A new limit variety of additively idempotent semirings

Abstract

We establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. Applying this condition, we prove that the six-element additively idempotent semiring $S{R}_6$ has no finite basis for its identity. Furthermore, we provide a complete description of the subvariety lattice of the variety $\mathsf{V}(S{R}_6)$ generated by $S{R}_6$, showing that it forms a four-element chain. Our results demonstrate that $\mathsf{V}(S{R}_6)$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. Moreover, $S{R}_6$ is the smallest known example of an additively idempotent semiring generating a limit variety.