Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory

Springer Monographs in Mathematics Dietlinde Lau Function Algebras on Finite Sets A Basic Course on Many-Valued Logic...

Author: Dietlinde Lau

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Springer Monographs in Mathematics

Dietlinde Lau

Function Algebras on Finite Sets A Basic Course on Many-Valued Logic and Clone Theory

With 42 Figures and 46 Tables

123

Dietlinde Lau Institute for Mathematics University of Rostock Universitätsplatz 1 18055 Rostock, Germany e-mail: [email protected]

Library of Congress Control Number: 2006929534

Mathematics Subject Classification (2000): 03B50, 08Axx, 08A40, 08A30, 08A05, 06A15 ISSN 1439-7382 ISBN-10 3-540-36022-0 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-36022-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author using a Springer TEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

44/3100YL - 5 4 3 2 1 0

To my mother, Brigitte Lau

Preface

Functions (or operations), which are deﬁned on ﬁnite sets, occur in almost all ﬁelds of mathematics. For more than 80 years, algebras (so-called function algebras), whose universes are such functions, have been studied. Particularly in Mathematical Logic, in Universal Algebra (more precise in the Clone Theory), and in parts of Computer Science, certain knowledge about these algebras are subject of the fundamental knowledge. Currently only one book has been published about function algebras, apart from certain monographs or dissertations of speciﬁc themes, survey articles and books that contain sections about function algebras or clones. This book has been written by R. P¨ oschel und L. A. Kaluˇznin in the German language and gives a very good overview about the results achieved up to 1979. During the last 26 years, many new results have been obtained; however, a new book about function algebras is overdue. The aim of the present book is to introduce the reader to the theory of function algebras and to give the latest state of research for some selected ﬁelds. The author would like to acquaint the reader with proof of the fundamental theorems and the diﬀerent proof methods, to enable research in the ﬁeld of the function algebras. This book is self-contained. All necessary fundamental concepts and facts are introduced, but some background knowledge about linear and abstract algebra would be helpful for readers. In the following Introduction, the reader ﬁnds short summaries of the 26 chapters of this book. The adjoined section Preliminaries explains abbreviations and some general symbols, which are more or less standard, and gives some facts from basic mathematics. Part I of this book introduces the reader to Universal Algebra to provide almost every knowledge concerning other ﬁelds of mathematics. Moreover, this part of Universal Algebra informs the reader that many of the following results

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Preface

of function algebras reply to questions that arise upon studying an algebra. The structure of this book enables the reader to skip the ﬁrst part and immediately start reading Part II Function Algebras. The author provides the reader with new proofs concerning classic results of the theory of function algebras. The remaining proofs are adapted to the style of the book. The theorems from Sections 14.10, 15.4, 18.2, 18.3, and from Chapter 17 have not yet been published. Small mistakes from the original papers (including the papers of the author) have been corrected in this book without referring to the original mistakes. During the writing process I have tried to solve open mathematics questions and problems, which I recognized during my study of the corresponding literature. In some cases colleagues helped me. In such a case their names are mentioned in the relevant places in the book. I would like to thank my colleagues as well as the authors of the articles I referred to in my book. Especially, I would like to thank my doctoral thesis supervisors, Prof. G. Burosch (R¨overshagen), and Prof. V. B. Kudrjavcev (Moscow). I received ﬁrst information on the complexity of themes in this book from their lectures. I owe Prof. I. G. Rosenberg (Montreal) much gratitude, as I learned many proof methods while studying his papers. I thank my colleagues Prof. Dr. K. Denecke (Potsdam), Prof. Dr. L. Haddad (Kingston), and Prof. Dr. R. P¨ oschel (Dresden) for their fruitful cooperation during many years. In particular, Prof. Haddad was a very great aid during the ﬁnishing of the book. I also owe him the organization and the ﬁnancing (by the Natural Sciences and Engineering Council of Canada) of a linguistic correction of my text by Ms Eryn Kirkwood (from RedInk Editors, Ottawa ON, Canada). Ms Kirkwood deserves my special thanks.

Rostock, June 2006

Dietlinde Lau

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Part I Universal Algebra 1

Basic Concepts of Universal Algebra . . . . . . . . . . . . . . . . . . . . . . . 1.1 Universal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples of Universal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Gruppoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.9 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.10 Semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.11 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.12 Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.13 Function Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 27 28 28 28 28 28 29 29 29 29 30 30 30 31

2

Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Two Deﬁnitions of a Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Examples for Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Isomorphic Lattices and Sublattices . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Complete Lattices and Equivalence Relations . . . . . . . . . . . . . . .

35 35 39 39 41

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3

Hull Systems and Closure Operators . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Some Properties of Hull Systems and Closure Operators . . . . . . 46

4

Homomorphisms, Congruences, and Galois Connections . . . 4.1 Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Congruence Relations and Factor Algebras of Algebras . . . . . . . 4.3 Examples for Congruence Relations and Some Homomorphism Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Congruences on Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Congruences on Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Galois Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 56 56 58 59

5

Direct and Subdirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Subdirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6

Varieties, Equational Classes, and Free Algebras . . . . . . . . . . . 6.1 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Terms, Term Algebras, and Term Functions . . . . . . . . . . . . . . . . . 6.3 Equations and Equational Classes . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Connections Between Varieties and Equational Deﬁned Classes 6.6 Deductive Closure of Equation Sets and Equational Theory . . . 6.7 Finite Axiomatizability of Algebras . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 76 78 81 82 84

Part II Function Algebras 1

Basic Concepts, Notations, and First Properties . . . . . . . . . . . 91 1.1 Functions on Finite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.2 Operations on PA , Function Algebras . . . . . . . . . . . . . . . . . . . . . . 94 1.3 Superpositions, Subclasses, and Clones . . . . . . . . . . . . . . . . . . . . . 96 1.4 Generating Systems for PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 1.5 Some Applications of the Function Algebras . . . . . . . . . . . . . . . . 104 1.5.1 Classiﬁcation of Universal Algebras . . . . . . . . . . . . . . . . . . 104 1.5.2 Propositional Logic and First Order Logic . . . . . . . . . . . . 105 1.5.3 Many-Valued Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 1.5.4 Information Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 1.5.5 Classiﬁcation of Combinatorial Problems . . . . . . . . . . . . . 118

2

The Galois-Connection Between Function- and Relation-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.2 Diagonal Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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2.3 Elementary Operations on Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.4 Relation Algebras, Co-Clones, and Derivation of Relations . . . . 127 2.5 Some Operations on Rk Derivable from the Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 2.6 The Preserving of Relations; Pol, Inv . . . . . . . . . . . . . . . . . . . . . . 130 2.7 The Relations χn and Gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 2.8 The Operator ΓA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.9 The Galois Theory for Function- and Relation-Algebras . . . . . . 135 2.10 Some Modiﬁcations of the P ol-Inv-Connection . . . . . . . . . . . . . . 137 2.10.1 Galois Theory for Finite Monoids and Finite Groups . . . 137 2.10.2 Galois Theory for Iterative Function Algebras . . . . . . . . . 139 2.11 Some Connections Between the Relation Operations . . . . . . . . . 142 3

The Subclasses of P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.1 Deﬁnitions of the Subclasses of P2 and Post’s Theorem . . . . . . . 145 3.2 A Proof for Post’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.2.1 The Subclasses A of P2 with A ⊆ L and A ⊆ S . . . . . . . . 149 3.2.2 The Subclasses of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.2.3 The Subclasses of S, Which Are Not Subsets of L . . . . . 155 3.2.4 A Completeness Criterion for P2 . . . . . . . . . . . . . . . . . . . . 156

4

The Subclasses of Pk Which Contain Pk1 . . . . . . . . . . . . . . . . . . . 159

5

The Maximal Classes of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.1 Introduction, a Rough Description of the Maximal Classes . . . . 163 5.2 Deﬁnitions of the Maximal Classes of Pk . . . . . . . . . . . . . . . . . . . 165 5.2.1 Maximal Classes of Type M (Maximal Classes of Monotone Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.2.2 Maximal Classes of Type S (Maximal Classes of Autodual Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.2.3 Maximal Classes of Type U (Maximal Classes of Functions, Which Preserve Non-Trivial Equivalence Relations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.2.4 Maximal Classes of Type L (Maximal Classes of Quasi-Linear Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.2.5 Maximal Classes of Type C (Maximal Classes of Functions, Which Preserve Central Relations) . . . . . . . . . 173 5.2.6 Maximal Classes of Type B (Maximal Classes of Functions, Which Preserve h-Universal Relations) . . . . . 174 5.3 Proof of the Maximality of the Classes Deﬁned in Section 5.2 . 179 5.4 The Number of the Maximal Classes of Pk . . . . . . . . . . . . . . . . . . 183 5.5 Remarks to the Maximal Classes of Pk (l) . . . . . . . . . . . . . . . . . . . 188

6

Rosenberg’s Completeness Criterion for Pk . . . . . . . . . . . . . . . . 191 6.1 Proof of Completeness Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

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7

Further Completeness Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.1 A Criterion for Sheﬀer-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.2 A Completeness Criterion for Surjective Functions . . . . . . . . . . . 216 7.3 Fundamental Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8

Some Properties of the Lattice Lk . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.1 Cardinality Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.2 On the Cardinalities of Maximal Sublattices of Lk . . . . . . . . . . . 224 8.3 Some Strategies for the Determination of Sublattices of Lk . . . . 229

9

Congruences and Automorphisms on Function Algebras . . . 233 9.1 Some Basic Concepts and First Properties . . . . . . . . . . . . . . . . . . 234 9.2 Congruences on the Subclasses of P2 . . . . . . . . . . . . . . . . . . . . . . . 235 9.3 Characterization of the Non-Arity Congruences . . . . . . . . . . . . . 238 9.4 About the Number of the Congruences on a Subclass of Pk . . . 243 9.5 A Criterion for the Proof of the Countability of Con A for Certain A ⊆ Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 9.6 Congruences on Some Classes of Linear Functions . . . . . . . . . . . 250 9.7 Congruences on the Maximal Classes of Pk . . . . . . . . . . . . . . . . . 256 9.8 Congruences on Subclasses of [Pk1 ] . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.9 Congruences on Some Subclasses of Pk,l . . . . . . . . . . . . . . . . . . . . 273 9.10 Some Further General Properties of the Congruences and the l-Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 9.11 The Connection Between Clone Congruences and Fully Invariant Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.12 Automorphisms of Function Algebras . . . . . . . . . . . . . . . . . . . . . . 285

10 The Relation Degree and the Dimension of Subclasses of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.1 The Deﬁnition of the Relation Degree and of the Dimension of a Subclass of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.2 The Dimensions and Relation Degrees of Post’s Classes . . . . . . 293 10.3 Further Examples of the Dimension and Relation Degree of Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11 On Generating Systems and Orders of the Subclasses of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 11.1 Some General Properties of Generating Systems and Bases . . . 308 11.2 The Orders and Sheﬀer-Functions of the Classes of Type C1 , S or U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.3 Orders of the Classes of Type L, C, B . . . . . . . . . . . . . . . . . . . . . . 314 11.4 The Order of P olk for ∈ Mk and k ≤ 7 . . . . . . . . . . . . . . . . . . 319 11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 11.6 Classiﬁcations and Basis Enumerations in Pk . . . . . . . . . . . . . . . 332

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12 Subclasses of Pk,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 12.2 Some Properties of the Inverse Images . . . . . . . . . . . . . . . . . . . . . 337 12.3 On the Number of the B-projectable Subclasses of Pk,2 , B ⊆ P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12.4 The Pl -projectable and the P oll {α}-projectable Subclasses of Pk,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 12.5 The Maximal and the Submaximal Classes of Pk,2 . . . . . . . . . . . 354 12.6 The Classes A with M ∩ T0 ∩ T1 ⊆ prA or L ∩ T0 ∩ S ⊆ prA or prA = M ∩ S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 13 Classes of Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 13.1 Some Properties of the Subclasses of Ud That Contain rd . . . . . 384 13.2 The Subclasses of Linear Functions of Pk with k ∈ P . . . . . . . . . 387 13.3 A Survey of Further Results on Linear Functions . . . . . . . . . . . . 390 14 Submaximal Classes of P3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 14.1 A Survey of the Submaximal Classes of P3 . . . . . . . . . . . . . . . . . . 400 14.2 Some Declarations and Lemmas for Sections 14.3–14.9 . . . . . . . 408 14.3 Proof of Theorem 14.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 14.4 Proof of Theorem 14.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 14.5 Proof of Theorem 14.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 14.6 Proof of Theorem 14.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 14.7 Proof of Theorem 14.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 14.8 Proof of Theorem 14.1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 14.9 Proof of Theorem 14.1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 14.10On the Cardinality of L↓3 (A) for Submaximal Clones A . . . . . . . 425 15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 15.1 The Lattice of Subclasses of P3 of Linear Functions . . . . . . . . . . 433 15.2 The Subsemigroups of (P31 ; ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 15.3 Classes of Quasilinear Functions of P3 . . . . . . . . . . . . . . . . . . . . . . 456 15.3.1 Some Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 15.3.2 Subclasses of L0,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 15.3.3 The Subclasses of L0,1 ∪ L0,2 That Are Not Subclasses of L0,1 or L0,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 15.3.4 The Remaining Subclasses of L . . . . . . . . . . . . . . . . . . . . . 463 15.4 The Subclasses of [O1 ∪ {max}] . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 15.4.1 Some Descriptions of the Class M . . . . . . . . . . . . . . . . . . . 464 15.4.2 Some Lemmas and a Rough Partition of the Subclasses of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 15.4.3 The Subclasses of [M 1 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 15.4.4 The Subclasses of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 15.4.5 The Subclasses of M ∩ P ol3 {(0, 2)} . . . . . . . . . . . . . . . . . . 482

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15.4.6 The Remaining Subclasses of M . . . . . . . . . . . . . . . . . . . . . 488 16 The Maximal Classes of a∈Q P olk {a} for Q ⊆ Ek . . . . . . . . . 499 16.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 16.2 Results of Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 16.3 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 16.4 Proof of Theorem 16.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 17 Maximal Classes of P olk El for 2 ≤ l < k . . . . . . . . . . . . . . . . . . . 515 17.1 Notations, Deﬁnitions, and Some Lemmas . . . . . . . . . . . . . . . . . . 515 17.2 Results of Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 17.3 Maximality Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 17.4 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 17.6 Classes Describable by Relations of Rmax (Pl ) ∪ Rmax (Pk ) . . . . 549 18 Further Submaximal Classes of Pk . . . . . . . . . . . . . . . . . . . . . . . . . 555 18.1 The Maximal Classes of P olk s for s ∈ Sk . . . . . . . . . . . . . . . . . 555 18.2 Some Maximal Classes of a Maximal Class of Type U . . . . . . . . 561 18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)}) . . . . . . . . . . . . 573 18.3.1 Deﬁnitions of the U -Maximal Classes . . . . . . . . . . . . . . . . 573 18.3.2 Proof of the U -Maximality of the Classes Deﬁned in 18.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 18.3.3 Proof of the Completeness Criterion for U . . . . . . . . . . . . 584 19 Minimal Classes and Minimal Clones of Pk . . . . . . . . . . . . . . . . 589 19.1 Minimal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 19.2 The Five Types of Minimal Clones . . . . . . . . . . . . . . . . . . . . . . . . . 590 20 Partial Function Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 20.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 20.2 One-Point Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 20.3 Description of Partial Clones by Relations . . . . . . . . . . . . . . . . . . 604 3 . . . . . . . . . . . . . . . . . . 606 2 and P 20.4 The Maximal Partial Classes of P k . . . . . . . . . . . . . . . . . . . . . . . . 614 20.5 The Completeness Criterion for P k . . . . . . . . 616 20.6 Some Properties of the Maximal Partial Clones of P 20.7 Intervals of Partial Clones That Contain a Maximal Clone . . . . 619 20.8 Intervals of Boolean Partial Classes . . . . . . . . . . . . . . . . . . . . . . . . 627 20.9 On Congruences of Partial Clones . . . . . . . . . . . . . . . . . . . . . . . . . 628 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

Introduction

The present book deals with a subarea of the Discrete Mathematics. We study functions, which are deﬁned on ﬁnite sets, and we study the composition of these functions. Such functions are used, for example, in Computer Science (in particular, in the Switching Theory and in the Theory of Automata), in Mathematical Logic, and in Universal Algebra (in particular, the Clone Theory).1 In other words, we choose an arbitrary ﬁnite set A and study an algebra, whose universe is the set PA

(or Pk := P{0,1,2,..,k−1} )

of all n-ary mappings (n ∈ N), which maps the Cartesian power An (of all ordered n-tuples of elements from A) into A, and whose operations are the so-called superposition operations that are described as follows:2 – permutation of variables – identiﬁcation of variables – adding of ﬁctitious variables and – substitution of variables of a function by functions Denote F (n) the set of all n-ary functions of F ⊆ PA . Moreover, let Ω be the set of all superposition operations described above. Then, PA := (PA ; Ω) is called (full) function algebra (on A). The universe of a subalgebra of PA is called closed set or a subclass (or brieﬂy a class) of PA . If F is a subclass of Pk then F is also called a subclass of the k-valued logic. The set of all closed sets of Pk together with the set inclusion forms a lattice Lk . 1 2

See, for this purpose, also Section 1.5 of Part II. One ﬁnds an exact deﬁnition of these operations in Section 1.2 of Part II.

2

Introduction

The closed sets of P2 were already determined in the papers [Pos 20] and [Pos 41] by E. L. Post. For over 50 years, many papers have dealt with function algebras or closed sets of Pk for arbitrary k. One ﬁnds a survey of the essential articles, which were published up to the year 1978, on the topic in [P¨ os-K 79] with 730 references and in [Ros 77] with 464 references. To give the reader a ﬁrst impression of the problems handled in the theory of the function algebras, some explanations are subsequently given to the completeness problem3 , which stood in the center of a line of investigations in many articles: One ﬁnds a criterion, to decide if a set of functions that belong to Pk is suﬃcient for the construction of any arbitrary other function of Pk (by means of the superposition operations). A general answer to that is given by the following criterion, which was formulated by E. L. Post in 1921, ﬁrst, for Boolean functions, i.e., for functions of P2 : A set F ⊆ Pk is complete in Pk if and only if F is a subset of no maximal class of Pk . A closed set M ⊂ Pk is said to be maximal in Pk , if M can not be properly extended to a closed proper subset of Pk . With regard to such and similar question formulations, one naturally deals with the structure of the subclasses of Pk or with the lattice of the subclasses of Pk . The ﬁrst and most important result in this direction is Post’s pioneering description of L2 ([Pos 41], see Figure 1), now known as Post lattice. One can describe the many Post’s classes with the aid of the classes P2 , M , S, L, Ta (1) Ta,µ , D ∪ C, K ∪ C, [P2 ], C, (a ∈ {0, 1}, µ ∈ N \ {1}) and through formation of certain intersections of these classes, where T0 , T1 , M , S, and L are exactly the maximal classes of P2 .4 For some time, it was thought that L3 is equally simple. Then, Ju. I. Janov and A. A. Muˇcnik showed in the year 1959 the existence of subclasses of Pk for k ≥ 3 with inﬁnite and without basis, respectively, and this implies the existence of a non-countable inﬁnite set of subclasses in Lk (see also [Ehr 55]). Thus, there are as many closed sets in Pk as there are subsets of Pk for k ≥ 3. In contrast to k = 2, therefore, it seems hard to give an eﬀective description of Lk for k ≥ 3. Nevertheless, one could determine the maximal classes of Pk for arbitrary k. S. V. Jablonskij determined all 18 maximal classes of P3 in [Jab 54] and [Jab 58]. Moreover, he extended his results by describing several types of maximal classes of Pk . This work was continued by V. V. Martynyuk [Mar 60], E. Ju. Zacharova [Zac 67], R. A. Bairamov [Bai 67] ˇ ˇ 67]. In [Zac-K-J (some results incorrect) and V. L. Rvaˇcev/ L. I. Sljarov [Rva-S 71] it has been reported that late A. I. Mal’tsev had all 82 maximal sets of P4 . I. G. Rosenberg published the missing maximal classes for Pk in [Ros 65] and 3 4

In Section 1.5 of Part II, it is shown that this problem is a mathematical way to describe problems that occur during the construction of electronic circuits. See Chapter 3 of Part II for details.

Introduction

3

he proved in the book [Ros 70a] that there can not be any further maximal classes for arbitrary k, whereby the general completeness problem for Pk was solved. Rosenberg’s description of the maximal classes is based on the idea of a function preserving a relation; i.e., he could prove that every maximal class has the form P olk , where is a certain h-ary relation on Ek , 1 ≤ h ≤ k, k ≥ 3 and P olk is the set of all functions of Pk , which preserve the relation . To ﬁnd the maximal ones among classes of the form P olk , he designed a sieve method, which eliminates at each step every relation for which a relation σ is found in the list such that P olk ⊆ P olk σ. The process terminates when the candidate list contains only relations such that P olk can be proved to be maximal. This approach was twofold in scope. To discover the maximal classes and, at the same time, to prove that one has a full list of them. The idea to describe classes through relations kept on being developed to a Galois-theory for function algebras and relation algebra by V. G. Bodnarˇcuk, L. A. Kaluˇznin, V.N. Kotov, B. A. Romov (in [Bod-K-K-R 69]). The interesting subclasses of Pk are not only the maximal ones, and so the Galois theory was and is an important aid for the study of subclasses of Pk . Namely, one can classify ﬁnite universal algebras and also combinatorial problems with the aid of the elements of Lk (see Section 1.5 of Part II). These and many other applications presuppose, however, precise knowledge about the subclasses of Pk . Therefore, the object of this book is to explain some methods to the ﬁnding of subclasses of Pk and to give a survey of subclasses and their properties, which were determined in the last years. To familiarize the reader from the beginning also with the algebraic side of the function algebras and to show that the concepts introduced for function algebras are only special cases of more general concepts mostly from the Universal Algebra, we begin with an introduction to the Universal Algebra in the ﬁrst part of this book. One ﬁnds supplements to this short introduction to the Universal Algebra in the books [Bur-S 81], [Coh 65], [Gr¨ a 68], [Ihr 93] (or [Ihr 2003]), [McK-M-T 87], [Wer 78] and [Lau 2004], volume 2. In Part II, we deal only with the function algebras. The construction of the book is chosen in a way that one can immediately begin reading Part II and, if one needs concepts and facts from Part I, one can reference these. Unlike Part I, which is strongly linearly structured in that each chapter is based on the preceding chapter, after studying the ﬁrst two chapters of Part II, all successive studies can also be studied. We concentrate in Part II on the following topics: • • • • •

Basic concepts and notations Galois-connection between function algebras and relation algebras Post’s results on the subclasses of P2 the maximal classes of Pk completeness criterions for Pk (in particular, the Rosenberg’s completeness criterion)

4

Introduction

• congruences on subclasses of Pk • complexity measures for generating systems (for example as order or relation degree and dimension) of certain subclasses of Pk • subclasses of linear functions of Pp (p prime number) • a survey on subclasses of Pk , whose functions have at most two diﬀerent values • submaximal classes of P3 • the description of all ﬁnite and countably inﬁnite sublattices of the depth 1 or 2 of the lattice of all subclasses of P3 • submaximal classes of Pk , which are subsets of maximal classes described by unary relations • a survey of results to maximal classes of Pk for arbitrary k • minimal clones • partial function algebras. One ﬁnds completions to this book and a survey on further topics of the function-algebras-theory in [P¨ os-K 79] and [Ros 84]. Subsequently the content of this text is described in brief without the necessary concepts and notations.

Part I Chapter 1 begins with the deﬁnition of a universal algebra (brieﬂy: algebra) as a pair (A;F) consists of a nonempty set A and a set F of certain operations on A and gives numerous examples of algebras (among that also the function algebras). In addition, one ﬁnds the very important concept of the subalgebra in this chapter. Chapter 2 compiles needed order concepts within the framework of an introduction to the lattice theory. The usual deﬁnitions of a lattice are indicated, and the calculation in the lattice theory illustrates by means of the proof of a few classical theorems of the lattice theory. Chapter 3 generalizes observations, which one can make when one examines more closely such concepts as subalgebra or linear hull of subsets of a vector space (or another closure operators of the classical algebra), to the concepts hull system and closure operator. Then, it is shown that these concepts deliver, roughly, the same. In addition, certain combinations to the lattice theory are given, and the lattices, formed from the subalgebras of an algebra, are uniquely characterized through certain properties. Chapter 4 combines properties of such important concepts as the homomorphic and isomorphic mappings between universal algebras, which the readers surely know from the classical algebra here, and which are deﬁned obviously for universal algebras. It is shown, how homomorphic mappings are

Introduction

5

ultimately determined by congruence relations (i.e., by equivalence relations, which are compatible with the operations of algebra). The (general) homomorphism theorem keeps on being proved and is shown by examples, as this theorem can be improved for concrete algebras. A section of Chapter 4 deals with Galois connections between set systems that are continued later for function algebras in Chapter 2 of the second part. Chapter 5 shows how one can form new algebras from given algebras through direct or subdirect products and how one can recognize whether a given algebra is isomorphic to an algebra formed in this way. More precisely: we deal with the following two questions: Which algebras are smallest constituents of given algebras? How can one reduce a given algebra to its smallest constituents? Answers to these questions are given by theorems found by G. Birkhoﬀ. We prove these theorems in Chapter 5. In particular, we prove the representation theorems for algebras: Each ﬁnite algebra is isomorphic to a direct product of directly irreducible algebras. Each algebra is isomorphic to a subdirect product of subdirectly irreducible algebras. Chapter 6 deals with classes of algebras of the same type and some theorems on such classes, which were found also by G. Birkhoﬀ. At ﬁrst we introduce so-called varieties as classes of algebras, which are closed in respect to the formation of subalgebras, homomorphic images and direct products. We then come to a method for the construction of algebra classes that strongly diﬀers from the ﬁrst method at ﬁrst sight: Based upon certain equations from variables and operation symbols of a certain type τ , we form the class of all algebras of type τ that these equations fulﬁll. The result is so-called equational classes. We will see, however, that there is a close connection between the two methods of the algebra class construction: class of algebras is equational deﬁned if and only if it is a variety. In the section on equational classes, we will also treat such concepts as the conclusion of an equational set. In addition, we treat ways to receive such conclusions. In Chapter 9 of Part II, we show that the results on congruences of a subclass F ⊆ PA imply results on subvarieties of the variety, which is generated by the algebra (A; F ) and vice versa.

6

Introduction

Part II Chapter 1 begins with the precise deﬁnition of function algebras (PA ; Ω), where Ω is an inﬁnite set of operations, which describe the above superposition operations exactly, on PA ﬁrst. It is shown then how one can receive the operations of the set Ω through ﬁve elementary operations (so-called “Mal’tsevOperationen”) ζ, τ, ∆, ∇ and by means of composition. The basis of further investigations is, then, the (full iterative) function algebra PA := (PA ; ζ, τ, ∆, ∇, ) bzw. Pk := PEk mit Ek := {0, 1, ..., k − 1}. A universe of a subalgebra of Pk is called subclass (or more brieﬂy, class) in the following. If [T ] denotes the set of all functions of PA , which one can form from the functions of T ⊆ PA by means of superposition operations, then we have [F ] = F for arbitrary subclass F of PA . (n) n A mapping deﬁned by ei : EA −→ A, (x1 , x2 , ..., xn ) → xi , where i ∈ {1, 2, ..., n}, is called a projection. A clone is a subclass of PA , which contains all projections of PA . In Chapter 1 there are notations, concepts . . . that are used in the following chapters repeatedly. In addition, it is shown that the set PA and some subsets of PA can be formed from binary functions of PA by means of superposition operations. At the end of Chapter 1, we brieﬂy show how one can use the results of this book and how certain investigations are motivated in the theory of function algebras. Chapter 2 provides tools (like the the concepts of “a function preserves a relation” and “a relation is an invariant for a function”) by which classes, dealt in later chapters, can be eﬀectively described. The set Rk of all n-ary relations on Ek (n = 1, 2, 3, ...) is deﬁned and operations over the set Rk so that Rk with these operations forms a relation algebra Rk . Each universe of a subalgebra of Rk is called a co-clone of Rk . The main result of Chapter 2 is the proof that the lattice of the clones of Pk is antiisomorphic to the lattice of all co-clones of Rk . The basis of this result is the Galois-connection (P ol, Inv), where P ol : P(Rk ) −→ P(Pk ) and Inv : P(Pk ) −→ P(Rk ) are mappings deﬁned by P olk Q := {f ∈ Pk | f preserves each relation of Q}, Invk F := { ∈ Rk | is an invariant for each function f of F } for arbitrary Q ⊆ Rk and arbitrary F ⊆ Pk . The Galois-connection (P ol, Inv) makes it possible to consider the function algebras instead of relation algebras and vice versa. Moreover, one can handle function algebras and relation algebras with equal signiﬁcance, as it happens in [P¨ os-K 79], for example. We deal, however, with the clones in this book and proceed only with the co-clones if the proof methods developed for function algebras are not suﬃcient.

Introduction

T0 r T0,2 r

r T0,3 r

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r pp pp p rpp

Pr2

r

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r

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7

rT1,2 r rT1,3 r pp pp p rpp

pp pp p rpp T1,∞

r

D

r C1

∅

Fig. 1. The Post’s lattice

The connection between algebras and relations was introduced by M. Krasner ([Kra 46] and [Kra 68/69]). He has completely developed the special Galois-connection between subsets of the permutation groups Sn := {f ∈ (1) Pk | f is bijective} and subsets of Rk . Later, Krasner also developed a sim(1) ilar theory for subsemigroups of Pk . Krasner’s theory was generalized by V. G. Bodnarˇcuk, L. A. Kaluˇznin, V. N. Kotov, and B. A. Romov in [Bod-K-K-R 69]. Chapter 3 gives all subclasses from P2 (the so-called Post’s classes) and proves the completeness of the list. In addition, smallest generating systems (“bases”) and the orders of the Post’s classes are determined. The results of Chapter 3 were found by E. L. Post (see [Pos 20] and [Pos 41]). Figure 1 gives the Hasse-diagram of the lattice of the subclasses of P2 . A knot of the graph without denotation represents a class that can be described as an intersection of classes that lie above it. Chapter 4 describes all classes that contain all unary functions of Pk . There exist k + 1 of such classes, which form a chain in the lattice of all subclasses of Pk . The results of Chapter 4 were found by G. A. Burle (see [Bur 67]). Chapter 5 begins with a coarse description of the maximal classes of Pk . The idea to describe classes this way was indicated already in [Kuz 59] by

8

Introduction

A. V. Kuznecov. Then classes, from which we show later (in Section 5.3) that they are maximal classes of Pk , are deﬁned. (In Chapter 6, the proof is all maximal classes are determined.) As in [Ros 70] the classes are divided in Section 5.2 into six types and characterized by their invariants (relations). We denote the relation sets needed in this case with with Mk , Sk , Uk , Lk , Ck and Bk . Subsequently a maximal class is called a maximal class of the type X, when it can be described with the aid of a relation from set Xk , where X ∈ {M, S, U, L, C, B}. Next to the deﬁnitions of the maximal classes of Pk one ﬁnds the proofs of some properties of the maximal classes needed in later chapters, and the derivation of a recursion formula for the number determination of the maximal classes of Pk . Chapter 6 gives the proof found by I. G. Rosenberg and published in [Ros 70a] that the classes deﬁned in Section 5.2 are the only maximal classes of Pk for arbitrary k. In some parts, the original proof can be abbreviated by using proof ideas from some papers (for example from [Qua 82] and [P¨ os-K 79]). As already explained, there is also a solution of the completeness problem for Pk through the description of maximal classes of Pk . Chapter 7 shows how one can derive some further completeness criteria for speciﬁc question formulations from the Rosenberg’s completeness criterion 6.1. First of all we will handle a criterion for Sheﬀer functions which was found by G. Rousseau. 5 Then we will show how one can reduce the conditions from Theorem 6.1 if one considers only surjective functions. Finally, we deal with criteria indicate under which conditions a set (⊆ Pk ) consisting of certain unary functions and a Slupecki-function 6 is complete in Pk . Chapter 8 explains the qualitative diﬀerences between the lattice L2 of the subclasses of P2 and the lattice Lk of the subclasses of Pk for k ≥ 3. While L2 is countable-inﬁnite, Lk has the cardinality of the continuum (denoted by c) for k ≥ 3. This results from the fact that, for every k ≥ 3, the set Pk has a subclass with an inﬁnite basis. One ﬁnds examples of such classes not only in Chapter 8, but also in Sections 12.3 and 14.10. Moreover, we deal with cardinality statements about chains and antichains (in Lk ) and with the embedding of Lk into Lk in Chapter 8. Then the question is clariﬁed, which cardinalities the lattices L↓k (A) 7 for maximal classes A of Pk have. In the last section of Chapter 8, the reader ﬁnds “strategies” for the determination of “manageable” sublattices of Lk . In addition, two examples of theorems prove how one can determine certain sections of Lk . Chapter 9 deals with homomorphic mappings from a subclass of Pk onto a 5 6 7

A Sheﬀer function is a function that every function of Pk can be formed from by means of the superposition operations. This is a function f ∈ Pk \ [Pk1 ] with Im(f ) = Ek . L↓k (A) denotes the set of all subclasses of A.

Introduction

9

subclass of Pk , As generally accepted,8 one can characterize every homomorphism from a class A through a certain congruence relation9 on A. After some basic concepts are deﬁned, all congruences on the subclasses of P2 are determined in Chapter 9. It is the aim of the sections following then, to specify the general homomorphism theorem for function algebras and to ﬁnd statements on the number of congruences on a subclass of Pk through determination of some general properties of the congruences on subclasses of Pk . Then, all congruences are determined for selected classes (among other things, these are the maximal classes of Pk and certain classes from linear functions). Criteria with which one can ﬁnd out whether on a partial class of Pk only trivial congruences exist are in addition derived. A further section deals with the connection between clone congruences and the fully invariant congruences on free algebras. The theorem found in this case has interesting inferences and is a bridge between certain investigations in the Universal Algebra and certain investigations in the theory of the function algebras. At the end of this chapter, one can ﬁnd some results on automorphisms. It is proven that Pk , the subclasses of P2 , and the maximal classes of Pk have only inner automorphisms. The starting point and the basis of subsequent results are derived from an article by A. I. Mal’tsev (see [Mal 66]). Important contributions to the topic of Chapter 9 are also performed by V. V. Gorlov and I. A. Mal’tsev. Chapter 10 deals with the relation degree and the dimension of subclasses A of Pk . The relation degree of A is the smallest number h so that the class A is unambiguously described by a relation set whose elements have the arity h at most. The dimension of A is the smallest arity of a relation that characterizes the class A unambiguously. Chapter 10 begins by investigating the connections between these two complexity measures of the relational description of classes. After that, the relation degrees and dimensions of the Post’s classes, which were found by G. N. Blochina in [Blo 70], are proven. In Section 10.3, one can ﬁnd further classes for which one knows the relation degree or the dimension. Chapter 11 deals with a further measure (the so-called order of a class), with whose aid the complexity of a description of a subclass of Pk can be characterized. If the class A ⊆ Pk is ﬁnitely generated, we call the smallest number r with [A(n) ] = A the order of A. In the case that the subclass A ⊆ Pk is not ﬁnitely generated we write ord A = ∞. Section 11.1 shows that one can determine the order easily for a class, if one knows the subclasses of the treated class. Therefore, order statements are often the “by-products” of considerations, which were actually used for determining certain classes or 8 9

See also Chapter 4 of Part I. Those ones are the equivalence relations, which are compatible with the superposition operations on the considered class.

10

Introduction

sublattices of Pk . So, for example, the orders of the Post’s classes result from the considerations of Chapter 3 for veriﬁcation of the Post’s lattice. Further order statements are found in Chapters 13 and 14. Therefore, in Chapter 11, only the statements on the maximal classes that are described in Chapter 5 are determined. It is proven that, for k ≥ 3, each maximal class of the type S, U, L, B or C1 has the order 2 and that, for the order of a class of the type Ch , where h ≥ 2, the arity h of the descriptive relation of this class is an upper bound. If M ⊂ Pk is a maximal class of the type M, we can only prove that M has the order 2 for the cases 2 ≤ k ≤ 7 and for case that the descriptive binary relation of M has certain properties. For k ≥ 8 there are maximal classes of the type M that do not have a ﬁnite order. To prove this fact, we give an example, published by G. Tardos in [Tar 86]. If one has a ﬁnite generating class A, it is an interesting problem to clarify which cardinalities are possible for the bases of this class A. In Section 11.2 we show that the proof for ord P olk ≤ 3, where ∈ C1k ∪ Uk ∪ Sk , implies the existence of functions f with [f ] = P olk . Following corresponding notation for functions from Pk , such function f is called a Sheﬀer function for P olk . The last section explains shortly, as one can determine the cardinalities of the possible bases of the class A, if ord A < ∞. Furthermore, some basic ideas of [Miy 71], [Miy-S-L-R 86], [Miy 88], and [Sto 87] are explained on basis classiﬁcations. For more information on the topic, we direct the reader to [Miy 88] and [Sto 87] of M. Miyakawa and I. Stojmenovic. Chapter 12 summarizes the subclasses of the class Pk,2 := {f ∈ Pk | f has only values of the set {0, 1} }, which were found by G. Burosch, J. Dassow, N. Gr¨ unwald, W. Harnau, and the author. When one restricts the domain of a function f (n) ∈ Pk,2 to the set E2n , a homomorphic mapping pr (“projection”) from Pk,2 onto P2 can be deﬁned. Since the image prA of a subclass A of Pk,2 is a subclass of P2 and the subclasses of P2 are known, one can hope to ﬁnd of certain properties of the inverse images (⊆ Pk,2 ) through the known properties of the images (⊆ P2 ). This hope conﬁrmed itself in a certain sense (see, e.g., Theorem 12.2.5 and Theorem 9.7.6). Conversely, Pk,2 , k ≥ 3, also reﬂects the negative properties of Pk because the examples of classes from Section 8.1.1 with inﬁnite and without bases are subclasses of Pk,2 . Further, the functions of Pk,2 are important since they can be interpreted as predicates. Chapter 12 is organized as follows: Section 12.1 contains the basic concepts and notations. Section 12.2 contains results on inverse images of subclasses of P2 (with respect to the above projection). The remaining sections deal with the determination of cardinality and with the determination of the elements

Introduction

r r r r pp pp pp p g

r pp pp pp p g

r r

r

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g

g

Fig. 2. A survey on |Nk (B)| for B ∈ L2

of the set Nk (B) := {A | A is a subclass of Pk,2 with pr A = B} for arbitrary subclass B of P2 . In Section 12.3, one can ﬁnd some structure statements about the lattice of all subclass of Pk,2 . It is also clariﬁed in this section whether the set Nk (B) is ﬁnite or inﬁnite or has the cardinality of continuum. Section 12.4 generalizes some statements from Section 12.3 for Pk,l with 2 ≤ l < k. In Section 12.5, the maximal and the submaximal classes of Pk,2 are determined. Then, the investigations are continued from Section 12.3 for k = 3, i.e., for many classes B ⊆ P2 with |N3 (B)| ≤ ℵ0 the elements of the set N3 (B) are determined. Figure 2 shows a survey of the obtained results. This ﬁgure shows the Post’s graph, where the knots of this graph are diﬀerently labeled. If a class B of P2 is marked in this graph through g, this means that the set Nk (B) has the cardinality of the continuum. The second marker w means that |Nk (B)| ≥ ℵ0 and |N3 (B)| = ℵ0 are valid. For the remaining classes B whose knots do not have any marker the set Nk (B) is ﬁnite for arbitrary k ∈ N. Chapter 13 deals with classes of linear functions of Pk , i.e., with closed subsets of the set

12

Introduction

Lk :=

n≥1

{f (n) ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 +

n

ai · xi (mod k)}.

i=1

The lattice of the subclasses of Lk belongs to the earliest and best investigated sublattices of Lk . For the case that k is a prime number, all subclasses, which are no subsets of [L1k ], were determined by A. A. Salomaa in [Sal 64]. The results of [Sal 64] were proven by J. Bagyinszki and J. Demetrovics in [Bag-D 82] and complemented with the remaining subclasses of [L1k ], p ∈ P. ´ Szendrei. For examMany results about linear functions were obtained by A. ple, she proved in [Sze 78] that Lk has only ﬁnitely many subclasses, if k is square-free. In addition, she showed that an arbitrary class has, at most, the order 2 (or 3), if k is quare-free and k is an odd number (or an even number), respectively. For the case that k is not square-free, one easily ﬁnds a class that does not have any ﬁnite basis, whereby the set Lk has inﬁnitely-many subclasses in this case. Section 13.1 starts with properties of certain subclasses of the set (n) := ∈ Lk | ∃a0 , ..., an ∈ Ek ∃j ∈ {1, ..., n} : Ud n≥1 {f f (x) = a0 + aj · xj + d · i=1,i=j ai · xi }. This set is closed, if d is a divisor of k (notation: d | k). Then, with the aid of the results from the ﬁrst section new proofs are indicated for the theorems of [Sal 64] and [Bag-D 82] in Section 13.2. Chapter 13 ends with a survey on further results about linear and quasi-linear functions. Chapter 14 is the ﬁrst chapter in this book of this book that deals only with submaximal classes. A subclass (or a subclone) of Pk is called submaximal if it is covered by a maximal class (clone). The concept submaximal class was introduced by I. G. Rosenberg in [Ros 74]. In [Ros 74] one ﬁnds also the ﬁrst results about submaximal classes of Pk (see Chapter 17 for details). The submaximal classes are interesting not only because of their position in the second layer below Pk , but also because further completeness criteria for Pk result from a list of all submaximal classes. For arbitrary k, the full list of the maximal classes of a maximal class A of Pk is only known if A has the type S ([Ros-S 84], see Section 18.1), C1 (see Chapters 16 and 17) or A = P olk , 2 ∪ {(k − 1, k − 1)} (see Section 18.3). There are, however, where := Ek−1 some papers, in which one ﬁnds submaximal classes for speciﬁc k or only such submaximal classes that contain certain functions. In Section 14.1, one ﬁnds a complete description of all submaximal classes of P3 and some remarks about generalizations of the indicated theorems. The following papers provide a basis for this description: [Mac 79], [Mar-D-H 80], [Sal 64], [Bag-D 82] and [Lau 82a]. The theorems from Section 14.1 are proven then in Sections 14.2 – 14.9. In Section 14.10, we will prove that there are 5 submaximal classes with ﬁnitely many subclasses; 7 have countably many

Introduction

13

subclasses, and the remaining 146 submaximal classes have uncountably many subclasses. All elements of the lattices L↓3 (A), where A is a submaximal class with|L↓3 (A)| ≤ ℵ0 are determined in Chapter 15. Chapter 15 gives all ﬁnite or countably inﬁnite sublattices of depth 1 or 2 of the lattice of the subclasses of P3 , where the ﬁnite cases are easy conclusions from Chapter 13. We say that the lattice L↓k (A) of subclasses of A = [A] ⊆ Pk has the depth t, if t is the least integer for which there are some classes A1 , ..., At−1 ∈ Lk with A ⊂ A1 ⊂ A2 ⊂ ... ⊂ At−1 ⊂ Pk . For k = 3 by the Theorems 13.2.3 and 8.1.6 there exist ﬁnite and countably inﬁnite sublattices of the depth 1 or 2. In Chapter 15 these sublattices shall be determined exactly. The ﬁnite lattices L↓3 (L3 ) of depth 1 are found in Section 15.1. This lattice is a conclusion from Chapter 13. In addition, by Section 14.10, this lattice is the only ﬁnite lattice of depth 1 and, further, this lattice contains all ﬁnite sublattices of L3 of depth 2. Because of Theorem 14.10.1, there are exactly 7 submaximal classes with |L↓3 (A)| = ℵ0 . One obtains these classes through formation of isomorphic pictures of the following two classes: (1) L3 = [P3 ] ∪ n≥1 {f (n) ∈ P3 | ∃f0 , f1 , ..., fn ∈ P31 : f (x1 , ..., xn ) = f0 (f1 (x1 ) + f2 (x2 ) + ... + fn (xn ) (mod 2))} and M :=

{f (n) ∈ P3 | ∃f1 , ..., fn ∈ O1 : f (x1 , ..., xn ) = f1 (x1 ) ∨ ... ∨ fn (x)},

n≥1

where O1 is the set of all unary monotonous functions (in respect to the total order 0 < 1 < 2) of P3 and x ∨ y := max{x, y}. Since L3 contains also all subclasses, which are generated from unary functions of P3 . First of all, the 1299 subsemigroups of the semigroup (P31 ; ) are determined in Section 15.2. The list of these semigroups is then an important aid in Section 15.3 during the determination of the remaining subclasses of L3 . In Section 15.4 one ﬁnds all subclasses of M. Chapter 16 supplies the description (coming from [Sze 91] or [Lau 82b, 95a]) of all maximal classes of the subclass P olk {a} TQ := a∈Q

of Pk for arbitrary Q with ∅ = Q ⊆ Ek , k ≥ 2. With the aid of these classes, a completeness criterion for TQ can be formulated easily. This criterion implies

14

Introduction

necessary and suﬃcient conditions for whether a ﬁnite algebra is semi-primal and has only trivial subalgebras. If |Q| = 1, then TQ is a maximal class of Pk and the maximal classes of TQ , given in Chapter 15, are submaximal classes of Pk . Moreover, if |Q| ≥ 2, we prove that every maximal class of TQ is an intersection of TQ with certain maximal classes of Pk or P olk {a} (a ∈ Q). Chapter 17 continues the investigations of Chapter 16 and generalizes Theorem 14.1.3. For arbitrary k, l ∈ N with 2 ≤ l ≤ k − 1 all maximal classes of P olk El are determined, where 9 relation sets are needed. It is important to note that, for the description of the maximal classes of P olk {a}, a ∈ Ek , one needs only 6 relation sets. With the help of the maximal classes of P olk El , one can easily give a completeness criterion for P olk El . The proofs given in Chapter 17 resemble those ones from Chapter 6, i.e., the results of this chapter were achieved with the means developed by I. G. Rosenberg in [Ros 70a]. Chapter 18 gives a survey (partial without proof) over further submaximal classes found until now. In supplement to Chapters 16 and 17, all submaximal classes of a maximal class of the type S are described in this chapter. It is shown, then, how one can prove the special case k ∈ P of this general description easily. The rest of this chapter deals with submaximal classes of Pk that lie below a maximal class of the type U. In Section 18.2, one can ﬁnd some maximal classes of P olk , where ∈ Uk is arbitrary. Then, in Section 18.3, the list is completed from Section 18.2 to the list of all maximal classes 2 ∪ {(k − 1, k − 1)} . of P olk for = Ek−1 Chapter 19 Chapter 19 deals with classes of Lk , which are either direct predecessors of the empty set (so-called minimal classes) or which are direct predecessors of the set of all projections (so-called minimal clones). Consequently, it is not diﬃcult to determine the minimal classes. However, for the minimal clones, only partial results can be given. In Chapter 19, one ﬁnds a description of all minimal classes. Moreover, Rosenberg’s classiﬁcation of minimal clones is proven. At the end of Chapter 19, one ﬁnds a survey of further results about minimal clones and the description of all partial minimal clones. Chapter 20 deals with partial function algebras. A partial n-ary function k the set on Ek is a mapping from a subset of Ekn into Ek , n ∈ N. Let P of all such functions with n ∈ N. Then one can introduce certain modiﬁed k of all partial functions on Ek . Then, the Mal’tsev-operations over the set P set Pk , together with these operations, forms a so-called (full) partial function k ; τ, ζ, ∆, ∇, ), which can similarly be examined like the function algebra (P algebra (Pk ; τ, ζ, ∆, ∇, ). The choice of results on partial function algebras in this chapter focuses on questions already treated for Pk in previous chapters.

Introduction

15

After a composition of some basic concepts in Section 20.1, Section 20.2 shows k is isomorphic to a certain sublattice that the lattice of all partial clones of P of the lattice of all clones of Pk+1 . Thus, one gets many properties of the partial clones from the properties of the clones, which were already found. However many results on clones that may be helpful to ﬁnd certain partial clones with the aid of the above-mentioned isomorphism are missing. One k with the help could, for example, not solve the completeness problem for P of an isomorphism. In Section 20.3, we show how to describe partial clones by relations. Sections 20.4 and 20.5 deal with the maximal partial clones, with whose help, as for Pk , the completeness problem of the partial logic is soluble. We will 3 has exactly 58 maximal partial clones. In 2 has exactly 8 and P prove that P k for arbitrary Section 20.5, there is a complete list of all maximal clones of P k ∈ N, found by L. Haddad and I. G. Rosenberg. The list is given without proof. In Section 20.6, we determine those descriptive relations of the maximal clones of Pk , which are also descriptive relations of the maximal partial clones k . In addition, we survey those papers that deal with the determination of of P the orders of maximal partial clones. Section 20.7 deals with the determinak | C = [C] ∧ C ∩ Pk = A}, tion of the cardinality of the set I(A) := {C ⊆ P where A is an arbitrary maximal clone of Pk . We will prove that, if A has the type U, S or C, I(A) is a ﬁnite set. On the other hand, the set I(A) has the cardinality of continuum if A has the type L or B. For the type M we can give only partial results. Section 20.8 surveys of the cardinalities of the sets I(A), where A is an arbitrary subclass of P2 . In the last section we determine the congruences on the maximal partial k has exactly 4 congruences, whereas clones. It is proven, particularly, that P a maximal partial clone has exactly 4, 8, or 10 pairwise distinct congruences. Finally, one can ﬁnd some technical references: As already noted, the chapters of this book are not continuously numbered, except for the chapters of the ﬁrst and second part. References to parts, chapters, sections, lemmas, theorems, ... are given through their numbers. If a reference of the ﬁrst or second part is missing, then the part discussed is referred to. About the basics of Universal Algebra there are many publications and books. Therefore, only some references for the theorems are given in Part I of this text. In Part II of the book, references for a theorem are left out only if one can prove the theorem easily and if there are several references for this theorem. Normally, one ﬁnds references at the beginning of the chapters and with the theorems, whereby it is also clear that the lemmas appertaining to these theorems are based on considerations from the quoted references. For some few theorems, the proofs are taken over from the quoted references directly.

16

Introduction

However new and shorter proofs are presented particularly for theorems that come from older books. The end of a proof (or of a statement with easy proof) is marked by . Some book parts that can be ignored during the ﬁrst reading diﬀer from the remaining text through a smaller writing.

Preliminaries

We assume that the reader has some knowledge about the basic concepts from the set theory1 , linear algebra, classical algebraic structures and mathematical logic. The following is a list of terms to which we assume the reader is familiar. 1) Logical Symbols So that we can write down mathematical statements quickly and correctly, we use the following symbols from mathematical logic:

symbol

denotation for

∧

and

∨

or

¬

not

=⇒

implies; if - then

⇐⇒

if and only if; iﬀ

:= :⇐⇒

equal by deﬁnition equivalent by deﬁnition

∃

there exists; it exists at least

∃!

it exists exactly

∀

for all

In order to save parentheses, we write instead of ∃x ( E ) (read: “There exists an x with the property E”) shortly ∃x : E. We arrange analogous one for 1

A naive theory of sets is suﬃcient for our purposes.

18

Preliminaries

formulas that contain the sign ∀. In addition, ∀x ∃y ... stands for ∀x (∃y (...)), and so forth. 2) Symbols and Concepts of the Set Theory In dealing with sets, we use the following standard notations: membership (∈), nonmembership (∈), set-builder notations ({.... | ...........} 2 ), the empty set (∅), (⊆), proper inclusion (⊂), intersection (∩ inclusion (⊆), noninclusion and ), union (∪ and ) and diﬀerence (\). The power set of a set A is the set {B | B ⊆ A} of all subsets of A. It is denoted by P(A). N, N0 , Z, Q, R, C denote respectively the set {1, 2, 3, ....} of all natural numbers, the set N ∪ {0}, the set of all integers, the set of all rational numbers, the set of all real numbers, and the set of all complex numbers. For n ∈ N, we write the order n-tuples (brieﬂy n-tuples) in the form (x1 , x2 , ..., xn ). For two n-tuples x := (x1 , x2 , ..., xn ) and y := (y1 , y2 , ..., yn ) is x = y iﬀ xi = yi for all i ∈ {1, 2, ..., n}. A × B is the set of all 2-tuples (a, b) with a ∈ A and b ∈ B and is called the (Cartesian or direct) product of the sets A and B. For n ∈ N, Π1n Ai (or Πi∈I Ai , where I := {1, 2, ..., n}) denotes the set of all (a1 , a2 , ..., an ) with ai ∈ Ai for all i ∈ I. 3 Further, let An := {(a1 , ..., an ) | ∀i ∈ {1, 2, ..., n} : ai ∈ A} be the direct n-powers of the set A, n ∈ N. A correspondence F of A into B is a subset of A × B, where D(f ) := {a ∈ A | ∃b ∈ B : (a, b) ∈ F } is the domain of F and Im(f ) := {b ∈ b | ∃a ∈ A : (a, b) ∈ F } is the image (or range) of F . If D(F ) = A, then F is a correspondence from F . If Im(F ) = B, then we say that F is a correspondence onto B. The inverse (or converse) F −1 of a correspondence F ⊆ A × B is given by F −1 := {(b, a) ∈ B × A | (a, b) ∈ F }. If F ⊆ A × B and G ⊆ B × C, then the correspondence product F 2G is deﬁned by F 2G := {(a, c) ∈ A × C | ∃b ∈ B : (a, b) ∈ F ∧ (b, c) ∈ G}. We remark that (F 2G)−1 = G−1 2F −1 and 2 is associative. A mapping (or map) f from A into B is a subset of A × B such that for each a ∈ A there is exactly one b ∈ B with (a, b) ∈ f . If f is a mapping from A into B, then we write f : A −→ B and, if (a, b) ∈ f , f (a) = b, or a → b. A mapping f : A −→ B is called injective (or is an injection) iﬀ f (a) = f (a ) implies a = a for all a, a ∈ A. The mapping f : A −→ B is surjective (or is a surjection) iﬀ Im(f ) = B. The mapping f : A −→ B is bijective (or is a bijection) iﬀ is both injective and surjective. Let A be a set and n ∈ N. Then, an n-ary relation on A (or an n-ary 2 3

For example, we write S := {x | x ∈ A ∧ P (x)} or brieﬂy S := {x ∈ A | P (x)} instead of “S is the set of all x ∈ A with the property P (x)”. For arbitrary I, we will deﬁne Πi∈I Ai in Part I, Section 5.

Preliminaries

19

relation over A) is a subset of An . A 2-ary , 3-ary or 4-ary relation on A is called respectively a binary, ternary or quaternary relation on A. Two sets A and B have the same cardinality (in symbol |A| = |B|) iﬀ there exists a bijection from A onto B. A set A is inﬁnite, iﬀ there is a subset B ⊂ A with |A| = |B|. A set A is called ﬁnite iﬀ A is not inﬁnite. If A is a ﬁnite set, then |A| denotes the number of elements of A, i.e., |∅| = 0 and |x ∪ {x}| = |x| + 1 (or n = |{0, 1, 2, ..., n − 1}| for all n ∈ N0 , where n is called a (ﬁnite) ordinal). We put ℵ0 := |N| and c := |R|. The set A is called countable if |A| = ℵ0 . If |A| = c, we say that A has the cardinality of the continuum. It is well-known that ℵ0 is the least inﬁnite ordinal and that N and R do not have ∞ the same cardinality. Further, it holds |Q| = ℵ0 , |C| = c, |P(N)| = c and | n=1 An | = ℵ0 if |An | = ℵ0 for all n ∈ N.

Part I

Universal Algebra

Part I Universal Algebra

23

The Universal Algebra is a mathematical discipline, which examines so-called universal algebras, where the classic algebraic structures – such as groups, rings, and lattices – are special cases of such algebras. Roughly said, a universal algebra is nothing other than a nonempty set together with a family of ﬁnite-ary operations over this set. Although the concept universal algebra was used already at the end of the 19th century, the development of the theory Universal Algebra started only in the 1930’s. This development has two historical roots. The ﬁrst root is the classical algebra and the studies of properties that have all algebraic structures (such as semigroups, groups, rings, etc.). The second root is the Mathematical Logic. In the last decades, the Universal Algebra has continued the classic examinations and has also developed new areas. An introduction to the bases of the Universal Algebra follows. The topic of the following six chapters is based on what is needed later, during the investigation of the function algebras. The ﬁrst ﬁve chapters show that the Universal Algebra is a general theory of the algebraic partial disciplines. Chapter 6 treats problems with an impression of the mode of operation of the algebra and that have partial also contact points to the mathematical logic. 1 Additions to the topics can be found in [Lau 2004] 2 . In detail one can study the Universal Algebra with the aid of the following books [Bur-S 81], [Coh 65], [Den-T 96], [Den 2003], [Gr¨ a 68], [Ihr 2003], [McKM-T 87], [Wer 78]. The book [McK-M-T 87] can particularly be recommended. In the preface, we have already noted that to understand the second part of the text need not be read absolutely. Thus, readers can select for themselves the required terms and theorems from Part I, during the reading of Part II. One night notice, after reading Part II, that some examinations are special cases of general problems.

1 2

To realize this, reads [Bur-S 81] and [Ric 78]. There, one ﬁnds a continuation to the chapters of Universal Algebra of this book and additional chapters.

1 Basic Concepts of Universal Algebra

1.1 Universal Algebras First, we deﬁne the concept of an n-ary partial operation: Let A be a nonempty set, n ∈ N and ∅ ⊂ ⊆ An . An n-ary partial operation is a mapping f from into A. The number n is called arity of f and the arity of f is also denoted by af . To denote the arity of f we also write f (n) or brieﬂy f n , since for content-related reasons, the interpretation is impossible of f n as Cartesian product in the following. If (a1 , ..., an ) ∈ then let f (a1 , ..., an ) be the image of (a1 , ..., an ) under an n-ary operation f . The set we call domain of f , and we denote the domain of f by D(f, A) or brieﬂy by D(f ). Let Im(f ) be the set {f (a1 , ..., an ) | (a1 , ..., an ) ∈ D(f )}, which is called image or range of f . (n) (n) For n ∈ N let c∞ be the n-ary partial operation with D(c∞ ) = ∅.1 n If = A and n ≥ 1, then f is called an n-ary operation on A. If D(f (n) ) ⊂ An we call f (n) a proper partial operation. A nullary operation f on A is an element f of A with af := 0, D(f ) := ∅ and Im(f ) := {f }. Example The operation ◦ of a group (see also Section 1.2) is a binary operation. One can understand the formation of inverse elements in a group as a unary operation −1 and the identity e (unit element) of a group as a nullary operation. As is generally accepted, one can form compositions of mappings in the following manner: If f : M1 −→ M2 and g : M2 −→ M3 are mappings, then we denote the mapping 1

The necessity of such a deﬁnition is seen, if one forms superpositions of partial (1) functions (see Part II, Chapter 1). For example, it holds g2f = c∞ for the unary operations f , g with D(f ) := {a} and a ∈ Im(g).

26

1 Basic Concepts of Universal Algebra

f 2g : M1 −→ M3 , x → g(f (x)) by f 2g. It is well-known (or it is easy to see) that 2 is associative. Thus we can renounce the putting of brackets in the case of more than two compositions. Furthermore, we write g1 g2 ...gr instead of g1 2g2 2...2gr (gi : Mi −→ Mi+1 , i = 1, ..., r), if the operation 2 follows from the context. We notice that the above deﬁnition of 2 is a special case of the following deﬁnition, if one writes the mappings in the form of subsets of Cartesian products: Let R ⊆ A × B and Q ⊆ B × C. Then let R2Q := {(a, c) | ∃b ∈ B : (a, b) ∈ R ∧ (b, c) ∈ Q}. With the help of the partial operations, we can deﬁne the following concepts: A partial algebra is an ordered pair A := (A; F ), where A is an arbitrary nonempty set and F is a set of partial operations on A. The set A is called universe (or underlying set) of A. The elements of F are called (partial) fundamental operations of A. A universal algebra or brieﬂy an algebra is a partial algebra (A; F ), where every element of F is an operation, i.e., F does not have any proper partial operations. Here we usually only deal with algebras. Therefore, we often introduce the following concepts only for algebras. If F = {f1 , ..., fr } holds we write (A; F ) in the form (A; f1 , ..., fr ), where we often choose af1 ≥ af2 ≥ ... ≥ afr . We also use the notation (A; (fi )i∈I ), if F = {fi | i ∈ I} and I is a certain index set. A (partial) algebra (A; F ) is called ﬁnite, if A is a ﬁnite set, otherwise inﬁnite. In order to receive a ﬁrst classiﬁcation of the algebras, it oﬀers itself to carry out a classiﬁcation according to the arities of the (partial) operations of the algebras. For this purpose, we deﬁne the concept type of an algebra (A; (fi )i∈I ) as a sequence

1.2 Examples of Universal Algebras

27

τ := (afi | i ∈ I) of the arities of their fundamental operations, if I is a ﬁnite or countable set. In the case that I is uncountable, we deﬁne as type of A the mapping τ : I −→ N, i → afi . If I is a ﬁnite set, then (A; (fi )i∈I ) is called of ﬁnite type. If A = (A; f1 , ..., fr ), af1 ≥ af2 ≥ ... ≥ afr , then let τ = (af1 , af2 , ..., afr ). Two (partial) algebras (A; F ), (B; G) are of same type, iﬀ there is a bijection ϕ : F −→ G, f → g with af = ag. Examples2 1) A semigroup (H, ◦) is an algebra of type (2) (F := {◦}). 2) By the above remark, a group is an algebra of the type (2, 1, 0). 3) A lattice (see 1.2.11) is an algebra of type (2, 2). Because of existence of a bijection between the operations of two algebras A := (A; F ) and B := (B; G) of same type, we often denote the operation sets of A and of B with the same symbol (F or G). In analog mode we deal with the denotation of the elements. If it is necessary to denote the operations of two diﬀerent algebras A and B of the same type diﬀerently, then we write f A or f B (or also fA or fB ), respectively. Before we deal with further basic terms from the theory of the algebras, let’s deﬁne some speciﬁc algebras. We will study some of these speciﬁc algebras in detail later.

1.2 Examples of Universal Algebras We start with some “classical algebras”. In deﬁning these algebras, we try to use no existence quantor. 3 We call the equations, which are given for the deﬁnition of an algebra, axioms. As usual, we will try to use only a few (independent) axioms. We diﬀer from this principle in the following only in few places (see e.q. the deﬁnition of a lattice). 1.2.1 Gruppoids An algebra (A; ◦) of type (2) is called gruppoid. Thus, a gruppoid is not a diﬀerent one as a nonempty set together with a binary operation. 2 3

Section 1.2 contains the explanations to the concepts used. In Chapter 6, we see that this manner of description has interesting conclusions. In the following, we use the descriptions of the algebraic structures given in this section.

28

1 Basic Concepts of Universal Algebra

1.2.2 Semigroups A gruppoid (H; ◦) is called semigroup if the algebra (H; ◦) fulﬁlls the following axiom: (A)

∀x, y, z ∈ H : (x ◦ y) ◦ z = x ◦ (y ◦ z)

(associativity).

A semigroup is called commutative, if it satisﬁes the following condition in addition: (C) ∀x, y ∈ H : x ◦ y = y ◦ x. 1.2.3 Monoids An algebra (M ; ◦, e) of type (2, 0) is called monoid, if (M ; ◦) is a semigroup and (E) ∀x ∈ M : e ◦ x = x ◦ e = x holds. The (nullary) operation e is the the neutral element of M . 1.2.4 Groups A group is an algebra (G; ◦,−1 , e) of type (2, 1, 0), which fulﬁlls the above axioms (A), (E) for x, y, z, e ∈ G and (I)

∀x ∈ G : x ◦ x−1 = x−1 ◦ x = e .

x−1 is called the inverse element of x. A group, which (C) fulﬁlls in addition, is called Abelian (or commutative). It is usual in Abelean groups to use +, −x, 0 instead of ◦, x−1 , e (the additive notation). 1.2.5 Semirings An algebra (R; +, ·) of type (2, 2) is called a semiring if (R; +) is a commutative semigroup, (R; ·) is a semigroup, and if the following distributive laws hold: ∀x, y, z ∈ R : x · (y + z) = (x · y) + (x · z), (D1 ) ∀x, y, z ∈ R : (x + y) · z = (x · z) + (y · z). (D2 ) 1.2.6 Rings An algebra (R; +, ·, −, 0) of type (2, 2, 1, 0) is called ring if (R; +, −, 0) is an Abelean group and (R; +, ·) is a semiring. In order to save parentheses, we use the known rule subsequently “The dot bill is carried out before the stroke bill”. A unitary ring (or a ring with a unit element) is an algebra (R; +, ·, −, 0, 1) of type (2, 2, 1, 0, 0), where (R; +, ·, −, 0) is a ring and (E) holds with e = 1 and ◦ = ·.

1.2 Examples of Universal Algebras

29

1.2.7 Fields A partial algebra (K; +, ·, −,−1 , 0, 1) is called ﬁeld if the algebra (K; +, ·, −, 0, 1) is an unitary ring and (K\{0}; ·,−1 , 1) is an Abelean group. We remark that the operation −1 on K is only a partial operation, since 0−1 is not deﬁned. 1.2.8 Modules Let R := (R; +, ·, −, 0) be a ring. An algebra (M ; F ) where F := {+, −, 0}∪R, + is binary, 0 is a nullary and − and all r ∈ R are unary operations is called a R-module (or module over the ring R), if (M ; +, −, 0) is an Abelean group and if for all r, s ∈ R the following equations hold: (M1 ) (M2 ) (M3 )

∀x, y ∈ M : r(x + y) = r(x) + r(y), ∀x, y ∈ M : (r + s)(x) = r(x) + s(x), ∀x ∈ M : (r · s)(x) = r(s(x)).

Let M := (M ; +, −, (r)r∈R , 0) be the short notation for the R-module deﬁned above. Further, we say that M has the type (2, 1, (1)r∈R , 0). A module over a unitary ring (R; +, ·, −, 0, 1) is an above module that also satisﬁes the following condition: (M4 )

∀x ∈ M : 1(x) = x.

Three remarks to the above deﬁnitions: – Instead of the unary operations r ∈ R one can also deﬁne a mapping : R × M −→ M, (r, x) → r x, which fulﬁlls the axioms. However, a module would then not be a universal algebra. – A module has inﬁnite many operations if the ring R is inﬁnite. – The operation symbols +, −, 0 have two diﬀerent meanings: on the one hand, they describe the operations of the Abelean group (R; +, −, 0); on the other hand, they describe the operations of the Abelean group (M ; +, −, 0). 1.2.9 Vector Spaces Let K := (K; +, ·, −, 0, 1) be a ﬁeld. Then every K-module (V ; +, −, (k)k∈K , 0) is called K-vector space (or vector space over the ﬁeld K). 1.2.10 Semilattices A commutative semigroup (S; ◦), in which ∀x ∈ S : x ◦ x = x holds, is called semilattice.

30

1 Basic Concepts of Universal Algebra

1.2.11 Lattices A lattice is an algebra (L; ∨, ∧) of type (2, 2), in which for arbitrary x, y, z ∈ L it holds: (L1 ) x ∨ y = y ∨ x, x ∧ y = y ∧ x (L2 ) x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z (L3 ) x ∨ x = x, x ∧ x = x (L4 ) x ∨ (x ∧ y) = x, x ∧ (x ∨ y) = x

(commutativity), (associativity), (idempotency), (absorption).

A bounded lattice (or a lattice with 0 and 1) is an algebra (L; ∨, ∧, 0, 1) of type (2, 2, 0, 0) such that (L; ∨, ∧) is a lattice and furthermore the following equations hold for all x ∈ L: x ∧ 0 = 0, x ∨ 1 = 1.

(L5 )

A lattice is called distributive, if the following distributive laws hold: (DL1 ) (DL2 )

∀x, y, z ∈ L : x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), ∀x, y, z ∈ L : x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z).

Remark: One can show (with the help of (L1 )–(L4 )) that (DL1 ) and (DL2 ) are equivalent; that is, it suﬃces to require either (D1 ) or (D2 ) in the deﬁnition of a distributive lattice. 1.2.12 Boolean Algebras An algebra (B; ∨, ∧,− , 0, 1) of type (2, 2, 1, 0, 0) is called Boolean algebra if (B; ∨, ∧, 0, 1) is a bounded distributive lattice and the following equalities hold for all x ∈ B: x ∧ x = 0, x ∨ x = 1 x ∧ 0 = 0, x ∨ 1 = 1,

(B1 ) (B2 ) where x :=

−

(x).

1.2.13 Function Algebras One can choose the set of all operations deﬁned on A as a universe of an algebra and can then deﬁne operations on the operations on A. For the purpose of distinction we will subsequently replace the concept “operation (on the set A)” by the concept “function (on the set A)”. In Part II, the set A is always a ﬁnite set. Therefore, the following concepts become explained only for a speciﬁc k-element set Ek . Put Ek := {0, 1, ..., k − 1},

1.3 Subalgebras

31

k ∈ N\{1}. Let Pkn be the set of all n-ary functions f n , which map the n-fold n n Cartesian product Ek into Ek . Put Pk := n≥1 Pk . Elementary operations (called Mal’tsev-operations) over Pk are ζ, τ, ∆, ∇ (unary operations) and (a binary operation) deﬁned by (ζf )(x1 , ..., xn ) := f (x2 , x3 , ..., xn , x1 ), (τ f )(x1 , ..., xn ) := f (x2 , x1 , x3 , ..., xn ), (∆f )(x1 , ..., xn−1 ) := f (x1 , x1 , x2 , ..., xn−1 ) if n ≥ 2, ζf = τ f = ∆f := f if n = 1, (∇f )(x1 , ..., xn+1 ) := f (x2 , x3 , ..., xn+1 ), (f g)(x1 , ..., xm+n−1 ) := f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) (f n , g m ∈ Pk ). It holds the following (see Part II, Chapter 1): With the help of the Mal’tsev-operations, one can form the following operations (for arbitrary functions and arbitrary variables of the functions): – permutation of variables – identiﬁcation of variables – adding of ﬁctitious variables and – substitution of variables of a function by functions The set of all functions that can be obtained by a ﬁnite number of applications of the above operations from the functions of F ⊆ Pk is called the closure of F , and is denoted by [F ]. If F = [F ], then we say that F is closed or F is a (sub)class of Pk . The algebra (Pk ; ζ, τ, ∆, ∇, ) of type (1, 1, 1, 1, 2) is called (full) iterative function algebra. If A is a subclass of Pk then (A; ζ, τ, ∆, ∇, ) is called function algebra.

1.3 Subalgebras Let A = (A; F ) be a (partial) algebra and let B be a nonempty subset of A. The (partial) algebra B = (B; F ) of the same type as A is called (partial) subalgebra of A (or A is an extension of B) iﬀ it holds D(fB , B) ⊆ D(fA , A) for arbitrary f ∈ F and fA (x1 , ..., xaf ) = fB (x1 , ..., xaf ) for all (x1 , ..., xaf ) ∈ B af ∩ D(f, A). Then we write

32

1 Basic Concepts of Universal Algebra

B ≤ A. If A has a nullary operation f ∈ A and B is a subalgebra of A, then f also belongs to B. The following lemma summarizes easy consequences from the above deﬁnition of a subalgebra. Lemma 1.3.1 (a) A subset B of the universe A of an algebra (A; F ) together with the restrictions f|B of the operations f of A to B forms an algebra if and only if f (b1 , ..., baf ) ∈ B for all b1 , ..., baf , if af ≥ 1, and f ∈ B if af = 0 holds for arbitrary f ∈ F . (b) Let I bean arbitrary index set, Bi ≤ A for every i ∈ I and i∈I Bi = ∅. Then ( i∈I Bi ; F ) is a subalgebra of A. (c) For every nonempty subset T of the universe A of an algebra A = (A; F ) there exists exactly one smallest subalgebra T of A, which contains T and which one can describe as follows: T = (T ; F ) with T := {B | B ≤ A and T ⊆ B}.

The set T from Lemma 1.3.1, (c) is called generating system of the algebra T and the universe T of T is denoted by [T ]A or [T ]F or by [T ]f1 ,...,fr , if F = {f1 , ..., fr }, or simply by [T ]. Example For the algebra A = ({0, 1, 2, ..., k − 1}; f ) of the type (3) with f (x, y, z) = x + y − z (mod k) for arbitrary x, y, z ∈ A, it holds e.g. [{a}] = {a} for every a ∈ A and [{0, 1}] = A. If [T ] = T for a subset T ⊆ A of an algebra A = (A; F ), then we say that T is closed. The following lemma gives another possibility of the description of the set [T ], deﬁned above. Lemma 1.3.2 Let A = (A; F ) be an algebra, T be a subset of A and F 0 be the set of all nullary operations of F . Then, for T we can deﬁne recursively the following subsets Tn of A as follows: T0 := T ∪ F 0 Tn+1 := Tn ∪ {f (g1 , ..., gaf ) | f ∈ F \F 0 and {g1 , ..., gaf } ⊆ Tn } (n ∈ N0 ).

1.3 Subalgebras

Then [T ]A =

Tn .

n≥0

Proof. To prove “⊆” in (1.1), we have to show that

33

(1.1)

is the universe of n≥0 Tn a subalgebra of A. This would be shown if we have shown that n≥0 Tn is closed regarding the application of all operations of F \ F 0 . Now let f ∈ F \ F 0 and {g1 , ..., gaf } ⊆ n≥0 Tn be arbitrary. Since af ∈ N, there is an m ∈ N with {g1 , ..., gaf } ⊆ Tm . Consequently, f (g1 , ..., gaf ) belongs to Tm+1 , i.e., the set n≥0 Tn is closed. “⊇” follows from Tn ⊆ [T ]A for all n ≥ 0. This inclusion is easy to prove by induction on n, since [T ]A is the universe of a subalgebra of A.

The set S(A) := {B | B is subalgebra of A} ∪ {∅} we call brieﬂy set of all subalgebras of A. Per deﬁnitionem (essentially for technical reasons) is also the empty set a subalgebra of each algebra. Then (as the reader can easily verify) S(A) together with the operations ∧ : S(A) × S(A) −→ S(A), B1 ∧ B2 = (B1 ∩ B2 ; F ) ∨ : S(A) × S(A) −→ S(A), B1 ∨ B2 = ([B1 ∪ B2 ]F ; F ) forms a lattice.4

4

This property would not hold for all algebras, if the empty set was not an algebra.

2 Lattices

Lattices arise often in algebraic investigations. In the following we see that the concept “lattice” is always needed, if the elements of sets are ordered in a certain meaning. In addition, the lattice theory is an interesting branch of the Universal Algebra. For space reasons, only the most important basis concepts and some proof ideas of the lattice theory can be indicated here. For a secondary study of the lattice theory, refer to the books on the Universal Algebra and to [Bir 48], [Ern 82], [Sko 73] and [Dav-P 90].

2.1 Two Deﬁnitions of a Lattice There are two standard ways of deﬁning lattices. One of these ways was already given in Section 1.2.11: Deﬁnition (First Deﬁnition of a Lattice) Let L be a nonempty set on which the binary operations ∨ (called “join”) and ∧ (called “meet”) are deﬁned. (L; ∨, ∧) is called lattice if for arbitrary x, y, z ∈ L the following identities hold: (L1 a) x ∨ y = y ∨ x, (L1 b) x ∧ y = y ∧ x

(commutativity),

(L2 a) x ∨ (y ∨ z) = (x ∨ y) ∨ z, (L2 b) x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity),

(L3 a) x ∨ x = x, (L3 b) x ∧ x = x

(idempotency),

(L4 a) x ∨ (x ∧ y) = x, (L4 b) x ∧ (x ∨ y) = x

(absorption).

36

2 Lattices

Since one receives an axiom again by exchanging of ∨ and ∧ in above axioms, results from every equation, which the axioms imply, by exchanging of ∨ and ∧ a further valid equation. One names this procedure of deriving equations in lattices, using the duality principle of the lattice theory. In the following proofs we often use the equivalence x ∨ y = y ⇐⇒ x ∧ y = x,

(2.1)

which is a conclusion from the absorption laws and from the commutative laws. For the second deﬁnition of a lattice we need some concepts and notations: Deﬁnitions • A binary relation ≤ (⊆ A × A) is called a partial order on the set A, if it fulﬁlls the following conditions: (O1 ) ∀a ∈ A : a ≤ a (reﬂexivity), (O2 ) ∀a, b ∈ A : (a ≤ b and b ≤ a) =⇒ a = b (antisymmetry), (O3 ) ∀a, b, c ∈ A : (a ≤ b and b ≤ c) =⇒ a ≤ c (transitivity). • The pair (A; ≤), where ≤ satisﬁes the above conditions, we call partially ordered set or, brieﬂy, poset. In examples, we often declare the posets through order diagrams1 (or Hasse diagrams). • A poset P := (P ; ≤) with P ⊆ A × A, which besides fulﬁlls the condition (O4 ) ∀a, b ∈ A : a ≤ b or b ≤ a , is called totally ordered set or linearly ordered set or, brieﬂy, chain. We also use the notation a < b if a ≤ b and a = b is valid, where (L; ≤) is a poset and a, b ∈ L. We also write b ≥ a instead of a ≤ b. Deﬁnition Let Q be a subset of P , where P = (P ; ≤) is a poset. The element s ∈ P is said to be a supremum of Q (denoted by sup Q) iﬀ s has the following properties: (S1 ) ∀q ∈ Q : q ≤ s; (S2 ) ∀p ∈ P ((∀q ∈ Q : q ≤ p) =⇒ s ≤ p). Remark The supremum of Q does not exist in general for every subset Q of a poset P . Let e.g. P = {0, 1, 2, 3, 4} and let ≤ deﬁned by the following order diagram: 1

Here, the elements of the poset are represented as points in the plane and, if x < y and no z exists with x < z < y, we draw y higher up than x and connect x and y with a line segment.

2.1 Two Deﬁnitions of a Lattice

37

4r @ @

@r 3 HH H 0 r Hr 1 2 rH

Fig. 2.1

Then, sup{0, 1} does not exist, since for p ∈ {2, 3} 0 ≤ p and 1 ≤ p are valid, the elements 2 and 3 are not comparable in respect to ≤. Deﬁnition Let Q be a subset of P , where P = (P ; ≤) is a poset. The element i ∈ P is said to be an inﬁmum of Q (denoted by inf Q) iﬀ i has the following properties: (I1 ) ∀q ∈ Q : i ≤ q; (I2 ) ∀p ∈ P ((∀q ∈ Q : p ≤ q) =⇒ p ≤ i). Deﬁnition (Second Deﬁnition of a Lattice) A poset L := (L; ≤) is called lattice iﬀ for arbitrary a, b ∈ L both sup{a, b} and inf{a, b} in L exist. Some elementary properties of sup and inf in a lattice are summarized in the following lemma: Lemma 2.1.1 Let (P ; ≤) be a lattice by the second deﬁnition. Then, for arbitrary x, y, z, u, v ∈ P it holds: (a) x ≤ y =⇒ sup{x, z} ≤ sup{y, z}, (b) x ≤ y =⇒ inf{x, z} ≤ inf{y, z}, (c) (x ≤ y and u ≤ v) =⇒ sup{x, u} ≤ sup{y, v}, (d) (x ≤ y and u ≤ v) =⇒ inf{x, u} ≤ inf{y, v}. Proof. (a) and (b) are easy to check. (c): Let x ≤ y and u ≤ v. Then by (a) it holds sup{x, u} ≤ sup{y, u} and sup{y, u} ≤ sup{y, v}. Since ≤ is transitive, by this (c) follows. One can prove (d) analogously to (c). The next theorem shows how the two lattice deﬁnitions are associated. Theorem 2.1.2 It holds: (a) If (L; ∨, ∧) is a lattice by the ﬁrst deﬁnition, then one can deﬁne by a ≤ b :⇐⇒ a = a ∧ b

(2.2)

a partial order ≤, which together with L forms a lattice by the second deﬁnition.2

38

2 Lattices

(b) Conversely, if (L; ≤) is a lattice by the second deﬁnition, then one can deﬁne two binary operations ∨, ∧ by a ∨ b := sup{a, b} a ∧ b := inf{a, b}

and

and (L; ∨, ∧) is a lattice by the ﬁrst deﬁnition. Proof. (a): Let (L; ∨, ∧) be a lattice and ≤ is deﬁned by (2.2). Then a ∧ a = a

holds and thus a ≤ a for every a ∈ A. Hence ≤ is reﬂexive. If a ≤ b and b ≤ a, we have a = a ∧ b and b = b ∧ a. This and (L1 b) imply a = b. Thus ≤ is antisymmetric. Let now a ≤ b and b ≤ c. Then it holds a = a ∧ b and b = b ∧ c by deﬁnition of ≤. By this and by (L2 b) we have a = a ∧ (b ∧ c) = (a ∧ b) ∧ c = a ∧ c. Hence a ≤ c and thus ≤ is transitive. It remains to show that there exist sup{a, b} and inf{a, b} for all a, b ∈ L. For these let a, b ∈ L be arbitrary. By (L4 a) and (L4 b) then we have a = a ∧ (a ∨ b) and b = b ∧ (a ∨ b). Thus a ≤ a ∨ b and b ≤ a ∨ b. Let now a ≤ u and b ≤ u for a certain u ∈ L. Then the following equations hold: a = a ∧ u, b = b ∧ u, a ∨ u = (a ∧ u) ∨ u = u, b ∨ u = (b ∧ u) ∨ u = u, and (a ∨ b) ∨ u = (a ∨ u) ∨ (b ∨ u) = u ∨ u = u. If one uses the equation (a ∨ b) ∨ u = u, which just was proven, then one gets: (a ∨ b) ∧ u = (a ∨ b) ∧ ((a ∨ b) ∨ u) = a ∨ b. Therefore, we have a ∨ b ≤ u and sup{a, b} = a ∨ b holds by deﬁnition of sup. Analogously, one can show a ∧ b = inf{a, b}. For this, one mixed up ∧ in above considerations by ∨ and one replaced ≤ by ≥. Thus (L; ≤) is a lattice. (b): Let (L; ≤) be a lattice, a ∨ b := sup{a, b} and a ∧ b := inf{a, b}. Obviously, the so deﬁned operations ∨ and ∧ are commutatively and idempotent. To show the validity of the associative law sup{x, sup{y, z}} = sup{sup{x, y}, z} one can prove (under use of Lemma 2.1.1) sup{x, sup{y, z}} ≤ sup{sup{x, y}, z} and sup{x, sup{y, z}} ≥ sup{sup{x, y}, z}. Analogously, one can show the associativity of inf. Because of sup{x, inf{x, y}} ≥ x and sup{x, inf{x, y}} ≤ x it holds obvious sup{x, inf{x, y}} = x. Analogously, one can show another absorption law. Consequently, (L; ∨, ∧) is a lattice by the ﬁrst deﬁnition. 2

Because of (2.1) one also could have deﬁned the following: a ≤ b :⇐⇒ a ∨ b = b.

2.3 Isomorphic Lattices and Sublattices

39

Because of Theorem 2.1.2 we can use the ﬁrst or the second deﬁnition of an lattice subsequently according to requirement, where the statements of the Theorem 2.1.2 are used where appropriate. If one wants the above-mentioned duality principle on inequalities of the form ... ≤ ... enlarge, one has to deﬁne the exchanging of ≤ through ≥ as an additional replacement rule. In addition the implication (x ≤ y and u ≤ v) =⇒ (x ∨ u ≤ y ∨ v and x ∧ u ≤ y ∧ v),

(2.3)

resulting from Lemma 2.1.1 and from Theorem 2.1.2, will be an important aid in the below-given proofs.

2.2 Examples for Lattices 2.2.1 Let L := {0, 1}. Further, let ∨ be the conjunction on L and let ∧ be the disjunction on L. Obviously, (L; ∨, ∧) is a lattice. 2.2.2 Let L := N0 , let a ∨ b be the least common multiple and let a ∧ b be greatest common divisor of the integers a, b ∈ N0 . In this case, it is also easy to check that (L; ∨, ∧) is a lattice. 2.2.3 Examples for lattices by the second deﬁnition are: (P(A); ⊆), where A is an arbitrary nonempty set; (R; ≤), where ≤ is the usual order on R. One ﬁnds further examples in Section 2.4 and in the following chapters.

2.3 Isomorphic Lattices and Sublattices Deﬁnition Two lattices L1 , L2 are called isomorphic, iﬀ there exists a bijective mapping α from L1 onto L2 such that for arbitrary a, b ∈ L1 the following two equations hold: α(a ∨ b) = α(a) ∨ α(b)

and

α(a ∧ b) = α(a) ∧ α(b). The mapping α is also called an isomorphism. Deﬁnition Let (P1 , ≤) and (P2 ; ≤) be posets. A mapping α from P1 onto P2 is called order-preserving, if the following holds: ∀a, b ∈ P1 : a ≤ b =⇒ α(a) ≤ α(b). Figure 2.2 gives an example of an order-preserving mapping between the lattices L1 and L2 :

40

2 Lattices

r A

r A A A

- r A A

- r AAr

- r

AAr

- r L2

L1 Fig. 2.2

Theorem 2.3.1 Two lattices L1 and L2 are isomorphic iﬀ there is a bijective mapping α from L1 onto L2 such that both α and α−1 are order-preserving. Proof. “=⇒”: Let α be an isomorphism from L1 onto L2 . Then for all a, b ∈ L1 it holds: a ≤ b =⇒ a = a ∧ b =⇒ α(a) = α(a ∧ b) = α(a) ∧ α(b). Consequently, we have α(a) ≤ α(b), i.e., α is order-preserving. Let now c, d ∈ L2 arbitrary with c ≤ d. Then there exist a, b ∈ L1 with α(a) = c and α(b) = d. That α−1 is also order-preserving follows then from c ≤ d =⇒ α(a) ≤ α(b) =⇒ α(a) ∧ α(b) = α(a) =⇒ α(a ∧ b) = α(a) =⇒ a ∧ b = a =⇒ a ≤ b. “⇐=”: Let now α be a bijective mapping from L1 onto L2 with the property that both α and α−1 are order-preserving. Then for arbitrary a, b ∈ L1 we have a ≤ a ∨ b and b ≤ a ∨ b. Thus, α(a) ≤ α(a ∨ b) and α(b) ≤ α(a ∨ b). From this, it follows α(a) ∨ α(b) ≤ α(a ∨ b). Further, for arbitrary u ∈ L2 it holds α(a) ∨ α(b) ≤ u =⇒ α(a) ≤ u and α(b) ≤ u =⇒ a ≤ α−1 (u) and b ≤ α−1 (u) =⇒ a ∨ b ≤ α−1 (u) =⇒ α(a ∨ b) ≤ u. Consequently, α(a ∨ b) = sup{α(a), α(b)} and therefore α(a) ∨ α(b) = α(a ∨ b). Analogously, one can prove α(a) ∧ α(b) = α(a ∧ b).

Remark Figure 2.2 shows that one cannot leave the condition “α−1 orderpreserving” from Theorem 2.3.1.

2.4 Complete Lattices and Equivalence Relations

41

Theorem 2.3.2 For every poset (P ; ≤) there is a set system MP such that (P ; ≤) and (MP ; ⊆) are isomorphic; i.e., there exists a bijective mapping α from P onto MP with the property that α and α−1 are order-preserving. Proof. For every p ∈ P let M (p) := {x ∈ P | x ≤ p}. Then the set MP := {M (p) | p ∈ P } is a set system with the properties claimed in the theorem. To show this, we study the mapping α : P −→ MP . α is surjective by deﬁnition. From M (a) = M (b) it follows a ∈ M (b) and b ∈ M (a). Hence a ≤ b and b ≤ a, i.e., a = b. Thus α is injective. α is order-preserving, since a ≤ b and x ∈ M (a) imply x ≤ a ≤ b. Therefore, a ≤ b implies M (a) ⊆ M (b). Furthermore, M (a) ⊆ M (b) implies a ≤ b, i.e., α−1 is also order-preserving and thus α is an isomorphism. The following deﬁnition is a special case of a deﬁnition from Section 1.3: Deﬁnition

If L is a lattice and L is a subset of L with the property ∀x, y ∈ L : x ∨ y ∈ L and x ∧ y ∈ L ,

then (L ; ∨, ∧) is called a sublattice of (L; ∨, ∧). Remark Let (P ; ≤) and (Q; ≤) be posets with Q ⊆ P . Then, it is not valid in general that (Q; ∨, ∧) is a sublattice of the lattice (P ; ∨, ∧), where ∨, ∧ are deﬁned in Theorem 2.1.2. An example is the poset ({0, 1, 2, 3, 4}; ≤) deﬁned by the following order diagram: 4 r r3 @ @r2

1 r @ @r 0

Fig. 2.3

Obviously, ({0, 1, 2, 4}; ≤) is a poset, but ({0, 1, 2, 4}; ∨, ∧) is not a sublattice of ({0, 1, 2, 3, 4}; ∨, ∧), since 1 ∨ 2 = 3 ∈ {0, 1, 2, 4}. Deﬁnition A lattice L1 can be embedded into a lattice L2 , if there is a sublattice of L2 isomorphic to L1 .

2.4 Complete Lattices and Equivalence Relations Deﬁnition A poset is called complete iﬀ for every subset A of P both sup A (denoted by A) and inf A (denoted by A) exist.

42

2 Lattices

A lattice which is complete as poset is a complete lattice. Examples 1) Obviously, each complete poset is also a lattice. 2) Every ﬁnite lattice is complete. 3) (R; ≤) is not complete. One can ﬁnd further examples in Theorems 2.4.3 and 2.4.5. Theorem 2.4.1 For an arbitrary poset (L; ≤) with a greatest and a least element it holds:

∀A ⊆ L ∃ A ⇐⇒ ∀A ⊆ L ∃ A. Proof. “=⇒”: Let Ao := {x ∈ L | ∀a ∈ A : a ≤ x}. Since L has a greatest element, we have Ao = ∅. It is easy to check that then can prove “⇐=”.

A=

Ao holds. Analogously, one

Theorem 2.4.2 Let A be an algebra. Then the lattice (S(A); ⊆) of all subalgebras of A is a complete lattice. Proof. The proof follows from Theorem 2.4.1 and the fact that every intersection of subalgebras of A is also a subalgebra of A (see Lemma 1.3.1, (b)). The following is a short summary of the equivalence relations: Deﬁnitions A binary relation is called an equivalence relation on a (nonempty) set A, if fulﬁlls the following conditions: (E1 ) {(a, a) | a ∈ A} ⊆ (reﬂexivity); (E2 ) ∀a, b ∈ A : (a, b) ∈ =⇒ (b, a) ∈ (symmetry); (E3 ) ∀a, b, c ∈ A : ((a, b) ∈ and (b, c) ∈ ) =⇒ (a, c) ∈ (transitivity). Let Eq(A) be the set of all equivalence relations on A. By deﬁning of the operation

−1

: P(A × A) −→ P(A × A), → −1 by

−1 := {(b, a) | (a, b) ∈ } and of the binary operation 2 : P(A × A) × P(A × A) −→ P(A × A), (, ) → ◦ by

2 := {(a, c) | ∃b ∈ A : (a, b) ∈ and (b, c) ∈ },

one can also write down the above conditions (E2 ) and (E3 ) as follows: (E2 ) −1 = , (E3 ) 2 ⊆ .

2.4 Complete Lattices and Equivalence Relations

43

Instead of (a, b) ∈ we often write ab or (a, b) or a = b (mod ) or a ∼ b (mod ) (one say: a is equals (or equivalent) to b modulo ). For every set A there are two trivial equivalence relations: ∇A (:= κ1 ) := A2 (all relation)

and

∆A (:= κ0 ) := {(a, a) | a ∈ A} (identity or diagonale). Theorem 2.4.3 (Eq(A); ⊆) is a complete lattice for every nonempty set A. One can determine the inﬁmum and the supremum of an arbitrary subset T := {i | i ∈ I} of Eq(A) as follows: T = i∈I i , T = i0 ,...,it ∈I;t<ℵ0 i0 2i1 2...2it . In particular, it holds for the equivalence relations κ and µ: κ ∨ µ = κ ∪ (κ2µ) ∪ (κ2µ2κ) ∪ (κ2µ2κ2µ) ∪ ... (i.e., (a, b) ∈ κ∨µ ⇐⇒ ∃c1 , ..., cn ∈ A : (∀i ∈ {1, 2, ..., n − 1} : (ci , ci+1 ) ∈ κ or (ci , ci+1 ) ∈ µ) and (a, b) = (c1 , cn )). Proof. The statement “(Eq(A); ⊆) is a complete lattice” follows from Theorem 2.4.1 and from the easily veriﬁable fact that the intersection of arbitrarily many equivalence relations on A is an equivalence relation on A again. The remaining statements one can easily check.

Deﬁnitions Let ∈ Eq(A) and a ∈ A. The set [a] (:= a/) := {x ∈ A | (a, x) ∈ } is called equivalence class of . The set A/ := {a/ | a ∈ A} of all equivalence classes of is called factor set of A by . The proof of the following theorem can be found in many books with sections on basic mathematics concepts. Theorem 2.4.4 Let A be a nonempty set. Then, for arbitrary ∈ Eq(A) and arbitrary a, b ∈ A: (a) A = a∈A a/; (b) a/ = b/ ⇐⇒ a/ ∩ b/ = ∅.

44

2 Lattices

Deﬁnitions A partition π of the set A is a set of nonempty subsets Ai (i ∈ I) of A (called blocks of π) with the properties: (Z1 ) i∈I Ai = A and (Z2 ) ∀i, j ∈ I : Ai ∩ Aj = ∅ or Ai = Aj . Let Π(A) be the set of all partitions of A. Regarding the contents, the following theorem, written with the help of concepts of the lattice theory, is identical to the well-known Main Theorem over equivalence relations. Theorem 2.4.5 If the partial order ≤ on Π(A) is deﬁned by π ≤ π :⇐⇒ ∀b ∈ π ∃b ∈ π : b ⊆ b , then the lattices (Eq(A); ⊆) and (Π(A); ≤) are isomorphic. An order-preserving mapping α of Eq(A) auf Π(A) can be deﬁned as follows: Let κ ∈ Eq(A) and π := {bi | i ∈ I} ∈ Π(A). Then α(κ) := {x/κ | x ∈ A} and

α−1 (π) := {(x, y) | ∃i ∈ I : {x, y} ⊆ bi }.

3 Hull Systems and Closure Operators

This chapter generalized observations that can be made when examining such similar concepts as subalgebra or linear hull of subsets of a vector space (or another closure operators of the classical algebra), to the concepts hull system and closure operator. Then, it is shown that these concepts deliver roughly the same.

3.1 Basic Concepts Deﬁnitions Let A be a nonempty set and let P(A) be the power set of A. A subset H of P(A) is a hull system (or a closed set system) on the set A if and only if (i) A ∈ H, and (ii) B := {b | b ∈ B} ∈ H for every nonempty subset B of H.1 The elements of H are called hulls. Examples 1) The set of all closed subsets of an algebra A is a hull system on A. 2) The set systems P(A), {A} and {A} ∪ {E ∈ P(A) | E ﬁnite} are hull systems on A for every nonempty set A. 3) The set of all equivalence relations Eq(A) is a hull system on A2 , A = ∅. Deﬁnitions A mapping C : P(A) −→ P(A) is called closure operator on A if and only if the following three conditions hold for all X, Y ⊆ A: (i) X ⊆ C(X) (extensivity), (ii) X ⊆ Y =⇒ C(X) ⊆ C(Y ) (monotony), (iii) C(C(X)) = C(X) (idempotency). 1

Let

{b | b ∈ B} :=

b∈B

b := {x | ∀b ∈ B : x ∈ b}.

46

3 Hull Systems and Closure Operators

Sets of the form C(X) are called closed, and we say that C(X) is generated of X. Example By Section 1.3, for every algebra, the operator [...]A is a closure operator. Deﬁnition If a closure operator C fulﬁlls the following condition in addition (iv) ∀X ⊆ A : C(X) = {C(Y ) | Y ⊆ X and Y is ﬁnite}, then it is called algebraic closure operator.

3.2 Some Properties of Hull Systems and Closure Operators Hull systems and closure operators are generally the same: Theorem 3.2.1 (Main Theorem on Hull Systems and Closure Operators) (a) Let H be a hull system on A. For all X ⊆ A let CH (X) := {h ∈ H | X ⊆ h}.

(3.1)

Then CH : P(A) −→ P(A), X → CH (X) is a closure operator on A, and the closed sets of CH are exactly the hulls of H. (b) Conversely, let C be a closure operator on A. Then HC := {C(X) | X ⊆ A}

(3.2)

is a hull system on A, and the hulls of HC are exactly the closed sets of A. (c) For every hull system H on A it holds H(CH ) = H,

(3.3)

and for every closure operator C on A it holds C(HC ) = C.

(3.4)

Proof. (a): The extensivity of the operator deﬁned by (3.1) follows directly from the deﬁnition of CH (X). If X ⊆ Y , we have U := {h ∈ H | X ⊆ h} ⊇ V := {h ∈

3.2 Some Properties of Hull Systems and Closure Operators

47

H | Y ⊆ h}. Thus, CH (X) = {h ∈ H | X ⊆ h} ⊆ CH (Y ) = {h ∈ H | Y ⊆ h}. Therefore CH is monotone. To prove the idempotency of CH , ﬁrst we show Y ⊆ CH (Z) =⇒ CH (Y ) ⊆ CH (Z).

(3.5)

Y ⊆ CH (Z) implies {h ∈ H | Z ⊆ h} ⊆ {h ∈ H | Y ⊆ h} and from this CH (Y ) ⊆ CH (Z), i.e., (3.5) holds. Choosing in (3.5) Y = CH (Z), then CH (CH (Z)) ⊆ CH (Z) follows from (3.5). By the extensivity of CH we have CH (Z) ⊆ CH (CH (Z)). Thus, CH is also idempotent and therefore CH is a closure operator. Next we prove (3.6) X ∈ H ⇐⇒ CH (X) = X. The statement “=⇒” in (3.6) follows directly from the deﬁnition of CH . “⇐=” follows from the deﬁnition of CH and the assumption (ii) (see Section 3.1). Thus (a) is proven. (b): Let C : P(A) −→ P(A) be a closure operator and HC is deﬁned as in (3.2). The following equivalences are easy to check: ∀X ⊆ A (X ∈ HC ⇐⇒ C(X) = X).

(3.7)

The deﬁnition of C implies ⊆ A, i.e., A = C(A) ∈ HC . For arbitrary A ⊆ C(A) B∈B B ⊆ HC we have further B (= X∈B X) ∈ HC to show. Since for every we have B = C(B) and X∈B X ⊆ B, the extensivity ofC implies C( X∈B X) ⊆ C(B) = B for every B ∈ B. Then, by this we have C( X∈BX) ⊆ B∈B B. On the other hand X∈B X ⊆ C( X∈B X) also holds. Thus, C( B) = B ∈ HC is valid. (c): By using (3.7) and (3.6) it follows H(CH ) = H from X ∈ H(CH ) ⇐⇒ CH (X) = X ⇐⇒ X ∈ H. To prove C(HC ) = C, let Y ⊆ A be arbitrary. We have C(HC ) (Y ) = C(Y ) to show. By deﬁnition it holds that C(HC ) (Y ) = {Z | Z ∈ HC , Y ⊆ Z}. Consequently, by (3.7), C(HC ) (Y ) = {Z | C(Z) = Z, Y ⊆ Z}. Because of C(Y ) = C(C(Y )), we have {Z | C(Y ) = Z, Y ⊆ Z} ⊆ C(Y ). On the other hand it follows from Y ⊆ Z = C(Z) that C(Y ) ⊆ C(Z) = Z. This implies {Z | C(Z) = Z, Y ⊆ Z} ⊇ C(Y ). By summarizing, we obtain {Z | C(Y ) = Z, Y ⊆ Z} = C(Y ).

Theorem 3.2.2 (a) For every algebra A = (A; F ), [...]A is an algebraic closure operator on A. (b) Conversely, for every algebraic closure operator C on the set A there is a set F of operations on A such that {C(X) | X ⊆ A} is the set of all subalgebras of A = (A; F ). Proof. (a) follows from the description of the closure [T ]A of an arbitrary set T ⊆ A (see Lemma 1.3.1).

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3 Hull Systems and Closure Operators

(b): Let C be an algebraic closure operator. We have to deﬁne a certain set of operations F on A such that C(X) = [X]F holds for arbitrary X ⊆ A. For every ﬁnite n-element subset E := {e1 , ..., en } of A and for every e ∈ C(E) we deﬁne fE;e as follows: e, if {x1 , ..., xn } = E, n (x1 , ..., xn ) := fE;e x1 otherwise. We summarize the functions of type fE;e to the set F : n {fE;e | |E| = n, E ⊆ A, e ∈ C(E)}. F := n∈N

Then, by construction, we have C(E) = [E]F . Since C is an algebraic closure operator, from this C(X) = [X]F follows for every subset X ⊆ A.

Theorem 3.2.3 (a) Let C be a closure operator on A and let LC := {X ⊆ A | C(X) = X}. Then LC is a complete lattice with

C(Ai ) = C(Ai ) i∈I

and

i∈I

C(Ai ) = C(

i∈I

Ai ).

i∈I

(b) Every complete lattice is isomorphic to the lattice LC of all closed subsets of a certain set A with a closure operator C. Proof. (a) is easy to check. (b) Let L be a complete lattice. For X ⊆ L we deﬁne C(X) := {a ∈ L | a ≤ sup X}. Then C is a closure operator on L and the mapping α : L −→ LC , a → {b ∈ L | b ≤ a} is an isomorphic mapping from L into LC .

Deﬁnitions Let X, Y ⊆ A, let H be a hull system on A and let h ∈ H. • X is called a generating system (or a generating set) of h, if h = CH (X) holds. • h is ﬁnitely generated, if there is a ﬁnite set E ⊆ h with h = CH (E).

3.2 Some Properties of Hull Systems and Closure Operators

49

• X is CH -independent (or brieﬂy: independent), if for all x ∈ X it holds x ∈ CH (X\{x}). • X is called a CH -basis (or brieﬂy: basis) of h, if X is a CH -independent generating set of h. From general theorems about hull systems and closure operators, one gets the following statements (about algebras), which we will prove, only for the concrete hull system S(A) and the corresponding closure operator [...]A . Theorem 3.2.4 (a) Let A be a ﬁnitely generated algebra. Then, for every generating set T of A, there is a ﬁnitely generating set T ⊆ T of A. (b) Every ﬁnitely generated algebra has a ﬁnite basis. (c) Every algebra, which has not a basis, is not ﬁnitely generated. (d) Let A be an algebra that has a subalgebra A1 with countable inﬁnite basis. Then S(A) has a cardinality that is greater than or equal to the cardinality of the continuum. Proof. (a): Let A = (A; F ) be ﬁnitely generated. Then, there is a ﬁnite generating set B0 = {b1 , b2 , ..., br } ⊆ A of the universe A with [B0 ]A = A. Now let T ⊆ A be an arbitrary generating set of the algebra A. Because of [T ] = A and because of Lemma 1.3.2, one can get an arbitrary element a ∈ A from elements of T by successively applying the operations of F . In particular, this also holds for the elements b1 , ..., br of B0 . Obviously, the set of all elements of T needed for this is a ﬁnitely generating set T ⊆ T of A. (b): Let A = (A; F ) be a ﬁnitely generated algebra. Then by (a) there exists with every generating set T of A a ﬁnite subset T ⊆ T that is already a generating set for the given algebra. If t ∈ [T \{t}] for all t ∈ T , then T is a ﬁnite basis. Otherwise, if t1 ∈ [T \{t1 }], T := T \ {t1 } is a generating set for A. If now for all t ∈ T it holds t ∈ [T \{t}], then T is basis for A. In opposite case, we deal with T like with T by going over to T , etc. By the ﬁniteness of T , this process breaks oﬀ with the construction of a basis for A, whereby (b) is proven. (c) follows from (b). (d): Let T := {t1 , t2 , ...} be a countable inﬁnite basis of the subalgebra A1 of A. By deﬁnition, for all subsets T ⊂ T , we have, then, that a ∈ T \T implies a ∈ [T ]. Therefore, diﬀerent subsets T and T of T deﬁne diﬀerent subalgebras [T ] and [T ]. Consequently, the cardinality of the set S(A) is not less than the cardinality of the set P(T ). Now, it is well-known that P(T ) has the cardinality of continuum. Thus (c) holds.

4 Homomorphisms, Congruences and Galois Connections

It is clear that abstract algebraic properties of an algebra do not change if the elements and the operations of the algebra get other names. One can describe this fact mathematically using the concept isomorphism or isomorphic mapping. A generalization of this concept is homomorphism (or homomorphic mapping), with which one can describe similarities of algebras. Congruences and factor algebras are aids to determine homomorphisms. A Galois connection is a pair of mappings (σ, τ ) between two power sets P(A) and P(B), where the mappings τ and σ are antitone and extensive (see Section 4.4). With the aid of these mappings, one can deﬁne hull operators on A and on B. Then, these operators deﬁne closed set systems (⊆ P(A) or ⊆ P(B), respectively). Part of the usefulness of such Galois connections resides in the possibility of drawing conclusions about a closed set system on the basis of information about the other system. The Galois connection between function algebras and relation algebras, which we study in Chapter 2 of Part II, is an important aid in the solution of the completeness problem of the many-valued logic (see Part II, Chapter 5 and 6).

4.1 Homomorphisms and Isomorphisms Deﬁnitions Let (A; F ) and (B; G) be two universal algebras of the same type and F = {fi | i ∈ I} and G = {gi | i ∈ I} with afi = agi for every i ∈ I. We call the mapping ϕ : A −→ B a homomorphism or a homomorphic mapping from A into B, if ϕ is compatible with all f ∈ F , g ∈ G; i.e., it holds for all i ∈ I and all a1 , ..., aafi ∈ A: ϕ(fi (a1 , ..., aafi )) = gi (ϕ(a1 ), ϕ(a2 ), ..., ϕ(aafi )), if afi > 0, and ϕ(fi ) = gi for all fi ∈ F with afi = 0.

52

4 Homomorphisms, Congruences, and Galois Connections

If ϕ in addition is bijective from A onto B, then ϕ is called an isomorphism or an isomorphic mapping from A onto B. A homomorphism from an algebra A into A is called an endomorphism of A. An isomorphism from A onto A is an automorphism of A. Obviously, ϕ−1 is an isomorphism from B onto A, if ϕ is an isomorphism from A onto B. Therefore, we can say: “A and B are isomorphic”, we write in this case A∼ =B (read: “A isomorphic B”) for the identiﬁcation of this fact. It is easy to see that the composition of homomorphic mappings ϕ1 2ϕ2 (ϕ1 : A1 −→ A2 , ϕ2 : A2 −→ A3 ) is a homomorphism from A1 = (A1 ; F1 ) into A3 = (A3 ; F3 ). With the help of this property, one can show that the relation “∼ =” is an equivalence relation on sets of algebras. The following lemma summarizes further elementary properties of homomorphic mappings. Lemma 4.1.1 Let ϕ be a homomorphic mapping from A = (A; F ) into B = (B; G). Then it holds: (a) ϕ(A) is closed in respect to B; i.e., (ϕ(A); G) is a subalgebra of B, which is called homomorphic image of A by ϕ. (b) If A is a subalgebra of A, then (ϕ(A ); G) is a subalgebra of (ϕ(A); G). (c) If T ⊆ A is a generating system of A, then ϕ(T ) is a generating system of ϕ(A). (d) If (B ; G) is a subalgebra of (ϕ(A); G), then the so-called inverse image (ϕ−1 (B ); F ) with ϕ−1 (B ) := {a ∈ A | ϕ(a) ∈ B } is a subalgebra of A. (e) Let ψ be a homomorphism from A into B, which is identical with ϕ on a generating system of A. Then, ϕ(a) = ψ(a) for each a ∈ A.

4.2 Congruence Relations and Factor Algebras of Algebras Deﬁnitions A congruence relation (brieﬂy: congruence) or a kernel of a homomorphic mapping ϕ of an algebra A = (A; F ) is an equivalence relation κϕ on A that is induced by the homomorphism ϕ from A into B, that is, for all a, a ∈ A it holds: (a, a ) ∈ κϕ :⇐⇒ ϕ(a) = ϕ(a ). In the following, we also denote the relation κϕ with Ker ϕ. Let Con(A) be the set of all congruences of A. Examples The identity mapping idA with idA : A −→ A, a → a

4.2 Congruence Relations and Factor Algebras of Algebras

53

and the mapping ϕC with ϕC : A −→ {c}, a → c from A onto the 1-element algebra of the same type ({c}; G), where g(c, c, ..., c) = c for all g ∈ G, if ag > 0, and g = c, if ag = 0 holds, induce two so-called trivial congruences, the zero-congruence κ0 and the all-congruence (or one-congruence) κ1 : κ0 := {(a, a) | a ∈ A}, κ1 := A × A. Deﬁnition An algebra, which has only both congruences κ0 and κ1 , is called simple. Examples for simple algebras are groups of prime order and the ﬁelds. As the following lemma demonstrates, it is possible to describe the concept “congruence” without use of the concept “homomorphism”. Lemma 4.2.1 An equivalence relation κ on A is a congruence on the algebra A = (A; F ) if and only if it is compatible with all not nullary operations of F ; i.e., if for all f n ∈ F with n = af > 0 and arbitrary a1 , ..., an , a1 , ..., an ∈ A it holds: {(a1 , a1 ), ..., (an , an )} ⊆ κ =⇒ (f (a1 , ..., an ), f (a1 , ..., an )) ∈ κ. Proof. Let F := {fini | i ∈ I}, where I denotes a certain index set. “=⇒”: Let κ be a congruence on A. Then, by the deﬁnition of a congruence, there exists an algebra B := (B; G), where G := {gini | i ∈ I}, and a homomorphic mapping ϕ : A −→ B with the property

κ = {(a, a ) | ϕ(a) = ϕ(a )}.

We have to show that κ is compatible with all f ∈ F \ F 0 . To show this let f := fini ∈ F \F 0 be arbitrary. For the purpose of simpliﬁcation, we put n := ni and g := gini . Furthermore, let (a1 , a1 ), ..., (an , an ) ∈ κ be arbitrary. Since ϕ is a homomorphism, then we have

54

4 Homomorphisms, Congruences, and Galois Connections ϕ(f (a1 , ..., an )) = g(ϕ(a1 ), ..., ϕ(an )) = g(ϕ(a1 ), ..., ϕ(an )) = ϕ(f (a1 , ..., an )).

Consequently, (f (a1 , ..., an ), f (a1 , ..., an )) ∈ κ. “⇐=”: Conversely, let the equivalence relation κ on A be compatible with all operations of the algebra A = (A; F ). We have to show: There is an algebra B = (B; G) of the same type as A and a homomorphic mapping ϕ : A −→ B with the property {(x, y) | ϕ(x) = ϕ(y)} = κ. To construct the algebra B, we deﬁne: a/κ := {x ∈ A | (x, a) ∈ κ}

(a ∈ A)

(i.e., a/κ is the notation of the equivalence class in which lies a) and B := A/κ := {a/κ | a ∈ A} (i.e., B is the set of all equivalence classes of κ). Now, we are able to deﬁne the operations gini on the set B as follows: gini (a1 /κ, ..., ani /κ) := fini (a1 , ..., ani )/κ

(4.1)

(i.e., gini (a1 /κ, ..., ani /κ) is exactly the equivalence class in which fini (a1 , ..., ani ) lies). The above deﬁnition is possible, since fi is compatible with κ. Detailed: Choosing (a1 , a1 ), ..., (ani , ani ) ∈ κ, then it holds (a1 /κ, ..., ani /κ) = (a1 /κ, ..., ani /κ) and fini (a1 , ..., ani )/κ = fini (a1 , ..., ani )/κ. We put G := {gini | i ∈ I}. Then the mapping ϕ : A −→ B, a → a/κ is a homomorphic mapping from A onto B := (B; G), since by deﬁnition of κ ϕ(fini (a1 , ..., ani )) = fini (a1 , ..., ani )/κ holds and (by the compatibility of κ with fini and by the deﬁnition (4.1) of gi ∈ G further fini (a1 , ..., ani )/κ = gini (a1 /κ, ..., ani /κ) = gini (ϕ(a1 ), ..., ϕ(ani )) holds, which implies ϕ(fini (a1 , ..., ani )) = gini (ϕ(a1 ), ..., ϕ(ani )). Furthermore, by deﬁnition of ϕ and by the properties of an equivalence relation, it holds: {(x, y) ∈ A × A | ϕ(x) = ϕ(y)} = {(x, y) ∈ A × A | x/κ = y/κ} = κ.

4.2 Congruence Relations and Factor Algebras of Algebras

Deﬁnitions Lemma 4.2.1,

55

The above algebra, which was designed in the proof of the (A/κ; G)

of the so-called congruence classes of A modulo κ is called factor algebra of (A; F ) (or quotient algebra). The homomorphic mapping ϕ : A −→ A/κ, a → a/κ, deﬁned in the proof of Lemma 4.2.1, is called natural homomorphism (or quotient homomorphism) from A onto A/κ. We make following use of our agreement from the ﬁrst chapter according to which the operations of algebras of the same type are described equal. Only then, if distinctions are necessary at the denotation, do we indicate the operations. After these preparations, we can prove the following theorem (in generalization of analogous theorems over groups, rings, ...): Theorem 4.2.2 (General Homomorphism Theorem) For each homomorphism ϕ from an algebra A := (A; F ) into an algebra of the same type B := (B; F ) the algebra ϕ(A) := (ϕ(A); F ) is isomorphic to the factor algebra A/κϕ := (A/κϕ ; F ), where κϕ = {(a, a ) ∈ A × A | ϕ(a) = ϕ(a )}. Proof. It suﬃces to check that α : ϕ(A) −→ A/κϕ , ϕ(a) → a/κϕ is an isomorphism. Since a/κϕ = a /κϕ =⇒ (a, a ) ∈ κϕ =⇒ ϕ(a) = ϕ(a ), α is a mapping. Obviously, by deﬁnition, α is a surjection. The injectivity of α follows from the following: α(ϕ(a)) = α(ϕ(a )) =⇒ a/κϕ = a /κϕ =⇒ (a, a ) ∈ κϕ =⇒ ϕ(a) = ϕ(a ) (a, a ∈ A). Thus, α is bijective. To prove that α is a homomorphism, let f n ∈ F and let ϕ(a1 ), ..., ϕ(an ) ∈ ϕ(A) be arbitrary. Then it holds:

56

4 Homomorphisms, Congruences, and Galois Connections α(fϕ(A) (ϕ(a1 ), ..., ϕ(an ))) = α(ϕ(fA (a1 , ..., an )))

(since ϕ is a homomorphism)

= f (a1 , ..., an )/κϕ

(by deﬁnition of α)

= fA/κϕ (a1 /κϕ , ..., an /κϕ )

(since κϕ is compatible with f )

= fA/κϕ (α(ϕ(a1 )), ..., α(ϕ(an ))) (by deﬁnition of α), Thus our bijective mapping α is an isomorphism. The next lemma summarizes some elementary properties of congruences. The proof for this lemma is left to the reader. Lemma 4.2.3 It holds: (a) The intersection of arbitrary many congruences of an algebra A is also a congruence on A. (b) If κ is a congruence on (A; F ) and ϕ is a homomorphism deﬁned over A with κϕ ⊆ κ, then ϕ(κ) := {(ϕ(a), ϕ(a )) | (a, a ) ∈ κ} is a congruence on (ϕ(A); F ). (c) If π is a congruence on (ϕ(A); F ) and ϕ is a homomorphism deﬁned on (A; F ), then ϕ−1 (π) := {(a, a ) ∈ A × A | (ϕ(a), ϕ(a )) ∈ π} is a congruence on (A; F ), which κϕ includes. (d) If κ is a congruence on A = (A; F ) and B = (B; F ) is a subalgebra of A, then κ|B := {(b, b ) ∈ κ | b, b ∈ B} is a congruence on B. (e) The set ConA of all congruence relations of an algebra A is a complete lattice in respect to the inclusion ⊆ with inf{κj | j ∈ J} = j∈J κj , sup{κj | j ∈ J} =< j∈J κj >ConA , where {κj | j ∈ J} ⊆ ConA and < j∈J κj >ConA is the intersection of all congruences which contain j∈J κj .

4.3 Examples for Congruence Relations and Some Homomorphism Theorems In the following, we characterize the congruence relations by groups and rings more closely. In addition, we give the special homomorphism theorems that follow from these characterizations. 4.3.1 Congruences on Groups Let G = (G; ◦,−1 , e) be a group. A subgroup N of G is called a normal subgroup iﬀ

4.3 Examples for Congruence Relations and Some Homomorphism Theorems

57

∀x ∈ G : x ◦ N = N ◦ x holds. Let NG be the set of all normal subgroups of G. If N is a normal subgroup of G, one also writes N G. The following connections between normal subgroups and congruences on G can easily be checked: (a) For every κ ∈ ConG, e/κ is a normal subgroup of G, and for arbitrary a, b ∈ G it holds: (a, b) ∈ κ ⇐⇒ a ◦ b−1 ∈ e/κ. (b) If N is a normal subgroup then one can obtain by κN := {(a, b) | a ◦ b−1 ∈ N } a congruence on G with e/κ = N . Consequently, the mapping α : ConG −→ NG, κ → e/κ is an order-preserving bijection. Since for every homomorphism ϕ from G into a group G the neutral element e is mapped to the neutral element of G , the properties (a), (b), and Theorem 4.2.2 imply the following Theorem 4.3.1.1 (Homomorphism Theorem for Groups) For every homomorphism ϕ from a group G into a group G there exists as so-called kernel (notation: ker ϕ) a normal subgroup K of G, which consists of all the elements of G which are mapped to the neutral element of G . The group G is isomorphic to the factor group G/K = ({x ◦ K | x ∈ G}; ◦,−1 , K)), where (x ◦ K) ◦ (y ◦ K) := (x ◦ y) ◦ K for arbitrary x, y ∈ G. Conversely, if K is a normal subgroup of G, then K is the kernel of the natural homomorphism from G onto the factor group G/K (∀x ∈ G : x → x ◦ K). We notice that the concept kernel of a group homomorphism diﬀers from the general term kernel of a homomorphism. The corresponding is valid for the concept kernel from Theorem 4.3.2.1.

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4 Homomorphisms, Congruences, and Galois Connections

4.3.2 Congruences on Rings Let R = (R; +, −, 0, ·) be a ring. A subgroup I of (R; +, −, 0) is called an ideal of R iﬀ ∀x ∈ R : x · I ⊆ I ∧ I · x ⊆ I holds. Let IR be the set of all ideals of R. Connections between congruences on R and ideals of R are the following: For every κ ∈ ConR, 0/κ is an ideal of R and we have for arbitrary a, b ∈ R: (a, b) ∈ κ ⇐⇒ a − b ∈ 0/κ. If I is an ideal of R then one can obtain by κI := {(a, b) | a − b ∈ I} a congruence on R with 0/κI = I. Consequently, the mapping ConR −→ IR, κ → 0/κ is an order-preserving bijection. Further it holds: Theorem 4.3.2.1 (Homomorphism Theorem for Rings) For every homomorphism ϕ from a ring R into a ring R belongs as a kernel (notation: ker ϕ) an ideal I of R, which consists of all the elements of R which are mapped to the zero element of R . The ring R is isomorphic to the residue class ring R/I = ({x + I | x ∈ R}; +, −, I, ·), where (x + I) + (y + I) := (x + y) + I and (x + I) · (y + I) := (x · y) + I for arbitrary x, y ∈ R. Conversely, if I is an ideal of R then I is the kernel of the natural homomorphism from R onto R/I (∀x ∈ R : x → x + I). It is obvious idea, according to a homomorphism theorem for rings, to ﬁnd also a homomorphism theorem for ﬁelds. The next theorem shows, however, that such a theorem is only a trivial and special case of the general homomorphism theorem. Theorem 4.3.2.2 A ring R, which is also a ﬁeld, has only the two trivial ideals {0} and R. In other words: A ﬁeld has only the trivial congruences κ0 and κ1 . Proof. Let R be a ﬁeld and let κ be a congruence of R, which is diﬀerent from κ0 . Denote I the ideal belonging to κ. Since κ = κ0 , there is certain a, b ∈ R with a = b and (a, b) ∈ κ. Consequently, we have (c, 0) := (a − b, b − b) ∈ κ. Hence, I contains the invertable element c. Since I also contains all elements r · c for every r ∈ R (specially, r = r · c−1 with arbitrary r ∈ R) we have I = R. Thus, κ = κ1 .

4.4 Galois Connections

59

4.4 Galois Connections Deﬁnitions A Galois connection (or a Galois correspondence) between the sets A and B is a pair (σ, τ ) of mappings σ : P(A) −→ P(B) and τ : P(B) −→ P(A), such that for all X, X ⊆ A and all Y, Y ⊆ B the following conditions are fulﬁlled:

X ⊆ X =⇒ σ(X) ⊇ σ(X ) (antitony) (GC1) Y ⊆ Y =⇒ τ (Y ) ⊇ τ (Y ) X ⊆ τ (σ((X))) (GC2) (extensivity). Y ⊆ σ(τ ((Y ))) Let (P ; ≤) be a poset. The dual order ≤δ to the order ≤ is deﬁned by x ≤δ y :⇐⇒ y ≤ x. (P ; ≤) is called dual isomorphic (or antiisomorphic) to (Q; ≤), iﬀ (P ; ≤) is isomorphic to (Q; ≤δ ). The bijective mapping appertaining to this fact is called dual isomorphism (or antiisomorphism) . Theorem 4.4.1 Let the pair (σ, τ ) of mappings σ: P(A) −→ P(B) and τ : P(B) −→ P(A) be a Galois connection between A and B. Then it holds: (a) The mappings στ := σ2τ : P(A) −→ P(A) and τ σ := τ 2σ : P(B) −→ P(B) are hull operators on A or B, respectively. (b) The στ -closed sets are exactly the sets of the form τ (Y ), Y ⊆ B. The τ σ-closed sets are exactly the sets of the form σ(X), X ⊆ A. (c) Let Hστ and Hτ σ be the hull systems that are assigned to στ and τ σ, respectively. Then the lattices (Hστ ; ⊆) and (Hτ σ ; ⊆) are dual isomorphic, and σ and τ are inversely dual isomorphisms to each other of these lattices. Proof. (a): The extensivity and the monotony of στ and τ σ immediately follow from (GC1) and (GC2), respectively. Thus, for all X ⊆ A we have X ⊆ τ (σ(X)) and therefore σ(X) ⊇ σ(τ (σ(X))). On the other hand, σ(X) ⊆ (τ σ)(σ(X)) = σ(τ (σ(X))) follows from (GC2). Consequently, we have

60

4 Homomorphisms, Congruences, and Galois Connections σ(X) = σ(τ (σ(X)))

(4.2)

τ (Y ) = τ (σ(τ (Y ))).

(4.3)

and analogously Then the equations τ (σ(X)) = τ (σ(τ (σ(X)))) and σ(τ (Y )) = σ(τ (σ(τ (Y )))) follow from these. Thus, στ and τ σ are idempotent. (b): For a στ -closed set X we have X = τ (σ(X)); that is, X has the form X = τ (Y ) with Y := σ(X) ⊆ B. Conversely, a set of the form X := τ (Y ), Y ⊆ B, is στ -closed by (4.3). Analogously, one can prove the assertion for τ σ-closed sets. (c): Because of (b) and Theorem 3.2.1 it holds Hστ = {τ (Y ) | Y ⊆ B} and Hτ σ = {σ(X) | X ⊆ A}. Thus, we have σ(Hστ ) := {σ(τ (Y )) | Y ⊆ B} = Hτ σ and τ (Hτ σ ) := {τ (σ(X)) | X ⊆ A} = Hστ . By (GC1) σ and τ are antitone, and therefore the restrictions of these mappings to Hστ and Hτ σ have also this property. It follows from the idempotence of στ that στ on Hστ is the identical mapping. Analogously, one can see that τ σ is also the identical mapping on Hτ σ . Therefore, the mappings σ : Hστ −→ Hτ σ and τ : Hτ σ −→ Hστ are bijective mappings and it holds σ −1 = τ . Consequently, σ, τ are isomorphisms of the lattices (Hστ ; ⊆) and (Hτ σ ; ⊆δ )

An example of a Galois connection concludes this section. Further examples can be found in [Den-E-W 2004]. Theorem 4.4.2 Let A, B be nonempty sets and let R ⊆ A × B with R = ∅. The mappings σ : P(A) −→ P(B), τ : P(B) −→ P(A) are deﬁned by σ(X) := {y ∈ B | ∀x ∈ X : (x, y) ∈ R}, τ (Y ) := {x ∈ A | ∀y ∈ Y : (x, y) ∈ R}. Then the pair (σ, τ ) is a Galois connection between A and B. Proof. Because of symmetry of the assumptions, it suﬃces to show that σ is an antitone and τ σ is an extensive mapping. Let X ⊆ X ⊆ A. Then for every y ∈ σ(X ) we have: (x, y) ∈ R for all x ∈ X . Thus (by X ⊆ X ) we have also (x, y) ∈ R for all x ∈ X. Therefore σ(X ) ⊆ σ(X) holds. The inclusion X ⊆ τ (σ(X)) follows directly from τ (σ(X)) = {x ∈ A | ∀y ∈ σ(X) : (x, y) ∈ R} and from the deﬁnition of σ(X).

5 Direct and Subdirect Products

With the help of direct and subdirect products, it is possible to form new algebras with larger universes from given algebras. Of these constructions one immediately asks the following questions: Which algebras are smallest “constituents” of given algebras? How can one reduce a given algebra to its smallest “constituents”? First, we will show that every ﬁnite algebra is isomorphic to a direct product of directly irreducible algebras. Then, we prove that every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras.

5.1 Direct Products First, we consider direct products of two algebras. Deﬁnitions Let B = (B; F ) and C = (C; F ) be algebras of the same type τ. The algebra A := B × C of type τ is called direct product of the algebras B and C, iﬀ B × C is the universe of the algebra A and the operations of A are deﬁned as follows: If f ∈ F is nullary, then let fA := (fB , fC ). If af ≥ 1, let fA deﬁned as follows: fB×C ((b1 , c1 ), (b2 , c2 ), ..., (baf , caf )) := (fB (b1 , ..., baf ), fC (c1 , ..., caf )). Obviously, B and C are homomorphic images of the algebra B × C, since the (so-called projection-)mappings pr1 : B × C −→ B, (b, c) → pr1 (b, c) := b and pr2 : B × C −→ C, (b, c) → pr2 (b, c) := c are homomorphisms from B × C onto B or C, respectively. The kernels of these projection mappings (congruences on B × C)

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5 Direct and Subdirect Products

Ker pri := {((b, c), (b , c )) ∈ (B × C)2 | pri (b, c) = pri (b , c )} (i = 1, 2) are distinguished from other congruences by certain properties (see Theorem 5.1.2). We can describe these properties with the help of the following Deﬁnition Two equivalence relations κ, µ ∈ Eq(A) are called permutable, if κ2µ = µ2κ holds; i.e., ∀x, z ∈ A : (∃y ∈ A : (x, y) ∈ κ ∧ (y, z) ∈ µ) ⇐⇒ (∃y ∈ A : (x, y ) ∈ µ ∧ (y , z) ∈ κ). The following lemma gives some other deﬁnition possibilities for the above concept. In this lemma and in the following theorems, (Eq; ⊆), by Theorem 2.4.3, is a complete lattice and operations ∧ = ∩ (“inﬁmum”) and ∨ (“supremum”) are deﬁned on Eq (see Chapter 2). Lemma 5.1.1 For κ, µ ∈ Eq(A) the following statements are equivalent: (a) (b) (c) (d) (e)

κ and µ are permutable κ2µ ⊆ µ2κ µ2κ ⊆ κ2µ κ ∨ µ = κ2µ µ ∨ κ = κ2µ

Proof. Because of the commutativity of ∨, we have (d)⇐⇒(e). The equivalence (b)⇐⇒(c) is a conclusion from (a)⇐⇒(b). Thus, we have to show that (a)⇐⇒(b) and (a)⇐⇒(e) holds. (a)=⇒(b) is trivial. (b)=⇒(a): Let κ2µ ⊆ µ2κ. Then (by (κ2µ)−1 = µ−1 2κ−1 and by the symmetry of κ and µ): (κ2µ)−1 ⊆ (µ2κ)−1 =⇒ µ−1 2κ−1 ⊆ κ−1 2µ−1 . =µ2κ

=κ2µ

Hence, κ2µ = µ2κ. (a)=⇒(e): Let κ2µ = µ2κ. (e) is shown, if it is proven that κ2µ is the smallest equivalence relation on A, which contains κ ∪ µ. Because of ∀(x, y) ∈ κ (( (x, y) ∈ κ ∧ (y, y) ∈ µ) =⇒ (x, y) ∈ κ2µ), ∀(x, y) ∈ µ (( (x, x) ∈ κ ∧ (x, y) ∈ µ) =⇒ (x, y) ∈ κ2µ) we have κ ∪ µ ⊆ κ2µ. κ2µ is reﬂexive, since κ (or µ) is reﬂexive. The symmetry of κ2µ follows from (κ2µ)−1 = µ−1 2κ−1 = µ2κ = κ2µ. Because of (κ2µ)2(κ2µ) = κ2 (µ2κ) 2µ = (κ2κ) 2 (µ2µ) ⊆ κ2µ =κ2µ

⊆κ

⊆µ

κ2µ is also transitive. Thus, κ2µ is an equivalence relation on A. κ2µ is the smallest equivalence relation of Eq(A), which contains κ ∪ µ, since every

5.1 Direct Products

63

other equivalence relation, which contains κ ∪ µ, also contains the transitive closure of κ ∪ µ and therefore contains κ2µ. (e)=⇒(a): Let κ ∨ µ = κ2µ. Since κ ∨ µ is an equivalence relation, the compatibility of κ and µ follows from κ2µ = κ ∨ µ = (κ2µ)−1 = µ−1 2κ−1 = µ2κ.

Theorem 5.1.2 For algebras B and C of the same type and the projection mappings pr1 : B × C −→ B and pr2 : B × C −→ C it holds: (a) Ker pr1 ∧ Ker pr2 = κ0 (b) Ker pr1 ∨ Ker pr2 = κ1 (c) Ker pr1 and Ker pr2 are permutable Proof. Let ((b, c), (b , c )) ∈ Ker pr1 ∩ Ker pr2 be arbitrary. Then, pri (b, c) =

pri (b , c ) for i = 1, 2. This means, however, that b = b and c = c . Therefore, (a) is shown. For arbitrary b, b ∈ B, c, c ∈ C we have ( (b, c), (b, c ) ) ∈ Ker pr1 ∧ ( (b, c ), (b , c ) ) ∈ Ker pr1 . Hence, ( (b, c), (b , c ) ) ∈ κ2µ for all b, b ∈ B and c, c ∈ C and thus (Ker pr1 )2(Ker pr2 ) = κ1 . With the help of Lemma 5.1.1, the assertions (b) and (c) follow from this.

Now we will examine the conditions under which an algebra is a direct product of two other smaller algebras. Theorem 5.1.2 provides instructions on how to proceed. Theorem 5.1.3 Let A = (A; F ) be an algebra and let κ, µ ∈ ConA be two congruence relations with the following three properties: (a) κ ∧ µ = κ0 (b) κ ∨ µ = κ1 (c) κ and µ are permutable Then A is isomorphic to the direct product of A/κ and A/µ. An isomorphism ϕ: A −→ A/κ × A/µ is deﬁned by ∀a ∈ A : ϕ(a) := (a/κ, a/µ). Proof. ϕ is injective: Let ϕ(a) = ϕ(b). Then, a/κ = b/κ and a/µ = b/µ. From these (a, b) ∈ κ ∧ µ follows. Thus a = b by (a). ϕ is surjective: By (b) and (c) for every pair a, b there is a c ∈ A with (a, c) ∈ κ and (c, b) ∈ µ. Consequently, (a/κ, b/µ) = (c/κ, c/µ) = ϕ(c). ϕ is an isomorphism: For all fA ∈ F (afA =: n) and arbitrary a1 , ..., an ∈ A it holds ϕ(fA (a1 , ..., an )) = (fA (a1 , ..., an )/κ, fA (a1 , .., an )/µ) = (fA/κ (a1 /κ, ..., an /κ), fA/µ (a1 /µ, ..., an /µ)) = fA/κ×A/µ (ϕ(a1 ), ..., ϕ(an )).

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Deﬁnition An algebra A is called directly irreducible, iﬀ A ∼ = B×C implies |B| = 1 or |C| = 1 for all algebras B and C. Example Obviously, every ﬁnite algebra A with |A| ∈ P is directly irreducible. One ﬁnds further examples after Theorem 5.1.4. Theorem 5.1.4 An algebra A is directly irreducible iﬀ κ0 and κ1 are the only one pair of congruences of ConA, which satisfy the conditions (a)–(c) of Theorem 5.1.3. Proof. Let A be directly irreducible. Further, κ, µ ∈ ConA fulﬁll the conditions (a)–(c) of Theorem 5.1.3. Then, by Theorem 5.1.3, we have A ∼ = A/κ × A/µ. Thus, w.l.o.g., |A/κ| = 1. From this, κ = κ1 follows, and then µ = κ0 by (a). Conversely, Let κ0 and κ1 be the only one pair of congruences of A with the properties (a)–(c) of Theorem 5.1.3, and let A ∼ = B × C. Obviously, then κ0 and κ1 are the only one pair of congruence relations on B × C with (a)–(c). By Theorem 5.1.2 the kernels of the projection mappings pr1 and pr2 fulﬁll (a)–(c). Thus Ker pr1 = κ0 or Ker pr2 = κ0 holds and we have |C| = 1 or |B| = 1, respectively.

With the help of Theorem 5.1.4, one can prove the following statements:1 (a) Every simple algebra A; i.e., every algebra A with Con A = {κ0 , κ1 }, is directly irreducible. (b) If A is a Boolean algebra then A is directly irreducible ⇐⇒ |A| ≤ 2. (c) The residue class group (Zn ; +, −, 0) is directly irreducible iﬀ n is a prime number power. (d) A vector space V := (V ; +, −, K, 0) over the ﬁeld K is directly irreducible iﬀ |V | = 1 or V is 1-dimensional. One can generalize our deﬁnition of the direct product of two algebras in an obvious way for ﬁnite many algebras of the same type. As the next deﬁnitions demonstrate, it is also possible to form direct products of arbitrarily many algebras of the same type. Deﬁnitions The Cartesian product Πj∈J Aj of the sets Aj (j ∈ J) is the set of all mappings α from J into j∈J Aj with α(j) ∈ Aj for all j ∈ J. We write down the elements of Πj∈J Aj in the form (xj | j ∈ J). In analog mode too, one can deﬁne then the algebra (Πj∈J Aj ; (fi )i∈I ) as a direct product Πj∈J Aj of the algebras Aj (j ∈ J), where 1

The proof for the statements can be found in [Lau 2004], volume 2.

5.1 Direct Products

65

fi ((aj1 |j ∈ J), (aj2 |j ∈ J), ..., (aj,afi |j ∈ J)) := (fji (aj1 , aj2 , ..., aj,afi )|j ∈ J) for arbitrary ((aj1 |j ∈ J), ..., (aj,afi |j ∈ J) ∈ Πj∈J Aj , if afi > 0, and fi = (fji |j ∈ J), if afi = 0. If J = ∅, then let Πj∈J Aj be the 1-element algebra of the corresponding type. If Aj = A for all j ∈ J, we write AJ instead of Πj∈J Aj . If J = {1, 2, .., n}, one also uses the denotation A1 × ... × An for the direct product of the algebras Aj . Theorem 5.1.5 Every ﬁnite algebra is isomorphic to a direct product of direct irreducible algebras. Proof. Induction over the cardinality of the universes of the algebras: Obviously, every algebra A with |A| = 1 is directly irreducible. Now, let A be a ﬁnite algebra with |A| ≥ 2, and assume that the assertion is proven for all algebras A with |A | < |A|. If A is directly irreducible, we have to show nothing more. If, however, A ∼ = B × C with |B| > 1, |C| > 1 is valid, then we have |B| < |A| and |C| < |A|; i.e., one can ﬁnd direct irreducible algebras B1 , ..., Bm , C1 , ..., Cn with B∼ = B1 × ... × Bm , C∼ = C1 × ... × Cn . Thus A ∼ = B1 × ... × Bm × C1 × ... × Cn .

The direct products of more than two algebras have similar properties, like the direct products of only two algebras. For example, the j0 -th projection prj0 from Πj∈J Aj onto Aj with (aj | j ∈ J) → aj0 is a homomorphism for every j0 ∈ J. One can easily check the following property of direct products. Theorem 5.1.6 For every family2 ϕi : B −→ Ai , i ∈ I, of homomorphisms one can form a homomorphism ϕ: B −→ Πi∈I Ai by (ϕ(b))i := ϕi (b).

2

A family (ai | i ∈ I) of elements of a set A is a mapping ϕ : I → A, i → ai . This denotation is used around a certain selection of (not necessarily diﬀerent) elements from A to characterize. Then, I is called index set of the family (ai | i ∈ I).

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5 Direct and Subdirect Products

5.2 Subdirect Products Unlike ﬁnite algebras, inﬁnite algebras cannot always be represented by direct products of directly irreducible algebras. Example As we already noticed above, a Boolean algebra A is directly irreducible iﬀ |A| ≤ 2. Furthermore, one can easily prove that every 2-element Boolean algebra is isomorphic to the algebra B = ({0, 1}; ∨, ∧,− , 0, 1). An inﬁnite direct product of B is not countable. Consequently, the countable Boolean algebra C = (C; ∨ , ∧ , ¬) with C := {(a1 , a2 , ...) ∈ {0, 1}N | |{i ∈ N | ai = 0}| < ℵ0 or |{i ∈ N | ai = 1}| < ℵ0 }, (a1 , a2 , ...) ◦ (b1 , b2 , ...) := (a1 ◦ b1 , a2 ◦ b2 , ...) for ◦ ∈ {∨, ∧} and ¬(a1 , a2 , ...) := (a1 , a2 , ...) is not isomorphic to a direct product of directly irreducible algebras, however, it is a subalgebra of the direct product BN . In generalizing this example, we get the following new product concept: Deﬁnition Let the algebras Ai , i ∈ I, be of the same type. A subalgebra B of Πi∈I Ai is called a subdirect product of the Ai iﬀ prj (B) = Aj holds for all j ∈ I. Example

Obviously, every direct product is also a subdirect product.

Theorem 5.2.1 For a subdirect product B of the algebras Ai , i ∈ I, and the projection mappings prj : Πi∈I Ai −→ Aj it holds Ker(prj )|B = κ0 . j∈I

Proof. a = b.

If (a, b) ∈

j∈I

Ker(prj )|B , then aj = bj follows for all j ∈ I and therefore

By the above theorem and by the fact that all prj|B are surjective, the subdirect products are already characterized: Theorem 5.2.2 Let A be an algebra. For certain congruences κi ∈ ConA, i ∈ I, let κi = κ0 . i∈I

Then A is isomorphic to a subdirect product of the algebras A/κi , i ∈ I. By ϕ(a) := (a/κi | i ∈ I) an injective homomorphism ϕ: A −→ Πi∈I (A/κi ) is deﬁned, and ϕ(A) is a subdirect product of the algebras A/κi .

5.2 Subdirect Products

67

Proof. By Theorem 5.1.6 ϕ is a homomorphism. ϕ is also injective: Let ϕ(a) = ϕ(b). This implies a/κi = b/κi and thus (a, b) ∈ κi for all i ∈ I. Therefore, (a, b) ∈ κ i∈I i = κ0 ; i.e., a = b. Consequently, A and ϕ(A) are isomorphic. Further, by deﬁnition of ϕ we have prj (ϕ(A)) = A/κj for all j ∈ I. Hence, ϕ(A) is a subdirect product of the algebras A/κi . Deﬁnition An injective homomorphism (a so-called embedding) ϕ: A −→ Πi∈I Ai is called a subdirect representation of A, if ϕ(A) is a subdirect product of the algebras Ai . Example The mapping ϕ of Theorem 5.2.2 is a subdirect representation. Deﬁnition An algebra A is called subdirectly irreducible, if for every subdirect representation ϕ : A −→ Πi∈I Ai there exists a j ∈ I such that the mapping ϕ2prj : A −→ Aj is an isomorphism. Thus, an algebra is subdirectly irreducible if and only if one gets by with a single component in every subdirect representation. Theorem 5.2.3 An algebra A is subdirectly irreducible, if and only if the universe of A contains at most an element, or if (ConA\{κ0 }) = κ0 holds. The latter holds obviously iﬀ κ0 exactly has an upper neighbor in ConA: κ1

r @

ConA :

@

@ @

@ @

@

@r

(ConA\{κ0 })

r

κ0 Proof. W.l.o.g. let |A| ∈ {0, 1} in the following. Suppose, (ConA\{κ0 }) = κ0 . Put I := ConA\{κ0 }. Then, with the help of Theorem 5.2.2, one obtains a subdirect representation ϕ: A −→ Πκ∈I (A/κ). For every mapping prκ (κ ∈ I) and all a ∈ A it holds (ϕ2prκ )(a) = a/κ. By κ0 ∈ I, therefore ϕ2prκ : A −→ A/κ is not injective (i.e., is not an isomorphism). Thus, A is not subdirectly irreducible. Let now µ := (ConA\{κ0 }) = κ0 ; i.e., there exists (a, b) ∈ µ \ κ0 , and denote ϕ: A −→ Πi∈I Ai a subdirect representation of A. We have to show that there exists an

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5 Direct and Subdirect Products

i ∈ I so that ϕ2pri is an isomorphism from A onto Ai . Since ϕ is injective and a = b, there exists a j ∈ I with prj (ϕ(a)) = prj (ϕ(b)), whereby (a, b) ∈ Ker(ϕ2prj ). Thus, by (a, b) ∈ µ, we have µ ⊆ Ker(ϕ2prj ). Then, by deﬁnition of µ, Ker(ϕ2prj ) = κ0 holds; i.e., ϕ2prj is injective. Since ϕ(A) is a subdirect product, ϕ2prj is surjective. Therefore, ϕ2prj is an isomorphism. Consequently, A is subdirectly irreducible. With the help of Theorems 5.2.3 and 5.1.4, one can easily prove the following connection between the direct and the subdirectly irreducible algebras: A is subdirectly irreducible =⇒ A is directly irreducible.

(5.1)

The reversal of the statement (5.1) is not valid, because one can prove with the help of a 3-element lattice which is directly irreducible but which is not subdirectly irreducible. An essential aid for proving the following Theorem is Zorn’s Lemma which is indicated without proof here. This lemma is equivalent to the axiom of choice (see e.g. [Her 55]). Lemma 5.2.4 (Zorn’s Lemma) In every set system M with the property ∀T ⊆ M ((∀X, Y ∈ T ∃Z ∈ T : X ∪ Y ⊆ Z) =⇒

X ∈ M)

X∈T

(i.e., M is an inductively set system) there is a maximal element3 ; i.e., an element M ∈ M that is not contained in any proper subset of M. After these preparations we can prove the following theorem, published by G. Birkhoﬀ in 1944. Theorem 5.2.5 Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras. Proof. Let A be an algebra. One can easily see that, for every pair a, b ∈ A with a = b, the set Ma,b := {κ ∈ ConA | (a, b) ∈ κ} is an inductive set system. By Lemma 5.2.4 Ma,b has a maximal element Φ(a, b). In the lattice ConA the element Φ(a, b) has exactly an upper neighbor, namely Φ(a, b) ∨ Ω(a, b), where Ω(a, b) is the congruence relation generated by (a, b). It is easy to check that the factor algebra A/Φ(a, b) is isomorphic to an interval [Φ(a, b), κ1 ] := {κ ∈ ConA | Φ(a, b) ⊆ κ ⊆ κ1 } 3

Let (B; ≤) be a poset. A maximal element of the set A ⊆ B is then an element a ∈ A with a < x =⇒ x ∈ A for all x ∈ B.

5.2 Subdirect Products

69

of ConA. By Theorem 5.2.3 the algebra A/Φ(a, b) is subdirectly irreducible. From {Φ(a, b) | a, b ∈ A ∧ a = b} = κ0 and from Theorem 5.2.2, it follows that A is isomorphic to a subdirect product of subdirectly irreducible algebras (namely of the algebras A/Φ(a, b)).

6 Varieties, Equational Classes, and Free Algebras

In this Chapter, only certain classes1 of algebras of the same type shall be regarded. First we introduce so-called varieties as classes of algebras, which are closed in respect to formation of subalgebras, homomorphic images, and direct products. We then come to a method for constructing algebra classes that strongly diﬀers from the ﬁrst method at ﬁrst sight: Starting from certain equations from variables and operation symbols of a certain type τ , we form the class of all algebras of type τ that fulﬁll these equations. The result is an equational class. We will see, however, that there is a close connection between the two methods of algebra class construction: A class of algebras is equationally deﬁnable if and only if it is a variety. Free algebras are “the most general” algebras within a variety or an equational class (or an equationally deﬁnable class). In the section on equational classes, we will also address such concepts such as conclusion of an equational set. In addition, we investigate methods to receive such conclusions.

6.1 Varieties The following operators S, H, P , I map a class K of algebras of type τ to a class of algebras of the same type. Let S(K) be the class of all subalgebras of algebras aus K, H(K) be the class of all homomorphic images of algebras of K, P (K) be the class of all direct products of families of algebras of K, I(K) be the class of all algebras which are isomorphic to algebras of K. 1

The concept “class” is a generalization of the concept “set”. Informally speaking, a class is a collection so large that subjecting it to the operations admissible for sets would lead to logical contradictions.

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6 Varieties, Equational Classes, and Free Algebras

We denote the composition of the operators Y, X ∈ {H, S, P, I} by XY ; i.e., it holds XY (K) := X(Y (K)). It is easy to check that S, H and IP are hull operators; i.e., for all classes K and L of algebras of the same type we have: ∀X ∈ {S, H, IP } : K ⊆ X(K) ∧ (K ⊆ L =⇒ X(K) ⊆ X(L)) ∧ X(K) = X(X(K)). The operator P is not idempotent: For all A, B, C ∈ K it holds (A×B)×C ∈ P (P (K)), but (A×B)×C ∈ P (K) generally does not hold. However, (A × B) × C ∼ = A × B × C ∈ P (K) is right; i.e., (A × B) × C ∈ IP (K). Deﬁnitions • A class K of algebras of the same type is called under X ∈ {S, H, P } closed, if X(K) ⊆ K holds. • If the class K of algebras of the same type is closed under the three operators S, H and P , then the class K is called a variety. Examples (1) It is easy to see that the class of all groups is a variety. (2) Since the direct product of the ﬁeld Z2 with the ﬁeld Z2 because of ∀x, y ∈ Z2 : (0, 1) · (x, y) = (0 · x, 1 · y) = (0, y) = (1, 1) is not a ﬁeld, the class of all ﬁelds is not a variety. Lemma 6.1.1 For every class K of algebras of the same type it holds: (a) SH(K) ⊆ HS(K), (b) P S(K) ⊆ SP (K), (c) P H(K) ⊆ HP (K). Proof. (a): Let A ∈ SH(K); i.e., there is an algebra B ∈ H(K) with A ≤ B and B is homomorphic image of an algebra C ∈ K. Let ϕ: C −→ B be a surjective homomorphism. Then, for the subalgebra ϕ−1 (A) of C it holds ϕ(ϕ−1 (A)) = A. Consequently, we have A ∈ HS(K). (b): Let A ∈ P S(K). Then it holds A = Πi∈I Bi with Bi ≤ Ci ∈ K for all i ∈ I. Since obviously Πi∈I Bi is a subalgebra of Πi∈I Ci , we have A ∈ SP (K). (c): Let A ∈ P H(K). Then, A = Πi∈I Bi , where for every i ∈ I there are an algebra Ci and a surjective homomorphism ϕi : Ci −→ Bi . If prj : Πi∈I Ci −→ Cj is the projection mapping, then prj 2ϕj : Πi∈I Ci −→ Bj is a surjective homomorphism. By Theorem 5.1.6 we get by ϕ(c)j := (prj 2ϕj )(c) a homomorphism ϕ: Πi∈I Ci −→ Πi∈I Bi , which is also surjective. Consequently, we have A ∈ HP (K).

Theorem 6.1.2 A class K of algebras of the same type is a variety if and only if HSP (K) = K holds.

6.2 Terms, Term Algebras, and Term Functions

73

Proof. If K is a variety, then obviously HSP (K) = K. Let now HSP (K) = K. We have to show H(K) ⊆ K, S(K) ⊆ K and P (K) ⊆ K. It holds: H(K) = H(HSP (K)) = HSP (K), since H is idempotent. Thus, H(K) = K by assumption. Further we have: S(K) = S(HSP (K)) = SH(SP (K)) ⊆ HS(SP (K)) ⊆ HSP (K) by Lemma 6.1.1, (a) and since S is idempotent. Therefore, S(K) ⊆ K holds. Furthermore, P (K) = P HSP (K) ⊆ HP SP (K) ⊆ HSP P (K) ⊆ HSIP IP (K) = HSIP (K) ⊆ HSHP (K) ⊆ HHSP (K) = HSP (K) by Lemma 6.1.1 and since the operators IP and H are idempotent. Thus, P (K) ⊆ K holds. Consequently, K is a variety.

Every variety is exactly determined by its elements, which are subdirect irreducible algebras: Theorem 6.1.3 Every algebra of a variety K is isomorphic to a subdirect product of subdirect irreducible algebras of K. Proof. By Theorem 5.2.5, every algebra A is isomorphic to a subdirect product of subdirect irreducible algebras Ai , where every Ai is isomorphic to a factor algebra of A; i.e., it holds Ai ∈ H(A). If A belongs to a variety K, then Ai ∈ H(K) ⊆ K.

6.2 Terms, Term Algebras, and Term Functions In Section 1.1, we had agreed to describe the operations of algebras of the same type by the same notations, if it is clearly from the context to which algebra a given operation belongs. Since we will often make use of this convention, we generalize the concept “type of an algebra” as follows: Deﬁnitions A type of algebras is an ordered pair (F, τ ), where F is a set, whose elements are called operation symbols, and τ : F −→ N0 is a mapping, which assigns the arity af to every operation f ∈ F. Then, the algebra A = (A; F ) with F := {fA | f ∈ F} is called algebra of type (F, τ ). Let Fn be the set of the n-ary operation symbols of F . Now let (F, τ ) be a type of algebras and let X

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be a ﬁnite or countable inﬁnite set, whose elements are called variables, where X ∩ F0 = ∅. T (X) denotes the smallest set with the following two properties: (1) X ∪ F0 ⊆ T (X), (2) (f ∈ Fn ∧ {t1 , ..., tn } ⊆ T (X)) =⇒ f (t1 , ..., tn ) ∈ T (X). One observes that f (t1 , ..., tn ) is a syntactic expression (a symbol sequence) and not a function value. The elements of T (X) are called terms of type (F, τ ) over the alphabet X. Example Let F := {e, f }, τ (e) := 0, τ (f ) := 2 and X := {x, y, z}. Then T (X) = {e, x, y, z, f (e, e), f (e, x), ..., f (z, y), f (f (e, e), e), f (f (x, e), e), ..., f (f (x, z), f (z, y)), ........}. We agree that (u ◦ v) := f (u, v) for arbitrary u, v ∈ T (X) and, furthermore, we do without outer brackets. Then the set T (X) can be also written down as follows: {e, x, y, z, e ◦ e, e ◦ x, ..., z ◦ y, (e ◦ e) ◦ e, (x ◦ e) ◦ e, ..., (x ◦ z) ◦ (z ◦ y), ...}. T (X) is the universe of the so-called term algebra T(X) := (T (X); F ) of type (F, τ ), where for every f ∈ Fn (n ∈ N ∪ {0}) the operations of this algebra are deﬁned as follows: fT(X) := f, if n = 0, and ∀t1 , ..., tn ∈ T (X) : fT(X) (t1 , ..., tn ) := f (t1 , ..., tn ) for n ≥ 1. A part of the operation table of the operation fT(X) from the above example then looks as follows: u e e x ◦ (y ◦ e)

v e e◦x (x ◦ x) ◦ z

fT(X) (u, v) e◦e e ◦ (e ◦ x) (x ◦ (y ◦ e)) ◦ ((x ◦ x) ◦ z)

The following lemma follows directly from the deﬁnition of T(X):

6.2 Terms, Term Algebras, and Term Functions

75

Lemma 6.2.1 The term algebra T(X) is generated by X; i.e., it holds [X] = T (X). The following theorem gives an important property of the algebra T(X): Theorem 6.2.2 Let T(X) be the term algebra of type (F, τ ) over X. Then, for every algebra A of type (F, τ ) and for every mapping ϕ: X −→ A there is exactly one homomorphism ϕ : T(X) −→ A, which ϕ continues, i.e., for which ϕ| X = ϕ holds. Proof. For a given algebra A of type (F, τ ) and for a given mapping ϕ: X −→ A let ϕ: T (X) −→ A be a mapping with the properties: ∀x ∈ X : ϕ(x) := ϕ(x) and ∀f (t1 , ..., tn ) ∈ T (X)\X : ϕ(f (t1 , ...tn )) := fA (ϕ(t 1 ), ..., ϕ(t n )). Obviously, ϕ is deﬁned over T (X) by the above conditions, and it is the only possible continuation of ϕ over T (X).

Using the concepts from Part II, Chapter 1, we can brieﬂy deﬁne the following Deﬁnition The term functions of an algebra A = (A; F ) of type (F, τ ) are operations over A that can be formed by superposition from the fundamental operations of F and from the projections. Without using concepts from Part II, Chapter 1, we can deﬁne the term functions an algebra A of type (F, τ ) as follows: Let t be a term of type (F, τ ) over X = {x1 , ..., xn }, and for a1 , ..., an ∈ A let ϕa1 ,...,an : T(X) −→ A be the unique homomorphism with xi → ai , i = 1, 2, ..., n. Then we can deﬁne an n-ary operation tA : An −→ A by ∀a1 , ..., an ∈ A : tA (a1 , ..., an ) := ϕa1 ,...,an (t). These operations are identical with the term functions already deﬁned above. If t and TA are deﬁned as above, we say that the term t induces the term function tA . Let T F (A) be the set of all term functions of A. One can prove the next two lemmas with properties of T F (A) easily. Lemma 6.2.3 Let [..] be the subalgebra-hull-operator (see Chapter 3). Then for every algebra A and every subset B of A it holds: [B] = {tA (b1 , ..., bn ) | n ∈ N ∧ t ∈ T ({x1 , ..., xn }) ∧ {b1 , ..., bn } ⊆ B}.

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6 Varieties, Equational Classes, and Free Algebras

Lemma 6.2.4 The algebras A, B and the n-ary term t have the same type. Then, for every homomorphism ϕ: A −→ B and all a1 , ..., an ∈ A it holds: ϕ(tA (a1 , ..., an )) = tB (ϕ(a1 ), ..., ϕ(an )); i.e., the term functions react just like the fundamental operations in respect to homomorphisms.

6.3 Equations and Equational Classes In this section, let T (X) be the set of all terms of type (F, τ ). To show that the variables of the term t ∈ T (X) are of the set {x1 , ..., xn } ⊆ X, we write t < x1 , ..., xn > and set t = t < x1 , ..., xn > . The notation s := t < t1 , ..., tn > meant that the term s was formed from the term t by substituting every variable xi (1 ≤ i ≤ n) by ti in every place the variable xi in t appeared. We agree on an analogous notation for term functions. Deﬁnitions • The elements of T (X) × T (X) are called equations (or identities) over X and we write s ≈ t :⇐⇒ (s, t) ∈ T (X) × T (X). • An algebra A of type (F, τ ) fulﬁlls the equation s < x1 , ..., xn >≈ t < x1 , ..., xn > (or the equation s ≈ t holds in A), if for all a1 , ..., an ∈ A sA < a1 , ..., an >= tA < a1 , ..., an > is right. In this case, we also write A |= s ≈ t. Further, we set IdX (A) := {(s, t) ∈ T (X) × T (X) | A |= s ≈ t}.

6.3 Equations and Equational Classes

77

• Let be for Σ ⊆ T (X) × T (X) and classes K of algebras of the same type (F, τ ): A |= Σ :⇐⇒ (∀s ≈ t ∈ Σ : A |= s ≈ t). • The class

M od(Σ) := {A | A |= Σ}

is called the set of all models of Σ. • Conversely, for every class K of algebras of type (F, τ ) let IdX (K) := {(s, t) ∈ T (X) × T (X) | ∀A ∈ K : A |= s ≈ t} be the class of all equations over X that hold in all algebras of K. • A class K of algebras is equationally deﬁnable, if there exists a Σ ⊆ T (X) × T (X) with M od(Σ) = K. • A set Σ ⊆ T (X) × T (X) is called equational theory over X, if there is a class K of algebras with Σ = IdX (K). • An equation s ≈ t is called a conclusion of Σ ⊆ T (X) × T (X), if A |= s ≈ t holds for all A ∈ M od(Σ). Let ConsX (Σ) be the set of all conclusions of Σ; i.e., it holds ConsX (Σ) := IdX (M od(Σ)). Instead of Y (Z(..)), where Y, Z ∈ {M od, IdX , ConsX }, we write brieﬂy Y Z(..). The following theorem summarizes elementary properties of the sets deﬁned above and connections between the concepts just deﬁned. Theorem 6.3.1 For arbitrary Σ, Σ ⊆ T (X) × T (X) and arbitrary classes K, K of algebras of type F it holds: (1) Σ ⊆ Σ =⇒ M od(Σ ) ⊆ M od(Σ), K ⊆ K =⇒ IdX (K ) ⊆ IdX (K); (2) Σ ⊆ IdX M od(Σ), K ⊆ M odIdX (K); (3) M odIdX M od(Σ) = M od(Σ), IdX M odIdX (K) = IdX (K); (4) Σ ⊆ ConsX (Σ), Σ ⊆ Σ =⇒ ConsX (Σ) ⊆ ConsX (Σ ), ConsX ConsX (Σ) = ConsX (Σ); (5) K ⊆ M odIdX (K), K ⊆ K =⇒ M odIdX (K) ⊆ M odIdX (K ), M odIdX M odIdX (K) = M odIdX (K); (6) Σ is equational theory ⇐⇒ Σ = ConsX (Σ), K is equationally deﬁnable ⇐⇒ K = M odIdX (K).

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6 Varieties, Equational Classes, and Free Algebras

Proof. (1) and (2) immediately follow from the deﬁnitions of M od and IdX . (3): By (2) Σ ⊆ IdX M od(Σ) =: Σ holds. Thus, by means of (1), we have M odIdX M od(Σ) ⊆ M od(Σ). Conversely, we have also by (2): K := M od(Σ) ⊆ M odIdX M od(Σ). Therefore, M od(Σ) = M odIdX M od(Σ). Analogously, one can show IdX M odIdX (K) = IdX (K). (4) and (5) one can easily prove by means of (1)–(3). (6): Let Σ be an equational theory; i.e., there is a class K of algebras of type (F, τ ) with Σ = IdX (K). Then ConsX (Σ) = IdX M od(Σ)

assumption

=

(3)

IdX M odIdX (K) = IdX (K)

assumption

=

Σ.

Conversely, let Σ = ConsX (Σ). Then we have Σ = IdX M od(Σ); i.e., Σ is an equational theory. The statement over equational deﬁnable classes can be proven analogously.

If we neglect that the objects formed by classes need an exact deﬁnition, 2 then the following theorem is an immediate conclusion of the above theorem and of Theorem 4.4.1. Theorem 6.3.2 Let X be a countable inﬁnite set, Alg(F, τ ) the class of all algebras of type (F, τ ) over X and let T (X) be the set of all terms of type (F, τ ). Then the pair (IdX , M od) forms a Galois connection between P(T (X) × T (X)) and P(Alg(F, τ )). Furthermore, the lattice of all equational classes of Alg(F, τ ) is antiisomorphic to the lattice of all equational theories of type (F, τ ).

6.4 Free Algebras To deﬁne a free algebra, we need the following properties of T(X): Theorem 6.4.1 Let K be a class of algebras of type (F, τ ), and let T(X) be the term algebra of the same type over the alphabet X. Then (a) IdX (K) = {Ker ϕ | ∃A ∈ K : ϕ : T(X) −→ A is a homomorphic mapping}, (b) IdX (K) ∈ ConT(X). Proof. (a): Let s, t ∈ T (X ) with X := {x1 , ..., xn } ⊆ X. For every algebra A ∈ K and all a1 , ..., an ∈ A there is by Theorem 6.2.2 a homomorphism ϕ: T(X) −→ A with ϕ(xi ) = ai , i = 1, ..., n. For every ϕ it holds ϕ(s) = sA (a1 , ..., an ) and ϕ(t) = tA (a1 , ..., an ). Therefore we have (s, t) ∈ Ker ϕ for all ϕ : T(X) −→ A with A ∈ K if and only if for all A ∈ K and all a1 , .., an ∈ A the equation sA (a1 , ..., an ) = tA (a1 , ..., an ) holds. But this is equivalently with A |= s ≈ t for all A ∈ K. (b) follows directly from (a), since the intersection of congruences of an algebra is a congruence of the algebra, as is well-known. 2

See, for example, [Sch 74], Chapter II.

6.4 Free Algebras

79

By the above theorem, we can form the factor algebra: T(X)/IdX (K)

(6.1)

for an arbitrary class K of algebras of the same type and of a set X of variables. Deﬁnitions If the factor algebra (6.1)belongs to K, then T(X)/IdX (K) is called the free algebra of K with free generating set X. If (6.1) belongs to K, we describe (6.1) with FK (X). In case X = {x1 , ..., xn } we also write FK (x1 , ..., xn ) or brieﬂy FK (n), and, if X = {xi | i ∈ N}, we write FK (x1 , x2 , ...) or FK (ℵ0 ) (or FK (ω)). We notice, that, strictly speaking, the free algebra FK (X) is not generated by the set X but by the congruence classes x/IdX (K), x ∈ X. Nevertheless, one often writes x instead of x/IdX (K), since in a nontrivial class K of algebras (i.e., K contains not only 0- or 1-element algebras), the equation x/IdX (K) = y/IdX (K) implies x = y. The importance of the free algebras results from the following theorems with whose aid the main theorems to the equational theory are proven in Section 6.5. The factor algebra T(X)/IdX (K), in respect to the class K, has the same property as T(X) in respect to the class of all algebras of type (F, τ ) (see Theorem 6.2.2): Theorem 6.4.2 Let K be a class of algebras of type (F, τ ) and T(X) be the term algebra of the same type. Furthermore, let x := x/IdX (K) and X := {x | x ∈ X}. Then there is for every algebra A ∈ K and every mapping ϕ : X −→ A exactly one homomorphism ϕ : T(X)/IdX (K) −→ A, which continues ϕ; i.e., for which ϕ|X = ϕ holds. Proof. Let α : X −→ A be a mapping deﬁned by α(x) := ϕ(x). Then, by Theorem 6.2.2 there is a homomorphism α : T(X) −→ A, which continues α. For the homomorphism π : T(X) −→ T(X)/IdX (K) with Ker π = IdX (K) we have by Theorem 6.4.1, (a) that Ker π ⊆ Ker α holds; i.e., it holds: π(s) = π(t) =⇒ α(s) = α(t). By ϕ(π(t)) := α(t) one receives a well-deﬁned mapping ϕ : T (X)/IdX (K) −→ A. It is easy to see that ϕ is a homomorphism and that ϕ(x) = ϕ(x) for all x ∈ X holds. Because of [X] = [π(X)] = π[X] = π(T (X)) = T (X)/IdX (K) the mapping ϕ is uniquely deﬁned through the deﬁnition over X. Theorem 6.4.3 For every class K of algebras of the same type and every variable set X it holds T(X)/IdX (K) ∈ ISP (K). Proof. Let T := T (X) and κ := IdX (K). By Theorem 6.4.1 (a) and with the help of Lemma 17.4.1 from [Lau 2004], volume 2 one can prove {(Ker ϕ)/κ | ∃A ∈ K : ϕ : T −→ A is a homomorphic mapping} = ∆T /κ (= κ0 on T /κ). Because of Theorem 5.2.2, the algebra T/κ is isomorphic to a subdirect product of the algebras (T/κ)/((Ker ϕ)/κ)

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with ϕ : T −→ A, A ∈ K. For every such ϕ it holds (by the First Isomorphism Theorem 3 ) (T/κ)/((Ker ϕ)/κ) ∼ = T/(Ker ϕ) ∼ = ϕ(T) ∈ S(K). Therefore one gets altogether T/κ ∈ ISP (IS(K)) ⊆ ISP (S(K)) ⊆ ISP (K), where, obviously, the ﬁrst inclusion is valid and the second inclusion follows from Lemma 6.1.1, (b). An immediate conclusion from Theorem 6.4.3 is as follows: Theorem 6.4.4 For every class K of algebras of the same type (in particular for a variety K) which is closed in respect to the operators I, S and P , it holds T(X)/IdX (K) ∈ K; i.e., K contains a free algebra FK (X). Lemma 6.4.5 Every free algebra FK (X) of a variety is isomorphic to a subdirect product of certain algebras FK (E), where E ⊆ X is ﬁnite and E = ∅. Proof. For x ∈ X let x := x/IdX (K). Further, for every E ⊆ X let E := {e ∈ FK (X) | e ∈ E}. Denote U(E) the subalgebra of FK (X) which is generated by E. It is easy to check that U(E) and FK (E) are isomorphic. Therefore, it is suﬃcient to show that FK (X) is isomorphic to a subdirect product of the algebras U(E), where E ⊆ X is nonempty and ﬁnite. For every such E, one chooses a surjective mapping ϕE : X −→ E with (ϕE )|E = idE . Then, the homomorphic continuation ϕE is surjective, and it holds (ϕE )|U (E) = idU (E) . Every term has only ﬁnite many variables. Thus for every pair s, t ∈ FK (X), there is a ﬁnite subset E ⊆ X with = t we have even (s, t) ∈ Ker (ϕE ) because of ϕE (s) = s s, t ∈ U (E). In the case s and ϕE (t) = t. Therefore {Kern(ϕE ) | ∅ ⊂ E ⊆ X ∧ E is ﬁnite} = κ0 . By Theorem 5.2.2 FK (X) is also isomorphic to an subdirect product of FK (x)/Kern(ϕE ). Because of FK (X)/Kern(ϕE ) ∼ = U(E), the assertion follows. Theorem 6.4.6 For every variety K it holds K = HSP ({FK (n) | n ∈ N}) = HSP ({FK (ω)}). Proof. Every algebra A ∈ K is a homomorphic image of FK (X), if |X| ≥ |A| (one chooses a mapping ϕ : X −→ A and then one uses Theorem 6.4.2). Thus, the ﬁrst equality sign in our theorem follows from Lemma 6.4.5, and the second equality sign follows from the fact that FK (n) is isomorphic to a subalgebra of FK (ω) for all n ∈ N.

3

See for example [Wec 92], p. 140 or [Den-W 2002], Theorem 3.2.2 or Theorem 17.4.2 from [Lau 2004], volume 2.

6.5 Connections Between Varieties and Equational Deﬁned Classes

81

6.5 Connections Between Varieties and Equational Deﬁned Classes We need the following statement. Lemma 6.5.1 Let K be a class of algebras of the same type. Then it holds for an arbitrary alphabet X that: (a) ∀Op ∈ {H, S, P } : Op(K) ⊆ M odIdX (K); (b) M odIdX (K) is a variety. Proof. (a): Let Op = H. First, we will show that IdX (K) ⊆ IdX (H(K)) is right. Let s < x1 , ..., xn >≈ t < x1 , ..., xn > be an equation of IdX (K) with {x1 , ..., xn } ⊆ X. Then, this equation also holds in an arbitrary algebra B ∈ H(K): If namely ϕ(A) = B for a certain algebra A ∈ K and a surjective homomorphism ϕ, then for arbitrary b1 , ..., bn ∈ B there are a1 , ..., an ∈ A with the property sB < b1 , ..., bn > = sB < ϕ(a1 ), ..., ϕ(an ) > = ϕ(sA < a1 , ..., an >) = ϕ(tA < a1 , ..., an >) = tB < ϕ(a1 ), ..., ϕ(an ) > = tB < b1 , ..., bn >, i.e., s ≈ t ∈ IdX ({B}) holds, and thus we have IdX (K) ⊆ IdX (H(K)). Now, if one uses Theorem 6.3.1, (1), then one gets M odIdX (H(K)) ⊆ M odIdX (K). Furthermore, by Theorem 6.3.1, (2), we have H(K) ⊆ M odIdX (H(K)). Thus H(K) ⊆ M odIdX (K). Analogously one can prove (a) for Op ∈ {S, P }. (b): Let K ∗ = M odIdX (K). Then, by (a) and with the help of Theorem 6.3.1, (3) it holds for every Op ∈ {H, S, P }: Op(K ∗ ) ⊆ M odIdX (K ∗ ) = M od(IdX M odIdX (K)) = M odIdX (K) = K ∗ . Therefore, K ∗ is a variety.

Theorem 6.5.2 (First Main Theorem of the Equational Theory; [Bir 35]) A class K of algebras of the same type is a variety iﬀ it is equationally deﬁnable; i.e., it holds (by Theorem 6.3.1, (6) and Theorem 6.1.2): K = HSP (K) ⇐⇒ ∃X : K = M odIdX (K). Proof. “⇐=”: By Lemma 6.5.1, (a) it holds Op(K) ⊆ M odIdX (K) for every Op ∈ {H, S, P }. If now K = M odIdX (K), then this implies Op(K) ⊆ K. Thus K is a variety. “=⇒”: Let K be a variety. By Lemma 6.5.1, (b) the class K ∗ := M odIdX (K) is also a variety and we have for an arbitrary alphabet X:

82

6 Varieties, Equational Classes, and Free Algebras FK∗ (X) = T(X)/IdX (K∗ ) (by deﬁnition) = T(X)/IdX (K)

(since by Theorem 6.3.1, (3) : IdX (K ∗ ) = IdX M odIdX (K) = IdX (K))

= FK (X)

(by deﬁnition).

From that (with the aid of Theorem 6.4.6 and X := {x1 , x2 , ...}) we get the equations K = HSP ({FK (ω)}) = HSP ({FK∗ (ω)}) = K ∗ Therefore, K is equationally deﬁnable.

6.6 Deductive Closure of Equation Sets and Equational Theory With the following deﬁnition of the deductive closure, we generalize the usual procedure of deriving equations from already proven equations. At the end of this section, we will be able to prove that the deductive closure of an equation sets Σ is identical with the set of all conclusions of Σ. Deﬁnitions Let (F, τ ) be a type of algebras, T (X) is deﬁned as in Section 6.2 and Σ ⊆ T (X) × T (X). Then, the deductive closure D(Σ) of Σ is the smallest subset of T (X) × T (X) containing Σ such that the following ﬁve conditions hold: (R1) ∀p ∈ T (X) : p ≈ p ∈ D(Σ); (R2) ∀p, q ∈ T (X) : p ≈ q ∈ D(Σ) =⇒ q ≈ p ∈ D(Σ); (R3) ∀p, q, r ∈ T (X) : (p ≈ q ∈ D(Σ) ∧ q ≈ r ∈ D(Σ) =⇒ p ≈ r ∈ D(Σ)); (Rep) ∀f n ∈ F ∀{s1 ≈ t1 , ...., sn ≈ tn } ⊆ D(Σ) : f (s1 , ..., sn ) ≈ f (t1 , ..., tn ) ∈ D(Σ); (“replacement rule”); (Sub) ∀s < x1 , ..., xn >≈ t < x1 , ..., xn >∈ D(Σ) ∀t1 , ..., tn ∈ T (X) : s < t1 , ..., tn >≈ t < t1 , ..., tn >∈ D(Σ) (“substitution rule”). Σ ⊆ T (X) × T (X) is called deductively closed if D(Σ) = Σ holds. Obviously, every set Σ of equations with Σ = IdX (K), where K denotes a class of algebras of type (F, τ ), is deductively closed. In other words, if Σ is an equational theory of a class of algebras, then Σ is deductively closed. Furthermore, it holds that D(Σ) ⊆ ConsX (Σ). The aim of the following considerations is the proof that the reversals of the above two statements are also right. Exacter: It shall be shown that every deductively closed set of equations is

6.6 Deductive Closure of Equation Sets and Equational Theory

83

the equational theory of a certain class of algebras and that for every set Σ of equations, it holds ConsX (Σ) ⊆ D(Σ). Obviously, a deductively closed set Σ ⊆ T (X) × T (X) can also be characterized as follows: Because of (R1)–(R3) Σ is an equivalence relation, because of (Rep) Σ is a congruence on T (X), and because of (Sub) Σ is compatible with every endomorphism of T(X) (this is a homomorphism from T(X) into T(X)) (Proof: Let t1 , ..., tn ∈ T (X) be arbitrary. Then there exists an endomorphism ϕ of T(X) with ϕ(x1 ) = t1 , ϕ(x2 ) = t2 , ..., ϕ(xn ) = tn , and for every such endomorphism, it holds that ϕ(s) = s < t1 , ..., tn > and ϕ(t) = t < t1 , ..., tn >.). Deﬁnition A congruence relation κ of an algebra A is called fully invariant, if it is compatible with all endomorphisms of A; i.e., if for every endomorphism ϕ of A, (a, b) ∈ κ implies (ϕ(a), ϕ(b)) ∈ κ. The following two lemmas follow immediately from this deﬁnition and the above considerations. Lemma 6.6.1 A set Σ ⊆ T (X) × T (X) is deductively closed iﬀ Σ is a fully invariant congruence on T(X). Lemma 6.6.2 For every class K of algebras of the same type and every variable set X, IdX (K) is a fully invariant congruence on T(X). The reversal of Lemma 6.6.2 is also valid: Lemma 6.6.3 For every fully invariant congruence κ over T(X) it holds that IdX ({T(X)/κ}) = κ, i.e., for arbitrary s, t ∈ T (X) we have: (s, t) ∈ κ ⇐⇒ T(X)/κ |= s ≈ t. In other words, an arbitrary fully invariant congruence κ on T(X) is an equational theory of the algebra T(X)/κ. Proof. “=⇒”: Let s = s < x1 , ..., xn >, t = t < x1 , ..., xn > and (s, t) ∈ κ. For arbitrary t1 , ..., tn ∈ T (X) it holds because of full invariance of κ: (s < t1 , ..., tn >, t < t1 , ..., tn >) ∈ κ. Consequently, we have: sT(X)/κ < t1 /κ, ..., tn /κ >= tT(X)/κ < t1 /κ, ..., tn /κ >, i.e., the equation s ≈ t holds in T(X)/κ. “⇐=”: Let s ≈ t ∈ IdX (T(X)/κ). Then it holds sT(X)/κ < x1 /κ, ..., xn /κ >= tT(X)/κ < x1 /κ, ..., xn /κ > . Thus (sT(X) , tT(X) ) ∈ κ and (s, t) ∈ κ.

The following theorem is a conclusion from the Lemmas 6.6.2 and 6.6.3:

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Theorem 6.6.4 (Second Main Theorem of the Equational Theory; [Bir 35]) A set Σ ⊆ T (X) × T (X) is an equational theory iﬀ Σ is a fully invariant congruence on T (X). Because of Theorem 6.3.1, (6) and Lemma 6.6.1, one can also write Theorem 6.6.4 as follows: Theorem 6.6.5 (Completeness Theorem for the Equational Logic; [Bir 35]) For an arbitrary alphabet X and an arbitrary Σ ⊆ T (X) × T (X) it holds: (a) Σ = ConsX (Σ) ⇐⇒ D(Σ) = Σ; (b) D(Σ) = ConsX (Σ). Proof. (a): “=⇒”: Let Σ = ConsX (Σ). Then, we have D(Σ) ⊆ ConsX (Σ) = Σ and Σ ⊆ D(Σ). Thus D(Σ) = Σ. “⇐=”: Let D(Σ) = Σ. Then, by Theorem 6.6.1, Σ is a fully invariant congruence on T (X). With the help of Theorem 6.6.4 it follows from this that Σ is an equational theory. Therefore, by Theorem 6.3.1, (6) Σ = ConsX (Σ). (b): Let Σ1 := D(Σ). Then we have D(Σ1 ) = Σ1 , Σ ⊆ Σ1 and D(Σ) ⊆ ConsX (Σ) ⊆ ConsX (Σ1 ).

(6.2)

By D(Σ1 ) = Σ1 it follows from (a): ConsX (Σ1 ) = Σ1 . Then, because of the idempotency of D, we have: D(Σ) = D(D(Σ)) = D(Σ1 ) = ConsX (Σ1 ). This and (6.2) imply D(Σ) = ConsX (Σ).

6.7 Finite Axiomatizability of Algebras The reader needs knowledge of the other sections of this chapter, as well as some knowledge of Part II for this section. An old question in universal algebra is whether, for given algebra A of ﬁnite type, there is a ﬁnite set Σ ⊆ IdX (A) with D(Σ) = IdX (A), where X := {x1 , x2 , x3 , ...}. For the case that D(Σ) = IdX (A) holds for a ﬁnite set Σ, we say that A is ﬁnitely axiomatizable or ﬁnitely based. The following theorem is easy to prove.

6.7 Finite Axiomatizability of Algebras

85

Theorem 6.7.1 ([Lyn 51]) Let A := (A; F ), B := (B; G) be ﬁnite and equivalent algebras of ﬁnite types; i.e., |A| < ℵ0 , A = B, |F | < ℵ0 , |G| < ℵ0 and T F (A) = T F (B) 4 . Then A is ﬁnitely axiomatizable if and only if B is.

The next theorem was founded by G. Birkhoﬀ. Theorem 6.7.2 Let A be a ﬁnite algebra of ﬁnite type (F, τ ) and let Xn := {x1 , x2 , ..., xn } be a ﬁnite set of variables. Then IdXn (A) is ﬁnitely axiomatizable. Proof. Let κ := {(s, t) ∈ T (Xn )×T (Xn ) | A |= s ≈ t}. Then κ ∈ ConT(Xn ) by Theorem 6.4.2, (b). The congruence κ has only a ﬁnitely many equivalence classes ε1 , ..., εq , since A, F and Xn are ﬁnite, and since (s, t) ∈ κ iﬀ the induced term functions sA , tA satisfy sA = tA . For t ∈ T (X) we denote by #t the number of operation symbols (∈ F) occurring in t. In particular, if t ∈ X then #t = 0. Now, from each equivalence class εi (i ∈ {1, ..., q}) of κ, we choose one representative ri with #ri ≤ #t for all terms t ∈ εi . Set M := {r1 , ..., rq }∪{f (ri1 , ri2 , ..., riaf ) | f ∈ F\F0 , {ri1 , ..., riaf } ⊆ {r1 , ..., rq }}, m := max{#ϕ | ϕ ∈ M } and Σ := {s ≈ t ∈ IdXn (A) | #s ≤ m ∧ # t ≤ m}. By induction on α we prove that ∀α ∈ N0 : (s ≈ t ∈ IdXn (A) ∧ #s ≤ α ∧ #t ≤ α) =⇒ s ≈ t ∈ D(Σ) . =:S(α)

(6.3) (I) If α ≤ m, then the statement S(α) is obviously valid. (II) Assume, S(β) holds for certain β ≥ m. Let s ≈ t ∈ IdXn (A) be arbitrary with #s ≤ β + 1 and #t ≤ β + 1. First, we consider the case s := f (s1 , s2 , ..., saf ), t := g(t1 , t2 , ..., tag ),

(6.4)

where f, g ∈ F and s1 , ..., saf , t1 , ...tag ∈ T (Xn ). Then #s1 +...+#saf ≤ β and #t1 + ... + #tag ≤ β. By deﬁnition of r1 , ..., rq there exist u1 , u2 , ...., uaf , v1 , v2 , ..., vag , w ∈ {1, ..., q} with [s1 ]κ = [ru1 ]κ , ..., [saf ]κ = [ruaf ]κ , [t1 ]κ = 4

In other words, the operations of algebra A can be represented as superpositions over the operations of algebra B and vice versa (see Part II, Section 1.5.1).

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[rv1 ]κ , ...., [tag ]κ = [rvag ]κ , [f (s1 , ..., saf )]κ = [rw ]κ = [g(t1 , ..., tag )]κ . Then, by assumption, we have si ≈ rui , tj ≈ rvj ∈ D(Σ) and (by deﬁnition of Σ) f (ru1 , ..., ruaf ) ≈ rw , g(rv1 , ..., rvag ) ≈ rw ∈ Σ. With the aid of these equations and the rules (R1)–(R3), (Rep) and (Sub) we get Σ f (s1 , ..., saf ) ≈ f (s1 , ..., sn ) =⇒ Σ f (s1 , ..., saf ) ≈ f (ru1 , ..., ruaf ) =⇒ Σ f (s1 , ..., saf ) ≈ rw =⇒ Σ f (s1 , ..., saf ) ≈ g(rv1 , ..., rvag ) =⇒ Σ f (s1 , ..., saf ) ≈ g(t1 , ..., tag ), i.e., (s, t) ∈ D(Σ) in case (6.4). In the remaining cases (i.e., {s, t}∩Xn = ∅), one can prove the above in analog mode as well. Thus, (6.3) is right, whereby IdXn (A) is ﬁnitely axiomatizable. Remark: The above proof shows that one can choose, instead of Σ, the following ﬁnite set Σ : Σ := {x ≈ y | x, y ∈ Xm ∧ (x, y) ∈ κ}∪ {x ≈ r | x ∈ Xn ∧ r ∈ {r1 , ..., rq } ∧ (x, r) ∈ κ}∪ {f (g1 , ..., gaf ) ≈ g | f ∈ F ∧ {g1 , ..., gaf , g} ⊆ {r1 , ..., rq } ∧ (f (g1 , ..., gaf ), g) ∈ κ} As a conclusion of Theorems 6.7.1 and 6.7.2 we get: Theorem 6.7.3 Let A := (A; F ) be a ﬁnite algebra of ﬁnite type with [A] ⊆ (1) [PA ], i.e, the operations of F have at most an essential variable (see Part II, Chapter 1). Then A is ﬁnitely axiomatizable. The equational theory and parts of the mathematical logic deal with similar problems. Therefore, one can use sometimes results of the one theory for the other and vice versa. For this purpose, the next theorem provides an example. Because of the better survey, we agree with the following notation: If A = ({0, 1}; F ), f ∈ F deﬁned by f (x) := ¬x or f (x, y) := x ◦ y

(6.5)

(◦ ∈ {∨, ∧, ⇒}) and t is a term, whose operation symbols belong to F , then tˆ denotes a formula of P rop (see Part II, Section 1.5.2), which one obtains by (6.5).

6.7 Finite Axiomatizability of Algebras

87

Theorem 6.7.4 ([Lyn 51]) Let A = ({0, 1}; 0, 1, f1 , f2 , f3 , g) be an algebra of the type (0, 0, 1, 2, 2, 2) with f1 (x) := ¬x, f2 (x, y) := x ∧ y, f3 (x, y) := x ∨ y and g(x, y) := x ⇒ y (see Table 1.2 of Part II). Further, let T be a set of tautologies (⊆ P rop) with the property that every tautology has a derivation from T with the help of sub and modus ponens. 5 Then, the equations (1) x ⇒ x ≈ 1 ( g(x, x) ≈ 1 ), (2) 1 ⇒ x ≈ x ( g(1, x) ≈ x ), (3) (x ⇒ y) ⇒ y ≈ (y ⇒ x) ⇒ x ( g(g(x, y), y) ≈ g(g(y, x), x) ) and the equations of the type (4) t ≈ 1 f¨ ur every t ∈ T form a system Σ of equations with D(Σ) = IdX (A) (more precise: D(Σ) = {ˆ α | α ∈ IdX (A)}). Proof. First, we show that (ϕ ∈ P rop is a tautology) implies ( ϕ ≈ 1 ∈ D(Σ) ).

(6.6)

By assumption, we can derive a tautology ϕ from T with the aid of sub and modus ponens. Consequently, we have to prove that it is possible to copy the modus ponens through the rules (R1)–(R3), (Rep) and (Sub). Let σ ≈ 1, σ ⇒ τ ≈ 1 ∈ D(Σ). Then τ ≈ 1 ∈ D(Σ) follows from (2)

Sub

σ ≈ 1, σ ⇒ τ ≈ 1 1 ⇒ τ ≈ 1 τ ≈ 1. Thus (6.6) holds. Now, let α ≈ β ∈ IdX (A) be arbitrary; i.e., α ⇒ β and β ⇒ α are tautologies. We show α ≈ β ∈ D(Σ). By (6.6) we have that α ⇒ β ≈ 1 and β⇒α≈1

(6.7)

belong to D(Σ). Furthermore, by (3) and (Rep): (β ⇒ α) ⇒ α ≈ (α ⇒ β) ⇒ β ∈ D(Σ). Thus (β ⇒ α) ⇒ α ≈ (α ⇒ β) ⇒ β

(Sub),(6.7)

1⇒α≈1⇒β

(2),(Sub)

α ≈ β,

whereby α ≈ β ∈ D(Σ) is shown. R. L. Lyndon proven the following basic result with the aid of Theorem 6.7.1 and Post’s theorem (see Part II, Theorem 3.1.1): 5

see Part II, Section 1.5.2.

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6 Varieties, Equational Classes, and Free Algebras

Theorem 6.7.5 ([Lyn 51], without proof ) Every two-element algebra of ﬁnite type is ﬁnitely axiomatizable. Notice that J. Berman gave a short proof for the above theorem in [Ber 80] with the aid of theorems by Baker ([Bak 77]) and McKenzie ([McK 78]). R. C. Lyndon constructed a 7-element algebra of type (0, 2) whose equations are not ﬁnitely based (see [Lyn 54]). The smallest such example was found by V. L. Murskij: Theorem 6.7.6 ([Mur 65]; without proof ) The algebra ({0, 1, 2}; ◦), where ◦ is deﬁned by ◦ 0 1 2 0 0 0 0 , 1 0 0 1 2 0 2 2 is not ﬁnitely axiomatizable. Notice that all ﬁnite groups, rings and lattices are ﬁnitely axiomatizable (see [Oat-P 65], [Kru 73], and [McK 70], respectively). The result for lattices was considerably generalized by K. Baker ([Bak 77]), who proven that every ﬁnite algebra whose generated variety is congruence distributive is ﬁnitely axiomatizable. In [Per 69] was proven that the multiplicative semigroup of all 2 × 2matrices over a 2-element ﬁeld has a 6-element subsemigroup with no ﬁnite basis. One can ﬁnd further important results on the topic in [McK 78] and [Wil 2001].

Part II

Function Algebras

1 Basic Concepts, Notations, and First Properties

In this chapter, we begin by investigating multi-digit operations, which are deﬁned on a ﬁnite set A. We deﬁne some operations on the set of these operations. For the purpose of distinction, we subsequently replace the concept “operation (on the set A)” by the concept “function (on the set A)”.

1.1 Functions on Finite Sets Let A be a ﬁnite set with at least two elements. Often we choose the set Ek := {0, 1, 2, ..., k − 1}, k ≥ 2 instead of A in the following. We say that f is an n-ary (n-digit) function on A (or an n-ary function of the |A|-valued logic1 ), if f a mapping from the n-fold Cartesian product An into A, n ≥ 1. For technical reasons (see Section 1.3), we renounce that, in this section, we consider also nullary functions. Otherwise, we use the concepts introduced in Chapter 1 of Part I for operations: af , f n , D(f ), ... . Let PAn be the set of all n-ary functions on A 2 , n ≥ 1. Instead of PEnk we write also Pkn . Further let

1 2

One ﬁnds an explanation for this notation in Section 1.5. A confusion with the direct product is not possible for content-related reasons.

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1 Basic Concepts, Notations, and First Properties

PA :=

n≥1

PAn ,

F n := F ∩ PAn for every subset F of PA , Pk := n≥1 Pkn , PA,B := {f ∈ PA | Im(f ) ⊆ B}, Pk,l := PEk ,El , PA (l) := {f ∈ PA | |Im(f )| ≤ l}, Pk (l) := PEk (l), PA [l] := {f ∈ PA | |Im(f )| = l} and Pk [l] := PEk [l] (2 ≤ l ≤ k). With (x1 , ..., xn ) (brieﬂy x(n) ) or x we denote an arbitrary n-tuple of An or Ekn and usually we say that the xi (i = 1, 2, ..., n) are variables. If n = 2 or n = 3 we also write (x, y) or (x, y, z) instead of (x1 ..., xn ), respectively. We will deﬁne, subsequently, speciﬁc functions f n from Pk either through a table of the form Table 1.1 x1 x2 0 0 0 0 . . a1 a2 . . k−1 k−1

... xn ... 0 ... 1 ... . ... an ... . ... k − 1

f (x1 , x2 , ..., xn ) f (0, 0, ..., 0) f (0, 0, ..., 1) . f (a1 , a2 , ..., an ) . f (k − 1, k − 1, ..., k − 1)

or through formulas, for example, of the form ∀x ∈ Ekn : f (x1 , ..., xn ) := x1 + ... + xn (mod k),

(1.1)

on the (variable-) alphabet {x, y, z, x1 , x2 , ...}. We write often instead of (1.1) brieﬂy (1.2) f (x1 , ..., xn ) := x1 + ... + xn (mod k) or (if the arity of f is clear from the context or is without importance) we write still more brieﬂy “f is deﬁned by x1 + ... + xn (mod k) ”;

(1.3)

i.e., we do not distinguish between a function and the formula (or term) deﬁning it.3 3

One ﬁnds the concept “term” explained in Part I, Section 6.2.

1.1 Functions on Finite Sets

93

Two functions f n , g m ∈ PA are identical (we write f n = g m ) iﬀ n = m and f (x) = g(x) for all x ∈ An hold. Let f n ∈ PA and i ∈ {1, 2, ..., n}. Then we say that the i-th variable (or the i-th place) of the function f ∈ PA is essential, iﬀ there are n-tuples a = (a1 , ..., ai−1 , b, ai+1 , ..., an ) and a = (a1 , ..., ai−1 , c, ai+1 , ..., an ) such that b = c and f (a) = f (a ) hold. In the opposite case, one calls the i-th variable (or i-th place) of f ﬁctitious (or non-essential). If the i-th variable of f is not ﬁctitious, we say that f depends on the i-th variable. The function eni deﬁned by eni (x1 , ..., xn ) := xi (i ∈ {1, ..., n}) is called projection or also selector. Let JA (or Jk ) be the set of all projections of PA (or Pk ), respectively. A constant function (brieﬂy, constant) is a function cna deﬁned by cna (x1 , ..., xn ) := a, where a ∈ A. Notations for certain functions of P2 , the Boolean functions, are given in the following table, where it is deﬁned, as usual ◦(x, y) := x ◦ y if ◦ ∈ {∧, ∨, +, ⇒, ⇐⇒} and

−

(x) := x.

Table 1.2

x x x y x ∧ y x ∨ y x + y x ⇒ y x ⇐⇒ y 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 1 Instead of x ∧ y we also write x · y or we write xy brieﬂy. It can easily be shown that the above functions have the following properties: Theorem 1.1.1 It holds: (a) ∀◦ ∈ {∨, ∧, ⇐⇒, +} : x ◦ (y ◦ z) = (x ◦ y) ◦ z; (b) x ∨ x = x, x ∧ x = x, x ⇐⇒ x = 1, x ⇒ x = 1, x + x = 0, x ∨ 0 = x, x ∧ 1 = x, x ∨ 1 = 1, x ∧ 0 = 0;

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1 Basic Concepts, Notations, and First Properties

(c) ∀◦ ∈ {∨, ∧, +, ⇐⇒} : x ◦ y = y ◦ x; (d) x ∧ x = 0, x ∨ x = 1, x = x, x ∨ y = x ∧ y, x ∧ y = x ∨ y (“de Morgan’s laws”); (e) x ⇒ y = x ∨ y, x ⇒ y = x ∧ y; (f ) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z); (g) x ∧ (x ∨ y) = x, x ∨ (x ∧ y) = x.

1.2 Operations on PA, Function Algebras The “formula notation” of our functions from the ﬁrst section motivates the determination of the following operations on PA : – permutation of variables, – identiﬁcation of variables, – adding of ﬁctitious variables and – substitution of variables of a function by functions; which are called superposition operations on PA and which one can describe in diﬀerent way exactly. We give only two possibilities here. First we want describe the above operations through the following (inﬁnite many) partial operations: πs : PAn −→ PAn ∆t : PAn −→ PAr (r < n) ∇q : PAn −→ PAu (u > n) i : PAn × PAm −→ PAn+m−1 Let f n , g m be functions of PA , let s be a permutation on the set {1, 2, ..., n}, let t be a mapping from {1, 2, ..., n} onto {1, 2, ..., r} (r < n), let q be an injective mapping from {1, 2, ..., n} into {1, 2, ..., u} (u > n) and let i ∈ {1, 2, ..., n}. Then, πs f ∈ PAn , ∆t f ∈ PAr , ∇q f ∈ PAu , f i g ∈ PAm+n−1 are deﬁned by (πs f )(x1 , ..., xn ) := f (xs(1) , xs(2) , ..., xs(n) ) (“permutation of variables of f ”), (∆t f )(x1 , ..., xr ) := f (xt(1) , xt(2) , ..., xt(n) ) (“identiﬁcation of certain variables of f ”), (∇q f )(x1 , x2 , ..., xu ) := f (xq(1) , xq(2) , ..., xq(n) ) (“adding of certain ﬁctitious variables”)

1.2 Operations on PA , Function Algebras

95

and (f i g)(x1 , ..., xm+n−1 ) := f (x1 , ..., xi−1 , g(xi , ..., xi+m−1 ), xi+m , ..., xm+n−1 ) (“the replacement of the i-th variable of f through the function g and the changing of the denotation of variables of f ”). For partial operations α ∈ {πs , ∆t , ∇q , i } deﬁned above, one can continue to certain operations α on PA . For later investigations, however, it is better that we choose the minimal number of the operations on PA . Therefore, next we consider that there are ﬁve elementary operations (or Mal’tsevoperations) ζ, τ, ∆, ∇, on PA , with which we can form certain continuations of the partial operations πs , ∆t , ∇q , i for arbitrary s, t, q, i, n, m through composition. These operations were published by A. I. Mal’tsev in [Mal 66], and one can deﬁne these operations as follows for arbitrary f n , g m ∈ PA : max{1,n−1}

ζf n ∈ PAn , τ f n ∈ PAn , ∆f n ∈ PA

, ∇f n ∈ PAn+1 , f n g m ∈ PAm+n−1

and (ζf )(x1 , ..., xn ) := f (x2 , x3 , ..., xn , x1 ), (τ f )(x1 , ..., xn ) := f (x2 , x1 , x3 , ..., xn ), (∆f )(x1 , ..., xn−1 ) := f (x1 , x1 , x2 , ..., xn−1 ) if n ≥ 2, ζf = τ f = ∆f = f if n = 1, (∇f )(x1 , ..., xn+1 ) := f (x2 , x3 , ..., xn+1 ), (f g)(x1 , ..., xm+n−1 ) := f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ). To prove that one can describe the operation πs (over PAn ) by means of operations ζ, τ (over PA ), it suﬃces to show that the set Sn of all permutations on the set {1, 2, ..., n} is generating by the permutations (12...n) and (12) (given in cyclic description); that is, that [{(12...n), (12)}]2 = Sn holds, where (s2s )(x) := s (s(x)) for all s, s ∈ Sn . But, this follows from the fact that every permutation s ∈ Sn is a product of pairwise disjunct cycles: s = (i1 ...ip )(j1 ...jq )..., that every cycle is a product of transpositions (i1 ...ip ) = (i1 i2 )(i1 i3 )...(i1 ip ), and that (ij) = (1i)(1j)(i1) for arbitrary i > j, i > 1, (1i) = (12)(23)...(i − 1i)(i − 1, i − 2)...(21) and (i, i + 1) = (12...n)n−i+1 (12)(12...n)i−1

96

1 Basic Concepts, Notations, and First Properties

are valid. Evidently, then, the operations ∆t or ∇q or i can be created (in respect to 2) through the operations πs (s ∈ Sn ), ∆ or πs (s ∈ Sn ), ∇ or πs (s ∈ Sn ), respectively. Thus we proven the following lemma. Lemma 1.2.1 It holds: (a) For every permutation s ∈ Sn there exists an operation π ∈ [{ζ, τ }]2 with πs f = π s f for all f ∈ PAn . (b) For every mapping t from {1, 2, ..., n} onto {1, 2, ..., r} (r < n) there ex ∈ [{ζ, τ, ∆}]2 with ∆t f = ∆ t f for arbitrary f ∈ P n . ists an operation ∆ A (c) For every injective mapping q from {1, 2, ..., n} into {1, 2, ..., u} there ex q f for arbitrary f ∈ P n . q ∈ [{ζ, τ, ∇}]2 with ∇q f = ∇ ists an operation ∇ A (d) For every i ∈ {1, 2, ..., n}, there exists an operation i ∈ [{ζ, τ, }]2 with i g for arbitrary f ∈ PAn and arbitrary g ∈ PAm . f i g = f By means of the operations ζ, τ, ∆, ∇, we can describe the subject of investigation of this chapter. PA together with the operations e21 , ζ, τ, ∆, forms an algebra (PA ; e21 , ζ, τ, ∆, ) of the type (0, 1, 1, 1, 2), which is called (full) function algebra on A. A little changed form of the full function algebra is the so-called iterative (full) function algebra (PA ; ζ, τ, ∆, ∇, ) of the type (1, 1, 1, 1, 2). Since ∇f = f (τ e21 ) is valid, however, both algebras can be regarded as equivalent in a certain sense: If (S; e21 , ζ, τ, ∆, ) is a subalgebra of (PA ; e21 , ζ, τ, ∆, ), then (S; ζ, τ, ∆, ∇, ) is also a subalgebra of (PA ; ζ, τ, ∆, ∇, ). Conversely, if (T ; ζ, τ, ∆, ∇, ) is a subalgebra of (PA ; ζ, τ, ∆, ∇, ), then (T ∪JA ; e21 , ζ, τ, ∆, ) is a subalgebra of (PA ; e21 , ζ, τ, ∆, ). Therefore, we often deal only with the algebra PA = (PA ; ζ, τ, ∆, ∇, ).

1.3 Superpositions, Subclasses, and Clones A function f ∈ PA is called a superposition over F (⊆ PA ), if f can be obtained by a ﬁnite number of applications of the operations ζ, τ, ∆, ∇, from the functions of F . We describe a superposition f over F in the rarest cases f through a term over certain function symbols, ζ, τ, ∆, ∇, and parentheses. We use the variable alphabet {x, y, z, x1 , x2 , ...},

1.3 Superpositions, Subclasses, and Clones

97

certain function symbols, commas, and parenthesis instead of this. In some cases, where an equation for the precise deﬁnition of the function would be necessary formally, we are satisﬁed with the the right side of the deﬁning equation if the remaining information on the function results from the context. Further, if f ∈ Pkn , g1 , ...gn ∈ Pkm and the m-ary function h ∈ Pk is deﬁned by h(x1 , ..., xm ) := f (g1 (x1 , ..., xm ), g2 (x1 , ..., xm ), ..., gn (x1 , ..., xm )), then, we write brieﬂy h := f (g1 , ..., gn ). The set of all superpositions over F (⊆ PA ) is called hull or closure of F and it is denoted by [F ]. Obviously, [..] is a hull operator on the set PA . A set F ⊆ PA satisfying [F ] = F is called a closed set or a subclass or brieﬂy class of PA . We deﬁne that the empty set is also a closed set, i.e., ∅ = [∅]. One can form many examples of closed sets with the aid of the following concept: Let T ⊆ Pkm and f ∈ Pkn . Then we say that f preserves the set T iﬀ ∀g1 , ..., gn ∈ T : f (g1 , ..., gn ) ∈ T. It is easy to see that a projection preserves every set T ⊆ Pkm and that the set of all functions, which preserve the set T , is closed. The set F ⊆ PA is called a clone of PA , if F is closed and JA ⊆ F holds.4 Obviously, the subclasses of PA are exactly the universes of subalgebras of (PA ; ζ, τ, ∆, ∇, ) and clones are exactly the universes of subalgebras of (PA ; e21 , ζ, τ, ∆, ). Let LA be the set of all closed subsets of PA . Put Lk := LEk . (LA ; ⊆) is a lattice (see Part I, Chapter 2). Further, let L↓A (F ) := {F ∈ LA | F ⊆ F } and

L↑A (F ) := {F ∈ LA | F ⊆ F }.

Analogously, one can deﬁne L↑k (F ) and L↓k (F ) for A = Ek . Further, let LA (F ; G) := L↑A (F ) ∩ L↓A (G), where F, G ∈ LA and F ⊂ G. 4

If F is a class of PA , then JA ⊆ F iﬀ e11 ∈ F .

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If [G] = F (⊆ PA ), then G is called complete in F . In particular, if F = PA , we say, G is complete or G is a complete set. A closed set F is called a maximal subclass of the closed set F , if F ⊂ F and [F ∪ {f }] = F for every f ∈ F \F . If F = PA then, we say brieﬂy, F is a maximal class. The maximal classes of the maximal classes of PA are called submaximal classes. As usual, we call a subset F of F a generating system of F , if [F ] = F . A generating system F of F is called basis of the closed set F , if every proper subset of F is not a generating system of F . If a subclass F of PA has a ﬁnite generating system, then the order of F we denote with ord F . We understand from that, the smallest number with [F r ] = F . If F does not have any ﬁnite generating system, we write ord F = ∞.

1.4 Generating Systems for PA For the purpose of determining of generating systems for the set PA , we consider some descriptions (so-called “normal forms”) for an arbitrary function f n ∈ PA . These descriptions are superpositions over certain functions of PA , which are to be described easily and which have small arities. We use the following notations: 1 if x = a, ja (x) := 0 otherwise (a ∈ A) and

ja (x1 , ..., xn ) :=

1 if (x1 , ..., xn ) = a, 0 otherwise

(a ∈ An , n ∈ N). Theorem 1.4.1 (Representation Theorem for Functions of PA ) Let 0, 1 ∈ A and let ∧, ∨ be two binary associative5 operations on A with a ∧ 1 = a, 0 ∨ a = a ∨ 0 = a and a ∧ 0 = 0

(1.4)

for each a ∈ A. Then, for every function f n ∈ PA it holds: f (x) = (

m

fai (x) := )fa1 (x) ∨ fa2 (x) ∨ ... ∨ fam (x),

i=1

where An := {a1 , ..., am }, m := |A|n and

(1.5)

1.4 Generating Systems for PA

99

fai (x) := cf (ai ) (x1 ) ∧ jai (x) (i = 1, ..., m). Furthermore: jai (x) = jai1 (x1 ) ∧ jai2 (x2 ) ∧ ... ∧ jain (xn ), where ai := (ai1 , ..., ain ). Proof. The correctness of the equation (1.5) can be assured by checking that on both the left side and the right side of the formula, the same value stands for every x. If A = {0, 1}, the functions ∨, ∧ (= ·) are deﬁned as in Table 1.2, j0 (x) = x and j1 (x) = x, then one receives the disjunctive normal form (or DNF) of an arbitrary Boolean function f n ∈ P2 as an conclusion from (1.5): f (x1 , ..., xn ) = f (a1 , ..., an ) · xa1 1 · xa2 2 · ... · xann , (1.6) a∈E2n

where α

x :=

x if α = 0, x if α = 1

(α ∈ E2 ). If f = cn0 , then we can write f (x1 , ..., xn ) =

xa1 1 · xa2 2 · ... · xann

(1.7)

a∈E2n ,f (x)=1

instead of (1.6). For example, f (x, y, z) = x · y · z ∨ x · y · z ∨ x · y · z is the DNF for the ternary function f deﬁned by Table 1.3. Table 1.3

x 0 0 0 0 1 1 1 1 5

y 0 0 1 1 0 0 1 1

z f (x, y, z) 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1

The associativity can be renounced if the necessary parentheses are put in the following formulas.

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If A is an arbitrary ﬁnite set, then one can choose ∧ and ∨ as lattice operations with 0 = A and 1 = A. Then, by Theorem 2.1.2 of Part I, we have ∨(x, y) = sup {x, y} and ∧(x, y) = inf {x, y}, where is the partial order (appertaining to the lattice) over A with the greatest element 1 and the least element 0. If A = Ek , then the functions ∨ := + (mod k) and ∧ := · (mod k) also fulﬁll (1.4) and we get the following normal form for an arbitrary function f n ∈ Pk : f (a1 , ..., an ) · ja1 (x1 ) · ... · jan (xn ) (mod k) (1.8) f (x) = a∈Ekn

The following theorem results from the above considerations immediately: Theorem 1.4.2 It holds: (a) Let ∨ and ∧ be binary operations on A, which (1.4) fulﬁll. Then, {∨, ∧}∪ {c1a , ja1 | a ∈ A} is a generating system for PA . In particular, if A = E2 , then [{∨, ∧,− }] = P2 and (because of Theorem 1.1.1, (d)) [{∨,− }] = [{∧,− }] = P2 . (b) ord PA = 2.

Theorem 1.4.3 Let A = Ek and let k = pm be a prime number power. Then one can deﬁne operations + and · on Ek so that (Ek ; +, ·) is a ﬁeld with the neutral element o in respect to + and the neutral element e of the group (Ek \{o}; ·). Then, one can represent an arbitrary function f n ∈ Pk with the aid of these ﬁeld operations as follows: ai1 i2 ...in · xi11 · xi22 · ... · xinn (1.9) f (x) = (i1 ,...,in )∈Ekn

(x0 := e; ai1 i2 ...in ∈ Ek ). This representation is unique except the order of addends, i.e., the equality of the corresponding coeﬃcients results from the equality of two functions ∈ Pkn . Proof. The existence of a ﬁeld (Ek ; +, ·) for k = pm , p ∈ P and m ∈ N is well-known. (see for example [Lid-N 87] or [Lau 2004], volume 2).

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101

Every polynomial of the form (1.9) is uniquely represented through the sen quence of coeﬃcients. Thus there are k (k ) diﬀerent formulas of the form (1.9). n (kn ) , our theorem is proven if Since |Pk | = k f (x) = (i1 ,...,in )∈E n ai1 i2 ...in xi11 xi22 ...xinn and k f (x) = (i1 ,...,in )∈E n bi1 i2 ...in xi11 xi22 ...xinn k

implies ai1 ...in = bi1 ...in for all (i1 , ..., in ) ∈ Ekn . This is clear for (i1 , ..., in ) = (o, o, ..., o) (one forms f (o, ..., o)!). Let I := {xij | ij = o ∧ j ∈ {1, ..., n}} for the proof of ai1 ...in = bi1 ...in in the case (i1 , ..., in ) ∈ Ekn \{o}. If one identiﬁes now the variables in f from I with x and one replaces the remaining variables through c0 (x), then one receives a unary function f that can be represented as follows: (1.10) f (x) = a0 + a1 · x + a2 · x2 + ... + ar−1 · xr−1 or

f (x) = b0 + b1 · x + b2 · x2 + ... + br−1 · xr−1

(1.11)

with r−1 := |I|, ar−1 = ai1 ...in and br−1 = bi1 ...in for certain a0 , ..., ar−2 , b0 , ..., br−2 . If one forms now in (1.10) and (1.11) f (α1 ), f (α2 ), ..., f (αr ) for pairwise distinct α1 , α2 , ..., αr ∈ Ek , then one sees that both (a0 , ..., ar−1 )T as also (b0 , ..., br−1 )T is a solution of the matrix equation A · x = (f (α1 ), ..., f (αr ))T with ⎛ ⎞ 1 α1 α12 ... α1r−1 ⎜ 1 α2 α2 ... αr−1 ⎟ 2 2 ⎟ A := ⎜ ⎝ ................. ⎠. 1 αr αr2 ... αrr−1 But, because of det A = o,6 this is only possible for a0 = b0 , ..., ar−1 = br−1 . The next property of functions of Pk for k ≥ 3 is not only useful while determining from generating systems for Pk ; it also has interesting consequences, which we will deal with later. We need the following denotation:7 ιhk := {(a1 , ..., ah ) ∈ Ekh | |{a1 , ..., ah } | ≤ h − 1} (h ≥ 2), 3 δ{α,β} := {(a1 , a2 , a3 ) ∈ Ek3 | aα = aβ } (α, β ∈ {1, 2, 3}) and 3 δ{1,2,3} := {(x, x, x) | x ∈ Ek }. 6 7

This follows from the fact that A is a Vandermonde matrix (see for example [Lau 2004], volume 1). See also Chapter 2.

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Furthermore, for arbitrary ri := (r1i , r2i , ..., rhi ) ∈ Ekh , i = 1, 2, .., n, and f ∈ Pkn we put: f (r1 , ..., rn ) := (f (r11 , r12 , ..., r1n ), f (r21 , r22 , ..., r2n ), .., f (rh1 , rh2 , ..., rhn )). Theorem 1.4.4 Let f be an n-ary function of Pk , which is essentially dependent of at least two variables (w.l.o.g. of x1 and x2 ) and which has q pairwise distinct values. Then: 3 3 (a) q ≥ 3 =⇒ ∃r1 , ..., rn ∈ δ{1,2} ∪ δ{2,3} : f (r1 , ..., rn ) ∈ Ek3 \ι3k

(“Fundamental Lemma of Jablonskij”); (Jab 58]) (b) q ≥ 3 =⇒ ∃r1 , ..., rn ∈ ιqk : f (r1 , ..., rn ) ∈ Ekq \ιqk ; 3 3 3 3 ∪ δ{2,3} : f (r1 , ..., rn ) ∈ δ{1,3} \δ{1,2,3} . (c) q = 2 =⇒ ∃r1 , ..., rn ∈ δ{1,2}

Proof. (a), (c): Since f depends on the variable x1 essentially, there is an a := (a2 , ..., an ) ∈ Ekn−1 , so that Ta := {f (x, a2 , ..., an ) | x ∈ Ek } has at least two diﬀerent elements. We distinguish two cases: Case 1: |Ta | < q. In this case, one can ﬁnd a tuple c = (c1 , ..., cn ) with γ := f (c) ∈ Ta . Consequently, we have ⎞ ⎛ ⎞ ⎞ ⎛ ⎛ α f (a1 , a2 , ..., an ) a1 a2 ... an f ⎝ c1 a2 ... an ⎠ := ⎝ f (c1 , a2 , ..., an ) ⎠ = ⎝ β ⎠ c1 c2 ... cn γ f (c1 , c2 , ..., cn ) and |{α, β, γ}| = 3 for certain a1 ∈ Ek . Case 2: |Ta | = q. Since f also depends on x2 essentially, the function f1 (x1 , ..., xn−1 ) := f (d, x1 , ..., xn−1 ) is not a constant function for certain d ∈ Ek . Let now β := f (d, a2 , ..., an ). Because of f1 = cβ there are certain c2 , ..., cn and a γ with γ := f (d, c2 , ..., cn ) = β . Since |Ta | = q, then one can ﬁnd an a1 ∈ Ek with γ if q = 2, α := f (a1 , a2 , ..., an ) = α ∈ {β , γ } if q ≥ 3. Consequently, we have ⎞ ⎛ ⎞ α a1 a2 ... an f ⎝ d a2 ... an ⎠ = ⎝ β ⎠ . d c2 ... cn γ ⎛

1.4 Generating Systems for PA

103

(b) follows easy from (a). From the many conclusions from this theorem, we provide only the following, ﬁrst. Lemma 1.4.5 Let f n be a function of Pk , which is essentially dependent on two variables and which has q ≥ 3 distinguish values. Then: Pk,Im(f ) ⊆ [{f } ∪ Pk (q − 1)]. Proof. W.l.o.g. let Im(f ) = Eq . By Theorem 1.4.4, (b) there exist r1 , ..., rn ∈ ιqk with f (r1 , ..., rn ) = (0, 1, ..., q − 1)T and ⎞ ⎛ a02 ... a0n a01 ⎜ a11 a12 ... a1n ⎟ ⎟ (r1T , ..., rnT ) = ⎜ ⎝ ....................... ⎠. aq−1,1 aq−1,2 ... aq−1,n For an arbitrary function g m ∈ Pk,Im(f ) let gj (x1 , ..., xm ) = aij :⇐⇒ ∃i : g(x1 , ..., xm ) = i, (j = 1, 2, ..., n). Obviously, the functions g1 , ..., gn belong to Pk (q − 1) and it holds: g(x1 , ..., xm ) = f (g1 (x1 , ..., xm ), ..., gn (x1 , ..., xm )). Thus g ∈ [{f } ∪ Pk (q − 1)]. Subsequently, we declare another possibility of the characterization of functions of Pk that we need later during the description of a certain type of maximal classes of Pk . Lemma 1.4.6 Let Ai (i ∈ Ek ) be a partition of the set Ek and ai ∈ Ai (i ∈ Ek ). Furthermore, let y if ∃i ∈ Ek : x = ai and y ∈ Ai , x y := x otherwise. Then, an arbitrary function f n of Pk is a superposition over the functions z, gf , fi (i ∈ Ek ) deﬁned by z(x, y) := x y, ai ⇐⇒ f (x1 , ..., xn ) ∈ Ai , gf (x1 , ..., xn ) := f (x1 , ..., xn ) if f (x1 , ..., xn ) ∈ Ai , fi (x1 , ..., xn ) := otherwise ai (i ∈ Ek ), and f (x) = ((...((gf (x) f0 (x)) f1 (x)) ...) fk −1 (x)) n

is a representation of f .

(1.12)

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1 Basic Concepts, Notations, and First Properties

1.5 Some Applications of the Function Algebras It is the aim of this section to show by examples how results in function algebras can be used in other mathematical disciplines. In addition, some problems that motivate certain investigations with function algebras are explained. 1.5.1 Classiﬁcation of Universal Algebras In Part I, Chapter 1, we introduced the concept of the type of an algebra, and in Chapter 6 combinations (classes) of algebras of the same type, which fulﬁll certain equations. Such a decomposition of the algebras is, however, very coarse. The set MA of all ﬁnite algebras on the same universal A can be more ﬁnely decomposed with the aid of the following equivalence relation RA : The algebras (A; F ) and (A; G) with F, G ⊆ PA are called equivalent in respect to RA , iﬀ [F ] = [G] holds, i.e., iﬀ the operations of the one algebra can be represented as superpositions over the operations of the other algebra and vice versa. The equivalence classes (blocks) of the relation RA form a partition of the set MA , where the set of all algebras (A; F ) with F ∈ LA is a representative system of these equivalence classes. Reducts of (A; F ) are deﬁned to be algebras of the form (A; G) with G ⊆ [F ]. An algebra (A; F ) with [F ] = PA is called primal ([Fos 59]). For example, by Theorem 1.4.2, the two-element Boolean algebra is primal. The Rosenberg’s completeness criterion from Chapter 6 can be used to obtain further examples of primal algebras (see also Chapter 7). An algebra (A; F ) is called preprimal iﬀ [F ] is a maximal clone of PA ([Den 82], [Kno 85]). For a description of further generalizations of “primal algebras”, we need some concepts and results of Chapter 2 and the notations S(A) (the set of all subalgebras of the algebra A), Aut(A) (the set of all automorphisms on A), SubIso(A) (the set of all isomorphisms between subalgebras of A) and Con(A) (the set of all congruences of A). Let A := (A; F ) be an algebra, for which there is a relation set Q on A with [F ] = P olA Q. Then A is called • semiprimal iﬀ Q ⊆ Rk1 , i.e., iﬀ Q = S(A) (or with other words: iﬀ every operation on A which preserves all subalgebras of A belongs to [F ]) [Fos-P 64]; • demiprimal iﬀ Q = Aut(A) ([Qua 71]); • infraprimal (or demisemiprimal) iﬀ Q = S(A) ∪ Aut(A) [Qua 71]; • quasiprimal iﬀ Q = SubIso(A) ([Pix 71]); • hemiprimal iﬀ Q = Con(A) ([Fos 70]);

1.5 Some Applications of the Function Algebras

105

For further concepts and properties of the above-deﬁned algebras, refer to [Den 82], [Den-W 2002] and [Sze 86]. Since the present book describes many clones, one ﬁnds properties of the above-deﬁned algebras, in many places in this book. 1.5.2 Propositional Logic and First Order Logic A proposition is a “sentence” (of a natural or artiﬁcial language) for which it makes sense to ask whether it false (notation: 0) or true (notation: 1). At the basis of the concept “proposition” we have the two-value principle (also called principle of the excluded middle). This means that each proposition must be either false or true, there is no other possibility “in between”. The following are propositions: – Rostock is a city in Germany. – There are inﬁnite many prime numbers. – 2 · 3 = 5. – There exists extra-terrestrial life. The following are not propositions: – Two chickens on the way after the day before yesterday. – Everything that I say is false. – It is a beautiful day. – Be quiet! Propositions will be denoted by capital letters A, B, .... Instead of “A is an arbitrary proposition” we say “A is a proportional variable”. Therefore, a propositional variable takes the values 0 and 1. One can associate propositions (“sentences”) in the informal language in multiple ways with each other (for example through such conjunctions as “and”, “or”, “if – then”, ...). The result of this connection is normally, again, a proposition, whose value (0 or 1) is dependent from the values of associated single propositions. In the propositional logic, a part of the colloquial connections is modelled and deﬁned exactly (unlike the informal language). Since we abstract from the content of a proposition during the consideration of a proposition (i.e., we have interest only in the so-called truth value 0 or 1 of the proposition), proposition combinations are multi-digit functions on {0, 1}, i.e., Boolean functions (or function of the 2-valued logic). Interpretations of the Boolean functions that are deﬁned in Table 1.2 are, for example, • The negation of the proposition A: A (“not A”). A is true if and only if A is false. • The conjunction of the propositions A, B: A ∧ B (“A and B”). A ∧ B is true if and only if A as well as B are true. • The disjunction of the propositions A, B: A ∨ B (“A or B”). A ∨ B is true if and only if either A or B or both are true.

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1 Basic Concepts, Notations, and First Properties

• contravalence: A + B (“either A or B”). A + B is true if and only if either A or B is true. • equivalence: A ⇐⇒ B (“A if and only if B”). A ⇐⇒ B is true if and only if A and B have the same truth value. • implication: A ⇒ B (“A implies B”; “If A, then B”). A ⇒ B is false if and only if A has the value 1 and B has the value 0. To provide mathematics with a precise language, the mathematical logic creates an artiﬁcial, formal language. Next, we give a short introduction on proportional logic; that is, the logic that deals only with propositions. Later, we extend our treatment to the ﬁrst order logic, which also takes properties of individuals into account. The process of formalization of proportional logic consists of two stages: (i) present a formal language (ii) specify a procedure for obtaining valid or true propositions. The language of propositional logic has an alphabet consisting of • the set of proposition symbols At := {A, B, C, ..., A1 , B1 , C1 , ..., An , Bn , Cn , ...} (the elements of At are called atoms or atomic propositions) • the set of connectives J0 := {∧, ∨, ¬, ⇒, ⇔} • the set of auxilliary symbols {(, )} The set P rop is the smallest set X with the following three properties: • At ∪ {0, 1} ⊆ X • if α, β ∈ X, then (α ◦ β) ∈ X for all ◦ ∈ J0 • if α ∈ X, then (¬α) ∈ X Notice that P rop = T (At) (see Part I, Section 6.2). A mapping v : P rop −→ {0, 1} is called a valuation if v(0) = 0, v(1) = 1, v(¬α) = ¬v(α) and v(α ◦ β) = v(α) ◦ v(β) for all α, β ∈ P rop and all ◦ ∈ J0 . 8 If v(α) = 1 (or v(α) = 0) then α is true (or false) under v, respectively. If α = (β) ∈ P rop, then we only write β instead of α in the following. Obviously, we have: If v0 : At −→ {0, 1}, then there exists a unique valuation v such that v(α) = v0 (α) for all α ∈ At. α ∈ P rop is a tautology if v(α) = 1 for all valuations v. We write |= α (or ∅ |= α) for “α is a tautology”. If Σ ⊆ P rop and α ∈ P rop, then 8

Notice that the right sides of the equations are determined by Table 1.2.

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107

Σ |= α iﬀ for all valuations v: v(σ) = 1 for all σ ∈ Σ implies v(α) = 1. If Σ |= α then α is called a consequence of Σ. Further, set Cons(Σ) := {α ∈ P rop | Σ |= α}. It is a classical problem of the propositional logic to ﬁnd a system of axioms and rules with which one can determine all tautologies and all consequences from a set of propositions. In books about mathematical logic, one ﬁnds many solutions for this purpose. We declare a solution with proof here. The proof is chosen so that it can be used as proof for a corresponding theorem of the predicate logic as a basic idea. It is normal to write down the rule “R: if α1 , ..., αr ∈ P rop are derivable, then αr+1 is derivable” in the following form: α1 , ..., αr R αr+1 As an example, we give the substitution rule sub: Let α ∈ P rop, which contains x ∈ At (in symbol: α(..., x, ...) ). If α is a tautology, then one can form a tautology by replacing every occurrence of x by β ∈ P rop in α: α(..., x, ...); β ∈ F orm sub . α(..., β, ...) The Hilbert-type-calculus for the classical proportional logic is deﬁned by the following 13 axioms and a rule, where α, β, γ, σ, τ are arbitrary elements of P rop: (I) Axioms are all formulas of the form (A1) α ⇒ α (A2) α ⇒ (β ⇒ α) (A3) (α ⇒ β) ⇒ ((β ⇒ γ) ⇒ (α ⇒ γ)) (A4) (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)) (A5) α ⇒ (α ∨ β), β ⇒ (α ∨ β) (A6) (α ⇒ γ) ⇒ ((β ⇒ γ) ⇒ ((α ∨ β) ⇒ γ)) (A7) (α ∧ β) ⇒ α, (α ∧ β) ⇒ β (A8) (γ ⇒ α) ⇒ ((γ ⇒ β) ⇒ (γ ⇒ (α ∧ β))) (A9) ((α ∧ β) ∨ γ) ⇒ ((α ∨ γ) ∧ (β ∨ γ)), ((α ∨ γ) ∧ (β ∨ γ)) ⇒ ((α ∧ β) ∨ γ) (A10) ((α ∨ β) ∧ γ) ⇒ ((α ∧ γ) ∨ (β ∧ γ)), ((α ∧ γ) ∨ (β ∧ γ)) ⇒ ((α ∨ β) ∧ γ) (A11) (α ⇒ β) ⇒ (¬β ⇒ ¬α) (A12) (α ∧ ¬α) ⇒ β (A13) β ⇒ (α ∨ ¬α). 9 (II) Rules: There is only one rule: “from σ and σ ⇒ τ conclude τ ”, written as 9

One can also choose α, β, γ as three diﬀerent variables and the substitution rule sub as an additional rule.

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1 Basic Concepts, Notations, and First Properties

σ, σ ⇒ τ (“modus ponens”) τ A derivation of ϕ from Σ (⊆ P rop) is a ﬁnite sequence (ϕ1 , ϕ2 , ..., ϕn ) of formulas with ϕn = ϕ, where each ϕi (1 ≤ i ≤ n) is an axiom or is an element of Σ or is the result of the application of modus ponens on ϕu , ϕv (u, v < i). Let be the derivation operator deﬁned in this way, i.e., we write Σϕ iﬀ there is a derivation of ϕ from Σ. Put cons(Σ) := {ϕ | Σ ϕ} (the set of all derivation consequences from Σ). It is easy to check that cons(Σ) ⊆ Cons(Σ), since the axioms are tautologies and since, for each valuation v, v(σ) = 1 and v(σ ⇒ τ ) = 1 imply v(τ ) = 1. For proof that cons(Σ) ⊂ Cons(Σ) is false, we need the following notations and facts: Since P rop is a set of terms, one can form the term algebra Prop := (P rop; ∨, ∧, ⇒, ¬, 0, 1) of the type (2, 2, 2, 1, 0, 0). For arbitrary α, β ∈ P rop let α ≈Σ β iﬀ (Σ α ⇒ β and Σ β ⇒ α). Theorem 1.5.2.1 The relation ≈Σ has the following properties for arbitrary Σ: (a) (b) (c) (d)

≈Σ is a congruence of the algebra Prop. The factor algebra (P rop/≈Σ ; ∨, ∧, ¬, 0, 1) is a Boolean algebra. The set cons(Σ) is the 1 of the Boolean algebra (P rop/≈Σ ; ∨, ∧, ¬, 0, 1). There exists a homomorphic mapping ν from Prop onto Prop/≈Σ with ν(ϕ) = 1 iﬀ Σ ϕ.

Proof. (a): First we show that ≈Σ is an equivalence relation. By Σ α ⇒ α (see (A1)) is ≈Σ reﬂexive. The symmetry of ≈Σ follows from the deﬁnition of ≈Σ . One can prove the transitivity of ≈Σ with the aid of (A3) and the modus ponens as follows: Let α ≈Σ β and β ≈Σ γ be arbitrary, i.e., it holds: Σ α ⇒ β, Σ β ⇒ α, Σ β ⇒ γ, Σ γ ⇒ β. By means of the modus pones ( σ := α ⇒ β, σ ⇒ τ := (A3) ) and (A3) one obtains Σ (β ⇒ γ) ⇒ (α ⇒ γ). When one uses the modus pones (σ := β ⇒ γ) again, one receives Σ α ⇒ γ. Analogously, one can show Σ γ ⇒ α. Thus, ≈Σ is an equivalence relation. To prove the compatibility of ≈Σ with ⇒ we consider

1.5 Some Applications of the Function Algebras

109

ϕ ≈Σ ϕ , ψ ≈Σ ψ , ( ϕ, ϕ , ψ, ψ ∈ P rop), i.e., it holds Σ ϕ ⇒ ϕ , Σ ϕ ⇒ ϕ, Σ ψ ⇒ ψ , Σ ψ ⇒ ψ. (ψ ⇒ ψ ) ⇒ (ϕ ⇒ (ψ ⇒ ψ )) (α = ψ ⇒ ψ and τ = β in (A2)) and the modus ponens imply Σ ϕ ⇒ (ψ ⇒ ψ ). Furthermore, by (A4), we have ((ϕ ⇒ (ψ ⇒ ψ )) ⇒ ((ϕ ⇒ ψ) ⇒ (ϕ ⇒ ψ )). Hence we get Σ ((ϕ ⇒ ψ) ⇒ (ϕ ⇒ ψ )) with the aid of modus ponens. Analogously, one can show Σ ((ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ)). Consequently,

ϕ ⇒ ψ ≈Σ ϕ ⇒ ψ .

(1.13)

By (A3) we have Σ (ϕ ⇒ ϕ) ⇒ ((ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ )), whereby Analogously, and therefore, Then

Σ (ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ ). Σ (ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ ), ϕ ⇒ ψ ≈Σ ϕ ⇒ ψ .

(1.14)

ϕ ⇒ ψ ≈Σ ϕ ⇒ ψ ,

since ≈Σ is transitive and (1.13) and (1.14) are valid. Hence, ≈Σ is compatible with ⇒. To prove that ≈Σ is compatible with ¬ we assume α ≈Σ β; i.e., we have Σ α ⇒ β and Σ β ⇒ α. Then, by (A11) and the modus ponens, we get Σ ¬β ⇒ ¬α and Σ ¬α ⇒ ¬β, whereby ¬α ≈Σ ¬β. Because of this property of ≈Σ , we can deﬁne a partial order on Prop/≈Σ as follows: [ϕ]≈Σ ≤ [ψ]≈Σ iﬀ Σ ϕ ⇒ ψ, where [ϕ]≈Σ = [ψ]≈Σ means that ϕ ≈Σ ψ (i.e., Σ ϕ ⇒ ψ and Σ ψ ⇒ ϕ) holds. (By (A1) is ≤ reﬂexive; (A3) implies the transitivity of ≤. The antisymmetry follows from the deﬁnition of ≈Σ : I If Σ ϕ ⇒ ψ and Σ ψ ⇒ ϕ, then we have ϕ ≈Σ ψ.) Now (A5) shows that [α]≈Σ ≤ [α ∨ β]≈Σ and [β]≈Σ ≤ [α ∨ β]≈Σ .

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Assume there is a γ with [α]≈Σ ≤ [γ]≈Σ and [β]≈Σ ≤ [γ]≈Σ . By (A6) and the modus ponens, this implies [α ∨ β]≈Σ ≤ [γ]≈Σ , whereby sup([α]≈Σ , [β]≈Σ ) = [α]≈Σ ∨ [β]≈Σ = [α ∨ β]≈Σ holds. Analogously, inf ([α]≈Σ , [β]≈Σ ) = [α]≈Σ ∧ [β]≈Σ = [α ∧ β]≈Σ follows from (A7), (A8) and the modus ponens. Thus ≈Σ is also compatible with the operations ∨ and ∧. (b): (a) implies that Prop/≈Σ is a lattice. Because of the axioms (A9) and (A10) this lattice is distributive. Assume Σ ϕ and Σ ψ. Then [ϕ]≈Σ = [ψ]≈Σ , since Σ ψ ⇒ ϕ and Σ ϕ ⇒ ψ follows from this with the aid of (A2) and the modus ponens. Because of (A2), we have Σ (β ⇒ α) for all β ∈ P rop and all α ∈ P rop with Σ α, i.e., β ≤ α. Therefore, {α | Σ α} is the greatest element of the lattice Prop/≈Σ . By (A12) the smallest element is the equivalence class [α ∧ ¬α]≈Σ . Consequently, the algebra Prop/≈Σ also fulﬁlls the axioms (B1 ) (see Section 1.2.12). (B2 ) follows from the fact that ≈Σ is compatible with ∧, ∨ and ¬ with the aid of (A13) as follows: [α]≈Σ ∧ (¬[α]≈Σ ) = [α ∧ (¬α)]≈Σ = 0 (see above), [α]≈Σ ∨ (¬[α]≈Σ ) = [α ∨ (¬α)]≈Σ = 1 (because of [β]≈Σ ≤ [α ∨ ¬α]≈Σ for each β ∈ P rop by (A13)). Hence, Prop/≈Σ is a Boolean algebra. (c): We have shown above that the 1 of P rop/≈Σ contains all formulas derivable from Σ. Further, if α ∈ P rop belongs to the equivalence class 1 of P rop/≈Σ , then we get Σ ϕ ⇒ α for each ϕ with Σ ϕ by the above shown. Thus (by modus ponens) we have Σ α. Hence, (c) is proven. (d) By Part I, Section 4.1, the natural homomorphism ϕ : P rop −→ P rop/≈Σ , α → [α]/≈Σ fulﬁlls (d). Theorem 1.5.2.2 (Completeness Theorem of Proportional Logic) Let Σ ⊆ P rop. Then cons(Σ) = Cons(Σ).

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111

Proof. Obviously, cons(Σ) ⊆ Cons(Σ). Assume there exists an α ∈ Cons(Σ) \ cons(Σ). Then [α]/≈Σ = [1]/≈Σ by Theorem 1.5.2.1, whereby α is an element of a certain prime ideal (maximal ideal) of the Boolean algebra Prop/≈Σ .10 Consequently, there exists a homomorphism µ from Prop/≈Σ onto the two-element Boolean algebra {0, 1} with µ([α]/≈Σ ) = 0. Then, with the aid of Theorem 1.5.2.1, (d), it is easy to see that v := ν2µ is a valuation with v(α) = 0 and v(σ) = 1 for all σ ∈ Σ. But, this is a contradiction to α ∈ Cons(Σ). Next is a short introduction to predicate logic (or ﬁrst order logic). Predicate logic is a language to describe statements about algebraic structures (to given signature). A signature is a pair δ := ((ni )i∈I , (mj )j∈J ) with ni ∈ N0 , mj ∈ N for all i ∈ I and j ∈ J. A := (A; (fiA )i∈I , (RjA )j∈J ) is called a structure of signature δ, if (A; (fiA )i∈I ) an algebra of the type (ni )i∈I and RjA ⊆ Amj holds for all j ∈ J. We say “fiA is the interpretation of fi ” and “RjA is the interpretation of Rj ”. With the aid of a mapping P from Ah into {0, 1}, one can describe an h-ary relation R on a set A (i.e., R ⊆ Ah ) as follows: P (a1 , .., ah ) = 1 iﬀ (a1 , ..., ah ) ∈ R. Such a mapping P is called an h-ary predicate. Let P, Q be predicates on A. Then one can understand the predicates as propositions and form (¬P ) and (P ◦ Q) for each ◦ ∈ J0 . m Therefore, let Pj j be the mj -ary predicate with mj

Pj (a1 , .., amj ) = 1 iﬀ (a1 , ..., amj ) ∈ Rj

in the following. The alphabet of the ﬁrst order logic consists of the following symbols: • • • •

the set V ar := {x0 , x1 , x2 , ...} of variables m the set {Pj j | j ∈ J} of predicate symbols the set {fini | i ∈ I} of operation symbols the set of connectives J := {∧, ∨, ¬, ⇒, ⇔, ∃, ∀} (∃ and ∀ are called the existential and universal quantiﬁer) • the set of auxilliary symbols {(, )} As described in Part I, Section 6.2, we form the set of all terms T erm := T (V ar) and the term algebra Term = (T erm; (fi )i∈I ). Then, F ORM is the smallest set X with the following four properties:

10

See books on Boolean algebras or universal algebras.

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1 Basic Concepts, Notations, and First Properties

• Pj (t1 , ..., tmj ) ∈ X for all j ∈ J and all t1 , ..., tmj ∈ T erm (Pj (t1 , ..., tmj ) is called an atom); • if α, β ∈ X , then (α ◦ β) ∈ X for all ◦ ∈ J0 • if α ∈ X then (¬α) ∈ X • if α ∈ X then (∃xk α) ∈ X and (∀xk α) ∈ X for all k ∈ N0 If t ∈ T erm, then V ar(t) denotes the set of all elements of X, which occur in T . For ϕ ∈ F ORM the set V ar(ϕ) of all variables of ϕ is deﬁned by • V ar(Pj (t1 , ..., tmj )) := V ar(t1 ) ∪ ... ∪ V ar(tmj ) • V ar(¬α) := V ar(α) and V ar(α ◦ β) := V ar(α) ◦ V ar(β) for all ◦ ∈ J0 • V ar(∃xk α) = V ar(∀xk α) := V ar(α) ∪ {xk } (α, β ∈ F ORM ). The set f r(ϕ) of free variables of ϕ is deﬁned by • f r(ϕ) := V ar(ϕ) if ϕ is an atom • f r(¬α) := f r(α) and f r(α ◦ β) := f r(α) ◦ f r(β) for all ◦ ∈ J0 • f r(∀xk α) = f r(∃xk α) := f r(α)\{xk } (α, β ∈ F ORM ). The set bd(ϕ) of bound variables of ϕ is deﬁned by • bd(ϕ) := ∅ if ϕ is an atom • bd(¬α) := bd(α) and bd(α ◦ β) := bd(α) ◦ bd(β) for all ◦ ∈ J0 • bd(∀xk α) = bd(∃xk α) := bd(α) ∪ {xk } (α, β ∈ F ORM ). – ϕ is called open formula if bd(ϕ) = ∅. ϕ is a sentence or is closed, if f r(ϕ) = ∅. Obviously, V ar(ϕ) = f r(ϕ) ∪ bd(ϕ), but f r(α) ∩ bd(ϕ) need not be empty. Let u, u be mappings from V ar into A. Then we write u =xk u iﬀ u(xj ) = u (xj ) for all j = k. Let A := (A; (fi )i∈I , (Rj )j∈J ) be a structure of signature δ and let u : V ar −→ A be a mapping. For u, there is exactly one homomorphism u : T(Var) −→ (A; (fi )i∈I ), which u continues (see Part I, Theorem 6.2.2). In the following manner, one can interpret the elements of F ORM with the aid of A, u and u : An interpretation function (or valuation) is a mapping vA,u from F ORM into {0, 1} deﬁned by • vA,u (Pj (t1 , ..., tmj )) = 1 iﬀ ( u(t1 ), ..., u (tmj )) ∈ RjA • vA,u (¬α) := ¬(vA,u (α)) and vA,u (α ◦ β) := vA,u (α) ◦ vA,u (β) for all ◦ ∈ J0 • vA,u (∀xk ϕ) = 1 iﬀ v A,u (ϕ) = 1 for all u with u =xk u (i.e., vA,u (∀xk ϕ) = u , u=x u vA,u (ϕ)) k • vA,u (∃xk ϕ) = 1 iﬀ there is a u with u =xk u and vA,u (ϕ) = 1 (i.e., vA,u (∃xk ϕ) = u , u=x u vA,u (ϕ) ) k

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113

ϕ ∈ F ORM is satisﬁed by u : V ar −→ A iﬀ vA,u (ϕ) = 1. ϕ is true (or valid) in A, iﬀ vA,u (ϕ) = 1 for all u : V ar −→ A. A |= ϕ stands for “ϕ is true in A”. A |= ϕ stands for “ϕ is not true (or false) in A”. If A |= ϕ, we call A a model of ϕ. In general, if A |= σ for all σ ∈ Σ ⊆ F ORM , we call A a model of Σ. Notice that a closed formula of F ORM is either true or false in A. Formulas α, β ∈ F ORM are equivalent (in symbol α ≡ β) iﬀ vA,u (α) = vA,u (β) for every structure A and for every u : V ar −→ A. The following theorem is easy to prove: Theorem 1.5.2.3 Let ϕ, ψ ∈ F ORM . Then: (1) ¬∀xk ϕ ≡ ∃xk ¬ϕ, ¬∃xk ϕ ≡ ∀xk ¬ϕ; (2) if xk ∈ f r(ψ) then (Qxk ϕ ◦ ψ) ≡ Qxk (ϕ ◦ ψ) for all Q ∈ {∃, ∀} and ◦ ∈ {∧, ∨}; (3) (∀xk ϕ ∧ ∀xk ψ) ≡ ∀xk (ϕ ∧ ψ), (∃xk ϕ ∨ ∃xk ψ) ≡ ∃xk (ϕ ∨ ψ); (4) ∀xk ∀xl ϕ ≡ ∀xl ∀xk ϕ, ∃xk ∃xl ϕ ≡ ∃xl ∃xk ϕ. Next, we deﬁne the mappings repτ , tutτ and subτ from F ORM into F ORM , where τ is a mapping from V ar into T erm. With repτ , one can describe the replacement of variables by some terms in a formula from F ORM . The mapping tutτ renames the bound variables of a formula so that there are no variables of the formula anymore, which is free and bound. subτ is a combination of repτ and tutτ . Let τ : V ar −→ T erm be an arbitrary mapping and τ : T erm −→ T erm the unique homomorphic continuation of τ . Then the mapping repτ : F ORM −→ F ORM is deﬁned by • repτ (Pj (t1 , ..., tmj )) := Pj ( τ (t1 ), ..., τ(tmj )) • repτ (¬α) := ¬repτ (α) and repτ (α ◦ β) := repτ (α) ◦ repτ (β) for all ◦ ∈ J0 • if ϕ = Qxk α with Q ∈ {∀, ∃}, then repτ (ϕ) := Qxk repτ (α), where xk if xi = xk , τ (xi ) := τ (xi ) if xi = xk . For the mapping τ : V ar −→ T erm and ϕ ∈ F ORM let n(τ, ϕ) be the smallest number j0 ∈ N0 with xj ∈ f r(ϕ) ∪ V ar(τ (x)) for all x ∈ f r(ϕ). Let tutτ : F ORM −→ F ORM be deﬁned by • tutτ (ϕ) = ϕ if ϕ is an atom • tutτ (¬ϕ) = ¬(tutτ (ϕ)) and tutτ (ϕ ◦ ψ) = tutτ (ϕ) ◦ tutτ (ψ) for all ◦ ∈ J0 ; • for Q ∈ {∀, ∃} let tutτ (Qxk ϕ) = Qxj tutτ (repτk,j (ϕ)) with j := n(τ, ϕ) and xj if k = l, τk,j (xl ) := xl otherwise.

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1 Basic Concepts, Notations, and First Properties

For τ : V ar −→ T erm and arbitrary ϕ ∈ F ORM set subτ (ϕ) := repτ (tutτ (ϕ)). The following lemma gives some properties of the above mappings: Lemma 1.5.2.4 (without proof ) Let A be a structure of signature δ, u : V ar −→ A a mapping with the homomorphic continuation u : T erm −→ A. Then: (a) vA,u (ϕ) = vA,u (tutτ (ϕ)); (b) f r(ϕ) = f r(tutτ (ϕ)); (c) vA,τ 2u (ϕ) = vA,u (subτ (ϕ)). With the aid of the above mapping, we can describe the Hilbert-typecalculus for the classical predicative logic, which consists of following axioms and rules: (I) axioms are: (a) the axioms (A1) - (A13) of the classical propositional logic; (b) all formulas of the form (A14) (∀xk ϕ) ⇒ subτ (ϕ), where τ =xk id (i.e, τ replaces something at most for the variable xk ); (A15) ∃xk ϕ ⇒ ¬∀xk ¬ϕ, ¬∀xk ¬ϕ ⇒ ∃xk ϕ; (A16) subid (ϕ) ⇒ ϕ. (II) rules are: (a) the modus ponens; (b) ϕ ⇒ subτk,l (ψ) ϕ ⇒ ∀xk ψ,

where τk,l (xj ) :=

xl if j = k, xj otherwise

and xl ∈ V ar(ψ) ∪ f r(ϕ) (“∀-introductory-rule”); (c) ϕ subτ (ϕ) for all τ . Analogously to the proportional logic, one can deﬁne and cons(Σ) for Σ ∈ F ORM in respect to the above calculus. The completeness theorem of the ﬁrst order logic says that = |= holds, i.e., cons(Σ) = Cons(Σ) is valid for all Σ ⊆ F ORM and that there are axioms and rules, so that = |= was proven for the ﬁrst time by K. G¨ odel in his dissertation 1929. For the above calculus one can prove the completeness theorem for ﬁrst order

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logic similar to Theorem 1.5.2.2, where the Rasiowa-Sikorski-Tarski-Lemma is needed (see [Ric 78] for details). The famous incompleteness theorems of G¨odel say that second order logic 11 does not have a completeness theorem in general.

1.5.3 Many-Valued Logics The above considerations can be generalized when one assigns the “sentences” certain values from the set Ek (k ≥ 3). We receive then a so-called k-valued logic or many-valued logic. One ﬁnds a full introduction to the logic for example in [Got 89] or [Kre-G-S 88]. From a mathematical perspective it doesn’t matter which interpretations have the elements of Ek . Nevertheless, it is shown which interpretations of the elements of Ek , for example, are possible. According to these interpretations, one can select certain functions of Pk , with which one can form many-valued logics. For reasons of space limitations, we have excluded information. These functions can be found in [Men 85], where literature on the topic is also provided. 1.5.2.1 The three values of a three-valued logic can be interpreted as follows: “false”, “indeﬁnite”, “true”; “false”, “possible”, “true”; “false”, “undecidable”, “true” or “invalid”, “in part valid”, “full-valid”. Legally relevant actions can be divided in “punishable”, “prohibited but not punishable”, “allowed”. 1.5.2.2 The 4-valued logic seems particularly suitable for analyzing problem areas logically, in which there are two diﬀerent kinds of truth (or validity) and two diﬀerent kinds of falsehood (or nullity). Examples: “fact falsehood”, “legal nullity”, “legal validity”, “fact truth” ; “knowledge falsehood”, “belief falsehood”, “belief truth”, “knowledge truth”. 1.5.2.3 A legal interpretation of the 6-valued logic is, for example, 0: “logically false”, 1: “legally invalid”, 2: “false according to facts”, 3: “true according to facts”, 4: “legally valid”, 5: “logically true”. 11

In this logic, quantiﬁcations over elements of a structure as well at quantiﬁcations over partial sets and relations of the structure are allowed.

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1 Basic Concepts, Notations, and First Properties

1.5.2.4 Instead of Ek , one can also choose a ﬁnite subset of the real numbers x with 0 ≤ x ≤ 1. Then, with the aid of such a set, a probability logic can be formed: The value 1 corresponds to the certainty of the truth; the value 0.5 corresponds to the uncertainty, whether true or false; the value 0 corresponds to the certainty of the falsehood (or the impossibility); values between 1 and 0.5 correspond to degrees of higher probability; values between 0.5 and 0 correspond to degrees of low probability. We notice that one could also have chosen the interpretation of the Fuzzy Logic instead of the above-mentioned probability logic (see for example [BanG 90] or [Til 92]). The problems of all the above-mentioned logics correspond basically to those of the propositional logic. 1.5.4 Information Transformer The functions f n ∈ PA can be understood simply as mathematical models of objects that process information (see Figure 1.1). The “object” from Figure 1.1 receives the information x1 , ..., xn ∈ A at the entries, processed to the information f (x1 ..., xn ) ∈ A. In this case, we neglect the time which is needed for the workmanship. x1 x2 ... xn ??

? f

?x ) f (x1 , ..., n Fig. 1.1

Superpositions over functions of PA correspond with this model to the “assembling” of such objects. For example, one can describe the diagram x1

x2

x3

? ? f A U A ?

? g

? g(f (x1 , x2 ), x3 , x3 ) Fig. 1.2

1.5 Some Applications of the Function Algebras

117

through the formula g(f (x1 , x2 ), x3 , x3 ). In particular, Boolean functions are used for the mathematical description of electrical circuits or components in computers. This mathematical description is independent of the concrete technical realization (as, for example, relay contact circuits or transistors). Naturally, one receives the so-called completeness problem from these interpretations of the functions from PA : A necessary and suﬃcient criterion is searched in order to be able to decide whether a system of certain selected functions (which correspond to certain elementary elements) produces all functions from PA by means of superposition. One way of ﬁnding this criterion indicates the following theorem: Theorem 1.5.4.1 Let A be a subclass of Pk with the property that to every proper subclass A of A there exists a certain maximal class M of A with A ⊆ M . Furthermore, denote M the set of all maximal classes of A. Then for an arbitrary subset T of A it holds: [T ] = A ⇐⇒ ∀M ∈ M : T ⊆ M.

(1.15)

Proof. “=⇒”: Let [T ] = A. Suppose there is an M ∈ M with T ⊆ M . Then, a contradiction results; however, from that, immediately, A = [T ] ⊆ [M ] = M ⊂ A. “⇐=”: Let T ⊆ Pk be no subset of a maximal class of A. Suppose, [T ] ⊂ A. Then, by assumption, one can ﬁnd a certain maximal class M ∈ M with [T ] ⊆ M , a contradiction to the supposition. By Theorem 11.1.1, every ﬁnitely generated class fulﬁlls the conditions of Lemma 1.5.3.1 and has only ﬁnitely many maximal classes, whereby (1.15) supplies a criterion of the wanted kind if one knows the maximal classes of A. Further problems that result from interpreting the functions of PA as information transformers are the following: • Find minimal generating systems, where “minimal” is related to the number of the generating system’s functions or to the arities of the generating system’s functions. • Find algorithms to construct a minimal realization of a given function by certain elementary functions. One can ﬁnd further problems and their solutions in [Rin 84] and [P¨ os-K 79].

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1.5.5 Classiﬁcation of Combinatorial Problems In this section, we need some notations (Rk , P olk , Invk , ...) of Chapter 2. Further, we need some concepts of the algorithm theory:12 Algorithms are techniques for solving problems, wherein the word “problem” is used in a very general sense: A problem class consists of inﬁnitely many instances having a common structure. A problem, whose solution is either “yes” or “no”, is called a decision problem. Often, ﬁnding a solution algorithm for the solution of a mathematical problem does not suﬃce. Statements about the complexity of the algorithms are important for the applicability of algorithms in particular. The complexity of an algorithm is a function f (n), which gives the number of the arithmetic steps (e.g. value assignments, elementary arithmetic operations, comparison operations, ...) which are necessary to the estimating of a solution of a given problem with n master data. For g : N −→ R+ let O(g(n)) := {f (n) | (f : N −→ R+ ) ∧ (∃c > 0 ∀n ∈ N : f (n) ≤ c · g(n))} We say that the algorithm has the complexity O(g(n)), if the complexity f (n) of the algorithm belongs to O(g(n)). Generally one holds algorithms with a complexity O(nt ) for “good” and these algorithms are called polynomial algorithms. A decision problem with the complexity O(nt ), where n, t ∈ N, is called tractable. Let P be the class of all tractable problems. L denotes the class of all problems with the complexity O(log n). The class of decision problems for which a positive answer can be veriﬁed in polynomial time is denoted by NP (for “non-deterministic polynomial”). That is, we do not only require the answer “yes” or “no”, but the explicit speciﬁcation of a certiﬁcate which allows to verify the correctness of a positive answer. LP denotes the class of decision problems for which a positive answer can be veriﬁed in logarithmic time. Obviously, P ⊆ NP. Further, we know that L ⊆ NL ⊆ P. It is the greatest problem of the algorithm theory to decipher whether P = NP is valid. Most people believe, however, that P = NP. A problem is called NP-complete if is in NP and if the polynomial solvability of this problem would imply that all problems in NP are solvable in polynomial time as well. In other words: Each problem in NP can be “transformed” (in polynomial time) to the given NP-complete problem, such that 12

See e.g. [Pap 94] and [Jun 2005].

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a solution of the NP-complete problem also gives a solution to that other problem in NP. Therefore, if one ﬁnds a polynomial algorithm for an NPcomplete problem, this would imply that P = NP. Next, is a short introduction to some papers that deal with classiﬁcations of combinatorial problems. Deﬁnition A CSP (or a “constraint satisﬁcation problem”) is a triple (V, D, C), where • V is a set of variables • D is a set of values which can take the variables • C := {C1 , C2 , ..., Cq } with Ci := (si , τi ) for i = 1, ..., q, where si is an mi -ary tuple of variables and τi is an mi -ary relation on D (Ci is called constraint). A solution of the CSP is a mapping f : V −→ D with the property ∀ Ci := ((xi1 , xi2 , ..., ximi ), τi ) ∈ C : (f (xi1 ), f (xi2 ), ..., f (ximi )) ∈ τi . Many combinatorial problems can be described as CSP (see [Jea 97] 13 ) Example The problem SAT as CSP An instance of the standard propositional satisﬁability problem is speciﬁed by giving a formula on propositional logic, and asking whether there are values for the variables that make the formula true. For example, consider the formula t(x1 , x2 , x3 , x4 ) := (x1 ∨ x2 ∨ x3 ∨ x4 ) ∧ (x1 ∨ x2 ∨ x3 )∧ (x3 ∨ x4 ∨ x1 ) ∧ (x3 ∨ x2 ∨ x4 ) ∧ (x1 ∨ x3 ). The question is whether there are a1 , a2 , a3 , a4 ∈ {0, 1} with t(a1 , a2 , a3 , a4 ) = 1. This problem can be expressed as the CSP as follows: Put V := {x1 , x2 , x3 , x4 }, D := {0, 1}, C1 C2 C3 C4 C5 C

:= := := := := :=

((x1 , x2 , x3 , x4 ), D4 \ {(0, 0, 0, 0)}), ((x1 , x2 , x3 ), D3 \ {(1, 1, 0)}), ((x3 , x4 , x1 ), D3 \ {(1, 1, 0)}), ((x3 , x2 , x4 ), D3 \ {(1, 1, 0)}), ((x1 , x3 ), D2 \ {(1, 1)}), {C1 , c2 , C3 , C4 }.

The solutions of the above problem are 13

In this paper are 20 combinatorial problems expressed as CSP.

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1 Basic Concepts, Notations, and First Properties

x1 x2 x3 x4 f1 f2 f3 f4 f5 f6

: : : : : :

0 0 0 0 1 1

0 0 1 1 0 0

0 1 0 0 0 0

1 0 0 1 0 1

It is also possible to describe a CSP with the aid of the following notations: Deﬁnitions Let V be a nonempty set and let τ1 , τ2 , ..., τt be relations on V . Then, (V ; τ1 , τ2 , ..., τt ) is called relational structure. The mapping : {1, 2, ..., t} −→ N is the rank functions of the relational structure (V ; τ1 , τ2 , ..., τt ) if (i) is the arity of the relation τi (i = 1, 2, ..., t). A relational structure Σ is similar to a relational structure Σ if they have the same rank function. Let Σ := (V ; τ1 , ..., τt ) and Σ := (V ; τ1 , ..., τt ) be two similar relational structure, and let be their common rank function. A homomorphism from Σ into Σ is a mapping h : V −→ V such that ∀i ∈ {1, ..., t} : (a1 , .., ahi ) ∈ τi =⇒ (h(a1 ), ..., h(ahi )) ∈ τi . Let Hom(Σ, Σ ) be the set of all homomorphisms from Σ into Σ . Obviously, it holds that Theorem 1.5.5.1 ([Jea-C-P 98]) (a) Let P := (V, D, C) be a CSP with C := {(s1 , τ1 ), (s2 , τ2 ), ..., (sq , τq )}. Then, the set of all solutions of P is the set Hom(Σ, Σ ), where Σ := (V ; {s1 }, {s2 }..., {sq }) and Σ := (D; τ1 , τ2 , ..., τq ). Conversely: Let Σ := (V ; τ1 , ..., τt ) and Σ := (D; τ1 , ..., τt ) be two similar relational structures. Then, Hom(Σ, Σ ) is the set of all solutions of the CSP (V, D, S) with C :=

t

{(s, τi ) | s ∈ τi }.

i=1

The following theorem is our ﬁrst example of an application of P ol and Inv (see Chapter 2) in algorithm theory.

1.5 Some Applications of the Function Algebras

121

Theorem 1.5.5.2 Let D := Ek , Γ := {1 , ..., q } ⊆ Rk , V := {x1 , ..., xn } and let M be a set of mappings from V into Ek . Then it holds: There exists a CSP (V , D, C), where V ⊆ V and C = {(c1 , 1 ), ..., (cq , q )}, with the solution set S and S|V = M if and only if := {(f (x1 ), ..., f (xn )) | f ∈ M } ∈ Invk (P olk Γ ). The proof of the above theorem is explained by subsequent examples: We choose: D := E2 , 1 := {(0, 1, 1), (1, 0, 0), (1, 1, 1)}, 2 := {(0, 0), (1, 0)}, 3 := {(1, 1)}, Γ := {1 , 2 , 3 }, V := {x1 , x2 , x3 , x4 } and M := {f1 , f2 }, where x f1 (x) f2 (x) x1 0 1 x2 1 0 x3 1 1 x4 1 1 “⇐=”: Assume, := {(f (x1 ), ..., f (xn )) | f ∈ M } ∈ Invk (P olk Γ ). Then can be obtained by a ﬁnite number of applications of the elementary operations 3 (see Section 2.3). For example: ζ, τ, pr, ∧ and × from and δk;{1,2} = {(x1 , x2 , x3 , x4 ) ∈ E24 | ∃x5 ∈ E2 : (x1 , x5 , x3 ) ∈ 1 ∧ (x5 , x2 ) ∈ 2 ∧ (x5 , x4 ) ∈ 3 }. Therefore, V = {x1 , x2 , x3 , x4 , x5 } and (V , D, C), where C := {((x1 , x5 , x3 ), 1 ), ((x5 , x2 ), 2 ), ((x5 , x4 ), 3 )}, is a CSP with the solution set S = {f1 , f2 }, where x f1 (x) f2 (x) x1 0 1 x2 1 0 x3 1 1 x4 1 1 x5 1 1 and S|V = M = {f1 , f2 }. “=⇒”: Choose V := {x1 , x2 } and M := {g1 , g2 }, where x g1 (x) g2 (x) x1 0 1 x2 0 0 ({x1 , x2 , x3 , x4 , x5 }, E2 , C) with C := {((x1 , x5 , x3 ), 1 ), ((x5 , x2 ), 2 ), ((x5 , x4 ), 3 )} is a CSP and it holds S|V = M = {g1 , g2 }. Then

122

1 Basic Concepts, Notations, and First Properties

= {(0, 0), (1, 0)} = {(x1 , x2 ) ∈ E22 | ∃x3 ∈ E2 ∃x4 ∈ E2 ∃x5 ∈ E2 : (x1 , x5 , x3 ) ∈ 1 ∧ (x5 , x2 ) ∈ 2 ∧ (x5 , x4 ) ∈ 3 } Thus ∈ Invk P olk Γ (see Chapter 2). Next we show how one can use the above theorem for solving the following task: Put D := E2 , Γ

:= { ⊆ E22 | = ∅} =: {1 , ..., 15 },

x f1 (x) f2 (x) f3 (x) x1 0 0 0 M := {f1 , f2 , f3 }, where x2 0 0 1 x3 0 1 0 x4 1 0 0 Is there a CSP (V , E2 , C) with V ⊆ V , C = {((c1 , 1 )), ..., (c1515 )} and the solution set S which ones S|V = M is valid for? Because of the above theorem, this question is equivalent for the following question: := {(f (x1 , ..., f (x4 ))) | f ∈ M } = {(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0)} ∈ Inv2 P ol2 Γ ? With the help of the results of Chapter 3, it is easy to prove that P ol2 Γ =

15

P ol2 i = [h2 ],

i=1

where h2 (x, y, z) := (x ∧ y) ∨ (x ∧ z) ∧ (y ∨ z). Then, because of ⎛ ⎞ ⎛ ⎞ 0 0 0 0 ⎜0 0 1⎟ ⎜0⎟ ⎟ ⎜ ⎟ h2 ⎜ ⎝0 1 0⎠=⎝0⎠ 1 0 0 0 we have ∈ Inv2 P ol2 ; i.e., there is no CSP with the above properties. Deﬁnitions The general combinatorial problem (GCP) is the decision problem with: Instance: A pair (Σ1 , Σ2 ) of similar ﬁnite relational structures Σ1 and Σ2 . Question: Is there a homomorphism from Σ1 to Σ2 ? For any GCP instance P := (Σ1 , Σ2 ) a homomorphism from Σ1 to Σ2 will be called a solution of P .

1.5 Some Applications of the Function Algebras

123

The GCP (or CSP) is known to be NP-complete. However, certain restrictions may aﬀect the complexity of GCP (or CSP). One of the natural possibilities for restricting CSP is by limiting the relations that can appear in constraints (or relational structures). Deﬁnition Let Γ be a set of relations on the set D. Denote by CSP(Γ ) the subclass of CSP deﬁned by the following property: any constraint relation in any instance must belong to Γ . The next theorem is the basis for the following theorems. Theorem 1.5.5.3 ([Jea 98]; without proof ) Let Γ and Ψ be ﬁnite sets of relation on D (w.l.o.g. we put D := Ek ). If Ψ ⊆ Invk (P olk Γ ) then GCP(Ψ ) can be reduced in polynomial time to GCP(Γ ). In other words: For any ﬁnite set Γ ⊆ Rk , the complexity of GCP(Γ ) is determined, up to polynomial-time reductions, by P olk Γ . Theorem 1.5.5.4 ([Jea 98]; without proof ) Let D := Ek , k ≥ 2 and Γ ⊆ Rk . Then (1) GCP(Γ ) ⊆ P, if one of the following conditions is valid: (a) P olk Γ contains a constant function. (b) P olk Γ contains a binary function which is associative, commutative and idempotent. (c) r ∈ P olk Γ , where r(x, y, z) := x + y + z and (Ek ; +) is Abelian 2group. (2) If P olk Γ contains a ternary function ϕ deﬁned by y if y = z, ϕ(x, y, z) := x otherwise, then GCP(Γ ) ⊆ NL. (3) If each function of P olk Γ is either a projection or a semiprojection, then GCP(Γ ) is NP-complete. With the aid of the above theorem and Post’s description of all closed sets of Boolean functions, one can prove the following: Theorem 1.5.5.5 ([Sch 78]) For arbitrary Γ ⊆ R2 it holds: GCP(Γ ) is NP-complete, if P ol2 Γ ⊆ [P21 ]; in all other cases we have GCP(Γ ) ⊆ P. More can be found on the topic in [Jea 98], [Jea-C-P 98], [Bul-K-J 2000] and [Coh-J-G 2003].

2 The Galois-Connection Between Function- and Relation-Algebras

This section aims to develop a “suitable” means to describe function algebras or clones. I mean, suitable in the sense that “big” function algebras (or clones) can be described with an “expenditure” as small as possible. Analogously to other ﬁelds of algebra we introduce invariants for our function algebras. The two papers [Bod-K-K-R 68/69] of V. G. Bodnarˇcuk, L. A. Kaluˇznin, V. N. Kotov and B. A. Romov are basis of this so-called Pol-Inv theory (or Galois theory for function- and relation-algebras). These articles generalize results by M. Krasner on a Galois theory for groups (see [Kra 45], [Kra 68/69]). A full representation of the Pol-Inv theory with many applications can be found in the monograph [P¨ os-K 79] by R. P¨ oschel and L. A. Kaluˇznin.

2.1 Relations An h-ary relation on Ek is a subset of the h-fold Cartesian product Ekh of the set Ek , h ∈ N. The elements (a1 , a2 , ..., ah ) of (we say “h-tuple”) are written as columns ⎞ ⎛ a1 ⎜ a2 ⎟ ⎟ ⎜ ⎝ ... ⎠ , ah and then we also write

⎛

⎞ a1 ⎜ a2 ⎟ ⎜ ⎟ ⎝ ... ⎠ ∈ . ah

The relation is written often as a matrix whose columns are the elements of the relation. For example,

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2 The Galois-Connection Between Function- and Relation-Algebras

⎞ a1 a1 a1 := ⎝ ... ... ... ⎠ ah ah ah ⎛

instead of := {(a1 , ..., ah ), (a1 , ..., ah ), (a1 , ..., ah )}. We think of this matrix representation of if we subsequently talk about the length h and the width || of the relation as well as about rows from . Denote Rkh the set of all h-ary relations on Ek and let Rk := Rkh . h≥1

We remark that the empty set is an element of Rk . If Q ⊆ Rk then let Qh := Q ∩ Rkh .

2.2 Diagonal Relations The simplest relations, in a certain sense1 , are the diagonal relations (or diagonals) deﬁned as follows: For an arbitrary equivalence relation ε on {1, 2, ..., h} let h δk,ε := {(a1 , ..., ah ) ∈ Ekh | (i, j) ∈ ε =⇒ ai = aj }.

If h or k follows from the context, then we write only δε or δεh or δk,ε . Every element of the set h Dkh := { δk,ε | ε is an equivalence relation on {1, 2, ..., h} }.

is called a diagonal h-ary relation. Let Dk := {∅} ∪

Dkh

h≥1

be the set of all diagonal relations. h , we often declare this For the purpose of a more simple description of δk,ε relation in the form h δk;ε 1 ,...,εr or, brieﬂy, by δε1 ,...,εr , where ε1 , ..., εr are exactly the equivalence classes of ε, which have at least two elements. In particular, we have h δk; = Ekh

and h = {(x, x, ..., x) ∈ Ekh | x ∈ Ek }. δk;E k 1

See Theorem 2.5.1.

2.4 Relation Algebras, Co-Clones, and Derivation of Relations

127

2.3 Elementary Operations on Rk We deﬁne in this section some operation ζ, τ, pr, ∧ and × on Rk with whose aid we can form later more “complex” operations on Rk . We call, therefore, the operations ζ, τ, pr, ∧ and × the elementary operations on Rk .

Let ∈ Rkh and ∈ Rkh , where h, h ∈ N. Then, if = ∅ and = ∅, let ζ ∈ Rkh , τ ∈ Rkh , pr ∈ Rkh−1 for h ≥ 2 and pr = ∅ for h = 1, × ∈ Rkh+h and ∧ ∈ Rkh (only for h = h ) deﬁned by: ζ := {(a2 , a3 , ..., ah , a1 ) | (a1 , a2 , ..., ah ) ∈ } (cyclical exchanging of the rows), τ := {(a2 , a1 , a3 , ..., ah ) | (a1 , a2 , ..., ah ) ∈ } (exchange of the ﬁrst two rows) for h ≥ 2 and ζ = τ = for h = 1 or = ∅; pr := {(a2 , ..., ah ) | ∃a1 ∈ Ek : (a1 , a2 , ..., ah ) ∈ } (projection onto the 2th,..., h-th coordinate or the strike of the ﬁrst row) for h ≥ 2, × := {(a1 , ..., ah , b1 , ..., bh ) | (a1 , a2 , ..., ah ) ∈ ∧ (b1 , b2 , ..., bh ) ∈ } (Cartesian product of and ) and ∧ := {(a1 , ..., ah ) | (a1 , ..., ah ) ∈ ∩ } (intersection of the relations and ).

2.4 Relation Algebras, Co-Clones, and Derivation of Relations The algebra

3 , ζ, τ, pr, ∧, ×) Rk := (Rk ; δk;{1,2}

of type (0, 1, 1, 1, 2, 2) is called full relation algebra on Ek . Every subalgebra Q of Rk (in symbol Q ≤ Rk ) is a relation algebra on Ek . Let Q ⊆ Rk . Then, [Q] denotes the set of all relations of Rk that can be obtained by a ﬁnite number of applications of the elementary operations 3 ,i.e., [Q] is the universe of ζ, τ, pr, ∧ and × from the relations of Q and δk;{1,2} the smallest relation algebra, which contains Q. If [Q] = Q (⊆ Rk ) we say that Q is closed or Q is a co-clone of Rk . Further, we say a relation can be derived from the relation (or is -derivable), if ∈ [{}]. In this case we also write: .

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2 The Galois-Connection Between Function- and Relation-Algebras

2.5 Some Operations on Rk Derivable from the Elementary Operations We say that an operation on Rk is derivable from the elementary operations ζ, τ, pr, ∧ and × (or {ζ, τ, pr, ∧, ×}-derivable), if their eﬀect on an arbitrary relation ∈ Rk can also be described by the eﬀect of a ﬁnite com3 on . position of the elementary operations and δk;{1,2} The following is a small list of such derivable operations. In this case, always describes an h-ary and an h -ary relation of Rk . (O1 ): Permutation of coordinates (or permutations of rows). As is known, the permutations 1 2 ... h − 1 h 1 2 3 ... h and 2 3 ... h 1 2 1 3 ... h form a generating system for the symmetric group Sh (i.e., for the set of all permutations on {1, 2, ..., h} with the operation 2). Consequently one can realize all rearrangements of the rows of with the aid of ζ and τ . Thus, for every s ∈ Sh , σs () := {(as(1) , ..., as(h) ) | (a1 , ..., ah ) ∈ } is a {ζ, τ, pr, ∧, ×}-derivable operation. (O2 ): Projection onto the α1 -th, ..., αt -th coordinates (or deleting of rows). For {α1 , ..., αt } ⊆ {1, 2, .., h} it holds: prα1 ,...,αt () := {(aα1 , ..., aαt ) | (a1 , .., ah ) ∈ } = pr(pr(...(pr (σs ()))...)), h−t times where s ∈ Sh and s(α1 ) = h − t + 1, s(α2 ) = h − t + 2, ..., s(αt ) = h. In particular, we have prs(1),...,s(n) () = σs (). We remark that, in the case where is given in the form {(a0 , a1 , ..., ah−1 ) | ....}, we choose {α1 , ..., αt } ⊆ {0, 1, 2, .., h − 1} and deﬁne prα1 ,...,αt () in analog manner, as above.

2.5 Some Operations on Rk Derivable from the Elementary Operations

129

(O3 ): Identiﬁcation of coordinates. For i, j ∈ {1, 2, ..., h} and i = j let ∆i,j () := {(a1 , ..., aj−1 , aj+1 , ..., ah ) | (a1 , ..., aj−1 , ai , aj+1 , ..., ah ) ∈ } and ∆ := ∆1,2 . ∆i,j can be formed as follows: 3 ) = Ek and It is easy to prove that pr1 (δk;{1,2} h δk;{i,j} = pr1,...,i−1,h+1,i+1,...,j−1,h+2,j+1,...,k (1 ), 3 and i < j. Consequently, ∆i,j is where 1 := Ek × ... × Ek ×pr1,2 δk;{1,2} h−2 times derivable because of h ∆i,j = pr1,...,j−1,j+1,...,h ( ∧ δk,{i,j} ).

(O4 ): Doubling of coordinates (rows). One receives a doubling of the i-th row of as follows: νi () := {(a1 , ..., ai−1 , ai , ai , ai+1 , ..., ah ) | (a1 , ..., ah ) ∈ } 3 = pr1,...,i−1,h,h+1,,i,...,h−1 (∆i,h+1 ( × pr(δk;{2,3} )).

(O5 ): Adding of ﬁctitious coordinates. Let ∇ := {(a1 , ..., ah+1 )|a1 ∈ Ek ∧ (a2 , ..., ah+1 ) ∈ }. The ﬁrst coordinate of ∇ is a so-called ﬁctitious coordinate. Then, with the aid of (O1 ) one can derive the relation ∇i := {(a1 , ..., ai−1 , ai , ai+1 , ..., ah+1 ) ∈ Ekh+1 | (a1 , ..., ai−1 , ai+1 , ..., ah+1 ) ∈ } for i ∈ {1, ..., h + 1}. (O6 ): General composition (relation product). Let ◦t := {(a1 , ..., ah−t , bt+1 , ..., bh ) | ∃u1 , ..., ut ∈ Ek : (a1 , ..., ah−t , u1 , ..., ut ) ∈ ∧ (u1 , ..., ut , bt+1 , ..., bh ) ∈ } for t ∈ N with t ≤ h and t ≤ h . In particular, o := o1 . For h = h = 2, ◦ is the well-known relation product 2. The following theorem results from the above considerations.

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2 The Galois-Connection Between Function- and Relation-Algebras

Theorem 2.5.1 Every co-clone Q of Rk contains all diagonal relations and is closed in respect to – permutation of coordinates (rows), – projection onto coordinates (or deleting of rows), – identiﬁcation of coordinates (rows), – doubling of coordinates (rows), – adding of ﬁctitious coordinates, – (ﬁnite) intersection formation, – Cartesian products and – general composition.

2.6 The Preserving of Relations; Pol, Inv We say that a function f n ∈ Pk preserves the relation ∈ Rkh (or is invariant for f or is a invariant for f ), if ⎛ ⎛ ⎞ ⎞ f (a11 , a12 , ..., a1n ) a11 a12 ... a1n ⎜ f (a21 , a22 , ..., a2n ) ⎟ ⎜ a21 a22 ... a2n ⎟ ⎜ ⎟ ⎟ f⎜ ⎝ . . . . . . . . . . . . . . . ⎠ := ⎝ . . . . . . . . . . . . . . . . . . ⎠ ∈ ah1 ah2 ... ahn f (ah1 , ah2 , ..., ahn ) for all

⎛

⎞ ⎛ a11 a12 ⎜ a21 ⎟ ⎜ a22 ⎜ ⎟ ⎜ ⎝ ... ⎠ , ⎝ ... ah1 ah2

⎞

⎛

⎞ a1n ⎟ ⎜ ⎟ ⎟ , ..., ⎜ a2n ⎟ ∈ . ⎠ ⎝ ... ⎠ ahn

The empty set ∅ is preserved by every function f ∈ PA . Note that f preserves iﬀ is the universe of a subalgebra (A; f )h . By P olk or, brieﬂy, P ol we denote the set of all functions f ∈ Pk that preserve the relation . For Q ⊆ Rk , we put P olk . P olk Q := ∈Q

P olk or P olk Q is a short-cut of polymorphisms of or Q, respectively. The set of all relations ∈ Rk that are preserved from the function f ∈ Pk is Invk f. For A ⊆ Pk let

2.6 The Preserving of Relations; Pol, Inv

Invk A :=

131

Invk f

f ∈A

be the set of all invariants of A and let (Invk A)n := (Invk A) ∩ Rkh be the set of all n-ary invariants of A. If k can be seen from the context, we write Inv instead of Invk . Further notations used by us are P oln Q := (P ol Q)n (Q ⊆ Rk or Q ∈ Rk ) and Inv n A := (Inv A)n (A ⊆ Pk or A ∈ Pk ) for n ∈ N. Elementary connections between P ol and Inv are summarized in the following theorem. Theorem 2.6.1 For arbitrary A, B ⊆ Pk and arbitrary S, T ⊆ Rk , it holds: (a) A ⊆ B =⇒ Inv B ⊆ Inv A, S ⊆ T =⇒ P ol T ⊆ P ol S; (b) A ⊆ P ol Inv A, S ⊆ Inv P ol S; ((a) and (b) mean that the pair (P ol, Inv) of mappings P ol : P(Pk ) −→ P(Rk ), A → P olk A and P ol : P(Rk ) −→ P(Pk ), Q → Invk Q is a Galois connection (see Part I, 4.4) between the sets Pk and Rk .) (c) Inv P ol Inv A = Inv A, P ol Inv P ol S = P ol S; (d) A ⊆ P ol S ⇐⇒ S ⊆ Inv A; (e) P ol (S ∪ T ) = P ol S ∩ P ol T , Inv A ∪ B = Inv A ∩ Inv B. Proof. (a), (b), (d), and (e) are direct conclusions from the deﬁnitions of P ol and Inv. (c): Let S := Inv A. By (b) we have S ⊆ Inv P ol S. Conversely, it holds A ⊆ P ol Inv A and thus by (a): Inv P ol Inv A ⊆ Inv A. Therefore, Inv P ol Inv A = Inv A. Analogously, one can show that P ol Inv P ol S = P ol S holds.

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2 The Galois-Connection Between Function- and Relation-Algebras

Theorem 2.6.2 For every A ⊆ Pk and every Q ⊆ Rk the sets Inv A and P ol Q are closed (in respect to the operations deﬁned above, respectively); i.e., P ol Q is a clone of Pk and Inv A is a co-clone of Rk . Furthermore, it holds: Inv [A] = Inv A and P ol [Q] = P ol Q. In particular we have: Inv [{e21 }] = Rh and P ol Dk = Pk . Proof. Let , ∈ Inv A and f ∈ A be arbitrary. Then, f preserves the -derivable relations ζ, τ , pr, ∧ and × obviously. Thus {ζ, τ , pr, ∧ , × } ⊆ Inv A, whereby Inv A is closed. Now let f n , g m ∈ P ol Q and ∈ Q be arbitrary. Clear that, then the functions ζf, τ f and ∆f also preserve . Further, the relation is also an invariant of f g, because f (g(r1 , ..., rm ), rm+1 , ..., rm+n−1 ) ∈ holds for all r1 , ..., rm+n−1 ∈ , since g(r1 , .., rm ) ∈ . Inv [A] = Inv A follows from Inv [A] ⊆ Inv A (because of A ⊆ [A] and Theorem 2.6.1, (a)) and Inv A ⊆ Inv [A]. (By Theorem 2.6.1, (a) we have A ⊆ P ol Inv A indeed. Since P ol Inv A is closed, this implies [A] ⊆ P ol Inv A. If one uses Theorem 2.6.1, (a) again and then 2.6.1, (c), one receives Inv P ol Inv A = Inv A ⊆ Inv [A].) Analogously, one can prove P ol [Q] = P ol Q. One can easily checks the remaining statements of the theorem. A conclusion of Theorem 2.6.1, (a) and of the deﬁnition of (see Section 2.4) is the following Theorem 2.6.3 For arbitrary relations , ∈ Rk it holds: =⇒ P ol ⊆ P ol .

2.7 The Relations χn and Gn For arbitrary n ∈ N and k ∈ N\{1} denote χk;n or – if k can be seen from the context – χn the k n -ary relation, whose rows are just all (x1 , ..., xn ) ∈ Ekn that are arranged (we say “lexicographical”) unambiguously according to the following regulation:

2.7 The Relations χn and Gn

133

The tuple (x1 , ..., xn ) is before the tuple (y1 , ..., yn ), if the integer x1 · k n−1 + x2 · k n−2 + ... + xn−1 · k + xn is smaller than y1 · k n−1 + y2 · k n−2 + ... + yn−1 · k + yn . For example, the following is valid:

χ2;3

⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎜0 := ⎜ ⎜1 ⎜ ⎜1 ⎜ ⎝1 1

0 0 1 1 0 0 1 1

⎞ 0 1⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 1⎟ ⎟ 0⎠ 1

We denote the columns of χn with χ(1), ..., χ(n). Obviously, there is exactly a function fr ∈ Pkn with f (χn ) = r to every column n r ∈ Ekk . The relation n Gn (A) := {r ∈ Ekk |fr ∈ An }, which one can form with the aid of the functions fr , is called n-th graphic of A ⊆ Pk . The following theorem summarizes elementary properties of the relation Gn (A). Theorem 2.7.1 For an arbitrary clone A ⊆ Pk it holds: (a) ∀n ∈ N : Gn (A) ∈ Inv A; (b) f n ∈ An ⇐⇒ f n ∈ P ol Gn (A); (c) A ⊆ ... ⊆ P ol Gn (A) ⊆ P ol Gn−1 (A) ⊆ ... ⊆ P ol G2 (A) ⊆ P ol G1 (A); (d) A = n≥1 P ol Gn (A); (e) ∀ ∈ Inv A : ∈ [ n≥1 {Gn (A)}]; (f ) Inv A = [ n≥1 {Gn (A)}]. Proof. (a): Let g m ∈ A and r1 , ..., rm ∈ Gn (A) be arbitrary. Then g(r1 , ..., rm ) = g(fr1 , ..., frm ) = h(χn ), where

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2 The Galois-Connection Between Function- and Relation-Algebras

h(x1 , ..., xn ) := g(fr1 (x1 , ..., xn ), ..., frm (x1 , ..., xn )). Since h belongs to An , we obtain h(χn ) ∈ Gn (A). (b): If f n ∈ A, we have f n ∈ P ol for every ∈ Inv A. Consequently, f n ∈ P ol Gn (A) because of (a). On the other hand, f n ∈ Gn (A) implies the existence of a certain r ∈ Gn (A) with f n (χn ) = r; thus f = fr ∈ An is valid. (c) and (d) follow from (b) and from the clone properties. (e): Let ∈ Inv h A be arbitrary and t := ||. We show that can be derived from Gt (A) using the operation prα1 ,...,αh . Since Jk ⊆ A, we have χt ⊆ Gt (A). For every j ∈ {1, ..., h}, one can ﬁnd an αj so that the j-th row of is identical with the αj -th row of χt . Because of ∈ Inv h A, it follows prα1 ,...,αh Gt (A) = ∈ [{Gt (A)}]. (f): By (e) we have Inv A ⊆ [ n≥1 {Gn (A)}]. Further, because of (a) and Theorem 2.6.2, we have [ n≥1 {Gn (A)}] ⊆ Inv A. Hence, (f) is valid.

2.8 The Operator ΓA For arbitrary A ⊆ Pk denote ΓA a mapping from Rk into Rk , which is deﬁned for σ ∈ Rkh as follows: (2.1) ΓA (σ) := { ∈ Rk | ∈ Inv A ∧ σ ⊆ }. In the language of the Universal Algebra, ΓA (σ) is a subalgebra of (Ek ; A)h , which is generated by σ, or ΓA (σ) is the universe of the smallest subalgebra of the algebra (Ek ; A)h , which contains the set σ. Obviously, ΓA is a hull operator. Theorem 2.8.1 For an arbitrary clone A ⊆ Pk and every n ∈ N it holds: (a) ΓA (χn ) ∈ Inv A; (b) ΓA (χn ) = Gn (A); (c) An = {fr |r ∈ ΓA (χn )}. Proof. (a) follows from [Inv A] = Inv A (see Theorem 2.6.2) and the deﬁnition of ΓA (χn ). (b): Since every projection eni belongs to An , we have χn ⊆ Gn (A). Denote now an arbitrary k n -ary relation of Rk with χn ⊆ . If ∈ Inv A, then f (χn ) ∈ for every f ∈ An , i.e., Gn (A) ⊆ . Consequently, we have shown that Gn (A) ⊆ ΓA (χn ) holds. ΓA (χn ) ⊆ Gn (A) follows from Gn (A) ∈ Inv A (see Theorem 2.7.1, (a)). (c) is an easy conclusion from (b) and the deﬁnition of Gn (A).

2.9 The Galois Theory for Function- and Relation-Algebras

135

2.9 The Galois Theory for Function- and Relation-Algebras Theorem 2.9.1 Let A be a clone of Pk . Then A = P ol Inv A. Proof. By Theorem 2.6.1, (b) we have A ⊆ P ol Inv A. To prove that P ol Inv A ⊆ A let f n ∈ P ol Inv A be arbitrary. Because of Theorem 2.7.1, (a) we have that f ∈ P ol Gn (A) holds and (by Theorem 2.7.1, (b)) f ∈ An . Thus A = P ol Inv A. Theorem 2.9.2 Let Q be a co-clone of Rk . Then Q = Inv P ol Q. Proof. Let A := P ol Q. Because of Theorem 2.6.1, (b) we have Q ⊆ Inv A. To prove that Inv A ⊆ Q it is suﬃcient to show that Γ (χt ) ∈ Q for arbitrary t ∈ N, since [ t≥1 {ΓA (χt )}] = Inv A (see Theorem 2.7.1, (f) and Theorem 2.8.1, (b)). Let now γ := { ∈ Q|χt ⊆ }. t

Because of Ekk ∈ Q and the fact that Q is closed in respect to ∩, we have t χt ⊆ γ and γ is the smallest relation (∈ Qk ), which contains χt , in respect to cardinality. Further, we have ΓA (χt ) ⊆ γ, since γ ∈ Q ⊆ Inv A (see (2.1)). Consequently, our theorem is proven, if we can show ΓA (χt ) = γ. Suppose, ΓA (χt ) ⊂ γ. Then, there is a column r ∈ γ\ΓA (χt ). Because of At = {fs |s ∈ ΓA (χt )} (see Theorem 2.8.1, (c)) we have fr ∈ At . Consequently, there exists an m-ary relation β ∈ Inv A and certain columns r1 , ..., rm ∈ β with f (r1 , ..., rm ) ∈ β. Every row of the matrix (r1 , ..., rm ) is also a row of the matrix χt . Denote ij the number of a row of χt , which agrees with the j-th row of (r1 , ..., rm ) (j = 1, 2, ..., m). Let now k +m γ := pr1,2,...,kt (γ × β) ∩ δ{i . t t t 1 ,k +1},{i2 ,k +2},...,{im ,k +m} t

Since Q is closed, γ belongs to Q, and by construction of γ we have χt ⊆ γ ⊆ γ. Furthermore, we have r ∈ γ\γ , since r1 , ...., rt ∈ β, fr (r1 , ..., rt ) ∈ β and fr (χt ) = r ∈ γ. With γ we received a contradiction to the choice of γ. Thus, γ = ΓA (χt ). With the aid of Theorems 2.9.1 and 2.9.2, we can prove the important properties of the P ol-Inv-connection:

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2 The Galois-Connection Between Function- and Relation-Algebras

Theorem 2.9.3 (Theorem of V. G. Bodnarˇ cuk, L. A. Kaluˇ znin, V. N. Kotov and B. A. Romov; [Bod-K-K-R 68/69]) indextheorem of Bodnarˇcuk, Kaluˇznin,. Kotov and Romov Let L(Pk ) (or L(Rk )) be the set of all clones (or co-clones) of Pk (or Rk ) respectively. Then the mappings Inv : L(Pk ) −→ L(Rk ), A → Inv A and P ol : L(Rk ) −→ L(Pk ), Q → P ol Q are bijective mappings, which reverse the partial order ⊆, i.e., it holds ∀A, B ∈ L(Pk ) : A ⊆ B =⇒ Inv B ⊆ Inv A and ∀S, T ∈ L(Rk ) : S ⊆ T =⇒ P ol T ⊆ P ol S. In other words: The lattices (L(Pk ), ⊆) and (L(Rk ), ⊆) are antiisomorphic. Pk

Rk

@ @ @

@ @

A = P ol @ Inv A

XX

Inv B

@

Inv A

S = Inv P ol S

@

: XXX X X XX X z X P ol T

B

X y XX

XXX XXXX 9 X

P ol S

@

@ @ @

T

@

@

@ @

Jk

Dk

Fig. 2.1

Proof. By Theorem 2.6.2, the mappings Inv and P ol are mappings from L(Pk ) (or L(Rk )) into L(Rk ) (or L(Pk )), respectively. The surjectivity and the injectivity (and then the bijectivity) of these mappings are easy conclusions from Theorem 2.9.1 and Theorem 2.9.2 (with the help of Theorem 2.6.2). The “reversal property” of P ol and Inv (in respect to ⊆) was already given in Theorem 2.6.1, (a).

2.10 Some Modiﬁcations of the P ol-Inv-Connection

137

2.10 Some Modiﬁcations of the P ol-Inv-Connection 2.10.1 Galois Theory for Finite Monoids and Finite Groups It is well-known that every ﬁnite semigroup H = (H; ◦) is isomorphic to a 1 certain subsemigroup of (P|H| ; ) and every ﬁnite group G = (G; ◦) is isomorphic to a certain subgroup of the group (S|G| ; ), where Sk := Pk1 [k] for k ∈ N. 2

Furthermore we have Lemma 2.10.1.1 For an arbitrary subset A of [Pk1 ], the set A is a clone of [Pk1 ] if and only if A = [A]∇ holds and (A1 ; ) is a subsemigroup of (Pk1 ; ) with the unit element e. In other words, a clone A ⊆ [Pk1 ] is completely determined by the monoid (A1 ; ). Because of the above properties, it is possible to derive a Galois theory for semigroups and groups from the Galois theory for function algebras (see Section 2.9). For this purpose, we deﬁne two new operations ∨ and ¬ on Rk : For arbitrary , ∈ Rkh and h ∈ N we set ∨ := ∪ (union) and ¬ := Ekh \ (negation, complement). The following lemma supplies a reason for these new relation operations. Lemma 2.10.1.2 It holds: (a) ∀H ⊆ Pk1 : , ∈ Inv h H =⇒ ∨ ∈ Inv h H; (b) ∀G ⊆ Sk : ∈ Inv h G =⇒ ¬ ∈ Inv h G. Proof. (a): If f ∈ H ⊆ Pk1 , {, } ⊆ Inv H and r ∈ ∨ , then we have obvious r ∈ or r ∈ , whereby f (r) ∈ ∨ , since f preserves the relations and . Consequently, ∨ is also an invariant of f . (b): Let f ∈ G ⊆ Sk , ∈ Inv G and r ∈ ¬ be arbitrary. Set r := f (r). Then r belongs to ¬, since f −1 (r ) = r, f −1 ∈ [G] and Inv G = Inv [G] (see Theorem 2.6.2). Consequently, f preserves the relation ¬. An algebra of the form 3 (Q; δk;{1,2} , ζ, τ, pr, ×, ∧, ∨) 2

For proof, one can assume w.l.o.g. H = G = {0, 1, ..., k − 1}. Let now fa (x) := a ◦ x ∈ Pk1 for arbitrary a ∈ Ek . Then, the mapping α : Ek −→ Pk1 , a → fa is an isomorphic mapping from H (or G) onto the semigroup (or group) (A)

({f0 , f1 , ..., fk−1 }; ), since α(a ◦ b)(x) = fa◦b (x) = (a ◦ b) ◦ x = a ◦ (b ◦ x) = fa (fb (x)) = (fa fb )(x).

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2 The Galois-Connection Between Function- and Relation-Algebras

with Q ⊆ Rk is called Krasner-algebra of ﬁrst kind and an algebra of the form 3 , ζ, τ, pr, ×, ∧, ∨, ¬) (Q; δk;{1,2} with Q ⊆ Rk Krasner-algebra of second kind. In Theorem 2.10.1.4, we will see that the algebras just deﬁned are the right partners for semigroups (or groups) for a Galois-correspondence. Following our terminology from Section 2.4, we call this a clone, which is also closed, concerning the operation ∨, a co-monoid. Moreover, a co-monoid, which is closed concerning the operation ¬, is a co-group. The following lemma summarizes all auxiliary statements that are necessary to prove Theorem 2.10.1.4. Lemma 2.10.1.3 Let H, G ⊆ Pk1 and Q, T ⊆ Rk . Further, let H := (H; ) be a monoid, G := (G; ) be a group, Q be a co-monoid, and T be a co-group. Then (a)

H = P ol1 Inv H

(a’)

G = P ol1 Inv G

(b)

P ol Q ⊆ [Pk1 ]

(b’)

P ol T ⊆ [Sk ]

(c)

Q = Inv P ol1 Q

(c’)

T = Inv P ol1 T .

Proof. (a), (a’): Let A be a monoid of Pk1 (or a group of Sk ). Then, by Lemma 2.10.1.1, [A]∇ is a clone of Pk and, because of Theorem 2.6.2, we have [A]∇ = P ol Inv [A]∇ = P ol Inv A. Consequently, A = [A]1 = P ol1 Inv A. (b): Since the diagonal relations belong to Q and since Q is closed in respect to ∨, the relation 3 3 ∪ δk;{2,3} γ := δk;{1,2} belongs to Q. With the help of Theorem 1.4.4, it can easily be shown that the relation is preserved of no function of Pk , which depends on at least two variables essentially. Thus, P ol Q ⊆ [Pk1 ]. (b’): Through (b) we already proven P ol T ⊆ [Pk1 ]. Further, the diagonal relation ι2k := {(x, x)|x ∈ Ek } belongs to Q, whereby also ¬ι2k ∈ Q. Obviously the relation ¬ι2k = {(x, y)|x = y} is preserved, however, only from the permutations of Pk1 . Consequently, we have P ol T ⊆ [Sk ]. (c), (c’): If Q is a co-monoid (or is a co-group) of Rk , then Q is also a co-clone of Rk and it is valid (because of P ol1 Q ⊆ P ol Q, Theorem 2.6.1, (a) and Theorem 2.9.2): Q = Inv P ol Q ⊆ Inv P ol1 Q. By (b) (or (b’)) we have in addition P ol Q ⊆ [P ol1 Q], whereby Inv [P ol1 Q] ⊆ Inv P ol Q and (because of Inv P ol1 Q = Inv [P ol1 Q] by Theorem 2.6.2 and Inv P ol Q = Q by Theorem 2.9.2) Inv P ol1 Q ⊆ Q follow. One obtains the following theorem as an easy conclusion in analogy to the above theorem from Lemmas 2.10.1.2 and 2.10.1.3.

2.10 Some Modiﬁcations of the P ol-Inv-Connection

139

Theorem 2.10.1.4 ([Kra 45], [Kra 68/69]) Let M be the set of all monoids of the form (M ; ) with M ⊆ Pk1 and G be the set of all groups of the form (G; ) with G ⊆ Sk . Further, let K1 be the set of all co-monoids of Rk and K2 be the set of all co-groups of Rk . Then, the lattices (M; ⊆) and (K1 ; ⊆) and the lattices (G; ⊆) and (K2 ; ⊆) are antiisomorphic. In particular, for i ∈ {1, 2}, L1 := M and L2 := G are valid: The mappings P ol1 : Ki −→ Li , Q → P ol1 Q and Inv : Li −→ Ki , H → Inv H are bijective mappings, which reverse the partial order ⊆, i.e., it holds ∀H, G ⊆ Li : H ⊆ G =⇒ Inv G ⊆ Inv H and

∀S, T ⊆ Ki : S ⊆ T =⇒ P ol1 T ⊆ P ol1 S.

2.10.2 Galois Theory for Iterative Function Algebras Iterative function algebras are algebras of the form (Pk ; ζ, τ, ∆, ∇; ). The universes of subalgebras of these algebras Pk (k ∈ N\{1}) are called subclasses or, brieﬂy, classes of Pk . Unlike clones of Pk , there are classes of Pk that do not contain the projections. Nevertheless, for the classes of Pk , one can also ﬁnd certain relation algebras as partners for a Galois-correspondence. Subsequently, two possibilities for this purpose will be presented. For the ﬁrst possibility, we choose a certain class A of Pk that does not contain the function e11 and which, therefore, is not a clone. Let L↓k (A) be the lattice of all subclasses of A and let Lk (Jk , Jk ∪ A) be the lattice of all subclasses of Jk ∪ A, which have Jk as a subset. Moreover, let Lk (Inv(Jk ∪ A), Rk ) be the set of all co-clones C of Rk with Inv(Jk ∪ A) ⊆ C. Obviously, the mapping α : L↓k (A) −→ Lk (Jk , Jk ∪ A), T → Jk ∪ T is an isomorphism from L↓k onto Lk (Jk , Jk ∪ A). Since the elements of Lk (Jk , Jk ∪ A) are clones of Pk , the lattice Lk (Jk , Jk ∪ A) is antiisomorphic to the lattice Lk (Inv(Jk ∪ A), Rk ) by Theorem 2.9.3. Consequently, we have

140

2 The Galois-Connection Between Function- and Relation-Algebras

Theorem 2.10.2.1 Let A be a subclass of Pk . Further, put P olA Q := A ∩ P olk Q for arbitrary Q ⊆ Rk . Then the mappings Inv : L↓k (A) −→ Lk (Inv(Jk ∪ A)), T → Inv T, and

P olA : Lk (Inv(Jk ∪ A), Rk ) −→ L↓k (A), Q → P olA Q

are bijective mappings, and the lattices L↓k (A) and Lk (Inv(Jk ∪ A), Rk ) are antiisomorphic. As W. Harnau in [Har 83] was pointing, another possibility to characterize subclasses of Pk consists that one establishes relation pairs and relation-pairalgebras. Let Rphk := {(, ) ∈ Rkh × Rkh | ⊆ } and Rpk :=

Rphk .

h≥1

The elements of Rpk are called relation pairs. We say that f n ∈ Pk preserves the relation pair (, ) ∈ Rpk , if f (r1 , ..., rn ) ∈ for arbitrary r1 , ..., rn ∈ . By deﬁnition, each function of Pk preserves the pair (∅, ∅). If = , then the fact “f preserves the relation pair (, )” is identical with the fact “f preserves ”. Let F ⊆ Pk and Q ⊆ Rpk . Then, denote Invpk F the set of all relation pairs of Rpk , which are preserved from every function of F . Further, let P olpk Q be the set of all functions of Pk , which preserve each relation pair of Q. It can easily be shown that the following lemma holds: Lemma 2.10.2.2 ([Har 83]) (a) ∀Q ⊆ Rpk : P olpk Q = [P olpk Q]ζ,τ,∆,∇, ; (b) ∀(, ) ∈ Rpk : ([P ol(, )]e21 ,ζ,τ,∆, = P ol(, ) ⇐⇒ = ). Analogous to Section 2.3, one can deﬁne certain operations on the set Rpk : h+1 (“the doubling For α ∈ {ζ, τ, ∆, pr, ∼} (see Section 2.3, ∼ h := ∇ ∧ δ{1,2} of the ﬁrst row of ”)) and (, ), (µ, µ ) ∈ Rpk we set: α(, ) := (α, α ), (∇, ∇ ), if = ∅, ∇(, ) := (∇, ∅), if = ∅, and

2.10 Some Modiﬁcations of the P ol-Inv-Connection

141

(, ) × (µ, µ ) := ( × µ, × µ ).

Further, put E := h≥1 Ekh in the following. For each a ∈ E one can deﬁne two operations on Rpk as follows: (\{a}, ), if ⊆ \{a}, ν1,a (, ) := otherwise, (, ) and ν2,a (, ) :=

(, ∪ {a}), if a ∈ , otherwise (, )

((, ) ∈ Rpk ). 3 The importance of the above-deﬁned operations results from the following lemma, which one can easily check. Lemma 2.10.2.3 Let (, ) be a relation pair which is derivable from Q (⊆ Rpk ) by means of the operations ζ, τ, ∆, ∇, ∼, pr, ×, ν1,a , ν2,a (a ∈ E). Then P olpk Q ⊆ P olpk (, ). The algebra Rpk := (Rpk ; (∅, ∅), ζ, τ, ∆, pr, ∼, ×, (ν1,a )a∈E , (ν2,a )a∈E ) of the type (0,1,1,1,1,1,2,1,1,1,...) is called full relation-pair algebra on Ek . A relation-pair algebras is a subalgebra of this algebra. Further, a universe of a relation-pair algebra is called co-class. Then, the following theorem can be proven with some expenditure (ultimately similar to the proof of Theorem 2.9.3):

Theorem 2.10.2.4 ([Har 83]) Let L(Pk ) (or L(Rpk )) be the set of all classes (or co-classes) of Pk (or Rpk ), respectively. Then the mappings Invk : L(Pk ) −→ L(Rpk ), A → Invk A and P olpk : L(Rpk ) −→ L(Pk ), Q → P olpk Q 3

In [Har 83] so-called multi-operators dh and dv are ﬁrst deﬁned instead of the operations ν1,a , ν2,a (a ∈ E): dh (, ) := {(, µ) ∈ Rpk | ⊆ µ ⊆ }, dv (, ) := {(µ, ) ∈ Rpk | ⊆ µ ⊆ }. Then, relation-matrix-algebras, in which these multi-operators are describable through two operations, are introduced. It is shown how one can repair the “lack” with the multi-operators by going over to the relation-matrix-algebras with ﬁnite many operations.

142

2 The Galois-Connection Between Function- and Relation-Algebras

are bijective mappings with the properties ∀A, B ∈ L(Pk ) : A ⊆ B =⇒ Invk B ⊆ Invk A, ∀S, T ∈ L(Rpk ) : S ⊆ T =⇒ P olpk T ⊆ P olpk S; i.e., the lattices (L(Pk ); ⊆) and (L(Rpk ); ⊆) are antiisomorphic.

2.11 Some Connections Between the Relation Operations In this last section of Chapter 2, we want to clarify the order in which one can use the relation operations given in Section 2.3, to obtain all relations from the set [Q] (⊆ Rk ) as eﬀectively as possible. Let ∆ := {(x1 , x2 , ..., xh ) ∈ | x1 = x2 } ( ∈ Rkh ). This relation is obvious derivable from our elementary operations. For arbitrary α, α1 , ..., αt ∈ {ζ, τ, ∆, ∆ , ∇, pr} we use the following notations: α1 := α(), αi := α(αi−1 ) for i ∈ N, α1 α2 ...αt := α1 (...(αt−1 (αt ())...) ( ∈ Rk ). It can easily be shown that the Lemma 2.11.1 holds.

Lemma 2.11.1 For arbitrary relations ∈ Rkh , ∈ Rkh and ∈ Rkh it is valid:

(a) × = (ζ h ∆h ) ∩ (∇h ); 3 (b) ∅ = prh , δk;{1,2} = ∆ ∇3 ∅;

(c) ζ(pr) = pr(ζ(τ ())), τ (pr) = pr(ζ h−1 (τ (ζ()))), ∆ (pr) = pr(ζ h−1 (∆ (ζ()))), ∇(pr) = pr(τ ((∇()))); (d) ζ( ∧ ) = (ζ) ∧ (ζ ), τ ( ∧ ) = (τ ) ∧ (τ ), ∆ ( ∧ ) = (∇) ∧ (∇ ), if h = h ; (e) ( ∨ ) ∧ = ( ∧ ) ∨ ( ∧ ), if h = h = h .

2.11 Some Connections Between the Relation Operations

143

3 Theorem 2.11.2 Let Ω := {δk;{1,2} , ζ, τ, pr, ×, ∧}. Then for all Q ⊆ Rk it is valid:

(a) [Q]Ω = [Q]ζ,τ,∆ ,∇,pr,∧ ; (b) [Q]ζ,τ,∆ ,∇,pr,∧ = [ [[Q]ζ,τ,∆ ,∇ ]∧ ]pr ; (c) [Q]Ω∪{∨} = [Q]ζ,τ,∆ ,∇,pr,∧,∨ ; (d) [Q]ζ,τ,∆ ,∇,pr,∧,∨ = [ [ [ [Q]ζ,τ,∆ ,∇ ]∧ ]∨ ]pr . Proof. (a): Since ∆ and ∇ are Ω-derivable operations, we have [Q]ζ,τ,∆ ,∇,pr,∧ ⊆ [Q]Ω . The reversed inclusion [Q]Ω ⊆ [Q]ζ,τ,∆ ,∇,pr,∧ follows from Lemma 2.11.1, (a), (b). (b) follows from Lemma 2.11.1, (c), (d). One proves the statements (c) and (d) analogously to (a) and (b) using Lemma 2.11.1, (e). A translation of these observations into the language of mathematical logic can be found in [P¨ os-K 79], p. 63–68.

3 The Subclasses of P2

A basic result of many-valued logic is the description of all closed sets of Boolean functions given by E. L. Post in [Pos 20] and [Pos 41]. Since Post’s proof is long and rather complicated, revisions (for instance [Jab-G-K 70] and [Ugo 88]) and new proofs (for instance [Ber 80], [McK-M-T 87] and [Res-D 89] or [Den-W 2002]) have been published. The new proof methods of the last years mainly result from the fact that parts of Post’ results are special cases or conclusions of certain theorems of many-valued logic or universal algebra. In this chapter, we tried to verify Post’s results in an elementary way by working out some essential basic ideas.

3.1 Deﬁnitions of the Subclasses of P2 and Post’s Theorem We need some notations introduced in Chapter 1 to deﬁne the subclasses of P2 and to describe certain generated sets of this subclasses. We shall deﬁne functions of P2 by formulae over the alphabet {x, y, z, x1 , x2 , ...} and we use the usual symbols ∧ (“conjunction” or “multiplication modulo 2”), ∨ (“disjunction”), + (“addition modulo 2”) and − (“negation”). By ◦ ∈ {∧, ∨, +},

−

, cna (a ∈ E2 ), eni (1 ≤ i ≤ n), m3 , t2 , q 3 , r3 , hµ+1 (µ ∈ N) µ

we denote functions of P2 given by

146

3 The Subclasses of P2

◦(x, y) := x ◦ y, −

(x) := x,

cna (x1 , ..., xn ) := a, eni (x1 , ..., xn ) := xi , m(x, y, z) := x ∧ (y ∨ z), t(x, y) := x ∧ y, q(x, y, z) := x ∧ (y ∨ z), r(x, y, z) := x + y + z, µ+1 hµ (x1 , ..., xµ+1 ) := i=1 (x1 ∧ x2 ∧ ... ∧ xi−1 ∧ xi+1 ∧ ... ∧ xµ+1 ) = 1 if ∃i ∈ {1, ..., µ + 1} : x1 = ... = xi−1 = xi+1 = ... = xµ+1 = 1, 0 otherwise (h1 = ∨), respectively. We write x for (x1 , ..., xn ), α for (α, α, ..., α) (α ∈ E2 ) and often xy instead of x ∧ y. Finally, let x, if σ = 0, xσ := x, if σ = 1. The mapping δ : P2 −→ P2 , f n −→ (f δ )n with f δ (x1 , ..., xn ) := f (x1 , x2 , ..., xn ) is known to be an automorphism from P2 , which we shall use to describe isomorphic closed subsets of P2 (see Section 9.11), where Aδ := {f δ | f ∈ A}. We remark that δ is the unique non-trivial isomorphism from a subalgebra of P2 to a subalgebra of P2 (see Theorem 9.12.5). In the following, we deﬁne some closed subsets of P2 with the help of which we can describe all subclasses of P2 with the applications of ∩ and ∪: 0 0 1 • M := P ol 0 1 1 = n≥1 {f n ∈ P2 | ∀a, b ∈ E2n : a ≤ b =⇒ f (a) ≤ f (b)} (set of all non-decreasing monotone functions), • S := P ol =

0 1 1 0

n≥1 {f

n

∈ P2 | f (x1 , ..., xn ) = f (x1 , x2 , ..., xn )}

(set of all self-dual functions),

3.1 Deﬁnitions of the Subclasses of P2 and Post’s Theorem

• L :=

n≥1 {f

n

147

n ∈ P2 | ∃a0 , ..., an ∈ E2 : f (x) = a0 + Σn=1 ai · xi }

(set of all linear functions), • T0,µ := P olE2µ \{1} if µ ∈ N ( f n ∈ T0,µ ⇐⇒ (∀a1 , ..., aµ ∈ E2µ : (∀i ∈ {1, ..., µ} : f (ai ) = 1 and ai = (ai1 , ..., ain )) =⇒ ∃ j ∈ {1, ..., n} : a1j = a2j = ... = aµj = 1) ⇐⇒ (∀a1 , ..., aµ ∈ E2n ∃ j ∈ {1, ..., n} : ∀x ∈ {a1 , ..., aµ } : f (x) = xj ∧ f (x))), δ T1,µ := T0,µ = P olE2µ \{0},

Ta := Ta,1 , where a ∈ E2 , T0,∞ := µ≥1 T0,µ =

n≥1 {f

n

∈ P2 | ∃j ∈ {1, ..., n}∃f ∈ P2 : f (x) = xj ∧ f (x)},

δ , T1,∞ := T0,∞

• K := [∧] (set of all conjunctions), • D := K δ = [∨] (set of all disjunctions), • C := [c0 , c1 ] (set of all constant functions), • Ca := [ca ], a ∈ E2 , • I := [e11 ] (set of all projections), • I := [− ].

148

3 The Subclasses of P2

Theorem 3.1.1 (Post’s Theorem; [Pos 41]) (1) The set of all subclasses of P2 is countably inﬁnite. (2) The non-empty subclasses of P2 are P2 , S, M, L, Ta,µ , Ta,µ ∩ Ta , Ta,µ ∩ M, Ta,µ ∩ M ∩ Ta , K ∪ C, K ∪ Ca , K, D ∪ C, D ∪ Ca , D, S ∩ T0 , S ∩ M, S ∩ L, S ∩ L ∩ T0 , L ∩ Ta , I ∪ C, I ∪ C, I, I ∪ Ca , I, C, Ca , where a ∈ E2 and µ ∈ {1, 2, ..., ∞}. (The Hasse-diagram of these classes is given in Figure 3.1.) (3) In the set P2 , there exists exactly (a) 9 closed subsets of order 1: [P21 ], I ∪ C, I, I ∪ C0 , I ∪ C1 , I, C, C0 , C1 ; (b) 20 closed subsets of order 2: P2 , T0 , T1 , M, L, M ∩ T0 , M ∩ T1 , L ∩ T0 , L ∩ T1 , M ∩ T0 ∩ T1 , K ∪ C, K ∪ C0 , K ∪ C1 , K, D ∪ C, D ∪ C0 , D ∪ C1 , D, T0,∞ , T1,∞ ; (c) 20 closed subsets of order 3: S, S ∩ T0 , S ∩ M, S ∩ L, S ∩ L ∩ T0 , T0,2 , T1,2 , T0,2 ∩ T1 , T1,2 ∩ T0 , T0,2 ∩ M, T1,2 ∩ M, T0,2 ∩ M ∩ T1 , T1,2 ∩ T0 ∩ M, T0,∞ ∩ T1 , T1,∞ ∩ T0 , T0,∞ ∩ M, T1,∞ ∩ M, T0,∞ ∩ M ∩ T1 , T1,∞ ∩ M ∩ T0 , T0 ∩ T1 ; (d) 8 closed subsets of order µ + 1 (µ ≥ 3): Ta,µ , Ta,µ ∩ Ta , Ta,µ ∩ M, Ta,µ ∩ M ∩ Ta (a ∈ E2 ).

3.2 A Proof for Post’s Theorem

T0 r T0,2 r

r T0,3 r

pp pp pp T0,∞ pr

r pp pp pp pr

Pr2

r

r

r

r

rM T r1 r r r

r r pp pp pp pr

r

r pp pp pp pr

r

r r

S∩M

r r

K ∪C

r

rS

r

r

r

K

r C0

r

r

r

rL

r

r [P21 ] r

rI

I r C

r

pp pp pp pr

r r D∪C r r

pp pp pp pr

149

rT1,2 r rT1,3 r pp pp pp pr

pp pp pp prT1,∞

r

D

r C1

∅

Fig. 3.1. The Post Lattice

3.2 A Proof for Post’s Theorem For any subclass A of P2 , the following three cases are possible: Case 1: A ⊆ L and A ⊆ S, Case 2: A ⊆ L, Case 3: A ⊆ L and A ⊆ S. By following this case distinction, all subclasses of P2 and minimal generating subsets of these classes are determined as follows. The orders of the subclasses given are easy conclusions from the proven statements about generating sets of the subclasses. We start with: 3.2.1 The Subclasses A of P2 with A ⊆ L and A ⊆ S Theorem 3.2.1.1 The following holds: P2 = [◦,− ] for ◦ ∈ {∧, ∨}, M = [∨, ∧, c0 , c1 ], T0,µ = [hµ , t], T0,µ ∩T1 = [hµ , q], T0,µ ∩ M = [hµ , m, c0 ] ( T0 ∩ M = [∨, ∧, c0 ] ), T0,µ ∩ M ∩ T1 = [hµ , m], T0,∞ = [t], T0,∞ ∩ T1 = [q], T0,∞ ∩ M = [m, c0 ] and T0,∞ ∩ M ∩ T1 = [m], where µ ∈ N.

150

3 The Subclasses of P2

Proof. Let be f n ∈ P2 , for which µ + 1 distinct tuples a1 , ..., aµ+1 exist with f (a1 ) = ... = f (aµ+1 ) = 1. Then, we have f (x) = hµ (fa1 (x), fa2 (x), ..., faµ+1 (x)),

where fai (x) :=

(3.1)

0 if x = ai , f (x) otherwise

(i = 1, 2, ..., µ + 1). We call every function f n of a subclass A of P2 with {hµ , fa1 , ..., faµ+1 } ⊆ A and f (a1 ) = ... = f (aµ+1 ) = 1 for certain a1 , ..., aµ+1 ∈ E2n a reducible function. We denote by NA the set of all not reducible functions f of a class A. Obviously, the set NA ∪ {hµ } is a generating set for the class A, if hµ ∈ A and A ∩ {h1 , ..., hµ−1 } = ∅ for certain µ ≥ 1. The Table 3.1 gives an easily veriﬁable description of the set (NA )n for A ∈ {P2 , T0,m , M, T0,m ∩ M, T0,m ∩ T1 , T0,m ∩ M ∩ T1 }, m ∈ N, and the minimal µ with hµ ∈ A. 1 The functions gJ and mJ for J ⊆ E2n from Table 3.1 are deﬁned by 1 if x ∈ J, gJ (x) := 0 otherwise

and mJ (x) :=

1 if ∃a ∈ J : x ≥ a, 0 otherwise. Table 3.1

A

NAn

P2

{gJ | |J| ≤ 1}

1

M

{mJ | |J| ≤ 1}

1

T0,µ

{gJ | |J| ≤ µ and ∃t : gJ (x) = xt ∧ gJ (x)}

µ

T0,µ ∩ T1

{gJ ∈ NTn0,µ | 1 ∈ J}

µ

T0,µ ∩ M

{mJ | |J| ≤ µ and ∃t : mJ (x) = xt ∧ mJ (x)}

µ

T0,µ ∩ M ∩ T1 {mJ ∈ NTn0,µ ∩M | 1 ∈ J} 1

a minimal µ with hµ ∈ A

µ

One possible proof of Table 3.1 is as follows: We start with a ”partition” of an arbitrary f ∈ A in functions fai by (3.1), then we repeat this construction for the functions fai instead of f , if fai ∈ NA , etc. In case A ⊆ M be let all tuples ai minimal with respect to ≤ by this fai ∈ M .

3.2 A Proof for Post’s Theorem

151

Since g∅n = mn∅ = cn0 , n g{(a (x) = xσ1 1 ∧ xσ2 2 ∧ ... ∧ xσnn , 1 ,...,an )}

mn{0} = cn1 , mn{(a1 ,...,an )} (x) = xi1 ∧ xi2 ∧ ... ∧ xiν if {i1 , ..., iν } = {i | ai = 1} = ∅, we have

NP2 ⊆ [∧,− ],

NT0 ⊆ [c0 , ∧, t], = [t],

NM ⊆ [∧, c0 , c1 ], NM ∩T0 ⊆ [∧, c0 ] and NM ∩T0 ∩T1 ⊆ [∧]. Consequently, P2 = [∨, ∧,− ] (and P2 = [◦,− ] for ◦ ∈ {∨, ∧} by Morgan’s laws), T0 = [∨, t], M = [∨, ∧, c0 , c1 ], M ∩ T0 = [∨, ∧, c0 ] and M ∩ T0 ∩ T1 = [∨, ∧]. Furthermore, Table 3.1 implies the following: if A = T0,µ ∩ B with B ∈ T0,∞ ∩ B = {P 2 , T1 , nM, T1 ∩ M } and µ ≥ 2, then A = [{hnµ } ∪ (T0,∞ ∩ B)] and {f ∈ P | ∃i ∈ {1, 2, ..., n} ∃ f ∈ B : f (x) = x ∧ f (x)}. By this 2 i n≥1 fact, from a generating subset {f n , g m , ...} of B, we get a generating subset of the form {∧ f, ∧ g, ...} for the class T0,∞ ∩ B. Thus, T0,∞ = [t], since t(x, t(x, y)) = x ∧ y and P2 = [∧,− ]. T0,∞ ∩ T1 = [q] follows from T1 = T0δ = [∧, tδ ], tδ (x, y) = x ∨ y and ∧ ∧, ∧ tδ ∈ [q]. By T1 ∩ M = [∧, ∨, c1 ] and {∧ ∧, ∧ ∨, ∧ c11 = e22 } ⊆ [m] is T0,∞ ∩ T1 ∩ M = [m]. Finally, T0,∞ ∩ M = [m, c0 ], since T0,∞ ∩ M = (T0,∞ ∩ M ∩ T1 ) ∪ [c0 ]. Lemma 3.2.1.2 If A = [A] ⊆ P2 , ca ∈ A for certain a ∈ E2 and A ⊆ L, then A contains a binary non-linear function. Proof. By x◦y+x+y = x◦ y for {◦, ◦ } = {∧, ∨}, ∆(+c1 ) =− and Theorem 1.4.2 we have [◦, +, c1 ] = P2 for ◦ ∈ {∧, ∨}, i.e., every function g n ∈ P2 has a description of the form ai1 i2 ...iν ◦ xi1 ◦ xi2 ◦ ... ◦ xiν g(x) = a0 + {i1 ,i2 ,...,iν }⊆{1,2,...,n}

for some a0 , ai1 i2 ...iν ∈ E2 (see Theorem 1.4.3). Therefore, because A ⊆ L, there exists a function f n ∈ A with f (x) = a0 + x1 ◦ x2 ◦ ... ◦ xr +

n i=1

where r ≥ 2 and

ai ◦ xi +

ai1 ...iν ◦ xi1 ◦ ... ◦ xiν ,

i1 , ..., iν ∈ {1, 2, ..., n}, ν ≥ r, {i1 , ..., iν } ⊆ {1, 2, ..., r}

152

3 The Subclasses of P2

◦ :=

∨ if a = 1, ∧ if a = 0.

Our statement follows from f (x, y, y, ..., y , ca , ...., ca ) = b + x ◦ y + c ◦ x + d ◦ y ∈ A\L (r−1) times for some b, c, d ∈ E2 . Lemma 3.2.1.3 Let A be a subclass of P2 , which is not a subset of L and not a subset of S. Then, the function ∧ or the function ∨ belongs to A. Proof. It is easy to verify that ∨ or ∧ is a superposition over a binary nonlinear function of P2 . Thus, we have to show A2 ⊆ L. Since A ⊆ S, there exists an f ∈ A with 0 1 a f = , 1 0 a a ∈ E2 , i.e., f ∈ {f1 , f2 , ..., f8 } (see Table 3.2). Table 3.2

x 0 0 1 1

y f1 0 0 1 0 0 0 1 0

f2 1 1 1 1

f3 1 0 0 1

f4 0 1 1 0

f5 0 0 0 1

f6 1 0 0 0

f7 0 1 1 1

f8 1 1 1 0

The functions f5 , ..., f8 are non-linear. If f ∈ {f1 , ..., f4 } then ∆f is a constant function and A2 ⊆ L follows from Lemma 3.2.1.2. Lemma 3.2.1.4 Let A be a subclass of P2 with ∧ ∈ A. Then the following implications hold: (a) (∃a ∈ E2 : A ⊆ Ta ) =⇒ ca ∈ A, (b) (A ⊆ M and A ⊆ K ∪ C) =⇒ m ∈ A, (c) A ⊆ M =⇒ q ∈ A, (d) (A ⊆ K ∪ C and A ⊆ T0 ) =⇒ ∨ ∈ A, (e) (∃µ ∈ N : A ⊆ T0,µ and A ⊆ T0,µ+1 ) =⇒ hµ ∈ A. Proof. (a): If A ⊆ Ta there exists a g 1 ∈ A with g(a) = a, i.e., g ∈ {ca ,− }. ca ∈ A follows from ∧ ∈ A, x ∧ x = 0 and 0 = 1. (b): f denotes an n-ary function of A\(K ∪ C). Then there exist two tuple a = (a1 , ..., an ) and b = (b1 , ..., bn ) with the following properties: f (a) = f (b) = 1, a ≤ b, b ≤ a and f (c) = 0 for every c with c < a or c < b . Deﬁning functions gi3 ∈ A by

3.2 A Proof for Post’s Theorem

⎧ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y ⎪ ⎨ gi (x, y, z) :=

if if

ai bi ai bi

=

=

1 1 0 1

153

, ,

ai 1 ⎪ ⎪ z if = , ⎪ ⎪ b 0 ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ai 0 ⎪ ⎪ = ⎪ ⎩ yz if bi 0

(i = 1, 2, ..., n) we obtain x ∧ f (g1 (x, y, z), ..., gn (x, y, z)) = m(x, y, z) ∈ A, since f ∈ M . (c): By A ⊆ M we have a function h3 in A with 1 0 0 1 h = . 1 0 1 0 Consequently, h (x, y, z) := x ∧ h(x, y, z) ∈ A and h (x, y, z) ∈ {x ∧ y ∧ z, x(y+z+1), x∧z, x(y∨z)}. Since x∧(y∧z) = x(y∨z) and x(yz+z+1) = x(y∨z), it holds q ∈ A. (d): By (a), (b), (c) and m(x, y, z) = q(x, y, q(x, y, z)) it holds {c1 , m} ⊆ A and hence ∆(m c11 ) = ∨ ∈ A. (e): If A ⊆ T0,µ+1 , there is an f ∈ A with f (E2µ+1 \{1}) = 1, i.e., w.l.o.g.: ⎛ ⎞ ⎛ ⎞ 1 0 1 1 ... 1 0 0 0 ... 0 ...... 0 ⎜1⎟ ⎜ 1 0 1 ... 1 0 1 1 ... 0 ...... 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ f⎜ ⎜ 1 1 0 ... 1 1 0 1 ... 0 ...... 0 ⎟ = ⎜ 1 ⎟ . ⎝.⎠ ⎝ ................................ ⎠ 1 1 1 1 ... 0 1 1 1 ... 1 ...... 0 (µ+1) times Since ∧ ∈ A, we have f (x1 , ..., xµ+1 ) := f (x1 , ..., xµ+1 , x1 x2 , x1 x3 , ..., x1 x2 x3 , ..., x1 x2 ...xµ+1 ) ∈ A. By f ∈ T0,µ we have

f (x) =

hµ (x) if x = 1, a if x = 1

for a certain a ∈ E2 . If a = 1 then f = hµ ∈ A. If a = 0 and µ ≥ 2 then f (x1 , ..., xµ−1 , xµ xµ+1 , f (x1 , ..., xµ+1 )) = hµ (x) ∈ A. Finally, if a = 0 and µ = 1, we have f = + and h1 (x, y) = xy + x + z ∈ A.

154

3 The Subclasses of P2

Theorem 3.2.1.5 The subclasses A of P2 with A ⊆ S and A ⊆ L are P2 , M, Ta,µ , Ta,µ ∩ B, K, K ∪ Ca , K ∪ C, D, D ∪ Ca , D ∪ C, where a ∈ E2 , µ ∈ {∞, 1, 2, ...} and B ∈ {Ta , M, M ∩ Ta }. Proof. Denote A a subclass of P2 with A ⊆ S and A ⊆ L. By Lemma 3.2.1.3, we have that A obtains K or D. W.l.o.g. [∧] = K ⊆ A. If A ∈ {K, K ∪ C0 , K ∪ C1 , K ∪ C} then we can distinguish 8 cases, which are given in Table 3.3. Using Lemma 3.2.1.4 and Theorem 3.2.1.1, we see that A is a certain class, which is given in the last column of Table 3.3. Table 3.3

∃µ ∈ A ⊆ T1 A ⊆ M {∞, 1, 2, ...} : A ⊆ T0,µ ∧ A ⊆ T0,µ+1 − − − − − + − + − − + + + − − + − + + + − + + +

conclusions from the assumptions

A

{c1 , c0 , q} ⊆ A {c1 , c0 , m} ⊆ A {c1 , q} ⊆ A ⊆ T1 {c1 , m} ⊆ A ⊆ T1 ∩ M {hµ , c0 , q} ⊆ A ⊆ T0,µ {hµ , c0 , m} ⊆ A ⊆ T0,µ ∩ M {hµ , q} ⊆ A ⊆ T0,µ ∩ T1 {hµ , m} ⊆ A ⊆ T0,µ ∩ T1 ∩ M

P2 M T1 T1 ∩ M T0,µ T0,µ ∩ M T0,µ ∩ T1 T0,µ ∩ T1 ∩ M

(+ stands for the truth of the assertion in the ﬁrst row, − for the truth of the negated assertion. Furthermore, let T0,∞+1 := ∅ and h∞ := e11 ) 3.2.2 The Subclasses of L Obviously, all subclasses of [P21 ] [P21 ], I ∪ C, I, I ∪ C0 , I ∪ C1 , C, C0 , C1 , I, ∅ are also subclasses of L. With the help of these sets, we can determine all subclasses of L with the following. Lemma 3.2.2.1 Let L be subclass of L with L ⊆ [P21 ]. Then L = [L1 ∪ {r}] (r(x, y, z) = x + y + z). Proof. Let L = [L] ⊆ L and L ⊆ [P21 ]. Then, there is a function g ∈ L with g(x, y, z) = a + x + y + bz for some a, b ∈ E2 . Thus g(g(x, y, z), z, z) = r(x, y, z) ∈ L. By forming of the functions of the type r r ... r and by n identifying of variables in these functions, we get that every function i=1 bi xi

3.2 A Proof for Post’s Theorem

155

with b1 + ... + bn = 1 belongs to [r] ⊆ A. Now, denote f an n-ary function of n L and let f (x) = a0 + i=1 ai xi . Then, we have f (x, x1 , ..., xn ) := x + (a2 + ... + an )x1 +

n

ai xi ∈ [r]

i=2

and

f (x) = f (f (x1 , ..., x1 ), x1 , ..., xn ).

Therefore f ∈ [{r} ∪ L1 ] and [{r} ∪ L1 ] = L is proven. Looking for subsemigroups of (P21 ; ), which have the property to be preserved by r, we get the following Theorem 3.2.2.2 All subclasses of L are L, L ∩ T0 = [c0 , +], L ∩ T1 , L ∩ S = [− , r], L ∩ T0 ∩ S = [r], [P21 ], I ∪ C, I, I ∪ C0 , I ∪ C1 , I, C, C0 , C1 , ∅. 3.2.3 The Subclasses of S, Which Are Not Subsets of L Obviously, a function f n ∈ P2 (n ≥ 2) belongs to S iﬀ there exists a function F n−1 ∈ P2 with the property f (x1 , ..., xn ) = x1 F (x2 , ..., xn ) ∨ x1 F (x2 , ..., xn ),

(3.2)

where F (x2 , ..., xn ) := f (0, x2 , ..., xn ). Consequently, we can deﬁne a bijective mapping α of S := S\S 1 onto P2 as follows: α : f −→ F. Lemma 3.2.3.1 The mapping α has the following properties: τ, ∆, ∇ and (a) For the operations ζ, , deﬁned by )(x1 , ..., xn ) := f (x1 , x3 , x4 , ..., xn , x2 ), (ζf ( τ f )(x1 , ..., xn ) := f (x1 , x3 , x2 , x4 , ..., xn ), )(x1 , ..., xn−1 ) := f (x1 , x2 , x2 , x3 , ..., xn−1 ), (∆f (∇f )(x1 , ..., xn+1 ) := f (x1 , x3 , x4 , ..., xn+1 ) and (f g)(x1 , ..., xm+n−2 ) := f (x1 , g(x1 , ..., xm ), xm+1 , ..., xm+n−2 ) (n, m ≥ 2), it holds α( γ f ) = γ(α(f )) for every γ ∈ {ζ, τ, ∆, ∇} and α(f g) = α(f ) α(g), i.e., the algebra (S ; ζ, τ, ∆, ∇, ) is isomorphic to the algebra (P2 ; ζ, τ, ∆, ∇, ). (b) For every subclass A (= ∅) of S, α(A) is a subclass of P2 , and it holds α(A) ⊆ S, A ⊆ α(A) and α(A) ∩ S = A.

156

3 The Subclasses of P2

Proof. (a) is easy to check. (b): Let A be a subclass of S. By (a) we have that α(A) is also a closed set. Assume α(A) ⊆ S. Then F (x2 , ..., xn ) = F (x2 , ..., xn ) for every f n ∈ A. Thus by (3.2) we get that the variable x1 is ﬁctitious for every function f n ∈ A. However, this is not possible. Hence α(A) ⊆ S holds. Let f n ∈ A. Then ∇f ∈ A and therefore α(∇f ) = f ∈ α(A), i.e., A ⊆ α(A). If f n ∈ S ∩ α(A), we have ∆(α−1 f ) = f ∈ A and thus S ∩ α(A) ⊆ A. From this, it follows that A = S ∩ α(A), since A ⊆ α(A) and A ⊆ S. With the help of Lemma 3.2.3.1 and Theorem 3.2.1.5, it is not diﬃcult to determine the missing subclasses of S. It holds Theorem 3.2.3.2 (a) The sets S ∩ M and S ∩ T0 are the only proper subclasses of S, which are not subsets of L. (b) It holds that S = [h2 ,− ], S ∩ T0 = [h2 , r], S ∩ M = [h2 ]. Proof. (a): Let A be a subclass of S with A ⊆ L. Then it holds α(A) ⊆ L. By Lemma 3.2.1.2 and Lemma 3.2.1.3, we have {∨, ∧} ∩ α(A) = ∅. Consequently, the function h2 (= xyz ∨ x(y ∨ z)) or the function g(x, y, z) := x(y ∨ z) ∨ xyz belong to A. Since it holds g(g(x, y, z), y, z) = h2 (x, y, z), h2 ∈ A. Further, the functions α(h2 ) = ∧, x ∧ h2 (x, y, z) = x(y ∨ z) and c10 = α(e21 ) are elements of α(A). Thus, T0,2 ∩M ⊆ α(A) and by Lemma 3.2.3.1, (b) we have T0,2 ∩M ∩S ⊆ A. By Theorem 3.2.1.5, there exists only the following possibilities for A: S ∩ T0,2 , S ∩ M = S ∩ M ∩ T0 , S ∩ T0 and (a) follows from S ∩ T0,2 = S ∩ M . (This fact is easy to prove, for example, with the help of the relation product 2 and theproperty P ol ∩ P ol ⊆P ol2 : 1 0 0 0 1 0 0 1 Let 1 := , 2 := and 3 := , Then 2 21 = 3 0 1 0 1 0 0 1 1 and thus S ∩ T0,2 ⊆ S ∩ M . Conversely, S ∩ M ⊆ S ∩ T0,2 holds, since 3 21 = 2 .) (b) follows from Theorem 3.2.1.1 and Lemma 3.2.3.1. 3.2.4 A Completeness Criterion for P2 The following is a conclusion from Theorem 3.1.1: Theorem 3.2.4.1 (Completeness Criterion for P2 ) Let A ⊆ P2 . Then [A] = P2 ⇐⇒ ∀X ∈ {T0 , T1 , M, S, L} : A ⊆ X.

3.2 A Proof for Post’s Theorem

157

Without using of Theorem 3.1.1, one can prove Theorem 3.2.4.1 with the help of Theorem 1.5.4.1 and the following: Theorem 3.2.4.2 P2 has exactly ﬁve maximal classes: T0 , T1 , M, S, and L. Proof. It is easy to see that the sets T0 , T1 , M, S, L are pairwise distinct proper subclasses of P2 . Therefore, proof of the following suﬃces for the proof of our theorem: ∀A ⊆ P2 : ((∀K ∈ {T0 , T1 , M, S, L} : A ⊆ K) =⇒ [A] = P2 ) Let now A ⊆ P2 with {f0 , f1 , fM , fS , fL } ⊆ A, where f0 ∈ T0 , f1 ∈ T1 , fM ∈ M, fS ∈ S and fL ∈ L. If one identiﬁes all variables of f0 with each other, then one gets an unary function f0 ∈ [A] with f0 (0) = 1, i.e., f0 ∈ {c1 , e11 }. Case 1: f0 = c1 . In this case, it holds that f1 (c1 (x), ..., c1 (x)) = c0 (x) ∈ [A]. Since fM ∈ A\M , there are some (ai , bi ) ∈ {(0, 0), (0, 1), (1, 1)} (i = 1, 2, ..., n) with f (a1 , ..., an ) > f (b1 , ..., bn ). Consequently, the function e11 is a superposition over {fM , c0 , c1 } ⊆ [A]. Thus P21 belongs to [A]. By the proof of Lemma 3.2.1.2 (see also Theorem 1.4.3), we can describe the function fLn with the help of a so-called Shegalkin polynom. Since fL does not belong to L, we can assume w.l.o.g. that fL (x) = a0 + x1 · x2 · ... · xr +

n i=1

ai · xi +

ai1 ...iν · xi1 · ... · xiν ,

i1 , ..., iν ∈ {1, ..., n}, ν ≥ r, {i1 , ..., iν } ⊆ {1, ..., r}

holds for r ≥ 2. Now, we consider the function fL (x, y) := fL (x,

y, ..., y , c0 (x), ..., c0 (x)) (r−1) times

which has the form fL (x, y) = a + b · x + c · y + x · y for some a, b, c ∈ {0, 1}. It is easy to check that x · y is a superposition over {fL } ∪ P21 (⊆ A). Consequently, by Theorem 1.4.2, we have [A] = P2 . Case 2: f0 = e11 . Since fSn ∈ S, there exist some a1 , ..., an ∈ {0, 1} with fS (a1 , ..., an ) = fS (a1 , ..., an ). Thus, c1 is a superposition over {fS , e11 } and the Case 2 is put down to the Case 1.

4 The Subclasses of Pk Which Contain Pk1

We start with the deﬁnitions of some subclasses of Pk . We show later that these classes are all classes A of Pk with Pk1 ⊆ A. Let Ut := Pk (t) ∪ [Pk1 ] for t = 2, 3, ..., k. In particular, Uk = Pk . Further, let Lk be the set 1 {f n ∈ Pk | ∃a ∈ E2 ∃f0 ∈ Pk1 ∃f1 , ..., fn ∈ Pk,2 : [Pk1 ]∪ n≥1

f (x) = f0 (a + f1 (x1 ) + f2 (x2 ) + ... + fn (xn ) (mod 2))}.

For k = 2 Lk is the set L, already deﬁned in Chapter 3. The sets Ut and Lk can be described with the help of relations: Lemma 4.1 Let ιhk := {(a0 , ..., ah−1 ) ∈ Ekh | ∃i, j ∈ Ek : i = j ∧ ai = aj } and λk := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ Ek }. Then (a) Ut = P ol ιt+1 for each t ∈ {2, 3, ..., k − 1} k and (b) Lk = P ol λk . t+1 Proof. (a): Obviously, Ut is a subset of P ol ιt+1 k . The inclusion Ut ⊂ P ol ιk t+1 (i.e., there is a function (∈ P ol ιk ) that essentially depends on at least two variables and which has at least t + 1 diﬀerent values) is false because of

160

4 The Subclasses of Pk Which Contain Pk1

Theorem 1.4.4, (b). Thus, Ut = P ol ιt+1 k . (b): It is easy to check that Lk ⊆ P ol λk . For the proof of P ol λk ⊆ Lk , denote f n an arbitrary function from Pk,2 ∩ (P ol λk ). First, we will show that then f (x1 , 0, ..., 0) + f (0, x2 , ..., xn ) + f (0, 0, ..., 0) = f (x1 , ..., xn ) (mod 2)

(4.1)

holds. Suppose, for some x1 = a1 , ..., xn = an is this false. Then, we have ⎞ ⎛ ⎛ ⎞ a a a a1 0 ... 0 ⎜ 0 a2 ... an ⎟ ⎜ a b b ⎟ ⎟ ⎜ ⎟ f⎜ ⎝ 0 0 ... 0 ⎠ ∈ ⎝ b a b ⎠ a1 a2 ... an b b a for arbitrary a, b ∈ E2 . However, this is a contradiction to f ∈ P ol λk . Therefore, (4.1) holds and this implies: f (x1 , ..., xn ) = a + f1 (x1 ) + ... + fn (xn ) (mod 2), where a := (n − 1) · f (0, 0, ..., 0) (mod 2) and fi (x) := f (0, ..., 0,

x , 0, ..., 0). i-th place

Because of pr1,2,3 λk = ι3k , an arbitrary function g from (P ol λk )\[Pk1 ] has exactly only two diﬀerent values. Consequently, there exists a certain permutation s ∈ Sk with s(Im(g)) = {0, 1} such that g has the description s−1 (s g) and s g belongs to Pk,2 ∩ P ol λk . It is obvious then that g ∈ Lk . Deﬁnition Let E ⊆ Ek with |E| ≥ 2. Then we can deﬁne a mapping prE from Pk,E into PE as follows: prE f n = g m :⇐⇒ (n = m ∧ ∀a ∈ E n : f (a) = g(a)) (f n ∈ Pk,E , g m ∈ PE ). Lemma 4.2 Let f 2 be a function from Lk with f (x, y) = x + y (mod 2) for all x, y ∈ E2 , let g m be a function from Pk,2 with prE2 g ∈ L2 and let h ∈ Ut \Ut−1 . Then, (a) Lk = [Pk1 ∪ {f }], (b) U2 = [Pk1 ∪ {g}] and (c) ∀t ∈ {3, 4, ..., k} : Ut = [Ut−1 ∪ {h}]. Proof. (a) follows directly from the deﬁnition of Lk . 1 ] = P2 . Therefore, some binary (b): By Theorem 3.2.4.1, [{prE2 g} ∪ prE2 Pk,2

4 The Subclasses of Pk Which Contain Pk1

161

functions ∧ , + with prE2 ∧ = ∧ and prE2 + = + belong to [Pk1 ∪ {g}] and an arbitrary function ut of Pk,2 is a superposition over Pk1 ∪ {∧ , + }: u(a1 , ..., at ) · ja1 (x1 ) · ... · jat (xt ) (mod 2) u(x1 , ..., xt ) = (a1 , ..., at ) ∈ Ekt

(see (1.8) from Section 1.4). This implies (b). (c) is a consequence from Lemma 1.4.5.

Theorem 4.3 (Burle’s Theorem, [Bur 67]) The classes with

[Pk1 ], Lk , U2 , U3 , ..., Uk−1 , Pk [Pk1 ] ⊂ Lk ⊂ U2 ⊂ U3 ⊂ ... ⊂ Uk−1 ⊂ Pk

are the only subclasses of Pk which contain Pk1 . Proof. Suppose there exists a class A of Pk with Pk1 ⊂ A, which is diﬀerent from the classes of the above theorem. Then A contains a certain function f n , which has at least two essentially variables and at least l ≥ 2 different values. Consequently, by Theorem 1.4.4, (a), (c), there exists some a1 , ..., an , b1 , ..., bn , α, β, γ ∈ Ek with ⎛ ⎞ ⎛ ⎞ a1 a2 a3 ...an α f ⎝ a1 b2 b3 ...bn ⎠ = ⎝ β ⎠ , b1 b2 b3 ...bn γ where |{α, β, γ}| = 3 for l ≥ 3 and α = γ and α = β for l = 2. Then, a binary function f with f (x, y) := g0 (f (g1 (x), g2 (y), ..., gn (y))) and ⎛ ⎞ ⎛ ⎞ 0 0 0 f ⎝ 0 1 ⎠ = ⎝ 1 ⎠ 1 1 0 is a superposition over f and some g0 , ..., gn ∈ Pk1 . We distinguish two cases: Case 1: f (1, 0) = 0. In this case, the function prE2 f is nonlinear. Thus, by Lemma 4.2, (b) it holds U2 ⊆ A. Since we have assumed, however, A = Ut for each t ∈ {2, ..., k}, we obtain a contradiction with the aid of Lemma 4.2, (c). Case 2: f (1, 0) = 1. In this case, by Lemma 4.2, (a) Lk is a subset of A and, because of A = Lk , there is a function g ∈ A, which does not preserve λk . Then, one can form a function g ∈ A ∩ Pk,2 with prE2 g ∈ L2 as a superposition over g and some functions of Lk . Thus, by Case 1, we get a contradiction.

162

4 The Subclasses of Pk Which Contain Pk1

Consequently, the classes given in our theorem are the only subclasses of Pk that contain Pk1 . The claimed chain property is an immediate conclusion from the deﬁnitions of these classes. The claimed chain property is a direct conclusion from the deﬁnitions of these classes.

5 The Maximal Classes of Pk

5.1 Introduction, a Rough Description of the Maximal Classes A subclass A of Pk is maximal in Pk (or A is called maximal class)1 if and only if no further classes of Pk exist between A and Pk . In other words: A = [A] ⊆ Pk is maximal in Pk if and only if A = Pk and [A ∪ {f }] = Pk for each f ∈ Pk \A. One is interested in these classes not only for structural reasons, but particularly because one can solve a central problem of the Many-Valued Logic (the so-called Completeness Problem) with the aid of these classes (see Theorem 1.5.4.1). Inter alia, the following papers dealt with the determination and description of maximal classes: [Pos 41], [Jab 54], [Jab 58], [Ros 70;a], [Mart 60], [Lo 63;a– ˇ 70;b]. c], [Lo 64], [Zac 67], [Zac-K-J 69], [Bai 67] and [Sai One knows the maximal classes T0 , T1 , M, S, and L of P2 (see Theorem 3.2.4.2) through the papers [Pos 20], [Pos 41] by E. L. Post. Eﬀorts to determine all maximal classes of Pk for k ≥ 3 began more than 50 years ago. S. V. Jablonskij determined all 18 maximal classes of P3 in [Jab 54]. A. I. Mal’tsev proved how in the paper [Zac-K-J 69] was mentioned that P4 has exactly 82 maximal classes. I. G. Rosenberg was the ﬁrst that succeeded in the description of all maximal classes of Pk for each k ∈ N \ {1} (see [Ros 65]), and he proved in [Ros 70a] that the list of the given classes is complete. Rosenberg deﬁned six relation sets, which are subsequently denoted by us with Uk , Mk , Sk , Lk , Ck and Bk , and he proved that the set {P olk | ∈ Uk ∪ Mk ∪ Sk ∪ Lk ∪ Ck ∪ Bk } 1

One uses instead of “maximal class” the concept “precomplete class” in older papers.

164

5 The Maximal Classes of Pk

is exactly the set of all maximal classes of Pk . He could use that some other authors had shown already the maximality of some classes P ol . For example, for classes of the form P ol 1 with 1 ∈ Uk ∪ Sk ∪ C1k and ∈ Lk , where k is a prime number, it was proven in [Jab 58] that these classes are maximal classes. In [Mart 60], one ﬁnds maximal classes of the form P ol 2 with 2 ∈ Mk . For further details, we refer the reader to [Ros 70;a]. In Section 5.2, we will describe the maximal classes of Pk in the manner found from Rosenberg.2 Further, we give ﬁrst properties of the maximal classes, where these properties are consequences from the deﬁnitions more or less. Then, in Section 5.3, the maximality of the classes described in the theorem is proven. It turns out, in this case, that one manages with three basic ideas in the proof. Chapter 6 is dedicated to the proof of the completeness of the given set of maximal classes then. This most diﬃcult part of the determination of maximal classes orientates itself strongly onto the proof given in [Ros 70;a]. One could abbreviate, however, the proof through transfer and modiﬁcation of some ideas from the papers [Qua 82], [P¨ os- K 79] (p. 126-129) and [Lau 92a]. A ﬁrst coarse description of the maximal classes supplies the following theorem basically already proven by A. V. Kuznezov in 1959. Theorem 5.1.1 ([Kuz 59], [Ros 70;a] (3.2.5), [But 60]) The class Lk for k = 2 or Uk−1 (= P olk ιkk ) for k ≥ 3 is the only maximal class of Pk , which contains Pk1 . For every maximal class A of Pk , which is diﬀerent from L2 and Uk−1 , is valid: A = P olk G1 (A). Proof. Let A be an arbitrary maximal class of Pk . Then, the following two cases are possible: Case 1: A1 = Pk1 . Then, by Theorem 4.3, A is the set L2 for k = 2 or the set Uk−1 for k ≥ 3. Case 2: A1 ⊂ Pk1 . In this case, (A1 ; ) is a proper subsemigroup of (Pk1 ; ) which e := e11 contains, what one can prove as follows: Suppose, e ∈ A1 . Then A1 ∩ [Pk1 [k]] = ∅, since sk = e for all s ∈ Pk1 [k]. Consequently, we have A ⊂ Jk ∪ A = [Jk ∪ A] ⊂ Pk , which contradicts the presupposed maximality of A. Thus A is a clone, for which, by Theorem 2.7.1, (c) A ⊆ P olk G1 (A) ⊆ Pk 2

This is not the only means to describe maximal classes. In [Den-P 88] one can ﬁnd descriptions of maximal classes of Pk through hyperidentities. One ﬁnds more about hyperidentities in the book [Den-W 2000] of K. Denecke and S. L. Wismath.

5.2 Deﬁnitions of the Maximal Classes of Pk

165

holds. Because of A1 = Pk1 and the maximality of A, this is possible only for A = P olk G1 (A).

5.2 Deﬁnitions of the Maximal Classes of Pk The maximal classes are deﬁned with the aid of the relation sets Mk , Sk , Uk , Lk , Ck and Bk . Indeed, by Theorem 5.1.1, one can describe every maximal class of Pk for k ≥ 3 with the aid of a certain k-ary relation in the form P ol . If P ol is a maximal class, then, P ol = P ol isvalid for all -derivable non-diagonal k relation . Therefore, the elements of h=1 Rkh \Dkh are possible descriptive relations for a maximal class P olk . The subsequently deﬁned relations of Mk , Sk , Uk , Lk , Ck and Bk are (with few exceptions), with respect to the arity, minimally chosen relations, which one can use to describe the maximal classes (see Chapter 10). We say that a maximal class A is a class of type X, if there exist an X ∈ {M, S, U, L, C, B} and a ∈ Xk with A = P olk . 5.2.1 Maximal Classes of Type M (Maximal Classes of Monotone Functions) Let Mk be the set of all partial orders on Ek with a greatest and a least element. More exactly, a binary relation ∈ Rk belongs to Mk if and only if has the following four properties: 1) is reﬂexive (i.e., ι2k ⊆ ); 2) is antisymmetric (i.e., ∩ −1 = ι2k ); 3) is transitive (i.e., ◦ = ) and 4) there exist elements o (”least element”) and e (”greatest element”) in Ek with {(o , x), (x, e ) | x ∈ Ek } ⊆ . It can easily be shown (see proof of Lemma 6.1.6) that the elements o and e are uniquely determined. We write a ≤ b instead of (a, b) ∈ and a < b, if (a, b) ∈ \ι2k . Furthermore, for a, b ∈ Ekn we put a ≤ b :⇐⇒ ∀i ∈ {1, ..., n} : ai ≤ bi . Obviously, a function f n ∈ Pk preserves the relation ∈ Mk if and only if the following holds: ∀a, b ∈ Ekn : a ≤ b =⇒ f (a) ≤ f (b). Thus the functions of P ol are (non-decreasing) monotone functions. Obviously, we have:

166

5 The Maximal Classes of Pk

Lemma 5.2.1.1 ∀, ∈ Mk : (P ol = P ol ⇐⇒ = τ ∨ = ).

For k ∈ {2, 3} all classes of type M are 0 0 1 M = P ol2 , 0 1 1 0 1 2 0 0 1 P ol3 0 1 2 1 2 2 0 1 2 0 0 2 P ol3 0 1 2 2 1 1 0 1 2 2 2 0 P ol3 0 1 2 0 1 1

(0 < 1 < 2), (0 < 2 < 1) and (2 < 0 < 1).

For larger k it is better to give the Hasse diagram, which characterizes the relation in unambiguous way, instead of the elements of . Figures 5.1 and 5.2 give the possible “basic diagrams” (i.e., diagrams without node names) for k = 4 and k = 5; hence (Ek ; ≤ ) is a lattice for k ≤ 5. q

k=4:

q

q @ @q

q

q @ @q

q

Fig. 5.1

k=5:

q @ q @q

q @ @q

q @ @q

q @ @q q

Fig. 5.2

q A

q @ @q

q

q

q

A Aq

q q q

For k ≥ 6 there are, however, ∈ Mk , so that (Ek , ) is not a lattice. Figure 5.3 is an example.

5.2 Deﬁnitions of the Maximal Classes of Pk

167

r @ @

r @r HH H H r Hr @ @ @r Fig. 5.3

5.2.2 Maximal Classes of Type S (Maximal Classes of Autodual Functions) For an arbitrary permutation s ∈ Sk , let s := {(x, s(x)) | x ∈ Ek }. Functions, which preserve a relation of the form s , are called autodual (in respect to s). Let Sk be the set of all relations of the form s , where s is an arbitrary permutation on Ek with k/p cycles of the same prime length p. Some examples for maximal classes of type S are the following: 0 1 P ol2 , 1 0 0 1 2 , P ol3 1 2 0 0 1 2 3 4 5 , P ol6 1 0 3 2 5 4 0 1 2 3 4 5 . P ol6 1 2 0 4 5 3 Subsequently, we assume that an arbitrary permutation s with s ∈ Sk is deﬁned as follows: If k = l · p (p a prime number, l ≥ 1) and {a1,0 , ..., a1,p−1 , a2,0 , ..., a2,p−1 , ..., al,0 , ..., al,p−1 } = Ek , we put s(ar,i ) := ar,i+1 (mod p) (r = 1, 2, ..., l; i = 0, 1, ..., p − 1). Then, it holds

168

5 The Maximal Classes of Pk

sa (ar,i ) = ar,i+a (mod p) if a = 0, 1, .... By deﬁnition of s , we have for arbitrary function f n ∈ P ol s : s(f (x1 , x2 , ..., xn )) = f (s(x1 ), s(x2 ), ..., s(xn ))

(5.1)

or f (x1 , x2 , ..., xn ) = si (f (sp−i (x1 ), sp−i (x2 ), ..., sp−i (xn ))) (i = 0, 1, ..., p − 1). (5.2) This implies that f (ar,i , x2 , ..., xn ) = si (f (ar,0 , sp−i (x2 ), ..., sp−i (xn ))) (i = 0, 1, ..., p − 1; r = 1, 2, ..., l).

(5.3)

Let Skn be the set of all n-ary functions f ∈ Pk , for which there exists some (n − 1)-ary functions F1 , F2 , ..., Fl ∈ Pk ∪ Ek such that f (x) =

p−1 l

jar,i (x1 ) · si (Fr (sp−i (x2 ), ..., sp−i (xn ))),

(5.4)

r=1 i=0

where + is the addition modulo k, · is the multiplication modulo k and ja is deﬁned by 1 if x = a, ja (x) := 0 otherwise, for a ∈ E2 . We call the function Fr r-th component of f . Theorem 5.2.2.1 It holds: P ol s =

n≥1

Skn .

Proof. For every function f ∈ Pkn there exists a description of the form f (x) =

k−1

ji (x1 ) · f (i, x2 , ..., xn ) (mod k).

(5.5)

i=0

If f ∈ P ol s then it follows from (5.5) with the aid of (5.3): f (x) =

p−1 l

jar,i (x1 ) · si (f (ar,0 , sp−i (x2 ), ..., sp−i (xn ))).

r=1 i=0

Consequently, P ol s ⊆ Let now g ∈ Pkn with g(x) =

p−1 l r=1 i=0

n≥1

Skn .

jar,i (x1 ) · si (Gr (sp−i (x2 ), ..., sp−i (xn ))) ∈ Skn ,

(5.6)

5.2 Deﬁnitions of the Maximal Classes of Pk

169

where G1 , G2 , ..., Gl are some (n − 1)-ary functions of Pk . We have to show that g ∈ P olk s . For i ∈ Ep and 1 ≤ r ≤ l it holds: g(s(ar,i ), s(x2 ), ..., s(xn )) = g(ar,i+1 mod p , s(x2 ), ..., s(xn )) = si+1 (Gr (sp−(i+1) (s(x2 )), ..., sp−(i+1) (s(xn )))) = s(si (Gr (sp−i (x2 ), ..., sp−i (xn )))) = s(g(ar,i , x2 , ..., xn )), i.e., g ∈ P olk s . Thus, n≥1 Skn = P olk s . We will need the following lemma to determine the number of maximal classes of Pk later. Lemma 5.2.2.2 ∀s , s ∈ Sk : P olk s = P olk s ⇐⇒ ∃i ∈ {1, ..., k} : si = s . Proof. “=⇒”: Let s, s ∈ Sk and P olk s = P olk s . Then we have Im(s) = Im(s ) = Ek and s, s ∈ P olk s . Therefore, by Lemma 5.2.2.1, we can de are nullary functions, i.e., elscribe s in the form (5.4), where S0 , ..., Sp−1 ements of Ek . Thus, s is unique determined by s (ar,0 ) (r = 1, 2, ..., l). For every r ∈ {1, 2, ..., l}, s (ar,0 ) belongs to {ar,0 , ar,1 , ..., ar,p−1 }, since one receives a contradiction in choosing s in the opposite case, as follows:Suppose we have s (ar,0 ) = as,i with r = s for certain r, s, i. Set V := {as,i , s(ar,i ), ..., sp−1 (ar,i )}. Then, the permutation t with t(x) := s(x) for all x ∈ V and t(x) := x for x ∈ Ek \V belongs to P ol s and, therefore, also to P ol s . Because of s (ar,0 ) = as,i we have (ar,0 , as,i ) ∈ s and, by deﬁnition of t, this implies ar,0 ar,0 . = t as,i s(as,i ) Thus, (because of t ∈ P ol s ) as,i = s(as,i ), in contradiction to s ∈ Sk . Let now s (a1,0 ) = a1,i for certain i ∈ Ep . We have to show that s (ar,0 ) = ar,i for all r ∈ {2, 3, ..., l} holds. Suppose s (ar,0 ) = ar,j for certain j ∈ Ep \{i} and r ∈ {2, 3, ..., l}. Then, a function g 2 with g(a1,0 , ar,0 ) = g(a1,0 , ar,j−i (mod p) ) = ar,0 belongs to P olk s , for which we have, by (5.3), a1,0 ar,0 ar,0 g = ∈ s . a1,i ar,j ar,i Hence, the function g does not preserve the relation s , contrary to the assumption P olk s = P olk s . Thus we have s (ar,0 ) = ar,i for all r ∈ {1, 2, ..., l} and si = s holds. “⇐=”: If s, s ∈ Sk and si = s then s s and s s. Hence P olk s ⊆ P olk s and P olk s ⊆ P olk s by Theorem 2.6.3. Thus, P olk s = P olk s .

170

5 The Maximal Classes of Pk

5.2.3 Maximal Classes of Type U (Maximal Classes of Functions, Which Preserve Non-Trivial Equivalence Relations) Let Uk be the set of all non-trivial equivalence relations on Ek , i.e., for an arbitrary binary relation ∈ Rk we have: ∈ Uk ⇐⇒ (ι2k ⊆ ∧ −1 = ∧ ◦ = ∧ ∈ {ι2k , Ek2 }). For k = 2, classes of the type U do not occur. For k = 3 there are exactly three classes of the form P ol with ∈ U3 : 0 1 2 0 1 , P ol3 0 1 2 1 0 0 1 2 0 2 and P ol3 0 1 2 2 0 0 1 2 1 2 . P ol3 0 1 2 2 1 In the following, let be an arbitrary relation of Uk . Further, let Ai (i ∈ Ek ) be the equivalence classes (blocks) of this relation. By Lemma 1.4.6, an arbitrary function f n ∈ Pk is representable in the form (1.12). Since, in this case, the used functions fi (i ∈ Ek ) and the function z belong to P ol , the function f preserves the relation if and only if gf ∈ P ol is valid. One can easily show that the functions of the form gf (∈ P ol ) are completely determined through their values on the set {ai | i ∈ Ek }n . Hence, ({gf | f ∈ P ol }; ζ, τ, ∆, ∇, ) is isomorphic to (Pk ; ζ, τ, ∆, ∇, ). Let now q a unary function of Pk deﬁned by ∀i ∈ Ek : q(x) = ai :⇐⇒ x ∈ Ai . Then, the following lemma results from our above considerations. Lemma 5.2.3.1 ∀f n ∈ Pk : f ∈ P ol ⇐⇒ ∃hn ∈ P{a0 ...,ak −1 } : gf (x) = h(q(x1 ), ..., q(xn )).

In addition, the following is obviously valid: Lemma 5.2.3.2 ∀, ∈ Uk : P ol = P ol ⇐⇒ = .

5.2 Deﬁnitions of the Maximal Classes of Pk

171

5.2.4 Maximal Classes of Type L (Maximal Classes of Quasi-Linear Functions) The maximal classes, which are treated in this section, only occur if k is a prime number power pm . Therefore, we assume k = pm with p ∈ P in this section. Let G be the set of all p-elementary Abelean groups of the form (Ek , ⊕). All elements of such groups, which are diﬀerent from the neutral element, have the order p, i.e., if o denotes the neutral element of the group (Ek , ⊕) ∈ G, then p is the smallest number with x ⊕ x ⊕ ... ⊕ x = o p times for all x ∈ Ek \ {o}. Let Lk be the set of all relations λG with G := (Ek ; ⊕) ∈ G and λG := {(a, b, c, d) ∈ Ek4 | a ⊕ b = c ⊕ d}. If k is not a prime number power, we set Lk = ∅. One can also describe the maximal classes of the form P ol λG as follows: Since k is a prime number power, it is well-known that one can deﬁne certain operations + and · on Ek , so that the algebra GF (pm ) := (Ek ; +, ·) is a ﬁeld, where the additive group of this ﬁeld belongs to G. If one now chooses a suitable ﬁeld with the operations + := ⊕ and · to a group G := (Ek , ⊕) ∈ G, then (by Theorem 1.4.3) every function f n ∈ Pk is uniquely deﬁnable through a formula of the form f (x) = ai1 ...in · xi11 · ... · xinn (i1 ...,in )∈Ekn

with the aid of the chosen ﬁeld operations. A function g n , which is deﬁned through a formula of the speciﬁc form g(x) = a0 +

n m−1

j

aij · xpi ,

(5.7)

i=1 j=0

is called quasi-linear. In particular, if m = 1 (i.e., k is a prime number), then g is called a linear function as usual. The set of all quasi-linear functions g n (n = 1, 2, ...) of the form (5.7) is a closed set, as one can easily check with aid of the well-known fact i

i

i

∀i ∈ {0, 1..., m} : (x + y)p = xp + y p , where x, y ∈ GF (pm ). Further, it holds that

172

5 The Maximal Classes of Pk

Lemma 5.2.4.1 The following statements are equivalent: (a) f n ∈ P ol λG ; (b) f n is quasi-linear; (c) f (x1 + y1 , x2 + y2 , ..., xn + yn ) = f (x1 , ..., xn ) + f (y1 , ..., yn ) + f (o, ..., o). Proof. (a) ⇐⇒ (c): Because of (x, y, x + y, o) ∈ λG every f n ∈ P ol λG satisﬁes the condition (c). Conversely, if (c) holds and (xi , yi , ui , vi ) ∈ λG (i = 1, 2, ..., n), then f (x1 , ..., xn ) + f (y1 , ..., yn ) = f (x1 + y1 , ..., xn + yn ) + f (o, ..., o) = f (u1 + v1 , ..., un + vn ) + f (o, ..., o) = f (u1 , ..., un ) + f (v1 , ..., vn ). Thus f ∈ P ol λG . (a) ⇐⇒ (b): Since every quasi-linear function is obviously a superposition i over B := {x + y, xp , a · x, ca | i ∈ {1, .., m − 1} ∧ a ∈ Ek }, and by (5.8) all functions of B preserve the relation λG , the implication (b) =⇒ (a) is valid. In Theorem 5.3.6, we show that the set of all quasi-linear functions is maximal in Pk . Therefore, (a) =⇒ (b) is right. Lemma 5.2.4.2 Let G := (Ek ; +) and G := (Ek , + ) be elements of G. Then, the following statements are equivalent: (a) P ol λG = P ol λG . (b) There exists an element c ∈ Ek such that die mapping α : Ek −→ Ek , x → x + c is an isomorphism from G onto G. (c) λG = λG . Proof. Denote o and o the neutral elements of G and G , respectively. (a) =⇒ (b): Let P ol λG = P ol λG . Then, the function g 2 with g(x, y) = x + y belongs to P ol λG . Because of (x, o , o , x), (o , y, o , y) ∈ λG this implies g(x, y) = g(x, o ) + g(o , y) − g(o , o ) or

x + y = x + y − o .

(5.8)

With the aid of (5.8), one can verify that α(x) := x − o is an isomorphic mapping from G onto G. (b) =⇒ (a): Let α be an isomorphism from G onto G with α(x) := x + c for certain c ∈ Ek . Then, for arbitrary f n ∈ P ol λG by Lemma 5.2.4.1, we have f (o) = f (o + c) = f (o ) + f (c) − f (o) and

f (x + y) = f (x) + f (y) + f (c) − f (c) − f (o) − f (o).

It results from (5.9) and (5.10):

(5.9) (5.10)

5.2 Deﬁnitions of the Maximal Classes of Pk

173

f (x + y) = f (x) + f (y) − f (o ). Thus

α(f (x + y)) = (f (x) + c) + (f (y) + c) − (f (o ) + c) = α(f (x)) + α(f (y)) − α(f (o )) = α(f (x) + f (y) − f (o ))

holds because α is an isomorphism. Consequently, by Lemma 5.2.4.1, f preserves the relation λ; hence P ol λG ⊆ P ol λG . Since α−1 is also an isomorphism (from G onto G ), we have also P ol λG ⊆ P ol λG . Thus (a) holds. (a) ⇐⇒ (c) follows easily by means of (a) ⇐⇒ (b). The following representation of quasi-linear functions is better suited for some purposes (for example, determining subclasses of P ol λG ) than the above definition. It is well-known that every group G = (Ek ; +) of G is isomorphic to (W ; ⊕), where W := Epm and (x1 , ..., xm ) ⊕ (y1 , ..., ym ) := (x1 + y1 (mod p), ..., xm + ym (mod p)) for arbitrary (x1 , ..., xm ), (y1 , ..., ym ) ∈ Epm . One can assign a function F n ∈ PW to every function f n ∈ Pk (k = pm ) by means of the isomorphism α, as follows: F n (α(x1 ), ..., α(xn )) := α(f (x1 , ..., xn )). Let LnW be the set of all n-ary functions of PW , for which there are an a ∈ W and some matrices A1 , ..., An ∈ Epm×m such that F (X1 , ..., Xn ) = a + X1 · A1 + ... + Xn · An ,

(5.11)

where + and · are the usual matrix operations and Xi := (xi1 , ..., xin ) ∈ Epm (i = 1, ..., n). Then, we have: Lemma 5.2.4.3 ([Ros 70;a], 7.3.3) LW := n≥1 LnW is isomorphic to P ol λG for all G ∈ G. Proof. It is easy to check that each function of the form (5.11) fulﬁlls the condition (c) of Lemma 5.2.4.1, whereby LW is isomorphic to a subset of P ol λG . 2 LW ∼ = P ol λG follows from the fact that |LnW | = pm+n·m = (pm )1+n·m = k 1+n·m = |(P ol λG )n |. 5.2.5 Maximal Classes of Type C (Maximal Classes of Functions, Which Preserve Central Relations) An h-ary relation γ (1 ≤ h ≤ k − 1) auf Ek is called central, iﬀ γ has the three following properties:

174

5 The Maximal Classes of Pk

1) γ is totally reﬂexive and non-diagonal, i.e., it holds ιhk ⊆ γ = Ekh (ι1k := ∅); 2) γ is totally symmetric, i.e., for every permutation s on {1, 2, ..., h} it holds: (a1 , ..., ah ) ∈ γ =⇒ (as(1) , ..., as(h) ) ∈ γ; 3) there exists at least a central element c ∈ Ek , i.e., (a1 , ..., ah−1 , c) ∈ γ for all a1 , ..., ah−1 ∈ Ek . Let Chk be the set of all central h-ary relations on Ek and Ck := Chk . h≥1

Notice that

C1k = { | ∅ ⊂ ⊂ Ek }.

For k ∈ {2, 3}, only the following classes are deﬁned by central relations: P ol2 (0), P ol2 (1), P ol3 (a) (a ∈ E3 ), P ol3 (a b) if {a, b} ⊆ E3 and a = b, 0 1 2 a b a c if {a, b, c} = E3 . P ol3 0 1 2 b a c a Because of total reﬂexivity and total symmetry of the relations of Ck , Pk,{a1 ,...,ah } is a subset of P ol γ for every γ ∈ Ck and (a1 , ..., ah ) ∈ γ. With the aid of this property, one can easily prove that the following lemma holds: Lemma 5.2.5.1 ∀γ, γ ∈ Ck : P ol γ = P ol γ ⇐⇒ γ = γ . 5.2.6 Maximal Classes of Type B (Maximal Classes of Functions, Which Preserve h-Universal Relations) In the literature, one ﬁnds two possibilities for describing the deﬁning relations of the maximal classes of the type B. Both possibilities go back to papers from I. G. Rosenberg. We begin with the deﬁnition from [Ros 65]. At the end of this section, the reader ﬁnds the second description for classes of the type B, which uses so-called h-regular relation sets deﬁned in [Ros 70c]. One can represent every a ∈ Ehm (h ≥ 3, m ≥ 1) unique in the form a = a(m−1) · hm−1 + a(m−2) · hm−2 + ... + a(1) · h + a(0) ,

(5.12)

where a(m−1) , a(m−2) , ..., a(0) ∈ Eh are suitably chosen. h ⊆ Ehhm is called h-ary elementary, if it holds: The h-ary relation ξm (i)

(i)

(i)

h (a0 , ..., ah−1 ) ∈ ξm :⇐⇒ ∀i ∈ Em : (a0 , a1 , ..., ah−1 ) ∈ ιhk .

In particular, ξ1h = ιhk for m = 1. An h-ary relation on Ek is a homomorphic inverse image of an h-ary

5.2 Deﬁnitions of the Maximal Classes of Pk

175

relation on Ek , if there exists a mapping q from Ek onto Ek , such that, for all a1 , ..., ah ∈ Ek , it holds: (a1 , a2 , ..., ah ) ∈ ⇐⇒ (q(a1 ), q(a2 ), ..., q(ah )) ∈ . Let Bhk be the set of all homomorphic inverse images of the h-ary elementary h and put relation ξm k Bhk . Bk := h=3

Bhk

The elements of are also called h-universal relations. For k = 3 we have B = {ι33 }. To derive a description of the functions from the clone P ol with ∈ Bk without using relations, we assume that is an h-ary relation on Ek which h ⊆ Ehm in is a homomorphic inverse image of the h-ary elementary relation ξm m the following. In addition, let q be the mapping from Ek onto Eh which belongs to this relation because of deﬁnition. This mapping deﬁnes a partition on Ek in the blocks Ai := {x ∈ Ek | q(x) = i}, i ∈ Ehm . We select a representative ai from every one of these blocks Ai . Further, let r be a unary function with r(i) = ai for all i ∈ Ehm . As already given in Lemma 1.4.6, an arbitrary function f n from Pk is as a superposition over the functions z, gf , fi (i ∈ Ehm ), deﬁned by z(x, y) := x3y ( x3y = y if ∃i ∈ Ek : x = ai and y ∈ Ai ; x3y = x otherwise), gf (x1 , ..., xn ) := ai ⇐⇒ f (x1 , ..., xn ) ∈ Ai , f (x1 , ..., xn ) if f (x1 , ..., xn ) ∈ Ai , fi (x1 , ..., xn ) := otherwise ai (i ∈ Ek ), represents as follows: f (x) = ((...((gf (x)3f0 (x))3f1 (x))3...)3fhm −1 (x)).

(5.13)

Since an arbitrary function tn ∈ Phm is describable in the form t(x) = (t(x))(m−1) · hm−1 + (t(x))(m−2) · hm−2 + ... + ((t(x))(0)

(5.14)

(see also (5.12)), we can describe the function from (5.13) with the aid of (5.14) as follows: gf (x) = r(fm−1 (x) · hm−1 + fm−2 (x) · hm−2 + ... + f1 (x) · h + f0 (x)), (5.15)

where fi (x1 , ..., xn ) := (q(f (x1 , ..., xn )))(i) for i ∈ Ehm . Now, one can describe the functions of P ol without using relations, as follows:

176

5 The Maximal Classes of Pk

Theorem 5.2.6.1 ([Lau 78;a]) An n-ary function f ∈ Pk belongs to the class P ol if and only if for the function gf (see (5.15)) is valid: ∀ i ∈ {0, 1, . . . , m − 1} : either (|Im(fi )| ≤ h − 1) or there are j ∈ {1, . . . , n}, v ∈ Em , a permutation s on Eh

(5.16)

such that fi (x1 , . . . , xn ) = s((q(xj ))(v) ) . Proof. Obviously, every function with the properties of the above theorem belongs to the class P ol . We must prove that an arbitrary n-ary function f , which does not fulﬁll the conditions of the theorem, does not belong to P ol . Case 1: In the description (5.15), for gf , there is an i so that fi is dependent at least of two variables essentially and it holds Im(fi ) = Eh . Then, by Theorem 1.4.4, (b) there exists some h−tuples r1 , ..., rn ∈ ιhk with fi (r1 , ..., rn ) ∈ Ekh \ιhk . Consequently, f (r1 , ..., rn ) ∈ . Case 2: In the description (5.15), for gf , there is an i, so that fi is dependent essentially only of the variable xj (1 ≤ j ≤ n), |Im(fi )| = Eh and fi can not be described in the form fi (x1 , . . . , xn ) = s((q(xj ))(v) ). h 1 ) with fi (x1 , ..., xn ) = F (q(xj )) exists. Case 2.1: No function F ∈ (P ol ξm In this case, there are tuples a = (a1 , ..., an ), b = (b1 , ..., bn ) ∈ Ekn with fi (a) = fi (b) and (q(a1 ), q(a2 ), ..., q(an )) = (q(b1 ), ..., q(bn )). By Im(fi ) = Eh there exist tuples c2 , c3 , ..., ch−1 ∈ Ekn with {fi (a), fi (b), fi (c2 ), ..., fi (ch−1 )} = Eh . Consequently, (f (a), f (b), f (c2 ), ..., f (ch−1 )) ∈ . This and the fact that the columns of the matrix ⎞ ⎛ a ⎟ ⎜b ⎟ ⎜ ⎜ c2 ⎟ ⎟ ⎜ ⎠ ⎝ ... ch−1 belong to imply f ∈ P ol . h 1 Case 2.2: There exists a function F ∈ (P ol ξm ) with fi (x1 , ..., xn ) = F (q(xj )). With the aid of function F , one can deﬁne an m-ary function F ∗ ∈ Ph as follows: F ∗ (a(m−1) , a(m−2) , ..., a(0) ) := F (a(m−1) · hm−1 + a(m−2) · hm−2 + ... + a(0) ). If one can not describe the function fi as it is given in the theorem, then the function F ∗ must depend on at least two variables, essentially. With the aid

5.2 Deﬁnitions of the Maximal Classes of Pk

177

of Theorem 1.4.4, (b) this implies that f does not preserve . We illustrate the above theorem with the following Examples Let h = 3, m = 2, k = 11, q : E11 −→ E9 be deﬁned by q(x) := x + 1 (mod 9) for x ∈ E9 , q(9) = 4 and q(10) = 1. Further, let the two permutations s1 and s2 be deﬁned by x s1 (x) s2 (x) 0 1 2 1 0 0 2 2 1 For x = x(1) · 3 + x(0) ∈ E9 , let g(x) := s1 (x(0) ) and g (x) := s2 (x(1) ), i.e., x x(1) x(0) g(x) g (x) 0 0 0 1 2 1 0 1 0 2 2 0 2 2 2 3 1 0 1 0 4 1 1 0 0 5 1 2 2 0 6 2 0 1 1 7 2 1 0 1 8 2 2 2 1 Then, the ternary functions f, h ∈ P9 deﬁned by f (x1 , x2 , x3 ) := g(x1 ) · 3 + g (x3 ) (1)

(0)

(here f0 (x1 , x2 , x3 ) = g (x3 ) = s2 (x3 ) and f1 (x1 , x2 , x3 ) = g(x1 ) = s1 (x1 )) and h(x1 , x2 , x3 ) := g(x2 ) · 3 + f (x1 , x2 , x3 ), where Im(f ) ⊂ {0, 1, 2} and |Im(f )| ≤ 2, both belong to Pol ζ2 . The following is an example of a unary function f ∈ P11 (see last column below) that preserves the relation : x q(x) (q(x))(1) (q(x))(0) s1 ((q(x))(1) ) r(x) q(f (x)) f (x) 0 1 0 1 1 1 4 3 1 2 0 2 1 1 4 9 2 3 1 0 0 1 1 10 3 4 1 1 0 1 1 0 4 5 1 2 0 0 0 8 5 6 2 0 2 0 6 5 6 7 2 1 2 0 6 5 7 8 2 2 2 1 7 6 8 0 0 0 1 1 4 3 9 4 1 1 0 1 1 0 10 1 0 1 1 1 4 9

178

since

5 The Maximal Classes of Pk

q(f (x)) = s1 ((q(x))[1] ) · 3 + r(x).

Lemma 5.2.6.2 follows from the deﬁnition of the classes of the form P ol with ∈ Bk : Lemma 5.2.6.2 ∀, ∈ Bk : P ol = P ol ⇐⇒ = . Finally, we declare another possibility for describing of h-universal relations. We need the following concepts and notations: Let 3 ≤ h ≤ k, m ≥ 1, hm ≤ k, let ϑi for i ∈ Em be an equivalence relation on Ek and T := {ϑ0 , ..., ϑm−1 }. Then, T is called h-regular if T fulﬁlls the following two conditions: 1) For every i ∈ Em , the equivalence relation ϑi has exactly h equivalence classes. 2) Choosing for every i ∈ Em an equivalence class εi of the relation ϑi m−1 arbitrary, then i=0 εi = ∅. An h-ary relation ⊆ Ekh is called h-regular if there exists an h-regular set T := {ϑ0 , ..., ϑm−1 } such that for arbitrary a0 , ..., ah−1 ∈ Ek it holds: (a0 , ..., ah−1 ) ∈ ⇐⇒ (∀i ∈ Em ∃ {r, s} ⊂ Eh : r = s ∧ (ar , as ) ∈ ϑi ). Lemma 5.2.6.3 ([Ros 70c]) Let 3 ≤ h ≤ k, m ≥ 1, hm ≤ k and ∈ Ekh . Then, is h-universal iﬀ is h-regular. Proof. “=⇒”: Let be h-universal; i.e., there exists a surjective mapping q : Ek −→ Ehm with h (a0 , ..., ah−1 ) ∈ ⇐⇒ (q(a0 ), ..., q(ah−1 )) ∈ ξm .

With the aid of , one can deﬁne equivalence relations ϑi ⊆ Ek2 with i ∈ Em as follows: (a, b) ∈ ϑi :⇐⇒ (q(a))(i) = (q(b))(i) . It is easy to check that T := {ϑ0 , ..., ϑm−1 } is h-regular and the h-regular relation T deﬁned with the aid of T is identical with . “⇐=”: Let be h-regular and T := {ϑ0 , ..., ϑm−1 } denotes the corresponding h-regular set. Let εi,0 , εi,1 , ..., εi,h−1 be the equivalence classes of the equivalence relation ϑi for i ∈ Em . Every x ∈ Ek belongs to exactly one equivalence class of ϑi , which we describe with εi,ai (x) . With the help of the numbers ai (x) ∈ Eh one can deﬁne a mapping q as follows:

5.3 Proof of the Maximality of the Classes Deﬁned in Section 5.2

q : Ek −→ Ehm , x →

m−1

179

ai (x) · hi .

i=0

Obviously, q is surjective. Furthermore, we have (x0 , ..., xh−1 ) ∈ if and only if ({ai (x0 ), ..., ai (xh−1 )) ∈ ιhh for all i ∈ Em . Consequently, q is a mapping from Ek onto Ehm with h , (x0 , ..., xh−1 ) ∈ ⇐⇒ (q(x0 )), ..., q(xh−1 ) ∈ ξm

i.e., is h-universal.

5.3 Proof of the Maximality of the Classes Deﬁned in Section 5.2 An h-ary relation is called a strong relation iﬀ it fulﬁlls the following two conditions: 1 ∈ P ol : fr,s (r) = s \ιhk = ∅ and ∀r ∈ \ιhk ∀s ∈ ∃fr,s

(5.17)

(ιhk := ∅ for h = 1). A set T ⊆ Ekq is called -independent if for arbitrary pairwise diﬀerent tuples a1 , a2 , ..., ah of T (ai := (ai1 , ai2 , ..., aiq ) for i = 1, ..., h) there is an j ∈ {1, 2, ..., q} with (a1j , a2j , ..., ahj ) ∈ . Lemma 5.3.1 Let be an h-ary relation with the properties: (a) is a strong relation. (b) For every q ∈ N and for every -independent set T := {a1 , ..., a|T| } ⊆ Ekq it holds

|T |

{(f (a1 ), ..., f (a|T| )) | f ∈ (P ol)q } = Ek .

(5.18)

(c) (h = 1) ∨ (h = 2 ∧ ∩ ιhk ∈ {∅, ιhk }) ∨ (h ≥ 3 ∧ (∃A ⊆ Ek : |A| = h ∧ PEk ,A ⊆ P olk ). Then P ol is maximal in Pk . Proof. Let g m ∈ Pk \P ol be arbitrary; i.e., there are some s1 , ..., sm ∈ with g(s1 , ..., sm ) ∈ . Because of (a) for every r ∈ \ιhk there exists a function fr,si with fr,si (r) = si (i = 1, 2, ..., m). Consequently, the function hr deﬁned by hr (x) := g(fr,s1 (x), ..., fr,sm (x)) has the following properties: hr (r) ∈ and hr ∈ [P ol ∪ {g}]. Let {h1 , h2 , ..., ht } := {hr | r ∈ \ιhk }. It is easy to check that for h ∈ {1, 2} and for every n ∈ N the set Tn := {(x1 , ..., xn , h1 (x1 ), h1 (x2 ), ..., h1 (xn ), h2 (x1 ), h2 (x2 ), ..., n·(t+1)

h2 (xn ), ..., ht (x1 ), ht (x2 ), ..., ht (xn )) | x1 , ..., xn ∈ Ek } ⊆ Ek

180

5 The Maximal Classes of Pk

is -independent. Then, by (b), for every n-ary function f ∈ Pk one can ﬁnd an n · (t + 1)-ary function f with f (x1 , ..., xn ) = f (x1 , ..., xn , h1 (x1 ), ..., h1 (xn ), ..., ht (x1 ), ..., ht (xn )). Consequently, we have [P ol ∪ {g}] = Pk in the case h ∈ {1, 2}. One can handle the case h ≥ 3 in analog mode when one uses the -independent set Tn := {(x1 , ..., xn , h1 (g1 ), ..., h1 (gu )) | x1 , ..., xn ∈ Ek } with {g1 , ..., gu } := PEnk ,A and suitably chosen h1 instead of Tn . Theorem 5.3.2 ([Jab 58], [Mart 60], [Ros 70;a]) For every ∈ Mk ∪ Uk ∪ Sk ∪ Ck the clone P ol is maximal in Pk . Proof. Obviously, every ∈ Mk ∪ Uk ∪ Sk ∪ Ck satisﬁes the condition (c) of Lemma 5.3.1. In the following, we show that such relation also fulﬁlls the conditions (a) and (b). For this purpose, hn 1 (n1 ∈ {1, n}) denotes a function of Pk , which either fulﬁlls h (r) = s for arbitrary r (∈ \ιhk ) and s ∈ (see (5.17)), or which can take arbitrary values on an arbitrary -independent set T ⊆ Ekn (see (5.18)). If ∈ Uk ∪Sk ∪Ck , one can consider easily determination for h , so that h ∈ P ol is valid. (For example, if ∈ Ck and h (x) is not yet deﬁned, one sets h (x) = c, where c is a central element of . For ∈ Uk ∪ Sk one can use Lemma 5.2.3.1 and Theorem 5.2.2.1.) If h is a function of the form fr,s with r = Let ∈ Mk in thefollowing. α a ∈ and s = ∈ , then β b a if x < α, h := b otherwise, is a function of P ol. If h is supposed to be determined in the above manner through a -independent set T and (5.18), we put for all x ∈ Ekn \T : o if ∃a ∈ T : x < a, h := e otherwise, where o is the least element and e is the greatest element of Ek in respect to . The function h deﬁned in this way belongs to P ol, as one can prove as follows: Suppose h ∈ P ol. Then, there exists some a, b ∈ Ekn with {a, b} ⊆ T , a < b and t := (h (a), h (b)) ∈ {(α, o), (e, α), (e, o)} ⊆ Ek2 \ for a certain α ∈ Ek \{o, e}. If t = (α, o), we have a ∈ T and thus b ∈ T . Because of h (b) = o, this implies (by the deﬁnition of h ) the existence of a c ∈ T with b < c, whereby a < c for a, c ∈ T , in contradiction to the construction of

5.3 Proof of the Maximality of the Classes Deﬁned in Section 5.2

181

T . In similar way one can show that t ∈ {(e, α), (e, o)} is also not possible: If t = (e, α), then b ∈ T and a ∈ T ; hence h (a) = o by the deﬁnition of h , in contradiction to h (a) = e. If t = (e, o) then the following three cases are possible: Case 1: a ∈ T and b ∈ T . Because of b ∈ T and h (b) = o there exists a c ∈ T with b < c. By a < b; however, this implies a < c, what was impossible for tuples a, c ∈ T . Case 2: a ∈ T and b ∈ T . This case contradicts the deﬁnition of the function h : h (a) = o, since a ∈ T and a < b ∈ T . Case 3: a ∈ T and b ∈ T . Because of h (b) = o and b ∈ T there exists a c ∈ T with b < c. Then a < c is also valid, however; hence h (a) = o, in contradiction to the assumption h (a) = e. Thus our assertion results from Lemma 5.3.1. Theorem 5.3.3 ([Ros 70;a]) For every ∈ Bk the clone P ol is a maximal class of Pk . Proof. Let be deﬁned as in Section 5.2.6. Then, one can represent an arbitrary function f n ∈ Pk \P ol in the form (4.14), (5.15) (see also Lemma 1.4.6), where, by Theorem 5.2.6.1, there exists a function fi that takes all values of {0, 1, ..., h − 1} on some tuples r1 , ..., rn ∈ (see also the proof of Theorem 5.2.6.1). Since g(x) := (q(x))(i) ∈ P ol and Pk,B ⊆ P ol for all B := {b1 , .., bh } with (b1 , ..., bh ) ∈ , the function g f = fi and a function of Pk,h are superpositions over f and some functions of Pk,B ⊂ [P ol ∪ {f }]. Therefore, by Lemma 1.4.5, Pk,h ⊆ [P ol ∪ {f }]; hence all functions of Pk are superpositions over P ol ∪ {f } (see Lemma 1.4.6 and Theorem 5.2.6.1). Now, we come to the proof of the maximality of the classes of type L. For this purpose, let k = pm (p prime, m ≥ 1) and let G := (Ek ; +, ·) be a ﬁeld with (w.l.o.g.) the zero element 0 and the unit element 1. Let QG be the set of all quasi-linear functions (in respect to G) of Pk . We need two lemmas for proof of the Pk -maximality of QG . Lemma 5.3.4 ([Ros 70;a], 7.2.19) Let f n ∈ Pk \QG (k = pm ). Then [QG ∪ {f }] contains a function of the form h(x, y) =

m p −1

aij · xi · y j ,

i,j=0

which has at least a coeﬃcient ast = 0 with 0 < s, t ≤ pm−1 . Proof. By Theorem 1.4.3, one can represent f in the form (1.9). Because of f ∈ Pk \QG , the following two cases are possible:

182

5 The Maximal Classes of Pk

Case 1: There exists at least a coeﬃcient ai1 ...in with |{i1 , i2 , ..., in }\{0}| ≥ 2. Obviously, in this case, one receives the function h from the function f by identifying some variables and by substituting variables through c1 . Case 2: Each addend diﬀerent of zero in the formula (1.9) has the form i a0...0ij 0...0 · xjj and there exists a q with g(x) := f (c0 , ..., c0 ,

x , c0 , ..., c0 ) ∈ Pk \QG . q -th place

pm −1 Since c0 ∈ QG , we have g ∈ [QG ∪ {f }]. Now let g(x) := i=0 bi · xi and let d be the greatest index in such kind that d is not a power of p and bd = 0 holds. Then one can also write d in the form d = d1 · pt , where t ≥ 0 and d1 > 0 is relatively prime to p. pm −1 When one forms the function w(x, y) := g(x + y) = i,j=0 aij xi y j , so surrent

t

ders with the aid of the binomial of xd−p y p is theorem that the coeﬃcient d d identical with ad−pt ,pt = · bd . Next, we show that is relatively pt pt prime to p. Obviously, we have: d1 pt (d1 pt − 1)..(d1 pt − p)..(d1 pt − p2 )..(d1 pt − pt + 1) d d 1 pt . = = t t p p pt (pt − 1)......(pt − p)....(pt − p2 )...(pt − pt + 1) In the numerator and denominator of the above number, some factors, which one can divide by a power of p, stand. When one divides the above fraction by corresponding powers of p, one receives! a"fraction whose factors have the form a + b with p|a and p |b. Therefore, pdt is relatively prime to p. Hence (by bd = 0) ad−pt ,pt = 0. Lemma 5.3.5 ([Ros 70;a], 7.2.11) pm −1 Let h(x, y) = i,j=0 aij xi y j with ast = 0 for some 0 < s, t ≤ pm − 1. Then the function r(x, y) = xs · y t belongs to [QG ∪ {f }]. Proof. Let (u, v) ∈ Ep2m with (u, v) = (s, t). W.l.o.g. we can assume u = s. For a primitive element α of Ek (i.e. for an α ∈ Ek with {αi | 1 ≤ i ≤ pm − 1} = Ek \{0}, see e.g. [Lid-N 87]) we form the superposition g(x, y) := αu · h(x, y) − h(α · x, y) over {h(x, y), αu · x − y, α · x} ⊆ QG ∪ {f }. Then it holds g(x, y) :=

m p −1

(αu − αi ) · aij · xi · y j ,

i,j=0

and we put aij := (αu − αi ) · aij . Obviously, auj = 0. By the properties of a primitive element and because of u = s we have αu = αs and thus ast = 0.

5.4 The Number of the Maximal Classes of Pk

183

Furthermore, aij = 0, if aij = 0. Through repeated use of this construction for every (u, v) = (s, t), one obtains a function d with d(x, y) = dst xs y t and dst = 0. Let b ∈ Ek \{0} be the inverse element of dst . Since b · x ∈ QG , we have b · dst · xs · y t = xs · y t ∈ [QG ∪ {h}]. Theorem 5.3.6 ([Ros 70;a], 7.2.12) Let k = pm (p prime, m ≥ 1). Then, for every λG ∈ Lk (G := (Ek ; +, ·) is a ﬁeld) the clone P olk λG is maximal in Pk . Proof. W.l.o.g. let 0 be the zero element and let 1 be the unit element of G. In the proof of Lemma 5.2.4.1 we saw that QG ⊆ P olλG . Since P olλG = Pk and QG = [QG ] are also valid, we have only to show the Pk -maximality of QG . Let f ∈ Pk \QG be arbitrary. Then, by Lemmas 5.3.4 and 5.3.5, a function r(x, y) = xs · y t belongs to [QG ∪ {f }], where 0 < s, t ≤ pm − 1. One can also give s and t in the form s = h · pu and t = l · pv , where u, v ≥ 0 and i m h, l are relatively prime to p. Since xp ∈ QG (i ∈ Em ) and xp = x holds, m−u m−v we have w(x, y) := r(xp , yp ) = xh · y l ∈ [QG ∪ {f }]. If one forms now the superposition q(x, y) := w(x + 1, y + 1) over QG ∪ {f }, then one gets the following with the aid of the binomial theorem: (x + 1) · (y + 1) = h

l

m p −1

aij · xi · y j .

i,j=0

It is easy to check that a11

h l = · = h · l. 1 1

Since h and l are relatively prime to p, this implies a11 = 0 and, therefore, x · y belongs to [P olG ∪ {f }] because of Lemma 5.3.5, Thus, {x · y, x + y, a · x, ca (x) | a ∈ Ek } ⊆ [QG ∪ {f }] holds. Obviously, [QG ∪ {f }] = Pk results from that then because of Theorem 1.4.3.

5.4 The Number of the Maximal Classes of Pk Let Rmax (Pk ) := Mk ∪ Sk ∪ Uk ∪ Lk ∪ Ck ∪ Bk . Because of Lemmas 5.2.1.1 and 5.2.2.2 the following relation ∼ is an equivalence relations on Mk ∪ Sk : ∼ :⇐⇒ ∃X ∈ {M, S} : {, } ⊆ Xk ∧ P olk = P olk . When one selects exactly a representative from every equivalence class of ∼, ∼ one receives the relation sets M∼ k ⊆ Mk and Sk ⊆ Sk .

184

5 The Maximal Classes of Pk

Theorem 5.4.1 ([Ros 69]) Let ∼ ∼ (Pk ) := M∼ Rmax k ∪ Sk ∪ Uk ∪ Lk ∪ Ck ∪ Bk . ∼ Then for all , ∈ Rmax it holds:

P olk = P olk ⇐⇒ = .

∼

(5.19) ∼

Proof. Let ∈ Xk and ∈ Yk with X, Y ∈ {M , S , U, L, C, B}. If X = Y, then one can easily prove the assertion (by constructing functions f , g with f ∈ P olk \P olk and f ∈ P olk \P olk ). In the case X = Y the assertion (5.20) follows from Lemmas 5.2.1.1, 5.2.2.2, 5.2.3.2, 5.2.4.3, 5.2.5.1 and 5.2.6.2.

Theorem 5.4.2 ([Ros 73]) For k ≥ 2 there is exactly ∼ ∼ |Rmax | = |M∼ k | + |Sk | + |Uk | + |Lk | + |Ck | + |Bk |

(5.20)

maximal classes of Pk and it is valid: • |M∼ k|=

k·(k−1) 2

· o(k − 2),

where o(i) denotes the number of all diﬀerent partial order relations on an i-element set (o(0) := 1); 1 k! • |S∼ k|= k=m·p, p prime p−1 · m!·pm ; • |Uk | = z(k) − 2, where z(i) denotes the number of all partitions on a k-element set, which one can calculate as follows: k i=0

• |Lk | = • |Ck | =

(−1)i ·

i t=0

(k−1)! m

p( 2 ) ·(p−1)·(p2 −1)·...·(pm −1)

0

(−1)t ·

tk ; t! · (i − t)! k = pm , p prime,

, if otherwise;

1≤h
− 1,

! " ! " ! " k−1 k−2 0 where ch (k) := k1 · 2( h ) − k2 · 2( h ) + ... + (−1)k−1 · kk · 2(h) and !x" h := 0, if x < h); • |Bk | = 3≤h≤k b(k, h), where b(k, h) :=

m≥0,hm ≤k

for h ≥ 3.

m h h 1 t · ( (−1) · · (hm − t)k ) m! · (h!)m t=0 t m

5.4 The Number of the Maximal Classes of Pk

185

(Speciﬁc numbers are given in the following tables.3 )

k 2 3 4 5 6 7 8 ∼ |Rmax | 5 18 82 643 15 182 7 848 984 549 761 933 169 k |M∼ k| |S∼ k| |Uk | |Lk | |Ck | |Bk |

2 1 1 0 1 2 0

3 3 1 3 1 9 1

4 18 3 13 1 40 7

5 6 7 8 9 10 190 3285 88 851 3 640 644 6 35 120 105 1120 19 089 50 201 875 4398 6 0 120 30 840 0 355 11490 7 758 205 549 758 283 980 36 171 813 4012

∼ Proof. In Chapter 6, we prove that {P olk | ∈ Rmax } is the set of all maximal classes of Pk . Because of Theorem 5.4.1, we have to determine the numbers |Xk | with X ∈ {M∼ , S∼ , U, L, C, B}. Case 1: X = M∼ . Obviously, there are k · (k − 1) possibilities to choose a greatest element and a smallest element of a relation ∈ Mk . Consequently, we have |Mk | = k·(k−1) · o(k − 2). k · (k − 1) · o(k − 2); with the aid of Lemma 5.2.1.1: |M∼ k|= 2 Case 2: X = S∼ . Let k = p · m, p prime, m ≥ 1. Then k k−p k − 2p p k! · · · ... · = p p p p m! · pm

is the number of the possible permutations s on Ek , which have exactly m cycles of the length p. Because of Lemma 5.2.2.2 the p − 1 permutations exactly the same maximal class P olk s . Therefore, s, s2 , ..., sp−1 determine k! 1 | = we have |S∼ k k=m·p, p prime p−1 · m!·pm . Case 3: X = U. Obviously, |Uk | = z(k) − 2, where z(k) denotes the number of the possible partitions of a k-element set. Case 4: X = L. Let k = pm , p prime, m ≥ 1. First, we determine |G|, where (as in Section 5.2.4) G is the set of all elementar Abelean p-groups of the form (Ek ; ⊕). As is generally known, every group G ∈ G is isomorphic to an m-fold direct product V := (Zpm ; +) of the group (Zp ; + (mod p)). If s : Zpm −→ Ek is a bijective mapping, then one can deﬁne Gs := (Ek ; +s ) as follows: 3

To |M∼ 8 | see also [Com 66], [Rad 83] and [Roz 85].

186

5 The Maximal Classes of Pk

∀x, y ∈ Ek : x +s y := s(s−1 (x) + s−1 (y)). It is easy to check that Gs = Gt ⇐⇒ s−1 t is an automorphism holds for arbitrary bijective mappings s, t from Zpm onto Ek . Consequently, we have k! , |G| = |A| where A denotes the set of all automorphisms on V . Since |A| is identical with the number of the bases of V , we obtain |A| = (pm − 1) · (pm − p) · (pm − p2 ) · ... · (pm − pm−1 ) = p · p2 · ... · pm−1 · (pm − 1) · (pm−1 − 1) · ...(p2 − 1) · (p − 1) m = p( 2 ) · (pm − 1) · (pm−1 − 1) · ... · (p − 1). Furthermore, by Lemma 5.2.4.2, we have |Lk | = |G| k . This implies our assertion about |Lk |. Case 5: X = C. Let Ah be the set of all h-element subsets of Ek , 1 ≤ h ≤ k. Then one can assign an element a of P(Ah ) as follows to every totally reﬂexive, totally symmetric, h-ary relation in unique manner: {a0 , a1 , ..., ah−1 } ∈ a ⇐⇒ (a0 , a1 , ..., ah−1 ) ∈ \ιhk .

(5.21)

Consequently, if belongs to Chk and the set of all central elements of contains the set C, then by means of (5.21), the relation is unambiguously deﬁnite through a subset a of Th,C := {A ∈ Ah | A ∩ C = ∅}. Because of |Th,C | = !k−|C|" k−|C| and |P(T )| = 2( h ) one obtains the number of elements of Ch ∪ h,C

h

{Ekh } through use of the principle of the inclusion and exclusion: k−1 k−2 0 k k k · 2( h ) + ... + (−1)k−1 · · 2( h ) − · 2(h) , ch (k) := 2 1 k

where

!x" h

:= 0 for x < h. Consequently, we have |Ck | =

k−1

h=1 ch (k)

k

− 1.

Case 6: X = B. Let h ≥ 3 and m ≥ 1. A bijective mapping s from Ehm onto Ehm is called an automorphism of ξm , if ∀(a0 , a1 , ..., ah−1 ) ∈ ξm : (s(a0 ), s(a1 ), ..., s(a0 )) ∈ ξm . Let and be relations of Bhk ; i.e., there exists certain mappings q or q from Ek onto Ehm or Ehm with = q −1 (ξm ) and = q −1 (ξm ), respectively (see Section 5.2.6). Next, we show that the equivalence

5.4 The Number of the Maximal Classes of Pk

187

= ⇐⇒ m = m ∧ (5.22) (∃ automorphism s on ξm : ∀x ∈ Ek : q (x) = s(q(x)) ) holds. “=⇒”: If = , then it follows from the deﬁnition of the relations = q −1 (ξm ) or = q −1 (ξm ) that the mapping equivalences κα := {(x, y) ∈ Ek2 | α(x) = α(y)} for α ∈ {q, q } are identical. Then, κq = κq implies the right side of the equivalence (5.22). “⇐=” follows from (a0 , a1 , ..., ah−1 ) ∈ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

(q(a0 ), q(a1 ), ..., q(ah−1 )) ∈ ξm (s(q(a0 )), s(q(a1 )), ..., s(q(ah−1 ))) ∈ ξm (q (a0 ), q(a1 ), ..., q (ah−1 )) ∈ ξm (a0 , a1 , ..., ah−1 ) ∈ .

Denote b1 (k, h) the number of the mappings from Ek onto Ehm and denote b2 (h, m) the number of the automorphisms on ξm . Because of (5.22), we have |Bhk | =

b1 (k, h) . b2 (h, m)

(5.23)

With the aid of the principle of the inclusion and exclusion, one can calculate the number b1 (k, h) as follows:

m

h

hm b1 (k, h) = (−1) · t t=0 t

· (hm − t)k .

(5.24)

For the purpose of determining b2 (h, m) the following considerations follows directly from the deﬁnition of the automorphisms of ξm that these are some functions of P olk ξm . Then, we have (because of the bijectivity of the automorphisms and by Theorem 5.2.6.1) that each automorphism s of ξm has a representation of the form s(x) =

m−1

si (x( tx ) ) · hi ,

(5.25)

i=0

where s0 , s1 , ..., sm−1 are some permutations on Eh and {t0 , t1 , ..., tm−1 } = Em holds. Therefore, (5.26) b2 (h, m) = m! · (h!)m . Then, (5.23)–(5.26) imply our assertion. We notice that there are also asymptotic estimates for the number of all maximal classes of Pk . One can take further information on that from the book [P¨ os- K 79] (p. 111) or the original papers [Kuz 59] and [Zac K J 64].

188

5 The Maximal Classes of Pk

5.5 Remarks to the Maximal Classes of Pk(l) For A ⊆ Pk , l ∈ N, l ≤ k, ∈ Rk put A(l) := {f ∈ A | |Im(f )| ≤ l}, P ol := (P olk )(l), λk := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ Ek }. Let Mk (l) be the set of all partial orders on Ek with the property that every l-element subset of Ek has a least and a greatest element with respect to . Notice that Mk (k) = Mk . We say that the h-ary relation on Ek is an l-central relation if = Ekh is totally reﬂexive, totally symmetric, and for every l-element subset E of Ek there exists an element cE ∈ E with (a1 , ..., ah−1 , cE ) ∈ for all a1 , ..., ah−1 ∈ E. The element cE is called a central element of ∩ E h . Let Chk (l) be the set of all h-ary l-central relations on Ek and let Ck (l) := l−1 h h=1 Ck (l). Notice that Ck (k) = Ck . Since any unary relation is totally reﬂexive and totally symmetric, we have C1k (l) = { ⊂ Ek | k − l + 1 ≤ || < k}. For the next deﬁnition, we need some notations of Section 5.2.6 and χ() := {α1 , ..., ah } ⊆ Ek | (a1 , ..., ah ) ∈ }. Notice that a totally reﬂexive and totally symmetric h-ary relation is completely determined by the set χ(). Bhk (l) denotes the set of all h-ary non-diagonal relations with the following three properties: (1) ∀E ⊆ Ek : |E| = l =⇒ ∩ E h ∈ Bhk ∪ {E h }, (2) ∀E ⊆ Ek : |E| = l =⇒ (∃idE ∈ (P ol)1 : ∀x ∈ E : idE (x) = x); (3) ∀A, B ∈ χ() : (A = B =⇒ ∃gA,B ∈ (P ol)1 : gA,B (A) = B). Example: Let be k = 5, l = 4 and h = 3. Then, the totally reﬂexive and totally symmetric h-ary relation with χ() := {A1 , A2 , A3 }, A1 := {0, 1, 2}, A2 := {1, 2, 3}, A3 := {1, 2, 4} is an element of Bhk (l), since ∩ {0, 2, 3, 4}4 = {0, 2, 3, 4}4 , ∩ {0, 1, 3, 4}4 = {0, 1, 3, 4}4 and ∩ {1, 2, 3, 4}4 , ∩ {0, 1, 2, 4}4 and ∩ {0, 1, 2, 3}4 are homomorphic inverse images of the relation ι33 and the functions idt := idE5 \{t} (t ∈ E5 ) and gpq := gAp ,Aq (p = q, p, q ∈ {1, 2, 3}) satisfy the conditions (2) and (3) (see the below indicated tables).

5.5 Remarks to the Maximal Classes of Pk (l)

x 0 1 2 3 4 x 0 1 2 3 4

id0 id1 id2 id3 id4 3 0 0 0 0 1 0 1 1 1 2 2 0 2 2 3 3 3 4 3 4 4 4 4 3

g12 g13 g21 g23 g31 g32 1 1 2 4 2 3 2 2 0 1 0 1 3 4 1 2 1 2 1 1 2 4 2 3 1 1 2 4 2 3 l−1

Put Bk (l) :=

189

h=3

∅

Bhk (l) if

l > 3,

otherwise.

Let Hk (l) be the set of all l-ary totally reﬂexive and totally symmetric relations = Ekh that satisfy the above condition (3). Examples for such relations are, as follows: 1) ιlk ; 2) E := Ekl \ {(as(0) , as(1) , ..., as(l−1) ) | s permutation on El }, where E := {a0 , ..., ah−1 } and |E| = l; 3) every relation of Blk (see [Lau 88d]). For l = 2 we deﬁne Hk (2) := {Ek2 \ {(a, b), (b, a)} | (a, b) ∈ Ek2 \ ι2k }. Theorem 5.5.1 ([Lau 92a]; without proof ) Let A ⊆ Pk (l), 2 ≤ l < k and Rk (l) := Mk (l) ∪ Uk ∪ Ck (l) ∪ Bk (l) ∪ Hk (l). Then (a) [A] = Pk (l) ⇐⇒ ((∀ ∈ Rk (l) : A ⊆ P ol) ∧ (l = 2 =⇒ A ⊆ P olλk )). (b) P ol is Pk (l)-maximal for every ∈ Rk (l) and (P olλk )(2) is Pk (2)maximal. (c) The maximal classes of Pk (2) are sets of the form (P ol )(2) with ∈ C1k (2) ∪ Mk (2) ∪ Uk ∪ Hk (2) ∪ {λk }; i.e., every maximal class of P olk (l), which is not equal (P olk λ)(2), is a intersection of Pk (2) with a certain maximal class of Pk .

190

5 The Maximal Classes of Pk

We remark that for l ≥ 3 not every Pk (l)-maximal class is an intersection of Pk (l) with a certain maximal class of Pk . An example for this fact is the relation {(a, b, c) ∈ E53 | {a, b, c} ∈ {{0, 3, 4}, {1, 2, 3}}} ∈ C35 (4). Notice that the Pk (l)-maximality of the class P ol for ∈ Mk (l) was proven by V. Kolpakov in [Kol 74].

6 Rosenberg’s Completeness Criterion for Pk

Since Pk is ﬁnitely generated, every proper subclass of Pk belongs to a certain maximal class of Pk ; therefore, by Theorem 1.5.3.1, it follows that a subset T of Pk is complete (i.e., it holds [T ] = Pk ) if and only if T is not a subset of a maximal subclass of Pk . The aim of this chapter is to prove that the classes of Chapter 5 are all maximal classes of Pk :

Theorem 6.1 (Rosenberg’s Completeness Criterion for Pk ; [Ros 65], [Ros 70;a]) For an arbitrary subset T of Pk it holds: [T ] = Pk ⇐⇒ ∀ ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk : T ⊆ P ol.

(6.1)

6.1 Proof of Completeness Criterion By Section 4.9, the equivalence (6.1) follows from (6.2), which we still must prove: A is maximal in Pk =⇒ (6.2) ∃γ ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk : A ⊆ P olγ. Denote A an arbitrary maximal class of Pk in the following. To simplify the proof of (6.2), some assumptions over A are given. These assumptions result from the following considerations: • We show in Chapter 4 that there is exactly a Pk -maximal class that contains Pk1 . In addition, were shown that every other Pk -maximal class has k the form P ol, where ∈ h=1 Rkh .

192

6 Rosenberg’s Completeness Criterion for Pk

• For every -derivable relation ’, which is not diagonal, we have by Theorem 2.6.3 that P ol ⊆ P ol ⊂ Pk . Obviously, if P ol is Pk -maximal then for an arbitrary -derivable non-diagonal relation ’ it holds: P ol = P ol . • Every unary non-diagonal relation ∈ Rk (i.e., ∅ ⊂ ⊂ Ek ) describes a maximal class of Pk (see Theorem 5.3.2). Thus, w.l.o.g. we can assume that our maximal class A of Pk can be described by a certain h-ary relation : A = P ol (1 ≤ h ≤ k). Further, we assume w.l.o.g. that fulﬁlls the three conditions (I) P ol is Pk -maximal and (P ol)1 = Pk1 ;

k (II) For every -derivable h -ary relation ∈ h=1 Rkh it holds: (a) ∈ InvPk =⇒ h ≥ h (i.e., is a relation with minimal arity which describes the maximal class P ol); (b) (h = h = 2 ∧ = ∩ (τ )) =⇒ ∈ Dk2 ∪ {} (i.e., by ∩ (τ ) no other relation which describes P ol either is derivable. With other words: If possible, we choose a symmetrical relation for the description of P ol; (c) (h = h ≥ 2 ∧ ιhk ⊆ ⊆ ∧ ( is totally symmetric ∨ (h = 2 ∧ is antisymmetric) )) =⇒ ∈ {, Ekh } (i.e., if is totally reﬂexive and totally symmetric or reﬂexive and antisymmetric, then for the description of the class P ol we choose a binary relation which has the greatest cardinality.); (III) is not a unary relation, i.e., h ≥ 2. Now, we want to prove:

If fulﬁlls the obove conditions (I)–(III), then or a -derivable relation belongs to Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk . (6.3) (6.3) implies (6.2) and our Theorem 6.1 follows from (6.2), which was already mentioned. For k = 2 it is easy to check that only the relations 0 1 0 0 1 and 1 0 0 1 1 satisfy (I)–(III); hence (6.1) is clearly for k = 2. Therefore, let k≥3 be in the following. In the proof of (6.1) for k ≥ 3 we use the notations for relations and relation

6.1 Proof of Completeness Criterion

193

operations from Sections 2.1–2.5. In particular, ◦ denotes the relation product and let i := ◦ ◦ ...o .1 i times An h-tuple a from Ekh with h ≤ k we often give in the form (a0 , a1 , ..., ah−1 ), i.e., we also choose the indices from Ek . Remember that a relation ’ is called -derivable iﬀ can be obtained by a ﬁnite number of applications of the relation operations (see 2.3) from and the diagonal relations. The following lemma is easy to check (see also Theorem 2.11.2): Lemma 6.1.1 ([Ros 70;a], 4.1.2) Each relation of the form C := {(a0 , ..., at−1 ) | ∃at , ..., am·h−1 ∈ Ek : (∀i : (aci1 , ..., acih ) ∈ )}, where C := (cij )m,h is a matrix of type (m, h) with elements from Em·h , is a -derivable relation. We remark that every one of the following claims on -derivability of a certain relation almost follows from the above lemma. For the proof of (6.1), we distinguish two cases:

Case 1: h = 2. Figure 6.1 gives a survey on the possible subcases and on the following lemmas. For 2 ≤ i ≤ k, the relation σi () := {(a1 , ..., ai ) | ∃u ∈ Ek : {(a1 , u), ..., (ai , u)} ⊆ } is a -derivable relation and it holds: Lemma 6.1.2 σ2 () = Ek2 =⇒ (∀i ∈ {2, 3, ..., k} : σi () = Eki ). Proof. We prove the statement by induction on i ≥ 2. For i = 2 the statement holds by assumption. Assume, σt−1 = Ekt−1 is valid for certain t−1 ∈ {2, ..., k− 1}. Then, by deﬁnition of σt (), we have ιtk ⊆ σt (). If ιtk ⊆ σt () ⊂ Ekt , then σt () is no invariant of Pk and hence (by (I)) P ol = P olσt () holds. This is, however, a contradiction of the fact that Pk (2) ⊆ P olσt (), whereby P olσt () does not have any description by a binary relation. Therefore, σt () = Ekt . Lemma 6.1.3 Either is reﬂexive (i.e., ι2k ⊆ ) or there exists a -derivable relation ’ which belongs to Sk . 1

For content-related reasons, it is not possible to confuse the index i with the arity of .

194

6 Rosenberg’s Completeness Criterion for Pk

HH HH

HH j H

is reﬂexive

is not reﬂexive

@

6.1.4@

is symmetric

6.1.5

6.1.3

@ R @

is antisymmetric

6.1.6 6.1.7

? ∈ Uk ∪ C2k

?

?

∈ Mk

∈ Sk

Fig. 6.1. A survey about the proof in Case 1

Proof. The relation 1 := ∩ ι2k is -derivable. If we have ∅ ⊂ 1 ⊂ ι2k , then pr1 is a non-diagonal -derivable unary relation, in contradiction to (IIa). Thus, is reﬂexive or areﬂexive. Now, let be areﬂexive. Since pr0 = pr1 = Ek holds by (II), we have ι2k ⊆ ◦ ζ =: . Because of this, (I) and the fact that P ol does not contain any constants, it follows that ∈ {ι2k , Ek2 }. If = Ek2 , then by ◦ ζ = σ2 () and by Lemma 6.1.2, we get a contradiction to the areﬂexivity of . Therefore, = ι2k holds, i.e., is a ﬁxed-point free permutation on Ek . If has not k/t cycles of the same length, then there is an i with ∅ ⊂ i ∩ ι2k ⊂ Ek2 and we have pr(i ∩ ι2k ) ∈ C1k , in contradiction to (IIa). Thus, the permutation has cycles of the same length t. The length t is a prime number, since for t = p · q (p prime, q ≥ 2), the -derivable relation := q has only cycles of prime length p. Therefore ’ belongs to Sk . Lemma 6.1.4 Let be reﬂexive. Then is either symmetric (i.e., ∩ζ = ) or antisymmetric (i.e., ∩ ζ = ι2k ). Proof. Since ι2k ⊆ , ι2k ⊆ ∩ ζ ⊆ holds, and by (IIb) we have ∩ ζ = ι2k or ∩ ζ = . Lemma 6.1.5 Let be reﬂexive and symmetric. Then ∈ Uk ∪ C2k . Proof. It is easy to check that ι2k ⊂ ⊆ ◦ . Then by (IIc) the following three cases are possible: Case 1: ◦ = . Obviously, in this case, is transitive and ∈ Uk .

6.1 Proof of Completeness Criterion

195

Case 2: ◦ = Ek2 . Since ◦ = σ2 (), σk () = Ekk follows from Lemma 6.1.2. Hence, for Ek there is a u ∈ Ek with {(e, u) | e ∈ Ek } ⊆ . Thus the relation belongs to C2k . Lemma 6.1.6 Let be reﬂexive and antisymmetric. Then, for Ek , there exists (with respect to ) a greatest element e ∈ Ek and (with respect to ) a least element o ∈ Ek , i.e., it holds: {(x, e ), (o , x) | x ∈ Ek } ⊆ , and these elements are uniquely determined. Proof. Since is reﬂexive, we have ι2k ⊆ ⊆ σ2 (). It is obvious that the relation σ2 () is symmetric and thus ⊂ σ2 (). This and (IIc) imply σ2 () = Ek2 . Then, because of k ≥ 3 and with the aid of Lemma 6.1.2, we have σk () = Ekk . Hence we ﬁnd an e ∈ Ek with {(x, e ) | x ∈ Ek } ⊆ . Then e is a greatest element of Ek . The uniqueness of e is to see as follows: Assume there are two greatest elements a, a ∈ Ek . Then we have (a, a ) ∈ and (a , a) ∈ and thus a = a by the antisymmetry of . The existence of a unique least element can be shown analogously starting with the -derivable relation σi (ζ) instead of σi (). Lemma 6.1.7 Let be a reﬂexive and antisymmetric relation with the property that Ek has a (unique) least and a (unique) greatest element with respect to . Then is transitive, i.e., ∈ Mk . Proof. We consider the -derivable relation 2 . Since ⊆ 2 , by (IIc) only the cases 2 = (i.e., is transitive) or 2 = Ek2 are possible. Let’s assume ◦ = Ek2 and let o be least element of Ek with respect to . Then we have (a, o ) ∈ for a ∈ Ek \{o }, (a, o ) ∈ ◦ and there is a b ∈ Ek with (a, b) ∈ and (b, o ) ∈ . However, this cannot hold by the antisymmetry of and by the choose of o . Thus the case 2 = Ek2 cannot occur and 2 = holds. Summing up, we have in Case 1: ∈ Uk ∪ Mk ∪ C2k ∪ Sk .

Case 2: h ≥ 3. Figure 6.2 gives a survey about the proof in this case.

196

6 Rosenberg’s Completeness Criterion for Pk

Z

Z Z Z

Z

Z

=

Z ~ Z

h≥4

h=3

H

HH HH

HH

6.1.8

HH

HH

?

is not totally reﬂexive

HH

HH j?

satisﬁes (*)

is totally reﬂexive 6.1.9

6.1.10

6.1.11 ... 6.1.15

?

a relation λ ∈ Lk is -derivable and k = pm , p prime

?

is totally symmetric

@

∈ Ck

6.1.16@

@ R @

is strongly homogeneous 6.1.17 ... ?6.1.22 ∈ Bk

Fig. 6.2. A survey about the proof in Case 2

First, we give some conclusions of the assumptions (I)–(III) over .

(IV )

∀r ≤ h − 1 ∀i1 , ..., ir ∈ Eh : pri1 ,...,ir = Ekr .

Since, if ∩ ιhk = ∅, pr1,2 is not an invariant of Pk , we have by (IIa)

(V )

∩ ιhk = ∅.

Obviously, it follows from (IIa):

6.1 Proof of Completeness Criterion

(V I)

197

∀δ ∈ Dkh \{Ekh } : ∩ δ ∈ Dkh .

If \ιhk = ∅, then, by (V I), the relation is an primitive relation (i.e., a union of certain diagonal relations), which is preserved by all unary functions of Pk . Hence it follows from (I): (V II) \ιhk = ∅. Lemma 6.1.8 If 4 ≤ h ≤ k, then either is totally reﬂexive or satisﬁes the following condition: (*): h = 4 ∧ | | = k 3 ∧ 4 4 4 ∪ δ{0,2},{1,3} ∪ δ{0,3},{1,2} ⊆ . δ{0,1},{2,3} Proof. First, note that for every (a, b), (b, c), (c, d) ∈ Eh2 \ι2h with |{a, b, c, d}| = 4, the implications (h ≥ 4 ∧ (x0 , ..., xh−1 ) ∈ ∧ xa = xb ) =⇒ xb = xc

(6.4)

(h ≥ 5 ∧ (x0 , ..., xh−1 ) ∈ ∧ xa = xb ) =⇒ xc = xd

(6.5)

and do not hold by (IV ). Next we will prove (∃(a, b) ∈ Eh2 \ι2h : δ{a,b} ⊆ ) =⇒ (∀(c, d) ∈ Eh2 \ι2h : δ{c,d} ⊆ ).

(6.6)

For this, it is suﬃcient to show that δ{0,1} ⊆ =⇒ δ{1,2} ⊆

(6.7)

holds. Let δ{0,1} ⊆ . Then we have δ{0,1,2} ⊆ ∩ δ{1,2} . Furthermore, by (V I) ∩ δ{1,2} is diagonal. Thus ∩ δ{1,2} ∈ {δ{0,1,2} , δ{1,2} }. Since an implication of form (6.4) follows from ∩ δ{1,2} = δ{0,1,2} , δ{1,2} ⊆ holds and therefore (6.7) and (6.6). For the proof of the total reﬂexivity of , we consider the relation 1 := ∩ δ{0,1} , which by (V ) and (V I) is a nonempty diagonal relation. Since the implications (6.4) and (6.5) are not true for , we have if h ≥ 5, = δ{0,1} 1 ∈ {δ{0,1},{2,3} , δ{0,1} } if h = 4.

198

6 Rosenberg’s Completeness Criterion for Pk

By (6.6) we have only to examine the case h = 4 and δ{0,1},{2,3} ∪ δ{0,2},{1,3} ∪ {0,3},{1,2} ⊆ . Obviously, either satisﬁes the condition (*) and our proof is closed or it holds || > k 3 . Let || > k 3 . Therefore w.l.o.g. there exists certain ai ∈ Ek (i = 0, 1, 2, 3) with (a0 , a1 , a2 , a3 ) ∈ , (a0 , a1 , a2 , a3 ) ∈ , a3 = a3 .

(6.8)

Then the -derivable relation 2 := {(a1 , a2 , a3 , a3 ) | ∃a0 : (a0 , a1 , a2 , a3 ) ∈ ∧ (a0 , a1 , a2 , a3 ) ∈ } has the properties 2 ∩ δ{2,3} = δ{2,3} (since pr1,2,3 =

(6.9)

Ek3 ), δ{2,3} ⊂ 2 (by (6.8)) and 2 ⊂ Ek4 .

Thus 2 is not an invariant of Pk . Further, is totally reﬂexive by (6.9) and (6.6). Then, by deﬁnition of 2 , we have now ∩ δ{0,1} = δ{0,1},{2,3} , a contradiction to our assumptions on . Lemma 6.1.9 If is totally reﬂexive, then is totally symmetric. Proof. The total symmetry of follows from (IIb) via ιhk ⊆ s∈Sh s ⊆ , where Sh denotes the set of all permutations on Eh and s denotes the relation {(as(0) , ..., as(h−1) ) | (a0 , ..., ah−1 ) ∈ }. Consequently, for the proof of (6.1), it is suﬃcient to examine the following cases: 2.1. satisﬁes (*) or it holds: h = 3 and is not totally reﬂexive and 2.2. h ≥ 3 and is totally symmetric and totally reﬂexive.

Case 2.1: satisﬁes (*) or h = 3 and ∩ ιhk = ιhk . W.l.o.g. we can assume that there does not exist a -derivable relation which is non-diagonal and totally reﬂexive.

6.1 Proof of Completeness Criterion

199

Lemma 6.1.10 The ternary relation is either of the form (a) = σ ∪ δ{0,1,2} with the properties ∅ ⊂ σ ⊆ Ek3 \ι3k and for any a, b ∈ Ek there are exactly a c1 , exactly a c2 and exactly a c3 with (c1 , a, b), (a, c2 , b), (a, b, c3 ) ∈ ; or of the form (b) = σ ∪ δ{a,b} ∪ δ{a,c} with ∩ s = δ{a,b} , where ∅ ⊂ σ ⊆ Ek3 \ι3k , {a, b, c} = {0, 1, 2}, s(a) := b, s(b) := a, s(c) := c and s := {(as(0) , as(1) , as(2) ) | (a0 , a1 , a2 ) ∈ }. Proof. By (V I) and (V II) it is suﬃcient to examine w.l.o.g. the following cases: = σ ∪ δ{0,1,2} (6.10) = σ ∪ δ{0,1}

(6.11)

= σ ∪ δ{0,1} ∪ {0,2}

(6.12)

Ek3 \ι3k .

where ∅ ⊂ σ ⊆ First let be of the form (6.11). A -derivable relation in this case is the totally symmetric relation i := {(a1 , ..., ai ) | ∃a, b : (∀i : (ai , a, b) ∈ )} (i = 2, 3, ..., k). Obviously, this relation has the property ∀i ∈ {2, ..., k − 1} : i = Eki =⇒ i+1 totally reﬂexive.

(6.13)

By our assumption in Case 2.1 and by (I) it follows from the total symmetry and total reﬂexivity of i+1 that i+1 = Eki+1 . However, this cannot occur, since for i = k − 1 it holds (0, 1, ..., k − 1) ∈ k =⇒ ∃a, b ∈ Ek : {(0, a, b), ..., (b, a, b), ..., (k − 1, a, b)} ⊆ } and ∩ ι3k = δ{0,1} is not possible. Thus our relation is not of the form (6.11), since we can show that 2 = Ek2 : δ{0,1} ⊆ implies ι2k ⊆ 2 . By pr0,1 = Ek2 there are certain a, b (a = b) and c with (a, b, c) ∈ . Furthermore we have (b, b, c) ∈ . Thus (a, b) ∈ 2 and 2 = Ek2 by (IIa). If = σ ∪ δ{0,1,2} and assumed does not fulﬁll the condition of (a) (w.l.o.g. for c3 ), then the -derivable relation := {(b, c, c ) | ∃a : (a, b, c) ∈ ∧ (a, b, c ) ∈ } is a non-diagonal relation of the form (6.11), of which we have shown that it cannot describe a maximal class of Pk . If has the form (6.12), it must hold ∩ (τ ) = δ{0,1} , since, in the opposite case, the -derivable relation ∩ (τ ) has the form (6.11). Thus by Lemma 6.1.10, we can assume w.l.o.g. that is either of the form (6.10) or (6.12) and fulﬁlls the conditions of Lemma 6.1.10.

200

6 Rosenberg’s Completeness Criterion for Pk

Lemma 6.1.11 It is possible to derive from a quaternary non-diagonal relation λ with the following 8 properties: 1) λ ∩ δ{0,2} = δ{0,2},{1,3} = λ ∩ δ{1,3} 2) λ ∩ δ{0,3} = δ{0,3},{1,2} = λ ∩ δ{1,2} 3) ∩ ι3k = δ{0,1} ∪ δ{0,2} ⇒ δ{0,1},{2,3} ⊆ λ 4) λ ∩ (τ λ) = λ, where τ λ := {(b, a, c, d) | (a, b, c, d) ∈ λ} 5) λ ∩ (λ)s = λ, where (λ)s := {(d, c, b, a) | (a, b, c, d) ∈ λ} 6) ∀x, y, z, z , o ∈ Ek : (x, y, z, o) ∈ ∧ (x, y, z , o) ∈ =⇒ z = z 7) pr0,1,3 λ = Ek3 8) pr0,2,3 λ = Ek3 . Proof. If the condition (*) fulﬁlls, it is easy to check that the -derivable relation λ := ∩ τ () ∩ {(d, c, b, a) | (a, b, c, d) ∈ } has the properties 1)–8), since it is possible in the opposite case to derive a non-diagonal totally reﬂexive, ternary relation from λ, in contradiction to the assumptions in Case 2.1. Let now h = 3 and let be a relation of the form (6.10) or (6.12) in the following. A -derivable relation is λ11 if has the form (6.10), λ1 := λ12 if has the form (6.12), where λ11 := {(a, b, c, d) | ∃u, v : {(u, d, a), (d, v, b), (u, v, c)} ⊆ } and λ12 := {(a, b, c, d) | ∃u1 , ..., u4 , v1 , ..., v4 : {(u1 , a, c), (u2 , b, d), (u3 , a, d), (u4 , b, c), (v1 , a, c), (v2 , b, d), (v3 , a, d), (v4 , b, c), (b, d, u1 ), (a, c, u2 ), (b, c, u3 ), (a, d, u4 ), (d, b, v1 ), (c, a, v2 ), (c, b, v3 ), (d, a, v4 )} ⊆ }. With the help of the properties of , which are given in Lemma 6.10, it is easy to see that λ1 fulﬁlls the conditions 1)–3). Since δ{0,2},{1,3} ∪ δ{0,3},{1,2} ⊆ λ11 ∩ (τ λ11 ) ∩ (λ11 )s and δ{0,2},{1,3} ∪ δ{0,3},{1,2} ∪ δ{0,1},{2,3} ⊆ λ12 ∩ (τ λ12 ), ∩(λ12 )s

6.1 Proof of Completeness Criterion

201

the non-diagonal relation λ := λ1 ∩ (τ λ1 ) ∩ (λ1 )s has the properties 1)–5). Furthermore, for any i, j ∈ E4 , i = j, it holds pri λ = Ek and pri,j λ = Ek2 .

(6.14)

If λ does not fulﬁll 6), we can derive the non-diagonal relation λ := {(c, c , a) | ∃b, d : (a, b, c, d) ∈ λ ∧ (a, b, c , d) ∈ λ} from λ. However λ is of the form σ ∪ δ{0,1} with ∅ ⊂ σ ⊆ Ek3 \ι3k (by 1) and 2)), what is not possible by the proof of Lemma 6.1.10. Thus λ fulﬁlls 1)–6). If = σ∪δ{0,1,2} , the properties 7) and 8) follow from (6.14), from the property (a) of Lemma 6.1.10 and from the deﬁnition of λ1 . If has the form σ ∪ δ{0,1} ∪ δ{0,2} , then we have ι3k ⊆ pr0,α,3 λ (α ∈ {1, 2}) by 1)–3) and hence pr0,α,3 λ = Ek3 , since the case pr0,α,3 λ ⊂ Ek3 is a contradiction to our assumptions over . By Lemma 6.1.11( 6), 7)) with the aid of λ, we can deﬁne for a ﬁxed o ∈ Ek an operation +o on Ek as follows: x +o y = z :⇐⇒ (x, y, z, o) ∈ λ.

(6.15)

Lemma 6.1.12 The operation +o has the following properties: 1) (Ek ; +o ) is an Abelean group with the identity element o. 2) There exists a prime number p so that every element of Ek \ {o} has the order p and it holds k = pm for certain m ≥ 1. Proof. 1): All properties of +o (except the associativity), which we have to show, follow directly from Lemma 6.1.11. By 1) of Lemma 6.1.11 o is the identity element of (Ek , +o ). Because of 4)–8) for every a ∈ Ek there exists an inverse element a ∈ Ek with a +o a = o. The operation +o is kommutative by 4). For proof of the associativity of +o , we consider the λ-derivable relation λ := {(a, b, c) | (a +o b) +o c = a +o (b +o c)} = {(a, b, c) | ∃u, v, w : {(a, b, u, o), (u, c, v, o), (b, c, w, o), (a, w, v, o)} ⊆ λ}, which is totally reﬂexive by {(o, a, a), (a, o, a), (a, a, o) | a ∈ Ek } ⊆ λ and (II). However, this is only possible if λ = Ek3 by our assumptions over . Thus +o is associative. 2): A λ-derivable relation is µn := {(x, o) | x +o x +o ... +o x = o} n times = {(x, o)|∃ui : {(x, x, u1 , o), (u1 , x, u2 , o), ..., (un−2 , x, o, o)} ⊆ λ}.

202

6 Rosenberg’s Completeness Criterion for Pk

By o +o o = o for all o ∈ Ek this relation is reﬂexive. Assume there exists two elements of Ek \{o} with the orders n and m (= n) respectively. Then we have ι2k ⊂ µn ⊂ Ek2 , in contradiction to our assumptions over . Thus all elements (= o) of (Ek , +o ) have the same order. Now it is easy to show that this order is a certain prime number p: If n > 1 is not a prime number, there exist r, s ∈ N with n = r · s, 1 < r < n and 1 < s < n. For a b ∈ Ek \ {0} we set c := s · b := b +o b +o ... +o b = o. Then we have c = o and r · c = n · b = o, s times a contradiction to the above-proven fact that the order of b is n. k = pm for a certain m is a conclusion from well-known theorems about Abelian groups. Lemma 6.1.13 For every x, y, o, o1 ∈ Ek it holds: 1) x +o1 x = x +o x −o o1 2) x +o1 o = x −o o1 3) k ≥ 4 =⇒ x +o1 y = x +o y −o o1 . Proof. To prove i) (i ∈ {1, 2, 3}) we consider certain λ-derivable relations λi : λ1 := {(x, o1 , o) | x +o1 x = x +o x −o o1 } = {(x, o1 , o) | ∃u, v, w : {(x, x, u, o1 ), (x, x, v, o), (o1 , w, o, o), (v, w, u, o)} ⊆ λ}, λ2 := {(x, o, o1 ) | x +o1 o = x −o o1 } = {(x, o, o1 ) | ∃u, v : {(x, o, u, o1 ), (o1 , v, o, o), (x, v, u, o)} ⊆ λ} and λ3 := {(x, y, o, o1 ) | x +o1 y = x +o y −o o1 } = {(x, y, o, o1 ) | ∃u, v, w : {(x, y, u, o1 ), (o1 , v, o, o), (x, y, w, o), (w, v, u, o)} ⊆ λ}. If we can show that λ1 , λ2 and λ3 are totally reﬂexive relations, then we have λ1 = λ2 = Ek3 and λ3 = Ek4 for k ≥ 4 by our assumptions in Case 2.1 and the statements of Lemma 6.1.13 are proven. For all x, o ∈ Ek we have: – (x, x, o) ∈ λ1 , since x +x x = x = x +o x −o x. – (x, o1 , x) ∈ λ1 , because for v = x, w = u and Lemma 6.1.11, 5) the conditions of deﬁnition of λ1 hold. – (x, o, o) ∈ λ1 , since x +o x = x +o x −o o. Thus 1) is proven. For all x, o, o1 ∈ Ek it holds: – (o, o, o1 ) ∈ λ2 , because o +o1 o = o +o o −o o1 = o −o o1 by 1). – (o1 , o, o1 ) ∈ λ2 , since o1 +o1 o = o = o1 −o o1 . – (x, o, o) ∈ λ2 by x +o o = x = x −o o.

6.1 Proof of Completeness Criterion

203

Thus b) is proven). 3) follows from – (x, x, o, o1 ) ∈ λ3 by 1); – (o, y, o, o1 ) ∈ λ3 , since o +o1 y = y +o1 o = y −o o1 = o +o y −o o1 by 2); – (o1 , y, o, o1 ) ∈ λ3 by o1 +o1 y = y = o1 +o y −o o1 ; – (x, o, o, o1 ) ∈ λ3 , since (by 2)) x +o1 o = x −o o1 = x +o o −o o1 ; – (x, o1 , o, o1 ) ∈ λ3 since x +o1 o1 = x = x +o o1 −o o1 ; – (x, y, o, o) ∈ λ3 , since x +o y = x +o y −o o. Lemma 6.1.14 For every o ∈ Ek and all a, b, c, d ∈ Ek the following equivalence holds: (a, b, c, d) ∈ λ ⇐⇒ a +o b = c +o d. Proof. If k ≥ 4 then we have by deﬁnition of +o and by Lemmas 6.1.12 and 6.1.13: (a, b, c, d) ∈ λ ⇐⇒ a +d b = c ⇐⇒ a +d b = a +o b −o d = c ⇐⇒ a +o b = c +o d. For k = 3 our statement follows from Lemma 6.1.11, 1), 2), 5) and Lemma 6.1.13, 1). Then the following lemma is a conclusion from the Lemmas 6.1.11, 6.1.12, and 6.1.14: Lemma 6.1.15 The -derivable relation λ belongs to Lk . Proof. If k ≥ 4, then λ ∈ Lk (k = pm , p prime, m ≥ 1) follows from the Lemmas 6.1.9, 6.1.10 and 6.1.12. For k = p = 3 the relation λ := {(a, b, c, d) | ∃α : a +o b = α +o α ∧ c +o d = α +o α} is -derivable and belongs to Lk . Summing up: Our Case 2.1 is only possible if k is a prime power. Furthermore, we have for k = pm : P ol = P olλ for certain λ ∈ Lk .

Case 2.2: is totally reﬂexive and totally symmetric. Since Pk (h − 1) ⊆ P ol for every totally reﬂexive h -ary relation (h ≥ 2), it is not possible to ﬁnd a totally reﬂexive h -ary relation with P ol = P ol and h > h. Thus we can assume the following property for : (V III)

∀ ∈ [{} ∪ InvPk ]h : h < h ≤ k ∧ ιhk ⊆ =⇒ = Ekh .

204

6 Rosenberg’s Completeness Criterion for Pk

An h-ary, totally reﬂexive, and totally symmetric relation γ we will call strongly homogeneous, if the following holds: (∃v0 , ..., vh−1 ∈ Ek : (v0 , ..., vh−1 ) ∈ γ ∧ ∀i ∈ Eh ∀j ∈ Eh \{i} : (a0 , ..., aj−1 , vi , aj+1 , ..., ah−1 ) ∈ γ) =⇒ (a0 , a1 , ..., ah−1 ) ∈ γ.

(6.16)

Lemma 6.1.16 Either is central or strongly homogeneous. Proof. First we consider the -derivable relation t := {(a0 , ..., at−1 ) | ∃c : (∀i0 , ..., ih−2 ∈ Et : (ai0 , ..., aih−2 , c) ∈ ))}, for t ∈ {h, h + 1, ..., k}. The total symmetry of implies that t is totally symmetric. By choosing c = a0 for t = h, we see that ⊆ h . Thus by our assumption (IIc), only the following two cases are possible: Case 1: h = Ekh . Then we have h < k since, in the opposite case, = Ekk holds by deﬁnition of . Obviously, for h < k, the -derivable relation h+1 is totally reﬂexive and thus, by (V III), h+1 = Ekh+1 . By induction, it is easy to see that i = Eki for all i ∈ {h + 1, ..., k}. Then, ∈ Ck follows from k = Ekk . Case 2: = h . In this case, is homogeneous, i.e., has the property (∃v : ∀i ∈ Eh : (a0 , ..., ai−1 , v, ai+1 , ..., ah−1 ) ∈ ) =⇒ (a0 , a1 , ..., ah−1 ) ∈ . for arbitrary a0 , a1 , ..., ah−1 ∈ Ek . We consider the -derivable relation γt := {(a0 , ..., at−1 ) | ∃v0 , v1 , ..., vt−1 : (∀i1 , ..., ih ∈ Et : (vi1 , ..., vih ) ∈ ) ∧ (∀n ∈ Et ∀j1 , ..., jh−2 ∈ Et \{n} : (aj1 , ..., ajh−2 , an , vn ) ∈ )} (t ∈ {h, h + 1, ..., k}). Obviously, γt is totally symmetric and, if γt−1 = Ekt−1 , is totally reﬂexive. Furthermore, ⊆ γh for h = t. (This follows from the deﬁnition of γh by setting vα = aα for all α ∈ Eh .) = γh implies that is strongly homogeneous. Now by (IIc) we have γh ∈ {, Ekh }. Thus our lemma is proven if we can show that γh = Ekh cannot occur. Assume γh = Ekh and h < k. Then γh+1 is totally reﬂexive. Thus, γh+1 = Ekh+1 by (V III) and γk = Ekk via induction. Hence, for every a0 , ..., ah−1 ∈ Ek there exists certain v0 , ..., vh−1 ∈ Ek with (v0 , v1 , ..., vh−1 ) ∈

(6.17)

6.1 Proof of Completeness Criterion

205

and ∀n ∈ Eh ∀α1 , ..., αh−2 ∈ Ek \{an } : (α1 , ..., αh−2 , an , vn ) ∈ .

(6.18)

With the help of this and by induction, we can show that ∀t ≥ 0 ∀a0 , ..., at−1 ∈ Ek : (a0 , ..., at−1 , vt , vt+1 , ..., vh−1 ) ∈

(6.19)

holds: For t = 0, (6.19) follows from (6.17). Assume (6.19) holds for t = n. Then (6.19) for t = n + 1 is a conclusion from the induction hypothesis, the homogeneity of , the totally symmetry of and from (6.18) putting v = vn in (6.16) for the tuple (a0 , ..., an , vn+1 , ..., vh−1 ). By (6.19) and t = h, we get a contradiction to ⊂ Ekh : ∀a0 , ..., ah−1 ∈ Ek : (a0 , ..., ah−1 ) ∈ . Hence γh = Ekh cannot occur and thus γh = , i.e., is strongly homogeneous. By Lemma 6.1.16, we have to show for the proof of our theorem that, if is strongly homogeneous, the relation or a -derivable relation belongs to Bk . Thus we can assume that is subsequently a strongly homogeneous relation. For this relation, we will prove ﬁrst that is a homomorphic inverse image of a certain relation ξ and then that the relation ξ is isomorphic to a certain elementary h-ary relation ξm .2 Lemma 6.1.17 The relation ε := {(a, b) ∈ Ek2 | ∀a0 , ..., ah−3 ∈ Ek : (a0 , a1 , ..., ah−3 , a, b) ∈ } is an equivalence relation on Ek . Proof. By the total reﬂexivity and total symmetry of , it follows directly the reﬂexivity and symmetry of ε. To prove the transitivity of ε, let {(a, b), (b, c)} ⊆ ε. Choosing v0 = α0 , v1 = α1 , ..., vh−3 = αh−3 , vh−2 = b and vh−1 = b in (6.16) for the tuple (α0 , α1 , ..., αh−3 , a, c), we get (α0 , ...., αh−3 , a, c) ∈ for arbitrary α0 , ..., αh−3 ∈ Ek . Hence, (a, c) ∈ ε. By the equivalence relation ε, the set Ek is partitioned into certain (nonempty) equivalence classes Ai (i = 1, 2, ...., r), from which we choose a representative αi . With the help of the representative set V := {α1 , ..., αr } we can deﬁne a mapping F : Ek −→ V by F (a) = αi :⇐⇒ {a, αi } ⊆ Ai . 2

See Lemma 6.1.22 to the concept “isomorphic”.

206

6 Rosenberg’s Completeness Criterion for Pk

Lemma 6.1.18 Let be strongly homogeneous. Then (a) (a0 , ...., ah−1 ) ∈ ⇐⇒ (F (a0 ), ..., F (ah−1 )) ∈ (b) ((∀a0 , ..., ah−3 ∈ V : (a0 , ..., ah−3 , a, b) ∈ ) ∧ {a, b} ⊆ V ) =⇒ a = b. Proof. (a): First, we show that the equivalence (a, a1 , ..., ah−1 ) ∈ ⇐⇒ (b, a1 , ..., ah−1 ) ∈

(6.20)

holds for (a, b) ∈ ε and for all a0 , ..., ah−1 ∈ Ek . If (a, a1 , ..., ah−1 ) ∈ then (b, a1 , ...., ah−1 ) ∈ follows from the strong homogeneity of , choosing v0 = v1 = .... = vh−1 = a. Since ε is symmetric, we have also proven “⇐=” of (6.20). Now (a) follows from (6.20) because of (ai , F (ai )) ∈ ε (i ∈ Eh ), Lemma 6.1.17 and of the total symmetry of : (a0 , ...., ah−1 ) ∈ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

(F (a0 ), a1 , ..., ah−1 ) ∈ (F (a0 ), F (a1 ), a2 , ..., ah−1 ) ∈ ... (F (a0 ), ..., F (ah−1 )) ∈ .

(b) is a conclusion from the deﬁnitions of ε and V and from (a). Now put ξ := F () := {(F (a0 ), ..., F (ah−1 )) | (a0 , ..., ah−1 ) ∈ }. By Lemma 6.1.18, (a) is a homomorphic inverse image of this relation, i.e., = {(a0 , ..., ah−1 ) ∈ Ekh | (F (a0 ), ..., F (ah−1 )) ∈ ξ} holds. Since = Ekh , ζ = V h . Thus w.l.o.g. we can assume V = Er and (0, 1, ..., h − 1) ∈ V h \ξ. To show that ξ is isomorphic to a certain h-ary elementary relation ξm (⊆ Ehhm ), i.e., that there is a bijective mapping ϕ of Ehm onto V with the property ξ = {(ϕ(a0 ), ..., ϕ(ah−1 )) | (a0 , ..., ah−1 ) ∈ ξm } we consider the -derivable hi -ary graphic i

Ghi (P ol) := χhi ∪ {f (κ1 , ..., κhi ) | f ∈ (P ol)h }, where χhi := (κ1 , ..., κhi ) is the hi -ary abscissa over Ek (in matrix form; see Section 2.7), i ∈ {h, h + 1, ..., k}. Let j1 , ..., ji be the numbers of those rows of χhi for which Ai := prj1 ,...,ji χhi is a matrix form of the relation Ehi . Further, let µi := prj1 ,...,ji Ghi (P ol).

6.1 Proof of Completeness Criterion

207

Lemma 6.1.19 If is strongly homogeneous and (0, 1, 2, ..., h − 1) ∈ , then µi = Eki for every i ∈ {h, h + 1, ..., k}, i.e., the functions of P ol can take every value of Ek on every row of the matrix Ai . Proof. First, we remark that = Ekh \{(a0 , ..., ah−1 ) | {a0 , ..., ah−1 } = Eh } because of ∈ Ck . Further, if (b0 , ..., bh−1 ) ∈ , all functions g with Im(g) = {b0 , ..., bh−1 } belong to P olk . Suppose the lemma is false for i = h. Then, we can form an h-ary relation with ⊂ ⊂ Ekh with the help of the hi -th graphic of P ol by projection, contradicting (IIc). Since all functions of P ol, which have at most hi essential variables, belong i+1 it is easy to show by induction that it holds: to (P ol)h ∀i ∈ {h, ..., k − 1} : µi = Eki =⇒ ιi+1 ⊆ µi+1 . k Consequently, our lemma follows from (V III). By Lemma 6.1.19, we can ﬁnd an hk -ary function f ∈ P ol with the properties Im(f ) = V ⎛

⎞ F (0) ⎜ F (1) ⎟ ⎟. f (Ak ) = ⎜ ⎝ ⎠ ... F (k − 1)

and

Next we will give some properties of this function f , from which will follows ∈ Bk . k Recall, f ∈ P ol means on tuples of Ehh that f preserves the relation ιhh . k The elements z of Ehh we also give in the form (z[1], z[2], ..., z[hk ]) k

and, for every z ∈ Ehh , denote z t,a (t ∈ {1, 2, ..., hk }, a ∈ Eh ) k

an element of Ehh , which is deﬁned by a, if i = t, z t,a [i] := z[i] otherwise (i ∈ {1, 2, ..., hk }). k

Lemma 6.1.20 Let t ∈ {1, 2, ..., hk }, z ∈ Ehh and (f (z t,0 ), ..., f (z t,h−1 )) ∈ ξ. Then (a) f (z t,0 ) = ... = f (z t,h−1 ), k

(b) ∀w ∈ Ehh : f (wt,0 ) = ... = f (wt,h−1 ).

208

6 Rosenberg’s Completeness Criterion for Pk

Proof. (a): For proof, it is suﬃcient to show that (w.l.o.g.) f (z t,h−2 ) = f (z t,h−1 ). First, we prove k

∀b0 , ..., bh−3 ∈ Ehh : (f (b0 ), ..., f (bh−3 ), f (z t,h−2 ), f (z t,h−1 )) ∈ ξ.

(6.21)

If {b0 [t], b1 [t], ..., bh−3 [t]} = {0, 1, ..., h − 3}, then (6.21) holds, since in this case, all columns of the matrix ⎛ ⎞ b0 ⎜ b1 ⎟ ⎜ ⎟ ⎜ ... ⎟ ⎜ ⎟ B := ⎜ ⎟ h−3 ⎟ ⎜ bt,h−2 ⎝z ⎠ z t,h−1 belong to ιhh and f preserves . Let w.l.o.g. bi [t] = i for i ∈ Eh−2 . Substituting the i-th row of B by z t,j , where i = j, we obtain a matrix Bi,j whose columns belong to ιhh . Hence f (Bi,j ) ∈ ξ holds. Consequently, by the strong homogeneity of , choosing vj = f (z t,j ) (j ∈ Eh ) for the tuple (f (b0 ), ..., f (bh−3 ), f (z t,h−2 ), f (z t,h−1 )), we get (6.21). k Since {f (α) | α ∈ Ehh } = V , for all a0 , ..., ah−3 ∈ V there are certain k b0 , ..., bh−3 ∈ Ehh with f (bi ) = ai (i ∈ Eh−2 ). Thus by (6.21) we have ∀a0 , ..., ah−2 ∈ V : (a0 , ..., ah−3 , f (z t,h−2 ), f (z t,h−1 )) ∈ ξ. Then, with the help of Lemma 6.1.18, (b) it follows f (z t,h−2 ) = f (z t,h−1 ). (b): Because of (a) it is suﬃcient to show that (f (wt,0 ), ...., f (wt,h−1 )) ∈ ξ

(6.22)

holds. By (6.21), it follows (f (wt,0 ), ..., f (wt,h−3 ), f (z t,h−2 ), f (z t,h−1 )) ∈ ξ. It is easy to check that every exchange of an i-th row in ⎞ ⎛ wt,0 ⎜ wt,1 ⎟ ⎟ ⎜ ⎝ ... ⎠ wt,h−1 by wt,j for i ∈ Eh−3 or by z t,j for i ∈ {h − 2, h − 1} and i = j, gives a matrix M, whose columns belong to ιhh , and it holds f (M) ∈ ξ. Hence by the strong homogeneity of , choosing v0 = f (wt,0 ), ..., vh−3 = f (wt,h−3 )), vh−2 = f (z t,h−2 ) and vh−1 = f (z t,h−1 ), we get (6.22). Lemma 6.1.21 For the function f , there exists certain digits t1 , ..., tm , which k are exactly the essential digits of f|E hk (restriction of f to Ehh ), and f has h

k

diﬀerent values on any two diﬀerent tuples of Ehh , i.e., it holds

6.1 Proof of Completeness Criterion k

∀z, w ∈ Ehh : f (z) = f (w) ⇐⇒ ∀i ∈ {t1 , ..., tm } : z[i] = w[i].

209

(6.23)

Further, f has the properties |Im(f )| = hm and ∀r1 , ..., rhk ∈ Ehh : ({rt1 , ..., rtm } ⊆ ιhh =⇒ f (r1 , ..., rhk ) ∈ )

(6.24)

Proof. For every t ∈ {1, 2, ..., hk }, we have either f (z t,0 ) = f (z t,1 ) = ... = k f (z t,h−1 ) for every z ∈ Ehh or (f (z t,0 ), f (z t,1 ), ..., f (z t,h−1 )) ∈ for all z ∈ k Ehh by Lemma 6.1.20. Let T := {t1 , ..., tm } be the set of all t ∈ {1, 2, ..., hk }, for which (f (z t,0 ), ..., f (z t,h−1 )) ∈ holds. Now, we will show that the digits ti ∈ T have the properties of Lemma 6.1.21. First, let f (z) = f (w) for certain k z, w ∈ Ehh and assume there exists an i ∈ T with α := z[i] = w[i]. Then, the columns of the matrix ⎛ i,0 ⎞ z ⎜ ... ⎟ ⎜ i,α−1 ⎟ ⎜z ⎟ ⎜ ⎟ ⎜ A := ⎜ w ⎟ ⎟ ⎜ z i,α+1 ⎟ ⎜ ⎟ ⎝ ... ⎠ z i,h−1 belong to ιhh . Thus f (A) ∈ ξ and by f (w) = f (z) = f (z i,α ) it follows (f (z i,0 ), ..., f (z i,h−1 )) ∈ ξ, a contradiction to i ∈ T and to the deﬁnition of T . Therefore “=⇒” in (6.23) holds. Let now z[i] = w[i] for every i ∈ T and w.l.o.g. T = {1, 2, ..., m}. f (w) = f (z) is proven, if we can show that f (un−1 ) = f (un )

(6.25) k

holds for every tuple un := (z[1], ..., z[n], w[n + 1], w[n + 2], ..., w[h ]) and all n ∈ {m + 1, m + 2, ..., hk }, since f (um ) = f (w) and f (uhk ) = f (z). By n > m n,j and the deﬁnition of T , we have f (un,i n−1 ) = f (un−1 ) for every i, j ∈ Eh . As n,w[n] n,z[n] un−1 = un−1 and un−1 = un we have (6.25) and (6.23) is proven. k (6.23) and {f (x) | x ∈ Ehh } = V imply |V | = hm . Finally, we prove (6.24). Assume (6.24) is false. Then there exists certain k z0 , z1 , ..., zh−1 ∈ Ehh with (f (z0 ), f (z1 ), ..., f (zh−1 )) ∈ ξ and (w.l.o.g.) zi [1] = k i for all i ∈ Eh and 1 ∈ T . Let w ∈ Ehh . Then, all columns of the matrix ⎛ ⎞ w1,0 ⎜ ... ⎟ ⎜ 1,j−1 ⎟ ⎜w ⎟ ⎜ ⎟ 1,i ⎜ ⎟ Ci,j := ⎜ zj ⎟ ⎜ w1,j+1 ⎟ ⎜ ⎟ ⎝ ... ⎠ w1,h−1

210

6 Rosenberg’s Completeness Criterion for Pk

belong to ιhh for i = j. Thus f (Ci,j ) ∈ ξ. By choosing vi = f (zi ), for the tuple (f (w1,0 ), ..., f (w1,h−1 )) and i ∈ Eh from this and the strong homogeneity of it follows (f (w1,0 ), ..., f (w1,h−1 )) ∈ ξ. But this contradicts the deﬁnition of T and 1 ∈ T . Lemma 6.1.22 If is strongly homogeneous, then there exists for certain m, a bijective mapping ϕ from Ehm onto V with ξ = ϕ(ξm ) := {(ϕ(a0 ), ..., ϕ(ah−1 )) | (a0 , ..., ah−1 ) ∈ ξm } and it holds that = {(a0 , ..., ah−1 ) ∈ Ekh | (ϕ−1 (F (a0 )), ..., ϕ−1 (F (ah−1 ))) ∈ ξm }, i.e., ∈ Bhk . Proof. With the function f in Lemma 6.1.21, we can deﬁne a bijective mapping from Ehm onto V as follows: ϕ(a(m−1) hm−1 + a(m−2) hm−2 + ... + a(1) h + a(0) ) = f (z) :⇐⇒ (z[t1 ], z[t2 ], ..., z[tm ]) = (a(m−1) , a(m−2) , ..., a(0) )), where a ∈ Ehm . Let (a0 , ..., ah−1 ) ∈ ξm . Then there are certain z0 , ..., zh−1 ∈ k Ehh , where (z0 [i], ..., zh−1 [i]) ∈ ιhh for all i ∈ {1, 2, ..., hk }, with the property (ϕ(a0 ), ϕ(a1 ), ..., ϕ(ah−1 )) = (f (z0 ), f (z1 ), ..., f (zh−1 )). Since f ∈ P ol, this implies (ϕ(a0 ), ..., ϕ(ah−1 )) ∈ ξ. Hence ϕ(ξm ) ⊆ ξ. Assume, there exists a tuple (a0 , ..., ah−1 ) ∈ ξ\ϕ(ξm ). By deﬁnition of ϕ, we k ﬁnd certain z0 , ..., zh−1 ∈ Ehh with (f (z0 ), ..., f (zh−1 )) = (a0 , ..., ah−1 ), where (z0 [i], ..., zh−1 [i]) ∈ ιhh for certain i ∈ {t1 , ..., tm }. But this is contrary to (6.24). Thus ξ = ϕ(ξm ). The other statements follow from Lemma 6.1.18, (a) and the deﬁnition of ξ. To sum up, we have in Case 2.2: ∈ Ck ∪ Bk , and Theorem 6.1 is proven.

7 Further Completeness Criteria

In the literature, one ﬁnds many articles which deal with complete sets. With the aid of complete sets, it is easy to obtain completeness criteria. In this chapter, we discuss only three types of such criteria. First we handle a criterion for Sheﬀer functions, which was found by G. Rousseau. Then, we show how one can reduce the conditions from Theorem 6.1 if one considers only surjective functions. Finally, we deal with criteria that indicate under which conditions a set (⊆ Pk ) which consists of certain unary functions and a Slupecki-function is complete in Pk .

7.1 A Criterion for Sheﬀer-Functions Deﬁnitions A function f ∈ Pk is called a Slupecki-function, if f ∈ P olk ιkk , i.e., if f depends on at least two variables essentially and takes k different values. A function f is called a Sheﬀer-function of Pk , if every g ∈ Pk is a superposition over f , i.e., if (7.1) [f ] = Pk holds. Obviously, every Sheﬀer-function is a Slupecki-function. The function x | y := x ∨ y which fulﬁlls (7.1) for k = 2 was published by H. M. Sheﬀer (in [She 13]). This function and the function x ∧ y, which is dual to this function, are the only binary functions of P2 , which generate P2 (cf. [Zyl 25]). One can easily prove this with the aid of the following theorem.

212

7 Further Completeness Criteria

Theorem 7.1.1 Let f n ∈ P2 . Then [f ] = P2 ⇐⇒ (f ∈ T0 ∧ f ∈ T1 ∧ f ∈ S).

(7.2)

Proof. “=⇒” is trivial. “⇐=”: Because off (0, ..., 0) = 1 and f (1, ..., 1) = 0, we have f ∈ M . Since f 0 1 does not preserve , there are some a1 , ..., an ∈ E2 with f (a1 , ..., an ) = 1 0 f (a1 , ..., an ). Consequently, (f (0, ..., 0), f (a1 , ..., an ), f (a1 , ..., an ), f (1, ..., 1)) does not belong to α := {(a, b, c, d) ∈ E24 | a + b = c + d (mod 2)}, although (0, ai , ai , 1) ∈ α for every i ∈ {1, ..., n} is valid. Thus, we have also f ∈ P ol2 α. With the aid of Theorem 3.3.1, this implies [f ] = P2 . Lemma 7.1.2 Let f n ∈ Pk be a Slupecki-function which preserves a certain h-ary relation ∈ Rk . Then (a) (∃s ∈ Sk \{e11 } : = {(a, s(a)) | a ∈ Ek }) =⇒ (∃γ ∈ Sk ∪ C1k : f ∈ P olγ); (or: (∃s ∈ Sk \{e11 } : f (x1 , ..., xn ) = s−1 (f (s(x1 ), ..., s(xn ))) ) =⇒ ∃γ ∈ Sk ∪ C1k : f ∈ P olγ; (b) ∈ Mk =⇒ ∃γ ∈ C1k : f ∈ P olγ; (c) (2 ≤ h ≤ k − 1 ∧ ∈ Chk ) =⇒ ∃γ ∈ C1k : f ∈ P olγ; (d) (k = pm (p prime, m ≥ 1) ∧ ∈ Lk ) =⇒ ∃γ ∈ Uk ∪ Sk : f ∈ P olγ; (e) ∈ Bk =⇒ ∃γ ∈ Uk ∪ Sk : f ∈ P olγ. Proof. (a): In the proof of Lemma 6.1.3, we had shown that, from the relation s := {(a, s(a)) | a ∈ Ek } (s ∈ Sk \{e11 }), either a relation of C1k (in the case s ∩ ι2k = ∅) or a relation of Sk (in the case s ∩ι2k = ∅) are derivable, whereby we have f ∈ P olγ for a certain γ ∈ C1k ∪Sk . (b): Let o ∈ Ek be the least element in respect to ∈ Mk . Because of Im(f ) = Ek and f ∈ P ol, f (o, ..., o) = o holds. Thus we have f ∈ P ol{o}. (c): Let C be the set of all central elements of the relation ∈ Chk , 2 ≤ h ≤ k − 1. Then, because of (a0 , ..., ah−2 , c) ∈ for every c ∈ C and arbitrary a0 , ..., ah−2 ∈ Ek , it results from f ∈ P ol and Im(f ) = Ek that f (c1 , ..., cn ) ∈

7.1 A Criterion for Sheﬀer-Functions

213

C is valid for all c1 , ..., cn ∈ C. Consequently, f preserves a relation of C1k . (d): Let k = pm (p prime, m ≥ 1) and let f n be a quasi-linear function. By Lemma 5.2.4.3, f is isomorphic to a certain function fˆn ∈ PA , where A := Epm . One can describe this function fˆn as follows: fˆ(x1 , ..., xn ) = a + x1 · A1 + ... + xn · An , where a, x1 , ..., xm ∈ Epm , A1 , ..., An are some matrices of the type (m, m) over Ep and + and · are the usual matrix operations over the ﬁeld (Ep ; + (mod p), · (mod p)). We form fˆ1 (x) := fˆ(x, x, ..., x) = a + x · A, where A := A1 + ... + An , and distinguish two cases for A: Case 1: det(A − I) = 0. 1 Then there exists b ∈ Epm with b · (A − I) = −a or a + b · A = b. Consequently, we have fˆ(b, ..., b) = b and f preserves a relation of C1k . Case 2: det(A − I) = 0. In this case, there is a d ∈ Epm \{o}, o := (0, ..., 0), with d · (A − I) = o and it holds: fˆ(x1 + d, x2 + d, ..., xn + d) − d = a + (x1 + d) · A1 + ... + (xn + d) · An − d = a + d · (A1 + ... + An ) −d +x1 · A1 + ...xn · An =A = fˆ(x1 , ..., xn ).

o

Thus, fˆ preserves the relation {(x, s(x)) | x ∈ Epm } with s(x) := x + d. With the help of (a), this implies f ∈ P olγ for certain γ ∈ Sk . (e): Let ∈ Bk . Then, by deﬁnition, there exists a mapping q from Ek onto Ehm (m ≥ 1) with (a0 , ..., ah−1 ) ∈ =⇒ (q(a0 ), ..., q(ah−1 )) ∈ ξm , where ξm is the h-ary elementary relation on Ehm . Because of Theorem 5.2.6.1, one can describe f ∈ P ol with Im(f ) = Ek in the form (5.14), where gf has the form m−1 si ((q(xji ))α(i)) · hi (7.3) gf (x1 , ..., xn ) = i=0

with {j0 , ..., jm−1 } ⊆ {1, 2, ..., n}, s0 , s1 , ..., sm−1 are some permutations on Ek and α is a certain mapping from Em into Em . By means of (7.3), one can prove that the function f preserves the equivalence relation 1

I denotes the identity matrix of the type (m, m).

214

7 Further Completeness Criteria

{(a, b) ∈ Ek2 | q(a) = q(b)}.

(7.4)

Consequently, f belongs to P olγ for a certain γ ∈ Uk , if q is not bijective. We still have to examine the case that q is bijective. W.l.o.g. let q be the identical mapping in the following. Thus our function f has the form f (x1 , ..., xn ) =

m−1

si ((xji )(α(i)) ) · hi

(7.5)

i=0

and we have k = hm and m ≥ 2 (because of f ∈ P olk ιkk ). For the mapping α, two cases are possible: Case 1: α is not a permutation. In this case, there is a proper subset A of Ek , which is preserved from α. Then, by means of (7.5), it is easy to check that f preserves the non-trivial equivalence relation {(a, b) ∈ Ek2 | ∀i ∈ A : a(i) = b(i) }; hence f ∈ P olγ holds for a certain γ ∈ Uk . Case 2: α is a permutation. We form f1 (x) := f (x, x, ..., x) and set α1 x := α(x), αi x := α(αi−1 x), i ≥ 2. Then we have by assumption and (7.5): g(x) := (f1 f1 ... f1 ) = m times where x(α

m

i)

m−1

si (sαi (...(sαm−1 i (x(α

m

i)

))...)) · hi ,

i=0

= x(i) . The function g has the properties g −1 (x) =

m−1

−1 −1 (i) i s−1 αm−1 i (sαm−2 i (...(si (x ))...)) · h

i=1

and

g −1 (f (g(x1 ), g(x2 ), ..., g(xn ))) = f (x1 , x2 , ..., xn ).

Thus, if g = e11 then by (a) f preserves a certain relation of Sk and (e) was proven. If g = e11 then for every a ∈ Ek the function f preserves the element m−1

sαi (sα2 i (...(sαm−1 i (a(i) ))...)) · hi

i=0

of Ek , whereby f preserves a relation of C1k . The following generalization of Theorem 7.1.1 was found by G. Rousseau.

7.1 A Criterion for Sheﬀer-Functions

215

Theorem 7.1.3 (Characterization Theorem for Sheﬀer-Functions; [Rou 67]) Let f n ∈ Pk . Then [f ] = Pk ⇐⇒ ∀ ∈ C1k ∪ Uk ∪ S∼ k : f ∈ P ol.

(7.6)

Proof. Obviously, “=⇒” is valid. “⇐=”: Let f ∈ Pk be a function that does not preserve any relation of the set C1k ∪ S∼ k ∪ Uk . Then f is a Slupecki-function and, because of Theorem 6.1, it is suﬃcient to show for the proof of “⇐=” of (7.6) that ∀ ∈ Mk ∪ Lk ∪ Bk ∪

k−1

Chk : f ∈ P olk

(7.7)

h=2

holds. From the assumption that (7.7) is false, contradictions to the assumption result with the aid of the statements (b) - (e) from Lemma 7.1.2 in all cases to be scrutinized for . Theorem 7.1.4 ([Sch 69]) One can not reduce the conditions from Theorem 7.1.3; that is, these conditions are independent of each other. Proof. It was shown in [Sch 69] by P. Schoﬁeld that every class of the form P olk with ∈ C1k ∪ Sk ∪ Uk can be generated by one function (see Section 11.2). With the help of this property, one can easily prove that our assertion is valid: Assume there is a relation α ∈ C1k ∪ S∼ k ∪ Uk with [f ] = Pk ⇐⇒ ∀ ∈ (C1k ∪ Uk ∪ S∼ k ) \ {α} : f ∈ P ol. Furthermore, let f be a function of A with [f ] = A. Since A is not complete in Pk , there exists a maximal class B = A of Pk with f ∈ B. Then, f belongs to A ∩ B and we get A = [f ] ⊆ A ∩ B ⊆ B. Because of A = B, however, this is contradictory to the Pk -maximality of A.

The literary about concrete Sheﬀer-functions is very extensive. One ﬁnds a bibliography for this purpose for example in [P¨ os-K 79], p. 135, and in [Ros 77] (or [Ros 84]). The following theorem gives only one of the classical examples for Sheﬀer-functions, which was published in 1936. Theorem 7.1.5 (Theorem of Webb; [Web 36]) The function f (x, y) := min(x, y) + 1 (mod k) is a Sheﬀer-function for Pk .

216

7 Further Completeness Criteria

Proof. A possible proof is the construction of a known generating system os-K 79]. We prove the for Pk over the function f , which one can look up [P¨ assertion here with the aid of (7.6). Superpositions over f are the functions f1 := ∆f with f1 (x) = x + 1 (mod k) and min := f1 f1 ... f1 f . k−1 times Obviously, the function f1 does not preserve any relation ∈ C1k . Thus we have f ∈ P olk for all ∈ C1k . 0 b Let ∈ Sk . Then there are a, b ∈ Ek with {(0, a), (b, 0)} ⊆ and f = a 0 1 . Consequently, f does not preserve any relation ∈ Sk . 1 Since f1 ∈ P olk is false for every ∈ Uk , we assume that f1 ∈ P olk for certain ∈ Uk is valid. Then, the equivalence classes of are equipotent, and there exist two equivalence classes [a] and [a + 1] of with the following properties: [a] = [a + 1] , b ∈ [a] , a < b, k − 1 ∈ [a] . a b+1 a Then (a, b), (a + 1, b + 1) ∈ and min = ∈ . Conseb a+1 a+1 quently, the function f does not preserve any relation of Uk . By means of Theorems 7.1.3, it is already recognizable that the number of Sheﬀer-functions is quite great. The next theorem summarizes some statements about concrete numbers and certain convergence. Theorem 7.1.6 Let ϕn (k) be the number of all n-ary Sheﬀer-functions of Pk . Then (a) ϕn (2) = 22 (b) limn→∞

n

−2

ϕn (2) 2 2n

− 22

n−1

−1

;

= 14 ;

(c) ϕ2 (3) = 3774 ([Mar 54]); (d) limk→∞

ϕn (k) k kn

= limn→∞

ϕn (k) k kn

=

1 e

for k ≥ 3, [Bai 67a].

(e) For a large k, given a randomly selected f one can decide whether f ∈ Pkn , n ≥ 2, is a Sheﬀer-function with probability almost 1 by testing the condition f (x, x, ..., x) = x for every x ∈ Ek only; [Dav 68]. Proof. (a) and (b) are easy to check. One can ﬁnd proofs for (c) –(e) in [Mar 54] and [Dav 68].

7.2 A Completeness Criterion for Surjective Functions For the case that one only considers subsets A ⊆ Pk of functions with the range of values Ek (i.e., surjective functions), our general completeness criterion (7.1) can be simpliﬁed. It holds:

7.3 Fundamental Sets

217

Theorem 7.2.1 ([Ros 70c], [Ros 75]) For k = hm with h ≥ 3 and m ≥ 1 let B∗k := { ∈ Bk | ∃n : is isomorphic to ξn }. Then, for arbitrary A ⊆ Pk [k] we have: [A] = Pk ⇐⇒ ∀ ∈ C1k ∪ Sk ∪ Uk ∪ Lk ∪ B∗k : A ⊆ P olk .

(7.8)

Proof. “=⇒” is trivial. “⇐=”: Our theorem follows from Theorem 6.1 if we can show that A is not a k−1 subset of clones P olk for every ∈ Mk ∪ ( h=2 Chk ) ∪ (Bk \B∗k ). Suppose it holds: k−1 ∃ ∈ Mk ∪ ( Chk ) ∪ (Bk \B∗k ) : A ⊆ P olk . (7.9) h=2

We distinguish three cases: Case 1: ∈ Mk . Let o be the least element of Ek in respect to . Since each function of A has the range Ek , we have then that A is a subset of P olk {o}, in contradiction to our assumption. Thus Case 1 is not possible. Case 2: ∈ Chk , 2 ≤ h ≤ k − 1. In analog mode to the proof of Lemma 7.1.2 (c), one can show that every function of A preserves the set of the central elements of the relation in this case. However, this also contradicts our assumption. Case 3: ∈ Bk \B∗k . One can describe the functions of A, then, by means of formulas of the form (5.14), where gf has the form (7.3); hence every function of A preserves the nontrivial equivalence relation (7.4). Consequently, Case 3 is not possible and, with that, the assumption (7.9) is wrong.

7.3 Fundamental Sets By Chapter 4, the set Pk1 ∪ {f }, where f is a Slupecki-function, is complete in Pk . Consequently, a subset A of Pk is complete in Pk iﬀ Pk1 ∪ {f } ⊆ [A] holds. One can now attempt to improve this completeness criterion in such a way that the set Pk1 is replaced by certain subsets of Pk1 . Deﬁnitions A subset A of Pk1 with ∀f ∈ Pk \([Pk1 ] ∪ Pk (k − 1)) : 2 [A ∪ {f }] = Pk 2

E.i., f is a Slupecki-function.

218

7 Further Completeness Criteria

is called fundamental set in Pk . If in particular, A is a group (or a semigroup) in respect to the operation , then A is also called fundamental group (or fundamental semigroup), respectively. We need the following deﬁnition to describe some fundamental sets: Deﬁnition A semigroup H ⊆ Pk1 is called t-fold transitive, t ∈ {1, 2, ..., k}, if for arbitrary pairwise distinct elements a1 , ..., at ∈ Ek and for arbitrary pairwise distinct elements b1 , ..., bt ∈ Ek there exists an h ∈ H with h(ai ) = bi for all i ∈ {1, 2, ..., t}. One ﬁnds some examples of fundamental sets in the following theorem. Theorem 7.3.1 (1) The following sets are fundamental semigroups: (a) Pk1 \Pk1 [k]; (b) a set H ⊆ Pk1 with k ≥ 5, which is (k − 1)-fold transitive ([Mal 67]); (c) a set H ⊆ Pk1 with k ∈ {3, 4} and the property that H and also H ∩ Pk1 (k − 1) are (k − 1)-fold transitive ([Mal 67]). (2) There are fundamental groups in Pk only for k ≥ 5 and examples for fundamental groups are (a) Pk1 [k]; (b) the set of all functions of Pk1 , which are even permutations on Ek .3 Proof. The statements of the theorem are easy consequences of Rosenberg’s completeness criterion 6.1 (see [P¨os-K 79], p. 132 and p. 135). One can ﬁnd proof that does not use Theorem 6.1, for the above statements in [Jab-L 80], 1.2.4–1.2.6.

3

A permutation s ∈ Sn is called even, if the inversion number I(s), i.e., the number of all pairs (i, j) with i < j and s(i) > s(j), is even.

8 Some Properties of the Lattice Lk

It is the aim of this ﬁrst section above Lk to basically elaborate on the diﬀerences between L2 , which we already completely determined in Chapter 3, and Lk for k ≥ 3. In particular, we deal with cardinality statements (both about Lk , than also about chains and antichains in Lk ) and with the embedding of Lk into Lk . Furthermore, the question, which cardinalities have the lattices L↓k (A) for maximal classes A of Pk , is clariﬁed. In the last section of chapter, there are remarks about “strategies” during the determination of “manageable” sublattices of Lk . In addition, two examples of theorems are proven as one can determine certain subsets of Lk . Notice that one can ﬁnd further properties of Lk , for instance, in [Bul-K-S-S 95], [Bul-K-S-S-S 2001], [Bul 99a,b], and [Bul 2001].

8.1 Cardinality Statements We begin with proving the existence of a subclass C ⊆ Pk for k ≥ 3, which has an inﬁnite basis. This example was published by J. I. Janov and A. A. Mucnik in 1959. Let ⎧ ∃i ∈ I : (xi ∈ {1, 2} ∧ ⎨ 1 if n (∀j ∈ J ∪ (I\{i}) : xj = 2)), (x1 , ..., xn ) := gI,J (8.1) ⎩ 0 otherwise, where I and J are certain disjoint subsets of {1, 2, ..., n}. n with fn . Except the subsequently We also denote the functions g{1,2,...,n},∅ indicated tuples, the function takes only the value 0:

220

8 Some Properties of the Lattice Lk

⎞ ⎛ ⎞ 1 1 2 2 ... 2 2 ⎜ 2 1 2 ... 2 2 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ . . . . . . . . . . . . . . ⎟ ⎜ ... ⎟ ⎟ ⎜ ⎟ fn ⎜ ⎜ 2 2 2 ... 1 2 ⎟ = ⎜ 1 ⎟ . ⎟ ⎜ ⎟ ⎜ ⎝ 2 2 2 ... 2 1 ⎠ ⎝ 1 ⎠ 1 2 2 2 ... 2 2 ⎛

(8.2)

Let B := {fi | i ∈ N} and let C be the set

n {gI,J | I, J ⊆ {1, 2, ..., n} ∧ I ∩ J = ∅}.

(8.3)

(8.4)

n≥1

Lemma 8.1.1 For the above-deﬁned sets C and B is valid: (a) C is a subclass of Pk . (b) ∀i ∈ N : fi ∈ [B\{fi }]. (c) B is an inﬁnite basis of C. Proof. Obviously, set C is closed in respect to the operations ζ, τ, δ, ∇. The completeness of C in respect to follows from ⎧ n+n −1 ⎪ ⎪ ⎪ g{a+n −1 | a∈I},{a+n −1 | a∈J} if 1 ∈ I ∪ J ∧ n ≥ 2, ⎨ n+n −1 n if 1 ∈ J ∨ (n = 1 ∧ I = ∅), gI,J gIn ,J = c0n+n −1 = g∅,∅ ⎪ ⎪ ⎪ ⎩ g n+n −1 if 1 ∈ I ∧ n ≥ 1. I ,J ∪{a+n −1 | a∈J∪(I\{1})} In addition, it results from the above formulas that set B is a generating system for C that does not continue being reducible, because of

n ∈ [{gIn ,J ∈ C | |I | = |I|}]. fn = gI,∅

Theorem 8.1.2 Let N be a countable set and let L(N) := (P(N ); ⊆). Then, for k ≥ 3, one can embed the lattice L(N) into the lattice Lk ; i.e., there exists a bijective mapping α from P(N ) in Pk , for which it holds: ∀A, B ∈ P(N ) : A ⊆ B =⇒ α(A) ⊆ α(B). In particular, for all k ≥ k ≥ 2, the lattice Lk one can embed into the lattice in Lk . Proof. W.l.o.g. let N := N. Then, for every subset T of N, one can form a subclass CT := [{fi | i ∈ T }] of the class C deﬁned in (8.4). Because of Lemma 8.1.1, we have: T = T ⇐⇒ CT = CT . Since T ⊆ T =⇒ CT ⊆ CT , we have that α : P(N) −→ Pk , T −→ CT is an embedding of L(N) into Lk .

8.1 Cardinality Statements

221

With the aid of the above theorem we can prove the statements of the following two theorems easily. In particular, the theorem shows that the crucial importance, which has the 2 in many ﬁelds of the algebra, is also provable for the function algebras. So, P2 has countable-many subclasses, however, the set Pk for k ≥ 3 already continuum-many subclasses: Theorem 8.1.3 (Theorem About the Cardinality of Lk (Pk )) It holds: (a) |Lk (P2 )| = ℵ0 . ([Pos 41]) (b) |Lk (Pk )| = c for k ≥ 3. ([Jan-M 59]) Proof. (a) was proven in Chapter 3. (b): Obviously, the set Pk is countable inﬁnite. As is generally known, the set of all subsets of Pk has the cardinality of the continuum. Consequently, there are at least continuum-many subclasses of Pk . Thus, (b) follows from Theorem 8.1.2, since |P(N)| = c (see e.g. [Lau 2004], volume 1). Theorem 8.1.4 ([P¨ os-K 79]) For k ≥ 3 there are in Lk both chains and antichains of the cardinality of the continuum. Proof. Because of Theorem 8.1.2, it suﬃces to prove that there are a chain and an antichain with the cardinality of the continuum in L(N), |N | = ℵ0 . Let Q be the set of all rational numbers and R be the set of all real numbers. For arbitrary s ∈ R we put Ms := {q ∈ Q | q ≤ s}. Then, the set {Ms | s ∈ R} forms a chain of the cardinality c in the lattice L(Q). For the purpose of constructing an uncountable antichain, let H := {(m, i) | m ∈ N ∧ i ∈ {0, 1}} and HI := {(m, 0) | m ∈ I} ∪ {(m, 1) | m ∈ I}. Obviously, {HI | I ⊆ N} is an antichain of the cardinality c in the lattice L(H) (|H| = ℵ0 ). Because of the above theorems, the concrete determination of the elements of the lattice Lk seems for k ≥ 3 to be a hopeless task. Nevertheless, one can attempt to determine certain “manageable parts” of Lk or one can try to get a general idea of the position of the “not manageable parts” of the lattice Lk . The following theorem is useful for proving that certain sublattices of Lk are countable, which is a generalization of a theorem that was proven by I. A. Mal’tsev (see [Mal 73]).

Theorem 8.1.5 (Countability Criterion; [Lau 86]) Let A be a subclass of Pk (or an algebra with countable many elements). Furthermore, there exists an order relation ≤ on A with the following three properties:

222

8 Some Properties of the Lattice Lk

(a) f ≤ g =⇒ [f ] ⊆ [g], (b) every chain is well-ordered (in respect to ≤), i.e., every chain has a minimal element (in respect to ≤), (c) every antichain (in respect to ≤) has only ﬁnitely many elements. Then, A has at most only countable many diﬀerent subclasses (or the algebra A has at most only countable many diﬀerent subalgebras). Proof. Let B be a closed subset of A and let G be the set of all minimal elements of A\B (in respect to ≤). First, we prove that G determines the set B unambiguously. Let f be an arbitrary function of A. The following cases are possible: Case 1: There exists a function g ∈ G with f ≤ g and f = g. Then f ∈ B, since for f ∈ B the function g is not a minimal element of A\B. Case 2: There exists a function g ∈ G with g ≤ f . Because of g ≤ f and by assumption (a), we have that g is a superposition over f . Thus f ∈ A\B, since g ∈ A\B. Case 3: Every function of G is not comparable with the f (in respect to ≤), or G = ∅. Because of the assumption (b) and by Zorn’s Lemma, this case is only possible for f ∈ B. Consequently, G determines the class B unambiguously. The set G is an antichain (in respect to ≤) and a ﬁnite set because of assumption (c). Every subclass B of A is also deﬁnable by a ﬁnite subset G of A. Obviously, A contains countable many elements. Consequently, there are only countable many possibilities for G and, therefore, there are only countable many closed subsets of A. Let ◦ be an operation on Ek with the properties a) ∀ a, b, c ∈ Ek : a ◦ (b ◦ c) = (a ◦ b) ◦ c, b) ∀ a, b ∈ Ek : a ◦ b = b ◦ a, c) ∃ r ≥ 1 : (∀ a ∈ Ek : a ◦ a ◦. . . ◦ a = a). (1+r)times

The properties a) - c) are e.g. fulﬁlls, if (Ek ; ◦) is an Abelean group or x ◦ y := maxω (x, y) or x◦y := minω (x, y) (ω is a partial order relation, for which exists max or min). With the help of ◦ one can deﬁne a subset K(m, ◦) of Pk for an m ∈ N as follows: K n (m, ◦), (8.5) K(m, ◦) := n≥1

where

8.1 Cardinality Statements

223

f n ∈ K n (m, ◦) :⇐⇒ m ∃ f01 , f1n1 , . . . , ftnt ∈ i=1 Pki : f (x1 , . . . , xn ) = f0 (f1 (x1 , ..., xn1 ) ◦ f2 (xn1 +1 , ..., xn1 +n2 ) ◦ . . . ◦ fn (xn1 +...+nt−1 +1 , ..., xn )) (8.6) In general, the set K0 is not closed. K0 contains, however, through suitable choice of m and ◦ a line of interesting subclasses of Pk (as for example the set of linear or quasi-linear functions) and it is valid: Theorem 8.1.6 A subclass A of Pk , for which exist an m ∈ N and an operation ◦ which the above properties a) - c) fulﬁlls, with A ⊆ K(m, ◦), has only countable many subclasses. Proof. We will deﬁne a partial order relation ≤ on K(m, ◦), which fulﬁlls the conditions (a)–(c) from the Theorem 8.1.5, whereby our theorem follows from Theorem 8.1.5. Every function of f n ∈ K(m, ◦) with the form (8.6) is well-deﬁned by tuples ϕ(f ) := (f0 , f1 , ..., ft ). Two functions f1 , f2 with ϕi (fi ) := (fi,0 , fi,1 , ..., fi,ti ), i = 1, 2, are called -equivalent, iﬀ these functions fulﬁll the following three conditions: 1) f1,0 = f2,0 ; 2) {f1,1 , ..., f1,t1 } = {f2,1 , ..., f2,t2 }; 3) If f occurs in (f1,1 , ..., f1,t1 ) exactly q times, then there exists a certain integer s, so that f occurs in (f2,1 , ..., f2,t2 ) exactly (s + r · q) times, where r denotes the smallest integer for which ◦ fulﬁlls die condition c). Obviously, the factor set K(m, ◦)/ is ﬁnite. Set {K1 , K2 , ..., Ku } := K(m, ◦)/ . On the sets K1 , K2 , ..., Ku one can deﬁne a partial order relation ≤ as follows: For functions f1 , f2 ∈ Ki , i ∈ {1, 2, ..., u}, with ϕ(fi ) = (fi,0 , ..., fi,ti ) we write f1 ≤ f2 if and only if for every f , which occurs in (f1,1 , ..., f1,t1 ) j − 1 times and occurs in (f2,1 , ..., f2,t2 ) j2 times, it holds j1 ≤ j2 . This partial order relation satisﬁes the condition (a) of Theorem 8.1.5, since one can receive the function f from g with the aid of the operations ζ, τ, ∆ by the changing of the denotation of certain r variables and by the identifying of groups of variables with r · q + 1 (q ≥ 1) elements because of the property (c) of ◦ . Obviously, the property (b) of Theorem 8.1.5 is also valid. Thus it remains to be seen that every antichain in the set K(m, ◦) in respect to ≤ has only ﬁnite-many elements. m On i=1 Pk1 one can deﬁne a total order. For ϕ(f ) = (f0 , ..., ft ), we arrange (concerning this order) that fi0 is the smallest element of the set {f1 , ..., ft }; fi1 is the smallest element of the set {f1 , ..., ft }\{fi0 }; ... and fis is the greatest element of {f1 , ..., ft }. If fij occurs in (f1 , ..., ft ) exactly bj times, j = 0, 1, ..., s, then let α(f ) := (b0 , b1 , ..., bs ). Further, let Di := {α(f ) | f ∈ Ki }, i = 1, 2, ..., u. One can deﬁne a partial order relation on Di . We set α(fi ) ≤ α(f2 ) iﬀ f1 ≤ f2 . The partial ordered sets Ki and Di are isomorphic, i = 1, 2, ..., u. Conse-

224

8 Some Properties of the Lattice Lk

quently, for the end of our proof, it suﬃces to show that an arbitrary antichain C of Di is a ﬁnite set: Obviously, if C consists only of tuples of the length 1, then C is a ﬁnite set. Suppose our assertion is proven for tuples of the length s and all tuples of C have the length s + 1. We select a certain tuple (b0 ..., bs ) in C and decompose the set C to this tuple in ﬁnite many classes as follows: Let the ﬁrst class be the set of all tuples whose ﬁrst coordinate is 1. All functions with the ﬁrst coordinate 2 belong into the second class, ..., all functions with the ﬁrst coordinate b0 − 1 belong into the (b0 − 1)-th class. All the tuples whose second coordinate 1 is and which are not tuples of the already deﬁned classes, belong in the b0 -th class, and so forth. Since the tuples of C are not comparable (i.e., if (b11 , ..., b1s ) = (b21 , ..., b2s ) are elements of C then there exist some i = j with b1i < b2i and b1j > b2j ), all elements of C\{(b0 ..., bs )} are in the above-described classes contained. By assumption, every class has only ﬁnite many elements. Hence C is a ﬁnite set.

8.2 On the Cardinalities of Maximal Sublattices of Lk In this section, we determine the cardinality of the lattice L↓k (A), where A is an arbitrary maximal class of Pk . We start with a lemma that is a consequence of [Dem-H 83], Lemma 1. Lemma 8.2.1 Let k = 2 · l with l ≥ 2, let s := (0 1)(2 3)(4 5)....(2l − 2 2l − 1) be the cycle representation of a permutation s on E2l , s := {(x, s(x)) | x ∈ (n) E2l } and yi := (0, 0, ..., 0, 1 , 0, ..., 0) for 1 ≤ i ≤ n. Furthermore, let tn i

(n ≥ 4) be an n-ary function of P olk s with the following properties: (n)

(α) (∀i ≥ 1 : tn ( yi ) = 1) ∧ tn (0, 1, 1, ..., 1) = 0; (n)

(n)

(β) ∀x ∈ E4n \{ y1 , ..., yn } : x1 ∈ {0, 2} =⇒ tn (x) = 2; (γ) ∀x ∈ Ekn \E4n : x1 ∈ {0, 2, 4, ..., 2l − 2} =⇒ tn (x1 , ..., xn ) = x1 . Let n be the n-ary relation (n)

{ yi

| 1 ≤ i ≤ n} ∪ (E4n \{0, 1}n ).

Then (a) The functions tn with n ≥ 4 are unique determined by the conditions (α)–(γ) and belong to P olk s . (b) ∀n ≥ 4 : tn ∈ P olk n . (c) ∀m = n : tm ∈ P olk n .

8.2 On the Cardinalities of Maximal Sublattices of Lk

225

Proof. (a): Because of Theorem 5.2.2.1, an arbitrary n-ary function f n ∈ P olk s is uniquely determined by the function f (a, x2 , ..., xn ) with a ∈ {0, 2, 4, ..., 2l − 2}. It is easy to check that the conditions (α)–(γ) determine the functions tn (a, x2 , ..., xn ) for arbitrary a ∈ {2, 4, ..., 2l − 2}. (b) follows from ⎛ ⎞ ⎛ ⎞ 1 0 ... 0 1 ⎜ 0 1 ... 0 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ tn ⎜ ⎝ . . . . . . . . . ⎠ = ⎝ ... ⎠ . 0 0 ... 1 1 (c): Let n = m and let r1 , ..., rn ∈ n be arbitrary. If an ri belong to E4n \{0, 1}n (n) (n) then tm (r1 , ..., rm ) ∈ E4n \{0, 1}n . If {r1 , ..., rm } ⊆ { y1 , ..., yn }, then one can show (when one distinguishes the cases m < n and m > n) that tm (r1 , ..., rn ) ∈ n . For the following lemma, we need the description of a subset T of the set 0 1 2 , S3 := P ol3 1 2 0 which is the only maximal class of type S for k = 3. A function f n belongs to T if and only if f n fulﬁlls the following four conditions: 1) f ∈ S3 ∩ P ol3 {0, 1}. 2) The restriction of f onto E2n (we write pr f ) is a function of T1,∞ ∩ M ∩ T0 . In order to be able to formulate the third and fourth condition, we need one property of the functions g m ∈ T1,∞ ∩ M ∩ T0 , which is a conclusion of Chapter 3: Every such function g m can be described through a formula above a variable alphabet and through the signs ∨ (disjunction) and ∧ (conjunction), as follows: g(x1 , ..., xm ) = xi ∨ A1 ∨ ... ∨ At , where i ∈ {1, 2, ..., m} and Aj (1 ≤ j ≤ t) are formulas of the form xj1 ∧ xj2 ∧ ... ∧ xjlj (”conjunctions”, lj ≥ 1). Since x ∨ (x ∧ y) = x as is generally known, we can assume that the sets of the indexes of the variables of the conjunctions Aj in the formula for g are not contained pairwise in each other (we speak in this case from an abbreviated DNF for g). 3) If variables, which take values of E2 in the tuple a, exist in every conjunction of the abbreviated DNF of pr f , then f (a) ∈ E2 . 4) The abbreviated DNF of pr f has the form (pr f )(x1 , ..., xn ) = xi1 ∨ xi2 ∨ ...xij ∨ A1 ∨ ... ∨ Ap , where A1 , ..., Ap are conjunctions, which consist of at least two variables in each case, over the variable set {x1 , ..., xn }\{xi1 , ..., xij }. If there are tuples

226

8 Some Properties of the Lattice Lk

a1 := (a11 , ..., a1n ), ..., ar := (ar1 , ..., arn ) of E3n with f (x1 ) = ... = f (xr ) = 0 and every of the conjunctions A1 , ..., Ap contains variables xi1 , xi2 , ..., xij , which only take values of {0, 1} in every tuple a1 , ..., ar , then there exists an s ∈ {1, ..., j} with a1is = ... = aris = 0. The following theorem was proven by S. S. Marcenkov: Theorem 8.2.2 ([Mar 83]) The set T is a closed set and contains a subclass, which has an inﬁnite basis. Proof. One can ﬁnd the proof for [T ] = T in [Mar 83]. In the following, we construct a subclass of the form [Φ] with Φ := {ϕ2 , ϕ3 , ...} ⊆ T, where the functions ϕn have the property ϕn ∈ [Φ\{ϕn }] for every n ≥ 2; i.e., [Φ] is a subclass of T with inﬁnite basis. For every n ≥ 2 let ϕn be a function of T 2n , which has the following three properties: (pr ϕn )(x1 , ..., xn , y1 , ..., yn ) := x1 ∨ ... ∨ xn ∨ y1 · y2 · ... · yn

(8.7)

(· := ∧), ϕn (1, 0, ..., 0, 2, 0, ..., 0) = ϕn (0, 1, 0, ..., 0, 0, 2, 0, ..., 0) = ... = n

n

ϕn (0, 0, ..., 1, 0, 0, 0, ..., 2, 0) = ϕn (0, 0, ..., 0, 1, 0, 0, 0, ..., 2) = 0, n

(8.8)

n

and ϕn (a) = 1 holds for every tuple a := (a1 , ..., an , an+1 , ..., a2n ), which fulﬁlls the condition 1 ∈ {a1 , ..., an } =⇒ (a1 , ..., an ) ∈ {0, 1}n and which is diﬀerent from the tuples that occur in (8.8). Because of condition ϕn ∈ T ⊆ S, many more values of the function are given through the above determination. It is easy to check that one can deﬁne the function, so on remaining tuples, that ϕn ∈ T is valid. How this happens is not important for the below-indicated considerations. Some remarks on the deﬁnition of the functions ϕn : If the elements ai of the tuple a := (a1 , ..., an , an+1 , ..., a2n ) take only, at most, two diﬀerent values, then ϕn (a) is already determines by (8.7). Let An be the set of all tuples of E32n which occur in the condition (8.8). Let now a ∈ E32n \ An with |{a1 , ..., a2n }| = 3. Suppose a1 = ... = an . By ϕn ∈ S we can assume w.l.o.g. a1 = ... = an = 1. Then by 3) we have ϕn (a) ∈ E2 . Because of 4) the case ϕn (a) = 0 is not possible, so that ϕn (a) = 1 holds. Let |{a1 , ..., an }| = 2. W.l.o.g. {a1 , ..., an } = E2 . Then the conditions a ∈ An , 3) and 4) imply ϕn (a) = 1.

8.2 On the Cardinalities of Maximal Sublattices of Lk

227

If |{a1 , ..., an }| = 3, then the deﬁnition of the set T does not supply any determination for ϕn (a). Further, we remark that (a1 , ..., an ) ∈ E2n and ϕn (a) = 0 are only possible if either a ∈ An or a1 = ... = an = 0 holds. From that, the correctness of the fourth property results for the functions ϕn . One easily checks that the functions ϕn also fulﬁll the remaining three properties from the deﬁnition of set T . Suppose ϕn is a superposition over {ϕm | m = n}. We consider a formula V (x1 , ..., xn , y1 , ..., yn ) with function symbols of Φ\{ϕn } and with minimal complexity (related to the arity of their functional symbols), which describes the function v(x1 , ..., xn , y1 , ..., yn ), where v(x) = 0 for all x ∈ An and v fulﬁlls the equation (pr v)(x1 , ..., xn , y1 , ..., yn ) = x1 ∨ ... ∨ xn ∨ y1 · ... · yn . One can ﬁnd a such formula, since by assumption ϕn (x) = 0 for all x ∈ An and ϕn fulﬁlls (8.7). We remark that v ∈ T . Let V (x1 , ..., xn , y1 , ..., yn ) = ϕm (f1 (x1 , ..., xn , y1 , ..., yn ), ..., f2m (x1 , ..., xn , y1 , ..., yn )), where {f1 , ..., f2m } ⊂ [Φ\{ϕn }]. Suppose there exists a function fα ∈ {f1 , ..., f2m } with fα (x) = 0 for every x ∈ An . Obviously, pr v = pr f1 ∨ pr f2 ∨ ... pr fm ∨ ... ∨ pr f2m .

(8.9)

Consequently, the abbreviated DNF of the function pr ft has the form xi1 ∨ ... ∨ xis ∨ xj1 · ...xjt · yl1 · ... · ylu ,

(8.10)

where {i1 , ..., is } ∩ {j1 , ..., jt } = ∅ and u = n for t = 0. Let s < n. Let B be the set of all tuple x := (x1 , ..., xn , y1 , ..., yn ) ∈ An with xi1 = xi2 = ... = xis = 1. If t > 0 then xjt = 0 for each x ∈ B. If t = 0 then we have u = n, {l1 , ..., lu } = {1, ..., n} and for every yi of an arbitrary tuple (x1 , ..., xn , y1 , ..., yn ) ∈ B with i ∈ {i1 , ..., is } we have yi = 0. Consequently, in this case, every conjunction of the abbreviated DNF, which deﬁnes the function pr fα , contains a variable that has the value 0 in every tuple of B. Since fα (x) = 0 for every x ∈ B and fα ∈ T , this is a contradiction to the property 3), because each tuple of the set B has to stand no 0 at the id -th place for 1 ≤ d ≤ s. Consequently, we have: s = n, {i1 , ..., is } = {1, ..., n}, t = 0, u = n, {l1 , ..., lu } = {1, ..., n} and pr fα = x1 ∨ ... ∨ xn ∨ y1 · ... · yn , whereby the function fα has the properties of v, which contradict the presupposed minimality of the formula V . Consequently, there is no i ∈ {1, ..., n}

228

8 Some Properties of the Lattice Lk

with fi (x) = 0 for every x ∈ An . Moreover, the abbreviated DNF of the functions pr f1 , ..., pr fm do not have the form (8.10). Therefore, and by property 3), {fi (x) | x ∈ An } = E2 for every i ∈ {1, ..., n}. For each i ∈ {1, ..., m} we choose a tuple ci ∈ An with f (ci ) = 1. Set C := {c1 , ..., cm }. Then, for arbitrary c ∈ C is valid: (f1 (c), ..., fm (c)) ∈ E2m , 1 ∈ {f1 (c), ..., fm (c)}.

(8.11)

Assume there exists an i ∈ {m + 1, ..., 2n} with {fi (c) | c ∈ C} ⊆ E2 . Then one obtains a contradiction to the property 4) of the functions ϕm , if one considers tuples of the form (f1 (a), ..., f2n (a)) with a ∈ C. Consequently, 2 ∈ {fi (c) | c ∈ C} for every i ∈ {m + 1, ..., 2n}. Now let m < n. Because of the property 3) we have for each f ∈ T : f (a) = 2 if and only if every variable of f which occurs in a conjunction of the abbreviated DNF of pr f has the value 2 in the tuple a. Every tuple of C has, however, the value 2 only at exactly a place. Therefore the abbreviated DNF of every function of {pr fm+1 , ..., pr f2m } contains a certain element of {y1 , ..., yn } as a unary conjunction. By (8.9) the product of these variables induces a conjunction of the form yj1 · ...yjt , where t < n (because of m < n), in the abbreviated DNF of the function pr v. Consequently, the case m < n is not possible. Let m > n. Since C ⊆ An , there exists a tuple a with fi (a) = 2 for at least two i ∈ {m + 1, ..., 2m}. By (8.11) and by the deﬁnition of the function ϕm this implies ϕm (f1 (a), ..., f2m (a)) = 1, which is a contradiction to v(a) = 0, however. After these preparations, the main result of this section can be formulated and proven: Theorem 8.2.3 ([Dem-H 83], [Mar 83]) Let k ≥ 3 and let A be an arbitrary maximal class of Pk . Then |L↓k (A)| = c if and only if A is not a maximal class of the type L. If A of type L, then we have k = pm , where p ∈ P and m ∈ N, and it holds < ℵ0 if m = 1, |L↓k (A)| = ℵ0 otherwise. Proof. Because of Theorem 6.1, we can assume that A = P olk with ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk . For proof of |L↓k (A)| = c for ∈ Mk ∪ Uk ∪ Sk ∪ Ck ∪ Bk , showing that the class A has a subclass with inﬁnite basis suﬃces. If ∈ Mk ∪ Sk ∪ Lk there exists a 2-element subset E ⊂ Ek with Pk,E ⊂ A. Then, with the aid of the Lemma 8.1.1 one can ﬁnd a subclass of A with an

8.3 Some Strategies for the Determination of Sublattices of Lk

229

inﬁnite basis easily. If A has the type M, then one can change the example from the Lemma 8.1.1 in the following manner: Instead of 0, 1, 2 one chooses a, b, c of Ek with a < b < c and with the property ∀x ∈ Ek : b < x ≤ c =⇒ x = c. Consequently, A has also for ∈ Mk a subclass with an inﬁnite basis. If ∈ Sk , the construction of a corresponding example is more complicated. Let := s ∈ Sk , where s is deﬁned as in Section 5.2.2. If s has only cycles of the length 2, then P olk s has a subclass with a inﬁnite basis because of Lemma 8.2.1. In the following we assume that the cycles of s have a length ≥ 3. If k = 3 then |L↓3 (P ol3 s )| = c results from Theorem 8.2.2. If k > 3 then we can assume w.l.o.g. that the cycle representation of s contains a cycle of the form (0 1 2 ...). In generalization of our example from the proof of the Theorem 8.2.2 one can deﬁne then 2n-ary functions ψn , which agree with the above functions ϕn on the set E32n and which are otherwise so deﬁned that ψn ∈ P olk s holds and that [{ψ2 , ψ3 , ...}] has an inﬁnite basis. We notice that one can ﬁnd another proof for |L↓k (P olk s )| = c in the case of k ≥ 4 in [Dem-H 83]. The remaining statements of our theorem result from Chapter 5, Theorem 8.1.6 and Chapter 13. In Chapter 13 one can ﬁnd all subclasses from A for the case that ∈ Lk and k is a prime number. Further, one ﬁnds in Chapter 13 an example of a class from linear functions, which has no basis if k is not square-free. One can generalize this example easily then for an example of a class from quasi-linear functions without basis, if k = pm and m ≥ 2. For k = 3 we will show in Chapter 15 that there are exactly 7 submaximal classes B with B = L3 and |L↓3 (B)| = ℵ0 .

8.3 Some Strategies for the Determination of Sublattices of Lk Because of Theorem 8.1.3, there is little hope that one can ever describe all elements of Lk for k ≥ 3. Nevertheless, one can attempt to get a certain idea of the construction of lattice Lk through the restriction to certain “manageable” sublattices. For example, one can restrict oneself to ﬁnitely generated subclasses of Pk . Subsequently, an enumeration is declared by sets for which there is a line of article in the literature and which characterize certain “sections” of Lk quite well: (a) The lattice of the subclasses from linear or quasi-linear functions (see Chapter 13, 15.3 and e.g. [Sze 79], [Sze 80], [Sza-S 80], [Bag-D 82], [Sze 86], [Bul 98a,b]. (b) The set of all commutator sets or subclasses of autodual functions (e.g. [Har 74-76], [Mar 79], [Mar-D-H 80], [Csa-G 80], [Csa 80], [Mac-R 2004]) (c) subclasses of

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8 Some Properties of the Lattice Lk

Uω :=

{f n ∈ Pk | (∃i : xi = ω) =⇒ f (x1 , ..., xi , ..., xn ) = ω}

n≥1

(this set is isomorphic to the set of all partial functions of Pk−1 ) (see e.g. [Mal 66], [Ros 88], [Ros-H 87,89,91] and Chapter 20) (d) subclasses of Pk , which contain certain subsemigroups of (Pk1 ; ) (see e.g. Chapter 4, [Had-R 94], [Kro 99]) (e) subclasses of Pk , which contain a “near unanimity function” or the discriminator z, if x = y, t(x, y, z) := x, otherwise (see e.g. [Bak-P 75], [Wer 78], [Csa-G 81]). (f) The set of the inverse images of subclasses of Pl in respect to certain mappings (see Chapter 12). (g) The set of all intersections from already determined classes (e.g. the maximal classes). (In this connection one can use for example the results on basis classiﬁcations from [Miy 71], [Miy-S-L-R 87], [Miy 88], and [Sto 87].) (h) clones with certain properties (e.g. minimal clones, solidifyable clones etc.), see e.g. Chapter 19 and [Den-W 2000]. For some further chapters we still need two theorems, which are the basis for the determination of certain sublattices of Pk . The following theorem is an insigniﬁcant generalization of the Theorem of Baker-Pixley which is proven in analog mode to [Wer 78] here. Theorem 8.3.1 Let A be a subclass of Pk , which contains a “near unanimity function” dm for certain m ≥ 2, i.e., an m + 1-ary function with the property dm (x1 , ..., xm+1 ) = x, if there is an i ∈ {1, ..., m + 1} with x1 = ... = xi−1 = xi+1 = ... = xm+1 = x. Then, A = P olk Inv m A holds, i.e., for ﬁxed numbers m and k there are only ﬁnite-many subclasses A ∈ Lk with dm ∈ A. Proof. Obviously, A ⊆ P olk(Invkm A). Let f n ∈ P olk (Inv m A). Further, for T ⊆ Ekn denote fT a function of Pk , which agrees with the function f on tuples from T . We prove by induction on |T | =: t ≥ m that there exists a certain function fT ∈ A to every T ⊆ Ekn ; through that f ∈ A for t = k m would be shown and A = P olk (Invkm A) would be proven. I) t = m: Let T = {(ai1 , ai2 , ..., ain ) | i = 1, 2, ..., m},

8.3 Some Strategies for the Determination of Sublattices of Lk

⎛

231

⎞

a11 a12 ... a1n ⎜ a21 a22 ... a2n ⎟ ⎟ := ⎜ ⎝ ................. ⎠ am1 am2 ... amn and let be the least relation of Inv m A with ⊆ , i.e., ⎛ ⎛ ⎞ a11 a12 ... a1n a1 ⎜ ⎜ a2 ⎟ ⎜ ⎟ ∈ \ ⇐⇒ ∃q ∈ An : q ⎜ a21 a22 ... a2n ⎝ ................. ⎝ ... ⎠ am am1 am2 ... amn

it holds ⎞ ⎛

⎞ a1 ⎟ ⎜ a2 ⎟ ⎟=⎜ ⎟ ⎠ ⎝ ... ⎠ . am (8.12) Since f n preserves the m-ary invariants of A, we have f () ∈ . If f () belongs to , there is a certain function in A which agrees with the function f on tuples from T . If also on the other hand, f () ∈ \ is valid, then there is because of (8.12) also a certain function fT in A. Thus the above assertion was proven for t = m. II) t −→ t + 1: Suppose for every T ⊆ Ekn with |T | = t, t ≥ m, there is a function fT ∈ A with fT (a) = f (a) for all a ∈ T . Let now T = {a1 , a2 , ..., at+1 } and |T | = t + 1. Because of our assumption, there exist functions fi ∈ An with fi (a) = f (a) for all a ∈ T \{ai }, i = 1, 2, ..., m + 1. Since dm ∈ A, we have through dm (f1 (x), f2 (x), ..., fm+1 (x)) a function in A, which agrees with f on tuples aus T . In Chapter 12, we generalize this theorem for Pk,l . The last theorem of this chapter, which is a slight generalization of Lemma 3.2.3.1, shows that there are sublattices of the form L↓k (A) of Pk whose elements one can receive by means of intersection formations of A with classes that are not subsets of A. We will use this theorem in Chapters 14 and 18. In preparation for the announced theorem, some notations and properties are given: Let s be a permutation of Pk deﬁned by s(x) := x + 1 (mod k), let s := {(x, s(x)) | x ∈ Ek } and put Sk := P olk s . Analogously to the proof of Theorem 5.2.2.1, one can prove that a function f n ∈ Pk belongs to Sk if and only if there exists an (n − 1)-ary function F of Pk := Pk ∪ Ek 1 with the following property: f (x1 , ..., xn ) =

k−1

ji (x1 ) · si (F (sk−1 (x2 ), ..., sk−i (xn ))) (mod k)

(8.13)

i=0

(si (x) := x+i (mod k); F (x1 , ..., xn−1 ) := f (0, x1 , ..., xn−1 )). Because of (8.13) it is possible to deﬁne a bijective mapping α of Sk (⊆ Pk ) onto Pk as follows: α : f −→ F. 1

In this case, the elements of Ek are the nullary functions of Pk .

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8 Some Properties of the Lattice Lk

Theorem 8.3.2 The mapping α has the following properties: τ, ∆, ∇ and (a) For the operations ζ, deﬁned by )(x1 , ..., xn ) = f (x1 , x3 , x4 , ..., xn , x2 ), (ζf ( τ f )(x1 , ..., xn ) = f (x1 , x3 , x2 x4 , ..., xn ), )(x1 , ..., xn−1 ) = f (x1 , x2 , x2 , x3 , ..., xn−1 ), (∆f )(x1 , ..., xn+1 ) = f (x1 , x3 , x4 , ..., xn+1 ) und (∇f (f g)(x1 , ..., xm+n−2 ) = f (x1 , g(x1 , ..., xm ), xm+1 , ..., xm+n−2 ) (n, m ≥ 2), it holds α( γ f ) = γ(α(f )) for every γ ∈ {ζ, τ, ∆, ∇} and α(f g) = τ, ∆, ∇, ) is isomorphic to the algebra α(f ) α(g); i.e., the algebra (Sk ; ζ, (Pk ; ζ, τ, ∆, ∇, ). (b) For every subclass A (= ∅) of Sk , α(A) is a subclass of Pk , and it holds α(A) ⊆ Sk , A ⊆ α(A) and α(A) ∩ Sk = A. Proof. (a) is easy to check. (b): Let A be a subclass of Sk . By (a) we have that α(A) is also a closed set. Assume α(A) ⊆ Sk . Then we have F (x2 , ..., xn ) = si (F (sk−i (x2 ), ..., sk−i (xn ))) for every i ∈ {0, 1, ..., k − 1} and for every f n ∈ A. Thus by (8.13) we get that the variable x1 is ﬁctitious for every function f n ∈ A. However, this is not possible. Hence α(A) ⊆ Sk holds. Let f n ∈ A. Then ∇f ∈ A and therefore α(∇f ) = f ∈ α(A), i.e., A ⊆ α(A). If f n ∈ Sk ∩ α(A), we have ∆(α−1 f ) = f ∈ A and thus Sk ∩ α(A) ⊆ A. From this it follows that A = Sk ∩ α(A), since A ⊆ α(A) and A ⊆ Sk .

9 Congruences and Automorphisms on Function Algebras

As generally known, one can characterize the homomorphic mappings from the algebra A := (A; F ) into the algebra B through the congruences of A (see Chapter 4 of Part I). Therefore, we will not deal here with the homomorphisms of function algebras but only with the congruences on the subclasses on Pk . After some basic concepts are deﬁned, all congruences on the subclasses of P2 are determined in this chapter. It is the aim of the following sections, then, to specify the general homomorphism theorem for function algebras and to ﬁnd statements on the number of the congruences on a subclass of Pk by determining some general properties of the congruences on subclasses of Pk . Then, all congruences are determined for selected classes (among other things these are the maximal classes of Pk and certain classes from linear functions). Criteria with which one can discuss whether, on a subclass of Pk , only trivial congruences exist are also derived. A later section deals with the connection between clone congruences and the fully invariant congruences on free algebras. The theorem found in this case has interesting inferences and is a bridge between certain investigations in the Universal Algebra and certain investigations in the theory of the function algebras. At the end of this chapter, are some results on automorphisms. It is proven that Pk , the subclasses of P2 , and the maximal classes of Pk have only inner automorphisms. The starting point and the basis of the subsequently compiled results was an article of A. I. Mal’tsev from the year 1966 (see [Mal 66]). Important contributions to the topic of Chapter 9 are also performed by V. V. Gorlov and I. A. Mal’tsev.

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9 Congruences and Automorphisms on Function Algebras

9.1 Some Basic Concepts and First Properties Deﬁnition A congruence on a subclass A of Pk is an equivalence relation κ on A, which fulﬁlls the following condition: ∀(f, g), (s, t) ∈ κ ∀α ∈ {ζ, τ, ∆, ∇} : (αf, αg) ∈ κ) ∧ (f ∗ s, g ∗ t) ∈ κ. Let Con A be the set of all congruences on the class A ∈ Lk . In the following, we write f ∼ g (κ) instead of (f, g) ∈ κ and we call such functions f and g κ-congruent. Furthermore, let κ(n) := κ ∩ (An × An ). If one wants to realize whether an equivalence relation on a subclass of Pk is compatible with the operation , it is helpful to use the equivalence given in the following lemma. Lemma 9.1.1 Let A be a subclass of Pk and let κ be an equivalence relation on A. Then, the following two conditions are equivalent: (1) ∀(f, g), (u, v) ∈ κ : (f u, g v) ∈ κ (2) ∀(f, g) ∈ κ ∀t ∈ A : ((f t, g t) ∈ κ ∧ (t f, t g) ∈ κ). Proof. “(1) =⇒ (2)” follows from the reﬂexivity of κ. “(2) =⇒ (1)”: Let (f, g), (u, v) ∈ κ. Choosing t = u in (2) and then t = g, one receives: (f u, g u) ∈ κ and (g u, g v) ∈ κ. Since κ is transitive, this implies (1). Next to the trivial congruences κ0 := {(f, f ) | f ∈ A} and κ1 := A × A, the following congruence, which we want also to call trivial congruence, exists on every subclass of Pk : κa := {(f, g) ∈ A2 | af = ag}. Later we obtain the following theorem as a consequence of Lemmas 9.7.1 and 9.2.1: Theorem 9.1.2 ([Mal 66]) The (trivial) congruences κ0 , κa and κ1 are the only congruences on Pk .

9.2 Congruences on the Subclasses of P2

235

It is easy to check that ∀A ∈ Lk ∀κ ∈ Con A : (κa ⊂ κ ⊆ κ1 =⇒ κ = κ1 ). Consequently, the following partition of Con A, where A is a subclass of Pk , is reasonable: Con1 A := {κ ∈ ConA | κ ⊆ κa } (congruences of the ﬁrst kind, non-arity congruences) and

Cona A := {κ ∈ ConA | κ ⊆ κa } (congruences of the second kind, arity congruences).

9.2 Congruences on the Subclasses of P2 We take over the notations from Chapter 3. Further, for an arbitrary subclass A of P2 set: κc := {(f, g) ∈ κa ∩ (A × A) | ∃a ∈ E2 : f (x) = g(x) + a}, µ := {(f, g) ∈ κa ∩ (A × A) | f = g =⇒ {f, g} ⊆ C}, κa := (C0 × C0 ) ∪ (C1 × C1 ) (More generally, the relation κa is deﬁned in (9.1)). It can easily be shown that κc is a congruence on the class A, if A ⊆ L; µ is a congruence on A, if A = [A] ⊆ [P21 ]; and κa is a congruence on the set C. Lemma 9.2.1 Let A ⊆ Pk be a clone. Then κ1 is the only non-arity congruence on A. Proof. Let κ be a congruence on A with κ ⊆ κa . Then, there are certain f n , g m ∈ A with (f, g) ∈ κ and n > m. Consequently, we have (∆(e22 ∗ ∆n−2 f ), ∆(e22 ∗ ∆n−2 g)) = (e22 , e11 ) ∈ κ and therefore

t−1 1 t t−1 (e22 ∗ et−1 t−1 , e1 ∗ et−1 ) = (et , et−1 ) ∈ κ

for t = 2, 3, ... . Then, since κ is an equivalence relation, it follows that ∀s, t ∈ N : (ess , ett ) ∈ κ. This implies κ = κ1 , since for every function hq ∈ A (e22 ∗ h, e11 ∗ h) = (eq+1 q+1 , h) ∈ κ holds.

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9 Congruences and Automorphisms on Function Algebras

Lemma 9.2.2 Let A ⊆ Pk be a clone and κ ∈ Con A with κ ∩ {(e11 , c10 ), (e11 , c11 ), (e21 , e22 )} = ∅. Then κa is the only arity congruence on A. Proof. Assume κ is an arity congruence on the clone A and f n is an arbitrary function of A. Put e := e11 and ca := c1a for a ∈ {0, 1}. Then, (e, ca ) ∈ κ implies (e∗f, ca ∗f ) = (f n , cna ) ∈ κ and (e21 , e22 ) ∈ κ implies (e21 (f (x), x1 ), e22 (f (x), x1 )) = (f (x), en1 (x)) ∈ κ. Since κ is an equivalence relation, κ = κa follows from that. With the aid of Lemmas 9.2.1 and 9.2.2, one can easily prove the following lemma. Lemma 9.2.3 The congruences κa , µ and κc are the only nontrivial congruences on a subclass of [P21 ]. Lemma 9.2.4 Let A be a subclass of P2 that contains ∧ or ∨. Then A has only trivial congruences. Proof. W.l.o.g. let ∧ ∈ A. Suppose there is a nontrivial congruence κ on A. Then, because of Lemma 9.2.1, we have κ0 ⊂ κ ⊂ κa and there exists κ-congruent functions f n , g n ∈ A with f = g. For f and g we distinguish two cases: Case 1: {∆n−1 f, ∆n−1 g} ∩ {c0 , c1 , e} = ∅. Because of (f, g) ∈ κ \ κ0 one can form diﬀerent unary κ-congruent functions f , g when one identiﬁes variables and replaces certain variables of f, g by the function h ∈ {c0 , c1 , e} ∩ A. W.l.o.g. (f , g ) ∈ {(e, c0 ), (e, c1 ), (e, e), (c0 , c1 )}. If (f , g ) = (e, e) ∈ κ, then, by ∧ ∈ A, the tuple (e, c0 ) = (e ∧ e, e ∧ e) belongs to κ. If (c0 , c1 ) ∈ κ then (e ∧ c0 , e ∧ c1 ) = (c0 , e) ∈ κ. Consequently, we have {(e, c0 ), (e, c1 )} ∩ κ = ∅ and by Lemma 9.2.2 κ = κa , in contradiction to the assumption. Case 2: ∆n−1 f = ∆n−1 g = e. Since f = g, there is an a ∈ E2n \{0, 1} with (w.l.o.g.) f (a) = 0 and g(a) = 1. When one identiﬁes certain variables of the functions f and g, one receives from (f, g) ∈ κ the existence of two binary κ-congruent functions f , g with f (0, 0) = f (0, 1) = 0, f (1, 1) = 1 and

g (0, 0) = 0, g (0, 1) = g (1, 1) = 1,

i.e., (f , g ) ∈ {(∧, e22 ), (∧, ∨), (e21 , e22 ), (e21 , ∨)}. If (∧, e22 ) ∈ κ then (τ ∧, τ e22 ) = (∧, e21 ) and thus (by the transitivity and symmetry of κ) we have (e22 , e21 ) ∈ κ. Analogously, (e21 , ∨) ∈ κ implies (e22 , e21 ) ∈ κ. If (∧, ∨) ∈ κ then x = (x ∧ y) ∨ x ∼ (x ∧ y) ∧ x = x ∧ y (κ). Therefore, (e21 , e22 ) ∈ κ in Case 2. A contradiction from this results with the aid of Lemma 9.2.2.

9.2 Congruences on the Subclasses of P2

237

Lemma 9.2.5 Let A be a subclass of S, which contains the function h2 deﬁned by h2 (x, y, z) := xy ∨ xz ∨ yz. Then A has only trivial congruences. Proof. Assume κ is a nontrivial congruence on A. Then, by Lemma 9.2.1, there exists two κ-congruent functions f n , g n ∈ A with f (a) = 0 and g(a) = 1 for a certain a ∈ E2n . If a ∈ {0, 1} then e belongs to A and {∆n−1 f, ∆n−1 g} = {e, e} holds, i.e., (e, e) ∈ κ. This implies x = h2 (x, y, y) ∼ h2 (x, y, y) = y (κ) and then (e21 , e22 ) ∈ κ. Therefore, by Lemma 9.2.2, κ = κa holds, if a ∈ {0, 1}. If a does not belong to {0, 1}, one can form two binary κ-congruent functions f and g when one identiﬁes certain variables of f and of g. Since S 2 = {e21 , e22 , e21 , e22 }, this implies (e21 , e22 ) ∈ κ and thus κ = κa by Lemma 9.2.2. This is, however, a contradiction to the assumption. Theorem 9.2.6 Let A ⊆ L be a subclass, which contains the function r deﬁned by r(x, y, z) = x + y + z. Then, κc is the only nontrivial congruence on A. Proof. Let κ be an arity congruence on A. The following two cases are possible: Case 1: κ0 ⊂ κ ⊆ κc . In this case, there exists κ-congruent functions f and g with f (x) = g(x) + 1. Consequently, because of r ∈ A, we have : x1 = f (x) + f (x) + x1 ∼ g(x) + f (x) + x1 = x1 + 1 (κ), i.e., (e, e) ∈ κ. Therefore, κ = κc . Case 2: κ ⊆ κc . Then, there are two κ-congruent functions f n , g n with f (x) = a0 + a1 x1 + ... + an xn , g(x) = b0 + b1 x1 + ... + bn xn and (w.l.o.g.) a1 = 0 and b1 = 1. Thus r(y, f (x, y, ..., y), f (y, y, ..., y)) = y ∼ r(y, g(x, y, ..., y), g(y, y, ..., y)) = x (κ). Consequently, κ = κa by Lemma 9.2.2. The following theorem is a consequence of the Post’s graph and of the six above lemmas: Theorem 9.2.7 (Congruence Theorem for P2 , [Gor 73]) The only nontrivial congruences on a subclass of P2 are µ (on classes A with A ⊆ [P21 ]), κc (on classes B with B ⊆ L) and κa (on C). An arbitrary subclass (= ∅) of P2 has at least two and at most ﬁve diﬀerent congruences. The congruence lattices are given in Figure 9.1.

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9 Congruences and Automorphisms on Function Algebras

κ1 q

κ1 q

κ1 q

κa q

κa q

κa q

κ1 q @ @q κ a κa q @

κc q

κc q

µ q

κ0 q

κ0 q

µ q

κ0 q

κ0 q [P21 ]

L, L ∩ S

I ∪C

C

κ1 q

κ1 q

κ0 q

κa q κ0 q

C0 , C1

in all remaining cases

Fig. 9.1. Congruence lattices of the subclasses of P2

9.3 Characterization of the Non-Arity Congruences In this section let A be a subclass of Pk . Deﬁnition We say that two functions f m and g n of A ⊆ Pk are associated (in A), or we say that f is associated with g (in A), written f 1 g, iﬀ there exist functions u1 , u2 , ..., um+n ∈ A1 with f (u1 (x), u2 (x), ..., um (x)) = g(un+1 (x), un+2 (x), ..., um+n (x)). Obviously, each function f ∈ A is associated with each function αf for α ∈ {ζ, τ, ∆, ∇}) and with f g, where g ∈ A. Let p0 , pt+1 ∈ A. We deﬁne an equivalence relation κa on A as follows: p0 ∼ pt+1 (κa ) :⇐⇒ ∃p1 , p2 , ..., pt ∈ A : p0 1 p1 1 p2 1 ... 1 pt 1 pt+1 .

(9.1)

9.3 Characterization of the Non-Arity Congruences

239

Lemma 9.3.1 The relation κa deﬁned by (9.1) is a congruence on A = [A] ⊆ Pk . Proof. Obviously, κa is an equivalence relation on A. To prove the compatibility of κa with the superposition operations, let p0 ∼ pt+1 (κa ) and q0 ∼ qs+1 (κa ) be arbitrary, i.e., there exist functions pi (i = 1, 2, ..., t), qj (j = 1, 2, ..., s) of A with p0 1 p1 1 p2 1 ... 1 pt 1 pt+1 and q0 1 q1 1 q2 1 ... 1 qs 1 qs+1 . Because of αp0 1 p0 1 p1 1 p2 1 ... 1 pt 1 pt+1 1 αpt+1 (α ∈ {ζ, τ, ∆, ∇}) we have αp0 ∼ αpt+1 (κa ). Furthermore, p0 q0 ∼ pt+1 qs+1 (κa ), since p0 q0 1 p0 1 p1 1 p2 1 ... 1 pt+1 1 pt+1 qs+1 . Thus κa is a congruence on A. Lemma 9.3.2 Every equivalence relation κ with κa ⊆ κ is a congruence on A. Proof. Let f ∼ g (κ) and u ∼ v (κ) be arbitrary. Then f ∼ αf (κa ) g ∼ αg (κa ) (κ) (α ∈ {ζ, τ, ∆, ∇}) and

f u ∼ f (κa ) g v ∼ g (κa ) (κ)

.

Because of κa ⊆ κ we have thus αf ∼ αg (κ) and f u ∼ g v (κ). Lemma 9.3.3 Let κ be a non-arity congruence on a subclass A of Pk . Then κa ⊆ κ. Proof. Let h1 ∈ A1 be arbitrary and let hi be the function ∇i−1 h1 with n ∈ N\{1}. We prove (9.2) ∀i, j ∈ N : hi ∼ hj (κ).

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Since κ is a non-arity congruence, there exist κ-congruent functions f n and g m of A with n > m. Consequently, we have h2 (∆n−2 f ) = h3 ∼ h2 (∆n−2 g) = h2 (κ), h2 = ∆h3 ∼ ∆h2 = h1 (κ), ∇h3 = h4 ∼ ∇h2 = h3 (κ), etc. Thus, (9.2) holds. Next we prove ∀f n ∈ A ∀g1 , ..., gn ∈ A1 : f (x1 , ..., xn ) ∼ f (g1 (x1 ), ..., gn (x1 )) (κ).

(9.3)

Let f n ∈ A be arbitrary and set f1 := ∇f . Then: f2 := f1 h1 ∼ f1 hn+1 =: f3 (κ) and f4 := ∆f2 ∼ ∆f3 =: f5 (κ), where f4 (x1 , ..., xn ) = f (x1 , ..., xn ) and f5 (x1 , ..., x2·n ) = f (xn+1 , ..., x2·n ). Consequently, we have for all g1 , ..., gn ∈ A1 : (ζ(...((ζ((ζ((ζ n+1 f5 ) gn )) gn−1 )) gn−2 )...) g1 ∼ (ζ(...((ζ((ζ((ζ n+1 f4 ) gn )) gn−1 )) gn−2 )...) g1 (κ) which can be written more brieﬂy in the form f4 (g1 (x1 ), ..., gn (x1 )) = f (g1 (x1 ), ..., gn (x1 )) ∼ f5 (g1 (x1 ), ..., gn (x1 ), x1 , ..., xn ) = f (x1 , ..., xn ) (κ). Thus (9.3) is right. Finally, we prove the following fact: ∀f, g ∈ A : (f ∼ g (κa ) =⇒ f ∼ g (κ)). If f ∼ g (κa ), there are functions p1 , ..., pt+1 ∈ A with p0 := f 1 p1 1 p2 1 ... 1 pt 1 pt+1 := g. Then, by deﬁnition (9.1) and by property (9.3), we obtain p0 := f ∼ p1 ∼ p2 ∼ ... ∼ pt ∼ pt+1 := g (κ), i.e., f ∼ g (κ). Thus κa ⊆ κ.

(9.4)

9.3 Characterization of the Non-Arity Congruences

241

Theorem 9.3.4 (I. A. Mal’tsev’s Theorem, [Mal 76]) Let A be a subclass of Pk . Then A has only ﬁnite many non-arity congruences. These congruences are equivalence relations on A with κa ⊆ κ. In particular, for arbitrary subclasses A of Pk it holds: Con1 A = {κ1 } ⇐⇒ κa = κ1 and

1 ≤ |Con1 A| ≤ µ(|A1 |),

where µ(n) denotes the number of possible equivalence relations on an nelement set. Proof. The statements of our theorem result directly from Lemmas 9.3.1– 9.3.3 and the fact that every function f ∈ A is κa –congruent to a certain unary function of A. We still notice that the above theorem is not valid if the operation ∇ is renounced. One ﬁnds an example for this purpose in [Gor 73]. The relation κ1 is the only non-arity relation on many classes. In Lemma 9.2.1, it was shown that this is valid for all clones. In generalizing Lemma 9.2.1 we get: Lemma 9.3.5 ([Mal 76]) Let A be a subclass of Pk . If there is a unary function u ∈ A with (a) ∀f ∈ A : u f = f (i.e., “u is a left unit”) or (b) ∀g ∈ A1 : g u = u (i.e., “u is a right zero”), then κ1 is the only non-arity congruence on A. Proof. By Theorem 9.3.4 it is suﬃcient to show that κa = κ1 . (a) : If u is a left unit then for arbitrary f n ∈ A we have f (u, u, ..., u) := u1 1 u u1 = u1 . Consequently, f 1 u and therefore κa = κ1 . (b) : If u is a right zero of A then ∀ f n ∈ A : f (u, u, ..., u) = u and u u = u. This implies f 1 u. Therefore κa = κ1 . When one proves κa = κ1 , one receives the next lemma as a conclusion of Theorem 9.3.4. Lemma 9.3.6 If the subclass A ⊆ Pk contains all constant functions on Ek and

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∀ f1 , ft+1 ∈ A ∃f2 , ..., ft ∈ A : ∀i ∈ {1, 2, ..., t} : Im(fi ) ∩ Im(fi+1 ) = ∅, holds, then κ1 is the only non-arity congruence on A. Finally, we solve the problem how the structure of classes is which ones have the maximum number of possible non-arity congruences (see Theorem 9.3.4). Lemma 9.3.7 Let A be a subclass of Pk and let σ be a relation on A deﬁned by (9.5) (f n , g m ) ∈ σ :⇐⇒ {f, g} ⊆ A ∧ ∆n−1 f = ∆m−1 g. Then: (a) The relation σ is a congruence on A if and only if A has the property ∀ r, s ∈ A1 : r s = r. (1) κa

(1) κ0 . 1 1

(b) κa = σ ⇐⇒ = (c) There are exactly µ(|A |) fulﬁlls the condition (9.6).

(9.6)

non-arity congruences on A if and only if A

Proof. Obviously, σ ⊆ κa

(9.7)

holds for the relation deﬁned by (9.5). (a): “=⇒”: Let σ be a congruence on A. Then, by Theorem 9.3.4 and (9.7), we have σ = κa . In particular, it holds (f 1 , g 1 ) ∈ σ ⇐⇒ f = g ∈ A1 .

(9.8)

Suppose (9.6) is false, i.e., there are certain r, s ∈ A1 with r s =: t = r. Because of r (s h) = t h for arbitrary h ∈ A1 , we have that r and t are associated in A. Because of r = t and σ = κa , this is, however, a contradiction to the equivalence (9.8). Therefore (9.6) holds. “⇐=”: Fulﬁll A the condition (9.6). Obviously, σ is an equivalence relation on A, which is compatible with all unary superposition operations. We have to show therefore only still the compatibility from σ with . Let (f n , g m ), (sp , tq ) ∈ σ be arbitrary. Then ∆n−1 f = ∆m−1 g =: f1 and ∆p−1 s = ∆q−1 t =: s1 . This implies (∆n+p−1 (f s))(x) = f (s1 (x), x, ..., x) (9.6)

= f (s1 ((s1 (x)), s1 (x), ..., s1 (x))

(9.6)

= f (s1 (x), s1 (x), ..., s1 (x)) = g(s1 (x), s1 (x), ..., s1 (x))

(9.6)

= g(s1 (s1 (x)), s1 (x), ..., s1 (x))

(9.6)

= g(s1 (x), x, ..., x) = (∆m+q−1 (g t))(x).

1

See Theorem 9.3.4.

9.4 About the Number of the Congruences on a Subclass of Pk

243

Thus (f s, g t) ∈ σ. (b): Obviously, “=⇒” holds. (1) (1) “⇐=”: Let κa = κ0 . The following two cases are possible: Case 1: A fulﬁlls (9.6). Because of (a) the relation σ in this case is a congruence, which is identical with κa because of (9.7) and Theorem 9.3.4. Case 2: A does not fulﬁll (9.6). (1) (1) In the proof of (a), we have shown that a contradiction results from κa = κ0 . (c) follows from Theorem 9.3.4 and (b). Lemma 9.3.8 Let κ be a non-arity congruence on A = [A] ⊆ Pk . Further, let µ := κ ∩ (A1 × A1 ). (µ is a congruence of the semigroup (A1 ; ).) Then, for every equivalence class F of µ it holds: ∀g ∈ A1 : F g ⊆ F. Proof. Suppose µ has a certain equivalence class F , for which there are f1 , g1 ∈ F with f1 g1 ∈ F . Let now f2m+1 := (∇f1 )hm and f3p+1 := (∇f1 )tp , where (hm , tp ) ∈ κ and m < p. (Such h and t exist, since κ ⊆ κa .) Then, (f2 , f3 ) ∈ κ, (∆p−1 f2 , ∆p−1 f3 ) = (f1 , ∇f1 ) ∈ κ and (f1 g1 , (∇f1 ) g1 ) ∈ κ. This implies (∆(f1 g1 ), ∆((∇f1 ) g1 )) = (f1 g1 , f1 ) ∈ κ, contrary to the assumption. We will see in Theorem 9.8.8 that, to every congruence on (A1 ; ), which fulﬁlls the condition of Lemma 9.3.8, there is exactly a non-arity congruence on [A1 ].

9.4 About the Number of the Congruences on a Subclass of Pk Denote ki (A) (i = 1, 2) the cardinality of the set of all congruences i-th kind on the subclass A of Pk . Further let k(A) := k1 (A) + k2 (A). The next theorem is a direct conclusion from the results of Section 9.2: Theorem 9.4.1 For every closed subset A of P2 it holds: k1 (A) ∈ {1, 2}, k2 (A) ∈ {1, 2, 3, 4} and k(A) ∈ {2, 3, 4, 5}. For k ≥ 3 the determination of all congruences is more complicated on an arbitrary class A ⊆ Pk . The following theorem mediates a ﬁrst idea of that. Theorem 9.4.2 Let k ≥ 3. For every integer n ≥ 2 there is a subclass An of Pk with k(An ) = n. Furthermore there exist certain closed sets B and C of Pk with k(B) = ℵ0 and k(C) = c.

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Proof. Obviously, a closed set of Pk has at least two diﬀerent congruences and at most continuum-many. Next, we give a class C with continuum-many diﬀerent congruences. Let the set C be deﬁned as in Section 8.1. In Lemma 8.1.1 it was proven that C is closed and that C has an inﬁnite basis. With the aid of properties of functions of C (given in the proof of Lemma 8.1.1) one can easily prove that the equivalence relation deﬁned by sn ∼ tm (κN ) :⇐⇒ {s, t} ⊆ C ∧ n = m ∧ (s = t ∨ {s, t} ⊆ {gI,J ∈ C | |I| ∈ N ∨ J = ∅ ∨ I = J = ∅})

is a congruence on C for every N ⊆ N. Furthermore, it holds that κN = κN for N = N . Therefore, by |P(N)| = c, there are continuum-many congruences on C. Next we determine the congruences on the closed set At , t ≥ 0, which is the set of all functions n , |J| ≤ t, n ≥ 1. gJn := g∅,J For all r with 0 ≤ r ≤ t, the equivalence relation πr deﬁned by

gJn ∼ gJn :⇐⇒ n = n ∧ (J = J ∨ (|J| ≤ r ∧ |J | ≤ r) ) is obvious a congruence on the class At . In particular, we have π0 = κ0 and πt = κa on At . Obviously, κ0 = κa and κ1 are the only congruences on A0 . If t ≥ 1 and κ is a congruence on At with πs ⊂ κ ⊆ κa , 0 ≤ s ≤ t − 1, then there exist functions gJn1 , gJn2 ∈ At with gJn1 ∼ gJn2 (κ), J1 = J2 and |J1 ≥ s+1. If |J2 | ≤ s, we have gJn1 ∼ cn0 (κ) and thus gJm ∼ cm 0 (κ) with |J| ≤ |J1 | and m ≥ 1, since one can form all functions gjm as superpositions on gJ1 . Therefore, in this case πs+1 ⊆ κ. If |J2 | ≥ s + 1 then w.l.o.g. one can assume 1 ∈ J1 and 1 ∈ J2 . Hence, we obtain gJn1 c10 = gJn1 ∼ cn0 = gJn2 c10 (κ), i.e., we have also in this case πs+1 ⊆ κ. In summarizing we see that κ0 , π1 , π2 ,..., πt−1 and κa are the only arity congruences on At . Because of Lemma 9.2.1, κ1 is the only non-arity congruence on At . Consequently, for t ≥ 0, At is a class with exactly t + 2 diﬀerent congruences. It is easy to see that the set of all congruences on the closed set B := t≥0 At has the cardinality ℵ0 . Each class A with |Con(A)| ∈ {ℵ0 , c} deﬁned in the above proof has no basis or an inﬁnite basis, respectively. A connection between ord A = ∞ and |Con(A)| ∈ {ℵ0 , c} does not exist, however, as the examples show from the following two lemmas. Lemma 9.4.3 For each k ≥ 3 there is a subclass A of Pk with ord A = ∞ and Con A = 3.

9.4 About the Number of the Congruences on a Subclass of Pk

245

Proof. Proving the lemma for k = 3 suﬃces. We consider the class {f ∈ P3,2 | ∀a ∈ E2n : f (a) = 0}, A := n≥1

for which we prove in Chapter 12 that it does not have any ﬁnite generating system. Subsequently, we will show that A has only trivial congruences. Let κ be a congruence on A with κ = κa . Then, the following two cases are possible: Case 1: κ0 ⊂ κ ⊆ κa . In this case, there are n-ary κ-congruent functions f, g ∈ A with f = g. W.l.o.g. we can assume that n = 3, f (0, 1, 2) = 0 and g(0, 1, 2) = 1. Consequently, we have f (c0 (x), j2 (x), x) = c10 (x) ∼ j2 (x) = g(c0 (x), j2 (x), x) (κ). Let tm be an arbitrary m-ary function of A. Then the (2 · m)-ary function ht deﬁned by t(x1 , ..., xm ) if xm+1 = ... = x2·m = 0, ht (x1 , x2 , ..., x2·m ) := 0 otherwise belongs to A. From that we receive ht (x1 , ..., xm , c0 (x1 ), ..., c0 (xm )) = t(x1 , ..., xm ) ∼ cm 0 (x1 , ..., xm ) = ht (x1 , ..., xm , j2 (x1 ), ..., j2 (xm )) (κ). This implies κ = κa immediately. Case 2: κ ⊆ κa . Since A has a right zero, κ = κ1 results in this case from Lemma 9.3.5. Without use of Lemma 9.3.5, this can be shown as follows: If κ ⊆ κa then there are κ-congruent functions f n , g m ∈ A with n > m. Therefore, c10 (∆n−1 f ) = c20 ∼ c10 = c10 (∆n−1 g) (κ). This implies that all constant functions of A are κ-congruent to each other. Furthermore, for a binary function h ∈ A with h(0, 2) = 1 we have: ∆((τ (h c10 )) c10 ) = c10 (x) ∼ r(x, y) := ∆((τ (h c20 )) c10 ) (κ), where r(0, 2) = 1. Because of (c10 , c20 ) ∈ κ, this implies (c20 , r) ∈ κ and thus κ ∩ κa = κ0 . Then κ = κ1 follows from our considerations for Case 1 and from the κ-congruence of the functions cn0 for arbitrary n. Hence A has only trivial congruences. The class deﬁned in the following lemma was published by V. L. Murskij in [Mur 65] as an example for a class without a ﬁnite basis of identities.

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Lemma 9.4.4 Let m be deﬁned by x, if (x, y) ∈ {(1, 2), (2, 1), (2, 2)}, m(x, y) := 0 otherwise a binary function of P3 . congruences.

2

Then, the subclass M := [m] has inﬁnite-many

Proof. For f n ∈ M we denote with χ(f ) the set of all tuples a of E3n with f (a) ∈ {1, 2}. Further, let Krn := {f ∈ P3n | ∀N ⊆ χ(f ) : (|N | ≤ r =⇒ (∃i : ∀(a1 , ..., an ) ∈ N : ai = 2} and Kr :=

Krn .

n≥1

One can easily prove the following statement: ∀f n ∈ M ∃i : a ∈ χ(f ) =⇒ f (a) = ai .

(9.9)

For r ≥ 1, we consider the equivalence relation µr on M deﬁned by f n ∼ g m (µr ) :⇐⇒ n = m ∧ (f = g ∨ {f, g} ⊆ Krn ) and show that it is a congruence on M . For this purpose, we choose arbitrary functions f n , g n , pm , q m of M with (f, g), (p, q) ∈ µr . Then we have (αf, αg) ∈ µr for every α ∈ {ζ, τ, ∆, ∇}. For the proof of (f p, g q) ∈ µr , it suﬃces to show that (f ∈ Kr ∨ p ∈ Kr ) ∧ (f depends essentially of x1 ) =⇒ f p ∈ Kr (9.10) holds for arbitrary f, p ∈ M . Let the ﬁrst place of the function f be essential. We distinguish two cases: Case 1: f ∈ Kr . Let a1 , ..., as ∈ χ(f p), 1 ≤ s ≤ r, aj := (aj,1 , ..., aj,n+m−1 ), j = 1, 2, ..., s, p(aj,1 , ..., aj,m ) =: bj . Because of f ∈ Kr we have either b1 = ... = bs = 2 or there exists a u ≥ m + 1 with a1,u = ... = as,u = 2. If b1 = ... = bs = 2, there is by (9.9) a v with 1 ≤ v ≤ m and av,1 = ... = av,s = 2. Consequently, f p belongs to Kr . Case 2: p ∈ Kr . In this case, since by assumption the ﬁrst place of f is essential, (f n pm )(a) ∈ {1, 2} for a ∈ E3m+n−1 is only possible, if p(a1 , ..., am ) ∈ {1, 2}. Consequently, we have χ(f p) ⊆ {a ∈ E3m+n−1 | (a1 , ..., am ) ∈ χ(p)} and this implies f p ∈ Kr . Thus (9.10) was proven and µr is a congruence on M . One can 2

We notice that the function m can be interpreted as a binary partial projection. In this case m(a, b) = 0 stands instead of “m(a, b) is not deﬁned”.

9.4 About the Number of the Congruences on a Subclass of Pk

247

prove that the congruences of the form µr are diﬀerent pairwise as follows: Let xy := m(x, y). The functions tn and sn deﬁned by t(x1 , ..., xn ) := ((((((x1 ((x2 x1 ))(x3 x1 ))(x3 x2 ))(x4 x1 ))...(x4 x3 ))...)(xn xn−1 )(xn x1 ))(x3 x2 )) ...(xn xn−1 )) x1 if x ∈ {(1, 2, ..., 2), (2, 1, 2, ..., 2), ..., (2, ..., 2, 1), (2, ..., 2)}, = 0 otherwise and s(x1 , ..., xn ) := t(m(x1 , x1 ), ..., m(xn , xn )) 2 if x1 = ... = xn = 2, = 0 otherwise belong to M , and it holds (tn , sn ) ∈ µn−1 and (tn , sn ) ∈ µn for arbitrary n ≥ 1. Consequently, there are inﬁnite-many congruences on M . With the aid of Lemma 9.10.1 and [Oat-W 80], we will later prove that the above-deﬁned class M has continuum-many congruences. The following theorem gives examples of classes A whose congruences are completely determined by the congruences on certain sets B with A ⊆ B. We use this theorem in Section 9.6. Theorem 9.4.5 Let S be the set of all functions f n ∈ Pk for which f (x1 , x2 , ..., xn ) = s−1 (f (s(x1 ), s(x2 ), ..., s(xn )) holds, i.e., S = P olk {(x, s(x)) | x ∈ Ek }. Further, let α be the isomorphism deﬁned in Theorem 8.3.2. Then, |Con A| ≤ |Con α(A)|. More precisely: Let A (= ∅) be a subclass of S, let κ be a congruence on A and let α(κ) be a relation on α(A) deﬁned by (F, G) ∈ α(κ) :⇐⇒ (α−1 F, α−1 G) ∈ κ. Then (a) α(κ) is a congruence on A; (b) α(κ)/A = κ, i.e., by means of restriction on A one can get the congruences on A from the congruences on α(A). Proof. Since κ is a congruence on A, κ is compatible with the operations ˆ τ ,∆, ˆ∇ ˆ and ˆ ζ,ˆ ∗ . This and Theorem 8.3.2, (a) imply our statement (a). By Theorem 8.3.2, (b) we have A ⊂ α(A). Therefore, α(κ)/A is a congruence on A. Now let f and g be arbitrary functions of A. If (f, g) ∈ κ then

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(∇f, ∇g) ∈ κ and by deﬁnition of α(κ); further, we have (α(∇f ), α(∇g)) = (f, g) ∈ α(κ)/A , i.e., κ ⊆ α(κ)/A . If (f, g) ∈ α(κ)/A then (α−1 f, α−1 g) ∈ κ. Since f, g ∈ S, it holds α−1 f = ∇f and α−1 g = g. Consequently, (∇f, ∇g) ∈ κ and (∆(∇f ), ∆(∇g)) = (f, g) ∈ κ, i.e., α(κ)/A ⊆ κ. This implies (b).

9.5 A Criterion for the Proof of the Countability of Con A for Certain A ⊆ Pk It is possible to choose a congruence relation on an algebra (in particular, a function algebra) as a universe of another algebra, as follows: Let A be the universe of a subalgebra of Pk = (Pk ; ζ, τ, ∆, ∇, ∗). Then, one can deﬁne for arbitrary (f, g), (p, q) ∈ A × A: ∀ α ∈ {ζ, τ, ∆, ∇} : α(f, g) := (αf, αg), (f, g) ∗ (p, q) := (f ∗ p, g ∗ q), σ(f, g) := (g, f ) and (f, q) if g = p, ((f, g), (p, q)) := (f, g) otherwise. Then each congruence on A is a universe of a subalgebra of the algebra (A × A; ζ, τ, ∆, ∇, ∗, σ, , (f, f )f ∈A ); i.e., it holds Con A = Sub(A × A; ζ, τ, ∆, ∇, ∗, σ, , (f, f )f ∈A ). (The tupel (f, f ), where f ∈ A is arbitrary, are certain nullary operations of the above deﬁned algebra.) In particular, we have for the arity congruences: An × An ; ζ, τ, ∆, ∇, ∗, σ, , (f, f )f ∈A ). Cona A = Sub( n≥1

In order to be able to prove the countability of Con A for some A ⊆ Pk later with the aid of Theorem 8.1.5, we show ﬁrst that class A fulﬁlls the conditions of Theorem 8.1.5. Denote ◦ an operation on Ek with the properties a) ∀ a, b, c ∈ Ek : a ◦ (b ◦ c) = (a ◦ b) ◦ c, b) ∀ a, b ∈ Ek : a ◦ b = b ◦ a, c) ∃ r ≥ 1 : (∀ a ∈ Ek : a ◦ a ◦. . . ◦ a = a). (1+r)−times

The properties a)–c) e.g. are fulﬁlled, if (Ek , ◦) is an Abelean group or x ◦ y := maxω (x, y) or x◦y := minω (x, y) (ω is a partial order relation, for which exists max or min, respectively).

9.5 A Criterion for the Proof of the Countability of Con A for Certain A ⊆ Pk

249

With the help of ◦ one can deﬁne the following subset of Pk : K0 := {f n ∈ Pk | ∃ f0 , f1 , . . . , fn ∈ PE1 : f (x1 , . . . , xn ) = n≥1

f0 (f1 (x1 ) ◦ f2 (x2 ) ◦ . . . ◦ fn (xn ))}. In general, the set K0 is not closed; it contains, however, some closed sets in the case of suitable choice of ◦ (see Theorem 9.5.2). In analog manner to the proof of |Sub A| ≤ ℵ0 for all A = [A] ⊆ K0 in Theorem 8.1.6 one can show: Theorem 9.5.1 ([Lau 90]) Each subclass A of Pk , for which exists an operation ◦ with the properties a)–c) and A ⊆ K0 , has at most countable-many congruences. Proof. Since, by Theorem 9.3.4, every subclass of Pk has only ﬁnite-many non-arity congruences, it is suﬃcient to show that the set Con ( n≥1 An × An ; ζ, τ, ∆, ∇, ∗, σ, ) is ﬁnite or countable. For this purpose we deﬁne an or≤, which fulﬁlls the conditions 1)–3) from Theorem 8.1.5, on der relation K0∗ := n≥1 K0n × K0n . Then, our theorem follows from Theorem 8.1.5. Each pair (f, g) ∈ K0n × K0n with f (˜ x) = f0 (f1 (x1 ) ◦ . . . ◦ fn (xn )) and g(˜ x) = g0 (g1 (x1 ) ◦ . . . ◦ gn (xn )) is well-deﬁned by the tuples ϕ(f, g) := ((f0 , g0 ), (f1 , g1 ), . . . , (fn , gn )). Two pairs (f, g) and (p, q) with ϕ(f, g) = ((f0 , g0 ), . . . , (fn , gn )) and ϕ(p, q) = ((p0 , q0 ), . . . , (pm , qm )) are called ∼–equivalent, if they fulﬁll the three following conditions: (1) (f0 , g0 ) = (p0 , q0 ), (2) {(f1 , g1 ), . . . , (fn , gn )} = {(p1 , q1 ), . . . , (pm , qm )}, (3) If (fi , gi ) occurs in ((f1 , g1 ), . . . , (fn , gn )) exactly u times, then there exists a certain integer v so that (fi , gi ) occurs in ((p1 , q1 ), . . . , (pm , qm )) exactly (u + v · r) times, where r denotes the smallest integer for which ◦ fulﬁlls the condition c). Obviously, the factor set K0∗/∼ is ﬁnite. Set {K1 , ..., Kt } := K0∗/∼ . On the sets K1 , . . . , Kt one can deﬁne an order relation ≤ as follows: For (f n , g n ), (pm , q m ) ∈ Ki (i = 1, . . . , t) with ϕ(f, g) = ((f0 , g0 ), . . . , (fn , gn )) and ϕ(p, q) = ((p0 , q0 ), . . . , (pm , qm )) we write (f, g) ≤ (p, q) if and only if for every (fi , gi ), which occurs in ((f1 , g1 ), . . . , (fn , gn )) j1 times and occurs in ((p1 , q1 ), . . . , (pm , qm )) j2 times, it holds j1 ≤ j2 . This order relation ≤ fulﬁlls the condition 1) of Theorem 8.1.5, since one can receive (f, g) from (p, q) with the aid of the operations ζ, τ, ∆, if (f, g)≤(p, q) is valid (because of property c) of ◦). Obviously the property 2) of Theorem 8.1.5 is also valid. Thus, it remains to show that every antichain in the set K0∗ in respect to 1 1 ≤ has only ﬁnite-many elements. On PE × PE one can deﬁne a total order. For ϕ(f, g) = ((f0 , g0 ), . . . , (fn , gn )), we arrange (concerning this order) that

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9 Congruences and Automorphisms on Function Algebras

(fi0 , gi0 ) is the smallest element of the set {(f1 , g1 ), . . . , (fn , gn )}; (fi1 , gi1 ) is the smallest element of the set {(f1 , g1 ), . . . , (fn , gn )}\ {(fi0 , gi0 )}, . . . and (fis , gis ) is the the greatest element of {(f1 , g1 ), . . . , (fn , gn )}. If (fij , gij ) occurs in {(f1 , g1 ), . . . , (fn , gn )} exactly bj times, j = 0, 1, . . . , s, then let α(f, g) := (b0 , b1 , . . . , bs ). Further, let Di := {α(f, g)| (f, g) ∈ Ki }, i = 1, 2, . . . , t. We set α(f, g) ≤ α(p, q) iﬀ (f, g)≤(p, q). The ordered sets Ki and Di are isomorphic. In the proof of Theorem 8.1.6, it was shown that every antichain C ⊆ Di is a ﬁnite set. Therefore, ≤ also fulﬁlls the condition 3) of Theorem 8.1.5. Some consequences from Theorem 9.5.1 are summarized in the following: Theorem 9.5.2 If A is a subclass of one of the sets (a) Lk (see Chapter 4), (b) P olk λ, λ ∈ Lk (see Section 5.2.4), (c) Ln (see Chapter 14) or (d) [O1 ∪ {max}] (see Chapter 15), then |Con A| ≤ ℵ0 . More precise statements about Con A for the above cases (a)–(c) are treated in the next section.

9.6 Congruences on Some Classes of Linear Functions Let R = (R; +, ·) be a unitary ring and let M = (M ; +, ·) be a left module on R. Then one can deﬁne a closed subset LM of PM as follows: LM :=

{f n ∈ PM | ∃ a0 ∈ M ∃ a1 , . . . , an ∈ R : f (x) = a0 +

n

ai · xi }.

i=1

n≥1

By [Sze 80], for every subclass A of LM that contains the function r deﬁned by r(x, y, z) := x + y − z, there exists a uniquely deﬁned subring TA of R and an uniquely deﬁned submodule NA of R × M on TA with A=

{f n ∈ LM | ∃ a0 ∈ M ∃ a1 , . . . , an ∈ TA : f (x) = a0 +

n i=1

n≥1

(1 − a1 − a2 − · · · − an , a0 ) ∈ NA } . The sets TA and NA are deﬁned by TA := {a ∈ R | ax + (1 − a)y ∈ A}, NA := {(1 − a, b) ∈ R × M | ax + b ∈ A1 }.

ai xi ∧

9.6 Congruences on Some Classes of Linear Functions

251

For the purpose of determining of the congruences on A, let I be an arbitrary ideal of TA , and let U be a suitably elected submodule of NA with ∀α, β ∈ TA ∀u, v ∈ NA : (α − β ∈ I ∧ u − v ∈ U =⇒ α · u − β · v ∈ U ). (9.11) With the help of I and U , one can deﬁne a relation κ(I, U ) (⊆ κa ) as follows: n n (a0 + i=1 ai xi , b0 + i=1 bi xi ) ∈ κ(I, U ) :⇐⇒ ∀i ∈ {1, . . . , n} : ai − bi ∈ I ∧ (1 − a1 − · · · − an , a0 ) − (1 − b1 − · · · − bn , b0 ) ∈ U. Lemma 9.6.1 κ(I, U ) is a congruence on A. Proof. It can easily be shown that κ(I, U ) is an equivalence relation and that κ(I, U ) is compatible with the operations ζ, τ, ∆ and ∇. Let now (f n , g n ), (sm , tm ) ∈ κ(I, U ) be arbitrary with f (x) = a0 + a1 x1 + · · · + an xn , g(x) = b0 +b1 x1 +· · ·+bn xn , s(x) = co +c1 x1 +· · ·+cm xm and t(x) = d0 + d1 x1 + · · · + dm xm . Then, bi = ai + αi , dj = cj + βj for certain αi , βj ∈ I (i = 1, . . . , n, j = 1, . . . , m), (f ∗ s)(x) = a0 + a1 c0 + a1 c1 x1 + . . . + a1 cm xm + a2 xm+1 + . . . + an xm+n−1 , (g ∗ t)(x) = b0 + b1 d0 + b1 d1 x1 + . . . + b1 dm xm + b2 xm+1 + . . . + bn xm+n−1 , a1 ci − b1 di = −α1 βi − α1 ci − a1 βi ∈ I (i = 1, . . . , m) and (1 − a1 c1 − . . . − a1 cm − a2 − . . . − am , a0 + a1 c0 ) −(1 − b1 d1 − . . . − b1 dm − b2 − . . . − bm , b0 + b1 d0 ) = a1 (1 − c1 − . . . − cm , c0 ) − b1 (1 − d1 − . . . − dm , d0 ) +(1 − a1 − . . . − an , a0 ) − (1 − b1 − . . . − bn , b0 ) ∈ U by (9.11) and (f, g), (s, t) ∈ κ. Thus κ(I, U ) is a congruence on A. We need some denotation and the following three lemmas to prove that on A next to the trivial congruences only congruences of the type κ(I, U ) still exist. The function qa (for a ∈ R) is deﬁned by qa (x, y) := a · x + (1 − a) · y. Lemma 9.6.2 Let κ ⊆ κa be a congruence on A, let f (x) = a0 + a1 x1 + . . . + an xn ∈ A and let g(x) = b0 + b1 x1 + . . . + bn xn ∈ A. Then (f, g) ∈ κ ⇐⇒ ∀i ∈ {1, . . . , n} : (qai , qbi ) ∈ κ ∧ (∆n−1 f, ∆n−1 g) ∈ κ with ∆i+1 f := ∆(∆i f ), ∆1 f := ∆f . Proof. Let h ∈ {f, g}, h(x1 , . . . , xn ) = α0 + α1 x1 + . . . + αn xn , and denote rn the (n + 2)–ary function of [{r}] which is deﬁned by rn (x1 , . . . , xn+2 ) :=

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9 Congruences and Automorphisms on Function Algebras

x1 +x2 +. . .+xn+1 −n·xn+2 . Then, the statement of our lemma easily follows from the following identities: qαi (x, y) = r(h(y, . . . , y, x , y, . . . , y), y, h(y, . . . , y)) and i

h(x) = rn (qα1 (x1 , x1 ), qα2 (x2 , x1 ), . . . , qαn (xn , x1 ), (∆n−1 h)(x1 ), x1 ).

Let QA := {qa ∈ A2 | a ∈ TA }. One can deﬁne two operations ⊕, on QA as follows: (qa ⊕ qb )(x, y) := r(qa (x, y), qb (x, y), y) and (qa qb )(x, y) := qa (qb (x, y), y). Because of qa ⊕ qb = qa+b and qa qb = qa·b , one can easily prove that the following holds: Lemma 9.6.3 The algebra (QA ; ⊕, ) is isomorphic to (TA ; +, ·). For arbitrary g, h ∈ A1 and arbitrary α ∈ TA we set now (g ⊕ h)(x) := r(g(x), h(x), x) and (α g)(x) := qa (g(x), x). With the help of the mapping ϕ : A1 −→ NA , ax + b −→ (1 − a, b) one can easily prove the following lemma: Lemma 9.6.4 The left module (A1 ; ⊕, ) and (NA ; +, ·) over TA are isomorph. Theorem 9.6.5 Let A be a subclass of LM with r ∈ A. Then A has only the congruences κ0 , κa , κ1 and such of the type κ(I, U ). Proof. Let κ be a congruence on A with κ = κ1 . By r ∈ A, A is a clone. Therefore, κ ⊆ κa (see Lemma 9.2.1 or Lemma 9.3.5). With the aid of Lemmas 9.6.2 and 9.6.3 (or 9.6.4) one can see that κ determines a congruence µκ (or νκ ) on TA (or NA ), respectively: (qa , qb ) ∈ κ ⇐⇒ (a, b) ∈ µκ (or (ax + b, cx + d) ∈ κ ⇐⇒ ((1 − a, b), (1 − c, d)) ∈ νκ ).

(9.12) (9.13)

Then there is an ideal I of TA and a submodule U of NA with the properties

9.6 Congruences on Some Classes of Linear Functions

(a, b) ∈ µκ ⇐⇒ a − b ∈ I, (u, v) ∈ νκ ⇐⇒ u − v ∈ U.

253

(9.14) (9.15)

To prove (9.11) let a, b ∈ TA , let u = (1 − α, β), v = (1 − γ, δ) ∈ NA with a − b ∈ I and let u − v ∈ U . Then, it holds {(qa , qb ), (αx + β, γx + δ)} ⊆ κ. Consequently, we have (qa (αx + β, x), qb (γx + δ, x)) = ((aα + 1 − a)x + aβ, (bγ + 1 − b)x + bδ) ∈ κ. Therefore, ((1 − (aα + 1 − a), aβ), (1 − (bγ + 1 − b), bδ)) = ((a − aα, aβ), (b − bγ, bδ)) ∈ νκ and thus a · (1 − α, β) − b · (1 − γ, δ)) ∈ U , i.e., (9.11) holds. By Lemma 9.6.2 and (9.12)–(9.15), we have κ = κ(I, U ). Next, we determine a property of the congruences on the closed subset LM ;id := n≥1 {f n ∈ LM | ∃a0 , . . . , an : n f (x) = a0 + i=1 ai xi ∧ a1 + a2 + . . . + an = 1} (a set of so-called idempotent functions) of LM . It is easy to check that LM ;id = LM ∩ P olM {(x, x + 1) | x ∈ M } holds. By Theorem 8.3.2 and with the aid of the bijective mapping α : f n −→ F n−1 from LM ;id := LM ;id \L1M onto LM , where F (x1 , . . . , xn−1 ) := f (0, x1 , . . . , xn−1 ), we can characterize the subclasses of LM ;id and receive as a consequence of Theorem 9.4.5: Theorem 9.6.6 Let A (= ∅) be a subclass of LM ;id , let κ be a congruence on A and let α(κ) be a relation on α(A) deﬁned by (F, G) ∈ α(κ) :⇐⇒ (α−1 F, α−1 G) ∈ κ. Then: (a) α(κ) is a congruence on A; (b) α(κ)/A = κ, i.e., by means of restriction on A one can get the congruences on A from the congruences on α(A). From Theorems 9.6.5 and 9.6.6 some consequences can be drawn (with the aid of known properties of the groups, ﬁelds and vector spaces) for the classes from Chapter 13 and from [Sze 80] and [Sza-S 81]. Here only a few examples are given without proof.

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9 Congruences and Automorphisms on Function Algebras

Corollary 9.6.7 (1) Let R = M and let R be a ﬁnite ﬁeld. Then, by [Sze 80], each subclass A of LM with r ∈ A and A ⊆ LM ;id has the form (up to isomorphic) LW ∩P ol V ( = n≥1 {f n | ∃ a0 ∈ V ∃ a1 , . . . , an ∈ W : f (x) = a0 + a1 x1 + . . . + an xn }), where W is a subﬁeld of R and V is a subspace of the vector space R on W . Then, for every nontrivial congruence κ on A there are an I ∈ {{0}, W } and a subspace U of V with the property ∀ α − β ∈ I ∀ a − b ∈ U : a · α − b · β ∈ U, and it holds that κ=

n n + i=1 ai xi , b0 + i=1 bi xi ) | a0 − b0 ∈ U ∧ ∀ i ∈ {1, . . . , n} : ai − bi ∈ I}.

n≥1 {(a0

In particular, κc := {(f, g) ∈ κa | ∃ a ∈ M : f = g + a} is the only nontrivial congruence on LM . (2) Let L be a maximal class of quasi-linear functions of Pk with k = pm , p prime, m ≥ 1. Then κc is the only nontrivial congruence on L (see Section 5.2.4 and [Lau 81]). By Lemma 5.2.4.3, L is isomorphic to a certain set LM , where R is the ring of all m × m–matrices on Ep and M is the vector space of all m × 1–matrices on Ep . (3) Choosing R = M = (Ek ; + mod k, · mod k), so there are only the following arity congruences on LM : n n κs,t := n≥1 {(a0 + i=1 ai xi , b0 + i=1 bi xi ) | s |a0 − b0 ∧ ∀ i ∈ {1, . . . , n} : t | ai − bi }, where s and t are arbitrary divisors of k with s | t. Notice that Con A is not a ﬁnite set for every subclass A of LM . If, for example, one ﬁnds an element z in R\{0} with z 2 = 0, then the closed set Z := {f n ∈ LM | ∃ a1 , . . . , an ∈ [{0, z}]+ : f (x) = a1 x1 + . . . + an xn } n≥1

has inﬁnite-many congruences. For the purpose of describing of some of these relations, let vare (f ) be the number of the essential variables of f ∈ Z and let

9.6 Congruences on Some Classes of Linear Functions

255

(f n , g n ) ∈ χi :⇐⇒ f = g ∨ (n = m ∧ vare (f ) ≤ i ∧ vare (g) ≤ i) (f, g ∈ Z, i ∈ N). It is easy to check that, for every i ∈ N, the relation χi is a congruence on Z. With the aid of Theorem 9.3.4 and Theorem 9.5.1, |Con Z| = ℵ0 results from that. Finally, we determine all congruences on the subclasses of Lk (see Chapter 13) for k ∈ P. We start with the determining of the congruences on certain subclasses of [Pk1 ] for arbitrary k. The following lemma is easily provable: Lemma 9.6.8 Let C ⊆ {c10 , c11 , ..., c1k−1 }, let G be a subgroup of Sk := (Pk1 [k]; ), whose elements preserve C, let U be a normal subgroup of the group G and let µ be an equivalence relation on C, which is preserved from all functions of G. Then the relation κU,µ on [G ∪ C] deﬁned by f n ∼ g m (κU,µ ) :⇐⇒ n = m ∧ (∃i ∃f , g ∈ C ∪ G : f (x) = f (xi ) ∧ g(x) = g (xi ) ∧ (f U = g U ∨ (f , g ) ∈ µ)) is a congruence on [G ∪ C]. Theorem 9.6.9 Let C ⊆ {c10 , c11 , ..., c1k−1 }, let G be a subgroup of Sk := (Pk1 [k]; ) and let G ⊆ P olk {a | ca ∈ C}. Then, there are except for κ1 and κa only the congruences of the type κU,µ on [C ∪ G], where U is an arbitrary normal subgroup of G and µ is an arbitrary equivalence relation on C, which is preserved from the functions of G. Proof. Let κ be a nontrivial congruence on [C ∪ G]. Because of Lemma 9.2.1, κ ⊆ κa . Discussing the following three cases suﬃces: Case 1: There exist κ-congruent functions f n and g n with ∆n−1 f ∈ G and ∆n−1 g ∈ C. Then, ∆n−1 f ∼ ∆n−1 g (κ) and thus e11 ∼ c1a := ∆n−1 g (κ). Consequently, we 1 have for every function hm ∈ [C ∪ G]: e11 h = h ∼ cm a = ca h (κ). Hence, κ = κa , in contradiction to the assumption. Case 2: There exist κ-congruent functions f n and g n , which essentially depend on diﬀerent variables. W.l.o.g. let f (x1 , ..., xn ) = f1 (x1 ), g(x1 , ..., xn ) = g1 (x2 ), {f1 , g1 } ⊆ G and (f, g) ∈ κ. For the inverse functions f1−1 and g1−1 appertaining to the functions f1 and g1 is valid then: f (f1−1 (x1 ), g1−1 (x2 ), x2 , ..., x2 ) = e21 (x1 , x2 )

∼ g(f1−1 (x1 ), g1−1 (x2 ), x2 , ..., x2 ) = e22 (x1 , x2 ) (κ).

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9 Congruences and Automorphisms on Function Algebras

Therefore, for arbitrary m-ary functions s and t ∈ [C ∪ G]: e21 (s(x), t(x)) = s(x) ∼ t(x) = e22 (s(x), t(x)) (κ); i.e., in this case, we have also κ = κa , in contradiction to the assumption. Case 3: Two arbitrary κ-congruent functions of [C ∪ G] are both either constant functions or the functions depend of the same variables essentially. In this case, the congruence κ is uniquely determined by κ/G and κ/C . As is generally known, a congruence of a group G causes a partition of this group in cosets, which can be formed with the aid of a normal subgroup U of this group, i.e., f, g ∈ G are congruent iﬀ f U = g U . Obviously, κ/C is an equivalence relation on C, which is preserved by G. Consequently, we have κ = κU,κ/C . Theorem 9.6.10 Let k ∈ P and let A ⊆ Lk (see Chapter 13). Then there are only ﬁnite-many congruences on A. The nontrivial congruences on A are (1) κc , if A ∈ { Lk , Lk ∩ P olk {(x, x + 1) | x ∈ Ek } }, (2) κµ := {(f n , g m ) ∈ A × A | (∆n−1 f, ∆m−1 g) ∈ µ} and κµa := {(f n , g m ) ∈ A × A | n = m ∧ (∆n−1 f, ∆n−1 g) ∈ µ} for each equivalence relation µ on A1 , if A ⊆ [{c0 , c1 , ..., ck−1 }]; (3) congruences of the type κU,µ (see Lemma 9.6.8), if A ⊆ [L1k ] and A ⊆ [{c0 , c1 , ..., ck−1 }]. Proof. Our theorem results from the description of the subclasses of Lk (see Chapter 13) and from Theorems 9.6.5, 9.6.6 and 9.6.9.

9.7 Congruences on the Maximal Classes of Pk The following two lemmas are auxiliary statements to the proof of Theorem 9.7.3, in which all maximal classes are given that have only trivial congruences. Not only do results on the congruences of certain maximal classes follow from Lemma 9.7.1, but also those on further classes (see for example Theorem 9.10.8). The statement (a) of the following lemma was already proven by A. I. Mal’tsev in [Mal 66]. Lemma 9.7.1 For ω, u ∈ Ek and ∈ Mk , where o is the smallest element of Ek in respect to and e is the greatest element of Ek in respect to , let 1 , m1,u and t2ω be functions deﬁned by c1ω,u , qω,u ω if x = ω, u if x = u, cω,u (x) := qω,u (x) := u otherwise, ω otherwise, o if ω if x = ω, x ≤ u, m,u (x) := tω (x, y) := y otherwise. e otherwise,

9.7 Congruences on the Maximal Classes of Pk

257

Furthermore, let A be a subclass of Pk , which fulﬁlls the following two conditions: (a) ∃ω ∈ Ek : ( u∈Ek {cω,u , qω,u }) ∪ {tω } ⊆ A; (b) ∃ ∈ Mk ∃ω ∈ {o , e } : ( u∈Ek {cu , m,u }) ∪ {tω } ⊆ A. Then, A has only trivial arity congruences. Proof. Denote κ an arbitrary congruence on A with κ0 ⊂ κ ⊆ κa . Then there exist two diﬀerent functions rn , sn ∈ A and a tuple a := (a1 , ..., an ) ∈ Ekn with r(a) = s(a) and (r, s) ∈ κ. First, assume A fulﬁlls (a). We distinguish two cases for A: Case 1: A ⊆ P olk {ω}. Then there is a function f 1 ∈ A with f (ω) := α = ω and we have f cω,ω = cα ∈ A. Thus because of cω,u cα = cu (u ∈ Ek ), all constant functions of Pk belong to A. Consequently, (r, s) ∈ κ implies the κ-congruence of certain constant functions c1a , c1b with a = b and a = ω. Thus we obtain ca = qω,a ca ∼ qω,a cb = cω (κ) and

∀g n ∈ A : tω (ca (x1 ), g(x1 , ..., xn )) = g(x1 , ..., xn ) ∼ tω (cω (x1 ), g(x1 , ..., xn )) = cnω (x1 , ..., xn ) (κ).

(9.16)

Hence κ = κa . Case 2: A ⊆ P olk {ω}. In this case, we obtain r(cω,a1 (x1 ), ..., cω,an (xn )) = cω,a ∼ s(cω,a1 (x1 ), ..., cω,an (xn )) = cω,b (κ), qω,a cω,a = cω,a ∼ qω,a cω,b = cω (κ) and tω (cω,a (x), x) = x ∼ tω (cω (x), x) = cω (x) (κ). Hence κ = κa is also valid in Case 2. Finally, assume A fulﬁlls the condition (b) for a certain ∈ Mk . Since the constant functions belong to A, (r, s) ∈ κ implies (c1a , c1b ) ∈ κ for certain a = b. If b = ω ∈ {o , e }, one obtains κ = κa by means of (9.16). Then one can reduce the remaining cases to the case already handled through m,α ca ∼ m,α cb (κ) with suitably chosen α. Lemma 9.7.2 Let s ∈ Sk and let κ be a congruence on P olk s . If there are certain κ-congruent functions f n , g n ∈ P olk s with f = g, then κa ⊆ κ. Proof. Deﬁne the relation s as in Section 5.2.2. Then the functions f , g have representations of the form (5.4) (see Chapter 5). Since f = g, we can assume

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9 Congruences and Automorphisms on Function Algebras

w.l.o.g. that F1 = G1 holds for the components F1 , G1 , i.e., there exists an (n − 1)-tuple a := (a2 , ..., an ) with F1 (a) =: c = d := G1 (a). Let qa (x) :=

p−1 l

jar,i (x) · si (a),

r=1 i=0

u(x) :=

p−1 l

jar,i (x) · si (a1,i )

r=1 i=0

and h(x1 , ..., xm+1 ) :=

p−1 l

jar,i (xi ) · si (Hr (sp−i (x2 ), ..., sp−i (xm+1 ))),

r=1 i=0

where Hr (c, x1 , ..., xm ) = H1,r (x1 , ..., xm ), Hr (d, x1 , ..., xm ) = H2,r (x1 , ..., xm ), r = 1, 2, ..., l, and H1,1 , H1,1 , ..., H1,l , H2,1 , ..., H2,l are arbitrary functions of Pk . Since (f, g) ∈ κ, we have f (u(x), qa2 (x), ..., qan )) = qc (x) ∼ g(u(x), qa2 (x), ..., qan )) = qd (x) (κ) so that h(x1 , qc (x1 ), x2 , ..., xm ) = l p−1 i p−i (x2 ), ..., sp−i (xm ))) r=1 i=0 jr,i (x1 ) · s (H1,r (s ∼ h(x1 , qd (x1 ), x2 , ..., xm ) = l p−1 i p−i (x2 ), ..., sp−i (xm ))) (κ). r=1 i=0 jr,i (x1 ) · s (H2,r (s Therefore κ = κa . Theorem 9.7.3 ([Mal 66], [Kol 74], [Lau 79a;81]) Let ∈ (Ck \{ ∈ C1k | || ≥ 2}) ∪ Mk ∪ Sk . Then, the maximal class P olk has only trivial congruences. Proof. Because of Lemma 9.2.1, every nontrivial congruence of a maximal class is an arity congruence. With that, our theorem results for ∈ Ck \{ ∈ C1k | || ≥ 2} from Lemma 9.7.1, when one chooses ω as a central element of and one proves that A := P olk fulﬁlls the condition (a) of Lemma 9.7.1. If ∈ Mk then our assertion also results of Lemma 9.7.1, since P olk fulﬁlls the condition (b) of Lemma 9.7.1. If ∈ Sk then we obtain |Con P olk | = 3 from Lemma 9.7.2. Subsequently, we will prove that, in Theorem 9.7.4, all maximal classes with only trivial congruences were determined. We begin with determining the congruences on the remaining maximal classes of the type C.

9.7 Congruences on the Maximal Classes of Pk

259

Theorem 9.7.4 Let be a unary relation on Ek with 2 ≤ || ≤ k − 1. Then the equivalence relation κ0, deﬁned by f n ∼ g m (κ0, ) ⇐⇒ n = m ∧ {f, g} ⊂ P olk ∧ ( ∀x ∈ n : f (x) = g(x) ), is the only nontrivial congruence on P olk . Proof. W.l.o.g. let := El , where 2 ≤ l ≤ k − 1. Since A := P olk El is a clone, the nontrivial congruences on A are arity congruences. Let κ ∈ Cona A\{κ0 }. Then the following two cases are possible: Case 1: κ0 ⊂ κ ⊆ κ0,El . Then there exist κ-congruent functions f n , g n ∈ P olk El , which are identical on tuples of Eln and which have diﬀerent values on a certain tuple a := (a1 , a2 , ..., an ) ∈ Ekn \Eln . Elements of P olk El are certain unary functions pa,b with pa,b (a) = b, a ∈ Ek \El , b ∈ Ek . Also elements of P olk El are the functions cai (x) if ai ∈ El , hi (x) := pa,ai (x) if ai ∈ Ek \El (i = 1, 2, ..., n; a ∈ Ek \El ). Consequently, it holds fa (x) := f (h1 (x), ..., hn (x)) ∼ ga (x) := g(h1 (x), ..., hn (x)) (κ), where fa (a) = ga (a) for all a ∈ Ek \El . For arbitrary κ0,El -congruent functions rm and sm of P olk El , there exists an (m + (k − l) · m)-ary function t with t(x1 , ..., xm , fl (x1 ), fl+1 (x1 ), ..., fk−1 (x1 ), ..., fl (xm ), fl+1 (xm ), ..., fk−1 (xm )) = r(x1 , ..., xm ) and t(x1 , ..., xm , gl (x1 ), gl+1 (x1 ), ..., gk−1 (x1 ), ..., gl (xm ), gl+1 (xm ), ..., gk−1 (xm )) = s(x1 , ..., xm ). Consequently, we have (r, s) ∈ κ and hence κ = κ0,El . Case 2: κ ⊆ κ0,El and κ ⊆ κa . In this case, there are certain κ-congruent functions that have diﬀerent values on a certain tuple of Eln . It is easy to show that this implies two diﬀerent constants ca and cb of P olk El are κ-congruent. Further, for arbitrary m-ary functions u and v of P olk El one can ﬁnd an (m + 1)-ary function w ∈ P olk El , for which w(x1 , ..., xm , a) = u(x1 , ..., xm ) and w(x1 , ..., xm , b) = v(x1 , ..., xm ) hold. Consequently, we have u(x) = w(x, ca (x1 )) ∼ v(x) = w(x, cb (x1 )) (κ), i.e., κ = κa holds. Thus P olk El has only four diﬀerent congruences.

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9 Congruences and Automorphisms on Function Algebras

Theorem 9.7.5 Let ∈ Uk and let Z := {ε1 , ..., εr } be the set of all equivalence classes of the equivalence relation . Then P olk has exactly four diﬀerent congruences. The only nontrivial congruence on P olk is the congruence κZ deﬁned by f n ∼ g m κZ :⇐⇒ n = m ∧ {f, g} ⊂ P olk ∧ (∀a ∈ Ekn ∃i ∈ {1, 2, ..., r} : {f (a), g(a)} ⊆ εi ). Proof. Denote κ be an arbitrary nontrivial congruence on P olk . Then, by Lemma 9.2.1, κ0 ⊂ κ ⊆ κa . Our theorem is proven if we can show the following facts: (∃f n , g n ∈ P olk ∃a ∈ Ekn : (f, g) ∈ κ ∧ (f (a), g(a)) ∈ \ι2k ) =⇒ κZ ⊆ κ, (∃f n , g n ∈ P olk ∃a ∈ Ekn : (f, g) ∈ κ ∧ (f (a), g(a)) ∈ ) =⇒ κ = κa . (9.17) If two diﬀerent functions f n , g n of P olk are κ-congruent, then it is easy to see that two diﬀerent constant functions ca , cb are also κ-congruent. For these constant functions, the following two cases are possible: Case 1: (a, b) ∈ \ι2k . Denote s and t two arbitrary m-ary functions of P olk with the property ∀a ∈ Ekm ∃i ∈ {1, 2, ..., r} : {s(a), t(a)} ⊆ ε. The following (m + 1)-ary function h with s(x1 , ..., xm ) if xm+1 = a, h(x1 , ..., xm+1 ) := t(x1 , ..., xm ) otherwise belongs to P olk . Thus h(x1 , ..., xm , ca (x1 )) = s(x) ∼ t(x) = h(x1 , ..., xm , cb (x1 )) (κ)

(9.18)

is valid. Hence, κZ ⊆ κ. Case 2: (a, b) ∈ . In this case, let s and t be two arbitrary m-ary functions of P olk . Then, a certain (m + 1)-ary function h with h(x1 , ..., xm , a) = s(x1 , ..., xm ) and h(x1 , ..., xm , b) = t(x1 , ..., xm ) belongs to P olk . Consequently, we obtain (s, t) ∈ κ as in (9.18). Thus κ = κa and (9.17) hold. The following theorem was already proven in Section 9.6 (see Corollary 9.6.7, (2).):

9.7 Congruences on the Maximal Classes of Pk

261

Theorem 9.7.6 Let k be a prime number power and λ ∈ Lk . Then {(f n , g n ) | ∃a ∈ Ek : f (x) = g(x) + a} κc := n≥1

is the only nontrivial congruence on P olk λ. Finally, denote an h-ary relation on Ek , which is a homomorphic inverse image of an elementary h-ary relation ξm ⊆ Ehm (h ≥ 3, m ≥ 1, k ≥ hm ). We use the same notations and representations for the functions of P olk as in Section 5.2.6. In addition, one can use Lemma 5.2.6.1 to deﬁne the following equivalence relation κN on P olk : Deﬁnition Let N be a normal subgroup of the symmetric group (Sh ; ). For arbitrary f n , g m ∈ P olk let (f n , g m ) ∈ κN if and only if n = m and for each i ∈ Em at least one of the following conditions is fulﬁlled: (a) |Im(fi )| ≤ h − 1 and |Im(gi )| ≤ h − 1; (b) there exist s1 , s2 ∈ Sh , j ∈ {1, 2, ..., n} and t ∈ Em so that fi (x1 , ..., xn ) = s1 ((q(xj ))(t) ), gi (x1 , ..., xn ) = s2 ((q(xj ))(t) ) and s1 N = s2 N. Lemma 9.7.7 The above-deﬁned equivalence relation κN is a congruence on P olk . Proof. Let (f n , g n ), (rm , sm ) ∈ κN be arbitrary. Obviously, (αf, αg) ∈ κN holds for every α ∈ {ζ, τ, ∆, ∇}. It remains to show that u := f s ∼ g t =: v (κN ), i.e., ui = fi r and vi := gi s satisfy (a) or (b) from the deﬁnition of κN for each i ∈ Em . By Lemma 5.2.6.1 the following cases are possible: Case 1: |Im(fi )| ≤ h − 1 and |Im(gi )| ≤ h − 1. Then |Im(ui )| ≤ h − 1 and |Im(vi )| ≤ h − 1, i.e., (a) is satisﬁed. Case 2: fi (x1 , ..., xn ) = s1 ((q(xj ))(t) ), gi (x1 , ..., xn ) = s2 ((q(xj ))(t) ), where s1 , s2 ∈ Sh , s1 N = s2 N , j ∈ {1, 2, ..., n} and t ∈ Em . Case 2.1: j = 1. Then ui (x1 , ..., xn+m−1 ) = s1 ((q(xj+m−1 ))(t) ) and vi (x1 , ..., xm+n−1 ) = s2 ((q(xj+m−1 ))(t) ), i.e., (b) is satisﬁed. Case 2.2: j = 1. Then ui (x1 , ..., xm+n−1 ) = s1 (rt (x1 , ..., xm ) and vi (x1 , ..., xm+n−1 ) = s2 (st (x1 , ..., xm )). Case 2.2.1: |Im(rt )| ≤ h − 1 and |Im(st )| ≤ h − 1. Then |Im(ut )| ≤ h − 1 and |Im(vt )| ≤ h − 1, i.e., (a) is satisﬁed. Case 2.2.2: rt (x1 , ..., xm ) = s3 ((q(xl ))(p) , st (x1 , ..., xm ) = s4 ((q(xl ))(p) , {s3 , s4 } ∈ Sh , s3 N = s4 N , l ∈ {1, 2, ..., m} and p ∈ Em . Then we have ui (x1 , ..., xm+n−1 ) = s1 (s3 (q(xl ))(p) )) and vi (x1 , ..., xm+n−1 ) =

262

9 Congruences and Automorphisms on Function Algebras

s2 (s4 (q(xl ))(p) )). Moreover, from the assumptions and from the well-known properties of normal subgroups we know that (s1 s3 ) N = (s1 (s3 N ) = (s1 (s4 N ) = s1 (N s4 ) = (s1 N ) s4 = (s2 N ) s4 = (s2 s4 ) N. Hence (b) is satisﬁed. Lemma 9.7.8 Let κ be a congruence on P olk and let e denote the unit of the symmetric group Sh . Then κ0 ⊂ κ ⊆ κa =⇒ κ{e} ⊆ κ. Proof. Let κ0 ⊂ κ ⊆ κa . Clearly, T := {f ∈ Pk | |Im(q(f ))| ≤ h − 1} ⊆ P olk . It is easy to see that we have only trivial congruences on T . From κ ⊂ κ ⊆ κa we get the existence of two diﬀerent unary constants that are κ-congruent. Consequently, all functions of the same arity of T are mutually κ-congruent. Then, with the help of Section 5.2.6, we have κ{e} ⊆ κ. Lemma 9.7.9 Let κ be a congruence on P olk , let N be a normal subgroup of the symmetric group Sh and let κN ⊂ κ ⊆ κSh . Then there exists a normal subgroup N of Sh with N ⊂ N ⊆ Sh so that κN ⊆ κ. Proof. W.l.o.g. we can assume that = ξm , i.e., k = hm and q is the identity mapping. Because of κN ⊂ κ ⊆ κSh there exist certain κ-congruent n-ary functions f , g with (t) (t) fi (x1 , ..., xn ) = s1 (xj ), gi (x1 , ..., xn ) = s2 (xj ) for certain i, t ∈ Em , j ∈ {1, 2, ..., n} and s1 , s2 ∈ Sh with s1 N = s2 N . The functions u(x) := x(i) and v(x) := x(0) ·ht belong to P olξm . Consequently, since (f, g) ∈ κ, we have: ((∆n−1 (u f )) v)(x) = s1 (x(0) ) ∼ ((∆n−1 (u g)) v)(x) = s2 (x(0) ) (κ).

(9.19)

It is well-known that the symmetric group Sh has only congruences of type κN , where N is a normal subgroup of Sh . Therefore (9.19) together with s1 N = s2 N imply the existence of a normal subgroup N so that N ⊂ N ⊆ Sh and ∀s, s ∈ Sh : (s N = s N =⇒ s(x(0) ) ∼ s (x(0) ) (κ) ). Then, by replacing the variable x by the function x(t) ∈ P ol ξm and with the aid of the operations ∇, ζ, τ , we get: (t)

(t)

s N = s N =⇒ a(x1 , ..., xr ) := s(xj ) ∼ s (xj ) =: b(x1 , ..., xr ) (κ) for every r ≥ 1, j ∈ {1, 2, ..., r}, t ∈ Em and s, s ∈ Sh . Hence κN ⊆ κ.

9.7 Congruences on the Maximal Classes of Pk

263

Lemma 9.7.10 Let κ be a congruence on P olk with the properties κ ⊆ κSh and κ ⊆ κa . Then κ = κa . Proof. W.l.o.g. let = ξm . Since κ ⊆ κSh and κ ⊆ κa , there exist κcongruent functions f n , g n with (f, g) ∈ κSh , i.e., there exists an i ∈ Em with (fi , gi ) ∈ κSh . We distinguish the following cases: Case 1: |Im(fi )| ≤ h − 1 and |Im(gi )| = h. Then by Lemma 9.7.8 we have (fi , cna ) ∈ κ for every a ∈ Ek . Hence gi (x, ..., x) =: s1 (x(t) ) ∼ ca (x) (κ) and (r) · ht )(t) ))) = s2 (x(r) ) s2 (s1 (s−1 1 ((x (r) ∼ ca (s1 (s−1 · ht )(t) ))) = ca (x) (κ) 1 ((x

for every s2 ∈ Sh , r ∈ Em and a ∈ Ek . Using the operations ∇, τ , ζ we get ∀s ∈ Sh ∀j ∈ {1, ..., n} ∀t ∈ Em ∀a ∈ Ek : (t)

p(x1 , ..., xn ) := s(xj ) ∼ cna (x1 , ..., xn ) (κ).

(9.20)

Moreover, since all n-ary functions f with |Im(f )| ≤ h − 1 are mutually κcongruent, κ = κa follows from (9.20) in the ﬁrst case. (t) (t) Case 2: fi (x1 , ..., xn ) = s1 (xj ), gi (x1 , ..., xn ) = s2 (xl ) and (j, t) = (l, r) (j, l ∈ {1, 2, ..., n}; s1 , s2 ∈ Sh ; t, r ∈ Em ). Case 2.1: j = l. Then, because of (fi , gi ) ∈ κ, fi (x, ..., x, c0 (x), x, ..., x) = s1 (x(t) ) l−1

∼ fi (x, ..., x, c0 (x), x, ..., x) = cs2 (0) (x) (κ) l−1

and thus by Case 1: κ = κa . Case 2.2: t = r. Since (fi , gi ) ∈ κ, we have (s1 (x(t) ), s2 (x(r) )) ∈ κ. Therefore s1 ((x(0) · ht )(t) ) = s1 (x(0) ) ∼ s2 ((x(0) · hr )(t) ) = cs2 (0) (x) (κ). Hence, again by Case 1, κ = κa .

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9 Congruences and Automorphisms on Function Algebras

Theorem 9.7.11 Let V be the four group in S4 and let Ah be the alternating group of Sh . Furthermore, let e be the unit of Sh . For h = 3 and h ≥ 5 the maximal class P olk with ∈ Bhk has only the nontrivial congruences κ{e} , κAh and κSh . A maximal class P olk with ∈ B4k has the nontrivial congruences κ{e} , κA4 , κS4 and, in addition, the nontrivial congruence κV on P olk . Proof. Our claim follows from Lemmas 9.7.7–9.7.10 and from well-known theorems in group theory.

q κ1 q κa if ∈ Ck \{ ∈ C1k | || ≥ 2} ∪ Mk ∪ Sk q κ0

q κa q κ0,

q κ1

q κ1

q κ1 if ∈ { ∈ C1k | || ≥ 2}

q κa q κZ

q κ0

q κ0

q κ1

q κ1

q κa

q κa

q κa q κc

if ∈ Lk

q κ0

q κS4

q κSh q κAh if ∈

if ∈ Uk

k h=3,h =4

Bhk

q κA4 if ∈ B4k

q κ{e}

q κV

q κ0

q κ{e} q κ0

Fig. 9.2. Congruence lattices of the maximal classes P olk of Pk

9.8 Congruences on Subclasses of [Pk1 ]

265

When one summarizes the above Theorems 9.7.3–9.7.6 and 9.7.11, one sees that Theorem 9.7.12 (Congruence Theorem for Maximal Clones; [Lau 79a;81]) For a maximal class A := P olk , ∈ Rmax (Pk ) (see Chapter 5 and 6) of Pk it holds |Con A| ∈ {3, 4, 6, 7} and |Con A| = 3 ⇐⇒ ∈ (Ck \{ ∈ C1k | || ≥ 2}) ∪ Mk ∪ Sk |Con A| = 4 ⇐⇒ ∈ { ∈ C1k | || ≥ 2} ∪ Uk ∪ Lk k |Con A| = 6 ⇐⇒ ∈ h=3, h=4 Bhk |Con A| = 7 ⇐⇒ ∈ B4k . The possible congruence lattices of a maximal class P olk of Pk are given in Figure 9.2.

9.8 Congruences on Subclasses of [Pk1 ] In this section, we describe the unary functions f ∈ Pk as follows: Let κf := {(a, b) ∈ Ek2 | f (a) = f (b)}

(9.21)

be the mapping equivalence of f . Denote F1 , ..., Fr the equivalence classes of this equivalence relation. Then there are some a1 , ..., ar ∈ Ek with ∀i ∈ {1, 2, ..., r} : ∀x ∈ Fi : f (x) = ai ,

and we write: f=

F1 F2 ... Fr a1 a2 ... ar

.

Before we consider some general properties of the congruences on subclasses of [Pk1 ], we deal ﬁrst with the congruences of the semigroup (Pk1 ; ). We will see later that these congruences induce the congruences on the class [Pk1 ] (see Theorem 9.8.4). Deﬁnition

Let r ∈ {2, 3, ..., k} and

µr (k) := {(f, g) ∈ Pk1 × Pk1 | f = g ∨ (|Im(f )| < r ∧ |Im(g)| < r }. (9.22) Furthermore, let N be a normal subgroup of the symmetric group S{1,2,...,r} . For arbitrary f, g ∈ Pk1 let (f, g) ∈ µr,N (k) if and only if the functions f and g fulﬁll the following conditions:

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9 Congruences and Automorphisms on Function Algebras

(1) (f, g) ∈ µr (k). (2) |Im(f )| = |Im(g)| = r and f , g are deﬁned by F1 F2 ... Fr F1 F2 ... Fr f= ,g= , a1 a2 ... ar ai1 ai2 ... air where

1 2 ... r i1 i2 ... ir

∈N

holds. If k results from the context, we only write µr (or µr,N ) instead of µr (k) (or µr,N (k)). We remark that µr = µr,{e} holds, where e is the unit of the group S{1,2,...,r} . Lemma 9.8.1 The above-deﬁned relation µr,N (k) is a congruence on the semigroup (Pk1 ; ). Proof. With the help of the group properties of N one can see that µr,N is an equivalence relation. To prove the compatibility of µr,N with , it suﬃces to prove the statement ∀(f, g) ∈ µr,N ∀h ∈ Pk1 : (h f, h g) ∈ µr,N ∧ (f h, g h) ∈ µr,N (9.23) (see Lemma 9.1.1). If (f, g) ∈ µr or |Im(h)| < r, then (9.23) is obviously valid. Let (f, g) ∈ µr,N \µr and |Im(h)| ≥ r in the following. Consequently, we can assume F1 F2 ... Fr F1 F2 ... Fr 1 2 ... r f= ,g= , ∈ N. a1 a2 ... ar ai1 ai2 ... air i1 i2 ... ir If bi := h(ai ), i = 1, 2, ..., r, then F1 F2 ... Fr F1 F2 ... Fr and h g = . hf = b1 b2 ... br bi1 bi2 ... bir Thus (h f, h g) ∈ µr,N . Let Hi := {x ∈ Ek | f (h(x)) = ai }, i = 1, 2, ..., r. If there exists i with Hi = ∅, then we have (f h, g h) ∈ µr . If Hi = ∅ for all i, then H1 H2 ... Hr H1 H2 ... Hr and g h = , f h= a1 a2 ... ar ai1 ai2 ... air whereby (f h, g h) ∈ µr,N holds.

9.8 Congruences on Subclasses of [Pk1 ]

267

The next lemma gives some properties of congruences of (Pk1 ; ) that we need, to prove Theorem 9.8.3. Lemma 9.8.2 Let κ be a congruence of the semigroup (Pk1 ; ). Then: (a) κ = κ0 =⇒ (∀a, b ∈ Ek : (ca , cb ) ∈ κ); (b) ( (f, ca ) ∈ κ ∧ |Im(f )| ≥ 2 ) =⇒ µ|Im(f )|+1 ⊆ κ; (c) ( (f, g) ∈ κ ∧ |Im(f )| > |Im(g)| ) =⇒ µ|Im(f )|+1 ⊆ κ; (d) ( (f, g) ∈ κ ∧ |Im(f )| = |Im(g)| ≥ 2 ∧ Im(f ) = Im(g) ) µ|Im(f )|+1 ⊆ κ; (e) ( (f, g) ∈ κ ∧ Im(f ) = Im(g) ∧ κf = κg ) =⇒ µ|Im(f )|+1 ⊆ κ;

=⇒

3

(f ) ( (f, g) ∈ κ ∧ Im(f ) = Im(g) ≥ 2 ∧ κf = κg ∧f = g ) =⇒ µ|Im(f )| ⊆ κ; Proof. (a) and (b) are easy to check. (c): Let (f, g) ∈ κ and |Im(f )| > |Im(g)|. If |Im(g)| = 1, (c) follows from (b). Therefore we can assume |Im(g)| ≥ 2. In addition, consider only the case Im(g) ⊂ Im(f ) suﬃces, since one can ﬁnd an h ∈ Pk1 with h f = f , h g =: g and Im(f ) ⊂ Im(g ). Therefore, let Im(g) = {a1 , ..., au } and Im(f ) = {a1 , ..., au , ..., ar }. Then, with the help of the functions x ∈ {au+1 , ..., ar }, au+1 if p(x) := x otherwise

and tj (x) := we obtain

a1 if x ∈ {a1 , ..., aj }, x otherwise,

fu+1 := p f ∼ p g = g (κ) fu+2−j := tj fu+1 ∼ tj g =: gu+1−j (κ),

for j = 2, 3, ..., u, where the indices of the functions agree with the cardinalities of their ranges of values. We can rewrite the above as follows: g1 ∼ f2 , g2 ∼ f3 , g3 ∼ f4 , ... gu ∼ fu+1 , fu+1 ∼ f (κ). This, (b) and the transitivity of κ imply our assertion (c). (d): Let a ∈ Im(g), b ∈ Im(g)\Im(f ) and a if x = b, q(x) := x otherwise. Then (f, g) ∈ κ implies q f = f ∼ q g =: g (κ), where |Im(g )| < |Im(f )|. Thus, (d) follows from (c). (e): If Im(f ) = Im(g) and κf = κg , there exist a, b ∈ Ek with (a, b) ∈ κf and (a, b) ∈ κg . Then, with the help of the above deﬁned function q, we have 3

κf , κg denote the mapping equivalences of f or g, respectively (see (9.21)).

268

9 Congruences and Automorphisms on Function Algebras

f q = f ∼ g q =: g (κ) with |Im(g )| < |Im(f )|. Hence (e) follows from (c). (f): Let (f, g) ∈ κ, f = g, r ≥ 2 and F1 F2 ... Fr Fi1 Fi2 ... Fir f= and g = , a1 a2 ... ar a1 a2 ... ar 1 2 ... r where denotes a certain permutation. If r = 2 then (f) follows i1 i2 ... ir from that because of (a). If r > 2 then there exist indices u, v with {u, v} = {iu , iv } so that Fu ∪ Fv = Fiu ∪ Fiv holds. Then, with the help of the function au if x ∈ {au , av }, w(x) := x otherwise. and through f1 := w f ∼ w g =: g1 (κ), one gets that two functions f1 , g1 are κ-congruent, where |Im(f1 )| = |Im(g1 )| = r − 1 and the mapping equivalences κf1 and κg1 are diﬀerent. Consequently, (f) follows from (e). Theorem 9.8.3 (A. I. Mal’tsev’s Theorem; [Mal 52]) (a) The only nontrivial congruences on (Pk1 ; ) are congruences of the form κr,N , where r ∈ {2, 3, ..., k} and N is an arbitrary normal subgroup of the symmetric group S{1,2,...,r} . (b) Let Sr be the group of all permutations on the set {1, 2, ..., r}, and let Ar (⊂ Sr , r ≥ 2) be the alternating group and let V (⊂ S4 ) be the four group. Furthermore, denote Con(k) the set of all congruences of the semigroup (Pk1 ; ). Then: Con(2) = {κ0 , µ2 , µ2,S2 , κ1 }; Con(3) = {κ0 , µ2 , µ2,S2 , µ3 , µ3,A3 , µ3,S3 , κ1 }; Con(4) = {κ0 , µ2 , µ2,S2 , µ3 , µ3,A3 , µ3,S3 , µ4 , µ4,V , µ4,A4 , µ3,S4 , κ1 }; Con(k) = {κ0 , µ2 , µ2,S2 , µ4,V , κ1 } ∪ r∈{3,4,5,...,k} {µr , µr,Ar , µr,Sr } for k ≥ 5. Proof. (a): Let κ (= κ0 ) be a congruence of the semigroup (Pk1 ; ). Then, by Lemma 9.8.2, (b)–(f), we have an r ∈ {2, ..., k − 1} with µr ⊆ κ ⊂ µr+1 . To prove that a normal subgroup N with κ = µr,N exists, we consider the functions F1 F2 ... Fr F1 F2 ... Fr f := f := a1 a2 ... ar aα(1) aα(2) ... aα(r) F1 F2 ... Fr F1 F2 ... Fr g := g := b1 b2 ... br bβ(1) bβ(2) ... bβ(r) of Pk1 . Further, let

h1 (x) :=

9.8 Congruences on Subclasses of [Pk1 ]

bi if ∃i : x = ai , h2 (x) := x otherwise,

269

aγ(i) if ∃i : x = ai , x otherwise,

where γ is an arbitrary permutation from Sr . Now, let F1 , ..., Fr be a partition of Ek and let N be the set of all permutations α ∈ Sr for which (f, f ) ∈ κ holds. If α, β ∈ N , then we have (f, f ), (g, g ) ∈ κ and therefore also g = h1 f ∼ h1 f (κ) and (because of transitivity) h1 f ∼ g (κ); thus, according to the deﬁnition of N the following is valid: α(1) α(2) ... α(r) = β α−1 ∈ N. β(1) β(2) ... β(r) Therefore N is a group. The fact that N is a normal group results from (f, f ) ∈ κ =⇒ h2 f ∼ h2 f (κ) γ(1) ... γ(r) =⇒ = γ α γ −1 ∈ N. γ(α(1)) ... γ(α(r)) For another partition Q1 , ..., Qr of Ek and for arbitrary functions Q1 Q2 ... Qr Q1 Q2 ... Qr g := , q := d1 d2 ... dr dδ(1) dδ(2) ... dδ(r) with δ ∈ Sr we must still show that (q, q ) ∈ κ if and only if δ ∈ N . If (q, q ) ∈ κ then δ ∈ N follows from (q h3 , q h3 ) ∈ κ, where h3 (Fi ) ∈ Qi (i = 1, ..., r). If δ ∈ N then there are κ-congruent functions h and h with F1 F2 ... Fr F1 F2 ... Fr h := , h := d1 d2 ... dr dδ(1) dδ(2) ... dδ(r) and (q, q ) ∈ κ follows from (h h4 , h h4 ) ∈ κ, where h4 (Qi ) ∈ Fi (i = 1, 2, ..., r). (b) follows from well-known results over the normal subgroups of the symmetric group. The following concept comes from the paper [Mal 76] by I. A. Mal’tsev. Deﬁnitions Let A be a subclass of [Pk1 ], which contains a certain constant ca . Furthermore, let α be a congruence on the semigroup (A1 ; ). Then, the α-induced relation κα is deﬁned as follows: For arbitrary (f n , g m ) ∈ A × A let (f n , g m ) ∈ κα iﬀ n = m and the functions f and g fulﬁll one of the two following conditions: (1) (∆n−1 f, ca ) ∈ α (or (∆n−1 g, ca ) ∈ α); (2) ∃i ∈ {1, ..., n} ∃f1 , g1 ∈ A1 : ( f (x1 , ..., xn ) = f1 (xi ) ∧ g(x1 , ..., xn ) = g1 (xi ) ∧ (f1 , g1 ) ∈ α ). We say that the congruence κ on A is induced from the congruence α on Pk1 iﬀ κ = κα . The following theorem is a special case of Theorem 9.8.8:

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9 Congruences and Automorphisms on Function Algebras

Theorem 9.8.4 ([Mal 76]) Let ca ∈ A = [A] ⊆ [Pk1 ]. Then (a) Each relation, induced from a congruence on the semigroup (A1 ; ), is a congruence on A. (b) All nontrivial arity congruences on A are given from the nontrivial congruences on A1 . The next theorem is a consequence from the above theorem and Theorem 9.8.3: Theorem 9.8.5 ([Mal 76]) On [Pk1 ] there are exactly

⎧ if k = 2, ⎨5 8 if k = 3, ⎩ 3 · k if k ≥ 4

diﬀerent congruences. The nontrivial congruences on [Pk1 ] are induced from the congruences of the type µr,N (k) (see Lemma 9.8.1) on Pk1 . The rest of this section is the summary of an unpublished manuscript by B. Strauch. Deﬁnition and Declarations Let H be a subsemigroup of (Pk1 ; ) and let µ be a congruence on H. Further, let {H1 , ..., Hl } be the partition correlated to the relation µ, where the indexes of the blocks Hi are so chosen that the following is valid: ∀i ∈ {1, 2, ..., s} ∀h ∈ H : Hi h ⊆ Hi

(9.24)

∀i ∈ {s + 1, ..., l} ∃h ∈ H : Hi h ⊆ Hi .

(9.25)

and We set s = 0 if there is not any block with the property (9.24). For functions f n ∈ [Pk1 ], we also write f instead of ∆n−1 f and put ess(f ) := i, if f (x1 , ..., xn ) = f (xi ) with f ∈ Pk1 [1]. If f n is a constant, then let ess(f ) := 0. With the help of µ, one can deﬁne the following relations on [H]:

9.8 Congruences on Subclasses of [Pk1 ]

271

αµ := µ ∪ {(f n , g m ) ∈ [H] × [H] | n = m ∧ (f , g ) ∈ µ ∧ (ess(f ) = ess(g) ∨ ∃i ∈ {1, ..., s} : {f , g } ⊆ Hi ) }, βµ := µ ∪ {(f n , g m ) ∈ [H] × [H] | n = m ∧ (f , g ) ∈ µ ∧ ess(f ) = ess(g)}, γµ := {(f n , g m ) ∈ [H] × [H] | n = m ∧ (f , g ) ∈ µ }. Lemma 9.8.6 Let H and µ be deﬁned as above. Then (a) ∀f ∈ Hs+1 ∪ ... ∪ Hl : ∀g ∈ H1 ∪ ... ∪ Hs : f g ∈ H1 ∪ ... ∪ Hs ; (b) (∃t ∈ {1, ..., l} ∃a ∈ Ek : ca ∈ Ht ) =⇒ t ∈ {1, 2, ..., s}. Proof. (a): Let f and g be given as in (a) and let (q, f g) ∈ µ. (a) is proven if we can show ∀h ∈ H : (q h, q) ∈ µ. (9.26) (9.26) results, however, from q h ∼ f (g h) ∼ f g ∼ q (µ) because of g ∈ H1 ∪ ... ∪ Hs . (b): If {ca , f } ⊆ Hi then for every h ∈ H: ca h = ca ∼ f h (µ). Therefore, f h ∈ Hi and thus i ∈ {1, 2, ..., s}. Lemma 9.8.7 Let H and µ be deﬁned as above. Then (a) αµ is a congruence on [H] with α|H = µ. (b) βµ is a congruence on [H] with β|H = µ, iﬀ H does not contain any constant function or the blocks H1 , ..., Hs of µ consist only of constant functions. (c) αµ = βµ iﬀ s = 0 or the blocks H1 , ..., Hs of µ consist only of constant functions. (d) γµ is a congruence on [H] with γ|H = µ, iﬀ s = l holds, i.e., iﬀ every block Hi of µ fulﬁlls the condition (9.24). Proof. (a): We start with proof that αµ is an equivalence relation on [H]. The reﬂexivity and the symmetry of αµ result immediately from the deﬁnition of αµ . One can prove the transitivity of αµ as follows: Let (f n , g n ), (g n , hn ) ∈ αµ be arbitrary. Then (f , g ), (g , h ) ∈ µ and (by the transitivity of µ) (f , h ) ∈ µ. In the case ess(f ) = ess(g) = ess(h) we have (f, h) ∈ αµ . Otherwise, we have s ≥ 1 and there exists an i ∈ {1, ..., s} with {f , g } ⊆ Hi or {g , h } ⊆ Hi . As already was proven, the functions f , g , h are µ-equivalent. Therefore {f , g , h } ⊆ Hi and thus (f n , hn ) ∈ αµ . Next we show the compatibility of αµ with the superposition operations. For the unary operations, this is clear. Let (f, g) ∈ αµ and t ∈ [H] be arbitrary. To prove the compatibility of αµ with , it is suﬃcient to show that (tf, tg), (f t, g t) ∈ αµ holds. One can check (f t, g t) ∈ αµ easily by deﬁning αµ . If t is a constant function or ess(t) ≥ 2 holds, then also (t f, t g) ∈ αµ . Therefore, we can assume ess(t) = 1. Then (t f ) = t f ∼ (t g) = t g (µ). If ess(f ) = ess(g) then also ess(t f ) = ess(t g). This implies (t f, t g) ∈ αµ . In the case ess(f ) = ess(g), there is an i ∈ {1, ..., s} with {f , g } ⊆ Hi .

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Let Hr be the block to which the functions t f and t g belong. Then, r ≤ s and thus (t f, t g) ∈ αµ follows from the following: Let q ∈ Hr and h ∈ H be arbitrary. Because of {q, t f } ⊆ Hr and {f h, f } ⊆ Hi it holds: q h ∼ (t f ) h = t (f h) ∼ t f ∼ q (µ). Consequently, we have Hr h ⊆ Hr for arbitrary h ∈ H. Hence, αµ is a congruence on [H]. One checks the remaining statements of the lemma by means of Lemmas 9.8.6 and 9.3.8. Theorem 9.8.8 Let H := (H; ) be a semigroup with H ⊆ Pk1 . Then, one can describe the congruences on [H] as follows: (1) αµ with µ ∈ Con H; (2) βµ with µ ∈ Con H, if H does not contain any constant function or the equivalence classes of µ, which (9.24) fulﬁlls, consist only of constant functions; (3) γµ with µ ∈ Con H and every equivalence class of µ fulﬁlls the condition (9.24). Proof. First, let κ be an arbitrary arity congruence on [H] and let µ := κ ∩ (H × H). Obviously, µ is a congruence of the semigroup (H; ) and we have (by the compatibility of κ with the operations ∇, ζ, τ ) βµ ⊆ κ. If βµ = κ, then, by Lemma 9.8.7, µ and H fulﬁll the conditions given in (2). Therefore, let βµ ⊂ κ. Then there are some κ-congruent functions f n , g n with f (x1 , ..., xn ) = f (xi ), g(x1 , ..., xn ) = g (xj ) and i < j. Consequently, it holds f1 (x, y) := f (x, ..., x, y, y, ..., y) = f (x) ∼ i

g1 (x, y) := g(x, ..., x, y, y, ..., y) = g (y)

(κ)

i

Then, for arbitrary q ∈ H with (q, f ) ∈ κ, we have q1 (x, y) := q(x) ∼ f1 (x, y) ∼ g1 (x, y) (κ). Thus

q h = ∆(q1 h) ∼ ∆(g1 h) = g (κ)

for every h ∈ H. Consequently, the equivalence class (in respect to µ), to which the functions f and g belong, fulﬁlls the condition (9.24). Thus κ ⊆ αµ holds. To prove αµ ⊆ κ let (rm , sm ) ∈ αµ . If ess(r) = ess(s) then obviously, (r, s) ∈ κ. In the case ess(r) = ess(s), it is suﬃcient to consider (w.l.o.g.) m = 2, r(x, y) = r (x) and s(x, y) = s (y). By deﬁnition of the relation αµ the equivalence class F of µ, to which r and s belong, has the property F H ⊆ F.

(9.27)

9.9 Congruences on Some Subclasses of Pk,l

273

With the help of the κ-congruent functions f1 , g1 we have r f1 ∼ s g1 (κ). Because of (9.27), {(r f , r ), (s g , s )} ⊆ µ is valid. Thus {(r f1 , r), (s g1 , s)} ⊆ κ. Then, the transitivity of κ implies (r, s) ∈ κ. Therefore αµ ⊆ κ and our statement on the arity congruences on [H] was proven. Finally, let κ be an arbitrary non-arity congruence on [H] and let µ be the restriction of κ onto H. Then, because of Lemma 9.3.8, the inclusion κ ⊆ γµ is valid and the relation µ has the properties mentioned in our theorem. To prove γµ ⊆ κ, it is suﬃcient to show that an arbitrary function hnn ∈ [H] with hnn (x1 , ..., xn ) = h (xn ) is κ-congruent to h (= h11 ) . If κ ⊆ κa there are κ-congruent functions f 2 , g 1 ∈ [H]. Consequently, we have h2 = ∆((∇h ) f ) ∼ ∆((∇h ) g) = h1 (κ), ∇h2 = h3 ∼ ∇h1 = h2 (κ), etc. Thus hn ∼ h (κ) and our theorem was proven.

9.9 Congruences on Some Subclasses of Pk,l In this section, we determine all congruences on a class A of Pk,l , which can be described with the aid of the homomorphism pr−1 (see Chapter 12): ∃B ∈ L↑l (Jl ) : A = pr−1 B,

(9.28)

i.e., A is an inverse image of a clone of Pl . Denote A the set of the subclasses of Pk,l of the form (9.28). Deﬁnition Let U1 , U2 , ..., Ut be some nonempty subsets of Ek \El and Tn (U1 , U2 , ..., Ut ) := {(a1 , a2 , ..., an ) ∈ Ekn | ∃ i ∈ {1, 2, ..., t} : Ui ⊆ {a1 , ..., an }}. Let κU1 ,...,Ut be a relation on a subclass of Pk,l deﬁned by (f n , g m ) ∈ κU1 ,...,Ut :⇐⇒ n = m ∧ (∀a ∈ Ekn \Tn (U1 , ..., Ut ) : f (a) = g(a) ). If U = {U1 , ..., Ut } we write brieﬂy κU instead of κU1 ,...,Ut . One can prove the following lemma easily.

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Lemma 9.9.1 (a) The above-deﬁned relation κU is a congruence on an arbitrary subclass of Pk,l . (b) Let U1 , U2 and U3 subsets of Ek \El . Then (1) κU1 ⊆ κU1 ,U2 , (2) κU1 ⊆ κU2 ⇐⇒ U2 ⊆ U1 , (3) κU1 ,U2 ⊆ κU3 ⇐⇒ U3 ⊆ U1 ∩ U2 , (4) (κU1 ⊆ κU2 ∧ κU2 ⊆ κU1 ) ⇐⇒ (U1 ⊆ U2 ∧ U2 ⊆ U1 ), (5) κU1 ,U2 = κU1 ⇐⇒ U1 ⊆ U2 . Theorem 9.9.2 ([Mal 79]) Let A ∈ A and let L be the set of all congruences of the form κU on A (see above). Then, (L; ⊆) is a free distributive lattice with k − l generators. Proof. For i = 1, 2 let Ui := {Ui1 , Ui2 , ..., Uiti } and let κUi be a congruence on A ∈ A. We write κU1 ≤ κU2 , if U1 ⊆ U2 and if for every U2i ∈ U2 there exists a certain U1j ∈ U1 with U1j ⊆ U2i . It results from the deﬁnition of the congruences of the type κU that U1 ≤ U2 implies κU1 = κU2 . Consequently, during the description of the congruences of the form κU , we can restrict ourselves to such sets U with the following property: ∀U ∀U ((U ∈ U ∧ U ⊆ U ⊆ Ek \El ) =⇒ U ∈ U). We call such sets J-closed. The set of all J-closed subsets of P(Ek \El ) forms a lattice. Consequently, because of (U1 ≤ U2 ∧ U2 ≤ U1 ) =⇒ κU1 = κU2 , and (U1 , U2 are J-closed ∧ U1 ⊆ U2 ) =⇒ κU1 ⊆ κU2 there is an isomorphism from the lattice of the congruences of the type κU onto the lattice of the J-closed subsets of Ek \El . Hence, our theorem follows from [Bir 48] (p. 146, Theorem 13). As a direct consequence of Lemma 9.3.5, we get: Lemma 9.9.3 Let A ∈ A. Then, congruence κ1 is the only non-arity congruence on A. Lemma 9.9.4 Let A ∈ A, ∅ = U ⊆ Ek \El , κ a congruence on A and let f n , g n be κ-congruent functions of A, for which there exists an a ∈ (U ∪ El )n ∩ Tn (U ) with f (a) = g(a). Then, κU ⊆ κ. Proof. Let a := (a1 , a2 , ..., an ). We can assume w.l.o.g. {a1 , a2 , ..., ar } = U , |U | = r and {ar+1 , ..., an } ⊆ El . The functions x if x ∈ El , ta (x) := a otherwise

9.9 Congruences on Some Subclasses of Pk,l

275

(a ∈ El ) belong to A. Consequently, we have f (x1 , ..., xr , tar+1 (x1 ), ..., tan (x1 )) =: f (x1 , ..., xr ) ∼ g(x1 , ..., xr , tar+1 (x1 ), ..., tan (x1 )) =: g (x1 , ..., xr ) (κ) and

f (a1 , ..., ar ) = g (a1 , ..., ar ).

Let f1 and f2 be arbitrary m-ary functions of A, which are identical on tuples of Ekm \Tm (U ), m ≥ r. We have to show that (f1 , f2 ) ∈ κ holds. We start with same notations: Let I = (i1 , ..., ir ) be a variation4 of the numbers 1, 2, ..., m, xi := (xi1 , ..., xir ) and let I1 , I2 , ..., It be all possible variations of the numbers 1, 2, ..., m taken m! ). r at a time (i.e., t = (m−r)! The function h(x1 , ..., xm , z1 , ..., zt ) := (x1 , ..., xm ) ∈ Tm (U ) ∧ (∀i : zi = f (xIi )), f1 (x1 , ..., xm ) if f2 (x1 , ..., xm ) otherwise. belongs to A. Consequently, we have h(x1 , ..., xm , f (xI1 ), ..., f (xIt )) = f1 (x1 , ..., xm ) ∼ h(x1 , ..., xm , g (xI1 ), ..., g (xIt )) = f2 (x1 , ..., xm ) (κ). Thus κU ⊆ κ. Lemma 9.9.5 Let A ∈ A, let κ be a congruence on A and let U1 , U2 , ..., Ut , Ut+1 ⊆ Ek \El . Then (κU1 ,U2 ,...,Ut ⊆ κ ∧ κUt+1 ⊆ κ) =⇒ κU1 ,U2 ,...,Ut ,Ut+1 ⊆ κ. Proof. If there is an i ∈ {1, ..., t} with Ai ⊆ At+1 , then by Lemma 9.9.1, (b) we have: κU1 ,U2 ,...,Ut = κU1 ,U2 ,...,Ut ,Ut+1 ⊆ κ. Therefore, we can assume that no U ∈ {U1 , ..., Ut } is a subset of At+1 . Furthermore, let f n and g n be arbitrary functions of A, which are identical on all tuples of Ekn \Tn (U1 , ..., Ut , Ut+1 ). Because of κU1 ,...,Ut ⊆ κ we have f (x1 , ..., xn ) ∼ f (x1 , ..., xn ) :=

g(x1 , ..., xn ) if (x1 , ..., xn ) ∈ Tn (U1 , ..., Ut ), f (x1 , ..., xn ) otherwise

(κ). Since κUt+1 ⊆ κ, it holds further that f ∼ g (κ). 4

Each arrangement of r diﬀerent objects of a set with n diﬀerent elements into a particular order is called a variation of n elements taken r at a time.

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9 Congruences and Automorphisms on Function Algebras

Thus (f, g) ∈ κ. Consequently, κU1 ,...,Ut ,Ut+1 ⊆ κ. We come now to the congruences on A ∈ A which are completely determined by the congruences on prl A. So that we can better distinguish the congruences on prl A from the congruences on A, we denote the congruences on prl A with large Greek alphabetic characters. In particular, let K0 , Ka and K1 be the trivial congruences on prl A. One can check the following lemma easily. Lemma 9.9.6 Let A ∈ A and let K be a congruence on prl A. Then the relation deﬁned by (f, g) ∈ prl−1 K :⇐⇒ (prl f, prl g) ∈ K is a congruence on A. In particular, it holds: prl−1 K1 = κ1 and prl−1 Ka = κa . Lemma 9.9.7 Let A ∈ A and let κ be a congruence on A. If there exist κ-congruent functions f n , g n ∈ A, for which there is a tuple a := (a1 , a2 , ..., an ) ∈ Eln with f (a) = g(a), then prl−1 K0 ⊆ κ. Proof. The functions ta , a ∈ El , which we have deﬁned in the proof of Lemma 9.9.4, belong to A. Consequently, (f, g) ∈ κ implies f (x) := f (ta1 (x), ..., tan (x)) ∼ g(ta1 (x), ..., tan (x)) =: g (x) (κ). Because of f (a) = g(a) we have f (c) = g (c) for all c ∈ Ek \El . With the aid of Lemma 9.9.4 it follows that κ{l} , κ{l+1} , ..., κ{k−1} ⊆ κ. Then, by Theorem 9.8.4, we have: κ{l},{l+1},...,{k−1} = prl−1 K0 ⊆ κ. Lemma 9.9.8 Let κ be a congruence on A ∈ A. Then prl κ = K0 =⇒ κ = prl−1 (prl κ). Proof. Obviously, K := prl κ is a congruence on prl A and we have κ ⊆ prl−1 K. If prl κ = K1 then there exists some functions f n , g m ∈ A with (f, g) ∈ κ and n = m. Thus, by Lemma 9.9.3, κ = κ1 (= prl−1 K1 ). If prl κ = K1 then κ ⊆ κa and because of prl κ = K0 there exist some κ-congruent functions rt , st , which are not identical to a certain tuple of Elt . Then, by Lemma 9.9.7, we have prl−1 K0 ⊆ κ. One can deduce prl−1 K ⊆ κ from that and from prl κ = K easy. Hence prl−1 K = κ. The above lemmas can be summarized as follows: Theorem 9.9.9 ([Lau 77]) Let A ∈ A. Then, a nontrivial congruence on A is either of the type prl−1 K, where K is a congruence on prl A, or it has the form κU1 ,U2 ,...,Ut , where U1 , ..., Ut are some nonempty subsets of Ek \El , which are pairwise incomparable (in respect to the inclusion). 5

9.9 Congruences on Some Subclasses of Pk,l

277

Since we determined all congruences on subclasses of P2 in Section 9.2, can also characterize all congruences on A ∈ A for l = 2 with the aid of the above theorem. As examples, we give only the congruence lattices of Pk,2 for k ∈ {3, 4, 5}, subsequently (in the Figures 9.3–9.5). r κ1 r κa r κ{2} r κ0 Fig. 9.3. Congruence lattice of P3,2

r κ1 r κa r κ{2},{3} P PPP P

r κ{2} P PP PP

PP PPr κ {3} PP PP r κ{2,3} r κ0

Fig. 9.4. Congruence lattice of P4,2

5

See also Theorem 9.9.2.

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9 Congruences and Automorphisms on Function Algebras

r κ1 r κa r κ{2},{3},{4} P PPP PP PP rPκ Pr κ r κ {2},{3} PP{2},{4} {3},{4} P P PP P P P PPP PPP P r κ Pr κ{4},{2,3} r κ {2},{3,4} {3},{2,4} r κ{2}

r κ{3}

r κ{4}

r κ r κ{2,3},{2,4} r κ P PP{2,3},{3,4} {2,4},{3,4} PP P P P P P PPP PPP Pr κ{3,4} r κ{2,3} r κ{2,4} P P PP PP PP PP r κ{2,3,4} r κ0 Fig. 9.5. Congruence lattice of P5,2

9.10 Some Further General Properties of the Congruences and the l-Classes In this section, we derive conditions with which one can realize whether a subclass of Pk has only trivial congruences. During the derivation of these conditions, we receive some general properties of the congruences as by-products. Most of the following lemmas and theorems were published by V. V. Gorlov in [Gor 77;1,2] and [Gor 79]. In the quoted papers by Gorlov proofs can be found that are excluded subsequently. Lemma 9.10.1 Let κ be a nontrivial congruence of the class A of Pk . Then (a) e11 ∈ A =⇒ κ0 ⊂ κ ⊂ κa , (k) (b) κ ∈ Cona A =⇒ κ(k) = κ0 , (k) (c) (κ ∈ Con1 A ∧ (∃f, g ∈ A1 : f g = f )) =⇒ κ(k) = κ0 , (n) 1 (n) (d) e1 ∈ A =⇒ (∀n ≥ 2 : κ = κa ). Proof. (a) follows from Lemma 9.2.1. (b): Because of (a) there are two diﬀerent κ-congruent functions f n , g n ∈ A.

9.10 Some Further General Properties of the Congruences and the l-Classes

279

By using of the superposition operations ζ, τ , ∆, ∇ one receives obviously k-ary functions f , g with (f , g ) ∈ κ and f = g . Consequently, (b) holds. (c) follows from Theorem 9.3.4 and Lemma 9.3.7. (2) (d) Proving the contention for n = 2 suﬃces. Suppose κa = κ(2) . Then we 2 2 have (e1 , e2 ) ∈ κ. Therefore, by Lemma 9.2.2, κ = κa . This is, however, a contradiction to our assumption. With the help of an arbitrary arity congruence κ of a subclass A ⊆ Pk one can deﬁne two further arity congruences: µ (9.29) σn,κ := µ∈Cona A; κ(n) =µ(n)

and πn,κ := {(f p , g q ) ∈ A2 | p = q ∧ ∀(f1 , g1 ), ..., (fp , gp ) ∈ κ(n) ∪ {(eni , eni ) | i ∈ {1, 2, ..., n}} : (f (f1 , ..., fp ), g(g1 , ..., gp )) ∈ κ(n) }, (9.30) which are completely deﬁned through κ(n) , n ∈ N. Some properties of these relations that are easily proven are summarized in the following lemma. Lemma 9.10.2 Let A be a subclass of Pk and let κ ∈ Cona A. Then (n) (n) (a) σn,κ = πn,κ = κ(n) , ⊆ σn+1,κ ⊆ κ ⊆ πn+1,κ ⊆ πn,κ , (b) σn,κ (c) κ = r≥1 πr,κ . The following lemma is a consequence of the above and of Lemma 9.10.1. Lemma 9.10.3 For every nontrivial arity congruence κ of a subclass A of Pk , it holds: κ0 ⊂ σk,κ ⊆ κ ⊆ πk,κ ⊂ κa . If A is in addition a clone, then κ0 ⊂ σk,κ ⊆ κ ⊆ π2,κ ⊂ κa .

Since there are only ﬁnitely many possibilities for σk,κ and πk,κ , we obtain the following theorem as a consequence of Lemma 9.10.3 and Theorem 9.3.4: Theorem 9.10.4 The congruence lattice (or the lattice of the arity congruences) of a subclass of Pk has only ﬁnitely many atoms and dual atoms. Deﬁnition Let A be a subclass of Pk . An equivalence relation µ on An , n ∈ N, is called an n-congruence, iﬀ there exists a congruence relation κ ∈ Cona A with κ(n) = µ.

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Lemma 9.10.5 Let A be a subclass of Pk and let µ an equivalence relation on An . Then µ is an n-congruence if and only if µ fulﬁlls the following condition: ∀ f m ∈ A ∀ (u0 , v0 ), ..., (un , vn ), .., (um , vm ) ∈ µ ∪ {(eni , eni ) | i ∈ {1, 2, ..., n}} : (f (u1 , ..., um ), f (v1 , ..., vm )) ∈ µ ∧ (u0 (u1 , ..., un ), v0 (v1 , ..., vn )) ∈ µ. (9.31) Proof. If µ is a congruence, then (9.31) obviously holds. Assume µ fulﬁlls (9.31). The equivalence relation πµ deﬁned by (f r , g s ) ∈ πµ :⇐⇒ r = s ∧ ∀(u1 , v1 ), ..., (ur , vr ) ∈ µ ∪ {(eni , eni ) | i ∈ {1, 2, ..., n}} : (f (u1 , ..., ur ), g(v1 , ..., vr )) ∈ µ) (n)

is a congruence on A with πµ = µ (see [Gor 77a]). Thus, µ is an n-congruence on A. With the help of Theorem 9.3.4, Lemma 9.3.7, and Lemma 9.10.5, easily one can prove the following theorem. Theorem 9.10.6 (k)

(a) A subclass A of Pk has only trivial congruences if and only if κ0 and (k) κa are the only k-congruences on A and there exist some f, g ∈ A1 with f g = f . (k) (k) (b) A clone A of Pk has only trivial congruences if and only if κ0 and κa are the only k-congruences on A. Next, we introduce a notation for classes that are minimal classes concerning the following property: Each class B, which fulﬁlls A ⊂ B, has only trivial congruences. Deﬁnition A subclass A of Pk is called a limit class (brieﬂy: l-class6 ), if it fulﬁlls the following two conditions: (1) ∀B ∈ Lk : (A ⊂ B =⇒ Con B = {κ0 , κa , κ1 }); (2) ∀C ∈ Lk : (C ⊂ A =⇒ ∃C ∈ L↑k (C) : Con C = {κ0 , κa , κ1 }). Lemma 9.10.7 ([Gor 77]) (a) Let A be a subclass of Pk , which is not a clone. Then the clone A ∪ Jk has some nontrivial congruences. (b) Each l-class is a clone. Proof. (a): Since there are maximal classes (clones) with nontrivial congruences, which contain all constant functions, we can assume that |An | ≥ 2 for n ≥ 2 holds. Then, one can check that the equivalence relation 6

In [Gor 77] the notation M -class was used instead of l-class.

9.10 Some Further General Properties of the Congruences and the l-Classes

281

κ := {(f, f ) | f ∈ Jk } ∪ {(f n , g n ) | {f, g} ⊂ A} is a nontrivial congruence on A. (b) follows directly from (a). The following theorem gives some examples of l-classes. Theorem 9.10.8 (a) For k = 2 there are exactly four l-classes. These classes are L, K, D and M ∩ S. (b) A maximal class P olk of Pk is an l-class if and only if ∈ { ∈ C1k | || ≥ 2} ∪ Uk ∪ Lk ∪ Bk . (c) The class [Uω ∪ {e11 }] with ω ∈ Ek and Uω := {f n ∈ Pk | ∀a ∈ Ekn \(Ek \{ω})n ) : f (a) = ω} n≥1

is an l-class. (d) Among the submaximal classes of P3 no l-classes exist. Proof. (a) follows from Section 9.2. The statements (b) and (c) can be proven with the aid of the Theorems of Section 9.7, the Lemmas 9.7.1 and 9.10.1, and Theorem 9.10.9. One can ﬁnd a proof for (d) in [Gor-L 82].

Theorem 9.10.9 (Gorlov’s Theorem; [Gor 77]) (1) Let A ⊆ Pk be an l-class. Then A fulﬁlls one of the following conditions: (a) |Con A| ≥ 4 and there exists a 2 · k k -ary relation on Ek with A = P olk . k (b) |Con A| = 3 and ord A ≤ k 2·k . (2) For every k ≥ 2 there are only ﬁnitely many l-classes in Lk . Proof. (1): For A, the following two cases are possible: Case 1: A has a nontrivial congruence κ. Then, by Theorem 9.10.6, κ(k) is a nontrivial equivalence relation on Ak . If one now forms the 2 · k k -ary relation := {(Gk (f ), Gk (g)) | (f, g) ∈ κ(k) }

(9.32)

(see Chapter 2), then one can see that A ⊆ P olk holds and that there exists the nontrivial congruence {(f r , g s ) ∈ (P ol ) × (P ol ) | r = s ∧ ∀(u1 , v1 ), ..., (ur , vr ) ∈ κ(k) : (f (u1 , ....ur ), g(v1 , ..., vr )) ∈ κ(n) } on P olk (see also Lemma 9.10.5). Since A is an l-class, this is only possible for A = P olk . Thus A fulﬁlls the condition (a).

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9 Congruences and Automorphisms on Function Algebras

Case 2: A has only trivial congruences. k Suppose, we have A := [A2·k ] ⊂ A. Since A is an l-class, there exists a class B ⊆ Pk with A ⊂ B and |Con B| ≥ 4. Consequently, there is a nontrivial equivalence relation of the form κ(k) on B k . This relation deﬁnes a 2 · k k -ary relation , given by (9.32), with B ⊆ P olk . In addition, because of A ⊂ B, one can prove A ⊆ P olk as follows: Suppose, there is a function f n ∈ A k that does not preserve (n > k (2k ) ). Then there is some r1 , ..., rn ∈ with f (r1 , ..., rn ) ∈ . We form the function f (x) := f (xi1 , ..., xin ) ∈ [f ] with ∀j, l ∈ {1, ..., n} : (ij = il ⇐⇒ rj = rl ). Then, because || ≤ k 2k , f belongs indeed to A , however, it does not preserve the relation , in contradiction to A ⊂ B ⊆ P olk . Consequently, A ⊆ P olk . But this is a contradiction to our assumption “A is an l-class and has only trivial congruences”. Thus A fulﬁlls the condition (b) and, therefore, (1) is proven. (2) is a consequence of (1). k

9.11 The Connection Between Clone Congruences and Fully Invariant Congruences Let A = (A; W ), W ⊆ PA , be an algebra, let VA be the variety generated by A and let F(X) (:= (F (X); W )) be the free algebra with countable ranges 7 of VA (see Part I). For F (X) we use the following model: X := {x1 , x2 , ...}, F (X) := n≥1 {f (xi1 , ..., xin ) | f ∈ T (n) (A) ∧ {xi1 , ..., xin } ⊆ X} (Therefore, F (X) consists of all terms that can be formed from X and the functional symbols of (T(A) = [W ∪ JA ]) and for arbitrary wn ∈ W and arbitrary fi (xi,1 , ..., xi,mi ) ∈ F (X), i = 1, ..., n + 1, it holds f1 (x1,1 , ..., x1,m1 ) = f2 (x2,1 , ..., x2,m2 ) :⇐⇒ ∀ a1,1 , ..., a2,m2 ∈ A : f1 (a1,1 , ..., a1,m1 ) = f2 (a2,1 , ..., a2,m2 )

(9.33)

and w(f1 (x1,1 , ..., x1,m1 ), ..., fn (xn,1 , ..., xn,mn )) = fn+1 (xn+1,1 , ..., xn+1,mn+1 ) :⇐⇒ ∀a1,1 , ..., an+1,mn+1 ∈ A : w(f1 (a1,1 , ..., a1,m1 ), ..., fn (an,1 , ..., an,mn )) = fn+1 (an+1,1 , ..., an+1,mn+1 ), 7

For example, X is countably inﬁnite.

9.11 The Connection Between Clone Congruences and Fully Invariant Congruences

where ai,j = al,k ⇐⇒ xi,j = xl,k (see Chapter 6 of Part I). The fully invariant congruences8 of F (X) are of interest to us here, since the lattice of these congruences is anti-isomorphic to the lattice of the subvarieties of VA . The following theorem is a well-known result of the category theory, which follows from the representation of a variety by an algebraic theory (a certain category). One ﬁnds complete information on this category-theory background in [Vog 85]. Theorem 9.11.1 ([Schw 84]) Through (t(x1 , ..., xn ), u(x1 , ..., xn )) ∈ κF ⇐⇒ (t, u) ∈ κ

(9.34)

every binary relation κF on F(X) (∈ VA ) is assigned to a relation κ ⊆ κa on T (A) and vice versa. A binary relation κF is a fully invariant congruence F(X) if and only if the relation κ on T (A) is an arity congruence of T (A); i.e., the lattice of the fully invariant congruences of F(X) is isomorphic to the lattice of the arity congruences of T (A) or antiisomorphic to the subvariety lattice of VA . Proof. Since f ∈ T (A) implies ∇f ∈ T (A) and because of (9.34), every relation on F (X) is completely characterized through the tuples (t(x1 , ..., xn ), u(x1 , ..., xn )) (t, u ∈ T (n) (A), n = 1, 2, ...). Thus (9.34) describes a bijection between binary relations on F (X) and arity relations on T (A). Let κ ⊆ κa be a congruence relation on T (A) and let κF be the relation on F (X) deﬁned by (9.34). Because of (w(t1 , ..., tr ), w(u1 , ..., ur )) ∈ κ for (ti , ui ) ∈ κ, i = 1, 2, ..., r, and arbitrary w ∈ W , κF is compatible with the operations of W , i.e., κF is a congruence relation on F (X). κF would be fully invariant if in addition (t(x1 , ..., xn ), u(x1 , ..., xn )) ∈ κF implies (ϕ(t(x1 , ..., xn )), ϕ(u(x1 , ..., xn ))) ∈ κF for every endomorphism ϕ on F (X). Let ϕ be an endomorphism on F (X) with ϕ(xi ) = ti (x1 , ..., xri ), i = 1, 2, ..., and (t(x1 , ..., xn ), u(x1 , ..., xn )) ∈ κF . Then (ϕ(t(x1 , ..., xn ), ϕ(u(x1 , ..., xn ))) = (t(ϕ(x1 ), ..., ϕ(xn )), u(ϕ(x1 ), ..., ϕ(xn ))) = (t(t1 (x1 , ..., xr1 ), ..., tn (x1 , ..., xrn )), u(t1 (x1 , ..., xr1 ), ..., tn (x1 , ..., xrn ))) ∈ κF , since (t, u) ∈ κ implies (t(t1 , ..., tn ), u(t1 , ..., tn )) ∈ κ. Thus κF is a fully invariant congruence on F(X) if κ is a congruence on T (A). 8

These are equivalence relations on F(X), which are compatible not only with the operations of the algebra F(X) but also with every endomorphism of F(X).

283

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9 Congruences and Automorphisms on Function Algebras

Let κF be a fully invariant congruence on F(X) and let (f n , g n ), (um , v m ) ∈ κ in the following. We have to prove that (f u, g v) ∈ κ and (αf, αg) ∈ κ for α ∈ {ζ, τ, ∆}. Obviously, one can continue every mapping from X into F (X) to an endomorphism on F (X). We consider the endomorphisms ϕ1 , ..., ϕ4 with ϕ1 (x1 ) = x2 , ϕ1 (x2 ) = x3 , ..., ϕ1 (xn−1 ) = xn , ϕ1 (xn ) = x1 ϕ2 (x1 ) = x2 , ϕ2 (x2 ) = x1 , ϕ2 (x3 ) = x3 , ..., ϕ2 (xn ) = xn , ϕ3 (x1 ) = ϕ3 (x2 ) = x1 , ϕ1 (x3 ) = x2 , ..., ϕ3 (xn ) = xn−1 , ϕ4 (x1 ) = u(x1 , ..., xm ), ϕ4 (x2 ) = xm+1 , ..., ϕ4 (xn ) = xm+n−1 . Since, by assumption, (f (x1 , ..., xn ), g(x1 , ..., xn )) ∈ κF and κF is fully invariant, for i = 1, 2, 3, 4 the tuples (ϕi (f (x1 , ..., xn )), ϕi (g(x1 , ..., xn ))) = (f (ϕi (x1 ), ..., ϕi (xn ), g(ϕi (x1 ), ..., ϕi (xn ))) ⎧ (f (x2 , ..., xn , x1 ), g(x2 , ..., xn , x1 )) if i = 1, ⎪ ⎪ ⎪ ⎪ if i = 2, ⎨ (f (x2 , x1 , x3 , ..., xn ), g(x2 , x1 , x3 , ..., xn )) = (f (x1 , x1 , x2 , ..., xn−1 ), g(x1 , x1 , x2 , ..., xn−1 )) if i = 3, ⎪ ⎪ (f (u(x1 , ..., xm ), xm+1 , ..., xm+n−1 ), ⎪ ⎪ ⎩ if i = 4, g(u(x1 , ..., xm ), xm+1 , ..., xm+n−1 )) belong to κF and, therefore, {(ζf, ζg), (τ f, τ g), (∆f, ∆g), (f u, g u)} ⊆ κ. In addition, since κF is compatible with the operations of W , we have (because of (u(x1 , ..., xm ), v(x1 , ..., xm )) ∈ κF ) (g(u(x1 , ..., xm ), xm+1 , ..., xm+n−1 ), g(v(x1 , ..., xm ), xm+1 , ..., xm+n−1 )) ∈ κF and (by transitivity) (f (u(x1 , ..., xm ), xm+1 , ..., xm+n−1 ), g(v(x1 , ..., xm ), xm+1 , ..., xm+n−1 )) ∈ κF . Thus (f u, g v) ∈ κ. From this, together with the above considerations, we get that κ is a congruence on T (A). The remaining statements of the theorem are obvious or result from Part I. Analogous to the above proof and with the aid of Lemma 9.10.5, the following theorem can be proven. Lemma 9.11.2 Let F (n) be the free algebra with n generators of VA . Through (f (x1 , ..., xn ), g(x1 , ..., xn )) ∈ µF (n) ⇐⇒ (f, g) ∈ µ (f, g ∈ T (n) (A) ) every fully invariant congruence on F (n) corresponds unambiguously with an n-congruence on T (A) and conversely.

9.12 Automorphisms of Function Algebras

285

9.12 Automorphisms of Function Algebras An automorphism of an algebra A is an isomorphic mapping from A onto A. In this section, we denote the image α(f ) of a function f ∈ A, where α denotes a mapping on A, with f α . An automorphism α of a subclass A ⊆ Pk is called an inner automorphism onto A, if there exists a bijective mapping ϕ from Ek onto Ek with the following property: ∀f n ∈ A : f α (x1 , ..., xn ) = ϕ(f (ϕ−1 (x1 ), ..., ϕ−1 (xn )) or f α (ϕ(x1 ), ..., ϕ(xn )) = ϕ(f (x1 , ..., xn )). In the following, we put together some properties of automorphisms of subclasses of Pk , and we prove, for some classes, that they have only inner automorphisms. In particular, we prove that all automorphisms of subclasses of P2 are inner automorphisms. This is not valid anymore for classes Pk , however, for k ≥ 3. The basis of these proofs is [Mal 66], from which we use many details here. In addition, we generalize some results of [Mal 66]. Lemma 9.12.1 Let A be a subclass of Pk and let α be an automorphism onto A. Then for arbitrary f ∈ A it holds: (a) (b) (c) (d)

a f = a (f α ). xi is a ﬁctitious variable of f iﬀ xi is a ﬁctitious variable of f α . f is a constant function iﬀ f α is a constant function. f is a permutation iﬀ f α is a permutation.

Proof. (a): Suppose there exists a function f ∈ A with a(f ) = a(f α ). We distinguish two cases. Case 1: n := a f < a(f α ) =: m. Then ∆n−1 f = ∆n f . Consequently, ∆n−1 f α = ∆n f α . The last equation is not, however, possible because of a(∆n−1 f α ) = m − n + 1 and a(∆n f α ) = m − n. Case 2: a f > a(f α ) =: m. In this case, we have ∆m−1 f α = ∆m f α and thus (∆m−1 f )α = (∆m f )α , in contradiction to the bijectivity of the mapping α. Therefore, a(f ) = a(f α ) for all f ∈ A. (b): W.l.o.g. let i = 1. Obviously, x1 is a ﬁctitious variable of f iﬀ ∇(∆f ) = f . The equation ∇(∆f ) = f holds iﬀ ∇(∆f α ) = f α . Hence (b) is right. (c) follows directly from (b). (d): If A contains a permutation, then all projections belong to A. The function e := e11 is the only function of A, for which f e = e f holds for all f ∈ A1 . Consequently, we have also f α eα = eα f α for all f ∈ A1 . Thus eα = e.

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Since for every permutation s ∈ A there exists a t with s s ... s = e, it t times holds sα sα ... sα = e, whereby sα is a permutation. t times Theorem 9.12.2 Let A be a subclass of Pk and let T be a subset of Ar , r ≥ 1, which has the following three properties: (1) Every function of A preserves the functions of T . (2) There exists a c ∈ Ekr and some functions g0 , g1 , ..., gk−1 ∈ T with gi (c) = i for every i ∈ Ek . (3) For each automorphism α onto A it holds T α = T and α|T is an inner automorphism onto T . Then each automorphism onto A is an inner automorphism. Proof. Let α be an automorphism onto A. By assumption (3), we have T α = T and that α|T is an inner automorphism onto T . The permutation ϕ appertaining to this inner automorphism deﬁnes an isomorphism βϕ from A onto Aβϕ : ∀f n ∈ A : f βϕ (x) = ϕ−1 (f (ϕ(x1 ), ..., ϕ(xn ))). We show that the mapping γ := α2βϕ−1 is the identical mapping, whereby βϕ = α is valid and the claim “α is an inner automorphism” follows: Let f n ∈ A and let a = (a1 , ..., an ) ∈ Ekn be arbitrary. By assumption (2), for every ai ∈ Ek , there exists a function gai ∈ T with gai (c) = ai , i = 1, ..., n. Because of (1) there is a function g ∈ T with the properties: f (ga1 (x), ..., gan (x)) = g(x) and g(c) = f (a). Consequently, we have g γ = f γ (gaγ1 , ..., gaγn ). Because the functions of T are ﬁxed points of the mapping γ, this implies g = f γ (ga1 , ..., gan ). Thus we have f γ (a) = g(c) = f (a) for arbitrary a ∈ Ekn , i.e., f = f γ holds for every f ∈ A. With the aid of the above theorem, we can prove the three following theorems. Theorem 9.12.3 ([Mal 72b]) If a subclass A ⊆ Pk contains all constants of Pk , then each automorphism onto A is an inner automorphism. Proof. The claim follows from Theorem 9.12.2, if one chooses T = {c0 , c1 , ..., ck−1 } and uses Lemma 9.12.1, (c). The following theorem generalizes Theorem 2 from [Mal 66].

9.12 Automorphisms of Function Algebras

287

Theorem 9.12.4 Let ∅ = E ⊆ Ek , a if x ∈ E, ta,b (x) := b otherwise, T := {ta,b | a ∈ E ∧ b ∈ Ek }, S := Pk1 [k]∩P olk E and let A be a subclass of Pk with S ∪T ⊆ A . Then, each automorphism onto A is an inner automorphism. Proof. If |E| = 1 then Theorem 9.12.4 follows from the proof of Theorem 2 in [Mal 66]. If A ⊆ P olk E, then by S ∪ T ⊆ A, we have that A contains all constant functions of Pk . Consequently, by Theorem 9.12.3, A has only inner automorphisms in this case. Thus we can assume A ⊆ P olk E in the following. Let α be an automorphism onto A. We show that T fulﬁlls the assumptions (1)–(3) of Theorem 9.12.2. Because of A ⊆ P olk E, it holds (1). (2) is also valid. Further, we have for arbitrary g ∈ A1 (∀s ∈ S : g s = g) ⇐⇒ g ∈ T.

(9.35)

Thus g ∈ T implies g s = g for all s ∈ S and therefore also g α sα = g α . Because the function sα belongs to S because of Lemma 9.12.1, (d), we obtain g α ∈ T from (9.35) Consequently, T = T α . Next we show that, for the set T1 := {ta,b | a ∈ E ∧ b ∈ Ek \E}, the equation T1α = T1 holds. Obviously, we have for all ta,b ∈ T : ta,b ta,b = tc,c ⇐⇒ a = c ∧ {a, b} ⊆ E.

(9.36)

α α Therefore, ta,b ∈ T \T1 implies ta,b ta,b = ta,a and then tα a,b ta,b = ta,a . Since is a constant by Lemma 9.12.1, we have t ∈ T \T because of (9.36). tα a,b 1 a,a Consequently, T1α = T1 . If one restricts the functions of T1 ∪ S to E or Ek \E, one obtains the sets (T1 ∪S)|E ⊆ PE or (T1 ∪S)|Ek \E ⊆ PEk \E , respectively. Since the set (T1 ∪S)|E or (T1 ∪ S)|Ek \E contains all constant functions of PE or PEk \E , respectively, α|(T1 ∪S)|E or α|(T1 ∪S)|(Ek \E) is an inner automorphism onto the semigroup (T1 ∪ S; ), respectively. Thus there exists a permutation ϕ onto Ek with

∀h ∈ T1 ∪ S : hα = ϕ−1 h ϕ.

(9.37)

Finally we show that (9.37) also holds for every function h ∈ T \T1 , whereby the condition (3) of Theorem 9.12.2 and then our theorem are proven: Let ta,b ∈ T \T1 , tα a,b = ta ,b and let s ∈ S, where s has only the ﬁxed points a and b. Then s ta,b = ta,b and sα ta ,b = ta ,b .

(9.38)

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9 Congruences and Automorphisms on Function Algebras

(9.38) implies that a and b are ﬁxed points of the permutation sα . Because of sα = ϕ−1 s ϕ the permutation sα has only the ﬁxed points ϕ−1 (a) and ϕ−1 (b). Consequently, {a , b } = {ϕ−1 (a), ϕ−1 (b)}. Thus for a = b, we have −1 ta,a ϕ is valid. From that and from ta,b ta,a = ta,a shown that tα a,a = ϕ α α and ta ,b ta,a = ta,a , then, a = ϕ−1 (a) and b = ϕ−1 (b) hold. Consequently, we have ta ,b = ϕ−1 ta,b ϕ. Theorem 9.12.5 ([Lau 79;b]) All automorphisms on a maximal class of Pk are inner automorphisms. Proof. With the aid of Theorems 9.12.3 and 9.12.4, it is easy to see that all maximal classes of Pk have only inner automorphisms, except for the maximal classes of the type S. Let A ⊆ Pk be a maximal class of type S, i.e., it holds A = P olk s , where s, s and the functions of A are deﬁned as in Section 5.2.2 in the following. Denote T the set of all functions of the form fa (x) :=

p−1 l

jar,i (x) · si (a), (a ∈ Ek ).

r=1 i=0

Our theorem is proven if we can show that T fulﬁlls the assumptions of Theorem 9.12.2. Obviously, T fulﬁlls the ﬁrst two conditions of Theorem 9.12.2. To prove (3) from Theorem 9.12.2 denote α an automorphism onto P olk s . First we prove T α = T . Obviously, for every function g ∈ (P olk s )1 and every function fa ∈ T there exists a function h ∈ (P olk s )1 with fa = h g. Consequently, we have (9.39) faα = hα g α . If the function also fulﬁlls the condition g α ∈ T , then we have hα g α ∈ T and (by (9.39)) faα ∈ T for arbitrary a ∈ Ek . Thus T α = T . Next we derive some properties of the function fa : For all r ∈ {1, 2, ..., l} and i ∈ Ep we have far,0 far,i = far,i and thus faαr,0 faαr,i = faαr,i , i.e., for every r there exists an r with faαr,0 = far,0 . Further, it holds far,i = (...((far,1 far,1 ) far,1 )... far,1 ). i times For faαr,1 = far,br (br ∈ Ep \{0}) this implies faαr,i = far,i·br (mod p) . Because of far,0 fa1,1 = far,1 we have faαr,0 faα1,1 = faαr,1 = far,br = far,0 = far,b1 , i.e., br = b1 =: b holds for arbitrary r ∈ {1, 2, ..., l}. In summary fa1,b 1 we get: (9.40) faαr,i = far,i·b (mod p) ,

9.12 Automorphisms of Function Algebras

289

where r, r ∈ {1, 2, ..., l}, i ∈ Ep and b ∈ Ep \{0}. Because of (9.40), one can deﬁne a permutation ϕ onto Ek as follows: ϕ(ar,i ) := ar,i·b (mod p) . This permutation has the property ∀au,v ∈ Ek : faαu,v (ϕ(x)) = ϕ(fau,v (x)), which one can prove as follows: Let x = ar,i (r ∈ {1, 2, ..., l}, i ∈ Ep ). Then we have (ar,i·b (mod p) ) = si·b (au ,v·b (mod p) ) = faαu,v (ϕ(ar,i )) = fau,v·b (mod p) au ,(v+i)·b (mod p) and ϕ(fau,v (ar,i )) = ϕ(si (au,v )) = ϕ(au,v+i (mod p) ) = au ,(v+i)·b (mod p) . Hence α is an inner automorphism onto T , and Theorem 9.12.2 implies that P olk s has only inner automorphisms. Theorem 9.12.6 ([Gor-L 83a]) Each automorphism on a subclass of P2 is an inner automorphism. Proof. Let A = [A] ⊆ P2 . By Chapter 4, the following cases are possible: Case 1: A1 = {ca }. Then A is the class [ca ], which has obviously only inner automorphisms. Case 2: {c0 , c1 } ⊆ A1 . In this case, our theorem follows from Theorem 9.12.3. Case 3: A1 ∈ {{c0 , e11 }, {c1 , e11 }, {e11 , e11 }}. In this case, we obtain our claim from Theorem 9.12.2, when we elect T = A1 and when we use Lemma 9.12.1, (d). Case 4: A1 = {e11 }. By Chapter 3, considering only the following possibilities for A suﬃces: Case 4.1: A ∈ {I, K, L ∩ T0 ∩ S, S ∩ M }. In this case, there exists a function f n ∈ A with [f ] = A and n = ord A (see Chapter 3). Then, with the help of Lemma 9.12.1, (b) and by the properties of the functions of A, one can prove that f α = f holds for each automorphism α onto A. Because of [f ] = A this implies that α is the identical mapping, which is obviously an inner automorphism. Case 4.2: A ∈ {S ∩ T0 , M ∩ T0 ∩ T1 , T0 ∩ T1 } ∪ m∈{2,3,...,∞} {T0,m ∩ T1 ∩ M, T0,m ∩ T1 }. Choosing T = A2 , A fulﬁlls the conditions (1) and (2) of Theorem 9.12.2. We prove that T = A2 also fulﬁlls condition (3): If A ∈ {M ∩ T0 ∩ T1 , T0 ∩ T1 } then A2 = {∧, ∨, e21 , e22 }; if A = S ∩ T0 then A2 = {e21 , e22 } and if A ∈ {T0,m ∩ T1 ∩ M, T0,m ∩ T1 } then A2 = {∧, e21 , e22 }. By Lemma 9.12.1, (b) we have (e2i )α = e2i , i ∈ {1, 2}, for each automorphism

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9 Congruences and Automorphisms on Function Algebras

α onto A. When one considers this, one convinces oneself that α|A2 is an inner automorphism onto T for each automorphism α onto A. Consequently, A fulﬁlls every condition of Theorem 9.12.2 in Case 4.2. Thus A has only inner automorphisms. Theorem 9.12.7 ([Gor-L 83a]) If k ≥ 3 then there are subclasses of Pk , whose automorphisms are not all inner automorphisms. Proof. Here is an example: Let x if x ∈ E2 , x if x ∈ E2 , u(x) := and v(x) := 0 otherwise, 1 otherwise. By Lemma 9.12.1, (a), (b) and Theorem 9.8.4, all automorphisms onto the set A := [u, v, c0 ] are induced by the automorphisms of the semigroup (A1 ; ). It is easy to check that the mapping α : u → v, v → u, c0 → c0 is an automorphism of (A1 ; ). Suppose α is an inner automorphism. Then there is a permutation ϕ onto Ek with ∀g ∈ {u, v, c0 } : g α = ϕ−1 g ϕ.

(9.41)

Because of cα 0 = c0 we have ϕ(0) = 0. (9.41) implies ϕ u = v ϕ and then ϕ(0) = v(ϕ(0)) and ϕ(0) = v(ϕ(2)). Because of deﬁnitions of ϕ and v this is only possible if ϕ(0) = 1, in contrary to the above statement ϕ(0) = 0. Thus α is not an inner automorphism.

10 The Relation Degree and the Dimension of Subclasses of Pk

This chapter deals with the relation degree and the dimension of subclasses A of Pk . The relation degree of A is the smallest number h so that class A is unambiguously described by a relation set whose elements have the arity h at most. The dimension of A is the smallest arity of a relation that characterizes class A unambiguously. We begin with investigating the connections between these two complexity measures of the relational description of classes. Then we prove the relation degrees and dimensions of Post’s classes, which were found by G. N. Blochina in [Blo 70]. In Section 10.3, there are further classes for which is known the relation degree or the dimension.

10.1 The Deﬁnition of the Relation Degree and of the Dimension of a Subclass of Pk In this section we deal with two possibilities for evaluating the complexity of relational descriptions of subclasses of Pk . Denote ar(), ∈ Rkh \{∅}, the arity h of the relation . Further, for Q ⊆ Rk , let ⎧ if Q = ∅, ⎪ ⎨ 0, armax (Q) := max{ar() | ∈ Q}, if the maximum exists, ⎪ ⎩ ∞ otherwise. The relation degree d(A) of a subclass A of Pk is the smallest of the possible numbers armax (Q) with A = P olk Q, i.e., ⎧ ⎨ min{armax (Q) | Q ⊆ Rk ∧ P ol Q = A} if ∃h ∈ N ∃Q ⊆ Rkh : P ol Q = A, d(A) := ⎩ ∞ otherwise. If one complements the above deﬁnition with the demand |Q| = 1, the result is the deﬁnition of the dimension of a subclass A of Pk :

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10 The Relation Degree and the Dimension of Subclasses of Pk

dim A :=

min{ar() | ∈ Rk ∧ P ol = A} if ∃ ∈ Rk : P ol = A, ∞ otherwise.

The following lemma gives some elementary connections from d and dim. Lemma 10.1.1 Let A be a subclass of Pk . Then (a) d(A) = dim A = 0 ⇐⇒ A = Pk . (b) d(A) ≤ dim A. (c) d(A) < ∞ ⇐⇒ dim A < ∞. (d) If ∞ = d(A) < dim A, then there are certain proper predecessors A1 , ..., Ar of A, which are pairwise not contained in each other (i.e., A ⊂ Ai , Ai = [Ai ] and Ai ⊆ Aj for i = j and i, j = 1, 2, ..., r), with the following properties: A = A1 ∩ A2 ∩ ... ∩ Ar , r ≥ 2 and d(A) = max{d(Ai ) | i = 1, ..., r}. (e) If A has exactly a direct predecessor B (i.e., A ⊂ B = [B] and no subclass B exists with A ⊂ B ⊂ B), then d(A) = dim A. Proof. (a) follows from P ol ∅ = Pk and (b) results immediately from the deﬁnitions of d and dim. (c) follows from P ol{1 , 2 , ..., t } = P ol 1 × 2 × ... × t . (d): If d(A) = ∞ and d(A) < dim A, there are certain relations 1 , ..., r with the properties r A = P ol {1 , ..., r } = i=1 P ol i , r ≥ 2, ∀i ∈ {1, .., r} : A = P ol {1 , ..., r }\{i } and d(A) = armax {1 , ..., r }. If one now sets Ai := P ol i (i = 1, .., r), most of the properties of the sets Ai mentioned in the lemma are immediately clear. The rest results from A ⊂ Ai for all i = 1, ..., r because of d(A) < dim A. (e) follows from (b) and (d).

10.2 The Dimensions and Relation Degrees of Post’s Classes

293

10.2 The Dimensions and Relation Degrees of Post’s Classes The basic results of this section come from the paper [Blo 70] by G. N. Blochina. Our aim is to prove the values given in Table 10.1. In the ﬁrst column, all subclasses A of P2 are listed (except for the isomorphic classes). Next to the values d(A) and dim(A), which we will subsequently prove, one can also ﬁnd some descriptive relations for subclasses A. The correctness of these statements can easily be proven in the following manner: First, A ⊆ P ol holds for the corresponding of the third column. (For most classes this is clear according to deﬁnition. For every remaining class, the statement is a conclusion from the given construction of with the aid of a certain n-ary graphic Gn (A) 1 .) Then, one proves that for every direct predecessor B of A there exists an f ∈ B 2 with f ∈ P ol . The equation P ol = A and an upper bound for dim A (≥ d(A)) result from the proven facts. After Table 10.1, one ﬁnds the lower bounds for dim A, which are identical to the upper bounds and the proofs for the remaining statements of this table. Table 10.1

A P2 T0 S M T0,2 T0 ∩ T1

1 2

dim A with A = P ol 0 ∅ 1 {0} # $ 0 1 2 1 0 # $ 0 0 1 2 0 1 1 # $ 0 0 1 2 0 1 0 # $ 0 2 1

d(A) 0 1

E23 \{1} 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0

3

T0,3

3

M ∩ T0

3

S ∩ T0

3

S∩M

3

See Section 2.7. It is suﬃcient to consider a basis of B instead of B.

2 2 2 1

2 2 2

294

10 The Relation Degree and the Dimension of Subclasses of Pk

A M ∩ T0,2 T1 ∩ T0,2 L ∩ T0 K ∪C M ∩ T0,2 T1 ∩ T0,2 L ∩ T0 K ∪C I I ∩ C0 I [P21 ] I ∪C M ∩ T0 ∩ T1 L L∩S L ∩ S ∩ T0 K ∪ C0

dim A with A = P ol d(A) 0 0 0 1 0 0 1 0 3 2 0 1 1 1 0 0 1 0 1 0 3 2 1 1 1 0 0 1 1 0 1 0 1 = pr2,3,4 G2 (A) 3 3 0 1 1 0 0 1 0 1 0 0 0 1 = pr2,1,3 G2 (A) 3 3 0 0 1 1 0 0 0 1 0 0 1 0 3 2 0 1 1 1 0 0 1 0 1 0 3 2 1 1 1 0 0 1 1 0 1 0 1 = pr2,3,4 G2 (A) 3 3 0 1 1 0 0 1 0 1 0 0 0 1 = pr2,1,3 G2 (A) 3 3 0 0 1 1 0 1 1 1 0 1 = pr4,6,7 G3 (A) 3 3 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 3 3 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 = pr4,6,7 G3 (A) 3 3 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 1 = pr1,2,3 G2 (A) 3 3 1 1 ⎛0 1 0 0 ⎞ 0 0 0 1 ⎝ 0 1 0 1 ⎠ = G2 (A) 4 3 1 0 0 1 1 1 0 1 ⎛ ⎞ 0 0 0 ⎝0 0 1⎠ 4 2 0 1 1 1 1 1 ⎛ ⎞ 0 0 0 1 1 0 1 1 ⎝ 0 0 1 1 0 1 0 1 ⎠ = G2 (A) 4 4 0 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 ⎛ ⎞ 1 0 0 0 0 1 1 1 ⎝ 0 1 0 0 1 0 1 1 ⎠ = pr1,2,3,5 G3 (A) 4 4 0 0 1 0 1 1 0 1 1 1 1 0 ⎛0 0 0 1⎞ 0 0 0 0 ⎝ 0 0 1 1 ⎠ = pr1,2,4,6 G3 (A) 4 3 0 1 1 0 1 0 1 0 ⎛ ⎞ 0 0 0 0 ⎝ 0 1 0 0 ⎠ = pr1,5,6,7 G3 (A) 4 3 0 1 1 0 0 1 0 1

10.2 The Dimensions and Relation Degrees of Post’s Classes

A K ∪ C1 T0,4 T0,3 ∩ T1 T0,3 ∩ M T0,2 ∩ M ∩ T1

K T0,t (t ≥ 5)

295

dim A ⎛ with A = P d(A) ⎞ol 0 0 0 1 ⎝ 0 1 0 1 ⎠ = G2 (A) 4 3 1 0 0 1 1 1 1 1 4 E24 \{(1, 1, 1, 1)} 4 ⎛ ⎞ 0 0 0 1 0 1 1 ⎝ 0 0 1 0 1 0 1 ⎠ = pr4,6,7,8 G3 (A) 4 3 0 1 0 0 1 1 0 1 1 1 1 1 1 1 ⎛ ⎞ 0 0 0 0 1 0 1 1 ⎝ 0 0 0 1 0 1 0 1 ⎠ = pr4,6,7,8 G3 (A) 3 4 0 0 1 0 0 1 1 0 1 1 1 1 ⎛0 1 1 1⎞ 0 0 0 1 ⎝ 0 0 1 0 ⎠ = pr3,5,7,8 G3 (A) 4 2 0 1 1 1 1 1 1 1 ⎛ ⎞ 0 0 0 0 ⎜1 0 0 0⎟ 5 3 ⎝ 1 0 1 0 ⎠ = pr1,5,6,7,8 G3 (A) 1 1 0 0 1 1 1 1 t E2t \{(1, 1, ..., 1)} =: αt t

T0,t−1 ∩ T1

t

αt−1 × (1) =: βt

t−1

T0,t−1 ∩ M

t

βt ∪ {(0, 0, ..., 0)} =: γt

t−1

T0,t−2 ∩ M ∩ T1 (t ≥ 5)

t

γt−1 × (1)

t−2

T0,∞

∞

∞

T0,∞ ∩ T1

∞

∞

T0,∞ ∩ M

∞

∞

T0,∞ ∩ M ∩ T1

∞

∞

C0

∞

∞

C

∞

∞

Lemma 10.2.1 Let M := {Ca , C, Ta,∞ , Ta,∞ ∩ Ta , Ta,∞ ∩ M, Ta,∞ ∩ M ∩ Ta | a ∈ E2 }. Then dim A = ∞ for every A ∈ M. Proof. Since e11 ∈ P ol holds for arbitrary , the contention is valid apparently for sets Ca and C. Assume for A ∈ M\{C0 , C1 , C} there is an h-ary relation with r columns, for which A = P ol holds. If one identiﬁes the variable xi with the variable xj (i = j) in the function , one receives a function of the form hr+1 r

296

10 The Relation Degree and the Dimension of Subclasses of Pk

xi ∧ g(x1 , ..., xi−1 , xi+1 , ..., xr )

(10.1)

with g ∈ M ∩ T0 ∩ T1 . Since all functions of the form (10.1) belong to A, hr preserves the relation , in contradiction to hr ∈ A. Lemma 10.2.2 Let (a) dim A = 0 ⇐⇒ (b) dim A = 1 ⇐⇒ (c) dim A = 2 ⇐⇒

A be a subclass of P2 . Then A = P2 ; A ∈ {T0 , T1 }; A ∈ {M, S, T0 ∩ T1 , T0.2 , T1,2 }.

Proof. The statements (a) and (b) are trivial. One easily checks that only classes of the set {P2 , Ta , M, S, T0 ∩ T1 , Ta,2 |a ∈ E2 } are describable through binary relations on E2 . As a direct consequence from Lemma 10.2.2 and the remarks before Table 10.1, we see that Lemma 10.2.3 If A ∈ {M ∩ Ta , S ∩ Ta , S ∩ M, L ∩ Ta , K ∪ C, I, I ∪ I, I ∪ Ca , [P21 ], M ∩ Ta,2 , Ta ∩ Ta,2 |a ∈ E2 }, then dim A = 3. Lemma 10.2.4 If b ≥ 2 then dim T0,b ≥ b. Proof. Let b ≥ 2. Assume there is an h-ary relation with T0,b = P ol , 1 ≤ h < b and does not contain any double rows. Because of c0 ∈ T0,b , has not an 1-column. Since [{x ∧ y}] = n≥1 {f n ∈ P2 | ∃i ∃g n−1 ∈ P2 : f (x1 , ..., xn ) = xi ∧ g(x1 , ..., xi−1 , xi−1 , ..., xn )} ⊆ T0,b , is valid in addition apparently: ⎛ ⎞ a1 ⎜ a2 ⎟ ⎜ ⎟ ⎝ ... ⎠ ∈ =⇒ (∀b1 , ..., bh ∈ E2 ah

⎞ ⎛ a1 b1 ⎜ b2 ⎟ ⎜ a2 ⎟ ⎜ :⎜ ⎝ ... ⎠ ≤ ⎝ ... bh ah ⎛

⎞

⎛

⎞ b1 ⎜ ⎟ ⎟ ⎟ =⇒ ⎜ b2 ⎟ ∈ ) ⎝ ... ⎠ ⎠ bh (10.2) ∈ T0,b−1 \T0,b , there is certain r1 , ..., rb ∈

Since T0,b−1 ⊇ T0,b = P ol and hb−1 with hb−1 (r1 , ..., rb ) ∈ . Denote z1 , ..., zt exactly the rows of (r1 ..., rb ) on which hb−1 takes the value 1. Because of deﬁnition of the class T0,b−1 , the matrix ⎛ ⎞ z1 ⎜ z2 ⎟ ⎜ ⎟ ⎝ ... ⎠ zt contains an 1-column. Thus (r1 , ..., rb ) has a column ≥ hb−1 (r1 , ..., rb ), in contradiction to (10.2) and hb−1 (r1 , ..., rb ) ∈ .

10.2 The Dimensions and Relation Degrees of Post’s Classes

297

Lemma 10.2.5 If b ≥ 2 and a ∈ {0, 1} then dim M ∩ Ta,b ≥ b + 1. Proof. Let b ≥ 2 and w.l.o.g. let a = 0. Further, let := {(x1 , ..., xb+1 ) ∈ E2b+1 \{(1, 1, ...1)} | xb+1 = 1} ∪ {(0, 0, ..., 0)} Then M ∩ T0,b = P ol . Assume there is an r-ary relation ‘ with 1 ≤ r ≤ b and P ol‘ = M ∩ T0,b . Since hb−1 ∈ M ∩ T0,b , there exists some aij ∈ E2 with ⎛ ⎞ ⎛ ⎞ a11 a12 ... a1b d1 ⎜ a21 a22 ... a2b ⎟ ⎜ d2 ⎟ ⎟ ⎜ ⎟ hb−1 (A) := hb−1 ⎜ ⎝ ... ... ... ... ⎠ = ⎝ ... ⎠ ∈ . ar1 ar2 ... arb dr W.l.o.g. let d1 = ... = ds = 0 and ds+1 = ... = db = 1. Since (0, ..., 0) ∈ , only the following two cases are possible: Case 1: s = 0. Since hb−1 (x1 , ..., xb ) = 1 ⇐⇒ (∃i : xi ∈ E2 ∧ (∀j = i : xj = 1))

(10.3)

and (1, ..., 1) ∈ (because of c1 ∈ M ∩ T0,b ), we have ⎛ ⎞ 0 1 ... 1 1 ⎜ 1 0 ... 1 1 ⎟ ⎜ ⎟ ⎜ . . . . . . . . . . . ⎟ ⊆ A ⊆ . ⎜ ⎟ ⎝ 1 1 ... 0 1 ⎠ 1 1 ... 1 0 With the help of xy ∈ M ∩ T0,b this implies = E2r \{(1, ..., 1)}, in contradiction to P ol = M ∩ T0,b . Case 2: 1 ≤ s ≤ b − 1. Because of (10.3), w.l.o.g., we can assume ⎛ ⎞ a11 a12 ... a1s a1,s +1 ... a1b ⎜ ............................. ⎟ ⎜ ⎟ ⎜ as1 as2 ... ass as,s +1 ... asb ⎟ ⎜ ⎟ 1 ... 1 ⎟ A=⎜ ⎜ 0 1 ... 1 ⎟. ⎜ 1 0 ... 1 ⎟ 1 ... 1 ⎜ ⎟ ⎝ ............................. ⎠ 1 1 ... 0 1 ... 1 This yields

⎛

a1b ⎜ ... ⎜ ⎜ asb hb (A, ⎜ ⎜ 1 ⎜ ⎝ ... 1

⎞

⎛

d1 ⎜ ... ⎟ ⎜ ⎟ ⎜ ⎟ ⎟) = ⎜ ds ⎜ ds+1 ⎟ ⎜ ⎟ ⎝ ... ⎠ dr

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

298

10 The Relation Degree and the Dimension of Subclasses of Pk

in contradiction to hb ∈ P ol and (d1 , ..., dr ) ∈ . Lemma 10.2.6 Let A = [A] ⊂ T1 and x ∧ y ∈ A. If A = P ol , then contains an 1-row. Proof. The relation cannot contain a 0-column because of A ⊂ T1 . If does not contain a 1-row then there are certain aij ∈ E2 with ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 a12 a1h ⎜ a21 ⎟ ⎜ 0 ⎟ ⎜ a2h ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a31 ⎟ , ⎜ a32 ⎟ , ..., ⎜ a3h ⎟ ∈ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ... ⎠ ⎝ ... ⎠ ⎝ ... ⎠ ah1 ah2 0 Since x ∧ y ∈ A, contains then a 0-column, however, contrary to the above remark. Lemma 10.2.7 dim T0,b ∩ T1 ≥ b + 1 for b ≥ 2. Proof. Denote a relation of smallest arity with P ol = T0,b ∩ T1 . W.l.o.g., by Lemma 10.2.6, we can assume = (1) × , where obviously T0,b ∩ T1 ⊂ P ol and P ol ⊆ T1 hold. It results, then, from the structure of Post’s graph that P ol = T0,b holds. This and Lemma 8.2.4 imply dim T0,b ∩ T1 ≥ b + 1. Analogously, one can prove the following lemma. Lemma 10.2.8 dim T0,b ∩ M ∩ T1 ≥ b + 2 for b ≥ 2. Lemma 10.2.9 dim K ∪ C1 ≥ 4. Proof. By Lemma 10.2.6, every relation with P ol = K ∪ C1 contains an 1-row. If one scrutinizes the possibilities for the h-ary relations with h ≤ 3, one receives P ol = K ∪ C1 . Lemma 10.2.10 dim K ∪ C0 ≥ 4. Proof. dim K ∪ C0 ≥ 3 follows from Lemma 10.2.2. Assume that P ol = K ∪ C0 holds for a certain ternary relation . Obviously, cannot contain an 1-column. Hence, one also does not ﬁnd any r1 , r2 , r3 ∈ with r1 (r2 ∨r3 ) ∈ . Thus, T0,∞ ∩M ∩T1 ⊆ P ol holds, in contradiction to K ∪C0 ⊂ T0,∞ ∩M ∩T1 . Lemma 10.2.11 dim K ≥ 5. Proof. Obviously, dim K ≥ 3. Assume an h-ary relation (h ∈ {3, 4}) exists without double rows and with P ol = K. The relation does not contain certainly a 0-column nor an 1-column. Furthermore, w.l.o.g. by Lemma 10.2.6, we have = (1) × , where P ol ⊃ K. If one scrutinizes the possibilities for P ol with the aid of the Post’s graph and of Lemma 10.1.1, one receives a contradiction.

10.2 The Dimensions and Relation Degrees of Post’s Classes

299

Lemma 10.2.12 dim L ≥ 4. Proof. Because of Lemma 10.2.2, class L cannot be described with the aid of a unary or binary relation. Assume there is a ternary relation with L = P ol . Because of c0 , c1 , x ∈ L we have ⎛ ⎞ 0 1 ⎝0 1⎠⊆ 0 1 and

⎛

⎞ ⎛ ⎞ a a ⎝ b ⎠ ∈ =⇒ ⎝ b ⎠ ∈ . c c

Furthermore, can not contain any double rows. This implies that the relation has three linearly independent columns. Hence, because of x + y ∈ L, we have = E23 , in contradiction to L = P ol . Lemma 10.2.13 dim L ∩ S ≥ 4. Proof. Proving dim L ∩ S = 3 suﬃces. Assume that L ∩ S = P ol holds for a certain ternary relation . Because of c0 , c1 ∈ L ∩ S and x ∈ L ∩ S we have (a, a, a) ∈ for a ∈ E2 and ⎛ ⎞ ⎛ ⎞ a a ⎝ b ⎠ ∈ =⇒ ⎝ b ⎠ ∈ . c c Then, w.l.o.g., has the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 0 1 1 0 0 1 0 1 1 = ⎝ 0 1 ⎠ , 2 = ⎝ 0 1 1 0 ⎠ or 3 = ⎝ 0 1 1 0 0 1 ⎠ . 0 1 0 1 0 1 0 1 0 1 1 0 Since P ol 1 = P ol 2 = S and P ol 3 = [{x}], we obtain a contradiction to the supposition. Lemma 10.2.14 dim L ∩ S ∩ T1 ≥ 4. Proof. Since x + y + z ∈ L ∩ S ∩ T1 , for all columns s of the relation with P ol = L ∩ S ∩ T1 we have: s, s + 1 ∈ =⇒ ∀t ∈ : s + t + (s + 1) = t + 1 ∈ (s + 1 := (s1 + 1, ..., sh + 1) if s := (s1 , ..., sh )). Because of x + 1 ∈ L ∩ S ∩ T1 this implies s ∈ =⇒ s + 1 ∈ . By this and by (a, a, a) ∈ (a ∈ E2 ), if is ternary, then can have at most three diﬀerent columns. Now one easily checks that the clone L∩S ∩T1 cannot be described by a such ternary relation.

300

10 The Relation Degree and the Dimension of Subclasses of Pk

Lemma 10.2.15 dim I ∪ C ≥ 4. Proof. Because of Lemma 10.2.2, it is suﬃcient to show that dim I ∪ C = 3 holds. Since the classes [P21 ], K ∪ C and D ∪ C are the direct predecessors of I ∪ C, every ternary relation with P ol = I ∪ C has w.l.o.g. the following four properties: ⎛ ⎞ 0 1 ⎝ 0 1 ⎠ ⊆ ⊂ E23 , 0 1 ⎛ ⎞ ⎛ ⎞ a a ∃a, b, c ∈ E2 : ⎝ b ⎠ ∈ ∧ ⎝ b ⎠ ∈ (by x ∈ P ol ), c ⎞ c ⎛ ⎞ ⎛ 0 1 0 ⎝ 1 0 ⎠ ⊆ ∧ ⎝ 0 ⎠ ∈ (by x ∧ y ∈ P ol ), ⎛1 1⎞ ⎛1⎞ 1 0 1 ⎝ 0 1 ⎠ ⊆ ∧ ⎝ 1 ⎠ ∈ (by x ∨ y ∈ P ol ). 0 0 0 Apparently there is no relation that fulﬁlls all four of these conditions. Thus dim I ∪ C ≥ 4. Now, in summarizing, we obtain the statements of the following theorem as consequences of Lemmas 10.2.1–10.2.15 and from the above remarks (before Table 10.1).

Theorem 10.2.16 For an arbitrary subclass A of P2 and t ≥ 6 the following holds: dim(A) = 0 ⇐⇒ A = P2 , dim(A) = 1 ⇐⇒ A ∈ {T0 , T1 }, dim(A) = 2 ⇐⇒ A ∈ {M, S, Ta,2 , T0 ∩ T1 | a ∈ E2 }, dim(A) = 3 ⇐⇒ A ∈ { M ∩ Ta , S ∩ T0 , S ∩ M, M ∩ Ta,2 , Ta ∩ Ta,2 , L ∩ Ta , K ∩ C, I, I ∪ Ca , I, [P21 ]}, dim(A) = 4 ⇐⇒ A ∈ { I ∪ C, M ∩ T0 ∩ T1 , L, L ∩ S, L ∩ S ∩ T0 , K ∪ Ca , Ta,4 , Ta ∩ Ta,3 , M ∩ Ta,4 , Ta ∩ M ∩ Ta,2 | a ∈ E2 }, dim(A) = 5 ⇐⇒ A ∈ {K, Ta,5 , Ta ∩ Ta,4 , M ∩ Ta,4 , Ta ∩ M ∩ Ta,3 | a ∈ E2 }, dim(A) = t ⇐⇒ A ∈ {Ta,t , Ta,t−1 ∩ Ta , Ta,t−1 ∩ M, Ta,t−2 ∩ M ∩ Ta | a ∈ E2 }, dim(A) = ∞ ⇐⇒ A ∈ {C, Ca , Ta,∞ , Ta,∞ ∩ Ta , Ta,∞ ∩ M, Ta,∞ ∩ M ∩ Ta | a ∈ E2 }.

10.3 Further Examples of the Dimension and Relation Degree of Classes

301

Theorem 10.2.17 For an arbitrary subclass A of P2 and t ≥ 5 the following holds: (a) d(A) = 0 ⇐⇒ A = P2 , d(A) = 1 ⇐⇒ A ∈ {T0 , T1 , T0 ∩ T1 }, d(A) = 2 ⇐⇒ A ∈ { M, S, Ta,2 , S ∩ T0 , M ∩ Ta , S ∩ M, Ta,2 ∩ Ta , Ta,2 ∩ M, M ∩ T0 ∩ T1 , Ta,2 ∩ M ∩ Ta | a ∈ E2 }, (b) d(A) = 3 ⇐⇒ A ∈ { L ∩ Ta , K ∪ C, D ∪ C, I, I ∪ I, I ∪ Ca , [P21 ], Ta,3 , I ∪ C, K ∪ Ca , D ∪ Ca , Ta,3 ∩ M, Ta,3 ∩ Ta , Ta,3 ∩ M ∩ Ta , L ∩ S ∩ T0 , K, D | a ∈ E2 }, (c) d(A) = 4 ⇐⇒ A ∈ {L, L ∩ S, Ta,4 , Ta,4 ∩ M, Ta,4 ∩ Ta , Ta,4 ∩ M ∩ Ta | a ∈ E2 }, (d) d(A) = t ⇐⇒ A ∈ {Ta,t , Ta,t ∩ Ta , Ta,t ∩ M, Ta,t ∩ M ∩ Ta | a ∈ E2 }, (e) d(A) = ∞ ⇐⇒ A ∈ {C, Ca , Ta,∞ , Ta,∞ ∩Ta , Ta,∞ ∩M, Ta,∞ ∩M ∩Ta | a ∈ E2 }. Proof. It is easy to check that the functions of M ∩ S preserve every unary and binary relation on E2 . Consequently, we have for an arbitrary subclass A of P2 : d(A) ≤ 2 ⇐⇒ S ∩ M ⊆ A. Thus, (a) follows from the deﬁnitions of the given classes, and d(A) ≥ 3 holds for the remaining classes A. (b) results from the descriptive relations of the classes given in Table 10.1 or from their deﬁnitions, with the aid of P ol ∩ P ol = P ol × = P ol {, }. (c) and (d) one can easy prove by means of the Post’ graph and with the aid of (a), (b) and Lemma 10.1.1. (e) follows from Lemma 10.1.1, (c) and Theorem 10.2.16.

10.3 Further Examples of the Dimension and Relation Degree of Classes We come now to the dimensions and relation degrees of some subclasses of Pk . The following theorem supplies simple examples ﬁrst. Theorem 10.3.1 ([Ros 70;a]) Let A be a maximal class of Pk (k ≥ 2), i.e., there exists a ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk with A = P ol (see Chapter 5). Then (a) d(A) = dim A;

302

10 The Relation Degree and the Dimension of Subclasses of Pk

(b) d(A) = 1, if ∈ C1k ; d(A) = 2, if ∈ C2k ∪ Mk ∪ Sk ∪ Uk ; (c) d(A) = h, d(A) = k, (d) d(A) = 3, d(A) = 4,

if if if if

h ∈ {3, 4, ..., k − 1} and ∈ Chk ∪ Bhk ; A = P olιkk and k ≥ 3; ∈ Lk , k = pm and p = 2; ∈ Lk , k = pm and p = 2.

Proof. (a) follows from Lemma 10.1.1, (e). (b) results immediately from the deﬁnitions of the corresponding classes P ol . (c): Since every h-ary relation with ∈ Chk ∪ Bhk for h ≤ k − 1 and = ιkk for h = k contains the set ιhk , all functions, which take at most h − 1 diﬀerent values, belong to P ol. Consequently, one cannot describe the class P ol in this case with the aid of (h − 1)-ary relations. (d): Let k = pm (p prime, m ≥ 1), (Ek , +, ·) a ﬁeld with the zero element 0 and the unit element 1, = {(a, b, c, d) ∈ Ek4 | a + b = c + d} (i.e. ∈ Lk ), A := P ol, dim A = q and denote λ a certain q-ary relation with P olλ = A. Let be given the relation λ in the form of a matrix, whose rows are denoted with z1 , ..., zq . Obviously, λ does not have any double rows and q ≥ 2 holds. Further, we can interpret the rows (and/or the columns) of λ in natural manner as elements of a vector space V|λ| (or Vq ) over the ﬁeld (Ek , +, ·), where the operations are deﬁned as usual in matrix vector spaces. First we prove that dimA ≥ 3 for p = 2 and dim A = 4 for p = 2. Obviously, all q-ary functions f of the form f (x1 , ..., xq ) = a1 x1 + ... + aq xq belong to A for arbitrary a1 , a2 , ..., aq ∈ Ek . If the rows z1 , ..., zq are linear independent (i.e., there exist certain q linear independent columns s1 , ..., sq ∈ λ ⊆ Ekq ), then for every s ∈ Ekq there is a certain function f ∈ A with f (x1 , ..., xn ) = a1 x1 + ... + aq xq and f (s1 , ..., sq ) = s. Hence, λ = Ekq holds, in contradiction to A = P olλ ⊂ Pk . For the proof of dim A ≥ 3, if p = 2, and dim A = 4, if p = 2, we have only to show that the rows z1 , ..., zq (q = 2 for p = 2, q ∈ {2, 3} for p = 2) are linear independent. If q = 2 and if we assume that z1 = α · z2 (for certain α ∈ Ek \{0}), then a contradiction follows from g(x) := x + 1 ∈ A and from a 0 a a+1 ∀ ∈ λ\ :g = ∈ λ. α·a 0 α·a α·a+1 Consequently, for arbitrary p, we have dim A ≥ 3 and pri,j λ = Ek2 for all i, j ∈ {1, 2, ..., q} with i = j. Now let q = 3 and p = 2. Obviously, the columns

10.3 Further Examples of the Dimension and Relation Degree of Classes

⎛

⎞

⎛

⎞

⎛

303

⎞

1 1 c s1 := ⎝ 0 ⎠ s2 := ⎝ b ⎠ s3 := ⎝ 1 ⎠ a 0 0 belong to λ for certain a, b, c ∈ Ek . Then, because of x + 1, x2 ∈ A, we have ⎞ ⎛ ⎞ ⎛ 0 1 s4 := ⎝ 1 ⎠ ∈ λ, s5 := ⎝ 0 ⎠ ∈ λ. a+1 a2 Now one easily checks that s1 , s4 , s3 if a = 0, s1 , s4 , s2 if a = 1 and s1 , s5 , s3 if a ∈ {0, 1} are linear independent. Consequently, dim A = 4 holds in the case p = 2. Since ∆λ = {(a, b, c) ∈ Ek3 | a + a = b + c} obviously is not a diagonal relation, for p = 2, A ⊆ P ol∆λ and A is maximal in Pk , we have dimA = 3 in the case p = 2. It is surely a hopeless venture to determine the relation degree for all subclasses of Pk . Nevertheless, there are rather precise estimates for d(A) or dim(A) if A is an element of a certain sublattice of Lk . Subsequently, two examples are given for this fact. The following theorem is a direct consequence of generalizing the Baker-Pixley Theorem (see Theorem 8.3.1). Theorem 10.3.2 If a subclass A of Pk contains a certain n-ary function h with the properties h(x, y, ..., y) = h(y, x, y, ..., y) = ... = h(y, ..., y, x) = y for arbitrary x, y ∈ Ek and n ≥ 3, then d(A) ≤ n − 1 holds. As a consequence of theorems on linear and quasi-linear functions from Chapter 13 we get: Theorem 10.3.3 Let k = pm (p prime, m ≥ 1), (Ek , +, ·) a ﬁeld and λ := {(a, b, c, d) ∈ Ek4 | a + b = c + d}. Then, for every subclass A ⊆ P olk λ, which contains the function r with r(x, y, z) = x + y − z, the following holds: d(A) = 3 for p = 2, 3 ≤ d(A) ≤ 4 for p = 2, 3 ≤ dimA ≤ 6 for p = 2 and 3 ≤ dimA ≤ 7 for p = 2.

304

10 The Relation Degree and the Dimension of Subclasses of Pk

In preparation for Theorem 10.3.7 with statements about subclasses of [Pk1 ], we subsequently prove three lemmas. 3 3 Lemma 10.3.4 Let µ := δ{1,2} ∪ δ{2,3} . Then P olµ = [Pk1 ].

Proof. Obviously, P olµ ⊆ [Pk1 ]. “=” follows from Theorem 1.4.4, (a), (c). Lemma 10.3.5 Let

ν := {(x, y) ∈ Ek2 | x = y}.

Then P olν = [Pk1 [k]]. Proof. Obviously, (P olν)1 = Pk1 [k]. Assume there exists an n-ary function f ∈ P ol = ν that depends on at least two variables essentially. Then f takes exactly k diﬀerent values and, by Theorem 1.4.4, (c) one can ﬁnd certain ri := (r1i , ..., rki ) ∈ ιkk (i = 1, ..., n) with f (r1 , ..., rn ) ∈ Ekk \ιkk . If one chooses ai ∈ Ek \{r1i , ..., rki } (i = 1, ..., n) now, there is so a j ∈ {1, ..., k} with f (rj1 , ..., rjn ) = f (a1 , ..., an ). Since (rji , ai ) ∈ ν for all i ∈ {1, ..., n}, f does not preserve the relation ν, contrary to the assumption. Thus P olν = [Pk1 [k]]. Lemma 10.3.6 Let k ≥ 3 and denote ω the binary relation {(0, 1), (1, 0), (0, x), (x, 1), (y, y + 1) | x ∈ Ek \{0, 1} ∧ y ∈ Ek \{0, 1, k − 1}}. Then for an arbitrary function f n ∈ Pk the following holds: f n ∈ P olω =⇒ ∃i : f n = eni .

(10.4)

(I.e., the projections are the only functions of Pk , which preserve ω.) Proof. We write a ≺ b instead of (a, b) ∈ ω. Furthermore, for arbitrary a, b ∈ Ekn let be: a ≺ b :⇐⇒ ∀i ∈ {1, ..., n} : ai ≺ bi . Denote f an arbitrary n-ary function of P olω in the following. Then the function f|{0,1} belongs to M ∩ S, since one can derive the relations 0 1 0 0 1 and 1 0 0 1 1 from ω as follows: 0 0 1 0 1 2 ω ∩ (τ ω) = , pr1 (ω ∩ (τ ω)) = E2 , (ω ◦ ω) ∩ E2 = 0 1 1 1 0 Because of f|{0,1} ∈ M ∩ S one can ﬁnd (w.l.o.g.) two tuples a := (0, 0, ..., 0, 1 , 1, ..., 1) and i

b := (1, 0, ..., 0, 1 , 1, ..., 1) i

10.3 Further Examples of the Dimension and Relation Degree of Classes

305

with f (a) = 0 and f (b) = 1. Now we want to prove the following statements: f|{0,1} = en1 ,

(10.5)

∀α ∈ {0, 1} : f (α, x2 , ..., xn ) = α,

(10.6)

∀β ∈ Ek \{0, 1} : f (β, x2 , ..., xn ) = β.

(10.7)

For this purpose we consider the tuples u := (d, 1, ..., 1, 0 , 0, ..., 0), i

v := (1, 0, ..., 0, d , d, ..., d), i

w := (0, d, ..., d, 1 , 1, ..., 1) i

for a certain d ∈ Ek \{0, 1} and z := (0, 1, 1, 1, ..., 1). Then a ≺ u ≺ b and u ≺ v ≺ w ≺ u. Consequently, we have f (u) ∈ Ek \{0, 1}, f (v) = 1 and f (w) = 0. Then, by v ≺ z, f (z) = 0 holds. Since f|{0,1} ∈ S ∩ M , (10.5) follows obviously from f (z) = 0. Furthermore, for an arbitrary tuple x ∈ Ekn there are tuples y, y with y ≺ x ≺ y and ⎧ ⎨ (1, 1) if x1 = 0, (y1 , y1 ) = (0, 0) if x1 = 1, ⎩ (0, 1) if x1 ∈ Ek \{0, 1}. Because of f (y) ≺ f (x) ≺ f (y ) and by (10.5) the statement (10.6) and f (x) ∈ Ek \{0, 1} if x1 ∈ Ek \{0, 1}

(10.8)

result from that. For every tuple x ∈ Ekn with x1 ∈ Ek \{0, 1} one can ﬁnd k − 2 tuples x2 , x3 , ..., xk−1 with the properties xi = (i, xi2 , ..., xin ) (i = 2, 3, ..., k − 1), xx1 = x and x2 ≺ x3 ≺ ... ≺ xk−1 . (To a given tuple x one can construct the tuples xi as follows, for example, xx1 −1,j := xj =: xx1 +1,j for xj ∈ {0, 1}, xx1 −1,j = 0 and xx1 +1,j = 1 for xj ∈ Ek \{0, 1} (j = 2, ..., n), etc.) Then f (x2 ) ≺ f (x3 ) ≺ ... ≺ f (xk−1 ) and f (xi ) ∈ Ek \{0, 1} (i = 2, ..., k − 1) because of (8). This is, however, possible only for f (xi ) = i for all i ∈ {2, ..., k− 1}. Thus (10.4) is right and our lemma was proven.

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10 The Relation Degree and the Dimension of Subclasses of Pk

Theorem 10.3.7 (1) Let A be a subclass of [Pk1 ] with e11 ∈ A. Then (a) d(A) = 3 and 3 ≤ dimA ≤ 4 for k = 2; (b) 2 ≤ d(A) ≤ k and 2 ≤ dimA ≤ k + 3 for k ≥ 3. (2) dim[Pk1 ] = d([Pk1 ]) = 3 and d(A) = dimA =

3 if k = 2, 2 if k ≥ 3

for every A ∈ {[Pk1 [k]], [e11 ]}. Proof. For k = 2, we have the statements of the theorem already proven (see Theorems 10.2.16 and 10.2.17). Let k ≥ 3 in the following. Then, obviously, d(A) ≥ 2 for all classes A ⊆ [Pk1 ] with e11 ∈ A. By Lemma 10.3.4 and because of A1 = (P olG1 (A))1 we have, furthermore, for every A ∈ Vk (e11 , [Pk1 ]): A = P ol{G1 (A), µ} = P olG1 (A) × µ. Thus (b) holds. The statements of (2) follow from Lemmas 10.3.5 and 10.3.6 and from the fact that only the diagonal relations of the binary relations preserve all functions of Pk1 .

11 On Generating Systems and Orders of the Subclasses of Pk

If A = [A] ⊆ Pk is ﬁnitely generated, we call the smallest number r with [Ar ] = A the order of A. In the case that the subclass A ⊆ Pk is not ﬁnitely generated, we write ord A = ∞. During the study of subclasses of Pk or during the solution of problems in the multi-valued logics, one often deals with constructing generating systems for the considered classes. Statements about the order of the considered classes often result from these investigations. Therefore, one ﬁnds results about the order of subclasses of Pk not only in this chapter, but also in other chapters of this book. First, some general statements about the orders of subclasses of Pk are declared in this chapter. Then we occupy ourselves with the orders of the maximal classes of Pk . It turns out, in this case, that only certain maximal classes of the type M do not have any ﬁnite order. If ∈ Mk , we can prove ord P olk = 2 only for k ≤ 7 and for the case where (Ek ; ) is a lattice. By means of an example proven by G. Tardos in [Tar 86], we will be able to show, however, that for k ≥ 8 there are classes of the type M with the order ∞. If one has a ﬁnite generating class A, it is an interesting problem to clear which cardinalities are possible for the bases of this class A. In Section 11.2, we show only that the proof for ord P olk ≤ 3, where ∈ C1k ∪ Uk ∪ Sk , implies the existence of functions f with [f ] = P olk . Following corresponding notation for functions from Pk , such function f is called a Sheﬀer-function for P olk . We notice, through the theorems of Section 12.2, Schoﬁeld’s Theorem 7.1.4 is also proven. The last section explains shortly, as one can determine the cardinalities of the possible bases of the class A, if ord A < ∞. Further, some basic ideas of [Miy 71], [Miy-S-L-R 86], [Miy 88] and [Sto 87] are explained about basis classiﬁcations. For more information on the topic, refer to the books [Miy 88] and [Sto 87] by M. Miyakawa and I. Stojmenovic.

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11 On Generating Systems and Orders of the Subclasses of Pk

11.1 Some General Properties of Generating Systems and Bases The following theorem is a consequence of [Coh 65, Th. 5.4]: Theorem 11.1.1 ([Jab 74]) Let A be a subclass of Pk . Then the following conditions are equivalent: (1) A is ﬁnitely generated. (2) A has only ﬁnitely many maximal classes (with other words: The lattice L↓k (A) is dualatomar). (3) For every chain of the form A1 ⊆ A2 ⊆ A3 ⊆ ... ⊆ An ⊆ ... ⊆ A with ∪n≥1 An = A there exists an n ∈ N with An = A. Theorem 11.1.2 (a) Each subclass of P2 is ﬁnitely generated. For every n ∈ N exists a subclass A ⊆ P2 with ord A = n. (b) For k ≥ 3 there are subclasses of Pk that do not have any basis or have an inﬁnite basis ([Jan-M 59]). Proof. (a) follows from Chapter 3. (b): Let k ≥ 3. One can ﬁnd a subclass of Pk with an inﬁnite basis in Lemma 8.1.1. The set A := [{f1 , f2 , f3 , ...}], where 1 if x1 = ... = xn = 2, fn (x1 , ..., xn ) := 0 otherwise, is an example of a class without basis, since fn fm = cm+n−1 and ∆fn+1 = fn 0 for arbitrary m, n ∈ N. The following theorem is easy to check. Theorem 11.1.3 (a) If A is a maximal class of B ⊆ Pk , then ord A < ∞ =⇒ ord B < ∞. (b) For arbitrary subclasses A, B of Pk it holds: ord[A ∪ B] ≤ max{ord A, ord B}.

11.1 Some General Properties of Generating Systems and Bases

309

As the following theorem shows, no connection exists between the existence of a ﬁnite relation degree (see Chapter 10) and the existence of a ﬁnite order of a subclass of Pk . Theorem 11.1.4 ([P¨ os-K 79]) There are subclasses A, B, C of Pk with the properties: (a) d(A) = ∞ and ord A < ∞. (b) d(B) < ∞ and ord B = ∞. (c) d(C) = ∞ and ord C = ∞. Proof. An example for a class A is the set T0,∞ (see Chapter 3). (b) follows from Theorem 11.5.5. One can ﬁnd an example of a class C in the book [P¨ os-K 79], p. 97. Despite the above theorem, the juxtaposition of the orders and the relation degrees of classes is quite interesting (see [P¨os-K 79], p. 94–97). The main aim of the following sections is to prove the following theorem. Theorem 11.1.5 (Theorem on the Orders of the Maximal Classes) (1) Let ∈ Mk . If 2 ≤ k ≤ 7 and (Ek ; ) is a lattice, then ord P olk = 2. For k ≥ 8 there is a certain ∈ Mk with ord P olk = ∞. (2) For every ∈ Sk it holds: ord P olk :=

3 if k = 2, 2 otherwise.

(3) If ∈ Uk ∪ Lk ∪ Bk , then P olk = 2. (4) Let γ be an h-ary relation of Ck . Then 2 ≤ ord P olk γ ≤ max(2, h). For k = 3, this theorem, already, was proven in 1962 by M. N. Gnidenko (see e.g. [Gni 65]). We need the following theorem for proof of Theorem 11.1.5. Theorem 11.1.6 Let T ⊆ Ek with |T | ≥ 2, Pk,T := {f ∈ Pk | W (f ) ⊆ T } and 2 ≤ l ≤ k. Then (a) ord Pk,T = 2, (b) ∃f ∈ Pk,T : [f ] = Pk,T ,

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11 On Generating Systems and Orders of the Subclasses of Pk

(c) ord Pk (l) = 2. Proof. In the case that T = Ek is valid, the above theorem is, already, proven(see Theorems 1.4.2 and 7.1.6). If |T | ≤ k − 1, then we can assume T = El w.l.o.g., through which the statement (a) follows from Lemma 12.2.1 and (b) from Theorems 12.2.7 and 7.1.6. (c) follows from (a).

11.2 The Orders and Sheﬀer-Functions of the Classes of Type C1 , S or U The aim of this section is to prove that all maximal classes of type X ∈ {C1 , S, U} have the order 2 for k ≥ 3 and that for every one of these classes, Sheﬀer-functions exist. We remark that all other maximal classes do not have any Sheﬀer-functions (see Theorem 7.1.4). Theorem 11.2.1 ([Sch 69], [Kud 70]) Let ∈ C1k . Then (a) ord P olk = 2 (b) ∃f : [f ] = P olk . Proof. W.l.o.g. let = El with 1 ≤ l ≤ k − 1. Further, let A := P olk El . We distinguish two cases: Case 1: l = 1. In this case, A is the set of all 0-preserving functions of Pk , and it is possible to describe an arbitrary function f n ∈ A with the help of functions x ∨ y := max(x, y), x · y := min(x, y) (in respect to the order relation 0 < 1 < ... < k − 1), b if x = a, ja;b (x) := 0 otherwise (a, b ∈ Ek ) as follows: f (x1 , x2 , ..., xn ) =

ja1 ;f (a) (x1 )·ja2 ;f (a) (x2 )·...·jan ;f (a) (xn ).

a = (a1 , ..., an ) ∈ Ekn \{(0, 0, ..., 0)} f (a) = 0

Consequently, B := {∨, ·} ∪

{ja;b (x) · jc;b (y)} (⊆ A)

a,b∈{1,2,...,k−1}, c∈Ek

is a generating system for A and hence ord A = 2. To prove (b) for the ﬁrst case let

11.2 The Orders and Sheﬀer-Functions of the Classes of Type C1 , S or U

ja,b;t (x, y) := ja;t (x) · jb;t (y) = and

t if (x, y) = (a, b), 0 otherwise,

la;t (x, y, z) := j0;k−1 (x) · ja;t (y) · ja;t (z) =

311

t if (x, y, z) = (0, a, a), 0 otherwise.

With the help of these functions, a function g ∈ A can be described as follows: g(..., xa,b;t , ya,b;t , ..., ua;t , va;t , wa;t , ...) :=

ja,b;t (xa,b;t , ya,b;t ) (a, b, t) 2 ∈ Ek ×{1, 2, ..., k −1} a = b

∨

la;t (ua;t , va;t , wa;t ). (a, t) 2 ∈ {1, 2, ..., k − 1}

If one identiﬁes all variables in the function g, then one obtains the constant function c0 . One obtains an arbitrary binary function of A if one replaces certain variables of the form ua;t by c0 in g and identiﬁes certain other variables of g. Thus ord A in Case 1. Case 2: l ≥ 2. Let f n be an arbitrary function of A and let f1n ∈ Pk,l with f (a) = f1 (a) for all a ∈ Eln . With the help of f1 , the above-deﬁned functions ∨, ·, ja,b and the function l − 1 if x ∈ El , l(x) := 0 otherwise, one can describe f as follows: f (x1 , x2 , ..., xn ) = f1 (x1 , ..., xn ) · l(x1 ) · ... · l(xn ) ∨ ja1 ;f (a) (x1 ) · ja2 ;f (a) (x2 ) · ... · jan ;f (a) (xn ). a = (a1 , ..., an ) ∈ Ekn \Eln f (a) = 0

Consequently, for every set B1 ⊆ Pk,l ⊂ A with [prl B1 ] = Pl , the set B := B1 ∪ {∨, ·, l} ∪ {ja;b (x) · jc;b (y)} (⊆ A) a∈Ek \El , b,c∈Ek

is a generating system for A. From this, Lemma 12.2.1 and Theorem 1.4.2 it follows that ord A = 2. Since Pk,l has a Sheﬀer-function tm (see Theorems 12.2.7 and 7.1.6), one can prove analogously to the ﬁrst case that the function h with

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11 On Generating Systems and Orders of the Subclasses of Pk

h(x1 , ...., xm , xm+1 , ..., xa,b;t , ya,b;t , ..., ua;t , va;t , wa;t , ...) := t(x1 , ..., xm ) · l(xm+1 ) ∨ g(..., xa,b;t , ya,b;t , ..., ua;t , va;t , wa;t , ...) is a Sheﬀer-function for A. Theorem 11.2.2 ([Sch 69]) Let k ≥ 3 and ∈ Uk . Then (a) ord P olk = 2 (b) ∃f : [f ] = P olk . Proof. The nontrivial equivalence relation on Ek decomposes the set Ek in the (pairwise distinct) equivalence classes A0 , A1 , ..., Ak −1 (k < k). We select a representative ai from every equivalence class Ai and deﬁne the function q ∈ Pk1 by ∀i ∈ Ek : q(x) = ai :⇐⇒ x ∈ Ai . Let now f n be an arbitrary function of P olk . With the help of functions z, fi , gf ∈ P olk deﬁned in Lemma 1.4.6, one can describe f as follows: f (x) = ((...((gf (x) f0 (x)) f1 (x))... fk −1 (x))

(11.1)

Furthermore, by Section 5.2.3, there exists a function h ∈ P{a0 ,a1 ,...,ak } with gf (x) = h(q(x1 ), ..., q(xn )).

(11.2)

It is easy to check that the set of all functions of the form (11.2) is isomorphic to Pk ; thus the functions of this set are superpositions over the binary functions of this set. Consequently, (11.1) and ord Pk,Ai = 2 (see Theorem 11.1.6) implies ord P olk = 2. Let p2 be a Sheﬀer-function for P{a0 ,a1 ,...,ak −1 } and let rim be a Sheﬀerfunktion for Pk,Ai (i = 0, 1, ..., k − 1) 1 . Then one can form the functions p(q(x), q(y)), ri (x1 , ..., xm ) (i = 0, 1, ..., k − 1) and z as superpositions over the function s(x, y, x0,1 , x0,2 , ..., x0,m , x1,1 , ..., x1,m , ..., xk −1,1 , ..., xk −1,m ) := (...((p(q(x), q(y)) r0 (x0,1 , ..., x0,m )) r0 (x1,1 , ..., x1,m ))... rk −1 (xk −1,1 , ..., xk −1,m )) (11.3) as follows: 1

See Theorem 12.2.7 for the existence of such Sheﬀer-functions.

11.2 The Orders and Sheﬀer-Functions of the Classes of Type C1 , S or U

313

When one carries out only replacements of the variables x or y, one can copy the formation of superpositions in P{a0 ,a1 ,...,ak −1 } over s. Therefore one obtains functions of the form (11.3) as superpositions over s, where p is an 2 arbitrary function of P{a } . Because of 0 ,a1 ...,a k −1

(...((cai (q(x1 ), q(x2 )) r0 (x1 , ..., xm )) r0 (x1 , ..., xm ))... rk −1 (x1 , ..., xm )) = ri (x1 , ..., xm ) k −1 in particular Pk,Ai ⊆ [s] results. If one inserts functions of i=0 Pk,Ai into the function s, one also obtains the functions z and p(q(x), q(y)) as superpositions over s. Thus we have [s] = P olk and (b) holds. Theorem 11.2.3 ([Sch 69], [Lau 78a]) For arbitrary s ∈ Sk is valid: (a) 3 if k = 2, ord P olk s = 2 otherwise. (b) ∃f : [f ] = P olk s . Proof. Let s ∈ Sk be deﬁned as in Section 5.2.2. Since our assertion was prove in Chapter 3 for k = 2, we can assume k ≥ 3 in the following. (a): Because of Chapter 7, all functions g(x1 , x2 ) :=

p−1 l

jar,i (x1 ) · si (G(sp−i (x2 )) (mod k),

r=1 i=0

with G ∈ Pk1 and a function h(x1 , x2 , x3 ) :=

p−1 l

jar,i (x1 ) · si (H(sp−i (x2 ), sp−i (x3 ))) (mod k),

r=1 i=0

where H ∈ Pk \[Pk1 ] and |Im(H)| = k, form a generating system for the functions of the form u(x1 , x2 , ..., xm ) :=

p−1 l

jar,i (x1 ) · si (U (sp−i (x2 ), ..., sp−i (xm ))) (mod k)

r=1 i=0

(11.4) (U ∈ Pkm−1 , m ≥ 2). Let now f n be an arbitrary function of P olk s (n ≥ 3). We show that f ∈ [(P olk 3 )2 ] is valid. For this purpose, we choose in (11.4) the function U as follows: Fr (x2 , ..., xn ) if x1 = ar,0 , r = 1, 2, ..., l, U (x1 , ..., xm ) := 0 otherwise,

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11 On Generating Systems and Orders of the Subclasses of Pk

where F1 , ..., Fl are the components of f from (5.4). Furthermore, let the function v be deﬁned by v(x) :=

p−1 l

jar,i (x) · si (ar,0 ) ∈ (P olk s )1 .

r=1 i=0

Then u(x1 , v(x1 ), x2 , ..., xn ) = f (x1 , ..., xn ) holds. Consequently, we have [(P olk s )2 ∪ {h}] = P olk s . Next we give a function that is a superposition over (P olk s )2 and which fulﬁlls the conditions listed above for the function h. For this purpose, we choose a binary function w with the components ⎧ x = a1,0 ∧ p > 2, ⎪ ⎪ a1,1 if ⎨ x = a1,0 ∧ p = 2, a2,0 if W1 (x) = ... = Wl (x) = W (x) := if x = a1,1 , a ⎪ 1,0 ⎪ ⎩ x otherwise. One now easily checks that W w ∈ Pk \ [Pk1 ] and |Im(W w)| = k are valid. (b): With the aid of the above considerations, we can easily prove that the function t(x, x1 , y1 , x2 , y2 , ...xl , yl ) :=

p−1 l

jar,i (x)·si (T (sp−i (xr ), sp−i (yr ))) (mod k),

r=1 i=0

where [T ] = Pk , is a Sheﬀer-function for P olk s .

11.3 Orders of the Classes of Type L, C, B The following theorem results immediately from Section 5.2.4: Theorem 11.3.1 ([Ros 70a]) For every ∈ Lk is valid ord P olk = 2.

Theorem 11.3.2 ([Lau 78a]) For every ∈ Bk is valid ord P olk = 2. Proof. Subsequently, we describe the functions from the clone P olk ( ∈ Bk ) as in Section 5.2.6. Because of (5.14) and (5.16), an arbitrary function f n ∈ P olk is a superposition over the functions r(fi ), z, fj (i = 0, 1, ..., m−1, j = 0, 1, 2, ..., hm − 1) and m H(x1 , ..., xm ) := r( q(xi ))(0) · hm−i ). i=1

11.3 Orders of the Classes of Type L, C, B

315

Every function of the form r(fi ) or fj belongs either to [(P olk )1 ] or to [(P olk )2 ] because of Theorem 11.1.6. Consequently, we must still prove H ∈ [(P olk )2 ]. The following functions belong to P olk : t Ht (x1 , ..., xt ) := r( i=1 q(xi ))(0) · hm−i ) t Ht (x1 , x2 )) := r( i=1 q(x1 ))(m−i) · hm−i + (q(x2 ))(0) · hm−t−1 ) (1 ≤ t ≤ m − 1). Because of Ht Ht = Ht+1 the function H (= Hm ) is a superposition over (P olk )2 . For the rest of this section, let γ be an h-ary central relation on Ek with 2 ≤ h < k, and let C be the set of the central elements of γ. The short proof of the following theorem comes from R. P¨ oschel. Theorem 11.3.3 The clone P olk γ has a ﬁnite generating system. ! " Proof. Let c ∈ C. The following ( |γ|+1 + |γ| + 1)-ary function belongs to 2 P olk γ: h(x1 , x2 , ..., x|γ|+1 , (yij )i,j=1,2...,|γ|+1;i<j ) ⎧ ∃i, j : ((i < j) ∧ (xi = xj ) ∧ ⎨ yij if (∀s, t : ((s < t ∧ xs = xt ) ⇒ yst = yij ))), = ⎩ c otherwise. We prove by induction on the arity n of an arbitrary function f n ∈ P olk γ that [(P olk γ)|γ| ∪ {h}] = P olk γ holds. Obviously, our assertion is valid for all functions f n with n ≤ |γ|. Assume the assertion holds for all m-ary function of P olk γ, where m ≥ |γ|. Let f m+1 ∈ P olk γ be arbitrary. Denote fij the function, which can be formed from f by identifying of the i-th variable with the j-th variable, i = j, i = 1, 2, .., |γ| + 1. By induction assumption, we have fij ∈ [(P olk γ)|γ| ∪ {h}]. It is easy to check that f (x1 , x2 , ..., xm+1 ) = h(x1 , ..., x|γ|+1 , (fij (x))i,j=1,2,...,|γ|+1;i|j ). Thus f ∈ [(P olk γ)|γ| ∪ {h}]. Theorem 11.3.4 ([Lau 78a]) It holds = 2 if k − |C| ≤ h, ord P olk γ ≤ h otherwise. If γ fulﬁlls the conditions k − |C| > h and (a0 , ..., ah−1 ) ∈ γ =⇒ ((a0 , ..., ah−1 ) ∈ ιhk ∨ (∃i ∈ Eh : ai ∈ C)), then ord P olk γ = # h2 $ + 1.

2

(11.5)

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11 On Generating Systems and Orders of the Subclasses of Pk

Proof. Below is a procedure comprising three steps. This procedure shows that an arbitrary function, which takes q ≥ 2 diﬀerent values, is a superposition over the set (P olk γ)h ∪ (P olk γ ∩ (Pk (q − 1))). The iteration of this procedure takes back the construction of an arbitrary function to the construction of functions that take at most h − 1 diﬀerent values. Since all such functions preserve the relation γ, we have ord P olk γ ≤ h by Theorem 11.1.6, (c). Let f n be an arbitrary function of P olk γ with n > 2 and let c be any ﬁxed element of the set C, i.e., c is a central element of the relation γ. 1.) To dismantle o f , we need the n-ary functions f1 , f2 and the binary function g of P olk γ: f (x1 , ..., xn ) if f (x1 , ..., xn ) ∈ C, f1 (x1 , ..., xn ) := c otherwise, c if f (x1 , ..., xn ) ∈ C, f2 (x1 , ..., xn ) := f (x1 , ..., xn ) otherwise, ⎧ x1 ∈ C ∧ x2 = c, ⎨ x1 if x1 = c ∧ x2 ∈ C, x2 if g(x1 , x2 ) := ⎩ c otherwise. We get: f (x) = g(f1 (x), f2 (x)), where f2 ∈ Pk,C ⊂ [(P olk γ)2 ] by Theorem 11.1.6, (a). 2.) In the second step the function f1 is dismantled, where we use functions ha ∈ (P olk γ)2 , a ∈ Ek \C, deﬁned as follows: ⎧ (x1 = c ∧ x2 = a) ∨ (x1 = a ∧ x2 = c), ⎨ a if x1 = x2 , ha (x1 , x2 ) := x1 if ⎩ c otherwise. Let Ea := {a ∈ Ekn | f1 (a) = a}. Further, for a ∈ Ea we put: ⎧ if x = a, ⎨a if x ∈ Ea \{a}, fa (x1 , ..., xn ) := c ⎩ f1 (x) otherwise. If Ea = {a1 , ..., as }, then there exists the following representation for f1 : f1 (x) = ha (...ha (ha (fa1 (x), fa2 (x)), fa2 (x))..., fas (x)). Similar to f1 , one can dismantle the functions fa . In this case, we choose a b ∈ Ek \(C ∪ {a}), put Eb := {b1 , ..., bt } and for b ∈ Eb we deﬁne ⎧ if x = b, ⎨b if x ∈ Eb \{b}, fa,b (x1 , ..., xn ) := c ⎩ fa (x) otherwise. 2

x denotes the greatest integer z with z ≤ x.

11.3 Orders of the Classes of Type L, C, B

317

Then the function fa has the representation fa (x) = hb (...hb (hb (fa,b1 (x), fa,b2 (x)), fa,b3 (x))...fa,bt (x)). Then, one dismantles the functions in analogous manner fa,b , and so forth. As a result, the function f1 is dismantled in certain functions ha;α ∈ P olk γ deﬁned by α if x = a, ha;α (x1 , ..., xn ) := c otherwise, where α ∈ (Ek \C) ∪ {c} and a ∈ Ekn . 3.) In this step, we construct representations for the n-ary functions u ∈ P olk γ, which are deﬁned by di if x = di , i = 1, 2, ..., q, u(x1 , ..., xn ) := c otherwise, where {d1 , ..., dq } ⊆ Ek \C, di = dj for i = j and di = (di1 , ..., dim ); i, j = 1, 2, ..., q. If q ≤ h − 1 and for the case that q = h and (d1 , ..., dq ) ∈ γ, we have u ∈ [(P olk γ)2 ]. Consequently, the following three cases must still be examined: Case 1: q = h and (d1 , ..., dq ) ∈ γ. In this case, there exists a j (1 ≤ j ≤ n) with (d1j , d2j , ..., dhj ) ∈ γ, since u ∈ P olk γ. Put d1 if x ∈ {d1 , d2 , ..., dh }, u1 (x1 , .., xn ) := c otherwise, and

u2 (x1 , x2 ) :=

di if x1 = dij ∧ x2 = d1 , i = 1, 2, ..., h, c otherwise.

Because of u1 , u2 ∈ P olk kγ and u(x) = u2 (xj , u1 (x)) we have u ∈ [(P olk γ)2 ]. Case 2: There is a set {e1 , e2 , ..., eh } ⊆ {d1 , ..., dq } with (e1 , ..., eh ) ∈ γ\ιhk . The functions ⎧ ei if x1 = ... = xi−1 = xi+1 = ... = xh = ei , ⎪ ⎪ ⎨ xi = c, i = 1, 2, ..., h, v(x1 , ..., xh ) := if x x ⎪ 1 1 = ... = xh ∈ {d1 , .., dq }\{e1 , ..., eh } ⎪ ⎩ c otherwise, and

vi (x1 , ..., xn ) :=

c if u(x1 , ..., xn ) = ei , u(x1 , ..., xn ) otherwise

(i = 1, 2, ..., h) belong to P olk γ. It is easy to check that u(x) = v(v1 (x), v2 (x), ..., vh (x)) holds, where |Im(vi )| < |Im(u)|.

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Case 3: q > h and for an arbitrary set {e1 , e2 , ..., eh } ⊆ {d1 , ..., dq } it holds: (e1 , e2 , ..., eh ) ∈ γ \ ιhk . We need the functions ⎧ d2i−1 if x1 = ... = x[ h +1 = d2i−1 , i = 1, 2, .., # h2 $, ⎪ ⎪ 2 ⎪ ⎪ ⎪ d2i if x1 = ...xi−1 = xi+1 = ... = x h +1 = d2i , ⎪ 2 ⎪ ⎪ ⎪ xi = d2i−1 , i = 1, 2, .., # h2 $, ⎨ if h odd ∧ x1 = ... = x h +1 = dh , w(x1 , ..., x h +1 ) := dh 2 2 ⎪ ⎪ ⎪ d h+1 if x1 = ... = x h = dh+1 ∧ x h +1 = dh , ⎪ 2 2 ⎪ ⎪ ⎪ if x1 = ... = x h +1 ∈ {dh+2 , ..., dq }, x1 ⎪ ⎪ 2 ⎩ c otherwise, d2i−1 if u(x) ∈ {d2i−1 , d2i }, wi (x1 , ..., xn ) := u(x) otherwise (i = 1, 2, ..., # h2 $) and w h +1 (x1 , ..., xn ) := 2

dh if u(x) ∈ {dh , dh+1 }, u(x) otherwise,

of P olk γ for the dismantling of the function u. It holds: u(x) = w(w1 (x), w2 (x), ..., w h +1 (x)), 2

where |Im(wi )| < |Im(u)|. The iteration of the above procedure proves the contention 2 if k − |C| ≤ k, ord P olk γ = ≤ h otherwise. If k − |C| > h and the second case is not possible for any function f ∈ P olk γ,i.e., γ fulﬁlls the condition (11.1), it results from the above procedure then that ord P olk γ ≤ # h2 $ + 1 is valid. We still have to show that w ∈ h [(P olk γ) 2 holds, if γ fulﬁlls the condition (11.1). h +1

Denote A a matrix whose rows are just the tuples of Ek 2 on which the function w takes the values d1 , d2 , ..., dh+1 . It is easy to check that certain rows a1 , ..., ah (ai := (ai1 , ..., air ), i = 1, 2, ..., h, r ≤ # h2 $) are found in every matrix, which can be formed from the matrix A by deleting at least a column, so that is valid for every i ∈ {1, 2, ..., r}: (ai1 , a2i , ..., air ) ∈ ιhk . Consequently, an arbitrary ( h2 + 1)-ary function of P olk γ, which depends on at most # h2 $ places essentially, takes either a value from the set C on at least a tuple of A or two diﬀerent rows of A, on which the function takes the very same value, exist. Further, every # h2 $-ary function of P olk γ can take only then h + 1 diﬀerent values from the set Ek \C on diﬀerent tuples b1 , ...,bh+1 , if there is at least a column that contains h+1 diﬀerent values of the set Ek \C among the columns of the matrix

11.4 The Order of P olk for ∈ Mk and k ≤ 7

⎛

319

⎞

b1 ⎜ b2 ⎟ ⎜ ⎟. ⎝ ... ⎠ bh+1 Consequently, we have w ∈ (P olk γ) 2 and ord P olk γ = # h2 $ + 1, if γ fulﬁlls the condition (11.1). h

11.4 The Order of P olk for ∈ Mk and k ≤ 7 In this section let be a relation of Mk . The following lemma generalizes a theorem from [Jab 58], p. 83. Lemma 11.4.1 Let E ⊆ Ek with |E| ≥ 2. Further, let be a partial order relation on E with the following properties: ⊆ and (E; ) is a lattice. Then {f ∈ P olk | Im(f ) ⊆ E} ⊆ [(P olk )2 ]. Proof. Obviously, has a least element o and has a greatest element e. It is easy to check that there exists a unary function i with Im(i) = E and i(a) = a for all a ∈ E. We show that an arbitrary n-ary function f ∈ Pk,E ∩ P olk is a superposition over (sup (i(x1 ), i(x2 ))), (inf (i(x1 ), i(x2 ))) and the functions a if x ≥ b, mb,a (x) := o otherwise (a ∈ E, b ∈ Ek ). For all a := (a1 , a2 , ..., an ) ∈ Ekn is valid: f (a) if x ≥ a, fa (x1 , ..., xn ) := o otherwise, = inf (ma1 ,f (a) (x), ma2 ,f (a) (x), ..., man ,f (a) (x)) ∈ [(P olk )2 ]. Then one can represent the function f n as follows: f (x) = sup (fa1 (x), fa2 (x), ..., fakn (x)), where {a1 , ..., akn } = Ekn . Consequently, f ∈ [(P olk )2 ]. A consequence of Lemma 11.4.1 is Theorem 11.4.2 If (E; ) a lattice then ord P olk = 2.

Theorem 11.4.3 If k ≤ 7 then ord P olk = 2.

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Proof. Let k = 6 and the partial order relation deﬁned by Figure 11.1. We will prove the theorem only for this relation. In the remaining cases, the relation fulﬁlls the conditions of Theorem 11.4.2 or one can lead the proof similarly to the proof which is subsequently given. 5 s @ @

3 s @s 4 HH H H 1 s Hs 2 @ @ @s 0 Fig. 11.1

We start with the deﬁnitions of some functions of (P olk )2 : 0 if x ∈ E3 , x if x ∈ E3 , i1 (x) := i2 (x) := x otherwise, 5 otherwise, ⎧ x = 1, ⎨ 2 if x = 2, e1 (x) := 1 if ⎩ otherwise,

⎧ x = 3, ⎨ 4 if x = 4, e2 (x) := 3 if ⎩ x otherwise,

supi (x1 , x2 ), i = 0, 1, 2 and infj (x1 , x2 ), j = 3, 4, 5, where the relations i and j are deﬁned in Figure 11.2. 0 :

5 s @ 3 s @s 4 @ 0 @s @ @s 2 1 s

3 :

1 :

5 s @ 3 s @s 4 @ 1 @s

5 s @ 3 s @s 4 @ 2 @s

s0

s0

s2

s1

3 s

5 :

s5

s5

s 3@ @s 2

s 4@ @s 2

3 s @

4 s

1 s @ @s 0

4 :

2 :

1 s @ @s 0

Fig. 11.2

s4

@s 5@ @s 2 1 s @ @s 0

11.4 The Order of P olk for ∈ Mk and k ≤ 7

321

Let f n ∈ P ol6 be arbitrary. We prove by induction on n that f n ∈ [(P ol6 )2 ] holds. For n = 1, 2 our assertion is trivial. Assume the assertion is valid for n − 1 ≥ 2 and we show that it is valid for n. If |Im(f )| ≤ 5, then, by Lemma 11.4.1, f ∈ [(P ol6 )2 ]. Let |Im(f )| = 6 in the following. We show through a dismantling procedure that f n is a superposition over certain binary functions and certain functions g m with |Im(g)| ≤ 5 or m ≤ n − 1. Because of induction assumption and Lemma 11.4.1, our theorem would then be proven. 1.) During the ﬁrst dismantling of function f , we use, the following functions of P ol6 : sup (x1 , i1 (x2 )), = f (x) if f (x) ∈ E3 , f1n (x) ≤ f (x) otherwise, where we assume the validity of ¬∃f1 ∈ P ol6 : = f (x) if f (x) ∈ E3 , ∃a : f1 (a) < f1 (a) ∧ f1 ((x) , ≤ f (x) otherwise,

and f2n (x)

:=

0 if f (x) ∈ E3 , f (x) otherwise.

We get f (x) = sup (f1 (x), i1 (f2 (x))). 2.) Because of f2 ∈ [(P ol6 )2 ] we must still ﬁnd only a dismantling of the kind described above for the function f1 . Let Ni := {a ∈ E6n | f1 (a) = i ∧ ¬∃a ∈ E6n : a < a ∧ f1 (a ) = i} (i = 1, 2), N1 := {a1 , a2 , ..., as },

N2 := {b1 , b2 , ..., bt },

N := N1 ∪ N2 ,

and for N ⊂ N let fN be an n-ary function of P ol6 with ⎧ if (∃a ∈ N ∩ N1 : x ≥ a) ∧ (¬∃b ∈ N ∩ N2 : x ≥ b), ⎪ ⎪=1 ⎪ ⎪ ⎪ ⎪ ⎨=2 if (¬∃a ∈ N ∩ N1 : x ≥ a) ∧ (∃b ∈ N ∩ N2 : x ≥ b), fN (x) : ⎪ ⎪ ≤ f1 (x) if (∃a ∈ N ∩ N1 : x ≥ a) ∧ (∃b ∈ N ∩ N2 : x ≥ b), ⎪ ⎪ ⎪ ⎪ ⎩ =0 otherwise, where for an arbitrary a ∈ E6n we assume that fN (a) = 5 is valid if and only if there is not any function fN ∈ P ol6 with fN (a) ∈ {3, 4}. We get the following description for f1 : f1 (x) = sup1 (fN (1) (x), fN (2) (x), ..., fN (1) (x))

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11 On Generating Systems and Orders of the Subclasses of Pk

and fN (i) (x) = sup2 (fN (i,1) (x), fN (i,2) (x), ..., fN (i,t) (x)) (i = 1, 2, ..., s). 3.) We search dismantling for the functions fi,j := fN (i,j) , 1 ≤ i ≤ s, 1 ≤ j ≤ t. Let Si,j := {c ∈ E6n | ai < c ∧ bj < c ∧ (¬∃c : ai < c < c ∧ bj < c < c)}. Case 1: |Si,j | = 1 or |{fi,j (a) | a ∈ Si,j | = 1. In this case, we have |Im(fi,j )| ≤ 5. Then by Lemma 11.4.1: fi,j ∈ [(P olk )2 ]. Case 2: |Si,j | > 1 and |{fi,j (a) | a ∈ Si,j }| > 1. Let ai = (ai1 , ai2 , ..., ain ) and bj := (bj1 , bj2 , ..., bjn ). W.l.o.g. we can assume that ⎧ if p = 1, 2, ..., u, ⎪ ⎪ = (1, 2) ⎨ if p = u, u + 1, ..., u + v, (aip , bjp ) = (2, 1) ⎪ ⎪ ⎩ ∈ {(1, 2), (2, 1)} if p = u + v + 1, ..., n, where u ≥ 1 or v ≥ 1 because of |Si,j | > 1. Case 2.1: u + v =: m < n. Let c := (sup (ai,m+1 , bj,m+1 ), sup (ai,m+2 , bj,m+2 ), ..., sup (ai,n , bj,n )), ⎧ 1 if x ∈ {1, 3, 4, 5}m \{3, 4, 5}m , ⎪ ⎪ ⎨ 2 if x ∈ {2, 3, 4, 5}m \{3, 4, 5}m , g(x1 , ..., xm ) :=

x ∈ {1, 3, 4, 5}m , fi,j (x1 , ..., xm , c ) if ⎪ ⎪ ⎩ 0 otherwise, and

⎧ 1 ⎪ ⎪ ⎨ 2 h(x1 , ..., xn ) := 4 ⎪ ⎪ ⎩ 0

if x ≥ ai ∧ x ≥ bj , if x≥ ai ∧ x ≥ bj , if x ≥ ai ∧ x ≥ bj , otherwise.

We get fi,j (x) = inf3 (g(x1 , ..., xu , e1 (xu+1 , ..., e1 (xm ), h(x1 , ..., xn )), i.e., fi,j ∈ [(P ol6 )2 ] by induction assumption and Lemma 11.4.1. Case 2.2: u + v = n. Let Ti,j := {a ∈ E6n | fi,j (a) ∈ {3, 4} ∧ (¬∃a ∈ E6n : a > a ∧ fi,j (a ∈ {3, 4}}. Case 2.2.1: Ti,j ⊆ {3, 4}n . For

11.4 The Order of P olk for ∈ Mk and k ≤ 7

323

{d ∈ Ti,j | fi,j (d) = 4} =: {d1 , ..., dl }, with dr := (dr1 , ..., drn ), r = 1, 2, ..., l and x if a = 4, qa (x) := e2 (x) if a = 3, we obtain fi,j (x) = inf4 (inf3 (qd11 (x1 ), ..., qd1u (xu ), qd1,u+1 (e1 (xu+1 ), ..., qd1n (e1 (xn ))), inf3 (qd21 (x1 ), ..., qd2u (xu ), qd2,u+1 (e1 (xu+1 ), ..., qd2n (e1 (xn ))), ..., ................................... inf3 (qdl1 (x1 ), ..., qdlu (xu ), qdl,u+1 (e1 (xu+1 ), ..., qdln (e1 (xn )))), i.e., fi,j ∈ [(P ol6 )2 ]. Case 2.2.2: Ti,j ⊆ {3, 4}n . Case 2.2.2.1: ¬(∃a := (a1 , ..., an ), (b1 , ..., bn ) ∈ Ti,j : fi,j (a) = 3 ∧ fi,j (b) = 4 ∧ (∀p ∈ {1, ..., n} : (ap , bq ) ∈ {(3, 4), (4, 3)}). Here one can show fi,j ∈ [P ol ]n−1 when one uses a procedure that results from the construction steps 1.), 2.) and 3.) (including Case 2.1) listed above, when one the following replacements are carried out: f through fi,j , 0 through 5, 1 through 3, 2 through 4, 3 through 2, 4 through 2, 5 through 0, ≤ through ≥ , i1 through i2 , e1 through e2 , inf through sup ... . Case 2.2.2.2: ∃a := (a1 , ..., an ), (b1 , ..., bn ) ∈ Ti,j : fi,j (a) = 3 ∧ fi,j (b) = 4 ∧ (∀p ∈ {1, ..., n} : (ap , bq ) ∈ {(3, 4), (4, 3)}. W.l.o.g. let u = n and (3, 4) if p = 1, 2, ..., y, (ap , bq ) = (4, 3) if p = y + 1, ..., n. We need the following functions of P ol6 : 5 if x = a, z1 (x1 , ..., xn ) := fi,j (x) otherwise, 5 if x = a ∧ fi,j (x) = 3, z2 (x1 , ..., xn ) := fi,j (x) otherwise, 5 if x = b, z3 (x1 , ..., xn ) := z2 (x) otherwise,

and z4 (x1 , ..., xn ) :=

5 if x = b ∧ z2 (x) = 4, z2 (x) otherwise,

= inf5 (x1 , ..., xy , e2 (xy+1 ), ..., e2 (xn )). We get

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11 On Generating Systems and Orders of the Subclasses of Pk

fi,j (x) = inf3 (z1 (x), z2 x)) and z2 (x) = inf4 (z3 (x), z4 (x)). Then, for the functions z1 and z3 , one can distinguish cases as above and repeat the constructions where appropriate. Therefore, one can reduce the Case 2.2.2.2 to the previous cases. Thus f ∈ [(P ol6 )2 ] holds. We still notice that one cannot transfer the methods from the proof of Theorem 11.4.3 for arbitrary relations ∈ Mk , since there are no binary functions in P ol8 with the properties used of function sup1 for k = 8 and for the relation , which is deﬁned in Figure 11.3. Suppose there exists a function f 2 ∈ P ol8 with f (1, 0) = f (0, 1) = 1 and f (x, x) = x for all x ∈ E8 . Then, f (1, 0) = 1, f (2, 2) = 2 and f (3, 3) = 3 implies f (3, 2) = 3. Further, f (0, 1) = 1, f (2, 2) = 2 and f (4, 4) = 4 implies f (2, 4) = 4. Consequently, we have f (3, 4) ∈ {5, 6, 7}. This is, however, a contradiction to f (5, 5) = 5, f (6, 6) = 6 and f ∈ P ol8 . 7 s @ @

5 s s6 @ HH H 3 s H s4 H HH H 1 s H Hs 2 @ @ @s 0 Fig. 11.3

In the next section, we show that the above-described class does not have a ﬁnite order.

11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated This section basically displays the content of the paper [Tar 86] by G. Tardos. The set E8 is written down in this section in the form: {0, α, α , β, β , γ, γ , 1}, On E8 we consider the partial order relation which is deﬁned by Figure 11.4. If from monotone functions the speech is subsequently, only functions are with that meant from P ol8 .

11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated

325

1 s @ @ γ s @s γ HH HH β s Hs β HH HH α s Hs α @ @ @s 0 Fig. 11.4

The signs <, >, ≥, ≤ refer to this order relation or are explained for tuples from E8n , on which the relation works coordinate wisely. The following deﬁnition is clariﬁed by means of Figure 11.5 which indicates a Hasse-diagram for certain n-tuples. A denotation of the form x; a at a knot of this diagram means that f (x) = a is valid for a corresponding n-ary function f. ;γ

s A A

s xm ; β

......

; γ

;γ

s

s A

; γ

s

A A A AAsx12 ; x10 A ;As @ @ @ @ ...... s @ @ xm ; β @ @ @ @s x13 ; ;s x11 A A A A A A AAs A As s s ; α

;α

;α

; α

Fig. 11.5

Deﬁnition Let ∅ = Q ⊆ E8n and f1 : Q −→ E8 . We call a sequence xm , ..., xm of pairwise distinct elements from the set E8n a zigzag for f1 if the following seven conditions hold: (a) m = 0 or m = 1 and m > m + 2. (b) xm , xm ∈ Q and xi ∈ E8n \Q for all i ∈ {m + 1, ..., m − 1}. (c) f1 (xm ) = β and f1 (xm ) = β . (d) ∀ 2i ∈ {m + 1, ..., m } : x2i > x2i−1 .

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11 On Generating Systems and Orders of the Subclasses of Pk

(e) ∀ 2i ∈ {m, m + 1, ..., m − 1} : x2i > x2i+1 . (f) ∀ 2i + 1 ∈ {m + 1, ..., m − 1} ∃y, y ∈ Q : α ∧ f1 (y ) = α .

y, y > x2i+1 ∧ f1 (y) =

(g) ∀ 2i ∈ {m+1, ..., m −1} ∃z, z ∈ Q : z, z > x2i ∧ f1 (z) = γ ∧ f1 (z ) = γ . Lemma 11.5.1 Let ∅ = Q ⊆ E8n and let f1 : Q −→ E8 be a mapping. Then there is a monotone function f : E8n −→ E8 with f|Q = f1 if and only if the following two conditions hold: (1) f1 is monotone; (2) there is no zigzag in Q for f1 . Proof. “=⇒”: The condition (1) is trivial. To prove (2) we assume by way of contradiction that xm , ..., xm ∈ Q is a zigzag for f1 according to the above deﬁnition and that f : E8n −→ E8 is a monotone extension of f1 . First, let m = 0 and m = 2q + 1, q ∈ N in the zigzag (see Figure 11.6). ; γ

;γ

s A

s

;γ

;α

; α

s A

; γ

s

A A x0 ; β AAsx2q ; s s x2 ; AA @ @ @ @ . @ @ . @ . @ @ @ @ @ @s x2q+1 ; β x1 @ ; s x3@ ; s A A A A A A AAs AAs s s ;α

A A

; α

Fig. 11.6

Since x1 , ..., x2q ∈ E8n and f1 preserves on these tuples, we have f1 (x1 ) = β. Continuation of this consideration supplies f1 (x2q ) = β, which is not possible because of x2q+1 < x2q and f1 (x2q+1 ) = β . For the other possibilities for m and m one shows in analog mode that the assumption of the existence of a zigzag for f1 supplies a contradiction. “⇐=”: Assume that the conditions (1) and (2) are satisﬁed. We construct a monotone function f : E8n −→ E8 extending the function f1 by deﬁning of the following sets Hx := f −1 (x) for each x ∈ E8 . For q ∈ E8n let f (q) := {f1 (y) | y ∈ Q ∧ y ≤ q}, f (q) := {f1 (z) | z ∈ Q ∧ z ≥ q}.

11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated

327

Furthermore, let H0 := {q ∈ E8n | f (q) ⊆ {0}}, H1 := {q ∈ E8n | f (q) ⊆ {1}}\H0 , Hα := {q ∈ E8n | f (q) ⊆ {0, α}}\(H0 ∪ H1 ), Hα := {q ∈ E8n | f (q) ⊆ {0, α }}\(H0 ∪ H1 ∪ Hα ), Hγ := {q ∈ E8n | f (q) ⊆ {1, γ}}\(H0 ∪ H1 ∪ Hα ∪ Hα ), Hγ := {q ∈ E8n | f (q) ⊆ {1, γ }}\(H0 ∪ H1 ∪ Hα ∪ Hα ∪ Hγ ) and H := E8n \(H0 ∪ H1 ∪ Hα ∪ Hα ∪ Hγ ∪ Hγ ). To partition the set H, we consider the elements of H as vertices of a nondirectional graph in which an edge joins the vertices x, y ∈ H if and only if x < y or y < x in E8n is valid. Then, graph G consists of certain connected maximal subgraphs (the so-called components) K1 ..., Kt , with which we deﬁne the sets Hβ and Hβ as follows: Hβ := {x ∈ H | ∃i ∈ {1, ..., t} : x ∈ Ki ∧ β ∈ {f1 (a) | a ∈ Ki }}, Hβ := H\Hβ . The subsets Hx for x ∈ E8 now form a partition of E8n , so they really determine a function f : E8n −→ E8 satisfying ∀x ∈ E8n ∀e ∈ E8 : f (x) = e :⇐⇒ x ∈ He . We prove that this function preserves the relation and that it agrees with the function f1 on E8n . One can check the monotony from f easily, using the following implications, which result from the deﬁnitions of the sets He , e ∈ E8 and which are valid for every a ∈ E8n : ∀e ∈ {α, α } : f (a) = e =⇒ e ∈ f (a) ∀e ∈ {γ, γ } : f (a) = e =⇒ e ∈ f (a). Finally we show f|Q = f1 . Obviously, for arbitrary z ∈ Q with f1 (z) = β we have f (z) = f1 (z). To show that f (z) = f1 (z) holds for elements z ∈ Q with f1 (z) = β as well, we must prove that no component G contains elements from both f1−1 (β) and f1−1 (β ). We use here the complicated condition (2). By the deﬁnition of H for every z ∈ H, we have f (z) ⊆ {0, α}, f (z) ⊆ {0, α }, f (z) ⊆ {1, γ}, f (z) ⊆ {1, γ }. By condition (1) these conditions are equivalent to

328

and

11 On Generating Systems and Orders of the Subclasses of Pk

β ∈ f (z) or β ∈ f (z) or {α, α } ⊆ f (z)

(11.6)

β ∈ f (z) or β ∈ f (z) or {γ, γ } ⊆ f (z).

(11.7)

Suppose a component K of the graph G contains a xm ∈ f1−1 (β) and a xm ∈ f −1 (β ). We choose xm and xm so that are the least distance among the possible vertices. So, there is a path between xm and xm . Take the shortest path. If x, y and z are three consecutive elements of this path then neither x < y < z nor x > y > z can hold since y would then be redundant. Thus, we can choose the indices so that the path be xm , ..., xm satisﬁes the conditions (a), (c), (d), and (e) in the deﬁnition of a zigzag. If m < 2i < m then there is no z ∈ Q with f1 (z) = β or f1 (z) = β and z ≥ x2i , since otherwise either z, x2i+1 , ..., xm or xm , ..., x2i−1 , z would be a shorter path in K from a element of f −1 (β) to an element f −1 (β ). Consequently, (b) holds. Condition (f) also holds because for x2i only {γ, γ } ⊆ f (x2i ) can occur among the possibilities in (11.3). Similarly, condition (g) holds. Hence xm , ..., xm is a zigzag for f1 contrary to (2). Next, for n ≥ 3, we shall deﬁne (n + 5)-ary relations µ0 , µ1 , ..., µn and µ on E8 , which we need to prove the following lemmas. C1

r

C2

Cn−2

r

r

C

rP P PP PP PP B r D2 r D4 r r B Pr D2n−4 @ @ @ @ @ @ @ @ @ @ @ @r D2n−3 D@ D D ..... D r @ 1 r 3 r 5 r 2n−5 aa Q !! AQ Q A ! Q aa Q aa A Q A !!! Q aAa Q ! A Q Q !! a Q A a Q!A! Q A Q A a a! a ! Q Q aa QAr r ! QA! A

A

Fig. 11.7. Poset Q0

Deﬁnitions Let n ≥ 3. Let Q0 (⊆ E8t ) Further, we take

3

the poset deﬁned by Figure 11.7.

Q0 := {A, A , B, B , C1 , C2 , ..., Cn−2 , C }, Q1 := Q0 \{B} and Q2 := Q0 \{B }. Let 3

Later we choose t = 2 · n.

11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated

329

T := (a, a , b, b , c1 , c2 , ..., cn , c ) ∈ E8n+5 . The index of c is always understood modulo n. Let fi : Q0 −→ E8 (i = 1, ..., n) be the following mapping: fi (A) := a,

fi (A ) := a ,

fi (B) := b,

fi (B ) := b ,

fi (Cj ) := ci+j (j = 1, ..., n − 2) and fi (C ) := c . (Clearly, the mapping fi depends on the choice of the tuple T .) We declare T ∈ µ0 iﬀ for each i ∈ {1, ..., n} both (fi )|Q1 and (fi )|Q2 can be extended monotonously to Q0 . Furthermore, for j ∈ {1, ..., n} let T ∈ µj iﬀ T ∈ µ0 and fj can be extended monotonously to a monotone mapping on Q0 . Finally, let n µj . (11.8) µ := j=1

Lemma 11.5.2 If T ∈ µ then T belongs to at least n − 2 of the relations µj with 1 ≤ j ≤ n. Proof. Let T := (a, a , b, b , c1 , ..., cn , c ) and l := |{i ∈ {1, ..., n} | T ∈ µi }|. T ∈ µ implies T ∈ µ0 . If T ∈ µi for all i ∈ {1, ..., n} then l = n. Thus, it is enough to consider the case T ∈ µ0 but T ∈ µi for some i ∈ {1, ..., n}. By the cyclical nature of the relations µi , we may assume T ∈ µ1 . It means that f1 cannot be extended monotonously to Q0 . Because of T ∈ µ0 both (f1 )|Q1 and (f1 )|Q2 have monotone extensions to Q0 . Therefore, f1 is monotone. So by Lemma 11.5.1, there is a zigzag xm , ..., xm in Q0 for f1 . Let H be the set of all elements of Q0 comparable with some xi (m < i < m ). Since xm , ...xm is a zigzag for (f1 )|(Q0 ∩H) as well, it follows that (f1 )|(Q0 ∩H) cannot be extended monotonously to Q0 . This implies that both B and B are in H. But xm+1 , ..., xm −1 is a series of distinct elements in Q0 \ Q0 such that every two consecutive elements are comparable. These two properties imply that {xi | m < i < m } = {Dj | 1 ≤ j ≤ 2n − 3}, where the correspondence is either in direct or reverse order depending on whether f1 (B) = β or f1 (B) = β . So, by condition (g) in the deﬁnition of a zigzag {a, a } = {fi (A), fi (A )} = {α, α }, and by condition (f) we have {ci , c } = {fi (Ci−1 ), fi (C )} = {γ, γ } for i = 2, ..., n − 1. W.l.o.g. we may assume that c = γ and ci = γ for all i = 2, ..., n − 1. Because of condition (c) of the deﬁnition of a zigzag we have

330

11 On Generating Systems and Orders of the Subclasses of Pk

{b, b } = {fi (B), fi (B )} = {β, β }. Using the fact that T ∈ µ0 , one can show that {c1 , cn } ⊆ {γ, γ , 1}. So, we have determined all tuples T ∈ µ0 \µ1 . By symmetry, we get similar characterizations for µ0 \µi for i = 2, ..., n. Applying them, we obtain: c1 = c2 = γ =⇒ l = 0 =⇒ T ∈ µ, ((c1 = γ ∧ cn = γ) ∨ (c1 = γ ∧ cn = γ)) =⇒ l = n − 2, (c1 = γ ∧ cn = γ) =⇒ l = n − 1. Lemma 11.5.3 Let n ≥ 3. For every l < E8 preserve the relation µ.

n 2

the l-ary monotone functions on

Proof. Let g be an l-ary function of P ol8 and let Ti ∈ µ for i = 1, 2, ..., l. Then by Lemma 11.5.2 we have ∀i ∈ {1, 2, ..., l} : |{j ∈ {1, ..., n} | Ti ∈ µj }| ≤ 2. Thus there is an index j ∈ {1, ..., n} with Ti ∈ µj for all i = 1, 2, ..., l. Since one can derive the relations µi (1 ≤ i ≤ n) from the t-th graphic Gt (P ol8 ) by means of the operation pr (projection), the relations mi belong to Inv8 (P ol8 ) (see Chapter 2). Therefore, function g preserves the relation µj , from which we receive g(T1 ..., Tl ) ∈ µj ⊆ µ. Lemma 11.5.4 For n ≥ 3 there exist 2n-ary monotone functions on E8 , which do not preserve the relation µ. Proof. We consider a matrix A of the type (n + 5, 2n), whose construction is given in the following table, where a, a , b, b , c1 , ..., cn , c are the rows of A and T1 , T2 , ..., T2n are the columns of A. Furthermore, let T := (α, α , β, β , γ, γ, ..., γ, γ )T be a column matrix of length n + 5, which is also deﬁned in the following table. a a b b c1 c2 c3 c4 . . . cn c

T1 α α β β 1 γ γ γ . . . γ γ

T2 α α β β γ 1 γ γ . . . γ γ

T3 α α β β γ γ 1 γ . . . γ γ

T4 α α β β γ γ γ 1 . . . γ γ

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

Tn Tn+1 Tn+2 Tn+3 Tn+4 α α α α α α α α α α β 1 1 1 1 β 1 1 1 1 γ 1 β γ γ γ β 1 β γ γ γ β 1 β γ γ γ β 1 . . . . . . . . . . . . . . . 1 β γ γ γ γ γ γ γ γ

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

T2n α α 1 1 β γ γ γ . . . 1 γ

T α α β β γ γ γ γ . . . γ γ

11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated

Let

331

Q := {a, a , b, b , c1 , ..., cn , c } ⊆ E82n

and f1 : Q −→ E8 be a function deﬁned by f1 (A) = f1 (T1 , T2 , ..., T2n ) := T. Clearly, f1 is monotone. Suppose that there is a zigzag xm , ..., xm ∈ E82n for f1 . Thus we have xm = b, xm = b , m < 2i + 1 < m =⇒ a < x2i+1 ∧ a < x2n+1 m < 2i < m =⇒ c > x2i ∧ (∃qi ∈ {1, 2, ..., n} : cqi > x2i ). Let p := % 12 (m − 1)&.4 Each j ∈ {1, ..., n} appears among the numbers q1 , ..., qp , since we could not otherwise choose the j-th coordinates of xm , ..., xm to satisfy the above conditions. Let l (1 ≤ l ≤ n) be the number that appears last from the beginning in the sequence q1 , ..., qp . Then there is an interval in this sequence with ﬁrst last element l − 1 and l + 1 or the reverse and containing no other element equal to l − 1, l or l + 1 considered again modulo n. However, in this case, we cannot choose the (n + l)-th coordinates of xm+1 , ..., xm −1 to satisfy the above conditions. The contradiction proves that there is no zigzag in E82n for f1 . So, by Lemma 11.5.1, we can extend f1 to a 2n-ary monotone function h with h(T1 , ..., Tn ) = T. Using that {T1 , ..., Tn , T } ⊆ µ0 and the characterization of µ0 \µi (1 ≤ i ≤ n) one can easily check that T1 , ..., Tn belong to µ but T ∈ µ. Thus the 2n-ary monotone function h does not preserve µ. Theorem 11.5.5 ([Tar 86]) does not have a ﬁnite order.

For k ≥ 8 there is a ∈ Mk , so that P olk

Proof. Let ∈ Mk deﬁned by the Hasse-diagram of Figure 11.4. Choosing a ﬁnite set A ⊂ P ol8 we have a number l, such that every element of A is at most l-ary. Taking n ≥ 2 · l and deﬁning µ corresponding to this n, we get that A ⊆ P ol8 {µ, } ⊆ P ol8 by Lemma 11.5.3, but (P ol8 )n ⊆ P ol8 {µ, } by Lemma 11.5.4. So the ﬁnite set A does not generate the whole clone. Presumably ord P olk = ∞ is valid for every relation , in whose diagram one can embed the diagram given in Figure 11.4. 4

x denotes the ﬂoor of x ∈ R, i.e., the largest integer which is ≤ x.

332

11 On Generating Systems and Orders of the Subclasses of Pk

11.6 Classiﬁcations and Basis Enumerations in Pk Let A be a class of Pk that is ﬁnitely generating. Then, A has only ﬁnite many A-maximal classes M1 , M2 , ..., Mt and it is valid for all B ⊆ A: [B] = A ⇐⇒ (∀i ∈ {1, ..., t} : B ⊆ Mi ). A generating system B := {f1 , ..., fr } of A with the following property is especially interesting: [B \ {fi }] = A for every i ∈ {1, ..., r}. Such a set B is called basis of A and the number r is the rank of B. It is brieﬂy explained in this section, as one can classify the bases of A and one can also determine, then, the ranks of the bases. With all maximal classes M1 , ..., Mt of the class A, can be classiﬁed the functions, by their membership in the A-maximal sets as follows: For f ∈ A we put 0 if f ∈ Mi , χi (f ) := 1 if f ∈ Mi (i = 1, 2, ..., t) and χ(f ) := (χ1 (f ), χ2 (f ), ..., χt (f )). We call χ(f ) the characteristic vector of f . We put f ≡ g iﬀ f, g ∈ A and χ(f ) = χ(g). Obviously, ≡ is an equivalence relation on A, and so it partitions A into pairwise disjoint nonempty sets (called equivalence classes or blocks). Let f ∈ A with χ(f ) = (a1 , ..., at ). Then, it is easy to see that the block [f ]≡ of ≡ has the form Ta1 ∩ Ta2 ∩ .... ∩ Tat , where, for i ∈ {1, ..., t}, Tai :=

Mi if ai = 0, A \ Mi for ai = 1.

If f ∈ B ⊆ A and χ(f ) = χ(g), then we have [B] = A ⇐⇒ [{g} ∪ (B \ {f })] = A. In other words, it suﬃces to study the completeness in A up to the equivalence relation ≡. Further, it is easy to see that ∀B ⊆ A : ([B] = A ⇐⇒ ( χ(f )) = (1, 1, 1, ..., 1)), f ∈B

where is the usual componentwise logical ∨ of the tuples (∈ E2t ). A set B := {f1 , ..., fr } ⊆ A is a basis of A iﬀ [B] = A and

11.6 Classiﬁcations and Basis Enumerations in Pk

∀j ∈ {1, ..., r} : (

χ(f )) = (1, 1, 1, ..., 1))

333

(11.9)

f ∈B\{fj }

Once we know all the characteristic vectors, we can ﬁnd all complete sets in A and all bases by a direct combinatorial check (which may be done by a simple computer program, provided t is not large (see [Sto 87])). If to α := (α1 , ..., αt ) ∈ E2t we associate the set Iα := {i | αi = 1} and if I1 , ..., Im are the subsets of {1, 2, ..., t} corresponding to the characteristic vectors, the completeness problem is reduced to listing irredundant coverings (i.e., no proper subset covers {1, 2, ..., t}). The study of classes also provides information on the classes of Pk , which are the intersections of families of A-maximal sets, which is of independent interest.5 The characteristic vectors can also be applied to seek the set of classes of functions that make an incomplete set complete. One ﬁnds many results above characteristic vectors and basis enumerations in the books [Mas 88] and [Sto 87]; e.g., classiﬁcation of A ∈ {P2 , P3 , Pk,2 , Lp }, p ∈ P, and classiﬁcations of the maximal classes of P3 and of Pk,2 . Subsequently, the concepts and ideas introduced above are only explained as an example. Let A = P2 . Then, by Theorem 3.2.4.2, we can say M1 := T0 , M2 := T1 , M3 := S, M4 := L, M5 := M. Then there are exactly 15 characteristic vectors (a1 , ..., a5 ) for P2 ([Jab 52], [Ibu-N-N 63], [Kri 65]): number of (a1 , ..., a5 ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5

a1 a2 a3 a4 a5 example for f with χ(f ) = (a1 , ..., a5 ) 1 1 1 1 1 x∧y 1 1 0 1 1 (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z) 0 1 1 1 1 x∧y 1 0 1 1 1 x∨y 1 1 0 0 1 x 1 0 1 0 1 x+y+1 0 1 1 0 1 x+y 0 0 1 1 1 x ∧ (y ∨ z) 1 0 1 0 0 c1 0 1 1 0 0 c0 0 0 1 1 0 x∨y 0 0 0 1 1 (x ∧ (y ∨ z)) ∨ (x ∧ y ∧ z) 0 0 0 1 0 (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z) 0 0 0 0 1 x+y+z 0 0 0 0 0 e11

E.g., for A = P3 with one exception the least nontrivial intersections are all minimal clones.

334

11 On Generating Systems and Orders of the Subclasses of Pk

With the aid of the above, one can classify the bases of P2 as follows: Theorem 11.6.1 ( [Ibu-N-N 63], [Kri 65]; without proof ) Let B := {f1 , ..., fr } be an arbitrary basis of P2 . Then r ∈ {1, 2, 3, 4} and for χ(B) := {χ(f1 ), ..., χ(fr )} is valid: |B| the sets of the numbers of the lements of χ(B) 1 {1} 2 {2, x}, where x ∈ {3, 4, 6, 7, 8, 9, 10, 11} {3, x}, where x ∈ {4, 5, 6, 9} {4, x}, where x ∈ {5, 7, 10} {5, x}, where x ∈ {8, 11} 3 {5, x, y}, where x ∈ {6, 7, 9, 10} and y ∈ {12, 13} {6, 7, x}, where x ∈ {8, 11, 12, 13} {6, 6, 10} {6, 10, x}, where x ∈ {11, 12, 13} {7, 8, 9} {7, 9, x}, where x ∈ {11, 12, 13} {9, 10, x}, where x ∈ {8, 12} 4 {9, 10, 11, 14}, {9, 10, 13, 14} In other words: There are exactly 42 aggregates6 for P2 . 1 aggregate has the rank 1, 17 aggregates have the rank 2, 22 aggregates have the rank 3 and 2 aggregates have the rank 4.

6

An aggregate is the set of all bases having the the same set of characteristic vectors.

12 Subclasses of Pk,2

In this chapter, we will to deal with subclasses of the class Pk,2 := {f ∈ Pk | W (f ) ⊆ {0, 1}}. When one restricts the domain of a function f n ∈ Pk,2 to the set E2n , a homomorphic mapping pr (”projection”) from Pk,2 onto P2 can be deﬁned. Since the image prA of a subclass A of Pk,2 is a subclass of P2 and the subclasses of P2 are known, one can hope +to ﬁnd certain properties of the inverse images (⊆ Pk,2 ) through the known properties of the images (⊆ P2 ). This hope conﬁrmed itself in a certain sense (see e.g. Theorem 12.2.5 and Theorem 9.7.6). On the other hand, Pk,2 , k ≥ 3 also reﬂects the negative properties of Pk because the examples of classes from Section 8.1.1 with inﬁnite and without bases are subclasses of Pk,2 . Further, the functions of Pk,2 are important since they can be interpreted as predicates. Some further applications are subsequently mentioned: Functions of P3,2 permit the description of a decision (values 0, 1) with abstention from voting (value 2). Special functions of P3,2 are of interest in the theory of noncorrect algorithms (see e.g. [Sch 78]). In [Eps-F-R 74] G. Epstein, G. Frieder and D. C. Rine mention that functions of Pk,2 are useful for describing logico-arithmetical branchings in programs where the arithmetical constants (mostly k > 2) are arguments and the two logical constants form the range. They also give a function of P3,2 which is used in control of real-time processes and in aeronautics. In 1973/1974 G. Burosch dealt with the set Pk,2 . Some years later he, J. Dassow, W. Harnau and the authoress continued the study of the subclasses of Pk,2 . The investigations of Pk,2 concerned the following problems:

336

12 Subclasses of Pk,2

(1) For a given subclass B of P2 determine the set of all subclasses of Pk,2 , which can be projected to B, i.e., the restrictions of the functions to arguments of E2 gives a function of B. (2) As completely as possible, construct the lattice of the subclasses of P3,2 . (3) For a given subclass of Pk,2 , decide whether it is ﬁnitely generated and construct a system of generators (if possible). A summary of the achieved results on the closed subsets of Pk,2 was published in [Bur-D-H-L 85]. Unfortunately, a proof could not be given in [Bur-D-H-L 85] for every theorem, since the extent of this proof was too great. After B. Cs´ak´ any showed the author how the Theorem of Baker and Pixley can be generalized (see Theorems 8.3.1 and 12.3.1), she found new and shorter proofs. The subsequently proofs to the theorems and lemmas without reference come mostly from the papers [Lau 88;a;b] (or [Lau 84c]) but also from [Bur-D-H-L 85] and [Lau 77;a;b]. In addition, the results of N. Gr¨ unwald from [Gr¨ u 83;a;b]which continued the investigations of the team Burosch-Dassow-Harnau-Lau. The chapter is organized as follows: Section 12.1 contains the basic concepts and notations. In Section 12.2, one can ﬁnd results on inverse images of subclasses of P2 (with respect to the above projection). The remaining sections deal with the determination of the cardinality and with the determination of the elements of the set Nk (B) := {A ∈ L↓k (Pk,2 ) | pr A = B} for subclasses B of P2 . In Section 12.3, one can ﬁnd some structure statements about L↓k (Pk,2 ). In addition, this section clariﬁes whether the set Nk (B) is ﬁnite or inﬁnite or has the cardinality of continuum. In Section 12.4, one can ﬁnd all subclasses A of Pk,l whose projection is the class Pl or P oll {α} (α ∈ El , 2 ≤ l < k). After that, in Section 12.5, the maximal and the submaximal classes of Pk,2 are determined. Then, the investigations are continued from Section 12.3 for k = 3, i.e., for many classes B ⊆ P2 with |N3 (B)| ≤ ℵ0 the elements of the set N3 (B) are determined.

12.1 Notations In this chapter, let 2 ≤ l < k and Pk,l as deﬁned in Section 1.1. In general, we deﬁne functions of Pk,l by formulae over an alphabet X := {x, x1 , x2 , ...}. In contrast, we deﬁne the Boolean functions over the alphabet Y := {y, y1 , y2 , ...}, and we use the notations for functions and closed sets of P2 from Chapter 3. By k1 and ja , where a ∈ Ek , we denote the unary functions of Pk,l given by

12.2 Some Properties of the Inverse Images

k1 (x) :=

1 if x ∈ E2 , 0 otherwise,

ja (x) :=

337

1 if x = a, 0 otherwise,

respectively. To characterize the subclasses of Pk,l we need the following homomorphism of Pk,l onto Pl , which we denote by prl or only with pr and which we call “projection” 1 ): For f n ∈ Pk,l and g m ∈ Pl let pr f n = g m if and only if n = m and f (a) = g(a) for all a ∈ Eln . If B ⊆ Pl , we call the subset pr−1 B := {f ∈ Pk,l | pr f ∈ B} of Pk,l inverse image of B. We say a subclass A of Pk,l is B-projectable iﬀ prl A = B. Denote Nk (B) the set of all B-projectable subclasses of Pk,l . By P olPk,l := Pk,2 ∩ P olk one can describe subclasses of Pk,l . If the index Pk,l can be seen from the context, we write only P ol instead of P olPk,l . . If B = P oll (⊆ Pl ), ⊆ Elh and a1 , ..., ar are some tuples of Ekh \Elh , then we also denote the closed set P olPk,l ( ∪ {a1 , ..., ar )} with B a1 ,...,ar . In particular, these notations will be used in Section 12.6 for l = 2 and B ∈ {T0 , T1 , M }. Furthermore, let 0 1 ... l − 1 a {a, b} ⊆ Ek . Za,b := P olPk,l 0 1 ... l − 1 b

12.2 Some Properties of the Inverse Images With the mapping pr, we can prove that some properties of the subclasses of Pl are transmitted to their inverse images in Pk,l . 1

n To avoid confusion with the notation for functions en i , the functions ei are called selectors at some places of this chapter.

338

12 Subclasses of Pk,2

Lemma 12.2.1 (a) Let A ⊆ Pk,l , [pr A] = Pl and b ∈ Ek . Then, the set A ∪ {ja | a ∈ Ek \{b}} is a generating system for Pk,l . (b) ord Pk,l = 2. Proof. (a): Since pr A is a generating system for Pl , there are binary functions h and g of [A] with (pr h)(y1 , y2 ) = y1 + y2 (mod l) and (pr g)(y1 , y2 ) = y1 · y2 (mod l). Furthermore, the constant functions c0 , c1 , ..., cl−1 and the function k−1 jb (x) = 1 + (l − 1) · ja (x) (mod l) a=0, a=b

belong to [A]. Further, any function f n of Pk,l has a representation of the form f (x1 , ..., xn ) = f (a1 , ..., an ) · ja1 (x1 ) · ja2 (x2 ) · ... · jan (xn ) (mod l), (a1 ,...,an )∈Ekn

(12.1) i.e., f belongs to [{g, h, c0 , ..., cl−1 , j0 , ..., jk−1 }] ⊆ [A]. (b) follows from (a) and Theorem 1.4.2, (b). Lemma 12.2.2 Let B be a subclass of Pl , which contains the set Jl of all projections (selectors) of Pl . Then, for every set A ⊆ Pk,l with [pr A] = pr B, the set A ∪ pr−1 Jl is a generating system for pr−1 B. Proof. Let f n ∈ pr−1 B. Since [pr A] = B, there exists a function f1n ∈ [A] with pr f1 = pr f . In pr−1 Jl one can ﬁnd the (n + 1)-ary function xn+1 if x ∈ Eln+1 , h(x) := f (x1 , ..., xn ) otherwise. Consequently, f (x1 , ..., xn ) = h(x1 , ..., xn , f1 (x1 , ..., xn )) is a superposition over A ∪ pr−1 Jl . By the above lemma, we obtain the following theorem. Theorem 12.2.3 If Jl ⊆ B ⊂ B ⊆ Pl and B is a maximal class of the class B then is also pr−1 B maximal in pr−1 B. It is easy to see that the above theorem does not hold if B does not contain Jl (see also Theorem 12.2.6). Lemma 12.2.4 The order of pr−1 Jl is 3.

12.2 Some Properties of the Inverse Images

Proof. Let

i(x) :=

339

x if x ∈ El , 0 otherwise.

Every n-ary function f of pr−1 Jl with pr f = eni can be represented in the following manner: f (x1 , ..., xn ) = i(xi ) · k1 (x1 ) · ... · k1 (xn )+ a∈Ekn \Eln f (a1 , ..., an ) · ja1 (x1 ) · ... · jan (xn ) (mod l). (12.2) We prove that (12.2) is a superposition over i(x1 ) · k1 (x2 ) (mod l), i(x1 ) + a · jq (x2 ) (mod l), i(x1 ) + i(x2 ) · jq (x3 ) (mod l),

(12.3)

i(x3 ) + i(x1 ) · jp (x2 ) · jq (x3 ) (mod l), (p, q ∈ Ek ; q > 1, a ∈ El ). First, we construct the function i(x) + a · ja1 (x1 ) · ja2 (x2 ) · ... · jan (xn ) (mod l)

(12.4)

for (a1 , ..., an ) ∈ Ekn \Eln , a ∈ El . W.l.o.g. let a1 ∈ Ek \El . The function i(x)+(i(x1 )+i(x3 ))ja2 (x2 )ja1 (x1 )ja1 (x1 ) = i(x) + i(x3 )ja1 (x1 )ja2 (x2 ) (mod l) is obtained from i(x) + i(x2 )ja1 (x1 ) if we substitute the variable x2 by i(x1 )+i(x3 )ja2 (x2 )ja1 (x1 ). By substitution of x3 by i(x1 ) + i(x4 )ja3 (x3 )ja1 (x1 ) we generate the function i(x) + i(x4 )ja1 (x1 )ja2 (x2 )ja3 (x3 ) (mod l). By iterated application of these constructions we obtain the function i(x) + i(xn+1 )ja1 (x1 )ja2 (x2 )...jan (xn ) (mod l).

(12.5)

The function (12.4) is generated from (12.5) by substituting xn+1 by i(x1 ) + a · ja1 (x1 ). Obviously, i(x) + f (a1 , ..., an )ja1 (x1 )ja2 (x2 )...jan (xn ) (mod l) (12.6) a∈Ekn \Eln

is a superposition of functions of type (12.4). If we substitute x in (12.6) by i(xi ) · k1 (x1 ) · ... · k1 (xn ) ∈ [i(x) · k1 (x1 )] we obtain the function (12.2); i.e., (12.3) is a generating system for pr−1 Jl and ord pr−1 Jl ≤ 3. Since the functions of (pr−1 Jl )2 preserve the relation

340

12 Subclasses of Pk,2

λl ∪ {(2, 2, 2, 2)} (see Lemma 4.1), the function i(x1 ) + j1 (x1 )j1 (x2 )j2 (x3 ) ∈ pr−1 Jl does not preserve this relation, however, we have ord pr−1 Jl = 3. Theorem 12.2.5 Let B ⊆ Pl be a clone. Then ord B ≤ ord pr−1 B ≤ max(3, ord B). Every inverse image (⊆ Pk,2 ) of a clone B ⊆ P2 is ﬁnitely generated. Proof. The statements of the theorem are consequences of Theorems 12.3.2 and 12.3.4 and of Chapter 3. The following theorem gives information on the order of the remaining inverse images (⊆ Pk,2 ) of subclasses (⊆ P2 ). Theorem 12.2.6 The inverse images pr−1 C0 , pr−1 C1 and pr−1 C have no ﬁnite basis. Proof. Let A be an inverse image of Pk,2 whose projection is generated by {ca , cb }, {a, b} ⊆ E2 . Then A contains the functions ⎧ ∃i, α : xi = 2 ∧ ⎨ α if x1 = ... = xi−1 = xi+1 = ... = xn = α ∈ E2 , f n (x1 , ..., xn ) := ⎩ a otherwise for all n ∈ N. A has no ﬁnite basis if we can prove that, for each n, the function f n is not a superposition of the (n − 1)-ary functions of A. It suﬃces to prove that f n has no representation by a formula h(x1 , ..., xn ) := g0 (g1 (x1 , ..., xn ), ..., gn−1 (x1 , ..., xn )),

(12.7)

where gi (0 ≤ i ≤ n) is a function of A or gi (1 ≤ i ≤ n) is deﬁned by gi (x1 , ..., xn ) = xj

(12.8)

for some j ∈ {1, 2, ..., n}. W.l.o.g. we can assume that g1 is not a function of form (12.8). Then, for α ∈ E2 βα := (g1 (2, α, ..., α), ..., gn−1 (2, α, ..., α)) is a tuple of E2n and thus g0 (β0 ) = g0 (β1 ). However by deﬁnition f n (2, α, ..., α) = α for every α ∈ E2 . Therefore, formula (12.7) does not deﬁne the function f n.

12.2 Some Properties of the Inverse Images

341

Theorem 12.2.7 (a) Let A ⊆ Pk,l be an inverse image whose projection prl A is a clone. Then A has a basis with exactly r elements if and only if pr A has such a basis. (b) ∃f ∈ Pk,l : [f ] = Pk,l . Proof. (a): If [{f1 , ..., fr }] = A then [{pr f1 , pr f2 , ..., pr fr }] = pr A holds. Let pr A = [{g1 , ..., gr }], where g1 denotes a function that stands in the construction formula of e11 “outside”; i.e., we have e11 (x) = g1 (t1 (x), ..., tn (x)) for certain t1 , ..., tn of [{g1 , ..., gr }]. Further, denote f1 , ..., fr certain functions of A with pr fi = gi , i = 1, ..., r and let IV f (x1 , ..., xn , x, ..., xp,q , ..., xp,q , ..., xp,q , ..., x p,q , ..., xp,q , ...) i(xp,q ) · jp (xp,q ) · jq (xp,q ) := f1 (x1 , ..., xn ) · k1 (x) + p, q

q≥l 0 ≤ p < q ≤ k−1

+

IV jp (x p,q ) · jq (xp,q ) (mod l). p, q q≥l 0 ≤ p < q ≤ k−1

We prove that {f, f2 , ..., fr } is a generating system for A. By Lemmas 12.2.2 and 12.2.4, it is suﬃcient to show that the function system (12.3) belongs to [{f, f2 , ..., fr }]. Due to the choice of the function f1 and by e11 ∈ prl A, the function IV (i(xp,q )jp (xp,q )jq (xp,q ) + jp (x i (x1 )k1 (x) + p,q )jq (xp,q )) p, q q≥l 0≤p
(12.9) with pr i = pr i is a superposition over the functions f, f2 , ..., fn . If one identiﬁes all variables = x1 and = x in (12.9) with x, one receives the function i (x1 )k1 (x). Therefore, the functions i(x) = i (x)k1 (x) and i(x1 )k1 (x2 ) = i (i(x1 ))k1 (x2 ) are also superpositions over f, f2 , ..., fn . By identifying certain variables and by substituting x1 by i(x) in (12.9), we obtain the functions i(x) + i(x1 )jp (x2 )jq (x3 ) and i(x) + jp (x1 )jq (x2 ), where q ≥ l and 0 ≤ p < q ≤ k − 1. This implies that the functions

342

12 Subclasses of Pk,2

i(x3 ) + i(x1 )jp (x2 )jq (x3 ), i(x1 ) + j0 (i(x2 ))jq (x2 ) = i(x1 ) + jq (x2 ) =: g(x1 , x2 ), g(...g(g(g (x1 , x2 ), x2 ), x2 )..., x2 ) = i(x1 ) + a · jq (x2 ) and a times i(x1 ) + i(x2 )j0 (i(x3 ))jq (x3 ) = i(x1 ) + i(x2 )jq (x3 ) for q ≥ l, 0 ≤ p < q ≤ k − 1 and a ∈ El are superpositions over {f, f2 , ..., fr }. Hence all functions of the system (12.3) belong to [{f, f2 , ..., fr }], and our theorem was proven. (b) follows from (a) with the aid of Theorem 7.1.6.

12.3 On the Number of the B-projectable Subclasses of Pk,2 , B ⊆ P2 In this section, dm denotes an (m + 1)-ary function with the property dm (x2 , x1 , ..., x1 ) = dm (x1 , x2 , x1 , ..., x1 ) = ... = dm (x1 , ..., x1 , x2 ) = x1 , where m ≥ 2. We call such a function a near unanimity function. Examples of P2 are the functions hm with m ≥ 2 (see Chapter 3). To generalize Theorem 8.3.1, we get: Theorem 12.3.1 Let A be a subclass of Pk,l whose projection prl A contains a “near unanimity function” dm for a certain m ≥ 2. Then A = P olPk,l Inv m A, i.e., for a ﬁxed m and for a ﬁxed k there exist only ﬁnite many subclasses A ⊆ Pk,l with dm ∈ prl A. Proof.2

Obviously, A ⊆ P ol(Inv m A).

Let f n ∈ P ol(Inv m A) be arbitrary. Further, denote fT for T ⊆ Ekn a function of Pk,l which is identical to f on tuples of T . We prove through induction on |T | =: t ≥ m that there is a certain function fT in A for every T ⊆ Ekn . f ∈ A for t = k m , with which A = P ol(Inv m A) would be proven. I) t = m: Let T = {(ai1 , ai2 , ..., ain ) | i = 1, 2, ..., m}, ⎛ ⎞ a11 a12 ... a1n ⎜ a21 a22 ... a2n ⎟ ⎟ := ⎜ ⎝ ................. ⎠ am1 am2 ... amn and let be the smallest relation of Inv m A with ⊆ , i.e., we have 2

The proof goes back to a manuscript by B. Cs´ ak´ any.

12.3 On the Number of the B-projectable Subclasses of Pk,2 , B ⊆ P2

⎛

⎛

⎞

⎞

⎛

343

⎞

a1 a11 a12 ... a1n a1 ⎜ a21 a22 ... a2n ⎟ ⎜ a2 ⎟ ⎜ a2 ⎟ n ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎝ ... ⎠ ∈ \ ⇐⇒ ∃q ∈ A : q ⎝ . . . . . . . . . . . . . . . . . ⎠ = ⎝ ... ⎠ . am am1 am2 ... amn am (12.10) Since f n preserves the m-ary invariants of A, f () ∈ holds. If f () belongs to , a certain function of A ∩ pr−1 [e11 ] agrees with the function f on tuples of T . On the other hand, if f () ∈ \, then there is also a function fT in A.Consequently, our assertion is proven for t = m. II) t −→ t + 1: Assume for every T ⊆ Ekn with |T | = t, t ≥ m, one can ﬁnd a function fT ∈ A with fT (a) = f (a) for all a ∈ T . Now, let T = {a1 , a2 , ..., at+1 } and |T | = t + 1. Because of our assumption, there are functions fi ∈ An with fi (a) = f (a) for all a ∈ T \{ai }, i = 1, 2, ..., m + 1. Since there is also a function dm ∈ A with prl dm = dm , the function dm (f1 (x), f2 (x), ..., fm+1 (x)) belongs to A. It is obvious that this function agrees with f on all tuples of T . As a direct consequence of the above proven theorem and Chapter 3, we get: Theorem 12.3.2 Let A be a clone of P2 with S∩M ⊆ A or Ta,m ∩Ta ∩M ⊆ A for certain m ∈ N \ {1} and a ∈ E2 . Then there are only ﬁnite many Aprojectable subclasses of Pk,2 . One can ﬁnd the exact description of the A-projectable classes from Theorem 12.3.2 in Sections 12.6 and 12.7. To prepare for the proofs in these sections, we derive some properties of the subclasses of Pk,2 , whose projections contain a certain function hm . m Lemma 12.3.3 Let A = i=1 P olPk,l i and i ⊆ Ekhi , i = 1, 2, ..., m. Then the following statements are equivalent: (1) prl A contains the selectors of Pl . (2) There exists a mapping t from Ek onto El with the ﬁxed points 0, 1, ..., l−1 and the property t(i ) := {(t(a1 ), t(a2 ), ..., t(aht )) | (a1 , ..., ahi ) ∈ i } . = i ∩ Elhi for every i ∈ {1, 2, ..., m}. (3) prl A =

m i=1

P oll (i ∩ Elhi ).

Proof. Obviously, if prl A contains the selectors, there is an e11 -projectable function t ∈ A (e11 ∈ Pl ) that has the properties given in (2). Thus (1) implies (2). The implication “(3) =⇒ (1)” is trivial. It remains to be proven “(2) =⇒ (3)”: Obviously, the relations i ∩ Elhi (i = 1, 2, ..., m) can be derived from the relations 1 , ..., m and the invariants of Pk,l . This implies m m A ⊆ i=1 P olPk,l (i ∩ Elhi ). Thus, prl A ⊆ i=1 P oll (i ∩ Elhi ).

344

12 Subclasses of Pk,2

m Now let g n ∈ i=1 (P oll i ∩ Elhi ) be arbitrary. The function f (x1 , ..., xn ) := g(t(x1 ), ..., t(xn )), where t fulﬁlls the conditions from (2), m preserves all relations 1 , ..., m and has the projection g. Consequently, i=1 P oll (i ∩ Elhi ) ⊆ prl A. Hence “(2) =⇒ (3)” is proven. Theorem 12.3.4 Let A be a subclass of Pk,2 , whose projection pr2 A (⊆ P2 ) is an intersection of at least two classes that are incomparable (with respect to inclusion) and for which there exists a certain m ∈ N \ {1} with hm ∈ pr2 A. Then, A can be described as an intersection of at least two incomparable classes of Pk,2 whose projections are diﬀerent from pr A. Proof. Let r ≥ 2 be the smallest number for which the function hr belongs to prA. By Theorem 12.3.1, there are certain r-ary relations 1 , ..., m with m A = i=1 P olPk,2 i . For proof of the theorem, it suﬃces that pr A ⊂ P olPk,2 i is correct for every i ∈ {1, 2, ..., m}. Obviously, A contains an e11 -projectable function. Consequently, by Lemma 12.3.3, pr P ol i = P ol2 (i ∩ E2r ). Further, one receives by scrutinizing the possibilities for pr A, where hr ∈ pr A, and considering the results (about the smallest arities of relations with pr A = P ol ) from Section 11.2 that pr A and P ol2 (i ∩ E2r ) are diﬀerent. Hence, it holds pr A ⊂ P oli , i = 1, 2, ..., m. The following procedure for the determination of all subclasses A ⊆ Pk,2 with M ∩ S ⊆ pr2 A or Ta,m ∩ Ta ∩ M ⊆ pr2 A (a ∈ E2 , m ∈ N\{1}) results from Theorems 3.1, 12.3.1, and 12.3.4 : (1) Determination of all A-projectable subclasses of Pk,2 , where A is P2 , T0 , T1 , M , S or Ta,m (a ∈ E2 , m ∈ N\{1}). (2) Formation of “permissible” intersections that is, only formation of such intersections whose projections contain a function hm . In the ﬁrst step of this procedure, because of Theorem r 12.3.1, one has to ﬁnd all hi -ary relations i , i = 1, 2, ..., r, with pr i=1 P ol i ⊇ B, B ∈ {T0 , T1 , M, S, T0,m , T1,m }. The following Lemma 12.3.5 and Theorem 12.3.6 give some helpful considerations for this purpose. hi Lemma 12.3.5 Let B be a clone of P l and let i be an hi -ary relation (⊆ Ek ) r for all i ∈ {1, ..., r}. Then B ⊆ pr i=1 P olPk,l i if and only if there is a mapping t from Ek onto El with the ﬁxed points 0, 1, ..., l − 1 and with the properties t(i ) = i ∩ Elhi and i ∩ Elhi ∈ Inv B for all i ∈ {1, ..., r}. r Proof. If B ⊆ pr i=1 P ol i , then there is an e11 -projectable function t ∈ r hi i=1 P ol i with the properties t(a) = a for all a ∈ El , t(i ) = i ∩ El hi and i ∩ El ∈ Inv B for all i ∈ {1, 2, ..., r}. Conversely, ifa mapping t r exists with these properties, then by Lemma 12.3.3, we have pr i=1 P ol i = r r r hi hi i=1 P ol (i ∩ El ). Thus, B ⊆ i=1 P ol (i ∩ El ) = pr i=1 P ol i holds hi because of i ∩ El ∈ Inv B, i = 1, ..., r.

12.3 On the Number of the B-projectable Subclasses of Pk,2 , B ⊆ P2

345

Theorem 12.3.6 Let 1 , ..., r be subsets of Ek2 \E22 , ⊆ E22 , = ∅ and P ol ∈ {P2 , T0 , T1 , S, M, T0,2 , T1,2 }, i.e., w.l.o.g. let 0 0 1 1 0 1 0 0 1 0 0 1 0 1 ∈ , , , , , , , 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1 0 1 0 1 , , . 0 1 1 0 0 1 1 1 0 r Then pr( i=1 P ol (∪i )) = P ol2 iﬀ fulﬁlls one of the following conditions: 0 0 1 1 0 1 0 1 0 1 (1) ∈ , ; 0 1 0 1 0 0 1 1 1 0 0 0 (2) = and there are no elements a1 , ..., aq , b1 , .., bq (q ≥ 2) with 0 ∈ 1 1 {a1 , b1 }, 1 ∈ {aq , bq }, {ai , bi } ∩ {ai+1 , bi+1 } = ∅ for all i ∈ {1, 2, ..., q − 1} and a1 a2 ... aq ⊆ 1 ∪ 2 ∪ ... ∪ r ; b1 b2 ... bq a (3) = , a ∈ E2 and 1 ∪ 2 ∪ ... ∪ r ⊆ (Ek \{a})2 ; a a a a a ... a (4) = , a ∈ E2 and 1 ∪ 2 ∪ ... ∪ r ⊆ Ek2 \ ; a a 2 3 ... k − 1 0 1 (5) = , (1 ∪ 2 ∪ ... ∪ r ) ∩ {(a, a) | a ∈ Ek } = ∅ and there are no 1 0 elements a1 , ..., aq , b1 , ..., bq , q ≥ 2, with the properties a1 ... aq ⊆ 1 ∪ 2 ∪ ... ∪ r ∧ b1 ... bq (∀i ∈ {1, ..., q − 1} : {ai , bi } = {ai+1 , bi+1 } ∧ {ai , bi } ∩ {ai+1 , bi+1 } = ∅) ∧ ((0 ∈ {a1 , b1 } ∧ 1 ∈ {aq , bq } ∧ q is even ∨ ({a1 , b1 } ∩ {aq , bq } = ∅ ∧ q is odd )); 0 0 1 (6) = and there are no elements a1 , ..., aq of Ek \E2 with 0 1 1 1 a1 a2 ... aq−2 aq−1 aq ⊆ 1 ∪ ... ∪ r . a1 a2 a3 ... aq−1 aq 0 Proof. One easily proves the above statements with the aid of Lemma 12.3.5. Subsequently, we deal with subclasses of Pk,2 , whose projection classes comprise linear functions. Theorem 12.3.7 Let L∩T0 ∩S ⊆ B ⊆ L. Then there exists at least countableinﬁnite-many B-projectable subclasses of Pk,2 for every k ≥ 3, i.e., |Nk (B)| ≥ ℵ0 . Further, for every B-projectable class A of Pk,2 and for every n ≥ 1, An has a generating system that consists of at most |A1 | + 2 functions of A.

346

12 Subclasses of Pk,2

Proof. |Nk (B)| ≥ ℵ0 for L ∩ T0 ∩ S ⊆ B ⊆ L follows from the fact that the sets n LB,r := n≥1 {f n ∈ Pk,2 | (∃ai , aI : f (x) = a0 + i=1 ai · j1 (xi )+ aI · i∈I j2 (xi ) ) ∧ pr f ∈ B} + I I ⊆ {1, ..., n} |I| ≤ r

for r = 1, 2, ... are closed and LB,r = LB,r holds for r = r . Now, let A ⊆ Pk,2 and L ∩ T0 ∩ S ⊆ pr A. Then there is a function g ∈ A with (pr g)(y1 , y2 , y3 ) = y1 + y2 + y3 . Thus the function r(x1 , x2 , x3 ) := i(x1 ) + i(x2 ) + i(x3 ) with i(x) := g(x, x, x) and pr i = e11 belongs to A. We will show that for every n ≥ 1 that An has a generating system of at most |A1 | + 2 functions of A. Let An = {f1 , f2 , ..., ft }. Then, by r ∈ A, the functions q(x1 , x2 , ..., xn·t , x) :=

t

ft (x(j−1)·n+1 , ..., xj·n ) + a · i(x)

j=1

and ps (x) :=

t

fj (x, x, ..., x) + (a + 1) · i(x),

j=1 j = s

belong to A, where

a=

0, if t odd, 1, if t even.

Then we have fs (x1 , ..., xn ) = r(q(x1 , ..., x1 ,

x1

((s − 1) · n + 1)-th place

, x2 , ...,

xn

, x1 , ..., x1 , ps (x1 ), x1 )

((s · n)-th place

for s = 1, 2, ..., t. Hence, every function of An is a superposition over A1 ∪{r, q}. Theorem 12.3.7 will be speciﬁed for k = 3 through Theorem 12.6.9. The following theorem states something about the cardinality Nk (B) of the remaining classes B ⊆ L. Theorem 12.3.8 Let ∅ = B ⊆ [P21 ] and k ≥ 3. Then there are continuummany B-projectable subclasses of Pk,2 .

12.3 On the Number of the B-projectable Subclasses of Pk,2 , B ⊆ P2

347

Proof. Denote N a subset of N\{1}. For every N , one can deﬁne a certain [P21 ]-projectable subset QN of Pk,2 , as follows inductively: 1 and QnN (n ≥ 2) be the set of all functions f n ∈ Pk,2 , which Let Q1N = Pk,2 fulﬁll the following conditions: (1) f ∈ [Qn−1 N ]ζ,τ,∇,∆ . (2) There exists a nonempty subset I of {1, 2, ..., n}, an a ∈ E2 , an n-ary function f1 ∈ [Qn−1 N ]ζ,τ,∇,∆ which depends on at least |I| variables ﬁctin with tiously, and a function t ∈ Pk,2 f (x) = (f1 (x) ·

k1 (xi ) ∨ t(x) ·

i∈I

a

g :=

j2 (xi ))a ,

i∈I

where

g if a = 1, g if a = 0.

(3) If n ∈ N , there are an h ∈ Q1N , an i of {1, ..., n} and an a ∈ E2 with f (x) = (h(xi )·

n

q=1

k1 (xq ) ∨

n

j2 (x1 )...j2 (xq−1 )j1 (xq )j2 (xq+1 )...f2 (xn ))a .

q=1

By (1), the set QN is a closed set with respect to the unary operations ζ, τ, ∇, ∆. One can show that the set QN is also closed with respect to the operation as follows: Let f n and g m be arbitrary functions of QN . Through induction on r := n+m, r ≥ 2, we will show that f g ∈ QN . If r = 2 (n = m = 1) then f g ∈ QN is obvious. Suppose for all n + m ≤ r − 1, f n g m ∈ QN holds for arbitrary f and g of QN . Now let n + m = r. f g ∈ QN is surely valid for n = 1. If n ≥ 2, then, by (1), it is suﬃcient to study the following cases: Case 1: f fulﬁlls (2). In this case, the function f g is not essentially dependent from all their variables or it also fulﬁlls the condition (2). Consequently, f g belongs to QN . Case 2: f fulﬁlls (3) and w.l.o.g. let be a = 1. In this case, we have for i = 1: (f g)(x) = (h g)(x1 , ..., xm )k1 (xm+1 )...k1 (xm+n−1 ) ∨ g(x1 , ..., xm )j2 (xm+1 )...j2 (xm+n−1 ). For i = 1 it holds: (f g)(x) = h(xi+m−1 )k1 (xm+1 )...k1 (xm+n−1 ) ∨ g(x)j2 (xm+1 )...j2 (xm+n−1 ). Hence, the function f g has the property (2) both for i = 1 and for i = 1 . Consequently, QN is closed. It follows from the deﬁnition of the sets of type QN and from the above proof

348

12 Subclasses of Pk,2

(for QN = [QN ]) that the sets QN , QN are pairwise diﬀerent for N = N . Therefore, there are continuum-many [P21 ]-projectable subclasses of Pk,2 . The following facts are easy to check: For every B with ∅ = B ⊆ [P21 ] and for arbitrary N ⊆ N\{1}, the set QN ∩ pr−1 B is a B-projectable subclass of Pk,2 and it holds QN ∩ pr−1 B = QN ∩ pr−1 B for all N = N with N, N ⊆ N\{1}. This implies Theorem 12.3.8. One can deal with the inﬁnite-digit relations deﬁned in the following lemma as with the ﬁnite-digit relations. Further, analogously to Chapter 2, one can introduce the same concepts as in Chapter 2 for the inﬁnite-digit relations. Lemma 12.3.9 ([Gr¨ u 84]) Let be E2∞ \{1} := {(a1 , a2 , a3 , ...) | ∀i ∈ N : ai ∈ {0, 1} ∧ ∃j ∈ N : aj = 0}, t := {(a1 , a2 , ..., at+1 , 2, 2, 2, ...) | ∃i ∈ {1, 2, ..., t + 1} : (ai = 2 ∧ ∀j ∈ {1, 2, ..., t + 1}\{i} : aj = 0)}, (t ∈ N), (q+1)

, q ∈ N, deﬁned by 1 if x ∈ {(2, 0, ..., 0),(0, 2, 0, ..., 0),...,(0, 0, ..., 0, 2)}, fq (x) := 0 otherwise; fq

At := P olPk,2 (E2∞ \{1}) ∪ t , (t ∈ N) and AT :=

Ai if T ⊆ N.

i∈T

Then (a) ∀i ∈ N : fi ∈ Ai ; (b) ∀i ∈ N ∀j ∈ N\{i} : fj ∈ Ai ; (c) ∀T, T ⊆ N : T = T =⇒ AT = AT . Proof. (a) follows directly from the deﬁnition of the functions fi and (c) is a conclusion from (b). Therefore, we only have to prove (b). Suppose for certain i, j ∈ N with i = j it holds fj ∈ Ai . Then there are some columns s1 , s2 , ..., sj+1 ∈ E2∞ \{1} ∪ i with fj (s1 , s2 , ..., sj+1 ) = 1.

(12.11)

Obviously, this is only valid if {s1 , ..., sj+1 } ⊆ E2∞ \{1}. The case i ⊆ {s1 , ..., sj+1 } is not possible, since in this case, the matrix (s1 , ..., sj+1 ) would have a row with at least a zero and no two, what (12.11) contradicts. Therefore we can presuppose i ⊆ {s1 , ..., sj+1 }. Then, because of i = j we have either

12.3 On the Number of the B-projectable Subclasses of Pk,2 , B ⊆ P2

349

{s1 , ..., sj+1 } ∩ (E2∞ \{1}) = ∅ or sq = sr for a certain q = r. In the ﬁrst case, one ﬁnds in (s1 ..., sj+1 ) (inﬁnite many) rows with the element 2 in at least two places and with an element of E2 . In the second case, there is a row with the element 2 in at least two places and with an element 0. In every one of these cases, a contradiction to (12.11) results with that from the deﬁnition of the functions fj . Theorem 12.3.10 ([Gr¨ u 84]) For every a ∈ E2 and every B ∈ {K, K ∪ C0 , D, D ∪ C1 , Ta,∞ ∩ Ta ∩ M, Ta,∞ ∩ M, Ta,∞ ∩ Ta , Ta,∞ } it holds |Nk (B)| = c. Proof. W.l.o.g. let a = 0 and K ⊆ B. Obviously, the sets AT , T ⊆ N, deﬁned in Lemma 12.3.9, belong to Nk (T0,∞ ). |Nk (T0,∞ )| = c results from that then because of Lemma 12.3.9, (c). Since pr fq ∈ K ⊆ B, statements analogous to (c) of Lemma 12.3.9 are valid for the sets AT ∩ pr−1 B with B ∈ {K, K ∪ C0 , T1 , M, M ∩ T1 }. This implies the remaining assertions of our theorem result. Theorem 12.3.11 For every B ∈ {K ∪ C, K ∪ C1 , D ∪ C, D ∪ C0 } it holds |Nk (B)| = c. Proof. Let n fn (x1 , ..., xn ) = i=1 j2 (x1 )...j2 (xi−1 )j1 (xi )j2 (xi+1 )...j2 (xn ), gn (x1 , ..., xn ) = j2 (x1 )j2 (x2 )...j2 (xn ), N ⊆ N\{1} AN := [{j1 (x1 )j1 (x2 )} ∪ {fi , gj | i ∈ N\(N ∪ {1}) ∧ j ∈ N\{1}}] and ⎛ ⎞ 1 2 ... 2 ⎜ 2 1 ... 2 ⎟ i i ⎟ i := ⎜ i ∈ N\{1}. ⎝ . . . . . . . . . ⎠ ∪ Ek \{1, 2, ...k − 1} , 2 2 ... 1 It is easy to check that the functions of A{j} preserve the relations i for all i ∈ N\{1, j}. On the other hand, the function fi does not preserve the relation j . Due to this property, there are continuum-many subclasses of the type AN with the projection K ∪ C0 . Since [AN ∪ {c1 }] = [{c1 }] ∪ AN is obviously valid, we have |Nk (K ∪ C)| = c too. Since the sets K ∪ C, D ∪ C and K ∪ C1 , D ∪ C0 are isomorphic, respectively, only the proof for |Nk (K ∪ C1 )| = c is still missing. Denote fn and gn functions deﬁned by fn (x) := fn (x) ∨ j1 (x1 )j1 (x2 )...j1 (xn )

350

or

12 Subclasses of Pk,2

gn (x) := gn (x) ∨ j1 (x1 )j1 (x2 )...j1 (xn )

and let AN := [{fi , gj | i ∈ N\(N ∪ {1}) ∧ j ∈ N}]

if N ⊆ N.

Obviously, all functions fn for n = i preserve the relation i := {1} × i ,

i ∈ N\{1}.

Furthermore, we have [AN ∪{c1 }] = [{c1 }]∪AN and pr([{c1 }]∪AN ) = K ∪C1 . Consequently, there are continuum-many K ∪ C1 -projectable subclasses of Pk,2 . With the above theorem, we are ready with a ﬁrst coarse survey over |Nk (B)| for all subclasses B of P2 . One ﬁnds more precise statements over |Nk (B)| for certain B in the following section and for k = 3 in Sections 12.6 and 12.7.

12.4 The Pl-projectable and the P oll{α}-projectable Subclasses of Pk,l Let R be the set of all relations of the form 0 1 2 ... l − 1 ∪ 0 1 2 ... l − 1 with ⊆ Ek2 \El2 and R(α) := {{α} ∪ | ⊆ Ek \El }. A subset Q of R is called permissible, iﬀ no elements a1 , ..., ar , b1 , ..., br with a1 ... ar ⊆ σ∈Q σ, {a1 , br } ⊆ El , a1 = br and {ai , bi } ∩ {ai+1 , bi+1 } = ∅ b1 ... br for all i ∈ {1, 2, ..., r} exist. Furthermore, a subset Q of R ∪ R(α) is called α-permissible, iﬀ Q ∩ R(α) = ∅, Q ∩ R is permissible and no elements a1 , ..., ar , b1 , ..., br and no relation a1 ... ar ∈ Q ∩ R(α) with ⊆ σ∈R∩Q σ, a1 ∈ El \{α}, br ∈ \{α} and b1 ... br {ai , bi } ∩ {ai+1 , bi+1 } = ∅ for all i ∈ {1, 2, ..., r} exist. With the help of Lemma 12.3.5, it is easy to prove that P olPk,l Q for Q ⊆ R is Pl -projectable iﬀ Q is permissible. Obviously, for Q ⊆ R ∪ R(α) we have pr(P olPk,l Q) = P oll {α} iﬀ Q is αpermissible. Denote Tn (Q) the set of all tuples of Ekn , on which all functions of P ol Q have the value α.

12.4 The Pl -projectable and the P oll {α}-projectable Subclasses of Pk,l

351

With the help of Q ⊆ R ∪ R(α), where Q is permissible or α-permissible, one can deﬁne an equivalence relation Eq(Q) on the set of all tuples with elements of Ek as follows: (a, b) ∈ Eq(Q) :⇐⇒ ∃n : {a, b} ⊆ Ekn ∧ ({a, b} ⊆ Tn (Q)∨ ({a, b} ⊆ Tn (Q) ∧ ∀f ∈ P ol Q : f (a) = f (b))). Let σ1 , σ2 be two equivalence relations. Then σ1 is ﬁner than σ2 , if (a, b) ∈ σ1 implies (a, b) ∈ σ2 for all a, b. We write for that σ1 = σ2 For two permissible (or α-permissible) sets Q1 , Q2 , we call Q2 a minimal coarsening of Q1 , if Eq(Q1 ) = Eq(Q2 ) holds and no permissible (or αpermissible) relation set Q exists with Eq(Q1 ) = Eq(Q ) = Eq(Q2 ). Theorem 12.4.1 Let Q be a permissible relation set. Then (1) The only maximal classes of P olPk,l Q are the following classes: (a) P olPk,l Q , where Q is an arbitrary minimal coarsening of Q, (b) prl−1 B ∩ P olPk,l Q, where B is an arbitrary maximal class of Pl . (2) The classes, deﬁned in (a) and (b), with diﬀerent deﬁnitions are pairwise incomparable (with respect to inclusion). (3) A set A ⊆ P olPk,l Q is P ol Q-complete if and only if A ⊆ M holds for every class M deﬁned above in (a) or (b). Proof. The second statement is easy to check and (1) follows from (2) and (3). Consequently, it is suﬃcient to prove the statement (3). A P ol Q-complete set A is obviously no subset of an arbitrary class deﬁned in (a) or (b). Let A ⊆ P ol Q be an arbitrary set with A ⊆ M for every class M , which is deﬁned in (a) or (b). We consider two n-tuples a := (a1 , a2 , ..., an ), b := (b1 , b2 , ..., bn ) with (a, b) ∈ Eq(Q) and we choose a minimal j, for which there exists i1 , ..., ij ∈ {1, 2, ..., n} with ((ai1 , ..., aij ), (bi1 , ..., bij )) ∈ Eq(Q). Now, 0 1 ... l − 1 ai1 ... aij := 0 1 ... l − 1 bi1 ... bij and Q := Q ∪ {}. If Q is permissible, then Q is a minimal coarsening of Q, and thus there is a function f ∈ A with f ∈ P ol Q . If Q is not permissible, pr P ol Q = Pl holds. Hence, one can ﬁnd a maximal class B of Pl with P ol Q ⊆ P ol Q ∩ pr−1 B. Therefore, there exists a function f ∈ A with f ∈ P ol Q . Obviously, for all a ∈ El , there is a ca -projectable function pa ∈ [A]. Because of pa pa = ca , all constant functions c0 , c1 , ..., cl−1 belong to [A]. Hence there is a j-ary function f1 ∈ [A] with f1 (ai1 , ..., aij ) = f1 (bi1 , ..., bij ) and, by this, there is a function f2 ∈ [A] with f2 (a1 , ..., an ) = f2 (b1 , ..., bn ). n ∈ [A] with the Now let n ≥ 1. We showed that there are functions g1n ..., gw

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12 Subclasses of Pk,2

following property: for every pair (a, b) ∈ Eq(Q) of tuples, there is a function gi with gi (a) = gi (b). Hence, for an arbitrary function f n ∈ P ol Q one can ﬁnd a function g ∈ Pl with g(g1 (x), ..., gw (x)) = f (x). Since A ⊆ prl−1 B∩P olPk,l Q, where B is an arbitrary maximal class of Pl , there is a g-projectable function in [A]. Consequently, f ∈ [A]. Thus A is P ol Q-complete. As a direct consequence of Theorem 12.4.1, we get: Theorem 12.4.2 The set of the closed sets A ⊆ Pk,l with prl A = Pl is identical with the set of all classes of the form P olPk,l Q with permissible Q.

Theorem 12.4.3 (Completeness Criterion for Pk,l , [Lau 75]) For all l with 2 ≤ l ≤ k − 1 it holds: (1) The maximal classes of Pk,l are exactly the following sets: (a) prl−1B, where B is anarbitrary maximal class of Pl ; 0 1 .. l − 1 i , 0 ≤ t ≤ k − 2, l ≤ i ≤ k − 1, t < i. (b) P ol 0 1 .. l − 1 t (2) A set T ⊆ Pk,l is Pk,l -complete if and only if T ⊆ B for all classes B deﬁned in (1). Proof. The statements (1) and (2) are consequences of Theorem 12.4.1. A proof that does not use Theorem 12.4.1 can be found in [Lau 75]. As we will see below, one can prove the following theorem similar to the above theorem. Theorem 12.4.4 Let Q be an α-permissible subset of R ∪ R(α). Then: (1) The only maximal classes of P olPk,l Q are sets of the type (a) P olPk,l Q , where Q is a minimal α-permissible coarsening of Q, and (b) pr−1 B ∩ P olPk,l Q, where B is a maximal class of P oll {α}. (2) The sets, deﬁned in (a) and (b), with diﬀerent deﬁnitions are pairwise incomparable (with respect to inclusion). (3) A set A ⊆ P olPk,l Q is P ol Q-complete if and only if A ⊆ M holds for every class M which is deﬁned above in (a) or (b). Proof. We prove only (3), “⇐=” here. The remaining statements of the theorem are either obvious or easily provable. Let A ⊆ P olPk,l Q and A ⊆ M for every class M which is deﬁned above in (a) or (b). Denote a := (a1 , ..., an ) an n-tuple, which does not belong to Tn (Q). Then there is a minimal j for i1 , ..., ij ∈ {1, 2, ..., n} with (ai1 , ..., aij ) ∈ Tj (Q). Now

12.4 The Pl -projectable and the P oll {α}-projectable Subclasses of Pk,l

353

we put := {α, ai1 , ..., aij } and Q := Q ∪ {}. If Q is α-permissible, then Q is a minimal coarsening of Q; thus in A there is a function f with f ∈ P ol Q . If Q is not permissible, then pr P ol Q = P oll {α}. Therefore, there exists a maximal class B of P ol {α} with P ol Q ⊆ P ol Q ∩ pr−1 B and, hence, there exists a function f ∈ A with f ∈ P ol Q in this case. Since cα ∈ [A], there is in [A] a j-ary function f1 with f1 (ai1 , ..., aij ) = α; thus there is also an n-ary function f2 ∈ [A] with f2 (a1 , ..., an ) = α. Next we consider two n-tuples a := (a1 , ..., an ), b := (b1 , ..., bn ) with (a, b) ∈ Eq(Q) and {a, b} ⊆ Tn (Q). Analogous to the considerations from proof of Theorem 12.3.1, one can prove that there is a function r ∈ [A] and that there are certain i1 , ..., ij ∈ {1, ..., n} with 0 1 ... l − 1 ai1 ... aij 0 1 ... l − 1 r ∈ . 0 1 ... l − 1 bi1 ... bij 0 1 ... l − 1 Because – as already was proven – functions s and t with s(ai1 , ..., aij ) = α and t(bi1 , ..., bij ) = α exist and because certain functions uβ with ai1 ... aij ai1 ... aij β uβ s ,t = , β ∈ El , bi1 ... aij bi1 ... aij β belong to [A], a certain function r with ai1 ... aij 0 1 ... l − 1 ∈ r bi1 ... bij 0 1 ... l − 1 is a superposition over r, u0 , ..., ul−1 . Consequently, there exists an n-ary function r ∈ [A] with r (a) = r (b). Now, let n ≥ 1. As proven above, there are functions q1 , ..., qv , g1 , ..., gw ∈ [A] with the following property: for every a ∈ Ekn \Tn (Q) there exists a qi with qi (a) = α and for every pair c, d ∈ Ekn \Tn (Q) with (c, d) ∈ Eq(Q) there exists a function gj with gj (c) = gj (d). Let f n ∈ P ol Q be arbitrary. Then one can ﬁnd a function g ∈ P oll {α} with g(q1 (x), ..., qv (x), g1 (x), ..., gw (x)) = f (x). Since pr[A] = P oll {α}, there exists in [A] a g-projectable function. Consequently, f ∈ [A] holds. Hence the set A is P ol Q-complete. As a consequence of Theorem 12.4.4, we get: Theorem 12.4.5 The only P oll {α}-projectable subclasses of Pk,l with α ∈ El are sets of the form P olPk,l Q, where Q are α-permissible subsets Q of R∪R(α).

354

12 Subclasses of Pk,2

12.5 The Maximal and the Submaximal Classes of Pk,2 A direct consequence of Theorem 12.4.1 is the following theorem, already proven in [Bur 74]. Theorem 12.5.1 The maximal classes of Pk,2 are exactly the following sets: (1) pr−1 B, where in P2 ; B is maximal 0 1 i (2) Zi,t := P ol , 0 ≤ t ≤ k − 2, 2 ≤ i ≤ k − 1, t < i. 0 1 t The above sets are all pairwise incomparable (with respect to ⊆), so that there are exactly 1 · (k − 2) · (k + 1) + 5 2 maximal classes of Pk,2 . Next we determine the submaximal classes of Pk,2 , i.e., we determine the maximal classes of the classes of Theorem 12.4.1. The following theorem is a special case of Theorem 12.4.1. This is easy to see considering the following facts: 0 1 i r 0 1 i 0 1 r P ol ⊆ P ol ∩ P ol 0 1 t s 0 1 t 0 1 s

and P ol

0 1 i i 0 1 t i

= P ol

0 1 i t 0 1 t t

for 0 ≤ t ≤ k − 2, 2 ≤ i ≤ k − 1, t < i. Theorem 12.5.2 (1) The maximal classes of Zi,t (0 ≤ t ≤ k − 2, 2 ≤ i ≤ k − 1, t < i) are exactly the following sets (a) Zi,t ∩ Zj,l , 0 ≤ l ≤ k − 2, 2 ≤ j ≤ k − 1, l < j, l = t or i = j, {0, 1} = {t, l} for i = j; 0 1 i j (b) P ol , 2 ≤ j ≤ k − 1, j = i for t ∈ E2 ; 0 1 t j −1 (c) Zi,t ∩ pr B, where B is a maximal class of P2 . (2) Zi,t has exactly 12 · (k − 1) · k + 2 maximal classes. A consequence from Theorem 12.4.4 is

12.5 The Maximal and the Submaximal Classes of Pk,2

355

Theorem 12.5.3 (1) pr−1 Tα (α ∈ E2 ) has exactly the following maximal classes: (a) pr−1 Tα ∩ pr−1 B, where B is Tα -maximal; (b) Zi,t ∩ pr−1 Tα , 0 ≤ t ≤ k − 2, 2 ≤ i ≤ k − 1, t < i, t = α; (c) P olPk,2 {α, i}, 2 ≤ i ≤ k − 1. (2) The classes listed in (a)– (c) are pairwise incomparable with respect to the inclusion, so that there are exactly 4 + 12 · (k − 2) · (k + 1) in pr−1 Tα maximal classes. (3) A set A ⊆ pr−1 Tα is pr−1 Tα -complete if and only if A ⊆ C holds for all classes C listed in (a)–(c).

Theorem 12.5.4 (1) pr−1 S has exactly the following maximal classes: (a) pr−1 B, where B is a maximal of S; (b) Zi,t ∩ pr−1 S,0 ≤ t <i < k, i ≥ 2; 0 1 i , 2 ≤ t < i < k. (c) S (i,t) := P ol 1 0 t (2) The classes listed in (a)– (c) are pairwise incomparable with respect to the inclusion, so that there are exactly 2 + (k − 1) · (k − 2) in pr−1 S maximal classes. (3) A set A ⊆ pr−1 S is pr−1 S-complete if and only if A ⊆ C holds for all classes C listed in (a)–(c). Proof. Let A be an arbitrary subset of pr−1 S with A ⊆ C for all classes C listed in (a)–(c). Then pr[A] = S holds. Now let f n ∈ pr−1 S be arbitrary. We show that f ∈ [A]. Let hi,t ∈ A\S (i,t) and fi,t ∈ A\Zi,t . Then, superpositions over these functions and over (suitably chosen) inverse images (∈ [A]) of functions of S are the of [A] with binary functions hi,t and fi,t

hi,t (1, i) hi,t (0, t)

=

1 1

and

(1, i) fi,t fi,t (1, t)

=

1 0

.

In [A] there is a inverse image l(x) of e11 ∈ S. It holds l(i) c l(i) c , = or = c l(t) c l(t) l(i) 1 W.l.o.g. we can assume that c = 1. In the ﬁrst case = we have l(t) 1

356

12 Subclasses of Pk,2

fi,t (x)

:=

fi,t (l(x), x)

∈ [A] and

(i) fi,t fi,t (t)

=

1 0

.

This implies (x), x) ∈ [A] and hi,t (x) := hi,t (fi,t

In the second case

l(i) l(t)

=

1 0

hi,t (i) hi,t (t)

=

1 1

.

we put

hi,t (x) := hi,t (l(x), x) and fi,t (x) := fi,t (hi,t (x), x),

where the above functions have the same properties as the functions in the i ﬁrst case. Therefore, there are certain functions of [A], which have on t 1 1 the values or . 0 1 Now we assign a tuple of the form w := (..., hi,t (a1 ), ..., hi,t (an ), ..., fi,t (a1 ), ..., fi,t (an ), ...)

to every n-tuple a := (a1 , ..., an ) ∈ Ekn , where i, t take all possible values with 0 ≤ t ≤ k − 2, 2 ≤ i ≤ k − 1, i > t (in a ﬁxed order). All tuples of the form w belong to a certain Cartesian product E2σ . Denote W the set of all tuples w deﬁned above. Next we prove: If w, w ∈ W were assigned to a, a ∈ Ekn (a := (a1 , ..., an )), respectively, then it holds: a = a =⇒ w = w

(12.12)

a = a ⇐⇒ w = w ,

(12.13)

and where a := (a1 , ..., an ), 0 := 1, 1 := 0, a := a for all a ∈ Ek \E2 . ..., a ). Then Let a = a , where a := (a1 , n ) and a:= (a 1 , ..., an there exists a c 0 c 1 j with c := aj = aj =: d. If , (12.12) is valid = or = d 1 d 0 certainly. If {c, d} = {0, 1} and w.l.o.g. c > d, then by deﬁnition of fc,d the statement (12.12) holds. If a = a then w = w , since all considered functions belong to pr−1 S. For a = a the two cases are possible:

12.5 The Maximal and the Submaximal Classes of Pk,2

357

a p has a column with a q p 0 0 1 1 ∈ , , , . 1 0 1 q 0 a q Case 2: The matrix has a column with q ∈ E2 . a q In the second case, w = w holds. In the ﬁrst case, one can w= w easily prove hi,t (i) 1 = . when one one assumes p > q considering the condition hi,t (t) 1 Case 1: The matrix

Let h be a mapping from W onto E2 deﬁned by h(w) = h(w1 , ..., wσ ) := f (a). Because of (12.12) and (12.13), an α-ary function h1 ∈ S can be found, whose restriction onto W is identical with h. Hence there is an inverse image h of h and hi,t . Thus in [A], and, by construction, f is a superposition over h , fi,t −1 f ∈ [A] and [A] = pr S; i.e., the third statement of Theorem 12.5.4 was proven. One checks the remaining statements of the theorem easily. Lemma 12.5.5 Let A ⊆ pr−1 M , [pr M ] = M and {k1 , j0 (x1 ) · ji (x2 ) | i = 2, 3, ..., k − 1} ⊆ [A]. Then [A] = pr−1 M . Proof. Obviously, for every function g ∈ M there exists a function g ∈ [A] with pr g = g. Consequently, certain inverse images of the conjunction, disjunction, n n = c0 (n ∈ [A], pr n = c0 ), i(x) · k1 (x) = j1 (x) (i ∈ [A], pr i = pr j1 ) and j0 (c0 (x)) · jp (x) = jp (x), 2 ≤ p ≤ k − 1 are superpositions over A. One can describe every function f n = cn0 of pr−1 M by f (x1 , .., xn ) = f1 (x1 , ..., xn ) · k1 (x1 ) · ... · k1 (xn ) ∨ (a1 , ..., an ) ja1 (x1 ) · ja2 (x2 ) · ... · jan (xn ),

(12.14)

∈ Ekn \E2n f (a1 , ..., an ) = 1

where f1 ∈ [A] and pr f1 = pr f . (12.14) is obvious a superposition over A and thus [A] = pr−1 M .

Theorem 12.5.6 (1) pr−1 M has exactly the following maximal classes: (a) pr−1 B, where B is maximal in M ; (b) Zi,t ∩ pr−1 M , 2 ≤ t < i < k;

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12 Subclasses of Pk,2

(0,r) (c) M , M (r,s) , 1≤ s < k, 2 ≤ r < k; 0 1 0 a (a,b) := P ol . M 0 1 1 b (2) The sets, deﬁned in a)–(c), with diﬀerent deﬁnitions are pairwise incomparable (with respect to inclusion) so that pr−1 M has exactly 4 + 32 (k − 1) · (k − 2) maximal classes. (3) A set A ⊆ pr−1 M is pr−1 M -complete if and only if A ⊆ T holds for every class T which is deﬁned in (a)–(c).

Proof. Obviously, if [A] = pr−1 M , then A is not a subset of the sets (a)–(c). Now let A be subset of pr−1 M with A ⊆ T for every class T deﬁned in (a)–(c). Then we have [pr A] = M and c0 , c1 ∈ [A]. Further, [A] contains functions fi,t , gr,s , qr and pr with the properties: fi,t ∈ Zi,t ,

gr,s ∈ M (r,s) ,

qr ∈ M (r,r) ,

pr ∈ M (0.r) .

, gr,s , qr and pr with Then certain unary or binary functions fi,t (i) = fi,t (t), fi,t 0 r 1 gr,s , = 1 s 0

qr

0 r 1 r

=

1 0

and

pr

0 0 1 r

=

1 0

,

2 ≤ t < i ≤ k − 1, 1 ≤ s ≤ k − 1, 2 ≤ r ≤ k − 1 are obvious superpositions over {c0 , c1 , fi,t , gr,s , qr , pr }. For the function (x), x), if fi,t (i) = 0, gi,t (fi,t fi,t (x) := gt,i (fi,t (x), x), if fi,t (i) = 1, (a) for a ∈ {i, t}. Thus w.l.o.g. we can assume for all (a) = fi,t it holds fi,t (i) = 1 and fi,t (i) = 0. permissible i, t in the following: fi,t Let i ∈ [A], pr i = pr j1 and let E be the set of all A ∈ Ek \E2 with i(a) = 0. (i(x), x) =: gr,1 (x), we have gr,1 (r) = 1 and gr,1 (1) = Then, for r ∈ E and gr,1 0. Consequently, i(x) ∨ gr,1 (x) = j0 (x) r∈E

and r−1

qr (i(x1 ), x2 )·gr,1 (i(x1 ), x2 )·(

fr,t (x2 ))·(

t=2

k−1

fi,r (x2 )) = j0 (x1 )·jr (x2 ) ∈ [A]

i=r+1

for r ≥ 2. Furthermore, it holds k−1

r=2

pr (j0 (c0 (x)) · jr (x), x) = k1 (x) ∈ [A].

12.5 The Maximal and the Submaximal Classes of Pk,2

359

Thus, we showed that the generating system for pr−1 M of Lemma 12.5.5 belongs to [A]. Therefore, the set A is pr−1 M -complete. One checks the remaining statements of the theorem easily. We come now to the maximal classes of pr−1 L. First we give some remarks on a possible description of functions of pr−1 L. Because of j0 = 1 + j1 (x) + ... + jk−1 (x) it follows from Section 12.1 that every function f n ∈ Pk,2 has an unambiguous description (up to the order of the summands) of the form aI1 ,...,Ik−1 · kI1 ,...,Ik−1 (x), (12.15) f (x) = a + I1 , ..., Ik−1 ⊆ {1, 2, ..., n}

where aI1 ,...,Ik−1 ∈ E2 , kI1 ,...,Ik−1 (x1 , ..., xn ) :=

k−1

ji (xq )

(12.16)

i=1 q∈Ii

and the sets I1 , ..., Ik−1 in (12.15) are pairwise disjunct. The functions kI1 ,...,Ik−1 with aI1 ,...,Ik−1 = 1 in (12.15) are called components of f . Let Kf be the set of all components of f . Denote kf,I an n-ary function of the form kI1 ,...,Ik−1 (x), kf,I (x) := I1 , ..., Ik−1 I1 ∪ ... ∪ Ik−1 = I kI1 ,...,Ik−1 ∈ Kf

where I ⊆ {1, 2, ..., n}. Lemma 12.5.7 Let {f, g, h} ⊆ pr−1 L, (pr g)(y1 , y2 ) = y1 + y2 and (pr h)(y) = y. Then every function of the form kf,I is a superposition over {c0 , f, g, h}. Proof. It is easy to see that the function f (x) := f (x) + f (0, 0, ..., 0) is a superposition over the functions f and h. Denote r the smallest number, for which f has a component with r essential variables. W.l.o.g. let f (x) = kf,{1,2,...,r} + f (x). Then it holds f (x1 , ..., xr , c0 , ..., c0 ) = kf,{1,2,...,r} (x) ∈ [{c0 , f, g, h}] and

f (x) = g(f (x), kf,{1,2,...,r} (x)) ∈ [{c0 , f, g, h}],

where Kf ⊂ Kf ⊆ Kf and every component of f has at least r essential variables. Through repetition of this construction, one receives the statement of the lemma.

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12 Subclasses of Pk,2

Theorem 12.5.8 (1) pr−1 L has exactly the following maximal classes: (a) pr−1 B, where B is a maximal class of L; (b) Zi,t ∩ pr−1 L, 2 ≤ t < i < k; (c) Lq := P olPk,l {(q, q, q, q), (a, b, c, d) | {a, b, c, d} ⊆ E2 ∧ a + b = c + d (mod 2)}, 2 ≤ q < k. (2) The classes listed in (a)–(c) are pairwise incomparable (with respect to the inclusion) so that there are exactly 4+ 12 (k−1)·(k−2) pr−1 L-maximal classes. (3) A set A ⊆ pr−1 L is pr−1 L-complete if and only if A ⊆ Q holds for all classes Q listed in (a)–(c). Proof. The statements (1) and (2) are consequences from (3). Since “=⇒” of (3) is trivial, we prove only (3), “⇐=”. Let A be a subset of pr−1 L with A ⊆ Q for all classes Q which are deﬁned in (a)–(c). Because of A ⊆ pr−1 B, B maximal in L, pr [A] = L holds. Consequently, c0 , c1 ∈ [A] and one can ﬁnd functions g, h ∈ [A] with (pr g)(y1 , y2 ) = y1 +y2 and (pr h)(y) = y. Since A is, in addition, not a subset of the sets Zi,t , 2 ≤ t < i ≤ k − 1, there are unary functions gi,t ∈ [A] with gi,t (i) = 1 and gi,t (t) = 0, t = i, 2 ≤ i ≤ k − 1, 2 ≤ t ≤ k − 1. Let now q ∈ {2, 3, ..., k − 1}. Since the functions of A not all preserve the relation ⎛ ⎞ 0 0 0 1 1 0 1 1 q ⎜0 0 1 1 0 1 0 1 q⎟ ⎜ ⎟ ⎝ 0 1 0 0 1 1 0 1 q ⎠, 0 1 1 0 0 0 1 1 q a function f with pr(f (x1 , ..., xn , q)) ∈ L belongs to [A]. Then by the maximality of L in P2 and by pr[A] = L, for every function um ∈ P2 there is a (m + 1)-ary function u ∈ [A] with u (y1 , ..., ym , q) = u(y1 , ..., ym ). Consequently, a function of the form k v(x1 , ..., xk ) = a + i=1 ai ji (xi ) + j1 (x1 ) · ... · j1 (xk−1 ) · jq (xk ) + k−1 p=2,p=q vp (x1 , ..., xk−1 ) · jp (xk ) belongs to [A] for some a, ai ∈ E2 and some functions vp ∈ Pk,2 . By Lemma 12.5.7 the functions of the type kv,I , I ⊆ {1, 2, ..., k}, are superpositions over A. If one adds all functions of the form kv,I with |I| ≤ k − 1 to v, one receives so the function

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

361

v(x1 , ..., xk ) := j1 (x1 ) · ... · j1 (xk−1 ) · jq (xk ) + aI1 ,...,Ik−1 · kI1 ,...,Ik−1 (x) · jp (xk ), I1 , ..., Ik−1 , p I1 ∪...∪Ik−1 = {1, ..., k−1} p ∈ {2, 3, ..., k − 1}\{q}

which also belongs to [A] and for which v (x1 , x2 , gq,2 (x3 ), gq,3 (x3 ), ..., gq,q−1 (x3 ), gq,q+1 (x3 ), ..., gq,k−1 (x3 ), x3 ) = j1 (x1 ) · j1 (x2 ) · jq (x3 ) ∈ [A] holds. With the help of (12.15) it is easy to see that every function of pr−1 L is a superposition over {c0 , c1 , g, h} ∪ {j1 (x1 )j1 (x2 )jq (x3 ) | q ∈ {2, 3, ..., k − 1}}. Thus, [A] = pr−1 L.

12.6 The Classes A with M ∩ T0 ∩ T1 ⊆ prA or L ∩ T0 ∩ S ⊆ prA or prA = M ∩ S In this and in the following section, we specify the cardinality statements of Section 12.3 about Nk (A) (A ∈ L2 ) for k = 3. In addition, we provide a concrete description of the elements of N3 (A) if |N3 (A)| ≤ ℵ0 is valid. Table 12.1 gives a ﬁrst survey of these statements. The statements of the last three rows of Table 12.1 were proven already in Theorems 12.3.8, 12.3.10, and 12.3.11. The cardinality statements of the eighth row are a consequence of Theorem 12.3.1. The remaining statements in the table result from the following theorems. Theorem 12.6.1 It holds: N3 (P2 ) = {P3,2 , Z2,0 , Z2,1 }; N3 (S) = {pr−1 S, pr−1 S ∩ Z2,0 , pr−1 S ∩ Z2,1 }; (2) N3 (Ta ) = {pr−1 Ta , Ta , pr−1 Ta ∩ Z2,0 , pr−1 Ta ∩ Z2,1 } (a ∈ E2 ); (2) (2) N3 (T0 ∩ T1 ) = {pr−1 (T0 ∩ T1 ), T0 ∩ pr−1 (T0 ∩ T1 ), T1 ∩ pr−1 (T0 ∩ T1 ), Z2,0 ∩ pr−1 (T0 ∩ T1 ), Z2,1 ∩ pr−1 (T0 ∩ T1 )}; (2) (2) (5) N3 (T0 ∩ S) = {pr−1 (T0 ∩ S), T0 ∩ pr−1 (T0 ∩ S), T1 ∩ pr−1 (T0 ∩ S), −1 −1 Z2,0 ∩ pr (T0 ∩ S), Z2,1 ∩ pr (T0 ∩ S)}.

(1) (2) (3) (4)

(One ﬁnds Hasse diagrams of some of the sets listed above in the Figure 12.1.)

362

12 Subclasses of Pk,2 Table 12.1 |N3 (B)| 3

B P2 , S Ta (a ∈ E2 )

4

T0 ∩ T1 , T0 ∩ S

5

M

23

M ∩ Ta (a ∈ E2 )

36

M ∩ T0 ∩ T1

49

Ta,2 (a ∈ E2 )

148

Ta,m , Ta,m ∩ Ta , Ta,m ∩ M, Ta,m ∩ Ta ∩ M, S ∩ M

< ℵ0

(a ∈ E2 , m ∈ {2, 3, ...}) L, L ∩ T0 , L ∩ T1 , L ∩ S, L ∩ T0 ∩ S Ta,∞ , Ta,∞ ∩ Ta , Ta,∞ ∩ M, Ta,∞ ∩ Ta ∩ M (a ∈ E2 )

c

K ∪ C, K ∪ Ca , K, D ∪ C, D ∪ Ca , D (a ∈ E2 )

c

[P21 ],

c

I ∪ C, I, I ∪ Ca , I, C, Ca (a ∈ E2 )

pr −1 r P2

@

r Z2,0

ℵ0

pr −1 r T0

@

@

@ @r Z2,1

r

(2) T0

pr

−1

pr −1 Tr0 ∩ T1

@

@

@ @r

T0 ∩ Z2,1

rpr−1 T0 ∩ Z2,0

@

@(2) r pr −1T0 ∩ T1 ∩ T0(2) @r −1

pr

r

T0 ∩ T1 ∩ T1

pr −1 T0 ∩ T1 ∩ Z2,0

r

pr −1 T0 ∩ T1 ∩ Z2,1

Fig. 12.1

Proof. The intersections of pr−1 B, B ⊆ P2 , with Z2,0 or Z2,1 are always the smallest B-projectable subclasses of P3,2 . Thus, (1) is a consequence from Theorem 12.5.1 and (2) is a consequence from Theorem 12.5.4. By Theorem 12.5.3, the maximal classes of pr−1 Ta belonging to N3 (Ta ) are just the sets (2) Z2,a ∩ pr−1 Ta and Ta = P olP3,2 {a, 2}. With the aid of Theorem 12.4.4, it is easy to see that Z2,a ∩ pr−1 Ta is the only Ta -projectable maximal class of (2) Ta . From that and from the above remarks, (3) follows. By Theorem 12.3.4, the statements (4) and (5) are consequences from (1)–(3).

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

363

We come now to the M -projectable subclasses A of P3,2 . Because of h2 ∈ M , Theorem 12.3.1, and Lemma 12.3.5 one can ﬁnd some 1 , ..., m ⊆ E32 \E22 and an A with pr A = P2 for such class A so that & %m 0 0 1 P ol ∪ i ∩ A (12.17) A= 0 1 1 i=1

holds. Because of Theorem 12.3.6, (6) must be valid for the relations 1 , ..., m in addition: 2 1 (12.18) ⊆ 1 ∪ 2 ∪ ... ∪ m . 0 2 We call relations 1 , ..., m with the above property (12.18) M -permissible. The following lemma gives some properties of M -projectable subclasses of P3,2 , which we need to prove Theorem 12.6.3. Lemma 12.6.2 Let be the relations 1 , ..., m ⊆ E32 \E22 M -permissible and let a ∈ E2 . Then: 0 0 1 0 0 1 (1) 1 ⊆ 2 =⇒ P ol ∪ 2 ⊆ P ol ∪ 1 ; 0 1 1 0 1 1 m a 2 0 0 1 (2) ⊆ 1 ∪ ... ∪ m =⇒ i=1 P ol ∪ i ⊆ pr−1 M ∩ Z2,a ; 2 a 0 1 1 0 0 1 1 0 0 1 0 1 (3) P ol = P ol ; 0 1 1 2 2 0 1 1 2 0 0 1 0 2 0 0 1 0 0 0 1 2 (4) P ol = P ol ∩ P ol ; 0 1 1 1 0 1 1 2 1 0 1 1 2 0 0 1 1 2 0 0 1 0 1 2 (5) P ol = P ol ; 0 1 1 2 2 0 1 1 2 2 2 0 0 1 2 2 0 0 1 2 0 0 1 2 (6) P ol = P ol ∩ P ol ; 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 2 2 0 0 1 2 2 2 (7) P ol = P ol ; 0 1 1 0 2 0 1 1 0 1 2 0 0 1 0 2 2 0 0 1 0 2 0 0 1 2 2 (8) P ol = P ol ∩ P ol . 0 1 1 2 1 2 0 1 1 2 2 0 1 1 1 2 Proof. The statement (1) is trivial. Since ⎧ 0 0 1 0 2 ⎪ ⎪ ⎨ 0 1 1 2 0 0 0 1 2 0 0 1 a ◦ =: = 0 1 1 a 0 1 1 2 ⎪ 0 0 1 1 2 ⎪ ⎩ 0 1 1 2 1 0 1 0 1 2 and ( ∩ τ ) ◦ = , we have 0 1 0 1 a 0 0 1 2 0 0 1 a P ol ∩ ⊆ pr−1 M 0 1 1 a 0 1 1 2

2 2 2 2

2 if a = 0, 1 0 if a = 1 2

∩ Z2,a .

364

12 Subclasses of Pk,2

With the aid of (1), statement (2) results. (3) follows from (1) and 0 0 1 0 0 1 1 0 0 1 0 1 ◦ = . 0 1 1 0 1 1 2 0 1 1 2 2 Because of (1) and 0 0 1 2 0 0 1 0 0 0 1 2 0 ◦ = 0 1 1 1 0 1 1 2 0 1 1 1 2 (4) holds. (5) follows from (1) and 0 0 1 1 2 0 0 1 1 2 0 0 1 0 1 2 ◦ = . 0 1 1 2 2 0 1 1 2 2 0 1 1 2 2 2 The classes from (6) 1, 1 → 0, 2 → 2). The last statement of 0 0 1 0 0 1 1 2

or (7) are isomorphic to those from (3) or (4) (0 → the lemma follows from (1) and 2 0 0 1 2 2 0 0 1 0 2 2 ◦ = . 2 0 1 1 1 2 0 1 1 2 1 2

Theorem 12.6.3 The M -projectable subclasses of P3,2 are exactly the following: (1) pr−1 M , (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

0 0 1 0 M := P ol , 0 1 1 2 0 0 1 2 M (22) := P ol , 0 1 1 2 0 0 1 2 M (21) := P ol , 0 1 1 1 0 0 1 1 M (12) := P ol , 0 1 1 2 M (02) ∩ M (22) , M (02) ∩ M (21) , M (21) ∩ M (22), 0 0 1 2 (20) M := P ol , 0 1 1 0 M (12) ∩ M (22) , 0 0 1 0 2 M (02)(22) := P ol , 0 1 1 2 2 M (02) ∩ M (22) ∩ M (21) , (02)

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

(13) M (21)(22) := P ol (14) (15) (16) (17) (18) (19) (20) (21) (22) (23)

0 0 1 2 2 0 1 1 1 2

M (20) ∩ M (22) , M (12) ∩ M (02)(22) , M (21) ∩ M (02)(22) , M (02) ∩ M (21)(22) , M (20) ∩ M (21)(22) , 0 M (02)(12)(22) := P ol 0 0 M (02)(21)(22) := P ol 0 0 M (20)(21)(22) := P ol 0 −1 pr M ∩ Z2,1 , pr−1 M ∩ Z2,0

0 1 0 1 0 1

1 1 1 1 1 1

0 2 0 2 2 0

,

1 2 2 1 2 1

2 , 2 2 , 2 2 , 2

(see Figure 12.2).

pr−1 M

r @ @

@ @ M (2,1) M r M r @r @ @ @ @ @ @ @ @ @ (2,0) @ @r M (1,2) r @r r @rM @ @ @ @ @ @ (0,2),(2,2) (2,1),(2,2) @ M M @ @r @r r r r A A A A A A A A A A A A A AAr A r AAr Ar A A A A A r r M (0,2),(2,1),(2,2) r M (2,1),(2,0),(2,2) M (0,2),(1,2),(2,2) S S S S S S SS SS r r (0,2)

(2,2)

pr−1 M ∩ Z2,1

pr−1 M ∩ Z2,0

Fig. 12.2

365

366

12 Subclasses of Pk,2

Proof. Because of Lemma 12.6.2 (1), 0 0 1 0 0 1 1 0 0 1 0 1 ◦ = 0 1 1 0 1 1 2 0 1 1 2 2

or

0 0 1 2 0 1 1 0

0 0 1 0 0 1 2 2 ◦ = 0 1 1 0 1 1 0 1

we have M (12) ⊆ M (02) or M (20) ⊆ M (21) . With the help of Table 12.2 one can prove the other inclusions (non-inclusions) of the classes (1) - (23). In Table 12.2, the sign + (or −) shows whether a function fi (i = 1, ..., 10) belongs (or does not belong) to a class of the left column of the table, respectively.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

pr−1 M M (02) M (22) M (21) M (12) M (02) ∩ M (22) M (02) ∩ M (21) M (21) ∩ M (22) M (20) M (12) ∩ M (22) M (02)(22) M (02) ∩ M (22) ∩ M (21) M (21)(22) M (20) ∩ M (22) M (12) ∩ M (02)(22) M (21) ∩ M (02)(22) M (02) ∩ M (21)(22) M (20) ∩ M (21)(22) M (02)(12)(22) M (02)(21)(22) M (20)(21)(22) pr−1 M ∩ Z21 pr−1 M ∩ Z20

j1 + + + + − + + + + − + + + + − + + + − + + − +

Table 12.2 j0 j2 k1 f1 + + + + + + − + + + + − + − + − + + − + + + − − + − − − + − + − − − + − + + − − + + − − + − − − + − + − − − + − + + − − + − − − + − − − − − + − + + − − + − − − − − + − + − − − − − − −

f2 + − − + − − − − + − − − − − − − − − − − − − −

f3 + + + − + + − − − + − − − − − − − − − − − − −

f4 + + − + − − + − − − − − − − − − − − − − − − −

f5 + + + + − + + + − − + + − − − + − − − − − − −

f6 + + + + + + + + − + − + + − − − + − − − − − −

f7 + + + + − + + + + − − + − + − − − − − − − − −

f8 + + + + − + + + − − − + + − − − + − − − − − −

The functions f1 –f10 from the above table are deﬁned as follows: f1 (x1 , x2 ) := j0 (x1 )j2 (x2 ), f2 (x1 , x2 ) := j1 (x1 )j2 (x2 ), f3 (x1 , x2 ) := k1 (x1 )j2 (x2 ), f4 (x1 , x2 ) := j0 (x1 )j2 (x2 ) ∨ j1 (x2 ),

f9 + + + − + + − − − + + − − − + − − − − − − − −

f10 + − + + − − − + + − − − + + − − − + − − − − −

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

367

f5 (x1 , x2 ) := j1 (x1 )j1 (x2 ) ∨ j2 (x1 )j2 (x2 ), f6 (x1 , x2 ) := k1 (x1 )j0 (x2 ) ∨ j2 (x1 )j1 (x2 ), f7 (x1 , x2 ) := k1 (x1 )k1 (x2 ) ∨ j2 (x1 )j2 (x2 ), f8 (x1 , x2 ) := k1 (x1 )j2 (x2 ) ∨ j1 (x2 ), f9 (x1 , x2 ) := j1 (x1 )j2 (x2 ), f10 (x1 , x2 ) := j0 (x1 )j2 (x2 ).

Thus it remains to show that no further classes are described by (12.17) than the ones listed above. By Theorem 12.6.1, only the sets P3,2 , Z2,0 , and Z2,1 are possible for A in (12.17). Since pr−1 M ∩ Z2,0 and pr−1 M ∩ Z2,1 are minimal M -projectable classes of P3,2 , the M -projectable classes diﬀerent from these classes are described as follows m 0 0 1 P ol ∪ i , (12.19) 0 1 1 i=1

where the relations 1 , ..., m are M -permissible subsets of E32 \E22 . With the aid of the statement (2) of Lemma 12.6.2 and considering (12.18), one can be convinced that the relations i , i = 1, ..., m, must be from the set

0 1 2 2 2 0 1 0 2 0 2 1 2 , , , , , , , , , 1 2 21 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 0 1 2 0 2 2 2 2 2 , , , , , . 0 1 0 2 1 2 2 2 2 2 1 2 0 1 2

Because of Lemma 12.6.2, (3)–(8) we can assume that the relations 1 , ..., m belong to

0 2

1 2 2 2 0 2 2 2 0 1 2 2 2 2 , , , , , , , , . 2 0 1 2 2 2 1 2 2 2 2 0 1 2

Now it is not diﬃcult to ﬁnd all classes, which are describable through (12.19), in a step-by-step way. Because of Lemma 12.6.2, (1) or Theorem 12.5.6, the largest M -projectable classes lying below pr−1 M are the sets M (02) , M (22) and M (21) . Then (with the help of Lemma 12.6.2, (2)) possible intersections of these sets and the next largest classes of the form (12.19) with m = 1 belong to layers 1–4 in Figure 12.2. As intersections of classes of the fourth layer (i.e., as next classes of the form (3) with m = 1), only the classes of the ﬁfth layer are possible, and so forth. Theorem 12.6.4 Let a ∈ E2 . Then N3 (M ∩ Ta ) = {A ∩ pr−1 M ∩ Ta | A ∈ N3 (M )}∪ (2) ∪{A ∩ Ta | A ∈ N3 (M ) ∧ A ⊆ M (12) ∧ A ⊆ M (20) } and |N3 (M ∩ Ta )| = 36.

368

12 Subclasses of Pk,2

Proof. W.l.o.g. let a = 0. Put A3 := M ∩ T0 . It is easy to check that the 36 sets given in the theorem are all A3 -projectable and pairwise diﬀerent (see Figure 12.3). By Theorem 12.3.4, for every class A of N3 (A3 ) there are certain sets A1 , A2 with A = A1 ∩ A2 , A1 ∈ N3 (M ), and A2 ∈ N3 (T0 ). Because of Theorem 12.3.6, (3) the sets of N3 (A3 ) diﬀerent from Z2,0 ∩ pr−1 A3 and Z2,1 ∩ pr−1 A3 belong to the set (2)

{A ∩ pr−1 A3 , A ∩ T0

| A ∈ N3 (M )\{Z2,0 ∩ pr−1 M, Z2,1 ∩ pr−1 M }}

(2)

(20)

(2)

Since pr(T0 ∩ M (12) ) = M ∩ T0 and pr(M0 ∩ pr−1 (M ∩ T0 )) ⊆ T0 is obviously valid, the set N3 (A3 ) agrees with the set given in Theorem 12.6.4, and it holds |N3 (A3 )| = 36 by Theorem 12.6.3.

Theorem 12.6.5 Exactly 49 subclasses of P3,2 are M ∩ T0 ∩ T1 -projectable, and it holds N3 (M ∩ T0 ∩ T1 ) = {A ∩ pr−1 (M ∩ T0 ∩ T1 ) | A ∈ N3 (M )}∪ (2)

{A ∩ Ta

| a ∈ E2 ∧ A ∈ N3 (M ) ∧ A ⊆ M (12) ∧ A ⊆ M (20) }

(see Figure 12.4, where A4 := M ∩ T0 ∩ T1 ). Proof. Analogous to the considerations in the proof of Theorem 12.6.4 the statements from Theorem 12.6.5 are a consequence of Theorems 12.3.4, 12.6.1, and 12.6.3, where one must consider that (2)

∩ M (12) ∩ pr−1 T1 ) = M ∩ T0 ∩ T1

(2)

∩ M (20) ∩ pr−1 T0 ) = M ∩ T0 ∩ T1 .

pr(T0 and

pr(T1

Fig. 12.3

pr−1 A3 ∩ Z2,0

PP @ PP PP @ PP PP @ (2) PP −1 −1 (0,2) −1 (2,2)@ −1 (2,1) pr A3 ∩ M pr A ∩ M pr A 3 3 ∩M PPrpr A3 ∩ T0 @P r r r P P PP @PPP @PPP @ PP @ PP PP@ PP @ PP PP P PP @ PP @ PP @ PP@ P P −1 (1,2) P P @ @ pr A3 ∩ M PP PPr PPr @P @P @r r rP r r PP PP @ @PPP @ @ P P @ @ PP PP@ PP @ PP PP PP @ PP PP @ PP @ @ −1 (0,2),(2,2) −1 (2,1),(2,2) @ @ P P P pr A3 ∩ M PP @ PP @ pr A3 ∩ M PP Pr @r @P P P @r @rpr−1 A3 ∩ M (2,0) r r r r P P A @ A PPP A PPP PPP @ A PP PP@ A @ A P PPP PP P P PP PP @ PP A A @ A PP @ PP PP A A @ A P PP PPr P @ Ar @r A r r r r Aaa @ @ a A @ @ aa A aa @ @ aa A @ @ a a A rP pr−1 A3 ∩ M (0,2),(2,1),(2,2) @r r @r ar pr−1 A3 ∩ M (0,2),(1,2),(2,2) S @ PPP PP @ S PP S PP @ PP @ S PP @r S r rpr−1 A3 ∩ M (2,0),(2,1),(2,2) @ pr−1 A3 ∩ Z2,1 @ @ @ @r

pr−1r A3

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S 369

−1

pr −1 A4 ∩ Z2,1

Fig. 12.4

pr −1 A4 ∩ Z2,0

pr r A4 P PP @ PP PP @ PP −1 @ −1 (2) (2) (0,2) −1 (2,2) pr −1 A4 ∩ T pr −1 A4 ∩ T0 pr A4 ∩ M pr A4 ∩ M pr A4 ∩ M (2,1) P PP 1 @ r r r r r PP PP PP @ @ @ @ PP PPP PPP @ P@ PP PP PP @ @ @ PPP PPP PPP@ @ @ PP PPr PPr @r r @r @r r r @ r P PP PPP @ P P @ P @ @ @ @ P P P PP P PP @ @ @ @ PP @ PP PPP @ @ @ PP@ PP@ P @ @ −1 (0,2),(2,2) −1 (2,1),(2,2) P ∩M ∩M PPr PP PP @rpr−1 A4 ∩ M (2,0) @rpr A4 P r r @r @r r @ r @ rpr A4 P pr −1 A4 ∩ M (1,2) r P P P P APP @ PP@ APP @ A P @ A PP PPP P@ @ PP A @ A PPP PPP@ PPP @ PPr PP PPr A A @r ra r r @r r @ r r ! ! @ A!a @ @ !a aa @ @ @ !! A aa ! @ A @ @ ! a aar @r A @r!! @r r r r P PP (0,2),(2,1),(2,2) @ @ pr −1 A 4 ∩M P PP @ PP @ @ PP@ PP rpr−1 A4 ∩ M (2,0),(2,1),(2,2) @ @r r pr −1 A4 ∩ M (0,2),(1,2),(2,2) r @ @ @ @ @ @ @r @r

370 12 Subclasses of Pk,2

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

371

Now we come to the determining of sets N3 (A) with A ∈ {L, L ∩ T0 , L ∩ T1 , L ∩ S, L ∩ T0 ∩ S}. As already shown in Section 12.5 while preparing proof of Theorem 12.5.8, there are some a, ai , aI,J of E2 with f (x) = a +

n i=1

ai j1 (xi ) +

aI,J · kI,J (x)

(12.20)

I, J I ∪ J ⊆ {1, ..., n} I ∩J =∅ J = ∅

for every function f n ∈ pr−1 L. The functions

kI,J (x) := ( j1 (xi )) · ( j2 (xj )) i∈I

j∈J

with aI,J = 1 in (12.20) are called components of f . Let Kf bethe set of all components of f . Further, let K := f ∈pr−1 L Kf and KM := f ∈M Kf , M ⊆ P3,2 . It is easy to check that the subsequently deﬁned subsets of pr−1 L are closed and are ﬁnitely generated by the given subsets: ⎛ ⎞ 0 0 0 1 1 0 1 1 2 ⎜0 0 1 1 0 1 0 1 2⎟ ⎟ L2 := P ol ⎜ ⎝0 1 0 0 1 1 0 1 2⎠ 0 1 1 0 0 0 1 1 2 = {f ∈ pr−1 L | Kf ⊆ {kI,J ∈ K | |I| ≤ 1}} = [j1 (x1 ) + j1 (x2 ), c1 , j1 (x1 ) · j2 (x2 )], L2,r := {f ∈ pr−1 L | Kf ⊆ {kI,J ∈ K | I = ∅ ∧ |J| ≤ 4}} if r < ∞, [j1 (x1 ) + j1 (x2 ), c1 , j2 (x1 ) · ... · j2 (xr )] ' ( = {j1 (x1 ) + j1 (x2 ), c1 } ∪ q≥1 j2 (x1 ) · ... · j2 (xq )} if r = ∞ (1 ≤ r ≤ ∞), Z2,0 ∩ pr−1 L n = n≥1 {f n ∈ P3,2 | ∃a, a1 , ..., an ∈ E2 : f (x) = a + i=1 ai · j1 (xi )}, = [j1 (x1 ) + j1 (x2 ), c1 ], Z2,1 ∩ pr−1 L = n≥1 {f n ∈ P3,2 | ∃a, a1 , ..., an ∈ E2 : n f (x) = a + i=1 ai · (j1 (xi ) + j2 (xi ))}, = [j0 (x1 ) + j0 (x2 ), c1 ].

372

12 Subclasses of Pk,2

Further put K := {kI,J ∈ K | J = ∅ ∧ I ∩ J = ∅}, K1 := {kI,J ∈ K | |I| ≥ 1}, K2 := {kI,J ∈ K | |I| ≤ 1}, K3 := K1 ∩ K2 = {kI,J ∈ K | |I| = 1}, K0,r := {k∅,J ∈ K | |J| ≤ r},

1 ≤ r ≤ ∞.

Lemma 12.6.6 Let f n ∈ P3,2 , pr f n = cn0 , f n = cn0 and Kf ⊆ K0,∞ . Further, let r be the smallest number for which there is a function k∅,J ∈ Kf with |J| = r. Then k∅,{1,...,r} (x) = j2 (x1 ) · ... · j2 (xr ) ∈ [f, j1 ]. Proof. By assumption, in Kf there is a function k∅,J with J ⊆ J for all k∅,J ∈ Kf \{k∅,J }. Consequently, we have f (h1 (x1 ), ..., hn (xn )) = k∅,J (x),

if hi (x) :=

j1 (x) if i ∈ J, x if i ∈ J

and i = 1, ..., n. Lemma 12.6.7 Let f n ∈ P3,2 , pr f n = cn0 and Kf ⊆ K0,∞ . Then there is an a ∈ E2 with (1) j1 (x1 )j2 (x2 ) + a · j2 (x2 ) ∈ [f, j1 ]; (2) j1 (x1 )j1 (x2 )j2 (x3 ) + a · j2 (x3 ) ∈ [f, j1 ], if furthermore Kf ∩ (K1 \K2 ) = ∅ holds. Proof. First we prove (1) through induction on the arity n of f . n = 2: Since Kf ⊆ K0,∞ we can assume w.l.o.g. f 2 (x1 , x2 ) = j1 (x1 )j2 (x2 ) + α · j2 (x1 )j1 (x2 ) + β · j2 (x1 ) + γ · j2 (x2 )+ (α, β, γ, δ ∈ E2 ). +δ · j2 (x1 )j2 (x2 ) Thus we have f (j1 (x1 ), x2 ) = j1 (x1 )j2 (x2 ) + γ · j2 (x2 ), i.e., (1) is right for n = 2. n − 1 −→ n: Suppose (1) holds for all (n − 1)-ary functions f with pr f = cn−1 0 and Kf ⊆ K0,∞ , n > 2. Now let f be an arbitrary n-ary function with pr f = cn0 and Kf ⊆ K0,∞ . Then there are (n − 1)-ary functions g, h and q with (w.l.o.g.) f (x) = j1 (x1 ) · g(x2 , ..., xn ) + j2 (x1 ) · h(x2 , ..., xn ) + q(x2 , ..., xn ),

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

373

where g = cn−1 . Consequently, 0 f (j1 (x1 ), x2 , ..., xn ) = j1 (x1 ) · g(x2 , ..., xn ) + q(x2 , ..., xn ) =: f (x). Case 1: Kg ⊆ K0,∞ . With the aid of the proof of Lemma 12.6.6, it is easy to see that j1 (x1 )j2 (x2 )+ a · j2 (x2 ) is a superposition over {f , j1 } for certain a ∈ E2 . Case 2: Kg ⊆ K0,∞ . By induction assumption, the function j1 (x2 )j2 (x3 ) + b · j2 (x3 ) is a superposition on g and j1 for certain b ∈ E2 . Thus we can construct the function f (x1 , x2 , x3 ) := j1 (x1 )(j1 (x2 )j2 (x3 ) + b · j2 (x3 )) + q (x2 , x3 ), where q denotes a c0 -projectable function, as a superposition over f and j1 (under the given conditions). For b = 1 we have f (x1 , x2 , x2 ) = j1 (x1 )j2 (x2 ) + a · j2 (x2 ),

a := q (2, 2).

If b = 0 then we construct the function f (x1 , x2 , x3 ) := f (x1 , j1 (x2 ), x3 ) = j1 (x1 )j1 (x2 )j2 (x3 ) + α · j1 (x2 )j2 (x3 ) + β · j2 (x3 ) ﬁrst. Consequently, f (x2 , x1 , x2 ) = j1 (x1 )j2 (x2 ) + β · j2 (x2 ) if α = 1, and

f (x1 , x1 , x2 ) = j1 (x1 )j2 (x2 ) + β · j2 (x2 )

in the case α = 0. Hence (1) holds for each n-ary function f ∈ P3,2 with pr f = cn0 and Kf ⊆ K0,∞ . Proof for (2) is obtained through the transfer of proof for (1). Lemma 12.6.8 Let B ∈ {L, L ∩ T0 , L ∩ S, L ∩ T0 ∩ S}, A ⊆ Z2,0 ∩ pr−1 B and [pr A] = B. Then (1) (2) (3) (4) (5) (6) (7)

pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 ), j1 (x) + j2 (x1 )}], L2 ∩ pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j2 (x2 ), j1 (x) + j2 (x1 )}], (2) T0 ∩ pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 )}], (2) T0 ∩ L2 ∩ pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j2 (x2 )}], L2,r ∩ pr−1 B = [A ∪ {j 1 (x) + j2 (x1 )j2 (x2 )...j2 (xr )}], r ≥ 1, L2,∞ ∩ pr−1 B = [A ∪ q≥1 {j1 (x) + j2 (x1 )j2 (x2 )...j2 (xq )}], Z2,0 ∩ pr−1 B = [A].

Proof. The proof results from deﬁnitions of the considered classes.

374

12 Subclasses of Pk,2

Theorem 12.6.9 Let B ∈ {L, L ∩ T0 , L ∩ S, L ∩ T0 ∩ S}. Then the following B-projectable classes (⊆ P3,2 ) only exist: (1) (2) (3) (4) (5) (6) (7)

pr−1 B, L2 ∩ pr−1 B, L2,r ∩ pr−1 B, r = 1, 2, ..., L2,∞ ∩ pr−1 B, Z2,a ∩ pr−1 B, a ∈ E2 , (2) Ta ∩ pr−1 B, a ∈ E2 , if B ∩ Ta = B, (2) Ta ∩ L2 ∩ pr−1 B, a ∈ E2 , if B ∩ Ta = B

(see Figure 12.5). q

pr −1 L

qL2 qL2,∞ q q q qL2,r q q q qL2,2

q

pr −1 L ∩ Z2,0

qL2,1 @ @ @q pr −1 L ∩ Z

2,1

Fig. 12.5

Proof. Let A be a B-projectable subclass of P3,2 . Then, j1 or j1 + j2 belongs to A. W.l.o.g. we can assume j1 ∈ A, since s j1 s = j1 + j2 holds for s(x) := 2x+1 (mod 3) and thus all considerations are in the case j1 ∈ A isomorphically to those of the case j1 + j2 ∈ A. Consequently, we have Z2,0 ∩ pr−1 B ⊆ A and j1 (x1 ) + j1 (x 2 ) + j1 (x3 ) ∈ A. For KA (= f ∈A Kf ) the following cases are possible: Case 1: KA ⊆ K0,∞ (i.e., A ⊆ L2,∞ ). Case 1.1: KA = ∅. In this case we have: A = Z2,0 ∩ pr−1 B. Case 1.2: It exists an r ≥ 1 with KA ⊆ K0,r−1 and KA ⊆ K0,r . Then, there is a function f n ∈ A with Kf ∩ K0,r = ∅. Consequently, the function f1 (x, x) := j1 (x) + f (j1 (x1 ), ..., j1 (xn )) + f (x) = j1 (x) + f1 (x) with

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

f1 (x) :=

375

k∅,J (x).

k∅,J ∈Kf

belongs to A. By Lemma 12.6.6, one can construct the function j2 (x1 )...j2 (xt ), where t := min |J|, k∅,J ∈Kf

f1

and j1 . Consequently, j1 (x) + j2 (x1 )....j2 (xt ) ∈ A as a superposition over for t ≤ t. A superposition over these functions and f1 is also f2 (x, x) := j1 (x) + f1 (x) + k∅,J ∈ Kf k∅,J (x) |J| ≤ t

= j1 (x) +

f2 (x)

with f2 (x) :=

k∅,J ∈ Kf k∅,J (x). |J| > t

If f2 = cn0 , then we have the function j2 (x1 )...j2 (xs ), where s :=

min

k∅,J ∈Kf , |J|>t

|J|,

by substituting certain variables by j1 and by possibly changing the numbering of the variables from f2 . Consequently, the function k∅,J (x) f3 (x, x) := j1 (x) + k∅,J ∈ Kf |J| > s

belongs to A, etc. By iterated application of these constructions we obtain the function j1 (x) + j2 (x1 )...j2 (xr ) ∈ A. Hence, by Lemma 12.6.8, (5), we have A = L2,r ∩ pr−1 B. Case 1.3: KA ∩ K0,r = ∅ for all r ≥ 1. Because of the considerations from Case 1.2 and Lemma 12.6.8, (6) is valid: A = L2,∞ ∩−1 B. A = L2,∞ ∩−1 B. Case 2: KA ⊆ K2,∞ . In this case, there is a function f n ∈ A with Kf ∩ K1 = ∅. Then, the function f1 (x, x) := j1 (x) + f (x) + f (j1 (x1 ), ..., j1 (xn )) = j1 (x) + f1 (x) with

f1 (x) := j1 (x) +

kI,J (x)

kI,J ∈Kf

belongs to A. With the aid of Lemma 12.6.6, it is easy to see that the function

376

12 Subclasses of Pk,2

f1 (x1 , x2 ) := j1 (x1 )j2 (x2 ) + a · j2 (x2 ), a ∈ E2 is a superposition over {j1 , f1 }. Therefore, the function g(x, x1 , x2 ) := j1 (x) + j1 (x1 )j2 (x2 ) + a · j2 (x2 ) is a superposition on {f1 , j1 }, i.e., g ∈ A. Consequently, g(g(x, x1 , x2 ), x2 , x2 ) = j1 (x) + j1 (x1 )j2 (x2 ) (2)

belongs to A. Then, by Lemma 12.6.8, we have T0 ∩ L2 ∩ pr−1 B ⊆ A. If (2) A = T0 ∩ L2 ∩ pr−1 B then the following three cases are possible: Case 2.1: A ⊆ L2 and pr A ⊆ T0 . In this case, there is a function p ∈ A with p(0) = 1. For (pr p)(y) = y it holds j1 (x) + j1 (x1 )j2 (x1 ) = j1 (x) + j2 (x1 ) ∈ A. If pr p = c1 then we also obtain j1 (x) + j2 (x1 ) = j1 (x) + p(p(x1 ))j2 (x1 ) ∈ A. Consequently, by Lemma 12.6.8, A = L2 ∩ pr−1 B. (2)

Case 2.2: A ⊆ L2 , pr A ⊆ T0 and pr A ⊆ T0 . In this case, there is a function r2 ∈ A with r(0, 0) = 0 and r(0, 2) = 1, and it holds that j1 (x) + r(j1 (x1 ), x1 ) + r(j1 (x1 ), j1 (x1 )) = j1 (x) + j2 (x1 ) ∈ A. Therefore, by Lemma 12.6.8, A = L2 ∩ pr−1 B. Case 2.3: A ⊆ L2 . Denote q an n-ary function of A, which does not belong to L2 . Then, q1 (x, x) := j1 (x) + q(x) + q(j1 (x1 ), ..., j1 (xn )) = j1 (x) + q1 (x), where

q1 (x) :=

kI,j (x),

kI,J ∈Kq

is a function of A. By Lemma 12.6.6, a function of the form j1 (x1 )j1 (x2 )j2 (x3 ) + a · j2 (x3 ) is a superposition over {q1 , j1 }. Thus, q2 (x, x1 , x2 , x3 ) := j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 ) + a · j2 (x3 ) ∈ A and therefore q2 (q2 (x, x1 , x2 , x3 ), x3 , x3 ) := j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 ) ∈ A.

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S (2)

377

(2)

Thus by Lemma 12.6.8, we have T0 ∩ pr−1 B ⊆ A. If a = T0 ∩ pr−1 B, then there is either a function p ∈ A with p(0) = 1, or a function r ∈ A with r(0, 0) = 0 and r(0, 2) = 1. As already shown in the above Case 2.1 or 2.2, j1 (x) + j2 (x1 ) ∈ A results. Thus by Lemma 12.6.8, (1), A = pr−1 B holds. Next we study the set N3 (T0,m ) with m ∈ N\{1}. By Theorem 12.3.1 (or 12.3.2), for every m ∈ N\{1}, we know that N3 (T0,m ) is a ﬁnite set. In the following, we determine the elements of N3 (T0,m ) and we show that |N3 (T0,2 )| = 148 holds. Since the description of the elements of the other sets N3 (A) with S ∩ M ⊆ A or Ta,m ∩ Ta ∩ M ⊆ A requires a lot of place, we must renounce this description here and refer the reader to the dissertation by N. Gr¨ unwald (see [Gr¨ u 84]). In the dissertation by N. Gr¨ unwald, one also ﬁnds the proofs left out in the following. Lemma 12.6.10 For every set P olP3,2 (E2m \{1} ∪ j )) T := A ∩ ( j∈J (2)

with A ∈ {T0 , P3,2 }, j ⊆ E3m \E2m , and every set B := {g ∈ K | g has the value 1 on at most m tuples} ∪ {g1m+1 } with pr g1m+1 = hm (∈ P2 ; see Chapter 3) it holds [B] = T . Proof. Let f n ∈ T and let a1 , ..., ar be the tuples of E3n , on which the function f n has the value 1. By induction on r, we prove f n ∈ [B]: For r ≤ m the assertion is obviously right. For r > m we assume that all functions of T , which have the value 1 on less than r tuples, belong to [B]. By deﬁnition of T , the functions 0 if x = ai , n fi (x) := f n (x) otherwise (i = 1, ..., m + 1) belong to T . Thus, by assumption, we have f1 , ..., fn ∈ [B]. Then our assertion follows from f n (x) = g1 (f1 (x), f2 (x), ..., fm+1 (x)) ∈ [B]. Lemma 12.6.11 Let W1,m := P olP3,2 (E3m \{1}), W2,m := P olP3,2 (E3m \{1, 2}m ). Then (a) For arbitrary sets Ai ⊆ Wi,m (i ∈ {1, 2}) with [pr Ai ] = T0,m the set B1 := A1 ∪ {j1 (x1 ) · j2 (x2 ), j1 (x), j1 (x1 ) · j0 (x2 )}

378

12 Subclasses of Pk,2

is a generating system for W1,m and B2 := A2 ∪ {j1 (x1 ) · j2 (x2 ), j1 (x), j1 (x1 ) · j0 (x2 )} is a generating system for W2,m . (b) The set pr−1 T0,2 ∩ Z2,0 is maximal in W1,m . (c) The set pr−1 T0,2 ∩ Z2,1 is maximal in W2,m . (d) For all A ∈ N3 (T0,m )\{pr−1 T0,2 ∩Z2,0 , pr−1 T0,2 ∩Z2,1 } it holds W1,n ⊆ A. Proof. See [Gr¨ u 84]. Lemma 12.6.12 Let A be a set of the following form P ol ((E2m \{1}) ∪ i ) mit i ⊆ (E3m \(E2m ∪ {2})), m ≥ 2.

(12.21)

i∈I

Then (1) A is T0,m -projectable and the only maximal classes of A that belong to N3 (T0,m ) are the following classes: (2) (a) A ∩ T0 ; (b) A ∩ P ol ((E2m \{1} ∪ δ), where δ fulﬁlls the following conditions: δ ⊆ (E3m \(E2m ∪ {2}, ∃i ∈ I : δ ⊆ i , ∀ε ⊂ δ∃i ∈ I : εi . (2) If B ⊆ A, [pr B] = T0,m , B ⊆ Z2,i ∩ pr−1 T0,m , i = 0, 1, and if B is not contained in the sets of the above type (a) or (b), then [B] = A holds. (3) If special A = W2,m , then Z2,1 ∩ pr−1 T0,m is the only T0,m -projectable maximal class of W2,m . Proof. See [Gr¨ u 84]. Lemma 12.6.13 Denote A a set of the form (2) P ol ((E2m \{1}) ∪ i ) mit i ⊆ (E3m \E2m , m ≥ 2. T0 ∩

(12.22)

i∈I

Then: (1) A is T0,m -projectable, and the only maximal classes of A, which belong to N3 (T0,m ), are classes of the following form: A ∩ P ol ((E2m \{1} ∪ δ), where δ fulﬁlls the following conditions: δ ⊆ E3m \E2m , ∃i ∈ I : δ ⊆ i , ∀ε ⊂ δ∃i ∈ I : εi .

(12.23)

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

379

(2) If B ⊆ A, [pr B] = T0,m , B ⊆ Z2,i ∩ pr−1 T0,m , i = 0, 1, and B ⊆ C for every class C that was described in (1), then it holds [B] = A. (3) If in particular A = W1,m then Z2,0 ∩pr−1 T0,m is the only T0,m -projectable maximal class of W1,m . Proof. See [Gr¨ u 84]. Theorem 12.6.14 ([Gr¨ u 84]) There are only ﬁnitely many subclasses of P3,2 , which are T0,m -projectable. One receives a concrete description of these subclasses with the aid of Lemmas 12.6.12 and 12.6.13. Proof. See [Gr¨ u 84]. In the following table are explanations to notations that we need to describe the T0,2 -projectable subclasses of P3,2 . Table 12.3

i 1 2 3 4 5 6 7 8 9

τi 1 2 0 2 2 2 1 2 2 1 1 2 2 2 1 0 2 2 1 2 2 0 2 2 0 2 2 0 0 2

i 10 11 12 13 14 15 16 17 18 19

1 2 1 2 2 1 2 1 2 1 0 2 1 2 2 1 1 2 2 1 2 2 1 0 1 2 0 2 1 2

2 1 2 1 2 0 0 2 2 0 2 0 0 2 0 2 0 2 2 0

τi

0 2 2 2 2 2 2 2 0 2 2 2 2 0 2 2 2 2 2 2

Theorem 12.6.15 ([Gr¨ u 83;a;b]) 0 0 1 ∪ τi for i = 1, 2, ..., 19. Let τ0 := {0, 2} and τi := 0 1 0

380

12 Subclasses of Pk,2

There are exactly 148 T0,2 -projectable subclasses of P3,2 that can be described as follows: 1): pr−1 T0,2 , 2)–5): P ol τi with i ∈ {0, 1, 2, 3}, 6)–10): P ol {τi , τj } with (i, j) ∈ {(1, 3), (1, 2), (2, 3), (1, 0), (2, 0)}; 11)–12): P ol {τi , τj , τk } with (i, j, k) ∈ {(1, 2, 3), (1, 2, 0)}; 13)–18): P ol τi with i ∈ {4, 5, 6, 7, 8, 9}; 19)–30): P ol {τi , τj } with (i, j) ∈ {(4, 3), (4, 2), (4, 0), (5, 2), (6, 3), (6, 0), (7, 3), (7, 0), (8, 1), (9, 0), (9, 1), (9, 3)}; 31)–34): P ol {τi , τj , τk } with (i, j, k) ∈ {(4, 2, 3), (4, 2, 0), (9, 1, 3), (9, 1, 0)}; 35)–49): P ol {τi , τj } with (i, j) ∈ {(4, 5), ((4, 6), (4, 7), (4, 8), (4, 9), (5, 6), (5, 7), (5, 8), (5, 9), ((6, 7), (6, 8), (6, 9), (7, 8), (7, 9), (8, 9)}; 50)–63): P ol {τi , τj , τk } with (i, j, k) ∈ {(4, 5, 2), (4, 6, 0), (4, 7, 0), (4, 6, 3), (4, 7, 3), (4, 9, 0), (4, 9, 3), (6, 7, 0), (6, 7, 3), (6, 9, 4), (6, 9, 3), (7, 9, 0), (7, 9, 3), (8, 9, 1)}; 64)–69): P ol τi with i ∈ {10, 11, 12, 13, 14, 15}; 70)–75): P ol {τi , τj } with (i, j) ∈ {(10, 0), (10, 3), (11, 2), (14, 0), (14, 3), (15, 1)}; 76)–95): P ol {τi , τj } with (i, j) ∈ {(10, 5), (10, 8), (10, 9), (14, 0), (11, 6), (11, 7), (11, 9), (12, 4), (12, 7), (12, 9), (13, 4), (13, 7), (13, 9), (14, 4), (14, 5), (14, 8), (15, 4), (15, 5), (15, 7)}; 96)–99): P ol {τi , τj , τk } with (i, j, k) ∈ {((10, 9, 0), (10, 9, 3), (14, 4, 0), (14, 4, 3)}; 100)–114): P ol {τi , τj } with (i, j) ∈ {(10, 11), (10, 12), (10, 13), (10, 14), (10, 15), (11, 12), (11, 13), (11, 14), (11, 15), (12, 13), (12, 14), (12, 15), (13, 14), (13, 15), (14, 15)}; 115)–117): P ol τi with i ∈ {16, 17, 18}; 118)–119): P ol {τi , τj } with (i, j) ∈ {(16, 0), (16, 3)}; 120)–123): P ol {τi , τj } with (i, j) ∈ {(16, 8), (16, 5), (17, 9), (18, 4)}; 124): P ol {τ16 , τ8 , τ5 }; 125)–132): P ol {τi , τj } with (i, j) ∈ {(16, 15), (16, 12), (16, 13), (16, 11), (17, 15), (17, 14), (18, 11), (18, 10)}; 133)–134): P ol {τi , τj , τk } with (i, j, k) ∈ {(16, 15, 5), (16, 11, 8)}; ∈ {(16, 15, 12), (16, 15, 14), 135)–142): P ol {τi , τj , τk } with (i, j, k) (16, 15, 11), (16, 12, 13), (16, 12, 11), (16, 13, 11), (17, 15, 14), (18, 11, 10)}; 143)–144): P ol {τi , τj } with (i, j) ∈ {(16, 17), (16, 18), (17, 18)}; 146): P ol τ19 ; 147)–148): Z2,1 ∩ pr−1 T0,2 , Z2,0 ∩ pr−1 T0,2 . Proof. The following ﬁve statements are direct consequences from Lemmas 12.6.12 and 12.6.13:

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S (02)(20)

(1) T0,2

:= P ol

0 1 0 0 2 0 0 1 2 0

381

has exactly the following T0,2 -projectable

maximal classes: (2) (02)(20) a) T0 ∩ T0,2 ; 0 1 0 1 (02)(20) b) T0,2 ∩ P ol ; 0 0 1 2 c) Z2,1 ∩ pr−1 T 0,2 . 2 0 2 1 0 1 0 and A = (2) If A := P ol ∪ i with i ⊆ 0 2 1 2 0 0 1 0 1 0 0 2 P ol , then A has exactly the following T0,2 -projectable 0 0 1 2 0 maximal classes: (2) a) T0 ∩ A; 0 1 0 b) A ∩ P ol ∪ δ , where δ fulﬁlls the conditions 0 0 1 2 0 2 1 δ⊆ 0 2 1 2 ∃i ∈ I : δ ⊆ i ∀ε ⊂ δ ∃i ∈ I : ε ⊆ i . 0 1 0 2 0 2 1 2 (2) then Z2,0 ∩ pr−1 T0,2 is the only (3) If A = T0 ∩ P ol 0 0 1 0 2 1 2 2 maximal class of A, which is T0,2 -projectable. 0 1 0 2 0 2 1 2 (2) ∪ i with i ⊆ and (4) If A := T0 ∩ P ol 0 0 1 0 2 1 2 2 0 1 0 2 0 2 1 2 A = P ol , then A has exactly the following T0,2 0 0 1 0 2 1 2 2 projectable maximal classes: 0 1 0 A ∩ P ol ∪ δ , where δ fulﬁlls the following conditions: 0 0 1

2 0 2 1 2 0 2 1 2 2 ∃i ∈ I : δ ⊆ i ∀ε ⊂ δ ∃i ∈ I : ε ⊆ i . δ⊆

(5) There is not any proper subset of Z2,i ∩ pr−1 T0,2 , i = 0, 1, whose projection agrees with T0,2 . With the aid of the above statements, our theorem can be proven as follows: 0 0 1 −1 One obtains the T0,2 -projectable maximal classes of pr T0,2 = P ol 0 1 0 by means of (2). Then, one obtains the T0,2 -projectable maximal classes of these classes by means of (1)–(4), etc.

13 Classes of Linear Functions

In this chapter, the elements of the clone Lk :=

n≥1

{f ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + n

n

ai · xi (mod k)}.

i=1

and the elements of a generalization of the set Lk are called linear functions. The lattice of the subclasses of Lk belongs to the earliest and best investigated sublattices of Lk . For the case that k is a prime number, all subclasses, which are no subsets of [L1k ], were determined by A. A. Salomaa in [Sal 64]. The results of [Sal 64] were also proven by J. Bagyinszki and J. Demetrovics in [Bag-D 82] and complemented with the remaining subclasses of [L1k ], p ∈ P. ´ Szendrei. For examMany results about linear functions were obtained by A. ple, she proven in [Sze 78] that Lk has only ﬁnitely many subclasses, if k is square-free.1 In addition, she showed in [Sze 78] that an arbitrary class has at most the order 2 (or 3), if k is square-free and k is an odd number (or an even number), respectively. One ﬁnds in [Sze 80] a short and ﬁne determination of the subclasses not contained in [L1p ] of Lp (p prime). Similarly Theorem 13.2.1 is proven here. For the case that k is not square-free one easily ﬁnd a class that does not have any ﬁnite basis (see Lemma 13.3.7). In Section 13.1, we start with properties of certain subclasses of the set Ud := n≥1 {f n ∈ Lk | ∃a0 , ..., an ∈ Ek ∃j ∈ {1, ..., n} : f (x) = a0 + aj · xj + d · i=1,i=j ai · xi }. This set is closed, if d is a divisor of k (notation: d | k). Then, with the aid of the results from the ﬁrst section new proofs are given for the theorems of [Sal 64] and [Bag-D 82]. Section 13.3 gives a survey of further results, which were ´ Szendrei and the author, to classes from linear found by A. A. Bulatov, A. functions. 1

Through that, a presumption of A. A. Salomaa from [Sal 64] was conﬁrmed.

384

13 Classes of Linear Functions

13.1 Some Properties of the Subclasses of Ud That Contain rd In this section, let d ∈ Ek be a divisor of k. Further, the ternary function rd is deﬁned by rd (x, y, z) := x + d · y − d · z (mod k). Lemma 13.1.1 For every subclass T of Ud , which contains the function rd , it holds: (a) T = [T 1 ∪ {rd }], (b) T = Lk ∩ P olk {(a, a + b) | a + bx ∈ T 1 }. Proof. (a): Let f n ∈ T be arbitrary and w.l.o.g. f (x1 , ..., xn ) = a0 + a1 x1 + d ·

n

ai xi .

i=2

Obviously, one can obtain every function of the type g(x1 , ..., xm ) := x1 + d ·

m

bi xi with b2 + ... + bm = 0 (mod k)

i=2

by identifying variables of a function of the form rd rd ... rd . In particular, the function f (x, y, x1 , ..., xn ) := x + d(−(a2 + ... + am )y +

n

ai xi )

i=1

is a superposition over rd . Consequently, by f (x1 , ..., xn ) = f (f (x1 , ..., x1 ), x1 , x2 , ..., xn ), we have f ∈ [T 1 ∪ {rd }]. (b): Because of (a), T = Lk ∩P olk G1 (T ) holds. The matrix G1 (T ) is, however, unambiguously determined by the ﬁrst two rows (x = 0, x = 1). Thus we have T = Lk ∩ pr0,1 G1 (T ), where pr0,1 G1 (T ) = {(a, a + b) | a + bx ∈ T 1 }. n Lemma 13.1.2 Let f (x1 , ..., xn ) = a0 + a1 x1 + d · i=2 ai xi and a1 ' k = a2 ' k = 1. Then, the function rd is a superposition over f . Proof. Let f1 (x, y, z) := f (x1 , x2 , x3 , ..., x3 ). Then, the functions fi (x, y, z) := fi−1 (f1 (x, z, z), y, z), i = 2, 3, ... are superpositions over f . Since a2 ' k = 1, there exists a t with at1 = 1 (mod k). Consequently, ft (x, y, z) = a + x + a2 dy + bdz for certain a, b ∈ Ek .

13.1 Some Properties of the Subclasses of Ud That Contain rd

385

The functions ft,1 := ft and ft,i (x, y, z) := ft,i−1 (ft (x, y, z), y, z), i = 2, 3, ... belong to [{f }]. Since a2 ' k = 1, there exists an s with a2 s = 1 (mod k). Consequently, ft,s (x, y, z) = as + x + dy + sbdz is valid. Next we form the superpositions ft,s,1 := ft,s and ft,s,i (x, y, z) := ft,s,i−1 (ft,s (x, y, z), y, z), i = 2, 3, ... For i = k we obtain ft,s,k (x, y, z) = x+dy +(k−1)dz, i.e., rd is a superposition over f . Subsequently, the subclasses of Lk that contain the function r1 are determined. Obviously, a subset A ⊆ L1k determines a subclass T of the form T = [T 1 ∪{r1 }] if and only if [A ∪ {r1 }]1 = A holds. Further we have: Lemma 13.1.3 (a) If [A ∪ {r1 }]1 = A for a subset A of L1k , then the binary relation A := {(a, a + b − 1) | a + bx ∈ A} is a subgroup of the direct product (Ek ; +)2 . (b) Conversely, if γ is a subgroup of (Ek ; +)2 , then [Aγ ∪ {r1 }]1 = A, where Aγ := {a + bx ∈ L1k | (a, a + b − 1) ∈ γ}. Proof. (a): Let (a, a + b − 1), (a , a + b − 1) ∈ A be arbitrary, i.e., the functions a + bx and a + b x belong to A. Then, by assumption, r1 preserves the functions x, a + bx and a + b x of A. Consequently, r1 (a + bx, a + b x, x) = a + a + (b + b − 1)x ∈ A holds and therefore (a + a , a + a + b + b − 2) ∈ A . Thus, by deﬁnition of , we have (a, a + b − 1) + (a , a + b − 1) ∈ A for arbitrary (a, a + b − 1), (a , a + b − 1) ∈ A . Hence, A is a subgroup of (Ek ; +)2 . (b): We have to show that Aγ is closed in respect to the operation and that the function r1 preserves the functions of Aγ . Let f (x) := a + bx, g(x) := a + b x and h(x) := a + b x be arbitrary of Aγ . Then the function (f g)(x) = a + a b + bb x belongs to Aγ , since (a, a + b − 1) ∈ γ, (a , a + b − 1) ∈ γ and (a, a + b − 1) + b · (a , a + b − 1) = (a + a b, a + a b + bb − 1) ∈ γ.

386

13 Classes of Linear Functions

Further r1 (a + bx, a + b x, a + b x) = (a + a − a ) + (b + b − b )x holds. Since (a + a − a , a + a − a + b + b − b − 1) = (a, a + b − 1) + (a , a + b − 1) − (a , a + b − 1) and γ is a group, the function (a + a − a ) + (b + b − b )x belongs to Aγ . Consequently, we have [Aγ ∪ {r1 }]1 = Aγ . When we summarize Lemmas 13.1.1–13.1.3, or as a consequence of [Sze 80], Lemma 4.3, we obtain: Theorem 13.1.4 The lattice of the closed subsets of Ud (d|k), which contain the function rd , is ﬁnite for arbitrary k ∈ N and is isomorphic to the subgroup lattice of the group (Ek ; +)2 for d = 1. One can describe the subgroups of the group (Ek ; +)2 easily. The subsequently given description is a special case of Theorem 4.3.1 from [Sco 64], p. 71: Theorem 13.1.5 (a) Let γ be a subgroup of (Ek ; +)2 . Then γ1 := {a | ∃b ∈ Ek : (a, b) ∈ γ}, γ2 := {b | ∃a ∈ Ek : (a, b) ∈ γ}, N2 := {b | (0, b) ∈ γ} N1 := {a | (a, 0) ∈ γ}, are subgroups of (Ek ; +), Ni ⊆ γi (i = 1, 2) and the mapping α from the factor set γ1 /N1 onto the factor set γ2 /N2 with α(a + N1 ) = b + N2 :⇐⇒ (a, b) ∈ γ is an isomorphism from γ1 /N1 onto γ2 /N2 . (b) Conversely: Let γ1 , γ2 , N1 , N2 be subgroups of (Ek ; +) with Ni ⊆ γi (i = 1, 2) and let α be an isomorphism from γ1 /N1 onto γ2 /N2 . Then γ := {(a, b) | a ∈ γ1 ∧ b ∈ γ2 ∧ α(a + N1 ) = b + N2 } is a universe of a subgroup of (Ek ; +)2 . With the help of Theorem 13.1.5 and with well-known theorems about cyclic groups, one can easily determine the cardinality of Lk ([{r1 }], Lk ). For this purpose, let ϕ be the Euler function (i.e., ϕ(n) is the number of all q ∈ {1, 2, ..., n − 1} with n ' q = 1) and let t(q, k) be the number of all n ∈ N with n|k and q|n.

13.2 The Subclasses of Linear Functions of Pk with k ∈ P

Theorem 13.1.6 It holds: |Lk ([{r1 }], Lk )| =

387

ϕ(q) · (t(q, k))2 .

q q ∈ N, q|k

Proof. Because of Theorem 13.1.4, we have to determine only the number of the subgroups of (Ek ; +)2 to the proof. Let γ ⊆ Ek2 be a subgroup of (Ek ; +)2 . This holds (by Theorem 13.1.5) if and only if there exist subgroups γi , Ni (Ni ⊆ γi , i = 1, 2) of (Ek ; +) and an isomorphism α from γ1 /N1 onto γ2 /N2 with γ = {(a, b) | α(a+N1 ) = b+N2 }. Moreover, it holds |γ1 /N1 | = |γ2 /N2 | =: q and q|k. Since (Ek ; +) is a cyclic group, the groups γi , Ni and γi /Ni (i = 1, 2) are cyclic. Therefore, for ﬁxed q and k, there are exactly t(q, k) possibilities for the choice of γi /Ni (i ∈ {1, 2}). Further, if g is a generating element of γ1 /N1 , then the mapping α is completely determined by α(g), where α(g) is a generating element of γ2 /N2 . As is generally known, a cyclic group of the order q has exactly ϕ(q) generating elements. Thus, for determining the mapping α there are exactly ϕ(q) possibilities. Consequently, for γ with |γ1 /N1 | = |γ2 /N2 | = q there are exactly ϕ(q) · (t(q, k))2 possibilities, whereby our assertion is proven. We notice that one can prove further properties of the elements of Lk ([{r1 }], Lk ) with the aid of Lemma 13.1.1. For example, every subclass T of Lk with r1 ∈ T is ﬁnitely axiomatizable (see [McK 78]) and such a class has only ﬁnite many congruences, which are determined through the congruences on [{r1 }] and of (T 1 ; ) (see Chapter 9).

13.2 The Subclasses of Linear Functions of Pk with k ∈ P In this section, let p be an arbitrary prime number. Because of Lemma 13.1.2, every class A ⊆ Lp with A ⊆ [L1p ] contains the function r1 . Therefore, by Lemma 13.1.3 and Theorem 13.1.4, such a class A is determined by a certain subgroup of the group (Ek ; +)2 . Obviously, by Theorem 13.1.5, there are only the following p + 3 subgroups of (Ek ; +)2 : Ep2 , {(0, 0)}, {(0, x) | x ∈ Ep }, and {(x, t · x) | x ∈ Ep }, if t ∈ Ep . Consequently, by Section 13.1, we have

Theorem 13.2.1 ([Sal 64], [Bag-D 82], [Sze 80]) Lp has exactly p + 3 subclasses which are not subsets of [L1p ]:

388

13 Classes of Linear Functions

Lp Lp ∩ P olp {(0, 1)} n (= n≥1 {f n ∈ Lp | f (x1 , ..., xn ) = i=1 ai xi ∧ a1 + ... + an = 1 } = [r1 ]), Lp ∩ n P olp {(x, x + 1) | x ∈ Ep } (= n≥1 {f n ∈ Lp | f (x1 , ..., xn ) = a0 + i=1 ai xi ∧ a1 + ... + an = 1 }), Lp ∩ P olp {0} and Lp ∩ P olp {(x, tx + 1) | x ∈ Ep } (= Lp ∩ P olp {(1 − t)−1 }), where t ∈ Ep \{1}. Since one obtains an arbitrary subclass A of [L1p ] from a subsemigroup A1 of (L1p ; ) by means of [A1 ]ζ,τ,∇ , it is suﬃcient to determine the subsemigroups of (L1p ; ) for the description of the remaining subclasses of Lp . Let A be an arbitrary subsemigroup of (L1p ; ). Then A = A ∪ CA , where A is either the empty set or a subgroup of the group S := ({ax+b | a ∈ Ep \{0}}; ) and CA ⊆ {ca | a ∈ Ep } =: C. If A = ∅ then the set Ua := {a | ∃b : ax + b ∈ A } is a subgroup of the group G := (Ep \{0}; ·). Now, one can easily see that for an arbitrary subgroup U of G, G/U =: {N1 , N2 , ..., Np−1 }, α ∈ Ep and 1 I ⊆ {1, 2, ..., p−1 |U | } the following subsets of Lp are closed in respect to : Tα,U := {ax + (1 − a)α | a ∈ U } (⊆ L1p ∩ P olp {α}), SU := {ax + b | a ∈ U ∧ b ∈ Ep }, Tα,U ∪ Cα,U,I with Cα,U,I := {cα+n (mod p) | n ∈ i∈N Ni } and Cα,U,∅ := {cα }, Tα,U ∪ Cα,U,I ∪ {cα } and SU ∪ C. Theorem 13.2.2 ([Bag-D 82]) An arbitrary subsemigroup of (L1p ; ) is either a subset of C or has the form Tα,U , SU , Tα,U ∪ Cα,U,I , Tα,U ∪ Cα,U,I ∪ {cα } or SU ∪ C. Proof. Let A be an arbitrary subsemigroup of (L1p ; ) with A = A ∪ CA (A ⊆ S, CA ⊆ C). The following cases are possible: Case 1: CA = ∅. 1.1.: A ⊆ L1p ∩ P ol{α} for a certain α ∈ Ep . In this case, A has the form Tα,U , since L1p ∩ P ol{α} is a cyclic group. 1.2.: For all α ∈ Ep it holds A ⊆ L1p ∩ {α}. Then A = {x} and A contains a function x + a (a = 0) or at least two functions of the form f (x) := bx + (1 − b)β, g(x) := cx + (1 − c)γ with c, b ∈ Ep \{0, 1} and β = γ, since a function ax + b with a ∈ Ep \{0, 1} has exactly a ﬁxed point. Then, in the ﬁrst case, all functions x, x + 1, ..., x + p − 1 belong to A , and in the second case, (f −1 g f g −1 )(x) = x + d ∈ A with

13.2 The Subclasses of Linear Functions of Pk with k ∈ P

389

d = 0 (because of β = γ). Thus, A = SU for certain U ≤ G. Case 2: CA = ∅. Because of considerations for Case 1, set A is either a subset of C or of the form SU ∪C or Tα,U ∪CA , where the functions of Tα,U preserve CA . The latter is valid if and only if CA has the form Cα,U ;I or Cα,U ;I ∪ {cα } for certain I. Theorem 13.2.3 ([Bag-D 82]) Lp has exactly 2 · t(p − 1) + 2p · (2 − p) + p + 3 + 2 · p ·

2h

h h | (p − 1)

closed subsets (including ∅), where t(p − 1) is the number of all divisors n ∈ N of p − 1. Proof. Because of Theorem 13.2.1, we only have to determine the number of all subsemigroups which were described in Theorem 13.2.2. Since Tα,U ⊆ L1p ∩P ol{α}, L1p ∩P ol{α}∩P ol{β} = {x} for α = β and U is a subgroup of the cyclic group G = (Ep \{0}; ·), there is exactly p · t(p − 1) − p + 1 subsemigroups of the form Tα,U . Obviously, there exist 2·t(p−1) sets of the form SU and SU ∪C (U ≤ G). Since there are exactly 2h possibilities for the choice of I (⊆ {1, 2, ..., h}, h := p−1 |U | ), one receives (considering the equivalence Tα,U ∪ Cα,U,I = Tβ,V,J ∪ Cβ,V,J ⇐⇒ (α, U, I) = (β, V, J) ∨ (U = V = {1} ∧ I = J) the following number of the subsemigroups of (L1p ; ) of the form Tα,U ∪Cα,U,I or Tα,U ∪ Cα,U,I ∪ {cα }: (2h+1 − 1) − (p + 1) · (2p − 1) p· h h | (p − 1)

= 2 · p · ((

h h | (p − 1)

2h ) − p · t(p − 1) − (p − 1) · (2p − 1).

By adding, one immediately obtains the number given in our theorem. Next, we provide some statements about the order of the subclasses of Lp . One ﬁnds generating systems for these classes in [Bag-D 82]. Theorem 13.2.4 ([Bag-D 82]) For an arbitrary subclass A ⊆ Lp it holds:

390

13 Classes of Linear Functions

⎧ 1, if A ⊆ L1p , ⎪ ⎪ ⎨ 0 1 ord A = 3, if p = 2 and A ∈ {L2 , L2 ∩ P ol2 , [r1 ]}, 1 0 ⎪ ⎪ ⎩ 2 otherwise. Proof. For p = 2 the above statements were proven in Chapter 3. Because of Lemmas 13.1.1 and 13.1.2, we have [A1 ∪{r1 }] = A and therefore 1 < ord A ≤ 3 for arbitrary A ∈ L↓p (Lp ) with A ⊆ [L1p ]. If p = 2 then (by Lemma 13.1.2) r1 ∈ [∆r1 ]. This implies our theorem.

13.3 A Survey of Further Results on Linear Functions In this section, k ≥ 2 is arbitrary. Let n n { ai xi ∈ Lk | ai = 1} Lk,id := n≥1 i=1

i=1

the set of all idempotent functions of Lk . For d|k we set Ik,d := n≥1 {f n ∈ Lk,id | ∃i ∈ {1, ..., n} ∃a1 , ..., an ∈ Ek : f (x) = ai xi + d · j∈{1,...,i−1,i+1,...n} aj xj } Theorem 13.3.1 ([Sze 76], [Sze 82]; without proof ) Any non-trivial closed subset A of Lk,id has a unique representation of the form m Ik,di , (13.1) A= i=1

where m ≥ 1 and d1 , ..., dm (> 1) are pairwise relatively prime proper divisors of k. If k is odd, then any subclass of Lk,id has a order ≤ 2. If, in turn, k is even, then the order of any subclass is at most 3 and is equal to 3 if and only if in the representation (13.1) of A d1 · ... · dm is odd. To determine subclasses of Lk , the following lemmas are useful. We notice that the proof (in [Sze 78]) for Lemma 13.3.2 uses Theorem 13.3.1 and that the proof (in [Lau 88a]) for Lemma 13.3.3 requires Lemma 13.3.2, so that the proof of Theorem 13.3.1 forms the basis of the proofs for most statements of this section. Lemma 13.3.2 ([Sze 78]; without proof ) Let f (x) := a0 +

n

ai xi (mod k) ∈ Lk

i=1

with 0 ∈ {a1 , ..., an }, n ≥ 2 and a1 ' ... ' an ' k = 1. Then

13.3 A Survey of Further Results on Linear Functions

391

(a) rq1 ·...·qn ∈ [{f }], where qi := a1 ' a2 ' ... ' ai−1 ' ai+1 ' ... ' an ' k, i = 1, 2, ... ([Sze 78], Basic Lemma); (b) f ∈ [[f ]id ∪ [f ]1 ], where [A]id := [A] ∩ Lk,id for arbitrary A ⊆ Lk ([Sze 78], Corollary 3.1). Lemma 13.3.3 ([Lau 88a]; without proof ) Let A be a subclass of Lk which is no subset of Wq := Uq ∪ Qt t;t|q;t=1

for every divisor q = 1 of k. Then the function r1 belongs to A. For every k, d ∈ N d||k is a short notation for “d|k and d ' kd = 1”. For lack of a better name, d will be called a full divisor of k. The following lemma is easy to prove. Lemma 13.3.4 Let k, d ∈ N with d||k. Furthermore dZk denotes the set {d · z (mod k) | z ∈ Zk }. Then: (a) dZk := (dZk ; + (mod k), · (mod k)) is a ring which is isomorphic to Zk/d . (b) dZk has a unit which will be denoted by ed . (c) ed + en/d = 1 (mod k). For α, d ∈ Ek with d|k let Qd,α :=

{a0 · α +

n≥1

n

ai xi (mod k) | a0 +

i=1

n

ai xi ∈ Qd }.

i=1

Lemma 13.3.5 ([Sze 78]) Let k, d ∈ N with d||k. Then: (a) The mapping ψd : Lk/d −→ Qd,ed , a0 +

n

ai xi → ed · (a0 +

i=1

n

ai xi )

i=1

is an isomorphism. (b) The mapping ϕd : Qd −→ Qd,ed , a0 +

n i=1

is a homomorphism.

ai xi → ed · a0 +

n i=1

ai xi

392

13 Classes of Linear Functions

(c) For any f ∈ Qd there exists a unique constant cd (f ) such that f = ϕd (f ) + cd (f ) (mod k). Moreover, if f, g ∈ Qd , the constant cd (f ) has the following properties: ∀α ∈ {ζ, τ, ∆, ∇} : cd (αf ) = cd (f ) cd (f g) = cd (f ). Proof. (a), (b), and (c) are easy to check with the aid of Lemma 13.3.4. Theorem 13.3.6 ([Sze 78]) Let k ∈ N\{1} be square-free. Then, for arbitrary subclass of Lk , 2 if k is odd, ord A ≤ 3 otherwise, i.e., there are only a ﬁnite number of subclasses of Lk . Proof. Let f n ∈ Lk \ [L1k ] be arbitrary. Set A := [f ]. To show our theorem, it suﬃces to prove the following statement: k is odd, [A2 ] if f∈ (13.2) 3 [A ] otherwise. W.l.o.g. we can assume that f is an n-ary function with n ≥ 4 and deﬁned by f (x) := a0 +

n

ai · xi (mod k),

i=1

where 0 ∈ {a1 , a2 , ..., an }. For q := a1 ' a2 ' ... ' an ' k the following two cases are possible: Case 1: q = 1. In this case, we have [f ] = [[f ]id ∪ [{f }]1 ] by Lemma 13.3.2, (b), whereby our theorem follows from Theorem 13.3.1. Case 2: q = 1. With the aid of Lemma 13.3.5, one can reduce this case to the ﬁrst case (see [Sze 78] for details). We notice that the paper [Sze 78] contains precise representations of the subclasses of Lk , where k is square-free. Lemma 13.3.7 If k ∈ N is not square-free, then there exists a subclass of Lk that is not ﬁnitely generated and does not have a basis. Proof. If k can not be represented as a product of diﬀerent prime numbers, then there is a t ∈ Ek \{0} with t2 = 0 (mod k). Then, the set

13.3 A Survey of Further Results on Linear Functions

Qt :=

{f n ∈ Lk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + t ·

n

393

ai · xi (mod k)}

i=1

n≥1

is a subclass of Lk , whose functions f n , g m for arbitrary n, m have the property that the number of the essential variables of f g is at most n−1. Therefore, Qt does not have a ﬁnite generating system. Since by Theorem 8.1.6 |L↓k (Lk )| ≤ ℵ0 holds, every class of linear functions that does not have a ﬁnite generating system, does not have a basis. The following theorem is a consequence of Theorems 13.3.6, 3.2.2.2, 13.2.3, and 8.1.6, Lemma 13.3.7, and [Lau-S 90] (for k = 6): Theorem 13.3.8 For arbitrary k ∈ N \ {1} it holds: < ℵ0 if k is square-free, |Lk (Lk )| = ℵ0 otherwise. In particular, we have: k 2 3 4 5 6 7 8 9 |L↓k (Lk )| 15 38 ℵ0 319 7524 470 ℵ0 ℵ0

A solution for the completeness problem for Lk , where k ∈ N is arbitrary, follows from αs 1 α2 Theorem 13.3.9 ([Lau 88a]; without proof ) Let k = pα (pi = 1 p2 ...ps j; pi ∈ P, αi ∈ N for i = 1, ..., s). Then, Lk has exactly pj for i = s s + 2s − 1 + i=1 pi pairwise diﬀerent maximal classes: n (1) Sp := n≥1 {f n ∈ Lk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + i=1 ai xi (mod k) ∧ n i=1 ai = 1 (mod p)}, where p ∈ {p1 , ..., ps }; (2) Wq := Uq ∪ t|q,t=1 Qt , where q is a square-free divisor of k;

(3) Ta,p := Lk ∩ P olk {x ∈ Ek | p|(x − a)}, where a ∈ Ek and p ∈ {p1 , ..., ps }. As a consequence from the above theorem and from Lemma 13.3.3 we get: Lemma 13.3.10 Let A be an subclass of Lk . Then either r1 ∈ A or there exists a square-free divisor q of k with A ⊆ Wq . Since all subclasses of Lk that contain r1 were determined in Section 13.1, we need all subclasses of the maximal classes of the type Wq for a complete description of L↓k (Lk ). For k = 4 the following theorem gives a rough description of the missing classes.

394

13 Classes of Linear Functions

Theorem 13.3.11 ([Lau 88a]; without proof ) It holds: (a) |L↓4 ([L14 ])| = 189. (b) |L4 ([r1 ], L4 )| = 15. (c) Exactly 20 subclasses of L4 contain the function r2 , but not the function r1 . (d) A subclass of L4 , which contains neither r1 nor r2 , is a subset of Q2 ∪[L14 ] and there exist a subclass B of [L14 ] and some t0 , t1 , t2 , t3 ∈ N ∪ {∞} with A = B ∪ K0,t0 ∪ K1,t1 ∪ K2,t2 ∪ K3,t3 ,

where Ka,r :=

∅

r

if r = 1,

[{a + 2 · i=1 xi (mod 4)}] if r > 1 (a ∈ E4 , r ∈ N) and Ka,∞ := r≥2 Ka,r . We notice that the lattice Lk (Lk ) for k = p2 (p prime) was studied in detail in [Bul-I 2002] and [Bul 2002]. To describe further results about classes of linear function, we generalize the set Lk as follows: Let R = (R; +, ·, −, 0, 1) be a ﬁnite unitary ring and let M := (M ; +, −, (r)r∈R , 0) be a ﬁnite faithful2 module over R. Further, we consider the mapping deﬁned by : R × M −→ M, (r, x) → r x := r(x), which fulﬁlls the axioms (M1 )–(M4 ) (see Part I, Section 1.2.8). Then one can deﬁne a closed subset LM of PM as follows: LM :=

n≥1

{f n ∈ PM | ∃ a0 ∈ M ∃ a1 , . . . , an ∈ R : f (x) = a0 +

n

ai xi }.

i=1

It is usual to call the elements of LM also linear functions over the (left) module M. We remark that one can deﬁne analogously a set LM for the case that M is a right module. (the ring of all (m, m)-matrices over the ﬁeld Zp ; p If one sets R = Zm×m p (the module of all (m, 1)-matrices over R), then LM prime) and M = Zm×1 p is isomorphic to the set of all quasi-linear functions of Ppm (see Section 5.2.4). This matrix representation of the quasi-linear function was used in [Sze-S 81] to describe all subclasses that contain certain sets of unary quasi-linear functions: 2

A module M over a ring R is faithful iﬀ {r ∈ R | ∀m ∈ M : r(m) = 0} = {0}.

13.3 A Survey of Further Results on Linear Functions

395

Theorem 13.3.12 ([Sze-S 81]; without proof ) Let p prime, m ∈ N, pm ≥ 3, and M = Zm×1 . Moreover, set R = Zm×m p p I := {(i, j) ∈ N20 | 0 ≤ j ≤ i ≤ k − 1} ∪ {(−1, 0), (k − 1, k)}, H(r,s) := [{A · x ∈ L1M | rg(A) ≤ r ∨ rg(A) = k}] ∪ n≥2 {f n ∈ LM ∩ P olM {0} | dim(Im(f )) ≤ s}, where (r, s) ∈ I; 3 H(r,s) ; c := {f + c ∈ LM | f ∈ H(r,s) ∧ c ∈ M }, where (r, s) ∈ I. Then: (a) H(r,s) ⊆ H(r ,s ) iﬀ r ≤ r and s≤ s . n } (b) H(k−1,k) = LM ∩ P olM {0} = n≥1 { i=1 Ai · xi | A1 , ..., An ∈ Zm×m p and H(k−1,k);c = LM . (c) The sets H(r,s) , (r, s) ∈ I, are the only subclasses of LM ∩ P olM {0} containing (LM ∩ P olM {0})1 . (d) The sets H(r,s);c , (r, s) ∈ I, are the only subclasses of LM containing L1M . For the case p = m = 2, one can ﬁnd in [Kro-R 2003] all subclasses of quasi-linear selfdual4 functions, which contain all unary quasi-linear selfdual functions. In the case that R = M is a ﬁnite ﬁeld F, one can ﬁnd results on L↓F (LF ) in [Sze 80]. Among other things, the following theorem was proven in [Sze 80]: Theorem 13.3.13 (without proof ) For any ﬁnite ﬁeld F, the class LF has only ﬁnitely many subclasses. Every subclass A of LF with A ⊆ [L1F ] contains the function x + y − z and is a class of the following form (a) or (b): n (a) I(E, S) := n≥1 {a0 + i=1 ai xi | {a1 , ..., an } ⊆ E ∧ (∃s, s ∈ S : n a0 = s − ( i=1 ai ) s }, where E is a universe of a subﬁeld of F and S = V + a for an a ∈ F and a subspace V of F, considered as a vector space over E. n n (b) I(E, S0 ) := n≥1 {a0 + i=1 ai xi | {a1 , ..., an } ⊆ E ∧ i=1 ai = 1 ∧ a0 ∈ S0 }, where E is a universe of a subﬁeld of F and S0 is a subspace of F, considered as a vector space over E. Theorem 13.3.14 ([Sze 80], without proof ) Let k = pm (p prime, m ≥ 1), F := (Ek ; +, ·) a ﬁeld and let QLF be the set of all quasi-linear functions of Pk (see Section 5.2.4). Further, we set n LF := n≥1 {f n ∈ QLk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + i=1 ai · xi }, d

LF ;d := [LF ∪ {xp }] where d ∈ N is a divisor of k. 3 4

rg(A) denotes the range of the matrix A and dim V is the dimension of the vector space V . We remark that Im(f ) is a vector space, if f (x) = n i=1 Ai · xi . with respect to a ﬁxed-point-free permutation π of order p

396

13 Classes of Linear Functions

(In particular, LF ;1 = QLk and LF ;k = LF .) Then for any subclass A ⊂ Pk with LF ⊆ A, there exists a divisor d of k such that A = LF ;d . Theorem 13.3.15 ([Sze 80], without proof ) Let M be a faithful unitary module over a unitary ring R. Then for any subclass A of LM containing the operation x+y −z there exists a unique subring T of R and a unique T-submodule N of T × M such that A=

n≥1

{m +

n i=1

ri xi | {r1 , ..., rn } ⊆ T ∧ (1 −

n

ri , m) ∈ N }

i=1

(see also Section 9.6). Next we give some results by A. A. Bulatov about the elements of L↓M (LM ), where the unitary ring R is commutative (i.e., · is kommutative). The following theorem is the basis for a classiﬁcation of the elements of L↓M (LM ). Theorem 13.3.16 ([Bul 98a]; without proof ) Let R be kommutative. Then: (1) The minimal classes 5 of L↓M (LM ) are exactly the following classes: (a) JM ; (b) [ca ], where a ∈ M ; (c) [ε x + (1 − ε) b], where b ∈ M , ε ∈ R \ {0, 1} and ε · ε = ε (e.g., ε is an idempotent of the ring R). (2) The minimal clones, which are elements of L↓M (LM ), are exactly the following clones: (a) JM ∪ A, where A is a minimal class of the form (b) or (c) (see (1)); (b) [x + b], where b ∈ M is of prime additive order; (c) [α x + b], where b ∈ M , α ∈ R \ {1} and there exists a prime number p with αp = 1 and (1 + α + α2 + ... + αp−1 ) b = 0; (d) [x − r y + r z], where r ∈ R \ {0} with r2 = 0 and r is of prime additive order; (e) [x + y − z], if char R is a prime number.6 In [Bul 98b], it was shown that any element of L↓M (LM ) can be assigned to one of four types, deﬁned below. The deﬁnitions of these types use properties of coeﬃcients of functions and, in addition, some special functions. Classes of the ﬁrst type are those containing a function ε x + (1 − ε) y where ε is an idempotent of R (ε = 0, 1). Similarily, classes of the second type are those not of the ﬁrst type and containing the function ε x + (1 − ε) b where ε is n an idempotent of R (ε = 0, 1) and b ∈ M . A function f = i=1 αi xi + a is called primitive, if there exists j ∈ {1, ..., n} such that αj is an invertible 5 6

See Chapter 19. If there is a least positive integer n with x + x + ... + x = 0 for all x ∈ R, then n times the ring R is said to have characteristic n (notation char R).

13.3 A Survey of Further Results on Linear Functions

397

element and αi is a nilpotent element7 whenever i = j. Classes of the third type consist of primitive functions and functions whose coeﬃcients are nilpotent elements. Finally, a class belongs to the fourth type if it contains the Mal’tsev function x + y − z and is not of the ﬁrst type. ´ Szendrei in [Sze 80] (see The classes of the forth type were described by A. Theorem 13.3.15). Classes of the ﬁrst and second type were studied in [Bul 98a]. The paper [Bul 98b] deals with the classes of primitive functions and with the lattice of classes of the third type. In particular, A. A. Bulatov completely describes the lattice of classes of primitive functions in [Bul 98b]. We need the following notations for the last theorem of this section: In [P¨ os-K 79], the direct product of functions and subclasses of Pk was deﬁned in the following way. Let a set A := A1 ×A2 be Cartesian product of the sets A1 (n) (n) and A2 . The direct product of the functions f1 ∈ PA1 and f2 ∈ PA2 is (n)

the function f ∈ PA

deﬁned by the equality

f ((b1 , c1 ), ..., (bn , cn )) = (f (b1 , ..., bn ), f (c1 , ..., cn )) for arbitrary b1 , ..., bn ∈ A1 , c1 , ..., cn ∈ A2 . In this case, we shall write f = f1 ⊗ f2 . The set (n)

(n)

C = {f1 ⊗ f2 | f1 ∈ C1 , f2 ∈ C2 , n ∈ N} is the direct product of the classes C1 ⊆ PA1 , C2 ⊆ PA2 and is denoted by C1 ⊗ C2 . It is well known that, for any idempotent ε ∈ R \ {0, 1}, we can represent the module M as the direct sum of two submodules (ε M) ⊕ ((1 − ε) M) (this is the so-called Peirce decomposition). Further, each function f (x) := n a + i=1 f = f1 ⊗ f2 where f1 (x) = nαi xi ∈ LM can be representedas n ε a + i=1 αi xi and f2 (x) = (1 − ε) a + i=1 αi xi are functions on the sets εM and (1−ε)M , respectively. To show this take (b1 , c1 ), . . . , (bn , cn ) ∈ M = (ε M ) × ((1 − ε) M ). Then f ((b1 , c1 ), . . . , (bn , cn )) = f (b1 + c1 , . . . , bn + cn ) n = a+ αi (bi + ci ) i=1

= εa+

n i=1

7

εαi (bi + ci ) + (1 − ε) a +

n

(1 − ε)αi (bi + ci )

i=1

An element a of a ring R = (R; +, ·, −, 0) is called nilpotent iﬀ there exists an n ∈ N with an = 0.

398

13 Classes of Linear Functions

= εa+

n

αi bi + (1 − ε) a +

i=1

n

αi ci

i=1

= (f1 (b1 , . . . , bn ), f2 (c1 , . . . , cn )). n Conversely, the function f = f1 ⊗ f2 = (b + c) + i=1 i + (1 − ε)ζi ) xi (εγ n corresponds to the given pair of functions f = b + γi xi ∈ LεM , 1 i=1 n f2 = c + i=1 ζi xi ∈ L(1−ε)M . The modules ε M , (1 − ε) M can be considered as εR- and (1 − ε)R-modules, respectively. So, LM = LM1 × LM2 where M1 = ε M is an εR-module and M2 = (1 − ε) M is an (1 − ε)Rmodule. Let C be a class of all functions preserving the kernels of projections os-K 79] (3.3.4, 3.3.5, p. 85) it was noted that from A onto A1 and A2 . In [P¨ the interval LA1 ×A2 (JA1 × JA2 , C) can be decomposed into the direct product [JA1 , PA1 ]×[JA2 , PA2 ]. Using this decomposition we obtain the next statement. Theorem 13.3.17 ([Bul 98a]; without proof ) For any idempotent ε ∈ R, (a) if the function ε x + (1 − ε) y belongs to a subclass C of LM , then C = C1 ⊗ C2 where C1 , C2 are subclasses of LεM and L(1−ε)M , respectively and L1 , L2 both contain all projections; (b) LM ([ε x + (1 − ε) y], LM ) ∼ = LεM (JεM , LεM ) × L(1−ε)M (J(1−ε)M , L(1−ε)M ). For a proper ideal I of R, let LIM be the set of all functions f ∈ LM whose coeﬃcients belong to I. Further, let LP M be the set of all primitive functions of LM . For a ring R let idemp(R) be the set of all idempotents = 0. Moreover, put n n KM (R , M ) = { i xi + a | n ≥ 1, 1 , . . . , n ∈ R , (1 − i , a) ∈ M } i=1

i=1

where R is a unitary subring of R and M is a submodule of R -module R × M . Theorem 13.3.18 ([Bul 98a]; without proof ) The set of all maximal clones of LM comprise exactly the following clones: (a) (LIεM ∪ LP εM ) ⊗ L(1−ε)M where ε ∈ idemp(R); (b) KεM (R , R × ε M ) ⊗ L(1−ε)M where ε ∈ idemp(R), R is a maximal unitary subring of εR without proper idempotents; (c) KεM (εR, M ) ⊗ L(1−ε)M where ε ∈ idemp(R) is a minimal idempotent and M is a maximal submodule of εR × εM .

14 Submaximal Classes of P3

A subclass (or a subclone) of Pk is called submaximal if it is covered by a maximal class (clone).1 The concept submaximal class was introduced of I. G. Rosenberg in [Ros 74]. In [Ros 74] one also ﬁnds the ﬁrst results about submaximal classes of Pk (see Chapter 17 for details). In general, the submaximal classes seem to be interesting for the following reasons.2 The largely unknown lattice Lk has intervals with antichains of cardinality c situated far down from the top. It is not unreasonable to assume that the lattice Lk is nicer near the top, and therefore the submaximal clones are good candidates. The problem of determining certain submaximal clones also came up in the study of shortest maximal chains in the lattice (see [Sze 83]). Given a maximal clone M , one can ask for a completeness criterion for M : under what conditions does the clone [F ] generated by some F ⊆ M coincide with M ? In the case that M is ﬁnitely generated, a full list of clones maximal in M would provide a general criterion, because then [F ] if and only if F is contained in no clone maximal in M . An application could be a characterization of Sheﬀer functions for M . V. B. Kudrjavcev and P. Schoﬁeld proven that they exist exactly for maximal clones of the form P olk with ∈ C1k ∪ Sk ∪ Uk (see Chapter 7), but the examples in the proofs have many variables. It would be interesting to have simple criteria of type Theorem 7.1.3, which, in its turn, could lead to the question: what is the minimum number of functional values 1

2

It is easy to see that every submaximal class is a clone: Assume there is a maximal class M ⊂ Pk and a submaximal class S ⊂ M with Jk ∩ S = ∅ and [S ∪ {f }] = M for all f ∈ M \ S. Then, M = S ∪ Jk , since M is a clone (see Theorem 5.1.1) and Jk = [e] for all e ∈ Jk . However, M = S ∪ Jk with S ∩ Jk = ∅ is not possible, since S ∪ Jk = Pk (k − 1) ∪ [Pk1 ] and every maximal class A (= Pk (k − 1) ∪ [Pk1 ]) of Pk contains an idempotent function g with g ∈ Jk . For example, if ∈ Mk and o is the smallest element of , then the function f 2 , deﬁned by f (o, x) = f (x, o) = o for all x ∈ Ek and f (x, y) = x otherwise, is idempotent and belongs to (P olk ) \ Jk . With the help of the theorems from Chapter 5, one can easily ﬁnd idempotent functions for the other possible cases. The following is an indirect quotation from [Ros-S 85].

400

14 Submaximal Classes of P3

whose knowledge can guarantee that a function is a Sheﬀer function? Finally, the submaximal clones may be of interest on their own, e.g., as a source of examples and counter-examples. For arbitrary k, the full list of the maximal classes of a maximal class A of Pk is only known, if A has the type S ([Ros-S 84], see Section 18.1), C1 (see 2 ∪ {(k − 1, k − 1)} (see Chapters 16 and 17) or A = P olk , where := Ek−1 Section 18.3). There are, however, some papers in which one ﬁnds submaximal classes for speciﬁc k or only such submaximal classes that contain certain functions. In Section 14.1, one ﬁnds a complete description of all submaximal classes of P3 and some remarks about generalization of the given theorems. The papers [Mac 79], [Mar-D-H 80], [Sal 64], [Bag-D 82], and [Lau 82a] form the basis of this description. The theorems from the ﬁrst section are proven then in Sections 14.2–14.9. In Section 14.10, we will prove that there are 5 submaximal classes with ﬁnitely many subclasses, 7 with countably many subclasses, and the remaining 146 with uncountably many subclasses. All elements of the lattices L↓3 (A), where A is a submaximal class with|L↓3 (A)| ≤ ℵ0 are determined in Chapter 15.

14.1 A Survey of the Submaximal Classes of P3 The following theorem was proven 1958 by Jablonskij and is a special case of Rosenberg’s Theorem 6.1. Theorem 14.1.1 ([Jab 58]) P3 has exactly 18 maximal classes: (1) P ol3 {0}

(2) P ol3 {1}

(3) P ol3 {2}

(4) P ol3 {0, 1}

(5) P ol3 {0, 2} 0 1 (7) P ol3 1 2 0 1 (9) P ol3 0 1 0 1 (11) P ol3 0 1 0 1 (13) P ol3 0 1 0 1 (15) P ol3 0 1

2 0 2 2 2 2 2 2 2 2

0 2 0 1 1 0 1 0

2 0 0 2 1 2 0 1

1 2 0 2 1 2 2 1

(6) P ol3 {1, 2} 0 1 (8) P ol3 0 1 0 1 (10) P ol3 0 1 0 1 (12) P ol3 0 1 0 1 (14) P ol3 0 1 0 1 (16) P ol3 0 1

(17) P ol3 {(a, b, c, d) ∈ E34 | a + b = c + d (mod 3)} (18) P ol3 {(a, b, c) ∈ E33 | |{a, b, c}| ≤ 2}

2 2 2 2 2 2 2 2 2 2

0 1 1 2 0 2 0 1 2 0

1 0 2 1 0 1 1 0 0 2

2 1 0 2 2 1

2 0 1 2

14.1 A Survey of the Submaximal Classes of P3

401

In the following theorems, one ﬁnds all A-maximal classes for every maximal class A from Theorem 14.1.1. The next two theorems are special cases of general statements about the maximal classes of P olk E with 1 ≤ |E| ≤ k − 1 (see Chapters 16 and 17). Theorem 14.1.2 ([Lau 82a]) Let {a, b, c} := E3 . Then Ta := P ol3 {a} has exactly the following 12 maximal classes: (1) Ta ∩ P ol3 {b}

(2) Ta ∩ P ol3 {c}

(3) Ta ∩ P ol3 {a, b} (5) Ta ∩ P ol3 {b, c} a b (7) Ta ∩ P ol3 a b a b (9) Ta ∩ P ol3 a b a a b a (11) P ol3 a b a c

c c c c c a

a b a c

b a a b

a c c a c b

(4) Ta ∩ P ol3 {a, c} a b c b c (6) T ∩ P ol3 a b c c b a b c a a b (8) Ta ∩ P ol3 a b c b c c a b c (10) P ol3 a c b a a b a c b c (12) P ol3 a b a c a c b

Theorem 14.1.3 ([Lau 82a]) Let {a, b, c} := E3 . Then Ta,b := P ol3 {a, b} has exactly the following 15 maximal classes: (1) Ta,b ∩ P ol3 {a} (2) Ta,b ∩ P ol3 {b} a b a (3) Ta,b ∩ P ol3 a b b a b (4) Ta,b ∩ P ol3 ⎛b a a a a b ⎜a a b b ⎜ (5) Ta,b ∩ P ol3 ⎝ a b a a a b b a (6) Ta,b ∩ P ol3 {c} 0 1 2 a (7) Ta,b ∩ P ol3 0 1 2 b 0 1 2 a (8) Ta,b ∩ P ol3 0 1 2 c a b a (9) P ol3 a b c a b b (10) P ol3 a b c a a b b a (11) P ol3 a b a b c a a b b b (12) P ol3 a b a b c

b a b a

a b b a

b a a b

b a c b c a c b

⎞ b b⎟ ⎟ b⎠ b

402

14 Submaximal a a b (13) P ol3 ⎛a b a a b a (14) P ol3 ⎝ a b a ⎛a b b a b b (15) P ol3 ⎝ a b a a b a

Classes of P3 b a c b c b c a c b ⎞ b a b a b b b a a b⎠ a c c c c ⎞ a a b b a a b b a b a b a b⎠ a b a b b c c

The following theorem is a special case of a theorem from [Lar 93], in which B. Larose determined all maximal classes of P olk , where is a total order on Ek and 2 ≤ k ≤ 5. Theorem 14.1.4 ([Mac 79]) Let E3 := {a, b, c} and let min, max be deﬁned by x a a a b b b c c c Then O := P ol

y max(x, y) min(x, y) a a a b b a c c a a b a . b b b c c b a c a b c b c c c

0 1 2 a a b 0 1 2 b c c

has exactly the following 13 maximal

classes: (1) O ∩ P ol{a}

(2) O ∩ P ol{c}

(3) O ∩ P ol{a, b} (5) O ∩ P ol{b, c} 0 (7) O ∩ P ol3 0 0 (9) O ∩ P ol3 0 (11) O ∩ P ol3 ι33

1 2 b c 1 2 c b

1 2 b b b c 1 2 a b c b

(4) O ∩ P ol{a, c} 0 (6) O ∩ P ol3 0 0 (8) O ∩ P ol3 0 0 (10) O ∩ P ol3 0

1 2 a b 1 2 b a

1 2 a b a c 1 2 b a c a 1 2 c a c b 1 2 a c b c

⎛

0 (12) P ol3 ⎝ 0 0 ⎛ 0 (13) P ol3 ⎝ 0 0

1 2 0 0 1 2 0 0 1 2 1 2 1 2 0 0 1 2 1 2 1 2 1 2

14.1 A Survey of the Submaximal Classes of P3 ⎞ 1 0 0 1 1 1 2 2 ⎠ = [{min} ∪ O1 ] 3 2 0 0 1 ⎞ 1 1 2 2 2 0 0 1 ⎠ = [{max} ∪ O1 ] 4 2 1 2 2

403

The following theorem is a special case of theorems from [Ros-S 84] and [Lau 84]: Theorem 14.1.5 ([Mar-D-H 80]) 0 1 2 S := P ol3 has exactly two maximal classes: 1 2 0 (1) S ∩ P ol3 {0} (2) S ∩ P ol3 λ3 . The following theorem follows from Section 13.2: Theorem 14.1.6 ([Bag-D 82]) L3 := P ol3 λ3 has exactly 5 maximal classes: (1) L3 ∩ P ol{0}. (2) L3 ∩ P ol{1}. (3) L3 ∩ P ol{2}. 0 1 2 (4) L3 ∩ P ol3 1 2 0 (5) [(L3 )1 ].

Theorem 14.1.7 ([Lau 82a]) Let E3 := {a, b, c}. Then C := P ol3

0 1 2 a b a c 0 1 2 b a c a

has exactly the fol-

lowing 7 maximal classes: (1) C ∩ P ol{a} (2) C ∩ P ol{a, b} 3

4

The equality of these two sets can be proven as follows: It is easily checked that [{min} ∪ O1 ] ⊆ P ol3 (...) is valid. Then one proves that [{min} ∪ O1 ] is maximal in O (see for this purpose also [Mac 79]). The equality of these two sets results from the considerations to the statement (12) and the fact that the classes (12) and (13) are isomorphic.

404

14 Submaximal Classes of P3

(3) C ∩ P ol{a, c} (4) C ∩ P ol{b, c} 0 (5) C ∩ P ol3 0 0 (6) C ∩ P ol3 0 ⎛ a (7) C ∩ P ol3 ⎝ b c

1 2 a a 1 2 b c

1 2 a b a c b 1 2 b a c a c

⎞ b a c a b a a b b a b c a a c c a c a c a a a b a b a b b a c a c a c c⎠ c b b a a a b a b b b a a c a c c c

Theorem 14.1.8 ([Lau 82a])

Let E3 := {a, b, c}. Then U := P ol3

0 1 2 a b 0 1 2 b a

13 maximal classes: (1) U ∩ P ol{c} (2) U ∩ P ol{a, b} (3) U ∩ P ol{a, c} (4) U ∩ P ol{b, c} 0 1 2 a b a c (5) U ∩ P ol3 0 1 2 b a c a 0 1 2 b a b c (6) U ∩ P ol3 0 1 2 a b c b 0 1 2 a (7) P ol3 0 1 2 b a c b c (8) P ol3 c a c b 0 1 2 a b a b (9) P ol3 0 1 2 b a c c ⎛ 0 1 2 a a b b c c a (10) P ol3 ⎝ 0 1 2 a a b b c c b 0 1 2 b c a c a b c ⎛ a a a a b b b b a b (11) P ol3 ⎝ a a b b a a b b a b a b a b a b a b c c ⎛ ⎞ a a a b b a b b c ⎜a a b b a b a b c⎟ ⎟ (12) P ol3 ⎜ ⎝a b a a b b a b c⎠ a b b a a a b b c

⎞ b a⎠ c ⎞ c c c c c c⎠ a b c

has exactly the following

14.1 A Survey of the Submaximal Classes of P3 ⎛ (13)

a ⎜a c c

P ol3 E24 ∪ ⎜ ⎝

a b c c

b a c c

b b c c

a c a c

a c b c

b c a c

b c b c

a c c a

a c c b

b c c a

b c c b

c a a c

c a b c

c b a c

c b b c

c a c a

c a c b

c b c a

c b c b

c c a a

c c a b

c c b a

c c b b

405 ⎞ c c⎟ ⎟. c⎠ c

Theorem 14.1.9 ([Lau 82a]) Let P3 (2) := {f ∈ P3 | |Im(f )| ≤ 2}. The set P ol3 ι33 (= P3 (2) ∪ [P31 ]) has exactly 5 maximal classes: (1) P3 (2) ∪ [{s1 , s2 }] (2) P3 (2) ∪ [{s1 , s3 }] (3) P3 (2) ∪ [{s1 , s6 }] (4) P3 (2) ∪ [{s1 , s4 , s5 }] (5) n≥1 {f n ∈ P3 |∃fi ∈ P31 : f (x1 , ..., xn ) = f0 (f1 (x1 ) + ... + fn (xn )) (mod 2)} ∪ [P31 ]. (The deﬁnitions of the functions s1 , ..., s6 are given in Section 15.2, Table 15.1.)

Table 14.1 gives a summary of the above-described submaximal classes, where {a, b, c} := E3 and {α, β, γ} := E3 . The submaximal classes that can be described as intersections of maximal clones are labelled with numbers 1–17. The other classes are labelled with numbers 18–43 in the order that they occur in Theorems 14.1.2–14.1.9. With the aid of Table 14.1 and with a b a 0 1 2 a c = P ol{a, b} ∩ P ol3 , 5 {a, b, c} = E3 , P ol3 a b c 0 1 2 c a one can prove the following theorem: Theorem 14.1.10 P3 has exactly 158 submaximal classes. 5

This equationcan be proven as follows: a b a Because of ∆ = {a, b} and a b c a b c a b a 0 1 2 a c ◦ = it holds a b a a b c 0 1 2 c a a b a 0 1 2 a c ⊆ P ol{a, b} ∩ P ol3 . P ol3 a b c 0 1 2 c a a b a 0 1 2 a c is valid, we Since, in addition, ∆ {a, b} × = P ol3 a b c 0 1 2 c a have also 0 1 2 a c a b a P ol{a, b} ∩ P ol3 ⊆ P ol3 . 0 1 2 c a a b c

406

14 Submaximal Classes of P3 Table 14.1 Submaximal classes of P3

i

A

1 P ol{a} ∩ P ol{b} 2 P ol{a} ∩ P ol{a, b} 3 P ol{a} ∩ P ol{b, c} # $ 1 2 b c 4 P ol{a} ∩ P ol 0 #0 1 2 c b $ 1 2 a b a c 5 P ol{a} ∩ P ol 0 # 0 1 2 b a c $a 1 2 a a b 6 P ol{a} ∩ P ol 0 0 1 2 b c c 7 P ol{a} ∩ P olλ3 # $ 1 2 8 P ol{0} ∩ 0 1 2 #0 $ 1 2 a b 9 P ol{a, b} ∩ P ol 0 #0 1 2 b a$ 1 2 a c 10 P ol{a, b} ∩ P ol 0 #0 1 2 c a $ 1 2 α α β 11 P ol{a, b} ∩ P ol 0 #0 1 2 β γ γ $ 1 2 α β α γ 12 P ol{a, b} ∩ P ol 0 0$ 1 2 β# α γ α # 1 2 a b ∩ P ol 0 1 2 a a 13 P ol 0 #0 1 2 b a$ #0 1 2 b c 1 2 a b ∩ P ol 0 1 2 a b 14 P ol 0 0# 1 2 b a #0 1 2 b a $ 1 2 a a b ∩ P ol 0 1 2 α 15 P ol 0 0 1 2 β 0 1 2 b c c 0 1 2 a a b 3 16 P ol 0 1 2 b c c ∩ P olι3 # $ 1 2 ∩ P olλ 17 P ol 0 3 1 2 0 # $ a b c 18 P ol a c b # $ a b a c 19 P ol a #a b a c a $ a b a c b c 20 P ol a # a b a $c a c b b a 21 P ol a # a b $b 22 P ol ab ab ⎛ ⎞ a a a b b a b b a b b a b a b⎠ 23 P ol ⎝ a a b a a b b a b # a b b a a$ a b b a b b a 24 P ol a #a b a b c $ a b b a c b c 25 P ol a a b a b c a c b

6

6

ni (A) 3 6 3 3 3 6 3 1 3 6 9 b c a c β α

$ $ c a $ α γ γ α

ni (A) denotes the number of possibilities for A in case i.

9 6 6 9 3 1 3 3 3 3 3 3 6 3

14.1 A Survey of the Submaximal Classes of P3

i 26 27 28 29 30 31 32 33 34 35 36

%

a b a b a b a b P ol a b a b b a a b %a b b a c c c c a b b a a b b a P ol a b a b a b a b a b a a b a b b [{max} ∪ O1 ] [{min} ∪ O1 ] [(L3 )1 ] # $ 1 2 a P ol 0 #0 1 2 b$ P ol ac ac cb cb # $ 1 2 a b a b P ol 0 %0 1 2 b a c c 0 1 2 a a b b c P ol 0 1 2 a a b b c %0 1 2 b c a c a a a a a b b b b P ol a a b b a a b b ⎛a b a b a b a b a a a b b a b b a b b a b a b ⎝ P ol a a b a a b b a b a b b a a a b b

37 ⎛P olE24 ∪ a ⎜ a ⎜ ⎝ c c

a b c c

#

38 P ol # 39 P ol 40 P ol

b a c c

b b c c

a c a c

a c b c

b c a c

b c b c

0 1 2 a a 0 1 2 b c

a c c a

$

a c c b

b c c a

b c

b a c

a c b

c a b

a a a

b a a

ni (A) 3

a b a b c c

& 3 3 3 1 3 3

c a c b b c a b a b c⎞ c c c⎠ c c b c c b

c a a c

c a b c

3

&

b a c & c c c c c c a b c

3 3 3

c b a c

c b b c

c a c a

c a c b

c b c a

c b c b

c c a a

c c a b

c c b a

c c b b

⎞ c ⎟ c ⎟ c ⎠ c

3 3

0 1 2 a b a c b 0 1 2 b a c a c

#a

A

&

407

a b a

a a b

b b a

$ 3 b a b

a b b

b b b

c a a

a c a

a a c

c c a

c a c

a c c

$ c c c

3

41 P3 (2) ∪ [X], X ∈ {{s1 , s2 }, {s1 , s3 }, {s1 , s6 }}

3

42 P3 (2) ∪ [{s1 , s4 , s5 }]

1

43

n≥1 {f

n

∈ P3 |∃fi ∈ P31 : f (x1 , ..., xn ) = f0 (f1 (x1 ) + f2 (x2 ) + ... + fn (xn )) (mod 2)} ∪ [P31 ]

1

408

14 Submaximal Classes of P3

14.2 Some Declarations and Lemmas for Sections 14.3–14.9 In this chapter, we use the notations from Chapter 15, Table 15.1 for the unary functions of P3 . We prove Theorems 14.1.2–14.1.9 as follows in the next sections: Suppose in the theorem a list is indicated with subclasses (1), (2), ..., (rT ) for the class T . To prove that this list indicates all submaximal classes of T , we will show that every subset A ⊆ T , which is not contained in any subclass of the list, is a generating system for T . Then, it remains to show that the listed classes (1), (2), ..., (rT ) are proper subsets of the set T , and the set of all these classes is an antichain; i.e., in this set, no two elements are comparable in respect to ⊆. For this purpose, we give some functions (∈ T ) and a table that shows whether any function of these functions belongs to the indicated class. The sign + stands for “the function belongs to the class”. If the considered function does not belong to the class, we write -. We leave the readers to check the given table and, with the aid of the table, to prove that the classes (1), (2), ..., (rT ) are incomparable. If we assume in the proof that A ⊆ T and A is not a subset of the class (i), then there exists a function fi ∈ A which does not belong to the class (i). Since we can choose [A] instead of A, we can assume w.l.o.g. that the following holds: (∗ ):

If the class (i) is given in the form T ∩ P ol3 or P ol3 , where := (σ1 , σ2 , ..., σm ), then fi (σ1 , σ2 , ..., σm ) ∈ /.

Now some lemmas are given that we subsequently need more often and which are consequences from some theorems of Chapters 3 and 12. Already in Chapter 3 (see Theorem 3.3.1) the following was proven: Lemma 14.2.1 Let A be an arbitrary subset of P2 . Then, [A] = P2 if and only if A ⊆ B for every class B of the following list: (1) P ol2 {0} (2) P ol2 {1} 0 1 (3) P ol2 1 0 0 0 1 (4) P ol2 0 1 1 ⎛ ⎞ 0 0 0 1 1 0 1 1 ⎜0 0 1 1 0 1 0 1⎟ ⎟ (5) P ol2 ⎜ ⎝ 0 1 0 0 1 1 0 1 ⎠. 0 1 1 0 0 0 1 1 Lemma 14.2.2 Let A be an arbitrary subset of T0 := P ol2 {0} ⊂ P2 . Then, [A] = T0 if and only if A ⊆ B for every class B of the following list: (1) T0 ∩ P ol2 {1}

14.2 Some Declarations and Lemmas for Sections 14.3–14.9

0 0 0 (3) T0 ∩ P ol2 0 ⎛ 0 ⎜0 (4) T0 ∩ P ol2 ⎜ ⎝0 0 (2) T0 ∩ P ol2

0 1 1 0 0 0 1 1

409

1 1 0 1 0 1 1 1 0 0 1 0

1 0 1 0

0 1 1 0

1 0 0 1

⎞ 1 1⎟ ⎟. 1⎠ 1

As a consequence of Theorem 12.4.3, we get: Lemma 14.2.3 Let A be an only if A ⊆ B for every class (1) P3,2 ∩ P ol3 {0} (2) P3,2 ∩ P ol3 {1} 0 1 (3) P3,2 ∩ P ol3 1 0 0 0 1 (4) P3,2 ∩ P ol ⎛0 1 1 0 0 0 1 1 ⎜0 0 1 1 0 (5) P3,2 ∩ P ol3 ⎜ ⎝0 1 0 0 1 0 1 1 0 0 0 1 0 (6) P3,2 ∩ P ol3 0 1 2 0 1 1 (7) P3,2 ∩ P ol3 . 0 1 2

arbitrary subset of P3,2 . Then [A] = P3,2 if and B of the following list:

0 1 1 0

1 0 0 1

⎞ 1 1⎟ ⎟ 1⎠ 1

The next lemma follows from Theorem 12.5.3: Lemma 14.2.4 Let A be an arbitrary subset of pr−1 T0 := P3,2 ∩ P ol3 {0} ⊂ P3 . Then [A] = pr−1 T0 if and only if A ⊆ B for every class B of the following list: (1) pr−1 T0 ∩ P ol3 {1} 0 0 1 −1 (2) pr T0 ∩ P ol3 0 1 1 0 1 0 (3) pr−1 T0 ∩ P ol3 0 0 1 ⎛ ⎞ 0 0 0 1 1 0 1 1 ⎜0 0 1 1 0 1 0 1⎟ ⎟ (4) pr−1 T0 ∩ P ol3 ⎜ ⎝0 1 0 0 1 1 0 1⎠ 0 1 1 0 0 0 1 1 (5) pr−1 T0 ∩ P ol3 {0, 2} 0 1 1 (6) pr−1 T0 ∩ P ol3 . 0 1 2

410

14 Submaximal Classes of P3

14.3 Proof of Theorem 14.1.2 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof, let A be an arbitrary subset of P ol{0}, which is not contained in any class ((1)–(12)) from the above list, which is given in Theorem 14.1.2. Then, there are some functions f1 , f2 , ..., f12 ∈ [A] with the above property (∗ ). First we prove that (P ol3 {0})1 ⊆ [A]. The function f1 belongs to {c0 , j2 , u2 , u1 , s2 , u5 }. If f1 ∈ {j2 , u1 } then c0 = f1 f1 ∈ [A]. If f1 = u2 then f2 f1 ∈ {c0 , j2 }. For f1 ∈ {s2 , u5 } we have f5 (x, f1 (x)) ∈ {c0 , j2 , u2 }. Therefore, c0 is a superposition over A. The function f3 (x) := f3 (co , x) ∈ {u1 , s2 , u5 } also belongs to [A]. Thus we have to distinguish three cases: Case 1: f3 = u1 . In this case j1 = f4 (c0 , u1 ) and a function f4 (x) = f4 (c0 , x) ∈ {j2 , j5 , s2 } belongs to [A]. Case 1.1: f4 = j2 . (x) := f11 (c0 , j2 , j1 , u2 , u1 ) ∈ [A] and f11 (x) := Then u1 j2 = u2 ∈ [A], f11 f11 (co , j1 , j2 , u1 , u2 ) ∈ [A], where {f11 , f11 } ∩ {s2 , j5 , u5 } = ∅. Because of f12 (c0 , j2 , j1 , u2 , u1 , x, s2 ) ∈ {j5 , u5 } and j2 u5 = j5 we can assume j5 ∈ [A]. Then j2 j5 = u5 and {f7 (c0 , j5 , u5 , j2 , j1 , u2 , u1 ), f7 (c0 , j5 , u5 , j1 , j2 , u1 , u2 )} = {s1 , s2 } ⊆ [A]. Therefore, (P ol3 {0})1 ⊆ [A] in Case 1.1. Case 1.2: f4 = s2 . Because of j1 s2 = j2 one can reduce this case to Case 1.1. Case 1.3: f4 = j5 . In this case, functions u5 = u1 j5 and f9 (c0 , j5 , u5 , u1 , j1 , x) ∈ {j2 , u2 , s2 } are superpositions over A. Therefore, because of j5 u2 = j2 , we have given either Case 1.1 or Case 1.2. Case 2: f3 = u5 . Here we can form the superpositions j5 = f4 (c0 , u5 ) and f6 (x1 , x2 ) := j5 (f6 (c0 , j5 (x1 ), u5 (x1 ), x1 , x2 )) 1 2 0 1 1 2 0 over A, where f6 ∈ . W.l.o.g. let f6 = . If 2 1 1 0 2 1 1 f6 (1, 1) = 1 we obtain Case 1 because of u5 (f6 (j5 (x), x)) = u1 . If f6 (1, 1) = 0 then it holds f6 (x, j5 (x)) = j2 , u5 j2 = u2 and f8 (c0 , j5 , u5 , j2 , u2 , x) ∈ {j1 , u1 , s2 }. Since u5 j1 = u1 and u2 s2 = u1 , we have also u1 ∈ [A] if f6 (1, 1) = 0, i.e., one can reduce Case 2 to Case 1. Case 3: f3 = s2 . Because of f10 (c0 , s2 , x) ∈ {j1 , j2 , j5 , u1 , u2 , u5 }, s2 ji = ui , i ∈ {1, 2, 5}, and u2 s2 = u1 , we obtain either Case 1 or Case 2. Consequently, (P ol{0})1 ⊆ [A] is proven. Next we prove that P3,2 ∩ P ol3 {0} is a subset of [A]. Since (P3,2 ∩ P ol3 {0})1 ⊆ [A] was already shown, we have to ﬁnd a subset of [A] ∩ P3,2 that is not a

14.3 Proof of Theorem 14.1.2

subset of P ol

0 0 1 0 1 0

, P ol

0 0 1 0 1 1

411

and P olλ2 := {(a, b, c, d) ∈ E24 | a+b =

c + d (mod 2)}, to be able to use Lemma 14.2.4. Obviously, the function f11 (x1 , x2 ) := j5 (f11 (c0 , x1 , x2 , u1 (x1 ), u1 (x2 ))) ∈ [A] ∩ P3,2 0 0 1 does not preserve the relation . For the function 0 1 0 f7 (x1 , x2 ) := f7 (c0 , f11 (x1 , x2 ), u1 (f11 (x1 , x2 )), x1 , x2 , u1 (x1 ), u1 (x2 )) ∈ [A] 1 0 1 we can assume w.l.o.g. that f7 = . If f7 (1, 1) ∈ {0, 2} we have 1 0 0 0 0 1 j1 f7 ∈ P ol3 ∪ P ol3 λ2 . If f7 (1, 1) = 1 then this is valid for j2 f7 0 1 1 instead of j1 f7 . Thus by Lemma 14.2.4, we have P3,2 ∩ P ol3 {0} ⊆ [A]. After these preparations we can easily show that an arbitrary function f n ∈ P ol3 {0} is a superposition over A. For this purpose, we need the functions q1 , q2 ∈ P3,2 deﬁned by 0 if f (x) ∈ {0, 1}, q1 (x) := 1 if f (x) = 2, 0 if f (x) ∈ {0, 2}, q2 (x) := 1 if f (x) = 1, Then f7 (q1 (x), q2 (x)) = f (x) ∈ [A] holds. Consequently, [A] = P ol3 {0}. As explained in Section 14.2, our theorem results from Table 14.2, where the binary functions h1 , h2 ,..., h8 are deﬁned in Table 14.3.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

c0 − − + + − + + + + + + +

j1 + − + + − − + − + − + +

s2 − − − − + + + − − + + +

u5 − + − + + + + + + − + −

Table 14.2 h1 h2 h3 + + + + + − − + − + − − − + + − − + − − − − − − − − + − − − − − − − − −

h4 − − − − − − − − − + − −

h5 + + + + + + − + − − − −

h6 − − + − − − + + − − − −

h7 + − − + − − − − − − + +

h8 − − − + − − − − − − − +

412

14 Submaximal Classes of P3

x1 0 0 0 1 1 1 2 2 2

x2 0 1 2 0 1 2 0 1 2

h1 0 1 2 2 1 0 2 0 2

Table 14.3 h2 h3 h4 h5 0 0 0 0 1 1 1 1 1 1 2 2 1 2 1 1 1 1 2 1 1 1 0 2 0 1 2 2 1 1 0 2 2 1 1 2

h6 0 0 1 0 0 1 1 1 1

h7 0 0 0 2 1 0 0 0 0

h8 0 1 0 2 0 0 0 0 0

14.4 Proof of Theorem 14.1.3 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof let A be an arbitrary subset of P ol{0, 1}, which is not contained in any class ((1)–(15)) from the above list, which is given in Theorem 14.1.3. Then there are some functions f1 , f2 , ..., f15 ∈ [A] with the above property (∗ ). With the help of Lemma 14.2.1, one can easily prove that every function of P2 is a restriction of a function of [A]. Therefore, a function g ∈ {c0 , j2 , u2 } and a function h ∈ {j0 , j4 , s3 } are superpositions over A. If g ∈ {c0 , j2 } then the functions g g = c0 and h c0 = c1 belong to [A]. If g = u2 then f6 := f6 g ∈ {c0 , c1 , j2 , j3 } and {f6 f6 , h f6 f6 } = {c0 , c1 }. Thus the constant functions c0 and c1 are superpositions over A. Next we prove that the remaining unary functions of P ol3 {0, 1} and all functions of P3,2 are also superpositions over A. For this purpose, we distinguish two cases for the function h: Case 1: h ∈ {j0 , j4 }. Then h := h h ∈ {j1 , j5 } and the functions fi := h fi , i ∈ {1, 2, 3, 4, 5} belong to [A], where fi ∈ P3,2 and fi does not belong to the class (i). Further, := h(f10 (c0 , c1 , x)) belong to [A]. the functions f9 := h(f9 (c0 , c1 , x)) and f10 } ⊆ P3,2 and that It is easy to check that {h, f9 , f10 {h, f9 , f10 } is no subset 0 1 0 0 1 1 and P ol3 . Consequently, by Lemma of the classes P ol3 0 1 2 1 1 2 14.2.3, every function of P3,2 is a superposition over A. := f11 (c0 , j2 , j0 , c1 , x) ∈ {v2 , s3 }. A function of [A] is also f11 Case 1.1: f11 = v2 . := f12 (c0 , j2 , j3 , c1 , v2 ) = u2 ∈ [A] and In this case, we have f12 {f15 (c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 , u2 , v2 ), f15 (c0 , c1 , j1 , j0 , j2 , j3 , j5 , j4 , u2 , v2 )} = {s1 , s2 }. Consequently, (P ol3 {0, 1})1 ⊆ [A]. = s2 . Case 1.2: f11

14.4 Proof of Theorem 14.1.3

First we form a ternary function f) 15 which ⎛ 0 0 ⎝1 0 f) 15 0 2

413

as a superposition over P3,2 ∪ {f15 }, for ⎞ ⎛ ⎞ 1 0 1⎠=⎝1⎠ 2 2

) ) holds. If f) 15 (0, 1, 0) = 0 then s3 (f15 (c0 , x, s3 )) = v2 ∈ [A]. If f15 (0, 1, 0) = 1 ) then f15 (j0 , x, s3 ) = v2 ∈ [A]. Consequently, we can reduce Case 1.2 to Case 1.1. Consequently, P3,2 ∪ (P ol3 {0, 1})1 ⊆ [A] was proven in Case 1. Case 2: h = s3 .

0 0 1 2 0 2 1 2 W.l.o.g. we can assume f8 = . Consequently, 0 1 2 2 0 2 1 1 f8 (x1 , x2, x3 ) :=f8 (c0 , c1, x1 , x2 , x3 , s3 (x2 ), s3 (x3 )) belongs to [A], and we 0 2 0 2 have f8 = . Then the functions f8 := f8 (s3 , c0 , x) and f8 := 2 2 0 1 f8 (x, s3 , c0 ) are superpositions over A, where {f8 , f8 } ∩ {j0 , j3 , j2 , j4 } = ∅. Thus, we have either Case 1 or j2 = s3 j3 and j3 = s3 j2 belong to [A]. A superposition over A is also a certain binary function q with the property ⎛ ⎞ ⎛ ⎞ 0 0 1 ⎜0 1⎟ ⎜0⎟ ⎟ ⎜ ⎟ q⎜ ⎝ 1 0 ⎠ = ⎝ 0 ⎠. 1 1 0 If {q(0, 2), q(2, 0), q(1, 2), q(2, 1), q(2, 2)} ∩ {0, 1} = ∅ then {q(c0 , x), q(x, c0 ), q(j2 , x), q(x, j2 ), q(x, x)} ∩ {j0 , j4 } = ∅, i.e., one can reduce Case 2 to Case 1 here. Therefore, we can assume q(a) = 2 for every a ∈ E32 \E22 in the following. := f14 (c0 , c1 , j2 , j3 , x, s3 , u2 , v2 ) ∈ Then q(j3 , x) = u2 , s3 u2 = v2 and f14 {j0 , j1 , j4 , j5 } belong to [A]. Since s3 j1 = j4 and s3 j5 = j0 , we have again Case 1. Hence P3,2 ∪ (P ol3 {0, 1})1 ⊆ [A]. 0 1 2 0 1 0 W.l.o.g. we can assume f7 = . Then f7 (x1 , x2 ) := 0 1 2 1 0 2 u2 (f7 (c0 , c1 , x1 , x2 , j0 (x2 ))) ∈ [A], where ⎞ ⎛ ⎞ ⎛ 0 0 0 ⎜0 1⎟ ⎜0⎟ ⎟ ⎜ ⎟ ⎜ ⎜1 0⎟ ⎜0⎟ ⎟ ⎜ ⎟ f7 ⎜ ⎜ 1 1 ⎟ = ⎜ 0 ⎟. ⎟ ⎜ ⎟ ⎜ ⎝2 0⎠ ⎝0⎠ 2 2 1

414

14 Submaximal Classes of P3

Further, f13 (x1 , x2 ) := u2 (f13 (c0 , j2 (x1 ), j2 (x2 ), c1 , x1 , x2 , s3 (x1 ), s3 (x2 ))) ∈ [A], where ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 0 2 ⎠ = ⎝ 2 ⎠. f13 2 0 2 It is easy to see that one obtains a binary function f15 with ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 1 0⎠=⎝1⎠ f15 2 0 2

as a superposition over P3,2 ∪ (P ol3 {0, 1})1 ∪ {f15 }. Let f n be an arbitrary function of P ol3 {0, 1}. We show f ∈ [A]. For this α1 , ..., α r ∈ E3n , β ∈ E3 ) be an n-ary function deﬁned purpose, let fα 1 ,...,αr ;β ( by β if x ∈ { α1 , ..., α r }, fα1 ,..., (x) := αr ;β 0 otherwise. Obviously, the functions of the type fα ;1 are superpositions over A for every = (α1 , ..., αn ) ∈ E3n and αi = 2 then fα ;2 ∈ [A] follows from α ∈ E3n . If α fα ;2 = f7 (xi , fα ;1 (x)). Furthermore, we have fα1 ,..., αr ;2 (x) = f13 (fα 1 ;2 (x), fα 2 , α3 ,..., αr ;2 (x)).

Consequently, all functions of the type fα1 ,..., α1 , ..., α r } ⊆ E3n \E2n αr ;2 with { are superpositions over A. Then f ∈ [A] follows from f (x) = f15 (fβ1 ,...,βs ;1 (x), fγ1 ,...,γt ;2 (x)),

1 , ..., βs } := {x ∈ E n | f (x) = 1} and {γ1 , ..., γt } := {x ∈ where {β 3 n E3 | f (x) = 2}. Consequently, [A] = P ol3 {0, 1}. Then our theorem follows from Table 14.4. The functions g1 , ..., g10 of Table 14.4 are deﬁned in Table 14.5. The function h3 is deﬁned by ⎧ x ∈ E33 \E23 , ⎨ 2 if x ∈ {(0, 1, 1), (1, 0, 1), (1, 1, 0), (1, 1, 1)}, h(x) := 1 if ⎩ 0 otherwise.

14.5 Proof of Theorem 14.1.4

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)

j0 − − − + + − + − − + + + + − +

j4 − − − + + − + − + − + + + − +

x1 0 0 1 1 0 1 2 2 2

u2 + − + − + + + + + − + − + + +

x2 0 1 0 1 2 2 0 1 2

s3 − − − + + + + + − − − − + + +

g1 0 0 0 0 0 2 0 2 0

g1 + − + − + − − + + − + − − + +

g2 1 1 1 1 1 2 1 1 1

Table g2 g3 − + + − + − − − + − − + − − + − − − + − − − + − + − + − + −

14.4 g4 g5 − + − + − + + + + + − + − − − − + − − + − − − − + + − − − −

Table 14.5 g3 g4 g5 g6 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 2 2 2 0 1 1 1 2 2 1 1 2 2 1 2 2

g6 − + − − − + + − − − − + + − − g7 0 1 1 1 2 2 2 2 2

g7 + + + − − + + + − + − + − + − g8 0 0 0 0 2 2 2 2 2

g8 + − + − + + + + + − + − − + −

g9 + + + + + + − − − − − − − + −

g10 + + + − − − + − + − + + + − +

415

h + + + + − + + + − − − − − + −

g9 g10 0 0 0 0 1 0 1 1 1 0 2 0 2 0 2 0 2 0

14.5 Proof of Theorem 14.1.4 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof, let A be an arbitrary subset of O, which is not contained in any class ((1)–(13)) from the above list, given in Theorem 14.1.4. Then, there are some functions f1 , ..., f13 ∈ [A] with the above property ( ). First we prove that O1 \{x} ⊆ [A] holds. For the unary function f1 , we have f1 (0) ∈ {1, 2}. Then, either f1 = c2 or we obtain c2 by forming f3 (x, f1 (x)). Consequently, {c0 , c1 , c2 } ⊆ [{f2 , f4 , f5 , c2 }] ⊆ [A]. W.l.o.g. we can assume 0 1 2 2 0 2 1 0 ∈ . f8 0 1 2 0 2 1 2 1 Since f8 is monotone, we have 0 1 2 0 0 1 0 0 ∈ . f8 0 1 2 0 2 1 2 1

416

14 Submaximal Classes of P3

Therefore, f8 (c0 , c1 , c2 , c0 , x, c1 , x) ∈ {j2 , j5 } is valid. Since we can assume w.l.o.g. 0 1 2 1 2 0 0 f10 ∈ , 0 1 2 2 1 1 2 and since f10 ∈ O, we have f10 (0, 1, 2, 0, 2) = 0 and f10 (0, 1, 2, 2, 2) ∈ {1, 2}. Thus f10 (c0 , c1 , c2 , x, c2 ) ∈ {j2 , u2 }. Since j5 u2 = j2 , j2 ∈ [A] holds. With the aid of functions f6 and f9 one can show analogously that v5 ∈ [A] also holds. Because of j5 = j2 v5 and v2 = v5 j2 we have j5 , v2 ∈ [A]. W.l.o.g. let 0 0 1 2 1 0 1 2 f7 ∈ . 0 1 2 0 1 2 1 2 Thus,

f7

0 1 2 1 0 1 1 0 1 2 1 1 2 1

∈

0 2

and therefore f7 (c0 , c1 , c2 , c1 , j5 , v5 , c1 ) = u5 ∈ [A]. Consequently, by u5 j2 = u2 , we have O1 \{x} ⊆ [A]. A conclusion of the Fundamental Lemma of Jablonskij (see Theorem 1.4.4, (a)) is the following fact: For the function f11 , there are some a, b, c ∈ E3 with the property (w.l.o.g.): ⎛ ⎞ ⎛ ⎞ 0 a c f11 ⎝ b c ⎠ ∈ ⎝ 1 ⎠ , 2 b d where (because of f11 ∈ O) (a, c) < (b, c) and (b, c) < (b, d). If one replaces the variables of function f11 by certain functions from O1 \{x}, one obtains a ∈ [A] with function f11 ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 2 0 ⎠ ∈ ⎝ 1 ⎠. f11 2 2 2 Now we can form gα (x, y) := f11 (u5 (x), u2 (y)), where ⎧ if (x, y) = (0, 2), ⎨α x ∈ {1, 2}, gα (x, y) = v2 (y) if ⎩ 0 otherwise.

Next, with the help of the function gα , we show that [A] ∩ {min, max} = ∅

(14.1)

is valid, where we use the notation x ∨ y := max(x, y) and x ∧ y := min(x, y). We distinguish three cases: Case 1: α = 0. In this case, we have g0 (x, y) = u5 (x) ∧ v2 (y) and

14.5 Proof of Theorem 14.1.4

417

g0 (u5 (g0 (x, v5 (y))), v2 (g0 (u2 (x), y))) = x ∧ y ∈ [A] holds. Case 2: a = 1. Then g1 (x, y) = (u5 (x) ∧ v2 (y)) ∨ j2 (y), whereby g1 (u5 (g1 (u2 (x), u2 (y))), u5 (j5 (g1 (x, u5 (y))))) = x ∨ y ∈ [A] holds. Case 3: a = 2. Since g2 (x, y) = (u5 (x) ∧ v2 (y)) ∨ u2 (y) is valid in this case, we have g2 (u5 (g2 (x, u5 (y))), v2 (g2 (u2 (x), y))) = x ∨ y ∈ [A] Thus (14.1) is proven. Since the classes (12) and (13) are isomorphic, we can assume w.l.o.g. that min ∈ [A]. The function f12 has the property ⎛ ⎞ ⎛ ⎞ 0 1 2 0 0 1 0 0 1 0 0 0 0 1 f12 ⎝ 0 1 2 0 0 1 1 2 2 ⎠ ∈ ⎝ 1 2 1 2 2 ⎠ . 0 1 2 1 2 2 0 0 1 1 1 2 2 2 (x, y) := f12 (c0 , c1 , c2 , x, u2 (x), v2 (x), y, u5 (y), v5 (y)), either If one forms f12 (x, y)) or the function u2 (f12 (x, y)) is identical to the the function u5 (f12 function g(x, y) := u5 (x) ∨ u5 (y). Therefore, we get x ∨ y = g(x, y) ∧ (v5 (g(u2 (x), u2 (y)))) ∈ [A]. Since (O1 \{x}) ∪ {∧, ∨} is a generating system of O (see Section 11.4), we have shown [A] = O. Then our theorem follows from Table 14.6, where g1 := ∧, g2 := ∨, g3 (x, y) := (j5 (x) ∧ j5 (y)) ∨ j2 (y) and g4 := u5 g3 .

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

c0 + − + + − + + + + + + + +

c1 − − + − + + + + + + + + +

c2 − + − + + + + + + + + + +

j2 + − + − − + + − + − + + +

Table j 5 u2 + + − + + + − + + − + + + − − + + + + − + + + + + +

14.6 u5 v2 + − + + − + + − + + + − − + + + − + + + + + + + + +

v5 − + − − + − + + − + + + +

g1 + + + + + + + − + + − + −

g2 + + + + + − + + + + − − +

g3 + − + − + + + − + − + − −

g4 + + − + + + − + − + + − −

418

14 Submaximal Classes of P3

14.6 Proof of Theorem 14.1.5 Let A be an arbitrary subset of S, which is no subset of P ol3 {0} and no subset of L. Then there are two functions f1 , f2 ∈ [A] with the above property (∗ ). To prove [A] = S it is suﬃcient to show that α([A]) := {f c0 | f ∈ [A]} = P3 holds (see Theorem 8.3.2). α([A]) = P3 is proven, if one can show that α([A]) ⊆ B for every class B from the list of Theorem 14.1.1. Obviously, we have f11 ∈ {s4 , s5 }, S 1 = {s4 , s5 , e11 } ⊆ [f1 ] and f1 is no element of a maximal class of P3 with the marking (1), ..., (6), (8), ..., (16) of Theorem 14.1.1. Because of α(e21 ) = c10 , α([A]) is no subset of a maximal class of P3 with the marking (7). Further, α(f2 ) ∈ B for each maximal class B of P3 with the marking (17) or (18). Consequently, α([A]) = P3 and therefore [A] = S. Then, our Theorem 14.1.5 follows from S ∩L ⊆ S ∩P ol3 {0} and S ∩L S ∩P ol3 {0}.

14.7 Proof of Theorem 14.1.7 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof, let A be an arbitrary subset of C, which is not contained in any class with the marking (1)–(7) from the above list, which is given in Theorem 14.1.7. Then there are some functions f1 , ..., f7 ∈ [A] with the above property ( ). First we show that {c0 , c1 , c2 } ⊂ [A]. Obviously, f1 ∈ {j0 , j4 , j3 , c1 , u0 , u4 , u3 , c2 }. We distinguish the following three cases: Case 1: f1 ∈ {c1 , c2 , j3 , u4 }. In this case f1 := f1 f1 = cα , α ∈ {1, 2} holds. Since f2 , f3 ∈ C and f2 (0, 1) = 2 or f3 (0, 2) = 1, we have f2 (1, 1) ∈ {0, 2} or f3 (2, 2) ∈ {0, 1}. Therefore, {c1 , c2 } or {c0 , cα } is a subset of [A]. With the aid of functions f2 , f3 , f4 (∈ [A]) one can prove {c0 , c1 , c2 } ⊂ [A] easily. Case 2: f1 = j0 . In this case, the functions j0 j0 = j5 and f2 (x) := f2 (j5 , j0 ) ∈ {c2 , u2 } belong to [A]. If f2 = c2 , then we obtain by Case 1 that {c0 , c1 , c2 } ⊂ [A]. If f2 = u0 then u5 = u0 u0 ∈ [A]. For the function f7 is valid: ⎞ ⎛ ⎞ ⎛ 0 1 0 2 0 1 0 0 1 1 0 1 2 0 0 2 2 0 2 1 2 f7 ⎝ 1 0 2 0 0 0 1 0 1 0 0 1 0 2 0 2 0 2 2 ⎠ ∈ ⎝ 1 2 ⎠ . 2 2 1 1 0 0 0 1 0 1 1 1 0 0 2 0 2 2 2 2 1 0 1 2 0 1 0 2 Since f7 preserves the relation , we have 0 1 2 1 0 2 0 f7 Consequently,

0 2 0 1 0 0 . . . . . 0 0 0 . . . . 0 2 0 1 0 1 1 . . . . . 1 2 2 . . . . 2

=

0 0

.

14.7 Proof of Theorem 14.1.7

419

j0 (f7 (u5 , u0 , j5 , j0 , j5 , j5 , ....., j5 , u5 , u5 , ...., u5 ) = c0 ∈ [A]. Therefore, one can reduce Case 2 to Case 1. Case 3: f1 ∈ {j4 , u0 , u3 }. It is easy to check that one can also reduce this case to the ﬁrst case analogously to Case 2. Thus {c0 , c1 , c2 } ⊂ [A] is proven. Since f6 ∈ C and

f6

holds, we have

f6

0 1 2 0 1 0 2 1 0 1 2 1 0 2 2 2 0 1 2 0 1 0 2 0 0 1 2 1 0 2 2 0

Consequently, the functions f6 (x)

=

=

2 1 0 0

.

:=

f6 (c0 , c1 , c2 , c0 , c1 , c0 , c2 , x) and 0 0 := f6 (c0 , c1 , c2 , c0 , c1 , c2 , c0 , x) belong to [A], where f6 = 1 2 0 0 0 1 2 0 0 and f6 . Therefore, we can assume w.l.o.g. f5 = = 2 1 0 1 2 1 2 1 . Then 0 p(x) := f5 (c0 , c1 , c2 , x, f6 (x)) ∈ {j0 , j4 }. f6 (x)

It is easy to check that f6 (0, 1, 2, 1, 0, 2, 0, 1) ∈ {0, 2}. Consequently, f6 (x) := f6 (c0 , c1 , c2 , c1 , c0 , c2 , c0 , x) ∈ {j2 , s2 }. Therefore, we have to distinguish the following two cases for f6 : Case 1: f6 = j2 . Then, the functions p j2 = j3 , f6 j2 = u2 , f6 j3 = u3 , f6 (c0 , c1 , c2 , j2 , j3 , u2 , u3 , x) = s2 , j2 s2 = j1 , j3 s2 = j4 , s2 j1 = u1 and s2 j4 = u4 are superpositions over A. Consequently, the function f7 (x1 , x2 ) := f7 (x1 , x2 , s2 (x1 ), s2 (x2 ), c0 , j1 (x2 ), j1 (x1 ), j2 (x1 ), j3 (x1 ), j4 (x1 ), j4 (x2 ), c1 , u1 (x1 ), u2 (x1 ), u3 (x1 ), j4 (x1 ), j4 (x2 ), c2 ) belongs to [A], where

⎛

⎞ ⎛ ⎞ 0 1 α f7 ⎝ 1 0 ⎠ = ⎝ α ⎠ 2 2 β

and {α, β} = {1, 2}. Since s2 ∈ [A], we can assume α = 1 and β = 2. Because of f7 ∈ C, we have f7 (0, 0) = f7 (0, 2) = f7 (2, 0) = 0 and {f7 (1, 2), f7 (2, 1)} ⊆

420

14 Submaximal Classes of P3

{0, 1}. Consequently, the function f7 (x1 , x2 ) := j2 (f7 (u1 (x1 ), u1 (x2 ))) ∈ [A] has the properties ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎜ ⎟ ⎜ ⎟ 0 1 ⎟ ⎜0⎟ f7 ⎜ ⎝1 0⎠=⎝0⎠ 1 1 1 and f7 ∈ P3,2 . Obviously, the set {j2 , p, c0 , c1 , f7 } is a subset of P3,2 , however no subset of the maximal classes of P3,2 (see Lemma 14.2.3). Consequently, the sets P3,2 and {s2 r | r ∈ P3,2 } = {r ∈ P3 | Im(r ) ⊆ {0, 2} } are subsets of [A]. With the aid of the functions 2 if (2, i) = (x1 , x2 ), ti (x1 , x2 ) := 0 otherwise, i ∈ {1, 2}, of [A] we can form a function t ∈ [A] by t(x1 , x2 ) := f7 (s2 (f7 (s2 (x2 ), t1 (x1 , x2 ))), t2 (x1 , x2 )) ⎧ (x1 , x2 ) = (2, 1), ⎨ 1 if 2 if (x1 , x2 ) = (2, 2), = ⎩ 0 otherwise. Let S and T be nonempty disjoint subsets of E3n with the property that for diﬀerent arbitrary tuples σ := (σ1 , ..., σn ) ∈ T and τ := (τ1 , ..., τn ) ∈ T there exists an i with {σi , τi } = {1, 2}. Then the function ⎧ x ∈ S, ⎨ 1 if x ∈ T, fS,T (x) := 2 if ⎩ 0 otherwise, belongs to C. [A] = C would be proven in Case 1, if we could show that every function of the form fS,T is an element of [A]. For σ := (σ1 , ..., σn ), τ := (τ1 , ..., τn ), {σi , τi } = {1, 2} and 2 if x ∈ {σ, τ }, gσ,τ (x) := 0 otherwise, we have

f{σ},{τ } (x) :=

t(g{σ},{τ } (x), xi ) if (σi , τi ) = (1, 2), t(g{σ},{τ } (x), s2 (xi )) if (σi , τi ) = (2, 1),

i.e., f{σ},{τ } ∈ [A]. Furthermore, fS,T (x) = f7 (f{σ},T (x), fS\{σ},T (x)), fS,T (x) = s2 (f7 (s2 (f{σ},T (x)), s2 (fS\{σ},T (x))))

14.8 Proof of Theorem 14.1.8

421

is valid. Consequently, the functions of the type fS,T are superpositions over A (see Section 11.2). Thus, [A] = C holds in Case 1. Case 2: f6 = s2 . If the above function p is the function j4 , then we can reduce Case 2 to Case 1 because of j4 j4 s2 = j2 . Consequently, we can assume p = j0 ∈ [A]. Then we have {j0 j0 = j5 , s2 j0 = u0 , s2 j5 = u5 } ⊆ A. Further, a function of [A] is also the function f7 (x1 , x2 , x3 ) := f7 (x1 , x2 , s2 (x1 ), s2 (x2 ), c0 , j0 (x1 ), j0 (x2 ), x3 , j0 (x3 ), j5 (x2 ), j5 (x1 ), c1 , u0 (x1 ), u0 (x2 ), u5 (x3 ), u0 (x3 ), u5 (x2 ), u5 (x1 ), c2 ), ⎛

⎞ ⎛ ⎞ 0 1 0 1 2 f7 ⎝ 1 0 0 ⎠ ∈ ⎝ 1 2 ⎠ . 2 2 1 2 1

where

Since f7 ∈ C, we have f7 (2, 0, 0) = 0. Thus j0 (f7 (x, j0 , c0 )) = j2 ∈ [A], i.e., one can reduce Case 2 to Case 1. We have proven [A] = C with that. Then our theorem follows from Table 14.7 and Table 14.8. Table 14.8 Table 14.7 (1) (2) (3) (4) (5) (6) (7)

c1 − + − + + + +

p1 + − − − − − −

p2 + + − − − − −

p3 + − + + + − −

p4 − − + + − + +

p5 + + + − + + −

p6 + + − − + − +

x1 0 0 0 1 1 1 2 2 2

x2 0 1 2 0 1 2 0 1 2

p1 0 0 1 0 2 0 1 0 0

p2 0 1 0 0 0 2 1 0 0

p3 0 2 0 0 2 1 2 2 2

p4 2 0 0 2 2 2 0 2 2

p5 0 1 0 1 1 0 0 0 2

p6 0 0 0 0 0 0 0 2 1

14.8 Proof of Theorem 14.1.8 7

W.l.o.g. we can assume a = 0, b = 1, and c = 2. Further, in this proof let A be an arbitrary subset of U , which is not contained in any class from the above list, which is given in Theorem 14.1.8. Then there are some functions f1 , ..., f13 ∈ [A] with the above property ( ). First we show {c0 , c1 , c2 } ⊂ [A]. 7

Essential parts of this proof come from the proof of the following theorem, found by W. Harnau: Let A ⊆ U be arbitrary. Then [A] = U if and only if U 1 \{u3 , v3 } ⊆ [A] and A ⊆ B for every class B with the marking (9), (11), (12), (13) in Theorem 14.1.8.

422

14 Submaximal Classes of P3

Because of f2 (0, 1) = 2 and f2 ∈ U we have f2 (x, x) ∈ {u3 , v3 , c2 }. Obviously, c0 , c1 and c2 are superpositions over {c2 , f1 , f3 , f4 }. If f2 (x, x) = c2 then we can assume either {c1 , u2 } ⊆ [A] or {co , v2 } ⊆ [A] because of u3 u3 = u2 , v3 v3 = v2 , f3 (u2 , u3 ) ∈ {j3 , c1 , v2 }, f4 (v2 , v3 ) ∈ {c0 , j2 , u2 }, j3 j3 = c1 , j2 j2 = c0 , v2 u3 = v3 , u2 v3 = u3 , f8 (u2 , u3 , v2 , v3 ) ∈ {c0 , c1 , c2 , j2 , j3 } and u3 c2 = c0 . Since u2 c1 = c0 , v2 c0 = c1 and f2 (c0 , c1 ) = c2 , we have {c0 , c1 , c2 } ⊂ [A]. Next we prove U 1 ∪ P3,2 ⊆ [A]. By deﬁnition of f9 we have 0 1 2 0 1 0 1 2 2 ∈ , f9 0 1 2 1 0 2 2 0 1 i.e.,

f9

0 1 2 1 0 1 1 0 1 2 1 0 0 0

=

2 2

and therefore f9 (c0 , c1 , c2 , c1 , c0 , x, x) ∈ {u3 , v3 }. Since v3 v3 = v2 , we have that either {u2 , u3 } or {v2 , v3 } is a subset of [A]. W.l.o.g. let 0 1 2 0 1 0 2 1 f5 = 0 1 2 1 0 2 0 2

and f6

0 1 2 1 0 1 2 0 1 2 1 0 2 1

=

0 2

,

i.e., we have f5 (c0 , c1 , c2 , c0 , c1 , u2 , u3 ) = v2 and f6 (c0 , c1 , c2 , c1 , c0 , v2 , v3 ) = u2 . Thus, because of v2 u3 = v3 and u2 v3 = u3 , the set {u2 , u3 , v2 , v3 } is a subset of [A]. Further, the function f7 with f7 (x) := f7 (c0 , c1 , c2 , x) ∈ {j0 , j4 , s3 } belongs to [A]. We distinguish two cases for f7 : Case 1: f7 ∈ {j0 , j4 }. In this case, the set {f7 , c0 , c1 , f7 (v2 ), f7 (f12 (x1 , ..., x8 , c2 ))} is a subset of [A] ∩ P3,2 , however, no subset of a maximal class of P3,2 (see Lemma 14.2.3). Consequently, P3,2 is a subset of [A]. Obviously, the functions s1 and s3 are superpositions over {f11 , c2 , u2 , u3 , v2 , v3 } ∪ P3,2 . Therefore, P3,2 ∪ U 1 ⊆ [A] in Case 1. Case 2: f7 = s3 . W.l.o.g. we can assume ⎛ ⎞ ⎛ ⎞ 0 1 2 0 0 1 1 2 2 0 1 1 1 f10 ⎝ 0 1 2 0 0 1 1 2 2 1 0 ⎠ ∈ ⎝ 0 0 ⎠ . 0 1 2 1 2 0 2 0 1 2 2 0 1 Since f10 ∈ U , we have f10 (0, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2) ∈ {0, 1}, i.e.,

14.8 Proof of Theorem 14.1.8

423

f10 (c0 , c1 , c2 , c0 , u2 , c1 , u2 , c1 , v2 , u3 , v3 , x, s3 ) ∈ {j0 , j4 }. Therefore, one can reduce Case 2 to Case 1. Thus P3,2 ∪ U 1 ⊆ [A]. with Then, a binary function f11 ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 1 0⎠=⎝1⎠ f11 2 0 2 is a superposition over A. It is ⎛ 0 ⎜ ⎜ 1 f13 ⎝ 2 2

easy to prove that a 6-ary function f13 with ⎞ ⎛ ⎞ 0 0 2 2 2 0 ⎜2⎟ 2 2 0 0 2⎟ ⎟=⎜ ⎟ 1 2 1 2 0⎠ ⎝2⎠ 2 1 2 1 1 2

∈ U , we have is a superposition over P3,2 ∪ U 1 ∪ {f13 }. Since f13 ⎛ ⎞ ⎛ ⎞ 0 0 0 2 2 2 0 ⎜ ⎟ ⎜2⎟ ⎜ 0 2 2 0 0 2 ⎟ f13 ⎝ = ⎜ ⎟. 2 0 2 0 2 0⎠ ⎝2⎠ 2 2 0 2 0 0 2

We form (x1 , x2 , x3 ) := u2 (f13 (x1 , x2 , x3 , u3 (x3 ), u3 (x2 ), u3 (x1 ))) ∈ [A]. f13 Because of {f13 (0, 0, 2), f13 (0, 2, 0), f13 (2, 0, 0)} ⊆ {0, 2} we can assume w.l.o.g. f13 (0, 2, 0) = f13 (2, 0, 0). Then: f13 (x1 , x2 , c0 ) ∈ {d1 , d2 }, where d1 and d2 are deﬁned in Table 14.9.

x1 0 0 1 1 0 1 2 2 2

x2 0 1 0 1 2 2 0 1 1

d1 0 0 0 0 0 0 0 0 2

d2 0 0 0 0 2 2 2 2 2

d3 0 0 0 1 0 1 0 1 0

Table 14.9 d4 min max 0 0 0 1 0 1 1 0 1 1 1 1 0 0 2 0 1 2 0 0 2 0 1 2 0 2 2

d5 0 1 1 0 2 2 2 2 0

d6 0 1 0 0 0 2 1 1 2

d7 2 2 2 2 0 1 2 2 2

Thus because of u5 (d2 (u3 (x1 ), u3 (x2 ))) = d1 (x1 , x2 ) and

d8 1 0 0 1 1 1 1 1 1

d9 0 0 0 1 2 2 2 2 2

424

14 Submaximal Classes of P3

u3 (d1 (u3 (x1 ), u3 (x2 ))) = d2 (x1 , x2 ), the functions d1 and d2 are superpositions over A. Further, we have min(x1 , x2 ) = f11 (d3 (x1 , x2 ), d1 (x1 , x2 )) ∈ [A]

and

max(x1 , x2 ) = f11 (d4 (x1 , x2 ), d2 (x1 , x2 )) ∈ [A]

(see Table 14.9). With that, it was shown that the basis functions, determined in [Gni 65] 8 for the class U , belong to [A]. Therefore, [A] = U . Then, our theorem follows from Tables 14.9 and 14.10, where the ternary function d ∈ U is deﬁned by 2 if x1 = x2 = 2 ∨ x1 = x3 = 2 ∨ x2 = x3 = 2, d(x1 , x2 , x3 ) := 0 otherwise.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

c1 − + − + + + + − + + + + +

c2 + − + + + + + − + + + + +

j2 − + − − + + + − + + + + +

u3 − − + − + − + + − + + + +

v3 − − − + − + + + − + + + +

Table s3 d5 + − + + − + − − − − − − − − + − + − + − + − + + + +

14.10 d6 d7 + + + − − + − + − − − − − + + − + − − + − − − + + −

d8 − + − + + + − − + − + + +

d9 + + + + + − + − + + − − −

d max min + + + + + + + + + − + + + − + − + + + + + + − − + + + + + − + − + + − − − − −

14.9 Proof of Theorem 14.1.9 Let A be an arbitrary subset of B := P olι33 , which is not contained in any class from the list, which is given in Theorem 14.1.9. Then, for each i ∈ {1, 2, ..., 5} there is a function fi ∈ A which does not belong to the class with marking (i). Obviously, S := {s1 , s2 , ..., s6 } is a subset of [{f1 , f2 , f3 , f4 }]. For the function f5 (x) := f (x, x, ..., x) ∈ [A], the following cases are possible: Case 1: f5 ∈ B 1 \{c0 , c1 , c2 }. In this case, one can easily prove (P3 (2))1 ⊆ [A] when one forms the superpositions of the form sf5 s and sf5 f5 s , where {s, s } ⊂ S. Consequently, P31 ⊆ [A]. Then, by Theorem 4.3, we have [P31 ∪ {f5 }] = P olι33 = [A]. 8

See Chapter 11.

14.10 On the Cardinality of L↓3 (A) for Submaximal Clones A

425

Case 2: f5 ∈ {c0 , c1 , c2 }. Obviously, we have {co , c1 , c2 } ⊆ [S ∪{f5 }] in this case. Since f5 ∈ P3 (2)\[P31 ], there exist some tuples a := (a1 , ..., an ) and a := (a1 , ..., ai−1 , ai , ai+1 , ..., an ) with ai = ai and f5 (a) = f5 (a ). Therefore, a unary function of B 1 \{c0 , c1 , c2 } belongs to [A]. Hence, one can reduce Case 2 to Case 1. Consequently, [A] = P olι33 is proven. Since one can easily prove that the classes with the marking (1)–(5) are Bmaximal, our theorem holds.

14.10 On the Cardinality of L↓3 (A) for Submaximal Clones A The aim of this section is to prove Theorem 14.10.1 ([Bul-L-S 96]) There are 5 submaximal clones (⊂ P3 ) with ﬁnitely many subclasses and 7 submaximal clones (⊂ P3 ) whose subclass lattice is inﬁnite but countable. The subclass lattices of the remaining 146 submaximal clones of P3 have the cardinality of continuum. More precise: Let F1 , ...., F43 be the 43 families of submaximal clones of P3 given in Table 14.1 and let A ∈ Fi with i ∈ {1, 2, ..., 43}. Then ⎧ 5 if ⎪ ⎪ ⎪ ⎪ ⎪ 8 if ⎪ ⎪ ⎨ ↓ 32 if |L3 (A)| = ⎪ ⎪ ⎪ ⎪ ℵ0 if ⎪ ⎪ ⎪ ⎩ c otherwise.

i = 17, i = 7, i = 30, i ∈ {28, 29, 43},

We need the following lemmas for proof of the above theorem. Lemma 14.10.2 Let n ∈ N \ {1, 2} and pn ∈ P3n be deﬁned by 1 if ∃i : (xi = 1 ∧ ∀i = j : xi = 2), pn (x1 , ..., xn ) := 0 otherwise. Let πn be the n-ary relation αn := {(1, 2, 2, ..., 2), (2, 1, 2, ..., 2), ..., (2, 2, , ..., 2, 1)} ∪ (E3n \{1, 2}n ). Then (a) ∀n ≥ 3 : pn ∈ P ol3 αn .

426

14 Submaximal Classes of P3

(b) ∀m = n : pm ∈ P ol3 αn . (c) (pi )i≥3 is an inﬁnite basis for the clone [{pi | i ≥ 3}]. Proof. (a) follows from ⎛

⎞ ⎛ ⎞ 1 2 2 ... 2 1 ⎜ 2 1 2 ... 2 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ pn ⎜ ⎝ . . . . . . . . . . . ⎠ = ⎝ ... ⎠ ∈ αn . 2 2 2 ... 1 1

(b) is easy to check by considering the two cases m < n and m > n. (c) Let j ≥ 3. By (b), we have that pi ∈ P ol3 αj for all i ≥ 3, i = j, and thus [{p3 , p4 , ..., }\{pj }] ⊆ P ol3 αj . Since pj ∈ P ol3 αj , we deduce that pj ∈ [{p3 , p4 , ..., }\{pj }]. Consequently, {p3 , p4 , ...} is an independent set of operations, proving (c). Lemma 14.10.3 Let n ∈ N \ {1} and qn ∈ P3n be deﬁned by 1 if ∀i : xi = 2, qn (x1 , ..., xn ) := pn (x) otherwise. Then {qi | i ≥ 2} is an inﬁnite basis for the clone [{qi | i ≥ 2}]. Proof. The proof can be found in [Jan-M 59] (see also Lemma 8.1.1). Lemma 14.10.4 Let n ∈ N \ {1} and rn ∈ P3n be deﬁned by ⎧ ∃i : (xi = 2 ∧ ∀j = i : xj = 1), ⎨ 1 if 2 if x1 = ... = xn = 2, rn (x1 , ..., xn ) := ⎩ 0 otherwise. Let n be the n-ary relation n := {(a1 , ..., an ) ∈ E3n | (∃i : ai = 2 ∧ ∀j = i : aj = 1) ∨ (∃i : ai = 0)}. Then (a) ∀n ≥ 2 : rn ∈ P ol3 n . (b) ∀n = m : rm ∈ P ol3 n . (c) {ri | i ≥ 2} is an inﬁnite basis for [{ri | i ≥ 2}]. Proof. (a) follows from ⎛

⎞ ⎛ ⎞ 2 1 1 ... 1 1 ⎜ 1 2 1 ... 1 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ rn ⎜ ⎝ ...................... ⎠ = ⎝ ... ⎠ . 1 1 1 ... 2 1 (b): Let m = n, m, n ≥ 2, A = (aij )n,m be an n×m matrix on E3 with columns a1 , ..., am ∈ n , where ai = (a1i , ..., ani ) (i = 1, ..., n). If aij = 0 for some i, j

14.10 On the Cardinality of L↓3 (A) for Submaximal Clones A

427

then rm (ai1 , ..., aim ) = 0 and by deﬁnition of n , rn (A) = rn (a1 , ..., an ) ∈ . So assume that a1 , ..., am ∈ {1, 2}n . We distinguish two cases: Case 1: m < n. At least one row of the matrix A consists of 1s, hence rm (A) ∈ n . Case 2: m > n. In this case, there are i = j with ai = aj . Hence, one row of the matrix A consists of 1s or contains at least two 2s, and so rm (A) ∈ n . Therefore (b) holds. (c) follows straightforward from (a) and (b). Lemma 14.10.5 Let n ∈ N \ {1, 2} and sn ∈ P3n be deﬁned by sn (x1 , ..., xn ) := x1 if ∃i : (xi = 1 ∧ ∀j = i : xj = 0) ∨ (xi = 0 ∧ ∀j = i : xj = 1), 2 otherwise. Let σn be the n-ary relation σn := {(a1 , ..., an , an+1 ) ∈ E3n | (∃i ∈ {1, 2, ..., n} : (xi = 1 ∧ ∀j = i : xi = 0) ∨ ai = 2}. Then (a) ∀n ≥ 3 : sn ∈ P ol3 σn . (b) ∀n = m : sm ∈ P ol3 σn . (c) {si | i ≥ 3} is an inﬁnite basis for [{si | i ≥ 3}]. Proof. (a) follows from ⎞ ⎛ ⎞ 1 1 0 0 ... 0 ⎜ 0 1 0 ... 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ sn ⎜ ⎜ ...................... ⎟ = ⎜ ... ⎟ . ⎝ 0 0 0 ... 1 ⎠ ⎝ 0 ⎠ 2 0 0 0 ... 0 ⎛

(b) and (c) are easy to verify. The following lemma was found by B. Strauch. Lemma 14.10.6 Let n ∈ N \ {1, 2} and let the n-ary function tn ∈ P3n be deﬁned by ⎧ ∃i : (xi ∈ E2 ∧ ∀j = i : xj = 0), ⎨ 0 if x = 1 ∨ (∃i : xi = 2), tn (x1 , ..., xn ) := 2 if ⎩ 1 otherwise. Moreover, let

428

14 Submaximal Classes of P3

αm := {(a1 , ..., am , a1 , ..., am ) ∈ E22·m | (∃i : ai = 1) ∧ ∀j = i : aj = 0)}, x := x + 1 (mod 2), βm := (E2m \{0}m ) × {1}m , γm := E32·m \E22·m , τm := αm ∪ βm ∪ γm . Then (a) ∀n ≥ 3 : tn ∈ P ol3 τn . (b) ∀m = n : tn ∈ P ol3 τm . (c) {tn | n ≥ 3} is an inﬁnite basis for [{tn | s ≥ 3}]. Proof. (a) follows from ⎛

⎞ ⎛ ⎞ 0 1 0 0 ... 0 0 ⎜ 0 1 0 ... 0 0 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ . . . . . . . . . . . . . . ⎟ ⎜ ... ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 ... 0 1 ⎟ ⎜ 0 ⎟ ⎟ = ⎜ ⎟ ∈ τn . ⎜ sn ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 1 ... 1 1 ⎟ ⎜ 1 ⎟ ⎜ 1 0 1 ... 1 1 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎝ . . . . . . . . . . . . . . ⎠ ⎝ ... ⎠ 1 1 1 1 ... 1 0

(b): Let m, n ≥ 3, m = n and r1 , ..., rn ∈ τm . If {r1 , ..., rn } ⊆ αm ∪ βm or {r1 , ..., rn } ⊆ βm , we have tn (r1 , ..., rn ) ∈ τm . So assume {r1 , ..., rn } ⊆ αm ∪βm and {r1 , ..., rn } ⊆ βm . It is easy to see that ri = rj for some i = j implies tn (r1 , ..., rn ) ∈ τm . Further, by the deﬁnition of tn tn (r1 , ..., rn ) = {o}m × {1}m

(14.2)

is possible only if tn (r1 , ..., rn ) ∈ τm . For pairwise distinct r1 , ..., rn ∈ αm ∪βm , the condition (14.2) can be valid only if m = n. Since we assume m = n, tn (r1 , ..., rn ) ∈ τm holds. (c) follows from (a) and (b). The following two lemmas were found by A. Bulatov. For these lemmas, we need some notations. For every n ∈ N let n := {1, 2, ..., n}. Furthermore, let

:=

and for n ≥ 3

0 1 0 2 1 0 2 0

(14.3)

14.10 On the Cardinality of L↓3 (A) for Submaximal Clones A

429

n := {(a1 , a2 , ..., an ) | ∃i : (ai ∈ {1, 2} ∧ (∀j = i : aj = 0)}∪({1, 2}n \{2}n ). (14.4) We want to prove that the co-clone [{} ∪ {m | m ≥ 3}]

(14.5)

has an inﬁnite basis. Lemma 14.10.7 For a ﬁnite index-set I and every i ∈ I let πi be an mi -ary relation of {} ∪ {m | m ≥ 3}. Moreover, let ϕi be a mapping of mi in n, i ∈ I. If the relation l , l ≥ 3 has the form l = {(a1 , ...., al ) ∈ E3l | ∃al+1 , ..., an ∈ E3 : ∀i ∈ I : (aϕi (1) , aϕi (2) , ..., aϕi (mi ) ) ∈ πi }

(14.6)

for a some n and if there exists an i ∈ I with πi = , then: (a) If a tuple (a1 , ..., an ) ∈ E3n fulﬁlls the condition ∀i ∈ I : (aϕi (1) , aϕi (2) , ..., aϕi (mi ) ) ∈ πi

(14.7)

(we say : “(a1 , ..., an ) is permissible”) and if aj = 2 for a some j ∈ n, then the n-tuple (a1 , ..., aj−1 , 1, aj+1 , .., an ) is also permissible. (b) ∃α ∈ I : ϕα (mα ) ⊆ l ∧ πα = . (c) Let ϕα (mα ) ⊆ l and πα = . Then ϕα is bijective and πα = l holds. Proof. (a) follows straightforwad from the observation that, if we replace a 2 with a 1 in a tuple of l or , we obtain again a tuple from l or , respectively. (b): Suppose that ∀i ∈ I : (πi = =⇒ ϕi (mi ) ⊆ l).

(14.8)

Because of (14.4) and (14.6), there exists a permissible n-tuple of the form a := (a1 , ..., al , al+1 , ..., an ) with a1 = 1 and a2 = ... = al = 2; i.e., ∀i ∈ I : (aϕi (1) , ..., aϕi (mi ) ) ∈ πi .

(14.9)

By (a) we can suppose w.l.o.g. that ∀i ∈ {l + 1, l + 2, ..., n} : ai ∈ {0, 1}. We now show that (14.8) and (14.9) imply the permissibility of the tuple b := (b1 , ..., bn ), where b1 = ... = bl = 2 and bi = ai for all i ∈ {l+1, l+2, ..., n}, which leads to the contradiction (2, 2, ..., 2) ∈ l . The following cases are possible for α ∈ I:

430

14 Submaximal Classes of P3

Case 1: 1 ∈ ϕα (mα ). In this case, the permissibility of b follows from (14.9). Case 2: 1 ∈ ϕα (mα ); i.e., there exists an i ≤ mα with ϕα (i) = 1. Case 2.1: πα = . W.l.o.g. let ϕα (1) = 1. By (14.8) we have ϕα (2) ∈ {3, ..., n}, hence (bϕα (1) , bϕα (2) ) ∈ {(2, 1), (2, 0)} ⊆ . Case 2.2: πα = s . Case 2.2.1: 0 ∈ {aϕα (1) , ..., aϕα (1) }. By (14.8) and the deﬁnition of s we obtain (aϕα (1) , ..., aϕα (i−1) , a1 , aϕα (i+1) , ..., aϕα (s) ) = (0, ..., 0, 1, 0, ..., 0).

(14.10)

Since ak = bk for all k ≥ 2 and b1 = 2, the inclusion (bϕα (1) , ..., bϕα (s) ) ∈ s follows from (14.10). Case 2.2.2: {aϕα (1) , ..., aϕα (s) } ⊆ {1, 2}. Since ϕα (mα ) ⊆ m there exists a j such that ϕα (j) ∈ m, bϕα (j) = aϕα (j) = 1 and {bϕα (1) , ..., bϕα (s) } ⊆ {1, 2}, hence (bϕα (1) , ..., bϕα (s) ) belongs to πα and therefore b is permissible as required. (c): By assumption, there is an s ≥ 3 with πα = s . Then mα = s. First we prove that ϕα is injective. Suppose ϕα is not injective. Then the following two cases are possible: Case 1: |ϕα (mα )| = 1, i.e., ϕα (s) = {j} for a some j ≤ m. Then for every permissible tuple a := (a1 , ..., as ) we have (aϕα (1) , aϕα (2) , ..., aϕα (s) ) = (aj , aj , ..., aj ) = (1, 1, ..., 1) which contradicts (14.6) because all elements of {0, 1, 2} occur in every row of the relation l . Case 2: ∃i1 , i2 , i3 ∈ s : i1 = i2 ∧ ϕα (i1 ) = ϕα (i2 ) ∧ ϕα (i3 ) = ϕα (i1 ). W.l.o.g. let ϕ(i1 ) = 1. Then for the tuple (a1 , ..., al ) := (1, 0, 0, ..., 0) ∈ l we obtain (aϕα (1) , aϕα (2) , ..., aϕα (s) ) ∈ s , which is not possible, because of aϕ(i1 ) = aϕα (i2 ) = 1 and aϕα (i3 ) = 0. Consequently, ϕα is injective. We now show that ϕα is surjective. Suppose that ϕα (mα ) ⊂ m. We may assume w.l.o.g. that 1 ∈ ϕα (mα ). Consider the tuple a := (a1 , a2 , ..., al ) = (1, 2, 2, ..., 2) ∈ l . Then (aϕα (1) , ..., aϕα (s) ) = (2, 2, 2, ..., 2) ∈ s , in contradiction with (14.6). We have shown that ϕα is a bijection. Combining this with the properties of l , we deduce that πα = l . Lemma 14.10.8 Let n ∈ N \ {1, 2} and let and n be as in (14.3) and (14.4). Then: (a) ∈ [{n | n ≥ 3}]. (b) ∀l ≥ 3 : l ∈ [{} ∪ {n | n ∈ N\{1, 2, l}}]. (c) The co-clone [{} ∪ {n | n ≥ 3}] has the inﬁnite basis {, 3 , 4 , ...}. Proof. (a) follows from the fact that the constant function c1 preserves the relation n for all n ≥ 3 but does not preserve the relation .

14.10 On the Cardinality of L↓3 (A) for Submaximal Clones A

431

(b): Suppose for an l ≥ 3 we have l ∈ [{} ∪ {n | n ∈ N\{1, 2, l}}]. Then, by Theorem 2.11.2,(a),(b) l = {(aϕ(1) , , ...., aϕ(l) ) ∈ E3l | ∃al+1 , ..., an ∈ E3 : ∀i ∈ I : (aϕi (1) , aϕi (2) , ..., aϕi (mi ) ) ∈ πi },

(14.11)

holds, where I is an index-set, πi are some mi -ary relations of {}∪{m | m ≥ 3, m = l} for all i ∈ I and ϕ : l −→ n and ϕi : mi −→ n are mappings. There must be at least a relation under the relations πi , which is diﬀerent from the relation , since in the opposite case, l ∈ [{}] holds and this implies P ol3 ∪ {c1 } ⊆ P ol 3 l , which by means of Theorem 14.1.8 0 1 2 1 2 [P ol3 ∪ {c1 }] = P ol3 ⊆ P ol3 l results from, which contra0 1 2 2 1 0 1 2 1 2 dicts c2 ∈ P ol3 l and c2 ∈ P ol3 . 0 1 2 2 1 It results now from the deﬁnition of the relation l that ϕ is an injective mapping. W.l.o.g. we can assume that ϕ(x) = x for all x ∈ l. Therefore (14.11) agrees with (14.6) and, with the help of Lemma 14.10.7, we have a contradiction to our assumption. (c) follows directly from (a) and (b). Proof of Theorem 14.10.1: The statements for the classes with the numbers 7, 17, or 30 of Table 14.1 are consequences from Chapter 13 (see also Theorem 15.1.1). For the classes with the number 28, 29, or 43, our assertion follows from Theorem 8.1.6 and Sections 15.3 and 15.4. 0 1 2 Since the class of Theorem 8.2.2 is a subclass of A8 = P ol3 {0}∩P ol3 1 2 0 we have |L↓ (A8 )| = c. The clones A ∈ Fj with j ∈ J := {9, 14, 24, 25, 27, 33, 35, 37, 39, 40, 41, 42} obviously satisfy the condition ∃a, b ∈ E3 : a = b ∧ P3,{a,b} ⊆ Ai . Hence for every j ∈ J there exists a subclass of A with inﬁnite basis such that the range of its functions is a 2-element set (see for instance Lemma 14.10.2). An example of a subclass of A ∈ F22 with inﬁnite basis is given in the proof of Theorem 12.3.8. Let A ∈ F32 with a = 1, b = 2 and c = 0. We have shown in Lemma 14.10.8 that there are uncountable many co-clones containing Inv3 A. This means that |L3 (J3 , A32 )| = c (see Chapter 2).

432

14 Submaximal Classes of P3

Finally, we use constructions given in Lemmas 14.10.2–14.10.6 to show that |L↓3 (A)| = c with A ∈ Fi for the remaining i. All results are presented in Table 14.11. Table 14.11 Let A ∈ Fi , where Considered cases Use the construci= for a, b, c, α, β, γ tion given in Lemma 14.10.j, where j = (w.l.o.g.)

1 2,4 3 5,19,20,21,23,26,31 6,13,16,36,38 10 11 11 11 12 12 15 18 34

a 0 0 2 0 0 0 0 0 0 0 0 0 2 0

b 2 1 0 1 1 2 1 2 1 1 2 1 0 1

c α 1 2 1 2 2 1 0 0 2 2 0 1 1 2 0 1 2

β γ

1 1 1 1 0 1

2 2 0 2 2 2

4 4 4 2 3 2 3 8 3 2 2 3 5 5

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

We say that the class A ∈ Lk (or the lattice L↓k (A) of subclasses of A = [A] ⊆ Pk ) has the depth t, if t is the least integer for which there are some classes A1 , ..., At−1 ∈ Lk with A ⊂ A1 ⊂ A2 ⊂ ... ⊂ At−1 ⊂ Pk . In particular, it holds that the maximal classes of Pk have the depth 1 and the submaximal classes of Pk have the depth 2. For k = 3 by Theorems 13.2.3 and 8.1.6, there are ﬁnite and countably inﬁnite sublattices of depth 1 or 2. In this chapter, these sublattices will be determined. The ﬁnite lattice L↓3 (L3 ) of depth 1 can be found in Section 15.1. This lattice is a conclusion from Chapter 13. In addition, by Section 14.10, this lattice is the only ﬁnite lattice of depth 1, and furthermore, this lattice contains all ﬁnite sublattices of L3 of depth 2. In Section 15.2 is a description of all subsemigroups (or subclasses) of (Ps1 ; ) (or [P31 ]), respectively. These subclasses are also subclasses of the submaximal class L of all quasilinear functions of P3 . The list of all subsemigroups of (Ps1 ; ) is then an important aid in Section 15.3 during the determination of the remaining elements of lattice L↓3 (L). Because of Theorem 8.1.6, a further countable sublattice of depth 2 is L↓3 ([O1 ∪ {max}]). This sublattice is given in Section 15.4. Except for isomorphic lattices, all countable sublattices of L3 of depth 2 are traced with that (see Section 14.10).

15.1 The Lattice of Subclasses of P3 of Linear Functions Let L3 be the set of all linear functions of P3 . As a consequence of Theorem 13.2.1, one can see that L↓3 (L3 ) has exactly 6 elements, which are not subsets of [L13 ]. The subclasses of [L13 ] is obtained from Theorem 13.2.2 or from Section 15.2. Thus it holds:

434

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Theorem 15.1.1 The class L3 of all linear functions of P3 has exactly 38 subclasses: 012 0 , L3 ∩ , [Hi ], L3 , L3 ∩ P ol3 {a}(a ∈ E3 ), L3 ∩ 120 1 where i ∈ {1, 2, 3, 4, 5, 6, 7, 8, 601, 602, 603, 604, 605, 606, 607, 608, 1201, 1202, 1203, 1204, 1232, 1233, 1234, 1235, 1263, 1264, 1265, 1266, 1294, 1295, 1297, 1298} (see Table 15.10 of Section 15.2).

15.2 The Subsemigroups of (P31 ; ) In this section, we determine the subclasses of [P31 ]. It suﬃces to describe all subsemigroups of (P31 ; ) for this. Since the subsemigroups of (P31 ; ) play a role in many investigations of P3 , it is particularly a question of clarifying the construction of these subsemigroups. To facilitate checking the following considerations without a lot of expenditure also by hand, some tables are given at the end of this section. The functions of P31 and their notations are given in Table 15.1. As usual the operation is deﬁned by (f g)(x) := f (g(x)) P31

(see Table 15.2). for all f, g ∈ The number statement of the following theorem was published by G. Wilde and Sh. Raney in 1972 without proof (as a result of a computer calculation). Theorem 15.2.1 ([Wil-R 72], [Lau 84a]) (P31 ; ∗) has exactly 1299 subsemigroups (including ∅), which are listed in Table 15.10. Proof.1 In preparation for the proof of the above theorem, some notations, which we use during the description of the subsemigroups of (P31 , ∗), are given. Let C := {c0 , c1 , c2 }, J := {c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 }, U := {c0 , c2 , u0 , u1 , u2 , u3 , u4 , u5 }, V := {c1 , c2 , v0 , v1 , v2 , v3 , v4 , v5 }, S := {s1 , s2 , s3 , s4 , s5 , s6 }. Starting from the possible subsemigroups Ji (i ∈ {−3, −2, −1, 0, 1, 2, ..., 41}) of J (= J41 ), given in Table 15.4, one can construct, with the aid of the following mappings, 1

Basically, the following proof comes from [Lau 84a], which required correction in some places. I owe K. Todorov (Soﬁa) and Anne Fearnley (Montreal) indications of the mistakes.

15.2 The Subsemigroups of (P31 ; )

435

ϕi : f → s−1 i ∗ f ∗ si (i ∈ {2, 3, ..., 6}) and the notation 1 ϕi (A) := { s−1 i ∗ f ∗ si | f ∈ A } (A ⊆ P3 )

the classes isomorphic to Ji . These isomorphic classes are given in Tables 15.5–15.9, where we use the notations Ui := ϕ2 (Ji ) and Vi := ϕ6 (Ji ) (i ∈ {−3, −2, −1, 0, 1, 2, ..., 41}). Furthermore we use the notations: S0 := ∅, S1 := {s1 }, S2 := {s1 , s3 }, S3 := {s1 , s2 }, S4 := {s1 , s6 }, S5 := {s1 , s4 , s5 }, S6 := S. To make the structure of the subsemigroups recognizable short, we assign a tuple τ (H) as follows to every subsemigroup H: τ (H) := (a, b, c, d) :⇐⇒ H ∩ J = Ja ∧ H ∩ U = Ub ∧ H ∩ V = Vc ∧ H ∩ S = Sd . Obviously, it holds H = H ⇐⇒ τ (H) = τ (H ). H always denotes an arbitrary subsemigroup of (P31 ; ∗). The number of possibilities for H in one of the below cases i for H is denoted with ni . In Cases 4.1 and 4.2 some possibilities are already included in other cases for H, so that we must change the number ni into the number ni , i.e., ni denotes the number of possibilities for H in Case i, which were not included in the preceded cases. In most of the following cases i for H, the numbers ni or/and ni and the possibilities for H are given. To check the following proof is an arduous matter that should be carried out only with the aid of a computer. A summary of all possibilities for H together with the corresponding characteristic tuples τ (H) can be found in Table 15.10. Case 1: H ⊆ C. Obviously, n1 = 8 and H = Hi (i = 1, ..., 8; see Table 15.10) is an arbitrary subset of C. Case 2: H ⊆ C and H ⊆ A ∈ {J, U, V }. Case 2.1: H ⊆ J. It is not hard to check that n2.1 = 41 and H ∈ {J1 , J2 , ..., J41 } (or H = Ht , t ∈ {9, 10, ..., 49}, see Table 15.10).

436

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Case 2.2: H ⊆ U . By ϕ2 (J) = U and by Case 2.1, we have n2.2 = 41, and H is one of the following sets: Ui := ϕ2 (Ji ) with i ∈ {1, 2, ..., 41} (or H = Ht for t ∈ {50, 51, ..., 90}). Case 2.3: H ⊆ V . By ϕ6 (J) = V and by Case 2.1, we have n2.3 = 41, and for H only, the following sets are possible: Vi := ϕ6 (Ji ) with i ∈ {1, 2, ..., 41} (or H = Hi , i ∈ {91, 92, ..., 131}). Case 3: H ⊆ J ∪ U , H ⊆ J and H ⊆ U . Case 3.1: H ⊆ C ∪ {j1 , j4 , u2 , u3 } = J25 ∪ U25 . The possibilities for H are: J3 ∪ {c0 , c2 }, {c0 , c1 } ∪ U3 , J12 ∪ {c0 , c2 }, {c0 , c1 } ∪ U12 , J25 ∪ {c0 , c2 }, {c0 , c1 } ∪ U25 and Jp ∪ Uq , where (p, q) ∈ {(3, 3), (3, 12), (12, 3), (12, 12), (3, 25), (25, 3), (12, 25), (25, 12), (25, 25)} (or H = Hi with i ∈ {132, 133, ..., 146}); i.e., we have n3.1 = 15. Case 3.2: H ∩ J ⊆ {c0 , c1 , j1 , j4 } and H ∩ U ⊆ {c0 , c2 , u2 , u3 }. For H ∩ U only the following sets are possible: {c2 }, {c0 , c2 }, {c0 , u2 } = U3 , {c0 , c2 , u2 } = U12 and {c0 , c2 , u2 , u3 } = U25 . Case 3.2.1: H ∩ U = {c2 }. Obviously, H = J8 ∪ {c2 } ( = H147 ). Case 3.2.2: H ∩ U = {c0 , c2 }. In this case, H ∩ J is an arbitrary subsemigroup of J, which contains {c0 , c1 }, but no subset is of J25 , i.e., it holds n3.2.2 = 19 and H = {c0 , c2 } ∪ Ji with i ∈ {13, 14, 15, 22, 23, 24, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41} (or H = Ht with t ∈ {148, 149, ..., 166}). Case 3.2.3: H ∩ U = U3 . In this case, we have n3.2.3 = 17 and H = U3 ∪ Ji with i ∈ {4, 13, 14, 16, 18, 23, 24, 27, 28, 30, 33, 34, 36, 38, 39, 40, 41} (or H = Ht with t ∈ {167, 168, ..., 183}). Case 3.2.4: H ∩ U = U12 . It is easy to check that n3.2.4 = 13 and H = U12 ∪ Ji with i ∈ {13, 14, 23, 24, 27, 28, 33, 34, 36, 38, 39, 40, 41} (or H = Ht with t ∈ {184, 185, ..., 196}). Case 3.2.5: H ∩ U = U25 . In this case, H ∩ J is an arbitrary subsemigroup of J, which contains {j2 , j3 }. Thus, we have n3.2.5 = 7 and H = U25 ∪ Ji with i ∈ {27, 33, 36, 38, 39, 40, 41} (or H = Ht with t ∈ {197, 198, ..., 203}). In summary, we get: n3.2 = 57. Case 3.3: H ∩ J ⊆ {c0 , c1 , j1 , j4 } and H ∩ U ⊆ {c0 , c1 , u2 , u3 }. In this case, H is isomorphic to a subsemigroup of J ∪ U which fulﬁlls the conditions of Case 3.2. Consequently, we have n3.3 = 57 and H = ϕ2 (Hi ) with i ∈ {147, 148, ..., 203} (or H = Ht with t ∈ {204, 205, ..., 260}). We remark that ϕ2 (Ja ∪ Ub ) = Jb ∪ Ua . Case 3.4: H ∩ J ⊆ {c0 , c1 , j1 , j4 } and H ∩ U ⊆ {c0 , c1 , u2 , u3 }. Then there are two functions f and g of H with 0 02 0 01 f ∈ and g ∈ . 1 20 2 10

15.2 The Subsemigroups of (P31 ; )

437

This implies |H ∩ J| = |H ∩ U | and, if (H\C) ∩ J = {ji | i ∈ I} with I ⊆ {0, 1, ..., 5}, (H\C)∩U ∈ {{ui | i ∈ I}, {u5−i | i ∈ I}}. Examining the possible cases yields n3.4 = 19 and H = Jp ∪ Uq with (p, q) ∈ {(2, 2), (5, 5), (8, 8), (9, 9), (15, 15), (16, 16), (17, 18, (18, 17), (22, 22), (23, 23), (24, 24), (26, 28), (28, 26), (30, 30), (34, 34), (37, 38), (38, 37), (39, 39), (41, 41)} . Summing up, we get n3 = 148. Case 4: H ⊆ A ∪ B ∈ {J ∪ V, U ∪ V }, H ⊆ A and H ⊆ B. Case 4.1: A = J and B = V . By ϕ3 (J ∪ U ) = J ∪ V we have n4.1 = 148, and H is isomorphic to a subsemigroup which we have already determined in Case 3. Hence, one receives a list of the subsemigroups with the aid of the results from the third case, where one has to consider ϕ3 (Ja ∪ Ub ) = ϕ3 (Ja ) ∪ ϕ5 (Jb ) (see Table 15.7 and 15.9). Twenty-three of the 148 subsemigroups were already determined, so that we have n4.1 = 125. 2 In Table 15.10 (for the Case 4.1), 148 sets are given, where every already listed set, is characterized by their ﬁrst number; this ﬁrst number is given in bold point. Case 4.2: A = U and B = V . By ϕ4 (J ∪ U ) = U ∪ V , we have n4.2 = 148 and H is isomorphic to a subsemigroup that we have already determined in Case 3. While listing the subsemigroups, one notices that ϕ4 (Ja ∪ Ub ) = ϕ4 (Ja ) ∪ Vb holds. With the aid of Table 15.10 one sees that 48 of the subsemigroups were determined in previous cases. Thus n4.2 = 102. Case 5: H ⊆ J ∪ U ∪ V , H ⊆ J ∪ U , H ⊆ J ∪ V and H ⊆ U ∪ V . Case 5.1: H ∩ (U ∪ V ) ⊆ C ∪ {u2 , u3 , v2 , v3 }. For H ∩ (U ∪ V ), only the sets U3 ∪ V8 = {c0 , c1 , u2 , v2 }, U12 ∪ V15 = {c0 , c1 , c2 , u2 , v2 } and U25 ∪ V22 = {c0 , c1 , c2 , u2 , u3 , v2 , v3 } come into consideration. Case 5.1.1: H ∩ (U ∪ V ) = U3 ∪ V8 . Then, all the subsets Ji of J which contain the two functions c0 , c1 and which have the properties j0 ∈ Ji ∨ j4 ∈ Ji =⇒ {j2 , j3 } ⊆ Ji , j1 ∈ Ji =⇒ j3 ∈ Ji , j5 ∈ Ji =⇒ j2 ∈ Ji . are possible for H ∩ J. Thus n5.1.1 = 11 and H = Ji ∪ U3 ∪ V8 with i ∈ {13, 14, 24, 27, 28, 33, 36, 38, 39, 40, 41}. Case 5.1.2: H ∩ (U ∪ V ) = U12 ∪ V15 . With the aid of Case 5.1.1, we obtain n5.1.2 = 11 and H = Ji ∪ U12 ∪ V15 , i ∈ {13, 14, 24, 27, 28, 33, 36, 38, 39, 40, 41}. Case 5.1.3: H ∩ (U ∪ V ) = U25 ∪ V22 . Then H ∩ J is a subsemigroup of J, which contains j2 and j3 . Consequently, n5.1.3 = 7 and H = Ji ∪ U25 ∪ V22 , i ∈ {27, 33, 36, 38, 39, 40, 41}. In summary, we obtain n5.1 = 29. 2

This was not taken into account in [Lau 84a]!

438

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Case 5.2: H ∩ (J ∪ V ) ⊆ C ∪ {j1 , j4 , v1 , v4 }. Because of ϕ2 (C ∪ {j1 , j4 , v1 , v4 }) = U25 ∪ V22 , we have n5.2 = 29 and H = ϕ2 (A), where A is a set which we have already determined in Case 5.1. More exactly: If A = Ja ∪ Ub ∪ Vc then ϕ2 (A) = Jb ∪ Ua ∪ ϕ5 (Jc ). Case 5.3: H ∩ (J ∪ U ) ⊆ C ∪ {j0 , j5 , u0 , u5 }. This case is also isomorphic to Case 5.1. One can obtain all 29 possibilities for H with the aid of the mapping ϕ6 from the sets determined in Case 5.1, where ϕ6 (Ja ∪ Ub ∪ Vc ) = Jc ∪ ϕ4 (Jb ) ∪ Va must be pointed out. Case 5.4: H fulﬁlls none of the conditions from Cases 5.1, 5.2, or 5.3. In this case, there are functions f, g, h ∈ H with 0 0212 0 0112 1 0102 f ∈ ,g ∈ and h ∈ . 1 2021 2 1021 2 1020 Since certain functions ja , ub , vc for certain a, b, c belong to H, one easily checks that C ⊂ H and, if (H\C) ∩ J = {ji | i ∈ I}, it holds (H\C) ∩ U ∈ {{ui | i ∈ I}, {u5−i | i ∈ I}} and (H\C) ∩ V ∈ {{vi | i ∈ I}, {v5−i | i ∈ I}}. Examining the possible cases using the results of Case 2 yields n5.4 = 7 and H = Jp ∪ Uq ∪ Vr with (p, q, r) ∈ {(24, 24, 26), (26, 28, 24), (28, 26, 28), (37, 38, 39), (38, 37, 38), (39, 39, 37), (41, 41, 41)}. Consequently, n5 = 94. Case 6: S ∩ H = ∅. Since S as is well-known has 6 subgroups, the following cases are possible: Case 6.1: S ∩ H = {s1 }. Then we have H = A ∪ {s1 }, where A is one of the 600 subsemigroups of J ∪ U ∪ V determined above. Case 6.2: S ∩ H = {s1 , s3 }. In this case, H\S is one of the following 31 sets: ∅, {c2 }, {c0 , c1 }, C, J27 , J32 , J37 , J, J27 ∪ {c2 }, J37 ∪ {c2 }, C ∪ J, U1 ∪ V2 , U3 ∪ V8 , U6 ∪ V5 , U12 ∪ V15 , U10 ∪ V9 , U25 ∪ V22 , U21 ∪ V16 , U29 ∪ V23 , U31 ∪ V30 , U35 ∪ V34 , U38 ∪ V39 , U ∪ V , J27 ∪ U3 ∪ V8 , J27 ∪ U12 ∪ V15 , J27 ∪ U25 ∪ V22 , J37 ∪ U38 ∪ V39 , J ∪ U12 ∪ V15 , J ∪ U3 ∪ V8 , J ∪ U25 ∪ V22 , J ∪ U ∪ V . Case 6.3: S ∩ H = {s1 , s2 }. In this case, n6.3 = 31 and the possibilities for H be determined with the results from Case 6.2 and with the mapping ϕ6 . Case 6.4: S ∩ H = {s1 , s6 }. In this case, n6.4 = 31 and one can determine the possibilities for H, using Case 6.2 and the mapping ϕ2 . Case 6.5: S ∩ H ∈ {{s1 , s4 , s5 }, S}. For H\S only the sets ∅, C and J ∪ U ∪ V are possible, whereby n6.5 = 6. Consequently, we have n6 = 699. In summary, we get that (P31 ; ∗) has exactly 1299 diﬀerent subsemigroups. We remark that one ﬁnds further information about the given subsemigroups in [Bij-T 91].

15.2 The Subsemigroups of (P31 ; )

439

Table 15.1

x j0 (x) j1 (x) j2 (x) j3 (x) j4 (x) j5 (x) u0 (x) u1 (x) u2 (x) u3 (x) u4 (x) u5 (x) 0 1 0 0 1 1 0 2 0 0 2 2 0 1 0 1 0 1 0 1 0 2 0 2 0 2 2 0 0 1 0 1 1 0 0 2 0 2 2 x v0 (x) v1 (x) v2 (x) v3 (x) v4 (x) v5 (x) s1 (x) s2 (x) s3 (x) s4 (x) s5 (x) s6 (x) 0 2 1 1 2 2 1 0 0 1 1 2 2 1 1 2 1 2 1 2 1 2 0 2 0 1 2 1 1 2 1 2 2 2 1 2 0 1 0

Table 15.2

f ∗g j0 j1 j2 j3 j4 j5 u0 u1 u2 u3 u4 u5 v0 v1 v2 v3 v4 v5 s2 s3 s4 s5 s6

ji j5−i ji c0 c1 j5−i ji u5−i ui c0 c2 u5−i ui v5−i vi c1 c2 v5−i vi ui j5−i vi u5−i v5−i

ui j5−i c0 ji j5−i c1 ji u5−i c0 ui u5−i c2 ui v5−i c1 vi v5−i c2 vi ji vi j5−i v5−i u5−i

vi s2 s3 c0 j0 j1 j5−i j2 j0 ji j1 j2 j5−i j4 j3 ji j3 j5 c1 j5 j4 c0 u0 u1 u5−i u2 u0 ui u1 u2 u5−i u4 u3 ui u3 u5 c2 u5 u4 c1 v0 v1 v5−i v2 v0 vi v1 v2 v5−i v4 v3 vi v3 v5 c2 v5 v4 v5−i s1 s5 ui s4 s1 u5−i s3 s6 ji s6 s2 j5−i s5 s4

s4 j2 j0 j1 j4 j5 j3 u2 u0 u1 u4 u5 u3 v2 v0 v1 v4 v5 v3 s6 s2 s5 s1 s3

s5 j1 j2 j0 j5 j3 j4 u1 u2 u0 u5 u3 u4 v1 v2 v0 v5 v3 v4 s3 s6 s1 s4 s2

s6 j2 j1 j0 j5 j4 j3 u2 u1 u0 u5 u4 u3 v2 v1 v0 v5 v4 v3 s4 s5 s2 s3 s1

440

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3 Table 15.3

ϕ2 (f ) = ϕ3 (f ) = ϕ5 (f ) = ϕ4 (f ) = ϕ6 (f ) = f s2 ∗ f ∗ s2 s3 ∗ f ∗ s3 s4 ∗ f ∗ s5 s5 ∗ f ∗ s4 s6 ∗ f ∗ s6 c0 c0 c1 c1 c2 c2 c1 c2 c0 c2 c0 c1 c2 c1 c2 c0 c1 c0 j0 u0 j4 v1 u3 v3 j1 u2 j5 v2 u5 v4 j2 u1 j3 v0 u4 v5 j3 u4 j2 v5 u1 v0 j4 u3 j0 v3 u0 v1 j5 u5 j1 v4 u2 v2 u0 j0 v1 j4 v3 u3 u1 j2 v0 j3 v5 u4 u2 j1 v2 j5 v4 u5 u3 j4 v3 j0 v1 u0 u4 j3 v5 j2 v0 u1 u5 j5 v4 j1 v2 u2 v0 v5 u1 u4 j2 j3 v1 v3 u0 u3 j0 j4 v2 v4 u2 u5 j1 j5 v3 v1 u3 u0 j4 j0 v4 v2 u5 u2 j5 j1 v5 v0 u4 u1 j3 j2 s1 s1 s1 s1 s1 s1 s2 s2 s6 s6 s3 s3 s3 s6 s3 s2 s6 s2 s4 s5 s5 s4 s4 s5 s5 s4 s4 s5 s5 s4 s6 s3 s2 s3 s2 s6

15.2 The Subsemigroups of (P31 ; ) Table 15.4 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Ji {c0 , c1 } {c1 } {c0 } ∅ {j1 } {j5 } {c0 , j1 } {c0 , j2 } {c0 , j5 } {c1 , j1 } {c1 , j3 } {c1 , j5 } {j0 , j5 } {j1 , j4 } {j1 , j5 } {c0 , c1 , j1 } {c0 , c1 , j2 } {c0 , c1 , j3 } {c0 , c1 , j5 } {c0 , j1 , j2 } {c0 , j1 , j5 } {c0 , j2 , j5 } {c1 , j1 , j3 } {c1 , j1 , j5 } {c1 , j3 , j5 } {c0 , c1 , j0 , j5 } {c0 , c1 , j1 , j2 } {c0 , c1 , j1 , j3 } {c0 , c1 , j1 , j4 } {c0 , c1 , j1 , j5 } {c0 , c1 , j2 , j3 } {c0 , c1 , j2 , j5 } {c0 , c1 , j3 , j5 } {c0 , j1 , j2 , j5 } {c1 , j1 , j3 , j5 } {j0 , j1 , j4 , j5 } {c0 , c1 , j1 , j2 , j3 } {c0 , c1 , j1 , j2 , j5 } {c0 , c1 , j1 , j3 , j5 } {c0 , c1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j4 , j5 } {c0 , c1 , j0 , j2 , j3 , j5 } {c0 , c1 , j1 , j2 , j3 , j4 } {c0 , c1 , j1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 }

Table 15.5 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Ui := ϕ2 (Ji ) {c0 , c2 } {c2 } {c0 } ∅ {u2 } {u5 } {c0 , u2 } {c0 , u1 } {c0 , u5 } {c2 , u2 } {c2 , u4 } {c2 , u5 } {u0 , u5 } {u2 , u3 } {u2 , u5 } {c0 , c2 , u2 } {c0 , c2 , u1 } {c0 , c2 , u4 } {c0 , c2 , u5 } {c0 , u2 , u1 } {c0 , u2 , u5 } {c0 , u1 , u5 } {c2 , u2 , u4 } {c2 , u2 , u5 } {c2 , u4 , u5 } {c0 , c2 , u0 , u5 } {c0 , c2 , u2 , u1 } {c0 , c2 , u2 , u4 } {c0 , c2 , u2 , u3 } {c0 , c2 , u2 , u5 } {c0 , c2 , u1 , u4 } {c0 , c2 , u1 , u5 } {c0 , c2 , u4 , u5 } {c0 , u2 , u1 , u5 } {c2 , u2 , u4 , u5 } {u0 , u2 , u3 , u5 } {c0 , c2 , u2 , u1 , u4 } {c0 , c2 , u2 , u1 , u5 } {c0 , c2 , u2 , u4 , u5 } {c0 , c2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u1 , u4 , u3 , u5 }

441

442

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3 Table 15.6 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Vi := ϕ6 (Ji ) {c2 , c1 } {c1 } {c2 } ∅ {v4 } {v2 } {c2 , v4 } {c2 , v5 } {c2 , v2 } {c1 , v4 } {c1 , v0 } {c1 , v2 } {v3 , v2 } {v4 , v1 } {v4 , v2 } {c2 , c1 , v4 } {c2 , c1 , v5 } {c2 , c1 , v0 } {c2 , c1 , v2 } {c2 , v4 , v5 } {c2 , v4 , v2 } {c2 , v5 , v2 } {c1 , v4 , v0 } {c1 , v4 , v2 } {c1 , v0 , v2 } {c2 , c1 , v3 , v2 } {c2 , c1 , v4 , v5 } {c2 , c1 , v4 , v0 } {c2 , c1 , v4 , v1 } {c2 , c1 , v4 , v2 } {c2 , c1 , v5 , v0 } {c2 , c1 , v5 , v2 } {c2 , c1 , v0 , v2 } {c2 , v4 , v5 , v2 } {c1 , v4 , v0 , v2 } {v3 , v4 , v1 , v2 } {c2 , c1 , v4 , v5 , v0 } {c2 , c1 , v4 , v5 , v2 } {c2 , c1 , v4 , v0 , v2 } {c2 , c1 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v1 , v2 } {c2 , c1 , v3 , v5 , v0 , v2 } {c2 , c1 , v4 , v5 , v0 , v1 } {c2 , c1 , v4 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v5 , v0 , v1 , v2 }

Table 15.7 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ϕ3 (Ji ) = Jt {c1 , c0 } {c0 } {c1 } ∅ {j5 } {j1 } {c1 , j5 } {c1 , j3 } {c1 , j1 } {c0 , j5 } {c0 , j2 } {c0 , j1 } {j4 , j1 } {j5 , j0 } {j5 , j1 } {c1 , c0 , j5 } {c1 , c0 , j3 } {c1 , c0 , j2 } {c1 , c0 , j1 } {c1 , j5 , j3 } {c1 , j5 , j1 } {c1 , j3 , j1 } {c0 , j5 , j2 } {c0 , j5 , j1 } {c0 , j2 , j1 } {c1 , c0 , j4 , j1 } {c1 , c0 , j5 , j3 } {c1 , c0 , j5 , j2 } {c1 , c0 , j5 , j0 } {c1 , c0 , j5 , j1 } {c1 , c0 , j3 , j2 } {c1 , c0 , j3 , j1 } {c1 , c0 , j2 , j1 } {c1 , j5 , j3 , j1 } {c0 , j5 , j2 , j1 } {j4 , j5 , j0 , j1 } {c1 , c0 , j5 , j3 , j2 } {c1 , c0 , j5 , j3 , j1 } {c1 , c0 , j5 , j2 , j1 } {c1 , c0 , j3 , j2 , j1 } {c1 , c0 , j4 , j5 , j0 , j1 } {c1 , c0 , j4 , j3 , j2 , j1 } {c1 , c0 , j5 , j3 , j2 , j0 } {c1 , c0 , j5 , j3 , j2 , j1 } {c1 , c0 , j4 , j5 , j3 , j2 , j0 , j1 }

t −3 −1 −2 0 2 1 8 7 6 5 4 3 10 9 11 15 14 13 12 21 20 19 18 17 16 25 29 28 22 26 27 24 23 31 30 32 36 35 34 33 37 39 38 40 41

15.2 The Subsemigroups of (P31 ; ) Table 15.8 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ϕ4 (Ji ) = Ut {c2 , c0 } {c0 } {c2 } ∅ {u5 } {u2 } {c2 , u5 } {c2 , u4 } {c2 , u2 } {c0 , u5 } {c0 , u1 } {c0 , u2 } {u3 , u2 } {u5 , u0 } {u5 , u2 } {c2 , c0 , u5 } {c2 , c0 , u4 } {c2 , c0 , u1 } {c2 , c0 , u2 } {c2 , u5 , u4 } {c2 , u5 , u2 } {c2 , u4 , u2 } {c0 , u5 , u1 } {c0 , u5 , u2 } {c0 , u1 , u2 } {c2 , c0 , u3 , u2 } {c2 , c0 , u5 , u4 } {c2 , c0 , u5 , u1 } {c2 , c0 , u5 , u0 } {c2 , c0 , u5 , u2 } {c2 , c0 , u4 , u1 } {c2 , c0 , u4 , u2 } {c2 , c0 , u1 , u2 } {c2 , u5 , u4 , u2 } {c0 , u5 , u1 , u2 } {u3 , u5 , u0 , u2 } {c2 , c0 , u5 , u4 , u1 } {c2 , c0 , u5 , u4 , u2 } {c2 , c0 , u5 , u1 , u2 } {c2 , c0 , u4 , u1 , u2 } {c2 , c0 , u3 , u5 , u0 , u2 } {c2 , c0 , u3 , u4 , u1 , u2 } {c2 , c0 , u5 , u4 , u1 , u0 } {c2 , c0 , u5 , u4 , u1 , u2 } {c2 , c0 , u3 , u5 , u4 , u1 , u0 , u2 }

Table 15.9 t −3 −1 −2 0 2 1 8 7 6 5 4 3 10 9 11 15 14 13 12 21 20 19 18 17 16 25 29 28 22 26 27 24 23 31 30 32 36 35 34 33 37 39 38 40 41

i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ϕ5 (Ji ) = Vt {c1 , c2 } {c2 } {c1 } ∅ {v2 } {v4 } {c1 , v2 } {c1 , v0 } {c1 , v4 } {c2 , v2 } {c2 , v5 } {c2 , v4 } {v1 , v4 } {v2 , v3 } {v2 , v4 } {c1 , c2 , v2 } {c1 , c2 , v0 } {c1 , c2 , v5 } {c1 , c2 , v4 } {c1 , v2 , v0 } {c1 , v2 , v4 } {c1 , v0 , v4 } {c2 , v2 , v5 } {c2 , v2 , v4 } {c2 , v5 , v4 } {c1 , c2 , v1 , v4 } {c1 , c2 , v2 , v0 } {c1 , c2 , v2 , v5 } {c1 , c2 , v2 , v3 } {c1 , c2 , v2 , v4 } {c1 , c2 , v0 , v5 } {c1 , c2 , v0 , v4 } {c1 , c2 , v5 , v4 } {c1 , v2 , v0 , v4 } {c2 , v2 , v5 , v4 } {v1 , v2 , v3 , v4 } {c1 , c2 , v2 , v0 , v5 } {c1 , c2 , v2 , v0 , v4 } {c1 , c2 , v2 , v5 , v4 } {c1 , c2 , v0 , v5 , v4 } {c1 , c2 , v1 , v2 , v3 , v4 } {c1 , c2 , v1 , v0 , v5 , v4 } {c1 , c2 , v2 , v0 , v5 , v3 } {c1 , c2 , v2 , v0 , v5 , v4 } {c1 , c2 , v1 , v2 , v0 , v5 , v3 , v4 }

t −3 −1 −2 0 2 1 8 7 6 5 4 3 10 9 11 15 14 13 12 21 20 19 18 17 16 25 29 28 22 26 27 24 23 31 30 32 36 35 34 33 37 39 38 40 41

443

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

444

Table 15.103 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 3

τ (Hi ) (0, 0, 0, 0) (−1, −1, 0, 0) (−1, 0, −1, 0) (0, −2, −2, 0) (−3, −1, −1, 0) (−1, −3, −2, 0) (−2, −2, −3, 0) (−3, −3, −3, 0) (1, 0, 0, 0) (2, 0, 0, 0) (3, −1, 0, 0) (4, −1, 0, 0) (5, −1, 0, 0) (6, 0, −2, 0) (7, 0, −2, 0) (8, 0, −2, 0) (9, 0, 0, 0) (10, 0, 0, 0) (11, 0, 0, 0) (12, −1, −2, 0) (13, −1, −2, 0) (14, −1, −2, 0) (15, −1, −2, 0) (16, −1, −2, 0) (17, −1, −2, 0) (18, −1, 0, 0) (19, 0, −2, 0) (20, 0, −2, 0) (21, 0, −2, 0) (22, −1, −2, 0) (23, −1, −2, 0) (24, −1, −2, 0) (25, −1, −2, 0) (26, −1, −2, 0) (27, −1, −2, 0) (28, −1, −2, 0) (29, −1, −2, 0) (30, −1, −2, 0) (31, 0, −2, 0) (32, 0, 0, 0) (33, −1, −2, 0) (34, −1, −2, 0) (35, −1, −2, 0) (36, −1, −2, 0) (37, −1, −2, 0) (38, −1, −2, 0) (39, −1, −2, 0) (40, −1, −2, 0) (41, −1, −2, 0) (0, 1, 0, 0) (0, 2, 0, 0) (−1, 3, 0, 0) (−1, 4, 0, 0) (−1, 5, 0, 0) (0, 6, −1, 0) (0, 7, −1, 0) (0, 8, −1, 0) (0, 9, 0, 0) (0, 10, 0, 0) (0, 11, 0, 0)

Hi ∅ {c0 } {c1 } {c2 } {c0 , c1 } {c0 , c2 } {c1 , c2 } {c0 , c1 , c2 } {j1 } {j5 } {c0 , j1 } {c0 , j2 } {c0 , j5 } {c1 , j1 } {c1 , j3 } {c1 , j5 } {j0 , j5 } {j1 , j4 } {j1 , j5 } {c0 , c1 , j1 } {c0 , c1 , j2 } {c0 , c1 , j3 } {c0 , c1 , j5 } {c0 , j1 , j2 } {c0 , j1 , j5 } {c0 , j2 , j5 } {c1 , j1 , j3 } {c1 , j1 , j5 } {c1 , j3 , j5 } {c0 , c1 , j0 , j5 } {c0 , c1 , j1 , j2 } {c0 , c1 , j1 , j3 } {c0 , c1 , j1 , j4 } {c0 , c1 , j1 , j5 } {c0 , c1 , j2 , j3 } {c0 , c1 , j2 , j5 } {c0 , c1 , j3 , j5 } {c0 , j1 , j2 , j5 } {c1 , j1 , j3 , j5 } {j0 , j1 , j4 , j5 } {c0 , c1 , j1 , j2 , j3 } {c0 , c1 , j1 , j2 , j5 } {c0 , c1 , j1 , j3 , j5 } {c0 , c1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j4 , j5 } {c0 , c1 , j0 , j2 , j3 , j5 } {c0 , c1 , j1 , j2 , j3 , j4 } {c0 , c1 , j1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 } {u2 } {u5 } {c0 , u2 } {c0 , u1 } {c0 , u5 } {c2 , u2 } {c2 , u4 } {c2 , u5 } {u0 , u5 } {u2 , u3 } {u2 , u5 }

Case 1

2.1

2.2

If the sets Hi are unions of certain sets, repeated constants are not removed in the descriptions of the sets to make the construction of the sets Hi recognizable.

15.2 The Subsemigroups of (P31 ; ) i 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

τ (Hi ) (−1, 12, −1, 0) (−1, 13, −1, 0) (−1, 14, −1, 0) (−1, 15, −1, 0) (−1, 16, −1, 0) (−1, 17, −1, 0) (−1, 18, 0, 0) (0, 19, −1, 0) (0, 20, −1, 0) (0, 21, −1, 0) (−1, 22, −1, 0) (−1, 23, −1, 0) (−1, 24, −1, 0) (−1, 25, −1, 0) (−1, 26, −1, 0) (−1, 27, −1, 0) (−1, 28, −1, 0) (−1, 29, −1, 0) (−1, 30, −1, 0) (0, 31, −1, 0) (0, 32, 0, 0) (−1, 33, −1, 0) (−1, 34, −1, 0) (−1, 35, −1, 0) (−1, 36, −1, 0) (−1, 37, −1, 0) (−1, 38, −1, 0) (−1, 39, −1, 0) (−1, 40, −1, 0) (−1, 41, −1, 0) (0, 0, 1, 0) (0, 0, 2, 0) (0, −2, 3, 0) (0, −2, 4, 0) (0, −2, 5, 0) (−2, 0, 6, 0) (−2, 0, 7, 0) (−2, 0, 8, 0) (0, 0, 9, 0) (0, 0, 10, 0) (0, 0, 11, 0) (−2, −2, 12, 0) (−2, −2, 13, 0) (−2, −2, 14, 0) (−2, −2, 15, 0) (0, −2, 16, 0) (0, −2, 17, 0) (0, −2, 18, 0) (−2, 0, 19, 0) (−2, 0, 20, 0) (−2, 0, 21, 0) (−2, −2, 22, 0) (−2, −2, 23, 0) (−2, −2, 24, 0) (−2, −2, 25, 0) (−2, −2, 26, 0) (−2, −2, 27, 0) (−2, −2, 28, 0) (−2, −2, 29, 0) (0, −2, 30, 0) (−2, 0, 31, 0) (0, 0, 32, 0) (−2, −2, 33, 0) (−2, −2, 34, 0) (−2, −2, 35, 0)

Hi {c0 , c2 , u2 } {c0 , c2 , u1 } {c0 , c2 , u4 } {c0 , c2 , u5 } {c0 , u2 , u1 } {c0 , u2 , u5 } {c0 , u1 , u5 } {c2 , u2 , u4 } {c2 , u2 , u5 } {c2 , u4 , u5 } {c0 , c2 , u0 , u5 } {c0 , c2 , u2 , u1 } {c0 , c2 , u2 , u4 } {c0 , c2 , u2 , u3 } {c0 , c2 , u2 , u5 } {c0 , c2 , u1 , u4 } {c0 , c2 , u1 , u5 } {c0 , c2 , u4 , u5 } {c0 , u2 , u1 , u5 } {c2 , u2 , u4 , u5 } {u0 , u2 , u3 , u5 } {c0 , c2 , u2 , u1 , u4 } {c0 , c2 , u2 , u1 , u5 } {c0 , c2 , u2 , u4 , u5 } {c0 , c2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u1 , u4 , u3 , u5 } {v4 } {v2 } {c2 , v4 } {c2 , v5 } {c2 , v2 } {c1 , v4 } {c1 , v0 } {c1 , v2 } {v3 , v2 } {v4 , v1 } {v4 , v2 } {c2 , c1 , v4 } {c2 , c1 , v5 } {c2 , c1 , v0 } {c2 , c1 , v2 } {c2 , v4 , v5 } {c2 , v4 , v2 } {c2 , v5 , v2 } {c1 , v4 , v0 } {c1 , v4 , v2 } {c1 , v0 , v2 } {c2 , c1 , v3 , v2 } {c2 , c1 , v4 , v5 } {c2 , c1 , v4 , v0 } {c2 , c1 , v4 , v1 } {c2 , c1 , v4 , v2 } {c2 , c1 , v5 , v0 } {c2 , c1 , v5 , v2 } {c2 , c1 , v0 , v2 } {c2 , v4 , v5 , v2 } {c1 , v4 , v0 , v2 } {v3 , v4 , v1 , v2 } {c2 , c1 , v4 , v5 , v0 } {c2 , c1 , v4 , v5 , v2 } {c2 , c1 , v4 , v0 , v2 }

445 Case

2.3

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

446 i 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190

τ (Hi ) (−2, −2, 36, 0) (−2, −2, 37, 0) (−2, −2, 38, 0) (−2, −2, 39, 0) (−2, −2, 40, 0) (−2, −2, 41, 0) (3, −3, −1, 0) (−3, 3, −2, 0) (12, −3, −3, 0) (−3, 12, −3, 0) (25, −3, −3, 0) (−3, 25, −3, 0) (3, 3, 0, 0) (3, 12, −1, 0) (12, 3, −2, 0) (12, 12, −3, 0) (3, 25, −1, 0) (25, 3, −2, 0) (12, 25, −3, 0) (25, 12, −3, 0) (25, 25, −3, 0) (8, −2, −3, 0) (13, −3, −3, 0) (14, −3, −3, 0) (15, −3, −3, 0) (22, −3, −3, 0) (23, −3, −3, 0) (24, −3, −3, 0) (26, −3, −3, 0) (27, −3, −3, 0) (28, −3, −3, 0) (29, −3, −3, 0) (33, −3, −3, 0) (34, −3, −3, 0) (35, −3, −3, 0) (36, −3, −3, 0) (37, −3, −3, 0) (38, −3, −3, 0) (39, −3, −3, 0) (40, −3, −3, 0) (41, −3, −3, 0) (4, 3, 0, 0) (13, 3, −2, 0) (14, 3, −2, 0) (16, 3, 0, 0) (18, 3, 0, 0) (23, 3, −2, 0) (24, 3, −2, 0) (27, 3, −2, 0) (28, 3, −2, 0) (30, 3, 0, 0) (33, 3, −2, 0) (34, 3, −2, 0) (36, 3, −2, 0) (38, 3, −2, 0) (39, 3, −2, 0) (40, 3, −2, 0) (41, 3, −2, 0) (13, 12, −3, 0) (14, 12, −3, 0) (23, 12, −3, 0) (24, 12, −3, 0) (27, 12, −3, 0) (28, 12, −3, 0) (33, 12, −3, 0)

Hi {c2 , c1 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v1 , v2 } {c2 , c1 , v3 , v5 , v0 , v2 } {c2 , c1 , v4 , v5 , v0 , v1 } {c2 , c1 , v4 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v5 , v0 , v1 , v2 } {c0 , j1 , c2 } {c0 , c1 , c0 , u2 } {c0 , c1 , j1 , c0 , c2 } {c0 , c1 , c0 , c2 , u2 } {c0 , c1 , j1 , j4 , c0 , c2 } {c0 , c1 , c0 , c2 , u2 , u3 } {c0 , j1 , c0 , u2 } {c0 , j1 , c0 , c2 , u2 } {c0 , c1 , j1 , c0 , u2 } {c0 , c1 , j1 , c0 , c2 , u2 } {c0 , j1 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j4 , c0 , u2 } {c0 , c1 , j1 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u3 } {c1 , j5 , c2 } {c0 , c1 , j2 , c0 , c2 } {c0 , c1 , j3 , c0 , c2 } {c0 , c1 , j5 , c0 , c2 } {c0 , c1 , j0 , j5 , c0 , c2 } {c0 , c1 , j1 , j2 , c0 , c2 } {c0 , c1 , j1 , j3 , c0 , c2 } {c0 , c1 , j1 , j5 , c0 , c2 } {c0 , c1 , j2 , j3 , c0 , c2 } {c0 , c1 , j2 , j5 , c0 , c2 } {c0 , c1 , j3 , j5 , c0 , c2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 } {c0 , c1 , j1 , j2 , j5 , c0 , c2 } {c0 , c1 , j1 , j3 , j5 , c0 , c2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 } J ∪ {c0 , c2 } {c0 , j2 , c0 , u2 } {c0 , c1 , j2 , c0 , u2 } {c0 , c1 , j3 , c0 , u2 } {c0 , j1 , j2 , c0 , u2 } {c0 , j2 , j5 , c0 , u2 } {c0 , c1 , j1 , j2 , c0 , u2 } {c0 , c1 , j1 , j3 , c0 , u2 } {c0 , c1 , j2 , j3 , c0 , u2 } {c0 , c1 , j2 , j5 , c0 , u2 } {c0 , j1 , j2 , j5 , c0 , u2 } {c0 , c1 , j1 , j2 , j3 , c0 , u2 } {c0 , c1 , j1 , j2 , j5 , c0 , u2 } {c0 , c1 , j2 , j3 , j5 , c0 , u2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , u2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , u2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , u2 } J ∪ {c0 , u2 } {c0 , c1 , j2 , c0 , c2 , u2 } {c0 , c1 , j3 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , c0 , c2 , u2 } {c0 , c1 , j1 , j3 , c0 , c2 , u2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 }

Case

3.1

3.2.1 3.2.2

3.2.3

3.2.4

15.2 The Subsemigroups of (P31 ; ) i 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255

τ (Hi ) (34, 12, −3, 0) (36, 12, −3, 0) (38, 12, −3, 0) (39, 12, −3, 0) (40, 12, −3, 0) (41, 12, −3, 0) (27, 25, −3, 0) (33, 25, −3, 0) (36, 25, −3, 0) (38, 25, −3, 0) (39, 25, −3, 0) (40, 25, −3, 0) (41, 25, −3, 0) (−2, 8, −3, 0) (−3, 13, −3, 0) (−3, 14, −3, 0) (−3, 15, −3, 0) (−3, 22, −3, 0) (−3, 23, −3, 0) (−3, 24, −3, 0) (−3, 26, −3, 0) (−3, 27, −3, 0) (−3, 28, −3, 0) (−3, 29, −3, 0) (−3, 33, −3, 0) (−3, 34, −3, 0) (−3, 35, −3, 0) (−3, 36, −3, 0) (−3, 37, −3, 0) (−3, 38, −3, 0) (−3, 39, −3, 0) (−3, 40, −3, 0) (−3, 41, −3, 0) (3, 4, 0, 0) (3, 13, −1, 0) (3, 14, −1, 0) (3, 16, 0, 0) (3, 18, 0, 0) (3, 23, −1, 0) (3, 24, −1, 0) (3, 27, −1, 0) (3, 28, −1, 0) (3, 30, 0, 0) (3, 33, −1, 0) (3, 34, −1, 0) (3, 36, −1, 0) (3, 38, −1, 0) (3, 39, −1, 0) (3, 40, −1, 0) (3, 41, −1, 0) (12, 13, −3, 0) (12, 14, −3, 0) (12, 23, −3, 0) (12, 24, −3, 0) (12, 27, −3, 0) (12, 28, −3, 0) (12, 33, −3, 0) (12, 34, −3, 0) (12, 36, −3, 0) (12, 38, −3, 0) (12, 39, −3, 0) (12, 40, −3, 0) (12, 41, −3, 0) (25, 27, −3, 0) (25, 33, −3, 0)

Hi {c0 , c1 , j1 , j2 , j5 , c0 , c2 , u2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 } J ∪ {c0 , c2 , u2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 , u3 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 } J ∪ {c0 , c2 , u2 , u3 } {c1 , c2 , u5 } {c0 , c1 , c0 , c2 , u1 } {c0 , c1 , c0 , c2 , u4 } {c0 , c1 , c0 , c2 , u5 } {c0 , c1 , c0 , c2 , u0 , u5 } {c0 , c1 , c0 , c2 , u2 , u1 } {c0 , c1 , c0 , c2 , u2 , u4 } {c0 , c1 , c0 , c2 , u2 , u5 } {c0 , c1 , c0 , c2 , u1 , u4 } {c0 , c1 , c0 , c2 , u1 , u5 } {c0 , c1 , c0 , c2 , u4 , u5 } {c0 , c1 , c0 , c2 , u2 , u1 , u4 } {c0 , c1 , c0 , c2 , u2 , u1 , u5 } {c0 , c1 , c0 , c2 , u2 , u4 , u5 } {c0 , c1 , c0 , c2 , u1 , u4 , u5 } {c0 , c1 , c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c1 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c1 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c1 } ∪ U {c0 , j1 , c0 , u1 } {c0 , j1 , c0 , c2 , u1 } {c0 , j1 , c0 , c2 , u4 } {c0 , j1 , c0 , u2 , u1 } {c0 , j1 , c0 , u1 , u5 } {c0 , j1 , c0 , c2 , u2 , u1 } {c0 , j1 , c0 , c2 , u2 , u4 } {c0 , j1 , c0 , c2 , u1 , u4 } {c0 , j1 , c0 , c2 , u1 , u5 } {c0 , j1 , c0 , u2 , u1 , u5 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 } {c0 , j1 , c0 , c2 , u2 , u1 , u5 } {c0 , j1 , c0 , c2 , u1 , u4 , u5 } {c0 , j1 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , j1 } ∪ U {c0 , c1 , j1 , c0 , c2 , u1 } {c0 , c1 , j1 , c0 , c2 , u4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 } {c0 , c1 , j1 , c0 , c2 , u2 , u4 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 } {c0 , c1 , j1 , c0 , c2 , u1 , u5 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u5 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , u5 } {c0 , c1 , j1 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c1 , j1 } ∪ U {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 }

447 Case

3.2.5

3.3

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

448 i 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 147 280 150 281 151 282 283 284 285 286 287 288 289 290 291 132 149 148 134 136 157 156 154 155 153 152 161 160 159 158 162 164 163 165 166 292 293 294 295 296 297

τ (Hi ) (25, 36, −3, 0) (25, 38, −3, 0) (25, 39, −3, 0) (25, 40, −3, 0) (25, 41, −3, 0) (2, 2, 0, 0) (5, 5, 0, 0) (8, 8, −3, 0) (9, 9, 0, 0) (15, 15, −2, 0) (16, 16, 0, 0) (17, 18, 0, 0) (18, 17, 0, 0) (22, 22, −3, 0) (23, 23, −3, 0) (24, 24, −3, 0) (26, 28, −3, 0) (28, 26, −3, 0) (30, 30, 0, 0) (34, 34, −3, 0) (37, 38, −3, 0) (38, 37, −3, 0) (39, 39, −3, 0) (41, 41, −3, 0) (8, −2, −3, 0) (−3, −1, 8, 0) (15, −3, −3, 0) (−3, −3, 15, 0) (22, −3, −3, 0) (−3, −3, 22, 0) (8, 0, 8, 0) (8, −1, 15, 0) (15, −2, 8, 0) (15, −3, 15, 0) (8, −1, 22, 0) (22, −2, 8, 0) (15, −3, 22, 0) (22, −3, 15, 0) (22, −3, 22, 0) (3, −3, −1, 0) (14, −3, −3, 0) (13, −3, −3, 0) (12, −3, −3, 0) (25, −3, −3, 0) (29, −3, −3, 0) (28, −3, −3, 0) (26, −3, −3, 0) (27, −3, −3, 0) (24, −3, −3, 0) (23, −3, −3, 0) (36, −3, −3, 0) (35, −3, −3, 0) (34, −3, −3, 0) (33, −3, −3, 0) (37, −3, −3, 0) (39, −3, −3, 0) (38, −3, −3, 0) (40, −3, −3, 0) (41, −3, −3, 0) (7, 0, 8, 0) (14, −2, 8, 0) (13, −2, 8, 0) (21, 0, 8, 0) (19, 0, 8, 0) (29, −2, 8, 0)

Hi {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , u5 } {c0 , c1 , j1 , j4 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c1 , j1 , j4 } ∪ U {j5 , u5 } {c0 , j5 , c0 , u5 } {c1 , j5 , c2 , u5 } {j0 , j5 , u0 , u5 } {c0 , c1 , j5 , c0 , c2 , u5 } {c0 , j1 , j2 , c0 , u2 , u1 } {c0 , j1 , j5 , c0 , u1 , u5 } {c0 , j2 , j5 , c0 , u2 , u5 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 } {c0 , c1 , j1 , j2 , c0 , c2 , u2 , u1 } {c0 , c1 , j1 , j3 , c0 , c2 , u2 , u4 } {c0 , c1 , j1 , j5 , c0 , c2 , u1 , u5 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 , u5 } {c0 , j1 , j2 , j5 , c0 , u2 , u1 , u5 } {c0 , c1 , j1 , j2 , j5 , c0 , c2 , u2 , u1 , u5 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 } J ∪U {c1 , j5 , c2 , c1 } {c0 , c1 , c1 , v2 } {c0 , c1 , j5 , c2 , c1 } {c0 , c1 , c2 , c1 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 } {c0 , c1 , c2 , c1 , v3 , v2 } {c1 , j5 , c1 , v2 } {c1 , j5 , c2 , c1 , v2 } {c0 , c1 , j5 , c1 , v2 } {c0 , c1 , j5 , c2 , c1 , v2 } {c1 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j5 , c1 , v2 } {c0 , c1 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v3 , v2 } {c0 , j1 , c2 } {c0 , c1 , j3 , c2 , c1 } {c0 , c1 , j2 , c2 , c1 } {c0 , c1 , j1 , c2 , c1 } {c0 , c1 , j1 , j4 , c2 , c1 } {c0 , c1 , j3 , j5 , c2 , c1 } {c0 , c1 , j2 , j5 , c2 , c1 } {c0 , c1 , j1 , j5 , c2 , c1 } {c0 , c1 , j2 , j3 , c2 , c1 } {c0 , c1 , j1 , j3 , c2 , c1 } {c0 , c1 , j1 , j2 , c2 , c1 } {c0 , c1 , j2 , j3 , j5 , c2 , c1 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j3 , c2 , c1 } {c0 , c1 , j0 , j1 , j4 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j3 , j5 , c2 , c1 } J ∪ {c2 , c1 } {c1 , j3 , c1 , v2 } {c0 , c1 , j3 , c1 , v2 } {c0 , c1 , j2 , c1 , v2 } {c1 , j3 , j5 , c1 , v2 } {c1 , j1 , j3 , c1 , v2 } {c0 , c1 , j3 , j5 , c1 , v2 }

Case

3.4

4.1

15.2 The Subsemigroups of (P31 ; ) i 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362

τ (Hi ) (28, −2, 8, 0) (27, −2, 8, 0) (24, −2, 8, 0) (31, 0, 8, 0) (36, −2, 8, 0) (35, −2, 8, 0) (33, −2, 8, 0) (39, −2, 8, 0) (38, −2, 8, 0) (40, −2, 8, 0) (41, −2, 8, 0) (14, −3, 15, 0) (13, −3, 15, 0) (29, −3, 15, 0) (28, −3, 15, 0) (27, −3, 15, 0) (24, −3, 15, 0) (36, −3, 15, 0) (35, −3, 15, 0) (33, −3, 15, 0) (39, −3, 15, 0) (38, −3, 15, 0) (40, −3, 15, 0) (41, −3, 15, 0) (27, −3, 22, 0) (36, −3, 22, 0) (33, −3, 22, 0) (39, −3, 22, 0) (38, −3, 22, 0) (40, −3, 22, 0) (41, −3, 22, 0) (−1, −3, 3, 0) (−3, −3, 14, 0) (−3, −3, 13, 0) (−3, −3, 12, 0) (−3, −3, 25, 0) (−3, −3, 29, 0) (−3, −3, 28, 0) (−3, −3, 26, 0) (−3, −3, 27, 0) (−3, −3, 24, 0) (−3, −3, 23, 0) (−3, −3, 36, 0) (−3, −3, 35, 0) (−3, −3, 34, 0) (−3, −3, 33, 0) (−3, −3, 37, 0) (−3, −3, 39, 0) (−3, −3, 38, 0) (−3, −3, 40, 0) (−3, −3, 41, 0) (8, 0, 7, 0) (8, −1, 14, 0) (8, −1, 13, 0) (8, 0, 21, 0) (8, 0, 19, 0) (8, −1, 29, 0) (8, −1, 28, 0) (8, −1, 27, 0) (8, −1, 24, 0) (8, 0, 31, 0) (8, −1, 36, 0) (8, −1, 35, 0) (8, −1, 33, 0) (8, −1, 39, 0)

Hi {c0 , c1 , j2 , j5 , c1 , v2 } {c0 , c1 , j2 , j3 , c1 , v2 } {c0 , c1 , j1 , j3 , c1 , v2 } {c1 , j1 , j3 , j5 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c1 , v2 } {c0 , c1 , j1 , j3 , j5 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c1 , v2 } J ∪ {c1 , v2 } {c0 , c1 , j3 , c2 , c1 , v2 } {c0 , c1 , j2 , c2 , c1 , v2 } {c0 , c1 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j2 , j5 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c2 , c1 , v2 } {c0 , c1 , j1 , j3 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c2 , c1 , v2 } J ∪ {c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c2 , c1 , v3 , v2 } {c0 , c1 , j2 , j3 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c2 , c1 , v3 , v2 } J ∪ {c2 , c1 , v3 , v2 } {c0 , c2 , v4 } {c0 , c1 , c2 , c1 , v0 } {c0 , c1 , c2 , c1 , v5 } {c0 , c1 , c2 , c1 , v4 } {c0 , c1 , c2 , c1 , v4 , v1 } {c0 , c1 , c2 , c1 , v0 , v2 } {c0 , c1 , c2 , c1 , v5 , v2 } {c0 , c1 , c2 , c1 , v4 , v2 } {c0 , c1 , c2 , c1 , v5 , v0 } {c0 , c1 , c2 , c1 , v4 , v0 } {c0 , c1 , c2 , c1 , v4 , v5 } {c0 , c1 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , c2 , c1 , v4 , v0 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , c2 , c1 , v3 , v4 , v1 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 } ∪ V {c1 , j5 , c1 , v0 } {c1 , j5 , c2 , c1 , v0 } {c1 , j5 , c2 , c1 , v5 } {c1 , j5 , c1 , v0 , v2 } {c1 , j5 , c1 , v4 , v0 } {c1 , j5 , c2 , c1 , v0 , v2 } {c1 , j5 , c2 , c1 , v5 , v2 } {c1 , j5 , c2 , c1 , v5 , v0 } {c1 , j5 , c2 , c1 , v4 , v0 } {c1 , j5 , c1 , v4 , v0 , v2 } {c1 , j5 , c2 , c1 , v5 , v0 , v2 } {c1 , j5 , c2 , c1 , v4 , v0 , v2 } {c1 , j5 , c2 , c1 , v4 , v5 , v0 } {c1 , j5 , c2 , c1 , v4 , v5 , v0 , v1 }

449 Case

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

450 i 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 204 352 207 355 208 356 405 406 407 408 409 410 411 412 413 414 206 205 135 137 214 213 211

τ (Hi ) (8, −1, 38, 0) (8, −1, 40, 0) (8, −1, 41, 0) (15, −3, 14, 0) (15, −3, 13, 0) (15, −3, 29, 0) (15, −3, 28, 0) (15, −3, 27, 0) (15, −3, 24, 0) (15, −3, 36, 0) (15, −3, 35, 0) (15, −3, 33, 0) (15, −3, 39, 0) (15, −3, 38, 0) (15, −3, 40, 0) (15, −3, 41, 0) (22, −3, 27, 0) (22, −3, 36, 0) (22, −3, 33, 0) (22, −3, 39, 0) (22, −3, 38, 0) (22, −3, 40, 0) (22, −3, 41, 0) (1, 0, 1, 0) (6, 0, 6, 0) (3, −3, 3, 0) (10, 0, 10, 0) (12, −3, 12, 0) (21, 0, 21, 0) (20, 0, 19, 0) (19, 0, 20, 0) (25, −3, 25, 0) (29, −3, 29, 0) (28, −3, 28, 0) (26, −3, 24, 0) (24, −3, 26, 0) (31, 0, 31, 0) (35, −3, 35, 0) (37, −3, 39, 0) (39, −3, 37, 0) (38, −3, 38, 0) (41, −3, 41, 0) (−2, 8, −3, 0) (−1, −3, 3, 0) (−3, 15, −3, 0) (−3, −3, 12, 0) (−3, 22, −3, 0) (−3, −3, 25, 0) (0, 8, 3, 0) (−2, 8, 12, 0) (−1, 15, 3, 0) (−3, 15, 12, 0) (−2, 8, 25, 0) (−1, 22, 3, 0) (−3, 15, 25, 0) (−3, 22, 12, 0) (−3, 22, 25, 0) (−2, 3, −2, 0) (−3, 14, −3, 0) (−3, 13, −3, 0) (−3, 12, −3, 0) (−3, 25, −3, 0) (−3, 29, −3, 0) (−3, 28, −3, 0) (−3, 26, −3, 0)

Hi {c1 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } {c1 , j5 , c2 , c1 , v4 , v5 , v0 , v2 } {c1 , j5 } ∪ V {c0 , c1 , j5 , c2 , c1 , v0 } {c0 , c1 , j5 , c2 , c1 , v5 } {c0 , c1 , j5 , c2 , c1 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v5 , v2 } {c0 , c1 , j5 , c2 , c1 , v5 , v0 } {c0 , c1 , j5 , c2 , c1 , v4 , v0 } {c0 , c1 , j5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v4 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j5 } ∪ V {c0 , c1 , j0 , j5 , c2 , c1 , v5 , v0 } {c0 , c1 , j0 , j5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j0 , j5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j0 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 } ∪ V {j1 , v4 } {c1 , j1 , c1 , v4 } {c0 , j1 , c2 , v4 } {j1 , j4 , v4 , v1 } {c0 , c1 , j1 , c2 , c1 , v4 } {c1 , j3 , j5 , c1 , v0 , v2 } {c1 , j1 , j5 , c1 , v4 , v0 } {c1 , j1 , j3 , c1 , v4 , v2 } {c0 , c1 , j1 , j4 , c2 , c1 , v4 , v1 } {c0 , c1 , j3 , j5 , c2 , c1 , v0 , v2 } {c0 , c1 , j2 , j5 , c2 , c1 , v5 , v2 } {c0 , c1 , j1 , j5 , c2 , c1 , v4 , v0 } {c0 , c1 , j1 , j3 , c2 , c1 , v4 , v2 } {c1 , j1 , j3 , j5 , c1 , v4 , v0 , v2 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 , v4 , v0 , v2 } {c0 , c1 , j0 , j1 , j4 , j5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 , v3 , v4 , v1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } J ∪V {c2 , u5 , c2 , c1 } {c0 , c2 , c2 , v4 } {c0 , c2 , u5 , c2 , c1 } {c0 , c2 , c2 , c1 , v4 } {c0 , c2 , u0 , u5 , c2 , c1 } {c0 , c2 , c2 , c1 , v4 , v1 } {c2 , u5 , c2 , v4 } {c2 , u5 , c2 , c1 , v4 } {c0 , c2 , u5 , c2 , v4 } {c0 , c2 , u5 , c2 , c1 , v4 } {c2 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u0 , u5 , c2 , v4 } {c0 , c2 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v1 } {c2 , u2 , c1 } {c0 , c2 , u4 , c2 , c1 } {c0 , c2 , u1 , c2 , c1 } {c0 , c2 , u2 , c2 , c1 } {c0 , c2 , u2 , u3 , c2 , c1 } {c0 , c2 , u4 , u5 , c2 , c1 } {c0 , c2 , u1 , u5 , c2 , c1 } {c0 , c2 , u2 , u5 , c2 , c1 }

Case

4.2

15.2 The Subsemigroups of (P31 ; ) i 212 210 209 218 217 216 215 219 221 220 222 223 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 281 354 353 283 285 362 361 359 360 358 357 366 365 364 363 367

τ (Hi ) (−3, 27, −3, 0) (−3, 24, −3, 0) (−3, 23, −3, 0) (−3, 36, −3, 0) (−3, 35, −3, 0) (−3, 34, −3, 0) (−3, 33, −3, 0) (−3, 37, −3, 0) (−3, 39, −3, 0) (−3, 38, −3, 0) (−3, 40, −3, 0) (−3, 41, −3, 0) (0, 7, 3, 0) (−1, 14, 3, 0) (−1, 13, 3, 0) (0, 21, 3, 0) (0, 19, 3, 0) (−1, 29, 3, 0) (−1, 28, 3, 0) (−1, 27, 3, 0) (−1, 24, 3, 0) (0, 31, 3, 0) (−1, 36, 3, 0) (−1, 35, 3, 0) (−1, 33, 3, 0) (−1, 39, 3, 0) (−1, 38, 3, 0) (−1, 40, 3, 0) (−1, 41, 3, 0) (−3, 14, 12, 0) (−3, 13, 12, 0) (−3, 29, 12, 0) (−3, 28, 12, 0) (−3, 27, 12, 0) (−3, 24, 12, 0) (−3, 36, 12, 0) (−3, 35, 12, 0) (−3, 33, 12, 0) (−3, 39, 12, 0) (−3, 38, 12, 0) (−3, 40, 12, 0) (−3, 41, 12, 0) (−3, 27, 25, 0) (−3, 36, 25, 0) (−3, 33, 25, 0) (−3, 39, 25, 0) (−3, 38, 25, 0) (−3, 40, 25, 0) (−3, 41, 25, 0) (−3, −1, 8, 0) (−3, −3, 13, 0) (−3, −3, 14, 0) (−3, −3, 15, 0) (−3, −3, 22, 0) (−3, −3, 23, 0) (−3, −3, 24, 0) (−3, −3, 26, 0) (−3, −3, 27, 0) (−3, −3, 28, 0) (−3, −3, 29, 0) (−3, −3, 33, 0) (−3, −3, 34, 0) (−3, −3, 35, 0) (−3, −3, 36, 0) (−3, −3, 37, 0)

Hi {c0 , c2 , u1 , u4 , c2 , c1 } {c0 , c2 , u2 , u4 , c2 , c1 } {c0 , c2 , u2 , u1 , c2 , c1 } {c0 , c2 , u1 , u4 , u5 , c2 , c1 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 } {c0 , c2 , u2 , u1 , u5 , c2 , c1 } {c0 , c2 , u2 , u1 , u4 , c2 , c1 } {c2 , c1 , v3 , v4 , v1 , v2 , c2 , c1 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 } U ∪ {c2 , c1 } {c2 , u4 , c2 , v4 } {c0 , c2 , u4 , c2 , v4 } {c0 , c2 , u1 , c2 , v4 } {c2 , u4 , u5 , c2 , v4 } {c2 , u2 , u4 , c2 , v4 } {c0 , c2 , u4 , u5 , c2 , v4 } {c0 , c2 , u1 , u5 , c2 , v4 } {c0 , c2 , u1 , u4 , c2 , v4 } {c0 , c2 , u2 , u4 , c2 , v4 } {c2 , u2 , u4 , u5 , c2 , v4 } {c0 , c2 , u1 , u4 , u5 , c2 , v4 } {c0 , c2 , u2 , u4 , u5 , c2 , v4 } {c0 , c2 , u2 , u1 , u4 , c2 , v4 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , v4 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , v4 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , v4 } U ∪ {c2 , v4 } {c0 , c2 , u4 , c2 , c1 , v4 } {c0 , c2 , u1 , c2 , c1 , v4 } {c0 , c2 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u1 , u5 , c2 , c1 , v4 } {c0 , c2 , u1 , u4 , c2 , c1 , v4 } {c0 , c2 , u2 , u4 , c2 , c1 , v4 } {c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 } U ∪ {c2 , c1 , v4 } {c0 , c2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 , v1 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } U ∪ {c2 , c1 , v4 , v1 } {c0 , c1 , v2 } {c0 , c2 , c2 , c1 , v5 } {c0 , c2 , c2 , c1 , v0 } {c0 , c2 , c2 , c1 , v2 } {c0 , c2 , c2 , c1 , v3 , v2 } {c0 , c2 , c2 , c1 , v4 , v5 } {c0 , c2 , c2 , c1 , v4 , v0 } {c0 , c2 , c2 , c1 , v4 , v2 } {c0 , c2 , c2 , c1 , v5 , v0 } {c0 , c2 , c2 , c1 , v5 , v2 } {c0 , c2 , c2 , c1 , v0 , v2 } {c0 , c2 , c2 , c1 , v4 , v5 , v0 } {c0 , c2 , c2 , c1 , v4 , v5 , v2 } {c0 , c2 , c2 , c1 , v4 , v0 , v2 } {c0 , c2 , c2 , c1 , v5 , v0 , v2 } {c0 , c2 , c2 , c1 , v3 , v4 , v1 , v2 }

451 Case

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

452 i 369 368 370 371 452 453 454 455 456 457 458 459 450 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512

τ (Hi ) (−3, −3, 38, 0) (−3, −3, 39, 0) (−3, −3, 40, 0) (−3, −3, 41, 0) (0, 8, 4, 0) (−2, 8, 13, 0) (−2, 8, 14, 0) (0, 8, 16, 0) (0, 8, 18, 0) (−2, 8, 23, 0) (−2, 8, 24, 0) (−2, 8, 27, 0) (−2, 8, 28, 0) (0, 8, 30, 0) (−2, 8, 33, 0) (−2, 8, 34, 0) (−2, 8, 36, 0) (−2, 8, 38, 0) (−2, 8, 39, 0) (−2, 8, 40, 0) (−2, 8, 41, 0) (−3, 15, 13, 0) (−3, 15, 14, 0) (−3, 15, 23, 0) (−3, 15, 24, 0) (−3, 15, 27, 0) (−3, 15, 28, 0) (−3, 15, 33, 0) (−3, 15, 34, 0) (−3, 15, 36, 0) (−3, 15, 38, 0) (−3, 15, 39, 0) (−3, 15, 40, 0) (−3, 15, 41, 0) (−3, 22, 27, 0) (−3, 22, 33, 0) (−3, 22, 36, 0) (−3, 22, 38, 0) (−3, 22, 39, 0) (−3, 22, 40, 0) (−3, 22, 41, 0) (0, 1, 2, 0) (0, 6, 5, 0) (−3, 3, 8, 0) (0, 10, 9, 0) (−3, 12, 15, 0) (0, 21, 16, 0) (0, 20, 18, 0) (0, 19, 17, 0) (−3, 25, 22, 0) (−3, 29, 23, 0) (−3, 28, 24, 0) (−3, 26, 28, 0) (−3, 24, 26, 0) (0, 31, 30, 0) (−3, 35, 34, 0) (−3, 37, 38, 0) (−3, 39, 37, 0) (−3, 38, 39, 0) (−3, 41, 41, 0) (13, 3, 8, 0) (14, 3, 8, 0) (24, 3, 8, 0) (27, 3, 8, 0) (28, 3, 8, 0)

Hi {c0 , c2 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c2 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c2 } ∪ V {c2 , u5 , c2 , v5 } {c2 , u5 , c2 , c1 , v5 } {c2 , u5 , c2 , c1 , v0 } {c2 , u5 , c2 , v4 , v5 } {c2 , u5 , c2 , v5 , v2 } {c2 , u5 , c2 , c1 , v4 , v5 } {c2 , u5 , c2 , c1 , v4 , v0 } {c2 , u5 , c2 , c1 , v5 , v0 } {c2 , u5 , c2 , c1 , v5 , v2 } {c2 , u5 , c2 , v4 , v5 , v2 } {c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c2 , u5 , c2 , c1 , v4 , v5 , v2 } {c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c2 , u5 } ∪ V {c0 , c2 , u5 , c2 , c1 , v5 } {c0 , c2 , u5 , c2 , c1 , v0 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 } {c0 , c2 , u5 , c2 , c1 , v4 , v0 } {c0 , c2 , u5 , c2 , c1 , v5 , v0 } {c0 , c2 , u5 , c2 , c1 , v5 , v2 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v2 } {c0 , c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c2 , u5 } ∪ V {c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c2 , u0 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c2 , u0 , u5 } ∪ V {u2 , v2 } {c2 , u2 , c2 , v2 } {c0 , u2 , c1 , v2 } {u2 , u3 , v3 , v2 } {c0 , c2 , u2 , c2 , c1 , v2 } {c2 , u4 , u5 , c2 , v4 , v5 } {c2 , u2 , u5 , c2 , v5 , v2 } {c2 , u2 , u4 , c2 , v4 , v2 } {c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c2 , u4 , u5 , c2 , c1 , v4 , v5 } {c0 , c2 , u1 , u5 , c2 , c1 , v4 , v0 } {c0 , c2 , u2 , u5 , c2 , c1 , v5 , v2 } {c0 , c2 , u2 , u4 , c2 , c1 , v4 , v2 } {c2 , u2 , u4 , u5 , c2 , v4 , v5 , v2 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 , v4 , v5 , v2 } {c2 , c1 , v3 , v4 , v1 , v2 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v3 , v4 , v1 , v2 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } U ∪V {c0 , c1 , j2 , c0 , u2 , c1 , v2 } {c0 , c1 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j1 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j2 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j2 , j5 , c0 , u2 , c1 , v2 }

Case

5.1.1

15.2 The Subsemigroups of (P31 ; ) i 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576

τ (Hi ) (33, 3, 8, 0) (36, 3, 8, 0) (38, 3, 8, 0) (39, 3, 8, 0) (40, 3, 8, 0) (41, 3, 8, 0) (13, 12, 15, 0) (14, 12, 15, 0) (24, 12, 15, 0) (27, 12, 15, 0) (28, 12, 15, 0) (33, 12, 15, 0) (36, 12, 15, 0) (38, 12, 15, 0) (39, 12, 15, 0) (40, 12, 15, 0) (41, 12, 15, 0) (27, 25, 22, 0) (33, 25, 22, 0) (36, 25, 22, 0) (38, 25, 22, 0) (39, 25, 22, 0) (40, 25, 22, 0) (41, 25, 22, 0) (3, 13, 3, 0) (3, 14, 3, 0) (3, 24, 3, 0) (3, 27, 3, 0) (3, 28, 3, 0) (3, 33, 3, 0) (3, 36, 3, 0) (3, 38, 3, 0) (3, 39, 3, 0) (3, 40, 3, 0) (3, 41, 3, 0) (12, 13, 12, 0) (12, 14, 12, 0) (12, 24, 12, 0) (12, 27, 12, 0) (12, 28, 12, 0) (12, 33, 12, 0) (12, 36, 12, 0) (12, 38, 12, 0) (12, 39, 12, 0) (12, 40, 12, 0) (12, 41, 12, 0) (25, 27, 25, 0) (25, 33, 25, 0) (25, 36, 25, 0) (25, 38, 25, 0) (25, 39, 25, 0) (25, 40, 25, 0) (25, 41, 25, 0) (8, 8, 13, 0) (8, 8, 14, 0) (8, 8, 24, 0) (8, 8, 27, 0) (8, 8, 28, 0) (8, 8, 33, 0) (8, 8, 36, 0) (8, 8, 38, 0) (8, 8, 39, 0) (8, 8, 40, 0) (8, 8, 41, 0) (15, 15, 13, 0)

Hi {c0 , c1 , j1 , j2 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c0 , u2 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , u2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , u2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , u2 , c1 , v2 } J ∪ {c0 , u2 , c1 , v2 } {c0 , c1 , j2 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } J ∪ {c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } J ∪ {c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , j1 , c0 , c2 , u1 , c2 , v4 } {c0 , j1 , c0 , c2 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u1 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u1 , u5 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u1 , u4 , u5 , c2 , v4 } {c0 , j1 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u5 , c2 , v4 } {c0 , j1 } ∪ U ∪ {c2 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 } ∪ U ∪ {c2 , c1 , v4 } {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 } ∪ U ∪ {c2 , c1 , v4 , v1 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 } {c1 , j5 , c2 , u5 , c2 , c1 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v2 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c1 , j5 , c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c1 , j5 , c2 , u5 } ∪ V {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 }

453 Case

5.1.2

5.1.3

5.2

5.3

454 i 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 600 + t

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3 τ (Hi ) (15, 15, 14, 0) (15, 15, 24, 0) (15, 15, 27, 0) (15, 15, 28, 0) (15, 15, 33, 0) (15, 15, 36, 0) (15, 15, 38, 0) (15, 15, 39, 0) (15, 15, 40, 0) (15, 15, 41, 0) (22, 22, 27, 0) (22, 22, 33, 0) (22, 22, 36, 0) (22, 22, 38, 0) (22, 22, 39, 0) (22, 22, 40, 0) (22, 22, 41, 0) (24, 24, 26, 0) (26, 28, 24, 0) (28, 26, 28, 0) (37, 38, 39, 0) (38, 37, 38, 0) (39, 39, 37, 0) (41, 41, 41, 0) (..., 1)

1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227

(0, 0, 0, 2) (0, −2, −1, 2) (−3 − 1, −2, 2) (−3, −3, −3, 2) (27, −1, −2, 2) (32, 0, 0, 2) (37, −1, −2, 2) (41, −1, −2, 2) (27, −3, −3, 2) (37, −3, −3, 2) (41, −3, −3, 2) (0, 1, 2, 2) (−3, 3, 8, 2) (0, 6, 5, 2) (−3, 12, 15, 2) (0, 10, 9, 2) (−3, 25, 22, 2) (0, 21, 16, 2) (−3, 29, 23, 2) (0, 31, 30, 2) (−3, 35, 34, 2) (−3, 38, 39, 2) (−3, 41, 41, 2) (27, 3, 8, 2) (27, 12, 15, 2) (27, 25, 22, 2) (37, 38, 39, 2)

1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238

(41, 12, 15, 2) (41, 3, 8, 2) (41, 25, 22, 2) (41, 41, 41, 2) (0, 0, 0, 3) (−1, −1, 0, 3) (−2, −2, −3, 3) (−3, −3, −3, 3) (−2, −2, 27, 3) (0, 0, 32, 3) (−2, −2, 37, 3)

Hi Case {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 } ∪ V {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 } ∪ V {c0 , c1 , j1 , j3 , c0 , c2 , u2 , u4 , c2 , c1 , v4 , v2 } 5.4 {c0 , c1 , j1 , j5 , c0 , c2 , u1 , u5 , c2 , c1 , v4 , v0 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 , u5 , c2 , c1 , v5 , v2 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u0 , u2 , u3 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v3 , v4 , v1 , v2 } J ∪U ∪V Ht ∪ {s1 } 6.1 (t ∈ {1, 2, ..., 600}) {s1 , s3 } 6.2 {c2 , s1 , s3 } {c0 , c1 , s1 , s3 } {c0 , c1 , c2 , s1 , s3 } {c0 , c1 , j2 , j3 , s1 , s3 } {j0 , j1 , j4 , j5 , s1 , s3 } {c0 , c1 , j0 , j1 , j4 , j5 , s1 , s3 } J ∪ {s1 , s3 } {c0 , c1 , j2 , j3 , c2 , s1 , s3 } {c0 , c1 , j0 , j1 , j4 , j5 , c2 , s1 , s3 } J ∪ {c0 , c2 , s1 , s3 } {u2 , v2 , s1 , s3 } {c0 , c1 , u2 , v2 , s1 , s3 } {c2 , u2 , c2 , v2 , s1 , s3 } {c0 , c2 , u2 , c2 , c1 , v2 , s1 , s3 } {u2 , u3 , v3 , v2 , s1 , s3 } {c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 , s1 , s3 } {c2 , u4 , u5 , c2 , v4 , v5 , s1 , s3 } {c0 , c2 , u4 , u5 , c2 , c1 , v4 , v5 , s1 , s3 } {c2 , u2 , u4 , u5 , c2 , v4 , v5 , v2 , s1 , s3 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 , v4 , v5 , v2 , s1 , s3 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v0 , v1 , s1 , s3 } U ∪ V ∪ {s1 , s3 } {c0 , c1 , j2 , j3 , c0 , u2 , c1 , v2 , s1 , s3 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , c2 , c1 , v2 , s1 , s3 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 , s1 , s3 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v 0 , v 1 , s 1 , s 3 } J ∪ {c0 , c2 , u2 , c2 , c1 , v2 , s1 , s3 } J ∪ {c0 , u2 , c1 , v2 , s1 , s3 } J ∪ {c0 , c2 , u2 , u3 , c2 , c1 , v2 , v3 , s1 , s3 } J ∪ U ∪ V ∪ {s1 , s3 } {s1 , s2 } 6.3 {c0 , s1 , s2 } {c2 , c1 , s1 , s2 } {c0 , c1 , c2 , s1 , s2 } {c2 , c1 , v5 , v0 , s1 , s2 } {v3 , v4 , v1 , v2 , s1 , s2 } {c2 , c1 , v3 , v4 , v1 , v2 , s1 , s2 }

15.2 The Subsemigroups of (P31 ; ) i 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258

τ (Hi ) (−2, −2, 41, 3) (−3, −1, 27, 3) (−3, −1, 37, 3) (−3, −3, 41, 3) (2, 2, 0, 3) (8, 8, −3, 3) (5, 5, 0, 3) (15, 15, −3, 3) (9, 9, 0, 3) (22, 22, −3, 3) (16, 16, 0, 3) (23, 23, −3, 3) (30, 30, 0, 3) (34, 34, −3, 3) (39, 39, −3, 3) (41, 41, −3, 3) (8, 8, 27, 3) (15, 15, 27, 3) (22, 22, 27, 3) (39, 39, 37, 3)

1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289

(15, 15, 41, 3) (8, 8, 41, 3) (22, 22, 41, 3) (41, 41, 41, 3) (0, 0, 0, 4) (−2, 0, −2, 4) (−1, −3, −1, 4) (−3, −3, −3, 4) (−1, 27, −1, 4) (0, 32, 0, 4) (−1, 37, −1, 4) (−1, 41, −1, 4) (−2, 27, −3, 4) (−2, 37, −3, 4) (−3, 41, −3, 4) (1, 0, 1, 4) (6, 0, 6, 4) (3, −3, 3, 4) (12, −3, 12, 4) (10, 0, 10, 4) (25, −3, 25, 4) (21, 0, 21, 4) (29, −3, 29, 4) (31, 0, 31, 4) (35, −3, 35, 4) (38, −3, 38, 4) (41, −3, 41, 4) (3, 27, 3, 4) (12, 27, 12, 4) (25, 27, 25, 4) (38, 37, 38, 4)

1290 1291 1292 1293 1294 1295 1296 1297 1298 1299

(12, 41, 12, 4) (3, 41, 3, 4) (25, 41, 25, 4) (41, 41, 41, 4) (0, 0, 0, 5) (−3, −3, −3, 5) (41, 41, 41, 5) (0, 0, 0, 6) (−3, −3, −3, 6) (41, 41, 41, 6)

Hi V ∪ {s1 , s2 } {c0 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c2 , c1 , v3 , v4 , v1 , v2 , s1 , s2 } {c0 , c2 } ∪ V ∪ {s1 , s2 } {j5 , u5 , s1 , s2 } {c1 , j5 , c2 , u5 , s1 , s2 } {c0 , j5 , c0 , u5 , s1 , s2 } {c0 , c1 , j5 , c0 , c2 , u5 , s1 , s2 } {j0 , j5 , u0 , u5 , s1 , s2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , s1 , s2 } {c0 , j1 , j2 , c0 , u2 , u1 , s1 , s2 } {c0 , c1 , j1 , j2 , c0 , c2 , u2 , u1 , s1 , s2 } {c0 , j1 , j2 , j5 , c0 , u2 , u1 , u5 , s1 , s2 } {c0 , c1 , j1 , j2 , j5 , c0 , c2 , u2 , u1 , u5 , s1 , s2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , s1 , s2 } J ∪ U ∪ {s1 , s2 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v3 , v4 , v 1 , v 2 , s 1 , s 2 } {c0 , c1 , j5 , c0 , c2 , u5 } ∪ V ∪ {s1 , s2 } {c1 , j5 , c2 , u5 } ∪ V ∪ {s1 , s2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 } ∪ V ∪ {s1 , s2 } J ∪ U ∪ V ∪ {s1 , s2 } {s1 , s6 } {c1 , s1 , s6 } {c0 , c2 , s1 , s6 } {c0 , c1 , c2 , s1 , s6 } {c0 , c2 , u1 , u4 , s1 , s6 } {u0 , u2 , u3 , u5 , s1 , s6 } {c0 , c2 , u0 , u2 , u3 , u5 , s1 , s6 } U ∪ {s1 , s6 } {c1 , c0 , c2 , u1 , u4 , s1 , s6 } {c1 , c0 , c2 , u0 , u2 , u3 , u5 , s1 , s6 } {c0 , c1 } ∪ U ∪ {s1 , s6 } {j1 , v4 , s1 , s6 } {c1 , j1 , c1 , v4 , s1 , s6 } {c0 , j1 , c2 , v4 , s1 , s6 } {c0 , c1 , j1 , c2 , c1 , v4 , s1 , s6 } {j1 , j4 , v4 , v1 , s1 , s6 } {c0 , c1 , j1 , j4 , c2 , c1 , v4 , v1 , s1 , s6 } {c1 , j3 , j5 , c1 , v0 , v2 , s1 , s6 } {c0 , c1 , j3 , j5 , c2 , c1 , v0 , v2 , s1 , s6 } {c1 , j1 , j3 , j5 , c1 , v4 , v0 , v2 , s1 , s6 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 , v4 , v0 , v2 , s1 , s6 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v3 , v5 , v0 , v2 , s1 , s6 } J ∪ V ∪ {s1 , s6 } {c0 , j1 , c0 , c2 , u1 , u4 , c2 , v4 , s1 , s6 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , c2 , c1 , v4 , s1 , s6 } {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , c2 , c1 , v4 , v1 , s1 , s6 } {c0 , c1 , j0 , j2 , j3 , j5 , co , c2 , u0 , u2 , u3 , u5 , c2 , c1 , v3 , v5 , v 0 , v 2 , s 1 , s 6 } {c0 , c1 , j1 } ∪ U ∪ {c2 , c1 , v4 , s1 , s6 } {c0 , j1 } ∪ U ∪ {c2 , v4 , s1 , s6 } {c0 , c1 , j1 , j4 } ∪ U ∪ {c2 , c1 , v4 , v1 , s1 , s6 } J ∪ U ∪ V ∪ {s1 , s6 } {s1 , s4 , s5 } {c0 , c1 , c2 , s1 , s4 , s5 } J ∪ U ∪ V ∪ {s1 , s4 , s5 } {s1 , s2 , s3 , s4 , s5 , s6 } {c0 , c1 , c2 , s1 , s2 , s3 , s4 , s5 , s6 } J ∪ U ∪ V ∪ {s1 , s2 , s3 , s4 , s5 , s6 }

455 Case

6.4

6.5 6.6

456

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

15.3 Classes of Quasilinear Functions of P3 that are not Subsets of [P31 ] The elements of the set L := [P31 ] ∪

n≥1 {f

n

1 ∈ P3 | ∃a ∈ E2 ∃f0 ∈ Pk1 ∃f1 , ..., fn ∈ P3,2 : f (x) = f0 (a+f1 (x1 )+f2 (x2 )+...+fn (xn ) (mod 2))},

are called quasilinear functions in this section. The set L agrees with the set L3 from Chapter 4. In Lemma 4.1, it was proven that L = P ol3 λ3 with λ3 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E3 } holds and that L is a submaximal class of P3 . L was examined already by I. A. Mal’tsev in [Mal 72] and [Mal 73b]. In [Mal 72], one also ﬁnds proof that L has exactly countable-inﬁnite-many subclasses (see Theorem 8.1.5). As in [Lau 85] we will determine all subclasses of L with the aid of the 1299 subsemigroups of (P31 ; ). Further, we determine the order of theses subclasses. After introducing some notations in Section 15.3.1, we determine in Section 15.3.2 the subclasses of L whose functions take only values from the set {0, 1}. With the aid of an isomorphic mapping, we obtain all subclasses of L whose functions take only values from the set {0, 2}. In Section 15.3.3, all subclasses of L, whose functions take only values from {0, 1} or {0, 2}, are determined then as follows: We begin with the proof of certain (necessary and suﬃcient) criteria with which one can easily determine whether the union of certain classes, which are given in Section 15.3.3, is again a class. Then we determine the remaining classes, which do not fulﬁll the conditions of the criteria. With the aid of the subclasses of {f ∈ L | Im(f ) ⊆ {0, 1} ∨ Im(f ) ⊆ {0, 2}}, the remaining classes are easily described, as shown in Section 15.3.4. Notably, another kind of derivation of the subclasses of {f ∈ L | Im(f ) ⊆ {0, 1} ∨ Im(f ) ⊆ {0, 2}} in [Dem-M 89] can be found. In [Dem-M 89], one can also ﬁnd rough drafts of lattices from such classes. 15.3.1 Some Notations We arrange to write only + instead of + (mod 2). Denote L the set of all linear functions of P2 , and let La,b := {f ∈ L | Im(f ) ⊆ {a, b}}, {a, b} ⊆ E3 , a = b. Let pra,b be a mapping from La,b onto L (⊆ P2 ), which is deﬁned by

15.3 Classes of Quasilinear Functions of P3

457

pra,b f n = F n :⇐⇒ ∀x ∈ {a, b}n : g(f (x1 , ..., xn )) = F (g(x1 ), ..., g(xn )), a 0 g = , f n ∈ La,b , and F ∈ L. b 1

where

With the aid of an arbitrary subclass A of L (see Theorem 3.2.2) one can describe a subclass of La,b by −1 pra,b A := {f ∈ La,b | pra,b f ∈ A}.

Further, let Za,b be the notation for the set bcb P ol3 , {a, b, c} = E3 . bca For the description of certain isomorphic classes, we use the automorphisms ϕi : L −→ L, f n → s−1 i (f (si (x1 ), ..., si (xn ))), (i = 1, 2, ..., 6) from Section 15.2. Obviously, one can describes every subclass A of L in the form A = (A ∩ L0,1 ) ∪ (A ∩ L0,2 ) ∪ (A ∩ L1,2 ) ∪ (A ∩ [P31 [3]]). We determine, therefore, the subclasses of L0,1 in Section 15.3.2 and then in Section 15.3.3 the subclasses of L0,1 ∪ L0,2 that are not contained in L0,1 or L0,2 . With the aid of these results, it is easy to determine the remaining subclasses of L in Section 15.3.4. 15.3.2 Subclasses of L0,1 The following lemma results from the deﬁnition of the class L0,1 and the facts j0 = 1 + j1 + j2 , j3 = 1 + j2 , j4 = 1 + j1 , j5 = j1 + j2 immediately. Lemma 15.3.2.1 It holds: L0,1 = n≥1 {f n ∈ P3 | ∃a0 , ..., an , b1 , ..., bn ∈ E2 : n f (x) = a0 + i=1 (ai j1 (xi ) + bi j2 (xi ))}.

With the aid of this lemma, one can see that the identities −1 A= {f n ∈ L0,1 | ∃a0 , ..., an , b1 , ..., bn ∈ E2 : pr0,1 n n≥1 f (x) = a0 + i=1 (a i j1 (xi ) + bi j2 (xi )) ∧ n (pr0,1 f )(y) = a0 + i=1 ai yi ∈ A}

458

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

and −1 Z2,c ∩pr0,1 A=

{f n ∈ L0,1 | ∃a0 , ..., an , c ∈ 2 : E n n≥1 f (x) = a0 + i=1 (a i (j1 (xi ) + cj2 (xi ))) ∧ n (pr0,1 f )(y) = a0 + i=1 ai yi ∈ A}

hold, where c ∈ E2 and A denotes a closed subset of L. The sets {f n ∈ L0,1 | ∃a1 , ..., an ∈E2 : Bc,r := n n≥1 f (x) = c + i=1 ai j2 (xi ) ∧ (at most r of the ai are 1) } and Br := B0,r ∪ B1,r , where r ∈ {∞, 1, 2, ...}, are also subclasses of L0,1 . Obviously, Bc,∞ and B∞ do not have any basis. Therefore, they are also not ﬁnitely generated. Lemma 15.3.2.2 Let A be a subclass of L0,1 , which contains a function −1 (pr0,1 A) and i(x1 ) + j2 (x2 ) with i ∈ {j1 , j5 }. Then, it holds that A = pr0,1 A = [A ∪ {j1 (x1 ) + j2 (x2 )}] for every A with A ⊆ A and [pr0,1 A ] = pr0,1 A. −1 −1 Proof. Obviously, A ⊆ pr0,1 (pr0,1 A). Let f n ∈ pr0,1 (pr0,1 A) and f (x) = n a0 + i=1 (ai j1 (xi ) + bi j2 (xi )). We want to show that f belongs to A. Be cause n of pr0,1 f ∈ pr0,1 A there is a function f ∈ A with f (x) = a0 + (a j (x ) + b j (x )). A superposition over i(x ) + j (x ) 1 2 2 is obviously i 2 i i=1 i 1 i n the function q(x, x1 , ..., xn ) = i(x) + i=1 (bi + bi )j2 (xi ). Consequently, we −1 (pr0,1 A). have f (x) = q(f (x), x1 , ..., xn ) ∈ A and therefore A = pr0,1 Since f ∈ [A ] holds for every A with [pr0,1 A ] = pr0,1 A, the equation A = [A ∪ {j1 (x1 ) + j2 (x2 )}] also results from the one shown above.

Lemma 15.3.2.3 Let A be a subclass of L0,1 and pr0,1 A ⊆ [L1 ]. Then, −1 −1 (pr0,1 A) or Z2,0 ∩ pr0,1 (pr0,1 A) or the set Z2,1 ∩ A is either the set pr0,1 −1 pr0,1 (pr0,1 A). n Proof. For f (x) = a0 + i=1 (ai j1 (xi ) + bi j2 (xi )) denote Ch(f ) the set {(ai , bi ) | i = 1, 2, ..., n}. We distinguish the following three cases for A: Case 1: There exists an a ∈ E2 so that Ch(f ) ⊆ {(1, a), (0, 0)} for every f ∈ A holds. 01a In this case, f ∈ A preserves the relation , and we have A = 012 −1 pr0,1 (pr0,1 A) ∩ Z2,a . Case 2: A contains a function f with (0, 1) ∈ Ch(f ). W.l.o.g. let (a1 , b1 ) = (0, 1). Then, f (x1 , x2 ) := f (x1 , x2 , ..., x2 ) = a0 + j2 (x1 ) + r(x2 ), where r ∈ {c0 , j1 , j2 , j5 }. If r ∈ {j1 , j5 } then we have f (x1 , f (x2 , x2 )) = j2 (x1 ) + i(x2 ) ∈ A, i ∈ {j1 , j5 }; thus Lemma 15.3.2.3 follows from Lemma 15.3.2.2. If r ∈ {c0 , j2 } then we obtain f (x) :=

15.3 Classes of Quasilinear Functions of P3

459

f (x, f (x, x)) ∈ {j2 , j3 }. Since pr0,1 A ⊆ [L1 ], an inverse image h of the function H with H(y1 , y2 , y3 ) = y1 + y2 + y3 belongs to A because of Lemma 3.2.2.1. Consequently, h(x1 , f (x2 ), f (f (x2 ))) = i(x1 ) + j2 (x2 ), i ∈ {j1 , j5 }, is a function of A. Therefore, Lemma 15.3.2.3 also follows from Lemma 15.3.2.2 for r ∈ {c0 , j2 }. Case 3: A contains a function f with {(1, 0), (1, 1)} ⊆ Ch(f ). W.l.o.g. let (a1 , b1 ) = (1, 0) and (a2 , b2 ) = (1, 1). Then the function f (x1 , x1 , x2 , ..., x2 ) fulﬁlls the condition of Case 2. Theorem 15.3.2.4 Let A be a subclass of L0,1 with A ⊆ [L10,1 ] ∪ B∞ . Then −1 −1 −1 A ∈ {pr0,1 (pr0,1 A), Z2,0 ∩ pr0,1 (pr0,1 A), Z2,1 ∩ pr0,1 (pr0,1 A)}.

Proof. If A ⊆ [L10,1 ] ∪ B∞ , there is a function f ∈ A with pr0,1 f ∈ [P21 ]. Let w.l.o.g. n (ai j1 (xi ) + bi j2 (xi )) ∈ A, f (x) = a0 + i=1

where a1 = 1 and (a2 , b2 ) = (0, 0). The following two cases are possible: Case 1: (a2 , b2 ) = (0, 1). One can obtain the function f (x1 , x2 , x3 ) = a0 + j1 (x1 ) + b1 j2 (x1 ) + j2 (x2 ) + p(x3 ) with p ∈ {c0 , j1 , j2 , j5 } by identifying the variables x3 , ..., xn from f . Then f (f (x1 , x2 , x2 ), x1 , x2 ) = i(x1 ) + j2 (x2 ), i ∈ {j1 , j5 }. −1 Thus by Lemma 15.3.2.2, A = pr0,1 (pr0,1 A). Case 1: a2 = 1. In this case, we have pr0,1 A ⊆ [L1 ] and our theorem follows from Lemma 15.3.2.3.

Theorem 15.3.2.5 The subclasses (= ∅) of [L10,1 ] ∪ B∞ are exactly [c0 ], [c1 ], [c0 , c1 ], [Ji ], [Ja ] ∪ Br , [Jb ] ∪ B0,r ∪ B1,s , [Jc ] ∪ B0,r , [Jd ] ∪ B1,r , where {r, s} ⊂ {∞, 2, 3, ...}, 1 ≤ i ≤ 41, a ∈ {27, 33, 36, 38, 39, 40, 41}, b ∈ {27, 33, 36, 40}, c ∈ {4, 13, 16, 18, 23, 27, 28, 30, 33, 34, 36, 40} and d ∈ {7, 14, 19, 21, 24, 27, 29, 31, 33, 35, 36, 40}. Proof. One obtains the statements of the theorem easily by means of the properties of the functions of [L10,1 ] ∪ B∞ and the results from Section 15.2 about the subsemigroups of (P31 ; ). Lemma 15.3.2.6 Let A be a subclass of L0,1 and pr0,1 A ⊆ [L1 ]. Then there is a function h3 ∈ A with h(x1 , x2 , x3 ) := i(x1 ) + i(x2 ) + i(x3 ), i ∈ {j1 , j5 }, and it holds A = [A1 ∪ {h}].

460

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Proof. By Lemma 3.2.2.1 we have pr0,1 A = [(pr0,1 A)1 ∪ {pr0,1 h}]. If A = −1 −1 (pr0,1 A), a ∈ E2 , then Lemma 15.3.2.6 holds. If A = pr0,1 (pr0,1 A) Z2,a ∩ pr0,1 then j1 (x1 ) + j2 (x2 ) ∈ A. By Lemma 15.3.2.2 and the above remark, we have A = [A1 ∪ {h, j1 (x1 ) + j2 (x2 )}], where h ∈ A is an inverse image of the function H 3 with H(y1 , y2 , y3 ) = y1 + y2 + y3 . Because of {j1 , j5 } ⊆ A, we can assume h(x1 , x2 , x3 ) = j1 (x1 ) + j1 (x2 ) + j1 (x3 ), and it holds h(x1 , x2 , j5 (x2 )) = j1 (x1 ) + j1 (x2 ) ∈ [A1 ∪ {h}]; i.e., we have A = [A1 ∪ {h}] −1 (pr0,1 A). Since there are no other possibilities because in the case A = pr0,1 of Theorem 15.3.2.4, our Lemma 15.3.2.6 is valid. Theorem 15.3.2.7 Let A be a subclass of L0,1 . Then ⎧ 2, if A ⊆ [L10,1 ] ∪ B∞ ∧ (C ∩ A1 = ∅ ∨ pr0,1 A ⊆ [L1 ]), ⎪ ⎪ ⎨ 3, if pr0,1 A ⊆ [L1 ] ∧ C ∩ A1 = ∅, ord A = 1 1 ⎪ ⎪ ⎩ t, if A ⊆ [L0,1 ] ∪ B∞ ∧ A ∩ (Bt \[A ]) = ∅ ∧ 1 (∀ r ≥ t : A ∩ (Br \[A ]) = ∅). Proof. By Theorem 15.3.2.4, the following three cases are possible for A: −1 Case 1: A = Z2,a ∩ pr0,1 (pr0,1 A), a ∈ E2 . In this case, ord A = ord pr0,1 A. Then, the statements of the theorem follow from Theorem 3.1.1. −1 (pr0,1 A) and A ⊆ [L10,1 ] ∪ B∞ . Case 2: A = pr0,1 By Lemma 15.3.2.6 and 15.3.2.2, it holds in this case: ≤ 3 if pr0,1 A ⊆ [L1 ], ord pr0,1 A ≤ ord A = 2 if pr0,1 A ⊆ [L1 ]. As is generally known, only the classes of L, which do not contain any constant functions and which are not subsets of [L1 ], have the order 3. If a constant function belongs to pr0,1 A then the function h from Lemma 15.3.2.6 is obviously a superposition over binary functions of A. Thus the statements of the theorem hold in Case 2. Case 3: A ⊆ [L10,1 ∪ B∞ . In this case, the order of A follows from Theorem 15.3.2.5 and from [L10,1 ] ∪ Br ⊂ [L10,1 ] ∪ Br+1 , 2 ≤ r ≤ ∞. One can obtain the following lemma as a direct consequence from Theorem 15.3.2.7: Theorem 15.3.2.8 The only ﬁnitely generated subclasses of L0,1 are the classes [Ja ]∪B∞ , [Jb ]∪B0,∞ , [Jc ]∪B1,∞ , [Jd ]∪B0,∞ ∪B1,s and [Jd ]∪B0,s ∪B1,∞ , where a ∈ {27, 33, 36, 38, 39, 40, 41}, b ∈ {4, 13, 16, 18, 23, 27, 28, 30, 33, 34, 36, 40}, c ∈ {7, 14, 19, 21, 24, 27, 29, 31, 33, 35, 36, 40}, d ∈ {27, 33, 36, 40} and 2 ≤ r, s ≤ ∞.

15.3 Classes of Quasilinear Functions of P3

461

15.3.3 The Subclasses of L0,1 ∪ L0,2 That Are Not Subclasses of L0,1 or L0,2 The aim of this section is ﬁrst of all the derivation of a necessary and suﬃcient criterion with which one can ﬁnd out whether a set A1 ∪ A2 (A1 ⊆ L0,1 , A2 ⊆ L0,2 ) is closed. Put A A := {f g | f ∈ A ∧ g ∈ A }. Obviously, it holds: Lemma 15.3.3.1 Let A1 and A2 be subclasses of L with A1 ⊆ L0,1 and A2 ⊆ L0,2 . Then, A1 ∪ A2 is closed if and only if A1 A2 ⊆ A1 and A2 A1 ⊆ A2 . Lemma 15.3.3.2 Let A1 and A2 be subclasses of L, A1 ⊆ L0,1 , A2 ⊆ L0,2 , (A1 ∪ A2 )1 a subsemigroup of (P31 ; ) and {i, j} = {1, 2}. Then (a) A11 ⊆ {c0 , c1 , j1 , j4 } =⇒ A1 A2 ⊆ A1 , (b) A12 ⊆ {c0 , c2 , u2 , u3 } =⇒ A2 A1 ⊆ A2 , (c) pr0,i Ai ⊆ [L1 ] =⇒ Ai Aj ⊆ Ai , (d) (pr0,1 A1 ⊆ [L1 ] ∧ j1 (x1 ) + j2 (x2 ) ∈ A1 ∧ A12 ⊆ {c0 , c2 , u2 , u3 }) =⇒ A1 A2 ⊆ A1 . Proof. (a): By Section 15.3.2 and by the assumptions over A1, A 1 is asub 0 0 class of Z2,0 ∩ L0,1 = [c1 , j1 (x1 ) + j1 (x2 )]. Thus, because of j1 , = 2 0 we have A1 A2 = A1 . (b): By assumption 15.3.2, A2 is a subset of ϕ2 (Z2,0 ∩L0,1 ). Thus, and Section 0 0 , we have A2 A1 ⊆ A2 . = because of u2 1 0 (c): Let w.l.o.g. pr0,1 A1 ⊆ [L1 ]. Then, because of Lemma 15.3.2.6, A1 = [A11 ∪ {h}], where h ∈ A1 is an arbitrary inverse image of the function H(y1 , y2 , y3 ) = y1 + y2 + y3 . Suppose, A1 A2 ⊆ A1 holds. Then, A1 := [A1 ∪ (A1 A2 )] is a closed set with A1 ⊃ A1 and A1 = [(A1 )1 ∪ {h}]. Consequently, we have (A1 )1 ⊃ A11 , i.e., there is a function g ∈ A1 and there are certain functions p1 , ..., pn ∈ (A1 ∪ A2 )1 with g (x) := g(p1 (x), p2 (x), ..., pn (x)) ∈ A11 . However, we have also g(x1 , ..., xn ) = g0 (g1 (x1 ), g2 (x2 ), ..., gn (xn )) for certain functions g0 ∈ [{h}] and g1 , ..., gn ∈ A11 . Therefore, we obtain g = g0 (g1 p1 , g2 p2 , ..., gn pn ). Moreover, by assumption, gi pi ∈ (A1 ∪A2 )1 , i = 1, 2, ..., n, and A1 = [A1 ]. Therefore, we have g ∈ A1 , contrary to the assumption. (d): Let pr0,1 A1 ⊆ [L1 ], j1 (x1 ) + j2 (x2 ) ∈ A1 and A12 ⊆ {c0 , c2 , u2 , u3 }. Then, by Section 15.3.2, A1 = [A11 ∪ {j1 (x1 ) + j2 (x2 )}] and A2 ⊆ ϕ2 (Z2,0 ∩ L0,1 ) = [c2 , u2 (x1 ) ⊕ u2 (x2 )] with xy x⊕y 00 0 02 2 . 20 2 22 0

462

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

0 0 = , j2 (a ⊕ u2 (x1 ) ⊕ u2 (x2 ) ⊕ ... ⊕ um (xm )) = 2 0 j2 (a) + j2 (x1 ) + j2 (x2 ) + ... + j2 (xm ) and (A1 ∪ A2 )1 is closed in respect to , it follows A1 A2 ⊆ A1 .

Since j1

Theorem 15.3.3.3 Let A1 and A2 be subclasses of L, A1 ⊆ L0,1 , A2 ⊆ L0,2 , A1 ⊆ [L10,1 ] ∪ B∞ ∪ ϕ2 (Z2,0 ∩ L0,1 ), A2 ⊆ ϕ2 ([L10,1 ] ∪ B∞ ) ∪ (Z2,0 ∩ L0,1 ) and A11 ⊆ {c0 , c1 , j1 , j4 } or A12 ⊆ {c0 , c1 , u2 , u3 }. Then, A1 ∪ A2 is closed if and only if (A1 ∪ A2 )1 is a subsemigroup of (P31 ; ). Proof. If A1 ∪ A2 is closed, then (A1 ∪ A2 )1 is obviously a subsemigroup of (P31 ; ). Let (A1 ∪ A2 )1 be a closed set in respect to . Then the following three cases are possible: Case 1: (A1 ∪ A2 )1 ⊆ {c0 , c1 , c2 , j1 , j4 , u2 , u3 }. In this case, by Lemma 15.3.3.2, (a), (b), and Lemma 15.3.3.1, A1 ∪ A2 is closed. Case 2: A11 ⊆ {c0 , c1 , j1 , j4 } and A12 ⊆ {c0 , c2 , u2 , u3 }. Because of Lemma 15.3.3.2, (b) it holds that A2 A1 ⊆ A2 . Since A1 is not a subset of [L10,1 ] ∪ B∞ , we have pr0,1 A1 ⊆ [L1 ] or pr0,1 A1 ⊆ [L1 ] and j1 (x1 ) + j2 (x2 ) ∈ A1 . With the aid of Lemma 15.3.3.2, (c), (d): A1 A2 ⊆ A1 . Thus, because of Lemma 15.3.3.1, A1 A2 is closed. Case 3: (A11 ⊆ {c0 , c1 , j1 , j4 } and (A12 ⊆ {c0 , c2 , u2 , u3 }. Because of ϕ2 ({c0 , c1 , j1 , j4 }) = {c0 , c2 , u2 , u3 } the classes that fulﬁll the conditions of the third case are isomorphic to such classes that fulﬁll the conditions of the second case. Therefore, Theorem 15.3.3.3 is also valid in the third case. Theorem 15.3.3.4 Let A1 and A2 be subclasses of L, A1 ⊆ L0,1 , A2 ⊆ L0,2 , A1 ∪ A2 ⊆ [L1 ], A11 ⊆ {c0 , c1 , j1 , j4 } and A12 ⊆ {c0 , c2 , u2 , u3 }. Then, A1 ∪ A2 is closed if and only if pr0,1 A1 ⊆ [L1 ] and pr0,2 A2 ⊆ [L1 ] hold and (A1 ∪ A2 )1 is closed in respect to . Proof. If pr0,i Ai ⊆ [L1 ] for i = 1, 2 and (A1 ∪ A2 )1 is a subsemigroup of (P31 ; ), then Theorem 15.3.3.2, (c) and Lemma 15.3.3.1 imply that A1 ∪ A2 is closed. } Let A1 ∪ A2 be a subclass of L. Since, by assumption, A11 ⊆ {c0 , c1 ,j1 , j4 0 0 1 and A12 ⊆ {c0 , c2 , u2 , u3 }, there is a function p1 ∈ A11 with p ∈ 2 10 0 0 2 and a function q ∈ A12 with p ∈ . Because A1 ∪ A2 is closed, we 1 20 have p A2 ⊆ A1 and q A1 ⊆ A2 . Obviously, from this and the assumption A1 ∪ A2 ⊆ [L1 ], it follows that A1 ⊆ [L1 ] and A2 ⊆ [L1 ]. Suppose pr0,1 A1 ⊆

15.3 Classes of Quasilinear Functions of P3

463

[L1 ]. Then, by Section 15.3.2, a function t(x1 , x2 ) = g(x1 )+j2 (x2 ) with g ∈ A11 belongs to A1 . Therefore the function if g ∈ {j0 , j1 , j4 , j5 }, t(x1 , q(x2 )) t (x1 , x2 ) := t(q(x1 ), q(x2 )) if g ∈ {j2 , j3 }, belongs to A1 and it holds pr0,1 t ∈ [L1 ] obviously, contrary to the assumption. Consequently, pr0,1 A1 ⊆ [L1 ]. Similarly, one can show that pr0,2 A2 ⊆ [L1 ]. To complete the description of all subclasses of L0,1 ∪ L0,2 , which are not subclasses of L0,1 , L0,2 or [L1 ], we are only still missing the subclasses of [L10,1 ] ∪ B∞ ∪ ϕ2 (Z2,0 ∩ L0,1 ) and of ϕ2 (L10,1 ∪ B∞ ) ∪ (Z2,0 ∩ L0,1 ) because of Theorems 15.3.3.3 and 15.3.3.4. Since the two sets to be examined are isomorphic, determining only the subclasses of one suﬃces, as in the following theorem. Theorem 15.3.3.5 The subclasses of [L10,1 ] ∪ B∞ ∪ ϕ2 (Z2,0 ∩ L0,1 ), which are not subclasses of [L1 ], L0,1 or L0,2 , are the following classes: A ∪ A , where A ∈ {[c0 , c1 ], [Ja ] ∪ B∞ , [Jb ] | a ∈ {27, 33, 36, 38, 39, 40, 41}, b ∈ {3, 12, 25} } and A ∈ { ϕ2 (Z2,0 ∩ L0,1 ), ϕ2 ([c0 , j1 (x1 ) + j1 (x2 )]) }, [Jc ] ∪ B0,∞ ∪ ϕ([c0 , j1 (x1 ) + j1 (x2 )]), c ∈ {4, 13, 16, 18, 23, 28, 30, 34}, [Jd ] ∪ B1,∞ ∪ ϕ2 ([c0 , j1 (x1 ) + j1 (x2 )]), d ∈ {14, 24}, and [A1 ] ∪ Br , [Je ] ∪ B0,r ∪ [c0 , u2 ], [Jf ] ∪ B1,r ∪ [c0 , u2 ], [Jg ] ∪ Bα,r ∪ Bβ,r−1 ∪ [A2 ], [Jg ] ∪ B0,r ∪ B1,s ∪ [c0 , u2 ], where e ∈ {4, 13, 16, 18, 23, 27, 28, 30, 33, 34, 36, 40}, f ∈ {7, 14, 19, 21, 24, 27, 29, 31, 33, 35, 36, 40}, g ∈ {27, 33, 36, 40}, 2 ≤ r, s ≤ ∞, {α, β} = E2 , A1 is a subsemigroup of J ∪ U25 with (A1 \{c0 }) ∩ U25 = ∅, which contains {j2 , j3 }, and A2 ∈ { [c2 ], [c0 , c2 , u2 ] }. Proof. The theorem follows from Sections 15.2 and 15.3.2. 15.3.4 The Remaining Subclasses of L Obviously, Lemma 15.3.3.1 implies the following Lemma 15.3.4.1 Let A1 , A2 and A3 be subclasses of L with A1 ⊆ L0,1 , A2 ⊆ L0,2 and A3 ⊆ L1,2 . Then A1 ∪ A2 ∪ A3 is closed if and only if A1 ∪ A2 , A1 ∪ A3 and A2 ∪ A3 are closed sets and (A1 ∪ A2 ∪ A3 )1 is a semigroup. Since the sets L0,1 ∪L0,2 , L0,1 ∪L1,2 , L0,2 ∪L1,2 are isomorphic, one has a complete description of the subclasses of L0,1 ∪L0,2 ∪L1,2 through Lemma 15.3.4.1

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

464

and Sections 15.3.2 and 15.3.3. The following theorem now gives information about the missing subclasses of L. Theorem 15.3.4.2 The subclasses of L, which are not subsets of [L]1 and which contain at least a permutation, are A1 ∪ [s1 ] (A1 is a subclass of L0,1 ∪ L0,2 ∪ L1,2 ), A2 ∪[s1 , si ] (i ∈ {2, 3, 6}, A2 is a subclass of L0,1 ∪L0,2 ∪L1,2 with si A2 ⊆ A2 , A12 ∪ {s1 , si } is a subsemigroup of (P31 ; ), L0,1 ∪ L0,2 ∪ L1,2 ∪ [s1 , s4 , s5 ] and L. Proof. The theorem follows from Section 15.2 and from the properties of the functions of L.

15.4 The Subclasses of [O 1 ∪ {max}] 15.4.1 Some Descriptions of the Class M Some functions of P3 , which are used in Sections 15.4.2–15.4.6, are deﬁned in the following two tables (see also Table 15.1). Table 15.12

Table 15.11 x 0 1 2

j2 0 0 1

j5 0 1 1

u2 0 0 2

u5 0 2 2

v2 1 1 2

v5 1 2 2

s1 0 1 2

x 0 0 0 1 1 1 2 2 2

y x ∨ y := max(x, y) 0 0 1 1 2 2 0 1 1 1 2 2 0 2 1 2 2 2

Our object of investigation is the set M := [{c0 , c1 , c2 , j2 , j5 , u2 , u5 , v2 , v5 , max}], which, by [Mac 79] (see Theorem 14.1.4), is a maximal class of the class 012001 O := P ol3 . 012122 M can also be described in the form [O1 ∪ {max}] or

n≥1

{f n ∈ P3 | ∃f1 , ..., fn ∈ O1 : f (x1 , ..., xn ) = f1 (x1 ) ∨ ... ∨ fn (xn )}.

15.4 The Subclasses of [O1 ∪ {max}]

465

15.4.2 Some Lemmas and a Rough Partition of the Subclasses of M One checks the following lemma easily: Lemma 15.4.2.1 (a) For every function f n ∈ M there is exactly a representation of the following form: f (x1 , ..., xn ) = f1 (x1 ) ∨ ... ∨ fn (xn )

(15.1)

fi (x) := f (0, ..., 0, x, 0, ..., 0)

(15.2)

with i−1

(i = 1, ..., n). (b) If the function f is given in the form (15.1) with (15.2), for every g ∈ M 1 it holds: (g f )(x1 , ..., xn ) = (g f1 )(x1 ) ∨ ... ∨ (g fn )(xn ). We agree to represent functions f n of M in the form (15.1) in which the functions fi are given by (15.2). Further, denote F (f ) the set of all functions fi of (15.2) that describe the function f . Let numf (fi ) be the number of occurrence of function fi in (15.1) for f , where fi is deﬁned by (15.2). If f arises from the context, instead of F (f ), we will write only F , and instead of numf (fi ), we will write only num(fi ). Lemma 15.4.2.2 is the basis for the following theorems about bases and generating systems for subclasses of M . Lemma 15.4.2.2 Let f n ∈ M , n ≥ 2, f (x1 , ..., xn ) := f1 (x1 ) ∨ ... ∨ fn (xn ),

(15.3)

F := {f1 , ..., fn }, {c0 , c1 , c2 } ∩ F = ∅ (i.e., all variables of f are essential) and denote num(g) the number of functions g ∈ F that occur in (15.3). Then (a) f ∈ [[{f }]2 ] if and only if the function f fulﬁlls at least one of the following conditions: 1) s1 ∈ F and u5 ∈ F ; 2) num(s1 ) = 1 and F ∈ {{s1 , u5 }, {s1 , u2 , u5 }}; 3) F ⊆ {u2 , u5 }; 4) F ∈ {{j5 }, {j2 , u5 }, {j5 , u2 }, {j5 , u5 }, {j2 , j5 , u2 }, {j2 , j5 , u5 }, {j2 , u2 , u5 }, {j5 , u2 , u5 }, {j2 , j5 , u2 , u5 }}; 5) num(j5 ) = 1 and F = {j2 , j5 }; 6) num(v2 ) = 1 and F = {v2 , v5 }; 7) num(u2 ) ≥ 2 and F = {j2 , u2 }; 8) n = 2 and F ∈ {{j2 }, {v5 }, {j2 , u2 }}.

466

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

(b) f ∈ [[{f }]3 ] and f ∈ [[{f }]2 ] if and only if the function f fulﬁlls at least one of the following conditions: 1) num(s1 ) ≥ 2 and u5 ∈ F ; 2) num(s1 ) = 1, u5 ∈ F and F ∈ {{s1 , u5 }, {s1 , u2 , u5 }}; 3) num(j5 ) ≥ 2 and F = {j2 , j5 }; 4) num(u2 ) = 1 and F = {j2 , u2 }; 5) num(v2 ) ≥ 2 and F = {v2 , v5 }; 6) n = 3 and F ∈ {{j2 }, {v5 }}. (c) f ∈ [[{f }]n−1 ] and n ≥ 4 if and only if F ∈ {{j2 }, {v5 }, {j2 , u2 }} and num(u2 ) ≤ 1. Proof. Since, by assumption, all variables of f are essential, we have F ⊆ {j2 , j5 , u2 , u5 , v2 , v5 }. If {v2 , v5 } ∩ F = ∅ then F ⊆ {v2 , v5 }. Thus for f the following cases are possible: Case 1: s1 ∈ F . Case 1.1: u5 ∈ F . Let w.l.o.g. f1 = s1 . The functions ht (x, y) := f (x, ..., x, y, x, ..., x) = x ∨ ft (y) t−1

are superpositions over f for every t ∈ {2, 3, ..., n}. Then one can obtain function f as a superposition over these functions as follows: (...((hn hn−1 ) hn−2 )... h3 ) h2 . Therefore, f ∈ [[{f }]2 ] in Case 1. Case 1.2: u5 ∈ F . Let w.l.o.g. f1 = s1 and f2 = u5 . Then the ternary functions x ∨ u5 (y) ∨ fi (z) (i = 3, ..., n) are superpositions over f , and these functions form a generating system for f . Therefore, f ∈ [[{f }]3 ]. This is reducible to f ∈ [[{f }]2 ] iﬀ num(s1 ) = 1 and F ∈ {{s1 , u5 }, {s1 , u2 , u5 }}. Case 2: F ⊆ {j2 , j5 , u2 , u5 }. Case 2.1: F = {j2 }. Because of j2 j2 = c0 , we have j2 (x1 ) ∨ ... ∨ j2 (xr ) ∈ [{j2 (x1 ) ∨ ... ∨ j2 (xr−1 )}] for all r ≥ 2. Case 2.2: F = {j5 }. In this case, we have f ∈ [{j5 (x) ∨ j5 (x2 )}], i.e., f ∈ [[{f }]2 ]. Case 2.3: F = {j2 , j5 }. By (...((g g) g)...) g, where g(x, y) := j5 (x) ∨ j2 (y), one can obtain all functions of the form j5 (x1 ) ∨ j2 (x2 ) ∨ ... ∨ j2 (xt ) for t ≥ 2. Thus f ∈ [[{f }]2 ], if num(j5 ) = 1. If num(j5 ) ≥ 2 then j5 (x) ∨ j5 (y) ∨ j2 (z) belongs to [{f }]3 and we have f ∈ [[{f }]3 ] and f ∈ [[{f }]2 ]. Case 2.4: F ⊆ {u2 , u5 }.

15.4 The Subclasses of [O1 ∪ {max}]

467

Since up uq = uq for all p, q ∈ {2, 5}, it follows from Lemma 15.4.2.1, (b) that f ∈ [[{f }]2 ]. Case 2.5: F = {j2 , u2 }. Because of j2 u2 = j2 , u2 j2 = c0 , u2 u2 = u2 and u2 (x) ∨ j2 (x) = u2 we have in this case: num(u2 ) ≥ 2 =⇒ {j2 (x) ∨ u2 (y), u2 (x) ∨ u2 (y)} ⊆ [{f }]2 =⇒ f ∈ [[{f }]2 ] and num(u2 ) = 1 =⇒ f ∈ [[{f }]n−1 ]. Case 2.6: F = {j2 , u5 }. Since ∆n−1 f = u5 , the functions u5 and j2 (u5 (x)) ∨ u5 (y) = j5 (x) ∨ u5 (y) belong to [{f }]2 . Thus by j5 u5 = j5 and u5 j5 = u5 , functions of the form j5 (x1 ) ∨ ... ∨ j5 (xr ) ∨ u5 (xr+1 ) ∨ ... ∨ u5 (xm ) are superpositions over [{f }]2 for arbitrary m > r ≥ 1. From this and by j5 (j2 (x1 ) ∨ u5 (x2 )) ∨ u5 (x2 ) = j2 (x1 ) ∨ u5 (x2 ) ) follows then that f ∈ [[{f }]2 ]. Case 2.7: F = {j5 , u2 }. If num(u2 ) = 1 or num(j5 ) ≥ 2, we have n = 2 or h(x, y) := j5 (x) ∨ j5 (y) ∨ u2 (z) ∈ [{f }]3 . Thus, h(x, y, y) = j5 (x) ∨ y ∈ [{f }]2 . Then functions of the form x1 ∨j5 (x2 )∨...∨j5 (xt ) (t ≥ 2) are superpositions over h. Replacing x1 by j5 (x1 ) ∨ u2 (x2 ) ∈ [{f }]2 and then identifying variables shows that f ∈ [[{f }]2 ] in the case num(u2 ) = 1. If we have num(u2 ) ≥ 2 and num(j5 ) = 1, then the function j5 (x) ∨ u2 (y) ∨ u2 (z) belongs to [{f }]3 and, therefore, the function x ∨ u2 (y) also belongs to [{f }]3 . From this, f ∈ [[{f }]2 ]. Case 2.8: F = {j5 , u5 }. Because of j5 (j5 (x1 )∨u5 (x2 ))∨u5 (j5 (x3 )∨u5 (x4 )) = j5 (x1 )∨j5 (x2 )∨u5 (x3 )∨ u5 (x4 ) we have f ∈ [[{f }]2 ]. Case 2.9: F = {j2 , j5 , u2 }. In this case, j2 (x) ∨ y and j5 (x) ∨ u2 (y) are some superpositions over f . Thus f ∈ [[{f }]2 ] is an easy conclusion from our considerations of Cases 1 and 2.7. Case 2.10: F = {j2 , j5 , u5 }. Some superpositions over f are u5 = ∆n−1 f , j2 (x) ∨ u5 (y) and j5 (x) ∨ u5 (y). Hence, and by Case 2.8, we have f ∈ [[{f }]2 ]. Case 2.11: F = {j2 , u2 , u5 }. Then, the functions u5 = ∆n−1 f , u2 (x) ∨ u5 (y), j2 (x) ∨ u5 (y) and j2 (u5 (x)) ∨ u5 (y) = j5 (x) ∨ u5 (y) are superpositions over f . Using considerations from Cases 2.4, 2.6, and 2.8, we obtain f ∈ [[{f }]2 ]. Case 2.12: F = {j5 , u2 , u5 }. Then the functions x∨u5 (y), u2 (x)∨u5 (y) and j5 (x)∨u5 (y) are superpositions over f . By Cases 2.4 and 2.8, we get f ∈ [[{f }]2 ]. Case 2.13: F = {j2 , j5 , u2 , u5 }. Since j2 (x) ∨ u5 (y), j5 (x) ∨ u5 (y), u2 (x) ∨ u5 (y) ∈ [{f }]2 , one can obtain f ∈ [[{f }]2 ] by proceeding to the above cases analogously. Case 3: F ⊆ {v2 , v5 }.

468

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Case 3.1: F = {v2 }. Obviously, f ∈ [{v2 (x) ∨ v2 (y)}], i.e., f ∈ [[{f }]2 ]. Case 3.2: F = {v5 }. Because of v5 v5 = c2 we have f ∈ [[{f }]n−1 ]. Case 3.3: F = {v2 , v5 }. This case resembles Case 2.3. Thus we have: num(v2 ) = 1 =⇒ f ∈ [[{f }]2 ] and

num(v2 ) ≥ 2 =⇒ f ∈ [[{f }]3 ] ∧ f ∈ [[{f }]2 ].

Next, we want to consider a rough partition of the lattice L3 (M ). Let R := {f ∈ M | |Im(f )| ≤ 2}. Then every subclass T of M has the form (T ∩ R) ∪ (T ∩ P ol3

0 ), 2

(15.4)

since all functions of M with |Im(f )| ≥ 3 have the property f (0, ..., 0) = 0 and f (2, ..., 2) = 2. Thus one can obtain all subclasses of M if every subclass T that fulﬁlls exactly one of the following conditions (I)–(IV ) is determined: (I) T ⊆ [M 1 ]; (II) T ⊆ R and T ⊆ [M1 ]; 0 , T ⊆ R and T ⊆ [M 1 ]; (III) T ⊆ M ∩ P ol3 2 0 (IV ) ∃T1 ∈ L3 (R)\{∅} ∃T2 ∈ L3 (M ∩ P ol3 )\L3 (R) : 2 1 T = T1 ∪ T2 ∧ T ⊆ [M ]. We determine the subclasses of M in Sections 15.4.3–15.4.6, in compliance with the above partition (I)–(IV) of the subclasses. First, we determine the maximal classes of M . Theorem 15.4.2.3 M has exactly 8 maximal classes. These classes are: (1) M ∩ P ol{0} = {f ∈ M | F (f ) ⊆ {c0 , j2 , j5 , u2 , u5 , s1 }}; (2) M ∩ P ol{2} = {f ∈ M | F (f ) ∩ {c2 , u2 , u5 , v2 , v5 , s1 } = ∅}; (3) M ∩ P ol{1, 2} = {f ∈ M | F (f ) ∩ {c1 , c2 , j5 , u5 , v2 , v5 , s1 } = ∅};

(4) M ∩ P ol

0120 0121

15.4 The Subclasses of [O1 ∪ {max}]

469

= {f ∈ M | F (f ) ⊆ {c0 , c1 , j2 , j5 , u2 , s1 } ∨ F (f ) ⊆ {c1 , c2 , v2 }}; 0121 (5) M ∩ P ol 0122 = {f ∈ M |F (f ) ⊆ {c0 , c1 , j5 , u5 , s1 } ∨ F (f ) ⊆ {c1 , c2 , v2 , v5 }}; 01201 (6) M ∩ P ol 01222 = {f ∈ M |F (f ) ⊆ {c0 , c2 , u2 , u5 , s1 } ∨ F (f ) ⊆ {c1 , v2 , v5 }}; 01201 (7) M ∩ P ol 01212 = {f ∈ M | F (f ) ⊆ {c0 , c1 , j2 , j5 , s1 } ∨ F (f ) ⊆ {c1 , c2 , v2 , v5 }}; (8) [{s1 }] ∪ R := [{s1 }] ∪ {f ∈ M | |Im(f )| ≤ 2}. Proof. Since M = [M 1 ∪ {max}] and max ∈ T for all classes T , which are deﬁned by (1)–(7), it is easy to prove that the classes (1)–(7) are M -maximal. One can show the M -maximality of [{s1 }] ∪ R as follows: Let f ∈ M \([{s1 }] ∪ R). Since c0 ∈ R, one of the following 9 functions is a superposition over R ∪ {f }: x ∨ y, x ∨ g(y) (g ∈ {j2 , j5 , u2 , u5 }), h1 (x) ∨ h2 (y) (h1 ∈ {j2 , j5 }, h2 ∈ {u2 , u3 }). Through substitution of x, y by certain functions of {j2 , j5 , u2 , u5 }, one can reduce these 9 cases to the case: t(x, y) := j5 (x)∨u2 (y) ∈ [R∪{f }]. Because of j5 (t(x, u5 (y)) = j5 (x)∨j5 (y) and u5 (j2 (x)∨j2 (y)) = u2 (x)∨u2 (y) it holds x∨y = t(j5 (x)∨j5 (y), u2 (x)∨u2 (y)). Hence [R ∪ {f }] = M and [{s1 }] ∪ R is M -maximal. We still have to show that the class M does not have any further maximal classes. Suppose T ⊂ M is an M -maximal class that is diﬀerent from the 8 M -maximal classes listed above. Then, there is for every i ∈ {1, 2, ..., 8}, a certain function fi ∈ T that does not belong to the class (i). Consequently, it holds f2 ∈ {c0 , c1 , j2 , j5 } for f2 (x) := f2 (x, x, ..., x), and f3 (a1 , a2 , ..., an ) = 0 for certain a1 , a2 , ..., an ∈ {1, 2}. Hence, c0 ∈ T , since j2 j2 = c0 and f3 (g1 (x), ..., gn (x)) = c0 , if f2 ∈ {c1 , j5 } and x if ai = 2, gi (x) := f2 (x) if ai = 1 (i = 1, 2, ..., n). Some unary functions ft with f4 ∈ {u5 , v5 }, f5 ∈ {j2 , u2 }, f6 ∈ {j2 , j5 } and f7 ∈ {u2 , u5 } are superpositions over {ft , c0 } (t = 4, 5, 6, 7). It is easy to check that {j2 , j5 , u2 , u5 } ⊂ [{f4 , f5 , f6 , f7 }] holds. In the above proof of the M -maximality of [{s1 }] ∪ R, we have shown already that max ∈ [{j2 , j5 , u2 , u5 , f8 }] holds. Since in addition ui ∨ c1 = vi (i = 2, 5), v2 (c0 ) = c1 and v5 (c1 ) = c2 , we have M 1 ∪{max} ⊆ T . Consequently, T = M , in contradiction to T ⊂ M .

470

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

15.4.3 The Subclasses of [M 1 ] Since all subclasses of [P31 ] were already determined in Section 15.2, one obtains the following theorem as a consequence of Theorem 15.2.1: Theorem 15.4.3.1 [M 1 ] has exactly 190 pairwise distinct subclasses Hi . These classes Hi are deﬁned in Table 15.13 through Hi1 for i ∈ {1, 2, ..., 95}, and through H95+i := Hi ∪ [{s1 }] for i = 1, 2, ..., 95.

i 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 7O 73 76 79 82 85 88 91 94

Hi1 ∅ {c2 } {c1 , c2 } {j2 , c0 } {j2 , c0 , c1 } {j2 , j5 , c0 , c1 } {u2 , c0 } {u5 , c2 } {u5 , c0 , c2 } {u2 , u5 , c0 , c2 } {v2 , c2 } {v2 , c1 , c2 } {j5 , c1 , c2 } {j2 , j5 , c0 , c1 , c2 } {u2 , c0 , c1 , c2 } {v2 , c0 , c1 } {v2 , v5 , c0 , c1 , c2 } {j5 , u5 , c0 } {j5 , u5 , c0 , c1 , c2 } {j2 , j5 , u2 , c0 , c1 } {j2 , j5 , u2 , u5 , c0 , c1 , c2 } {j5 , v2 , c1 } {j5 , v2 , c0 , c1 , c2 } {j2 , j5 , v2 , c0 , c1 } {j5 , v2 , v5 , c0 , c1 , c2 } {u2 , v2 , c0 , c1 } {u5 , v5 , c2 } {u5 , v2 , v5 , c2 } {u2 , u5 , v2 , v5 , c2 } {j2 , u2 , v2 , c0 , c1 , c2 } {j2 , j5 , u2 , v2 , c0 , c1 } {j5 , u5 , v2 , v5 , c0 , c1 , c2 }

i 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95

Table 15.13 Hi1 {c0 } {c0 , c1 } {c0 , c1 , c2 } {j5 , c0 } {j5 , c0 , c1 } {u2 } {u5 , c0 } {u2 , u5 } {u2 , u5 , c0 } {v2 } {v2 , c1 } {v2 , v5 , c2 } {j2 , c0 , c1 , c2 } {u2 , c0 , c1 } {u5 , c0 , c1 , c2 } {v5 , c0 , c1 , c2 } {j5 , u5 } {j2 , u2 , c0 , c1 } {j2 , u2 , c0 , c1 , c2 } {j2 , j5 , u2 , c0 , c1 , c2 } {j2 , v2 , c0 , c1 } {j5 , v2 , c0 , c1 } {j5 , v5 , c1 , c2 } {j2 , j5 , v2 , c0 , c1 , c2 } {j2 , j5 , v2 , v5 , c0 , c1 , c2 } {u2 , v2 , c2 } {u5 , v5 , c1 , c2 } {u5 , v2 , v5 , c1 , c2 } {u2 , u5 , v2 , v5 , c0 , c1 , c2 } {j5 , u5 , v5 , c1 , c2 } {j2 , j5 , u2 , v2 , c0 , c1 , c2 } {j2 , j5 , u2 , u5 , v2 , v5 } ∪ H81

i 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93

Hi1 {c1 } {c0 , c2 } {j5 } {j5 , c1 } {j2 , j5 , c0 } {u5 } {u2 , c2 } {u2 , c0 , c2 } {u2 , u5 , c2 } {v5 , c2 } {v5 , c1 , c2 } {v2 , v5 , c1 , c2 } {j5 , c0 , c1 , c2 } {u5 , c1 , c2 } {u2 , u5 , c0 , c1 , c2 } {v2 , c0 , c1 , c2 } {j2 , u2 , c0 } {j5 , u5 , c1 , c2 } {j2 , j5 , u2 , c0 } {j2 , j5 , u2 , u5 , c0 } {j2 , v2 , c0 , c1 , c2 } {j5 , v2 , c1 , c2 } {j5 , v5 , c0 , c1 , c2 } {j5 , v2 , v5 , c1 , c2 } {u2 , v2 } {u2 , v2 , c0 , c1 , c2 } {u5 , v5 , c0 , c1 , c2 } {u5 , v2 , v5 , c0 , c1 , c2 } {j2 , u2 , v2 , c0 , c1 } {j5 , u5 , v5 , c0 , c1 , c2 } {j5 , u5 , v2 , v5 , c1 , c2 }

15.4 The Subclasses of [O1 ∪ {max}]

471

15.4.4 The Subclasses of R

Theorem 15.4.4.1 The class J1 := {f ∈ M | Im(f ) ⊆ {0, 1}} has exactly the following (countable inﬁnite-many) subclasses, which are not subclasses of [M 1 ]: J1 , J2 := J1 ∩ P ol{0} = {f ∈ J1 | f ∈ [{c1 }]}, J3 := J1 ∩ P ol{1} = {f ∈ J1 | f ∈ [{c0 }]}, J4 := J1 ∩ P ol{0} ∩ P ol{1}, J5 := {f ∈ J1 | numf (j5 ) ≤ 1}, J6 := J2 ∩ J5 , J7 := J3 ∩ J5 , J8 := J4 ∩ J5 , J9 := {f ∈ J1 | f ∈ [{j5 }] ∨ numf (j5 ) = 0}, J10 := {f ∈ J9 | f ∈ [{c1 }]}, J11 := {f ∈ J9 | f ∈ [{j5 }]}, J12 := {f ∈ J9 | f ∈ [{c1 , j5 }], J9,r := {f ∈ J9 | numf (j2 ) ≤ r}, J10,r := J10 ∩ J9,r , J11,r := J11 ∩ J9,r , J12,r := J12 ∩ J9,r , J13 := {f ∈ J1 | numf (j2 ) = 0}, J14 := J13 ∩ J2 , J15 := J13 ∩ J3 , J16 := J13 ∩ J4 , where r = 1, 2, 3, 4, .... . Proof. With the aid of the Hasse diagram of the above classes (see Figure 15.1) and the generating systems of these classes (see Table 15.14), one can prove the theorem.

472

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Jr1 J r3

J2 r

Jr5

J4 r

J6

r

rJ7 r J8 rJ9

J10 r

rJ11 J12 r J9,r r

J10,r r

rJ11,r r J12,r

J13 r J14 r

rH16 rJ15

r J16

H15 r H11

rH14 rH13

r

rH10 rH5 r H12

H9

r

r H2 r ∅ = H1

Fig. 15.1. The subclasses of J1

rH3

15.4 The Subclasses of [O1 ∪ {max}]

473

Table 15.14 A J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 J9,r J10,r J11,r J12,r J13 J14 J15 J16

generating system (or basis, if it exists) for A {j5 (x) ∨ j5 (y), j2 , c1 } {j5 (x) ∨ j5 (y), j2 } {j5 (x) ∨ j5 (y), j2 (x) ∨ j5 (y), c1 } {j5 (x) ∨ j5 (y), j2 (x) ∨ j5 (y)} {j5 (x) ∨ j2 (y), c0 , c1 } {j5 (x) ∨ j2 (y), c0 } {j5 (x) ∨ j2 (y), c1 } {j5 (x) ∨ j2 (y)} {j5 , c1 , j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {j5 , j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {c1 , j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {j5 , c1 , j2 (x1 ) ∨ ... ∨ j2 (xr )} {j5 , j2 (x1 ) ∨ ... ∨ j2 (xr )} {c1 , j2 (x1 ) ∨ ... ∨ j2 (xr )} {j2 (x1 ) ∨ ... ∨ j2 (xr )} {j5 (x) ∨ j5 (y), c0 , c1 } {j5 (x) ∨ j5 (y), c0 } {j5 (x) ∨ j5 (y), c1 } {j5 (x) ∨ j5 (y)}

A1 {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j5 , c1 } {j5 } {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j5 , c1 } {j5 } {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j2 , c0 , c1 } {j2 , c0 } {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j2 , c0 , c1 } {j2 , c0 } {j5 , c0 , c1 } {j5 , c0 } {j5 , c1 } {j5 }

In analog mode, the following two theorems can be proven when one uses Figures 15.2 and 15.3 and Tables 15.15 and 15.16. Theorem 15.4.4.2 The class U1 := {f ∈ M | Im(f ) ⊆ {0, 2}} has exactly the following 14 pairwise diﬀerent subclasses, which are not subclasses of [M 1 ]: U1 , U2 := U1 ∩ P ol{0} = {f ∈ U1 | f ∈ [{c2 }]}, U3 := {f ∈ U1 | F (f ) ⊆ {c0 , c2 , u2 }}, U4 := {f ∈ U1 | F (f ) ⊆ {c0 , c2 , u5 }}, U5 := U1 ∩ P ol{2} = {f ∈ U1 | f ∈ [{c0 }]}, U6 := {f ∈ U3 | f ∈ [{c2 }]}, U7 := {f ∈ U4 | f ∈ [{c2 }]}, U8 := {f ∈ U3 | f ∈ [{c0 }]}, U9 := {f ∈ U1 | f ∈ [{c2 }] ∨ numf (u5 ) ≥ 1}, U10 := {f ∈ U4 | f ∈ [{c0 }]}, U11 := {f ∈ U1 | f ∈ [{c0 , c2 }]}, U12 := {f ∈ U3 | f ∈ [{c0 , c2 }]}, U13 := {f ∈ U9 | f ∈ [{c2 }]}, U14 := {f ∈ U4 | f ∈ [{c0 , c2 }]}.

H4

H22

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

U10

474

H6

H26

U12

H17

∅

U13

U11

H24 H25

H28

H20

U7

U3

Fig. 15.2. The subclasses of U1

H2

H19

U6

U2

U1

U14

H18

H27

H23

U4

U8

H21

U5

U9

q q q BB

B q

B

B q

B

B q q

B

B

B q

B

B

B q q

B

B q B q

B q q

B

B q q q

Bq @ J J @ q J @ J @ J q q @ J @ J @ q q J A@ A @ J A @ A Jq A @ A J A @ A J A A @ J A A @q q J A A @ @ J A A @ J A A q @ J A A H HH @ A J A HH J @A A HH Aq @Aq Jq

15.4 The Subclasses of [O1 ∪ {max}]

475

Table 15.15 A U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14

Basis for A {u2 (x) ∨ u2 (y), u5 , c0 , c1 } {u2 (x) ∨ u2 (y), u5 , c0 } {u2 (x) ∨ u2 (y), c0 , c2 } {u5 (x) ∨ u5 (y), c0 , c2 } {u2 (x) ∨ u2 (y), u5 , c2 } {u2 (x) ∨ u2 (y), c0 } {u5 (x) ∨ u5 (y), c0 } {u2 (x) ∨ u2 (y), c2 } {u2 (x) ∨ u5 (y), c2 } {u5 (x) ∨ u5 (y), c2 } {u2 (x) ∨ u2 (y), u5 } {u2 (x) ∨ u2 (y)} {u2 (x) ∨ u5 (y)} {u5 (x) ∨ u5 (y)}

A1 {u2 , u5 , c0 , c2 } {u2 , u5 , c0 } {u2 , c0 , c2 } {u5 , c0 , c2 } {u2 , u5 , c2 } {u2 , c0 } {u5 , c0 } {u2 , c2 } {u5 , c2 } {u5 , c2 } {u2 , u5 } {u2 } {u5 } {u5 }

Theorem 15.4.4.3 The class V1 := {f ∈ M | Im(f ) ⊆ {1, 2}} has exactly the following (countable inﬁnite-many) subclasses, which are not subclasses of [M 1 ] : V1 , V2 := V1 ∩ P ol(2) = {f ∈ V1 | f ∈ [{c1 }]}, V3 := {f ∈ V1 | f ∈ [{c1 , c2 , v2 }] ∨ numf (v5 ) ≥ 1}, V4 := {f ∈ V3 | f ∈ [{c1 }]}, V5 := {f ∈ V1 | f ∈ [{c1 , c2 , v2 }] ∨ numf (v2 ) ≤ 1}, V6 := {f ∈ V5 | f ∈ [{c1 }]}, V7 := {f ∈ V1 | f ∈ [{c2 , v2 }] ∨ F (f ) ⊆ {c1 , v5 }}, V8 := {f ∈ V7 | f ∈ [{c1 }]}, V9 := {f ∈ V7 | f ∈ [{v2 }]}, V10 := {f ∈ V7 | f ∈ [{c1 , v2 }]}, V7,r := {f ∈ V7 | numf (v5 ) ≤ r}, V8,r := {f ∈ V7,r | f ∈ [{c1 }]}, V9,r := {f ∈ V7,r | f ∈ [{v2 }]}, V10,r := {f ∈ V7,r | f ∈ [{c1 , v2 }]}, V11 := {f ∈ V3 | f ∈ [{v2 }]}, V12 := V5 ∩ V11 , V13 := {f ∈ V11 | f ∈ [{c1 }]}, V14 := {f ∈ V12 | f ∈ [{c1 }]}, V15 := {f ∈ V1 | F (f ) ⊆ {c1 , c2 , v2 }}, V16 := {f ∈ V15 | f ∈ [{c1 }]}, V17 := {f ∈ V15 | f ∈ [{c2 }]}, V18 := {f ∈ V15 | f ∈ [{c1 , c2 }]}, where r = 1, 2, 3, 4, ... .

476

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Vr1 rV3

V2 r

V11 r

rV5

r V4 V6

rV7

r r

V8

V12 r V9 r

V10 r

V7,r+1 r V8,r+1 r

rV9,r+1 rV10,r+1 V7,r r

V8,r r

rV9,r r V10,r

r V15 V16 r

H36 r

V17 r H35 r V18

r

r

H31 H29

r r H30 r

r

r H4

rH33 rH7 r H3

r ∅ Fig. 15.3. The subclasses of V1

rV13 r V14

15.4 The Subclasses of [O1 ∪ {max}]

477

Table 15.16 A V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V7,r V8,r V9,r V10,r V11 V12 V13 V14 V15 V16 V17 V18

Generating system (or basis, if it exists) for A {v2 (x) ∨ v2 (y), v5 , c1 } {v2 (x) ∨ v2 (y), v5 } {v2 (x) ∨ v2 (y) ∨ v5 (z), v2 , c1 } {v2 (x) ∨ v2 (y) ∨ v5 (z), v5 } {v2 (x) ∨ v5 (y), v2 , c1 } {v2 (x) ∨ v5 (y), v2 } {v2 , c1 , v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {v2 , v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {c1 , v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {v2 , c1 , v5 (x1 ) ∨ ... ∨ v5 (xr )} {v2 , v5 (x1 ) ∨ ... ∨ v5 (xr )} {c1 , v5 (x1 ) ∨ ... ∨ v5 (xr )} {v5 (x1 ) ∨ ... ∨ v5 (xr )} {v2 (x) ∨ v2 (y) ∨ v5 (z), c1 } {v2 (x) ∨ v5 (y), c1 } {v2 (x) ∨ v2 (y) ∨ v5 (z)} {v2 (x) ∨ v5 (y)} {v2 (x) ∨ v2 (y), c1 , c2 } {v2 (x) ∨ v2 (y), c2 } {v2 (x) ∨ v2 (y), c1 } {v2 (x) ∨ v2 (y)}

A1 {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v5 , c1 , c2 } {v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v5 , c1 , c2 } {v5 , c2 } {v5 , c1 , c2 } {v5 , c1 , c2 } {v5 , c2 } {v5 , c2 } {v2 , c1 , c2 } {v2 , c2 } {v2 , c1 } {v2 }

Theorem 15.4.4.4 Class J1 ∪U1 has exactly the following subclasses (ordered with respect to equality of the unary functions), which are not subclasses of J1 or U1 or [(J1 ∪ U1 )1 ]: J3 ∪ H4 , J7 ∪ H4 , J15 ∪ H4 ; J11 ∪ H6 , J11,r ∪ H6 ; J13 ∪ H6 ; J1 ∪ H6 , J5 ∪ H6 , J9 ∪ H6 , J9,r ∪ H6 ; H5 ∪ U6 ; H3 ∪ U9 , H3 ∪ U10 ; H5 ∪ U3 ; H5 ∪ U4 ; H5 ∪ U1 ; J4 ∪ U13 , J16 ∪ U14 ; J12 ∪ H19 , J12 ∪ U6 , J12,r ∪ H19 ; J14 ∪ U7 ; J11 ∪ H19 , J11 ∪ U6 , J11,r ∪ H19 ; J3 ∪ U9 ; J13 ∪ U4 ; J11 ∪ H24 , J11,r ∪ H24 , J11 ∪ U3 ; J10 ∪ H19 , J10,r ∪ H19 , J10 ∪ U6 , J6 ∪ H19 , J6 ∪ U6 , J2 ∪ H19 , J2 ∪ U6 ;

478

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

J9 ∪ H19 , J9,r ∪ H19 , J5 ∪ H19 , J1 ∪ H19 , J9 ∪ U6 , J5 ∪ U6 , J1 ∪ U6 ; J9 ∪ H24 , J9,r ∪ H24 , J5 ∪ H24 , J1 ∪ H24 , J5 ∪ U3 , J9 ∪ U3 , J1 ∪ U3 ; J2 ∪ U2 ; J1 ∪ U1 ; where r = 1, 2, 3, .... Proof. Let T be an arbitrary subclass of J1 ∪ U1 , which was not described in Theorems 15.4.3.1, 15.4.4.1, or 15.4.4.2. Then, obviously T1 := T ∩ J1 and T2 := T ∩ U1 are subclasses of J1 or U1 with T = T1 ∪ T2 . Further, [T 1 ] is a subclass H of [M 1 ] with H 1 = T 1 . The possibilities for H follow from Theorems 15.4.3.1, 15.4.4.1, and 15.4.4.2. These possibilities imply the possibilities for T1 and T2 , which are given in Table 15.17. When one selects the sets T1 ∪ T2 ⊆ [M 1 ], which are closed, one receives the statement of the theorem.

Table 15.17 [T 1 ] H37 H38 H39 H40 H41 H42 H43 H44 H45 H50 H51 H52 H53 H54 H55 H56 H57 H58 H59 H60 H61

Possibilities for T ∩ J1 T ∩ U1 J3 , J7 , J15 H4 J11 , J11,r H6 J13 H6 J1 , J5 , J9 , J9,r H6 H5 U6 H3 U9 , U10 H5 U3 H5 U4 H5 U1 H9 , J4 , J8 , J16 H18 , U13 , U14 H10 , J12 , J12,r H19 , U6 H11 , J14 H20 , U7 H13 , J11 , J11,r H19 , U6 H12 , J3 , J7 , J15 H22 , U9 H14 , J13 H25 , U4 H13 , J11 , J11,r H24 , U3 H15 , J2 , J6 , J10 , J10,r H19 , U6 H16 , J1 , J5 , J9 , J9,r H19 , U6 H16 , J1 , J5 , J9 , J9,r H24 , U3 H15 , J2 , J6 , J10 , J10,r H26 , U2 H16 , J1 , J5 , J9 , J9,r H28 , U1

Theorem 15.4.4.5 Class J1 ∪ V1 has exactly the following subclasses, which are not subclasses of J1 or V1 or [(J1 ∪ V1 )1 ]: J3 ∪ H7 , J7 ∪ H7 , J15 ∪ H7 ; J11 ∪ H7 , J11,r ∪ H7 ; J13 ∪ H7 ; J1 ∪ H7 , J5 ∪ H7 , J9 ∪ H7 , J9,r ∪ H7 ;

15.4 The Subclasses of [O1 ∪ {max}]

479

H5 ∪ V17 ; H5 ∪ V9 , H5 ∪ V9,r , H5 ∪ V11 , H5 ∪ V12 ; H5 ∪ V15 ; H5 ∪ V1 , H5 ∪ V3 , H5 ∪ V5 , H5 ∪ V7 , H5 ∪ V7,r ; J11 ∪ H32 , J11 ∪ V17 , J11,r ∪ H32 ; J11 ∪ H34 , J11 ∪ V15 , J11,r ∪ H34 ; H12 ∪ V17 , J3 ∪ H32 , J3 ∪ V17 , J7 ∪ H32 , J7 ∪ V17 , J15 ∪ H32 , J15 ∪ V17 ; H14 ∪ V17 , J13 ∪ H32 , J13 ∪ V17 ; H12 ∪ V15 , J3 ∪ H34 , J3 ∪ V15 , J7 ∪ H34 , J7 ∪ V15 , J15 ∪ H34 , J15 ∪ V15 ; H14 ∪ V15 , J13 ∪ H34 , J13 ∪ V15 ; H12 ∪V9 , H12 ∪V11 , H12 ∪V12 , H12 ∪V9,r , J15 ∪V9 , J15 ∪V12 , J15 ∪V11 , J7 ∪V12 , J7 ∪ V11 , J3 ∪ V11 ; H14 ∪ V9 , H14 ∪ V11 , H14 ∪ V12 , H14 ∪ V9,r , J13 ∪ V9 , J13 ∪ V11 , J13 ∪ V12 ; J1 ∪ H32 , J1 ∪ V17 , J5 ∪ H32 , J5 ∪ V17 , J9 ∪ H32 , J9 ∪ V17 , J9,r ∪ H32 ; J1 ∪ H34 , J1 ∪ V15 , J5 ∪ H34 , J5 ∪ V15 , J9 ∪ H34 , J9 ∪ V15 , J9,r ∪ H34 ; H12 ∪ V1 , H12 ∪ V3 , H12 ∪ V5 , H12 ∪ V7 , H12 ∪ V7,r , J15 ∪ V7 , J15 ∪ V5 , J15 ∪ V3 , J15 ∪ V1 , J7 ∪ V5 , J7 ∪ V3 , J7 ∪ V1 , J3 ∪ V3 , J3 ∪ V1 ; H14 ∪ V1 , H14 ∪ V3 , H14 ∪ V5 , H14 ∪ V7 , H14 ∪ V7,r , J13 ∪ V1 , J13 ∪ V3 , J13 ∪ V5 , J13 ∪ V7 ; J1 ∪ V1 ; where r = 1, 2, 3, ... . Table 15.18 [T 1 ] H37 H38 H39 H40 H46 H47 H48 H49 H62 H63 H64 H65 H66 H67 H68 H69 H70 H71 H72 H73 H74

Possibilities for T ∩ J1 T ∩ V1 H12 , J3 , J7 , J15 H7 H13 , J11 , J11,r H7 H14 , J13 H7 H16 , J1 , J5 , J9 , J9,r H7 H5 H32 , V17 H5 H33 , V9 , V9,r , V11 , V12 H5 H34 , V15 H5 H36 , V1 , V3 , V5 , V7 , V7,r H13 , J11 , J11,r H32 , V17 H13 , J11 , J11,r H34 , V15 H12 , J3 , J7 , J15 H32 , V17 H14 , J13 H32 , V17 H12 , J3 , J7 , J15 H34 , V15 H14 , J13 H34 , V15 H12 , J3 , J7 , J15 H33 , V9 , V9,r , V11 , V12 H14 , J13 H33 , V9 , V9,r , V11 , V12 H16 , J1 , J5 , J9 , J9,r H32 , V17 H16 , J1 , J5 , J9 , J9,r H34 , V15 H12 , J3 , J7 , J15 H36 , V1 , V3 , V5 , V7 , V7,r H14 , J13 H36 , V1 , V3 , V5 , V7 , V7,r H16 , J1 , J5 , J9 , J9,r H36 , V1 , V3 , V5 , V7 , V7,r

480

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Proof. Let T be an arbitrary subclass of J1 ∪ V1 , which was not described in Theorems 15.4.3.1, 15.4.4.1, or 15.4.4.3. Then obviously T1 := T ∩ J1 and T2 := T ∩ V1 are subclasses of J1 or V1 with T = T1 ∪ T2 . Further, [T 1 ] is a subclass H of [M 1 ] with H 1 = T 1 . The possibilities for H follow from Theorems 15.4.3.1, 15.4.4.1, and 15.4.4.3. These possibilities imply the possibilities for T1 and T2 , which are given in Table 15.18. When one selects the sets T1 ∪ T2 ⊆ [M 1 ], which are closed, one receives the statement of the theorem.

Theorem 15.4.4.6 Class U1 ∪ V1 has exactly the following subclasses, which are not subclasses of U1 or V1 or [(U1 ∪ V1 )1 ]: U6 ∪ H3 ; U9 ∪ H7 ; U3 ∪ H7 ; U4 ∪ H7 ; U1 ∪ H7 ; H2 ∪ V17 ; H6 ∪ V9 , H6 ∪ V9,r , H6 ∪ V11 , H6 ∪ V12 ; H6 ∪ V15 ; H6 ∪ V1 , H6 ∪ V3 , H6 ∪ V5 , H6 ∪ V7 , H6 ∪ V7,r ; U12 ∪ V18 ; U6 ∪ V17 ; U8 ∪ V16 ; U3 ∪ V15 ; H22 ∪ V10 , H22 ∪ V10,r , H22 ∪ V13 , H22 ∪ V14 , U9 ∪ V13 ; H22 ∪ V9 , H22 ∪ V9,r , H22 ∪ V11 , H22 ∪ V12 , U9 ∪ V11 ; H25 ∪ V9 , H25 ∪ V9,r , H25 ∪ V11 , H25 ∪ V12 , U4 ∪ V9 ; H22 ∪ V2 , H22 ∪ V4 , H22 ∪ V6 , H22 ∪ V8 , H22 ∪ V8,r , U9 ∪ V2 , U9 ∪ V4 ; H22 ∪ V1 , H22 ∪ V3 , H22 ∪ V5 , H22 ∪ V7 , H22 ∪ V7,r , U9 ∪ V1 , U9 ∪ V3 ; H25 ∪V1 , H25 ∪V3 , H25 ∪V5 , H25 ∪V7 , H25 ∪V7,r , U4 ∪V1 , U4 ∪V3 , U4 ∪V5 , U4 ∪V7 ; U5 ∪ V2 ; U1 ∪ V1 ; where r = 1, 2, 3, ... . Proof. Let T be an arbitrary subclass of U1 ∪ V1 , which was not described in Theorems 15.4.3.1, 15.4.4.2, or 15.4.4.3. Then obviously T1 := T ∩ U1 and T2 := T ∩ V1 are subclasses of U1 or V1 with T = T1 ∪ T2 . Further, [T 1 ] is a subclass H of [M 1 ] with H 1 = T 1 . The possibilities for H follow from Theorems 15.4.3.1, 15.4.4.2, and 15.4.4.5, which are given in Table 15.19. When one selects the sets T1 ∪ T2 ⊆ [M 1 ], which are closed, one receives the statement of the theorem.

15.4 The Subclasses of [O1 ∪ {max}]

481

Table 15.19 [T 1 ] H41 H42 H43 H44 H45 H46 H47 H48 H49 H75 H76 H77 H78 H79 H80 H81 H82 H83 H84 H85 H86

Possibilities for T ∩ U1 T ∩ V1 H19 , U6 H3 H22 , U9 H7 H24 , U3 H7 H25 , U4 H7 H28 , U1 H7 H2 H32 , V17 H6 H33 , V9 , V9,r , V11 , V12 H6 H34 , V15 H6 H36 , V1 , V3 , V5 , V7 , V7,r H17 , U12 H29 , V18 H19 , U6 H32 , V17 H21 , U8 H31 , V16 H24 , U3 H34 , V15 H22 , U9 H30 , V10 , V10,r , V13 , V14 H22 , U9 H33 , V9 , V9,r , V11 , V12 H25 , U4 H33 , V9 , V9,r , V11 , V12 H22 , U9 H35 , V2 , V4 , V6 , V8 , V8,r H22 , U9 H36 , V1 , V3 , V5 , V7 , V7,r H25 , U4 H36 , V1 , V3 , V5 , V7 , V7,r H27 , U5 H35 , V2 , V4 , V6 , V8 , V8,r H28 , U1 H36 , V1 , V3 , V5 , V7 , V7,r

Theorem 15.4.4.7 Class J1 ∪ U1 ∪ V1 has the following subclasses, which are not subclasses of J1 ∪ U1 , J1 ∪ V1 , U1 ∪ V1 or [M 1 ]: J11 ∪ H19 ∪ H32 , J11,r ∪ H19 ∪ H32 , J11,r ∪ U6 ∪ V17 ; J11 ∪ H24 ∪ H34 , J11,r ∪ H24 ∪ H34 , J11 ∪ U3 ∪ V15 ; H12 ∪ H22 ∪ V9 , H12 ∪ H22 ∪ V9,r , H12 ∪ H22 ∪ V11 , H12 ∪ H22 ∪ V12 , J3 ∪ U9 ∪ V11 ; H14 ∪ H25 ∪ V9 , H14 ∪ H25 ∪ V9,r , H14 ∪ H25 ∪ V11 , H14 ∪ H25 ∪ V12 , J13 ∪ U4 ∪ V9 ; J9 ∪ H19 ∪ H32 , J9,r ∪ H19 ∪ H32 , J5 ∪ H19 ∪ H32 , J1 ∪ H19 ∪ H32 , J5 ∪ U6 ∪ V17 , J1 ∪ U6 ∪ V17 , J9 ∪ U6 ∪ V17 ; J9 ∪ H24 ∪ H34 , J9,r ∪ H24 ∪ H34 , J5 ∪ H24 ∪ H34 , J1 ∪ H24 ∪ H34 , J1 ∪ U3 ∪ V15 , J5 ∪ U3 ∪ V15 , J9 ∪ U3 ∪ V15 ; H12 ∪ H22 ∪ V1 , H12 ∪ H22 ∪ V3 , H12 ∪ H22 ∪ V5 , H12 ∪ H22 ∪ V7 , H12 ∪ H22 ∪ V7,r , J3 ∪ U9 ∪ V3 , J3 ∪ U9 ∪ V1 ; H14 ∪ H25 ∪ V1 , H14 ∪ H25 ∪ V3 , H14 ∪ H25 ∪ V5 , H14 ∪ H25 ∪ V7 , H14 ∪ H25 ∪ V7,r , J13 ∪ U4 ∪ V1 , J13 ∪ U4 ∪ V3 , J13 ∪ U4 ∪ V5 , J13 ∪ U4 ∪ V7 ; J1 ∪ U1 ∪ V1 ; where r = 1, 2, 3, .... Proof. It is easy to check that a class T of the form T1 ∪ T2 ∪ T3 with T1 ∈ L3 (J1 )\{∅}, T2 ∈ L3 (U1 )\{∅} and T3 ∈ L3 (V1 )\{∅} is closed if and only if the sets Ti ∪ Tj are closed for all i, j ∈ {1, 2, 3} and i = j. Thus, our theorem

482

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

follows from Theorems 15.4.4.4–15.4.4.6 with the aid of Theorem 15.4.3.1, in which one can ﬁnd the possibilities for T 1 . 15.4.5 The Subclasses of M ∩ P ol3 {(0, 2)} To receive a coarse partition of the lattice of the subclasses of 0 A1 := M ∩ P ol3 2 = {f ∈ M | f ∈ [{c0 , c1 , c2 }] ∧ F (f ) ∩ {v2 , v5 } = ∅ ∧ ({j2 , j5 } ∩ F (f ) = ∅ =⇒ {s1 , u2 , u5 } ∩ F (f ) = ∅)}, we determine the maximal classes of A1 ﬁrst. Lemma 15.4.5.1 A1 has exactly three maximal classes: (1) A2 := A1 ∩ P ol3 {0, 1} = {f ∈ A1 | u5 ∈ F (f )}; (2) A3 := A1 ∩ P ol3 {1, 2} = {f ∈ A1 | {j2 , u2 } ∩ F (f ) = ∅ =⇒ {s1 , j5 , u5 } ∩ F (f ) = ∅}; (3) B1 := {f ∈ A1 | F (f ) ⊆ {c0 , u2 , u5 , s1 }}. Proof. One can conclude from Theorem 15.4.2.2 that A1 A2 A3 B1

= [{max, x ∨ j2 (y), u2 , u5 }], = [{max, x ∨ j5 (y), u2 }], = [{max, x ∨ j2 (y), x ∨ u2 (y), u5 }], = [{max, u2 , u5 }].

With the aid of the above statements, it is easy to prove the A1 -maximality of A2 , A3 , and B. Denote now T an arbitrary subset of A1 , which is not a subset of X for all X ∈ {A2 , A3 , B}. Then there are some functions qi (i = 1, 2, 3) with q1 ∈ T \ B1 , q2 ∈ T \ A2 and q3 ∈ T \ A3 . By identifying the variables in the functions q2 and q3 , one obtains the functions u5 and u2 . The function q1 has at least two essential variables and it holds that F (q1 )∩{j2 , j5 } = ∅. By identifying certain variables of q1 and substituting certain variables of q1 through the functions u2 , u5 we obtain the functions j5 (x) ∨ u2 (y) and j5 (x) ∨ u5 (y). In the proof of Lemma 15.4.2.2, we showed that an arbitrary function t ∈ M with F (t) ∈ {{j5 , u2 }, {j5 , u5 }} is a superposition over the above-constructed functions. Then, by identifying variables in a function t4 ∈ M with F (t) = {j5 , u2 } and numt (j5 ) = numt (u2 ) = 2, we obtain max ∈ [{q1 , q2 , q3 }]. Since in addition x ∨ (j5 (u2 (y)) ∨ u2 (x)) = x ∨ j2 (y) holds, we have [{q1 , q2 , q3 }] = A3 , whereby our lemma is proven. Obviously, it holds I1 := A2 ∩ A3 = A1 ∩ P ol3 {1},

15.4 The Subclasses of [O1 ∪ {max}]

483

and the functions f of I1 are idempotent; i.e., f (x, x, ..., x) = x. Subsequently, we determine the elements of L3 (A1 )\(L3 (R) ∪ L3 ([M 1 ]), which belong to B1 and I1 , and then the remaining elements of L3 (A1 )\(L3 (R) ∪ L3 ([M 1 ]) (see Figure 15.4). With the aid of Theorems 15.4.3.1 and 15.4.4.2, one obtains a complete description of L3 (A1 ).

Theorem 15.4.5.2 B1 has exactly the following 20 subclasses, which are not subsets of [M 1 ] or U1 : B1 = [{max, u2 , u5 }], B2 := {f ∈ B1 | F (f ) ∩ {s1 , u5 } = ∅} = [{max, x ∨ u2 (y), u5 }], B3 := {f ∈ B1 | numf (s1 ) ≤ 1} = [{x ∨ u2 (y), x ∨ u5 (y), u2 }], B4 := {f ∈ B1 | u5 ∈ F (f )} = [{max, u2 }], B5 := {f ∈ B2 | numf (s1 ) ≥ 2 =⇒ u5 ∈ F (f )} = [{x ∨ y ∨ u5 (z), x ∨ u2 (y)}], B6 := {f ∈ B2 | u2 ∈ F (f ) =⇒ u5 ∈ F (f )} = [{max, u2 (x) ∨ u5 (y)}], B7 := B5 ∩ B6 = [{x ∨ y ∨ u5 (z), u2 (x) ∨ u5 (y), s1 }], B8 := {f ∈ B1 | u2 ∈ F (f )} = [{max, u5 }], B9 := B2 ∩ B3 = [{x ∨ u2 (y), u5 }], B10 := B2 ∩ B4 = [{max, x ∨ u2 (y)}], B11 := B7 \[{s1 }], B12 := B7 ∩ B8 = [{x ∨ y ∨ u5 (z), s1 }], B13 := B7 ∩ B3 = [{x ∨ u5 (y), u2 (x) ∨ u5 (y), s1 }], B14 := B3 ∩ B4 = [{x ∨ u2 (y), u2 }], B15 := B11 ∩ B12 = [{x ∨ y ∨ u5 (z)}], B16 := B12 ∩ B13 = [{x ∨ u5 (y), s1 }], B17 := B9 ∩ B13 = [{x ∨ u5 (y), u2 (x) ∨ u5 (y)}], B18 := B9 ∩ B14 = [{x ∨ u2 (y)}], B19 := [{max}], B20 := [{x ∨ u5 (y)}] (See Figure 15.5). Proof. Except for the functions f with u5 ∈ F (f ) and numf (s1 ) ≥ 2, for which f ∈ [[{f }]3 ] is valid, we have by Lemma 15.4.2.2 f ∈ [{f }]2 for all other functions f ∈ B1 \[M 1 ]. Consequently, one can describe an arbitrary subclass B of B1 in the form of a closure of a certain subset of {x ∨ y ∨ u5 (z), max, x ∨ u2 (y), x ∨ u5 , u2 (y) ∨ u5 (y), u2 , u5 , s1 }. When one examines the possible cases for B (⊆ U1 or ⊆ [M 1 ]) with the aid of Figure 15.5, one obtains our theorem.

r

r

r

r

r

r

A15,r−1

r

A15,r

Fig. 15.4

r r

r

r

r

r

r

A25

A24

r

A20

r

A14,1

A19

r

r

A15,1

A18

r A23

A17

r

A16 A22

r A37

A21

r

A14,r−1

A35

A34

r

A14

r A36

r

see Figure 15.5

A32 r

A31

r

r

A14,r

r A33

r

A15

B1

A30

A29

A28

A27

r

r

A13

r

A12

see Figure 15.6

A26 r

A11

A10

r

r

A9

r r

A8

I1 r

A7

A6

r

A5

A3 r

A4

A2

A1

484 15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

r

r

r

r

15.4 The Subclasses of [O1 ∪ {max}]

485

B1

q Q

Q

Q Q Q

Q Q Q B4 B2 B3 Q Qq q q P

J @PPP PP J

@ P B5 B6 P

PP J @ @ q X q XX PP JP

X X X P XXX

J PP B7 B8 PP B10 J

XXXXqB9 qP qP Pq PP J

A @PPP P P J

P @ B PPP PP A B11 B13

12 14 P JB A P @ P Jq q ` q q P

` PP ` ``` @ APP @ @ `` A PPP @``` @ @B B16 ``` B17 B PP B19 AA18 @ @ ` @ 15 q @ q q @q ` Pq @ @ @ @ B20

Fig. 15.5

Theorem 15.4.5.3 I1 has exactly 17 subclasses, which are not subsets of [M 1 ]: I1 = [{max, u2 (x) ∨ j5 (y)}], I2 := [{max, x ∨ j5 (y), x ∨ u2 (y)}], I3 := [{x ∨ j5 (y), j5 (x) ∨ u2 (y)}], I4 := [{x ∨ u2 (y), j5 (x) ∨ u2 (y)}], I5 := [{max, x ∨ j2 (y), x ∨ j5 (y)}], I6 := [{max, x ∨ j2 (y), x ∨ u2 (y)}], I7 := [{x ∨ j2 (y), j5 (y) ∨ u2 (y)}], I8 := [{x ∨ j5 (y), x ∨ j2 (y)}], I9 := [{x ∨ j2 (y), x ∨ u2 (y)}], I10 := [{max, x ∨ j5 (y)}], I11 := [{max, x ∨ j2 (y)}], B10 = [{max, x ∨ u2 (y)}], I12 := [{j5 (x) ∨ u2 (y)}], I13 := [{x ∨ j5 (y)}], I14 := [{x ∨ j2 (y)}], B18 = [{x ∨ u2 (y)}], B19 = [{max}]. Proof. Because of Theorem 15.4.2.2, every subclass of I1 has a generating system from binary functions of I1 . Consequently, one obtains the subclasses

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

486

of I1 , which are not subclasses of [M 1 ], through closure of the subsets of {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), j5 (x) ∨ u2 (y)}. (During the forming of these classes one notices that the equations x ∨ j5 (j5 (x) ∨ u2 (y)) = x ∨ j2 (y) and u2 (x) ∨ j5 (x ∨ u2 (y)) = x ∨ j2 (y) are valid.) The Figure 15.6 gives the Hasse diagram of the classes constructed in this manner. I1

q P P PP PP P

PP

PPI2 @ @ @ I6 I3 I4 I5 q q q @q P P PP @PPP @ P PP PP @ @ P PP P @ I7 PP I8 PP I9 I10 @ I11 B10 P P q @q q q @P q q PP @ @ PP @ PP @ PP @ @ @ @q q q PP q q

I12

I13

I14

B18

B19

Fig. 15.6

Theorem 15.4.5.4 The following classes are the only subclasses of A2 that are not contained in I1 , B1 or [M 1 ]: A2 = [{max, x ∨ j5 (y), u2 }], A4 := [{max, j2 (x) ∨ u2 (y)}], A5 := [{x ∨ u2 (y), j5 (x) ∨ u2 (y), u2 }], A6 := [{x ∨ j5 (y), u2 }], A7 := [{x ∨ j2 (y), x ∨ u2 (y), u2 }], A8 := [{j5 (x) ∨ u2 (y), u2 }], A9 := [{x ∨ u2 (y), j2 (x) ∨ u2 (y)}], A10 := [{x ∨ j2 (y), u2 (x) ∨ u2 (y)}], A11 := [{s1 , j2 (x) ∨ u2 (y), u2 (x) ∨ u2 (y)}], A12 := [{x ∨ j2 (y), u2 }], A13 := A11 \[{s1 }], A14 := [{u2 (x1 ) ∨ j2 (x2 ) ∨ ... ∨ j2 (xn ) | n ∈ N\{1}}],

15.4 The Subclasses of [O1 ∪ {max}]

487

A14,r := [{f ∈ A10 | numf (j2 ) ≤ r}], A15 := A14 ∪ [{s1 }], A15,r := A14,r ∪ [{s1 }], where r = 1, 2, ... . Proof. Let A be a subclass of A2 , which is not a subset of I1 , B1 or [M 1 ]. Because of A ⊆ I1 = A2 ∩ A3 , u2 belongs to A. Then, by A ⊆ [M 1 ] and A ⊆ B1 , we have j2 (x) ∨ u2 (y) ∈ A. Thus A contains the class A14,1 . One can verify the remaining statements of our theorem easily with the aid of the above-noted generating systems of the classes, Figure 15.4, and Theorem 15.4.2.2. Theorem 15.4.5.5 The following classes are the only subclasses of A3 that are not contained in I1 , B1 or [M 1 ]: A3 = [{max, j5 (x) ∨ u2 (y), u5 }], A16 := {f ∈ A1 | u5 ∈ F (f )} = [{j5 (x) ∨ u5 (y), j2 (x) ∨ u5 (y)}], A17 := {f ∈ A16 | {j2 , u2 } ∩ F (f ) = ∅} = [{u5 (x) ∨ j5 (y), x ∨ y ∨ u5 (z)}], A18 := {f ∈ A17 | numf (s1 ) ≤ 1} = [{x ∨ j5 (y) ∨ u5 (z)}], A19 := {f ∈ A18 | numf (s1 ) = 1 =⇒ j5 ∈ F (f )} = [{x∨u5 (y), j5 (x)∨u5 (y)}], A20 := {f ∈ A19 | numf (s1 ) = 0} = [{j5 (x) ∨ u5 (y)}], A21 := A16 ∪ [{s1 }], A22 := A17 ∪ [{s1 }], A23 := A18 ∪ [{s1 }], A24 := A19 ∪ [{s1 }], A25 := A20 ∪ [{s1 }], A26 := I3 ∪ A16 , A27 := I2 ∪ A16 , A28 := I5 ∪ A16 , A29 := B10 ∪ A16 , A30 := I8 ∪ A16 , A31 := I10 ∪ A17 , A32 := B19 ∪ A16 , A33 := I13 ∪ A17 , A34 := B18 ∪ A16 , A35 := B19 ∪ A17 , A36 := I13 ∪ A18 , A37 := I13 ∪ A20 . Proof. Let T be a subclass of A3 with T ⊆ I1 , T ⊆ B1 and T ⊆ [M 1 ]. It follows from T ⊆ I1 , T ⊆ B1 , T ⊆ [M 1 ] and j2 u5 = j5 that the functions u5 and j5 (x) ∨ u5 (y) belong to T . Hence we have A20 ⊆ T . Since for an arbitrary function f ∈ A3 either f (1, ..., 1) = 1 or f (1, ..., 1) = 2 holds, and the case f (1, ..., 1) = 2 and f ∈ A3 is only possible, if u5 ∈ F (f ),

488

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

the class T has the form (T ∩ I1 ) ∪ (T ∩ A16 ).

(15.5)

First we show the possibilities for T ∩ A16 . Because of j2 (u5 (y)) ∨ u5 (j2 (x) ∨ u5 (y)) = u2 (x) ∨ u5 (y) and j5 (u2 (x) ∨ u5 (y)) ∨ u5 (y) = j2 (x) ∨ u5 (y) we have j2 (x) ∨ u5 (y) ∈ T ⇐⇒ u2 (x) ∨ u5 (y) ∈ T.

(15.6)

Further, it holds that: j5 (x1 ) ∨ j5 (x2 ) ∨ u5 (x3 ) ∨ u5 (x4 ) ∈ [{j5 (x) ∨ u5 (y)}] This implies x ∨ y ∨ u5 (z) = j5 (g(z, x)) ∨ j5 (g(z, y)) ∨ u5 (g(x, z)) ∨ u5 (g(y, z)) ∈ [{j5 (x) ∨ u5 (y), g}]

(15.7)

for every g(x, y) ∈ {j2 (x) ∨ u5 (y), u2 (x) ∨ u5 (y)}. With the aid of (15.6) and (15.7), it is easy to check that the classes A16 , ..., A20 are the only possibilities for T if T ⊆ A16 holds. The remaining possibilities for T can be obtained with the help of (15.5) and Theorem 15.4.5.3, when one determines the classes I ∈ L3 (I1 )\{∅} and A ∈ {A16 , ..., A20 } with I ∪ A = [I ∪ A].

15.4.6 The Remaining Subclasses of M The following lemma is the basis for determining the subclasses of M that are still missing. Lemma 15.4.6.1 Let T be a subclass of M , which is not a subset of [M 1 ], R or A1 . Furthermore, let T1 := T ∩ R and T2 := T ∩ P ol

0 . 2

Then T fulﬁlls one of the two following conditions: (a) T ∈ {T1 ∪ [{s1 }] | T1 ∈ L3 (R)\L3 ([M 1 ])}. (b) T2 ∩ {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), x ∨ u5 (y), j2 (x) ∨ u2 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} = ∅.

15.4 The Subclasses of [O1 ∪ {max}]

489

Proof. For T the following two cases are possible: Case 1: T2 ⊆ [M 1 ]. 0 1 Because of T ⊆ R and (T ∩P ol ) ⊆ {u2 , u5 , s1 }, this case is only possible 2 for s1 ∈ T2 . Since T2 \[{s1 }] ⊆ R and classes of the form T1 ∪ [{s1 }] are closed for all T1 ∈ L3 (R), T fulﬁlls the condition (a). Case 2: T2 ⊆ [M 1 ]. By Section 15.4.5, T fulﬁlls the condition (b). In the following, denote T a subclass of M with the properties: T ∩ R = ∅ and T ∩ {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), x ∨ u5 (y), j2 (x) ∨ u2 (y), = ∅. j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} Furthermore, we use the notations: T1 := T ∩ R and T2 := T ∩ P ol

0 (= T ∩ A1 ). 2

Theorem 15.4.6.2 If T ⊆ J1 ∪ A1 then T is one of the following classes: 1) H2 ∪ B19 , J12 ∪ I11 , J14 ∪ I10 , J2 ∪ I5 (classes that contain the set {c0 , max}); 2) J2 ∪ I8 , J14 ∪ I13 , Ji ∪ I14 (i ∈ {1, 2, 5, 6, 11, 12}) (classes T with c0 ∈ T , max ∈ T and {x∨j5 (y), x∨j2 (y)}∩T = ∅); 3) H3 ∪ I14 , J7 ∪ I14 , J3 ∪ I14 (classes T with c1 ∈ T and c0 ∈ T ); 4) J16 ∪ I13 , J16 ∪ I10 , J8 ∪ I4 , J8 ∪ I7 , J8 ∪ I14 , J4 ∪ I5 , J4 ∪ I1 , J4 ∪ I8 , J4 ∪ I3 , J4 ∪ I14 (classes that do not contain constant functions). Proof. Because of x∨c1 (x) = v2 (x) the set J1 ∪A1 is not closed. To determine all closed subsets T of J1 ∪ A1 , we distinguish the following cases: Case 1: c0 ∈ T1 . Then for every function f ∈ T , we have F (f ) ⊆ T 1 . Thus T2 is a subset of I1 , and we have {max, x ∨ j5 (y), x ∨ j2 (y)} ∩ T = ∅ and {j5 (x) ∨ u2 (y), x ∨ u2 (y)} ∩ T = ∅. If {max, x ∨ j5 (y)} ∩ T = ∅, T does not contain c1 , since x ∨ c1 (x) = v2 (x) ∈ T . Case 1.1: max ∈ T2 . In this case, the class T is also describable in the form T = [T 1 ∪[max}]. With

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15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

the aid of Theorems 15.4.3.1, 15.4.4.1, 15.4.5.3 and Table 15.14, one obtains the classes given in 1). Case 1.2: max ∈ T2 . Case 1.2.1: x ∨ j5 (y) ∈ T2 . If one examines the possibilities that arise from Theorems 15.4.4.1 and 15.4.5.3 for T , then one sees that only the sets J14 ∪ I13 and J2 ∪ I8 are closed. Case 1.2.2: x ∨ j5 (y) ∈ T2 and x ∨ j2 (y) ∈ T2 . In this case, we have T2 = I14 and T1 contains J12 . Then, the possibilities for T are: Ji ∪ I14 , where i ∈ {1, 2, 5, 6, 11, 12}). Case 2: c0 ∈ T1 and c1 ∈ T1 . Because of x ∨ c1 (x) = v2 (x), c1 ∨ ui = vi (i = 2, 5) and j2 (c1 ) ∨ u2 = u2 , only classes T are possible with {max, x ∨ j5 (y), x ∨ u2 (y), x ∨ u5 (y), j2 (x) ∨ u2 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} ∩ T = ∅. Hence, by Theorem 15.4.4.1 and Section 15.4.5, T1 ∈ {H3 , J7 , J3 } and T2 = I14 , where every possibility supplies a closed class, which is given in 3). Case 3: {c0 , c1 } ∩ T = ∅. In this case, by Theorem 15.4.4.1, T1 ∈ {H9 , J16 , J8 , J4 }. Further, we have {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), j5 (x) ∨ u2 (y)} ∩ T = ∅ and {x ∨ u5 (y), j2 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} ∩ T = ∅. Thus T1 = H9 is not possible. We must only continue to examine, therefore, the following three cases: Case 3.1: T1 = J16 . Then, T can not contain the functions x ∨ j2 (y), x ∨ u2 (y) and j5 (x) ∨ u2 (y); thus T2 ∈ {I13 , B19 , I10 } and T ∈ {J16 ∪ I13 , J16 ∪ I10 }. Case 3.2: T1 = J8 . In this case, we have T ∩ {max, x ∨ j5 (y)} = ∅ and T2 ⊆ I4 . By scrutinizing the possibilities that result, one receives T ∈ {J8 ∪ I4 , J8 ∪ I7 , J8 ∪ I14 }. Case 3.3: T1 = J4 . The following cases are still possible, then: Case 3.3.1: max ∈ T2 . Then T contains the class J4 ∪ I5 . Because of j5 (x) ∨ u2 (y) ∈ [{j5 , x ∨ u2 (y)}] the set J4 ∪ I2 is not closed, and only the classes T with T = J4 ∪ I5 or T = J4 ∪ I1 fulﬁll the conditions of this case. Case 3.3.2: max ∈ T2 . Case 3.3.2.1: x ∨ j5 (y) ∈ T . Since x ∨ j5 (j5 (x) ∨ j2 (y)) = x ∨ j2 (y), T contains the class I8 and it holds that T ∈ {J4 ∪ I8 , J4 ∪ I3 }. Case 3.3.2.2: {j5 (x) ∨ u2 (y), x ∨ u2 (y)} ∩ T = ∅ and x ∨ j5 (y) ∈ T . Because of (j5 (x) ∨ j5 (y)) ∨ u2 (x) = x ∨ j5 (y), this case is not possible. Case 3.3.2.3: x ∨ j2 (y) ∈ T and {x ∨ j5 (y), x ∨ u2 (y), j5 (x) ∨ u2 (y)} ∩ T = ∅. Only the class J4 ∪ I14 can be T in this case.

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491

Theorem 15.4.6.3 If T ⊆ U1 ∪ A1 , then T is exactly one of the following classes: 1) H2 ∪ B19 , H6 ∪ B19 , U6 ∪ B4 , U7 ∪ B8 , U3 ∪ B4 , U4 ∪ B8 , U2 ∪ B1 , U1 ∪ B1 (classes that contain {c0 , max}; 2) U1 ∪ B3 , U2 ∪ B3 , U3 ∪ B14 , U6 ∪ B14 , U4 ∪ B16 , U7 ∪ B16 (classes T with c0 ∈ T and max ∈ T ); 3) H4 ∪ B19 , H4 ∪ B10 , U5 ∪ B1 , U8 ∪ B4 , U9 ∪ B2 , U9 ∪ B6 , U10 ∪ B8 (classes T with {c2 , max} ⊆ T and c0 ∈ T ); 4) H4 ∪ B18 , U5 ∪ B3 , U8 ∪ B14 , U9 ∪ B5 , U9 ∪ B9 (classes T with {c2 , x ∨ u2 (y)}, c0 ∈ T and max ∈ T ); 5) U9 ∪ Bi (i ∈ {7, 11, 13, 17}), U10 ∪ Bt (t ∈ {12, 15, 16, 20}) (classes T with {c2 , x ∨ u5 (y)} ⊆ T and {c0 , max, x ∨ u2 (y)} ∩ T = ∅). Proof. Since U1 \[{c0 , c1 }] ⊆ A1 , we have T ∩ {c0 , c2 } = ∅. Case 1: c0 ∈ T1 . In this case, it holds that F (f ) ⊆ T 1 for all f ∈ T . Because of Section 15.4.5, then, we have either B18 or B19 or B20 as a subset of T . Case 1.1: max ∈ T2 (B19 ⊆ T ). Then [T 1 ∪ {max}] = T , and because of Theorems 15.4.3.1, 15.4.4.2, and 15.4.5.2, for T there are only the possibilities given in 1). Case 1.2: max ∈ T2 . If x ∨ u2 (y) ∈ T2 (i.e., B18 ⊆ T2 ) then u2 belongs to T1 , and by Theorem 15.4.5.2, we have T2 ∈ {B3 , B14 }. Thus T ∈ {U6 ∪ B14 , U3 ∪ B14 , U2 ∪ B3 , U1 ∪ B3 }. In the case x∨u5 (y) ∈ T2 (i.e., B20 ⊆ T2 ) and x∨u2 (y) ∈ T , only T1 ∈ {U4 , U7 } and T2 = B16 are possible (because of c0 ∈ T ). Therefore, in Case 1.2, only the classes given in 2) are possible. Case 2: c0 ∈ T1 and c2 ∈ T1 . Since jr (c2 ) ∨ us = vs and x ∨ jr (c2 ) = v2 (r, s ∈ {2, 5}), we have T ∩ {jr (x) ∨ us (y), x ∨ jr (y) | r, s ∈ {2, 5}} = ∅ and, by Section 15.4.5, T contains either B18 , B19 or B20 . Case 2.1: max ∈ T2 . Then T1 belongs to {H4 , U5 , U8 , U9 , U10 } and T2 belongs to {B1 , B2 , B4 , B6 , B8 , B10 , B19 }. In 3) those classes of the form T1 ∪ T2 are given, which are closed. Case 2.2: max ∈ T and x ∨ u2 (y) ∈ T . In this case, by Theorem 15.4.5.2, we have T2 ∈ {B3 , B5 , B9 , B14 , B18 }. Then T1 belongs to the set {H4 , U5 , U8 , U9 }. When one considers the possible cases, one receives the classes given in 4). Case 2.3: {max, x ∨ u2 (y)} ∩ T = ∅ and x ∨ u5 (y) ∈ T .

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15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

Then it holds that T2 ∈ {U9 , U10 } and T2 ∈ {B7 , B11 , B12 , B13 , B15 , B16 , B17 , B20 }. The closed sets of the form T1 ∪ T2 are given in 5). Theorem 15.4.6.4 If T ⊆ V1 ∪ A1 , then T is exactly one of the following classes: 1) V1 ∪ I5 , V1 ∪ I12 , V1 ∪ B19 , V15 ∪ Is (s ∈ {2, 5, 6, 10, 11}), V15 ∪ B10 , V15 ∪ B19 , V17 ∪ It (t ∈ {2, 5, 6, 10, 11}), V17 ∪ B10 , V17 ∪ B19 (classes that belong {c1 , max}); 2) V15 ∪ B18 , V15 ∪ I9 , V17 ∪ B18 , V17 ∪ I9 (classes T with {c1 , x ∨ u2 (y)} ⊂ T and {max, x ∨ j5 (y)} ∩ T = ∅); 3) V ∪ I8 (V ∈ {H32 , H34 , V1 , V3 , V15 , V17 }), V ∪ I13 (V ∈ {H32 , H34 , V1 , V3 , V5 , V7 , V15 , V17 }) (classes T with {c1 , x ∨ j5 (y)} ⊂ T and {max, x ∨ u2 (y)} ∩ T = ∅); 4) V ∪ I14 (V ∈ {H3 , H32 , H34 , V15 , V17 }) (classes T with x ∨ j2 (y) ∈ T and {max, x ∨ u2 (y), x ∨ j5 (y)} ∩ T = ∅); 5) V2 ∪ B19 , V10 ∪ B19 , V13 ∪ B19 , V2 ∪ I13 , V4 ∪ I13 , V8 ∪ I13 , V2 ∪ I8 , V4 ∪ I8 , V2 ∪ I10 , V2 ∪ I5 (classes T with {c2 , v5 } ⊂ T and c1 ∈ T ); 6) H4 ∪ B10 , H4 ∪ B19 , H31 ∪ Ir (r ∈ {3, 7, 8, 12, 13, 14}), V16 ∪ Bs (s ∈ {10, 18, 19}), V16 ∪ It (t ∈ {1, 2, 4, 5, 6, 9, 10, 11}) (classes T with c2 ∈ T and T ∩ {c1 , v5 } = ∅); 7) H29 ∪ Ir (r ∈ {3, 7, 8, 12, 13, 14}), V18 ∪ Bs (s ∈ {10, 18, 19}), V18 ∪ It (t ∈ {1, 2, 4, 5, 6, 9, 10, 11}) (classes T with T ∩ {c1 , c2 } = ∅). Proof. Because of U1 ∩ A1 = ∅, the set A1 ∪ V1 is not closed and, therefore, T ⊂ V1 ∪ A1 . Since u5 ∈ A for every A ∈ {B20 , A20 }, u2 ∈ A14,1 and u2 ∈ B14 , T2 is a certain subclass of I1 . Case 1: c1 ∈ T1 . Then, because of j5 ∨ u2 (c1 ) = j5 , we have j5 (x) ∨ u2 (y) ∈ T2 . Case 1.1: max ∈ T2 . Since x∨c1 = v2 (x), it holds that V17 ⊆ T1 . With the aid of Theorems 15.4.4.3 and 15.4.5.3, one obtains only the possibilities given in 1) for T .

15.4 The Subclasses of [O1 ∪ {max}]

493

Case 1.2: max ∈ T2 . By Theorem 15.4.5.3 and because of j5 (x) ∨ u2 (y) ∈ T , only the following three cases are possible: Case 1.2.1: x ∨ u2 (y) ∈ T2 . Because of c1 ∨u2 = v2 it holds V17 ⊆ T1 . Further, because x∨u2 (v5 ) = u5 (x), v5 ∈ T . Hence, T ∈ {V15 ∪ B18 , V15 ∪ I9 , V17 ∪ B18 , V17 ∪ I9 }. Case 1.2.2: x ∨ u2 (y) ∈ T2 and x ∨ j5 (y) ∈ T2 . Because of Theorem 15.4.5.3, this case is only possible for T2 ∈ {I8 , I13 }. With the aid of Theorem 15.4.4.3, this implies that T is a class given in 3). Case 1.2.3: T2 = I14 (= [{x ∨ j2 (y)}]). Because of x ∨ j2 (v5 (y)) = x ∨ j5 (y), the function v5 does not belong to T1 . With the aid of Theorem 15.4.4.3 this implies that T is a class which is given in 4). Case 2: c1 ∈ T1 and c2 ∈ T1 . We distinguish two cases: Case 2.1: v5 ∈ T1 . Because of j5 (x) ∨ u2 (v5 (x)) = u5 (x) ∈ T and x ∨ u2 (v5 (x)) = u5 (x) ∈ T , T2 cannot contain j5 (x) ∨ u2 (y) or x ∨ u2 (y). Further, we have: x ∨ j2 (y) ∈ T =⇒ x ∨ j2 (v5 (y)) = x ∨ j5 (y) ∈ T. Thus T2 ∈ {B19 , I5 , I8 , I10 , I13 } and V10 ⊆ T1 . Because of c1 ∈ T , this implies T1 ∈ {V2 , V4 , V6 , V8 , V10 , V13 , V14 }. The possibilities resulting for T are given in 5). Case 2.2: v5 ∈ T1 . Then, because of Theorem 15.4.4.3, T1 ∈ {H4 , H31 , V16 }. The possibilities resulting for T are given in 6). Case 3: {c1 , c2 } ∩ T = ∅. Because of v5 v5 = c2 , v5 ∈ T . Thus, either T1 = H29 = [{v2 }] or T1 = V18 = [{v2 (x) ∨ v2 (y)}]. With the aid of Theorem 15.4.5.3, this implies that T is a class which is given in 7).

Theorem 15.4.6.5 If T ⊆ (J1 ∪ U1 ) ∪ A1 , T1 ⊆ J1 ∪ A1 and T1 ⊆ U1 ∪ A1 , then T is exactly one of the following classes: 1) (J4 ∪ U13 ) ∪ Ai (i ∈ {3, 16, 21, 26, 30}), (J16 ∪ U14 ) ∪ Am (m ∈ {17, 18, 19, 20, 22, 23, 24, 25, 33, 35, 36, 37}), (J12 ∪ U6 ) ∪ Ap (p ∈ {4, 7, 9, 10, 11, 13}), (J12 ∪ H19 ) ∪ An (n ∈ {12, 14, 15}), (J12 ∪ H19 ) ∪ As,r (s ∈ {14, 15}), r ∈ N), (J12,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J14 ∪ U7 ) ∪ Aq (q ∈ {17, 18, 19, 20, 22, 23, 24, 25, 33, 35, 36, 37}),

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15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

(J10,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J10 ∪ H19 ) ∪ At (t ∈ {12, 14, 15}), (J10 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J16 ∪ H19 ) ∪ Ai (i ∈ {8, 12, 14, 15}), (J16 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J10 ∪ U6 ) ∪ Am (m ∈ {11, 13}), (J6 ∪ U6 ) ∪ An (n ∈ {5, 10, 11, 13}), (J2 ∪ H19 ) ∪ Ap (p ∈ {6, 12, 14, 15}), (J2 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J2 ∪ U6 ) ∪ Aq (q ∈ {2, 10, 11, 13}), (J2 ∪ U2 ) ∪ A1 (classes T with {c0 , c1 } ∩ T = ∅); 2) (J11 ∪ H19 ) ∪ Ai (i ∈ {12, 14, 15}), (J11 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J11 ∪ U6 ) ∪ Am (m ∈ {10, 11, 13}), (J11,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J9 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J9,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J5 ∪ H19 ) ∪ An (n ∈ {12, 14, 15}), (J5 ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r ∈ N), (J1 ∪ H19 ) ∪ Ap (p ∈ {12, 14, 15}), (J1 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J9 ∪ U6 ) ∪ Aq (q ∈ {11, 13}), (Ji ∪ U6 ) ∪ At (i ∈ {1, 5}, t ∈ {10, 11, 13}) (classes T with c1 ∈ T and c2 ∈ T ). Proof. Case 1: c2 ∈ T1 . Case 1.1: c1 ∈ T1 . In this case, T is a subset of (J2 ∪ U2 ) ∪ A1 . Table 15.20 indicates the possibilities for T1 that result from Theorem 15.4.4.4: Table 15.20 T11 1 H50 1 H51 1 H52 1 H57 1 H60

T1 J4 ∪ U13 , J16 ∪ U14 J12 ∪ H19 , J12 ∪ U6 , J12,r ∪ H19 J14 ∪ U7 J10 ∪ H19 , J10,r ∪ H19 , J10 ∪ U6 , J6 ∪ H19 , J6 ∪ U6 , J2 ∪ H19 , J2 ∪ U6 J2 ∪ U2

Then, by Section 14.4.5, we have T ∩ {j2 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} = ∅.

15.4 The Subclasses of [O1 ∪ {max}]

495

Consequently, with the aid of Theorems 15.4.5.4 and 15.4.5.5, one obtains the classes of 1) as possibilities for T , where these classes are sorted after the cases that result from Table 15.20. Case 1.2: c1 ∈ T1 . Because of T ∩ R ⊆ V1 , this case is only possible for T ∩ {max, x ∨ u2 (y), x ∨ u5 (y), x ∨ j5 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} = ∅. Consequently, we have A14,1 ⊆ T2 ⊆ A10 or T2 = A16 . Because of Theorem 15.4.4.4, T1 can be only a class from Table 15.21. Table 15.21 T21 1 H14 1 H53 1 H58

T2 H5 ∪ U6 J11 ∪ H19 , J11 ∪ U6 , J11,r ∪ H19 J9 ∪ H19 , J9,r ∪ H19 , J5 ∪ H19 , J1 ∪ H19 , J9 ∪ U6 , J5 ∪ U6 , J1 ∪ U6

When one scrutinizes the cases resulting from that, one receives the classes of 2). Case 2: c2 ∈ T1 . Then, by T ∩ R ⊆ V1 , we have {j2 (x) ∨ u2 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y), x ∨ j5 (y)} ∩ T = ∅ and T2 ⊆ B1 . Thus T contains x ∨ u2 (y) or x ∨ u5 (y). If T ∩ J1 = [{c1 }], then we have T ∩ {j2 , j5 } = ∅. However, by T2 ⊆ B1 , this cannot be possible. Equally, the case T ∩ J1 = [{c1 }] is not possible, since v2 ∈ [{c1 , x ∨ u2 (y)}] and v5 ∈ [{c1 , x ∨ u5 (y)}]. Thus the second case cannot occur.

Theorem 15.4.6.6 If T ⊆ (J1 ∪ V1 ) ∪ A1 , T1 ⊆ J1 ∪ A1 and T1 ⊆ V1 ∪ A1 , then T is exactly one of the following classes: 1) (H14 ∪ Vi ) ∪ B19 (i ∈ {1, 15, 17}), (J11 ∪ Vm ) ∪ I11 (m ∈ {15, 17}), (J13 ∪ Vn ) ∪ I10 (n ∈ {1, 15, 17}), (J1 ∪ Vp ) ∪ I5 (p ∈ {1, 15, 17}) (classes that contain {c0 , max}); 2) (J13 ∪ V ) ∪ I13 (V ∈ {H32 , H34 , V1 , V3 , V5 , V7 , V17 }), (J11 ∪ V ) ∪ I14 (V ∈ {H32 , V17 , H34 , V15 }), (Ji ∪ V ) ∪ I14 (i ∈ {1, 5}, V ∈ {H32 , V17 , H34 , V15 }), (J1 ∪ V ) ∪ I8 (V ∈ {H32 , V17 , H34 , V15 , V1 }) (classes that contain c0 but not max); 3) (J15 ∪ V15 ) ∪ I10 , (J3 ∪ V15 ) ∪ I5 , J3 ∪ V15 ) ∪ I1 , (J7 ∪ V15 ) ∪ I4 ,

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15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

(J15 ∪ H34 ) ∪ I15 , (J15 ∪ V15 ) ∪ I15 , (J3 ∪ H34 ) ∪ I10 , (J3 ∪ V15 ) ∪ I10 , (J7 ∪ H34 ) ∪ I7 , (J7 ∪ V15 ) ∪ I7 , (J3 ∪ H34 ) ∪ I14 , (J3 ∪ V15 ) ∪ I14 , (J7 ∪ H34 ) ∪ I14 , (J7 ∪ V15 ) ∪ I14 , (J15 ∪ V1 ) ∪ I10 , (J3 ∪ V1 ) ∪ I5 , (J3 ∪ V1 ) ∪ I1 , (J3 ∪ V1 ) ∪ I8 , (J15 ∪ V7 ) ∪ I13 , (J15 ∪ V5 ) ∪ I13 , (J15 ∪ V3 ) ∪ I13 , (J15 ∪ V1 ) ∪ I13 (classes that contain c2 but not c0 ); 4) (J15 ∪ V17 ) ∪ I10 , (J15 ∪ V17 ) ∪ I5 , (J15 ∪ V17 ) ∪ I1 , (J7 ∪ V17 ) ∪ I4 , (J7 ∪ V17 ) ∪ I1 , (J15 ∪ H32 ) ∪ I13 , (J15 ∪ V17 ) ∪ I13 , (J3 ∪ H32 ) ∪ I8 , (J3 ∪ V17 ) ∪ I3 , (J7 ∪ H32 ) ∪ I12 , (J7 ∪ H32 ) ∪ I7 , (J7 ∪ H32 ) ∪ I14 , (J7 ∪ V17 ) ∪ I14 , (J3 ∪ H32 ) ∪ I14 , (J3 ∪ V17 ) ∪ I14 (classes T with {c0 , c1 , c2 } ∩ T = {c1 }). Proof. Because of T ∩ U1 = ∅, we have {j2 (x) ∨ u2 (y), j5 (x) ∨ u5 (y), x ∨ u5 (y)} ∩ T = ∅, Thus T2 ⊆ I1 . Then the following cases are possible: Case 1: c0 ∈ T1 . Then {j5 (x) ∨ u2 (y), x ∨ u2 (y)} ∩ T = ∅. Case 1.1: max ∈ T2 . Because of T = [T 1 ∪ {max}], T is one of the classes given in 1) (see also Theorems 15.4.3.1 and 15.4.5.3). Case 1.2: max ∈ T2 . In this case, by Theorem 15.4.5.3, T2 belongs to {I13 , I8 , I14 } and it holds that 1 1 1 , H67 , H73 }, T2 = I13 =⇒ T11 ∈ {H65 1 1 1 1 1 T2 = I14 =⇒ T1 ∈ {H62 , H63 , H70 , H71 }, 1 1 1 1 T2 = I8 =⇒ T1 ∈ {H70 , H71 , H74 }.

From this and from Theorem 15.4.4.5, we get the possibilities given in 2) for T. Case 2: c0 ∈ T1 . Then, because of T ∩ J1 = ∅ and T ∩ V1 = ∅, the function c1 belongs to T . Case 2.1: c2 ∈ T1 . 1 1 , H72 }. Because of Theorems 15.4.4.5 and 15.4.5.3, the possiThen T11 ∈ {H66 bilities 3) for T follow. Case 2.2: c2 ∈ T1 . 1 . Then, with the help of Theorems 15.4.4.5 and In this case, we have T11 = H64 15.4.5.3, one obtains the possibilities given in 4) for T .

15.4 The Subclasses of [O1 ∪ {max}]

497

Theorem 15.4.6.7 If T ⊆ (U1 ∪ V1 ) ∪ A1 , T1 ⊆ U1 ∪ A1 and T1 ⊆ V1 ∪ A1 , then T is exactly one of the following classes: 1) (H2 ∪ V17 ) ∪ B19 , (H6 ∪ V15 ) ∪ B19 , (H6 ∪ V1 ) ∪ B19 , (U6 ∪ V17 ) ∪ B4 , (U3 ∪ V15 ) ∪ B4 , (U4 ∪ V1 ) ∪ B8 , (U1 ∪ V1 ) ∪ B1 (classes that contain max); 2) (U6 ∪ V17 ) ∪ B14 , (U3 ∪ V15 ) ∪ B14 , (U1 ∪ V1 ) ∪ B3 (classes that contain x ∨ u2 (y) but not max); 3) A ∪ Bi (A ∈ {U4 ∪ V9 , U4 ∪ V1 , U4 ∪ V3 , U4 ∪ V5 , U4 ∪ V7 }, i ∈ {16, 20}) (classes T with T ∩ {max, x ∨ u2 (y)} = ∅ and x ∨ j5 (y) ∈ T ). Proof. Because of T ⊆ V1 ∪ A1 , we have c0 ∈ T1 . T ∩ V1 = [{c2 }] implies c1 ∈ T1 . Hence, T2 ⊆ B1 . By Theorem 15.4.5.2, only the following cases are possible: Case 1: max ∈ T2 . Then T = [T 1 ∪ {max}] and one obtains the possibilities given in 1) of T with the aid of Theorems 15.4.3.1, 15.4.4.6, and 15.4.5.2. Case 2: max ∈ T2 . Then {x ∨ u2 (y), x ∨ u5 (y)} ∩ T = ∅. Case 2.1: x ∨ u2 (y) ∈ T2 . Every class T that satisﬁes this condition is given in 2). Case 2.2: x ∨ u5 (y) ∈ T and x ∨ u2 (y) ∈ T . 1 1 1 , H84 , H86 } and the possibilities for T are given Then T11 is an element of {H81 in 3). Theorem 15.4.6.8 If T ⊆ R ∪ A1 , T1 ⊆ J1 ∪ U1 ∪ A1 , T1 ⊆ J1 ∪ V1 ∪ A1 and T1 ⊆ U1 ∪ V1 ∪ A1 , then T is exactly one of the following classes: 1) (J11 ∪ U6 ∪ V17 ) ∪ A4 , (J11 ∪ U3 ∪ V5 ) ∪ A4 , (J1 ∪ U6 ∪ V17 ) ∪ A2 , (J1 ∪ U3 ∪ V15 ) ∪ A2 , (J13 ∪ U4 ∪ V1 ) ∪ A31 , (J1 ∪ U1 ∪ V1 ) ∪ A1 (classes that contain {c0 , max}); 2) (J11 ∪ U6 ∪ V17 ) ∪ A7 , (J11 ∪ H19 ∪ H32 ) ∪ A12 , (J11 ∪ U6 ∪ V17 ) ∪ A10 , (J11 ∪ H19 ∪ H32 ) ∪ A (A ∈ {A14 , A15 , A14,r , A15,r | r ∈ N}), (J11 ∪ U6 ∪ V17 ) ∪ Ai (i ∈ {11, 13}), (J11,r ∪ H19 ∪ H32 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r) 1 ); (classes T with max ∈ T and T11 = H87 3) (J11 ∪ U3 ∪ V15 ) ∪ A7 , (J11 ∪ H24 ∪ H34 ) ∪ A12 , (J11 ∪ U3 ∪ V17 ) ∪ A10 , (J11 ∪ H24 ∪ H34 ) ∪ A (A ∈ {A14 , A15 , A14,r , A15,r | r ∈ N}), (J11 ∪ U3 ∪ V15 ) ∪ Ai (i ∈ {11, 13}),

498

4) 5)

6)

7)

8)

15 Finite and Countably Inﬁnite Sublattices of Depth 1 or 2 of L3

(J11,r ∪ H24 ∪ H34 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r) 1 ); (classes T with max ∈ T and T11 = H88 (J13 ∪ U4 ∪ V9 ) ∪ Ai (i ∈ {19, 20, 24, 25, 36, 37}) 1 ); (classes T with max ∈ T and T11 = H90 (J5 ∪ U6 ∪ V17 ) ∪ A5 , (J1 ∪ H19 ∪ H32 ) ∪ A6 , (J5 ∪ H19 ∪ H32 ) ∪ A12 , (J1 ∪ H19 ∪ H32 ) ∪ A12 , (J5 ∪ U6 ∪ V17 ) ∪ A10 , (J1 ∪ U6 ∪ V17 ) ∪ A10 , (J5 ∪ H19 ∪ H32 ) ∪ A8 , (Ji ∪ U6 ∪ V17 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {11, 13}), (Ji ∪ H19 ∪ H32 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {14, 15}), (J9,r ∪ H19 ∪ H32 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (Ji ∪ H19 ∪ H32 ) ∪ As,r (i ∈ {1, 5, 9}, s ∈ {14, 15}, r ∈ N) 1 ); (classes T with max ∈ T and T11 = H91 (J5 ∪ U3 ∪ V15 ) ∪ A5 , (J1 ∪ H24 ∪ H34 ) ∪ A6 , (Ji ∪ H24 ∪ H34 ) ∪ A12 (i ∈ {1, 5}), (Ji ∪ U3 ∪ V15 ) ∪ A10 (i ∈ {1, 5}), (J5 ∪ H24 ∪ H34 ) ∪ A8 , (Ji ∪ U3 ∪ V15 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {11, 13}), (Ji ∪ H24 ∪ H34 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {14, 15}), (J9,r ∪ H24 ∪ H34 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (Ji ∪ H24 ∪ H34 ) ∪ As,r (i ∈ {1, 5, 9}, s ∈ {14, 15}, r ∈ N) 1 ); (classes T with max ∈ T and T11 = H94 (J13 ∪ U4 ∪ Vi ) ∪ At (i ∈ {1, 3, 5}, t ∈ {18, 19, 23, 24, 36}), (J13 ∪ U4 ∪ Vi ) ∪ At (i ∈ {1, 3, 5, 7}, t ∈ {20, 25, 37}) 1 ); (classes T with max ∈ T and T11 = H94 M 1 ). (class T with max ∈ T and T11 = H91

Proof. Because of T ⊆ (J1 ∪ V1 ) ∪ A1 , we have c0 ∈ T1 . Case 1: max ∈ T2 . Then T = [T 1 ∪ {max}] and one obtains the possibilities for T given in 1) with the aid of Theorem 15.4.3.1. Case 2: max ∈ T2 . Since c0 ∈ T , we have T11 = Hi1 with i ∈ {87, 88, 90, 91, 92, 94, 95} by Theorems 15.4.3.1 and 15.4.4.7. With the aid of Theorems 15.4..4.7, 15.4.5.2– 15.4.5.5, one obtains the possibilities given in 2)–8) for T .

16 The Maximal Classes of Q ⊆ Ek

a∈Q

P olk {a} for

In this section we describe all maximal classes of the subclass TQ := P olk {a} a∈Q

of Pk for arbitrary Q with ∅ = Q ⊆ Ek , k ≥ 2. With the aid of these classes, a completeness criterion for TQ can easily be formulated. This criterion implies necessary and suﬃcient conditions regarding whether a ﬁnite algebra is semiprimal and has only trivial subalgebras. Moreover, if |Q| ≥ 2, we prove that every maximal class of TQ is an intersection of TQ with a certain maximal classes of Pk or P olk {a} (a ∈ Q).Presumably, something similar is valid for the maximal classes of TQ := ∈Q P olk , where Q ⊆ P(Ek ) and |Q | ≥ 2. For k = 3 this presumption was proven in [Lau 95b].

16.1 Notations We say that a relation is derivable from the relation with the aid of Invk TQ (or brieﬂy is -derivable), if ∈ [{} ∪ Invk TQ ] (see Section 2.4). In this case we also write {} ∪ TQ or brieﬂy

.

The following lemma provides the basis of later proofs and summarizes some well-known statements that can easily be checked. Lemma 16.1.1 (a) ∀, ∈ Rk : (P ol ⊆ TQ ∧ ({} ∪ Invk TQ ) =⇒ P ol ⊆ TQ ∩ P ol ); (b) Invk Pk = Dk (see Chapter 2);

500

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

(c) For every relation ∈ Invk TQ there are some relations 1 , ..., r ∈ {{a} | a ∈ Q} ∪ Dk and a relation ∈ Dk with = (1 × 2 × ... × r ) ∩ . Next we deﬁne some relation sets that we need to describe of the maximal classes of TQ (Q ⊆ Ek ). In this case, we also use the notations from Chapter 5. ⎧ { ∈ Mk | a is greatest or smallest element of Ek in respect to ⎪ ⎪ ⎪ ⎪ }, ⎪ ⎪ ⎪ ⎪ ⎨ if Q = {a}, Mk;Q := { ∈ Mk | a is greatest (smallest) and b is smallest (greatest) ⎪ ⎪ ⎪ element of Ek in respect to }, if Q = {a} and ⎪ ⎪ ⎪ ⎪ a = b, ⎪ ⎩ ∅ otherwise; ⎧ ⎪ ⎨ { ∈ Uk | ∀x ∈ Ek ∀q ∈ Q : (x, q) ∈ =⇒ x = q}, Uk;Q := if |Q| ≤ k − 2, ⎪ ⎩ ∅ otherwise; ⎧ ⎪ ⎨ { ∈ Sk | ∀x, y ∈ Ek : (x, y) ∈ ∧ x ∈ Q =⇒ y ∈ Q}, if |Q| = s · p, k = t · p, p prime, Sk;Q := ⎪ ⎩ ∅ otherwise; Pk,Q := { {(x, s(x)) | x ∈ Ek } | s is a permutation on Ek with exactly one ﬁxed point (∈ Q); all proper cycles of s have the same prime number length, and s preserves Q }; ⎧ 3 { {(a, b, c) ∈ Ek | a +G b = c} | (Ek ; +G ) is an elementar Abelean ⎪ ⎪ ⎪ 2-group with the neutral element ⎨ Lk;Q := q }, ⎪ ⎪ if k = 2m , m ≥ 1 and Q = {q}, ⎪ ⎩ ∅ otherwise; Ck;Q := (C1k \{{q} | q ∈ Q}) ∪

k−1

h=2 {

∈ Chk | ∀q ∈ Q : q is a central element of }

(in particular, we have Ck;Ek = C1k \{{q} | q ∈ Ek }) and ⎧ 2 2 2 ⎪ ⎨ { (\ιk ) ∪ {(q, q)} | ∈ Ck;Q ∪ {Ek }}, if Q = {q}, Nk;Q := { ⊆ E 2 |∃q ∈ Q : {(x, q), (q, x) | x ∈ Ek } ⊆ ∧ τ ρ = ρ ∧ k ⎪ ⎩ ∩ ι2k = {(q, q)} ∧ ∩ ((Q\{q}) × (Ek \{q})) = ∅} , if 2 ≤ |Q|. Obviously, Nk;Q ⊆ a∈Q Nk;{a} . An element c ∈ Q with {(x, c), (c, x) | x ∈ Ek \{c}} ⊆ (∈ Nk;Q ) is called central element of . If one considers an arbitrary other ﬁnite set A instead of Ek , then the relation

16.2 Results of Chapter 16

501

sets MA , MA;Q , UA , UA;Q , SA , SA;Q , PA;Q , LA , LA;Q , CA , CA;Q , NA;Q and BA for Q ⊆ A can be deﬁned when one replaces the set Ek by A in the above deﬁnitions or in the deﬁnitions of Chapter 5.

16.2 Results of Chapter 16 Our aim is to prove the following Theorem 16.2.1 ( [Sze 91], [Lau 82b], [[Lau 95a]) Let Rmax (TQ ) := Mk;Q ∪ Uk;Q ∪ Sk;Q ∪ Lk;Q ∪ Ck;Q ∪ Nk;Q ∪ Pk;Q . Then {TQ ∩ P olk | ∈ Rmax (TQ )} is the set of all maximal classes of TQ for ∅ = Q ⊆ Ek . The following theorem is a direct conclusion from the above theorem and from the fact that TQ is ﬁnitely generating (see Lemma 16.3.1): Theorem 16.2.2 (Completeness Criterion for TQ ) For an arbitrary subset M of TQ is valid: [M ] = TQ ⇐⇒ ∀ ∈ Rmax (TQ ) : M ⊆ TQ ∩ P olk .

The next theorem is a special case of Theorem 16.2.2: Theorem 16.2.3 (Completeness Criterion for the Class of all Idempotent Functions of Pk ) For an arbitrary subset M of TEk ( = n≥1 {f n ∈ Pk | f (x, ..., x) = x}) with k ≥ 3 is valid: [M ] = TEk ⇐⇒ ∀ ∈ C1k;Ek ∪ Sk;Ek ∪ Pk;Ek ∪ Nk;Ek : M ⊆ TEk ∩ P olk . Theorem 16.2.1 can also be formulated in the language of the Universal Algebra: 1 F A ﬁnite algebra (A; F ) (F ⊆ PA ) is called semi-primal, if [F ] = P olA InvA holds (see [Fos-P 64], [Den 82], [P¨ os-K 79], p. 143) or [Den-W 2002]. Let 1 F is the set of all uniSub(A) be the set of all subalgebras of A. Then InvA verses of algebras of Sub(A). If Sub(A)\{A} contains only 1-element algebras and Q := {a | ({a}; F ) ∈ Sub(A)} holds, then every ∈ Inv F ∩(PA;Q ∪Sk;Q )

502

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

deﬁnes a non-trivial automorphism s (s(a) = b :⇐⇒ (a, b) ∈ ) of the algebra A and the relations of Uk;Q are some non-trivial congruences of A. Then the following theorem follows from Theorem 16.2.1: Theorem 16.2.4 ([Sze 91]) Let A = (A; F ) be a ﬁnite algebra with the property that (SubA)\{A} contains only 1-element algebras. Then the following conditions are equivalent: (a) (A; F ) is semi-primal with {a ∈ A | ({a}; F ) ∈ SubA} = Q. (b) A has no proper automorphisms, is simple (i.e., A has only trivial congruences) and the direct product Ah of A for h = 2, 3, ..., |A|−1 has no subalgebra whose universe is an element of the set MA;Q ∪LA;Q ∪NA;Q ∪CA;Q .

16.3 Some Lemmas Lemma 16.3.1 TQ = [TQ3 ] for all Q with ∅ = Q ⊆ Ek . Proof. For k = 2, our assertion is valid by Chapter 3. Let k ≥ 3. Then the below-deﬁned functions ∨, ·, w, ra;b , if {a, b} ⊂ Ek , qa,b;c , if a = b or {a, b} ⊆ Q and {a, b, c} ⊆ Ek , belong to TQ : x ∨ y := max(x, y), x · y := min(x, y) in respect to the total order 0 < 1 < 2 < . . . < k − 1; x if x = y ∈ Q, w(x, y) := 0 otherwise; b if x = a, ja;b (x) := 0 otherwise; ra;b (x, y, z) := x ∨ ja;b (y) · z; qa,b;c (x, y, z) := x ∨ ja;c (y) · jb;c (z). We show that the above-deﬁned functions form a generating system for TQ . ... w with The function wn := w w n − 1 times x if x1 = x2 = ... = xn = x ∈ Q , wn (x1 , ..., xn ) = 0 otherwise

(n ≥ 1) is a superposition over w. Let f n be an arbitrary function of TQ , which is diﬀerent from wn . Then one can represent f as follows:

16.3 Some Lemmas

f (x1 , ..., xn ) = wn (x1 , ..., xn )∨ ∈

a = (a1 , ..., an ) Ekn \{(q, q, ..., q) | q f (a) = 0

503

ja1 ;f (a) (x1 ) · ... · jan ;f (a) (xn ). ∈ Q}

Therefore, f is a superposition over the set B := {wn } ∪ {x ∨ ja1 ;b (x1 ) · ... · jan ;b (xn ) | (a1 , ..., an ) ∈ Ekn \{(q, q, ..., q) | q ∈ Q} ∧ b ∈ Ek }. An arbitrary function of B\{wn } is generated from functions of the type ra;b and qa,b;c , since ra1 ,...,ai ,ai+1 ;b (x, x1 , ..., xi , xi+1 , y) := ra1 ,...,ai ;b (x, x1 , ..., xi , rai+1 ;b (x, xi+1 , y)) = x ∨ ja1 ;b (x1 ) · ... · jan ;b (xn ) · y (i = 1, ..., n) and ra1 ,...,an ;b (x, x1 , ..., xn , qai ,aj ;b (x, xi , xj )) = x ∨ ja1 ;b (x1 ) · ... · jan ;b (xn ). Consequently, [T 3 ] = T. The next lemma is a conclusion of the above lemma: Lemma 16.3.2 For every Q with ∅ = Q ⊆ Ek , the lattice of the subclasses of TQ is dual atomar and TQ has only ﬁnite-many maximal classes. Since the maximal classes of TQ are clones1 , one can easily show the following Lemma 16.3.3 For every maximal class M of TQ there exists an h-ary relation M with M = P olk M and 1 ≤ h ≤ k 3 . Lemma 16.3.4 Let ∅ = Q ⊆ Ek . Then (a) ∀ ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Bk ∪ (Ck \{{q} | q ∈ Q}) : TQ ⊆ P olk ; (b) Vk↑ (TQ ) = {TQ | Q ⊆ Q} (T∅ := Pk ); (c) ∀ ∈ a∈Q (Pk;{a} ∪ Nk;{a} ) : TQ ∩ P olk ⊂ TQ . Proof. (a) and (c) are easy to check. (b): It is suﬃcient to prove the following statement for ∅ = Q ⊆ Ek and f ∈ Pk : (a ∈ Q ∧ f (a, a, ..., a) = a) =⇒ TQ\{a} ⊆ [TQ ∪ {f }]. Let f (a, a, ..., a) = a for a certain a ∈ Q, Q ⊆ Ek and f (x) := f (x, x, ..., x). To prove TQ\{a} ⊆ [TQ ∪{f }], let g m ∈ TQ\{a} be arbitrary. Then the function hm+1 deﬁned by g 1

One can prove this analogous to the proof of footnote 1 of Chapter 14.

504

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

⎧ x, if x1 = ... = xm+1 = x ∈ Q, ⎪ ⎪ ⎨ u, if x1 = ... = xm = u ∈ Q\{a} ∧ hg (x1 , .., xm+1 ) := xm+1 = f (u), ⎪ ⎪ ⎩ g(x1 , ..., xm ) otherwise, belongs to TQ and g(x1 , ..., xm ) = hg (x1 , ..., xm , f (x1 )) is valid. Therefore, g ∈ [TQ ∪ {f }] holds. Lemma 16.3.5 For every relation γ ∈ R := Mk ∪ Uk ∪ Sk ∪ Lk ∪ Bk ∪ (Ck \{{q} | q ∈ Q}) there exists a γ ∈ [{γ} ∪ Invk TQ ], which belongs to Mk;Q ∪ Uk;Q ∪ Sk;Q ∪ Lk;Q ∪ Ck;Q ∪ (Pk;{q} ∪ Nk;{q} ). q∈Q

Proof. Examining the case γ ∈ R\(Mk;Q ∪ Uk;Q ∪ Sk;Q ∪ Ck;Q )

(16.1)

suﬃces. Further, we can assume that γ is h-ary, where h ≥ 2 because of (16.1). First we form the (h − 1)-ary relation γa := pr1,2,...,h−1 (∆({a} × γ))

(16.2)

for a ∈ Q. If h = 2 (i.e., γ ∈ (C2k \C2k;Q )∪(Mk \Mk;Q )∪(Uk \Uk;Q )∪(Sk \Sk;Q )), then we have (16.3) γa = {x ∈ Ek | (a, x) ∈ γ}. Because of (16.1), it is easy to check that there is an a ∈ Q such that the relation γa belongs to C1k;Q = {E ⊂ Ek | |E| ≥ 2 ∨ ∃b ∈ Ek \Q : E = {b}}: If γ ∈ Mk \Mk;Q , then for this purpose, one can choose a ∈ Q \ {o, e}, where o is the smallest element and e is the greatest element of Ek in respect to γ. If γ ∈ Uk \Uk;Q , then an a ∈ Q of an at least 2-element equivalence class of γ fulﬁlls the above condition. If γ ∈ Sk \Sk;Q , then there exists an a ∈ Q and a b ∈ Ek \Q with (a, b) ∈ γ; thus γa = {b} ∈ C1k;Q . If γ ∈ C2k \C2k;Q , there is an a ∈ Q, which is no central element of γ. Consequently, we also have γa ∈ C1k;Q in this case. Now, let h ≥ 3 and γ ∈ Lk , i.e., γ ∈ Chk ∪ Bhk . Choose a ∈ Q \ C, if γ ∈ Ck and C is the set of all central elements of γ. For γ ∈ Bhk let a ∈ Q be arbitrary. Then the relation γa deﬁned by (16.2) is an (h − 1)-ary reﬂexive, totally symmetric relation with the same central elements as γ and the new central element a. Since a is no central element of γ, there exist d1 , ..., dh−1 ∈ Ek with (a, d1 , ..., dh−1 ) ∈ γ. Consequently, we have γa = Ekh−1 and γa is a central relation. Through repetitions of the above construction, one obtains a γ-derivable relation of Ck;Q . Finally, let γ ∈ Lk , where k = pm , p prime and m ≥ 1. Then there is an elementar Abelean p-group (Ek ; +) with γ = {(x, y, u, v) ∈ Ek4 | x+y = u+v}. By Lemma 5.2.4.2, we can assume w.l.o.g. that the neutral element o of the p-group (Ek ; +) belongs to Q. For p = 2 we now form the γ-derivable relation

16.3 Some Lemmas

505

5 γ := pr3,4 (δ{0,1,2} ∩ ({o} × γ)) = {(x, y) ∈ Ek2 | x + y = o}.

Obviously, γ ∈ Pk;{o} , since for p = 2 the equation x + x = o is valid only if x = o. If p = 2, then one can form the relation γ := pr1,2,3 (∆({o} × γ)) = {(x, y, z) ∈ Ek3 | x + y = z} ∈ [{γ} ∪ Invk TQ ]. Thus γ ∈ Lk;{o} in the case {o} = Q. If |Q| ≥ 2 there is an a ∈ Q\{o} and the γ-derivable relation 4 γ := pr0,1 ((γ × {a}) ∩ δ{2,3} ) = {(x, a − x) | x ∈ Ek }

belongs to Sk . Further, we have that either γ ∈ Sk;Q is valid or (see above) it is possible to derive from γ a relation of C1k;Q . Lemma 16.3.6 Let |Q| ≥ 2 and γ ∈ a∈Q Pk;{a} ∪ Nk;{a} . Then there exists a ∈ [{γ} ∪ Invk TQ ] with ∈ Pk;Q ∪ Nk;Q ∪ C1k;Q . Proof. First let γ ∈ ( a∈Q Pk;{a} )\Pk;Q . Then there is (b, c) ∈ γ with b ∈ Q and c ∈ Ek \Q. The γ-derivable relation γb := pr1 (∆({b} × γ)) = {x ∈ Ek | (b, x) ∈ γ}

(16.4)

belong to Ck;Q (because of γb = {c}). Finally, let γ ∈ ( a∈Q Nk;{a} )\Nk;Q ; i.e., γ has the following properties: • ∃q ∈ Q : ι2k ∩ γ = {(q, q)} ∧ {(x, q), (q, x) | x ∈ Ek } ⊆ γ; • γ is symmetric and • ∃b ∈ Q \ {q} ∃c ∈ Ek \{b, q} : (b, c) ∈ γ. We form the γ-derivable relation γb (see (16.4)). Then we have {q, c} ⊆ γb and b ∈ γb . Thus γb ∈ C1k;Q . Lemma 16.3.7 Let A := P olk be TQ -maximal, where ∈ Rkt . Moreover, Vk↑ (A)\{A} = Vk↑ (TQ ). Then |Q| = 1 is valid and there exists a relation γ ∈ Pk;Q ∪ Nk;Q with A ⊆ P olk γ. Proof. Because of Lemma 16.3.4, the relation has the following properties: (I) If |Q| ≥ 2, then there is a relation γ ∈ [{} ∪ Invk TQ ] with γ ∈ (Rmax (Pk )\{{q} | q ∈ Q}) ∪ Rmax (T{a} ). a∈Q

In particular, it follows from (I) that t ≥ 2. W.l.o.g. we can assume the following three properties of :

506

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

(II) does not have any double rows. (III) Every -derivable (t−1)-ary relation belongs to Invk TQ (see 16.1.1,(c)). (IV) No -derivable t-ary relation with the properties P olk ⊂ TQ and | | < || exists. From assumptions (II) - (IV), some further properties of the relation follow: (V) does not have a constant row. (Suppose, has a constant row (a, a, ..., a). Because of (I) we have a ∈ Q and it is valid = pr0,...,i−1 × {a} × pri+1,...,t−1 for certain i ∈ Et . Then, with the help of (III) and 16.1.1,(c) one can prove that is an invariant of TQ , contrary to our assumptions about .) A direct consequence from (V) and (III) is: (VI) ∀i ∈ Et : pr0,...,i−1,i+1,...,t−1 = Ekt−1 . Next we show that t (VII) (t ≥ 3 ∨ |Q| = 1) =⇒ ∩ δ{0,1,...,t−1} ∈ {{(q, q, ..., q)} | q ∈ Q}; t (t = 2) =⇒ ∩ δ{0,1} ∈ {∅, {(q, q)} | q ∈ Q}

holds. By (III) we have t t ∩ δ{0,1,...,t−1} ∈ {∅, {(q, q, ..., q)}, δ{0,1,...,t−1} | q ∈ Q}. t t ∩ δ{0,1,...,t−1} = δ{0,1,...,t−1} is not possible, since A = P olk ⊂ TQ and TQ contains at most a constant. t = ∅, the equation pr0,...,t−2 = Ekt−1 (see (VI)) If Q = {a} and ∩ δ{0,1,...,t−1} t+1 implies prt (({a} × ) ∩ δ{0,...,t−1} ∈ C1k;{a} , a contradiction to (I). t = ∅ that prt−2,t−1 ( ∩ In the case that t ≥ 3, it follows from ∩ δ{0,1,...,t−1} t δ{0,...,t−2} ) ∈ Invk TQ , which contradicts (III). Therefore (VII) holds.

As generally known (see Chapter 6), the relation σi () := {(a1 , ..., ai ) ∈ Eki | ∃u ∈ Ek : {(a1 , u), ..., (ai , u)} ⊆ } is derivable from the relation for t = 2, and it is valid that (VIII) (t = 2 ∧ ◦ (τ ) = Ek2 ) =⇒ ∀i ≥ 2 : σi () = Eki . For the relation , the following three cases are possible: Case 1: t = 2. We consider the -derivable relation ∩ (τ ). Case 1.1: ∩ (τ ) ∈ {∅, {(q, q)}} for a certain q ∈ Q. Then is antisymmetric. Because of (VI) the relation ◦(τ ) has the property ι2k ⊆ ◦ (τ ). Case 1.1.1: ◦ (τ ) = ι2k . Because of pr0 = pr1 = Ek (see (VI)) we have || ≥ k. If || > k, there exists a, b, c ∈ Ek with (a, c), (b, c) ∈ and a = b. Thus (a, b) ∈ ◦ (τ ),

16.3 Some Lemmas

507

which is not possible because of ◦ (τ ) = ι2k . Therefore || = k, and has the form {(x, s(x)) | x ∈ Ek }, where s = e11 is a permutation on Ek , which has at most a ﬁxed point (namely q). Assume the permutation has proper cycles of diﬀerent length. Let r (≥ 2) be the length of a smallest proper cycle. Then we have pr0 (( ◦ ◦ ... ◦ ) ∩ ι2k ) ∈ C1k;Q , r times a contradiction to (I). Therefore, all proper cycles of s have the very same length l. Suppose l = p · m, p prime and m ≥ 2. Then ◦ ◦ ... ◦ has proper m times cycles of the length p; i.e., a relation of the set Sk ∪ Pk;{q} is derivable from the relation . But, this contradicts the condition (I) for |Q| ≥ 2 or ∈ Sk . Therefore, Q = {q} and A ⊆ P olk γ for a certain γ ∈ Pk;{q} in Case 1.1.1. Case 1.1.2: ι2k ⊂ ◦ (τ ) ⊂ Ek2 . In this case, we see that the relation = ◦ (τ ) is not an invariant of TQ and that P olk ⊆ TQ holds, which is not possible because of Vk↑ (A)\{A} = Vk↑ (TQ ). Case 1.1.3: ◦ (τ ) = Ek2 . By (VIII) we have σk () = Ekk . Consequently, there exists a u ∈ Ek with (x, u) ∈ for all x ∈ Ek . Because of ∩ ι2k ∈ {∅, {(q, q)}} (q ∈ Q), this is only possible for ∩ ι2k = {(q, q)} and u = q. If (τ ) ◦ = Ek2 is valid, we can prove {(x, q) | x ∈ Ek } ⊆ τ in analog mode too, through which we receive a contradiction to the antisymmetry of the relation . Therefore ι2k ⊆ (τ ) ◦ ⊂ Ek2 . Consequently, Case 1.1.3 is reducible to Cases 1.1.1 and 1.1.2. Case 1.2: {(q, q)} ⊂ ∩ (τ ) ⊂ . Because of condition (III) this case is not possible. Case 1.3: ∩ (τ ) = . In this case, is symmetric. Further, we have ι2k ⊆ ◦ . We distinguish three cases: Case 1.3.1: ◦ = ι2k . Then is a permutation with at most a ﬁxed point q (if ∩ ι2k = {(q, q)}), and every proper cycle of has the length 2 because of symmetry of , i.e., ∈ Sk ∪ Pk;{q} . Thus, as in Case 1.1.1, we obtain a contradiction to the condition (I) either or |Q| = 1 and A ⊆ P olk γ for certain γ ∈ Pk;Q are valid. Case 1.3.2: ι2k ⊂ ◦ ⊂ Ek2 . This case can be excluded as Case 1.1.2. Case 1.3.3: ◦ = Ek2 . Because of ◦ = ◦ (τ ) = Ek2 and by (VIII), we have σk () = Ekk ,; i.e., there is u ∈ Ek with {(x, u) | x ∈ Ek } ⊆ . Consequently, ∩ ι2k = {(q, q)} and u = q. This and the symmetry of implies that q is a central element of . Therefore, belongs to Nk;{q} . Because of (I), this is only possible for |Q| = 1.

508

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

Case 2: t = 3. 3 = {(q, q, q)} for certain q ∈ Q (see Since pr0,1 = Ek2 (by (VI)), ∩ δ{0,1,2} (VII)) and (III) is valid, we have ∆ = Ek ×{q}, i.e., (a, a, q) ∈ for all a ∈ Ek . Analogously, one can prove that the tuples (a, q, a) and (q, a, a) belong to for every a ∈ Ek . Next we prove that is totally symmetric. Assume the relation is not totally symmetric. Then the -derivable relation {(as(0) , as(1) , as(2) ) | (a0 , a1 , a2 ) ∈ } := s s is permutation on E3 is totally symmetric. Because of ⎛ ⎞ a a q ⎝ a q a ⎠ ⊆ ⊂ (a = q) q a a the relation is not, however, an invariant of TQ . This is a contradiction to the condition (IV). Next we prove (IX) ∀a, b ∈ Ek : {(a, b, c), (a, b, c )} ⊆ =⇒ c = c . Suppose there are a, b, c, c with {(a, b, c), (a, b, c )} ⊆ , and c = c . Then the -derivable relation 6 ) 1 := pr0,2,5 (( × ) ∩ δ{0,3},{1,4} = {(x, y, z) | ∃u ∈ Ek : {(x, u, y), (x, u, z)} ⊆ } 3 has the property P olk 1 ⊆ TQ because of δ{0,1,2} ⊆ 1 . Consequently, 1 is a diagonal relation. Further, it is easy to check that (a, c, c ) ∈ 1 and (d, q, q) ∈ 1 for all d ∈ Ek hold. Therefore, 1 is the diagonal relation Ek3 . For our relation , this means that for arbitrary (x, y, z) ∈ Ek3 there is a u with (x, u, y) ∈ and (x, u, z) ∈ . Then, when one chooses x = y = q and z = q and considers the total symmetry of the relation , there exists an r with r = q and (q, q, r) ∈ . Above ∆ = Ek × {q} was, however, proven. Therefore, our assumption c = c was false. With that, (IX) is valid. Now we consider the -derivable relation

2 := ◦ = {(a, b, c, d) | ∃u ∈ Ek : {(a, b, u), (u, c, d)} ⊆ }. 4 Because of {(a, a, q), (q, a, a) | a ∈ Ek } ⊆ we have δ{0,1,2,3} ⊆ 2 and thus P olk 2 ⊆ TQ . Obviously, the relation 2 does not have any double rows. Because of our assumptions about the relation , however, this is possible only for 2 = Ek4 . Then, by deﬁnition of 2 , we have {(a, b, u), (u, c, a)} ⊆ for arbitrary a, b, c ∈ Ek , and certain u. Since is totally symmetric, however,

16.3 Some Lemmas

509

{(a, u, b), (a, u, c)} ⊆ , where b = c is possible, contrary to (IX). Therefore, Case 2 is not possible. Case 3: t ≥ 4. Because of pr0,1,...,t−2 = Ekt−1 (see (VI)), (II), (III) and (VII) we have ∩ t−1 t = δ{0,...,h−3} × {q} for certain q ∈ Q. Since pr0,...,h−3,h−1 = Ekt−1 δ{0,...,h−3} t−2 t is also valid, one can analogously prove ∩ δ{0,...,h−3} = δ{0,..,h−3} × {q} × Ek , contrary to that just shown. Therefore, there is no t-ary relation with the above-demanded properties and t ≥ 4. An equivalence relation ∼ is deﬁned by ∼ :⇐⇒ TQ ∩ P olk = TQ ∩ P olk on a set A ⊆ Rk . We select a representative from every equivalence class of ∼ now and obtain a certain subset of A, which we denote with A∼ , where A ∈ {Pk;Q , Mk;Q , Sk;Q } in the following. Lemma 16.3.8 Let ∅ = Q ⊆ Ek . Then (a) ∀, ∈ Mk;Q : ∼ ⇐⇒ ( = τ ∨ = ); (b) ∀, ∈ Pk;Q ∪ Sk;Q : ∼ ⇐⇒ (∃t : = ◦ ◦ ... ◦ ); t times ∼ ∼ (c) ∀, ∈ M∼ k;Q ∪Sk;Q ∪Pk;Q ∪Uk;Q ∪Nk;Q ∪Lk;Q ∪Ck;Q : ∼ ⇐⇒ = . Proof. The statements (a) and (b) are direct conclusions from the proof of corresponding statements about relations of Mk ∪ Sk (see Chapter 5). To prove (c), we agree that o denotes the smallest element of Ek and that e denotes the greatest element of Ek (in respect to ∈ Mk;Q ). Because of (a), we can assume w.l.o.g. that o ∈ Q. Now, let and be two diﬀerent relations ∼ ∼ of M∼ k;Q ∪ Sk;Q ∪ Pk;Q ∪ Uk;Q ∪ Nk;Q ∪ Lk;Q ∪ Ck;Q . To prove TQ ∩ P olk ⊆ TQ ∩ P olk

(16.5)

we distinguish the following 9 cases: ∼ Case 1: {, } ⊆ C1k;Q ∪ Uk;Q ∪ P∼ k;Q ∪ Sk;Q . If |Q| ≥ k − 1 then Uk;Q = Pk;Q = Sk;Q = ∅ and (16.5) is obviously valid for the relations , . Let now |Q| ≤ k − 2. Denote ω the relation {(q, q) | q ∈ Q} ∪ (Ek \Q)2 of Uk;Q . It is easy to check that the set A(γ) := {f n ∈ PEk \Q | ∃f ∈ TQ ∩P olk γ : (∀a ∈ (Ek \Q)n : f (a) = f (a))} n≥1

is a maximal class of PEk \Q for γ ∈ (C1k;Q \{γ | γ ⊆ Q ∨ γ = Ek \Q}) ∪ ∼ 1 (Uk;Q \{ω}) ∪ P∼ k;Q ∪ Sk;Q and that A(γ) = PEk \Q holds for γ ∈ {γ ∈ Ck | γ ⊆ Q ∨ γ = Ek \Q} ∪ {ω}. With the aid of Chapters 5 and 6, our assertion (16.5) results from that for ∈ {γ ∈ C1k | γ ⊆ Q ∨ γ = Ek \Q} ∪ {ω}. If ∈ {γ ∈ C1k | γ ⊆ Q ∨ γ = Ek \Q} ∪ {ω}, then it is also easy to prove that

510

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

there is a | |-ary function f1 ∈ TQ ∩ P olk with f1 ( ) ∈ . Consequently, (16.5) is also valid for the remaining relations of the Case 1. 1 Case 2: ∈ C1k;Q and ∈ M∼ k;Q ∪ (Ck;Q \Ck;Q ) ∪ Nk;Q ∪ Lk;Q . The following | |-ary function f2 with the properties f2 ( ) ∈ , f2 (q, ..., q) = q for all q ∈ Q and f2 (a) = c for the remaining tuples a, where c ∈ , belongs to TQ ∩ P olk . Thus (16.5) holds. ∼ ∼ 1 Case 3: ∈ M∼ k;Q ∪ Uk;Q ∪ Pk;Q ∪ Sk;Q and ∈ (Ck;Q \Ck;Q ) ∪ Nk;Q ∪ Lk;Q . Let be an h -ary relation. Further, let q ∈ Q and a ∈ Ek \Q. In addition, if ∈ Nk;Q , then we choose q ∈ Q so that (q, q) ∈ holds. Then the h -tuples (q, a, a, . . . , a), (a, q, a, . . . , a), . . . , (a, a, . . . , a, q) belong to , though not all belong to . Consequently, there exists an h -ary function f3 ∈ TQ ∩ P olk with ⎞ ⎛ q a a . . . a ⎟ ⎜ ⎜a q a . . . a⎟ ⎟∈ ⎜ f3 ⎜ ⎟ / . . . . . . . . ⎠ ⎝ a a a . . . q Hence (16.5) holds in Case 3. ∼ ∼ Case 4: ∈ P∼ k;Q ∪ Sk;Q and ∈ Mk;Q . This case can occur only for |Q| ≤ 2. Further, we have S∼ k;Q = ∅, if |Q| = 1 or k = 3. It is easy to check that (TQ ∩ P olk ) ⊆ (TQ ∩ P olk )1 is valid for k ≥ 3. Case 5: ∈ Uk;Q and ∈ M∼ k;Q . For o ∈ Q the binary function f4 deﬁned by y if x = o , f4 (x, y) := o otherwise, belongs to TQ ∩ P olk . However, f4 does not preserve the relation with the smallest element o , since (f4 (o , e ), f (α, e )) = (e , o ) if α ∈ Ek \{o , e } . 1 Case 6: ∈ M∼ k;Q and ∈ Ck;Q . Let a ∈ Ek \ and

ta (x, y) :=

x if x = y ∈ Q, a otherwise.

The function ta ∈ TQ preserves and does not preserve . Therefore, (16.5) holds.

16.3 Some Lemmas

511

∼ ∼ Case 7: ∈ M∼ and ∈ M∼ k;Q ∪ Uk;Q ∪ Pk;Q ∪ Sk;Q . k;Q r | | , {r, s} ⊂ Ek and f5 a | |-ary function deﬁned by Let = s if ⊆ , ∈ α f5 ( ) := β ∈ \ if ⊂

and, if r < s and hence α < ⎧ o ⎪ ⎪ ⎪ ⎨ α f5 (a) := ⎪ β ⎪ ⎪ ⎩ e

β, if a = (o , o , ..., o ), if o < a < s, if a = s, otherwise,

and, if s and r are incomparable in respect to , ⎧ o if a < r ∨ a < s, ⎪ ⎪ ⎪ ⎨ α if a = r, f5 (a) := ⎪ β if a = s, ⎪ ⎪ ⎩ e otherwise. It is easy to check that f5 ∈ TQ ∩ P olk and f5 ∈ TQ ∩ P olk . ∼ Case 8: ∈ (Ck;Q \C1k;Q ) ∪ Nk;Q and ∈ Uk;Q ∪ M∼ k;Q ∪ Ck;Q ∪ Pk;Q ∪ Sk;Q ∪ Nk;Q ∪ Lk;Q . Let c be a central element of . Further, let f6 be an | |-ary function deﬁned by if ⊆ , ∈ f6 ( ) ∈ \ if ⊂ and f6 (q, q, ..., q) = q for all q ∈ Q and f6 (a) = c for the remaining tuples | | a ∈ Ek . Then f6 ∈ TQ ∩ P olk and f6 ∈ TQ ∩ P olk . Therefore, (16.5) holds. Case 9: ∈ Lk;Q . Then |Q| = 1 and k = 2m , m ≥ 1. W.l.o.g. let Q = {0}. As shown in Chapter 5, one can deﬁne the relation with the aid of an elementary Abelean 2-group G, where 0 is the neutral element of G. It follows from the proof of the Lemma 16.3.5 that the relation is derivable from a relation {0} × λ, where λ ∈ L2m . Since ◦ = λ is in addition valid, we have T{0} ∩ P olk λ = P olk . Therefore, by Chapter 5, every n-ary function f ∈ P olk is a quasi-linear function of the form n m−1 j ai,j · x2i , f (x1 , ..., xn ) = i=1 j=0

where a1,0 , ..., an,m−1 ∈ E2m and +(= +G ) and · are the operations of a certain ﬁeld (E2m ; +, ·) with G = (E2m ; +).

512

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

Case 9.1: ∈ C1k;Q . Let a ∈ and b ∈ . Then the quasi-linear unary function f7 deﬁned by f7 (x) := (a · b−1 ) · x does not preserve because of f7 (b) = a, i.e., (16.5) holds. Case 9.2: ∈ P∼ k;Q . Let 1 the unit of the multiplicative group of the ﬁeld (E2m ; +, ·). Since the order of this group is odd, we have a2 = 1 for all a ∈ E2m \{0, 1}. The unary function f8 deﬁned by f8 (x) := x2 belongs to P olk , but it does not preserve , since (by deﬁnition of ) there is an a ∈ Ek \{1} with (1, a) ∈ and (f8 (1), f8 (a)) = (1, a2 ) ∈ because of a2 = a and because of the above. Case 9.3: ∈ Uk;Q . The binary function f9 with f9 (x, y) : = x + y belongs to P olk . One can prove f9 ∈ P olk as follows: Let (a, b) ∈ and a = b. Then 0 a a f9 = ∈ , b a a+b since a = b implies a + b = 0. Case 9.4: ∈ M∼ k;Q . In this case, the above-deﬁned function f9 does not preserve the relation , since 0 a a f9 = ∈ / e e 0 holds for a ∈ Ek \{0}. Case 9.5: ∈ Lk;Q . For arbitrary λ ∈ L22 ,0 we have {(0, x, x), (x, 0, x), (x, x, 0) | x ∈ Ek } ⊆ λ. Because of = , there are certain a and b of Ek with (a, b, a +G b) ∈ and (a, b, a +G b) ∈ . Consequently, the ternary function f10 ∈ P olk deﬁned by f10 (x, y, z) := x +G y +G z does not preserve the relation , since ⎞ ⎛ ⎛ ⎞ 0 a 0 a ⎜a a b⎟ f10 ⎝ ⎠ = ⎝ b ⎠ a 0 b a +G b holds. Case 9.6: ∈ (Ck;Q \C1k;Q ) ∪ Nk;Q . Let be an h -ary relation and (a1 , . . . , ah ) ∈ . Then the h -ary function f11 deﬁned by f11 (x1 , . . . , xh ) := a1 · x1 +G a2 · x2 +G · · · +G ah · xh does not preserve the relation , ⎛ 1 0 0 ⎜ ⎜0 1 0 f11 ⎜ ⎜. . . ⎝ 0 0 0

since, if 1 is the unit ⎞ ⎛ . . . 0 a1 ⎟ ⎜ . . . 0⎟ a2 ⎟ = ⎜ ⎜ . . . .⎟ ⎝ · ⎠ . . . 1 ah

of the ﬁeld (Ek ; +G , ·), ⎞ ⎟ ⎟ ⎟ ⎠

16.4 Proof of Theorem 16.2.1

513

holds and because 0 is a central element of . Therefore, (16.5) holds in Case 9.6. Since TQ ∩ P olk ⊆ TQ ∩ P olk implies = , (c) is proven.

16.4 Proof of Theorem 16.2.1 Put Rmax (TQ ) := Mk;Q ∪ Uk;Q ∪ Sk;Q ∪ Lk;Q ∪ Ck;Q ∪ Pk;Q ∪ Nk;Q . Let A be an arbitrary maximal class of TQ with ∅ = Q ⊆ Ek . We will prove by induction on r := |Q| that ∃ ∈ Rmax (TQ ) : A ⊆ TQ ∩ P olk

(16.6)

holds. r = 1 : If Vk↑ (A)\{A} = Vk↑ (TQ ), then (16.6) follows from Lemma 16.3.7. If Vk↑ (A)\{A} = Vk↑ (TQ ), then there exists a maximal class B1 of Pk with B1 = P olk and A = TQ ∩ B1 because of Lemma 16.3.4, (b). Then, by Chapter 6, there is a relation γ ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Bk ∪ Ck \{Q} with γ ∈ Invk A. Therefore, (16.6) follows from Lemma 16.3.5. r − 1 −→ r : Suppose, (16.6) holds for all Q ⊆ Ek with 1 ≤ |Q | ≤ r − 1. Let Q (⊆ Ek ) be an r-element set. Because of r = |Q| ≥ 2 and Lemma 16.3.7, we have Vk↑ (A)\{A} = Vk↑ (TQ ). Consequently, there exists a subclass B of Pk with (16.7) A = B ∩ TQ and B ⊆ TQ . We distinguish two cases for B: Case 1: ∀a ∈ Q : B ⊆ P olk {a}. Since A = Pk and because of Chapter 6, there is a γ ∈ Rmax (Pk )\{{a} | a ∈ Q} with B ⊆ P olk γ. Then this, (16.7), and Lemma 16.3.4, (a) imply A = B ∩ TQ ⊆ TQ ∩ P olk γ ⊂ TQ . Because γ ∈ Invk A, the statement (16.6) results through Lemmas 16.3.5 and 16.3.6. Case 2: ∃Q ⊂ Q : B ⊆ TQ ∧ (∀b ∈ Q\Q : B ⊆ P olk {b}). Because of B ∈ Vk↑ (TQ ), we have B ⊂ TQ . Thus there exists a maximal class B2 of TQ with B ⊆ B2 (see Lemma 16.3.2). By induction assumption, there is a ∈ Rmax (TQ ) with B2 ⊆ TQ ∩ P olk . Because of Lemma 16.3.4, (a), (c) the we have TQ ∩ P olk = TQ . This implies B2 = TQ ∩ P olk . Furthermore, following is valid: ∈ Mk ∪ Sk ∪ Uk ∪ (Ck \{{q} | q ∈ Q}) ∪ a∈Q (Pk;{a} ∪ Nk;{a} ) (notice ∀b ∈ Q\Q : B ⊆ P olk {b}). This and Lemmas 16.3.5 and 16.3.6 imply (16.6). Consequently, (16.6) is proven. It is easy to see that TQ ∩ P olk = TQ holds for every ∈ Rmax (TQ ). Therefore, Theorem 16.2.1 follows from (16.6) and Lemma 16.3.7.

17 Maximal Classes of P olk El for 2 ≤ l < k

This chapter continues the investigations of Chapter 16 and generalizes Theorem 14.1.3. For arbitrary k, l ∈ N with 2≤l ≤k−1 all maximal classes of P olk El are determined. With the help of these maximal classes, one can easily give a completeness criterion for P olk El . The proofs given in this chapter resemble those ones from Chapter 6; that is, the results of this chapter were achieved with the means which were developed by I. G. Rosenberg in [Ros 70a].

17.1 Notations, Deﬁnitions, and Some Lemmas A relation , for which there are certain t1 , ..., tn ∈ {l, k} and a relation δ ∈ Dk with = (Et1 × Et2 × ... × Eth ) ∩ δ, is called El -diagonal. For relations ∈ Dk ∪ {ιhk }, we agree with the following brief notations: (t1 , t2 , ..., tn ) := ∩ (Et1 × Et2 × ... × Eth ) (t1 , ..., th ∈ {l, k}) and [t] := (l, l, ..., l, k, k, ..., k). t times In this chapter, we say that a relation is -derivable if one can receive by a ﬁnite number of applications of the relation operations (see Section 2.3) from the relations of {} ∪ Invk P olk El . In this case, we also write {} ∪ Invk P olk El

17 Maximal Classes of P olk El for 2 ≤ l < k

516

or

∈ [{} ∪ Invk P olk El ].

In the following proofs of this chapter, we assume that the reader knows the statements of the following lemma. The lemma contains statements that were already proven or can easily be proven. Lemma 17.1.1 It holds: (a) ∀ ∈ Rk : P olk El ∩ P olk = P olk El × ; (b) (∃i ∈ El : pri = El ) =⇒ P olk El × = P olk ; (c) ∀, ∈ Rk : ((P olk ⊆ P olk El ∧ ({} ∪ Invk P olk El )) =⇒ P olk ⊆ P olk El ∩ P olk ); (d) Invk Pk = Dk (see Theorem 2.6.2); (e) A relation belongs to Invk P olk El if and only if is El -diagonal. In addition to those notations introduced in Chapter 5, we need the following for relations and relation sets: Uk;El := { ∈ Uk | ∩ El2 ∈ {ι2l , El2 } ∧ a∈El {x | (a, x) ∈ } ∈ {El , Ek }}. Sk;El := {{(x, s(x)) | x ∈ Ek } | s is a permutation on Ek with exactly l ﬁxed points (∈ El ), and all proper cycles of s have the same prime length }; Put (1)

Ck;El := { ∈ C1k | ∩ El ∈ {∅, El } ∧ = El }, Chk;El := { ∈ Chk | each element of El is a central element of } (2 ≤ h ≤ k − 1); ZC2k;El := { ∈ C2k | ∩ El2 = ι2l ∧ (∀x ∈ Ek ∃αx ∈ El : (x, αx ) ∈ ) ∧ (∀a, b ∈ Ek \El : ((∃u ∈ El : {(a, u), (u, b)} ⊆ ) =⇒ (a, b) ∈ )))} 0 1 2 2 2 0 1 (For example: If k = 3 and l = 2 then belongs to ZCk;El .) 0 1 2 0 1 2 2 and Ck;El := ZC2k;El ∪

k−1 h=1

Chk;El .

Let Zk;El be the set of all binary relations that fulﬁll the following four conditions: 1) ⊂ El × Ek ; 2) ∩ El2 = ι2l ;

17.1 Notations, Deﬁnitions, and Some Lemmas

517

3) {x | ∃y : (x, y) ∈ } = El ∧ {y | ∃x : (x, y) ∈ } = Ek ; 4) ∃c ∈ Ek \El : El × {c} ⊆ . One ﬁnds connections between the sets ZCk;El and Zk;El explained in Lemma 17.3.5. Let Nk;El be the set of all binary relations of the form (\ι2k ) ∪ El2 , where either = Ek2 or ∈ C2k;El 1 holds. An h-ary relation is called (l; r)-central if fulﬁlls the following ﬁve conditions: 1) ⊂ Elr × Ekh−r , 2 ≤ h ≤ k − 1, 1 ≤ r ≤ h − 1; 2) is totally (l; r)-reﬂexive, i.e., ιhk [r] ⊆ ; 3) is totally (l; r)-symmetric, i.e., for every permutation s ∈ Sh [r] := {f | f is permutation on Eh ∧ ∀a ∈ Er : f (a) ∈ Er } it holds: (a0 , a1 , ..., ah−1 ) ∈ =⇒ (as(0) , as(1) , ..., as(h−1) ) ∈ . 4) is a weakly (l; r)-central relation, i.e., {(a1 , ..., ar , b1 , ..., bh−r ) ∈ Elr × Ekh−r | ∃i ∈ {1, ..., h − r} : bi ∈ El } ⊆ ; 5) has a central element c ∈ El , i.e., for arbitrary a1 , ..., ar ∈ El and arbitrary b1 , ..., bh−r ∈ Ek it holds: c ∈ {a1 , ..., ar } =⇒ (a1 , ..., ar , b1 , ..., bh−r ) ∈ . h We denote the set of all h-ary (l; r)-central relations with Ck;E [r]. l Further, let k−1 h−1 h Ck;E [r]. Ck;El := l h=2 r=1

Let Hk;El be the set of all ternary relations with the following six properties: 1) 2) 3) 4) 5) 6)

⊆ El2 × Ek ; ∩ El3 = {(x, x, y) | x, y ∈ El }; ∩ ι3k = {(x, x, z) | x ∈ El ∧ z ∈ Ek }; ∃u ∈ Ek \El : El2 × {u} ⊆ ; is totally (l; 2)-symmetric; is strongly (l; 2)-homogeneous, i.e., ∀x, y ∈ El ∀z ∈ Ek : (∃α, β ∈ El : {(x, α, z), (β, y, z), (α, β, z)} ⊆ ) =⇒ (x, y, z) ∈ .

1

I.e., is symmetric and ι2k ∪ (El × Ek ) ∪ (Ek × El ) ⊆ .

17 Maximal Classes of P olk El for 2 ≤ l < k

518

With the help of the notations given in the deﬁnition of an (l; r)-central relation and with γr,s := {(a1 , ..., ar , b1 , ..., bs ) ∈ Elr × Eks | (∃i ∈ {1, ..., s} : bi ∈ El ) ∨ (a1 , ..., ar , b1 , ..., bs ) ∈ ιhk [r]} one can deﬁne the following relations: An h-ary relation is called (l; 2)-universal relation (3 ≤ h ≤ k − l + 2) if fulﬁlls the following six conditions: 1) 2) 3) 4) 5)

⊂ El2 × Eks , s := h − 2, 3 ≤ h ≤ k − l + 2. is totally (l; 2)-reﬂexive. is totally (l; 2)-symmetric. is a weakly (l; 2)-central relation. The relation b := {(a1 , a2 ) | (a1 , a2 , b1 , ..., bs ) ∈ }

is an equivalence relation for every b := (b1 , ..., bs ) ∈ (Ek \El )s \ιsk . 6) s ∈ {1, k − l} and = γ2;s or γ2;s ⊂ . Let h Bk;E [2] l

be the set of all h-ary (l; 2)-universal relations. In the following deﬁnitions, we use notations introduced in Section 5.2.6. An h-ary relation is called an (l; r)-universal if fulﬁlls the following six conditions: 1) 2) 3) 4) 5)

⊂ Elr × Eks , s := h − r, 3 ≤ r ≤ l, 1 ≤ s ≤ k − l. is totally (l; r)-reﬂexive. is totally (l; r)-symmetric. is a weakly (l; r)-central relation. For every b := (b1 , ..., bs ) ∈ (Ek \El )s \ιsk with Elr × {b} ⊆ it holds b := {(a1 , ..., ar ) | (a1 , ..., ar , b1 , ..., bs ) ∈ } ∈ Bl , i.e., there exists an m[b] ∈ N and a mapping qb : El −→ Erm[b] with the property ∀a1 , ..., ar ∈ El : ((a1 , ..., ar , b1 , ..., bs ) ∈ ⇐⇒ ∀i ∈ Em[b] : ((qb (a1 ))(i) , ..., (qb (ar ))(i) ) ∈ ιrr ).

17.2 Results of Chapter 17

519

6) If s ≥ 2 then: For every tuple (α, β) := (α1 , ..., αr , β1 , ..., βs ) ∈ (Elr × Eks )\ there is an n-ary function f(α,β) that preserves the relation , and there is a matrix A(α,β) of the type k ×n, whose columns belong to {α1 , ..., αr }l × {β1 , ..., βs }k−l with the property ⎛ ⎞ qβ (0) ⎜ qβ (1) ⎟ ⎜ ⎟ ⎜ ⎟ ... ⎜ ⎟ ⎜ qβ (l − 1) ⎟ ⎜ ⎟. f(α,β) (A(α,β )) := ⎜ ⎟ l ⎜ ⎟ ⎜ l+1 ⎟ ⎜ ⎟ ⎝ ⎠ ... k−1 The simplest relations, which fulﬁll the above six conditions, are the abovedeﬁned relations γr;s with (r, s) ∈ {(l, 1), (l, k − l)} (see also the Lemmas 17.5.13 and 17.5.14). Let Bk;El [r] be the set of all h-ary (l; r)-universal relations for r ≥ 3 and put h Bk;El := Bk;E [r]. l 3 ≤ h ≤ k, 2 ≤ r ≤ l 1≤h−r ≤k−l

17.2 Results of Chapter 17 The aim of Chapter 17 is to prove the following theorem: Theorem 17.2.1 Let 2 ≤ l ≤ k − 1, Rmax (Pl ) := Ml ∪ Sl ∪ Ul ∪ Ll ∪ Cl ∪ Bl and Rmax (P olk El ) := Rmax (Pl ) ∪ Uk;El ∪ Ck;El ∪ Sk;El ∪ Nk;El ∪ Zk;El ∪ Ck;Ek ∪ Hk;El ∪ Bk;El . Then {P olk El × | ∈ Rmax (P olk El )} is the set of all maximal classes of P olk El . The next theorem follows from the above theorem and from the fact that P olk El is ﬁnitely generated (see Lemma 17.4.1).

520

17 Maximal Classes of P olk El for 2 ≤ l < k

Theorem 17.2.2 (Completeness Criterion for P olk El ) For an arbitrary subset M of P olk El it holds: [M ] = P olk El ⇐⇒ ∀ ∈ Rmax (P olk El ) : M ⊆ P olk × El .

The following theorem (basically given in [Ros 74]) is also a conclusion from Theorem 17.2.1. Theorem 17.2.3 Let M be an arbitrary subset of P olk El with (P olk El )1 ⊆ M . Then, [M ] = P olk El if and only if M ⊆ P olk for all of the following list: {(a, b, c, d) ∈ E24 | a + b = c + d (mod 2)} if l = 2, λl := ιll if l ≥ 3, µ := {(x, y) ∈ Ek2 | {x, y} ⊆ El ∨ x = y} (∈ Uk;El ), κh := ιhk ∪ {(a1 , ..., ah ) ∈ Ekh | ∃i ∈ {1, ..., h} : ai ∈ El } (∈ Ck;El ) if 2 ≤ h ≤ k − l and κh = Ekh , κ∗ := (El × Ek ) ∪ (Ek × El ), γr,s if (s, t) ∈ {(2, 1), (l, 1), (2, k − l), (l, k − l)}. In other words: P olk El ⎧ ⎨4 nk,l := k − l + 4 ⎩ k−l+6

has exactly if (k, l) = (3, 2), if (k, l) ∈ {(k, 2), (k, k − 1)} and k ≥ 4, otherwise,

maximal classes that contain (P olk El )(1) : P olk 1 (1 ∈ {λl , κ∗ , γ2,1 , γl,1 , γ2,k−l , γl,k−l }), P olk 2 × El (2 ∈ {µ, κh | κh = Ekh ∧ 2 ≤ h ≤ k − l}).

17.3 Maximality Proofs The following lemma is a conclusion of Chapter 6: Lemma 17.3.1 For every ∈ Rmax (Pl ) the clone P olk is a maximal clone of P olk El , 2 ≤ l ≤ k − 1. Lemma 17.3.2 Let ∈ C1k;El \C1l , i.e., ∅ ⊂ ⊂ Ek , \El = ∅ and ∩ El ∈ {∅, El }. Then, P olk × El is maximal in P olk El .

17.3 Maximality Proofs

521

Proof. The unary functions fa (a ∈ ) with 0 if x ∈ El , fa (x) := a otherwise, belong to P olk × El . Let now f n ∈ P olk El \P olk be arbitrary. Then there exist certain a1 , ..., an ∈ with f (a1 , ..., an ) =: α ∈ Ek \. Consequently, we have f∗ (x) := f (fa1 (x), ..., fan (x)) ∈ [P olk × El ∪ {f }], where f∗ (β) ∈ for every β ∈ \El . For every function g m ∈ P olk El , one ﬁnds a 2 · m-ary function hg with ⎧ g(x1 , ..., xm ) if (xm+1 , ..., x2·m ) = (f∗ (x1 ), ..., f∗ (xm )), ⎪ ⎪ ⎪ ⎨0 if (xm+1 , ..., x2·m ) ∈ Elm and hg (x1 , ..., x2·m ) := (xm+1 , ..., x2·m ) = (f∗ (x1 ), ..., f∗ (xm )), ⎪ ⎪ ⎪ ⎩ a otherwise (a ∈ ) in P olk × El . Consequently, g(x1 , ..., xm ) = hg (x1 , ..., xm , f∗ (x1 ), ..., f∗ (xm )) is a superposition over P olk × El ∪ {f }. Lemma 17.3.3 Let be a relation of Uk;El and let [a] be the equivalence class {x ∈ Ek | (a, x) ∈ }, a ∈ Ek . Then is valid: (1) fulﬁlls exactly one of the following conditions: (a) ∩ El2 = ι2l ∧ a∈El [a] = El ; (b) ∩ El2 = ι2l ∧ a∈El [a] = Ek ; (c) ∩ El2 = El2 ∧ a∈El [a] = El . (2) P olk × El is maximal in P olk El . Proof. (1) follows from the deﬁnition of the set Uk;El and from the fact that the conditions ∩ El2 = ι2l and a∈El [a] = Ek only of the relation Ek2 , which does not belong to Uk , are fulﬁlled. (2): Let := ∩ (Ek \El )2 and := ∩ (El × (Ek \El )). If fulﬁlls (a) or (b) of (1), then = ∅ and for arbitrary a := (a1 , a2 ) ∈ and b := (b1 , b2 ) ∈ we have that the function ⎧ x ∈ El , ⎨ 0 if x = a1 , fa;b (x) := b1 if ⎩ b2 otherwise, belongs to P olk × El . If fulﬁlls the condition (c), then one can ﬁnd exactly an element of El in each equivalence class of , through which, for every a := (a1 , a2 ) ∈ ∪ and all b ∈ , the function

522

17 Maximal Classes of P olk El for 2 ≤ l < k

⎧ x ∈ El ∧ [α] = [a1 ] ∧ α ∈ El , ⎨ α if x = a1 , fa;b (x) := b1 if ⎩ b2 otherwise,

preserves the relation . Let f ∈ P olk El \P olk be arbitrary. Then, certain unary functions ha with ha (a) ∈ for all a ∈ ∪ are superpositions over f and functions of the . We brieﬂy denote all functions of this form with h1 , ..., form fa;b or fa;b ht . If one considers a matrix of the type (k m , t · (m + 1)), whose rows are (in arbitrary sequence) of the form y := (x1 ..., xm , h1 (x1 )..., h1 (xm )..., ht (x1 )..., ht (xm )) (x1 ..., xm ∈ Ek ), then one sees that each two rows that are diﬀerent and do not consist only of elements of the set El , contain at least a column of Ek2 \. In analogous mode to the proof of the maximality of P olk in Pk (see Theorem 5.3.2) one can easily prove that, for every m-ary function g ∈ P olk El , there is a function hg ∈ P olk ×El with hg (y) = g(x1 ..., xm ), whereby [P olk × El ∪ {f }] = P olk El is valid. Analogous to the proof of Lemma 17.3.3 or analogous to the proof of similar statements about maximal classes (see Section 5.3) one can prove the following lemma. 2 [1] ∪ C2k;El then P olk × El is Lemma 17.3.4 If ∈ Sk;El ∪ Nk;El ∪ Ck;E l maximal in P olk El .

Lemma 17.3.5 (1) Let be a relation of ZC2k;El ; i.e., it holds that (a) ι2k ⊆ ⊂ Ek2 and is symmetric; (b) ∩ El2 = ι2l ; (c) ∃c ∈ Ek \El : Ek × {c} ⊆ ; (d) ∀x, y, z ∈ Ek : {(x, z), (y, z)} ⊆ =⇒ (x, y) ∈ . Then := ∩ (El × Ek ) belongs to Zk;El , i.e., has the following properties: (e) ⊂ El × Ek , pr0 = El and pr1 = Ek ; (f ) ∩ El2 = ι2l ; (g) ∃c ∈ Ek \El : El × {c} ⊆ . (2) P olk × El = P olk ∩ (El × Ek ) is valid for every ∈ ZC2k;El . Proof. (1) one can easily check. (2): Because of (1), we have P olk × El ⊆ P olk ∩ (El × Ek ) for every ∈ ZCk;El l . To prove the reverse inclusion, we show = (τ ) ◦ , where := ∩ (El × Ek ): Let (a, b) ∈ (τ ) ◦ . Then there exists a c ∈ El with (a, c) ∈ τ and (c, b) ∈ . Therefore, (a, b) ∈ because of (d) and the symmetry of . Thus we have (τ ) ◦ ⊆ .

17.3 Maximality Proofs

523

Let (a, b) ∈ . Because of (e) there exist certain xa , xb ∈ El with (xa , a) ∈ and (xb , b) ∈ . Moreover, because of (g) we have (xa , c) ∈ and (xb , c) ∈ , through what (xa , xb ) ∈ holds because of (d). However, this is only possible for xa = xb because of (b). Therefore we have (a, xa ) ∈ τ and (xa , b) ∈ through what (a, b) ∈ (τ ) ◦ holds. One can describe classes of the form P olk × El with ∈ ZCk;El through relations of the set Zk;El . Consequently, the P olk El -maximality of these classes surrenders through the following lemma. Lemma 17.3.6 Let ∈ Zk;El . Then P olk is maximal in P olk El . Proof. The following unary functions belong to P olk : α if x ∈ El , f(α,β) := β otherwise ((α, β) ∈ ),

⎧ x = c, ⎨ 0 if x ∈ (El \{c}) ∪ {d}, g(c,d) (x) := 1 if ⎩ c otherwise

((c, d) ∈ (El × Ek )\) and h(u,v) (x) :=

0 if u ∈ El ∪ {u}, w otherwise

((u, v) ∈ Ek2 ; u = v; v, w ∈ El and (0, w) ∈ ). Further, P olk has the following property: For every set T of t-tuples with ∀a := (a1 , ..., at ), b := (b1 , ..., bt ) ∈ T : a = b =⇒ ∃i ∈ {1, ..., n} : (ai , bi ) ∈ {(0, 1), (1, 0)}

(17.1)

and for every determination f (a) ∈ Ek for the t-tuples a of T , where a ∈ T ∩ Eln implies f (a) ∈ El , there is a certain determination for the remaining tuples x of Ekn , so that f belongs to P olk . If x ∈ Eln then one can set f (x) = w, where El × {w} ⊆ , with that f ∈ P olk . If x ∈ Elt , then (because of El2 ∩ = ι2k ) maximally a tuple of T can inﬂuence the value f (x), with which the value f (x) is also easily deﬁned in this case. Let now f n ∈ P olk El \P olk be arbitrary. Then there are certain (a1 , ..., an ), (b1 , ..., bn ) with (ai , bi ) ∈ for all i ∈ {1, ..., n} and (f (a1 , ..., an ), f (b1 , ..., bn )) ∈ . Thus a unary function f1 with c α := f1 ∈ (El × Ek )\, d β where (α, β) ∈ \ι2l is arbitrary, is a superposition over the functions f(αi ,βi ) and f .

524

17 Maximal Classes of P olk El for 2 ≤ l < k

Further, if one forms f2 (x) := g(c,d) (f1 (x)), then the function f2 has the property α 0 f2 = . β 1 Consequently, the unary functions of the form f2 h(u,v) belong to [P olk ∪ {f }], where {(f2 h(u,v) (u), (f2 h(u,v) (v)} = E2 for arbitrary u, v ∈ Ek with u = v and v ∈ El . Let {r1 , ..., rq } := {(u, v) ∈ Ek2 | u = v ∧ v ∈El }and denote sr1 ,..., srq 0 certain functions of [P olk ∪ {f }] with sri (ri ) = . Then 1 T := {(x1 , ..., xm , sr1 (x1 ), ..., sr1 (xm ), ..., srq (x1 ), ..., srq (xm )) | x1 , ..., xm ∈ Ek } is a set of tuples with the property (17.1). Thus, by our above considerations, for each m-ary function g ∈ P olk El there is a certain function hg ∈ P olk with g(x1 , ..., xm ) = hg (x1 , ..., xm , sr1 (x1 ), ..., srq (xm )), whereby [P olk ∪ {f }] = P olk El is proven. Lemma 17.3.7 For every ∈ Hk;El , P olk is a maximal class of P olk El . Proof. Let ∈ Hk;El . Then fulﬁlls the conditions 1)–6) of the deﬁnition of the elements of Hk;El , which we refer more often to in the following considerations. We begin with some easily proven auxiliary statements: 1) Let (a, b, c) ∈ (El2 × Ek )\ with c ∈ El . Then, because of 6), there exists a unary function fa,b,c ∈ P olk with ⎛ ⎞ ⎛ ⎞ a 0 fa,b,c ⎝ b ⎠ = ⎝ 1 ⎠ c 0 (for x ∈ Ek \(El ∪ {c}) one can set f (x) = u, where {u} ⊆ (see 4)); for E l × a 0 x ∈ El , the value f (x) is determined either by f = or is arbitrary b 1 of El ). 2) For every a ∈ Ekn , the n-ary function ga deﬁned by ⎧ x = a, ⎨ 1 if x ∈ Eln \{a}, ga (x1 , ..., xn ) := 0 if ⎩ u otherwise, where u fulﬁlls 4), belongs to P olk . 3 with c ∈ Ek \El and for each α := 3) For each α := (a, b, c) ∈ \δk;{0,1} (a , b , c ) ∈ there is a unary function tα;α ∈ P olk with ⎛ ⎞ ⎛ ⎞ a a tα;α ⎝ b ⎠ = ⎝ b ⎠ . c c

17.3 Maximality Proofs

525

Let now f n ∈ P olk El \P olk be arbitrary. Then there are certain a := (a1 , ..., an ), b := (b1 , ..., bn ), c := (c1 , ..., cn ) ∈ Ekn with (ai , bi , ci ) ∈ for i = 1, ..., n and ⎛ ⎞ ⎛ ⎞ a a f ⎝ b ⎠ = ⎝ b ⎠ ∈ . c c By replacing certain variables of f through functions deﬁned in 3), one obtains a unary function f with ⎛ ⎞ ⎛ ⎞ a a f ⎝ b ⎠ = ⎝ b ⎠ c c 3 for arbitrary (a, b, c) ∈ \δk;{0,1} with c ∈ El . If c ∈ Ek \El , then the function f := f(a ,b ,c ) f has the property ⎛ ⎞ ⎛ ⎞ a 0 f ⎝ b ⎠ = ⎝ 1 ⎠ . c 0

If c ∈ El , then w.l.o.g. we can assume (a , b , c ) = (0, 1, 0). Thus with the help of functions deﬁned in 2), we obtain functions of the form 1 if x = a, qan (x) := 0 otherwise, for every a ∈ Ekn , n ∈ N. Let now g m ∈ P olk El be arbitrary. To prove gm ∈ [P olk ∪ {f }], we set Ekm = {a1 , ..., akm } and consider the (m + k m )-tuples of the form yi := (ai , 0, ..., 0,

1 , 0, ..., 0) (m+1)-th place

for i = 1, ..., km . Then, because of 4) and 6), there is an (m + k m )-ary function hg ∈ P olk with hg (yi ) = g(ai ) for each i ∈ {1, ..., km }. Consequently, g is a superposition over {hg } ∪ {qa | a ∈ Ekm } and therefore also over P olk ∪ {f }. h Lemma 17.3.8 For every ∈ Ck;E [r] ∪ Chk;El (3 ≤ h ≤ k − 1, 1 ≤ r ≤ h − 1) l the clone P olk × El is maximal in P olk El . h Proof. First let ∈ Ck;E [r] and let f ∈ P olk El \P olk be arbitrary. l Every function f1 of P olk El with Im(f1 ) = {a1 , ..., ah } and (a1 , ..., ah ) ∈ belongs to P olk . Hence, for a certain b := (b1 , ..., bh ) ∈ (Elr × Eks )\, where s := h−r, each function of P olk El with the image {b1 , ..., bh } is a superposition m ∩ P olk El and let c ∈ El over P olk ∪ {f }. Let {h1 , h2 , ..., hq } := Pk;{b 1 ,...,bh } be a central element of . We choose an m-ary function g ∈ P olk El arbitrary

526

17 Maximal Classes of P olk El for 2 ≤ l < k

and put y := (x1 , ..., xm , h1 (x1 , ..., xm ), ..., hq (x1 , ..., xm )). Obviously, the (m · (q + 1))-ary function hg deﬁned by and hg (y) := g(x1 , ..., xm ) hg (y ) := c for the remaining tuples y preserves the relation . Thus the function g is a superposition over P olk ∪ {f }, whereby our Lemma is proven for (l; r)-central relations . Analogously, one can lead the proof for the central relations of Chk;El . h Lemma 17.3.9 For every ∈ Bk;E [2] with 3 ≤ h ≤ k − l + 2 the clone P olk l is maximal in P olk El .

Proof. Let f ∈ P olk El \P olk be arbitrary. To prove [P olk ∪{f }] = P olk El , we distinguish the following cases: Case 1: = γ2,1 . Each unary function of P olk El and each function f1 with Im(f1 ) = {a, b, c}, where (a, b, c) ∈ γ2,1 , preserves the relation γ2,1 . Thus each function of P olk El with the image {0, 1, l} is a superposition over P olk ∪ {f }. Let now g m ∈ P olk El be arbitrary. Set Elm = {a1 , ..., alm }. The (m + lm )-ary function hg with hg (x1 , ..., xm , 0, ..., 0,

1 , 0, ..., 0) := g(ai ) (i+m)-th place for all x1 , ..., xm ∈ El , i = 1, 2, .., lm , hg (y1 , ..., ym , l, l, ..., l) := g(y1 , ..., ym ) for all (y1 , ..., ym ) ∈ Ekm \Elm , m and hg (z) := 0 for the remaining tuples z ∈ Ekm+l

belongs to P olk . By replacing certain variables of the function fg through certain functions with the image {0, 1, l}, one obtains the function g; that is, g ∈ [P olk ∪ {f }] is proven. Case 2: = γ2,k−l and k − l ≥ 2. Clearly, each unary function of P olk El and each function h ∈ P olk El with |Im(h)| ≤ k − l + 1 preserves . Thus {g ∈ P olk El | |Im(g)| ≤ h − l + 2} ⊆ [P olk ∪ {f }]. Consequently, [P olk ∪ {f }] = P olk El for l = 2. If l ≥ 3, one can prove the P olk El -maximality of P olk with the help of the l-ary function t ∈ P olk deﬁned by ⎞ ⎞ ⎛ ⎛ 0 1 0 0 ... 0 ⎟ ⎜ ⎜ 0 1 0 ... 0 ⎟ ⎟ ⎜ 1 ⎟ ⎜ ⎜ . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎟ ⎜ ... ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 0 0 ... 1 ⎟ t⎜ ⎟=⎜ l−1 ⎟ ⎜ 0 ⎜ ⎟ ⎜ l l l ... l ⎟ ⎜ l ⎟ ⎟ ⎜ ⎝ . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎠ ⎝ ... ⎠ k−1 k − 1 k − 1 k − 1 ... k − 1 and t(x) = 0 otherwise.

17.3 Maximality Proofs

527

Case 3: γr,s ⊆ ⊂ El2 × Eks . If s = 1, one can similar to Case 1 prove [P olk ∪{f }] = P olk El , when w.l.o.g. one assumes El2 × {l} ⊆ and (0, 1) ∈ εl := {(a, b) | (a, b, l) ∈ } and if one changes the deﬁnition of the function hg as follows: m

∀(x, y) := (x1 , .., xm , y1 , ..., ylm ) ∈ Elm+l ∀i : ((∀j ∈ {1, .., lm }\{i} : (yj , 0) ∈ εl ) ∧ (yi , 1) ∈ εl ) =⇒ hg (x, y) := g(ai ). Let s ≥ 2. Because of assumption, for every b := (b1 , ..., bs ) ∈ (Ek \El )s \, we have that εb := {(x, y) | (x, y, b1 , ..., bs ) ∈ } is an equivalence relation on El with at least two equivalence classes. For tuples yi := (yi1 , ..., yit ) ∈ Elt (i = 1, 2), we deﬁne: y1 ∼εb y2 :⇐⇒ ∀j : (y1j , y2j ) ∈ εb . Since f does not preserve the relation , there are certain r1 , ...rn ∈ and an α := (a1 , a2 , b1 , ..., bs ) ∈ (El2 × Eks )\ with f (r1 , ..., rn ) = α. Hence, each function h ∈ P olk with Im(h) ⊆ {a1 , a2 , b1 , ..., bs } is a superposition over P olk ∪ {f }. Let the m-ary function g ∈ P olk El be arbitrary. Then the m · (q + 1)-ary function hg deﬁned by y := (x1 , ..., xm ), h1 (x1 , ..., xm ), ..., hq (x1 , ..., xm )) ({h1 , ..., hq } := {h ∈ (P olk El )m | Im(f ) = {a1 , a2 , b1 , ..., bs }}, hg (y) := g(x1 , ..., xm ), ∀y ∈ Elm : (y ∼εb y =⇒ hg (y ) = hg (y), h(y ) := 0 for the remaining tuples y preserves . Therefore, g(x) = hg (x, h1 (x), ..., hq (x)) ∈ [P olk ∪ {f }] holds. h Lemma 17.3.10 For every ∈ Bk;E [r] with 3 ≤ r ≤ l and 1 ≤ s := h − r ≤ l k − l, the clone P olk is maximal in P olk El .

Proof. Let f n ∈ P olk El \P olk be arbitrary. Then, there are some r1 , ..., rn ∈ and an α := (a1 , ..., ar , b1 , ..., bs ) ∈ (Elr × Eks )\ with f (r1 , ..., rn ) = α. Consequently, each function h ∈ P olk El with Im(h) ⊆ {a1 , ..., ar , b1 , ..., bs } is a superposition over P olk ∪ {f }. We distinguish the following two cases: Case 1: s = 1. Because of (a1 , ..., ar , b1 ) ∈ , the relation b1 := {(x1 , ..., xr ) | (x1 , ..., xr , b1 ) ∈ } belongs to Brl . First let (a1 , ..., ar ) := (0, 1, ..., r − 1), l := rq and (i)

(i)

b1 := {(x1 , ..., xr ) | ∀i ∈ Eq : (a1 , ..., ar ) ∈ ιrr }

528

17 Maximal Classes of P olk El for 2 ≤ l < k

(notations: see Section 5.2.6). Let the m-ary function g ∈ P olk El be arbitrary. Then the (q + m)-ary function hg deﬁned by (0)

q−1 yi+1 · hq−i hg (y1 , ..., yq , x1 , ..., xm ) := Σi=0

if (y1 , ..., ym , x1 , ..., xm ) ∈ Elq+m , hg (b1 , b1 , ..., b1 , x1 , ..., xm ) := g(x1 , ..., xm ) if (x1 , ..., xm ) ∈ Ekm \Elm and hg (z) := 0 for the remaining tuples z ∈ Ekq+m preserves . Moreover, the functions gj (j ∈ Eq ) deﬁned by (g(x1 , ..., xm ))(j) if (x1 , ..., xm ) ∈ Elm , gf (x1 , ..., xm ) := otherwise, b1 are superpositions over P olk ∪ {f }, since the images of these functions are subsets of {0, 1, ..., r −1, b1 }. Thus, because of g(x) = hg (gq−1 (x), ..., g0 (x), x), x := (x1 , ..., xm ), g is a superposition over P olk ∪ {f } in the above case. If b is a homomorphic inverse image of an r-ary elementary relation, then this case can be reduced to the above case, using the constructions from Section 5.2.6. Case 2: s ≥ 2. Since fulﬁlls the condition 6) of the deﬁnition of the elements of Bhk;El [r] in this case, an arbitrary function of P olk is a superposition over {f(a1 ,...,as ,b1 ,...bs ) } ∪ Pk;{a1 ,...,ar ,b1 ,...,bs } ⊆ [P olk ∪ {f }] (see also Lemma 1.4.6).

17.4 Some Lemmas It is subsequently shown in analog mode to corresponding statements about Pk that every maximal class A of Pk can be described with the aid of a relation A in the form P olk A . Lemma 17.4.1 P olk El has a generating system of functions of (P olk El )2 . Proof. For a, b ∈ Ek let ja;b be a unary function of Pk deﬁned by b if x = a, ja;b (x) := 0 otherwise. The function ja;b preserves El (i.e., ja;b ∈ P olk El ) iﬀ b ∈ El or a ∈ Ek \El . Also, the following functions belong to P olk El :

17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes

529

x ∨ y := max(x, y), x · y := min(x, y) (in respect to the total order 0 < 1 < 2 < ... < k − 1) and ra;b (x, y) := ja;b (x) · y. Then one can describe an arbitrary function f n ∈ P olk El as follows: f (x1 , ..., xn ) = ja1 ;f (a) (x1 ) · ... · jan ;f (a) (xn ) n a = (a1 , ..., an ) ∈ Ek where f (a) ∈ El , if (a1 , ..., an ) ∈ Eln . Consequently, {∨, ·} ∪ {ja;b | a ∈ Ek \El ∧ b ∈ El } ∪ {ra;b | a, b ∈ Ek } is a generating system for P olk El . The following lemma is a conclusion of the above lemma. Lemma 17.4.2 The lattice of the subclasses of P olk El is dual atomar, and P olk El has only ﬁnitely many maximal classes. Since one can easily prove that all maximal classes of P olk El are clones, the following is also valid: Lemma 17.4.3 For every maximal class M of P olk El there exists at most a k 2 -ary relation M with M = P olk M . Obviously, the only maximal classes of P olk El describable through relations from Rl are the classes of the form P olk with ∈ Rmax (Pl ). Possible descriptive relations for the maximal classes of P olk El are relations of the set Rmax (Pl ) ∪ Rmax (Pk ), which we will scrutinize in Section 17.6. We begin, however, by determining those maximal classes A of P olk El , which are not representable in the form P olk × El with ∈ Rmax (Pl ) ∪ Rmax (Pk ).

17.5 Not Through Relations of Rmax(Pl) ∪ Rmax(Pk) Describable Classes In this section, let A be an arbitrary maximal class of P olk El with prl A = Pl L↑k (A)

and

= {A, P olk El , Pk }.

Then, by Lemma 17.4.2, there is a relation ∈ Rkh with h ≤ k 2 and A = P olk El ∩ P olk . The assumptions above A and are equivalent to

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17 Maximal Classes of P olk El for 2 ≤ l < k

(I)

A ∈ {P olk γ ∩ P olk El | γ ∈ Rmax (Pl ) ∪ Rmax (Pk )}

(I )

No relation from the set Rmax (Pl ) ∪ Rmax (Pk ) is derivable from relations of {} ∪ Invk P olk El .

or

Since Rk1 \{El } ⊆ Rmax (Pk ), (I ) implies (I )

h ≥ 2 holds and no relation of the set C1k \{El } is derivable from .

W.l.o.g. we can assume the following properties of the relation :

(II)

⊆ Elr × Eks (r + s = h; 0 ≤ r ≤ h; 1 ≤ s ≤ h).

(III)

A is not describable through a relation ⊆ Elr × Ekh−r with r > r. In other words: Every relation ∈ [{} ∪ Invk P olk El ] with ⊆ Elr × Ekh−r and r > r belongs to Invk P olk El .

(IV )

Every h -ary relation ∈ [{} ∪ Invk P olk El ] with h < h belongs to Invk P olk El .

(V )

If possible, choose a symmetric relation for the description of A, i.e., for every permutation s ∈ Sh and

s := {(as(0) , as(1) , ..., as(h−1) ) | (a0 , ..., ah−1 ) ∈ } is valid: ∩ s ∈ {} ∪ Invk P olk El . (V I)

From the possible (l; r)-symmetric, h-ary relations , which describe A, the relation is that relation with the greatest cardinality, i.e., ∀ ⊆ Elr × Eks : (A = P olk El × ∧ ( is (l; r)-symmetric)) =⇒ | | ≤ ||.

The following is a conclusion from (II)–(IV):

17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes

(V II)

531

∀i ∈ {0, 1, ..., h − 1} : pr0,1,...,i−1,i+1,...,h−1 = pr0,1,...,i−1,i+1,...,h−1 Elr × Eks . (In particular, has no double rows.)

h h Lemma 17.5.1 ∩ δk;{0,1,...,h−1} = δl;{0,1,...,h−1} . h h h Proof. By (IV ) we have ∩ δk;{0,1,...,h−1} ∈ {∅, δl;{0,1,...,h−1} , δk;{0,1,...,h−1} }. h h ∩ δk;{0,1,...,h−1} = δk;{0,1,...,h−1} cannot occur, since A ⊂ P olk El and P olk El h does not contain all constant functions of Pk . If ∩ δk;{0,1,...,h−1} = ∅, then h+1 the -derivable relation := pr0,h (El × ) ∩ δk;{0,1,...,h−1} = {(x, y) ∈ El × Ek | (x, x, ..., x, y) ∈ El × } has the following properties: – ∩ ι2k = ∅; – ∩ El2 = ∅ =⇒ ∩ El2 ∈ Invk P olk El ; – ∩ El2 = ∅ =⇒ pr1 ∈ C1k \{El }. Thus because of our assumptions (I ) and (I ), the case just seen cannot h occur for ∩ δk;{0,...,h−1} . Hence our assertion is valid.

In the following, we distinguish two cases for , and we start with

Case 1: h = 2. Because of (I ) we have pr0 ∈ {El , Ek } and pr1 = Ek in this case. As is generally accepted, the relations σi () := {(a1 , ..., ai ) | ∃u ∈ Ek : {(u, a1 ), ..., (u, ai )} ⊆ } (i = 2, 3, ...) are -derivable for h = 2. Lemma 17.5.2 Let (τ ) ◦ = Ek2 . Then σi () = Eki for all i ≥ 2. Proof. We prove the statement by induction on i ≥ 2. For i = 2 the statement holds by assumption, since σ2 () = (τ ) ◦ . Assume, σi () = Eki , i ≥ 2. ⊆ σi+1 (). The case σi+1 () = Then, by deﬁnition of σi+1 (), we have ιi+1 k Eki+1 implies P olk ∩ P olk El ⊂ P olk σi+1 () ∈ {P olk El , Pk }, contrary to (I). Therefore, we have σi+1 () = Eki+1 . By Lemma 17.5.1, the -derivable relation ∩ (τ ) has the property ι2l = ι2k ∩ ∩ (τ ). Thus, by (V ), ∈ {ι2l , , El2 } holds; i.e., we must examine three cases for , and do so in the following three lemmas. Lemma 17.5.3 Let ∩ (τ ) = ι2l . Then there exists a -derivable relation γ ∈ Sk;El whose proper cycles have a length ≥ 3.

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17 Maximal Classes of P olk El for 2 ≤ l < k

Proof. First we form the -derivable relation := (τ ) ◦ = {(x, y) | ∃u ∈ Ek : {(u, x), (u, y)} ⊆ }. Because of pr1 = Ek , we have ι2k ⊆ and || ≥ k. The following two cases are possible: Case 1: = ι2k . If || > k or pr0 = El , there exist a, b, c ∈ Ek with (c, a), (c, b) ∈ and a = b, whereby (a, b) ∈ with a = b follows, contrary to our assumption in the ﬁrst case. Therefore || = k, and it holds that = {(x, s(x)) | x ∈ Ek } for a certain permutation s ∈ Sk \{id} with the ﬁxed points 0, 1, ..., l − 1 (Sk denotes the set of all permutations on Ek and id is the identity permutation). Because of ∩ (τ ) = ι2k , every proper cycle of has at least the length 3. If has proper cycles with diﬀerent lengths, and if r ≥ 3 is the smallest length of the proper cycles, then the relation ∆(( ◦ ◦ ... ◦ ) ∩ ι2k ) is a -derivable relation of C1k \{El }, contrary to (I ). Therefore, all proper cycles of have the same length q ≥ 3. The length q is a prime number, since for q = p · m (p prime, m ≥ 2), the -derivable relation ◦ ◦ ... ◦ m times has only proper cycles of the length p. Consequently, it is possible to derive from a relation of Sk;El , which has proper cycles of the length ≥ 3. Case 2: ι2k ⊂ . In this case, we have {c0 , c1 , ..., ck−1 } ⊆ P olk , whereby = Ek2 is valid because of (I ). With the aid of Lemma 17.5.2, this implies σk () = Ek2 . Therefore, has a central element u, i.e., {(u, x) | x ∈ Ek } ⊆ . Because of ∩ ι2k = ι2l , u belongs to El . Consequently, ι2l ⊂ ∩ El2 and hence (by(I )) ∩ El2 = El2 , contrary to our assumption ∩ (τ ) = ι2l . Thus Case 2 is not possible. Lemma 17.5.4 If ∩ (τ ) = , then either ∈ Nk;El or it is possible derive from a permutation of the set Sk;El , whose proper cycles have the same length 2. Proof. The symmetry of and pr1 = Ek imply pr0 = Ek . Consequently, ι2k ⊆ ◦ . Because of (I ), this is only possible if ◦ ∈ {ι2k , Ek2 }; i.e., we have to distinguish the following two cases. Case 1: ◦ = ι2k . Since ◦ = (τ ) ◦ , one can show (analogous to the proof of Lemma 17.5.3, Case 1) that there is a -derivable relation γ ∈ Sk;El . Because of the symmetry of , every proper cycle of has the length 2.

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Case 2: ◦ = Ek2 . Since ◦ = (τ ) ◦ , Lemma 17.5.2 implies σk () = Ek2 . Therefore, there is a u ∈ Ek with {(u, x) | x ∈ Ek } ⊆ . Because of ∩ ι2k = ι2l (see Lemma 17.5.1) u belongs to El . Hence ι2l ⊂ ∩ El2 , whereby El2 ⊆ (because of our assumptions in this section). Now, if one forms := ∩ (El × Ek ), then El2 ⊂ ⊆ El × Ek . Since the case ⊂ El × Ek is a contradiction to our assumption (III), we have El × Ek ⊆ and then (because of symmetry of ) ∈ Nk;El . 2 Lemma 17.5.5 If ∩ (τ ) = El2 , then r = 1 and ∈ Ck;E . l

Proof. Suppose, r = 0. By (III) and by ∩(τ ) = El2 , for 0 := ∩(El ×Ek ) and 1 := ∩(Ek ×El ) the following cases are to study (w.l.o.g.): 0 = 1 = El2 or 0 = El × Ek and 1 = El2 . In both cases, the -derivable relation i := ◦ ◦ ... ◦ for every i ∈ N has the property that i ∩ (Ek × (Ek × El )) = El2 i times holds. Furthermore, there is an i with ι2l ⊂ i ∩ ι2k . Consequently, if r = 0, then there is a -derivable relation that describes a class B with B ⊆ P olk El and B = Pk . This is, however, excluded from this section. Therefore, we can set r = 1 in the following; i.e., ⊆ El × Ek holds. If one forms the relation (τ ) ◦ , one can see that Ek × El ⊆ (τ ) ◦ and ι2k ⊆ (τ ) ◦ are valid. By our assumptions over , this is only possible if (τ ) ◦ = Ek2 holds. Because of Lemma 17.5.2, this implies that there is a u ∈ El with {(u, x) | x ∈ Ek } ⊆ . 2 . Consequently, ∈ Ck;E l Let’s summarize: In Case 1 our maximal class A has the form: 2 P olk with ∈ Sk;El ∪ Nk;El ∪ Ck;E . l

Case 2: h ≥ 3. Lemma 17.5.6 Let s ≥ 2 and Elh ⊆ . Then s−1

Elr × Eki × El × Eks−i−1 ⊆ .

i=0

Proof. First let r = 0, i.e., pri = Ek for every i ∈ Eh . Because of Lemma 17.5.1 and (V II), we have ∩ δk;{1,2,...,h−1} = δk;{1,2,...,h−1} [1] and therefore pri ( ∩ (El × Ekh−1 )) = Ek for every i ∈ {1, 2, ..., h − 1}. Consequently, by (III), ∩ (El × Ekh−1 ) can be only the relation El × Ekh−1 . Analogously, one can prove Eki × El × Ekh−i−1 ⊆ for i ∈ {1, ..., s − 1}. Thus our assertion is valid for r = 0. Let now r ≥ 1. Because of (V II), we have pri ( ∩ (Elr+1 × Ekh−r−1 )) = Ek for i ∈ {r + 1, ..., h − 1}. In analog mode, one can prove that our assertion for r ≥ 1 and h − r ≥ 2 follows.

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17 Maximal Classes of P olk El for 2 ≤ l < k

Lemma 17.5.7 Let h = 3. Then for is valid either (a) r = 2 and ∩ ι3k = δk;{0,1} [2] or (b) r ∈ {1, 2} and is (l; r)-reﬂexive, i.e., ι3k [r] := ι3k ∩ (Elr × Ek3−r ) ⊆ . Proof. We must study the following three cases: Case 1: r = 0. (V II) and Lemma 17.5.1 imply ∩ δk;{1,2} = δk;{0,1} [1].

(17.2)

Analogously, one can show δk;{0,1} (k, k, l) ∪ δk;{0,2} (k, l, k) ⊆ . Consequently, we have ι3l ⊆ ∩Elh , which can be possible because of (I ) only for ∩El3 = El3 . With the help of Lemma 17.5.6, this implies El × Ek2 ∪ Ek × El × Ek ∪ Ek2 × El ⊆ .

(17.3)

Now, if one forms the relation := ∆, then this relation has the properties El × Ek ∪ Ek × El ⊆ and

= Ek2

(by (17.3))

(by (17.2)),

whereby is no invariant of P olk El , contrary to (IV ). Therefore, the case r = 0 does not occur. Case 2: r = 1. Since by (V I) pr1,2 = Ek2 is valid, we have 3 3 ∩ E33 ∈ {El3 , δl;{0,1} , δl;{0,2} }.

(17.4)

In addition, as in Case 1, (17.2) is valid. Because of (17.2) and (17.4), the relation ∩ El3 does not have any double rows. Hence, by (I ), ∩ El3 = El3 follows. Then with the help of Lemma 17.5.6 and (17.2), it follows that ι3k [1] ⊆ . Case 3: r = 2. Because of pr1,2 = El × Ek , we have that (17.4) is valid and, in addition, that (17.5) δk;{0,1} [2] ⊆ holds. In analog mode to the considerations from Case 2, either ∩ El3 = El3 3 or ∩ El3 = δl;{0,1} results. If El3 ⊆ then, because of (17.5), the relation 3 3 then ∩ ι3k = δk;{0,1} [2] must be valid, is (l; 2)-reﬂexive. If ∩ El3 = δl;{0,1} whereby our Lemma is proven. Lemma 17.5.8 For h ≥ 4, ιhk ∩ (Elr × Eks ) ⊆ holds, i.e., is (l; r)-reﬂexive.

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Proof. One can lead the proof analogously to the proof of Lemma 6.1.8: Because of (V II), for arbitrary (a, b), (b, c), (c, d) ∈ Eh2 \ι2h with |{a, b, c, d}| = 4 the following implications are not valid: (h ≥ 4 ∧ (x0 , ..., xh−1 ) ∈ ∧ xa = xb ) =⇒ xb = xc (h ≥ 5 ∧ (x0 , ..., xh−1 ) ∈ ∧ xa = xb ) =⇒ xc = xd .

(17.6) (17.7)

Next we prove ∃(a, b) ∈ Eh2 \ι2h : δ{a,b} [r] ⊆ =⇒ (∀(c, d) ∈ Eh2 \ι2h : δ{c,d} [r] ⊆ ). (17.8) Let δ{a,b} [r] ⊆ ({a, b} ⊆ Ek , a = b), {b, c} ⊆ Ek and c ∈ {a, b}. To prove (17.8), it is suﬃcient to show that δ{b,c} [r] ⊆ is valid. Because of δ{a,b} [r] ⊆ , we have δ{a,b,c} [r] ⊆ . The assumption (IV ) implies that ∩ δ{b,c} [r] is El diagonal. Consequently, we have ∩ δ{b,c} [r] ∈ {δ{a,b,c} [r], δ{b,c} [r]}. The case ∩ δ{b,c} [r] = δ{a,b,c} [r] leads to an implication of the form (17.6). Therefore, ∩ δ{b,c} [r] = δ{b,c} [r] is only possible, whereby (17.8) is right. Hence, it remains to show that δ{a,b} [r] ⊆ holds. W.l.o.g. let (a, b) = (0, 1). Set 1 := ∩δ{0,1} [r]. Because of (V II) and (IV ), the relation 1 is a nonempty El -diagonal relation. Consequently, since implications (5) and (6) are false, we have: if h ≥ 5, = δ{0,1} [r] ∈ {δ{0,1},{2,3} [r], δ{0,1} [r]} if h = 4. Thus, because of (17.8), our Lemma were proven for h ≥ 5 or h = 4 and 1 = δ{0,1},{2,3} [r]. Hence only the case h = 4 and ∩ δ{a,b} [r] ∈ {δ{0,1},{2,3} [r], δ{0,2},{1,3} [r], δ{0,3},{1,2} [r]}.

(17.9)

must still be examined. Because of pr0,1,2 = pr0,1,2 Elr × Eks and because of (IV ) we have || > |Elr × Eks−1 |. Consequently, there are certain a0 , a1 , a2 , a3 , a3 ∈ Ek with (a0 , a1 , a2 , a3 ), (a0 , a1 , a2 , a3 ) ∈ and a3 = a3 .

(17.10)

Then, the relation 2 := {(x1 , x2 , x3 , x3 ) ∈ Elr × Ek4−s | ∃x0 ∈ Ek : (x0 , x1 , x2 , x3 ) ∈ ∧ (x0 , x1 , x2 , x3 ) ∈ }, which is derivable from and El , has the properties: 2 ∩ δ{2,3} [r] = δ{2,3} [r] (since pr1,2,3 = pr1,2,3 Elr × Ek4−r ), δ{2,3} [r] ⊂ 2 (because of (17.10)) and

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17 Maximal Classes of P olk El for 2 ≤ l < k

2 ⊂ Elr × Ek4−r (By assumption (IV) we have: (a0 , a1 , a1 , a3 ) ∈ =⇒ a0 = a3 . Consequently: ((a1 , a2 , a3 , a3 ) ∈ 2 ∧ a1 = a2 ) =⇒ a3 = a3 .) Thus is no invariant of P olk El . In addition, because of the above facts, 2 is a totally (l; r)-reﬂexive relation. Then, by deﬁnition of 2 , we have ∩δ{1,2} [r] = δ{0,3},{1,2} [r], contrary to (17.9). Because of Lemmas 17.5.7 and 17.5.8, the following three cases are possible for h ≥ 3: 3 3 Case 2.1: h = 3, ∩ El3 = δl;{0,1} and ∩ ι3k = δk;{0,1} [2].

The investigation of this case concludes with minimal expenditure, with the aid of the following lemma. Lemma 17.5.9 In Case 2.1, relation has the following properties: (a) ∃u ∈ Ek \El : El2 × {u} ⊆ ; (b) is (l; 2)-symmetric; (c) is strongly (l; 2)-homogeneous, i.e., ∀x, y ∈ El ∀z ∈ Ek : (∃α, β ∈ El : {(x, α, z), (β, y, z), (α, β, z)} ⊆ ) =⇒ (x, y, z) ∈ . Proof. (a): The relation γt := {(a1 , b1 , a2 , b2 , ..., at , bt ) ∈ El2·t | ∃u ∈ Ek : {(a1 , b1 , u), (a2 , b2 , u), ..., (at , bt , u)} ⊆ } 3 [2] ⊆ , we have is a -derivable relation. Because of δk;{0,1}

{(x1 , x1 , x2 , x2 , ..., xt , xt ) ∈ El2·t | x1 , ..., xt ∈ El } ⊆ γt . Further, {(y1 , y2 , y1 , y2 , ..., y1 , y2 ) ∈ El2·t | y1 , y2 ∈ El } ⊆ γt holds, since pr0,1 = El2 . Consequently, the relation γt does not have any double rows, which is possible according to the assumption only for γt = El2·t . For t = l · (l − 1), this implies that there exists a u ∈ Ek with Ek × {u} ⊆ . The element u belongs to Ek \El , since the opposite case u ∈ El supplies a 3 . contradiction to ∩ El3 = δl;{0,1} (b): Let := {(x, y, z) ∈ El2 × Ek | {(x, y, z), (y, x, z)} ⊆ }. Obviously, the 3 ∪ (El2 × {u}), which is not El -diagonal, is a subset of . By relation δl;{0,1} (V ) this is only possible for = , whereby is (l; 2)-symmetric. (c): If one forms the -derivable relation 1 := {(a1 , a2 , b) ∈ El2 × Ek | ∃x1 , x2 ∈ El : {(a1 , x1 , b), (x2 , a2 , b), (x1 , x2 , b)} ⊆ },

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then one can see that 1 is an (l; 2)-symmetric relation with ⊆ 1 because of (b). By (V I) this is only possible for 1 ∈ {, El2 ×Ek }. Suppose, 1 = El2 ×Ek . Then, in particular, (0, 1, 0) ∈ 1 \. This implies the existence of certain 3 x1 , x2 ∈ El with (0, x1 , 0), (x2 , 1, 0), (x1 , x2 , 0) ∈ . Because of ∩El3 = δl;{0,1} the ﬁrst two conditions are only fulﬁlled for x1 = 0 and x2 = 1, contrary to (0, 1, 0) ∈ . Consequently, 1 = and, therefore (c) is valid. Hence in Case 2.1 we have

3 . ∈ Hk;E l

Case 2.2.: h ≥ 3 and is (l; r)-reﬂexive. The following lemma summarizes some consequences from our assumptions in this case and from the properties obtained from . Lemma 17.5.10 (a) is (l; r)-symmetric. (b) Elh ⊆ . (c) For s ≥ 2, is a weakly (l; r)-central relation, i.e., s−1

Elr × Eki × El × Eks−i−1 ⊆ .

i=0

(d) r ≥ 1. Proof. (a): We consider the -derivable relation s , := s∈Sh [r]

where s := {(as(0) , as(1) , ..., as(h−1) ) | (a0 , a1 , ..., ah−1 ) ∈ }. By assumption, ιhk [r] ⊆ , where ιhk [r] is not El -diagonal. By (V ) this is only possible if = , whereby is (l; r)-symmetric. (b) follows from ιhl ⊆ and (I ). (c) follows from (b) and Lemma 17.5.6. (d): The case r = 0 (and therefore ιhk ⊆ ) cannot occur because of Lemma 17.5.1. Subsequently, we need the following property of set A more often. Lemma 17.5.11 It is not possible that the maximal class A (= P olk El × ) is describable with the aid of an h -ary totally (l; r)-reﬂexive and totally (l; r)symmetric relation with h > h. Proof. Let be an h -ary totally (l; r)-reﬂexive and totally (l; r)-symmetric relation, where h > h. Set A = P olk El × . Then A contains every function f ∈ P olk El with

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17 Maximal Classes of P olk El for 2 ≤ l < k

Im(f ) ⊆ T ∈ {{a1 , .., ar , b1 , ..., bh −r } | (a1 , ..., ar , b1 , ..., bh −r ) ∈ }. Then, because of h < h , it is not possible to describe A by a relation ⊆ Elr × Ekh−1 , contrary to our assumptions. Case r = 1 cannot occur, as shown by the following lemma: h Lemma 17.5.12 If r = 1 then ∈ Ck;E [1]. l

Proof. Because of Lemma 17.5.10,(a),(c) we must only show that has an (l; 1)-central element of El . The following relation is a -derivable relation: γt := {(a1 , ..., at ) ∈ El × Ekt−1 | ∃c ∈ El : ∀i1 , ..., ih−1 ∈ {1, ..., t} : (c, ai1 , ..., aih−1 ) ∈ }. First we show that γh = El × Ekh−1 holds. For this purpose, let (a1 , ..., ah ) be an arbitrary element of El × Ekh−1 . By (V II) for (a2 , ..., ah ) there is a certain α ∈ El with (α, a2 , ..., ah ) ∈ . Since is weakly (l, 1)-central because of Lemma 17.5.10,(c), we have (α, ai−1 , ..., aih−1 ) ∈ for all {i1 , ..., ih−1 } ⊆ {1, 2, ..., h} and 1 ∈ {i1 , ..., ih−1 }. Consequently, (a1 , ..., ah ) belongs to γh , whereby γh = El × Ekh−1 holds. Because of γh = El × Ekh−1 it is easy to prove that the relation γh+1 is (l; 1)-reﬂexive, which is valid only for γh+1 = El × Ekh because of Lemma 17.5.11. Therefore, by induction, we have γk = El × Ekk−1 . The existence of an (l; 1)-central element follows immediately from that. We can, therefore, always presuppose that r ≥ 2. By Lemma 17.5.10, the relation γr,s := {(a1 , ..., ar , b1 , ..., bs ) ∈ Elr × Eks | (∃i ∈ {1, ..., s} : bi ∈ El ) ∨ (a1 , ..., ar , b1 , ..., bs ) ∈ ιr+s k [r]} is a subset of . Next we clarify for which r, s the equation = γr,s is valid. Lemma 17.5.13 The relation γr,s has the following properties: (a) (4 ≤ r ≤ l ∧ 1 ≤ s ≤ k − l) =⇒ γr,s ∈ [{γr−1,s } ∪ Invk P olk El ]; (b) (2 ≤ r ≤ l ∧ 3 ≤ s ≤ k − l) =⇒ γr,s ∈ [{γr,s−1 } ∪ Invk P olk El ]; (c) (3 ≤ r ≤ l ∧ s = 1) =⇒ γl,1 ∈ [{γr,1 } ∪ Invk P olk El ]; (d) (r = 2 ∧ 2 ≤ s ≤ k − l) =⇒ γ2,k−l ∈ [{γ2,s } ∪ Invk P olk El ]; (e) (3 ≤ r ≤ l ∧ 2 ≤ s ≤ k − l) =⇒ γl,k−l ∈ [{γr,s } ∪ Invk P olk El ]; Proof. (a): Obviously, the (r + s)-ary relation α1 := {(a1 , ..., ar , b1 , ..., bs ) ∈ Elr × Eks | ∃u ∈ El : (a1 , a2 , ..., ar−2 , u, b1 , ..., bs ) ∈ γr−1,s ∧ (∀j1 , ..., jr−4 ∈ {1, 2, .., r − 2} : (u, aj1 , aj2 , ..., ajr−4 , ar−1 , ar , b1 , ..., bs ) ∈ γr−1,s )}

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539

is γr−1,s -derivable. Consequently, to prove (a), it suﬃces to show the equation α1 = γr,s : Let x := (a1 , ..., ar , b1 , ..., bs ) ∈ γr,s be arbitrary. The following table shows that γr,s ⊆ α1 . i 1 2 2.1 2.2 2.3 3

Case i for x ∃i ∈ {1, ..., s} : bi ∈ El ∃{i, j} ⊆ {1, ..., r} : i < j ∧ ai = aj {i, j} ⊆ {1, 2, ..., r − 2} i ∈ {1, 2, ..., r − 2} ∧ j ∈ {r − 1, r} i=r−1 ∧ j =r ∃{i, j} ⊆ {1, ..., s} : i < j ∧ bi = bj

u ∈ El such that x ∈ α1 u ∈ El u = ar−1 u = aj u = a1 u ∈ El

Suppose there is a tuple (x1 , ..., xr , y1 , ..., ys ) ∈ α1 \γr,s . Since the elements of this tuple are pairwise diﬀerent, an element u ∈ El , for which (x1 , x2 ..., xr−2 , u, y1 ...ys ) ∈ γr−1,s holds, must belong to the set {x1 ..., xr−2 }. Analogously, one can show that the remaining conditions of the deﬁnition of α1 are only fulﬁlled if u ∈ {xr−1 , xr }. Therefore, no element u that fulﬁlls all conditions from the deﬁnition of α1 exists, whereby α1 = γr,s is proven. (b): The relation α2 := {(a1 , ..., ar , b1 , ..., bs ) ∈ Elr × Eks | ∃u1 , u2 ∈ El : ∀i1 , ..., ir−2 ∈ {1, ..., r} : (u1 , a1 , ai1 , ..., air−2 , b1 , ..., bs−2 , bs−1 ) ∈ γr,s−1 ∧ (u2 , a2 , a11 , ..., air−2 , b1 , ..., bs−2 , bs ) ∈ γr,s−1 ∧ (u1 , u2 , ai1 , ..., air−2 , b2 , b3 , ..., bs−1 , bs ) ∈ γr,s−1 } is an γr,s−1 -derivable relation. Analogous to the proof of (a), one can show that α2 = γr,s . We only prove here that γr,s ⊆ α2 is valid. However, this follows from the following table, where x := (a1 , ..., ar , b1 , ..., bs ) ∈ γr,s is arbitrary. i 1 2 3 4 4.1 4.2 4.3 5

Case i for x ∃i ∈ {1, ..., s − 2} : bi ∈ El bs−1 ∈ El bs ∈ El ∃{i, j} ⊆ {1, ..., s} : i < j ∧ bi = bj {i, j} ⊆ {1, 2, ..., s − 2, s − 1} {i, j} ⊆ {1, 2, ..., s − 2, s} {i, j} ⊆ {2, 3, ..., s − 1, s} ∃{i, j} ⊆ {1, ..., r} : i < j ∧ ai = aj

u1 , u2 ∈ El such that x ∈ α1 u1 = u2 u2 = a2 u1 = a1 u1 u1 u1 u1

= u2 = a2 = u2 = a1 = a1 , u2 = a2 = a1 , u2 = a2

The statements (c)–(e) follow from (a) and (b). Then Lemma 17.5.13, (c)–(e) implies Lemma 17.5.14 If = γr,s then (r, s) ∈ {(2, 1), (l, 1), (2, k − l), (l, k − l)}.

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17 Maximal Classes of P olk El for 2 ≤ l < k

Note: As is generally known, every relation that is preserved from all unary functions of Pk , is representable through the union of certain diagonal relations on Ek . Then, as one can easily prove, every relation that is preserved from all unary functions of P olk El is representable through the union of certain El -diagonal relations on Ek . Consequently, because of Lemma 17.5.14 (and Lemmas 17.3.9 and 17.3.10) all maximal classes of P olk El , which all functions of (P olk El )1 contain and fulﬁll the conditions of this section and of Case 2, were determined. We say that an h-ary, totally reﬂexive and totally symmetric relation γ (⊆ Elr × Eks ) is strongly (l; r)-homogeneous, if the following is valid: ∀a0 , ..., ar−1 ∈ El ∀b0 , ..., bs−1 ∈ Ek : ( (∃v0 , ..., vr−1 ∈ El : (v0 , ..., vr−1 , b0 , ..., bs−1 ) ∈ ∧ (∀i ∈ Er ∀j ∈ Er \{i} : (a0 , ..., aj−1 , vi , aj+1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ )) =⇒ (a0 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ) Lemma 17.5.15 is either an (l; r)-central or a strong (l; r)-homogeneous relation. Proof. First, for t ∈ {r, r + 1, ..., l}, we consider the -derivable relation t := {(a0 , ..., at−1 , b0 , ..., bs−1 ) ∈ Elt × Eks | ∃c ∈ El : (∀i0 , ..., ir−2 ∈ Et : (ai0 , ..., air−2 , c, b0 , ..., bs−1 ) ∈ ))}. Since is totally (l; r)-symmetric, this relation is also totally (l; r)-symmetric. If one put c = a0 for t = r, then one can see that ⊆ r holds. Therefore, because of our assumption (V I), the following two cases are possible: Case 1: r = Elr × Eks . This case is only possible for r < l, since, in the opposite case, the deﬁnition of t implies = Ell × Eks . For r < l the -derivable relation r+1 is obvious totally (l; r + 1)-reﬂexive, whereby r+1 = Elr+1 × Eks follows with the aid of Lemma 17.5.11. One easily shows by induction that i = Eli × Eks is valid for h [r]. every i ∈ {r + 1, ..., l}. Then, l = Ell × Eks implies ∈ Ck;E l Case 2: = r . In this case, the relation is (l; r)-homogeneous; that is, it fulﬁlls the condition (∃v ∈ El : ∀i ∈ Er : (a0 , ..., ai−1 , v, ai+1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ) =⇒ (a0 , a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ for arbitrary a0 , a1 , ..., ar−1 ∈ El and arbitrary b0 , ..., bs−1 ∈ Ek . The total (l; r)-symmetric relation

17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes

541

γt := {(a0 , ..., at−1 , b0 , ..., bs−1 ) | ∃v0 , v1 , ..., vt−1 ∈ El : (∀i1 , ..., ir ∈ Et : (vi1 , ..., vir , b0 , ..., bs−1 ) ∈ ) ∧ (∀n ∈ Et ∀j1 , ..., jr−2 ∈ Et \{n} : (aj1 , ..., ajr−2 , an , vn , b0 , ..., bs−1 ) ∈ )} (t ∈ {r, r + 1, ..., l}) is -derivable and, if γt−1 = Elt−1 × Eks , totally (l; t)reﬂexive. In particular, for r = t we have ⊆ γr . (To see this, one chooses vα = aα in the deﬁnition of γr for every α ∈ Er .) And = γr means that is strongly (l; r)-homogeneous. Now, by assumption (V I), we have that γr ∈ {, Elr × Eks } holds. Thus our lemma would be proven, if we could show that the case γr = Elr × Eks does not occur. Suppose γr = Elr × Eks and r < l. Then γr+1 is totally (l; r + 1)-reﬂexive and therefore γr+1 = Elr+1 × Eks because of Lemma 17.5.11. Thus, one can prove by induction that γl = Ell × Eks . Therefore, for arbitrary a0 , ..., ar−1 ∈ El and b0 , ..., bs−1 ∈ Ek , there are certain v0 , ..., vr−1 ∈ El with (v0 , v1 , ..., vr−1 , b0 , ..., bs−1 ) ∈

(17.11)

and ∀n ∈ Er ∀α1 , ..., αh−2 ∈ El \{an } : (α1 , ..., αr−2 , an , vn , b0 , ..., bs−1 ) ∈ . (17.12) By induction we show that for arbitrary a0 , ..., ar−1 ∈ El and b0 , ..., bs−1 ∈ Ek is valid: ∀t ≥ 0 : (a0 , ..., at−1 , vt , vt+1 , ..., vr−1 , b0 , ..., bs−1 ) ∈ .

(17.13)

For t = 0, (17.13) holds because of (17.11). Assume (17.13) is valid for t = n. Then, for t = n + 1 the statement (17.13) follows from this assumption, from the (l; r)-homogeneousness of (considering (17.12)) and from the total (l; r)-symmetry of , when one chooses v = vn for the tuple (a0 , ..., an , vn+1 , ..., vr−1 , b0 , ..., bs−1 ). (17.13) implies (a0 , ..., ar−1 , b0 , ..., bs−1 ) ∈ for arbitrary a0 , ..., ar−1 ∈ El and b0 , ..., bs−1 ∈ Ek , contrary to ⊂ Elr × Eks . Consequently, our assumption γr = Elr × Eks is false and therefore is strongly (l; r)-homogeneous. To conclude our proof in this section, because of Lemma 17.5.15, it remains to show that if is strongly (l; r)-homogeneous the relation or a relation derivable from belongs to Bk;El [r]. Because of Lemma 17.5.10, we have only to prove conditions 5)–6) from the deﬁnition of an (l; r)-universal relation. Thus we can assume that is a strong (l; r)-homogeneous relation in the following. Lemma 17.5.16 For every b := (b0 , ..., bs−1 ) ∈ (Ek \El )s the relation εb := {(a, b) ∈ El2 | ∀a0 , ..., ar−3 ∈ El : (a0 , a1 , ..., ar−3 , a, b, b0 , ..., bs−1 ) ∈ } is an equivalence relation on El .

542

17 Maximal Classes of P olk El for 2 ≤ l < k

Proof. By the total (l; r)-reﬂexivity and total (l; r)-symmetry of , the reﬂexivity and symmetry of εb follows directly. To prove the transitivity of εb let {(a, b), (b, c)} ⊆ εb . Choosing v0 = α0 , v1 = α1 , ..., vr−3 = αr−3 , vr−2 = b and vr−1 = b in the deﬁnition of a strong (l; r)-homogeneous relation for the tuple (α0 , α1 , ..., αh−3 , a, c, b0 , ..., bs−1 ) we get (α0 , ...., αr−3 , a, c, b0 , ..., bs−1 ) ∈ for arbitrary α0 , ..., αr−3 ∈ El , whereby (a, c) ∈ εb . The following lemma is a direct consequence from Lemmas 17.5.10, 17.5.14, and 17.5.16: Lemma 17.5.17 If r = 2 then the strong (l; 2)-homogeneous relation beh longs to Bk;E [2]. l From the above lemma, we can assume that r≥3 in our further considerations. By the equivalence relation εb , the set El is partitioned into certain (nonempty) equivalence classes Ai [b] (i = 1, 2, ..., t[b]), from which we choose a representative αi [b]. With the help of the representative set Vb := {α1 [b], ..., αt[b] [b]}, we can deﬁne a mapping Fb : El −→ Vb by Fb (a) = αi [b] :⇐⇒ {a, αi [b]} ⊆ Ai [b]. Lemma 17.5.18 Let be strongly (l; r)-homogeneous. Then for every b := (b0 , ..., bs−1 ) ∈ (Ek \El )s \ιsk is valid: (a) (a0 , ...., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ (Fb (a0 ), ..., Fb (ar−1 ), b0 , ..., bs−1 ) ∈ ; (b) ((∀a0 , ..., ar−3 ∈ Vb : Vb ) =⇒ a = b.

(a0 , ..., ar−3 , a, b, b0 , ..., bs−1 ) ∈ ) ∧ {a, b} ⊆

Proof. (a): First we show that the equivalence (a, a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ (b, a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ (17.14) is valid for (a, b) ∈ εb and for arbitrary a0 , ..., ar−1 ∈ El . If (a, a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ , then (b, a1 , ...., ar−1 , b0 , ..., bs−1 ) ∈ follows from the strong (l; r)-homogeneousness of , choosing v0 = v1 = .... = vr−1 = a. Since εb is symmetric, we have also proven “⇐=” of (17.14). Now (a) follows from (ai , Fb (ai )) ∈ εb (i ∈ Er ), (17.14), Lemma 17.5.16 and of the total (l; r)-symmetry of :

17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes

543

(a0 , ...., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ ⇐⇒ (Fb (a0 ), a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ (Fb (a0 ), Fb (a1 ), a2 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ .. . (Fb (a0 ), ..., Fb (ar−1 ), b0 , ..., bs−1 ) ∈ .

⇐⇒

(b) is a conclusion from the deﬁnitions of εb and Vb and from (a). Now put ξb := Fb (b ) := {(Fb (a0 ), ..., Fb (ar−1 )) | (a0 , ..., ar−1 , b0 , ..., bs−1 ) ∈ }. By Lemma 17.5.18, (a) b is a homomorphic inverse image of this relation; i.e., it holds that b = {(a0 , ..., ar−1 ) ∈ Elr × Eks | (Fb (a0 ), ..., Fb (ar−1 )) ∈ ξb }. Because of = Elr ×Eks there is at least a b ∈ (Ek \El )s \ιsk with Elr ×{b} ⊆ . Thus w.l.o.g. we can assume Vb = Et[b] and (0, 1, ..., r − 1, b0 , ..., bs−1 ) ∈ (Vbr × {b})\ξb . To prove that has the properties 5) and (only for s ≥ 2) 6) from the deﬁnition of an (l; r)-universal relation, we consider the -derivable q-ary graphic Gq (P ol) := χq ∪ {g(κ1 , ..., κq ) | g ∈ (P ol)q }, where χq := (κ1 , ..., κq ) is the q-ary abscissa over Ek (in matrix form, see Section 2.7) and q ∈ N. Let m1 , ..., mi , n1 , ..., nj be the numbers of those rows of χri ·sj for which Ai,j := prm1 ,...,mi ,n1 ,...,nj χri ·sj is a matrix form of the relation Eri × {b0 , ..., bs−1 }j (r ≤ i ≤ l; s ≤ j ≤ k − l). Further, let µi,j := prm1 ,...,mi ,n1 ,...,nj Gri ·sj (P ol). Lemma 17.5.19 If is strongly (l; r)-homogeneous and (0, 1, 2, ..., r−1, b0 , ..., bs−1 ) ∈ , then µi,j = Eli × Ekj for all i, j with r ≤ i ≤ l and j = 1, if s = 1, and s ≤ j ≤ k − l. Proof. First we remark that = (Elr × Eks )\{(a0 , ..., ar−1 , b0 , ..., bs−1 ) | {a0 , ..., ar−1 } = Er }

544

17 Maximal Classes of P olk El for 2 ≤ l < k

h because of ∈ Ck;E [r]. Furthermore, if (β0 , ..., βr−1 ) ∈ , every function g l with Im(g) = {β0 , ..., βh−1 } belongs to P ol. Suppose the lemma is false for (i, j) = (r, s). Then we can form an h-ary relation with ⊂ ⊂ Elr × Eks with the help of the rr · ss -ary graphic of P ol by projection, contradicting (V I). Since all functions of P ol with at most ri · sj essential variables belong to i+1 j i j+1 (P ol)r ·s or (P ol)r ·s , it is easy to show by induction that the following holds:

∀i ∈ {r, ..., l − 1} : µi,j = Eli × Ekj =⇒ ιki+j+1 [i + 1] ⊆ µi+1,j and ∀i ∈ {s, ..., k − l − 1} : µi,j = Eli × Ekj =⇒ ιki+j+1 [i] ⊆ µi,j+1 , if s > 1. Consequently, our lemma follows from Lemma 17.5.11.

Let νs :=

rl if s = 1, rl · sk−l if s ≥ 2.

Because of Lemma 17.5.19, we can ﬁnd a νs -ary function fb ∈ P ol for s ≥ 2 with den properties Im(fb ) = Vb ∪ (Ek \El ) and

⎛

⎞ Fb (0) ⎜ Fb (1) ⎟ ⎜ ⎟ ⎜ ⎟ ... ⎜ ⎟ ⎜ Fb (l − 1) ⎟ ⎜ ⎟. fb (Al,k−l ) = ⎜ ⎟ l ⎜ ⎟ ⎜ l+1 ⎟ ⎜ ⎟ ⎝ ⎠ ... k−1

In the case that s = 1, an analogous statement is valid for the matrix Al,1 : ⎞ ⎛ Fb (0) ⎜ Fb (1) ⎟ ⎟ ⎜ ⎟. ... fb (Al,1 ) = ⎜ ⎟ ⎜ ⎝ Fb (l − 1) ⎠ b0 Therefore, fulﬁlls condition 6) of the deﬁnition of an (l; r)-universal relation. Let w0 , ..., ws−1 be certain rows of Al,k−l or Al,1 with f (wi ) = bi (i = 0, ..., s − 1).

17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes

545

Next we give some properties of the function fb on νs -ary tuples of Erνs , from h [r]. which will follow that ∈ Bk;E l We also give elements z of Erνs in the form (z[1], z[2], ..., z[νs ]) and, for every z ∈ Erνs , denote z t,a

(t ∈ {1, 2, ..., νs }, a ∈ Er )

an element of Erνs , which is deﬁned by a, if i = t, z t,a [i] := z[i] otherwise (i ∈ {1, 2, ..., νs }). Lemma 17.5.20 Let t ∈ {1, 2, ..., νs }, z ∈ Erνs and (fb (z t,0 ), ..., fb (z t,r−1 )) ∈ ξb . Then (a) fb (z t,0 ) = ... = fb (z t,r−1 ), (b) ∀w ∈ Erνs : fb (wt,0 ) = ... = fb (wt,r−1 ). Proof. (a): For proof, it is suﬃcient to show that w.l.o.g. fb (z t,r−2 ) = fb (z t,r−1 ). First we prove ∀β0 , ..., βr−3 ∈ Erνs : (fb (β0 ), ..., fb (βr−3 ), fb (z t,r−2 ), fb (z t,r−1 )) ∈ ξb . (17.15) If {β0 [t], β1 [t], ..., βr−3 [t]} = {0, 1, ..., r − 3}, then (17.15) holds, since in this case all columns of the matrix ⎞ ⎛ β0 ⎜ β1 ⎟ ⎟ ⎜ ⎜ ... ⎟ ⎟ ⎜ ⎜ βr−3 ⎟ ⎜ t,r−2 ⎟ ⎟ B := ⎜ ⎜ z t,r−1 ⎟ ⎟ ⎜z ⎟ ⎜ ⎜ w0 ⎟ ⎟ ⎜ ⎝ ... ⎠ ws−1 belong to ιhk [r] and fb preserves the relation . Let w.l.o.g. βi [t] = i for i ∈ Er−2 . Substituting the i-th row of B by z t,j , where i = j, we obtain a matrix Bi,j the columns of which belong to ιhk [r], whereby fb (Bi,j ) ∈ ξb × {b} holds. Consequently, by the strong (l; r)-homogeneousness of , choosing vj = fb (z t,j ) (j ∈ Er ) for the tuple (fb (β0 ), ..., fb (βr−3 ), fb (z t,r−2 ), fb (z t,r−1 ), b0 , ..., bs−1 ), we get (17.15). Since {fb (x) | x ∈ Erνs } = Vb , there are certain β0 , ..., βr−3 ∈ Erνs with fb (βi ) = ai (i ∈ Er−2 ) for arbitrary a0 , ..., ar−3 ∈ Vb . Consequently, by (17.15) we have

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17 Maximal Classes of P olk El for 2 ≤ l < k

∀a0 , ..., ar−2 ∈ Vb : (a0 , ..., ar−3 , fb (z t,r−2 ), fb (z t,r−1 ), b0 , ..., bs−1 ) ∈ ξb × {b}. Because of Lemma 17.5.18, (b), this implies fb (z t,r−2 ) = fb (z t,r−1 ). (b): Because of (a), it is suﬃcient to show that (fb (wt,0 ), ...., fb (wt,r−1 )) ∈ ξb

(17.16)

holds. (17.15) implies (fb (wt,0 ), ..., fb (wt,r−3 ), fb (z t,r−2 ), fb (z t,r−1 )) ∈ ξb . Now it is easy to check that any exchange of an i-th row in ⎛ ⎞ wt,0 ⎜ ... ⎟ ⎜ t,r−1 ⎟ ⎜w ⎟ ⎜ ⎟ ⎜ w0 ⎟ ⎜ ⎟ ⎝ ... ⎠ ws−1 by wt,j for i ∈ Er−3 or by z t,j for i ∈ {r − 2, r − 1} and i = j gives a matrix M, whose columns belong to ιhk [r] and it holds f (M) ∈ ξb × {b}. Hence, by the strong (l; r)-homogeneousness of , choosing v0 = fb (wt,0 ), ..., vr−3 = fb (wt,r−3 ), vr−2 = fb (z t,r−2 ), vr−1 = fb (z t,r−1 ), we get (17.16). Lemma 17.5.21 For the function fb , there are certain digits t1 , ..., tm[b] that are exactly the essential digits of (fb )|Erνs (restriction of fb to Erνs ); i.e., it holds that ∀z, w ∈ Erνs : fb (z) = fb (w) ⇐⇒ ∀i ∈ {t1 , ..., tm[b] } : z[i] = w[i]. (17.17) Furthermore, f has the properties |Im(fb )| = rm[b] and ∀r1 , ..., rνs ∈ Err × {b} : ({rt1 , ..., rtm[b] } ⊆ ιhk [r] =⇒ fb (r1 , ..., rνs ) ∈ ) (17.18) Proof. For every t ∈ {1, 2, ..., νs }, we have either fb (z t,0 ) = fb (z t,1 ) = ... = fb (z t,r−1 ) for every z ∈ Erνs or (fb (z t,0 ), fb (z t,1 ), ..., fb (z t,r−1 ), b0 , ..., bs−1 ) ∈

17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes

547

for all z ∈ Erνs by Lemma 17.5.20. Let T := {t1 , ..., tm[b] } be the set of all t ∈ {1, 2, ..., νs }, for which (fb (z t,0 ), ..., fb (z t,r−1 ), b0 , ..., bs−1 ) ∈ holds. Now we will show that the digits ti ∈ T of fb have the properties of Lemma 17.5.21. First let fb (z) = fb (w) for certain z, w ∈ Erνs and assume there exists an i ∈ T with α := z[i] = w[i]. Then the columns of the matrix ⎛ i,0 ⎞ z ⎜ ... ⎟ ⎜ i,α−1 ⎟ ⎟ ⎜z ⎟ ⎜ ⎜ w ⎟ ⎜ i,α+1 ⎟ ⎟ ⎜z ⎟ A := ⎜ ⎜ ... ⎟ ⎜ i,r−1 ⎟ ⎟ ⎜z ⎟ ⎜ ⎜ w0 ⎟ ⎟ ⎜ ⎝ ... ⎠ ws−1 belong to ιhk [r], whereby fb (A) ∈ ξb × {b} holds. Because of fb (w) = fb (z) = fb (z i,α ) follows (fb (z i,0 ), ..., fb (z i,r−1 ), b0 , ..., bs−1 ) ∈ ξb , contrary to i ∈ T and the deﬁnition of T . Therefore “=⇒” in (17.17) holds. Let now z[i] = w[i] for every i ∈ T and w.l.o.g. T = {1, 2, ..., m[b]}. fb (w) = fb (z) is proven if we can show that fb (un−1 ) = fb (un )

(17.19)

holds for every tuple un := (z[1], ..., z[n], w[n + 1], w[n + 2], ..., w[νs ]) and all n ∈ {m + 1, m + 2, ..., νs }, since fb (um ) = fb (w) and fb (uνs ) = fb (z). n,j By n > m[b] and the deﬁnition of T , we have fb (un,i n−1 ) = fb (un−1 ) for arbin,w[n] n,z[n] trary i, j ∈ Er . As un−1 = un−1 and un−1 = un , we have (17.19), whereby (17.17) is proven. From (17.17) and {fb (x) | x ∈ Erνs } = Vb follows |Vb | = rm[b] . Finally we prove (17.18). Assume (17.18) is false. Then there are certain z0 , z1 , ..., zr−1 ∈ Erνs with (fb (z0 ), fb (z1 ), ..., fb (zr−1 ), b0 , ..., bs−1 ) ∈ ξb × {b} and (w.l.o.g.) zi [1] = i for every i ∈ Er and 1 ∈ T . Let w, z ∈ Erνs be arbitrary. Then each column of the matrix

548

17 Maximal Classes of P olk El for 2 ≤ l < k

⎛

Ci,j

w1,0 ⎜ ... ⎜ 1,j−1 ⎜w ⎜ 1,i ⎜ z ⎜ 1,j+1 ⎜w := ⎜ ⎜ ... ⎜ 1,r−1 ⎜w ⎜ ⎜ w0 ⎜ ⎝ ... ws−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

belongs to ιhk [r] for i = j. Thus fb (Ci,j ) ∈ ξb × {b}. By choosing vi = fb (z 1,i ) for the tuple (fb (w1,0 ), ..., fb (w1,r−1 ), b0 , ..., bs−1 ) and i ∈ Er , (fb (w1,0 ), ..., fb (w1,r−1 ), b0 , ..., bs−1 ) ∈ ξb × {b} follows from this and the strong (l; r)-homogeneousness of , contrary to the deﬁnition of T and 1 ∈ T . Lemma 17.5.22 Let be strongly (l; r)-homogeneous, b ∈ (Ek \El )s \ιsk and Elr × {b} ⊆ . Then for certain m[b] ∈ N there is a bijective mapping ϕb from Erm[b] onto Vb with ξb = ϕb (ξm[b] ) := {(ϕb (a0 ), ..., ϕb (ar−1 )) | (a0 , ..., ar−1 ) ∈ ξm[b] } (ξm[b] denotes an r–ary elementary relation) and it holds that −1 b = {(a0 , ..., ar−1 ) ∈ Elr | (ϕ−1 b (F (a0 )), ..., ϕb (F (ar−1 ))) ∈ ξm[b] }; h i.e., ∈ Bk;E [r]. l

Proof. Subsequently, we denote m[b] with m. Because of Lemma 17.5.21, a bijective mapping from Erm onto Vb is deﬁned by ϕb (a(m−1) hm−1 + a(m−2) hm−2 + ... + a(1) h + a(0) ) = fb (z) :⇐⇒ (z[t1 ], z[t2 ], ..., z[tm ]) = (a(m−1) , a(m−2) , ..., a(0) ), where a ∈ Erm . Let now (a0 , ..., ar−1 ) ∈ ξm . Then there are certain z0 , ..., zr−1 ∈ Erνs with the following properties: (z0 [i], ..., zr−1 [i]) ∈ ιrr for every i ∈ {1, 2, ..., νs } and (ϕb (a0 ), ϕb (a1 ), ..., ϕb (ar−1 )) = (fb (z0 ), fb (z1 ), ..., fb (zr−1 )). Then, because of fb ∈ P ol, we have (ϕb (a0 ), ..., ϕb (ar−1 ), b0 , ..., bs−1 ) ∈ ξb × {b} and therefore ϕb (ξm ) ⊆ ξb . Suppose there exists

17.6 Classes Describable by Relations of Rmax (Pl ) ∪ Rmax (Pk )

549

(a0 , ..., ar−1 ) ∈ ξb \ϕb (ξm ). Then, by deﬁnition of ϕb , there exist certain z0 , ..., zr−1 ∈ Erνs and an i ∈ {t1 , ..., tm } with (fb (z0 ), ..., fb (zr−1 )) = (a0 , ..., ar−1 ) and (z0 [i], ..., zr−1 [i]) ∈ ιrr . But this is contrary to (17.18). Thus ξb = ϕb (ξm ). The remaining assertions follow from Lemma 17.5.18, (a) and the deﬁnition of ξb . Summing up, we obtain in Case 2.2, h h ∈ Ck;E [r] ∪ Bk;E . l l

17.6 Classes Describable by Relations of Rmax(Pl) ∪ Rmax(Pk) The aim of this section is to prove the following: Theorem 17.6.1 Let be an h-ary relation of Rmax (Pl ) ∪ Rmax (Pk ). Then P olk × El is a maximal class of P olk El if and only if ∈ Rmax (Pl ) ∪ Uk;El ∪ Ck;El . With the help of Sections 5.2 and 5.4, one can easily prove the following lemma: Lemma 17.6.2 It holds: (a) ∀ ∈ Rmax (Pk )\Rmax (Pl ) : prEl (P olk × El ) = Pl =⇒ P olk × El is not maximal in P olk El . (b) ∀, ∈ Rmax (Pk ) : P olk = P olk ∧ prEl P olk × El = prEl P olk × El = Pl =⇒ P olk × El = P olk × El . Lemma 17.6.3 For every relation ∈ Mk ∪ Sk ∪ Lk , there is a certain -derivable relation which belongs to Rmax (Pl ) ∪ Ck;El . Proof. Let be an arbitrary h-ary relation of Mk ∪ Sk ∪ Lk . Then the following relations are -derivable: := ∩ Elh and El := pr1,...,h−1 (∆(El × )). We distinguish two cases for : Case 1: ∈ Mk ∪ Sk . Then the following three cases are possible for the relation : Case 1.1: = ∅. This case is possible only for ∈ Sk and the relation El ( = {x ∈ Ek | ∃a ∈ El : (a, x) ∈ } ) belongs to C1k;El , since (by = ∅) (a, b) ∈ for all a ∈ El implies b ∈ El .

550

17 Maximal Classes of P olk El for 2 ≤ l < k

Case 1.2: ∅ ⊂ ⊂ ι2l or ι2l ⊂ ⊂ El2 . Then, ∈ Rl \Dl , and because of Chapter 6, one can derive a relation of Rmax (Pl ) from the relations of { } ∪ Dl . Case 1.3: = ι2l . This case is possible only for ∈ Mk . Then, all elements of El are incomparable in respect to the partial order relation , and both the smallest element o and the greatest element e of Ek (in respect to ) belong to the set Ek \El . Since e ∈ El and o ∈ El , we have El ⊆ El and El ⊂ El ⊂ Ek for the above deﬁned relation El , obviously. Consequently, El ∈ C1k;El and our assertion also holds in Case 1.3. Case 2: ∈ Lk . In this case, k = pm (p prime, m ≥ 1) and = {(a, b, c, d) ∈ Ek4 | a + b = c + d}, where (Ek ; +) is an elementar Abelean p-group. W.l.o.g. (see Lemma 5.2.4.2) we can assume that the element 0 ∈ El is the neutral element of (Ek ; +). Then the relation is not El -diagonal because of {(a, 0, 0, a), (a, 0, a, 0), (0, a, 0, a), (0, a, a, 0) | a ∈ El } ⊆ ⊆ El4 and (0, 1, 1, 1) ∈ . Consequently, by Chapter 6, one can derive a relation of Rmax (Pl ) from relations of {} ∪ Dl . Lemma 17.6.4 For every ∈ Bhk (3 ≤ h ≤ k) the clone P olk × El is not maximal in P olk El . Proof. Let ∈ Bhk . We consider the -derivable relations r := ∩ (Elr × Ekh−r ) (0 ≤ r ≤ h). Since ιhk ⊆ and h ≥ 3, the sets ιhl and {(x1 , ..., xh ) ∈ Ekh | |{x1 , ..., xh }| ≤ 2} are subsets of . Consequently, the relations r do not have any double rows for every r ∈ {0, 1, ..., h}, and it holds that ∀i ∈ Eh : pri r = pri Elr × Ekh−r . Therefore, r ∈ Invk P olk El implies r = Elr × Ekh−r . Let r∗ be the smallest number, for which is valid r∗ ∈ Invk P olk El and r ∈ Invk P olk El for all r < r∗ . The following cases are possible: Case 1: r∗ = 0. Because of the above remarks, we have 1 = El × Ekh−1 ⊆ , whereby every element c ∈ El is a central element of the relation , which cannot, however, be for the relation ∈ Bk . Case 2: r∗ = 1. In this case, the relation 1 is totally (l; 1)-reﬂexive, totally (l; 1)-symmetric, weakly (l; 1)-central and diﬀerent from El × Ekh−1 . In addition, we can assume that for every (a2 , ..., ah ) ∈ Ekh−1 there exists a c ∈ El with (c, a2 , ..., ah ) ∈ 1 .

17.6 Classes Describable by Relations of Rmax (Pl ) ∪ Rmax (Pk )

551

Namely, if this is false, we can derive a relation of Ck by forming pr1,...,h−1 1 , whereby P olk ×El is not a maximal class of P olk El because of Lemma 17.6.2. Let γt := {(a1 , ..., at ) ∈ El × Ekt−1 | ∃c ∈ El : ∀i2 , ..., ih ∈ {1, 2, ..., t} : (c, ai2 , ..., aih ) ∈ 1 }. This relation has the property (γt = El ×Ekt−1 ∧ t ≥ h) =⇒ γt+1 = El ×Ekt ∨ (∃ ∈ Ck : γt ) (17.20) as one can prove: Because of γt = El × Ekt−1 and the above properties of 1 , the relation γt+1 is strongly (l; 1)-reﬂexive, strongly (l; 1)-symmetric and := pr1,...,t+1 γt+1 = Ekt , then γt+1 ∈ Ck . weakly (l; 1)-central. Thus, if γt+1 t t If γt+1 = Ek , then for every (a2 , ..., at+1 ) ∈ Ek there is an α ∈ El with (α, a2 , ..., at+1 ) ∈ γt+1 and we can show γt+1 = El × Ekt , as follows: Let (a1 , a2 , ..., at+1 ) ∈ El × Ekt . Then there exists an α ∈ El with (α, a2 , ..., at+1 ) ∈ γt+1 . Because of the deﬁnition of γt+1 , this implies ∃c ∈ El ∀i2 , ..., ih ∈ {2, ..., t + 1} : (c, ai2 , ...aih ) ∈ 1 . In addition, we have ∀i2 , ..., ih ∈ {1, 2, ..., t+1} : (∃q ∈ {2, ..., h} : iq = 1) =⇒ (c, ai2 , ..., aih ) ∈ 1 , since 1 is a weakly (l; 1)-central relation. Therefore (a1 , a2 , ..., at+1 ) ∈ γt+1 and γt+1 = El × Ekt is proven, i.e., (17.20) holds. Analogous to the just carried out considerations or to the proof of Lemma 17.5.12, one can show γh = El × Ekh−1 . From that and from (17.20), it follows that a relation of Ck is -derivable. Consequently, P olk × El is not maximal in P olk El . Case 3: r∗ ≥ 2. We consider the (proper subclass) A := P olk r∗ of P olk El , where P olk ×El ⊆ A holds, since r∗ is -derivable. Next we show that P olk × El = A (and with that, our assertion) through the construction of a function f ∈ A, which does not preserve. For this purpose, let T be a matrix, which is a matrix representation of the relation \Elh . Then, the |T |-ary function f deﬁned by f (T ) = α for certain α ∈ Ekh \ and |T | f (x) = 0 for the remaining tuples x ∈ Ek preserves r∗ , but does not preserve . Lemma 17.6.5 Let ∈ Uk . Then the class P olk × El is P olk El -maximal if and only if ∈ Uk;El . Proof. Let [a] := {x ∈ Ek | (a, x) ∈ }. If ∈Uk;El , then we have either := ∩ El2 ∈ {ι2l , El2 } (and therefore ∈ Ul ) or a∈El [a] ∈ {El , Ek }. In the ﬁrst case a relation of Ul ⊆ Rmax (Pl ) is -derivable. In the second case, the -derivable relation El := pr1 (∆(El × )) = {x ∈ Ek | ∃a ∈ El : (a, x) ∈ } belongs to C1k;El . Our assertion results from that and from Lemma 17.3.3.

552

17 Maximal Classes of P olk El for 2 ≤ l < k

Lemma 17.6.6 Let ∈ Ck . Then, P olk × El is P olk El -maximal if and only if ∈ C1l ∪ Ck;El . Proof. If is a unary relation and ∈ C1l ∪C1k;El , then the -derivable relation ∩ El belongs to C1l ∪ C1k;El . Therefore, our assertion for unary relation follows from Lemmas 17.3.1 and 17.3.2. Let now ∈ Chk with 2 ≤ h ≤ k − 1 and let P olk × El be P olk El -maximal, where it is not possible to describe the class P olk × El with the aid of a relation of Rl \Dl . Because of Lemmas 17.3.5, and 17.3.6–17.3.8 it is suﬃcient proof to show that ∈ Ck;El is valid. Obviously, ∩ El2 ∈ {ι2l , El2 } if h = 2 and Elh ⊆ if h ≥ 3 result from our assumptions about the relation . Then the following cases are possible: Case 1: h = 2 and ∩ El2 = ι2l . Then every central element of belongs to Ek \El . We form 1 := ∩ (El × Ek ). This relation has the following properties: 1 ∩ El2 = ι2l , pr0 1 = El , 1 ⊂ El × Ek , and ∃c ∈ Ek \El : El × {c} ⊆ 1 {c} ∪ El ⊆ pr1 1 . If pr1 1 = Ek , then one can derive a relation of C1k;El from 1 and therefore from . Consequently, the -derivable relation 1 belongs to Zk;El . If there are a, c ∈ Ek \El and b ∈ El with {(a, b), (b, c)} ⊆ and (a, c) ∈ , then the binary function f deﬁned by ⎧ (x, y) = (a, b), ⎨ a if (x, y) = (b, c), f (x, y) := c if ⎩ b otherwise, preserves 1 , but does not belong to P olk ×El . Since, by assumption, P olk × El is P olk El -maximal, ∀a, c ∈ Ek ∀b ∈ El : {(a, b), (b, c)} ⊆ =⇒ (a, c) ∈ results. Hence ∈ ZC2k;El and our lemma follows from Lemmas 17.3.5 and 17.3.6. Case 2: h ≥ 2 and Elh ⊆ . Case 2.1: has a central element c which belongs to El . For the -derivable relation := ∩ (El × Ekh−1 ) the following two cases are possible: Case 2.1.1: = El × Ekh−1 . Then, ∈ Invk P olk El and it holds

17.6 Classes Describable by Relations of Rmax (Pl ) ∪ Rmax (Pk )

553

P olk × El ⊂ P olk ⊂ P olk El , since one can easily check that the h-ary function f with ⎛ ⎞ ⎛ ⎞ c l l ... l a1 ⎜ l c l ... l ⎟ ⎜ a2 ⎟ ⎟ ⎜ ⎟ f⎜ ⎝ . . . . . . . . . . ⎠ = ⎝ ... ⎠ ∈ l l l ... c ah and f (x) = c for the remaining tuples x of Ekh preserves the relation , however f does not preserve the relation . Case 2.1.2: = El × Ekh−1 . In this case, every element of El is a central element of and belongs to Ck;El . Case 2.2: Every central element of belongs to Ek \El . If 1 = El2 × Ekh−2 then the -derivable relation 1 := ∩ (El2 × Ekh−2 ) is not an invariant of P olk El and P olk × El ⊂ P olk 1 ⊂ P olk El , holds, since the h-ary function f with ⎛ ⎛ ⎞ ⎞ c 0 0 ... 0 a1 ⎜ l c 0 ... 0 ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎟ f⎜ ⎝ . . . . . . . . . . . ⎠ := ⎝ ... ⎠ ∈ l 0 0 ... c ah (c is a central element von ) f (x) := 0 for x ∈ Elh and f (x) := c for the remaining tuples x belongs to P olk 1 \P olk . Therefore, we can assume that El2 × Ekh−2 ⊆ or {(a1 , ..., ah ) ∈ Ekh | ∃i = j : {ai , aj } ⊆ El } ⊆ .

(17.21)

We consider the relation h := {(a0 , ..., ah−1 ) | ∃u ∈ El : ∀i ∈ Eh : (a0 , ..., ai−1 , u, ai+1 , ..., ah−1 ) ∈ }, which is totally symmetric. W.l.o.g. we can assume that for arbitrary a := (a1 , a3 , ..., ah ) ∈ Ekh−1 there is a ua ∈ El with (ua , a1 , a3 , ..., ah ) ∈ . Namely, if this is not valid, we have that the -derivable relation := pr1,...,h−1 ((El × Ekh−1 ) ∩ ) ; i.e., El is the set of all central elements of . belongs to Ch−1 k Because of symmetry of we have in addition (a1 , ua , a3 , ..., ah ) ∈ . Then,

554

17 Maximal Classes of P olk El for 2 ≤ l < k

with the total reﬂexivity of , (a1 , a1 , a3 , ..., ah ) ∈ h holds. Thus h is totally reﬂexive. Next we show that each element u ∈ El is a central element of h : Let (α, a2 , ..., ah ) ∈ El × Ekh−1 be arbitrary. Then, as already shown, there is a u ∈ El with (u, a2 , ..., ah ) ∈ . In addition, because of (17.21), we have (α, a2 , ..., ai−1 , u, ai+1 , ..., ah ) ∈ for each i ∈ {2, ..., h}. By deﬁnition of h , this implies (α, a2 , ..., ah ) ∈ h . Therefore, h is either a central relation with central elements of El or h = Ekh holds. Because of Lemma 17.6.2, we must continue to examine the case h = Ekh . We form the relation t := {(a1 , ..., at ) | ∃u ∈ El : ∀{i1 , ..., ih−1 } ⊆ {1, ..., t} : (u, ai1 , ..., aih−1 ) ∈ } (h + 1 ≤ t ≤ k). Then, t−1 = Ekt−1 implies ιtk ⊆ t (h + 1 ≤ t ≤ k). Therefore, we have either t = Ekt or t ∈ Invk P olk El . Since does not have any central elements of El , there is a tuple (α1 , α2 , ..., αh ) ∈ (El × Ekh−1 )\. Then the h-ary function g with ⎞ ⎛ ⎞ ⎛ c 0 0 ... 0 α1 ⎜ l c 0 ... 0 ⎟ ⎜ α2 ⎟ ⎟ ⎟ ⎜ g⎜ ⎝ . . . . . . . . . . . ⎠ := ⎝ ... ⎠ ∈ αh l 0 0 ... c (c is a central element of ) g(x) := α1 otherwise, does not preserve . However, g preserves the relation t for t ≥ h + 1. Thus P olk × El is P olk El -maximal only if t = Ekt holds for all t ∈ {h, ..., k}. It results, however, from k = Ekk that has a certain central element of El , which we had excluded in Case 2.2. Therefore, this case cannot occur. A direct conclusion from Lemmas 17.6.2–17.6.6 is Theorem 17.6.1, whereby the theorems of Section 17.2 are also proven.

18 Further Submaximal Classes of Pk

As already indicated in Chapter 14, there are few results about submaximal classes for arbitrary k. Supplementary to Chapters 16 and 17, all submaximal classes of a maximal class of the type S are described in this chapter. It is then shown how one can prove the special case k ∈ P of this general description. The rest of this chapter deals with submaximal classes of Pk which lie below a maximal class of the type U. In Section 18.2, one can ﬁnd some maximal classes of P olk , where ∈ Uk is arbitrary. Then, in Section 18.3, the list from Section 18.2 is completed to the list of all maximal classes of P olk for 2 = Ek−1 ∪ {(k − 1, k − 1)}.

18.1 The Maximal Classes of P olks for s ∈ Sk In this section, p always denotes a prime number. Further, let k := p · l with l ∈ N. Denote s a ﬁxed point free permutation on the set Ek , whose cycles have the same length p. Furthermore, let s := {(x, s(x)) | x ∈ Ek }. By Chapter 5 S := P olk s is a maximal class of type S. For a description of the maximal classes of S, we need the following concepts and deﬁnitions. Let θs := {(x, y) ∈ Ek2 | ∃i ∈ {0, 1, ..., p − 1} : y = si (x)}. It is easy to see that θs is an equivalence relation on Ek . An h-ary relation γ ∈ Rk is called θs -closed, if ∀(x1 , ..., xh ) ∈ γ ∀(y1 , ..., yh ) ∈ Rkh : (∀i ∈ {1, ..., h} : (xi , yi ) ∈ θs ) =⇒ (y1 , ..., yh ) ∈ γ.

556

18 Further Submaximal Classes of Pk

Obviously, a θs -closed relation ⊆ Ekh is the homomorphic inverse image of an h-ary relation on the factor set (quotient set) (Ek )/θs in respect to the mapping Ek −→ (Ek )/θs , x → x/θs . It is easy to check that the following statements are valid: – An equivalence relation is θs -closed if and only if θs ⊆ . – A central relation is θs -closed if and only if it is the homomorphic inverse image of a central relation deﬁned on (Ek )/θs . – An h-regular relation , which is deﬁned as in Section 5.2.1 1 , is θs -closed if and only if θs ⊆ ϑ0 ∩ ϑ1 ∩ ... ∩ ϑm−1 . An equivalence relation ε on Ek is called transversal to s, if s ∈ P olk ε and ε ∩ θs = ι2k ; i.e., the permutation s maps each equivalence class (block) of ε onto another equivalence class of ε. / µ whenever A unary relation µ on Ek is called transversal to s, if si (x) ∈ i ∈ {1, ..., p − 1} and x ∈ Ek . Let q, r be prime numbers and let n be the least positive integer with q n = 1 (mod r). Moreover, denote GF (q n ) a ﬁeld with the order q n .2 Let G(q, r) be the set of all mappings of the form GF (q n ) −→ GF (q n ), x → a · x + b with a, b ∈ GF (q n ) and ar = 1. It is easy to see that the algebra G(q, r) := (G(q, r); 2) is a group. P. P. Palfy proved the following fact (see [Ros-S 85]): Lemma 18.1.1 (without proof ) A ﬁnite group G has a maximal subgroup of order p (p prime) if and only if G is isomorphic to one of the groups listed below: (1) an Abelean group of order p · q, where q ∈ P; (2) G(p, q), where q ∈ P and p = 1 (mod q); (3) G(p, q), where q ∈ P \ {p}. Now we deﬁne permutation groups with the aid of groups of Lemma 18.1.1. For a group G := (G; ◦) whose order divides k = |G|·r, we consider a partition of Ek into |G|-element blocks A1 , ..., Ar and select arbitrary bijections ϕi : Ai −→ G, 1 2

See also Lemma 5.2.6.3. As generally known, there is (up to isomorphism) exactly one ﬁnite ﬁeld of order qn .

18.1 The Maximal Classes of P olk s for s ∈ Sk

557

1 ≤ i ≤ r. Then, with the help of the mappings ϕi , for every g ∈ G one can deﬁne a permutation πg on Ek as follows: ∀i ∈ {1, ..., r} ∀x ∈ Ai : πg (x) := ϕ−1 (g ◦ (ϕi (x))). It is easy to check that the set of all permutations of the form πg with g ∈ G together with the operation 2 forms a group, which will be called a semiregular representation of the group G. ´ Szendrei. The following theorem was found by I. G. Rosenberg and A. Theorem 18.1.2 (Rosenberg-Szendrei Theorem; [Ros-S 85]; without proof ) The following list describes all maximal classes of S := P olk s : (1) P olk {(x, π(x)) | π ∈ G}, where G is a semiregular representation of a group, which belongs to the ones described in Lemma 18.1.1, (1)–(3) and for which s ∈ G is valid; (2) S ∩ P olk λG , where λG ∈ Lk , G := (Ek , ⊕) is an elementary Abelean p-group (see Section 5.4) and there exists an element c ∈ Ek with s(x) = x ⊕ c; (3) S ∩ P olk ε, where ε ∈ Uk is either θs -closed or transversal to s. (4) S ∩ P olk γ, where γ ∈ Ck and γ is either θs -closed or a nonempty unary relation transversal to s; (5) S ∩ P olk , where ∈ Bk and is θs -closed. There are also certain k for which the description of the maximal classes of S is simple, as the following theorem shows: Theorem 18.1.3 ([Sze 84], [Lau 84b]) Let k = p ∈ P and s ∈ Sp . Then S := P olp s has exactly two maximal classes. One maximal class has the type (2) and the other maximal has the type (4) from Theorem 18.1.2. Choosing s(x) := x + 1 (mod p), one can describe these maximal classes as follows: S ∩ P olp {0} and S ∩ P olp λ with λ := {(a, b, c, d) ∈ Ek4 | a + b = c + d (mod p)}. The above theorem is a special case of the following theorem: Theorem 18.1.4 ([Lau 84b]) Let k ∈ N, s(x) := x + 1 (mod k), S := P olk {(x, s(x)) | x ∈ Ek } 3 and let Tk be the set of all divisors ∈ N of k. Then the following list gives all maximal classes of S: 3

S is a maximal clone of Pk iﬀ k ∈ P. For k ∈ P the set S is a subclone of a certain clone of the type U.

558

18 Further Submaximal Classes of Pk

(1) S ∩ P olk γr , where r ∈ Tk \ {1} and γr := {x ∈ Ek | r divides x}; (2) S ∩ P olk t , if k ∈ P, t ∈ Tk \ {1, k} and t := {(x, y) ∈ Ek | r divides (x − y)}; (3) S ∩ Lk , if k ∈ P and Lk := P olk {(a, b, c, d) ∈ Ek4 | a + b = c + d (mod k)} = n n i=1 ai · xi (mod k)}. n≥1 {f ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + Proof. We prove the above theorem with the aid of Theorem 6.1 and Theorem 8.3.2. Let α be the mapping deﬁned in Theorem 8.3.2. First we prove that the classes described in (1)–(3) are S-maximal. For this purpose, let r ∈ Tk \{1}, t ∈ Tk \{1, k} and A ∈ {P olk γr , P olk t , Lk } be arbitrary. By Chapter 5, the classes P olk γr , P olk t and Lk are maximal in Pk . Consequently, because of Theorem 8.3.2, (a), we have to show that α(S ∩ A) = A holds. Obviously, because of c0 ∈ A, we have α(S ∩ A) ⊆ A. For the proof of A ⊆ α(S ∩ A), we show that every function f n ∈ S with α(f ) ∈ A belongs to A. We distinguish three cases: Case 1: A = P olk γr . Let a1 , ..., an ∈ γr and f n ∈ S with α(f ) ∈ P olk γr be arbitrary. By deﬁnition of γr , bi := ai − a1 (mod k) is an element of γr for every i ∈ {1, ..., n}. Thus, because of α(f ) ∈ P olk γr , we have b := f (0, b2 , ..., bn ) ∈ γr and therefore b + a1 (mod k) = f (a1 , ..., an ) ∈ γr . Consequently, f preserves the relation γr , and α(S ∩ P olk γr ) = P olk γr is valid. Case 2: A = P olk t . Let (a1 , b1 ), ..., (an , bn ) ∈ t and f n ∈ S with α(f ) ∈ P olk t be arbitrary. Set ci := ai − a1 (mod k) and di := bi − b1 (mod k) for i = 1, ..., n. Then (ci , di ) ∈ t . Further, because of α(f ) ∈ P olk t , we have u 0 c2 ... cn ∈ t . := f 0 d2 ... dn v Since f ∈ S,

f

a1 a2 ... an b1 b2 ... bn

=

u + a1 (mod k) v + b1 (mod k)

∈ t

results. Consequently, f preserves the relation t and α(S ∩ P olk t ) = P olk t holds. Case 3: A = Lk . It is easy to check that S ∩ Lk = n≥1 {f n ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + n i=1 ai · xi (mod k) ∧ a1 + ... + an = 1 (mod k)}. This implies α(S ∩ Lk ) = Lk . Now we come to the completeness proof. For this purpose, we choose an arbitrary

18.1 The Maximal Classes of P olk s for s ∈ Sk

559

subset M of S with M ⊆ A for every class A deﬁned in (1)–(3). Our theorem is proven if we can show that [M ] = S holds. S 1 = {s, s2 , ..., sk−1 = e11 } ⊆ [M ] follows from the following considerations: Because of M ⊆ S ∩ P olk γk , there is a certain function f n ∈ M with a := f (0, 0, ..., 0) = 0. Consequently, ∆n−1 f = sa belongs to [M ]. If a and k are relatively prime, then [sa ]1 = S 1 . If a and k are not relatively prime, there is a t ∈ Tk \ {1} with [sa ]1 = {sx | x ∈ γt }. Because of M ⊆ S ∩ P olk γt , a certain function gtm with b := gt (a1 , ..., am ) ∈ γt for certain a1 , ..., am ∈ γt belongs to M . Consequently, we have gt (sa1 , ..., sam ) = sb ∈ [M ]. Further, there exists a t ∈ Tk \ {t} with [sa , sb ]1 = {sx | x ∈ γt }. If t = 1, then S 1 ⊆ [M ]. If t = 1, there is a function gt ∈ [M ], which does not preserve γt . Analogous to the above, one can form a function sc ∈ S 1 \ [sa , sb ] as a superposition over {gt , sa , sb }, and two cases are possible. The iteration of the construction above shows S 1 ⊆ [M ]. Because of Theorem 8.3.2, [M ] = S is proven if we can show that α([M ]) = Pk holds. By Theorem 6.1, α([M ]) = Pk is proven if one can show that for every relation ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk a function f ∈ [M ] that does not preserve the relation exists. Since α(sa ) = a and sa ∈ [M ] for a ∈ Ek , the constants 0, 1, ..., k − 1 belong to α([M ]). Further, [M ] ⊆ α([M ]) by Theorem 8.3.2, (b). Because of {a, sa | a ∈ Ek } ⊆ α([M ]) we have ∀ ∈ C1k ∪ Sk ∪ Mk : α([M ]) ⊆ P olk .

(18.1)

Let ∈ Chk with 2 ≤ h ≤ k−1, let c be a central element of and (a1 , ..., ah ) ∈ Ekh \. Then the function sa1 −c ∈ [M ] does not preserve because of ⎞ ⎛ ⎛ ⎞ a1 c ⎜ a2 − a1 + c (mod k) ⎟ ⎜ a2 ⎟ ⎟ ⎜ ⎜ ⎟ sa1 −c ⎜ ⎟ = ⎜ . ⎟. .. . ⎝ ⎠ ⎝ . ⎠ . ah ah − a1 + c (mod k) Consequently, we have ∀ ∈

k−1

Chk : α([M ]) ⊆ P olk .

(18.2)

h=2

Next we determine all such relations ∈ Uk that are preserved from the permutation s. In the following, let be an arbitrary equivalence relation on Ek and U := {x ∈ Ek | (0, x) ∈ }. Suppose, s ∈ P olk . Then because of (sk−a (a), sk−a (b)) = (0, b − a) and (sa (0), sa (b − a)) = (a, b), we have ∀(a, b) ∈ Ek2 : (a, b) ∈ ⇐⇒ (0, b − a) ∈ ⇐⇒ b − a ∈ U.

(18.3)

With the help of (18.3) and with the properties of equivalence relations, one can easily prove that U is a subgroup of the cyclic group (Ek ; + (mod k)). As is generally known, every subgroup of a cyclic group with k elements can be described with the aid of a divider of k. Hence, there exists a t with t ∈ Tk and U = {x ∈ Ek | t|x}. By (18.3), this implies = t . Consequently, if k ∈ P, s preserves only trivial equivalence relations and, if k ∈ P, s preserves only equivalence relations, deﬁned in (2). Therefore, by our assumptions of M , we have

560

18 Further Submaximal Classes of Pk ∀ ∈ Uk : α([M ]) ⊆ P olk .

(18.4)

Because of |Im(f )| = k for all f ∈ S, S 1 = L1p and M ⊆ S ∩ Lp for p ∈ P and, if k ∈ P, S 1 = (P olk t )1 and M ⊆ S ∩ P olk t for t ∈ Tk \ {1, k} α([M ]) ⊆ P olk ιkk

(18.5)

is obviously valid. Next, let ∈ Bhk be arbitrary with 3 ≤ h ≤ k − 1. Then by deﬁnition there is a surjective mapping q : Ek −→ Ehm with m ≥ 1 and (a1 , ..., ah ) ∈ ⇐⇒ (q(a1 ), ..., q(ah )) ∈ ξm , where ξm is an h-ary elementary relation on Ehm (see Chapter 5). If q is a bijective mapping, then with the aid of Theorem 5.2.6.1, one can prove that s does not preserve the relation . If q is not bijective, then the mapping equivalence σ := {(x, y) ∈ Ek2 | q(x) = q(y)} belongs to Uk . We have shown that a function fσn , which does not preserve the relation σ, belongs to α([M ]) ∩ S. Therefore, there are tuples (a1 , b1 ), ..., (an , bn ) ∈ σ with (c, d) := (fσ (a1 , ..., an ), fσ (b1 , ..., bn )) ∈ σ. Because of ιhh ∩ Ehhm ⊆ ξm , we have {(ai , bi , x3 , ..., xh ) | x3 , ..., xh ∈ Ek } ⊆ for all i = 1, ..., n. The deﬁnition of and the choice of the elements c and d imply the existence of certain u3 , ..., uh ∈ Ek with (c, d, u3 , ..., uh ) ∈ . Im(fσ ) = Ek holds because of fσ ∈ S. Consequently, there exist certain (ai , bi , ci3 , ..., cin ) ∈ , i = 1, ..., n, with ⎛ ⎞ ⎛ ⎞ c a 1 a2 . . . a n ⎜ b1 b2 . . . b n ⎟ ⎜ d ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ fσ ⎜ ⎜ c13 c23 . . . cn3 ⎟ = ⎜ u3 ⎟ . ⎝ ................ ⎠ ⎝ ... ⎠ c1h c2h . . . cnh uh Hence, fσ does not preserve the relation ∈ Bhk and ∀ ∈

k−1

Bhk : α([M ]) ⊆ P olk

(18.6)

h=3

is valid. It still remains to be proven that ∀λ ∈ Lk : α([M ]) ⊆ P olk λ.

(18.7)

For this purpose let λ ∈ Lk . Then there exist p ∈ P, m ∈ N with k = pm and an elementary Abelean p-group (Ek ; ⊕) with the property λ := {(a, b, c, d) ∈ Ek4 | a ⊕ b = c ⊕ d}. W.l.o.g.4 let 0 be the neutral element of the group (Ek ; ⊕). If k ∈ P, then either S 1 ⊆ P olk λ or λ = {(a, b, c, d) ∈ Ek4 | a + b = c + d (mod p)}. If k ∈ P, (18.6) results from that and then (with the aid of (18.1), (18.2), (18.4)–(18.6) and Theorems 8.3.2 and 6.1) our theorem. Let k ∈ P in the following. 4

See Section 5.2.4

18.2 Some Maximal Classes of a Maximal Class of Type U

561

To prove (18.7), we show ﬁrst that there is a relation γ ∈ Uk ∪ Mk ∪ Ck ∪ Bk with S ∩ P olk λ ⊆ S ∩ P olk γ. It is easy to check that the following three relations are invariants of S ∩ P olk λ: λ1 := {(a1 , ..., ap ) ∈ Ekp | a1 ⊕ a2 ⊕ ... ⊕ ap = 0}, λ2 := {(a1 , ..., ap , b1 , ..., bp ) ∈ Ek2·p | a1 ⊕ a2 ⊕ ... ⊕ ap = b1 ⊕ b2 ⊕ ... ⊕ bp } and, where + denotes the addition modulo k, := {(i, i + 1, i + 2, ..., i + p − 1) | i ∈ Ek }. For i ∈ Ek we set: ti := (i ⊕ (i + 1) ⊕ (i + 2) ⊕ ... ⊕ (i + p − 1), where + is the addition modulo k. Because of k = p the equations t0 = t1 = ... = tk−1 are not possible. If there is a j with tj = 0 (this is possible only for p = 2), then pr1 ( ∩λ1 ) ∈ C1k . If ti = 0 for all i ∈ Ek , there are certain r, s with tr = ts and r = s, whereby the relation := pr1,p+1 (( × ) ∩ λ2 ) is a binary reﬂexive non-diagonal relation. A conclusion of Chapter 6 is the fact that a relation γ ∈ Uk ∪ Mk ∪ Ck ∪ Bk is derivable from . Consequently, we obtain from the assumption α([M ]) ⊆ P olk λ a contradiction to the statements (18.1), (18.2), and (18.4)–(18.6). Our assumption was therefore false, and (18.7) is valid. Thus, the set α([M ]) is no subset of an arbitrary maximal class of Pk . Hence, α([M ]) = Pk is valid. This and Theorem 8.3.2 imply [M ] = S.

18.2 Some Maximal Classes of a Maximal Class of Type U Let be an arbitrary relation of Uk , where Ai (i ∈ Et ) denote the equivalence classes of this relation. In addition, a/ denotes the equivalence class of , which contains a ∈ Ek . One checks the following lemma easily (see Table 18.1):5 5

See also Lemma 1.4.6.

562

18 Further Submaximal Classes of Pk

Lemma 18.2.1 Let Ai (i ∈ Et ) be a partition the set Ek and ai ∈ Ai (i ∈ Et ). Furthermore let y for ∃i ∈ Et : x = ai and y ∈ Ai , x y := x otherwise. Then, an arbitrary function f n ∈ Pk is a superposition over the functions z, gf , fi (i ∈ Et ), deﬁned by z(x, y) := x y, gf (x1 , ..., xn ) := ai ⇐⇒ f (x1 , ..., xn ) ∈ Ai , f (x1 , ..., xn ) for f (x1 , ..., xn ) ∈ Ai , fi (x1 , ..., xn ) := otherwise ai (i ∈ Et ), and a representation of f is given by f (x) = ((...((gf (x) f0 (x)) f1 (x)) ...) ft−1 (x)).

(18.8)

Table 18.1

x f (x) gf (x) fi (x) gf f0 (gf f0 ) f1 .. }= a0 }= ai }= f (x) }= f (x) . }∈ A0 .. }= a1 }= ai }= a1 }= f (x) . }∈ A1 .. }= a2 }= ai }= a2 }= a2 . }∈ A2 .. .. .. .. .. .. . . . . . . .. }= ai }= f (x) }= ai }= ai . }∈ Ai .. .. .. .. .. .. . . . . . . .. }= at−1 . }∈ At−1 }= at−1 }= ai }= at−1 Remark

(18.8) is also valid, if one deﬁnes y for ∃i ∈ Et : {x, y} ⊆ Ai , x y := x otherwise

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

(18.9)

and if gf fulﬁlls the following condition: ∀x ∈ Ekn : (gf (x), f (x)) ∈ .

Let f n ∈ Pk be arbitrary and let (18.8) be a representation of f . Then f ∈ P olk if and only if gf ∈ P ol , since the functions fi (i ∈ Et ) and

18.2 Some Maximal Classes of a Maximal Class of Type U

563

the function z preserve . Obviously, every function of the form gf (∈ P ol ) is unambiguously characterized through its values on {ai | i ∈ Et }n . Consequently, ({gf | f ∈ P ol }; ζ, τ, ∆, ∇, ) is isomorphic to (Pt ; ζ, τ, ∆, ∇, ). Let q a unary function of Pk deﬁned by ∀i ∈ Et : q(x) = ai :⇐⇒ x ∈ Ai . Lemma 18.2.2 follows from the above considerations: Lemma 18.2.2 ∀f n ∈ Pk : f ∈ P ol ⇐⇒ ∃ hn ∈ P{a0 ...,at−1 } : gf (x) = h(q(x1 ), ..., q(xn )). 2 Since [Pk2 ] = Pk and [Pk,A ] = Pk,A , the fact

ord P olk = 2 follows from Lemmas 18.2.1 and 18.2.2 (see also proof of Theorem 11.2.2). In analog mode to a corresponding statement for Pk , the following lemma results from that then: Lemma 18.2.3 For every proper subclone T of P olk there is a maximal clone of P olk , which contains T . P olk has only ﬁnite many maximal clones. 2 For every maximal class M of P olk , there is a relation ψ ∈ Rkk with M = P olk ψ. W.l.o.g. let |A0 | ≥ |A1 | ≥ ... ≥ |At−1 | and A0 := {0, 1, ..., l − 1},

l≥2

in the following. For the purpose of deﬁning a homomorphism αϕ : Pk −→ Pt we need the mapping ϕ deﬁned by ϕ : Ek −→ Et , ∀i ∈ Et ∀x ∈ Ai : ϕ(x) := i. Obviously, αϕ : Pk −→ Pt , f → f αϕ ∀x1 , ..., xn ∈ Ek : f αϕ (ϕ(x1 ), ..., ϕ(xn )) := ϕ(f (x1 , ..., xn )) is a homomorphism. Next we describe maximal clones of

564

18 Further Submaximal Classes of Pk

U := P olk which contain all unary functions of U or which have the form P olk ∩ P olk , where P olk is Pk -maximal. Some of these clones can be described in the form P olk βi for certain indices i. Put λ := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ {0, 1}}, ιhi := {(a1 , ..., ah ) ∈ Eih | |{a1 , ..., ah }| ≤ h − 1}, −1 ϕ (λ) if t = 2, β1 := ϕ−1 (ιtt ) if t ≥ 3, λ := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | ∃i : |Ai | = 2 ∧ {a, b} ⊆ Ai }∪ {(x, x, x, x) | ∃j : Aj = {x}}, λ if l = 2, β2 := ι if l ≥ 3. For the purpose of describing a third type of maximal clones of U , we ﬁrst deﬁne the following relation for i1 , ..., ir ∈ N: γi1 ,...,ir := {(x1,1 , x1,2 , ..., x1,i1 , x2,1 , x2,2 , ..., x2,i2 , ..., xr,1 , xr,2 , ..., xr,ir ) ∈ Eki1 +...+ir | ∀j ∈ {1, ..., r} ∃q : {xj,1 , xj,2 , ..., xj,ij } ⊆ Aq }. Then we can deﬁne the relations βi1 ,...,ir , as follows: x := (x1,1 , x1,2 , ..., x1,i1 , x2,1 , x2,2 , ..., x2,i2 , ..., xr,1 , xr,2 , ..., xr,ir ) ∈ βi1 ,...,ir :⇐⇒ x ∈ γi1 ,...,ir ∧ (∃s ∈ {i1 , ..., ir } : |{xs,1 , xs,2 , ..., xs,ij }| ≤ ij − 1). For the case |A2 | = ... = |At | = 1 we need the following relations: σr,s := {(x1 , ..., xr , y1 , ..., ys ) ∈ Ekr+s | x1 = ... = xr ∨ (x1 , ..., xr ) ∈ ιrl ∨ ({x1 , ..., xr } = El =⇒ |{x1 , ..., xr , y1 , ..., ys }| ≤ r + s − 1} (2 ≤ r ≤ l, 1 ≤ s ≤ k − l). Lemma 18.2.4 Let ∅ ⊂ σ ⊂ Ek . Then P olk σ ∩ P olk is a maximal clone of U := P olk if and only if the following condition is valid: (∃I ⊂ {0, 1, ..., t − 1} : σ = Ai ) ∨ (∀j ∈ {0, 1, ..., t − 1} : σ ∩ Aj = ∅). i∈I

(18.10)

18.2 Some Maximal Classes of a Maximal Class of Type U

565

Proof. “=⇒”: Let P olk σ ∩ P olk be U -maximal and let σ := {y ∈ Ek | ∃x ∈ Ek : (x, x, y) ∈ σ × }. The following two cases are possible: Case 1: ∅ ⊂ σ ⊂ Ek . In this case, we have U ∩ P olk σ ⊆ U ∩ P olk σ ⊂ U and σ has the form ∃I ⊆ {1, 2, ..., t} : σ =

Ai ,

i∈I

since

a ∈ σ =⇒ (∀b ∈ a/ : b ∈ σ )

holds (a/ := {x ∈ Ek | (a, x) ∈ }). Case 2: σ = Ek . (Such a case occurs, for example, if k = 3, := {(0, 0), (1, 1), (2, 2), (0, 1), (1, 0)} and σ := {0, 2}.) It is easy to check that σ fulﬁlls the condition ∀j ∈ {0, 1, ..., t − 1} : σ ∩ Aj = ∅ in this case. “⇐=”: Let f n ∈ U \P olk σ, where σ has the property (18.10). Then there are a1 , ..., an ∈ σ with f (a1 , ..., an ) = a ∈ Ek \σ. Since the constant cs with s ∈ σ belongs to U ∩ P olk σ, we have ca ∈ [(U ∩ P olk σ) ∪ {f }]. Let now g m ∈ U be arbitrary. We show that g ∈ [(U ∩ P olk σ) ∪ {f }]. For this purpose, we consider a function hm+1 ∈ U with hg (x1 , ..., xm , a) = g g(x1 , ..., xm ). Our assertion is proven, ifwe can show that there is a such function hg in the set U ∩ P olk σ. If σ = i∈I Ai for a certain I ⊂ Et , we set g(x1 , ..., xm ), if xm+1 ∈ a/, hg (x1 , ..., xm+1 ) := s otherwise, where s ∈ σ. Obviously, hg ∈ U ∩ P olk σ. Let σ ∩ Aj = ∅ for all j ∈ Et . Then one must choose hg (a1 , ..., am , am+1 ) ∈ (b/) ∩ σ, if hg (a1 , ..., am , am+1 ) = b was chosen and (ai , ai ) ∈ for every i ∈ {1, ..., m+1} is valid. For the remaining tuples x, one can set e.g. hg (x) = s, where s ∈ σ, so that hg ∈ U ∩ P olk σ. Lemma 18.2.5 (1) Let γ ⊆ Eth and let P olt γ Pt -maximal. Furthermore, let −1 αϕ (γ) := {(a1 , ..., ah ) ∈ Ekh | (αϕ (a1 ), ..., αϕ (ah )) ∈ γ}. −1 Then P olk αϕ (γ) is U -maximal.

566

18 Further Submaximal Classes of Pk

(2) P olk β1 is U -maximal. (3) Let A be a subset of U with U 1 ⊆ A and A ⊆ P olk β1 . Then ∀f ∈ Pt ∃g ∈ [A] : αϕ (g) = f.

(18.11)

−1 Proof. (1): Let f ∈ U \P olk αϕ (γ) be arbitrary. Then we have obvious −1 [αϕ (P olk αϕ (γ) ∪ {f })] = Pt . Consequently, −1 (γ) : αϕ (g) = h. ∀h ∈ Pt ∃g ∈ P olk αϕ

(18.12)

−1 For arbitrary functions q ∈ U it follows from the deﬁnition of P olk αϕ (γ): −1 (γ) ⇐⇒ αϕ (q) ∈ P olt γ. q ∈ P olk αϕ

(18.13)

Case 1: {c0 , c1 , ..., ct−1 } ⊂ P olt γ. With the help of (18.13) and the fact that the constant functions c0 , ..., ct−1 of Pt belong to P olt γ, one gets: {} ∪

t−1

−1 Pk,Ai ⊆ P olk αϕ (γ).

(18.14)

i=0 −1 Then (with the help of Lemma 18.2.1), (18.12), and (18.14) imply [P olk αϕ (γ)∪ −1 {f }] = U , i.e., P olk αϕ (γ) is U -maximal. Case 2: γ ⊂ Et . In this case, our assertion follows from Lemma 18.2.4. Case 3: γ ∈ St . Because of (18.12), one can ﬁnd a unary function o with αϕ (o) = c0 in −1 (γ) ∪ {f })]. Since a unary function p with [αϕ (P olk αϕ

∀x ∈ A0 : p(x) = 0 −1 belongs to P olk αϕ (γ), the constant c0 = p o is a superposition over −1 P olk αϕ (γ) ∪ {f }. Let g ∈ U m be arbitrary. Then there is an (m + 1)-ary function hg with hg (0, x1 , ..., xm ) = g(x1 , ..., xm ) −1 −1 in P olk αϕ (γ), whereby g = hg c0 ∈ [P olk αϕ (γ)∪{f }] holds. Consequently, −1 P olk αϕ (γ) is U -maximal.

(2) is a special case of (1). (3): A conclusion of Rosenberg’s Theorem is [αϕ (A)] = Pt , i.e., (18.11) holds.

18.2 Some Maximal Classes of a Maximal Class of Type U

567

Lemma 18.2.6 For some i ∈ Et let |Ai | ≥ 2. Furthermore, let γ ⊆ Ahi be a relation that describes a maximal clone P olAi γ in PAi , where all constant functions of PAi belong to P olAi γ. Set t−1

γ := γ ∪

Ahj .

j=1, j=i

Then the clone P olk γ is U -maximal. Proof. W.l.o.g. let Ai := Er with r ≥ 2 and γ ∈ Ur ∪ Mr ∪ Lr ∪ Br ∪

r−1

Chr .

h=2

Obviously, P olk γ ⊂ U and {, c0 , c1 , ..., ck−1 } ∪ U0 ∪

t−1

Pk,Aj ⊆ P olk γ .

j=1, j=i

Elements of P olk γ are also all n-ary functions qg with g(x) if x ∈ Ern , qg (x) := 0 otherwise, for arbitrary g ∈ P olr γ. Moreover, all unary functions pd0 ,...,dr−1 deﬁned by pd0 ,...,dr−1 (x) := dj ⇐⇒ x ∈ Aj (j = 0, 1, ..., t − 1) belong to P olk γ for all d0 , d1 , ..., dt−1 ∈ Ai = Er . Let f ∈ U \P olk γ be arbitrary. To prove [{f } ∪ P olk γ ] = U , we have only to show that Pk,r ⊆ [{f } ∪ P olk γ ] is valid because of Lemma 18.2.2 and because of the above considerations. Obviously, there are certain (a1j , a2j , ..., ahj ) ∈ γ for j = 1, ..., m and certain (b1l , b2l , ..., bhl ) ∈ γ \γ for l = 1, ...n with ⎞ ⎛ ⎞ ⎛ α1 a11 a12 ... a1m b11 b12 ... b1n ⎜ a21 a22 ... a2m b21 b22 ... b2n ⎟ ⎜ α2 ⎟ h ⎟ ⎜ ⎟ f⎜ ⎝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎠ = ⎝ ... ⎠ ∈ Er \γ. ah1 ah2 ... ahm bh1 bh2 ... bhn αh By choosing certain m-ary functions f1 ..., fn from the m-ary function f ∈ Pk,r by means of

t−1 j=1, j=i

Pk,Aj one receives

f (x1 , ..., xm ) := qe11 (f (x1 , ..., xm , f1 (x1 , ..., xm ), ...., fn (x1 , ..., xm ))), 1

where

568

18 Further Submaximal Classes of Pk

⎞ ⎛ α1 a11 a12 ... a1m ⎜ ⎟ ⎜ α2 ⎜ a21 a22 ... a2m ⎟ =⎜ f ⎝ . . . . . . . . . . . . . . . . ⎠ ⎝ ... ah1 ah2 ... ahm αh ⎛

⎞ ⎟ ⎟ ∈ Erh \γ ⎠

holds, as a superposition over P olk γ ∪{f }. With the help of the completeness criterion for Pk,r (see Theorem 12.4.3 or the proof of the following lemma) one can easy prove that {f } ∪ {qg | g ∈ Pr } ∪ {pd0 ,...,dt−1 | d0 , ..., dr−1 ∈ Er } is a generating system for Pk,r , whereby Pk,r ⊆ [{f } ∪ P olk γ ] was shown. With that, a generating system for the clone U was proven in {f } ∪ P olk γ . Therefore, P olk γ is U -maximal. Lemma 18.2.7 (1) P olk β2 is U -maximal. (2) Let A be a subset of U with U 1 ⊆ A and A ⊆ P olk β2 . Furthermore, A fulﬁlls the condition (18.11). Then ∀i ∈ {0, 1, ..., t − 1} : Pk,Ai ⊆ [A].

(18.15)

Proof. (1): For proof, we use the completeness criterion for Pk,s from Chapter 12: Let pr : Pk,s −→ Ps be deﬁned by pr(f n ) = g m :⇐⇒ n = m ∧ ∀x ∈ Esn : f (x) = g(x). With the aid of the mapping pr, one can describe the following maximal classes of Pk,s : pr−1 M := {f ∈ Pk,s | pr(f ) ∈ M }, where M is an arbitrary maximal clone of Ps . Furthermore, let ζi,t := {(x, x) | x ∈ Es } ∪ {(i, t)} for all i, t with i < t ≤ k − 1 and s ≤ t. Then, for all A ⊆ Pk,s is valid: [A] = Pk,s ⇐⇒ A is no subset of every set of the form pr−1 M (where M is a maximal clone of Ps ) and A does not preserve every relation of the form ζi,t . 1 (This means that the set Pk,s ∪ {f }, where f ∈ Pk,s and pr(f ) ∈ L for s = 2 s and pr(f ) ∈ P ols ιs for s ≥ 3, is complete in Pk,s .) Obviously, U 1 ⊂ P olk β2 . Let f ∈ U \P olk β2 be arbitrary. Then a function g ∈ Pk,l with pr(g) ∈ L for l = 2 and with pr(g) ∈ P oll ιll for l ≥ 3 is a superposition over U 1 ∪ {f }. Since U 1 ⊆ P olk β2 in addition, we have

Pk,l ⊆ [P olk β2 ∪ {f }]

18.2 Some Maximal Classes of a Maximal Class of Type U

569

because of the completeness criterion, and also ∀i ∈ Et : Pk,Ai ⊆ [P olk β2 ∪ {f }]. It is easy to check that the function deﬁned in the remark on Lemma 18.2.1 preserves the relation β2 . Moreover, we have ∀h ∈ Pt ∃g ∈ P olk β2 : αϕ (g) = h. Consequently, the clone [P olk β2 ∪ {f }] contains a generating system for U , whereby (1) is proven. (2) follows from the proof for (1). Lemma 18.2.8 Let |A1 | = ... = |At−1 | = 1 (i.e., A0 = El is the only equivalence class of that contains at least two elements). Furthermore, let U0 := {f ∈ U | |Im(f ) ∩ El | ≤ 1}, U1,α := {f ∈ U | Im(f ) ⊆ El ∪ {α}}, U1 := α∈Ek \El U1,α . Then (a) ∀f ∈ U \P olk σl,1 : U1 ⊆ [{f } ∪ U 1 ∪ Pk,l ∪ U0 ]; (b) P olk σl,1 is U -maximal. Proof. The deﬁnition of σr,s , which one can ﬁnd before Lemma 18.2.4, implies σl,1 = {(x1 , ..., xl , y) ∈ Ekl+1 | x1 = ... = xl ∨ {x1 , ..., xl+1 } ⊆ El ∨ ({x1 , ..., xl } ∈ ιll } (a): Let f ∈ U \P olk σl,1 be arbitrary and M := {f } ∪ U 1 ∪ Pk,l ∪ U0 . Then, an n-ary function f1 with the property ⎞ ⎛ ⎞ ⎛ 0 0 0 a13 a14 · · · a1n ⎜ 0 1 a23 a24 · · · a2n ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 2 a33 a34 · · · a3n ⎟ ⎜ 2 ⎟ ⎟, ⎜ ⎜ ⎟ f1 ⎜ ⎟ ⎟=⎜ ⎜ ........................ ⎟ ⎜ ··· ⎟ ⎝ 0 l − 1 al3 al4 · · · aln ⎠ ⎝ l − 1 ⎠ l l 0 b 3 b4 · · · bn where (a1i , a2i , ..., ali ) (i = 3, 4, ..., n) are certain tuples of ιll and b3 , ..., bn are certain elements of Ek , is a superposition over M . The above property of f1 and f1 ∈ U implies that f1 has only values of El on tuples of Eln . Next we consider the function

570

18 Further Submaximal Classes of Pk

f1 := (f1 c0 )|El ∈ Pl , that is, we consider the restriction of f1 (c0 (x1 ), x2 ..., xn ) on (x1 ..., xn ) ∈ Eln . Then the following cases are possible: Case 1: x2 is the only essential variable of f1 . In this case, we have ⎛

0 0 0 0 ··· 0 ⎜0 1 0 0 ··· 0 ⎜ ⎜0 2 0 0 ··· 0 f1 ⎜ ⎜ ..................... ⎜ ⎝ 0 l − 1 0 0 ··· 0 l 0 b 3 b4 · · · bn

⎞ 0 ⎟ ⎜ 1 ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎟ ⎟=⎜ ⎟ ⎜ ··· ⎟ , ⎟ ⎟ ⎜ ⎠ ⎝l−1⎠ l ⎞

⎛

and for every α ∈ Ek \El , a certain binary function fα with the property ⎞ ⎞ ⎛ ⎛ 0 0 0 ⎜0 1 ⎟ ⎜ 1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜0 2 ⎟ ⎜ 2 ⎟ ⎟ ⎜ ⎟ ⎜ fα ⎜ ⎟ ⎟=⎜ ⎜ ....... ⎟ ⎜ ··· ⎟ ⎝0 l−1⎠ ⎝l−1⎠ α l 0 is a superposition over M . Let g be an arbitrary m-ary function of U1,α with α ∈ Ek \El . The m-ary functions g1 ∈ U0 and g2 ∈ Pk,l deﬁned by 0, if g(x) ∈ El , g1 (x) := l otherwise,

and g2 (x) :=

= f (x), if g(x) ∈ El , 0 otherwise,

belong to M , whereby g ∈ [M ] holds because of g(x) = fα (g1 (x), g2 (x)). Consequently, we have U1,α ⊆ [M ] and (since U 1 ⊆ M ) also U1 ⊆ [M ], i.e., our assertion (a) is proven in Case 1. Case 2: x2 and xi for certain i ∈ {3, .., n} are essential variables of f1 . In this case, the Fundamental Lemma of Jablonskij (see Theorem 1.4.4) implies the existence of some ci , di , eji ∈ El (i = 2, ..., n; j = 4, ..., l) with

18.2 Some Maximal Classes of a Maximal Class of Type U

⎛

0 c2 c3 c4 · · · cn ⎜ 0 c2 d3 d4 · · · dn ⎜ ⎜ 0 d2 d3 d4 · · · dn ⎜ f1 ⎜ ⎜ 0 e42 e43 e44 · · · e4n ⎜ ...................... ⎜ ⎝ 0 el2 el3 el4 · · · eln l 0 b3 b4 · · · bn

⎞

⎛ α0 ⎟ ⎟ ⎜ α1 ⎟ ⎜ ⎟ ⎜ α2 ⎟=⎜ ⎟ ⎜ ··· ⎟ ⎜ ⎟ ⎝ αl−1 ⎠ l

571

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and {α0 , ..., αl−1 } = El . First we show that Uα1 ,α2 ,l := {f ∈ U | Im(f ) ⊆ {α1 , α2 , l} } ⊆ [M ]

(18.16)

holds. For this purpose, we use the property ⎞ ⎛ ⎞ ⎛ 0 c2 d3 d4 · · · dn α1 f1 ⎝ 0 d2 d3 d4 · · · dn ⎠ = ⎝ α2 ⎠ . l 0 b 3 b 4 · · · bn l Let h be an arbitrary m-ary function of Uα1 ,α2 ,l . Then the m-ary functions h1 , h2 , ..., hn deﬁned by 0, if h(x) = α1 , h1 (x) := (∈ U0 ), l otherwise ⎧ h(x) = α1 , ⎨ c2 , if h(x) = α2 , (∈ Pk,l ), h2 (x) := d2 , if ⎩ 0 otherwise di , if h(x) ∈ {α1 , α2 } hi (x) := (∈ U0 ), bi otherwise (i = 3, ..., n), belong to [M ], and we have: h(x) = f1 (h1 (x), h2 (x), ..., hn (x)), whereby (18.16) is proven. Since U 1 ⊆ M , (18.16) implies {f ∈ U | Im(f ) ⊆ {a, b, c} } ⊆ [M ]. a,b∈El , c∈Ek \El

Then, analogously to the above and ⎛ 0 c2 c3 c4 ⎜ 0 c2 d3 d4 f1 ⎜ ⎝ 0 d2 d3 d4 l 0 b 3 b4

with the aid of ⎞ ⎛ · · · cn α0 ⎜ α1 · · · dn ⎟ ⎟=⎜ · · · dn ⎠ ⎝ α2 · · · bn l

⎞ ⎟ ⎟, ⎠

572

18 Further Submaximal Classes of Pk

one can show that {f ∈ U | Im(f ) ⊆ {α0 , α1 , α2 , l} } ⊆ [M ] is valid. Consequently, because of U 1 ⊂ M , we have {f ∈ U | Im(f ) ⊆ {β0 , β1 , β2 , γ} } ⊆ [M ]. β0 ,β1 ,β2 ∈El , γ∈Ek \El

Then (a) results by means of induction from what was shown till now. (b): It is easy to see that the binary function z deﬁned by x if x ∈ El , z(x, y) := y otherwise, preserves the relation σl,1 . Moreover, we obviously have: U 1 ∪ Pk,l ∪ U0 ⊆ P olk σl,1 .

(18.17)

Let f ∈ U \P olk σl,1 be arbitrary. Then, because of (a) and (18.17), U1 ⊆ [{f } ∪ P olk σl,1 ] holds. Let q be an arbitrary m-ary function of U . Then, one can form q as a superposition over z and the m-ary functions q1 , q2 deﬁned by q(x), if q(x) ∈ El , q1 (x) := (∈ U1 ), l otherwise

and q2 (x) :=

0, if q(x) ∈ El , (∈ U0 ), q(x) otherwise

as follows: q(x) = z(q1 (x), q2 (x)). From that and from what was shown till now [{f } ∪ P olk σl,1 ] = U , through which (b) is proven. Theorem 18.2.9 Let k = l + 1. Then U has exactly three maximal clones that have U 1 as a subset. These clones are P olk β1 , P olk β2 , P olk σl,1 .

(18.18)

Proof. The U -maximality of the given clones was proven in Lemmas 18.2.5, 18.2.7, and 18.2.8. Let A be an arbitrary subset of U which has U 1 as a subset and which is no subset of the clones given in (18.18). To prove our theorem, we have to show that [A] = U . Lemmas 18.2.5 and 18.2.7 and U 1 ⊆ U imply U0 ∪ Pk,l ⊆ A. Consequently, with the aid of Lemma 18.2.8, (a), we have that U1 = U is a subset of [A].

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

573

18.3 The Maximal Classes of P olk(Ek2-1 ∪ {(k-1, k-1)}) In this section let k ≥ 3, 2 := Ek−1 ∪ {(k − 1, k − 1)}

and U := P olk . The sets U0 := {f ∈ U | k − 1 ∈ Im(f ) ∧ |Im(f )| ≤ 2} and Ua,b,k−1 := {f ∈ U | Im(f ) ⊆ {a, b, k − 1} }. are subsets of U . 18.3.1 Deﬁnitions of the U -Maximal Classes For k = 3, the clone U has exactly 13 maximal classes, which were given in Chapter 13, Theorem 13.1.8. In generalization of these classes (for a = 0 and b = 1), we will deﬁne six types of classes from which in the following two sections it is shown that they are the only maximal classes of the clone U . Type I (homomorphic inverse images of the maximal classes of P2 ): With the help of the mapping ϕ : Ek −→ E2 , ∀x ∈ Ek−1 : ϕ(x) := 0, ϕ(k − 1) := 1. −1 one can deﬁne a homomorphic inverse image αϕ (γ) of a relation γ ⊆ E2h as follows: −1 αϕ (γ) := {(x1 , ..., xh ) ∈ Ekh | (ϕ(x1 ), ϕ(x2 ), ..., ϕ(xh )) ∈ γ}.

In the case that γ describes a maximal class of P2 , we obtain the following maximal classes of U : −1 U ∩ P olk αϕ ({0}), −1 U ∩ P olk αϕ ({1}), 0 1 −1 U ∩ P olk αϕ , 1 0 0 0 1 −1 , P olk αϕ ⎛0 1 1 0 0 1 1 0 ⎜ −1 ⎜ 0 1 1 0 1 P olk αϕ ⎝ 1 0 0 1 1 1 1 0 0 0

1 0 0 1

0 0 0 0

⎞ 1 1⎟ ⎟. 1⎠ 1

574

18 Further Submaximal Classes of Pk

For k = 3, these are the classes with the numbers (2), (1), (8), (9), and (13) of Theorem 13.1.8. Type II (U -maximal classes described by certain unary relations): Let σ be a subset of Ek , which has one of the following two properties: 1) 2)

σ = Ek−1 , {k − 1} ⊆ σ ⊂ Ek .

Then U ∩ P olk σ is maximal in U (see Lemma 18.2.4). For k = 3 there are exactly four maximal clones that are describable in this manner. Two of these classes were already recorded by means of type I. Type III (U -maximal classes that are determined by the maximal h classes of Pk−1 ; ﬁrst possibility): Let γ ⊆ Ek−1 , 2 ≤ h ≤ k − 1, be an h-ary relation, which describes by means of P olk−1 γ a maximal class of Pk−1 and which contains all constant functions of Pk−1 . Then P olk (γ ∪ {(k − 1, k − 1, ..., k − 1)}) is U -maximal. For k = 3 there are exactly two U -maximal clones of type III: 0 1 2 0 P ol3 , 0 1 2 1 ⎛ ⎞ 0 0 0 1 1 0 1 1 2 ⎜0 0 1 1 0 1 0 1 2⎟ ⎟ P ol3 ⎜ ⎝ 0 1 0 0 1 1 0 1 2 ⎠. 0 1 1 0 0 0 1 1 2 The U -maximality of the classes of type III follows from Lemma 18.2.6. Type IV (U -maximal classes that are determined by the maximal classes of Pk−1 ; second possibility): This type of U -maximal class occurs only for k ≥ 4, since for k = 3 the conditions (a) and (b) mentioned below h , 2 ≤ h ≤ k − 1, an h-ary relation with the cannot be met. Denote γ ⊆ Ek−1 following properties: (a) P olk−1 γ is a maximal class of Pk−1 ; (b) γ is totally reﬂexive and totally symmetric, i.e., we can assume γ ∈ Uk−1 ∪

k−2 h=2

Moreover, let

Chk−1 ∪

k−1 h=3

Bhk−1 .

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

575

h+1 γ := Ek−1 ∪ {(x, x, ..., x, y) | x, y ∈ Ek }

∪{(x1 , ..., xh , k − 1) | (x1 , ..., xh ) ∈ γ}. Then P olk γ is U -maximal by Lemma 18.3.2.4. Example Let k = 4 and 0 1 2 3 0 3 γ := . 0 1 2 3 3 0 It is well-known that P ol3 γ is a maximal class of P3 and that γ is a reﬂexive and symmetric relation, whereby the relation ⎛ ⎞ 0 1 2 3 0 3 γ := E33 ∪ {(x, x, y) | x, y ∈ E4 } ∪ ⎝ 0 1 2 3 3 0 ⎠ 3 3 3 3 3 3 describes a maximal class of U . Type V (U -maximal classes that are described by certain binary 2 central relations): Denote τ a binary central relation ⊆ Ek−1 (i.e., a binary reﬂexive and symmetric relation, which has at least a central element c ∈ Ek−1 2 with {(c, x) | x ∈ Ek−1 } ⊆ τ and which is diﬀerent from Ek−1 .) If τ fulﬁlls the two following conditions (a) ⊆ τ , (b) ∃c ∈ Ek−1 : c is central element of τ , then U ∩ P olk τ is U -maximal by Lemma 18.3.2.9. Examples For k = 3 there are exactly two maximal classes of this type: 0 1 2 0 1 0 2 U ∩ P ol3 0 1 2 1 0 2 0 0 1 2 1 0 1 2 U ∩ P ol3 . 0 1 2 0 1 2 1 Type VI (Two U -maximal classes that are described by ternary relations): For A ∈ {Ek−1 , {k − 1}} let 2 σA := (Ek−1 × A) ∪ {(x, x, y) | x, y ∈ Ek }.

The class P olk σEk−1 is U -maximal by Lemma 18.3.2.5 and the class P olk σ{k−1} is U -maximal by Lemma 18.3.2.7. Example For k = 3 these classes have the number (10) and (11) in Theorem 13.3.8.

576

18 Further Submaximal Classes of Pk

18.3.2 Proof of the U -Maximality of the Classes Deﬁned in 18.3.1 The U -maximality of a class of the type I, II or III follows from Lemmas 18.2.4–18.2.6. Lemma 18.3.2.1 Let M be a subset of U which has the two following properties: (1) The set {f n ∈ Pk−1 | ∃g n ∈ M ∩ P olk {k − 1} ∩ P olk Ek−1 : n ∀x ∈ Ek−1 : g(x) = f (x)} is complete in Pk−1 . (2) There are two diﬀerent elements a, b ∈ Ek−1 with Ua,b,k−1 := {f ∈ U | Im(f ) ⊆ {a, b, k − 1}} ⊆ [M ]. Then M is complete in U . Proof. Because of (1) a (k − 1)-ary function p with the property ⎞ ⎞ ⎛ ⎛ 0 b a a ... a a a ⎟ ⎜ ⎜ a b a ... a a a ⎟ ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜ a a b ... a a a ⎟ ⎟ ⎜ ⎜ ⎜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎟ ⎜ ... ⎟ ⎟ ⎜ ⎟ ⎜ =⎜ p⎜ ⎟ a a ... b a a ⎟ ⎟ ⎜k−4⎟ ⎜ a ⎟ ⎜k−3⎟ ⎜ a a a ... a b a ⎟ ⎟ ⎜ ⎜ ⎝ a a a ... a a b ⎠ ⎝k−2⎠ k−1 k − 1 k − 1 k − 1 ... k − 1 k − 1 k − 1 belongs to [M ]. Then one can form an arbitrary n-ary function u ∈ U as a superposition over {p} ∪ Ua,b,k−1 ⊆ [M ] as follows: The n-ary functions u0 , u1 , ..., uk−2 deﬁned by ⎧ if u(x) = i, ⎨b if u(x) ∈ Ek−1 \{i}, ui (x) := a ⎩ k − 1 if u(x) = k − 1 (i = 0, 1, ..., k − 2) belong to Ua,b,k−1 . Then u = p(u0 , u1 , ..., uk−2 ) ∈ [M ], whereby [M ] = U is proven. Therefore, M is complete in U . Lemma 18.3.2.2 Let M be a subset of U , which has the two following properties:

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

577

(1) The set n : g(x) = f (x)} {f n ∈ Pk−1 | ∃g n ∈ M ∩ P olk Ek−1 : ∀x ∈ Ek−1

is complete in Pk−1 . (2) There are two diﬀerent elements a, b ∈ Ek−1 with Pk,{a,b} := {f ∈ Pk | Im(f ) ⊆ {a, b} } ⊆ [M ]. Then Pk,k−1 ⊆ [M ]. Proof. The proof is similar to the proof of Lemma 18.3.2.1: Because of (1) a (k − 1)-ary function p with the property ⎞ ⎞ ⎛ ⎛ 0 b a a ... a a a ⎜ a b a ... a a a ⎟ ⎜ 1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ a a b ... a a a ⎟ ⎜ 2 ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ p⎜ ⎜ . . . . . . . . . . . . . . . . ⎟ = ⎜ ... ⎟ ⎜ a a a ... b a a ⎟ ⎜ k − 4 ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ a a a ... a b a ⎠ ⎝ k − 3 ⎠ k−2 a a a ... a a b belongs to [M ]. Then one can form an arbitrary n-ary function u ∈ Pk,k−1 as a superposition over {p} ∪ Pk,{a,b} ⊆ [M ], as follows: The n-ary functions u0 , u1 , ..., uk−2 deﬁned by b if u(x) = i, ui (x) := a if u(x) ∈ Ek−1 \{i} (i = 0, 1, ..., k − 2) belong to Pk,{a,b} . Then u = p(u0 , u1 , ..., uk−2 ) ∈ [M ], whereby Pk,k−1 ⊆ [M ] is proven.

2 2 Lemma 18.3.2.3 For each f ∈ U \P olk (ιk−1 k−1 ) and for each (a, b) ∈ Ek−1 \ιk−1 the set {f } ∪ U 1 ∪ Pk,k−1 ∪ Ua,b,k−1

is complete in U .

Proof. Let f ∈ U \P olk (ιk−1 k−1 ) be arbitrary. Then, w.l.o.g. there are certain k (a1i , a2i , ..., aki ) ∈ Ek−1 (i = 1, ..., m) and (b1j , b2j , ..., bk−1,j ) ∈ ιk−1 k−1 (j = 1, ..., n) with ⎞ ⎛ ⎞ ⎛ 0 a11 a12 ... a1m b11 b12 ... b1n ⎜ ⎟ ⎜ a21 a22 ... a2m b21 b22 ... b2n ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎜ ⎟ f ⎜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎟ = ⎜ ... ⎟ ⎟. ⎝ ak−1,1 ak−2,2 ... ak−1,m bk−1,1 bk−1,2 ... bk−1,n ⎠ ⎝ k − 2 ⎠ k−1 ak1 ak2 ... akm k − 1 k − 1 ... k − 1

578

18 Further Submaximal Classes of Pk

By replacing of some variables of f through certain functions of Pk,k−1 one

can form an n-ary function f ∈ U \P olk (ιk−1 k−1 ) with f (k−1, k−1, ..., k−1) = k − 1. Consequently, the set {f } ∪ {g ∈ U 1 | g(k − 1) = k − 1} fulﬁlls the assumption (1) of Lemma 18.3.2.1. Then with the help of Lemma 18.3.2.1, one can prove [{f } ∪ U 1 ∪ Pk,k−1 ∪ Ua,b,k−1 ] = U . k−2 Lemma 18.3.2.4 Let γ ∈ Uk−1 ∪ Bk−1 ∪ k=2 Chk−1 be an h-ary relation. Then P olk γ is U -maximal. Proof. Clearly, P olk γ is a proper subset of U . For each n-ary function g ∈ P olk−1 γ we deﬁne an n-ary function fg ∈ Pk by ⎧ n x ∈ Ek−1 , ⎨ g(x) if x = (k − 1, k − 1, ..., k − 1), fg (x) := k − 1 if ⎩ 0 otherwise. It is easy to check that fg belongs to P olk γ . Moreover, we have Pk,k−1 ⊆ P olk γ and there are two diﬀerent elements a, b ∈ Ek−1 with Ua,b,k−1 ⊆ P olk γ . More precisely: If γ ∈ Uk−1 ∪ C2k−1 one chooses the elements a and b so that (a, b) ∈ γ\ι2k−1 . If γ has an arity h ≥ 3, then one can choose the elements a and b arbitrary. Now let f ∈ U \P olk γ be arbitrary. Then, w.l.o.g. there are certain tuh+1 (i = 1, ..., n) and (b1j , b2j , ..., bhj ) ∈ γ ples (a1i , a2i , ..., ah+1,i ) ∈ Ek−1 (j = 1, ..., m) with ⎛ ⎞ ⎛ ⎞ a1 a11 a12 ... a1n b11 b12 ... b1m ⎜ ⎜ a21 ⎟ a22 ... a2n b21 b22 ... b2m ⎟ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎜ ⎟ ⎟ · f⎜ ⎟ = ⎜ ... ⎟ ⎝ ⎝ ah1 ⎠ ah2 ... ahn bh1 bh2 ... bhm ah ⎠ ah+1,1 ah+1,2 ... ah+1,n k − 1 k − 1 ... k − 1 k−1 h and (a1 , ..., ah ) ∈ Ek−1 \γ. By replacing some variables of f through certain functions of Pk,k−1 ⊆ P olk γ one can form a function f ∈ [{f } ∪ P olk γ ], which preserves the relations {k − 1} and Ek−1 but does not preserve the relation γ. Then, the maximality of P olk−1 γ in Pk−1 implies that the set {f } ∪ {fg | g ∈ P olk γ} fulﬁlls the condition (1) of Lemma 18.3.2.1. Since we have already proven that P olk γ fulﬁlls condition (2) of Lemma 18.3.2.1, the set {f } ∪ P olk γ is complete in U . Hence P olk γ is U -maximal.

Lemma 18.3.2.5 Let 3 σEk−1 := Ek−1 ∪ {(a, a, b) | k − 1 ∈ {a, b} ⊆ Ek }.

Then

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

579

(1) For every function f ∈ U \P olk σEk−1 there are two diﬀerent elements a, b ∈ Ek−1 with Ua,b,k−1 ⊆ [{f } ∪ Pk,k−1 ∪ U0 ]. (2) P olk σEk−1 is U -maximal. Proof. (1): Let f ∈ U \P olk σEk−1 be arbitrary. W.l.o.g., we can assume that there are certain (a1j , a2j , a3j ) ∈ 3 3 (j = 1, 2, ..., n) and certain (bi , bi , ci ) ∈ Ek3 \Ek−1 (i = 1, 2, ..., m) with Ek−1 the properties ⎞ ⎛ ⎛ ⎞ a a11 ... a1n b1 ... bm f ⎝ a21 ... a2n b1 ... bm ⎠ = ⎝ b ⎠ , a31 ... a3n c1 ... cm k−1 a = b and {a, b} ⊆ Ek−1 . We choose some functions fi ∈ Pk,k−1 and gj ∈ U0 with the properties ⎛ ⎞ ⎛ ⎞ 0 0 a1i fi ⎝ 0 1 ⎠ = ⎝ a2i ⎠ a3i k−1 0 (i = 1, ..., n) and

⎛

⎞ ⎛ ⎞ 0 0 bj gj ⎝ 0 1 ⎠ = ⎝ bj ⎠ cj k−1 0

(j = 1, ..., m). Thus the binary function f with f := f (f1 , f2 , ..., fn , g1 , ..., gm ) belongs to [{f } ∪ Pk,k−1 ∪ U0 ] and it holds that ⎛ ⎞ ⎛ ⎞ 0 0 a f ⎝ 0 1 ⎠ = ⎝ b ⎠ . k−1 0 k−1 Next we show that every function of Ua,b,k−1 := {q ∈ U | Im(q) ⊆ {a, b, k−1}} is a superposition over {f } ∪ U0 ∪ Pk,k−1 : Let q t ∈ Ua,b,k−1 be arbitrary. Then the t-ary function q1 with 0 if q(x) ∈ {a, b}, q1 (x) := k − 1 otherwise, belongs to U0 and the t-ary function q2 with 1 if q(x) = b, q2 (x) := 0 otherwise, belongs to Pk,k−1 . Because of

580

18 Further Submaximal Classes of Pk

q = f (q1 , q2 ) ∈ [{f } ∪ Pk,k−1 ∪ U0 ] we have Ua,b,k−1 ⊆ [{f } ∪ Pk,k−1 ∪ U0 ]. (2): Clearly, U = P olk σEk−1 . Since from σEk−1 the relation is derivable, P olk σEk−1 ⊂ U holds. Let f ∈ U \P olk σEk−1 be arbitrary. Since Pk,k−1 ∪U0 ⊆ P olk σEk−1 , above statement (1) implies that Ua,b,k−1 ⊆ [{f } ∪ P olk σEk−1 ] is valid for two certain elements a, b ∈ Ek−1 , whereby {f } ∪ P olk σEk−1 fulﬁlls the condition (2) of Lemma 18.3.2.1. Obviously, each n-ary function p ∈ U with ∀x ∈ Ekn : p(x) = k − 1 ⇐⇒ x = (k − 1, k − 1, ..., k − 1) is a function of P olk σEk−1 . Consequently, {f } ∪ P olk σEk−1 also fulﬁlls the condition (1) of Lemma 18.3.2.1. Therefore, the U -maximality of P olk σEk−1 follows from Lemma 18.3.2.1. Lemma 18.3.2.6 Denote max ∈ Pk the binary function that is deﬁned with respect to the order 0 < 1 < 2 < ...k − 1 as usual. Then, the set U := {max} ∪ Pk,k−1 ∪ U0 is complete in U . Proof. Obviously, Pk,k−1 ∪ U0 ⊆ U . Since the function max has the property max(x, y) = k − 1 ⇐⇒ k − 1 ∈ {x, y}, max belongs to U . Consequently, U is a subset of U . Let f n ∈ U be arbitrary. Then, the n-ary functions f1 and f2 deﬁned by f (x), if f (x) ∈ Ek−1 , f1 (x) := 0 otherwise,

and f2 (x) :=

0, if f (x) ∈ Ek−1 , k − 1 otherwise,

belong to U . Hence, the U -completeness of U follows from f = max(f1 , f2 ). Lemma 18.3.2.7 Let 2 × {k − 1}) ∪ {(a, a, b) | {a, b} ⊆ Ek }. σ{k−1} := (Ek−1

Then: (1) For every function f ∈ U \P olk σ{k−1} and every set M ⊆ U with [prEk−1 M ] = Pk−1 , where prEk−1 M := {g n ∈ Pk−1 | ∃g1 ∈ M ∩ P olk Ek−1 : n : g(x) = g1 (x) }, ∀x ∈ Ek−1 is valid:

Pk,k−1 ⊆ [{f } ∪ U0 ∪ M ∪ (U 1 ∩ P olk {k − 1})].

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

581

(2) P olk σ{k−1} is maximal in U . Proof. (1): Let f ∈ U \P olk σ{k−1} be arbitrary. W.l.o.g. we can assume that there are certain ai , bi ∈ Ek−1 (i = 1, 2, ..., r), cj ∈ Ek (j = 1, 2, ..., s) and dl , el ∈ Ek−1 (l = 1, ..., t) with the property ⎛ ⎞ ⎛ ⎞ a1 ... ar k − 1 ... k − 1 d1 ... dt a f ⎝ b1 ... br k − 1 ... k − 1 d1 ... dt ⎠ = ⎝ b ⎠ , k − 1 ... k − 1 c1 ... cs e1 ... et c where a = b and {a, b, c} ⊆ Ek−1 . Then, because ⎛ a1 ... ar k − 1 ... k − 1 d1 f ⎝ b1 ... br k − 1 ... k − 1 d1 k − 1 ... k − 1 c1 ... cs d1

of f ∈ U , we have ⎞ ⎛ ⎞ ... dt a ... dt ⎠ = ⎝ b ⎠ ... dt c

for a certain c ∈ Ek−1 . W.l.o.g. let c = a. We choose some unary functions fi ∈ U 1 ∩ P olk {k − 1} (i = 0, 1, ..., r) and gj ∈ U0 with the properties 0 if x = a, f0 (x) = 1 if x ∈ Ek−1 \{a}, ⎞ ⎛ ⎞ ⎛ ai 0 fi ⎝ 1 ⎠ = ⎝ bi ⎠ k−1 k−1 (i = 1, ..., r) and

⎛

⎞ ⎛ ⎞ 0 k−1 gj ⎝ 1 ⎠ = ⎝ k − 1 ⎠ k−1 cj

(j = 1, ..., s). Therefore, the unary function f with f := f0 (f (f1 , f2 , ..., fr , g1 , ..., gs , cd1 , ..., cdt )) belongs to [{f } ∪ (U 1 ∩ P olk {k − 1}) ∪ U0 ], where f ∈ Pk,2 (because of f ∈ U and f (k − 1) = k − 1) and ⎛ ⎞ ⎛ ⎞ 0 0 f ⎝ 1 ⎠ = ⎝ 1 ⎠ k−1 1 are valid. Now it is possible to prove that Pk,2 ⊆ [{f } ∪ U0 ∪ M ] holds with

582

18 Further Submaximal Classes of Pk

the help of the following completeness criterion for Pk,2 (see Theorem 12.4.3): A subset T ⊆ Pk,2 is a generating system for Pk,2 if and only if T fulﬁlls the following two conditions: (i) The set pr{0,1} T := {f n ∈ P2 | ∃f1n ∈ T : ∀x ∈ E2n : f (x) = f1 (x)} is a generating system for P2 . (ii) For each a ∈ Ek−1 and each b ∈ Ek \{0, 1} with a < b it is valid: 0 1 a . T ⊆ Pk,2 ∩ P olk 0 1 b Clearly, the set pr{0,1} {f g | g ∈ M } fulﬁlls the above condition (i) because of our assumption prEk−1 M = Pk−1 . For arbitrary a ∈ Ek−1 , the unary function qa with ⎧ if x = a, ⎨0 x = k − 1, qa (x) := k − 1 if ⎩ 1 otherwise, belongs to U 1 ∩ P olk {k − 1}. Then, for arbitrary a ∈ Ek−1 and b ∈ Ek \{0, 1} with a < b we have a 0 (f qa ) = , b 1 whereby {f } ∪ (U 1 ∩ P olk {k − 1}) also fulﬁlls the conditions of (ii). Thus by the completeness theorem for Pk,2 , Pk,2 ⊆ [{f } ∪ U0 ∪ M ∪ (U 1 ∩ P olk {k − 1})] is valid. This and Lemma 18.3.2.2 imply our assertion (1). (2): Clearly, P olk σ{k−1} = U . Further, it is easy to prove that is derivable from σ{k−1} . Hence P olk σ{k−1} ⊂ U . Let f ∈ U \P olk σ{k−1} be arbitrary. Obviously, U0 ⊆ P olk σ{k−1} and each n-ary function p ∈ U (n ∈ N) with the property n ∀x ∈ Ekn : p(x) = k − 1 ⇐⇒ x ∈ Ekn \Ek−1

belongs to P olk σ{k−1} . In particular, max ∈ P olk σ{k−1} . Then [{f } ∪ P olk σ{k−1} ] = U follows from statement (1) and Lemma 18.3.2.6. Lemma 18.3.2.8 For certain t ∈ {1, 2, ..., k − 2} we set C := {t, t + 1, ..., k − 2}. Furthermore let 2 α := {(x, x) | x ∈ Ek } ∪ Ek−1 ∪ {(x, y), (y, x) | x ∈ Ek , y ∈ C},

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

583

i.e., α is a binary central relation, where C is the set of all central elements of α and α has the property Ek2 \α = {(u, k − 1), (k − 1, u) | u ∈ Et }. Then max ∈ U ∩ P olk α. Proof. Because of Lemma 18.3.2.6, we have max ∈ U . Suppose max does not preserve the relation α. Then there are certain (a, b), (c, d) ∈ α and a u ∈ Et with a c u max = . b d k−1 W.l.o.g. we can assume that a = u and d = k − 1, i.e., c ≤ u and b ≤ k − 1. Then, because of u ∈ Et we have however c ∈ Et , contrary to (c, d) = (c, k − 1) ∈ α. Lemma 18.3.2.9 Let α ∈ C2k be a binary central relation with ⊆ α and the property that at least an element of Ek−1 is a central element of α. Then U ∩ P olk α is U -maximal. Proof. Because of ⊆ α and ∈ Dk , element k − 1 cannot be a central element of α. Therefore, w.l.o.g. we can assume that the set C := {t, t + 1, ..., k − 2} for a certain t ∈ {1, 2, ..., k − 2} is the set of all central elements of α, whereby we have 2 ∪ {(x, y), (y, x) | x ∈ Ek , y ∈ C} α = {(x, x) | x ∈ Ek } ∪ Ek−1

and

Ek2 \α = {(u, k − 1), (k − 1, u) | u ∈ Et }.

Notice that, with that, k − 2 is a central element of α. Let f n ∈ U \P olk α be arbitrary. Then there are certain (a1 , b1 ), ..., (an , bn ) ∈ α with a1 a2 ... an a f = b1 b2 ... bn k−1 and (a, k − 1) ∈ α. Since the functions tr,s (i = 1, ..., n) with (r, s) ∈ α and qa (a ∈ Ek−1 ) with r f¨ u x ∈ Ek−1 , tr,s (x) := s if x = k − 1, and

⎧ f¨ u x = a, ⎨b qa,b (x) := k − 2 if x ∈ Ek−1 \{a}, ⎩ k − 1 if x = k − 1,

for arbitrary b ∈ Ek−1 belong to U ∩ P olk α, the functions

584

18 Further Submaximal Classes of Pk

tb,k−1 = qa,b (f (ta1 ,b1 , ..., tan ,bn )) and tk−1,b = qa,b (f (tb1 ,a1 , ..., tbn ,an )) are superpositions over {f } ∪ (U ∩ P olk α) for all b ∈ Ek−1 . Consequently, U0 ⊆ [{f } ∪ (U ∩ P olk α)]. Moreover, max ∈ U ∩ P olk α (see Lemma 18.3.2.8) and (because of ⊆ α) Pk,k−1 ⊆ P olk α. Then the U -completeness of {f } ∪ (U ∩ P olk α) follows from Lemma 18.3.2.6. 18.3.3 Proof of the Completeness Criterion for U Let M be an arbitrary subset of U , which is not contained in any class from type I–VI. We show that U = [M ] results from this assumption when we prove the generating system from Lemma 18.3.2.3 in [M ]. Since M is no subset of a class of type I, it results from the completeness theorem for P2 that the following is valid: ∀g m ∈ P2 ∃g1m ∈ [M ] : αϕ (g1 ) = g.

(18.19)

In particular, (18.19) implies: ck−1 ∈ [M ].

(18.20)

With the aid of the functions of M , which do not belong to the clones of type II, one sees from (18.20) that the constant functions of Pk belong to [M ]: c0 , c1 , ..., ck−1 ∈ [M ].

(18.21)

Because of (18.19), there is a unary function in [M ] that is an inverse image of the function g ∈ [M ] with x g(x) 0 1 . 1 0 Let g be this function with x 0 1 2 . . k−2 k−1 where a is a certain element of Ek−1 . The relation

g (x) k−1 k−1 k−1 , . . k−1 a

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

585

2 Ek−1 ∪ {(k − 1, k − 1), (a, k − 1), (k − 1, a)}

describes a certain class of the type V. Thus there is a function q ∈ M that does not preserve this relation. W.l.o.g. we can assume b a1 a2 ... an k − 1 a k − 1 = , q k−1 b1 b2 ... bn k − 1 k − 1 a where a1 , ..., an , b1 , ..., bn are certain elements of Ek−1 . Because of q ∈ U , we have also b a1 a2 ... an k − 1 a k − 1 = . q k−1 a1 a2 ... an k − 1 k − 1 a Consequently, the functions tb,k−1 (x) := q(ca1 (x), ..., ca2 (x), ck−1 (x), g (g (x)), g (x)) and

tk−1,b (x) := q(ca1 (x), ..., ca2 (x), ck−1 (x), g (x), g (g (x)))

belong to [M ], where x tb,k−1 (x) 0 b 1 b 2 b . . . . k−2 b k−1 k−1

tk−1,b k−1 k−1 k−1 . . . k−1 b

With the aid of further functions of M , which do not belong to the clones of type V, in analog mode, we see that all functions of the form α if x ∈ Ek−1 , tα,β (x) := β if x = k − 1, belong to [M ] for all α, β ∈ Ek−1 with k − 1 ∈ {α, β}. Next we show that U0 := {f ∈ U | |Im(f )| ≤ 2 ∧ k − 1 ∈ Im(f )} ⊆ [M ]

(18.22)

holds. For this purpose, let f n be an arbitrary function of U0 with Im(f ) = {a, k−1}. Because of (18.19) there is an n-ary function f ∈ [M ] with the property ∀x ∈ Ekn : f (x) = k − 1 ⇐⇒ f (x) = k − 1. With the aid of the above function ta,k−1 ∈ [M ], one can prove that

586

18 Further Submaximal Classes of Pk

ta,k−1 f = f ∈ [M ]. Therefore, and because of (18.21), (18.22) is valid. If one forms superpositions over the constants and the functions of M , which do not belong to the clones of type III, one obtains functions gγm ∈ [M ] with x

gγ (x)

m ∈ Ek−1

}= gγ ∈ P olk−1 γ

otherwise

certain values

for all γ with the properties: (a) P olk−1 γ is maximal in Pk−1 and (b) c0 , ..., ck−1 ∈ P olk−1 γ. Hence, because of (18.21), there are functions gγ ∈ [M ] that do not fulﬁll (b). This and the completeness criterion for Pk−1 imply that the following function belongs to [M ]: x

y

s(x, y)

2 ∈ Ek−1

}= max(x, y) + 1 (mod k − 1)

k−1 k−1 otherwise

α certain values

We distinguish two cases for α := s(k − 1, k − 1): Case 1: α ∈ Ek−1 . By s ∈ [M ] the functions s := ∆s and

s := (s )k−1 ,

(s (x) = s(x, x))

(s (x) = s (s (s (...s (x)...))) ) k−1

belong to [M ]. Let β := s (k − 1) and γ ∈ Ek−1 \{β}. With the help of the completeness criterion for Pk,k−1 , one can easily prove that the set M1 := {s s, tβ,γ = s tk−1,γ } ⊆ [M ] is a generating system for Pk,k−1 . The completeness criterion for Pk,k−1 is a special case of Theorem 12.4.3 and says: An arbitrary subset T ⊆ Pk,k−1 is a generating system for Pk,k−1 , if and only if T fulﬁlls the following two conditions:

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

587

(i) The set n pr T := {f n) ∈ Pk−1 | ∃f1n ∈ T : ∀x ∈ Ek−1 : f (x) = f1 (x)}

is a generating system for Pk−1 . (ii) For all a ∈ Ek : T ⊆ Pk,k−1 ∩ P olk

0 1 ... k − 2 a 0 1 ... k − 2 k − 1

.

Because of s s ∈ Pk,k−1 , ∀x, y ∈ Ek−1 : (s s)(x, y) = max(x, y) + 1 (mod k − 1) and the fact that max(x, y)+1 is a Sheﬀer function for Pk−1 fulﬁlls (i). Because of a β tβ,γ = k−1 γ

6

the set T := M1

T := M1 also fulﬁlls the condition (ii). Thus Pk,k−1 ⊆ [M ] was proven in Case 1. Then Lemma 18.3.2.5 implies the existence of two diﬀerent elements a, b ∈ Ek−1 with Ua,b,k−1 ⊆ [M ].

(18.23)

Except for the unary function e ∈ U \U0 with x e(x) 0 1 . ∈ Ek−1 . k−2 k−1 k−1 we have proven that the other unary functions of U belong to [M ]. Next, with the aid of the completeness theorem for Pk−1 , we show that the functions of the form e are superpositions over {c0 , c1 , ..., ck−1 } ∪ U0 ∪ Pk,k−1 ⊆ [M ] and functions of M , which do not belong to the clones of type IV. 6

See Theorem 7.1.5.

588

18 Further Submaximal Classes of Pk

k−2 Let γ ∈ Uk−1 ∪Bk−1 ∪ h=1 Ck−1 be an h-ary relation; i.e., γ is a totally reﬂexive and totally symmetric relation that describes the maximal class P olk−1 γ of Pk−1 . Then, the relation h+1 ∪ {(x, x, ..., x, y) | x, y ∈ Ek } γ := Ek−1

∪{(x1 , ..., xh , k − 1) | (x1 , ..., xh ) ∈ γ} describes a maximal class of U of type IV. If one replaces the variables of a function fγ ∈ M \P olk γ by certain functions of the set {c0 , c1 , ..., ck−1 } ∪ U0 ∪ Pk,k−1 ⊆ [M ], one receives an n-ary function fγ ∈ [M ] with fγ (x)

x n ∈ Ek−1

*

= fγ (x) ∈ P olk−1 γ

(18.24)

k − 1 k − 1 ... k − 1 k−1 otherwise certain values k−2 for all γ ∈ Uk−1 ∪Bk−1 ∪ h=1 Ck−1 . Since function ta,k−1 belongs to U0 ⊆ [M ] for every a ∈ Ek−1 , there are also functions fγ ∈ [M ] with the property (18.24) for all γ ∈ Sk−1 ∪ C1k−1 . Because of (18.23), there is a function fγ ∈ [M ] with the property (18.24) for every γ ∈ Mk−1 ∪ Lk−1 . With the help of the completeness criterion for Pk−1 and the fact that the function fγ ∈ [M ] k−2 preserves the relation {k −1} for every γ ∈ Mk−1 ∪Sk−1 ∪Uk−1 ∪ h=1 Ck−1 ∪ k−1 1 1 h=3 Bk−1 , one can prove that U ∩ P olk {k − 1} ⊆ [M ]. Consequently, U ⊆ [M ]. To summarize, we have proven Pk,k−1 ∪ U0 ∪ Ua,b,k−1 ∪ U 1 ⊆ [M ] in Case

1. Since a function of U \P olk (ιk−1 k−1 ) belongs to M , this and Lemma 18.3.2.3 implies [M ] = U . Case 2: α = k − 1. Since function s preserves {k − 1} in this case and s is a Sheﬀer function for 2 Pk−1 on the tuples of Ek−1 , every unary function e with e(k − 1) = k − 1 is a superposition over M . Furthermore, [prEk−1 M ] = Pk−1 . Since we have proven that U0 ⊆ [M ] holds, the ﬁrst statement of Lemma 18.3.2.7 implies Pk,k−1 ⊆ [M ]. Therefore, there exists a function of [M ] that fulﬁlls the assumption of Case 1. Consequently, [M ] = U in Case 2, and we have proven the following theorem: Theorem 18.3.3.1 (Completeness Theorem for U ) (1) The clones deﬁned in Section 18.3.1 are the only maximal classes of U . (2) An arbitrary set M ⊆ U is U -complete if and only if there exists no class of the type I, II, ... or V I that has M as a subset.

19 Minimal Classes and Minimal Clones of Pk

In this chapter we deal with classes (of Lk ), which are either direct predecessors of the empty set (so-called minimal classes) or which are direct predecessors of the set of all projections (so-called minimal clones). It will turn out that it is not heavy to determine the minimal classes. However, for the minimal clones only partial results can be given.

19.1 Minimal Classes A subclass A of Pk is called a minimal class, if no subclass B = ∅ with B ⊂ A exists. Because of [∆n−1 f ] ⊂ [f ] for every function f ∈ Pk \[Pk1 ], every minimal classes has the order 1. If k = 2 then [c0 ], [c1 ] and [e11 ] are the only minimal classes. For arbitrary k one can easy check that Jk is a minimal class and, for every other minimal class A, there exists a unary function f ∈ Pk1 (k − 1) with the property f f = f and [f ] = A. Then a minimal class of P6 is e.g. x f (x) 0 1 1 1 2 1 3 3 4 4 5 4 By generalizing this example, one obtains the following theorem which is a conclusion from [P¨ os-K 79], Section 4.4.

590

19 Minimal Classes and Minimal Clones of Pk

Theorem 19.1.1 Pk contains exactly 1+

k−1 r=1

k · rk−r r

minimal classes. These are class Jk and the classes of the form [f ] with f ∈ Pk1 (k − 1) and f f = f . One can describe a function f ∈ Pk1 (k − 1) with the property f f = f as follows: Let θ be an equivalence relation on Ek , which is diﬀerent from ι2k (= κ0 ) and has exactly r equivalence classes. Further, denote a := (a1 , ..., ar ) an r-tuple with elements, which in pairs are not θ-equivalent. For such a θ und such an a, one can deﬁne the mapping faθ ∈ Pk by faθ (x) = ai :⇐⇒ x ∈ [ai ]θ . Then, for a function f ∈ Pk1 (k − 1), f f = f holds, if and only if there is an equivalence relation θ and a tuple a of the above form with f = faθ .

19.2 The Five Types of Minimal Clones A clone A ⊆ Pk is called minimal in Pk , iﬀ Jk is a maximal clone of A, i.e., it holds (19.1) ∀f ∈ A \ Jk : [Jk ∪ {f }] = A. Therefore, a basis of a minimal clone has at most two elements. Furthermore, it is easy to see that the following implication holds: (A is a minimal clone with ord A ≥ 2) =⇒ ∀f ∈ A \ Jk : A = [f ].

(19.2)

From Chapter 3 it follows directly: Theorem 19.2.1 P2 has exactly 7 minimal clones. These are the clones I ∪ C0 = [e11 , c0 ], I ∪ C1 = [e11 , c1 ], I = [e11 ], K = [∧], D = [∨], L ∩ S ∩ T0 = [r] and [S ∩ M ∩ T0 ] = [h32 ]. One has solved the problem of describing minimal clones of the form [f n ] ⊆ Pk for arbitrary k thus far only for n = 1.

Theorem 19.2.2 ([P¨ os-K 79], [Har 74]; without proof ) Pk has exactly k−1 r=1

k−2 k k · rk−r + · r r r=0

p·t=k−r,p∈P,t∈N

pt

(k − r)! · t! · (p − 1)

19.2 The Five Types of Minimal Clones

591

minimal clones of the order 1. A minimal clone A of the order 1 has either the form A = Jk ∪ [f ], where [f ] is a minimal class diﬀerent to Jk (see Theorem 19.1.1), or has the form A = [s], where s ∈ Pk1 [k] \ {e11 } is a permutation for which there is a prime number p with sp = e11 . The following theorem supplies the existence of at most ﬁnite many minimal clones (for a ﬁxed k) and a coarse division of the minimal clones. Theorem 19.2.3 (Rosenberg’s Classiﬁcation of the Minimal Clones; [Ros 82]) For every k ∈ N \ {1} there is only ﬁnite many minimal clones. If A = [Jk ∪ {f n }] is an arbitrary minimal clone of Pk , where [An−1 ] = Jk , then this clone fulﬁlls one of the following ﬁve conditions: (1) n = 1 and A is described in Theorem 19.2.2. (2) n = 2 and f is idempotent, i.e., f (x, x) = x holds for arbitrary x ∈ Ek . (3) n = 3 and ∀x, y ∈ Ek : f (x, x, y) = f (x, y, x) = f (y, x, x) = y,

(19.3)

i.e., f is a so-called ternary minority function. A minority function g 3 of Pk generates a minimal clone if and only if g(x, y, z) = x ⊕ y ⊕ z and (Ek ; ⊕) is an elementary 2-group. (4) n = 3 and ∀x, y ∈ Ek : f (x, x, y) = f (x, y, x) = f (y, x, x) = x, (5)

(19.4)

i.e., f is a so-called ternary majority function. n ∈ {3, 4, ..., k} and f is a semiprojection, i.e., there exists an i ∈ {1, ..., n} with f (a1 , ..., an ) = ai for every tuple (a1 , ..., an ) ∈ Ek with |{x1 , ..., xn }| ≤ n − 1.

1

Proof. Let A be an arbitrary minimal clone of the order n ∈ N. Because of Theorem 19.2.2 and by (19.2), we can assume w.l.o.g. that n ≥ 2 and that A = [f n ] for certain f ∈ Pk . Obviously, because of ord f ≥ 2 and the minimality of the clone A, it holds: ∀i ∈ {1, ..., n − 1} : ∆i f ∈ Jk

(19.5)

In particular this implies f (x, x, ..., x) = x, i.e., f is idempotent. Thus, (2) is proven for n = 2. Because of (19.5) we obtain for n = 3 that 1

´ The following statement is also mentioned Swierczkowski Lemma in the literature. ´ S. Swierczkowski published in [Swi 60] a theorem from which statement (5) of Theorem 19.2.3 results.

592

19 Minimal Classes and Minimal Clones of Pk

{f (x, x, y), f (x, y, x), f (y, x, x)} ⊆ {x, y}. Consequently, the following eight cases are possible: 1 f (x, x, y) = x f (x, y, x) = x f (y, x, x) = x

2 x x y

3 x y x

Case 4 5 x y y x y x

6 y x y

7 y y x

8 y y y

In Case 1, the function f fulﬁlls the condition (19.4). In Cases 2, 3, and 5 the function f is a semiprojection, i.e., f fulﬁlls the condition (5). Case 8 gives the condition (19.3). The remaining cases 4, 6, and 7 cannot occur for a function f which generates a minimal clone. This can be shown as follows: Suppose it holds Case 6; i.e., we have f (x, x, y) = f (y, x, x) = y and f (x, y, x) = x. Then the function g(x, y, z) := f (x, f (x, y, z), z) is a majority function. Since every ternary superposition t ∈ Jk over the majority function f is a majority function, too 2 , this is a contradiction to the minimality of clone A. In analog mode, the other cases can be led to a contradiction. Let now n = 4. Then, by assumption [A3 ] = Jk . Therefore, in particular ∆f ∈ Jk , i.e., we have f (x1 , x1 , x3 , x4 ) ∈ {x1 , x3 , x4 }. Then the following two cases are possible: Case 1: f (x1 , x1 , x3 , x4 ) = x1 . Then, because of [A3 ] = Jk it holds: xa := f (x1 , x2 , x1 , x4 ) ∈ {x1 , x2 , x4 }, xb := f (x1 , x2 , x3 , x1 ) ∈ {x1 , x2 , x3 }, xc := f (x1 , x2 , x2 , x4 ) ∈ {x1 , x2 , x4 }, xd := f (x1 , x2 , x3 , x2 ) ∈ {x1 , x2 , x3 }, xe := f (x1 , x2 , x3 , x3 ) ∈ {x1 , x2 , x3 }. If one puts x1 = x2 in the above equations and if one compares this with f (x1 , x1 , x3 , x4 ) = x1 , it follows {a, b, c, d, e} ⊆ {1, 2}. Since there exists a t ∈ {1, 2} with f (x1 , x2 , x1 , x1 ) = xt in addition, one obtain from the above equations: a = b = c = d = e = t. Thus f is a semiprojection in Case 1. 2

Use induction on the number of occurrences of f in t; see e.g. [Sza 83b] or [Qua 95].

19.2 The Five Types of Minimal Clones

593

Case 2: There exists an i ∈ {3, 4} with f (x1 , x1 , x3 , x4 ) = xi . If i = 3 then f (x1 , x2 , x3 , x3 ) = x3 . Consequently, we can continue analogously to the ﬁrst case with the proof. If i = 4, then we have f (x1 , x2 , x3 , x1 ) = x1 and one can show (as in the ﬁrst case) that f is a semiprojection. In analog mode to the case n = 4, one can examine the case n ≥ 5. It remains to show that a minority function g 3 , which generates a minimal clone, fulﬁlls condition (4). One can show this most easily with the aid of the ´ Szendrei in [Sze 87]: following statement, proven by A. Let k ≥ 2 and d ∈ Pk be a ternary function, which fulﬁlls the Mal’tsevcondition d(x, y, y) = d(y, y, x) = x. Then [d] is a minimal clone of Pk if and only if there exists an elementary Abelean p-group (Ek ; ⊕) with d(x, y, z) = x ) y ⊕ z. 3 Since a minority function g fulﬁlls the Mal’tsev-condition according to deﬁnition, the function g has the form given in the above statement. However, the function g(x, y, z) = x ) y ⊕ z is a minority function iﬀ p = 2.

Theorem 19.2.4 ([Pal 86]) For every t ∈ {1, 2, ..., k} there is a minimal clone of the order t. Proof. For t = 1 one can ﬁnd examples for minimal clones of the order 1 in Theorem 19.2.1. In [P¨ os-K 79] one ﬁnds proof that the binary function f with f (x, y) = x ◦ y, where the operation ◦ fulﬁlls the equations x ◦ x = x (idempotent law), (x ◦ y) ◦ z = x ◦ (y ◦ z) (associative law), (x ◦ y) ◦ x = x ◦ y (absorption law), generates a minimal clone. Let t ∈ {3, ..., k − 1}. Denote a1 , ..., at+1 pairwise distinct elements of Ek . The t-ary semiprojection f is deﬁned by at+1 if x1 = a1 ∧ {x2 , ..., xt } = {a2 , ..., at }, f (x1 , ..., xt ) := x1 otherwise. In [Pal 86] it was proven that [f ] is a minimal clone of the order t. Put x1 if |{x1 , ..., xk }| ≤ k − 1, g(x1 , ..., xk ) := x2 otherwise. Then it is easy to see that [g] is a minimal clone of the order k. Of the many results found in the literature on minimal clones, only some are given without proof. 3

One ﬁnds this statement also proven in [Qua 95].

594

19 Minimal Classes and Minimal Clones of Pk

Theorem 19.2.5 ([Csa 83b]; without proof ) P3 has exactly 84 minimal clones. One obtains every one of these clones by using an inner automorphism of P3 onto exactly one of the 24 following clones (under that 4 of the order 1, 12 of the order 2 and 8 of the order 3): [j2 ], [u2 ], [s2 ], [s4 ] (see Table 15.1), [bi ] and [mj ], where i ∈ {1, 2, ..., 12}, bi idempotent, j ∈ {1, 2, ..., 8}, mj majority function and x 0 1 0 2 1 2

y 1 0 2 0 2 1

b1 0 0 0 0 0 0

b2 0 0 0 2 0 2

x 0 0 1 1 2 2

y 1 2 0 2 0 1

z 2 1 2 0 1 0

b3 0 0 0 0 1 1

b4 0 0 0 0 1 2

b5 0 0 0 2 1 1

b6 0 0 0 2 1 2

b7 0 0 0 2 2 2

b8 0 1 0 2 0 0

b9 b10 b11 b12 0 0 0 2 1 2 0 2 0 0 2 1 2 1 2 1 0 1 1 0 2 2 1 0

m1 m 2 m 3 m 4 m 5 m 6 m 7 m 8 0 0 2 0 0 0 0 1 0 1 1 0 0 0 0 2 0 1 2 0 0 0 2 0 0 0 0 0 0 2 2 2 0 0 1 0 2 2 1 0 0 1 0 0 2 2 1 1

Note that the cardinality of the set of all subclasses of P3 , which contain a ﬁxed minimal clone of the order 1, can be found in [Pan-V 2000]. It was proven by B. Szczepara in his 210-page long Ph.D. thesis that there are exactly 2182 binary minimal clones for k = 4 (see [Szc 95]). All minimal clones, which are generated by majority functions of P4 , were determined by T. Waldhauser in [Wal 2000]. In [Lev-P 96] one can ﬁnd all minimal clones C of the order 2 with |C 2 | ∈ {3, 4, 6}. Furthermore, one can ﬁnd examples for minimal clones C of the order 2 with |C 2 | = 2t + 2 (t ≥ 1) or |C 2 | = 3t + 2 (t ≥ 2) in this paper. Up to isomorphic functions, all binary commutative functions, which generate minimal clones of the order 2, were determined in [Kea-S 99]. In [P¨ os-K 79] it was proven that Pk is the smallest clone, which contains all minimal clones of Pk . Further, it was proven that Jk is the intersection of all maximal clones of Pk . The following problem results from that: How many clones M1 , ..., Mt does one need so that at least t minimal (or maximal) t [ i=1 Mi ] = Pk (or i=1 Mi = Jk ) is valid, respectively? In [Zsa 92] it was proven that t ≤ 3 holds. The solution t = 2 for the above problem can be found in [Cz´e-H-K-P-S 2001]. A necessary and suﬃcient condition for [f, g] = Pk , if [f ] and [g] are minimal clones, can be found in [Ros-M 2001]. The answer 3 for the corresponding problems during the investigation of partial minimal (or maximal) clones was proven in [Had-M-R 2002].

19.2 The Five Types of Minimal Clones

595

One ﬁnds supplements to this chapter in the survey articles [Qua 95] and [Csa 2002]. In the next chapter, we deal with partial functions, for which we will handle similar problems as in the previous chapters. The next theorem shows that the problem of determining minimal partial clones can be reduced to the problem of determining minimal clones. Theorem 19.2.6 ([B¨ or-H-P 91]; without proof ) k . Then C is either a minimal clone of Pk or Let C be a partial clone of P C is generated by a partial n-ary projection with a nontrivial totally reﬂexive and totally symmetric domain D(e) ⊂ Ekn for certain n ≤ k. Denote t(k) the number of all minimal clones of Pk and m(k) the number of k . Then all partial minimal clones of P m(k) = t(k) +

k

k (2( i ) − 1).

i=1

In particular, we have m(2) = 11 and m(3) = 99, where t(2) = 7 and t(3) = 84 (see Theorems 19.2.1 and 19.2.5).

20 Partial Function Algebras

In Part I, Chapter 1, we introduced the concept of partial operation over a set A. By choosing A = Ek and replacing the concept of “operation” with the concept of “function”, we get the concept “partial function” over Ek . One k of can then introduce certain modiﬁed Mal’tsev-operations over the set P all partial functions on Ek . Then the set Pk together with these operations k ; τ, ζ, ∆, ∇, ), which can forms a so-called (full) partial function algebra (P be examined similar to the function algebra (Pk ; τ, ζ, ∆, ∇, ). The choice of results on partial function in this chapter focuses on questions that were already treated for Pk in the previous chapter. After a composition of some basic concepts in Section 20.1, Section 20.2 shows k is isomorphic to a certain sublatthat the lattice of all partial clones of P tice of the lattice of all clones of Pk+1 . Thus, one gets many properties of the partial clones from the properties of the clones that were already found. To ﬁnd certain partial clones, however, with the aid of the above-mentioned isomorphism, is not possible, because of the absence of results on clones. One k with the help could, for example, not solve the completeness problem for P of isomorphism. In Section 20.3, we show how one can describe partial clones by relations. Sections 20.4 and 20.5 deal with the maximal partial clones with whose aid, analogously to Pk , one can solve the completeness problem of the partial 3 has exactly 58 maximal 2 has exactly 8 and P logic. We will prove that P partial clones. In Section 20.5 one can ﬁnd the complete list of all maxik for arbitrary k ∈ N, which was found by L. Haddad and mal clones of P I. G. Rosenberg. The list is given without proof. In Section 20.6, we determine the descriptive relations of the maximal clones of Pk that are also k . In addition, a surdescriptive relations of the maximal partial clones of P vey those papers that deal with determining the orders of the maximal partial clones. Section 20.7 deals with determining the cardinality of the set k | C = [C] ∧ C ∩ Pk = A}, where A is an arbitrary maximal I(A) := {C ⊆ P clone of Pk . We prove, that, if A has the type U, S or C, I(A) is a ﬁnite set.

598

20 Partial Function Algebras

On the other hand, the set I(A) has the cardinality of continuum, if A has the type L or B. For the type M we can give only partial results. Section 20.8 gives a survey on the cardinalities of the sets I(A), where A is an arbitrary subclass of P2 . In last section, we determine the congruences on the maximal partial clones. It k has exactly 4 congruences, whereas a maximal is proven particularly that P partial clone has exactly 4, 8, or 10 pairwise distinct congruences.

20.1 Basic Concepts Let A be nonempty sets and let T be a proper subset of An . For an arbitrary mapping f from T into A and for (a1 , ..., an ) ∈ An \T , one can use the notation f (a1 , ..., an ) = ∞ with ∞ ∈ A, to indicate that f (a1 , ..., an ) is not deﬁned. Then D(f ) := {(x1 , ..., xn ) ∈ An | f (a1 , ..., an ) = ∞} is the domain of f . In the following, we use these notations for A = Ek and for functions which are deﬁned over subsets of Ekn . More exact: For a ﬁxated k ∈ N \ {1} let k := Ek ∪ {∞}. E An n-ary mapping f of the form k f : Ekn −→ E is called (n-ary) partial function. and the set of all n-ary partial functions let n k . P Furthermore, we put

k := P

n

k . P

n≥1

By the above deﬁnition, the functions of Pk are also partial functions. To k \ Pk , we call these the distinguish these functions from functions of the set P total functions of Pk . n n k , if g(a) ∈ {f (a), ∞} k is called subfunction of f ∈ P A function g ∈ P holds for all a ∈ Ekn . In the following, the notation g ⊆p f is used to indicated the fact that g is a subfunction of f . We take the notations introduced for total functions and we set

20.1 Basic Concepts

599

cn∞ (x1 , ..., xn ) := ∞, i.e., cn∞ is the notation for the n-ary function with empty domain, and C∞ := {cn∞ | n ∈ N}. The reduction of a function f n ∈ Pk to T ⊂ Ekn is a function deﬁned by f (x), if x ∈ T, g(x) := ∞ otherwise, which is also denoted with f|T . k . Now we consider the operations over P n m For arbitrary f , g ∈ Pk we deﬁne the unary Mal’tsev-operations analogous to Chapter 1 and the binary operation as follows: (ζf )(x1 , x2 , ..., xn ) := f (x2 , x3 , ..., xn , x1 ), (τ f )(x1 , x2 , ..., xn ) := f (x2 , x1 , x3 , ..., xn ), (∆f )(x1 , x2 , ..., xn−1 ) := f (x1 , x1 , x2 , ..., xn−1 ) for n ≥ 2, ζf = τ f = ∆f = f for n = 1, (∇f )(x1 , x2 , ..., xn+1 ) := f (x2 , x3 , ..., xn+1 ) and (f ∗ g)(x1 , ..., xm+n−1 ) := f (g(x1 , ..., xm ), xm+1 , ...., xm+n−1 ) if g(x1 , ..., xm ) ∈ Ek , ∞ otherwise. We adopt the concepts (such as closure, closed set, clone ...) and the notations (such as [...]) coupled with the Mal’tsev-operations from Chapter 1. If distinctions are required, we complete these concepts with the word “partial”. For k , which contains example,“partial clone” instead of “clone” if a closed set of P the set Jk of all projections of Pk , is meant. A partial clone C is called strong if it contains all subfunctions of its functions. If C is a clone of Pk , then let k | ∃g ∈ Pk : g|D(f ) = f }. Str(C) := {f ∈ P It is easy to check that Str(C) is a partial strong clone. As we see in Section 20.3, partial clones, like the clones of Pk , are describable k ) suitably chosen. with the help of relations (on E At ﬁrst we want to show, however, how one can embed the lattice of the k into the lattice of the subclones of Pk+1 isomorphically. subclones of P

600

20 Partial Function Algebras

20.2 One-Point Extension In this section, let A := Ek and B := Ek ∪ {∞}. The following mappings establish relations between P A := Pk and PB (isomorphic to Pk+1 ): n n + : P A −→ PB , f → f+ , n n − : PB −→ P A , g → g− n where f+ is the so-called extended function deﬁned by f (x) if x ∈ D(f ), f+ (x) := ∞ otherwise, n and g− , the so-called restricted function, is deﬁned by g(x) if x ∈ Ekn ∧ g(x) ∈ Ek , g− (x) := ∞ otherwise.

Furthermore, for subsets F ⊆ PA and G ⊆ PB we put: F+ := {f+ | f ∈ F } and G− := {g− | g ∈ G}. To distinguish the projections from PA of the projections from PB , we use the notation eni,X with X ∈ {A, B}, i.e., ∀x1 , ..., xn ∈ X : eni,X (x1 , ..., xn ) := xi (i ∈ {1, ..., n}) holds. Let JX be the set of all projections of the form eni,X . One can prove that, if f ∈ P A and D(f ) = ∅, ∇f+ ∈ F+ holds. In the following, the clones (⊆ PB ), which are formed by closing certain subsets of PB , are important to further considerations: k )+ ]. H := [(Jk )+ ] and U := [(P The next Lemma gives properties of the clones deﬁned above. Lemma 20.2.1 ([Ros 88]) (1) A function q t ∈ PB belongs to U , if and only if the following two statements hold: (a) q preserves ∞ and (b) for every essential place i of q and for arbitrary b1 , ..., bn ∈ B it holds q(b1 , ..., bi−1 , ∞, bi+1 , ...,t ) = ∞. (2) H is a minimal clone of PB .

20.2 One-Point Extension

601

Proof. (1): Denote U the set of all functions of PB , which fulﬁlls conditions (a) and (b) above. At ﬁrst we show that U is a clone. The equation U = U from which the assertion (1) immediately follows is then proven. Let f n , g m ∈ U be arbitrary. Then obviously τ f, ζf, ∇f ∈ U and ∆f ∈ U for n = 1. Let n > 1. If ∆ f depends on the ﬁrst place essentially, then f depends at least on the places 1 and 2 essentially. Consequently, we have (∆ f )(∞, x3 , ..., xn ) = f (∞, ∞, x3 , ..., xn ) = ∞ for all x3 , ..., xn ∈ B. If ∆f depends on the i-th place with 2 ≤ i ≤ n − 1 essentially, then f depends on the (i + 1)-th place essentially and (∆ f )(x1 , x3 , ..., xi−1 , ∞, xi+1 , ..., xn ) = f (x1 , x1 , x3 , ..., xi−1 , ∞, xi+1 , ..., xn ) =∞ holds for all x1 , x3 , .., , , xn ∈ B. Consequently, the function ∆ f also belongs to U in the case n > 1. To prove f g ∈ U , we put hm+n−1 := f g. Obviously h fulﬁlls the condition (a). Let h be dependent of the i-th place and let x := (x1 , ..., xm+n−1 ) ∈ B m+n−1 be arbitrary, where xi = ∞. Then the following two cases are possible: Case 1: i ∈ {1, ..., m}. In this case, the ﬁrst place of f and the i-th place of g is essential. Thus by assumption, we have g(x1 , ..., xm ) = ∞ and h(x) = f (∞, xm+1 , ..., xm+n−1 ) = ∞. Case 2: i ∈ {m + 1, ..., m + n − 1}. Since in this case f depends on the (i − m + 1)-th place essentially, h(x) = ∞ holds. It also holds that h = f g ∈ U . Thus U is a closed set. Since the function e21,B fulﬁlls conditions (a) and (b), U is a clone. It is easy to check that (PA )+ ⊆ U and U ⊆ U hold. For the missing proof of U ⊆ U , we consider an arbitrary function q t ∈ U with at least an essential place. Using the operations ζ, τ, ∆ we get from q a function q1 ∈ U that depends on all its places essentially and which belongs to (PA )+ . Since we can get again the function q by using the operations ∇, ζ, τ from q1 , we have q ∈ [(PA )+ ] = U . Thus U = U and (1) is proven. (2): It is easy to check that JB ⊂ [(e21,A )+ ] = H holds. To prove the maximality of JB in H we put b := |B| and := {(x1 , ..., xb ) ∈ B b | |{x1 , ..., xb }| = b}. Since (e21,A )+ ∈ P olB , we have H ⊆ P olB . Let q t ∈ H \JB be arbitrary. Since no constants belong to P olB , the function q has at least an essential place i. Suppose i is the only one essential place of q. Then we have q(x1 , ..., xi−1 , xi , xi+1 , ..., xn ) = g(xi , ..., xi , ..., xi ) = xi , i.e., q t = eti,B , in contradiction to q ∈ JB . Thus the function q has also at least two essential places. Using the operations ζ, τ, ∆ we get from q a binary

602

20 Partial Function Algebras

function g, which depends on both places essentially, preserves all elements of B, and belongs to H ⊂ U ; i.e., for all x ∈ B it holds: g(x, x) = x, g(x, ∞) = g(∞, x) = ∞, g(∞, ∞) = ∞. Consequently, g = (e21,A )+ and therefore H ⊆ [g] ⊆ H. Thus JB is a maximal clone of H. Lemma 20.2.2 For arbitrary f n , g m ∈ U holds: (1) α(f− ) = (α f )− for every operation α ∈ {ζ, τ, ∆, ∇}; (2) f− g− = h− , where hm+n−1 (x1 , ..., xm+n−1 ) := (e21 )+ (f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ), g(x1 , ..., xm )); (3) f− g− ⊆p (f g)− ; (4) f− g− = (f g)− , if the ﬁrst place of f is essential; (5) f− g− = (f g)− , if g− ∈ Pk . Proof. (1): For α ∈ {ζ, τ, δ} the assertion is obvious. For arbitrary (x1 , ..., xn+1 ) ∈ A is by deﬁnition (∇f )(x1 , ..., xn+1 ) = f (x2 , ..., xn+1 ). Furthermore (x1 , ..., xn+1 ) ∈ D((∇f )− ) holds iﬀ (x2 , ..., xn+1 ) ∈ D(f− ). Consequently, we have (∇f )− (x1 , ..., xn+1 ) = ((∇f )− )(x2 , ..., xn+1 ) for arbitrary (x2 , ..., xn+1 ) ∈ D(f− ). (b2): Let x ∈ Am+n−1 be arbitrary. It is easy to check that h(x) = ∞ iﬀ (f g)(x) = ∞. If h(x) = ∞, then h(x) = (f g)(x). Consequently, D(h− ) = {(x1 , ..., xm+n−1 ) ∈ Am+n−1 | (x1 , ..., xm ) ∈ D(g− ) ∧ (g− (x1 , ..., xm ), xm+1 , ..., xn+m−1 ) ∈ D(f− )} = D(f− g− ). Thus, for x := (x1 , ..., xm+n−1 ) ∈ D(h− ) we have g(x1 , ..., xm ) = g− (x1 , ..., xm ) and h(x) = (f g)(x) ∈ A. Hence h− = f− g− . (3): Let x := (x1 , ..., xm+n−1 ) ∈ D(f− g− ) be arbitrary. Then (x1 , ..., xm ) ∈ D(g− ) and (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) ∈ D(f− ). Thus (f g)(x) = ∞. Consequently, x belongs to D((f g)− ) and we have (f− g− )(x) = f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) = (f g)(x), i.e., f− g− ⊆p (f g)− . (4): Let the ﬁrst place of f be essential and let x := (x1 , ..., xm+n−1 ) ∈ D((f g)− ) be arbitrary. Because of f ∈ U it follows that g(x1 , ..., xm ) = ∞. Thus (x1 , ..., xm ) ∈ D(g− ) and (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) ∈ D(f− ). Hence x ∈ D(f− g− ) and by (3) we have f− g− = (f g)− . (5): Let g ∈ PA and let x := (x1 , ..., xm+n−1 ) ∈ Am+n−1 be arbitrary. Then, (x1 , ..., xm ) ∈ D(g− ) and, because of x ∈ D(f− g− ) ⇐⇒ (g(x − 1, ..., xm ), xm+1 , ..., xm+n−1 ) ∈ D(f− ) ⇐⇒ x ∈ D((f g)− ), the assertion follows with the help of (c).

20.2 One-Point Extension

603

Lemma 20.2.3 For arbitrary f, g ∈ P A it holds: (1) α(f+ ) = (α f )+ for every operation α ∈ {ζ, τ, ∆}; (2) ∇(f+ ) = (∇f )+ , if f ∈ C∞ ; (3) f+ g+ = (f g)+ . Proof. (1) and (3) are immediate conclusions from the deﬁnition of the mapping +. (2) is easy to check. Lemma 20.2.4 Let G ⊆ U ⊆ PB be a clone and let F ⊆ P A be a partial clone with G− ⊆ F . Then ([G])− ⊆ F. (20.1) Proof. Let f, g ∈ G be arbitrary. Then by assumption the function f− and g− belong to F . Because of Lemma 20.2.2, (1) it follows that (αf )− = α(f− ) ∈ F for every α ∈ {ζ, τ, ∆, ∇}. Let h := f g ∈ G. If the ﬁrst place of f is essential, then h− = (f g)− = f− g− ∈ F holds by Lemma 20.2.2, (4). If the ﬁrst place of f is ﬁctitious, then m m m hm+n−1 = f n g m = f n em 1,B . Since (e1,B )− = e1,A , we have h− = f− e1,A by Lemma 20.2.2, (5). Theorem 20.2.5 ([Ros 88], [B¨ or-P 90], [B¨ or 97]) (1) For every partial clone F ⊆ P A we have F = ([F+ ])− . (2) For every clone G ⊆ PB with H ⊆ G the set G− is a partial clone of P A with the property G = [(G− )]+ . (3) The mapping ϕ : LB (H; U ) −→ LA (JA ; P A ), G → G−

(20.2)

+ is a lattice isomorphism between the lattices LB (H, U) and LA (JA ; P A ), −1 where ϕ (F ) = [F+ ] holds for every F ∈ LA (JA ; PA ). Proof. (1): Because of f = (f+ )− for every f ∈ PA we have F ⊆ ([F ]+ )− . It follows from (F+ )− ⊆ F and Lemma 20.2.4 that ([F+ ])− ⊆ F . (2): Let rn and sm be two arbitrary functions of G− . Then there are two total functions un and v m of G with u− = r and v− = s. Because of Lemma 20.2.1, (a) we have αr = α(u− ) = (αu)− ∈ G− for all α ∈ {ζ, τ, ∆, ∇}. Furthermore, by Lemma 20.2.1, (b) the following is valid: (r s)(x1 , ..., xm+n−1 ) = (u− v)(x1 , ..., xm+n−1 ) = ((e21,A )+ (u(v(x1 , ..., xm ), xm+1 , ..., xm+n−1 ), v(x1 , ..., xm )))− . Thus r s ∈ G− . Consequently, G− is a partial clone. Let hn1 ∈ G ⊆ U be arbitrary. By using the operations ζ, τ, ∆ one can form

604

20 Partial Function Algebras

from h1 a function hm 2 , which depends on all its places essentially. Then, for the function h2 and for arbitrary j ∈ {1, ..., m} and arbitrary x1 , ..., xm ∈ B, it holds: h2 (x1 , ..., xj−1 , ∞, xj+1 , ...xm ) = ∞, i.e., h2 = ((h2 )−)+ ∈ (G− )+ . With the help of the operations ζ, τ, ∆, ∇, one can form the function h1 from the function h2 . Consequently, hn1 ∈ [(G− )+ ] and G ⊆ [(G− )+ ] hold. Let ht3 ∈ G− be arbitrary. Then there is a function ht4 ∈ G with (h4 )− = h3 . t Because of H ⊆ G we have (et+1 1,A )+ ∈ G. Then, for the function h5 with t h5 (x1 , ..., xt ) := (et+1 1,A )+ (h4 (x1 , ..., xt ), x1 , ..., xt )

it follows: h5 ∈ G and h5 = (h3 )+ . Thus (G− )+ ⊆ G. (3): Let G1 and G2 be two diﬀerent clones of PB with H ⊆ G1 ⊂ G2 ⊆ U . By (1) (G1 )− and (G2 )− are partial clones of LA [JA , PA ]. Obviously, (G1 )− ⊆ (G2 )− . Suppose, (G1 )− = (G2 )− . Then, by (2) it follows that G1 = [((G1 )− )+ ] = [((G2 )− )+ ] = G2 , in contradiction to G1 = G2 . Consequently, the mapping ϕ is an order-preserving bijective mapping from LB (H; U ) into LA (JA ; P A ). Let F ∈ LA [JA , PA ] be arbitrary. Then G := [F+ ] is a clone with H ⊆ G ⊆ U , since if e21,A ∈ F , then (e21,A )+ ∈ G and F+ ⊆ (PA )+ ⊆ U also hold. Further, we have ϕ(G) = ([F+ ])− = F by (1). Consequently, ϕ is surjective and ϕ−1 (F ) = [F+ ].

20.3 Description of Partial Clones by Relations h

k , h-ary relations (i.e., subsets of E k ), h ≥ 1, To describe closed subsets of P are suitable. We often write the elements of relations in the form of columns and we often give a relation in the form of a matrix, the columns of which are the elements of the relation. h k let The set of all h-ary relations over E k R and we put k := R

h

h

k . R

h≥1

k preserves an h-ary relation over E k , iﬀ We say that a function f ∈ P for all r1 , r2 , ..., rn ∈ with ri := (r1i , r2i , ..., rhi ), i = 1, 2, ..., n, it holds ⎛ ⎞ f (r11 , r12 , ..., r1n ) ⎜ f (r21 , r22 , ..., r2n ) ⎟ ⎜ ⎟ f (r1 , ..., rn ) := ⎜ ⎟ ∈ , .. ⎝ ⎠ . f (rh1 , rh2 , ..., rhn )

20.3 Description of Partial Clones by Relations

605

n

k \E h . where f (a) = ∞ is deﬁned for all a ∈ E k Let pP olk h

k . Furtherk that preserve the relation ⊆ E be the set of all functions of P more, h

k \E h )). pP OLk := pP olk ( ∪ (E k The following Lemma is easy to check. Lemma 20.3.1 For every relation ∈ Rkh is valid: (1) pP olk ρ and pP OLk ρ are partial clones. (2) Str(P olk ) ⊆ pP OLk . h

k , the following conditions are equivLemma 20.3.2 For each relation ∈ R alent: (1) pP olk is a partial clone; (2) e21 ∈ pP olk ; (3) If (a1 , . . . , ah ), (b1 , . . . , bh ) ∈ and (c1 , . . . , ch ) is deﬁned by ai if bi ∈ Ek , ci := ∞ if bi = ∞, (i = 1, ..., n), then (c1 , . . . , ch ) ∈ holds. Proof. Obviously, (1) ⇐⇒ (2) holds by deﬁnition. (2) ⇐⇒ (3) follows from ⎛ ⎞ ⎛ ⎞ a1 b1 c1 . .. ⎠ ⎝ .. ⎠ = . e21 ⎝ .. . . ah bh ch The following two lemmas give important properties that are needed to determine the maximal partial clones. 1 We need the following concept for the wording of the lemmas: The relation ∈ Rkh is called irredundant, iﬀ it fulﬁlls the following two conditions: 1) for all i, j with 1 ≤ i < j ≤ h, there is a tuple (a1 , ..., ah ) ∈ with ai = aj ; 2) No i ∈ {1, ..., h} exists, such that (a1 , ..., ah ) ∈ implies (a1 , ..., ai−1 , x, ai+1 , ..., ah ) ∈ for all x ∈ Ek . We say that for i = 1, ..., n the relations χi ⊆ {1, ..., t}hi cover the set {1, ..., t}, if for every x ∈ {1, ..., t} there exists an i ∈ {1, ..., n} such that x ∈ {a1 , ..., ahi } holds for at least a tuple (a1 , ..., ahi ) ∈ χi . 1

As one can gather from Sections 20.4 and 20.5, the maximal partial clones are all strong clones, with an exception.

606

20 Partial Function Algebras

k be a strong partial Lemma 20.3.3 ([Rom 81], without proof ) Let C ⊆ P of irredundant relations clone. Then there is a certain nonempty set M ⊆ R k with C = ∈M pP OLk . Lemma 20.3.4 (Representation Lemma of B. A. Romov, [Rom 81]; without proof ) Let αi ⊆ Ekhi for i = 1, ..., n and let β ⊆ Ekt be an irredundant relation. Then n pP OLk αi ⊆ pP OLk β i=1

if and only if there are certain (help-)relations χi ⊆ {1, ..., t}hi for i = 1, ..., n that cover the set {1, ..., t} and for which β = {(b1 , ..., bt ) ∈ Ekt | ∀j ∈ {1, ..., n} ∀(ij1 , ..., ijhj ) ∈ χj : (bij , ..., bij ) ∈ αj } 1

hj

holds.

2 and P 3 20.4 The Maximal Partial Classes of P Lemma 20.4.1 Let f be an n-ary function of P3 , which essentially depends on at least two variables (w.l.o.g. let x1 and x2 be the essential variables). Furthermore, let {a, b, c} = E3 and δI := {(a0 , a1 , a2 ) ∈ E33 , | ∀α, β ∈ I : aα = aβ }, I ⊆ Ek . Then 3 3 ∪ δ{1,2} : f (r1 , ..., rn ) ∈ E33 \ι33 ; (a) Im(f ) = E3 ⇒ ∃ r1 , ..., rn ∈ δ{0,1} 3 3 3 3 ∪ δ{b,c} : f (r1 , ..., rn ) ∈ δ{a,c} \δ{0,1,2} ; (b) |Im(f )| = 2 ⇒ ∃ r1 , ..., rn ∈ δ{a,b} 1 (c) Im(f ) = E3 ⇒ [{f } ∪ { g ∈ P3 | g(a) = g(b) ∨ g(b) = g(c) }] = P3 . Proof. (a) is a special case of the “fundamental lemma of Jablonskij” and (b) is an easy conclusion from this lemma (see Theorem 1.4.4). (c): W.l.o.g. let a = 0, b = 1 and c = 2. Obviously, then, we have { g ∈ P31 | g(0) = g(1) ∨ g(1) = g(2) } = {c0 , c1 , c2 , jα , uα , vα | α ∈ {0, 2, 3, 5}}. It is easy to check that this set of unary functions is not a subset of maximal classes of type M, U, S, C and L. Since Im(f ) = E3 and f ∈ P3 \[P31 ] hold, f does not preserve by (a) the relation ι33 . Thus (c) follows from the completeness criterion for P3 . k \(Pk ∪ [{c∞ }]) it holds [Pk ∪ {g}] = P k . Lemma 20.4.2 (a) For every g ∈ P k that contains Pk . (b) Pk ∪ [{c∞ }] is the only maximal class of P k \(Pk ∪ [{c∞ }]). Since the function g := e2 ∗ g has Proof. (a): Let g m ∈ P 2 k . Consequently, k + 1 diﬀerent values, we can assume w.l.o.g. Im(g) = E

2 and P 3 20.4 The Maximal Partial Classes of P

607

k ). there are k + 1 tuples ai := (ai1 , ai2 , ..., aim ) ∈ Ekm with g(ai ) = i (i ∈ E n Let f be an arbitrary function of Pk . Independently from f , one can deﬁne the following functions fj (j = 1, 2, ..., m): fj (b1 , ..., bn ) = aij :⇐⇒ f (b1 , ..., bn ) = i k ). Thus we have f (x) = g(f1 (x), ..., fm (x)). Hence (b1 , ..., bn ∈ Ek ; i ∈ E f ∈ [Pk ∪ {g}]. (b) follows directly from (a). 3 The next four lemmas deal with the maximality of certain subclasses of P (or Pk ) in P3 (or Pk ), respectively. The following statement (a) was proven in [Bur 67] (see also Chapter 4) and (b) of the following lemma was proven in [Rom 80] (or in [Lau 77], [Lau 88]). Lemma 20.4.3 Let 1 := {(a, a, b, b), (a, b, a, b) | a, b ∈ Ek }, 2 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ Ek }, i := {(a1 , a2 , ..., ai ) ∈ Eki | |{a1 , ..., ai }| ≤ i − 1 } (i = 3, ..., k). Then (a) The classes P olk i (i = 1, 2, ..., k) are the only proper subclasses of Pk that contain Pk1 . Furthermore, it holds that [Pk1 ] = P olk 1 ⊂ P olk 2 ⊂ ... ⊂ P olk k−1 ⊂ P olk k ⊂ Pk . k and the (b) The classes pP OLk i (i = 1, 2, ..., k) are maximal classes of P 1 only maximal classes of Pk that contain Pk . Lemma 20.4.4 Let := δ ∪ σ, where δ denotes a certain h-ary diagonal relation, which is diﬀerent from Ekh , and σ fulﬁlls the condition ∅ = σ ⊆ {(a1 , ..., ah ) ∈ Ekh | |{a1 , ..., ah }| = h }. There exists to every a := (a1 , ..., ah ) ∈ σ a certain equivalence relation εa on Ek with the following two properties: (1) For every i ∈ {1, ..., h} there exists exactly an equivalence class of εa which contains ai . (2) To every b ∈ one can ﬁnd in pP OLk a unary function ga,b with ga,b (a) = b and ga,b (x) = ga,b (y) for all (x, y) ∈ εa . k . Then pP OLk is a maximal class of P k . Let f ∈ P k \pP OLk . Since has the Proof. Obviously, pP OLk = P properties (1) and (2), one gets a certain unary function hr with hr (r) ∈ Ekh \ as a superposition over unary functions of pP OLk and f for every r ∈ σ. Let σ = {r1 , ..., rm }. One can ﬁnd in pP OLk certain functions, which arbitrary k on rows of the form values have of E (x1 , x2 , gr1 (x1 ), gr2 (x1 ), ..., grm (x1 ), gr1 (x2 ), gr2 (x2 ), ..., grm (x2 ))

608

20 Partial Function Algebras

and otherwise only have the value ∞. Consequently, arbitrary functions of Pk2 are superpositions over {f } ∪ pP OLk . Hence (by well-known properties of Pk ) it follows that Pk ⊆ [{f } ∪ pP OLk ]. Since pP OLk obtains functions with exactly k + 1 diﬀerent values, it follows from Lemma 20.4.2, (a) that k holds. Hence pP OLk is a maximal class of P k . [{f } ∪ pP OLk ] = P Table 20.1 i 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

39

τi {0} {2} {0, 2} 0 1 1 2 0 2 2 0 0 0 1 2 0 1 2 2 0 1 1 1 0 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 ⎛0 ⎞ 0 ⎝1⎠ ⎛2 ⎞ 0 2 ⎝1 1⎠ 2 0

i 2 4 6 8 10 12 14

2 1 0 1 1 2 0 2 0 1 0 2 1 0 0 1 0 1 0 1

16

18

20

2 0 0 2 1 2 2 1 0 2 0 2 1 0

2 1 1 2 1 2 2 1

22 24 26 28 30 32 34 36 38

40

τi {1} {0, 1} {1, 2} 0 2 0 1 1 0 1 2 2 1 0 2 1 1 0 1 0 1 0 2 0 2 1 2 0 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 2 ⎛0 1 ⎞ 0 0 ⎝1 2⎠ ⎛2 1⎞ 0 1 ⎝1 0⎠ 2 2

2 0 2 1 0 2 0 1 1 2 0 1 0 1 2 0 0 1 0 1 0 2

1 0 2 1 2 1 2 0 1 2 2 1 1 0 2 0

2 0 0 2 1 2

2 0 2 1

2 and P 3 20.4 The Maximal Partial Classes of P

i τ⎛i 41 ⎝ ⎛ 43 ⎝ ⎛ 45 ⎝ ⎛ 47 ⎝

i 49 50 51 52 53 54 55 56 57

0 1 2 0 0 0 0 0 0 0 0 0

1 2 0 1 1 1 1 1 1 1 1 1

2 0 1 2 2 2 2 2 2 2 2 2

Table 20.2 i τ⎛i ⎞ 0 0 ⎠ 42 ⎝ 1 2 ⎛2 1 ⎞ 0 0 1 1⎠ 44 ⎝ 0 1 2 ⎞ ⎛0 1 0 2 0 1 1 1⎠ 46 ⎝ 0 1 2 0 ⎞ ⎛0 1 0 1 2 0 1 1 2 0 ⎠ 48 ⎝ 0 1 2 0 1 0 1

1 0 2 2 2 2 2 2 2 2 2 2

1 2 0 0 1 2 0 1 2 0 1 2

2 0 1 0 2 1 1 0 2 0 2 1

609

⎞ 2 1⎠ 0 ⎞ ⎠ ⎞ ⎠ ⎞ 1 1 2 2 0 2 0 1⎠ 2 0 1 0

Table 20.3 τi { (0, 1, 2), (a, a, b) | a, b ∈ E3 } { (0, 1, 2), (a, b, a) | a, b ∈ E3 } { (0, 1, 2), (b, a, a) | a, b ∈ E3 } { (0, 1, 2), (1, 0, 2), (a, a, b) | a, b ∈ E3 } { (0, 1, 2), (2, 1, 0), (a, b, a) | a, b ∈ E3 } { (0, 1, 2), (0, 2, 1), (b, a, a) | a, b ∈ E3 } { (a, a, b, b), (a, b, a, b) | a, b ∈ E3 } { (a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E3 } { (a, b, c) ∈ E33 | |{a, b, c}| ≤ 2 }

Lemma 20.4.5 (1) For every ∈ {{0}, {1}, {(0, 1)}, {(0, 1), (1, 0)}, {(0, 0), (0, 1), (1, 1)}, λ2 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E2 }, G2 ([P21 ])}, the set pP Ol2 is 2 . a maximal class of P (2) The classes pP OL3 τi (i ∈ {1, 2, ..., 57}, see Tables 20.1 - 20.3) are maximal classes of P3 . Proof. (1): The maximality of pP OL2 for ∈ {λ2 , G2 ([P21 ])} was proven in Lemma 20.4.3. 2 : Let f n ∈ Next we show that pP OL2 {(0, 1)} is a maximal class of P n−1 1 P2 \ pP OL2 {(0, 1)} be arbitrary. Then ∆ f ∈ {c0 , c1 , e1 }. We distinguish two cases: Case 1: ca ∈ [f ] for certain a ∈ E2 . The unary function g with g(a) = a and g(a) = ∞ belongs to pP OL2 {(0, 1)}, whereby {c0 , c1 } ⊂ [{f } ∪ pP OL2 {(0, 1)}]. With the aid of Theorem 3.2.4.1, it is easy to prove that P2 = [(T0 ∩ T1 ) ∪ {c0 , c1 }] ⊆ [{f } ∪ pP OL2 {(0, 1)}].

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20 Partial Function Algebras

Further, we have pP OL2 {(0, 1)} ⊆ P2 ∪ [c∞ ]. Therefore, by Lemma 20.4.2, 2 holds, i.e., pP OL2 {(0, 1)} is a maximal class of [{f } ∪ pP OL2 {(0, 1)}] = P P2 . Case 2: e11 ∈ [f ]. In this case, by Theorem 3.2.4.1, P2 = [(T0 ∩T1 )∪{e11 }] ⊆ [{f }∪pP OL2 {(0, 1)}], whereby Case 2 is reducible to Case 1. The maximality of the other classes pP OL2 is easy to check. (2): For i ∈ {1, 2, ..., 54} one can easily prove the lemma with the help of Lemma 20.4.4. If i ∈ {55, 56, 57}, the above statement follows from Lemma 20.4.3. Theorem 20.4.6 ([Fre 66]) 2 has exactly 8 maximal classes: P pP OL2 σi , where σ0 := {0}, σ1 := {1}, σ2 := {(0, 1)}, σ3 := {(0, 1), (1, 0)}, σ4 := {(0, 0), (0, 1), (1, 1)}, σ5 := G2 ([P21 ]) = {(a, a, b, b), (a, b, a, b) | a, b ∈ E2 } and σ6 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E2 }, and P2 ∪ [c∞ ] = pP ol2 (Ek2 ∪ {(∞, ∞)}). Proof. Because of Lemmas 20.4.2 and 20.4.5, we must show that each subset 2 , which fulﬁlls A ⊆ pP OL3 τi for all i ∈ {0, 1, 2, ..., 6} and A ⊆ P2 ∪[c∞ ], A of P 2 . is complete in P Let A ⊆ P2 be an arbitrary set, for which there are f0 , ..., f6 , f7 ∈ A with fi ∈ pP OL2 σi (i = 0, 1, ..., 6) and f7 ∈ P2 ∪ [c∞ ]. 2 deﬁned by g(x) := f2 (x, x, ..., x). Then g ∈ [A] and g ∈ {c0 , c1 , e1 }. Let g ∈ P 1 Consequently, since c1 ∈ [f0 , c0 ], c0 ∈ [f1 , c1 ], {c0 , c1 , e11 } ⊂ [f3 , e11 ] and e11 ∈ [c0 , c1 , f4 ], we have P21 ⊂ [A]. Thus there exists a function h ∈ [{f5 }∪P21 ] ⊆ [A] with h ∈ P2 \ [P21 ]. We distinguish two cases: Case 1: h is non-linear. In this case, by Theorem 3.2.4.1, we have [P21 ∪ {h}] = P2 ⊆ [A]. Because of 2 . A ⊆ P2 ∪ [c∞ ] and Lemma 20.4.2, this implies [A] = P 1 Case 2: h ∈ [P2 ] is a linear function. Then it is easy to check that all binary linear functions belong to [A], whereby (by f6 ∈ A) a non-linear function of P2 is a superposition over A. Thus Case 2 is reducible to Case 1. Theorem 20.4.7 ([Lau 77], [Rom 80]) 3 has exactly 58 maximal classes. These classes are the sets pP OL3 τi (i = P 1, 2, 3, ..., 57), where τi are given in Tables 20.1–20.3, and the set P2 ∪ [c∞ ].

2 and P 3 20.4 The Maximal Partial Classes of P

611

Proof.2 Because of Lemmas 20.4.2 and 20.4.5, we must show that each subset M

3 , which fulﬁlls M ⊆ pP OL3 τi for all i ∈ {1, 2, ..., 57} and M ⊆ P2 ∪ [c∞ ], is of P 3 . complete in P 3 be an arbitrary set that fulﬁlls M ⊆ P2 ∪ [c∞ ] and M ⊆ pP OL3 τi for Let M ⊆ P all i ∈ {1, 2, ..., 57}). Consequently, there are functions fini ∈ M \ pP OL3 τi for all i = 1, 2, ..., 57) and f58 ∈ M \ (P2 ∪ [c∞ ]). If τi = (σ1 σ2 ... σmi ) (i ∈ {1, 2, ..., 57}) is an hi -ary relation, then we can assume w.l.o.g. ni = mi and fi (σ1 , σ2 , ..., σmi ) ∈ E3hi \τi . As already mentioned, we must show that [M ] = P3 . First we prove {c0 , c1 , c2 } ⊆ [M ].

(20.3)

The function f37 is unary and belongs to P31 \{s1 }. Then the following three cases are possible: Case 1: f37 = ca (a ∈ E3 ). Obviously, we have {c0 , c1 , c2 } ⊆ [{f37 , f1 , f2 , ..., f6 }] in this case. Case 2: |Im(f37 )| = 2. W.l.o.g. let Im(f37 ) = {0, 1}, i.e., f37 ∈ {j0 , j1 , ..., j5 }. Because of j2 ∗ j2 = c0 , j0 ∗ j0 = j5 , j3 ∗ j3 = c1 , j4 ∗ j4 = j1 and Case 1 it is suﬃcient to assume that f37 ∈ {j1 , j5 }. 2.1: f37 = j1 . We form f7 := f7 ∗ j1 ∈ {c0 , c1 , c2 , j4 , u1 , u4 , v1 , v4 }. Because of Case 1, u1 ∗ u1 = c0 , v1 ∗ v1 = v4 and u4 ∗ u4 = c2 , we can conﬁne ourselves to f7 ∈ {j4 , v4 }. 2.1.1: f7 = j4 . Putting the functions f37 (= j1 ) and f7 (= j4 ) into f10 provides a unary function (x) := f10 (j1 (x), j4 (x)) with f10 ∈ {c0 , c1 , c2 , u1 , u4 , v1 , v4 }. Because of u1 ∗ u1 = f10 = v4 . It c0 , u4 ∗ u4 = c2 and v1 ∗ v1 = v4 we still have to examine the case f10 (x) := f17 (j1 (x), j4 (x), v1 (x), v4 (x)), then f17 ∈ holds v4 ∗ j4 = v1 . If we form f17 {c0 , c1 , c2 , u1 , u4 } holds and f17 ∗ f17 is a constant function. With that we have reduced the Case 2.1.1 to the ﬁrst case. 2.1.2: f7 = v4 . (x) := f14 (j1 (x), v4 (x)) belongs to {c0 , c1 , c2 , j4 , u1 , u4 , v1 }. Because The function f14 of u1 ∗ u1 = c0 , u4 ∗ u4 = c2 , j1 ∗ v1 = j4 and by Case 2.1.1, we reduce Case 2.1.2 to the ﬁrst case. 2.2: f37 = j5 . In this case, one can proceed to Case 2.1 analogously using the function f16 instead of f17 and f13 instead of f14 . Case 3: |Im(f37 )| = 3 and f37 = s1 . With the help of functions f38 , ..., f42 , this case is reduced at ﬁrst to Case 1 or 2 and, therefore, to Case 1. Consequently, the constant functions belong to [M ]. Let f43 (x) := f43 (c0 (x), c1 (x), c2 (x), x). This function belongs to P31 ∩ [M ] and has is a permutation, then with the help of functions at least two diﬀerent values. If f43 2

The deﬁnitions of the following unary functions of P3 are in Chapter 15, Table 15.1.

612

20 Partial Function Algebras

f44 , ..., f48 , we can form a function with 2-element range. Thus w.l.o.g. we can ) = E2 in the following; i.e., f43 ∈ {j0 , j1 , ..., j5 }. assume Im(f43 Let Ma,b := {f ∈ P31 | f (a) = f (b)}. Next ∃ a, b ∈ E3 : a = b ∧ Ma,b ⊂ [M ]

(20.4)

shall be proven. Because of j0 ∗ j0 = j5 , j4 ∗ j4 = j1 and for reasons of the duality, ∈ {j1 , j2 }. we can assume that f43 Case 1: f43 = j1 . ∈ {j4 , u1 , u4 , v1 , v4 } as a superposition over the constant functions, One receives f19 j1 and f19 . Because of v1 ∗ v1 = v4 and f22 (c0 , c1 , c2 , j1 , j4 ) ∈ {u1 , u4 , v1 , v4 } we can ∈ {u1 , u4 , v4 }. assume f19 = u1 . 1.1: f19 (x) := f25 (c0 (x), c1 (x), c2 (x), j1 (x), u1 (x)) In this case, we can form a function f25 ∈ {j4 , u4 , v4 }. with f25 ∈ {j4 , u4 , v1 , v4 }. Further, we can assume w.l.o.g. f25 1.1.1: f25 = j4 . := f34 (c0 , c1 , c2 , j1 , j4 , It holds that u1 ∗j4 = u4 . Consequently, the unary function f34 u1 , u4 ) ∈ {v1 , v4 } is a superposition over M . Thus by v1 ∗ v1 = v4 and v4 ∗ j4 = v1 it holds: {c0 , c1 , c2 , j1 , j4 , u1 , u4 , v1 , v4 } = {f ∈ P31 | f (0) = f (2)} ⊂ [M ]. = u4 . 1.1.2: f25 := f23 (c0 , c1 , c2 , u1 , u4 ) we have f23 ∈ {j1 , j4 , v1 , v4 }. Because For the function f23 of v1 ∗ v1 = v4 and j4 ∗ j4 = j1 , we can assume f23 ∈ {j1 , v4 }. = j1 . 1.1.2.1: f23 When one substitutes u1 instead of u4 and u4 instead of u1 into f23 , one receives = j4 instead of f23 . Now one can continue the proof as in Case 1.1.1. f23 1.1.2.2: f23 = v4 . When one substitutes u1 instead of u4 and u4 instead of u1 into f23 , one receives the = v1 instead of f23 . Further, we have f36 := f36 (c0 , c1 , c2 , u1 , u4 , v1 , v4 ) ∈ function f23 = j1 , we substitute u1 instead of u4 , u4 instead of u1 , v1 instead of {j1 , j4 }. If f36 = j4 instead of f36 . v4 and v4 instead of v1 into f36 and we receive the function f36 Thus Case 1.1.2.2 was also reduced to Case 1.1.1. = v4 . 1.1.3: f13 := f31 (c0 , c1 , c2 , j1 , u1 , v4 ) ∈ {j4 , u4 , v1 } is a superposition over M . BeThen, f31 cause of j1 ∗ v1 = j4 , this case is reducible to Cases 1.1.1 and 1.1.2. = u4 . 1.2: f19 := f28 (c0 , c1 , c2 , j1 , u4 ) ∈ {j4 , u1 , v1 , v4 } or w.l.o.g. (by v1 ∗ v1 = v4 We can form f28 and u4 ∗ j4 = u1 ) f28 ∈ {u1 , v4 }. 1.2.1: f28 = u1 . In this case, one can continue the proof as in Case 1.1. = v4 . 1.2.2: f28 := f32 (c0 , c1 , c2 , j1 , v4 , u4 ) ∈ {j4 , u1 , v1 }. Because of v4 ∗ j4 = v1 Here we have f32 and u4 ∗ v1 = u1 we can assume f32 = u1 . Thus Case 1.2 is reducible to Case 1.1. 1.3: f19 = v4 . := f26 (c0 , c1 , c2 , j1 , v4 ). Then, f26 ∈ {j4 , u1 , u4 , v1 } or w.l.o.g. (by We form f26 v4 ∗ j4 = v1 ) f26 ∈ {u1 , u4 , v1 }. ∈ {u1 , u4 }. 1.3.1: f26 Continuation of the proof as in Cases 1.1 or 1.2.

2 and P 3 20.4 The Maximal Partial Classes of P

613

1.3.2: f26 = v1 . := f35 (c0 , c1 , c2 , j1 , It holds that j1 ∗v1 = j4 . Consequently, we can form a function f35 j4 , v1 , v4 ) ∈ {u1 , u4 }; i.e., Case 1.3.2 is reducible to Cases 1.1 and 1.2. Hence, we have proven (20.4) in Case 1. = j2 . Case 2: f43 Then f19 := f19 (c0 , c1 , c2 , j2 ) ∈ {j3 , u2 , u3 , v2 , v3 }. Because of u3 ∗ u3 = u2 and ∈ {j3 , u2 , v2 }. v3 ∗ v3 = v2 let w.l.o.g. f19 = j3 . 2.1: f19 := f22 (c0 , c1 , c2 , j2 , j3 ) and f22 := f22 (c0 , c1 , c2 , j3 , j2 ). Then {f22 , f22 } ∈ Set f22 {{u2 , u3 }, {v2 , v3 }} and {u2 , u3 , v2 , v3 } ⊆ [{f22 , f22 , j2 , j3 , f34 , f35 }]. Thus M0,1 ⊆ [M ] is proven. = u2 . 2.2: f19 := f25 (c0 , c1 , c2 , j2 , u2 ) ∈ {j3 , u3 , v2 , v3 }. Because of j2 ∗ u3 = j3 , Here we have f25 = v2 . Since f33 (c0 , c1 , c2 , j2 , u2 , v2 ) ∈ v3 ∗ v3 = v2 and Case 2.1 we can assume f25 {j3 , u3 , v3 }, j2 ∗ u3 = j3 and j2 ∗ v3 = j3 , Case 2.2 is reducible to Case 2.1. = v2 . 2.3: f19 Analogously to 2.2.

Thus (20.4) is proven. Therefore, w.l.o.g. we can assume M0,1 := { f ∈ P31 | f (0) = f (1) } = {c0 , c1 , c2 , j2 , j3 , u2 , u3 , v2 , v3 } ⊂ [M ].

(20.5)

By substituting the functions of M01 and identifying variables in f49 , one can form ∈ P31 with f49 (0) = f49 (1) and f49 = s1 . a function f49 Case 1: |Im(f49 )| = 2. In this case it is easy to check that M0,1 ∪ Ma,b ⊂ [M ] for a certain (a, b) ∈ {(0, 2), (1, 2)}. W.l.o.g. let (a, b) = (1, 2). Further, by (20.4) w.l.o.g. we can assume that, for certain α, β, γ, δ ∈ E3 , ⎛ ⎞ α γ ⎜α δ ⎟ 4 ⎟ f55 ⎜ ⎝ β γ ⎠ ∈ E3 \τ55 β δ holds. Then we can form the function f55 (x, y) := f55 (x), g2 (y)) (g1 g1 ∈ [M ], where 0 0 α and g2 are certain functions of M0,1 ∪ M1,2 with g1 = = and g2 1 1 β γ ∈ P3 \[P31 ]. . Obviously, f55 δ If |Im(f55 )| = 3, then it follows from Lemma 20.4.1, (c) that P3 ⊆ [M ]. This implies 3 with the help of Lemma 20.4.2, (a) and f58 ∈ P 3 \(P3 ∪ [{c∞ }]). [M ] = P Let )| = 2 (20.6) |Im(f55

be in the following. Because of Lemma 20.4.1, (b) there are certain a1 , a2 , a3 , b1 , b2 , b3 ∈ E3 with ⎛ ⎞ ⎛ ⎞ a 1 a2 a3 f55 ⎝ b1 a2 ⎠ = ⎝ b3 ⎠ , a1 b 2 b3

614

20 Partial Function Algebras

where a3 = b3 . By substituting functions of M0,1 ∪ M1,2 (⊂ [M ]) into f55 , one can form a unary function f55 with the property: M0,2 ⊆ [{f55 } ∪ M0,1 ∪ M1,2 ]. Then, with the help of function f57 ∈ M and Lemma 20.4.1, (a), it follows P31 ⊂ [M ]. By Lemma 20.4.3, (a) we have then P ol3 τ56 ⊆ [M ]. With the help of functions f56 and f57 of M , it is easy to prove that all functions of P3 are superpositions on M . Because of f58 ∈ M and Lemma 20.4.2, (a), this implies [M ] = P3 . )| = 3. Case 2: |Im(f49 In this case, f49 is a permutation = s1 . If f49 = s3 , then, because of (20.5), one can form a certain unary function g with g(0) = g(1) as a superposition over M and one continues to be able to use the proof as in the ﬁrst case. = s3 , then it is possible to form a unary function h ∈ [M0,1 ∪ {f52 }](⊂ [M ]), If f49 which is either a permutation ∈ {s1 , s3 } or h is a function with |Im(h)| = 2 and h ∈ M0,1 . Consequently, one can completely reduce the second case to the ﬁrst case.

Next, some remarks on partial Sheﬀer functions: k is called Sheﬀer iﬀ [f ] = P k . A partial function f ∈ P In [Had-R 91] all partial Sheﬀer functions for k = 2 and all binary Sheﬀer function for k = 3 were described. Further, the statement 2 ⇐⇒ [f ] = P (∀A ∈ {pP OL2 {0}, pP OL2 {1}, pP OL2 {(0, 1), (1, 0)}, P2 ∪ [c∞ ]} : f ∈ A) was proven. In [Had-L 2006] one ﬁnds the proof of the following criterion: 3 ⇐⇒ [f ] = P (∀A ∈ {pP OL3 τi | i ∈ {1, 2, 3, 4, 5, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 41, 42, 47, 48, 55, 56, 57}} ∪ {P2 ∪ [c∞ ]} : f ∈ A), where the relations τi are deﬁned in Tables 20.1–20.3. In [Had-L 2006] one ﬁnds also the proof that it is not possible to reduce the conditions from the above criteria; i.e., these conditions are independent of each other.

k 20.5 The Completeness Criterion for P In analog mode to the sixth chapter, one can show that a completeness critek . Subsequently, k can be found using the maximal partial classes of P rion for P the maximal partial classes are described in a form found by L. Haddad and I. G. Rosenberg. The following deﬁnitions are needed: Deﬁnitions Let Eqh be the set of all equivalence relations over {1, ..., h}. An h-ary relation ⊆ Ekh is called • areﬂexive, if ∩δε = ∅ for every ε ∈ Eqh , ε = ι2h , i.e., for all (x1 , . . . , xh ) ∈ we have xi = xj for all 1 ≤ i < j ≤ h.

k 20.5 The Completeness Criterion for P

615

• quasi-diagonal, if = σ ∪ δε , where σ is a nonempty areﬂexive relation, ε ∈ Eqh \ {ι2h } holds, and further = Ek2 for h = 2. Furthermore 1 := = 2 := :=

{(a, a, b, b), (a, b, a, b) | a, b ∈ Ek } δ{1,2},{3,4} ∪ δ{1,3},{2,4} , {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ Ek } δ{1,2},{3,4} ∪ δ{1,3},{2,4} ∪ δ{1,4},{2,3} .

In the following, denote an h-ary relation of the form =σ∪( (δε )), ε∈F

where σ is an areﬂexive h-ary relation and F ⊂ Eqh . Let Gσ : = {π ∈ Sh | σ ∩ σ (π) = ∅}, where Sh denotes the set of all permutations over the set {1, ..., h} and σ (π) := {(aπ(1) , ..., aπ(h) ) | (a1 , ..., ah ) ∈ σ}. The model of is the h-ary relation M () : = {(π(1), . . . , π(h)) | π ∈ Gσ } ( {(x1 , . . . , xh ) ∈ {1, . . . , h}h | ε∈F

(i, j) ∈ ε ⇒ xi = xj })

on the set {1, . . . , h}.3 Suppose h, F , and σ fulﬁll exactly one of the following ﬁve conditions: i) h ≥ 2, F = ∅ and σ = ∅, i.e., is a nonempty h-ary areﬂexive relation; ii) h ≥ 2, F = {ε}, where ε = ι2h , σ = ∅ and σ ∪ δε = Ek2 , i.e., is a trivial quasi-diagonale h-ary relation; iii) h = 4 and F = {{{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}}}, i.e., = σ ∪ 2 , where σ is an areﬂexive 4-ary relation (the empty set is possible); iv) h = 4 and F = {{{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}}, i.e., = σ ∪ 1 , where σ is an areﬂexive 4-ary relation (= ∅ is possible); v) h = 2, h ≤ k, F = {i, j} and = Ekh , i.e., is a totally reﬂexive 1≤ij≤h

and totally symmetric not trivial relation.4 We say that is coherent, iﬀ 3 4

In some papers the set Eh is elected instead of {1..., h} in deﬁning the model. For h = 1 we have ∅ ⊂ ⊂ Ek .

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20 Partial Function Algebras

(1) Gσ = {π ∈ Sh | σ (π) = σ} and π(ε) := {(π(x), π(y)) | (x, y) ∈ ε} = ε for all π ∈ Gσ , if fulﬁlls either the above condition i) or ii), Gσ = {π ∈ Sh | σ (π) = σ} ∪ {π ∈ Sh | π(F ) = F }, if fulﬁlls iii) or iv), Gσ = {π ∈ Sh | σ (π) = σ} = Sh , if fulﬁlls the condition v), and (2) for every nonempty subset σ of σ there exists a relational homomorphism γ : Ek → {1, . . . , h} of σ in M (), such that (γ(i1 ), . . . , γ(ih )) = (1, . . . , h) for at least an h-tuple (i1 , . . . , ih ) ∈ σ . Theorem 20.5.1 (Haddad-Rosenberg Theorem; [Had-R 89], [Had-R 92]; without proof) k there is a maximal parLet k ≥ 2. For every proper partial subclass A of P k , then either tial clone that contains A. If C is a maximal partial clone of P 2 C = Pk ∪ {f ∈ Pk | D(f ) = ∅} (= pP olk Ek ∪ {(∞, ∞)}) or C = pP OLk , where is one of the following relations: (1) an h-ary not trivial totally reﬂexive and totally symmetric relation with 1 ≤ h ≤ k; (2) an h-ary areﬂexive or quasidiagonal relation with h ≥ 2, which is coherent; (3) a quaternary relation 2 or 1 ; (4) a quaternary coherent relation σ ∪ τi , where i ∈ {1, 2} and σ = ∅ is a quaternary areﬂexive relation. max be the set of all relations (∈ Rk ) given in the above theorem. Then Let R the following theorem is a consequence of the above: k ; [Had-R 92]) Theorem 20.5.2 (Completeness Criterion for P k if and only if C ⊆ pP OLk for all ∈ R max and k . Then [C] = P Let C ⊆ P C ⊆ Pk ∪ {f ∈ Pk | D(f ) = ∅}.

20.6 Some Properties of the Maximal Partial Clones k of P We show ﬁrst that each maximal clone of Pk is a subset of exactly a k . Then we specify the relations ∈ Rmax := maximal partial clone of P Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk , with the property that pP OLk is a maximal k . partial clone of P A survey of the orders of the maximal partial clones forms the end of this section.

k 20.6 Some Properties of the Maximal Partial Clones of P

617

Next to the notations from the ﬁfth chapter, we still need the following notations for certain relation sets: Let Pk,p be the set of all ﬁxed-point-free permutations on Ek whose cycles have the same prim number length p. Let Sk,p := { {(x, s(x)) | x ∈ Ek } | s ∈ Pk,p } and let M k be the set of all ∈ Mk with the property that (Ek ; ) is a lattice. Theorem 20.6.1 For every maximal clone C ⊆ Pk there is exactly one maxk with C ∩ Pk = C. imal partial clone C ⊆ P k . Put Proof. Let C be a partial clone of P 2 := C n≥1 {f ∈ Pk | ∀f1 , . . . , fn ∈ C ∩ Pk :

(f (f1 , . . . , fn ) ∈ Pk =⇒ f (f1 , . . . , fn ) ∈ C) }. It was already shown Notice that C ∩ Pk2 = ∅. As C is a partial clone, C ⊆ C. is a partial clone with the following property: in [Fre 66] that C = P k . k =⇒ C C = P Consequently, if C is a maximal partial clone, then C = C.

(20.7)

Let M be a maximal clone of Pk and let C1 , C2 be two maximal partial clones )1 and C2 = C )2 . From M = Pk of Pk that contain M . Then, by (20.7) C1 = C 2 and [Pk ] = Pk (see Theorem 1.4.2), we have M 2 = (C1 ∩ Pk )2 = (C2 ∩ P2 )2 ⊂ Pk2 . )2 and by (20.7), C1 = C2 . we have C )1 = C By deﬁnition of C, Theorem 20.6.2 ([Had-L 2000]; without proof ) (1) Let M := Ck ∪ Mk ∪ Sk ∪ Uk ∪ Lk ∪ Bk and M1 := Sk \Sk,2 . For every k . ∈ M \ (Lk ∪ M1 ) the set pP OLk is a maximal partial clone of P (2) For s ∈ Pk,p with p ≥ 3 let so := {(x, s(x)) | x ∈ Ek } s2 := {(x, s(x), s2 (x), . . . , sp−1 (x)) | x ∈ Ek }. Then pP OLk s2 is a maximal partial clone that properly contains the partial clone pP OLk s0 .

618

20 Partial Function Algebras

We need the following notations for the statement still missing on classes of type L. Let p ∈ P, m ∈ N and W := Epm . As shown in Section 5.2.4, a maximal class of Ppm of type L is isomorphic to the set LW . The set LW one can also describe in the form P olW λ with λ := {(a, b, c, d) ∈ (Epm )4 | a ⊕ b = c ⊕ d}, where (a1 , ..., am ) ⊕ (b1 , ..., bm ) := (a1 + b1 , ..., am + bm ) and + is the addition modulo p. With the help of ⊕ and (α(a1 , ..., am ) := (α·a1 (mod p), ..., α·am (mod p)) for α ∈ Ep and (a1 , ..., am ) ∈ W ) we can deﬁne the following p-ary relation over W : λp := {(a, a ⊕ b, a ⊕ 2 b, ..., a ⊕ (p − 1) b) | a, b ∈ W }. Theorem 20.6.3 ([Had-L 2000]; without proof ) + For p ∈ P the partial clone pP OLW λp is a maximal clone of P W that properly contains the partial clone pP OLW λ. Now, we come to some theorems that deal with the ﬁnite generating of partial clones. 2 has partial subclasses, which are not It was shown already in [Fre 66] that P ﬁnitely generating. In [Lau 88], it was proven that there are maximal partial 2 that are not ﬁnitely generating. These are exactly the partial clones of P clones pP OL2 with ∈ {λ2 , G2 ([P21 ])} (see also [B¨or-H 97]). The order of 2 agrees with the order of each ﬁnite generating maximal partial clone C of P C ∩ P2 . We need the following concept for the following criterion about the ﬁnite generating of strong partial clones: Let r > 0 and let C be a clone of Pk . Then C is called r-separable, if for every n > 0 and every b ∈ Ekn there are certain n-ary functions g1 , ..., gr ∈ C such that the mapping g : Ekn −→ Ekr with g(a) := (g1 (a), ..., gr (a)) has the property g −1 (g(b)) = {b} for all b ∈ Ekn . Theorem 20.6.4 ([B¨ or-H 97]; without proof ) Let C ⊆ Pk be a clone. Then the partial clone Str(C) is ﬁnitely generated if and only if C is ﬁnitely generated and there is an r ∈ N such that C is r-separable. If C r-separable and ﬁnitely generated, then ord(Str(C)) ≤ max{ord(C), r}.

Theorem 20.6.5 ([Noz-L 97], [Had-L 2000]; without proof ) Let k ≥ 3. Then (1) For every ∈ Mk ∪ Uk ∪ C1k ∪ C2k it holds ord(pP OLk ) = 2.

20.7 Intervals of Partial Clones That Contain a Maximal Clone

619

(2) For every ∈ Chk with 3 ≤ h ≤ k − 1 it holds ord(pP OLk ) ≤ h. (3) For every ∈ Mk ∪ Bk it holds ord(Str(P ol )) = 2. (4) ord(pP OLk s2 ) = 2 (see Theorem 20.6.2).

20.7 Intervals of Partial Clones That Contain a Maximal Clone Since there are many results about subclasses of Pk , it is obvious to classify k after Pk ∩ C. For an arbitrary subclass A of Pk : the subclasses C of P k | [F ] = F ∧ F ∩ Pk = A}. I(A) := {F ⊆ P This section aims to obtain cardinality statements over the sets I(A) for the maximal clones A of Pk . First, however, some general properties of the set I(A): Lemma 20.7.1 ([Str 97]; without proof ) Let A ⊆ Pk be a clone. Then (1) I(A) contains the partial clones A ∪ C∞ and Str(A) as well all partial clones of the form pP OLk with P olk = A. (2) If A is ﬁnitely generated with ord A = n, then pP OLk Gn (A) is the greatest element in the lattice (I(A); ⊆). Now we obtain from (2) of the above lemma: Theorem 20.7.2 If A is a ﬁnitely generated clone on Ek of the order n, then I(A) is exactly the interval [A, pP OLk Gn (A)] of the lattice of all partial k . clones of P Lemma 20.7.3 Let A = [A] ⊆ Pk . Then (1) A is a maximal subclass of A ∪ C∞ . (2) If {c10 , . . . , c1k−1 } ⊂ A then A ∪ C∞ ⊆ B for all B ∈ I(A)\{A}. Proof. (1) is clear. (2): Let B ∈ I(A)\{A}. If B = A ∪ C∞ , we do not have to prove anything. Otherwise, there is a function f n ∈ B \(A ∪ C∞ ). Let f (a1 , . . . , an ) = ∞. Now c1∞ = f (c1a1 , . . . , c1an ) ∈ B and from this one can easily verify that C∞ ⊆ B. Theorem 20.7.4 ([Had-L-R 2002; without proof ) Let ∅ = ⊂ Ek , T := P olk and k | f (n ) = {∞} }. {f n ∈ P T∞ := n≥1

Then I(T ) consists exactly of the partial clones:

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20 Partial Function Algebras

T , T ∪ C∞ , pP olk , T ∪ T∞ , pP olk ∪ C∞ , pP olk ∪ T∞ , pPOL k . The partial clones are pairwise distinct for || > 1 whereas for || = 1 pP olk ∪ T∞ and pP OLk coincide. Their inclusions are shown in Figure 20.1. q pP OLk

q T ∪ T∞

q T ∪ C∞

q T

q pP olk ∪ T∞ q pP olk ∪ C∞ q pP olk

Fig. 20.1

Theorem 20.7.5 ([Had-L-R 2002]; without proof ) Let h ≥ 2 and ∈ Chk . Then the set I(P olk ) is a 3-element chain: P olk ⊂ (P olk ) ∪ C∞ ⊂ pP olk .

Theorem 20.7.6 ([Had-L-R 2002]; without proof ) Let ∈ Uk , X := P olk and 1 := ∪ {(∞, ∞)}. Then I(X) is a 4-element chain: X ⊂ X ∪ C∞ ⊂ pP olk 1 ⊂ P OLk . For clones M of type M, there are only partial results over I(M ). The following is one of these results: Theorem 20.7.7 ([Had-L-R 2002]; without proof ) Let ≤ ∈ Mk . Set M := P olk ≤, k }, ≤0 := ≤ ∪ {(∞, x) | x ∈ E k }, ≤1 := ≤ ∪ {(x, ∞) | x ∈ E 3 2 := {(x, y, z) ∈ Ek | x ≤ y ≤ z} ∪ {(∞, x, y), (x, y, ∞) | x, y ∈ Ek , x ≤ y}∪ ({∞} × Ek × {∞}), Mi := pP olk (≤i ) (i = 0, 1) and M2 := pP olk 2 . Then

20.7 Intervals of Partial Clones That Contain a Maximal Clone

621

{M, M ∪ C∞ , M0 , M1 , M2 , Str(M )} is the set of all partial clones from I(M ) included in Str(M ). Their inclusions are shown in Figure 20.2. Str(M ) q q M 2 @ @ @q M 1

M0 q @ @

@q M ∪ C ∞ q M

Fig. 20.2

Because of Str(P olk ≤) = P olk ≤ ⇐⇒ ≤∈ M k , Theorem 20.7.8 follows. Theorem 20.7.8 For every ∈ M k is |I(P olk )| = 6. Next we prove that I(A) is a ﬁnite set if A is a maximal class of type S. For this, we need some notations: Let p be a prime factor of k and s ∈ Pk,p a ﬁxed-point-free permutation on Ek comprising of cycles of the same length p. Set so := {(x, s(x)) | x ∈ Ek }, S := P olk so , s2 := {(x, s(x), . . . , sp−1 (x)) | x ∈ Ek } and Smax := pP OLk s2 . By Lemma 20.6.2, we have that Smax is the (unique) maximal partial clone containing the maximal clone P olk so . As usual, the powers of s are deﬁned recursively by setting s0 (x) := x and si+1 (x) := s(si (x)) for all x ∈ Ek . To describe functions of Smax , we deﬁne the following relations on Ekn : x =s y :⇐⇒ (∃i ∈ {0, 1, ..., p − 1} : si (x) = y), where si (x) = si ((x1 , . . . , xn )) := (si (x1 ), . . . , si (xn )). The relation =s is an equivalence relation of Ekn with rn := k n /p equivalence classes (or blocks) which we denote by U1 , . . . , Urn . Fix vi ∈ Ui (i = 1, . . . , rn ) and set Vn := {v1 , v2 , . . . , vrn }. Since for all f n ∈ S, b ∈ Ekn and i ∈ Ep

622

20 Partial Function Algebras

f (si (b)) = si (f (b)), each function f n ∈ S is fully determined by its values on Vn . Set p

k \E p | ∃α ∈ Ek ∀i ∈ Ep (ai = ∞ =⇒ ai = si (α)}. γ := {(a0 , . . . , ap−1 ) ∈ E k For i = 1, 2, . . . , p let the p-ary relation γi consist of all (a0 , a1 , . . . , ap−1 ) ∈ γ with exactly i coordinates ∞. Furthermore, set γ0 := s2 . For every I ⊆ Ep let τI := {(a0 , a1 , . . . , ap−1 ) ∈ γ | (∀i ∈ I : ai = ∞) ∧ (∀j ∈ Ep : \I aj = ∞)}. For every function f n ∈ Smax set χ (f ) := {(f (a), f (s(a)), . . . , f (sp−1 (a))) | a ∈ Vn } and χ(f ) := {(f (a), f (s(a)), . . . , f (sp−1 (a))) | a ∈ kn }. k with χ (f ) ⊆ χ(f ) and Notice that χ (f ) and χ(f ) are p-ary relations on E p

k \E p ) ∪ s2 , χ(f ) ⊆ (E k

(20.8)

and hence, in general, χ(f ) is not a subrelation of s2 . It is easy to check that χ(f ) = {(ai , ai+1 , . . . , ap−1 , a0 , a1 , . . . , ai−1 ) | (a0 , a1 , ..., ap−1 ) ∈ χ (f ), i ∈ Ep }. (20.9) p p 2 Moreover, for R ⊆ (Ek \Ek ) ∪ s and α ∈ {χ , χ}, set α−1 (R) := {g ∈ Smax | α(g) ⊆ R}. Then it holds that α−1 (R) = {g ∈ Smax | ∀ r1 , . . . , rn ∈ γ0 g+ (r1 , . . . , rn ) ∈ R}. We start with Lemma 20.7.9 Let f n ∈ Smax and I, I ⊆ Ep . Then (1) Gf := {g n ∈ Str (S) | D(g) = D(f )} ⊆ [S ∪ {f }], (2) χ−1

( χ (f ) ) ⊆ [S ∪ {f }],

20.7 Intervals of Partial Clones That Contain a Maximal Clone

623

(3) (α0 , α1 , . . . , αp−1 ) ∈ χ (f ) =⇒ χ−1

( {(α1 , α2 , . . . , αp−1 , α0 )} ∪ χ (f )) ⊆ [S ∪ {f }], (4) there is a function g ∈ [S ∪ {f }] with χ (g) = χ(f ), (5) χ−1 ( χ(f ) ) ⊆ [S ∪ {f }], (6) χ(f ) ∩ τI = ∅ =⇒ χ−1 (τI ∪ χ(f )) ⊆ [S ∪ {f }], (7) (χ(f ) ∩ τI = ∅ ∧ χ(f ) ∩ τI = ∅) =⇒ χ−1 (τI∪I ∪ χ(f )) ⊆ [S ∪ {f }], p (8) (p ∈ {2, 3} ∧ j ∈ {1, . . . , p} ∧ χ(f )∩γj = ∅) =⇒ χ−1 ( i=j γi ∪χ(f )) ⊆ [S ∪ {f }], 3

3

k \E 3 ∪χ(f )) ⊆ [S∪{f }]. k \(E 3 ∪γ)) = ∅) =⇒ χ−1 (E (9) (p = 3 ∧ χ(f )∩(E k k Proof. (1): Let g n ∈ Gf and g1 ∈ S with g1|D(f ) = g. Then g = e21 (g1 , f ) ∈ [S ∪ {f }]. (2): Let g m ∈ χ−1

(χ (f )) be arbitrary. Then there is for every v ∈ Vm a bv := (bv1 , bv2 , . . . , bvn ) ∈ χ (f ) with (g(v), g(s(v)), . . . , g(sp−1 (v))) = (f (bv ), f (s(bv )), . . . , f (sp−1 (bv ))). It is easy to check that the functions gim ∈ S (i = 1, 2, . . . , n) exist with gi (v) = bvi for all i ∈ {1, . . . , n} and v ∈ Vm . Then we have that g = f (g1 , . . . , gn ) and (2) hold. (3): Let a1 , . . . , aq ∈ Vn with ai = (ai1 , ai2 , . . . , ain ) (i = 1, . . . , q), χ (f ) = {(f (ai ), f (s(ai )), . . . , f (sp−1 (ai ))) | i = 1, . . . , q} and (f (aq ), f (s(aq )), . . . , f (sp−1 (aq ))) = (α0 , α1 , . . . , αp−1 ). Then there exists a t ∈ N, b1 , . . . , bq+1 ∈ Vt and functions gjt ∈ S (j = 1, 2, . . . , n) with gj (bi ) := aij (i = 1, 2, ..., q, j = 1, . . . , n) and gj (bq+1 ) := s(aqj ) (j = 1, . . . , n). Let g := f (g1 , . . . , gn ). So we have g ∈ [S ∪ {f }] and it is easy to check that χ (f ) ∪ {(α1 , α2 , . . . , αp−1 , α0 )} ⊆ χ(g). Consequently, (3) follows from (2). (4) follows from (20.9) and (3). (5) follows from (4) and (2). In the following, we can assume χ (f ) = χ(f ). (6): Let (a0 , a1 , . . . , ap−1 ) ∈ χ(f ) ∩ τI . Then there exists an α ∈ Ek with ∀i ∈ Ep (ai = ∞ =⇒ ai = si (α) .

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20 Partial Function Algebras

Obviously, there are a t ∈ N, a function q t ∈ S with χ (q) = γ0 and a function h2 ∈ S with y if x ∈ {α, s(α), . . . , sp−1 (α)}, h(x, y) := x otherwise. Then n+t n+t h1 := h(f (en+t , en+t ..., en+t n ), q(en+1 , . . . , en+t )) ∈ [S ∪ {f }] 1 2

and it is easy to check that τI ∪ χ(f ) ⊆ χ(h1 ). Thus by (5) we have χ−1 (τI ∪ χ(f )) ⊆ χ−1 (χ(h1 )) ⊆ [S ∪ {f }]; i.e., (6) holds. (7): Let a1 , . . . , aq ∈ Vn with ai = (ai1 , ai2 , . . . , ain ) (i = 1, . . . , q), χ(f ) = {(f (ai ), f (s(ai )), . . . , f (sp−1 (ai ))) | i = 1, . . . , q}, (f (aq−1 ), f (s(aq−1 )), . . . , f (sp−1 (aq−1 ))) ∈ τI and (f (aq ), f (s(aq )), . . . , f (sp−1 (aq ))) ∈ τI . Then there exists t ∈ N, b1 , . . . , bq+1 ∈ Vt and functions gjt , htj ∈ S (j = 1, 2, . . . , n) with gj (bi ) := hj (bi ) := aij (i = 1, 2, ..., q, j = 1, . . . , n) and gj (bq+1 ) := aq−1,j , hj (bq+1 ) := aq,j (j = 1, . . . , n). Let g := e21 (f (g1 , . . . , gn ), f (h1 , . . . , hn )). Then g ∈ [S ∪{f }], and it is easy to check that χ(f ) ⊆ χ(g) and χ(g)∩τI∪I = ∅. Then, it follows from (6): χ−1 (τI∪I ∪ χ(f )) ⊆ [S ∪ {g}] ⊆ [S ∪ {f }]. (8) follows from (3) and (5)–(7). (9): Let p = 3, (a, b, ∞) ∈ χ(f ), a, b ∈ Ek and s(a) = b. Furthermore, let t ∈ N, k t /3 ≥ k 2 and {(αi , βi ) | i = 1, 2, ..., k2 } := Ek2 . Then there are c1 , . . . , ck2 ∈

:= {(a, ci ), (b, s(ci )) | i = 1, 2, . . . , k2 } with Vt with x =s y for all x, y ∈ Vt+1

⊆ Vt+1 holds, there is a x = y. Since we can choose Vt+1 such that Vt+1 t+1 ∈ S with function g ⎧ ⎨ αi if x = a, x = ci , g(x, x) := βi if x = b, x = ci , ⎩

. x if x ∈ Vt+1 \Vt+1 Then

20.7 Intervals of Partial Clones That Contain a Maximal Clone

625

n+t n+t h := g(f (en+t , . . . , en+t n ), en+1 , . . . , en+t ) ∈ [S ∪ {f }] 1

and it is easy to check that Ek2 × {∞} ∪ χ(f ) ⊆ χ(h). Then by (3) and (8) it follows that (9) holds. Theorem 20.7.10 ([Had-L-R 2002]) The set I(S) is ﬁnite. Proof. Obviously, Smax =

χ−1 (R).

(20.10)

k p \E p )∪s2 R⊆(E k

By Lemma 20.7.9, (5) we have χ−1 (χ(f )) ⊆ [S ∪ {f }]

(20.11)

for every function f n ∈ Smax . Let G := {χ(f ) | f ∈ Smax }. By (20.8) G is a ﬁnite set. Let C ∈ I(S). Obviously, H := {χ(f ) | f ∈ C} ⊆ G is ﬁnite. Thus H = {χ(1 ), . . . , χ(h )} for certain 1 , . . . h ∈ C. Furthermore, it holds that C ⊆ χ−1 (χ(1 )) ∪ . . . ∪ χ−1 (χ(h )), where, by (20.11), χ−1 (χ(1 ))∪. . .∪χ−1 (χ(h )) ⊆ [S∪{1 }]∪. . .∪[S∪{h }] ⊆ [S∪{1 , . . . , h }] ⊆ C. Thus C = [S ∪ {1 , . . . , h }]. Consequently, the partial clone C ∈ I(S) is generated from S and from not more than |G| functions of Smax \ S. Now we determine exactly the set I(S) for the cases p = 2 and p = 3. We begin with the case p = 2. Str(S) q

S1 q @ @

q S ∪S 1 2 @ @ @q S 2 @q S ∪ C ∞ q S

Fig. 20.3

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20 Partial Function Algebras

Theorem 20.7.11 ([Had-L-R 2002]; without proof ) Let k ≥ 2, s ∈ Pk,2 and S = P olk so . Furthermore, let S1 := χ−1 (γ0 ∪ γ2 ) and S2 := S ∪ χ−1 (γ1 ∪ γ2 ). Then I(S) = {S, S ∪ C∞ , S1 , S2 , S1 ∪ S2 , Str(S)}, where Smax = pPOL k so = Str(S). The lattice (I(S)); ⊆) is given in Figure 20.3. Smax

r

Str(S) ∪ S5 r

H

HH

HH

H H r S3 ∪ S5 = S1 ∪ S3 ∪ S5

Str(S) r

r S1 ∪ S3

r S4 ∪ S5 = S 1 ∪ S4 ∪ S5

S ∪ S r 1 4 r S ∪ S5 = S1 ∪ S5 r r S1 S2 ∪ S4 H HH HH H H r S2 S4 r @ @ @ @r S ∪ C∞ r S3

rS Fig. 20.4

Theorem 20.7.12 ([Had-L-R 2002]; without proof ) Let s ∈ Pk,3 and S = Pol k so . Furthermore, let S1 := S ∪ χ−1 (γ1 ∪ γ2 ∪ γ3 ), S2 := S ∪ χ−1 (γ2 ∪ γ3 ), 3

k \ E 3 ), S3 := χ−1 (γ0 ∪ γ2 ∪ γ3 ), S4 := χ−1 (γ0 ∪ γ3 ) and S5 := χ−1 (E k where Str(S) = χ−1 (γ0 ∪ . . . ∪ γ3 ). Then

20.8 Intervals of Boolean Partial Classes

627

I(S) = {S, S ∪ C∞ , S2 , S4 , S2 ∪ S4 , S1 , S3 , S1 ∪ S4 , S1 ∪ S3 , S ∪ S5 , S4 ∪ S5 , S3 ∪ S5 , Str (S), Str (S) ∪ S5 , Smax }. The lattice (I(S); ⊆) is given in Figure 20.4. Theorem 20.7.13 ([Ale-V 94], [B¨ or-H 98], [Had-L 2003]) For every ∈ Lk ∪ Bk , the set I(P olk ) has the cardinality of continuum. Proof. For k = 2 (and therefore = {(a, b, c, d) | a + b = c + d (mod 2)}) the theorem was proven in [Ale-V 94]. If k is an arbitrary prime number power, then |I(P olk )| = c with ∈ Lk was proven in [Had-L 2003], where the proof of the general result includes the proof from [Ale-V 94]. or-H 98]. This result and Theorem 5.2.6.1 |I(P olk ιkk )| = c was shown in [B¨ were used in [Had-L 2003] to prove the remaining statements of the theorem.

20.8 Intervals of Boolean Partial Classes In continuation of the examinations of Section 20.7, one can ﬁnd cardinality statements over the set I(A) for arbitrary subclasses A of P2 in the following. For this we use the notations from Chapter 3. Theorem 20.8.1 ([Ale-V 94], [Str 97b]) Let A be a subclass of P2 with A ⊆ L or A ⊆ B ∈ {I ∪ C, D ∪ C, K ∪ C, T0,∞ , T1,∞ }. Then the set I(A) has the cardinality of continuum. Proof. For A = L, the theorem was proven by V. B. Alekzeev and L. L. Voronenko in [Ale-V 94]. By easy modiﬁcation of the proof from [Ale-V 94], one gets the statements of the theorem for A ∈ {L, L ∩ T0 , L ∩ T1 , L ∩ T0 ∩ T1 }. One ﬁnds proof of the remaining statements of the theorem in [Str 97b]. Theorem 20.8.2 ([Ale-V 94], [Str 97a], [Str 95], [Str 96], [Lau 2006]) Let A be a subclass of P2 with T0 ∩ T1 ∩ M ⊆ A or T0 ∩ S ⊆ A (or T1 ∩ S ⊆ A). Then I(A) is a ﬁnite set and it holds that

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20 Partial Function Algebras

A |I(A)| P2 3 Ta (a ∈ {0, 1}) 6 M 6 S 6 T0 ∩ T1 30 Ta ∩ M (a ∈ {0, 1}) 15 T0 ∩ T1 ∩ M 101 T0 ∩ S ? < 2000 For the remaining subclasses A of Pk , i.e., M ∩ Ta ∩ Ta,∞ ⊂ A ⊆ Ta,2 for certain a ∈ E2 or A = T0 ∩ M ∩ S, it holds |I(A)| ≥ ℵ0 . 2 } and therefore |I(P2 )| = 3 follows from Proof. I(P2 ) = {P2 , P2 ∪ C∞ , P Lemma 20.4.2. The statements of the theorem over maximal classes of P2 diﬀerent from L were proven by V. B. Alekzeev and L. L. Voronenko and by B. Strauch independently of each other. (see [Ale-V 94], [Str 94], [Str 97a]). Certain elements of the sets I(A) with A ∈ {T0 , T1 , S, M } were already determined in [Lau 88] 2 . On can ﬁnd the description at the determination of submaximal classes of P of the elements of I(A) with A ∈ {T0 , T1 , S, M } in Theorems 20.7.4, 20.7.7, and 20.7.11. The sets I(T0 ∩ T1 ) and I(T0 ∩ M ) were determined in [Str 97a]. One can ﬁnd I(M ∩ T0 ∩ T1 ) in [Str 95]. The ﬁniteness of the set I(S ∩ T0 ∩ T1 ) was proven in [Str 96]. In [Lau 2006] it was proven that every set I(A) with A ⊆ T0,2 has inﬁnitely many elements.

20.9 On Congruences of Partial Clones This section is a revised version of the papers [Lau-D 90] and [Lau-D 91]. For an arbitrary partial clone C, we deﬁne the following equivalence relations which – as A. I. Mal’tsev in [Mal 66] was already proving – are congruences k , are the only possible congruences (see Theorem over C and, for C = P 20.9.3): κ0 := {(f, f ) | f ∈ C} κ1 := C × C κa := {(f n , g m ) ∈ C × C | n = m} κ∞ := κ0 ∪ {(f n , g m ) ∈ C × C | D(f ) = D(g) = ∅}

20.9 On Congruences of Partial Clones

629

κ1 r @ @ κa r @r κ∞ κ0 r One is easily able to transfer many results from Chapter 9 (because of results of Section 20.2) to partial clones. Therefore, only the congruences on the maximal partial clones shall be determined. For this we need the following notations, which are introduced for relations ∈ Rkh . κ1 () := κ0 ∪ {(f m , g n ) ∈ κ1 | ∀r1 , ..., rmax(m,n) ∈ : k )h \E h }, {f (r1 , ..., rm ), g(r1 , ..., rn )} ⊆ (E k κa () := κ1 () ∩ κa , U () := {α ∈ Ek | ∀(a1 , ..., ah ) ∈ : α ∈ {a1 , ..., ah }}, µ0 () := {(f m , g m ) ∈ κa | ∀a ∈ (Ek \U ())m : f (a) = g(a)}, µ() := {(f m , g m ) ∈ κa | ∀r1 , ..., rm ∈ : k )h \E h ∨ {f (r1 , ..., rm ), g(r1 , ..., rm )} ⊆ (E k f (r1 , ..., rm ) = g(r1 , ..., rm )}. Obviously, the above relations are equivalence relations with the following Hasse-diagram: κ1 r @ @ κa r @r κ1 () ∪ µ() κa () ∪ µ() r r κ () ∪ µ0 () 1 κa () ∪ µ0 () r r κ1 () κa () r r κ∞ @ @ @r κ0 Since Ek \U () = , if is a unary relation, we have in this case µ() = µ0 (). If U () = ∅, then µ0 () = κ0 . If all constant functions and a function cn∞ belong to the partial clone pP OLk , then we have κ1 () = κ∞ and κa () ∪ µ() = κ0 . The following Lemma is easy to prove.

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20 Partial Function Algebras

Lemma 20.9.1 k . Then the relations κ0 , κa , κ1 , κ∞ are congruences on C. (1) Let C ⊆ P k . Then the relations κa (), κa () ∪ (2) Let ⊆ Ekh and C := pP OLk ⊆ P µ0 (), κa () ∪ µ() κ1 (), κ1 () ∪ µ0 (), κ1 () ∪ µ() are congruences on C. k of the In the following, it shall be proven that a maximal partial clone of P form pP OLk has only those congruences given above. Then, dependently of , one has 4, 8 or 10 pairwise distinct congruences per partial clone pP OLk . Lemma 20.9.2 Let C be a partial clone on Ek and κ a congruence of C. Then (a) κ ∩ (Pk × Pk ) ⊆ κa =⇒ κ = κ1 ; (b) (c1∞ ∈ C ∧ κ ⊆ κa ) =⇒ κ∞ ⊆ κ. Proof. (a): Let κ∩(Pk ×Pk ) ⊆ κa . Then there are functions f m , g n ∈ C ∩Pk with m < n and (f, g) ∈ κ. Consequently, we have f11 := ∆n−2 f ∼ g12 := ∆n−2 g (κ) and

e22 = e22 f1 ∼ e33 = e22 g (κ).

Thus (enn , e11 ) ∈ κ for all n ≥ 1 and e11 h = h ∼ e22 h = et+1 t+1 (κ) for all ht ∈ C, i.e., κ = κ1 . (b): Let κ ⊆ κa and c∞ ∈ C. Then there are two functions f m , g n , m < n, in C with (f, g) ∈ κ. Thus (∆n−2 g) c1∞ = c2∞ ∼ (∆n−2 f ) c1∞ = c1∞ (κ). Consequently, c2∞ er1 = cr+1 ∼ c1∞ er1 = cr∞ (κ) for arbitrary r ∈ N. ∞ Therefore, κ∞ ⊆ κ. Theorem 20.9.3 ([Mal 66]) Let C be a clone with Str({ca | a ∈ Ek }) ⊂ C. Then C has exactly the following four congruences: κa , κa , κ∞ , κ1 . Proof. Let κ = κ0 be a congruence of C. Then the following cases are possible: Case 1: κ0 ⊂ κ ⊆ κa . In this case, there are two functions f m , g n ∈ C and an n-tuple a := (a1 , ..., an ) with f (ca1 , ..., can ) =: cα , g(ca1 , ..., can ) =: g , g ∈ {cβ , c∞ }, {α, β} ⊆ Ek , (cα , g ) ∈ κ and cα = g .

20.9 On Congruences of Partial Clones

631

Since Str({ca | a ∈ Ek }) ⊂ C, there is a function h1 ∈ C with D(h) = {α} and h(α) = α. Thus we have h c1α ∼ h g = c1∞ (κ) and e11 = ∆(e22 cα ) ∼ ∆(e22 c1∞ ) = c1∞ (κ). We obtain e11 t = t ∼ cn∞ (κ) for all tn ∈ C. Thus κa ⊆ κ and κ = κa . Case 2: κ0 ⊂ κa . Since c∞ ∈ C, it follows from Lemma 20.9.2: κ∞ ⊆ κ. Suppose, κ = κ∞ . Then there are two functions f m , g n ∈ C with m = n, {f, g} ⊆ {cn∞ | n ∈ N} and (f, g) ∈ κ. W.l.o.g. we can assume that f = cm ∞ and m < n. Let a := (a1 , ..., am ) ∈ D(f ) and f (a) = α ∈ Ek . Then (...((∆((∆((∆(f ca1 )) ca2 )) ca3 )) ca4 ))... cam = f (ca1 , ..., cam ) = cα ∼ (...((∆((∆((∆(g ca1 )) ca2 )) ca3 )) ca4 ))... cam =: g1r (κ) with r := n − m + 1. W.l.o.g. let r = 2. If g1 = c2∞ , we obtain (c1α , c1∞ ) ∈ κ and analogously to Case 1, κa ⊆ κ. κa ⊆ κ and κ∞ ⊆ κ imply κ = κ1 . If g1 = c2∞ , there exists a (b1 , b2 ) ∈ Ek2 with g1 (b1 , , b2 ) ∈ Ek and thus we have c1α = c1α (τ (cα cb1 ) cb2 ) ∼ c1α (τ (g1 cb1 ) cb2 ) = c2α (κ). By Lemma 20.9.2, (a) we get κ = κ1 . Theorem 20.9.4 ([Lau-D 90]) max \{ ∈ R max | ∅ ⊂ Let C = Pk ∪ {cn∞ | n ∈ N} or C = pP OLk with ∈ R ⊂ Ek or is a coherent areﬂexive relation}, where Rmax denotes the set of all relations by which a maximal partial clone pP OLk is described (see 20.5). Then C has exactly the following four congruences: κ0 , κa , κ∞ , κ1 . max \{ ∈ R max | ∅ ⊂ ⊂ Ek or is a Proof. If C = pP OLk with ∈ R coherent areﬂexive relation}, our theorem follows from Theorem 20.9.3. It was proven in Chapter 9 (see Theorem 9.1.2) that Pk has only the congruences κ0 , κa , κ1 . From this and from Lemma 20.9.2 follows our theorem for C = Pk ∪ {cn∞ | n ∈ N}. Lemma 20.9.5 Let κ be a congruence of pP OLk . Then (a) κ0 ⊂ κ ⊆ κa =⇒ κa () ⊆ κ, (b) κ∞ ⊂ κ =⇒ κ1 () ⊆ κ, (c) κa () ⊂ κ ⊆ κ1 () =⇒ κ = κ1 (), (d) κa () ∪ µ0 () ⊂ κ ⊆ κ1 () ∪ µ0 () =⇒ κ = κ1 () ∪ µ0 (), (e) κa () ∪ µ() ⊂ κ ⊆ κ1 () ∪ µ() =⇒ κ = κ1 () ∪ µ(). Proof. (a): Let κ0 ⊂ κ ⊆ κa . Then there are functions f n , g n with (f n , g n ) ∈ κ and certain a := (a1 , ..., an ) ∈ Ekn , a, b ∈ Ek , a = b with f (a) =: a = b := g(a). Furthermore, pP OLk contains functions tα (α ∈ Ek ) and hβ,γ (β, γ ∈ Ek ) with

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20 Partial Function Algebras

tα (x, y) :=

and hβ,γ (x) :=

y if x = α, ∞ otherwise, γ if x = β, ∞ otherwise,

where, if is areﬂexive or {ca | a ∈ Ek } ⊂ pP OLk holds, β and γ are arbitrary elements of Ek ; otherwise we must choose, however, β ∈ {a | (a, a, ..., a) ∈ }. Thus we obtain f (hβ,a1 (x), ..., hβ,an (x)) = hβ,a (x) ∼ g(hβ,a1 (x), ..., hβ,an (x)) = hβ,b (x) (κ) and

ta (hβ,a (x), y) = tβ (x, y) ∼ ta (hβ,b (x), y) = c2∞ (x, y) (κ).

Let um be an arbitrary function from pP OLk with the property ∀(r11 , r21 , ..., rh1 ), ..., (r1n , r2n , ..., rhn ) ∈ ∃i ∈ {1, ..., h} : (ri1 , ri2 , ..., rin ) ∈ D(u). Then the function v m deﬁned by β if u(x) ∈ Ek , v(x) := ∞ otherwise, belongs to pP OLk . Consequently, we have tβ (v(x), u(x)) = u(x) ∼ c2∞ (v(x), u(x)) = cm ∞ (x) (κ), i.e., it holds κa () ⊆ κ. (b): For κ∞ ⊂ κ and κ ∩ κa = κ0 , we have κa () ⊆ κ because of (a), since the inclusion κa () ⊆ κ follows from κ ∩ κa = κ = κ0 and since the inclusion κa () ⊆ κ follows from κ ⊆ κ. κa () ⊆ κ and κ∞ ⊆ κ imply κ1 () ⊆ , since, if (f m , g n ) ∈ κ1 () (m = n), n n the inclusions (f m , cm ∞ ) ∈ κ and (c∞ , g ) ∈ κ follow, because of κa () ⊆ κ m n and (c∞ , c∞ ) ∈ κ by κ∞ ⊆ κ. By the transitivity of κ we get (f m , g n ) ∈ κ. If κ∞ ⊂ κ and κ ∩ κa = κ0 , there are two functions sl , tr (l = r) in pP OLk with (s, t) ∈ κ. We can assume l > r and s = cl∞ , i.e., there exists an element a := (a1 , ..., al ) with s(a) ∈ Ek . We distinguish the following two cases: Case 1: For all i ∈ {1, ..., l} it holds (ai , ..., ai ) ∈ . In this case we have {ca1 , ..., cal } ⊂ pP OLk . From (s, t) ∈ κ and κ∞ ⊆ κ the existence of two κ-kongruent constant functions with diﬀerent arities follows. Therefore, by Lemma 20.9.2, (a) we get (Pk ∩ pP OLk ) × (Pk ∩ pP OLk ) ⊆ κ, a contradiction to κ ∩ κa = κ0 .

20.9 On Congruences of Partial Clones

633

Case 2: There exists an i ∈ {1, ..., l} with (ai , ..., ai ) ∈ . W.l.o.g. we can assume (a1 , ..., a1 ) ∈ . Then the function h1a1 ,aj belongs to pP OLk for certain aj ∈ Ek , j = 1, ..., n. Further, we can assume a1 = al−r . (If a1 = al−r , then we can choose s := s e22 and t := t e2 and then use the operations ζ and τ to get functions s and t with (s , t ) ∈ κ, s (a) ∈ Ek , (a1 , ..., a1 ) ∈ and a1 = al−r ). Thus s1 := (....(ζ((ζ((ζs) ha1 ,a1 )) ha1 ,al−1 ha1 ,al−2 )) ...) ha1 ,al−r ) ∼ t1 := (....(ζ((ζ((ζt) ha1 ,a1 )) ha1 ,al−1 ha1 ,al−2 )) ...) ha1 ,al−r ) (κ) with s1 (x1 , ..., xl ) = s(xl−r+1 , ..., xl , ha1 ,al−r (x1 ), ..., ha1 ,al (xl−r )) and t1 (x1 , ..., xr ) = t(ha1 ,al−r+1 (x2 ), ..., ha1 ,al−1 (xr ), ha1 ,al (ha1 ,al−r (xl ))) = cr∞ (x), since ha1 ,al ha1 ,al−r = c1∞ and because of a1 = a1−r . From this and from κ∞ ⊆ κ, it follows that κ ∩ κa = κ0 , in contradiction to our assumption. Therefore (b) holds. (c): Let κa () ⊂ κ ⊆ κ1 (). Then κ ⊆ κa . By Lemma 20.9.2, (b) we have κ∞ ⊆ κ. From κ∞ ⊆ κ ⊆ κ1 () it follows that κ = κ∞ or κ = κ1 () (by (b)). Since the ﬁrst case is not possible, (c) holds. (d) and (e) follow from (c). Lemma 20.9.6 Let ∅ ⊂ ⊂ Ek or let be an h-ary relation on Ek with the properties h ≥ 2 and ∀(a1 , ..., ah ) ∈ ∃t ∈ Pk ∩ pP OLk : Im(t) ⊆ {a1 , ..., ah }.

(20.12)

Then for an arbitrary congruence κ of pP OLk , (a) (κ ⊆ κa () ∪ µ() ∧ κ ⊆ κa ) =⇒ κ = κa , (b) (κ ⊆ κ1 () ∪ µ() ∧ κ ⊆ κa ) =⇒ κ = κ1 . (c) (κ ⊆ κa () ∧ κ ⊆ κa ) =⇒ κa () ∪ µ0 () ⊆ κ. Proof. (a): κ ⊆ κa ∪µ() and κ ⊆ κa imply that there are functions (f n , g n ) ∈ κ and a set R := {(rj1 , rj2 , ..., rjn ) | i = 1, ..., h} with {(r1i , r2i , ..., rhi ) | i = 1, ..., n} ⊆ , and 1) R ⊆ D(f ), R ⊆ D(g) and f (rj1 , rj2 , ..., rjn ) = g(rj1 , rj2 , ..., rjn ) for certain j ∈ {1, ..., h} or 2) R ⊆ D(f ) and R ⊆ D(g). The ﬁrst case can be reduced to the second case as follows: Let e33 be a ternary function of pP OLk deﬁned by

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20 Partial Function Algebras

e33 (x, y, z) =

z if x = y, ∞ otherwise.

Then e33 (f (x), f (x), f (x)) = f (x) ∼ e33 (f (x), g(x), f (x)) =: g (x) (κ) with R ⊆ D(g ). Consequently, we can assume that R ⊆ D(f ) and R ⊆ D(g). Let (a1 , ..., ah ) ∈ . Then the functions t1 , ..., th with ⎛ ⎞ ⎛ ⎞ r1i a1 ⎜ a2 ⎟ ⎜ r2i ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ti ⎜ ⎜ . ⎟=⎜ . ⎟ ⎝ . ⎠ ⎝ . ⎠ ah rhi and ti (x) = ∞ for x ∈ {a1 , ..., ah } (i = 1, 2, ..., n) belong to pP OLk . Then, from (f, g) ∈ κ it follows f (t1 (x), ..., tn (x)) =: f (x) ∼ g(t1 (x), ..., tn (x)) =: g (x) (κ), where {r1 , ..., rh } ⊆ D(g ) and (f (a1 ), ..., f (ah )) ∈ for all (r1 , ..., rh ) ∈ . By Lemma 20.9.2 we have κa () ⊆ κ and thus (g , c1∞ ) ∈ κ and (f , c1∞ ) ∈ κ. (20.12), implies the existence of a function t1 ∈ Pk ∩ pP OLk with Im(t1 ) ⊆ {a1 , ..., ah }. Using this function and the fact that f = f t, we obtain another function of Pk ∩ pP OLk with (f , c1∞ ) ∈ κ. Thus (∆(e22 f ), ∆(e22 c1∞ )) = (e11 , c1∞ ) ∈ κ and κ = κa . (b): It is easy to check that κ ⊆ κ1 () ∪ µ() and κ ⊆ κa imply κ ∩ κa ⊆ κa () ∪ µ(). Hence (b) follows from (a) and κ∞ ⊆ κ. (c): Obviously, (c) holds, if U () = ∅. Thus we can assume U () = ∅. Because of Lemma 20.9.5, (a) we have to show that µ0 () ⊆ κ. Let κ be a congruence of pP OLk with κ ⊆ κa and κ ⊆ κa (). Then there are two diﬀerent nary functions f, g with (f, g) ∈ κ and ρj := (r1j , r2j , ..., rhj ) ∈ (j = 1, ..., n) with (ri1 , ri2 , ..., rin ) ∈ D(f ) for all i ∈ {1, ..., h} or (ri1 , ri2 , ..., rin ) ∈ D(g) for all i ∈ {1, ..., h}. We can assume that κ = κa . Then by (a) we have κ ⊆ κa () ∪ µ(), i.e., f (r1 , ..., rn ) = g(r1 , ..., rn ) holds. Because of f = g there exists an n-tuple a := (a1 , ..., an ) ∈ Ekn with f (a) = g(a). It was already shown in proof of (a) that we can assume a ∈ D(g) and a ∈ D(f ). Let (α1 , ..., αh ) ∈ and a ∈ U (). Then the functions h1 , ..., hn deﬁned by (hi (α1 ), ...hi (αh )) = ri , hi (a) = ai and hi (x) := ∞ otherwise belong to pP OLk . By this we have f (x) := f (h1 (x), ..., hn (x)) ∼ g(h1 (x), ..., hn (x)) =: g (x) (κ), where f (αi ) = g (αi ), f (a) = f (a1 , ..., an ) and a ∈ D(g ). Obviously, if is a unary relation or because of (20.12) (if is an areﬂexive relation and

20.9 On Congruences of Partial Clones

635

h ≥ 2) pP OLk contains a function t ∈ Pk with Im(t) ⊆ {α1 , ..., αn }. Thus the function t deﬁned by t(x) if x ∈ U (), t (x) := a otherwise, belongs to pP OLk and we get f t =: f ∼ g := g t (κ) and e22 (f (x), x) = e11 (x) ∼ e11 (x) := e22 (g (x), x) (κ) with x if x ∈ U (), e(x) := ∞ if x ∈ U (). Let pm be an arbitrary function of pP OLk and let p(x) if x ∈ (Ek \U ())m , pa (x) := a otherwise,

and p∞ (x) :=

p(x) if x ∈ (Ek \U ())m , ∞ otherwise.

From (e11 , e11 ) ∈ κ it follows e11 pa = pa ∼ e11 pa = p∞ (κ) and e22 (pa (x), p(x)) = p(x) ∼ e22 (p∞ (x), p(x)) = p∞ (x) (κ). Consequently, two m-ary functions p, q with p(x) = q(x) for x ∈ (Ek \U ())m are κ-kongruent, i.e., µ0 () ⊆ κ. Lemma 20.9.7 Let be an areﬂexive h-ary relation with h ≥ 2 and let (0, 1, ..., h − 1) ∈ . Further, for every (r1j , r2j , ..., rhj ) ∈ , j = 1, ..., n, there ∈ Pk ∩ pP OLk with are unary functions h1 , ..., hn (h1 (i), h2 (i), ..., hn (i)) = (ri1 , ri2 , ..., rin ) for all i ∈ {0, ..., h − 1}. Then for every congruence κ of pP OLk : (a) µ0 () ⊂ κ\κ1 () ⊆ µ() =⇒ µ() ⊆ κ; (b) κa () ∪ µ0 () ⊂ κ ⊆ κa () ∪ µ() =⇒ κ = κa () ∪ µ(); (c) κ1 () ∪ µ0 () ⊂ κ ⊆ κ1 () ∪ µ() =⇒ κ = κ1 () ∪ µ(). Proof. (a): Let κ be a congruence of pP OLk with µ0 () ⊂ κ\κ1 () ⊆ µ(). Then there are two κ-kongruent functions f n , g n , such that for certain h-tuple (r11 ..., rh1 ), ..., (r1n , ..., rhn ) ∈ there are an n-tuple (α1 , ..., αn ) ∈ Ekn and an element α ∈ Ek with f (α1 , ..., αn ) = α, (α1 , ..., αn ) ∈ D(g) and ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ a1 r11 ... r1n r11 ... r1n ⎜ . ... . ⎟ ⎜ . ... . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ f⎜ ⎝ . ... . ⎠ = ⎝ . ⎠ = g ⎝ . ... . ⎠ . rh1 ... rhn ah rh1 ... rhn By assumption, there are functions h1 , ...., hn ∈ Pk ∩ pP OLk with {(h1 (i), ..., hn (i)) | i ∈ Eh } = {(rj1 , rj2 , ..., rjn ) | j ∈ {1, 2, ..., h}}. Let tm be an arbitrary function of pP OLk . Then, with the help of

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20 Partial Function Algebras

Dt := ⎧ ⎛ a1 ⎪ ⎪ ⎨ ⎜ . m x ∈ Ek | ∃a1 , ..., ah : x ∈ {a1 , ..., ah } ∧ ⎜ ⎝ . ⎪ ⎪ ⎩ ah

⎫ ⎞ a1 ⎪ ⎪ ⎬ ⎜ . ⎟ ⎟ ⎟ ⊆ ∧ t⎜ ⎟∈ ⎝ . ⎠ ⎠ ⎪ ⎪ ⎭ ah ⎞

⎛

one can deﬁne the functions ti by ⎧ x ∈ Dt , ⎨ hi (t(x)) if if x∈ Dt ∧ x ∈ D(t), ti (x) := αi ⎩ ∞ otherwise (i = 1, 2, ..., n). These functions belong to pP OLk and we get u(f (t1 (x), ..., tn (x)) = t(x) ∼ t (x) = u(g(t1 (x), ..., tn (x)) (κ)

with u(x, y) :=

y if x ∈ {a1 , ..., ah , α}, ∞ otherwise,

and

t (x) :=

t(x) if x ∈ Dt , ∞ otherwise.

Thus µ() ⊆ κ. (b) and (c) follow from (a). Lemma 20.9.8 Let ⊆ Ekh be an areﬂexive coherent relation with h ≥ 2. Then (a) ∀(a1 , ..., ah ) ∈ ∃t ∈ Pk ∩ pP OLk : Im(t) ⊆ {a1 , ..., ah } and w.l.o.g. (b) one can assume that fulﬁlls the assumptions of Lemma 20.9.7. Proof. By deﬁnition of (see Section 20.5) we can assume that there is a subgroup G of Sh and a surjective function ϕ : Ek −→ {0, 1, ..., h − 1} with the following properties: = {(π(0), π(1), ..., π(h − 1)) | π ∈ G }, is symmetric in respect to every π ∈ G and ∀(a0 , ..., ah−1 ) ∈ : (ϕ(a0 ), ..., ϕ(ah−1 )) ∈ . For an arbitrary a := (a0 , ..., ah−1 ) ∈ we consider the function ϕa : {0, 1, ..., h − 1} −→ {a0 , ..., ah−1 } with ϕa (i) = ai for all i ∈ {0, ..., h − 1}. Further, let ha := ϕa ϕ. Then Im(ha ) = {a0 , ..., ah−1 } and ha ∈ pP OLk , since for every b := (b0 , .., bh−1 ) ∈ there exists a permutation πb ∈ G with ϕ(b) = (ϕ(b0 ), ..., ϕ(bh−1 )). Thus we obtain ha (b) = ϕa (ϕ(b)) = ϕa ((πb (0), ..., πb (h − 1))) = (aπb (0) , ..., aπb (h−1) ) ∈ because of symmetry of with respect to every permutation π ∈ G . Consequently, (a) and (b) hold.

20.9 On Congruences of Partial Clones

637

Theorem 20.9.9 ([Lau-D 91]) Let ∅ ⊂ ⊂ Ek or let ⊆ Ekh an areﬂexive coherent relation with h ≥ 2 and U () = ∅. Then pP OLk has exactly 8 congruences with the following congruence lattice: κ1 r @ @ κa r @r κ () ∪ µ() 1 κa () ∪ µ() r r κ1 () κa () r r κ∞ @ @ @r κ0

Proof. Since µ() = µ0 () for all with ∅ ⊂ ⊂ Ek and for all areﬂexive relations with U () = ∅, our theorem follows from Lemmas 20.9.2 and 20.9.5– 20.9.8. The following theorem also follows from Lemmas 20.9.2, 20.9.5–20.9.8: Theorem 20.9.10 ([Lau-D 91]) Let ⊆ Ekh be an areﬂexive relation with h ≥ 2 and U () = ∅. Then pP OLk has exactly 10 congruences, and the congruence lattice of pP OLk is given by

κa κa () ∪ µ() κa () ∪ µ0 () κa ()

κ1 r @ @ r @r κ () ∪ µ() 1 r κ () ∪ µ0 () r 1 κ () r r 1 r κ∞ r @ @ @r κ0

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Glossary

Ud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ∧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ¬ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 =⇒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ⇐⇒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 :=. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 :⇐⇒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∃! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 af . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 f (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 f n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 D(f, A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 D(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Im(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 (n) c∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 f 2g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 g1 g2 ...gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 (A; f1 , ..., fr ) . . . . . . . . . . . . . . . . . . . . . . . . 26 (A; (fi )i∈I ) . . . . . . . . . . . . . . . . . . . . . . . . . . 26 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 f A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 fA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (D1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (D2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (M1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

(M2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (M3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (M4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (L1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L5 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (DL1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (DL2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (B1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Pkn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 [T ]A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 [T ]F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 [T ]f1 ,...,fr . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 [T ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 S(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (L1 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L1 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L2 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L2 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L3 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L3 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L4 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L4 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (O1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

656

Glossary

(O2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (O3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (O4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 sup Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (S1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (S2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 inf Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (I1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (I 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (E1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (E2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (E3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Eq(A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 a = b (mod ) . . . . . . . . . . . . . . . . . . . . . . . 43 a ∼ b (mod ) . . . . . . . . . . . . . . . . . . . . . . . 43 ∇A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ∆A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Π(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 A∼ = B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ∼ = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Ker ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Con(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 NG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ker ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 IR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 (σ, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 (GC1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 (GC2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ≤δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 pr1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 pr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 S(K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 H(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 P (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 I(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 XY (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 (F, τ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 T (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 T(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

T F (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 t < x1 , ..., xn > . . . . . . . . . . . . . . . . . . . . . . 76 s := t < t1 , ..., tn > . . . . . . . . . . . . . . . . . . 76 s ≈ t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A |= s ≈ t . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 IdX (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 M od(Σ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .77 IdX (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ConsX (Σ) . . . . . . . . . . . . . . . . . . . . . . . . . . 77 T(X)/IdX (K) . . . . . . . . . . . . . . . . . . . . . . 79 FK (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 FK (x1 , ..., xn ) . . . . . . . . . . . . . . . . . . . . . . . 79 FK (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 FK (x1 , x2 , ...) . . . . . . . . . . . . . . . . . . . . . . . 79 FK (ℵ0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 FK (ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 R1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Rep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 tˆ, t ∈ P rop . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 PAn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Pkn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 F n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 PA,B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pk,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 PA (l). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 Pk (l) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 PA [l] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pk [l] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 x(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 JA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Jk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 cn a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ∧ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ∨ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x + y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ⇒ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ⇐⇒ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x · y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 xy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 πs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Glossary ∇q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 f (g1 , ..., gn ) . . . . . . . . . . . . . . . . . . . . . . . . . 97 [F ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Lk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↓A (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↑A (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↑k (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↓k (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 LA (F ; G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 ord F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ord F = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ιhk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 δ{α,β} 3 δ{1,2,3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 P rop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 |= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Cons(Σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 V ar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 F ORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 =xk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 vA,u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Rkh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Qh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δk,ε δε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δεh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 Dkh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Dk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 h . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δk;ε 1 ,...,εr δε1 ,...,εr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δk;

657

h δk;E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 k ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 × . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 ∧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 [Q] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 σs (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 prα1 ,...,αt () . . . . . . . . . . . . . . . . . . . . . . . . 128 ∆i,j () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 νi () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ∇i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ◦t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 P olk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 P ol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 P olk Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Invk f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Invk A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 P oln Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Inv n A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 χk;n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 χn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 χ(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Gn (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 ΓA (σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 P olA Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Rphk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Rpk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 α(, ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 ∇(, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 (, ) × (µ, µ ) . . . . . . . . . . . . . . . . . . . . . 141 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ν1,a (, ) . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ν2,a (, ) . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Rpk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 m(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . 146 t(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 q(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 r(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 hµ (x1 , ..., xµ+1 ) . . . . . . . . . . . . . . . . . . . . 146 xσ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 f δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

658

Glossary

T0,µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 T1,µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Ta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 T0,∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 T1,∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 ζ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Ut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Lk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 ιhk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 λk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 prE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Mk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 ≤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Sk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Skn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Fr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 ja (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Uk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Lk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 λG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 (W ; ⊕) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Ln W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 LW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 a(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 ξm Bhk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Bk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 z(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

gf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175 fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Rmax (Pk ) . . . . . . . . . . . . . . . . . . . . . . . . . . 183 ∼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 M∼ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 S∼ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 ∼ (Pk ) . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Rmax o(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 z(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 ch (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 b(k, h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 +o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 αi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 µi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 z[i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 z t,a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 z t,a [i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 ϕn (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 gI,J Con A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 f ∼ g (κ) . . . . . . . . . . . . . . . . . . . . . . . . . 234 κ(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 κa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Con1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Cona A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 κc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235 µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 κa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 f 1 g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 κa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 µ(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 ki (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 k(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 α(κ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 LM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 r(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 TA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 NA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 κ(I, U ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 qa (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Glossary QA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252 ⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 LM ;id . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 κc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254 κs,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 κU,µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 κ0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 κZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 κc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261 κN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 κf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 µr (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 µr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 µr,N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Con(k). . . . . . . . . . . . . . . . . . . . . . . . . . . . .268 κα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 ess(f ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270 αµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 βµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 γµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Tn (U1 , U2 , ..., Ut ) . . . . . . . . . . . . . . . . . . . 273 κU1 ,...,Ut . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 κU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Ka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 prl−1 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 σn,κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 πn,κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 F (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 f α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 ar(). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .291 armax (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . 291 d(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 dim A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Q1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 k1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 ja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 prl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .337 pr−1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Nk (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 P olPk,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

659

P ol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 B a1 ,...,ar . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Za,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 dm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 t(i ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 R(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Tn (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Eq(Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Kf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 kf,I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Kf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 KM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K0,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Ud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 ϕ(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 t(q, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Tα,U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 SU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Cα,U,I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Cα,U,∅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 LM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Ta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Ta,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 ϕi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .435 ϕi (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

660

Glossary

τ (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 La,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 pra,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 pra,b Za,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Bc,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Br . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 numf (fi ) . . . . . . . . . . . . . . . . . . . . . . . . . . 465 num(fi ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 (III). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .468 (IV ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Ji,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 A3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Ij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 Aj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 TQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Mk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Uk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Sk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Pk,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Lk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Ck;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Nk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 MA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 MA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 UA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

UA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 SA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 SA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 PA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 LA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 CA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 CA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 NA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 BA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Rmax (TQ ) . . . . . . . . . . . . . . . . . . . . . . . . . . 501 (t1 , t2 , ..., tn ) . . . . . . . . . . . . . . . . . . . . . . 515 [t]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 Uk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Sk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 (1) Ck;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Chk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 ZC2k;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Ck;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Zk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .516 Nk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 h [r] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Ck;E l Ck;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Hk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 γr,s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 h Bk;E [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 l Bk;El [r] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Bk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Rmax (Pl ) . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Rmax (P olk El ) . . . . . . . . . . . . . . . . . . . . . 519 (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 (I ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 εb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Ai [b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 αi [b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Vb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Fb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 ξb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543 Ai,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 µi,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 νs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 fb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 wi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 z[i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 z t,a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 un . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Glossary s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 θs .n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 P ⊆p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 f+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 f− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 en i,X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 JX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 k h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 R k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 R

661

pP olk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 pP OLk . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Eqh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 R κ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .628 κ1 (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .629 κa () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 U () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 µ() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 µ0 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

Index

A. I. Mal’tsev’s theorem, 268 adding of certain ﬁctitious variables, 94 adding of ﬁctitious coordinates (rows), 129 algebra, 26 semiprimal, 104 axiom of, 27 closed subset, 32 demiprimal, 104 demisemiprimal, 104 directly irreducible, 64 extension of, 31 factor, 55 ﬁnite, 26 ﬁnitely axiomatizable, 84 ﬁnitely based, 84 free, 79 fundamental operations, 26 generating system, 32 hemiprimal, 104 inﬁnite, 26 infraprimal, 104 of ﬁnite type, 27 partial, 26 preprimal, 104 primal, 104 quasiprimal, 104 quotient, 55 semiprimal, 501 set of all subalgebras, 33 simple, 53 subalgebra, 31 type, 26, 73

universal, 26 universe, 26 algebras of same type, 27 all relation, 43 all-congruence, see congruence antiisomorphic, 59 antiisomorphism, 59 arity, see operation arity congruence, 235 atom, 112 atomic proposition, 106 automorphism, 52 inner, 285 basis, 98 block, 44 Boolean algebra, 30 Cartesian product, 64 chain, 36 characterization theorem for Sheﬀerfunctions, 215 class, 97 B-projectable, 337 l-class, 280 inverse image, 337 maximal, 98 minimal, 589 of type B, 174 of type C, 173 of type L, 171 of type M, 165 of type U, 170 of type X, 165

664

Index

order, 98 submaximal, 98 class of algebras closed, 72 class of all models of Σ, 77 clone, 97 minimal, 590 strong, 599 closed set system, 45 closure, 97 deductive, 82 closure operator, 45 algebraic, 46 co-class, 141 co-clone, 127 co-group, 138 co-monoid, 138 complete, 98 completeness criterion for P2 , 156 for Pk , 191 for Pk,l , 352 for TQ , 501 for the class of all idempotent functions of Pk , 501 completeness problem, 117 completeness theorem for the equational logic, 84 completeness theorem of proportional logic, 110 composition, 25 general, 129 conclusion, 77 congruence, 52, 234 n-congruence, 279 congruence class, 55 fully invariant, 83 of the ﬁrst kind, 235 of the second kind, 235 theorem for maximal clones, 265 trivial, 53, 234 congruence relation, see congruence congruence theorem for P2 , 237 constant, 93 countability criterion, 221 cyclical exchanging of the lines, 127 deductive closure, 82 depth of a subclass, 433

diagonal, 126 diagonale, 43 dimension, 291 direct product, 61 of classes, 397 of functions, 397 DNF, 99 domain, see operation doubling of coordinates (rows), 129 dual isomorphic, 59 duality principle of the lattice theory, 36 element central, 174 greatest, 165 inverse, 28 least, 165 neutral, 28 elementary operations, 95 embedding, 67 endomorphism, 52 equation, 76 equational class, 77 equational theory, 77 equivalence class, 43 equivalence relation, 42 equivalence class, 43 ﬁner, 351 permutable, 62 transversal to s, 556 trivial, 43 exchange of the ﬁrst two rows, 127 factor algebra, 55 factor set, 43 family of sets, 65 ﬁctitious place of a function, 93 ﬁeld, 29 ﬂoor function, 331 free algebra, 79 free generating set, 79 function n-ary on A, 91 r-th component, 168 autoduale, 167 Boolean, 93 component, 371 components of f , 359

Index constant, 93 extended, 600 linear, 171 monotone, 165 near unanimity function, 342 partial, 598 preserves the relation , 130 quasi-linear, 171 quasilinear, 456 reducible, 150 reduction, 599 restricted, 600 semiprojection, 591 subfunction, 598 total, 598 function algebra, 30 full, 96 iterative full, 96 functions κ-congruent, 234 are associated, 238 identity of, 93 fundamental group, 218 fundamental lemma of Jablonskij, 102 fundamental operations, see algebra fundamental semigroup, 218 fundamental set, 218 fuzzy logic, 116 Galois connection, 59 Galois correspondence, 59 generating set, 48 generating system, 48, 98 Gorlov’s tqheorem, 281 graphic n-te of A, 133 group, 28 Abelian, 28 additive notation, 28 commutative, 28 semiregular representation, 557 gruppoid, 27 Haddad-Rosenberg theorem, 616 Hasse diagram, 36 Hilbert-type-calculus, 107 homomorphism, 51 kernel, 52 natural, 55

665

quotient, 55 homomorphism theorem for groups, 57 for rings, 58 general, 55 hull, 45, 97 hull system, 45 I. A. Mal’tsev’s theorem, 241 ideal, 58 identiﬁcation of certain variables of f , 94 identiﬁcation of coordinates, 129 identity, 43, 76 inductively set system, 68 information transformer, 116 intersection, 127 inverse element, 28 inverse image homomorphic, 174 isomorphic, 39 isomorphic lattices, 39 isomorphism, 39, 52 anti-, 59 dual, 59 kernel of a group homomorphism, 57 of a homomorphism, 52 of a ring homomorphism, 58 Krasner-algebra of ﬁrst kind, 138 of second kind, 138 lattice, 30 bounded, 30 complete, 42 distributive, 30 ﬁrst deﬁnition, 35 isomorphic, 39 second deﬁnition, 37 sublattice, 41 with 0 and 1, 30 left unit, 241 lexicographical order, 132 limit class, 280 main theorem of the equational theory ﬁrst, 81 second, 84

666

Index

majority function, 591 Mal’tsev-operations, 31, 95 mapping homomorphic, 51 isomorphic, 52 order-preserving, 39 projection-, 61 minority function, 591 module, 29 R-module, 29 over a unitary ring, 29 over the ring R, 29 modus ponens, 108 monoid, 28 neutral element, 28 normal form disjunctive, 99 normal subgroup, 56 operation n-ary partial, 25 arity, 25 domain, 25 elementary on Rk , 127 nullary, 25 range, 25 operation symbol, 73 order, 98 dual, 59 partial, 36 order diagram, 36 partition, 44 Peirce decomposition, 397 permutation of coordinates, 128 permutation of variables of f , 94 polymorphism, 130 poset, 36 antiisomorphic, 59 complete, 41 dual isomorphic, 59 Post’s theorem, 148 predecessor proper, 292 predicate, 111 preserve a relation pair, 140 preserving of a set, 97

preserving of relations, 130 product Cartesian, 127 projection, 93 onto the α1 -te, ..., αt -te coordinates, 128 onto the i-th coordinate, 127 projection mapping, see mapping proposition, 105 quotient algebra, 55 range, see operation reduct of an algebra, 104 relation (l; r)-homogeneous, 540 M -permissible, 363 θs -closed, 555 -derivable, 127, 499, 515 {ζ, τ, pr, ∧, ×}-derivable, 128 h-ary, 125 h-ary elementary, 174 h-regular, 178 h−universal, 175 (l; 2)-universal, 518 (l; r)-central, 517 (l; r)-universal, 518 areﬂexive, 614 central, 173 central element c ∈ El , 517 coherent, 615 derivable, 127 diagonal, 126 induced relation, 269 invariant of the function f , 130 irredundant, 605 length, 126 primitive, 197 quasidiagonal, 615 row, 126 strong, 179 strongly (l; r)-homogeneous, 540 strongly (l; 2)-homogeneous, 517 strongly homogeneous, 204 totally (l; r)-reﬂexive, 517 totally (l; r)-symmetric, 517 totally reﬂexive, 174 totally symmetric, 174 unary transversal to s, 556

Index

667

weakly (l; r)-central, 517 width, 126 relation algebra on Ek , 127 full, 127 relation degree, 291 relation pair, 140 relation pairalgebra full, 141 relation product, 129 relation set α-permissible, 350 -independent, 179 h-regular, 178 is closed, 127 minimal coarsening, 351 permissible, 350 relation-pair algebras, 141 replacement of the i-th variable of f through the function g and the changing of the denotation of variables, 95 replacement rule, 82 representation theorem for functions of PA , 98 residue class ring, 58 right zero, 241 ring, 28 ideal, 58 with unit element, 28 ring, unitary, 28 Rosenberg’s completeness criterion, 191

inﬁmum of a subset, 37 linearly ordered, 36 of all invariants, 131 of conjunctions, 147 of constant functions of P2 , 147 of diagonal relations, 126 of disjunctions, 147 of linear functions of P2 , 147 of monotone functions of P2 , 146 of projections of P2 , 147 of self-dual functions of P2 , 146 of the h-ary relations on Ek , 126 partially ordered, 36 partition, 44 supremum of a subset, 36 totally ordered, 36 Sheﬀer-function, 211 Sheﬀer-function for P olk , 10, 307 subclass, 97 congruence on, 234 depth, 433 dimension, 291 maximal, 98 relation degree, 291 subdirect product, 66 subdirect representation, 67 subdirectly irreducible, 67 substitution rule, 82, 107 superposition operations, 94 superposition over F , 96 Slupecki-function, 211

selector, 93 semigroup, 28 commutative, 28 semilattice, 29 semiprojection, 591 semiring, 28 set CH -basis, 49 CH -independent, 49 J-closed, 274 basis, 49 closed, 46, 97 complete, 98 complete in a class, 98 ﬁnitely generated, 48 generated, 46 independent, 49

tautology, 106 term, 74 induces a term function, 75 term algebra, 74 term function, 75 theorem of Webb, 215 theorem on the orders of the maximal classes, 309 theorem over the cardinality of Lk , 221 transitive t-fold, 218 tuple h-tuple, 125 universe, see algebra valuation, 106

668

Index

variable, 74 bound, 112 essential, 93 ﬁctitious, 93 free, 112 variety, 72

vector space over the ﬁeld K, 29 zero-congruence, see congruence zigzag, 325 Zorn’s Lemma, 68

Springer Monographs in Mathematics This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in ﬁelds of mathematical research that have acquired the maturity needed for such a treatment. They are sufﬁciently self-contained to be accessible to more than just the intimate specialists of the subject, and sufﬁciently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its ﬁeld, an SMM volume should also describe its relevance to and interaction with neighbouring ﬁelds of mathematics, and give pointers to future directions of research.

Abhyankar, S.S. Resolution of Singularities of Embedded Algebraic Surfaces 2nd enlarged ed. 1998 Alexandrov, A.D. Convex Polyhedra 2005 Andrievskii, V.V.; Blatt, H.-P. Discrepancy of Signed Measures and Polynomial Approximation 2002 Angell, T. S.; Kirsch, A. Optimization Methods in Electromagnetic Radiation 2004 Ara, P.; Mathieu, M. Local Multipliers of C*-Algebras 2003 Armitage, D.H.; Gardiner, S.J. Classical Potential Theory 2001 Arnold, L. Random Dynamical Systems corr. 2nd printing 2003 (1st ed. 1998) Arveson, W. Noncommutative Dynamics and E-Semigroups 2003 Aubin, T. Some Nonlinear Problems in Riemannian Geometry 1998 Auslender, A.; Teboulle M. Asymptotic Cones and Functions in Optimization and Variational Inequalities 2003 Banasiak, J.; Arlotti, L. Perturbations of Positive Semigroups with Applications 2006 Bang-Jensen, J.; Gutin, G. Digraphs 2001 Baues, H.-J. Combinatorial Foundation of Homology and Homotopy 1999 Böttcher, A.; Silbermann B. Analysis of Toeplitz Operators 2nd ed. 2006 Brown, K.S. Buildings 3rd printing 2000 (1st ed. 1998) Chang, K. Methods in Nonlinear Analysis 2005 Cherry, W.; Ye, Z. Nevanlinna’s Theory of Value Distribution 2001 Ching, W.K. Iterative Methods for Queuing and Manufacturing Systems 2001 Crabb, M.C.; James, I.M. Fibrewise Homotopy Theory 1998 Chudinovich, I. Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes 2005 Dineen, S. Complex Analysis on Inﬁnite Dimensional Spaces 1999 Dugundji, J.; Granas, A. Fixed Point Theory 2003 Ebbinghaus, H.-D.; Flum J. Finite Model Theory 2006 Elstrodt, J.; Grunewald, F. Mennicke, J. Groups Acting on Hyperbolic Space 1998 Edmunds, D.E.; Evans, W.D. Hardy Operators, Function Spaces and Embeddings 2004 Engler, A.; Prestel, A. Valued Fields 2005 Fadell, E.R.; Husseini, S.Y. Geometry and Topology of Conﬁguration Spaces 2001 Fedorov, Y.N.; Kozlov, V.V. A Memoir on Integrable Systems 2001 Flenner, H.; O’Carroll, L. Vogel, W. Joins and Intersections 1999 Gelfand, S.I.; Manin, Y.I. Methods of Homological Algebra 2nd ed. 2003 Griess, R.L. Jr. Twelve Sporadic Groups 1998 Gras, G. Class Field Theory corr. 2nd printing 2005 Hida, H. p-Adic Automorphic Forms on Shimura Varieties 2004 Ischebeck, F.; Rao, R.A. Ideals and Reality 2005 Ivrii, V. Microlocal Analysis and Precise Spectral Asymptotics 1998 Jech, T. Set Theory (3rd revised edition 2002) corr. 4th printing 2006 Jorgenson, J.; Lang, S. Spherical Inversion on SLn (R) 2001 Kanamori, A. The Higher Inﬁnite corr. 2nd printing 2005 (2nd ed. 2003) Kanovei, V. Nonstandard Analysis, Axiomatically 2005 Khoshnevisan, D. Multiparameter Processes 2002 Koch, H. Galois Theory of p-Extensions 2002 Komornik, V. Fourier Series in Control Theory 2005 Kozlov, V.; Maz’ya, V. Differential Equations with Operator Coefﬁcients 1999

Lau, D. Function Algebras on Finite Sets 2006 Landsman, N.P. Mathematical Topics between Classical & Quantum Mechanics 1998 Leach, J.A.; Needham, D.J. Matched Asymptotic Expansions in Reaction-Diffusion Theory 2004 Lebedev, L.P.; Vorovich, I.I. Functional Analysis in Mechanics 2002 Lemmermeyer, F. Reciprocity Laws: From Euler to Eisenstein 2000 Malle, G.; Matzat, B.H. Inverse Galois Theory 1999 Mardesic, S. Strong Shape and Homology 2000 Margulis, G.A. On Some Aspects of the Theory of Anosov Systems 2004 Miyake, T. Modular Forms 2006 Murdock, J. Normal Forms and Unfoldings for Local Dynamical Systems 2002 Narkiewicz, W. Elementary and Analytic Theory of Algebraic Numbers 3rd ed. 2004 Narkiewicz, W. The Development of Prime Number Theory 2000 Onishchik, A.L.; Sulanke, R. Projective and Cayley-Klein Geometries 2006 Parker, C.; Rowley, P. Symplectic Amalgams 2002 Peller, V. Hankel Operators and Their Applications 2003 Prestel, A.; Delzell, C.N. Positive Polynomials 2001 Puig, L. Blocks of Finite Groups 2002 Ranicki, A. High-dimensional Knot Theory 1998 Ribenboim, P. The Theory of Classical Valuations 1999 Rowe, E.G.P. Geometrical Physics in Minkowski Spacetime 2001 Rudyak, Y.B. On Thom Spectra, Orientability and Cobordism 1998 Ryan, R.A. Introduction to Tensor Products of Banach Spaces 2002 Saranen, J.; Vainikko, G. Periodic Integral and Pseudodifferential Equations with Numerical Approximation 2002 Schneider, P. Nonarchimedean Functional Analysis 2002 Serre, J-P. Complex Semisimple Lie Algebras 2001 (reprint of ﬁrst ed. 1987) Serre, J-P. Galois Cohomology corr. 2nd printing 2002 (1st ed. 1997) Serre, J-P. Local Algebra 2000 Serre, J-P. Trees corr. 2nd printing 2003 (1st ed. 1980) Smirnov, E. Hausdorff Spectra in Functional Analysis 2002 Springer, T.A.; Veldkamp, F.D. Octonions, Jordan Algebras, and Exceptional Groups 2000 Sznitman, A.-S. Brownian Motion, Obstacles and Random Media 1998 Taira, K. Semigroups, Boundary Value Problems and Markov Processes 2003 Talagrand, M. The Generic Chaining 2005 Tauvel, P.; Yu, R.W.T. Lie Algebras and Algebraic Groups 2005 Tits, J.; Weiss, R.M. Moufang Polygons 2002 Uchiyama, A. Hardy Spaces on the Euclidean Space 2001 Üstünel, A.-S.; Zakai, M. Transformation of Measure on Wiener Space 2000 Vasconcelos, W. Integral Closure. Rees Algebras, Multiplicities, Algorithms 2005 Yang, Y. Solitons in Field Theory and Nonlinear Analysis 2001 Zieschang, P.-H. Theory of Association Schemes 2005