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arXiv · 2507.08381

On the variety generated by all semirings of order two

Aifa Wang·Lili Wang
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Abstract

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.

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Aifa Wang, Lili Wang. 2025-07-11. On the variety generated by all semirings of order two. https://arxiv.org/abs/2507.08381

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