${\aleph }_0$-categoricity of semigroups

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.