Finite semigroups as categories, ordered semigroups or compact semigroups

Summary

Abstract

Algebraic semigroup theory has made great strides in the recent years. It is a remarquable fact that these new results did require the introduction of some auxiliary structures. The aim of this article is to emphasize the role of three of these tools: topology, partial order and categories. We survey in particular Reiterman's theorem on identities for pseudovarieties, an extension of Eilenberg's variety theorem to ordered semigroups and some decomposition theorems on semigroups that require the notion of a derived category.

I met Professor A. H. Clifford just once, in 1984. It was my first semigroup conference in the USA and I felt very honored when he told me very kindly how he enjoyed the way automata theory and semigroups mixed together. Ten years later, it was again a great honor for me to participate to the Tulane conference in his memory.

Introduction

Algebraic semigroup theory has made great strides in the recent years. It is a remarkable fact that these new results did require the introduction of some auxiliary structures. In this article, I would like to emphasize the role of three of these tools: topology, partial orders and categories. The three parts are relatively independent.

Identities and Topology

The results presented in this section are a good illustration of the following quotation of Marshall Stone [34]: ‘A cardinal principle of modern mathematical research may be stated as a maxim: “One must always topologize”.’ Varieties of finite semigroups are a good example where Stone's principle was applied successfully.