Introduction

The lambda calculus began as an attempt by Alonzo Church to give a foundation for mathematics based on functions rather than sets. As such, it inherently is a theory of functions. While interest in the theory was confined to logicians for quite some time, more recently it has received extensive attention from the theoretical computer science community, which took it up because of its emphasis on the computational aspects of functions. The lambda calculus has been a focus of research in the semantics of programming languages in recent years, and this area has become a major contributor to the development of the theory.

Category theory is another theory of functions, but one with quite different beginnings. It began in an attempt to abstract away from the details of specific areas of mathematics in order to provide a uniform setting in which results common to many areas could be formulated and proved. Category theory also plays a fundamental role in theoretical computer science, precisely because it provides a setting that is broad and rich enough to accommodate the many requirements needed to give mathematical models for programming languages.

It is only natural then that these two theories of functions should have a close relationship. One only has to consult works such as Asperti and Longo [2] and the earlier Lambek and Scott [8] to see ample evidence of the richness of the relationship between category theory and the typed lambda calculus.

The theory of algebraic semigroups began formally in the 1920s in the work of Suschkewitsch (see, e.g., [S]), although it traces it roots to even earlier work. In the 1960s, Alfred H. Clifford and Gordon B. Preston wrote a two-volume work on the algebraic theory of semigroups [CP] which played a fundamental role in laying out the basics of the theory. That treatise remains the most authoritative work in the area to this day, and its authors, particularly Professor Clifford, are viewed as the founding fathers of the area. Due to recent advances in theoretical computer science as well as other areas of application, algebraic semigroup theory has assumed a role of increasing importance in recent years. Because of his long association with Tulane University, whose mathematics department emerged as a center for research largely during Professor Clifford's most active years, and because Tulane long has been viewed as a national and international center for research in the algebraic and topological theory of semigroups, the Mathematics Department decided to host a conference commemorating Professor Clifford's contributions to the area.

The Conference on the Theory of Semigroups and Its Applications took place in March, 1994, but rather than seeking to reminisce over Professor Clifford's long and distinguished career, the object of the Conference was to identify the state of the theory at that time, and to look forward by identifying those areas of research where semigroup theory can play an important rôle.