Two nonfinitely based additively idempotent semirings of order four

TL;DR

This paper identifies conditions under which certain 4-element additively idempotent semirings are nonfinitely based, providing examples and implications for the structure of semiring varieties.

Contribution

It introduces two sufficient conditions for nonfinite basis in additively idempotent semirings and demonstrates their application to specific 4-element semirings, revealing complex variety structures.

Findings

Two specific 4-element semirings are nonfinitely based.

The variety lattice interval contains continuum many varieties.

The join of two finitely based varieties can be nonfinitely based.

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.