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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2410.00146 |
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| _version_ | 1866916417160347648 |
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| author | Faul, Peter F. Janelideze, Zurab Joubert, Gideo |
| author_facet | Faul, Peter F. Janelideze, Zurab Joubert, Gideo |
| contents | Cayley's theorem tells us that all groups $\mathbf{G}$ occur as subgroups of the group of automorphisms over some set $X$. In this paper we consider a `sort-of' converse to this question: given a set $X$ and some transformation group $\mathbf{S}$ over $X$, what are the possible group structures on $X$ that result in groups represented by $\mathbf{S}$? We solve this problem in the more general setting of faithful semigroups and observe that the solutions to this problem, which we term unrepresentations, have the structure of a heap. We study this phenomenon in depth and then move onto looking at particular classes of semigroups namely monoids, groups, inverse semigroups and Clifford semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Reverse Representation Problem Faul, Peter F. Janelideze, Zurab Joubert, Gideo Group Theory Rings and Algebras 20M30 Cayley's theorem tells us that all groups $\mathbf{G}$ occur as subgroups of the group of automorphisms over some set $X$. In this paper we consider a `sort-of' converse to this question: given a set $X$ and some transformation group $\mathbf{S}$ over $X$, what are the possible group structures on $X$ that result in groups represented by $\mathbf{S}$? We solve this problem in the more general setting of faithful semigroups and observe that the solutions to this problem, which we term unrepresentations, have the structure of a heap. We study this phenomenon in depth and then move onto looking at particular classes of semigroups namely monoids, groups, inverse semigroups and Clifford semigroups. |
| title | The Reverse Representation Problem |
| topic | Group Theory Rings and Algebras 20M30 |
| url | https://arxiv.org/abs/2410.00146 |