Pseudovarieties of semigroups

TL;DR

The paper surveys the theory of pseudovarieties of finite semigroups, emphasizing their role as algebraic invariants linked to regular languages via Eilenberg's correspondence and their profinite (Stone) duals. It outlines two main operator families on pseudovarieties (constructive vs generator-based) and surveys key issues such as locality, notable equations, irreducibility, and central problems like Krohn–Rhodes complexity and dot-depth. A substantial portion introduces the tameness framework, detailing how word problems, equation solvability, and topological separability combine to yield decidability results for semidirect and Mal’cev products, with extensive results across many pseudovarieties. The latter sections discuss the fine structure of relatively free profinite semigroups, including their connections to shift spaces and Schützenberger groups, and conclude with open problems and directions, highlighting the interplay between algebra, topology, and formal languages.

Abstract

The most developed aspect of the theory of finite semigroups is their classification in pseudovarieties. The main motivation for investigating such entities comes from their connection with the classification of regular languages via Eilenberg's correspondence. This connection prompted the study of various natural operators on pseudovarieties and led to several important questions, both algebraic and algorithmic. The most important of these questions is decidability: given a finite semigroup is there an algorithm that tests whether it belongs to the pseudovariety? Since the most relevant operators on pseudovarieties do not preserve decidability, one often seeks to establish stronger properties. A key role is played by relatively free profinite semigroups, which is the counterpart of free algebras in universal algebra. The purpose of this paper is to give a brief survey of the state of the art, highlighting some of the main developments and problems.