The contributions of Alfred Tarski to general algebra

Abstract

A distinctive feature of modern mathematics is the interaction between its various branches and the blurring of the boundaries between different areas. This is strikingly illustrated in the work of Alfred Tarski. He was a logician first and an algebraist second. His contributions to algebra can be divided into three (ill-defined and overlapping) categories, general algebra, the study of various algebraic structures arising from problems outside algebra, mostly in logic and set theory, and the use of concepts and techniques from logic in the study of algebraic structures. Even more roughly, these three categories could be labeled as pure algebra, applications of algebra to logic, and applications of logic to algebra.Before Tarski came to the United States in 1939, he had written a series of papers on both the axiomatic and the structural aspects of Boolean algebras, and his inclination to algebraize mathematical problems is well illustrated by his paper [38g], Algebraische Fassung des Massproblems. Many of his later investigations of various types of algebraic structures are inspired by work done in this earlier period. However, beginning around 1940 there is a much greater emphasis on the study of algebra in its various aspects.The paper [41], On the calculus of relations, is a landmark event in this respect. The object here was to find an axiomatic basis for the arithmetic of binary relations. The axioms that he chose are simple and natural (see Monk [1986]).