A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety

RESUMO

We present an explicit infinite equational basis for the six-element additively idempotent semiring $TR_6$ and prove that $TR_6$ is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by $TR_6$, showing that it forms a four-element chain. Our results demonstrate that the variety generated by $TR_6$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, $\mathsf{V}(TR_6)$ is the first explicit limit subvariety of the variety generated by the max-plus algebra $\mathbf{N}$. Moreover, $TR_6$ is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring $SR_6$, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by $SR_6$ and $TR_6$, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.