A characterisation of semigroups with only countably many subdirect products with ℤ

2024/04/28 by Ashley Clayton, Clayton, Ashley, Catherine Reilly +3
Computer Science · Mathematics · #20M05 #20M10 #20M17 #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Rings and Algebras (math.RA) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2404.18122

openalex publication_date 2024/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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