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arXiv · 2603.00015

Two nonfinitely based additively idempotent semirings of order four

Mengya Yue·Miaomiao Ren·Zidong Gao
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Abstract

We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{(4,545)}$ and $S_{(4,634)}$, whose additive reducts are chains, have no finite basis for their identities. Furthermore, we show that the interval $[\mathsf{V}(S_{(4,545)}),\mathsf{V}(S_{(4,634)})]$ in the lattice of semiring varieties contains \(2^{\aleph_0}\) distinct varieties. Consequently, the join of two finitely based additively idempotent semiring varieties is not necessarily finitely based. Moreover, we obtain the smallest example of a finitely based additively idempotent semiring $S$ whose extension $S^0$ (obtained by adjoining a new element) is nonfinitely based.

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Mengya Yue, Miaomiao Ren, Zidong Gao. 2026-01-31. Two nonfinitely based additively idempotent semirings of order four. https://arxiv.org/abs/2603.00015

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