Profinite Methods in Semigroup Theory

A highlight of the Conference held at the University of Nebraska-Lincoln in May 2000 was the proof of the decidability of the complexity of nite semigroups and automata, presented by Rhodes 78]. This problem had been open for more than thirty years and had inspired many active branches of semigroup theory. As is usually the case with old and diicult problems, the solution for the original question is given by a sequence of results, often very interesting in their own right, and the paper nally closing the problem actually states and proves a more speciic result, which completes the whole construction. A number of techniques and notions have come to play a role in the solution to this problem, which did not exist in the 1960s when the problem was stated, notably the notion of pseudovariety of nite semigroups in the 1970s, the introduction of category-theoretic tools in semigroup theory in the 1980s, and the introduction of proonite methods. This survey concentrates on the latter topic, which is especially interesting in the context outlined above since proonite semigroup theory provides the framework in which to formulate the precise questions which Rhodes answered, and makes it possible to understand the importance of the recently submitted paper by McCammond 55], another essential ingredient in the proof of the decidability of complexity. Our paper aims to speak to both specialists and non-specialists. For the latter it ooers an outlook on semigroup theory which will allow them to appreciate the way in which the study of relatively free proonite semigroups has enriched and modiied nite semigroup theory, and has provided the tools used to solve the complexity problem. The end of the paper, somewhat more technical, is intended to help the specialists of pseudovariety theory who were not directly involved in the most recent results in that direction, to get a general view of the present situation. The part of semigroup theory we are concerned with here is the theory of nite semigroups, and especially the classiication of nite semigroups in pseudovarieties. The introduction of proonite methods in the mainstream of semigroup theory is rather recent. Reiterman's seminal theorem was published in 1982, and Almeida's work developing the theory of implicit operations in the speciic framework of semigroup theory started in the mid 1980s. Important results started owing rapidly, and there are at present a number of surveys and basic references, some dated 3, …