The finite basis problem for matrix semirings over a two-element additively idempotent semiring

TL;DR

This paper classifies when matrix semirings over two-element additively idempotent semirings have a finite basis, revealing that they are finitely based precisely when the base semiring is not a distributive lattice.

Contribution

It provides a complete classification of the finite basis property for matrix semirings over two-element additively idempotent semirings, a previously unresolved problem.

Findings

Matrix semirings are finitely based iff the base semiring is not a distributive lattice.

The classification applies for all matrix sizes n ≥ 2.

The main theorem characterizes the finite basis property completely.

Abstract

We provide a complete classification of matrix semirings $\mathbf{M}_n(S)$ over two-element additively idempotent semirings $S$ with respect to the finite basis property.Our main theorem shows that for every integer $n \geq 2$,the semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ is distinct from a distributive lattice.