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-by-n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between 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 $D \neq J$forthesemigroupofall3-by-3matricesoverthetropicalsemiringwith$-\infty$,butincontrast,weshowthat$D = J$ for all full matrix semigroups over the finitary tropical semiring.