A characterisation of semigroups with only countably many subdirect products with $\mathbb{Z}$
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929329994203136 |
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| author | Clayton, Ashley Reilly, Catherine Ruškuc, Nik |
| author_facet | Clayton, Ashley Reilly, Catherine Ruškuc, Nik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterisation of semigroups with only countably many subdirect products with $\mathbb{Z}$ Clayton, Ashley Reilly, Catherine Ruškuc, Nik Group Theory Rings and Algebras 20M10, 20M05, 20M17 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. |
| title | A characterisation of semigroups with only countably many subdirect products with $\mathbb{Z}$ |
| topic | Group Theory Rings and Algebras 20M10, 20M05, 20M17 |
| url | https://arxiv.org/abs/2404.18122 |