PROFINITE METHODS IN SEMIGROUP THEORY

Abstract

Many recent results in finite semigroup theory make use of profinite methods, that is, they rely on the study of certain infinite, compact semigroups which arise as projective limits of finite semigroups. These ideas were introduced in semigroup theory in the 1980s, first to describe pseudovarieties in terms of so-called pseudo-identities: this is Reiterman's theorem, which can be viewed as the (much more complex) finite algebra analogue of Birkhoff's variety theorem. Soon, these methods were used in conjunction with virtually all the other approaches of finite semigroups, notably to study the decidability of product pseudovarieties. This paper surveys the contribution of profinite methods and the way they enriched and modified finite semigroup theory.