₀-categoricity of semigroups

Original · EN

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