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Preprint Number 1385

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1385. Victoria Gould, Thomas Quinn-Gregson
ℵ_0-categoricity of semigroups

Submission date: 15 February 2018


In this paper we initiate the study of ℵ_0-categorical semigroups, where a countable semigroup is ℵ_0-categorical if it is defined up to isomorphism by its first order theory. We show that ℵ_0-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. Conversely, we consider when ℵ_0-categoricity is implied by the ℵ_0-categoricity of the substructures.
We examine the relationship between ℵ_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 ℵ_0-categoricity of an E-unitary inverse semigroup with finite semilattice of idempotents in terms of that of the maximal group homomorphic image.

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Full text arXiv 1802.05703: pdf, ps.

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