## Gordon Preston on semigroups

In 1990 a conference was held in Melbourne, Australia, to pay tribute to the contributions of

G B Preston, Personal reminiscences of the early history of semigroups, in

Below we give a version of Preston's talk.

**Gordon Preston**. Gordon himself gave a lecture*Personal reminiscences of the early history of semigroups*and a version of this talk appeared in the conference*Proceedings*as:G B Preston, Personal reminiscences of the early history of semigroups, in

*Monash Conference on Semigroup Theory, Melbourne, 1990*(World Sci. Publ., River Edge, NJ, 1991), 16-30.Below we give a version of Preston's talk.

### Personal reminiscences of the early history of semigroups

by**G B Preston**

I have been asked to talk about the early development of semigroup theory, and what I saw of it and how I became interested in it. There must be many of you here who know as much, or more, of this early history than I do. So anything extra I have to offer must lie in how it looked to me when I became a research student.

It will perhaps help if I say something first about myself, about the mathematical background from which I looked out.

I took finals at Oxford in 1948 with a syllabus that I and most of my contemporaries regarded as impossibly out of date. We could use final examination papers set in 1910 to give us practice in answering the 1948 papers. There was no algebra - groups, rings, fields, etc. in the syllabus. The study of matrices did not involve the mention of vector spaces. Topological spaces had not been heard of - though some very popular lectures were given on topology by J H C Whitehead (Henry Whitehead). It is interesting to reflect that although we, the students, knew that this syllabus had stood still, neglecting most of modern mathematics - analysis was more up to date - we did not complain about it. We read, variously, mathematics right outside the syllabus, but it did not occur to us to suggest that the syllabus be changed. For example, our differential geometry lectures told us about two adjacent points on a curve, or two adjacent tangents to a curve, and my immediate contemporaries at Magdalen (my college at Oxford), Michael Barrett and Victor Guggenheim, and I, read the appropriate books that did this differential geometry properly. But we knew that answers using a rigorous language were not what our teachers wanted. So our answers were appropriately phrased.

There was another influence on me that was possibly as important. After spending one year at Oxford in 1943/4, I was called up for war service, volunteered for the navy, and was drafted to work for the foreign office at Bletchley Park. There, I was with a small group of about twenty mathematicians, assisted by about 250 WRENS, in what was called the Newmanry. M H A Newman, the MHAN, was the head of this group, and the assistant head was Shaun Wylie. Other members included Henry Whitehead, David Rees, Michael Crum, Donald Nichle - not a mathematician by training -, I J (Jack) Good. J A (Sandy) Green, Joe Gillis, Howard Campaign - a good poker player -. Philip Watson (Philip and I had come together from Oxford), A 0 L (Oliver) Atkin, and Michael Ashcroft. We had a research session, so far as I recall, on Monday afternoons and a research log book in which, at any time, ideas were recorded. This was my first experience of research - it was a mixture of algebra and statistics, or probability theory, and I greatly enjoyed it. I also got to know well P J (Peter) Hilton and A N (Alan) Turing. With Turing I spent uncountable hours playing Go, as also with David Rees.

When I became a research student my first supervisor was J H C Whitehead. I was, I believe, his 17th research student. Most of the Oxford professors had about 20 research students each, at that time. [I believe that, what appears to be the most common current view, that a supervisor can properly supervise only a small number of students, say 2 or 3, at any one time is a mistaken view. One's contemporaries as a research student are just as important, if not more important, than one's supervisor. To have twenty or so people, working on a set of inter-related problems, perhaps suggested by the supervisor, provides a milieu of competition and stimulus that no supervisor can replace.] He was available to see me at anytime, a gregarious, party loving man, who also met all of his research students, and others, in a weekly seminar. For the first year I worked with him on algebraic topology. He then went off to the States for a year and my new supervisor was E C (Edward) Thompson and, with him, I changed my subject to algebraic geometry, with a strong emphasis on commutative ring theory and with not so much geometry.

So far nothing to do with semigroups. My semigroup influence came because of the friends I had met at Bletchley Park. I used to go to the National Physical Laboratory at Teddington - this was while I was an undergraduate again from 1946 to 1948 - to keep in touch with the development of the pilot ACE (Automatic Computing Engine) that Alan Turing had been drafted there to develop. I was interested in the mathematical papers of my other Bletchley friends and, in particular, this was how I came to read David Rees's papers. I have no records, and my memory may be playing me false, but I believe the first paper I read on semigroups was his paper

*On semi-groups*from the

*Proceedings of the Cambridge Philosophical Society*, 1940, together with the small technical note that followed it,

*Note on semi-groups*, ibid., 1941.

Rees wrote his 1940 paper in the summer vacation after completing his undergraduate honours degree at Cambridge. One thing struck me about this paper, when I read it, and I wonder whether anyone else has had the same reaction. In 1939, just before David Rees did his work, A A Albert's book

*Structure of algebras*, principally about linear associative algebras, was published. The notation, the choice of letters of the alphabet, where relevant, appears to be closely followed by Rees in his paper. I assumed that, on going down from Cambridge, David Rees had read this book of Albert's and tried, what was a natural thing to do, to reproduce the ring structure theory for a system without the addition operation. But, when I asked him whether this was so, David Rees assured me that this was not the case, that he had not read Albert's book then. Still it is a curious coincidence.

Perhaps this ring theoretic notation was common in certain circles those days. Certainly there was one strong influence on algebra operating in Britain at that time, much of which did not see its way into print, and that was the lectures given at Cambridge by Philip Hall. Philip Hall lectured on algebra, in its widest sense, at Cambridge. I am not certain when these lectures began. I have often thought that Garrett Birkhoff, who went to work with Philip Hall at Cambridge in the early thirties, received the inspiration for his work on universal algebra from Philip Hall, especially from his paper

*A contribution to the theory of groups of prime-power order*, in the

*Proceedings of the London Mathematical Society*, in 1933. The work in this paper on commutators of different orders naturally suggests working with general

*n*-ary operators satisfying various laws.

One account, consisting largely of material from Hall's lectures, appeared in Paul Cohn's book

*Universal Algebra*, in 1965. My introduction to them came when Sandy Green, probably in 1951, lent me a copy of his notes on the lectures given one year by Philip Hall. I made a hand-written copy of them - this was before the days of the ubiquitous photocopier - which I still have. This was exciting and tremendous stuff. Semigroups were used quite naturally in Hall's lectures, at any rate for that year.

So far as I know the first substantial paper on semigroups was that of A K Suschkewitsch in 1928

*Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit*in

*Mathematische Annalen*

**99**pp. 30-50 (submitted 1 January 1927). In modern terminology the title can be construed as

*On finite semigroups which are not necessarily cancellative*. This remarkable paper determined the structure of finite semigroups containing no proper ideals and showed that each finite semigroup contained such a semigroup as its minimum ideal. The minimal left ideals and right ideals of this minimum ideal, and their group intersections, were explored and exhibited in what has become the familiar "egg-box" pattern. He also introduced the ideas of left groups and right groups. His results have been central to all further work in semigroups. He had the exhilarating advantage, to which Paul Cohn referred in his talk at Townsville [July 6, 1990. Invited talk to the annual meeting of the Australian Mathematical Society:

*Valuations on skew fields*] on skew fields, of being first in the field: the major basic results were just waiting there to be discovered.

An interesting detail about the Suschkewitsch paper is a footnote in which Suschkewitsch thanks Emmy Noether for one of his references.

There is perhaps another candidate for the first paper in semigroups, and that is the paper of H Brandt

*Über eine Verallgemeinerung des Gruppen Begriffes*, published in

**96**, pp. 360-366, of the same

*Mathematische Annalen*. This paper was submitted on 12 December 1925 and so certainly ante-dates Suschkewitsch's paper. What Brandt discussed in this paper is what has since been called either a Brandt groupoid or a Brandt semigroup. His paper deals with finite such groupoids only, although he comments that most results can be extended to countably infinite groupolds. (Why the restriction to countable? Of course I know there was a general worry about uncountable sets at that time, as sets that perhaps hid forbidding monsters.) He points out that a zero element can be adjoined to give a product where this is otherwise undefined; but says that he has not done this because in general it is of small advantage: "im allgemeinen doch von geringen Vorteil".

What he gave was the structure, in terms of its maximal subgroups, of a completely 0-simple inverse semigroup. By not using a zero element he seems to me to have missed a great chance. His view still prevails with many: for example, see Philip J Higgins' book "Categories and groupoids", Van Nostrand, 1973.

Brandt's paper is perhaps to be regarded as the first paper on category theory, not on semigroup theory.

A H Clifford made an early entry into semigroup theory. His 1933 paper

*A system arising from a weakened set of group postulates*(

*Annals of Mathematics*, pp. 865-871) characterized infinite left and right groups, and generalized Suschkewitsch's structure theory for finite left and right groups. Clifford was not aware of the prior work of Suschkewitsch.

His

*Semigroups admitting relative inverses*(

*Annals of Mathematics*, 1941, pp. 1037-1049) showed that semigroups which are unions of groups are disjoint unions of completely simple semigroups, an important early structure theorem, emphasising also the importance of completely simple semigroups, introduced by Rees in his 1940 paper.

Meanwhile Paul Dubrell had initiated an entirely different approach. In the papers I have mentioned to date, and in most of the others that had by then appeared, there was clearly an attempt to seek analogies, either with groups or with rings. Dubrell's 1941 paper

*Contribution à la théorle des demi-groupes*(

*Mémoires de l'Academle des Sciences de l'Institut de France*, 52pp.) was conceptually breaking entirely fresh ground. I found it difficult to understand properly what it was about, and it took me some time before I felt I had grasped it. Dubrell's paper became the foundation stone for a vigorous French school of "demigroupes". Robert Crolsot, who so tragically died in a skiing accident in the French Alps in 1961, was an outstanding and early member of this school.

David Rees wrote four papers on semigroups, as sole author, and also a joint paper with Sandy Green. His 1947 paper

*On the group of a set of partial transformations*, in

**22**1947, of the

*Journal of the London Mathematical Society*, pp. 281-284. (submitted 14 July 1947) was the first paper to discuss inverse semigroups. This paper begins with the definition of a general inverse semigroup of bijections between subsets of a set. with the one proviso. that it was required that this product of any two bijections in the set considered should never be the empty bijection. Rees showed the same dislike of the empty set as did Brandt, given the appropriate interpretation of what Brandt did. Except for this proviso, Rees was considering, in this paper, an arbitrary inverse subsemigroup of the symmetric inverse semigroup on a set

*X*. And, of course, this proviso does not exclude any inverse semigroup: an extra element added to the set

*X*ensures that all inverse subsemigroups of the symmetric inverse semigroup on

*X*are included under the scope of Rees's definition. Rees discovers that each element of his semigroup has a unique inverse in the semigroup, he discovers the natural partial ordering, containment of mappings, of the elements of the semigroup, and shows that it is compatible with both product and taking inverses. He then constructs a quotient group of his inverse semigroup, the maximum quotient group, as the image modulo the congruence defined by two elements being congruent if and only if they contain a common submapping. i.e. bijection, within the semigroup. He then uses this construction to prove Ore's theorem, namely that a cancellative semigroup

*S*for which the intersection of any two principal left ideals is non--empty, can be embedded in a group: under these conditions

*S*can be embedded in an inverse semigroup such that its quotient group contains

*S*.

Rees left a profound mark on semigroups in his four short papers. He also is responsible for giving the name "regular" to what we now call regular semigroups, as witness a footnote thanking Rees for suggesting this term in J.A. Green's paper

*On the structure of semigroups*(

*Annals of Mathematics*

**54**(1951), 163-172).

Green's paper was an important one. Sandy Green once told me that this paper, which came from his Ph.D. thesis, represented a specialisation to semigroups of results obtained in his thesis for universal algebras. I cannot guess how this was done, and I never enquired about the matter further. Perhaps there is someone here familiar with his thesis.

At about the time I became interested in semigroups there were about 50 papers, of which I was aware, published on semigroups. There were perhaps many more Russian papers of which I am not, and was not, aware for this was one country from which throughout the years there has been a major contribution to semigroup theory. Indeed the father of the subject, Suschkewitsch. was Russian. and wrote his paper from Voronesh. I understand he had moved there from Lithuania - am I right?

There being only 50 papers, but with the number growing fast,it was not too difficult to read all that had been written on the subject; one was immediately at the frontiers of knowledge. It was an exciting prospect.

My thesis was not written on semigroups. It was on universal algebra and I tried to extend properties of groups and rings to a more general context, for example some of the ideal theory of Noetherian rings. I added a chapter at the end of my thesis in which I tried to axiomatise the inverse semigroups that Rees had considered in his 1947 paper. My attempt was not successful: but I proved some results about the semigroups I had defined. My D. Phil. examiners were David Rees and Henry Whitehead. David gave me a list of detailed comments which Henry Whitehead would not let him ask me about in the oral examination - Henry was the chairman of examiners - because Henry had to rush off to captain a cricket team playing that afternoon. Henry asked me a number of questions himself, including some about my chapter on semigroups; and in my answers to his questions which were critical questions I was able apparently to show that he had failed to grasp properly the concepts involved. After this I was invited to come and watch the cricket game, which I took as a sign that I had passed my viva.

When I thought about Henry's comments during the next few days, I discovered that, despite their incorrectness, they contained the essence of what I was looking for. I had my axioms for inverse semigroups. My further work on inverse semigroups quickly followed and, at the end of 1953, I submitted three papers on inverse semigroups to the

*Journal of the London Mathematical Society*.

I thought this axiomatic characterization of semigroups of bijections which contained with each element its inverse was exciting. I had managed to show, in my first papers, that subsemigroups of an inverse semigroup, closed under inversion, were themselves inverse, that homomorphic images of inverse semigroups were inverse - indeed that, as for groups, a homomorphism with respect to multiplication was simultaneously one with respect to inversion. And it was clear that direct products of inverse semigroups were inverse. So we had, by Birkhoff's theorem, a variety. With the axiom definition I had given of inverse semigroups it would be possible to develop their theory much more easily than if we merely had to work all the time with bijections. Graham Higman, who identified himself as the referee of these papers, asked me if I had a set of identities which characterized the variety of inverse semigroups, and I provided him with such a set.

There was also, by me, a strong hope that inverse semigroups would provide a vehicle for studying the partial symmetries of a system and would provide a tool that would strengthen and amplify the information provided, on its full symmetries, by groups. Sometime later I suggested to L E (Leslie) Orgel. an old friend from undergraduate days who was a theoretical chemist at Cambridge, that they might be of use in the quantum theory of the molecule. He and a couple of others worked on this possibility in the late 50's, but eventually reported that it did not seem to work as well as might be expected. One missing item, at that time, was an adequate matrix representation theory for inverse semigroups.

I reported on my inverse semigroup results to the International Congress of Mathematicians at Amsterdam in 1954, where what I announced attracted a great deal of interest.

When my papers were in proof stage, in late 1954, I received a letter from Bernhard Neumann telling me about related work by E S Lyapin in

*Doklady Akademii Nauk*, (1953) volume 88, on the symmetric inverse semigroup which led me to V V Vagner's paper

*Generalized groups*in volume 84 (1952) of the same journal. Vagner had an equivalent set of axioms for inverse semigroups and gave the same representation by partial bijections as I gave in my third paper

*Representations of inverse semigroups*in the

*Journal of the London Mathematical Society*

**59**(1954), pp. 411-419.

One paper that I read about this time was by J Riguet:

*Travaux récents de Malcev, Vagner, Liapin sur la représentation des demi-groupes*, of 18 January 1954, in the

*Séminaire d'Algébre of the Faculté des Sciences de Paris*. Riguet gave a short survey of recent Russian work, and in particular he gave Vagner's axiomatisation of semigroups of bijections containing with each element its inverse. His axiomatic definition was:

*C*is a Vagner semigroup if

*C*is a semigroup in which any two idempotents commute and in which each element

*a*has a generalized inverse, i.e. an element

*x*such that

*axa*=

*a*and

*xax*=

*x*. Riguet gives Vagner's proof of his representation of such semigroups by partial bijections of a set. To read Lyapin's paper and Vagner's paper

*Generalised groups*, (

*Doklady Akademii Nauk*

**84**(1952), 1119-1122), I had to face the fact that I could not read Russian. I tackled the paper armed with a dictionary and no knowledge of either Russian or the Russian alphabet. I remember that it took me three hours to translate the first sentence - though whether this was of Lyapin's paper or of Vagner's I have now forgotton.

Riguet quotes Vagner as saying [

*Theory of generalized heaps and generalized groups*, (

*Matematicheskii Sbornik*

**74**(1953), 545-562): my translation from the French]:-

One comment I must make about the state of semigroups in the world of mathematics at that time is that it was not just a new area at its exciting beginning stages but it was also an area that had attracted some criticism from other mathematicians. It was said that semigroups were objects that were too simple to be interesting, and that useful mathematics would not stem from their study. I don't think I was aware of this attitude at the time I did my first work. I talked to mathematicians like Bernhard and Hanna Neumann, Graham Higman. Sandy Green, David Rees, Oystein Ore, and also many at the Amsterdam Conference who were most interested in what I had done. Graham Higman, for example, with whom I had close contact until I left England to come to Australia, kept up, as far as I could judge, with most of what was written on semigroups. He considered that semigroup theory, so he once told me, was in some ways more difficult than the developed areas of algebra, the difficulty being due to its youth and the consequent fact that there was a plethora of new concepts continually being introduced, almost a flood of them. The difficulty was not in working with them, but in deciding which were the more promising to pursue. Of technical difficulties, in the early days there were few, of conceptual difficulties many.The study by algebraic methods of the formal properties of operations on sets and on binary relations is beginning to take a greater and greater importance. Conversely, the use of algebraic methods for the study of operations on sets and relations leads to the construction of corresponding abstract algebraic theories. The abstract algebraic theories thus obtained have evidently a much greater importance than those that are the result of a purely formal generalization of an already existing theory by making changes to an axiom system, since it is possible to interpret such theories in terms of the theory of sets and consequently to apply them to other parts of mathematics.

I became aware that some mathematicians regarded semigroup theory, I think it is not too strong to say, with scorn, because its results were too superficial, only when I began talking to others working principally in semigroup theory. I have been told, since coming to this conference, that such a view prevailed for many years with many Russian mathematicians. Certainly one mathematician to whom such opinions were attributed was Kurosh; and he has comments in print to this effect. However Kurosh clearly changed his mind on this subject for, when he visited Melbourne in 1964, he chose to talk about inverse semigroups and Russian work on them, in his invited talk at Melbourne University.

I am inclined to think that there was nothing specially significant in this criticism of semigroups. Indeed I think it was principally exhibiting the attitudes of closed minds that were not open to new ideas, attitudes that have been expressed towards very many emerging new areas of mathematics at their birth. Consider the stormy weather that set theory had to endure. Recent examples are numerous. Mathematical logic had a hard fight to be recognised as a part of mathematics. John Crossley, for example, left the permanent job of fellow of All Soul's, at Oxford, surely one of the most enviable jobs for anyone to have, to come to Monash as a professor, because he saw a chance here, which he did not see at most universities in Britain, that mathematical logic would be treated as the important part of mathematics that it is.

Again graph theory, lattice theory and universal algebra have had a long and very rough ride. Saunders Mac Lane invented the name "Abstract Nonsense" for category theory, so he once said, to prevent its critics coming up with it first.

No, semigroups is in good company in the criticism it has received. If it had been totally inconsequential no-one would have bothered to criticise it.

Perhaps I could now respond to the request to say a word or two about the books that Clifford and I wrote.

The invitation I received from Clifford to go to Tulane sealed my mathematical interests. Until then I had come from a background in algebraic topology and algebraic geometry. As an undergraduate my principal skills had been in analysis and applied mathematics. Until I went to Tulane I was uncommitted as to any principal direction of my mathematical interests, although they seemed to have settled in algebra rather than one or another kind of geometry. At Tulane I met this enthusiasm and concentration upon semigroups, both algebraic and topological, and was infected by it. In preparation for going to Tulane I had read pretty well everything published on semigroups that had been written by then, except for the Russian literature, which I still could not read, except in an emergency.

One feature of the developing semigroup theory at this time, growing very fast in the early fifties, was that virtually no two authors agreed on their definitions and terminology. It was becoming most difficult to keep track of what were often minor, but essential, differences from paper to paper. You would try to apply a theorem you had learnt to a new situation, failing to note that a slight difference in the definitions made the theorem inapplicable. I felt it was time a book, indeed a treatise, appeared on semigroups, which would both unify present knowledge and also standardise terminology. I felt that Clifford was the man who should do it, and after I had been about six months at Tulane I proposed this to him.

He told me that he had a book in plan with several chapters completed. And he showed some of the early chapters of it to me. It was a good book, but not at all what I wanted him to write. It was written with Clifford's usual limpid clarity: but it was aimed at an undergraduate, or perhaps beginning graduate level, providing an excellent introduction to algebra via semigroups. At this distance, I may be misremembering. But this is certainly the impression I now have of the excellent chapters of this that I read.

I made a number of suggestions about what I thought was the basic material that should be introduced early in the book. The summer vacation was about to begin. Clifford was about to depart for Cape Cod for the summer, and he suggested that I should sketch in more detail what I had in mind.

At the end of the summer I had completed what became the essence of Chapters 2 and 3 of Volume 1. It was an attempt at giving an integrated account of the Green relations, the Schützenberger group and representations (we had just had what we called a "Schützenberger week" at which Schützenberger had lectured each day on his representations and associated matters), completely 0-simple semigroups, 0-minimal ideals, etc. It included results of mine on the Schützenberger representations that I afterwards published in my 1958 paper: "Matrix representations of semigroups".

I typed this up and circulated it for comment and, when Clifford came back, he liked it. I certainly had no thought at this time of being co-author with Clifford of the book I was proposing he write. Clifford eventually persuaded me to join him. I was now certainly committed to semigroups, if not previously committed, for some time to come. By the time I left Tulane to return to England, in 1958, we had drawn up plans for both volumes. We intended to include all significant work that had been done to date, that we knew of. We felt we could be flexible about volume 2 and include in it any new important results that had appeared by the time we got round to writing it. Clifford was given the final editorial decisions for volume 1, I for volume 2.

We spent a tremendous amount of time on notation and terminology. I still have - I am a bit of a squirrel - all the papers and letters that led up to the final books. I have not looked at them for 20 years. But I would not be surprised if say, about a tenth of our correspondence and discussions was about the choice of words. We succeeded in our aim of standardising terminology: for years afterwards writers just referred to Clifford and Preston for notation and terminology.

I hope I have dealt with what you expected me to deal with. Semigroups is stronger today than it has ever been and I look forward to being involved in its further growth.