardinality p+l
.
Examples. For a reflexive binary relation, the binary relation of cardinality 1 which takes the value ( - ) , i s a bound. For a reflexive, symmetric relation, we have the preceding bound plus the chain o f cardinality 2 . For a binary relation always (+) of cardinality 3 1 , we have the two preceding bounds plus the identity relation of cardinality 2 . For an i n f i n i t e relation always (t), we only have these three bounds. For a relation always (+) of f i n i t e cardinality p , we additionally have as a bound the relation always (+) of cardinality p t l . For a reflexive, antisymmetric relation, we have as a bound the relation of cardinality 1 taking the value (-), and the relation always (+) of cardinality 2 . For a reflexive, antisymmetric and comparable relation ( f o r any x, y the relation takes value (t) either f o r (x,y) or for (y,x) ) , we have the two preceding bounds plus the identity relation o f cardinality 2 . For a chain of cardinality 3 2 , we have the three preceding bounds, plus the If the chain is i n f i n i t e , then these reflexive binary cycle of cardinality 3 four bounds are the only possible. If the chain has f i n i t e cardinality p , then i n addition we have as a bound the chain of cardinality p t l .
.
A f i n i t e relation, a chain, a relation always ( t ) , an identity relation, a l l have f i n i t e l y many bounds (up t o isomorphism). A relation in which every f i n i t e relation of the same a r i t y i s embeddable, has no bound ( f o r example a rich relation). On the other hand, the consecutivity relation on the natural numbers, has infinit e l y many bounds, among which, f o r each integer p > , 3 , the binary cycle of cardinality p
.
THEORY OF RELATIONS
35 2
R be a relation. For any f i n i t e relation X , we have X embeddable i f f no bound of R i s embeddable in X . ( 2 ) Let R , S be two r el at i o n s , a t l e a s t one of which i s f i n i t e . Then RG S iff no bound of S i s embeddable in R
1.1.(1) Let
in
R
.
( 1 ) Follows from ch.4 5 8.2, since embeddability between f i n i t e relations i s a well-founded p a r t i a l ordering. However l e t us give a d i r ect proof. If X < R , then no bound of R i s embeddable in X Conversely, i f X 3 ; R , then e i t h e r X i s a bound of R . Or there e x i s t s a r e s t r i c t i o n X1 of X t o i t s base minus one element, such t h a t R Iterating t h i s , a f t e r a f i n i t e number h of s t ep s , we obtain a re stric tion Xh of X which i s a bound of R 0 0 ( 2 ) If R i s f i n i t e , we find ( 1 ) . Suppose t h at R i s i n f i n i t e and S f i n i t e , hence R $ S . Replace R by a r es t r i ct i o n R ' whose cardinality i s f i n i t e b u t S , hence R ' admits an s t r i c t l y greater than the cardinality of S . Then embedding of a bound of S , by our ( 1 ) . This bound i s thus embeddable in R 0 0
.
XB I .
.
R'4
.
On the other hand, i f R and S are both i n f i n i t e , then statement ( 2 ) does n o t necessary hold. 0 Take R t o be the chain of the natural numbers. Take a sequence of f i n i t e relations Ai ( i integer) with d i s j o i n t bases, and such tha t every f i n i t e binary relation is isomorphic t o an Ai . Take S t o be the common extension of the Ai , which takes the value (+) f o r every ordered pair whose terms belong t o the bases of two d i s t i n c t Ai . Then S has no bound; hence no bound of S i s embeddable in R , and y e t R i s non-embeddable in S . 0
f i n i t e or S f i n i t e , i f every bound of S i s a bound of R , Indeed no bound of S i s embeddable in R (2) Any two f i n i t e relations having the same bounds, are isomorphic. (3) If R i s f i n i t e and R < S , then there ex i s t s a bound of R which i s embeddable in S ; hence there e x i s t s a bound of R which i s n o t a bound of S Indeed S i s not embeddable in R , hence S admits an embedding of a t l e a s t a (2). bound of R , by the preceding 1.l. ( 4 ) If e i t h e r R o r S i s f i n i t e , and i f the s e t of bounds of S i s properly included in the s e t of bounds of R , then R < S Follows from ( 1 ) and ( 2 ) . 1.2.(1) For
then
R
.
.
RQ S
-
.
-
.
>/ 1 has a t l e a s t 2 bounds. ( 2 ) Every non-empty f i n i t e relation of a r i t y >/ 2 has a t l e a s t 4 bounds.
1.3.(1) Every f i n i t e relation of a r i t y
-
.
be f i n i t e ; by the preceding 1 . 2 . ( 3 ) , there e xists a bound A of R We have R By ch.5 5 1.3.(1), assuming t h at the a r i t y i s n o t zero, there e x is t s an extension S 7 R respecting the non-embeddability S . By the preceding 1.2.(3), there e x i s t s a bound B of R with B & S , hence non-isomorphic
0
(1) Let
R
AS .
A4
353
Chapter 12
.0
A
with
0 (2) Let
be a non-empty f i n i t e r e l a t i o n w i t h a r i t y
R
.
2
Keeping t h e two
bounds A, B a l r e a d y o b t a i n e d and u s i n g ch.5 § 1.3.(3), t h e r e e x i s t s a p r o p e r e x t e n s i o n R+ o f R r e s p e c t i n g A $ R+ and B * R + . By t h e p r e c e d i n g 1 . 2 . ( 3 ) , t h e r e e x i s t s a bound with
nor with
A
of
C
.
B
such t h a t
R
obtain another proper extension o f i s n o t isomorphic w i t h any o f 1.4.(1)
Let
There does n o t e x i s t any (2) Let
S
be a
e x i s t s no of
T
S
Then
of
t i n c t and T
o f cardinality
S
.
p+l
i s a bound o f
S
, we
R
D o f R which
be t h e b i n a r y c y c l e o f c a r d i n a l i t y
and add an element
S
which i s
S
b, c
p-monomorphic,
, for
p
Let
.
p+l
which t h e r e
be a r e s t r i c t i o n
R
and e v e r y bound o f
R
p
S
T(b,b')
= T(b,a)
=
whose base c o n t a i n s
Note t h a t t h e c y c l e
b
t o i t s base. To o b t a i n an T(b,a) S
i n the cycle
= T(a,c)
. Then
=
a, b, b '
. Let
+
are dis-
+ , which makes i t i m n o s s i b l e f o r any r e s t r i c t i o n , t o be isomorphic w i t h t h e c y c l e S . 0
a, b, b '
o f cardinality
S
a
i t i s necessary t h a t t h e r e e x i s t
I S 1 with
o f t h e base
be t h e element c o n s e c u t i v e t o
of
p
i s obviously
(p-1)-monomorphic;
thus
s a t i s f i e s t h e hypotheses o f o u r ( 2 ) .
S 0
.
p-1
and
(communicated by POUZET 1978).
R
two d i s t i n c t elements b'
, and
p-monomorphic e x t e n s i o n o f
(1) S t a r t w i t h t h e c y c l e
extension
0
p-monomorphic e x t e n s i o n o f c a r d i n a l i t y
i s a bound o f 0
B, C .
(p-1)-monomorphic r e l a t i o n o f c a r d i n a l i t y
with cardinality
S
A,
i s isomorphic n e i t h e r
C
A, B, C
and f i n a l l y a f o u r t h bound
R
be an i n t e g e r 3 3
p
, hence
R+
C,<
I t e r a t i n g t h i s and s t a r t i n g w i t h
( 2 ) Since
is
S
(p-1)-monomorphic,
p-1
i s isomorphic t o
R ; so t h a t
Let
A
S
be a bound o f
d i n g (1). Hence R
, since
is
S
i s a bound o f
S
A
Then
p-1
p
.
S
.
.
Thus
Finally A
p+l
, by o u r prece-
Every p r o p e r r e s t r i c t i o n o f
and i s embeddable i n
(p-1)-monomorphic.
s i n c e i t i s non-embeddable i n
R
with cardinality
S
cannot have c a r d i n a l i t y
has a t most c a r d i n a l i t y
A
has a t most c a r d i n a l i t y in
.
every r e s t r i c t i o n o f
S
, thus
i s non-embeddable i n
A
i s a bound o f
R
R
R
, with
S
,
f i n i t e . T h i s answers a problem o f GILLAM 1974.
F o r t h e case where
i s finite,
R
S
i n f i n i t e and
R
< S , with
t h e same p r o p e r
i n c l u s i o n o f t h e s e t o f bounds, we have a l r e a d y c o n s i d e r e d a f i n i t e c h a i n an i n f i n i t e c h a i n
,
.0
The p r e c e d i n g statement ( 2 ) completes 1.2.(4), by g i v i n g an exemple o f R < S such t h a t t h e s e t o f bounds o f S i s p r o p e r l y i n c l u d e d i n t h e s e t o f bounds of -
A
embeddable as w e l l
S
R
and
.
R and S i n f i n i t e , we have g i v e n i n 1.1 a counterexample w i t h R = c h a i n o f t h e n a t u r a l numbers and S w i t h o u t any bound, s o t h a t t h e empty s e t o f bounds
For of
S
in
S
i s p r o p e r l y i n c l u d e d i n t h e s e t o f bounds o f
.
R ;yet
R
non-embeddable
.
354
THEORY OF RELATIONS
1 . 5 . L e t R, S be two r e l a t i o n s o f t h e same a r i t y . Then t h e f o l l o w i n g t h r e e c o n d i tions are equivalent. (1) Every f i n i t e r e s t r i c t i o n o f R i s embeddable i n S ; i n o t h e r words, R younger t h a n S ( 2 ) No bound o f S i s embeddable i n R . ( 3 ) Every bound o f S admits an embedding o f a bound o f R
.
.
0
Assume t h e f i r s t c o n d i t i o n and l e t
, hence
A S S
A
triction
i s n o t a bound o f
A
of
. Conversely,
S
which i s non-embeddable i n
R
, hence
be a bound o f
A
, which
S
.
A,<
If
, then
R
also
i f there e x i s t s a f i n i t e res-
, then there e x i s t s a r e s t r i c -
S
. Thus
( 1 ) and ( 2 ) a r e e q u i v a l e n t . A i s non-embeddable in R Thus t h e r e e x i s t s a r e s t r i c t i o n o f A which i s a bound o f R . Conversely, i f t h e r e e x i s t s a bound A o f S which i s embeddable i n R , t h e n A admits no embedding o f any bound o f R . Thus ( 2 ) and (3) a r e e q u i v a l e n t . 0
tion of
A
of
R
Assume c o n d i t i o n ( Z ) , and l e t
i s a bound o f
S
be a bound o f
A
S ; hence
.
I n particular, i f
and
R
are f i n i t e , then t h e embeddability
S
l e n t t o t h e c o n d i t i o n t h a t no bound o f
i n l . l . ( Z ) ) , o r a g a i n e q u i v a l e n t t o t h e c o n d i t i o n t h a t e v e r y bound o f an embedding o f a bound o f
i s equiva-
R<S
R ( t h i s i s already
i s embeddable i n
S
S
admits
.
R
Another consequence: Let
R, S
iff
R
R
be two r e l a t i o n s o f t h e same a r i t y . Then
and
S
S
have t h e same age:
have t h e same bounds;
i f f no bound o f
i s embeddable i n
R
In particular, i f
R
and
S
R
S
, n o r i s any bound o f
are f i n i t e , then
have same bounds; i f f no bound o f embeddable i n
and
R
embeddable i n
S
R
and
S
a r e isomorphic i f f t h e y
i s embeddable i n
S
n o r i s any bound o f
R
S
.
Now we can achieve t h e r e f i n e m e n t o f 1 . 2 . ( 4 ) as f o l l o w s . If
R
and
S
a r e i n f i n i t e and
R
< S , then
p r o p e r l y i n c l u d e d i n t h e s e t o f bounds o f 0
Let
A
be a bound o f
exists a restriction B proper r e s t r i c t i o n o f be a bound o f
S
which i s n o t a bound o f
of
i s embeddable i n ble i n S .
A
s . 0
S
o f t h e same a r i t y , i f e v e r y f i n i t e r e s t r i c t i o n o f
, t h e n e v e r y bound of R
be a bound o f
proper r e s t r i c t i o n o f of
. .
.0
R
Let
cannot be
R
R By 1 . 5 . ( 1 ) and ( 3 ) , t h e r e A which i s a bound o f R Necessarily B i s a A , so t h a t B i s embeddable i n S and hence B cannot S
1 . 6 . Given two r e l a t i o n s R, S
0
t h e s e t o f bounds o f
.
S
A
. Assume
that
R
A
i s embeddable i n
i s a bound o f
S
i s non-embeddable i n
R
, hence
in
S
. Thus
o r i s embeddaS
A
.
Yet
every
i s a bound
.
Chapter 12
355
The converse i s f a l s e . Take R t o be an i n f i n i t e chain, and S t o be t h e chain of c a r d i n a l i t y 3 . Then every bound of R i s one of t h e four r e l a t i o n s w i t h cardinal i t i e s l , 2 , 3 considered a t t h e beginning of the present 5 l . Hence every bound of R i s a bound of S . Yet t h e chain of c a r d i n a l i t y 4 i s a bound of S and not a bound of R . Another example w i t h R and S f i n i t e . Take R t o be t h e binary r e f l e x i v e cycle of c a r d i n a l i t y 3 . There e x i s t f o u r bounds of R , u p t o isomorphism. These are: t h e binary r e l a t i o n of c a r d i n a l i t y 1 and value ( - ) ; t h e binary r e l a t i o n always (+) of c a r d i n a l i t y 2 ; t h e i d e n t i t y r e l a t i o n of c a r d i n a l i t y 2 ; and f i n a l l y the chain of c a r d i n a l i t y 3 . Take S t o be t h e common extension of these four bounds, taken w i t h d i s j o i n t bases, S taking the value (t) f o r every ordered p a i r whose terms belong t o the bases of two d i s t i n c t bounds. Then each bound of R i s embeddable i n S , y e t R i s non-embeddable i n S . The preceding example can be modified so a s t o make S i n f i n i t e : add t o our four bounds, a binary i n f i n i t e r e l a t i o n always (t). Even we can make both R and S i n f i n i t e : replace R by i t s extension t o an i n f i n i t e base, w i t h the value (+) f o r a l l new ordered p a i r s : then t h e r e remain f i n i t e l y many bounds of R and the preceding argument holds. 1 . 7 . We have already s a i d t h a t f o r a r e l a t i o n with f i n i t e c a r d i n a l i t y p , the bounds have a t most c a r d i n a l i t y p+l . For example a chain of c a r d i n a l i t y p admits as a bound the chain of c a r d i n a l i t y p + l Let us give examples where t h e maximum c a r d i n a l i t y of bounds i s a t most p Take the unary r e l a t i o n of c a r d i n a l i t y p+q , which takes t h e value (+) on p e l e ments and ( - ) on q elements. The bounds a r e t h e unary r e l a t i o n always (+) of c a r d i n a l i t y p+l , and the unary r e l a t i o n always (- ) of c a r d i n a l i t y q + l Take t h e binary cycle of c a r d i n a l i t y p This i s a (p-1)-monomorphic r e l a t i o n . This Moreover by 1 . 4 . ( 1 ) , t h e maximum possible c a r d i n a l i t y of bounds i s p number i s taken on by the c o n s e c u t i v i t y r e l a t i o n associated w i t h a chain of cardin a l i t y p : t h i s i s a bound of the given binary c y c l e . Take t h e p a r t i a l ordering formed of p component c h a i n s , each of c a r d i n a l i t y q ( p , q p o s i t i v e i n t e g e r s ) . Then a s bounds, we have the chain of c a r d i n a l i t y q+l , the i d e n t i t y of c a r d i n a l i t y p + l , t h e r e l a t i o n of c a r d i n a l i t y 1 and value ( - ) , the r e l a t i o n of c a r d i n a l i t y 2 always ( + ) . F i n a l l y , assuming t h a t p , q 2 , we have the r e f l e x i v e cycle of c a r d i n a l i t y 3 ; and the t h r e e r e l a t i o n s of c a r d i n a l i t y 3 , each having a s proper r e s t r i c t i o n s , an i d e n t i t y and two chains of c a r d i n a l i t y 2: i n a p o s i t i o n of convergence, divergence, o r succession. For p = q , the cardinal i t y of the base i s p2 b u t the maximum c a r d i n a l i t y of bounds is p + l (example due t o LOPEZ). A refinement of t h i s example i s obtained by replacing each element by an equivalence c l a s s of f i n i t e c a r d i n a l i t y r ; hence transforming t h e p a r t i a l ordering i n t o
.
.
.
.
.
356
THEORY OF RELATIONS
a pre-ordering. For p = q = r , t h e c a r d i n a l i t y of the base i s p3 , b u t t h e maximum c a r d i n a l i t y of bounds i s p + l . A second refinement i s obtained by repeating the pre-ordering and making i t i r r e f l e x i v e ( w i t h t h e value ( - ) on t h e diagonal). Then take two d i s j o i n t bases f o r the r e f l e x i v e and the i r r e f l e x i v e pre-orderings, and t h e i r extension t o t h e union of t h e two bases, with f o r example the value (+) f o r ordered p a i r s whose terms belong t o d i f f e r e n t bases. Now t h e new base has c a r d i n a l i t y Z.p3 , t h e maximum c a r d i n a l i t y of bounds remaining p + l (example due t o POUZET 1978). Problem. Obtain a base of f i n i t e c a r d i n a l i t y s t r i c t l y l a r g e r than t h e maximum c a r d i n a l i t y of bounds i s a t most p + l .
Z.p3 , while
1 . 8 . A b i r e l a t i o n RS can have f i n i t e l y many bounds, even though t h e f i r s t compohas i n f i n i t e l y many bounds, and t h e second component S has f i n i t e l y many bounds.
nent R
Take S t o be the chain of the natural numbers, and take R t o be t h e consec u t i v i t y r e l a t i o n a s s o c i a t e d with S : hence R(x,y) = + i f f y = x + 1 . Then R has as bounds a l l f i n i t e binary c y c l e s . However, a bound of RS has c a r d i n a l i t y a t most 3 ; so t h a t RS has f i n i t e l y many bounds. Indeed l e t A , B be two f i n i t e r e l a t i o n s such t h a t every r e s t r i c t i o n of AB w i t h c a r d i n a l i t y 3 i s embeddable i n RS . Then A i s a f i n i t e chain; and B takes the value (+) only f o r ordered p a i r s of consecutive elements (mod A) . T h u s AB i s embeddable in RS ; hence AB cannot be a bound.
Even more curious i s the f a c t t h a t t h e r e e x i s t two denumerable binary r e l a t i o n s each having i n f i n i t e l y many bounds; and y e t t h e i r concatenation only f i n i t e l y many bounds (POUZET 1978, unpublished).
R, S
For each f i n i t e set
F , consider a l l b i r e l a t i o n s
A8 based on
F
RS has
, such
that
f o r a l l x , y d i s t i n c t , i f A(x,y) # A(y,x) then B ( x , x ) # B(y,y) ; and s i m i l a r l y i f B(x,y) # B(y,x) then A ( x , x ) # A(y,y) . Let 'f be t h e c l a s s of these b i r e l a t i o n s . For any two f i n i t e d i s j o i n t s e t s F, G , two b i r e l a t i o n s belonging , one w i t h base F and the o t h e r w i t h base G , always admit a common to extension w i t h base F u G , again belonging t o f To see t h i s , f o r each element x in F and each y i n G , take t h e same value
.
A(x,y) = A(y,x) , and s i m i l a r l y B(x,y') = B(y,x) , which i s compatible w i t h the values A ( x , x ) , A(y,y), B ( x , x ) , B(y,y) already imposed by the preceding. Consequently, t h e r e e x i s t s a b i r e l a t i o n RS with denumerable base, i n which i s embeddable. Moreover, every bound of every b i r e l a t i o n AB of t h e c l a s s RS has c a r d i n a l i t y a t most 2 . Indeed, given a b i r e l a t i o n AB of f i n i t e cardinal i t y g r e a t e r than o r equal t o 3 , i f a l l i t s r e s t r i c t i o n s of c a r d i n a l i t y 2 belong to , then AB i t s e l f belongs t o , and hence i s embeddable i n RS .
e
Chapter 12
351
However, because o f the s i g n change i n S(x,x) when we pass from an element x t o an element y w i t h R(x,y) # R(y,x) , any f i n i t e c y c l e which i s a r e s t r i c t i o n even c a r d i n a l i t y . Thus R has i n f i n i t e l y many bounds, which are o f R has a l l b i n a r y c y c l e s o f odd c a r d i n a l i t y , w i t h a r b i t r a r y values f o r ,R(x,x) , f o r each element x i n such a cycle. S i m i l a r l y f o r S 0
.
1.9. L e t R, S be two r e l a t i o n s , each o f which i s f r e e l y i n t e r p r e t a b l e i n t h e other. Then t o each bound A R , o f c a r d i n a l i t y s t r i c t l y l a r g e r than the
of
-
maximum o f the a r i t i e s o f having the same base as
S
R and S , we can b i j e c t i v e l y associate a bound o f A
(POUZET 1978, unpublished).
3
L e t 'j, be a f r e e operator t a k i n g R i n t o S , and be a f r e e operator t a k i n g S i n t o R : see ch.9 5 3.1. L e t B = ( A ) . Then every proper r e s t r i c t i o n o f B i s embeddable i n 9 ( R ) = S . Moreover d, (B) = A , since both have the same proper r e s t r i c t i o n s , hence the same r e s t r i c t i o n s o f c a r d i n a l i t i e s l e s s
0
.
than o r equal t o the a r i t y o f A Hence B i s n o t embeddable i n S ; f o r otherwise 2 (8) = A would be embeddable i n &?, ( S ) = R 0 1.10. (1) A f i n i t e 0-ary r e l a t i o n A
.
can be a bound i f f A
i s non-empty. However,
.
i f p = Card A , then A i s a bound o f R i f f Card R = p-1 (2) For a r i t y 1 , a f i n i t e unary r e l a t i o n A can be a bound i f f
and i s always (+) o r always ( - ) . Indeed i f
p = Card A
A i s non-empty
and i f A
i s always (+)
f o r instance, then any unary r e l a t i o n ( f i n i t e o r i n f i n i t e ) which takes p-1 times t h e value (+) admits A as a bound. ( 3 ) For a r i t y n + 2 , a f i n i t e n-ary r e l a t i o n A can be a bound i f f A i s non-empty. Moreover there e x i s t s an i n f i n i t e r e l a t i o n R such t h a t A i s a
.
Indeed t h e s e t o f f i n i t e r e l a t i o n s i n which bound o f R i s an i n f i n i t e age: see ch.10 5 2.3, using ch.5 5 1.3.(2).
5 2 - PRESERVATIONOF
A
i s n o t embeddable
BOUNDS I N THE PASSAGE FROM A RELATION TO SOME
OF ITS EXTENSIONS Note f i r s t t h a t t h e n o t i o n o f bound i s immediately e x t e n d i b l e t o m u l t i r e l a t i o n s . 2.1. L e t
R
be a f i n i t e m u l t i r e l a t i o n and U
an extension o f
a bound o f
R
. Then
there e x i s t s
R t o i t s base augmented by one element, f o r which U i s again
a bound.
R i s a unary r e l a t i o n o f c a r d i n a l i t y p+q , t a k i n g the U i s , f o r example, the We replace R by i t s extension which r e l a t i o n always (+) o f c a r d i n a l i t y p + l takes t h e value (+) on p elements and (-) on q+l elements.
0 Suppose f i r s t t h a t
value (+) on p elements and (-) on q elements. Then
.
THEORY OF RELATIONS
358
The reader will immediately extend t h i s procedure t o the case of a unary multirelation, i . e . the concatenation of f i n i t e l y many unary relations with a common base. Suppose now t h a t R has a t l e a s t one component with a r i t y >, 2 . We shall argue in the case t h a t R i s i t s e l f a binary r e l a t i o n , the proof in the general case being an immediate extension of our argument. Let U be a bound of R . Add t o the base of R a new element a , and l e t Ra be the extension of R defined by Ra(a,x) = Ra(x,a) = Ra(a,a) = + f o r every x in the base I R 1 Similarly add b and define Rb by the analogous cond'itions w i t h ( - ) instead of (+). Then U i s e i t h e r a bound of Ra or a bound of
.
Rb . Indeed every proper r e s t r i c t i o n of U i s embeddable in R , hence in Ra and in Rb . On the other hand, U i s not embeddable in both: use the argument in ch.5 5 1 . 3 . ( 1 ) ( f a i t h f u l extension). 0 2 . 2 . Consider a f i n i t e s e t of f i n i t e relations A1,...,Ah a l l of the same a r i t y , and suppose t h a t there e x i s t relations of the same a r i t y , with a r b i t r a r i l y large f i n i t e c a r d i n a l i t y , each having the bounds A1, ...,Ah and possibly other bounds. Then there e x i s t s a denumerable relation having, among other ones, the bounds A1 ,. . .,Ah
.
Let Ri ( i integer) be an &-sequence of f i n i t e r e l a t i o n s , whose c a r d i n a l i t i e s has a t l e a s t the bounds A1, ...,Ah are s t r i c t l y increasing, such t h a t each Ri Let pi denote the cardinality of Ri ; we can assume t h a t R i has base { 1,2,. . ...,pi Moreover, since f o r each i , the relation Ri admits an embedding of a l l proper r e s t r i c t i o n s of A1 f o r instance, then l e t t i n g kl denote the sum of the c a r d i n a l i t i e s of these proper r e s t r i c t i o n s , we can suppose t h a t they are a l l embeddable in the r e s t r i c t i o n of Ri t o i1.2 ,...,kl) . Similarly f o r A2 , l e t k2 denote the sum of the c a r d i n a l i t i e s of a l l proper r e s t r i c t i o n s of A2 : they are a l l embeddable in the r e s t r i c t i o n of Ri t o { 1 , 2 ,..., and so f o r t h . There e x i s t s an i n f i n i t e sequence, extracted from the sequence of the Ri , which i s formed of relations having the same r e s t r i c t i o n S1 t o the singleton of 1 From t h i s f i r s t extracted sequence, we e x t r a c t a second sequence, formed of relations a l l having the same r e s t r i c t i o n S2 t o the pair 1,2) Iterating t h i s , we obtain, f o r each integer r , a relation S r based on { 1,2, ...,r} , where each Sr ( r + 2 ) i s an extension of Sr-l Let S denote the common extension of the Sr , based on a l l positive integers. Then A1 f o r instance, i s a bound of S . Indeed A1 i s n o t embeddable in S , since otherwise i t would be embeddable in SOM S r , hence in some Ri . Moreover,
0
.
1.
ktd;
.
.
.
each proper r e s t r i c t i o n of A1 i s embeddable in S , since i t i s embeddable in . Same argument the r e s t r i c t i o n of each Ri t o { 1 , 2 ,...,kl) , hence in S(kl) f o r A2,.
. .,Ah
which are thus also bounds of
S
.0
.
359
Chapter 12
2.3. Consider again the f i n i t e relations A1, ... ,Ah ; suppose t h a t f o r each integer p , there e x i s t s a relation with c a r d i n a l i t y g r e a t e r than or equal t o p , whose p . bounds are exactly A1,. . ,Ah plus possibly some bounds of c a r d i n a l i t i e s Then there e x i s t s a denumerable relation whose bounds are exactly A1, ...,Ah (suggested by PABION in 1970).
>
.
Let Ri ( i positive integer) be our f i n i t e r e l a t i o n s , which are l i s t e d by increasing values of p . We shall modify our construction in the preceding 2 . 2 , as follows. Take a sequence of a l l the f i n i t e relations U . ( j positive integer) with 3 the same a r i t y as the Ri , and l e t k br the f i n i t e cardinality of U j j' Replacing, i f necessary, each Ri by an isomorphic copy, we can suppose the following. For each i , i f U1 i s embeddable in Ri , then U1 i s embeddable in the r e s t r i c t i o n of Ri t o { 1 , 2 , ..., k l ) Again f o r each i , i f U 2 i s embeddable in Ri , then U 2 i s embeddable in the r e s t r i c t i o n of Ri t o the s e t {1,2 ,... , k l + k 2 ) ; and so forth. Now, construct relations Sr ( r integer) as in the preceding 2 . 2 , and then take t h e i r common extension S . Then A 1 , . . . , A h are a l l bounds of S . I t remains t o prove t h a t S has no other bound. Suppose t h a t B i s a bound of S d i f f e r e n t from A 1 , . . . , A h Then f i r s t l y , each proper r e s t r i c t i o n of B i s embeddable i n S , hence in the Sr f o r a l l r greater than some r ( 0 ) ; hence in a l l the Ri which extend S r(O) . Secondly B cannot be a bound of Ri , f o r i s u f f i c i e n t l y large, so t h a t the integer p associated by hypothesis with Ri i s larger than the cardinality of B . Hence B i s embeddable in a l l the Ri which extend S and whose r(O) index i i s s u f f i c i e n t l y large. Finally there e x i s t s r(1) >/ r ( 0 ) such t h a t B i s embeddable in a l l those Ri which extend S r(1) . From the f i r s t paragraph, there e x i s t s an integer k f o r which, i f B i s embeddable in Ri , then B i s s t i l l embeddable in the r e s t r i c t i o n Ri/{1,2 ,...,k ) . Hence B i s embeddable in Sm , where m i s the maximum of k and r ( 1 ) . Thus B i s embeddable in S : contradiction. 0
0
.
.
§
3 - WELL
MULTIRELATION:
REASSEMBLING
THEOREM
(FRASNAY)
&,
We say t h a t a multirelation R i s i f the s e t of i t s f i n i t e r e s t r i c t i o n s , when p a r t i a l l y ordered under embeddability, forms a well partial ordering. In other words, i f any s e t of f i n i t e r e s t r i c t i o n s of R , mutually incomparable under embeddability, i s f i n i t e .
For example, every chain i s well. Every t r e e i s well, by KRUSKAL's theorem (ch.5 5 2.3). The consecutivity relation on the natural numbers i s well. Indeed, each f i n i t e r e s t r i c t i o n can be represented by a f i n i t e sequence of positive integers, each
360
THEORY OF RELATIONS
integer i representing a component of i consecutive integers. Then the embeddab i l i t y between two f i n i t e sequences of integers, implies the embeddability between the two corresponding f i n i t e r e s t r i c t i o n s . Now i t suffices t o recall t h a t embeddabil i t y between f i n i t e sequences, or words of integers, i s a well partial ordering by HIGMAN's theorem ch.4 5 4.4.
p Given a natural number concatenating R with any mu1 t i re1 ation.
rf p-well.
3.1.
R
2
, we p
say t h a t a multirelation R i s p - e , i f upon unary relations with the same base, we obtain a well
p-well, then every multirelation freely interpretable in. R
Let S be freely interpretable in R , the base. I f S/F augmented by p unary augmented by p unary relations A ' on embeddability, then the same i s true f o r augmented by the A ' .
0
&
and l e t F, F ' be two f i n i t e subsets of relations A based on F , and S/F' F ' , are incomparable with respect t o R/F augmented by the A and R/F'
The consecutivity relation on the natural numbers i s well, b u t i s n o t 1-well. i , take a sequence of i+2 consecutive elements, and define a unary relation t o take the value ( - ) f o r the f i r s t and the l a s t element, and the value ( + ) between. 0
0 For each integer
Problems communicated by POUZET in 1972. (1) If a multirelation i s 1-well, then i s i t 2-well, and even p-well f o r every integer p ( 2 ) S t a r t with a multirelation R Take the concatenations RX with X an arbit r a r y unary relation; then the union of the ages of these concatenations, where R i s fixed and X varies. This union i s not in general an age. If t h i s union i s well partial ordered under embeddability, then R i s 1-well. Is the converse t r u e . Same question f o r 2 , 3 , ... unary relations added.
.
.
3.2. Every chain, and consequently every chainable multirelation, i s each integer p (POUZET 1 9 7 2 ) .
p-well f o r
t o see t h a t the multirelations ( A , B l , . . . , B p ) , where A i s a f i n i t e chain and the B are unary r e l a t i o n s , form, up t o isomorphism, a s e t which i s well p a r t i a l l y ordered under embeddability. To each multirelation, associate the f i n i t e cardinality h of i t s base. Then associate a word of length h , obtained by replacing each i = 1, ,h by the sequence of values B1(x) , ... , Bp(x) , where x designates the i t h element of the base, ordered modulo A . We say t h a t two of these sequences are considered t o be incomparable i f f they are d i s t i n c t . T h u s we have a well partial ordering of
0 I t suffices
...
i d e n t i t y , with
2p elements, or sequences, mutually incomparable. By HIGMAN's
36 1
Chapter 12
theorem (ch.4
5
4.4 f i n i t e case, p r o v a b l e i n ZF), t h e s e t o f words formed o f t h e
p r e c e d i n g elements, c o n s t i t u t e s a w e l l p a r t i a l o r d e r i n g , under t h e usual embeddabil i t y o f words. T h i s g i v e s a w e l l p a r t i a l o r d e r i n g , under e m b e d d a b i l i t y , o f o u r
f i n i t e multirelations. 0 3.3. THEOREMS ON FINITE NUMBER OF BOUNDS
Let
If
be a m u l t i r e l a t i o n ; denote by
R
g
R
2m-well,
then
t h e maximum o f i t s components' a r i t i e s .
m
has f i n i t e l y many bounds (POUZET 1972).
R
Consequently e v e r y c h a i n a b l e m u l t i r e l a t i o n has f i n i t e l y many bounds (FRASNAY 1965) 0 Let
al
be a bound o f
U
in
, and
E
let
t h e r e e x i s t s an
and
R
w i t h base
S< R
F ut a l i . Note t h a t when t h e a r i t y
m = 1
condition. For otherwise
S
I f t h e r e e x i s t s no such
relation:
El and
,..., Vm,A1
(S,V1
V
; where t h e
o n l y on
Consider t h o s e subsets
F o f El
E
, which
U
coincides w i t h
, then ,..., A),, A
on
f o r which and on
El
, t h e n no subset F , even empty, s a t i s f i e s t h i s
, hence U&
U
a s s o c i a t e t o t h e bound where
and
S,< R
.
R
the following multi-
U
coincides w i t h
t a k i n g t h e value (+) on
V1
a r e unary,
S
U
b e i n g t h e s i n g l e t o n unary r e l a t i o n o f
A1
an a r b i t r a r y element
.
would be i d e n t i c a l w i t h
and t h e
, and
El
F
. Take
U
be t h e base o f
E
- {al)
El = E
al
El
on and
V
; the other
t a k i n g always t h e value ( - ) .
A
Now suppose t h a t t h e r e e x i s t subsets
s a t i s f y i n g o u r p r e c e d i n g c o n d i t i o n . Take
F
F o f maximum c a r d i n a l i t y , say
such an
E2
.
This
i s a p r o p e r subset o f
E2
El
, t h e n we would have U,< R . L e t a2 be an a r b i t r a r y element o f El - E2 . Consider those subsets F o f E2 f o r which t h e r e e x i s t s an S,< R w i t h base E , which c o i n c i d e s w i t h U on El and on E2 u {al\ and on Fu{al,a2). Note t h a t when t h e a r i t y m = 2 , t h e n no such F , even empty, e x i s t s . F o r o t h e r , wise, t h i s would y i e l d t h a t S c o i n c i d e s w i t h U on t h e union E2u{al,a2) for if
E2 = El
contradicting the maximality o f relation: El
(S,V1
and on
E2u{al\
(+) o n l y on singleton o f
F
; where
E2u{al\ a2
E2
.
, t h e n a s s o c i a t e t o t h e bound U t h e f o l l o w i n g m u l t i ,..., Vm,A1 ,...,Am) where S 4 R and S c o i n c i d e s w i t h U on
I f t h e r e e x i s t s no such
V1
takes t h e v a l u e (+) o n l y on
A1
; the relation
, and
i s the singleton o f
V
f i n a l l y the other
A
and
El
al
, and V2 takes
Em
and elements
Thus t h e r e e x i s t s an E2 \al) tion S
al S,<
, on E3 u {al,a2) o f maximum a r i t y
i n E-El
,
R w i t h base
.. .
m
a2
in
El-E2
E
, which
i s the
a r e always ( - ) . E 1 3 E2 3 ...
I t e r a t i n g t h i s , we o b t a i n , a t most, a s t r i c t l y descending sequence
... 3
A2
and
,
... , a,
coincides w i t h
in U
Emm1-E, on
E,
.
, on
, , on E m u {al ,... ,am-1)- . Yet no m u l t i r e l a can s a t i s f y t h e p r e c e d i n g c o n d i t i o n s , and moreover
;
THEORY OF RELATIONS
362
U
coincide w i t h
on { a l,...,am).
.. ,a,)
Em u {al,.
For otherwise
, contradicting the maximality o f
Associate t o t h e bound
U
the multirelation
and
U
on
S
coincides w i t h
the relation
... , and
, on
El
E2 V i a l )
o n l y on
,
al
Emu{al,
... , and
...,am-1}
Am
or
...
coincides w i t h or
... o r
A1
. Yet
being the singleton am
.
coinciding with
S,(R
U
when
or
V1
i t i s impossible t h a t i n addition,
on t h e s e t o f t h e elements g i v i n g t h e v a l u e (+) t o
U
on
U
Em . ,..., Vm,A1 ,... ,Am) where S d R , ... , on Emu{al ,..., am)l, , and V2 o n l y on E2 u {al) ,
the singleton relation o f
takes t h e v a l u e (+)
Vm
or
El
would c o i n c i d e w i t h
; and f i n a l l y
I n e v e r y case, we see t h a t t h e r e e x i s t s an
V2
(S.V1
t a k i n g t h e v a l u e (+) o n l y on
V1
Vm
relation o f
S
S
o r Ap
A1
.
Am
Suppose now t h a t
is
R
Pm-well. Then t h e m u l t i r e l a t i o n s
p r e v i o u s l y asso-
(S,V,A)
c i a t e d t o t h e bounds, f o r m a w e l l p a r t i a l o r d e r i n g under e m b e d d a b i l i t y . Suopose t h a t t h e r e e x i s t i n f i n i t e l y many bounds. Then because o f t h i s w e l l p a r t i a l order i n g and by RAMSEY's theorem, t h e r e e x i s t s an w - s e q u e n c e o f m u l t i r e l a t i o n s a s s o c i a t e d t o t h e bounds, and such t h a t each m u l t i r e l a t i o n admits an (S,V,A) embedding o f each p r e c e d i n g one. More p r e c i s e l y , we can assume t h a t each (S,V,A) i s an e x t e n s i o n o f t h e p r e c e d i n g . Then t h e r e a r e a t most or
...
m elements f o r which
w -sequence e x t r a c t e d
takes t h e v a l u e (+). So t h e r e e x i s t s an
or
A1 Am f r o m t h e p r e c e d i n g one, f o r which a l l t h e bounds
have t h e same r e s t r i c t i o n t o
U
these elements. Let
U
be one o f these bounds,
base p r o p e r l y i n c l u d e s
i t s base, and l e t
E
. Then
E
U'/E
6R .
gnate t h e m u l t i r e l a t i o n a s s o c i a t e d w i t h
...,)A;
U
the m u l t i r e l a t i o n associated w i t h
t h e f i r s t one. The r e s t r i c t i o n
U'/E
Let
, and
be a n o t h e r bound whose
U'
similarly
(S',Vi,
desi-
...
...,Vh,Ai,
U ' : t h e second m u l t i r e l a t i o n extends
coincides w i t h
S'
, hence
on t h e s e t where
V1
takes t h e v a l u e ( + ) , on t h e s e t where
on t h e s e t where
Vm
t a k e s (+). Moreover,
o f those elements g i v i n g t h e v a l u e (+) t o
. ,Am)
. . ,Vm,A1,..
(S,V1,.
U'/E
A1
or
with
and
S
U
...
V2 takes (+), c o i n c i d e s w i t h U on t h e s e t A2
or
... o r
, ,
Am : c o n t r a -
diction. 0 3.4.
REASSEMBLING THEOREM
Let
G
be a group and
m
i t s a r i t y . There e x i s t s an i n t e g e r
n 3 m satisfying
the following. Given a s e t
o f cardinality
E
a l l the
F of AF
E
, given
>/
n
(finite or infinite);
and f o r each
AF w i t h base F , where t h e G-compatible; t h e n t h e r e e x i s t s a c h a i n based on E , which i s
subset
a chain
n-element
AF a r e m u t u a l l y G-compatible w i t h
(FRASNAY 1965; t h e a r i t y o f a group and t h e G - c o m p a t i b i l i t y a r e
d e f i n e d i n ch.11
0
3; o u r p r o o f uses u l t r a f i l t e r axiom; ZF s u f f i c e s i f
E
countable).
Chapter 12
0
Suppose f i r s t t h a t
and d e f i n e t h e
i s f i n i t e . Let
E
363
be t h e c h a i n o f t h e n a t u r a l numbers,
A
m-ary r e l a t i o n
R
on t h e s e i n t e g e r s , by s e t t i n g
i f f e i t h e r a t l e a s t two o f t h e
x
are i d e n t i c a l , o r i f the
x
,. . . ,xm)
R(xl
+
=
are a l l d i s t i n c t
xl,. . . ,xm i n i n c r e a s i n g o r d e r C l e a r l y e v e r y l o c a l automorphism o f t h e c h a i n A i s a
and t h e p e r m u t a t i o n which r e o r d e r s t h e sequence belongs t o t h e group R
is
A-chainable.
.
G
R ; i n o t h e r words
l o c a l automorphism o f
p ; and by t h e p r e c e d i n g theorem 3.3,
R
is
p - w e l l f o r each i n t e g e r
has f i n i t e l y many bounds.
R
F
of
Let
,let
E
be t h e
S
E
has c a r d i n a l i t y a t l e a s t
be a c h a i n on
AF
F
, where
m-ary r e l a t i o n based on
e i t h e r a t l e a s t two
x
n AF
the
. F o r each
are i d e n t i c a l , o r i f the
G-compatible
...,xm)
... ,xm
xl,
G ; where
taining
the value o f F
S
o f the
AF
onto a r e s t r i c t i o n o f
, so
n
no bound o f
i s embeddable i n
S
would be a bound o f of
R
chain
.
B
, with
. Let
x
according the chain AF
f o r each
t h e case t h a t Now suppose non-empty s e t
f
E
UD
.
R
S/F
,
.
F o r o t h e r w i s e some r e s t r i c t i o n o f
...,x m )
=
+
S
S
onto a r e s t r i c t i o n
i f f e i t h e r two
A x
into a are i d e n t i -
a r e d i s t i n c t and t h e p e r m u t a t i o n which r e o r d e r s these
B , belongs t o
.
G
n-element subset
F
It f o l l o w s t h a t of
E
. The
B
is
x
G-compatible
p r o o f i s now achieved f o r
i s finite.
E i n f i n i t e . F o r each f i n i t e subset D o f E , t h e r e e x i s t s a UD o f c h a i n s based on D , each b e i n g G-compatible w i t h t h e AF
, when r e s t r i c t e d t o
. For a D'
subset
, gives
D'
of
an element o f
0
based on
, whose r e s t r i c t i o n t o each f i n i t e s e t
E
F
S : o t h e r w i s e , such a
, hence i n R
lemma (ch.2 is
con-
c a r d i n a l i t y o f each bound o f
takes a r e s t r i c t i o n o f t h e c h a i n
S(xl,
g i v e n chains, s t i l l denoted by ging t o
. The
A
be an isomorphism f r o m
f-l
the r e s u l t t h a t
cal, o r i f a l l the with
R
The i n v e r s e f u n c t i o n
E
f o r any two
.
F
i s embeddable i n
R
bound would be embeddable i n a r e s t r i c t i o n It f o l l o w s t h a t
n-element
AF
S/F i s embeddable i n R : i t s u f f i c e s t o t a k e an
restriction
AF
i s less than
, belongs
n-element subset o f
G-compatibility o f the
does n o t depend on t h e chosen
, the
isomorphism f r o m R
d e s i g n a t e s an a r b i t r a r y
... ,xm . Because
xl,
F o r each
F
+ iff
=
a r e a l l d i s t i n c t and
t h e p e r m u t a t i o n which r e o r d e r s t h i s sequence a c c o r d i n g t o t h e c h a i n t o t h e group
assume
n-element subset
are mutually
, such t h a t S(xl,
E
. We
R
be s t r i c t l y g r e a t e r t h a n t h e c a r d i n a l i t i e s o f bounds o f
Let n a m that the f i n i t e set
A , hence
i s freely interpretable i n
R
Using 3.2 above, we see t h a t
D
, every UD,
c h a i n belon-
. By t h e
coherence
1.3, e q u i v a l e n t t o t h e u l t r a f i l t e r axiom), t h e r e e x i s t s a c h a i n
G-compatible w i t h a l l t h e
AF
D
belongs t o
.0
UD
. This
chain
3.5. CHAINABILITY THEOREM Let
m
be a p o s i t i v e i n t e g e r . There e x i s t s an i n t e g e r
m-ary r e l a t i o n w i t h c a r d i n a l i t y >/ p
, which
is
p a m such t h a t
(6p)-monomorphic,
e
x
i s chainable
(FRASNAY 1965; uses u l t r a f i l t e r axiom; ZF s u f f i c e s f o r a c o u n t a b l e r e l a t i o n ) .
THEORY OF RELATIONS
364
Consider a l l groups of a r i t y m , and l e t n be the maximum of the integers associated t o these groups in the preceding reassembling theorem. By ch.9 5 5 . 5 . ( 2 ) , there e x i s t s an integer pa n such t h at every m-ary re la tion with cardinality 3 p has a chainable r es t r i ct i o n with cardinality n Let R be a ( 4p)-monomorphic m-ary relation with base E o f cardinality a t le a s t equal t o p , Then a l l the r es t r i ct i o n s o f R with cardinality n are isomorphic, hence they are a l l chainable. To each n-element subset F of E , associate a chain AF based on F , such t h at the re stric tion R/F i s AF-chainable. Moreover, f o r any two n-element subsets F, F ' of E , take A F , t o be the image of AF under one of the isomorphisms of R/F onto R/F' . Let H be an m-element subset of the base E , and l e t F , F ' be two n-element subsets, each of which includes H . The permutation of H which takes AF/H i n to AF,/H i s an automorphism of R/H . Indeed, take the image H ' of H under the isomorphism from AF onto A F , ; then H ' i n t o H by preserving the order of elements (mod A F , ) and using chainability by A F , . Designate each element o f H by i t s rank (mod AF/H) . Then the group o f automorphisms of R/H becomes an m-ary permutation group. The preceding isomorphisms show t h a t G depends neither on H nor on the choice of the n-element s e t F including H . By the preceding, AF and A F , are G-compatible for any two n-element s e t s F and F ' Now apply the reassembling theorem: there ex i s t s a chain A based on E , which i s G-compatible with every AF . W e shall prove t h a t R i s A-chainable. Let H, H ' be two m-element subsets of E : we shall f i r s t prove t h a t the isomorphism from A/H onto A / H ' takes R/H i n t o R / H ' . We can assume t h a t 2m , hence t h a t there e x i st s an n-element subset F of E including both H and H ' . The desired isomorphism can be obtained by composing three isomorphisms: from A/H onto AF/H , from AF/H onto AF/H' , then from AF/H' o n t o A/H' The f i r s t and the t h i r d isomorphisms belong t o G , once each element i s designated by i t s rank (mod AF/H o r AF/H') . These are respectively an automorphism of R / H and an automorphism of R / H ' . The second i s an isomorphism from R/H onto R / H ' by the definition of AF . Now l e t K, K ' be two r-element subsets of E , with r s t r i c t l y le ss t h a n m Suppose f i r s t t h a t there ex i s t s an m-element subset H of E including both K and K ' Take an n-element s e t F including H ; transform K and K ' by the isomorphism of A/H onto AF/H (which belongs t o G and hence i s an automorphism of R/H ) ; we see t h at the isomorphism from A / K onto A / K ' takes R / K into R / K ' If no such m-element s e t H e x i s t s , then we take an m-element s e t H including K and another H ' including K ' , and by the isomorphism from A/H onto A/H' we are in the preceding case. 0
0
.
.
na
.
.
.
.
365
Chapter 12
3.6. Let m be an i n t e g e r , p t h e i n t e g e r 3 m defined by the preceding proposit i o n , and R an m-ary r e l a t i o n of c a r d i n a l i t y s t r i c t l y g r e a t e r than p . A s u f f i c i e n t (and necessary) condition f o r R t o be chainable is t h a t each rest r i c t i o n of R t o p + l elements i s chainable, o r even i s ( s p)-monomorphic (uses t h e u l t r a f i l t e r axiom; ZF s u f f i c e s i f R i s countable). This follows from t h e preceding proposition and from ch.9 § 6.1.
§
4 - REDUCTION
THRESHOLD, REASSEMBLING THRESHOLD, G - C H A I N
4.1. REDUCTION THRESHOLD Given an m-ary group G of permutations, we d e f i n e t h e reduction threshold of G denoted by s(G) , a s t h e l e a s t i n t e g e r s such t h a t the d i l a t e d group Gm+s i s i n d i c a t i v e : see ch.11 5 4.5. In p a r t i c u l a r s(G) = 0 i f f G i s i n d i c a t i v e . Given an i n t e g e r m , we define t h e m-ary reduction t h r e s h o l d , denoted by s(m) , a s being t h e maximum of t h e s(G) f o r a l l m-ary groups G . These d e f i n i t i o n s a r e due t o FRASNAY 1965, who o b t a i n s t h a t s ( 1 ) = s ( 2 ) = s ( 3 ) 2 = 0 , and s ( 4 ) = 2 , and f o r m >/ 5 , t h e i n e q u a l i t i e s 16 s(m) 5 (3m-8) - m + l ( I b i d . p . 493-494). The upper bound is improved t o s(m) 6 m-3 (again f o r m a 5 ) by HODGES, LACHLAN, SHELAH 1977. F i n a l l y i t i s proved by FRASNAY 1984 t h a t s(m) = m-3 f o r m 3 5 . More p r e c i s e l y , t h e value s(G) = in-3 i s reached by taking G t o be t h e group on { 1 , 2 , ,m} which preserves t h e e x t r e m i t i e s 1 and m Indeed t h e f i r s t d i l a t e d group 6"'' preserves 1 , 2 and m , m t l ; then Gm+2 preserves 1 , 2 , 3 and m , m + l , m + 2 and f i n a l l y t h e d i l a t e d group G2m-3 preserves 1,2,...,in-2 and m , m + l , ..., 2m-3 , hence i t i s t h e i d e n t i t y group, which i s obviously i n d i c a t i v e .
.
...
4.2. REASSEMBLING THRESHOLD Given an m-ary group G , we define t h e reassembling threshold of G , denoted by t ( G ) , as t h e l e a s t i n t e g e r t such t h a t n = m + t s a t i s f i e s t h e reassembling theorem 3.4. I f G i s not i n d i c a t i v e , then t(G)4 s(G) + 1 (FRASNAY 1965 p . 500). 2 However, f o r t h e i n d i c a t i v e group J 4 on {1,2,3,4) generated by ( 1 , 4 ) , ( 2 , 3 ) and the two t r a n s p o s i t i o n s ( 1 , 2 ) and ( 3 , 4 ) , we have s = 0 and t = 2 ( I b i d . p . 500). Given an i n t e g e r m , we define t h e w a r y reassembling t h r e s h o l d , denoted by t ( m ) , as being the maximum of t h e t ( G ) f o r a l l m-ary groups G We have t(1) = 0 , t ( 2 ) = t ( 3 ) = 1 , t ( 4 ) = 2 , and f o r m a 5 we have 14 t ( m ) 4 s(m) + 1 (FRASNAY 1965 p . 500 and JULLIEN 1966). Hence t(m)4m-2 f o r m a3 , i n view of the preceding improved upper bound of s(m) Finally, i t is proved by FRASNAY 1984 t h a t t ( m ) = s(m) + 1 = m - 2 f o r m 3 5 , by
.
.
,
THEORY OF RELATIONS
366
t a k i n g a g a i n t h e group on 41,. . . ,m} Problem. F o r each group (G,A)-CHAIN,
4.3.
, do
G
we have t h a t
A
w i t h base
E
, and an m-ary group
i n s p i r e d by CLARK, KRAUSS 1970, we d e f i n e t h e
R
. . ,xm
xl,.
.
s(G),< t ( G )
G-CHAIN
Consider a c h a i n relation
.
1 and m
which preserves
based on
and
E
G
.
F o l l o w i n g FRASNAY 1973,
as b e i n g t h e
(G,A)-*
A-chainable, such t h a t
...,)x,
R(xl,
s
a r e a l l d i s t i n c t and t h e r e e x i s t s a p e r m u t a t i o n
m-ary iff
= t
belonging t o
G
x ~ ( ~ )x <s ( 2 ) < ... < x s(m) modulo A . I n t h e p a r t i c u l a r case where G i s t h e b i n a r y group which reduces t o t h e i d e n t i t y ,
with
t h e n we have a g a i n t h e usual c h a i n is a
G-chain i f f
, more p r e c i s e l y t h e s t r i c t c h a i n < (mod
A
We say t h a t
R
there e x i s t s a chain
(G,A)-chain
( G-rangement i n FRASNAY's t e r m i n o l o g y ) .
A
such t h a t
S t a r t i n g from the chain w
o f the integers, consider the
denote by
m
G
group
i s t h e group o f automorphisms o f any r e s t r i c t i o n o f
Gn
the a r i t y o f
. Then
(G, w ) - c h a i n
we see t h a t , f o r each
, the
n a m
to
R
A).
i s the
R
R
and
dilated n
elements
( o b v i o u s l y we r e p l a c e each element by i t s r a n k , which i s an i n t e g e r ) . Consequently, becomes an i n d i c a t i v e group f o r
Gn
More g e n e r a l l y , c o n s i d e r an n a m R
, denote by G(n)
to
n
n &s(G)
m-ary r e l a t i o n
the
which i s LJ -chainable;
f o r every
n - a r y group o f automorphisms o f any r e s t r i c t i o n o f
elements. Then t h e r e e x i s t s an
~ r o u pG(n)
(reduction threshold). R
n
such t h a t , f r o m t h i s p o i n t on, t h e
i s i n d i c a t i v e (HIGMAN 1977).
Problem, posed by FRASNAY 1984. F o r
n >/ s(m)
, the
group
G(n)
i s i t always
indicative . 4.4. The reassembling t h r e s h o l d l e d FRASNAY 1965 ( p . 517) t o t h e f o l l o w i n g r e s u l t
(l), and t h e n POUZET 1981 (p. 307) t o t h e f o l l o w i n g ( 2 ) and (3). Let
G
be an
m-ary group. There always e x i s t s an
m-ary r e l a t i o n
f r e e l y i n t e r p r e t a b l e i n t h e c h a i n o o f t h e i n t e g e r s , such t h a t o f automorphisms o f t h e r e s t r i c t i o n o f
R
t o an a r b i t r a r y
ordered by i n c r e a s i n g v a l u e s ( f o r i n s t a n c e we can t a k e c h a i n ) . Then d e n o t i n g a g a i n by
(1) t h e maximum b(R) b(R)
\<
m
+
t(G)
t(G)
R
which i s
R G
i s t h e group
m-element s e t , when t o be t h e
(G,G,)-
t h e reassembling t h r e s h o l d :
o f c a r d i n a l i t i e s o f a l l bounds o f
; t h e l a t t e r v a l u e i s reached i f
R
R
i s the
satisfies (G,w)-chain:
see FRASNAY 1973 ; ( 2 ) t h e r e e x i s t s a bound o f
R
with cardinality
m ; hence
b(R)
b
m ;
or
(3) e i t h e r b(R) = m+t(G) b(R) = m and t ( G ) = 1 , hence m+t(G) = mtl . 0 (1) L e t 9 be a f r e e o p e r a t o r which t a k e s w i n t o R : see ch.9 5 3.1. L e t U be an w a r y r e l a t i o n , bound o f R w i t h c a r d i n a l i t y s t r i c t l y g r e a t e r t h a n mtt(G)
. Thus
restrictions o f
U
w i t h c a r d i n a l i t y a t most equal t o
m+t(G)
361
Chapter 12
.
(m+t(G))-element subset X of the base I U l s a t i s f y i n g t o U/X = ? ( C x ) . By our hypotheses, the i s t h e base of a chain Cx G-compatible. By the reassembling theorem 3.4, t h e r e e x i s t s a Cx a r e mutually chain C w i t h base I U I , which i s G-compatible w i t h each Cx . I t follows t h a t U = ( C ) , hence t h a t U i s embeddable i n R , so U i s not a bound: contradiction. 0 0 ( 2 ) Take t h e r e s t r i c t i o n H of R t o t h e f i r s t m i n t e g e r s . Preserve the values of H f o r each sequence of m terms f o r which a t l e a s t two terms a r e equal. However, change the value of H when the m terms a r e a l l d i s t i n c t , e i t h e r by taking always t h e value (+) ( i f i t i s not already t h e case f o r H ) , o r by always taking ( - ) . The r e l a t i o n H thus modified i s a bound of R w i t h cardinality m . 0 ( 3 ) I t s u f f i c e s t o prove t h a t n = Max(m+l,b(R)) s a t i s f i e s t h e i n e q u a l i t y n 3 m t t ( G ) , hence s a t i s f i e s t h e reassembling theorem. Take a s e t E of c a r d i n a l i t y a t l e a s t equal t o n , i n which each n-element subset X i s t h e base of a chain DX , where these chains a r e assumed t o be mutually G-compatible. Then t h e image r e l a t i o n s 7 ( O x ) a r e mutually compatible, i . e . they have a common r e s t r i c t i o n t o the i n t e r s e c t i o n of their bases. To see a r e a l l embeddable i n
R
Thus each
t h i s , use t h e i n e q u a l i t y n >/ m + l : we can always pass from an n-element s e t X t o another Y by a f i n i t e number of intermediate n-element s e t s Xi , each of which includes the i n t e r s e c t i o n XnY , and each of which has a t l e a s t m common elements with t h e preceding one. Thus f o r these successive Xi , t h e
P(DX) , then y(DX)and y(Dy)have i i n t e r s e c t i o n Xn Y . images
a common r e s t r i c t i o n t o t h e
.
Under t h e s e c o n d i t i o n s , t h e r e e x i s t s a common extension S of these ? ( O x ) Moreover, each r e s t r i c t i o n of S t o a t most n elements i s t h e image of a chain under 9 , hence i s embeddable i n R . Additionally n 3 b(R) , hence S admits no embedding of any bound of R I t follows t h a t S i s i t s e l f embeddable in R . Thus t h e r e e x i s t s a chain D w i t h base E , such t h a t S = P(D). This D i s G-compatible w i t h each DX , thus the reassembling theorem holds. 0
.
Problem. Existence of a " t e r a t o l o g i c a l " w -chainable m-ary r e l a t i o n R whose bounds have maximum c a r d i n a l i t y b ( R ) = m and y e t w i t h t ( G ) = 1 where G denotes t h e group of automorphisms of any r e s t r i c t i o n of R t o an m-element s e t .
4 . 5 . Given t h e i n t e g e r m , the maximum c a r d i n a l i t y of t h e bounds of a l l m - 3 chainable r e l a t i o n s with i n f i n i t e base is equal t o t h e m-ary - reassembling t h r e s hold, in i t s complete form m + t ( m ) (FRASNAY 1973; t h i s r e s u l t i s extended
-
THEORY OF RELATIONS
3 68
by POUZET 1981 t o chainable and almost chainable m u l t i r e l a t i o n s w i t h maximum a r i t y equal t o m ) . In view of the preceding p r o p o s i t i o n , i t s u f f i c e s t o c o n s t r u c t an m-ary r e l a t i o n R f r e e l y i n t e r p r e t a b l e i n w , which admits a t l e a s t one bound of c a r d i n a l i t y m + l . Take the ( G , d ) - c h a i n where G i s t h e i d e n t i t y ; i n o t h e r words, take t h e r e l a t i o n x l < x2 < . . . 4 xm (mod L S ) ) . Then consider t h e binary cycle C w i t h c a r d i n a l i t y m+l ; on t h e base of C , define t h e m-ary r e l a t i o n U which t a k e s t h e value (+) i f f x1,x2, ...,xm a r e consecutive modulo C : t h i s U i s a bound
0
of
R . 0
4 . 6 . I f an m-ary r e l a t i o n R (m+t(m))-monomorphic and i f i t s c a r d i n a l i t y i s i n f i n i t e , o r f i n i t e but s u f f i c i e n t l y l a r g e , then R is chainable (FRASNAY 1965 p. 508 prop. 12.1.1 and POUZET 1981 p . 311 proo. V.3.8). 0 By the f i n i t a r y form of RAMSEY's theorem (ch.3 5 1 . 3 ) , i f t h e c a r d i n a l i t y of the base i s i n f i n i t e o r f i n i t e b u t s u f f i c i e n t l y l a r g e , then t h e r e e x i s t s a r e s t r i c t i o n of R which is chainable and of c a r d i n a l i t y m + t ( m ) . By monomorphism, a l l r e s t r i c t i o n s with c a r d i n a l i t y l e s s than o r equal t o m + t ( m ) a r e chainable.
S t a r t i n g w i t h such a r e s t r i c t i o n , and extending t h e chain i n which i t i s f r e e l y i n t e r p r e t a b l e t o t h e chain w , we obtain an m-ary r e l a t i o n S , which i s f r e e l y i n t e r p r e t a b l e i n G, , and a l l of whose r e s t r i c t i o n s 'to a t most m + t ( m ) elements a r e embeddable in R . We s h a l l show t h a t every f i n i t e r e s t r i c t i o n of R i s embeddable i n S , which w i l l prove t h a t t h e s e r e s t r i c t i o n s a r e chainable, and consequently t h a t R i s i t s e l f chainable. Suppose t h a t every r e s t r i c t i o n of R t o a t most k elements i s embeddable in S , and suppose k >/ m + t ( m ) . I f t h e r e e x i s t s a r e s t r i c t i o n with c a r d i n a l i t y k + l which i s not embeddable i n S , then t h i s r e s t r i c t i o n i s a bound of S , c o n t r a d i c t i n g t h e preceding proposition. 0
§
5 - MONOMORPHISM
THRESHOLDS, C H A I N A B I L I T YTHRESHOLD
5.1. MONOMORPHISM THRESHOLD-PAIR, THRESHOLDS p AND q Given an i n t e g e r m , we define t h e monomorphism threshold-pair a s being an ordered p a i r ( p , q ) of i n t e g e r s such t h a t : (1) every m-ary, (6p)-monomorphic r q l a t i o n w i t h c a r d i n a l i t y a t l e a s t equal t o q i s chainable; ( 2 ) t h i s i s no longer true i f we replace p by p ' < p o r q by q ' < q Consequently we have two d i s t i n c t monomorphism thresholds: the threshold p(m) i s the l e a s t i n t e g e r p f o r which every m-ary p-monomorphic r e l a t i o n w i t h i n f i n i t e o r s u f f i c i e n t l y l a r g e f i n i t e c a r d i n a l i t y i s chainable
.
( r e c a l l t h a t , f o r a s u f f i c i e n t l y l a r g e cardinal of t h e base,
p-monomorphism
369
Chapter 12
( 4 p)-monomorphism by ch.9 5 6.3);
implies
the threshold
q(m)
i s the least integer
r e l a t i o n o f cardinal & q
q
f o r which e v e r y
m-ary monomorphic
i s chainable.
The e x i s t e n c e o f these t h r e s h o l d s f o l l o w s f r o m 3.5 above (see FRASNAY 1965 p. 513) By t h e p r e c e d i n g 4.6, we have
p(m)
\<
m
+
t(m) : t h e threshold
p
i s bounded by
t h e reassembling t h r e s h o l d . Moreover
p ( 1 ) = 1, p ( 2 ) = 3, p ( 3 ) = 4, p ( 4 ) = 6 s i n c e i t i s proved by CAMERON 1983 t h a t t h e o t h e r p o s s i b l e value 5 i s f a l s e (see 5.3 below). From t h e upper bound o f HODGES, LACHLAN, SHELAH 1977 (see 4.1),
i t follows that
,a
m >/ 3
\<
p(m)
2m
-
2
for
m )/3
.
F i n a l l y we have
p(m) = 2m-2
for
r e s u l t due t o FRASNAY 1984, unpublished.
For the threshold
, we
q
have
q ( 1 ) = 1, q ( 2 ) = 4
and
q ( m ) b m+2
m
for
+2
(FRASNAY 1965 p. 513). 5.2. CHAINABILITY THRESHOLD r(m) : the l e a s t integer
r
p , we d e f i n e t h e c h a i n a b i l i t y t h r e s h o l d such t h a t e v e r y m-ary r e l a t i o n a l l o f whose
restrictions o f cardinal
r
a r e chainable, and whose base i s i n f i n i t e o r
Following the notion o f the threshold
f i n i t e but s u f f i c i e n t l y large, i s i t s e l f chainable.
r(m),<
Clearly
p(m),(
.
+ t(m)
m
5.3. The f o l l o w i n g s t r e n g t h e n i n g o f t h e theorem on s e t - t r a n s i t i v e groups (ch.11
5
5.10) i s due t o CAMERON 1976, as i s t h e theorem i t s e l f , and i s more
e a s i l y r e o b t a i n e d by POUZET 1981 p . 323. Let
m
set
E
be a p o s i t i v e i n t e g e r and
.f
G
&
the closure o f
m, m+l,
under
G
...,
be a p e r m u t a t i o n group on a denumerable
m-adherence i s
Then t h e c o n c l u s i o n i n ch.11 i s not
G
p(m)-set t r a n s i t i v e b u t n o t
5
m-transitive,
then
u - s e t - t r a n s i t i v e f o r every integer
u
.
5.10 h o l d s , except f o r t h e symmetric group,
Q
since
G
on
E
and isomorphic w i t h t h e c h a i n o f t h e r a t i o n a l s , such t h a t t h e c l o s u r e
m - t r a n s i t i v e . More p r e c i s e l y , t h e r e e x i s t s a c h a i n
of
G
under
m-adherence i s e i t h e r t h e group o f automorphisms o f
group o f automorphisms o f i n t e r m e d i a c y (mod Q)
, or
based
, or
Q
the
t h e group o f automorphisms
o f t h e c y c l i c r e l a t i o n a s s o c i a t e d w i t h Q , o r f i n a l l y t h e group o f automorphisms o f t h e dihedral r e l a t i o n associated w i t h Q
.
These groups a r e a t most 3 - t r a n s i t i v e . So i t f o l l o w s t h a t f o r
m, m+l,...,
t i v e . The case where
in
= 4
= 5 or 6
m-transi-
i s i n t e r e s t i n g . Indeed t h e r e e x i s t s a group whose
c l o s u r e i s none o f t h e f o u r groups l i s t e d above, and which i s transitive yet not
, every
m 3 5
p ( m ) - s e t - t r a n s i t i v e group which i s 4 - t r a n s i t i v e i s a l s o
4 - t r a n s i t i v e (CAMERON 1983). Thus
by FRASNAY 1965, hence n e c e s s a r i l y
p(4) = 6
4
and
p(4) f 5 ; since
.
5-setp(4)
THEORY OF RELATIONS
370
§
6 - FINITELY BOUNDED
AGE, U N I V E R S A L CLASS, ANTIBOUND
6 . 1 . BOUND OF AN AGE
Given an age 4 , a f i n i t e relation A i s said t o be a = o f n o t belong t o (/t yet every proper r e s t r i c t i o n belongs t o i&. a bound of any representative relation of the age.
FINITELY BOUNDED AGE An age & i s said t o be of 4 . For example, the a l l f i n i t e consecutivity since every binary cycle
df i f f A Equivalently
does A is
f i n i t e l y bounded i f f there are only f i n i t e l y many bounds age of a l l f i n i t e chains i s f i n i t e l y bounded. The age of relations and t h e i r r e s t r i c t i o n s i s n o t f i n i t e l y bounded, i s a bound: see 5 1 above.
6.2. UNIVERSAL CLASS
Given an a r b i t r a r y f i n i t e s e t U of f i n i t e relations of given a r i t y , the universal class defined by U i s the s e t of those f i n i t e relations ( u p t o isomorphism) which do not admit an embedding of any element of U A f i n i t e or i n f i n i t e relation R i s said t o be a representative of the class i f f R does n o t admit an embedding of any element of U . We can always suppose t h a t a l l elements of U are mutually incomparable under embeddability: i t suffices t o remove each relation in U which has a proper r e s t r i c t i o n belonging t o U A f i n i t e relation A i s said t o be a bound of the universal class ‘4 i f f A does n o t belong t o h ! y e t every proper r e s t r i c t i o n belongs t o A!. By the previous removing, we reduce U t o the bounds. As a f i r s t example of a universal c l a s s , take any f i n i t e l y bounded age; for instance the age of a l l f i n i t e chains. Every f i n i t e intersection of unjversal classes ( o f a given a r i t y ) i s a universal
.
.
class.
I t suffices t o take the union o f the s e t s of bounds, and then t o remove those relations which have a proper r e s t r i c t i o n in t h i s union. Every f i n i t e union of universal classes (of a given a r i t y ) i s a universal c l a s s . 0 Let & and @ be two universal classes. A f i n i t e relation X belongs t o the union i f f , f o r each bound A of and each bound B of d3 , e i t h e r X does n o t admit any embedding of A , or of B . Equivalently, f o r each minimal common extension C of A and B , the relation X does not admit any embedding of C Finally note t h a t Card C i s a t most equal t o Card A + Card B , so t h a t
.
Chapter 12
37 1
.
there e x i s t only f i n i t e l y many such C 0 Now we have other examples of universal classes, by taking f i n i t e unions and f i n i t e intersections of f i n i t e l y bounded ages. By ch.10 5 2 . 1 , we see t h a t such unions and intersections are n o t necessarily ages. For example, we leave i t t o the reader t o show t h a t there e x i s t ten binary relations with cardinality 2 , up t o isomorphism. Now consider again the age mentioned in ch.10 § 2.1: take an a r b i t r a r y f i n i t e s e t E , then a subset F of E , and take the value (+) f o r ordered pairs in F , the value ( - ) i f a term in the pair belongs t o E-F . Then among the ten considered r e l a t i o n s , seven of them are the bounds of the considered age (which i s thus f i n i t e l y bounded). Given an a r i t y n 3 2 , every universal class i s an intersection of f i n i t e l y many f i n i t e l y bounded ages. Consequence of the f a c t t h a t , given a f i n i t e n-ary relation A , the universal class having the unique bound A i s an age, by ch.10 5 2.3. For a r i t y 1, the class formed of a l l unary f i n i t e relations always (+) and f i n i t e relations always ( - ) i s neither an age, nor even an intersection of ages. 6.3. TWO ELEMENTARY CHARACTERIZATIONS OF UNIVERSAL CLASSES
The reader will see the equivalence between our definition and the following charact e r i z a t i o n of universal classes by VAUGHT 1953: & i s universal i f f there e x i s t s an integer p ( i n f a c t p i s the maximum card i n a l i t y of bounds) such t h a t : if R i s a representative of & and i f S i d relation a l l of whose r e s t r i c t i o n s t o at.m_Tt p elements are_ embeddable in R , .-then S i s i t s e l f a representative of LA! Also l e t us mention the o r i g i n a l , logical definition due t o TARSKI: there e x i s t s a prenex formula having only universal q u a n t i f i e r s , and taking the value (t) only f o r representatives of the class. Note t h a t the minimal possible number of quantifiers i s equal t o the maximum cardinality of bounds. For the meticulous reader, note t h a t , when extending the usual truth-value t o the case where the base of relations can be empty, then the logical definition f o r universal classes f a i l s , and y e t the two other mentioned definitions subsist. The exceptional case concerns t h i s universal class which reduces t o the singleton of the 0-ary relation (O,+) with empty base (and similarly with the value ( - ) ) :
.
.
This i s an exceptional case where the representing formula cannot be written in .. prenex form: see FRAISSE 1982, p . 72. 6 . 4 . NON-TRIVIAL EXAMPLES OF UNIVERSAL CLASSES Given an integer m , the class whose representatives are a l l m-ary chainable relations i s universal; similarly f o r m-ary monomorphic relations (JEAN 1967; easy consequence of FRASNAY's theorems 3.3 t o 3.5 above).
THEORY OF RELATIONS
312
Given a permutation group G , t h e c l a s s whose r e p r e s e n t a t i v e s a r e a l l theG-chains i s u n i v e r s a l ; moreover t h e maximum c a r d i n a l i t y of bounds i s m + t ( G ) (FRASNAY 1979).
6.5. ANTIBOUND OF A UNIVERSAL CLASS Given a universal c l a s s L& , a f i n i t e r e l a t i o n A i s s a i d t o be an antibound of df i f f A belongs t o d y e t no proper extension of A belongs t o ( A . I n t h e p a r t i c u l a r case of a f i n i t e l y bounded age, only a t r i v i a l antibound can e x i s t : take f o r instance a f i n i t e r e l a t i o n A and the age of a l l r e s t r i c t i o n s of A : then A i t s e l f i s t h e only antibound. In t h e general case, consider f o r instance t h e q u a d r i r e l a t i o n Rn = (In,Cn,On,Un) of MALITZ, introduced i n ch.5 § 8.3. Take the universal c l a s s whose represent a t i v e s a r e those q u a d r i r e l a t i o n s R , a l l of whose r e s t r i c t i o n s t o a t most 3 elements a r e embeddable i n R7 , and consequently embeddable in each Rn ( n 2 7 ) . Then each Rn f o r n ..a 7 i s an antibound of our universal c l a s s . For f u r t h e r informations, see FRAISSE 1973 p. 98 t o 103.
.
31 3
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J . TUKEY 1940 - Convergence and uniformity in topology. Annals of Math. Studies vol . 2, Princeton University Press. Paul TURAN 1955 - On the theory of graphs. Colloquium Math. vol. 3 p. 19-30. S t a n i s l a s M. ULAM 1960 - A c o l l e c t i o n of mathematical problems. I n t e r s c i e n c e Tracts i n Pure and A p p l . Math. vol. 8, I n t e r s c i e n c e Publishers (Groningen), 150 p . Robert L. VAUGHT 1953 - Remarks on universal a r i t h m e t i c a l c l a s s e s . B u l l e t i n h e r . Math. SOC. vol. 59 p . 391. 1961 - Denumerable models of complete t h e o r i e s . I n f i n i s t i c Methods. Pergamon Press (Oxford) p. 303-321. Bartel van d e r WAERDEN 1931 - Modern a l g e b r a , vol. 11; ed. 1967, Heidelberger Taschenbucher vol . 23, Springer-Verlag ( B e r l i n ) , 300 p . K. WALKER 1971 - An upper bound f o r t h e Ramsey number M(5,4). Journal Combin. Theory vol. 11 p . 1-10. Earl Glen WHITEHEAD 1973 - The Ramsey number N(3,3,3,3;2). Discrete Math. vol. 4 p . 389-396. E.S. WOLK 1967 - P a r t i a l l y well-ordered s e t s and p a r t i a l o r d i n a l s . Fundamenta Math. v o l . 60 p . 175-186. 1968 - The topology o f a p a r t i a l l y well-ordered s e t . Ibid. vol. 62 p . 225-264. James YACKEL 1972 - I n e q u a l i t i e s and asymptotic bounds f o r Ramsey numbers. Journal Combin. Theory ( B ) vol. 13 p . 56-68. Ernst ZERMELO 1904 - Beweis dass j e d e Menge wohlgeordnet werden kann. Math. Annalen v o l . 59 p. 514-516. 1908 - Untersuchungen uber d i e Grundlagen d e r Mengenlehre. Ibid. vol . 65 p . 261-281. 1930 - Uber Grenzzahlen und Mengen bereiche. Fundamenta Math. vol. 16 p . 29-47.
Martin ZIEGLER 1980 ed. S . I . Adian, W . W .
M.M. ZUCKERMAN 1973 p. 289-294.
-
Algebraisch abgeschlossene Gruppen. Word Problems 11, Boone, G. Higman, North-Holland (Amsterdam), p . 449-576. Natural sums o f o r d i n a l s . Fundamenta Math. vol. 77
381
INDEX
ABBOTT, HANSON, 3-exerc. 1
absorption, 1-3.1; 1-3.3 a c c e s s i b l e c a r d i n a l , 2-6.7 a d d i t i v e l y f r e e set, 3-exerc. 1 adherence, n-adherence, adherent permutation, 11-2.3 age, 10-2 AJDUKIEWICZ, 1-1.2 almost chainable r e l a t i o n , 10-9 amalgamable age, amalgamable s e t , 11-1.5 amalgamation lemma, 2-2.2; amalgamation theorem (homogeneous re1 a t i o n s ) , 11-1.6 a l e p h , 1-6.1; aleph rank, 1-6.4 ALEXANDROFF, 1-exerc. 2 antibound, 12-5.4 a n t i c h a i n , 2-2.9 a r i t i e s of an o p e r a t o r , 9-3; comparison between a r i t i e s , 9-3.2 a r i t y , n-ary ( r e l a t i o n , m u l t i r e l a t i o n ) , 2-1 ARONSZAJN ( t r e e ) , 5-7.1 a s s i g n a b l e ( m u l t i r e l a t i o n ) , 9-3 ASSOUS, 8- 1.3 automorphism, 2-3; local automorphism, 9-1.7 axiom of a c c e s s i b i l i t y , 2-6.7 axiom of choice, 1-1.7; f o r f i n i t e s e t s , 1-2.10 axiom of dependent choice, 2-1.6 axiom of foundation, 1-2; 1-exerc. 1 axiom of i n f i n i t y , 1-2.4 axiom of maximal chain (HAUSDORFF-ZORN), 2-2.7 axiom of maximal chain of i n c l u s i o n , maximal chain on i n i t i a l i n t e r v a l s , 4-1.4 axiom of maximal i d e a l , 4-5 axiom of ordering, 2-3 axiom of reinforcement, 2-4.2 axiom o f trichotomy, 2-2.5 axiom of well-ordering, 2-2.5 axioms of ZF, 1-2.4 BACHMANN, 1-9
bad sequence, 4-2.2 b a r r i e r , 8-1 b a r r i e r sequence, good and bad b a r r i e r sequence, 8-2 base ( n o t a t i o n 1 I ) , 2-1 basic clopen set, 7-2.2 BAUMERT, 3-exerc. 1 BELL, SLOMSON, 2-exerc. 2 BENEJAM, 2-1.5 BERCOV, HOBBY, 11-2.8 BERNAYS, 1-exerc. 1 BERNSTEIN, 2-8.1 BERNSTEIN, SCHROOER, 1-1.4; f o r r e l a t i o n s , 5-1.1 b e t t e r p a r t i a l ordering, q - b e t t e r p a r t i a l ordering, _better quasi-ordering, 8-5.1 - b e t t e r p a r t i a l ordering ( w i t h respect t o b a r r i e r s ) , b e t t e r quasi-ordering, 8-6 b e t t e r p a r t i a l ordering of words, 8-6.9 betweenness r e l a t i o n = intermediacy, 2-8.6 bichain, 11-3.7 b i j e c t i o n = b i j e c t i v e f u n c t i o n , 1-1.2 binary r e l a t i o n , 2-1; binary c y c l e , 2-8.6
388
Index
b i r e l a t i o n , 2-1 BIRKHOFF. 4-1.8 b i v a l e n t - t a b l e , 9-9 BLASS, 1-7.2; 2-6.8 BOGART, 4-exerc. 2 BONNET, 4-5.3; 7-2.1 and 2.4; 7-4.8 and 4 . 9 ; 7-exerc. 4; 8-4.5 BONNET, POUZET, 4-1.3 and 1 . 5 ; 4-9.2; 7-4.8 and 4.9 bound: i n i t i a l i n t e r v a l , 4-8; bound o f a r e l a t i o n , 12-1
; 8-exet-c. 3
CALAIS, 11-1.5; 11-6.4 CAMERON, 11-5.10; 12-5.1 and 5.3 CANTOR, lemma, theorem, 1-1.5; normal orm, 1-3.5 C a n t o r ' s theorem f o r p a r t i a l o r d e r i n g s (DILWORTH, GLEASON), 5-2.6 canonical group, 11-5; 11-exerc. 2 c a r d i n a l , 1-5.4 c a r d i n a l e x p o n e n t i a t i o n ( a b ) , product x ) , sum (+), 1-5.5; 2-3.8 C a r t e s i a n product ( x ) , 1-1.2; 1-1.8 c h a i n = t o t a l o r d e r i n g , 2-2.3 chain a s s o c i a t e d t o a tree, 4-6.2 chain meeting every h e i g h t (POUZET), 4-4.5 chain sequence, 8-7.1 G-chain, (G,A)-chain, 12-4.3 (G-rangement = G-chain) c h a i n a b i l i t y t h r e s h o l d , 12-5.2 chainable r e l a t i o n , 9-5 CHANG, KEISLER, 10-1.4 c h o i c e axiom, choice s e t and f u n c t i o n , 1-1.7 choice axiom f o r f i n i t e s e t s , 1-2.10 CHUNG, 3-1.6 CLARK, KRAUSS, 12-4.3 c l o s e d c o f i n a l set = c l o s e d unbounded s e t , 1-exerc. 2 closed under embeddabi 1 i t y , 10-2 c l o s u r e ( t o p o l o g y ) , 7-2.2 c l o s u r e ( t r a n s i t i v e ) , 1-5.1 c o f i n a l , c o - i n i t i a l set and r e s t r i c t i o n , 2-5.1 c o f i n a l h e i g h t (Cofh), 2-7.3 the c o f i n a l h e i g h t is a c a r d i n a l (POUZET), 2-7.6 c o f i n a l i t y ( C o f ) , c o - i n i t i a l i t y , 2-5.4 o( = Sup o( . ) , 2-5.5 c o f i n a l sum i n an indecomposable o r d i n a l ( 1 c o f i n a l well-founded r e s t r i c t i o n , 2-5.1; 2-5.3; 2-7.1 COHEN, 1-2.4; 1-4.4; 2-exerc. 1 coherence lemma, 2-1.3 c o l o r , 3-1.1; 3-4.1 combinatorial lemmas on c o l o r s (POUZET), 3-4.1 t o 4.3 common e x t e n s i o n o f compatible r e l a t i o n s , 2-1.2 common 1-extension, 10-2.5 commutative or n a t u r a l sum and p r o d u c t , 1-9 comparable elements i n a p a r t i a l o r d e r i n g , 2-2 comparable r e l a t i o n , 5-2.7 comparison between a r i t i e s , 9-3.2 compatible r e l a t i o n s , 2-1.2; G-compatible c h a i n s , 11-3 component i n a m u l t i r e l a t i o n , 2-1 conjugate p a r t i a l o r d e r i n g s , 4-exerc. 3 conjunction o f p a r t i a l o r d e r i n g s , 4-7.3; c o n j u n c t i o n o f r e l a t i o n s , 2-1.7 c o n s e c u t i v i t y r e l a t i o n , 2-8.6 continuum-equipotent s e t , 1-4.1; continuum-equipotent r e l a t i o n , 2-1 continuum h y p o t h e s i s , 1-4.4; connection w i t h the e q u a l i t y G 2 = a l , 1-6.5 c o n t r a c t e d group, 11-3.7
z
Index
389
convergent f u n c t i o n , 7-exerc. 1 converse r e l a t i o n , 2-1.7 COROMINAS, 8-1.4; 8-4.4 countable set, countable axiom of choice, 1-2.5 couple = ordered p a i r , 1-1.2 covering by doublets, 6-4.5; f i n i t e number of c l a s s e s ( L A V E R ) , 8-4.5 covering by indecomposable chains, 6-4.3 c r i t e r i o n theorem f o r prehomogeneous r e l a t i o n s ( P O U Z E T ) , 11-7.9 c r i t e r i o n theorem f o r s a t u r a t e d r e l a t i o n s (POUZET, VAUGHT) , 10-7.6 c u t , 4-1.1; DEDEKIND's theorem, 1-4.6; 4-1.1 cycle o f c o n s e c u t i v i t y , 2-8.6 ; 3-cycle, g-6:5 c y c l i c t e r n a r y r e l a t i o n a s s o c i a t e d w i t h a chain, 2-8.6 decomposable chain = a d d i t i v e l y decomoosable chain, 6-3 decomposable o r d i n a l , 1-3.6 decomposition of a chain (homomorphic image), 2-3.6 decomposition of an indecomposable chain (HAGENDORF), 6-3.3 decomposition of a p a r t i a l ordering i n t o i d e a l s , 4-5.2 and 5.3 decreasing sequence of chains of r e a l s , DUSHNIK, MILLER, 5-5.2; SIERPINSKI, 5-5.3 DEDEKIND, 1-1.3 ( f i n i t e s e t ) ; 1-4.6; 4-1.1 d e f i n i t i o n by r e c u r s i o n , 1-2.11 DE JONGH, PARIKH, 4-9.2 DEMBOWSKI , 9-6.9 dense chain, 5-3.1; a-dense chain, 5-3.4 dense s e t of r e a l s , 1-4.5; dense s e t i n a c h a i n , 5-6 denumerable s e t , 1-2.5; denumerable r e l a t i o n , 2-1 denumerable subset axiom, 1-2.6 denumerablv SzDilra.in chain. 7-4.4 dependent choice axjom, 2-1 16 DEVLIN, JOHNSBRATEN, 5-6 d i l a t e d group, 11-3.2 DILWORTH, 4-exerc. 2 DILWORTH, GLEASON, 5-2.6 dimension of a p a r t i a l ordering, 4-7.3 d i r e c t e d p a r t i a l ordering = n e t , 4-5 d i r e c t e d under embeddability, 10-2 d i r e c t product of p a r t i a l o r t o t a l o r d e r i n g s , 4-7 d i s j u n c t i o n of r e l a t i o n s , 2-1.7 domain of a function (Dom), 1-1.2 domain chain, 8-7.1 doublet of indecomposable c h a i n s , 6-4.4 DUSHNIK, MILLER, 3-3.2 and 3.3; 4-7.3; 4-7.8; 5-5.2 n-element s e t , 1-2.3 ELLENTUCK, 3-6.1 embeddability (6 ) between r e l a t i o n s , 5-1; between sequences, 4-2 A-embeddability, 10-3.1 embeddability theorem f o r s c a t t e r e d chains ( L A V E R ) , 8-4.4 empty function, 2-3 equimorphic, equimorphism ( n o t a t i o n I ), 5-1 equipotence, 1-1.3 equivalence r e l a t i o n generated by a quasi-ordering, 2-2 (1,p)-equivalence, 10-8.5 ERDdS, 3-3.4; ERDOS, RADO, 3-3.5; 3-exerc. 2 and 3; 5-3.7 existence criterion, see "criterion" e x i s t e n t i a l l y c l o s e d , see "maximalist" exponentiation between c a r d i n a l s o r s e t s , notation ab , 1-1.6; 1-5.5 exponentiation between o r d i n a l s : ba , 1-3.3; 1-exerc. 3 extension of a f u n c t i o n , 1-1.3; extension of a r e l a t i o n , 2-1.1 1-extension, 10-1.4; common 1-extension, 10-2.5 extensive s e t , 11-6; G-extensive s e t , 11-7
390
Index
extensive b i v a l e n t t a b l e , 9-9 e x t r a c t e d sequence, 1-2.2 e x t r a c t i o n property, 4-9.3 e x t r a c t i o n theorem f o r w e l l p a r t i a l orderings, 4-3.4 f a i t h f u l extension o f a r e l a t i o n , 5-1.3; o f a chain, 5-4.5 FEFERMAN, 2-1.6 FELGNER , 2-4.2 f i c h e = i n d i c a t o r , 11-4.2 f i l t e r , 1-7.1 f i n a l i n t e r v a l , 2-2 f i n e r ( f i l t e r ) , 1-7.1 f i n i t e l y bounded age, 12-5.1 f i n i t e l y f r e e p a r t i a l ordering, 4-3 f i n i t e l y generated i n i t i a l i n t e r v a l ( F ) , 4-1.8 f i n i t e number of bounds f o r a chainable r e l a t i o n (FRASNAY), 12-3.3 f i n i t e set, 1-1.1; f i n i t e r e l a t i o n , f i n i t e m u l t i r e l a t i o n , 2-1 f i n i t e t r e e s (KRUSKAL's theorem), 5-2.3 f i n i t i s t r e l a t i o n , 10-8; c h a r a c t e r i z a t i o n , 10-8.5 f i n i t i s t s t r u c t u r e , 10-exerc. 2 f i x e d D o i n t lemma (KNASTER. TARSKI). 1-1.2 FODOR,' 1-exerc. 2 ' FOLKMAN, 3-1.6 forerunner b a r r i e r , forerunning, 8-3 and 3.1 forerunner b a r r i e r sequence , 8-3.3 foundation axiom, 1-2; 1-exerc. 1 FRAENKEL, 1-1.4; 1-2.4 FRAISSf, 10-2.5; 10-8.5; 10-exerc. 2 ; 12-6.3 and 6.5 FRASNAY, 11-4.5; 12-3.3 t o 3.5; 12-4.1 and 4.2: 12-5 FREDRJCKSON. 3-e&erc. 1 f r e e i n t e r p r e t a b i l i t y , 9-2 f r e e operator, 9-3 f r e e s e t ( i n a p a r t i a l o r d e r i n g ) , 2-2.9 f u n c t i o n , 1-1.2 fundamental rank , 1-5.2
GALVIN, 3-2.2; 3-6 ; 7-4.8; 8-exerc. 1 GALVIN, MAC KENZIE, 7-4.8 GALVIN, PRIKRY , 3-6.1 GARCIA, 3-1.6 generalized continuum hypothesis, 1-4.4; 2-exerc. 1 generated i n i t i a l i n t e r v a l , 4-1.8; 4-2.5; 4-4.3 GILLAM, 12-1.4 GIRAUD, 3-1.6 GLEASON, GREENWOOD, 3-1.4 GLEASON. see DILWORTH GBDEL, i-2.4; 1-4.7 GOLOMB, see BAUMERT good sequence, 4-2.2 graph = symmetric b i n a r y r e l a t i o n GRAVER, YACKEL, 3-1.5 g r e a t e r sequence , 4-2 GREENWOOD, see GLEASON GRINSTEAD, ROBERTS, 3-1.5 HAGENDORF, 5-1.2 and 1.3; 5-2.2; 5-4.1 t o 4.7; 6-3.8; 6-6.3; 6-exerc. 3; 8-4.6 and 4.7 HALPERN, LEVY, 2-2.8
5-5.2;
5-exerc. 1 t o 3; 6-3.3 t o 3.5;
Index HANSON, 3-1.6; see ABBOTT HARTOGS aleph, o r s e t , 1-6.2 HATCHER. 1-2.9 HAUSDORFF,6-2.1 HAUSDORFF, ZORN, 2-2.7; 2-exerc. 3 h e i g h t ( H t ) , 2-3.2 and 3.3 HENSON, 11-1.5 h e r e d i t a r i l y f i n i t e s e t , 1-5.1 h e r e d i t a r i l y indecomposable chain, 6-5 h e r e d i r a r i l y indecomposable chains: theorem (LAVER), 6-5.4 and 5.5 h e r e d i t a r i l v t r a n s i t i v e . 1-2.9 HESSENBERG," 1-9 HIGMAN, 4-4.1; 4-4.4 ; 8-5.3 HILL, IRVING, 3-1.6 h-indecomposable c h a i n = h e r e d i t a r i l y indecomposable chain, 6-5 HIRAGUCHI, 4-7.9 HIRSCHFELD, WHEELER, 10-7.3 HOBBY, see BERCOV HODGES, 2-6.8; 4-2.4; 10-7.3; 10-exerc. 2 HODGES, LACHLAN, SHELAH, 12-4.1; 12-4.6 homogeneous r e l a t i o n , p-homogeneous r e l a t i o n , 11-1 homogeneous system, 11-2.1 homomorphic image o f a c h a i n , 2-3.6 ; 8-7.2 homomorphic image o f a r e l a t i o n , 9-8
i d e a l , 4-5 i d e n t i c a l f i l t e r , 10-1.4 image c h a i n = homomorphic image, 2-3.6; 8-7.2 immediate e x t e n s i o n o f a c h a i n , 5-4.2; o f a r e l a t i o n , 5-1.2 i m m e d i a t e l y g r e a t e r = immediate e x t e n s i o n (up t o isomorphism) i n a c c e s s i b l e c a r d i n a l , 2-6.7 i n c i d e n c e m a t r i x , 3-5.1 i n c l u s i o n ( C_ ) , s t r i c t o r p r o p e r i n c l u s i o n ( c ) , 1-1 i n c l u s i o n among i n i t i a l i n t e r v a l s ( 3 ) , 4-1.3 i n c o m p a r a b i l i t y lemma f o r r e l - a g e s (POUZET), 10-3.7 incomparable c h a i n s o f r e a l s , 5-5.3 incomparable elements i n a p a r t i a l o r d e r i n g ( n o t a t i o n I ) , 2-2 i n c r e a s i n g f u n c t i o n on r e a l s , 2-8.3 i n c r e a s i n g number o f o r b i t s (LIVINGSTONE, WAGNER), 11-2.8 indecomposable chain, 6-3 indecomposable c h a i n : HAGENDORF's decomposition theorem, 6-3.3 indecomposable o r d i n a l , 1-3.6 indecomposable sequence, 8-5 i n d e x ( i n a sequence), 1-2.2 i n d i c a t i v e group (FRASNAY), 11-4; i n d i c a t o r ( = f i c h e ) , 11-4.2 Q - i n d i c a t i v e group, 11-5 i n d i v i s i b l e chain, HAGENDORF and POUZET's r e s u l t s , 6-6.3 i n d i v i s i b l e r e l a t i o n , 6-6 induced age, 10-3.3 induced c u t , 4-1.1 i n d u c t i o n ( t r a n s f i n i t e ) , 1-2.11 i n e x h a u s t i b l e r e l a t i o n , 10-5; i n e x h a u s t i b l e age, 10-5.4 i n e x t e n s i v e t a b l e , 9-9 infimum ( I n f ) , see "supremum" i n f i n i t e s e t , 1-1.1
3 91
392
Index
i n f - r e s t r i c t i o n of a b a r r i e r sequence, 8-2.2 i n i t i a l i n t e r v a l = i n i t i a l segment, 2-2 i n i t i a l i n t e r v a l generated by a minimal bad sequence, 4-2.5; 4-4.3 i n i t i a l i n t e r v a l of a sequence, 4-2.1 i n i t i a l i n t e r v a l s of a well p a r t i a l ordering (HIGMAN), 4-4.1 i n i t i a l i n t e r v a l s : ordering by i n c l u s i o n ( n o t a t i o n 3 ), 4-1.3 i n i t i a l i n t e r v a l theorem (GALVIN), 3-2.2 i n i t i a l l y maximal chain, 4-6.5 i n j e c t a b l e o r d i n a l , 1-6.2 i n j e c t i o n = i n j e c t i v e f u n c t i o n , 1-1.2 i n j e c t i v e f i l t e r , 10-1.1 i n j e c t i v e o p e r a t o r , 9-3.2 i n t e g e r = non-negative i n t e g e r = n a t u r a l number = f i n i t e o r d i n a l , 1-2.3 intermediacy = betweenness r e l a t i o n , 2-8.6 i n t e r p r e t a b i l i t y ( f r e e ) , 9-2 i n t e r v a l , 2-2 inverse function f-' ; inverse transformation ( f - ' ) " , 1-1.2 inverse p r o j e c t i o n modulo a f i l t e r , 10-1 inverted p a i r , 11-5.1 IRVING, 3-1.6 ISBELL, 3-1.6 i s o l a t e d element (topology) , 7-2.3 i s o l a t e d rel-age, i s o l a t i n g p a i r of f i n i t e r e l a t i o n s , 11-7.4 isomorphic, isomorphism, isomorphism type, 2-3 isomorphism (1ocal ) , 9-1.4 1-isomorphism, (1,p)-isomorphism, 10-1.8 A-isomorphism, 10-3.1 JEAN, 9-6.6 and 6.7; 12-6.4 JECH, 1-2.5 and 2.6; 2-1.6; 5-5 JENSEN, 5-5 JOHNSBPATEN, see DEVLIN JONSSON. 11-1.5 JORDAN, 9-6.7; 11-2.2 JULLIEN, 5-2.1; 5-4.5 and 4.7; 6-3.3;
6-3.6 and 3.7; 7-4.6 and 4.7; 12-4.2
KALBFLEISCH, 3-1.6 KANTOR, 3-5.1 KEISLER, see CHANG KERY, 3-1.5 kernel of a f i n i t i s t r e l a t i o n , 10-8.1; of a f i n i t i s t s t r u c t u r e , 10-exerc. 2 kernel of an almost chainable r e l a t i o n , 10-9.5 KLAUA, 1-9 KNASTER, 1-1.2 KONIG, theorem, 1-1.8; lemma, 4-4.5; 5-2.5 KRASNER, 7-exerc. 2 KRAUSS. 10-2 KRAUSSIsee CLARK KRUSKAL, 5-2.3 KURATOWSKI, 1-1.2; 2-2.7; 2-8.1 KUREPA, 4-6.1; 5-7.1 LACHLAN, see HODGES LARSON, 6-3.8 LAUCHLI, 2-2.3 LAVER, 6-5.4 and 5.5; 8-4.4; 8-7.3 t o 7.8 l e f t indecomposable chain, s t r i c t l y l e f t indecomposable chain, 6-3.1 length of a sequence, 1-2.2
Index lesser sequence, 8-7.1 "less than" relation among sequences, 4-2 LEVY, 1-2.5; see HALPERN lexicographically well-ordered set, lexicographic rank, 3-2.1 limit aleph, 1-6.3 limit ordinal, 1-2.4 linear independence lemma (KANTOR), 3-5.1 LIVINGSTONE, WAGNER, 11-2.8 local isomorphism, 9-1.4; local automorphism, 9-1.7 LOPEZ, 3-6; 5-8.3 and 8.4; 5-exerc. 1; 9-1.2; 9-9; 12-1.7
MAC KENZIE, see GALVIN MAC NEILLE completion, 4-exerc. 1 MALITZ, 5-8.3 ; 12-6.5 mapping = function, 1-1.2 MATHIAS, 2-4.2 matrix (incidence), 3-5.1 maximal chain, 2-2.6 maximal chain axiom (HAUSDORFF-ZORN), 2-2.7; 2-exerc. 3 maximal chain of inclusion, 4-1.2; 4-1.4 maximal free set or maximal antichain, 2-2.9 maximal ideal axiom, 4-5 maximal reinforced chain theorem (DE JONGH, PARIKH), 4-9.2 maximal rel-age , 10-3.5 maximalist extension theorem, 10-3.9 maximalist set, maximalist relation, 10-3.8 maximum (Max). maximal element. 2-2 MILLER,see DUSHNIK MILNER, 4-5.3 minimal bad sequence, 4-2.3 minimal bad barrier sequence (NASH-WILLIAMS), 8-2.2 minimum (Min), minimal element, 2-2 monochromatic set, 3-1.1; 3-1.3 monomorphic, p-monomorphic, ( 5 p)-monomorphic relation, 9-6 monomorphism threshold, 12-4.6 MOORE, 2-2.7 1-morphism, (1,p)-morphism, 10-1.8 -morphism, 10-4.5 MOSTOWSKI, 2-1.6 multicolor theorem (POUZET), 3-5.3 multirelation, 2-1
*
NASH-WILLIAMS, 3-2.4; 8-1.4; 8-2.1 and 2.2; 8-3.1; 8-6 natural number = integer (non-negative), 1-2.3 natural sum and product, 1-9 negation of a relation, 2-1.7 neighborhood, 4 -neighborhood, 6-2.2 ; neighborhood rank, 6-2.4 net = directed partial ordering, 4-5 von NEUMANN, 1-2 non-embeddability rank, non-richness rank, 10-4.7 normal form (CANTOR), 1-3.5
older relation, 10-1.2 A-older relation, 10-3.2; o( -older relation, 10-4.5 operator: free, 9-3; injective, 9-3.2; partial, 9-3.3
393
394
Index
o r b i t , 11-2.2 o r d e r a b l e s e t , 2-2.3 ordered p a i r , 1-1.2 o r d e r i n g , 2-2 ; o r d e r t y p e , 2-3 o r d e r i n g axiom, 2-3 o r d i n a l , 1-2.1 ; o r d i n a l w , 1-2.4; o r d i n a l w1 , 1-6.3 o r d i n a l e x p o n e n t i a t i o n , 1-3.3; 1-exerc. 3 ordinal-indexed sequence = o r d i n a l sequence, 1-2.2 o r d i n a l sum ( ) and product; o r d i n a l s : 1-3.2 and 3 . 3 ; c h a i n s : 2-3.6 and 3.7
PABION, 9-2.2; 11-7; 12-2.3 P A I L L E T , 9-2.4 ; 10-2.7
p a i r = unordered p a i r , 1-1 PARIKH, see DE JONGH
p a r t i a l o r d e r i n g , 2-2 p a r t i a l o r d e r i n g o f i n i t i a l i n t e r v a l s ( 3 ) , 4-1.1 p a r t i a l o r d e r i n g o f words, 4-2 p a r t i t i o n i n s l i c e s (BONNET, POUZET), 4-1.5 p a r t i t i o n lemmas: OUSHNIK, M I L L E R , 3-3.2; ERDOS, 3-3.4 p a r t i t i o n lemmas and theorems: DUSHNIK, M I L L E R , 3-3.2 and 3.3 p a r t i t i o n lemmas and theorems: ERDOS, RADO, 3-3.4 and 3.5; 3-exerc. 2 and 3 p e r f e c t b a r r i e r sequence, 8-2.1 PERLES, 4-exerc. 2 permutation, 1-1.2 PINCUS, 2-2.3 POIZAT, 4-3.3; 4-3.5 POUZET, 2-5.1 t o 5.3; 2-7.1 t o 7.6; 3-2.2; 3-4.2; 3-5.3; 4-3.5; 4-4.1 t o 4.5; 4-7.8; 4-10; 4-exerc. 4; 7-2.3; 7-3.1 t o 3.10; 8-1.3 and 1 . 7 ; 8-5.4; 8-6.7 and 6.8; 9-3.5; 9-6.3 t o 6 . 5 ; 9-7; 10-5.5 and 5.6; 10-6.5; 10-7.6; 10-9.9; 11-2.6 t o 2.10; 11-7.9; 11-8.3; 12-1.7 t o 1.9; 12-3.1 t o 3.3; 12-4.3; 12-4.7 POUZET, POUZET, POUZET. POWELL;
ROSENBERG, 3-exerc. 4 VAUGHT, 10-7.5 see BONNET
1-2.9 power, see "exponentiation"; power s e t ( T ), , 1-1 p r e d e c e s s o r , 1-2 prehomogeneous r e l a t i o n , 11-7; c r i t e r i o n theorem, 11-7.9 pre-ordering = q u a s i - o r d e r i n g preserved p a i r , 11-5.1 .
I
PRIKRY, see G A L V I N
p r i n c i p a l chain o r p r i n c i p a l t o t a l o r d e r i n g , 2-5.2 product: o r d i n a l s , 1-3.2; c a r d i n a l s ( x ) , 1-5.5; c h a i n s , 2-3.7 p r o f i l e o f a r e l a t i o n , 9-7 p r o j e c t i o n f i l t e r , 10-1 proper embedding = s t r i c t embedding ( < ) , 5-1 proper i n i t i a l i n t e r v a l , 2-2 pseudo-amalgamable age, 11-6.6 pseudo-amalgamation theorem ( C A L A I S ) , 11-6.6 pseudo-homogeneous r e l a t i o n , 11-6
Q = chain of the r a t i o n a l s , 2-2.3 q u a n t i f i e r ( u n i v e r s a l , e x i s t e n t i a l ) , 9-8 q u a s i - o r d e r i n g , 2-2
Index RAOO, 2-1.5; Rado's well p a r t i a l o r d e r i n g , 4-4.2; 8-5.2 RADO, see EROOS RAMSEY. 3-1.1 and 1.3
Ramsey number, 3-1.4; 3-3.1,. Ramsey set o r sequence (EROOS-RADD, followed by LOPEZ), 3-6.1 rang& (Rng) , 1-1:2 rank: fundamental, 1-5.2; lexicographic, 3-2.1; neighborhood, 6-2.4 ranking f u n c t i o n , 8-3.3 r a t i o n a l and real number, 1-4.5 reassembling theorem (FRASNAY), 12-3.4; t h r e s h o l d , 12-4.2 r e c u r s i o n , 1-2.11 reduced t r e e , 4-6.4 reduction theorem (FRASNAY), 11-4.6; t h r e s h o l d , 12-4.1 r e g r e s s i v e f u n c t i o n , 1-exerc. 2 r e g u l a r aleph o r c a r d i n a l , 2-6.1 reinforcement, 2-4.1; reinforcement axiom, 2-4.2 reinforcement of a well-founded ordering, 2-4.4 reinforcement of a well p a r t i a l ordering, 4-3.5; 4-exerc. 5 rel-age, 10-3.1 r e l a t i o n , 2-1 r e l a t i o n a l system, 11-2.1 re1 a t i ve r e s t r i c t i o n , isomorphism, embeddabi 1i t y , 10-3.1 r e p r e s e n t a t i v e r e l a t i o n ( i n an a g e ) , 10-2 r e s t r i c t i o n : f u n c t i o n , 1-1.3; r e l a t i o n , 2-1.1 r e t r o - o r d i n a l , 2-1.7 r i c h r e l a t i o n , 10-4; r i c h f o r i t s age, 10-6 r i c h f o r i t s age: e x i s t e n c e c r i t e r i o n (POUZET, VAUGHT), 10-7.6 r i g h t indecomposable, s t r i c t l y r i g h t indecomposable chain, 6-3.1 ROBERTS, s e e GRINSTEAD ROBINSON. 10-3.10: 10-7.3 ROSENBERG, see PO~JZET ROSENSTEIN, 2-8.2; 5-5.3; 8-3.4 and 3.5; 8-6.10 RUBIN, 1-1.7; 2-3.8 SABBAGH , 5-4.7
saturated scattered scheme of scheme of
r e l a t i o n , 10-7.3; e x i s t e n c e c r i t e r i o n , 10-7.6 chain, 6-1 foundation, 1-2.8 induction f o r f i n i t e s e t s , 1-1.1 SCHRODER, s e e BERNSTEIN SCHUR (number), 3-exerc. 1 SCOTT, 1-5.4 segment = i n t e r v a l separable elements, 4-1.2 sequence, ordinal-indexed sequence, o( -sequence, 1 - 2 . 2 w -sequence, 1-2.5 set-extensive subset, set-homogeneous r e l a t i o n , 11-8 s e t of f i l t e r s , 2-exerc. 2 set of i n j e c t i v e functions ( S I E R P I N S K I ) , 2-8.2 set-pseudo-homogeneous r e l a t i o n , 11-8 s e t - t r a n s i t i v e group, 11-2.2 SHELAH, see HODGES SHEPHERDSON, 2-6.7 S I E R P I N S K I , 2.8.2; 2-exerc. 1; 3-3.1; 5-4.4; 5-5.3; 7-exerc. 4 S i e r p i n s k i ' s p a r t i a l o r d e r i n g , 7-exerc. 4 S I K O R S K I , 1-9 S I L V E R , 3-6.1 SIMMONS, 10-7.3
s i n g u l a r aleph o r c a r d i n a l , 2-6.1 s l i c e s ( p a r t i t i o n ; BONNET, POUZET), 4-1.5
395
396
Index
SLOMSON, see BELL smaller sequence, 4-2 SOLOVAY, 5-6 sparse s e t (topology), 7-2.3 specification ( r e l - a g e ) , 10-3.3 SPECKER, 5-7.3 SPERNER, 3-exerc. 4 stationary s e t , 1-exerc. 2 s t r a t i f i e d partial ordering, 2-5.2 s t r i c t embedding ( < ) , 5-1 s t r i c t inclusion ( c ) , 1-1 s t r i c t l y r i g h t and s t r i c t l y l e f t indecomposable chain, 6-3.1 strongly amalgamable s e t , 11-1.5 strongly minimal bad sequence, 4-2.8 5trongly scattered chain, 6-2.3 subpotence, 1-1.3 successive elements ( i n a b a r r i e r ) , 8-1.5 successor aleph a+ , 1-6.3 successor b a r r i e r , 8-3.3 successor cardinal, 2-3.10 successor ordinal a + l , 1-2.4 successor s e t , 1-1 sum along a chain, d. -5um, 2-3.6 sum: ordinal sum (+), 1-3.1 and 2-3.6; cardinal sum (+), 1-5.5 SUPPES, 1-1.4 supremum ( S u p ) , 2-2; f o r ordinals, 1-2.1; f o r r e a l s , 1-4.6 SUSLIN, 5-6 Suslin chain. 5-6.1: Suslin t r e e . 5-6.3 system ( r e l a t i o n a l ) ; 11-2.1 SZPILRAJN, 2-4.2 Szpilrajn chain, 7-4
table ( b i v a l e n t ) , 9-9 t a i l = f i n a l interval o f a sequence, 8-5 TARSKI, 1-1.2; 2-3.8 t o 3.10;'2-6.5; 2-exerc. 2; 12-5.3 TENNENBAUM, 5-5 thresholds, 12-4.1 and 4 . 2 ; 12-4.6; threshold p a i r , 12-4.6 TODORCEVIC, 7-exerc. 2 topology on s e t s of integers, 1-8 topology on i n i t i a l i n t e r v a l s , 7-2.2 t o t a l l y ordered s e t (by membership), 1-2 t o t a l ordering = chain, 2-2.3 tournament, 5-2.7 t r a n s f i n i t e induction, 1-2.11 transformation f " , 1-1.2 t r a n s i t i v e closure, 1-5.1 t r a n s i t i v e group, n - t r a n s i t i v e group, 11-2.2 t r a n s i t i v e s e t , 1-2 transposition, 1-1.2 t r e e , 4-6 trichotomy f o r ordinals, 1-2.1; f o r cardinals, 2-2.5 t r i v i a l ul t r a f i l t e r , 1-7.2 TUKEY, 7-exerc. 1 n-tuple, 1-2.3 type (isomorphism), 2-3
Index u l t r a f i 1 t e r , ul t r a f i l t e r axiom, 1 - 7 . 2 unbounded s e t or sequence = without any upper bound universal c l a s s , 12-5.2 u n i v e r s a l l y S z p i l r a j n p a i r of r e l a t i o n s , 7-4.9 universal theory = age, 10-2 unordered p a i r = p a i r , 1-1 upper bound, 2-2
value (+) o r ( - ) , 2-1 VAUGHT, 12-5.3; see POUZET
WAGNER, see LIVINGSTONE WALKER, 3-1.6
weakening, 2-4.1 weakly i n a c c e s s i b l e aleph, 2-6.7 well-founded cofinal r e s t r i c t i o n (POUZET), 2-5.1 well-founded p a r t i a l ordering, well-ordering, 2-2.4 well m u l t i r e l a t i o n , 12-3 well-ordered s e t ( l e x i c o g r a p h i c a l l y ) , 3-2.1 well-ordering axiom, well-orderable s e t , 2-2.5 well p a r t i a l ordering, 4-3.2 ; theorem on i n i t i a l i n t e r v a l s (HIGMAN), 4-4.1 well p a r t i a l ordering of f i n i t e t r e e s (KRUSKAL), 5-2.3 well p a r t i a l ordering of words (HIGMAN), 4-4.4 well quasi-ordering, 4-3.2 WHEELER. see HIRSCHFELD WHITAKER, 1-1.4; 3-1.6 WHITEHEAD, 3-1.6 word = f i n i t e sequence, 1-2.3; p a r t i a l ordering of words, 4-2
YACKEL, 3-1.6; see GRAVER younger r e l a t i o n , 10-1.2 A-younger r e l a t i o n , 10-3.2; W -younger r e l a t i o n , 10-4.5
Z = chain of p o s i t i v e and negative i n t e g e r s , 2-2.3 ZAGUIA, 4-9.3 ZERMELO, 1-1.7; 1-2 Z F (axioms), 1-2.4 ZORN, see HAUSDORFF
397
