Nonfinitely based ai-semirings with finitely based semigroup reducts

TL;DR

This paper investigates the nonfinite axiomatisability of certain additively idempotent semirings with finitely based semigroup reducts, revealing minimal examples and extending group-theoretic results.

Contribution

It identifies the smallest nonfinitely based ai-semiring, demonstrates its NP-hard membership, and generalizes nonaxiomatizability results to broader classes of semirings.

Findings

The smallest nonfinitely based ai-semiring has 3 elements.

This semiring has NP-hard membership for its variety.

Nonabelian nilpotent subgroups imply nonfinite axiomatisability.

Abstract

We present some general results implying nonfinite axiomatisability of many additively idempotent semirings with finitely based semigroup reducts. The smallest is a $3$-element commutative example, which we show also has \texttt{NP}-hard membership for its variety. As well as being the only nonfinite axiomatisable ai-semiring on $3$-elements, we are able to show that its nonfinite basis property infects many related semirings, including the natural ai-semiring structure on the semigroup $B_2^1$. We also extend previous group-theory based examples significantly, by showing that any finite additively idempotent semiring with a nonabelian nilpotent subgroup is not finitely axiomatisable for its identities.