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The Algebraic Theory of Semigroups the Algebraic Theory of Semigroups http://dx.doi.org/10.1090/surv/007.1 The Algebraic Theory of Semigroups The Algebraic Theory of Semigroups A. H. Clifford G. B. Preston American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 20-XX. International Standard Serial Number 0076-5376 International Standard Book Number 0-8218-0271-2 Library of Congress Catalog Card Number: 61-15686 Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © Copyright 1961 by the American Mathematical Society. All rights reserved. Printed in the United States of America. Second Edition, 1964 Reprinted with corrections, 1977. The American Mathematical Society retains all rights except those granted to the United States Government. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 06 05 04 03 02 01 TABLE OF CONTENTS PREFACE IX NOTATION USED IN VOLUME I xm CHAPTER 1. ELEMENTARY CONCEPTS 1.1 Basic definitions ........ 1 1.2 Light's associativity test ....... 7 1.3 Translations and the regular representation .... 9 (Lemma 1.0-Theorem 1.3) 1.4 The semigroup of relations on a set . .13 (Lemma 1.4) 1.5 Congruences, factor groupoids and homomorphisms . 16 (Theorem 1.5-Theorem 1.8) 1.6 Cyclic semigroups . .19 (Theorem 1.9) 1.7 Units and maximal subgroups . .21 (Theorem 1.10-Theorem 1.11) 1.8 Bands and semilattices; bands of semigroups ... 23 (Theorem 1.12) 1.9 Regular elements and inverses; inverse semigroups . 26 (Lemma 1.13-Theorem 1.22) 1.10 Embedding semigroups in groups ..... 34 (Theorem 1.23-Theorem 1.25) 1.11 Right groups ......... 37 (Lemma 1.26-Theorem 1.27) 1.12 Free semigroups and generating relations; the bicyclic semi­ group .......... 40 (Lemma 1.28-Corollary 1.32) CHAPTER 2. IDEALS AND RELATED CONCEPTS 2.1 Green's relations ........ 47 (Lemma 2.1-Theorem 2.4) 2.2 ^-structure of the full transformation semigroup &"x on a set X 51 (Lemma 2.5-Theorem 2.10) 2.3 Regulars-classes 58 (Theorem 2.11-Theorem 2.20) vi TABLE OF CONTENTS 2.4 The Schutzenberger group of an ^-class .... 63 (Lemma 2.21-Theorem 2.25) 2.5 0-minimal ideals and 0-simple semigroups .... 66 (Lemma 2.26-Theorem 2.35) 2.6 Principal factors of a semigroup . .71 (Theorem 2.36-Corollary 2.42) 2.7 Completely 0-simple semigroups ...... 76 (Lemma 2.43-Corollary 2.56) CHAPTER 3. REPRESENTATION BY MATRICES OVER A GROUP WITH ZERO 3.1 Matrix semigroups over a group with zero .... 87 (Lemma 3.1-Theorem 3.3) 3.2 The Rees Theorem 91 (Theorem 3.4-Lemma 3.6) 3.3 Brandt groupoids ........ 99 (Lemma 3.7-Theorem 3.9) 3.4 Homomorphisms of a regular Rees matrix semigroup 103 (Lemma 3.10-Theorem 3.14) 3.5 The Schutzenberger representations ..... 110 (Lemma 3.15-Theorem 3.17) 3.6 A faithful representation of a regular semigroup . 117 (Lemma 3.18-Theorem 3.21) CHAPTER 4. DECOMPOSITIONS AND EXTENSIONS 4.1 Croisot's theory of decompositions of a semigroup . 121 (Lemma 4.1-Theorem 4.4) 4.2 Semigroups which are unions of groups .... 126 (Theorem 4.5-Theorem 4.11) 4.3 Decomposition of a commutative semigroup into its archi- medean components; separative semigroups 130 (Theorem 4.12-Theorem 4.18) 4.4 Extensions of semigroups ....... 137 (Theorem 4.19-Theorem 4.21) 4.5 Extensions of a group by a completely 0-simple semigroup; equivalence of extensions ...... 142 (Theorem 4.22-Theorem 4.24) CHAPTER 5. REPRESENTATION BY MATRICES OVER A FIELD 5.1 Representations of semisimple algebras of finite order . .149 (Lemma 5.1-Theorem 5.11) TABLE OF CONTENTS vii 5.2 Semigroup algebras . .158 (Lemma 5.12-Theorem 5.31) 5.3 Principal irreducible representations of a semigroup . .170 (Lemma 5.32-Theorem 5.36) 5.4 Representations of completely 0-simple semigroups . 177 (Theorem 5.37-Corollary 5.53) 5.5 Characters of a commutative semigroup . .193 (Lemma 5.54-Theorem 5.65) APPENDIX A ......... 207 BIBLIOGRAPHY 209 AUTHOR INDEX • 217 INDEX 219 PREFACE So far as we know, the term "semigroup" first appeared in mathematical literature on page 8 of J.-A. de Siguier's book, filaments de la Theorie des Groupes Abstraits (Paris, 1904), and the first paper about semigroups was a brief one by L. E. Dickson in 1905. But the theory really began in 1928 with the publication of a paper of fundamental importance by A. K. Susch- kewitsch. In current terminology, he showed that every finite semigroup contains a "kernel" (a simple ideal), and he completely determined the structure of finite simple semigroups. A brief account of this paper is given in Appendix A. Unfortunately, this result of Suschkewitsch is not in a readily usable form. This defect was removed by D. Rees in 1940 with the introduction of the notion of a matrix over a group with zero, and, moreover, the domain of validity was extended to infinite simple semigroups containing primitive idempotents. The Rees Theorem is seen to be the analogue of Wedderburn's Theorem on simple algebras, and it has had a dominating influence on the later development of the theory of semigroups. Since 1940, the number of papers appearing each year has grown fairly steadily to a little more than thirty on the average. It is in response to this developing interest that this book has been written. Only one book has so far been published which deals predominantly with the algebraic theory of semigroups, namely one by Suschkewitsch, The Theory of Generalized Groups (Kharkow, 1937); this is in Russian, and is now out of print. A chapter of R. H. Brack's A Survey of Binary Systems (Ergebnisse der Math., Berlin, 1958) is devoted to semigroups. There is, of course, E. Hille's book, Functional Analysis and Semi-groups (Amer. Math. Soc. Colloq. PubL, 1948), and the 1957 revision thereof by Hille and R. S. Phillips; but this deals with the analytic theory of semigroups and its application to analysis. The time seems ripe for a systematic exposition of the algebraic theory. (Since the above words were written, there has appeared such an exposition, in Russian: Semigroups, by E. S. Lyapin, Moscow, 1960.) The chief difficulty with such an exposition is that the literature is scat­ tered over extremely diverse topics. We have met this situation by con­ fining ourselves to a portion of the existing theory which has proved to be capable of a well-knit and coherent development. All of Volume 1 and the first half of Volume 2 center around the structure of semigroups of certain types (such as simple semigroups, inverse semigroups, unions of groups, ix X PREFACE semigroups with minimal conditions, etc.) and their representation by mappings or by matrices. The second half of Volume 2 treats the theory of congruences and the embedding of semigroups in groups, including a modest account of the active French school of semigroups (which they call "demi-groupes") founded in 1941 by P. Dubreil. In order to keep our book within reasonable bounds, moreover, we have construed the term "algebraic'' in a somewhat narrow sense: the semigroups under consideration are not endowed with any further structure. This has the effect of excluding not only topological semigroups, but ordered semi­ groups as well. Fortunately, a good account of lattice-ordered semigroups and groups is to be found in G. BirkhofTs Lattice Theory (Amer. Math. Soc. Colloq. Publ., 1940; revised 1948). It also excludes P. Lorenzen's generaliza­ tion of multiplicative ideal theory (see, for example, §5 of W. Krull's Ideal- theorie, Ergebnisse der Math., Berlin, 1935) to any commutative semigroup S with cancellation, in which S (or its quotient group) is endowed with a family of subsets called r-ideals, satisfying certain conditions analogous to those for closed sets in topology. The book aims at being largely self-contained, but it is assumed that the reader has some familiarity with sets, mappings, groups, and lattices. The material on these topics in an introductory text such as Birkhoff and MacLane, A Survey of Modern Algebra (New York, Revised Edition, 1953) should suffice. Only in Chapter 5 will more preliminary knowledge be required, and even there the classical definitions and theorems on the matrix representations of algebras and groups are summarized. We have included a number of exercises at the end of each section. These are intended to illuminate and supplement the text, and to call attention to papers not cited in the text. They can all be solved by applying the methods and results of the text, and often more simply than in the paper cited.
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