Semiring identities of finite inverse semigroups

TL;DR

This paper investigates the finite basis problem for certain finite additively idempotent semirings with inverse semigroup reducts, showing many lack a finite identity basis.

Contribution

It establishes that most such semirings, especially those with nontrivial rook monoid reducts, do not have a finite identity basis, advancing understanding of their algebraic identities.

Findings

Nontrivial rook monoid reducts lack finite identity basis

Almost all additively idempotent semirings with combinatorial inverse semigroup reducts lack finite basis

Provides new insights into the algebraic structure of finite inverse semigroups

Abstract

We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups.