Tropical matrix duality and Green's D relation

TL;DR

This paper characterizes Green's D relation for tropical matrices using a new duality concept between row and column spaces, establishing necessary and sufficient conditions and metric isometries.

Contribution

It introduces a novel duality variant for tropical matrices that provides a complete description of Green's D relation and related foundational results.

Findings

Complete description of Green's D relation for tropical matrices

A new duality theorem with a converse for tropical convex sets

The duality map is an isometry under the Hilbert projective metric

Abstract

We give a complete description of Green's D relation for the multiplicative semigroup of all n-by-n tropical matrices. Our main tool is a new variant on the duality between the row and column space of a tropical matrix (studied by Cohen, Gaubert and Quadrat and separately by Develin and Sturmfels). Unlike the existing duality theorems, our version admits a converse, and hence gives a necessary and sufficient condition for two tropical convex sets to be the row and column space of a matrix. We also show that the matrix duality map induces an isometry (with respect to the Hilbert projective metric) between the projective row space and projective column space of any tropical matrix, and establish some foundational results about Green's other relations.