A characterisation of semigroups with only countably many subdirect products with $\mathbb{Z}$

TL;DR

This paper characterizes when the direct product of the integer group with a finite semigroup has only countably many subdirect products or subsemigroups, linking these properties to regularity conditions of the semigroup.

Contribution

It establishes a precise criterion connecting the regularity of a finite semigroup to the countability of subdirect products and subsemigroups in its product with the integers.

Findings

Countably many subdirect products iff S is regular

Countably many subsemigroups iff S is completely regular

Characterization of semigroup properties via product substructure countability

Abstract

Let $\mathbb{Z}$ be the additive (semi)group of integers. We prove that for a finite semigroup $S$ the direct product $\mathbb{Z}\times S$ contains only countably many subdirect products (up to isomorphism) if and only if $S$ is regular. As a corollary we show that $\mathbb{Z}\times S$ has only countably many subsemigroups (up to isomorphism) if and only if $S$ is completely regular.