Died: 14 April 2015 in Oxford, England

**Gordon Preston** was brought up in Carlisle, in the north west of England. He won an open scholarship to Magdalen College Oxford in 1943 but although the war was drawing to close by this time, he could not complete his studies at Oxford but undertook war service. In common with other bright mathematics undergraduates, Preston went to Bletchley Park, Buckinghamshire, a government research centre north of London. The work there was all aimed at cracking German codes and Preston worked there with Alan Turing [3]:-

After spending one year at Oxford in1943/4, 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 about250WRENS, 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.

In 1946 Preston returned to Oxford to complete his undergraduate studies and he graduated with first class honours in mathematics in 1948 [3]:-

I took finals at Oxford in1948with a syllabus that I and most of my contemporaries regarded as impossibly out of date. We could use final examination papers set in1910to give us practice in answering the1948papers. 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. 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.

It was through the connections that he had made in Bletchley Park that Preston first became interested in semigroups [3]:-

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 from1946to1948- 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 Proceeding of the Cambridge Philosophical Society,1940, together with the small technical note that followed it, "Note on semi-groups", ibid.,1941.

Preston now continued to undertake research at Oxford on a part-time basis, first with Whitehead then with E C (Edward) Thomson as his advisor. He earned his living as an assistant mathematics master at Westminster School in London where he was appointed in 1948. He left this post in 1950 to take up a position teaching at the Royal Military College of Science at Shrivenham, which is close to Swindon and about 35 km south west of Oxford. He continued working towards his D.Phil. on a part-time basis and was awarded the degree in 1954 for his thesis *Some Problems in the Theory of Ideals*. He writes in [3]:-

[

In my thesis]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 his1947paper. My attempt was not successful: but I proved some results about the semigroups I had defined. My D. Phil. examiners were David Rees and 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.

Despite continuing to teach at the Royal Military College of Science, he soon had a number of papers in print. In 1954 alone, five of his papers were published, all in the *Journal* of the London Mathematical Society: *The arithmetic of a lattice of sub-algebras of a general algebra; Factorization of ideals in general algebras; Inverse semi-groups; Inverse semi-groups with minimal right ideals*; and *Representations of inverse semi-groups*.

Howie writes [2]:-

... when he wrote his three hugely influential papers laying the foundations of inverse semigroup theory it is not at all surprising that he was completely unaware of the closely similar work of Vagner in Russia. He did, however, quickly realise how important it was to make contact with Central and Eastern Europe, and had some retrospective amusing difficulties with the authorities at the Military College, to whom even harmless semigroupers in those countries were 'the enemy'.

Preston was released from his teaching duties at the Royal Military College of Science so that he could spend two years in the United States. He writes in [4]:-

I spent the years1956 - 58in New Orleans at the invitation of Al Clifford. ... Clifford and I drew up our plans and completed first drafts of several chapters of our two-volume treatise on semigroups during these two years. It proved the major mathematical task I have undertaken. The collaboration was a stimulating and happy one.

The first volume of Clifford and Preston's *The algebraic theory of semigroups* was published in 1961. He had returned to his teaching duties at Shrivenham after his time in New Orleans, but lived in Oxford where he participated in the mathematical life. Howie writes [3]:-

I met Gordon Preston first in1959, when I arrived in Oxford as a graduate student working under the supervision of Graham Higman. Gordon lived in Oxford - Shrivenham, his place of work, is just30km away - and he was a regular attender at the Higman algebra seminar. For the final year of my study at Oxford, when Higman was on sabbatical leave in Chicago, Preston was my supervisor, and it is a pleasure now to be able to pay public tribute to the quality of the help and encouragement I received from him during that period.

We mentioned above that the first volume of Clifford and Preston, *The algebraic theory of semigroups*, was published in 1961. Schwarz, in a review of the volume, writes:-

This is a well-arranged book furnishing a reasonably comprehensive account of a new field developed by a large number of algebraists very rapidly in the last twenty years. The book under review is the first volume; the second one will follow in the near future. After the book of E S Lyapin, 'Semigroups'(Russian) (1960)this is the second book on the subject in the last eighteen months. Before, there has been no systematic treatment on semigroups at all, with the exception of the book of Suchkewitsch, 'The theory of generalized groups'(1937)containing naturally a very limited number of results. The book under review and that of Lyapin were written at about the same time. As to the results they overlap, of course, at many places, but in the presentation and the emphasis on problems there are frequent differences. ... This excellent and clearly written book will be very useful both for study and orientation and for reference in further work.

Volume II of Clifford and Preston, *The algebraic theory of semigroups* was published in 1967. Schwarz again wrote a review:-

Volume II has been eagerly awaited by those who are working in semigroup theory and related subjects(e.g., automata theory). This volume ... deals with additional branches of the theory to which there was at most passing reference in Volume I. The manuscript was finished in1963, but notes on developments up to1967are included. ... The book as a whole is an excellent achievement. It is clearly written, contains all known main results to date, and will undoubtedly remain a source book for many years for all workers in this field.

By the time Volume II was published, Preston was professor at Monash University in Australia. He had been offered the chair of mathematics in this new university just outside Melbourne in 1963 and emigrated to Australia in the year to take up the position. He published papers over the next years such as *Chains of congruences on a completely 0-simple semigroup* (1965), *Matrix representations of inverse semigroups* (1969) and *Free inverse semigroups* (1973). Howie writes in [2]:-

Gordon Preston has an original and penetrating mind, and his list of publications, while not long by today's ridiculously inflated standards, contains many items of real quality, items that have greatly influenced the development of the subject and have stimulated other to activity.

**Article by:** *J J O'Connor* and *E F Robertson*

**April 2015**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Preston.html]