Nonfinite basis of finite ai-semirings whose varieties contain S7

Determine whether every finite additively idempotent semiring whose generated variety contains the semiring S7 is nonfinitely based.

Background

The semiring S7 is a three-element additively idempotent semiring that generates a nonfinitely based variety and is minimal in order among such examples. The paper establishes nonfinite basis results for several larger additively idempotent semirings and for broad intervals of subvarieties containing S7, but these results do not cover every finite additively idempotent semiring whose generated variety contains S7.

Consequently, the general inheritance question remains unresolved: it is not known whether the presence of S7 in the generated variety forces every finite additively idempotent semiring in that setting to be nonfinitely based.

References

Also, it can infect the nonfinitely based property to many other finite additively idempotent semirings, though it remains unknown if all finite additively idempotent semirings whose variety contains S7 are nonfinitely based.

— The finite basis problem for additively idempotent semirings that relate to S_7  (2501.19049 - Gao et al., 31 Jan 2025) in Section 1, Introduction, p. 1; reiterated in Section 5, Conclusion, p. 20