Green’s J-order and the rank of tropical matrices

We study Green’s J-order and J-equivalence for the semigroup of all n×n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between certain tropical convex sets. We establish connections between the J-order, isometries of tropical convex sets, and various notions of rank for tropical matrices. We also study the relationship between the relations J and D; Izhakian and Margolis have observed that D≠J for the semigroup of all 3×3 matrices over the tropical semiring with −∞, but, in contrast, we show that D=J for all full matrix semigroups over the finitary tropical semiring.