Inherently nonfinitely based additively idempotent semirings
We establish a sufficient condition for an additively idempotent semiring to be inherently nonfinitely based: if its generated variety is locally finite and every Zimin word is minimal in the additive order, then it is contained in no finitely based locally finite variety. For every positive integer $n$, we construct an infinite finitely generated flat semiring satisfying all identities in at most $n$ variables of every ai-semiring with this minimality property. We also characterize Zimin minimality by the membership of a countable flat factor semiring $\Finf$ in the generated variety, and prove that$\V(\Finf)$ is locally finite and inherently nonfinitely based. As applications, we show that the six-element ai-semirings $\A$ and $\B$ are inherently nonfinitely based. In contrast, we prove that the six-element ai-semiring $\overline{A_2^1}$ is not inherently nonfinitely based, although its multiplicative reduct $A_2^1$ is inherently nonfinitely based. To the best of our knowledge, this is the first example of an ai-semiring whose multiplicative reduct is inherently nonfinitely based while the ai-semiring itself is not.