Finite simple additively idempotent semirings

TL;DR

This paper advances the classification of finite simple additively idempotent semirings by characterizing them as semirings of join-morphisms of semilattices, completing the classification for those with an additively neutral element.

Contribution

It provides a new characterization of finite simple additively idempotent semirings as join-morphism semirings of semilattices, especially for those with an additively neutral element.

Findings

Complete classification of finite simple semirings with an additively neutral element.

Characterization of many cases of finite simple additively idempotent semirings.

Development of the theory of idempotent irreducible semimodules.

Abstract

Since for the classification of finite (congruence-)simple semirings it remains to classify the additively idempotent semirings, we progress on the characterization of finite simple additively idempotent semirings as semirings of join-morphisms of a semilattice. We succeed in doing this for many cases, amongst others for every semiring of this kind with an additively neutral element. As a consequence we complete the classification of finite simple semirings with an additively neutral element. To complete the classification of all finite simple semirings it remains to classify some very specific semirings, which will be discussed here. Our results employ the theory of idempotent irreducible semimodules, which we develop further.