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

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.

arXiv admin note: text overlap with arXiv:1706.02670