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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.08381 |
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| _version_ | 1866915383065182208 |
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| author | Wang, Aifa Wang, Lili |
| author_facet | Wang, Aifa Wang, Lili |
| contents | There are ten distinct two-element semirings up to isomorphism, denoted \( L_2, R_2, M_2, D_2, N_2, T_2, Z_2, W_2, Z_7 \), and \( Z_8 \) (see \cite{bk}). Among these, the multiplicative reductions of \( M_2, D_2, W_2 \), and \( Z_8 \) form semilattices, while the additive reductions of \( L_2, R_2, M_2, D_2, N_2 \), and \( T_2 \) are idempotent semilattices, commonly referred to as \emph{idempotent semirings}. In 2015, Vechtomov and Petrov \cite{vp} studied the variety generated by \( M_2, D_2, W_2 \), and \( Z_8 \), proving that it is finitely based. In the same year, Shao and Ren \cite{srii} examined the variety generated by the six idempotent semirings, demonstrating that every subvariety of this variety is finitely based.
This paper systematically investigates the variety generated by all ten two-element semirings. We prove that this variety contains exactly 480 subvarieties, each of which is finitely based. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the variety generated by all semirings of order two Wang, Aifa Wang, Lili Group Theory There are ten distinct two-element semirings up to isomorphism, denoted \( L_2, R_2, M_2, D_2, N_2, T_2, Z_2, W_2, Z_7 \), and \( Z_8 \) (see \cite{bk}). Among these, the multiplicative reductions of \( M_2, D_2, W_2 \), and \( Z_8 \) form semilattices, while the additive reductions of \( L_2, R_2, M_2, D_2, N_2 \), and \( T_2 \) are idempotent semilattices, commonly referred to as \emph{idempotent semirings}. In 2015, Vechtomov and Petrov \cite{vp} studied the variety generated by \( M_2, D_2, W_2 \), and \( Z_8 \), proving that it is finitely based. In the same year, Shao and Ren \cite{srii} examined the variety generated by the six idempotent semirings, demonstrating that every subvariety of this variety is finitely based. This paper systematically investigates the variety generated by all ten two-element semirings. We prove that this variety contains exactly 480 subvarieties, each of which is finitely based. |
| title | On the variety generated by all semirings of order two |
| topic | Group Theory |
| url | https://arxiv.org/abs/2507.08381 |