... at that time group theory was perhaps a household word, and I saw an opportunity to introduce some group theory into mathematical logic.In fact Church was only one year older than Foster, who became his first Ph.D. student, and he had begun to supervise Foster in the same year that he was awarded his own doctorate. There is some confusion in the literature as to whether Church or Veblen supervised Foster, but he explained the precise position in an interview:-
... I officially was working with Church, but Veblen was more insistent on guiding me through than Church was.Foster submitted his Ph.D. thesis Formal Logic in Finite Terms to Princeton University in 1930 and the degree was awarded in the following year. He wrote a paper Formal logic in finite terms, based on his thesis, which was published in 1931 in the Annals of Mathematics.
Foster married Else Wagner in 1930 after submitting his thesis. They travelled to Germany where Foster undertook a year of postdoctoral study at Göttingen. It was a year in which he fell in love with the style of teaching and research undertaken there, and it would influence him throughout the rest of his career. The Great Depression began in 1929 while Foster was working for his doctorate and by 1932, after he finished his studies in Germany, one quarter of the workers in the United States were unemployed. However Foster had been fortunate in these extremely difficult times:-
... at Princeton at that time, Joel Hildebrandt was scouting for the department at Berkeley, which was not very strong in the beginning years. Hildebrandt went scouting all over the country. When he was at Princeton he interviewed me along with some others. ... Eisenhart gave me a strong recommendation to Hildebrandt, and sometime later I got an offer from Berkeley.On their return to the United States the Fosters travelled by car across the country to Berkeley where Alfred continued his studies and also undertook some teaching.
Griffith Evans was appointed to the University of California at Berkeley in 1933 and was given the explicit mandate to revitalise and improve the mathematics department and set up a programme for graduate studies. In the summer of 1934 Evans began his task at Berkeley as chairman of the mathematics department, and one of his first acts was to appoint Foster to the Faculty. Except for several periods of study leave, mainly in Freiburg and Tubingen, Foster continued to work at Berkeley until he retired in 1971. Alfred and Else Foster brought up their two sons and two daughters in Berkeley.
Foster, as a student of Church, naturally began his research career working in mathematical logic. However through a natural interest in Boolean algebra and Boolean rings, he moved more towards an interest in algebraic structures. For example in 1941 he published Natural systems : the structure of abstract monotone sequences. In this paper he defines a "natural system, with unit", to be an abstract system (N, ), where N is a denumerable set and a binary operation (product), such that the product of two elements of N is unique and in N, the associative and commutative laws hold, there is a single unit, there is at least one prime, and unique factorisation into primes holds. In 1944, together with his colleague Benjamin Bernstein, he published Symmetric approach to commutative rings, with duality theorem : Boolean duality as special case in the Duke Mathematical Journal. Also in 1944 Foster published Natural orderings which, in his own words, was written:-
... to study [the] arithmetic [of positive integers] so as to make multiplication ... basic [instead of addition] and only sparingly ... admit new operations ... until it is definable.His work on generalising Boolean rings culminated in his classic paper The theory of Boolean-like rings which he published in the Transactions of the American Mathematical Society in 1946. Dieudonné explains what Foster means by a Boolean-like ring:-
A "Boolean-like" ring is a commutative ring H with unit element such that a + a = 0 for all a in H and ab(a + b + ab) = ab for any two elements a, b of H. The main properties of such rings areFoster went on to define the concept of a primal algebra generalising a Boolean algebra within the theory of varieties of universal algebras. In 1953 showed that the variety generated by a primal algebra has the same essential structure as the variety of Boolean algebras. He continued devoting his efforts to the structure theory of algebras that are generalizations of Boolean algebras and, more than ten years down the line in 1966, he published Families of algebras with unique (sub-)direct factorization. Equational characterization of factorization in Mathematische Annalen. Another paper Automorphisms and functional completeness in universal algebras. I. General automorphisms, structure theory and characterization which appeared in the same journal in 1989:-
a4 = a2, for all a in H;
the nilpotent elements of H form an ideal N such that the product of two elements of N is 0;
the idempotent elements of H form a Boolean subring J of H and every element of H can be expressed in one and only one way as a sum of an idempotent and of a nilpotent element.
... is a continuation of a long series of articles by [Foster] and his students which investigates unique factorization in certain classes of abstract algebras. As [Foster] remarks, previous articles have all considered algebras with trivial automorphism groups; this paper begins investigation of algebras with larger groups of automorphisms.Leon Henkin, John Kelley and Alden Pixley have given an excellent sketch of Foster's character and interests:-
Alfred Foster, though somewhat formal and socially shy, will be remembered as warm-hearted, good-humoured, and unconditionally generous, by all who knew him. Several of his former students, in recalling him, have pointedly used the word "gentleman" in describing his character. His teaching style was rather old-fashioned, in a very good sense, and was probably influenced by his admiration for the universities of Germany, which developed during his visits there. His former students were often surprised and flattered that he remembered them years after classroom contact, and that he had a continuing interest in their lives. ... Music and politics were of particular interest to him, and in the latter field he held deep and morally grounded convictions. On the other hand, and consistent with his personality, he did not engage in highly visible political action. Along with mathematics, Foster took the current great issues of science and human culture very seriously indeed. An important key to his character was that he never took himself nearly so seriously.Foster underwent surgery in the spring of 1994. By this time he was close to 90 years old and he never fully recovered. His wishes were that he be cremated and his ashes scattered at sea as indeed they were on 5 January 1995.
Article by: J J O'Connor and E F Robertson