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. We characterize Zimin minimality by the membership of a countable flat factor semiring $\Finf$ in the generated variety and prove that $\V(\Finf)$ is a minimal inherently nonfinitely based variety. We also obtain general restrictions on the additive order of finite inherently nonfinitely based ai-semirings. As applications, the six-element ai-semirings $\A$ and $\B$ are shown to be inherently nonfinitely based. Finally, we prove that the six-element ai semiring $\Abar$ is not inherently nonfinitely based and is finitely based. Its multiplicative reduct is inherently nonfinitely based; thus, to the best of our knowledge, $\Abar$ is the first finitely based ai-semiring with an inherently nonfinitely based multiplicative reduct.