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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.00893 |
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| _version_ | 1866910036791394304 |
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| author | Gao, Zidong |
| author_facet | Gao, Zidong |
| contents | This paper establishes the existence of a finitely based finite semiring whose variety contains a continuum of subvarieties; such a variety is said to be of type \(2^{\aleph_0}\). Using the homomorphism theory of Kneser graphs, we prove that the 3-element semiring \(S_{53}\) is the first known example with this property. Moreover, \(S_{53}\) belongs to the variety of the max-plus semiring \((\mathbb{N},\max,+)\), which therefore is also of type \(2^{\aleph_0}\). For the finitely based 4-element semiring \(B_0\), we demonstrate that its variety contains infinitely many subvarieties and suggest that \(B_0\) could be another potential example of type \(2^{\aleph_0}\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00893 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A finitely based finite semiring generates a variety with continuum many subvarieties Gao, Zidong Rings and Algebras This paper establishes the existence of a finitely based finite semiring whose variety contains a continuum of subvarieties; such a variety is said to be of type \(2^{\aleph_0}\). Using the homomorphism theory of Kneser graphs, we prove that the 3-element semiring \(S_{53}\) is the first known example with this property. Moreover, \(S_{53}\) belongs to the variety of the max-plus semiring \((\mathbb{N},\max,+)\), which therefore is also of type \(2^{\aleph_0}\). For the finitely based 4-element semiring \(B_0\), we demonstrate that its variety contains infinitely many subvarieties and suggest that \(B_0\) could be another potential example of type \(2^{\aleph_0}\). |
| title | A finitely based finite semiring generates a variety with continuum many subvarieties |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2603.00893 |