Congruence-simple multiplicatively idempotent semirings

TL;DR

This paper classifies finite multiplicatively idempotent congruence-simple semirings, showing they are either of size two or isomorphic to endomorphism semirings of a 2-element semilattice, and discusses the open question of their infinitude.

Contribution

It provides a classification of finite such semirings and explores conditions under which they are of size two or related to semilattice endomorphisms.

Findings

If $S$ has a multiplicatively absorbing element, then $|S|=2$.

Finite $S$ are either of size two or isomorphic to $End(L)$ for a 2-element semilattice.

The question of whether infinite such semirings exist remains open.

Abstract

Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question, whether $S$ can be infinite at all.