Factoriality and the Pin-Reutenauer procedure

TL;DR

This paper investigates the algebraic structure of finite semigroups with specific signatures and demonstrates that certain closure properties hold under these signatures, extending the Pin-Reutenauer procedure to a broader context.

Contribution

It establishes conditions under which free algebras are closed under factors and generalizes the Pin-Reutenauer procedure to finite semigroups.

Findings

Finitely generated free algebras are closed under factors in certain pseudovarieties.

The Pin-Reutenauer procedure applies to the pseudovariety of all finite semigroups.

A pseudovariety is full if and only if it satisfies the closure property.

Abstract

We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.