Locally countable pseudovarieties

TL;DR

This paper studies the class of profinite semigroups where all finitely generated closed subsemigroups are countable, exploring their properties, operations preserving this countability, and their relation to algebraic expressions and specific pseudovarieties.

Contribution

It introduces the concept of locally countable pseudovarieties, analyzes operations affecting local countability, and characterizes when certain pseudovarieties are locally countable.

Findings

Operations like local finiteness do not preserve local countability.

Countability of elements relates to expressibility via generators and idempotent powers.

The pseudovariety generated by finite ordered monoids with $1 \,\le\, x^n$ is locally countable only when n=1.

Abstract

The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressable in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J-classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality $1\le x^n$ is locally countable if and only if $n=1$.