Universal Algebra Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL Department of Mathematics, Erasmus University. Rotterdam, The Netherlands

Editorial Board:

R.W. BROCKETT, Harvard University. Cambridge, Mass., U.S.A. Yu. I. MANIN, Steklov Institute of Mathematics. Moscow, U.S.S.R. G.-c. ROTA, M.I.T" Cambridge. Mass .. U.S.A.

Volume 6 P.M. Cohn, F.R.S. Bedford College. University of London. London. England

Universal Algebra

D. REIDEL PUBLISHING COMPANY Dordrecht: Holland / Boston: U.S.A. / London: England Library of Congress Cataloging in Publication Data

Cohn, Paul Moritz. Universal algebra.

(Mathematics and its applications; v. 6) 1. Algebra, Universal. 1. Title. II. Series: Mathematics and its applications (Dordrecht) ; v. 6) QA251.C55 1981 512 80-29568 ISBN-13: 978-90-277-1254-7 e-ISBN-\3: 978-94-009-8399-1 001: 10.1007/978-94-009-8399-1

Published by D. Reidel Publishing Company, P.O. Box 17,3300 AA Dordrecht, Holland

Sold and distributed in the U.S.A. and Canada by Kluwer Boston Inc., 190 Old Derby Street, Hingham, MA 02043, U.S.A.

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All Rights Reserved Originally published in 1965 by Harper & Row Copyright © revIsed edition 1981 by D. Reidel Publishing Company, Dordrecht, Holland Soflcover reprint of the hardcover I st Edition 1981 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner To Deirdre, and her Parents Notes to the Reader

The central part of the book is Chapter II, and this may be read directly after 1.3, referring back to other parts of Chapter I when necessary. Much of the material in Chapters V and VI can be read after Chapter II, and most of Chapter VII can be read after Chapter IV. Any exceptions to these rules are usually indicated by a reference to the relevant chapter and section. All theorems, propositions, and lemmas are numbered in a single series; thus 'Theorem IV.3.5' refers to item 5 of Section 3 of Chapter IV. Cross-references within a single chapter omit the roman numeral. The end of a proof is indicated thus: I The Bibliography includes, besides the works referred to in the text, a few papers of general interest on universal algebra, but it is not intended to be ex• haustive in any direction. Practically all the items listed have appeared since 1900, and this fact is utilized by referring to an item by the author's name and the last two digits of the year of publication, with primes to distinguish papers published in the same year.

vi Contents

Notes to the Reader VI

Preface to the Revised Edition xi

Preface xiii

CHAPTER I: SETS AND MAPPINGS 1. The Axioms of Set Theory 1 2. Correspondences 9 3. Mappings and Quotient Sets 12 4. Ordered Sets 18 5. Cardinals and Ordinals 28 6. Categories and Functors 36

CHAPTER II: ALGEBRAIC STRUCTURES I. Closure Systems 41 2. O-Algebras 47 3. The Isomorphism Theorems 57 4. Lattices 63 5. The Lattice of Subalgebras 79 6. The Lattice of Congruences 86 viii Contents

7. Local and Residual Properties 99 8. The Lattice of Categories of Q-Algebras 104

CHAPTER III. FREE ALGEBRAS l. Universal Functors 108 2. Q-Word Algebras 116 3. Clones of Operations 126 4. Representations in Categories of Q-Algebras 132 5. Free Algebras in Categories of Q-Algebras 137 6. Free and Direct Composition of Q-Algebras 142 7. Derived Operators 145 8. Presentations of Q-Algebras 150 9. The Word Problem 154

CHAPTER IV. VARIETIES I. Definition and Basic Properties 161 2. Free Groups and Free Rings 165 3. The Generation of Varieties 169 4. Representations in Varieties of Algebras 180

CHAPTER V. RELATIONAL STRUCTURES AND MODELS I. Relational Structures over a Predicate Domain 188 2. Boolean Algebras 191 3. Derived Predicates 200 4. Closed Sentence Classes and Axiomatic Model Classes 205 5. Ultraproducts and the Compactness Theorem 209 6. The Model Space 213

CHAPTER VI. AXIOMATIC MODEL CLASSES I. Reducts and Enlargements 220 2. The Local Determination of Classes 222 3. Elementary Extensions 228 4. p-C1osed Classes and Quasivarieties 233 5. Classes Admitting Homomorphic Images 236 6. The Characterization of Axiomatic Model Classes 239 Contents ix

CHAPTER VII. APPLICATIONS 1. The Natural Numbers 247 2. Abstract Dependence Relations 252 3. The Division Problem for Semigroups and Rings 263 4. The Division Problem for Groupoids 279 5. Linear Algebras 283 6. Lie Algebras 289 7. Jordan Algebras 297

Foreword to the Supplements 310

CHAPTER VIII. CATEGORY THEORY AND UNIVERSAL ALGEBRA 1. The Principle of Duality 311 2. Adjoint Pairs of Functors 312 3. Monads 315 4. Algebraic Theories 318

CHAPTER IX. MODEL THEORY AND UNIVERSAL ALGEBRA 1. Inductive Theories 321 2. Complete Theories and Model Complete Theories 324 3. Model Completions 326 4. The Forcing Companion 328 5. The Model Companion 331 6. Examples 333

CHAPTER X. MISCELLANEOUS FURTHER RESULTS 1. Subdirect Products and Pullbacks 336 2. The Reduction to Binary Operations 338 3. Invariance of the Rank of Free Algebras 339 4. The Diamond Lemma for Rings 341 5. The Embedding of Rings in Skew Fields 342

CHAPTER Xl. ALGEBRA AND LANGUAGE THEORY I. Introduction 345 x Contents

2. Grammars 346 3. Machines 355 4. Transductions 364 5. Monoids 365 6. Power Series 371 7. Transformational Grammars 379

Bibliography and Name Index 381

List of Special Symbols 401

Subject Index 404 Preface to the Revised Edition

The present book was conceived as an introduction for the user of universal algebra, rather than a handbook for the specialist, but when the first edition appeared in 1965, there were practically no other books entir~ly devoted to the subject, whether introductory or specialized. Today the specialist in the field is well provided for, but there is still a demand for an introduction to the subject to suit the user, and this seemed to justify a reissue of the book. Naturally some changes have had to be made; in particular, I have corrected all errors that have been brought to my notice. Besides errors, some obscurities in the text have been removed and the references brought up to date. I should like to express my thanks to a number of correspondents for their help, in particular C. G. d'Ambly, W. Felscher, P. Goralcik, P. J. Higgins, H.-J. Hoehnke, J. R. Isbell, A. H. Kruse, E. J. Peake, D. Suter, J. S. Wilson. But lowe a special debt to G. M. Bergman, who has provided me with extensive comments. particularly on Chapter VII and the supplementary chapters. I have also con• sulted reviews of the first edition, as well as the Italian and Russian translations. In addition there are four new chapters. Chapter VIII deals with category theory, in so far as it affects our subject. The construction of monads (triples) is described, with free algebras as an illustration, and Lawvere's definition of algebraic theories is outlined. Chapter IX presents the various notions of algebraic closure developed in model theory, particularly the existential closure and A. Robinson's infinite forcing, and its applications in algebra. Chapter X contains a number of isolated remarks related to the main text, and the final

xi xii Preface to the Revised Edition

chapter is an article on algebraic language theory which appeared in 1975 in the Bulletin of the London Mathematical Society; I am grateful to the Society for permission to include it here. Although not directly concerned with our topic, it describes the links with automata theory, itself of considerable relevance in uni• versal algebra. Several friends have read the new chapters and have provided helpful com• ments; in addition to G. M. Bergman they are W. A. Hodges and M. Y. Prest, and I should like to thank them here. I am also grateful to the publisher, D. Reidel and Co., for accepting the book in their Mathematical Series, and to their staff, for their efficiency in seeing it through the press.

Bedford College. London P. M. COHN December, 1980 Preface

Universal algebra is the study of features common to familiar algebraic systems such as groups, rings, lattices, etc. Such a study places the algebraic notions in their proper setting; it often reveals connexions between seemingly different concepts and helps to systematize one's thoughts. The actual ideas involved are quite simple and follow as natural generalizations from a few special instances. However, one must bear in mind that this approach does not usually solve the whole problem for us, but only tidies up a mass of rather trivial detail, allowing us to concentrate our powers on the hard core of the problem. The object of this book is to provide a simple account of the basic results of universal algebra. The book is not intended to be exhaustive, or even to achieve maximum generality in places where this would have meant a loss of clarity. Enough background has been included to make the text suitable for beginning graduate students who have some knowledge of groups and rings, and, in the case of Chapter V, of the basic notions of topology. Only the final section of the book (VII.7) requires a somewhat greater acquaintance with representation theory. The discussion centres on the notion of an algebraic structure, defined roughly as a set with a number of finitary operations. The fact that the opera• tions are finitary may be regarded as characteristic of algeb,a, and its conse• quences are traced out in Chapter II. Those consequences, even more basic, that are independent of finitarity are treated separately in Chapter I. This chapter also provides the necessary background in set theory, as seen through the eyes of an algebraist.

xiii xiv Preface

One of the main tools for the study of general algebras is the notion of a/ree algebra. It is of particular importance for classes like groups and rings which are defined entirely by laws-i.e., varieties of algebras-and this has perhaps tended to obscure the fact that free algebras exist in many classes of algebras which are not varieties. To emphasize the distinction, free algebras are developed as far as possible without reference to varieties in Chapter III, while properties peculiar to varieties are treated separately in Chapter IV. These two chapters present the only contact the book makes with homological algebra, and a word should perhaps be said about the connexion. The central part of homological algebra is the theory of abelian categories; this is highly developed, but is too restrictive for our purpose and does not concern us here. The general theory of categories. though at an earlier stage of development, has by now enough tools at its disposal to yield the main theorems on the existence of free algebras, but in an account devoted exclusively to algebra these results are much more easily proved directly; in particular the hypotheses under which the theorems are obtained here are usually easier to verify (in the case of alge• bras) than the corresponding hypotheses found in general category theory. For this reason we have borrowed little beyond the bare definitions of category and functor. These of course are indispensable in any satisfactory account of free algebras, and they allow us to state our results concisely without taking us too far from our central topic. The notion of an algebraic structure as formulated in Chapter II is too nar• row even in many algebraic contexts and has to be replaced by that of a relational structure, i.e., a set with a number of finitary relations defined on it. Besides algebraic structures themselves, this also includes structures with operations that are many-valued or not everywhere defined. In recent years, relational structures satisfying a given system of axioms, or models, have been the subject of intensive study and many results of remarkable power and beauty have been obtained. With the apparatus of universal algebra all set up, this seemed an excellent opportunity for giving at least a brief introduction to the subject, and this forms the content of Chapters V-VI. The final chapter on applications is not in any way intended to be systema• tic; the aim was to include results which could be established by using the earlier chapters and which in turn illuminate the general theory, and which, moreover, are either important in another context (such as the development of the natural numbers in VILI or the representation theory of Lie and Jordan algebras in VI!. 5-7), or interesting in their own right (e.g., Ma1cev's embedding theorem for semigroups, VI!'3). Although the beginnings of our subject can be found in the last century (A. N. Preface xv

Whitehead's treatise with the same title appeared in 1898), universal algebra as understood today only goes back to the 1930's, when it emerged as a natural development of the abstract approach to algebra initiated by Emmy Noether. As with other fields, there is now a large and still growing annual output of papers on universal algebra, but a curiously large portion of the subject is still only passed on by oral tradition. The author was fortunate to make acquaintance with this tradition in a series of most lucid and stimulating lectures by Professor Philip Hall in Cambridge 1947-1951, which have exercised a much greater in• fluence on this book than the occasional reference may suggest. In other re• ferences an easily accessible work has often been cited in preference to the original source, and no attempt has been made to include remarks of an histori• cal character; although such an attempt would certainly have been well worth while, it would have delayed publication unduly. For the same reason the biblio• graphy contains, apart from papers bearing directly on the text, only a selection of writings on universal algebra. This was all the more feasible since a very full bibliography is available in Mathematical Reviews; besides, a comprehensive bibliography on universal algebra is available in G. Gratzer [79]. The book is based on a course of lectures which I gave at Yale University in 1961-1962. I am grateful to the audience there for having been such good listeners, and to the many friends who have performed the same office since then. In particular, D. E. Cohen and P. J. Higgins read parts of the manuscript and made many useful suggestions; J. L. MacDonald helped with the proof• reading; A. J. Bowtell and F. E. J. Linton checked through the whole text and brought a number of inaccuracies to my attention. To all of them I should like to express my warmest thanks. I am also grateful to Messrs. Harper and Row for their willingness to carry out my wishes and to their editor, Mr. John Cron• quist, for his help in preparing the manuscript for the press.

Queen Mary College, London P. M. COHN January, 1965