Inequalities for Gondran–Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran–Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.