$\aleph_0$-categoricity of semigroups

TL;DR

This paper introduces the concept of $eth_0$-categoricity in semigroups, exploring how it applies to substructures and various constructions, and characterizes it for certain inverse semigroups.

Contribution

It initiates the study of $eth_0$-categoricity in semigroups, analyzing its transfer to substructures and behavior under common semigroup constructions.

Findings

$eth_0$-categoricity transfers to maximal subgroups and principal factors.

Characterization of $eth_0$-categoricity for $E$-unitary inverse semigroups.

Analysis of $eth_0$-categoricity under direct sums, unions, and semidirect products.

Abstract

In this paper we initiate the study of $\aleph_0$-categorical semigroups, where a countable semigroup $S$ is $\aleph_0$-categorical if, for any natural number $n$, the action of its group of automorphisms Aut $S$ on $S^n$ has only finitely many orbits. We show that $\aleph_0$-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between $\aleph_0$-categoricity and a number of semigroup and monoid constructions, namely direct sums, 0-direct unions, semidirect products and $\mathcal{P}$-semigroups. As a corollary, we determine the $\aleph_0$-categoricity of an $E$-unitary inverse semigroup with finite semilattice of idempotents in terms of that of the maximal group homomorphic image.