  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 $D \neq
J$ for the semigroup of all 3-by-3 matrices over the tropical semiring with
$-\infty$, but in contrast, we show that $D = J$ for all full matrix semigroups
over the finitary tropical semiring.
