On the variety generated by all semirings of order two
Abstract
There are ten distinct two-element semirings up to isomorphism, denoted \( L2, R2, M2, D2, N2, T2, Z2, W2, Z7 \), and \( Z8 \) (see bk). Among these, the multiplicative reductions of \( M2, D2, W2 \), and \( Z8 \) form semilattices, while the additive reductions of \( L2, R2, M2, D2, N2 \), and \( T2 \) are idempotent semilattices, commonly referred to as idempotent semirings. In 2015, Vechtomov and Petrov vp studied the variety generated by \( M2, D2, W2 \), and \( Z8 \), proving that it is finitely based. In the same year, Shao and Ren 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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