Every additively idempotent semiring satisfying $xy\approx xz$ is finitely based
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914049194721280 |
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| author | Yue, Mengya Ren, Miaomiao |
| author_facet | Yue, Mengya Ren, Miaomiao |
| contents | We study the finite basis problem for additively idempotent semirings satisfying the identity $xy \approx xz$. Let $\mathbf{R}$ denote the variety of all such semirings. Yue et al. (2025, Algebra Universalis, DOI:10.1007/s00012-025-00908-5) established that $\mathbf{R}$ is finitely generated. In this paper, we show that the subvariety lattice of $\mathbf{R}$ forms a distributive lattice of order $10$. As a consequence, the variety $\mathbf{R}$ is a Cross variety, and every member of $\mathbf{R}$ is finitely based. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16711 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Every additively idempotent semiring satisfying $xy\approx xz$ is finitely based Yue, Mengya Ren, Miaomiao Group Theory We study the finite basis problem for additively idempotent semirings satisfying the identity $xy \approx xz$. Let $\mathbf{R}$ denote the variety of all such semirings. Yue et al. (2025, Algebra Universalis, DOI:10.1007/s00012-025-00908-5) established that $\mathbf{R}$ is finitely generated. In this paper, we show that the subvariety lattice of $\mathbf{R}$ forms a distributive lattice of order $10$. As a consequence, the variety $\mathbf{R}$ is a Cross variety, and every member of $\mathbf{R}$ is finitely based. |
| title | Every additively idempotent semiring satisfying $xy\approx xz$ is finitely based |
| topic | Group Theory |
| url | https://arxiv.org/abs/2509.16711 |