As the author explains, the subject of his book is the development of the algebraic theory of semigroups “from its earliest origins (in the 1920s) up to the foundation of the …journal Semigroup Forum in 1970” with a particular emphasis on the comparison of results and methods “of semigroup researchers in East and West, together with an investigation of the extent to which they were able to communicate with each other”. Since a good portion of the book is devoted to the work done prior to the descent of the Iron Curtain (which occurred after the end of World War II), its title is not quite accurate. The author’s suggestion that “semigroup theory might be termed ‘Cold War mathematics’ because of the time during which it developed” is also somewhat misleading for it could be interpreted as an assertion that semigroup theory was the only young mathematical discipline that was developing during that time period (which is hardly the case). Nevertheless the book’s striking title will certainly attract attention of its potential readers, so the author should not be faulted for making that choice (especially because communication between Soviet and Western scientists was extremely difficult under Stalin’s regime even before the start of the Cold War).
The main part of the book consists of twelve chapters (subdivided into sections and subsections) followed by a brief appendix containing basic definitions of semigroup theory to help those readers who may not be familiar with the relevant terminology. The book ends with a set of copious notes (which clarify and complement the material in the main part of the book), a very extensive bibliography, and detailed name and subject indices.
In Chapter 1 the author discusses the emergence of ‘abstract algebra’ at the beginning of the 20th century, traces the origin of the term ‘semigroup’ and changes in the meaning of that term during the first several decades of that century, and gives a broad overview of the development of the algebraic theory of semigroups. Chapter 2 contains a well-documented analysis of the difficulties of East-West communications in mathematics and other areas of science. Chapter 3 is devoted to A. K. Sushkevich, a pioneer of semigroup theory. After a biography of Sushkevich, the author discusses his major results and their impact on subsequent researchers. Only a few of his papers appeared in international mathematical journals and were readily available in the West. The foremost of them was ‘Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit’ [A. Suschkewitsch, Math. Ann. 99, 30-50 (1928; JFM 54.0151.04)] in which he showed that every finite semigroup contains a ‘kernel’ (that is, a simple ideal) and completely determined the structure of all finite simple semigroups. The author gave a detailed description of that paper in Section 6.3 and included a schematic representation of a ‘kernel’ in Figure 6.1 as well as on the cover of the book (that diagram is the source of the ‘eggbox picture’ so well known to all mathematicians working in semigroups). Although the author stated that “Sushkevich’s work had little or no influence on subsequent semigroup researchers”, he did observe that the 1928 paper mentioned above had an impact on a number of future researchers (most importantly, on Rees, Clifford, and Schwarz).
In Chapter 4 the author gives a short biography of A. H. Clifford and outlines his contributions to the study of unique factorization in semigroups (which was the topic of his PhD thesis written at Caltech under the supervision of E. Bell). One of Clifford’s main goals was finding a semigroup version of E. Noether’s theorem concerning unique factorization in a commutative integral domain. The author described Clifford’s major results and analyzed how his research was influenced by the ideas of Noether and those of Bell and Ward (another PhD student of Bell). Since subsemigroups of a group are cancellative, a natural question is whether any cancellative semigroup can be embedded in a group. Sushkevich erroneously claimed that the answer is yes. A counterexample to that claim was constructed by A. I. Maltsev who also found necessary and sufficient conditions (countably infinite in number) for the embeddability of a cancellative semigroup in a group (with no finite subset of those conditions being sufficient). More useful sufficient conditions were found by O. Ore. In Chapter 5 the author outlines the mentioned works of Sushkevich, Maltsev, Ore and some others.
In Chapter 6 the author discusses one of the most fundamental structure theorems of semigroup theory due to D. Rees. In Rees’s paper ‘On semi-groups’ [D. Rees, Proc. Camb. Philos. Soc. 36, 387-400 (1940; Zbl 0028.00401; JFM 66.1207.01)], Sushkevich’s structure theorem for finite simple semigroups was extended to the class of all completely \(0\)-simple semigroups. That theorem and the tool that Rees created for establishing it (a special matrix semigroup over a group) have served as a model for many subsequent researchers. Another influential work discussed in Chapter 6 is Clifford’s paper ‘Semigroups admitting relative inverses’ [A. H. Clifford, Ann. Math. (2) 42, 1037-1049 (1941; Zbl 0063.00920)] in which he studied semigroups that are unions of groups and completely determined the structure of such semigroups under the additional assumption that their idempotents commute.
Chapter 7 is devoted to the work of the French school of semigroup theory headed by P. Dubreil while Chapter 8 describes the activities of various other national schools in the 1940s and 1950s – Slovak created by S. Schwarz, American under the leadership of A. Clifford, Japanese originated with T. Tamura, and Hungarian headed by L. Rédei and O. Steinfeld. At the end of Chapter 8 the author turns to the work of British authors who followed the lead of D. Rees and describes, in particular, fundamental contributions of J. A. Green. The discussion of very important results of G. B. Preston and W. D. Munn is postponed until later chapters.
In Chapter 9 the author concentrates on the work of two prominent Soviet mathematicians – E. S. Lyapin in Leningrad (currently, St. Petersburg) and L. M. Gluskin in Kharkov (the latter was a student of Sushkevich). The stories of their lives during the war with Germany and the hardships that they experienced after the war under the Stalinist regime are fascinating. However the author’s main goal is to describe their mathematical results, and he does that in great detail. In addition to his own contributions, Lyapin created a large school of followers in Leningrad (and elsewhere in the USSR) and in 1960 published the first book devoted entirely to semigroup theory. Gluskin obtained important results in several areas of the theory of semigroups (especially, in his study of semigroups of transformations and densely embedded ideals).
The discovery and development of inverse semigroups, one of the most important generalizations of groups, is the subject of Chapter 10. Inverse semigroups were first introduced in 1952 by V. V. Wagner under the name ‘generalized groups’, and his short 1952 article was followed in 1953 by a very substantial 88-page treatise. Wagner’s papers appeared in major Soviet mathematical journals and were available in the West. However they were written in Russian and Western mathematicians had hard time reading and understanding them due to linguistic difficulties. Since there is still no English translation of Wagner’s 1953 paper, the author gives a fairly extensive overview of its results. Inverse semigroups were independently discovered by G. B. Preston who published several articles about them in 1954 (and coined the term ‘inverse semi-groups’), and the author describes major results of Preston’s papers. He also includes biographies of Wagner and Preston and compares the starting points and ideas which led both of them to the notion of an inverse semigroup.
Chapter 11 is devoted to matrix representations of semigroups. After outlining the early results of Sushkevich and Clifford the author turns to the work of two major contributors to this area of semigroup theory in the 1950s – W. D. Munn and J. S. Ponizovskii. Both of them started working on semigroup algebras and matrix representations of semigroups almost simultaneously. They independently obtained many similar results by using essentially the same methods and only later became aware that their investigations were very much parallel to one another. The author includes a biography of Munn and a brief one of Ponizovskii and describes in detail their respective results. Both of them contributed to other areas of semigroup theory – in particular, to inverse semigroups. Especially influential was the work of Munn. The author quotes Howie’s description of Munn as “arguably the most influential semigroup theorist of his generation” which, in my opinion, is an understatement – Douglas Munn was arguably one of the most influential semigroup theorists of all time.
In Chapter 12, the final chapter of the book, the author lists the monographs, conferences, and seminars on semigroups. In the first section of that chapter he outlines the content of Sushkevich’s “Theory of Generalized Groups” (1937), then gives detailed descriptions of Lyapin’s “Semigroups” (1960) and a very influential Clifford and Preston’s two-volume monograph “The Algebraic Theory of Semigroups” (1961, 1967), and finishes that section with a fairly complete list of other books on semigroup theory. In the second and third sections the author describes some (but certainly not all) seminars and conferences devoted to semigroups. In particular, the first international conference on semigroup theory was held in Smolenice (near Bratislava) in June of 1968. One of the byproducts of that meeting was the decision of its participants to create a special journal dedicated to the theory of semigroups, and before long Semigroup Forum was ‘born’ (its first issue appeared in 1970).
The author has done a thorough job describing the development of semigroup theory in various countries and illustrating the obstacles for communication and interaction between mathematicians due to political and linguistic barriers. His analysis is quite complete up to 1960. The decade from 1960 to 1970 is covered more selectively – the work of certain mathematicians is described in detail, contributions of some are mentioned only briefly, and important results of others are not discussed at all. For instance, Mario Petrich (an influential and by far the most prolific semigroup theorist who was one of the founding editors of Semigroup Forum and for a number of years one of its managing editors) is mentioned as the author of several books but his own results and papers are not discussed (except for one joint article with G. Lallement). Nevertheless the author’s overall historical account of the development of the algebraic theory of semigroups is very valuable and his book will be of interest not only to semigroup theorists but to other algebraists and mathematicians in general. An undertaking of that scope requires a huge amount of effort, and it is next to impossible to accomplish it without blemish. Thus it should not be surprising if the author’s exposition contained some inaccuracies, and I would be remiss if I did not correct those which I have noticed.
1) On page 102 the author writes that a brief overview of factorization in semigroups is given in Chapter 4 of the first volume of Jacobson’s “Lectures in Abstract Algebra” but it “was removed from subsequent editions of Jacobson’s” and even offers his explanation for that removal. In fact, the material in Section 2.14 of Jacobson’s “Basic Algebra I” is almost identical to that of §§1-3 of Chapter 4 of the first volume of “Lectures in Abstract Algebra”.
2) On page 156 the author states that the term ‘completely regular semigroup’ was introduced in 1973 by Petrich in “Introduction to Semigroups” (incidentally, the same mistaken assertion is contained in Section 4.8 of Howie’s “Fundamentals of Semigroup Theory”). Actually, that term was introduced by Lyapin in his book “Semigroups” in 1960 (this fact is confirmed on page 63 of the authoritative monograph “Completely Regular Semigroups” by Petrich and Reilly).
3) On page 235 the author writes: “Unfortunately, the concept of a densely embedded ideal does not seem to have been adopted in Western semigroup theory …”. In fact, that concept and the closely related notion of a dense extension have been studied in a number of papers by Petrich, Reilly, Pastijn and others. Some results of those papers were discussed in Petrich’s “Introduction to Semigroups” and in the 1985 monograph “Regular Semigroups as Extensions” by Petrich and Pastijn (not listed in the bibliography of the author’s book).
4) On page 298, speaking about the minimum group congruence on an inverse semigroup, the author claims that in Munn’s 1961 paper “it was first proved that such congruence exists”. In reality, contrary to the misleading statement in Section 5.12 of Howie’s “Fundamentals of Semigroup Theory”, it was Wagner who first showed that the minimum group congruence on an arbitrary inverse semigroup exists and found its convenient characterization in his seminal 1953 paper (Theorem 4.39). Apparently the author has overlooked that fact while reading Wagner’s paper (as noted in Section III.5 of Petrich’s “Inverse semigroups”, Wagner’s and Munn’s characterizations of the minimum group congruence on an inverse semigroup are obviously equivalent).