Two nonfinitely based additively idempotent semirings of order four
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911474351341568 |
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| author | Yue, Mengya Ren, Miaomiao Gao, Zidong |
| author_facet | Yue, Mengya Ren, Miaomiao Gao, Zidong |
| contents | We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{(4,545)}$ and $S_{(4,634)}$, whose additive reducts are chains, have no finite basis for their identities. Furthermore, we show that the interval $[\mathsf{V}(S_{(4,545)}),\mathsf{V}(S_{(4,634)})]$ in the lattice of semiring varieties contains \(2^{\aleph_0}\) distinct varieties. Consequently, the join of two finitely based additively idempotent semiring varieties is not necessarily finitely based. Moreover, we obtain the smallest example of a finitely based additively idempotent semiring $S$ whose extension $S^0$ (obtained by adjoining a new element) is nonfinitely based. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00015 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two nonfinitely based additively idempotent semirings of order four Yue, Mengya Ren, Miaomiao Gao, Zidong Rings and Algebras Group Theory We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{(4,545)}$ and $S_{(4,634)}$, whose additive reducts are chains, have no finite basis for their identities. Furthermore, we show that the interval $[\mathsf{V}(S_{(4,545)}),\mathsf{V}(S_{(4,634)})]$ in the lattice of semiring varieties contains \(2^{\aleph_0}\) distinct varieties. Consequently, the join of two finitely based additively idempotent semiring varieties is not necessarily finitely based. Moreover, we obtain the smallest example of a finitely based additively idempotent semiring $S$ whose extension $S^0$ (obtained by adjoining a new element) is nonfinitely based. |
| title | Two nonfinitely based additively idempotent semirings of order four |
| topic | Rings and Algebras Group Theory |
| url | https://arxiv.org/abs/2603.00015 |