Additively idempotent matrix semirings

Let [Formula: see text] be an additively idempotent semiring and [Formula: see text] be the semiring of all [Formula: see text] matrices over [Formula: see text]. We characterize the conditions of when the semiring [Formula: see text] is congruence-simple provided that the semiring [Formula: see text] is either commutative or finite. We also give a characterization of when the semiring [Formula: see text] is subdirectly irreducible for [Formula: see text] being almost integral (i.e. [Formula: see text] for all [Formula: see text]). In particular, we provide this characterization for the semirings [Formula: see text] derived from the pseudo MV-algebras.