A new limit variety of additively idempotent semirings

TL;DR

This paper proves that a specific six-element additively idempotent semiring generates a nonfinitely based limit variety, the smallest known example of such a structure, with a detailed subvariety lattice.

Contribution

It introduces a sufficient condition for nonfinite baseness in additively idempotent semirings and characterizes the subvariety lattice of the generated variety.

Findings

SR_6 has no finite basis for its identities.

The subvariety lattice of V(SR_6) is a four-element chain.

V(SR_6) is a limit variety, nonfinitely based with all proper subvarieties finitely based.

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 $SR_6$ has no finite basis for its identity. Furthermore, we provide a complete description of the subvariety lattice of the variety $\mathsf{V}(SR_6)$ generated by $SR_6$, showing that it forms a four-element chain. Our results demonstrate that $\mathsf{V}(SR_6)$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. Moreover, $SR_6$ is the smallest known example of an additively idempotent semiring generating a limit variety.