Translational hulls of semigroups of endomorphisms of an algebra
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911848439218176 |
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| author | Gould, Victoria Grau, Ambroise Johnson, Marianne Kambites, Mark |
| author_facet | Gould, Victoria Grau, Ambroise Johnson, Marianne Kambites, Mark |
| contents | We consider the translational hull $Ω(I)$ of an arbitrary subsemigroup $I$ of an endomorphism monoid $\mathrm{End}(A)$ where $A$ is a universal algebra. We give conditions for every bi-translation of $I$ to be realised by transformations, or by endomorphisms, of $A$. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of $I$ and the idealiser of $I$ within $\mathrm{End}(A)$, which in the case where $I$ is an ideal is simply $\mathrm{End}(A)$. We describe the connection between these conditions and work of Petrich and Gluskin in the context of densely embedded ideals. Where the conditions fail, we develop a methodology to extract information concerning $Ω(I)$ from the translational hull $Ω(I/{\approx})$ of a quotient $I/{\approx}$ of $I$. We illustrate these concepts in detail in the cases where $A$ is: a free algebra; an independence algebra; a finite symmetric group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14005 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Translational hulls of semigroups of endomorphisms of an algebra Gould, Victoria Grau, Ambroise Johnson, Marianne Kambites, Mark Rings and Algebras 20M12, 20M25, 20M30 We consider the translational hull $Ω(I)$ of an arbitrary subsemigroup $I$ of an endomorphism monoid $\mathrm{End}(A)$ where $A$ is a universal algebra. We give conditions for every bi-translation of $I$ to be realised by transformations, or by endomorphisms, of $A$. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of $I$ and the idealiser of $I$ within $\mathrm{End}(A)$, which in the case where $I$ is an ideal is simply $\mathrm{End}(A)$. We describe the connection between these conditions and work of Petrich and Gluskin in the context of densely embedded ideals. Where the conditions fail, we develop a methodology to extract information concerning $Ω(I)$ from the translational hull $Ω(I/{\approx})$ of a quotient $I/{\approx}$ of $I$. We illustrate these concepts in detail in the cases where $A$ is: a free algebra; an independence algebra; a finite symmetric group. |
| title | Translational hulls of semigroups of endomorphisms of an algebra |
| topic | Rings and Algebras 20M12, 20M25, 20M30 |
| url | https://arxiv.org/abs/2404.14005 |