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

TL;DR

This paper characterizes Green's J-order for tropical matrices, linking it to morphisms of tropical convex sets and various matrix ranks, and clarifies the relationship between J and D relations.

Contribution

It provides an exact characterization of the J-order in tropical matrices and explores the connections with convex set morphisms and matrix ranks, including the J=D relationship.

Findings

J-order characterized via tropical convex set morphisms

Connections established between J-order, isometries, and matrix ranks

D equals J for full matrix semigroups over the finitary tropical semiring

Abstract

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.