Finite basis problem for power semirings of finite semigroups

Determine precisely when the full power semiring of a finite semigroup is finitely based as an additively idempotent semiring.

Background

For a finite semigroup S, the full power semiring P(S)\mathcal{P}(S) consists of all subsets of S, with union as addition and setwise multiplication as multiplication. The finite basis problem asks for a classification of those power semirings whose identities admit a finite equational basis.

The paper establishes nonfinite basability for the particular power semiring P(S7,⋅)\mathcal{P}(S_7,\cdot), where S7S_7 is the unique nonfinitely based three-element additively idempotent semiring of order at most three, but explicitly presents this as only a partial answer to the general problem.

References

DolinkaProblem~6.5 and Jackson et al.Problem~7.2 posed the following general problem. When is the power semiring of a finite semigroup finitely based as an ai-semiring?

— Explicit equational bases for the power semirings of $S_7$  (2609.19957 - Yue et al., 17 Sep 2026) in Section 1, Problem 1 (labelled power-semiring)

It remains open whether bounded local obstructions for all prescribed-phase instances suffice for finite basability under $SD\subseteq C_2$.

— Finite bases for full power semirings of finite nilpotent semigroups  (2610.01329 - Wang et al., 1 Oct 2026) in Section 7, subsection “A five-element example and further questions,” final paragraph after Proposition 7.1

Closing this gap, and obtaining a classification for arbitrary finite semigroups, remain open.

— Finite bases for full power semirings of finite nilpotent semigroups  (2610.01329 - Wang et al., 1 Oct 2026) in Section 7, subsection “A five-element example and further questions,” final paragraph