# On the idempotent semirings such that $\mathcal{D}^\bullet$ is the least distributive lattice congruence

## TL;DR

This paper characterizes the least distributive lattice congruence on idempotent semirings and describes the structure of specific varieties where this congruence coincides with known relations.

## Contribution

It provides a general description of the least distributive lattice congruence on idempotent semirings and characterizes the varieties where this congruence matches specific known congruences.

## Key findings

- The least distributive lattice congruence $ heta$ on an idempotent semiring is characterized.
- Varieties $D^ullet, L^ullet, R^ullet$ are described where $ heta$ equals $ ext{D}^ullet, ext{L}^ullet, ext{R}^ullet$.
- Semirings in $D^ullet$ are spined products of semirings in $L^ullet$ and $R^ullet$ with respect to a distributive lattice.

## Abstract

Here we describe the least distributive lattice congruence $\eta$ on an idempotent semiring in general and characterize the varieties $D^\bullet, L^\bullet$ and $R^\bullet$ of all idempotent semirings such that $\eta=\mathcal{D}^\bullet, \mathcal{L}^\bullet$ and $\mathcal{R}^\bullet$, respectively. If $S \in D^\bullet [L^\bullet, R^\bullet]$, then the multiplicative reduct $(S, \cdot)$ is a [left, right] normal band. Every semiring $S \in D^\bullet$ is a spined product of a semiring in $L^\bullet$ and a semiring in $R^\bullet$ with respect to a distributive lattice.