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Mathematics Across the Iron Curtain: A History of the Algebraic Theory of Semigroups

Christopher Hollings
Publisher: 
American Mathematical Society
Publication Date: 
2014
Number of Pages: 
441
Format: 
Hardcover
Series: 
History of Mathematics 41
Price: 
109.00
ISBN: 
9781470414931
Category: 
Monograph
[Reviewed by
Henry Heatherly
, on
10/16/2014
]

This book gives a comprehensive history of the development of the algebraic theory of semigroups from its origins up through the founding in 1970 of an international journal dedicated to all aspects of semigroup theory, Semigroup Forum. One of the major themes is the comparison of the mathematics of semigroup researchers in the Soviet bloc and in the West, together with an examination of the extent which they were able to communicate with one another and the barriers to doing so.

The author makes it clear that the mathematical focus of the book is on the algebraic theory of semigroups. Topological semigroups, the semigroups used in analysis, or applications of semigroup theory are mentioned only in passing. The book is divided into twelve chapters and has an extensive bibliography (approximately 1400 references), both a name and a subject index, an appendix giving basic background theory on semigroups, and a section of thirty-five pages of notes.

Chapter 1 begins with a brief discussion on the state of abstract algebra in the early 20th century and then gives a narrative of the origins of the term “semigroup” and of the early development of semigroup theory. The term “semigroupe,” in French, was first used in print in 1904 by de Seguier in his book on group theory [4]. To de Seguier the term meant what we now call a “cancellative” semigroup. The concept was immediately picked up in the U. S. by L. E. Dickson, who gave it the English name “semi-group.” The terms “semi-group” or “semigroup” were used to denote at least four different, albeit closely related, concepts in the period 1905 to 1940. The first to use the term “semi-group” in the modern sense was Harold Hilton in his 1908 book on finite groups, [6, p. 51]. As late as the mid 1950s both the formats “semi-group” and “semigroup” were used to denote a algebraic system with an associative binary operation.

During the early period of development of semigroup theory, and for decades thereafter, there were two primary sources motivating the theory: as generalizations of group theory and from considering the multiplicative properties of a ring. Exemplary of the former is the work of Sushkevich on what he called “generalized groups” in the period 1918 to the late 1940s. Exemplary of the ring theory motivation is the 1930s work by Clifford on factorization in semigroups, somewhat in the spirit of the work on factorization in commutative rings done in the 1920s by Emmy Noether and others.

Chapter 3 is devoted to the life and works of A. K. Sushkevich (1889–1961), a Russian mathematician who lived most of his professional life in Kharkov in Ukraine. Hollings says “he was arguably the first person to try to construct a systematic theory for semigroups,” (p. 45). This opinion is shared by Clifford and Preston who wrote: “the theory really begins in 1928 with the publication of a paper of fundamental importance by A. K. Suschkevich,” [2, p. IX]. In 1937, Sushkevich published a monograph on his work up to that time: Theory of Generalized Groups (in Russian). By “generalized group” he meant a set with a binary operation which satisfies some, but not necessarily all, of the traditional group axioms. Except for in a few pages of his monograph, all his generalized groups are associative. Sushkevich continued publishing work on semigroup theory through 1948, but his work failed to have a major impact on the theory in the U.S.S.R. In part this may have been because much of his work was published in journals based in Kharkov and the difficulty in getting access to many regional journals in the U.S.S.R. during the Stalin era.

Sushkevich’s work was not well known to semigroup theorists in the West until the 1970s. Primarily this was due to the systematic problems of communication “across the Iron Curtain.” In Chapter 2 Hollings addresses at length these problems in East-West communications, and he returns to this problem often throughout the book.

The development of algebraic semigroup theory in the U.S.A. can be said to have started in the early 1930s with the work of Alfred H. Clifford (1908–1992) on unique factorization in a commutative monoid, which he called an “ovum.” This began with work leading to his 1933 Cal Tech Ph.D. thesis, Arithmetic in Ova. Even as he was finishing his thesis Clifford was working on another semigroup problem, one on generalization of group axiomatics. This resulted in a 1933 publication entitled “A system arising from a weakened set of group properties.” In 1941 he published a seminal work on the decomposition of semigroups into the union of groups or into the union of special types of semigroups. In 1994, G. B. Preston called Clifford’s 1941 paper “immensely influential,” [12, p. 6]. Hollings discusses this work of Clifford’s in section 6.6. There was a hiatus in Clifford’s work on semigroups from Spring 1942 until the end of World War II, while he served as an officer in the U. S. Navy.

The first major structure theorem for semigroups appeared in 1940 in a paper by David Rees entitled “On semi-groups.” This result became known as “Rees’s Theorem.” It gives the structure of completely 0-simple semigroups in terms of some special matrix semigroups, analogously to the Artin-Wedderburn Theorem for rings. Hollings discusses the Rees Theorem in detail in Chapter 6. Rees subsequently published four more papers on semigroups, but he abandoned the study of semigroups in the late 1940s and moved to commutative ring theory.

During World War II, Rees was part of the code breaking team at Bletchley Park. Several others who were or became semigroup researchers of note were at Bletchley Park at various times during the war. This included Preston, Green, and a U.S. Navy officer named Alfred H. Clifford, who was at Bletchley Park from May 1943 to June 1944.

In a survey paper in 2002, Howie identified what he described as “three seminal papers” in the development of semigroup theory, [8]. The three papers are: the Rees paper of 1940, Clifford’s paper of 1941, and a 1940 paper by the French mathematician Paul Dubreil. Dubreil used the French term “demi-groupes” to mean what is now known as a cancellative semigroup. This paper was very influential in France and world-wide. Clifford and Preston called the paper “ground-breaking,” [3, p. 174]. The French school of semigroup theory continued to thrive for decades. One prominent contributor was Marie-Louise Dubreil-Jacotin, (1905–1970), the wife of Paul Dubreil.

Active and influential “schools” of semigroup research emerged in several places in the late 1940s and the 1950s. Hollings discusses these in some detail, as well as the interactions between schools. Chapter 9 concerns what Hollings calls “The Post-Sushkevich Soviet School.” This emerged in the 1950s and became, arguably, second to none in the world. The most influential mathematician was G. S. Lyapin (1914–2005). He became interested in semigroups about 1939 and completed a doctoral dissertation on the subject at Leningrad State University in 1945: Elements of an abstract theory of systems with one operation.

Lyapin had been teaching at Leningrad State University since 1939, but in 1949 the Marxist-Leninist ideologues at the University declared semigroup theory to be one of three mathematical topics that were ideologically objectionable. Lyapin was brought in for public questioning and his work on semigroups was condemned for being “ideologically alien” and for “not bringing benefit to a socialist society.” He was dismissed from the University. Lyapin continued his research, however, taking a teaching job at Leningrad State Pedagogical Institute. He became the most influential semigroup theorist in the U.S.S.R. during the 1950s and 1960s. Lyapin’s work on semigroups was wide ranging and modern in flavor. Hollings discusses Lyapin’s work in detail in section 9.2.

Hollings says that “We cannot really speak of a British school of semigroup theory in the chosen time span,” (p. 185), but he discusses the work of three British mathematicians who were major figures in the development of semigroup theory in the 1950s: J. A. Green, Gordon Preston, and Douglas Munn. Each was influenced by Rees’s work of the 1940s.

Green obtained the Ph.D. at Cambridge in 1951 with a thesis entitled “Abstract algebra and semigroups.” Many of the results of Green’s thesis appeared in a seminal paper published the same year, “On the structure of semigroups.” Green initiated the study of regular semigroups and of the equivalence relations later called “Green’s relations.” (This terminology was introduced by Clifford and Preston in [2, Section 2.1]). Green carried the study of the structure of simple semigroups beyond what had been done by Sushkevich and Rees and initiated work on what are now called “bisimple semigroups.”

Preston burst on the semigroup scene in 1954 with the publication of a series of three papers on inverse semigroups. It wasn’t until he was finishing the third paper in the series that Preston found out that V. V. Wagner (Saratov, U.S.S.R) had published some similar work in 1952. Preston’s motivation for studying inverse semigroups came from the study of one-to-one partial mappings on a set. Inverse semigroups rapidly became one of the most important topics in semigroup theory, both in the East and the West. Two monographs on the subject ([9] and [11]) have appeared.

Walter Douglas Munn received the Ph.D. from Cambridge in 1955, with a thesis entitled “Semigroups and Their Algebras.” He was influenced by Rees’s 1940 paper and the papers of Clifford. He found out about inverse semigroups from conversations with Preston. His first publication was on semigroup algebras, in 1955. His next was on matrix representations of semigroups, in 1957. Many other semigroup papers by Munn followed, not only on these two topics, but on inverse semigroups and allied topics. On page 290 Hollings notes that in 1999 Howie described Munn as “arguably the most influential semigroup theorist of his generation”.

Hollings uses the phrase “Slovak school” as a convenient label for the body of mathematicians who were inspired to study semigroup problems by the work of the Slovak mathematician Stefan Schwarz (Czechoslovakia, 1914–1996). Schwarz originally worked in other areas of algebra and his first semigroup paper appeared in 1943. In the 1950s he published seminal and deep work on ideals in semigroups and on periodic semigroups. These ideas were picked up by others of the Slovak school and carried further, e.g., in the work of Kolibiarova, Sulka, Hrmova, Sik, and Jan Ivan.

Takayuki Tamura (1919–2009) stands out as the founder and leader of the Japanese school of semigroup research. His first paper on semigroup theory appeared in 1950. The semigroup papers by Tamura in the 1950s were on ordered semigroups, extensions of Sushkevich’s work, translations of semigroups, completely simple semigroups, semilattice decompositions of semigroups, and enumeration of finite semigroups. These topics were picked up by others in the Japanese school, some of them working jointly with Tamura. The notion of semilattice decompositions first appeared in work by Tamura and Naoki Kimura in the mid-1950s. Tamura and his colleagues did immense computational work on enumerating the number of non-isomorphic, non-anti-isomorphic semigroups of a given order. This problem was also being attacked actively in the U.S.A. at this time (1950s and 1960s). For details on this, see pp. 203–204. On this and on other semigroup topics there was considerable interaction with mathematicians in the U.S.A. From 1960 on Tamura worked at the University of California, Davis, but he never lost his close contact with the Japanese school of semigroup theory.

The Hungarian school of semigroup theory has its origins primarily in works of Lázló Rédei (1900–1980) and of his student, Otto Steinfeld (1924–1990). Rédei is well-known for his work in many areas of algebra and is credited with educating a whole generation of Hungarian algebraists. Rédei’s most important result in semigroup theory is his theorem that every finitely generated commutative semigroup is finitely presented. The proof of this result formed the bulk of his 1963 monograph The Theory of Finitely Generated Commutative Semigroups. Much of the semigroup influence on Steinfeld came from Rédei, but Steinfeld also drew inspiration from contact with Dubreil and Schwarz. Steinfeld is probably best known for introducing the concept of quasi-ideal in semigroups and in rings (see [14]). The 1950s saw the emergence of the Hungarian school of semigroup theory. Many of its members had been students of Rédei.

By the 1960s, the theory of semigroups had developed into a mature and expanding independent field of study. Illustrating the level reached by then were the establishment of on-going semigroup seminars and the accompanying centers of semigroup research, publication of substantial monographs on semigroups, and the holding of the first international conference on semigroups.

Lyapin’s semigroup seminar began soon after he joined the faculty at Leningrad State Pedagogical Institute. This seminar was at the heart of the Leningrad school of semigroup research. Wagner’s seminar in Saratov was organized by Boris Schein. This seminar was attended by semigroup theorists from all over the U.S.S.R., including Gluskin, Ponizovskii, and Shevrin. A volume of lectures from those seminars was eventually published. Around the Wagner seminar thrived the influential Saratov school. Another important semigroup seminar in the U.S.S.R. was Gluskin’s, in Kharkov.

Clifford arrived at Tulane University in New Orleans in 1955. He began a seminar on the algebraic theory of semigroups. In parallel with this was a seminar on topological semigroups, organized by Wallace, Mostert, and Hofmann. The confluence of these two highly active groups attracted semigroup theorists from all over the world and made Tulane a world center in semigroup theory for over two decades. Dubreil’s Paris seminar on algebra and number theory attracted many semigroup theorists from other countries. However, very few mathematicians from the Eastern Bloc participated. The Paris semigroup school was centered around the Dubreils.

In 1960 Lyapin published the Russian edition of the first monograph of semigroup theory since the Sushkevich monograph of 1937. While the latter tightly focused on Sushkevich’s own work, the Lyapin monograph covered a wide range of topics and cut across national boundaries. In 1963 the American Mathematical Society published the first English language edition, [10].

Preston visited Tulane in 1956–1958. While there the collaboration of Clifford and Preston on a two volume monograph on semigroups began. The first volume of The Algebraic Theory of Semigroups appeared in 1961 and the second volume in 1967. This work succeeded in unifying the theory and standardizing notation and terminology. The first volume, in particular, combines the best aspects of a monograph for researchers and a textbook for students. Exercises are included at the end of each section and some historical comments are given.

The first formal conference on semigroups in the West was the Conference on the Algebraic Theory of Machines, Languages, and Semigroups, held in California in 1966. On the other side of the Iron Curtain there were several conferences on semigroups in the 1960s. With the growth and maturity of semigroup theory, and with some thawing in the icy barrier between the West and the Soviet bloc, several thought the time was ripe for an international conference on semigroups. It appears that Schwarz in Czechoslovakia initiated the realization of this by sounding out Mostert, Hofman, and Clifford about the possibility of such a conference, to be held in Czechoslovakia. The conference began to take shape, with Schwarz as its driving force. The conference was held June 18–22, 1968, in Smolenice. It was sponsored by the Slovak Academy of Sciences. There were sixty participants, from eleven different countries. More details about this conference are given in section 12.3.1 of the book under review.

The Smolenice conference of 1968 helped stimulate the perceived need for an international journal devoted to semigroup theory. It appears that Hofman, Mostert, and Clifford were approached by several others to organize the publication of such a journal. The fruit of this labor appeared in 1970, with the first issue of Semigroup Forum. It was a pioneering journal in several respects. It was one of the early instances of a highly specialized mathematics journal, and it was innovative in being produced via photographic reproduction of the authors’ type-script. The managing editors of the journal were Clifford, Hofman, and Mostert. There were nineteen other editors from around the world. Hollings gives the full editorial board list in Table 12.10 and table of contents for the first issue of Semigroup Forum in Table 12.9. The journal provided, and still provides, a forum for all semigroup related material.

Mathematics Across the Iron Curtain is a mathematics book as well as a history book. Much of the book assumes an understanding of algebraic semigroup theory at the level of a standard introductory monograph on the subject, e.g., [7]. A first year graduate student or a senior level student who has had a solid first course in abstract algebra and who judiciously makes use of the background on semigroup theory in the Appendix on Basic Theory will find this book informative and illuminating.

Hollings has done a masterful job. The book is well written, both in telling the story and in explaining the mathematics involved. It is an important and valuable contribution to the history of mathematics in the 20th century. This book should be in the libraries of all research universities, and on the shelves of those interested in the history of abstract algebra, as well as those of semigroup researchers.

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References

1. A. H. Clifford, “Semigroups admitting relative inverses,” Ann. of Math., 42 (1941), 1037–1049.

2. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. I, Mathematical Surveys No. 7, Amer. Math. Soc., Providence, 1961.

3. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. II, Mathematical Surveys No. 7, Amer. Math Soc., Providence, 1967.

4. J. A. de Seguier, Théorie des Groupes Finis: Elements de la Théorie des Groupes Abstraits, Gauthiers-Villars, Paris, 1904.

5. Paul Dubreil, “Contribution a la théorie des demi-groupes,” Mem. Acad. Sci Inst. France, 63 (1941).

6. H. Hilton, An Introduction to the Theory of Groups of Finite Order, Clarendon Press, Oxford, 1908.

7. J. M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976.

8. J. M. Howie, “Semigroups, past, present, and future,” in Proceedings of the International Conference on Algebra and its Applications, W. Hemakul, ed., Dept. Math. Chulalongkorn Univ., Bangkok, Thailand, 2002.

9. M. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, Singapore, 1998.

10. F. S. Lyapin, Semigroups. Trans. Math. Monographs, Vol. 3, Amer. Math. Soc., Providence, 1963.

11. M. Petrich, Inverse Semigroups, John-Wiley & Sons, Inc., New York, 1984.

12. G. B. Preston, “ A. H. Clifford’s work on unions of groups,” in Semigroup Theory and its Applications: Proceedings of the 1994 Conference Commemorating the Work of Alfred H. Clifford, K. H. Hoffman and M . W. Mislove (eds.), LMS Lecture Note Series, No. 231, Cambridge Univ. Press, Cambridge, 1996, pp. 5–14.

13. D. Rees, “On semi-groups,” Proc. Cambridge Phil. Soc., 36 (1940), 387–400.

14. O. Steinfeld, Quasi-Ideals in Rings and Semigroups, Akademiai Kiado, Budapest, 1978.


Henry Heatherly is Emeritus Professor of Mathematics at the University of Louisiana, Lafayette. His current research interests are rings and semigroups. Each Spring semester he teaches a course on the history of mathematics.