A. H. Clifford: The first sixty-five years

Pkoto by JED Dien~

Semigroup Forum, Vol. 7 (1974), 4-9. BIOGRAPHICAL

ARTICLES

A. H. CLIFFORD: THE FIRST SIXTY-FIVE YEARS

D. D. Miller

Two papers in Volume 5 of this Forum, in which structure theorems are given for certain classes of bisimple semigroups,

testify to the current scientific

activity of the mathematician who, it is acknowledged on all sides, has done more than anyone else to bring the theory of semigroups to its present healthy state. With a very few others, notably A. K. Suschkewitsch and D. Rees, he may be counted as a founder of the theory. It is fitting that, as we honor him on the occasion of his slxty-flfth birthday, we review briefly the part of his career already accomplished:

forty years of scientif-

ic research and teaching, twice interrupted by calls to serve his country. Alfred Hoblitzelle Clifford was born ii July 1908 in St. Louis, Missouri, but in his youth the family moved to California.

He went east for his collegiate

education and in 1929 was graduated from Yale University. An accomplished pianist, he tried his hand at composition while at Yale and has resumed his piano studies in recent years. 4 © 1974 by Springer-Verlag New York Inc.

MILLER

Returning

to the west for graduate study, Clifford

entered the California

Institute of Technology,

was a pupil of E. T. Bell and Morgan Ward. with Bell's advice and encouragement, published paper

[i] I.

he wrote his first

on Arithmetic

ry of Abstract Multiplication,

and Ideal Theo-

and once more crossed the

country to become a member of the Institute

five years. Neumann,

treatise

for

It was the same year in which Einstein, von

and Hermann Weyl joined the Institute, to be his assistant

and Weyl from 1936

It was in this period that Weyl wrote his [A,35] on the classical groups,

of which he comments [of expression] plishment

for Advanced

New Jersey, where he remained

chose the young algebraist to 1938.

In 1932,

In 1933 he was awarded the doct-

orate, with a dissertation

Study in Princeton,

where he

in the preface

"If at least the worst blunders

have been avoided,

this relative accom-

is to be ascribed solely to the devoted col-

laboration of my assistant, even more valuable mathematical

Dr. Alfred H. Clifford,

for me than the linguistic,

criticisms."

irreducible

[4] and [5], on

induced in an invariant

representation

were his

It was also under Weyl's in-

fluence that Clifford did his work, representations

and

subgroup by an

of the whole group.

In those years of economic depression,

the newly

formed Institute was not only a hotbed of mathematical

Numbers in square brackets

refer to the bibliography

at the end of the article by G. B. Preston that follows this one.

I am grateful

to Preston and to W. D.

Munn for comments on earlier drafts of this article. 5

MILLER

creativity but also a shelter in which some of the best young mathematicians

in the country were shielded from

the pressure for "instant publication" that bore on their contemporaries

in the universities.

Always an

exceptionally clear expositor, with a taste for doing things decently and in order, Clifford did not rush into print with his dissertation, abstract

but, having published an

[3], allowed his thought to mature for another

four years, and then published in 1938 a revised and expanded version [6], his second contribution to the field in which he has become the leading authority. In 1938 Clifford took up an instructorship at the Massachusetts

Institute of Technology,

and three years

later (very quickly for those days!) was promoted to an assistant professorship.

While at M.I.T. he first

turned his attention to partially ordered groups, conceiving of them as unifying the work of Krull [A,34] on the arithmetical theory of fields and that of Kantorovlch [A,3] on partially ordered linear spaces.

But

while he was writing up his results for publication he was shown the proofsheets of Lorenzen's work [A,5], which went considerably beyond his own. published

He therefore

[7] only the statements of his theorems and

some illuminating remarks on their relation to the work of Lorenzen, Krull, and Kantorovich. Clifford's next paper, the seminal "Semlgroups admitting relative inverses"

[9], was destined to become

a landmark in the theory of semigroups.

It appeared

only two months before the attack on Pearl Harbor, and was followed quickly by the important work on matrix

MILLER

representations

of completely simple semigroups

[i0],

which was its author's last work in pure mathematics until the end of the war. In the spring of 1942 Lieut.

(j.g.) Clifford was

ordered to active duty in Washington.

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 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 return-

ing to inactive duty as a Lieutenant Commander, he took up a post as associate professor at the Johns Hopkins University. The nine years at Hopkins saw a return to research, resulting in a sequence of basic papers in the theory of semigroups,

and the birth of the Cliffords'

second son,

Karl, but they were interrupted in 1950 by a second call to active duty in the Naval Reserve when South Korea was invaded.

Returning to inactive duty as Commander in

1952, Clifford remained at Hopkins for three years before accepting the headship of the department of mathematics at Sophie Newcomb College of the Tulane University of Louisiana.

On sabbatical leave in 1961-62,

just after completion of the first volume [29] of the treatise he wrote in collaboration with G. B. Preston, Clifford and his family spent the year in Paris, where he pursued his researches and participated in Dubreil's seminar.

In the autumn and early winter

the northern hemisphere!)

(as viewed from

of 1972 he taught at Monash

MILLER

University in Australia.

Apart from these excursions,

and from lecturing at many universities and at international conferences, he has remained at Tulane since 1955, spending the academic year in New Orleans and the summer on Cape Cod.

A founding editor of this FORUM,

and a former editor of the Transactions of the American Mathematical Society, he has recently relinquished his administrative duties in order to divide his time between research and editorial work. An able and sympathetic teacher, Clifford has of course attracted students.

Writing their doctoral dis-

sertations under his direction at Johns Hopkins were S. G. Bourne and R. P. Rich, both in 1950, and, at Tulane, N. Kimura (1957), C. R. Storey, Jr. (1959), E. J. Tully, Jr. (1960), V. R. Hancock (1960), W. E. Clark (1964), F. D. Pedersen (1967), and A. J. Hulin

(1971). Even the briefest summary of Clifford's career up to this time must include more than the passing reference made above to the two-volume work The Algebraic Theory o_~f semigroups, written in collaboration with G. B. Preston, of which the first volume [29] appeared at the end of 1961 and the second volume [33] in 1967. Only two books on the subject had appeared earlier, both in Russian: Suschkewltsch's Theory of Generalized Groups [A,37] in 1937 and Lyapin's Semigroups [A,36] in 1960. An English translation of the former has been made by Hancock, but it is now of primarily historical interest and has not been published; Lyapln's book appeared in English translation [A,36] in 1963.

MILLER

The Clifford-Preston work, which goes beyond its predecessors

in breadth and depth, is both a treatise

and a textbook.

As a treatise it represents an immense

labor of collection,

selection,

and systematic organiza-

tion of theorems scattered through scholarly journals in many languages.

Yet it makes no claim to complete-

ness, and the bibliography

is deliberately limited to

the 264 items cited in the text.

Indeed, one of the

great merits of the work is the complex of judicious choices of material to be included.

As a textbook,

with its nearly five hundred exercises, with exemplary precision,

it is written

and appeals to students by

its carefully steered course between the boredom induced by too much detail and the frustration engendered by too little. 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.