On the idempotent semirings such that 𝒟• is the least distributive lattice congruence

Here we describe the least distributive lattice congruence [Formula: see text] on an idempotent semiring in general and characterize the varieties [Formula: see text], [Formula: see text] and [Formula: see text] of all idempotent semirings such that [Formula: see text], [Formula: see text] and [Formula: see text], respectively. If [Formula: see text], then the multiplicative reduct [Formula: see text] is a [left, right] normal band. Every semiring [Formula: see text] is a spined product of a semiring in [Formula: see text] and a semiring in [Formula: see text] with respect to a distributive lattice.