Strong nonfinite basability of the three-element flat semiring $S_7$

Abstract

We prove that the three-element flat semiring $S_7$ is strongly nonfinitely based within the class of additively idempotent semirings. More generally, let $\mathcal{N}$ be the subvariety of $\Var(S_7)$ defined by $x^{2}y\approx x^2$. For every finite ai-semiring $A$, each variety $\mathcal{W}$ satisfying $\mathcal N\subseteq\mathcal W\subseteq\Var(A)$ has infinite axiomatic rank.