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Christopher Hollings HISTORY OF MATHEMATICS Y VOLUME 41 Mathematics Across the Iron Curtain A History of the Algebraic Theory of Semigroups https://doi.org/10.1090/hmath/041 HISTORY OF MATHEMATICS v VOLUME 41 Mathematics Across the Iron Curtain A History of the Algebraic Theory of Semigroups Christopher Hollings American Mathematical Society Providence, Rhode Island Editorial Board June Barrow-Green Bruce Reznick Robin Hartshorne Adrian Rice, Chair 2010 Mathematics Subject Classification. Primary 01A60, 20-03. For additional information and updates on this book, visit www.ams.org/bookpages/hmath-41 Library of Congress Cataloging-in-Publication Data Hollings, Christopher, 1982– Mathematics across the Iron Curtain : a history of the algebraic theory of semigroups / Christopher Hollings. pages cm. — (History of mathematics ; volume 41) Includes bibliographical references and indexes. ISBN 978-1-4704-1493-1 (alk. paper) 1. Semigroups. 2. Mathematics—History—20th century. 3. Cold War. I. Title. QA182.H65 2014 512.27—dc23 2014008281 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the author. Printed in the United States of America. ∞ 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 http://www.ams.org/ 10987654321 191817161514 Contents Preface vii Languages ix Acknowledgements x Chapter 1. Algebra at the Beginning of the Twentieth Century 1 1.1. A changing discipline 1 1.2. The term ‘semigroup’ 3 1.3. An overview of the development of semigroup theory 7 Chapter 2. Communication between East and West 11 2.1. Communication down through the decades 14 2.2. Access to publications 29 Chapter 3. Anton Kazimirovich Sushkevich 45 3.1. Biography 47 3.2. The theory of operations as the general theory of groups 54 3.3. Generalised groups 58 3.4. Sushkevich’s impact 73 Chapter 4. Unique Factorisation in Semigroups 77 4.1. Postulational analysis 81 4.2. E. T. Bell and the arithmetisation of algebra 83 4.3. Morgan Ward and the foundations of general arithmetic 88 4.4. Alfred H. Clifford 91 4.5. Arithmetic of ova 93 4.6. Subsequent developments 103 Chapter 5. Embedding Semigroups in Groups 107 5.1. The theorems of Steinitz and Ore 111 5.2. Embedding according to Sushkevich 116 5.3. Further sufficient conditions 122 5.4. Maltsev’s immersibility conditions 124 5.5. Other embedding problems 133 Chapter 6. The Rees Theorem 135 6.1. Completely (0-)simple semigroups 137 6.2. Brandt groupoids 140 6.3. Sushkevich’s ‘Kerngruppen’ 145 6.4. Clifford’s ‘multiple groups’ 149 6.5. The Rees Theorem 151 6.6. Unions of groups and semigroups 156 v vi CONTENTS Chapter 7. The French School of ‘Demi-groupes’ 161 7.1. Paul Dubreil and Marie-Louise Dubreil-Jacotin 164 7.2. Equivalence relations 166 7.3. Principal equivalences and related concepts 171 7.4. Subsequent work 177 Chapter 8. The Expansion of the Theory in the 1940s and 1950s 183 8.1. The growth of national schools 184 8.2. The Slovak school 186 8.3. The American school 192 8.4. The Japanese school 201 8.5. The Hungarian school 206 8.6. British authors 208 Chapter 9. The Post-Sushkevich Soviet School 217 9.1. Evgenii Sergeevich Lyapin 220 9.2. Lyapin’s mathematical work 225 9.3. Lazar Matveevich Gluskin 238 9.4. Gluskin’s mathematical work 239 9.5. Other authors 246 Chapter 10. The Development of Inverse Semigroups 249 10.1. A little theory 251 10.2. Pseudogroups and conceptual difficulties 252 10.3. Viktor Vladimirovich Wagner 258 10.4. Wagner and generalised groups 259 10.5. Gordon B. Preston 268 10.6. Preston and inverse semigroups 269 Chapter 11. Matrix Representations of Semigroups 277 11.1. Sushkevich on matrix semigroups 281 11.2. Clifford on matrix semigroups 285 11.3. W. Douglas Munn 289 11.4. The work of W. D. Munn 291 11.5. The work of J. S. Ponizovskii 298 Chapter 12. Books, Seminars, Conferences, and Journals 303 12.1. Monographs 305 12.2. Seminars on semigroups 323 12.3. Czechoslovakia, 1968, and Semigroup Forum 326 Appendix. Basic Theory 333 Notes 337 Bibliography 373 Name Index 429 Subject Index 433 Preface A semigroup is a set that is closed under an associative binary operation. We might therefore regard a semigroup as being either a defective group (stripped of its identity and inverse elements) or a defective ring (missing an entire operation). Indeed, these are two of the original sources from which the study of semigroups sprang. However, to regard the modern theory of semigroups simply as the study of degenerate groups or rings would be to overlook the comprehensive and independent theory that has grown up around these objects over the years, a theory that is rather different in spirit from those of groups and rings. Perhaps most importantly, semigroup theory represents the abstract theory of transformations of a set: the collection of all not-necessarily-invertible self-mappings of a set forms a semigroup (indeed, a monoid: a semigroup with a multiplicative identity), but not, of course, a group. The development of the theory of semigroups from these various sources is the subject of this book. I chart the theory’s growth from its earliest origins (in the 1920s) up to the foundation of the dedicated semigroup-oriented journal Semigroup Forum in 1970. Since the theory of semigroups developed largely after the Second World War, it might be termed ‘Cold War mathematics’; a comparison of the mathematics of semigroup researchers in East and West, together with an investigation of the extent to which they were able to communicate with each other, is therefore one of the major themes of this book. I believe that semigroup theory provides a particularly good illustration of these problems in East-West communication precisely because it is such a young theory. We are not dealing here with a well-established mathematical discipline, to whose traditions and methods mathematicians in East and West were already privy and had in common when the Iron Curtain descended. Instead, the foundations of many semigroup-theoretic topics were laid independently by Soviet and Western mathematicians who had no idea that they were working on the same problems. Thus, different traditions and priorities were established by the two sides from the earliest days of the theory. The structure of the book is as follows. In Chapter 1, I set the scene by con- sidering the status of algebra within mathematics at the beginning of the twentieth century. I discuss the coining of the term ‘semigroup’ in 1904 and give an overview of the broad strokes of the subsequent development of the theory of semigroups. Chapter 2 is devoted to the major theme mentioned above: the East-West divide in mathematical research. I provide a general discussion of the extent to which scientists on opposite sides of the Iron Curtain were able to communicate with each other and the degree to which the publications of one side were accessible to the other. The description of the development of semigroup theory begins in Chapter 3 with a survey of the work of the Russian pioneer A. K. Sushkevich. I investigate his vii viii PREFACE influences, the types of problems that he considered, and the legacy of his work, with an attempt to explain his lack of impact on the wider mathematical community. Chapter 4 is the first of two chapters dealing with the semigroup-theoretic problems that arose in the 1930s by analogy with similar problems for rings. Thus, Chapter 4 concerns unique factorisation in semigroups, while Chapter 5 deals with the problem of embedding semigroups in groups. In these two chapters, we see how the investigation of certain semigroup-related questions began to emerge, although this was not yet part of a wider ‘semigroup theory’. The beginning of a true theory of semigroups is dealt with in Chapter 6, which concerns the celebrated Rees Theorem, together with a result proved by A. H. Clifford in 1941, which might be regarded as semigroup theory’s first ‘independent’ theorem: a result with no direct group or ring analogue. Paul Dubreil and the origins of the French (or, more accurately, Francophone) school of ‘demi-groupes’ are the subject of Chapter 7, while Chapter 8 concerns the expansion of semigroup theory during the 1940s and 1950s, both in terms of the subjects studied and also through the internationalisation of the theory. I thus indicate the major semigroup-theoretic topics that emerged during this period and also give an account of the various national schools of semigroup theory that developed. Chapter 8 marks something of a watershed in this book: the material appearing before Chapter 8 represents the efforts of the early semigroup theorists to build up their discipline, while that coming after may be regarded as being part of a fully established theory of semigroups. Chapter 9 concerns the development of the post-Sushkevich Soviet school of semigroup theory through the work of E.
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