Semigroup Congruences and Subsemigroups of the Direct Square
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911373148028928 |
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| author | Barber, Callum Ruškuc, Nik |
| author_facet | Barber, Callum Ruškuc, Nik |
| contents | We investigate semigroups $S$ which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on $S$. We call such $S$ a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups -- including periodic, stable, inverse and several well-known types of simple semigroups -- the only DSC members are groups. However, it turns out that there exist non-group DSC semigroups, which we obtain utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semigroup Congruences and Subsemigroups of the Direct Square Barber, Callum Ruškuc, Nik Rings and Algebras Group Theory 20M10, 08A30, 20M12 We investigate semigroups $S$ which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on $S$. We call such $S$ a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups -- including periodic, stable, inverse and several well-known types of simple semigroups -- the only DSC members are groups. However, it turns out that there exist non-group DSC semigroups, which we obtain utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple. |
| title | Semigroup Congruences and Subsemigroups of the Direct Square |
| topic | Rings and Algebras Group Theory 20M10, 08A30, 20M12 |
| url | https://arxiv.org/abs/2504.02428 |