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arXiv · 2404.18122

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

Ashley Clayton·Catherine Reilly·Nik Ruškuc
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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.

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Ashley Clayton, Catherine Reilly, Nik Ruškuc. 2024-04-28. A characterisation of semigroups with only countably many subdirect products with $\mathbb{Z}$. https://arxiv.org/abs/2404.18122

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