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 {text {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 Brandt semigroups, 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.