The finite basis problem for additively idempotent semirings that relate to S_7
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916592158244864 |
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| author | Gao, Zidong Jackson, Marcel Ren, Miaomiao Zhao, Xianzhong |
| author_facet | Gao, Zidong Jackson, Marcel Ren, Miaomiao Zhao, Xianzhong |
| contents | The $3$-element additively idempotent semiring $S_7$ is a nonnitely based algebra of the smallest possible order. In this paper we study the nite basis problem for some additively idempotent semirings that relate to $S_7$. We present a su cient condition under which an additively idempotent semiring variety is nonnitely based and as applications, show that some additively idempotent semiring varieties that contain $S_7$ are also nonnitely based. We then consider the subdirectly irreducible members of the variety $\mathsf{V}(S_7)$ generated by $S_7$. We show that $\mathsf{V}(S_7)$ contains exactly $6$ finitely based subvarieties, all of which sit at the base of the subvariety lattice, then invoke results from the homomorphism theory of Kneser graphs to verify that $\mathsf{V}(S_7)$ contains a continuum of subvarieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_19049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The finite basis problem for additively idempotent semirings that relate to S_7 Gao, Zidong Jackson, Marcel Ren, Miaomiao Zhao, Xianzhong Group Theory Combinatorics 16Y60, 03C05, 08B15 The $3$-element additively idempotent semiring $S_7$ is a nonnitely based algebra of the smallest possible order. In this paper we study the nite basis problem for some additively idempotent semirings that relate to $S_7$. We present a su cient condition under which an additively idempotent semiring variety is nonnitely based and as applications, show that some additively idempotent semiring varieties that contain $S_7$ are also nonnitely based. We then consider the subdirectly irreducible members of the variety $\mathsf{V}(S_7)$ generated by $S_7$. We show that $\mathsf{V}(S_7)$ contains exactly $6$ finitely based subvarieties, all of which sit at the base of the subvariety lattice, then invoke results from the homomorphism theory of Kneser graphs to verify that $\mathsf{V}(S_7)$ contains a continuum of subvarieties. |
| title | The finite basis problem for additively idempotent semirings that relate to S_7 |
| topic | Group Theory Combinatorics 16Y60, 03C05, 08B15 |
| url | https://arxiv.org/abs/2501.19049 |