# Alfred Hoblitzelle Clifford

### Quick Info

Born
11 July 1908
St Louis, Missouri, USA
Died
27 December 1992
New Orleans, Louisiana, USA

Summary
Alfred H Clifford was an American mathematician who worked in group theory and semigroup theory.

### Biography

Alfred Hoblitzelle Clifford, known as Al Clifford to fellow mathematicians, was born in St Louis, Missouri, but was brought up in California where his family moved when he was young. He attended Yale University, graduating in 1929 with a degree in mathematics but [4]:-
An accomplished pianist, he tried his hand at composition while at Yale ...
Clifford then went to the California Institute of Technology where he undertook research for a doctorate. His advisors were E T Bell and Morgan Ward. It was with Bell's advice that he wrote his first paper A system arising from a weakened set of group postulates in 1932. It was published in the Annals of Mathematics in the following year. Preston writes that it [6]:-
... must have appeared at the time to its author as an incidental piece of work, discussing a possible ambiguity in the formulation of the axioms for groups. In fact it turned out to be a remarkably fruitful start for the structure theory of semigroups.
In 1933 Clifford was awarded his doctorate for his dissertation entitled Arithmetic of Ova. In this work he considered the arithmetic and ideal theory of abstract multiplication. He later extended the work of his thesis and published the two papers Arithmetic and ideal theory of abstract multiplication (1934) and Arithmetic and ideal theory of commutative semigroups (1938). The first is essentially an abstract of his thesis while the second is an extended version containing deeper results which have arisen during the intervening years. After the award of his doctorate, Clifford became a member of the Institute for Advanced Study in Princeton. He remained there for five year and during this period, from 1936 to 1938, he was Weyl's assistant. This was the time when Weyl was writing The classical groups and he wrote in the Preface:-
If at least the worst blunders of expression have been avoided, this relative accomplishment is to be ascribed solely to the devoted collaboration of my assistant, Dr Alfred H Clifford, and even more valuable to me than the linguistic, were his mathematical criticisms.
Weyl's influence is clearly seen in Clifford's two papers in 1937 Representations induced in an invariant subgroup. In these he considered the representation induced on a normal subgroup by an irreducible representation of the group. In 1938 he was appointed as an Instructor in Mathematics at the Massachusetts Institute of Technology, being promoted to Assistant Professor in 1941. Rhodes writes [7]:-
It was during the writing of paper 'Semigroups admitting relative inverses' in 1941, after moving to M.I.T. as an assistant professor, a paper famous in the semigroup community about union of groups semigroups, that Clifford learned about Rees's Theorem determining the structure of completely 0-simple semigroups, generalizing the Wedderburn theory of rings. Rees's theory generalized much earlier results of Suschkiewitsch unknown to Clifford until 1941. The impact was profound, first because [Clifford's first paper] was a special case (in fact of the Suschkiewitsch paper), second, its application to paper 'Semigroups admitting relative inverses' in hand, where Clifford proved S is a union of groups if and only if S is a semilattice of completely simple semigroups (to which Rees structure theorem applies), and finally because of its intrinsic beauty and importance.
In the spring of 1942 he was called up for active duty to serve his country in World War II [4]:-
In the same year he married Alice Colt, an accomplished linguist who later served in the office of Strategic Services. Their first son, Harry, who was to die tragically in a motor car accident at the age of sixteen, was born in 1943. Clifford served with distinction for four years, in the European theatre and again in Washington. Upon returning to inactive duty as a Lieutenant Commander, he took up a post as an associate professor at the Johns Hopkins University.
After his period of active duty had ended, Clifford was appointed as an associate professor at Johns Hopkins University. Of course he had been unable to undertake mathematical research while on active duty and he published no papers between 1942 and 1948. Once he had returned to an academic life at Johns Hopkins he began again to produce important papers on semigroups. However the Korean War saw Clifford return to active duty in 1950 and he served until 1952 when he returned to Johns Hopkins. Rhodes writes [7]:-
... it was at this time that Clifford read J A Green's paper on the structure of semigroups introducing the J, L, R and D relations. The influence of this paper on Clifford was immense and quite profound. In the beginning he found it difficult to comprehend, especially the D relation. Clifford's 1953 paper 'A class of d-simple semigroups' on d-simple semigroups was his first reaction to Green and was really about inverse d-simple (one D class, and remember Green introduced D) semigroups. Preston was just in the stage of formulating modern inverse semigroups and the idea of inverse semigroups was "in the air".
In 1955 Clifford left Johns Hopkins and moved to New Orleans when he was appointed as Head of Mathematics at Sophie Newcomb College of Tulane University. He remained at Tulane for the rest of his career. Preston writes [5]:-
I spent the years 1956 - 58 in 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, 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 in 1963, but notes on developments up to 1967 are 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.
These two volumes have had an enormous impact on the development of semigroup theory. Miller writes in 1974 [4]:-
Clear in exposition, broad and deep in its coverage of the field, the book has had, and continues to have, a profound influence on the development of the theory of semigroups.
Rhodes describes Clifford's research following the publication of the second volume of 'Clifford and Preston' in [7]:-
By the very late 60's Clifford became interested in $B_{n}$, the semigroup of all binary relations on a finite set with $n$ elements, and reproved the Montague-Plemmons-Schein result that any finite group G was a maximal subgroup of $B_{n}$ for some $n$. This appeared in the first volume of Semigroup Forum, a mathematics journal which Clifford founded with Hoffman and Mostert in 1970 (it continues to be published by Springer-Verlag). ... he also continued his attack on the structure of bisimple-inverse semigroups (with Reilly) and bisimple orthodox semigroups as ordered pairs. In 1974, at the International Congress of Mathematicians in Vancouver, Clifford spoke on orthodox semigroups which are the union of groups.

In the mid 70's Clifford became very excited by the work of Nambooripad on the structure of regular semigroups in terms of their idempotent ordering and "sandwich matrices" and wrote several expository papers on Nambooripad structure theorem for regular semigroups. Clifford's final research, published in 1979, was influenced by universal algebra and concerned determining the structure of the free complexity regular (aka a "union of groups") semigroup on a set.
Preston sums up Clifford's contribution [6]:-
He has produced significant results in group theory and semigroup theory. Many of his results are of fundamental importance and, especially in semigroup theory, he initiated many techniques and approaches that are peculiar to, and are now part and parcel of, the theory.
After he retired, Clifford left New Orleans and returned to live in California. He enjoyed playing bridge, but also decided that he wanted to teach himself quantum mechanics. In early December 1992 Clifford suffered a stroke. He died later that month. Tulane University had given Clifford a Doctor of Science honoris causa in 1982. After his death they paid tribute to him saying that he:-
... gave to this institution to the fullest as a scholar, as a teacher, and as a benefactor.

### References (show)

1. Alfred Hoblitzelle Clifford : July 11, 1908-December 27, 1992, Semigroup Forum 46 (2) (1993), 137.
2. K H Hofmann and J D Lawson, Linearly ordered semigroups: historical origins and A H Clifford's influence, Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ. Press, Cambridge, 1996), 15-39.
3. D D Miller, Reminiscences of a friendship, Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ. Press, Cambridge, 1996), 1-2.
4. D D Miller, A H Clifford : the first sixty-five years, Collection of articles dedicated to Alfred Hoblitzelle Clifford on the occasion of his 65th birthday and to Alexander Doniphan Wallace on the occasion of his 68th birthday, Semigroup Forum 7 (1-4) (1974), 4-9.
5. G B Preston, A H Clifford's work on unions of groups, Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ. Press, Cambridge, 1996), 5-14.
6. G B Preston, A H Clifford : an appreciation of his work on the occasion of his sixty-fifth birthday, Semigroup Forum 7 (1-4) (1974), 32-57.
7. J Rhodes, The relationship of Al Clifford's work to the current theory of semigroups, Semigroup theory and its applications, New Orleans, LA, 1994 (Cambridge Univ. Press, Cambridge, 1996), 43-51.