The finite basis problem for endomorphism semirings of finite chains
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911641207046144 |
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| author | Gusev, Sergey V. Volkov, Mikhail V. |
| author_facet | Gusev, Sergey V. Volkov, Mikhail V. |
| contents | For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no finite identity basis. This, combined with earlier results by Dolinka (The finite basis problem for endomorphism semirings of finite semilattices with zero, Algebra Universalis 61, 441-448 (2009)), gives a complete solution to the finite basis problem for semirings of the form $\mathrm{End}(\mathcal{S})$ where $\mathcal{S}$ is a finite chain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01770 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The finite basis problem for endomorphism semirings of finite chains Gusev, Sergey V. Volkov, Mikhail V. Group Theory 16Y60, 08B05, 20M18 For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no finite identity basis. This, combined with earlier results by Dolinka (The finite basis problem for endomorphism semirings of finite semilattices with zero, Algebra Universalis 61, 441-448 (2009)), gives a complete solution to the finite basis problem for semirings of the form $\mathrm{End}(\mathcal{S})$ where $\mathcal{S}$ is a finite chain. |
| title | The finite basis problem for endomorphism semirings of finite chains |
| topic | Group Theory 16Y60, 08B05, 20M18 |
| url | https://arxiv.org/abs/2312.01770 |