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arXiv · 2610.12486

The finite basis problem for varieties generated by all semirings of a fixed order

Aifa Wang·Lili Wang
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Abstract

Let $\Avar_n$ and $\VV_n$ be the varieties generated by all $n$-element additively idempotent semirings (ai-semirings) and all $n$-element semirings with commutative addition, respectively, in the signature $(+,\cdot)$ without constants. We prove that both varieties are finitely based if and only if $n\leq2$. The five-element case follows from cyclic identities and a structural analysis of products of squares. For $n\geq6$, each variety contains a finite inherently nonfinitely based ai-semiring. We also prove $\VV_n\cap\AI=\Avar_n$ for every $n$, where $\AI$ is the variety of all ai-semirings.

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Aifa Wang, Lili Wang. 2026-09-19. The finite basis problem for varieties generated by all semirings of a fixed order. https://arxiv.org/abs/2610.12486

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