A class of idempotent semirings

The least lattice congruence on a semiring always contains the least semilattice congruence on the additive reduct. We shall investigate the class of all idempotent semirings for which these two congruences coincide. We find that this class of semirings forms a variety and specify several systems of identities which determine this variety. Various structural characterizations for the semirings which belong to this variety are given. Noticeable among these are the characterizations in terms of the natural partial orders or in terms of the D-relations of the two reducts. We identify important subvarieties and give structure theorems for special types of such semirings.