Profinite topologies

TL;DR

This chapter explores profinite topologies on semigroups, emphasizing their algebraic and topological properties, applications in language theory, and connections with symbolic dynamics, providing a comprehensive theoretical framework.

Contribution

It develops a general theory of profinite topologies within universal algebra, focusing on semigroups and their applications in solving membership problems and understanding their structure.

Findings

Profinite semigroups are linked to Boolean algebras of regular languages.

Solving systems of equations in pseudovarieties aids membership problem solutions.

Connections between large profinite semigroups and symbolic dynamics are established.

Abstract

Profinite semigroups are a generalization of finite semigroups that come about naturally when one is interested in considering free structures with respect to classes of finite semigroups. They also appear naturally through dualization of Boolean algebras of regular languages. The additional structure is given by a compact zero-dimensional topology. Profinite topologies may also be considered on arbitrary abstract semigroups by taking the initial topology for homomorphisms into finite semigroups. This text is the proposed chapter of the Handdbook of Automata Theory dedicated to these topics. The general theory is formulated in the setting of universal algebra because it is mostly independent of specific properties of semigroups and more general algebras naturally appear in this context. In the case of semigroups, particular attention is devoted to solvability of systems of equations with respect to a pseudovariety, which is relevant for solving membership problems for pseudovarieties. Focus is also given to relatively free profinite semigroups per se, specially "large" ones, stressing connections with symbolic dynamics that bring light to their structure.