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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.18588 |
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| _version_ | 1866908981066203136 |
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| author | Lyu, Simin Ren, Miaomiao Yue, Mengya |
| author_facet | Lyu, Simin Ren, Miaomiao Yue, Mengya |
| contents | We establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. Applying this condition, we prove that the six-element additively idempotent semiring $SR_6$ has no finite basis for its identity. Furthermore, we provide a complete description of the subvariety lattice of the variety $\mathsf{V}(SR_6)$ generated by $SR_6$, showing that it forms a four-element chain. Our results demonstrate that $\mathsf{V}(SR_6)$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. Moreover, $SR_6$ is the smallest known example of an additively idempotent semiring generating a limit variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18588 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new limit variety of additively idempotent semirings Lyu, Simin Ren, Miaomiao Yue, Mengya Rings and Algebras We establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. Applying this condition, we prove that the six-element additively idempotent semiring $SR_6$ has no finite basis for its identity. Furthermore, we provide a complete description of the subvariety lattice of the variety $\mathsf{V}(SR_6)$ generated by $SR_6$, showing that it forms a four-element chain. Our results demonstrate that $\mathsf{V}(SR_6)$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. Moreover, $SR_6$ is the smallest known example of an additively idempotent semiring generating a limit variety. |
| title | A new limit variety of additively idempotent semirings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2604.18588 |