The finite basis problem for additively idempotent semirings of order four, I

TL;DR

This paper investigates the finite basis problem for 4-element additively idempotent semirings with semilattice structure, identifying which are finitely based and which are not, among 58 such algebras.

Contribution

It classifies all 58 semirings of this type regarding their finite basis property, providing a complete characterization.

Findings

49 semirings are finitely based

9 semirings are nonfinitely based

Complete classification of these semirings' basis properties

Abstract

We study the finite basis problem for 4-element additively idempotent semirings whose additive reducts are semilattices of height 1. Up to isomorphism, there are 58 such algebras. We show that 49 of them are finitely based and the remaining ones are nonfinitely based.