Tropical matrix duality and Green's D relation

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.

21 pages

References in corpus (4)

• Duality and separation theorems in idempotent semimodules
• Enumerative tropical algebraic geometry in R2
• The Minkowski Theorem for Max-plus Convex Sets
• Reachability problems for products of matrices in semirings

Cited by in corpus (10)

• Pure Dimension and Projectivity of Tropical Polytopes
• Identities in Upper Triangular Tropical Matrix Semigroups and the Bicyclic Monoid
• Exact rings and semirings
• Diameters of commuting graphs of matrices over semirings
• Enumerating Polytropes
• Idempotent tropical matrices and finite metric spaces
• Minimal generating sets for matrix monoids
• Tropical matrices and group representations
• Polytropes and Tropical Eigenspaces: Cones of Linearity
• Linear isomorphisms preserving Green's relations for matrices over semirings