A nonfinitely based additively idempotent semiring of order four

TL;DR

This paper identifies conditions under which certain additively idempotent semirings are nonfinitely based and provides examples, including a specific 4-element semiring with unique properties.

Contribution

It establishes a sufficient condition for nonfinite basis in additively idempotent semirings and presents explicit examples satisfying this condition.

Findings

The semiring $S_{(4,124)}$ has no finite basis for its identities.

Several additively idempotent semirings satisfy the nonfinitely based condition.

The semiring $S_{(4,124)}$ has two minimal elements and two coatoms.

Abstract

We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element semiring $S_{(4,124)}$ whose additive reduct has two minimal elements and two coatoms. Consequently, these semirings have no finite basis for their identities.