Idempotent tropical matrices and finite metric spaces

TL;DR

This paper explores the relationship between tropical idempotent matrices and finite metric spaces, revealing how tropical polytopes can realize various metric and asymmetric distance functions, with implications for tropical algebra and geometry.

Contribution

It establishes new connections between tropical idempotents and metric space realizations, including the structure of maximal subgroups in tropical matrix semigroups.

Findings

Every n-point metric space is realized by the Hilbert projective metric on tropical polytopes.

Every n-point asymmetric distance function is realized by a residuation operator on tropical polytope vertices.

The maximal subgroup of tropical matrices associated with a metric space is a product of the real numbers and the space's isometry group.

Abstract

There is a well known correspondence between the triangle inequality for a distance function on a finite set, and idempotency of an associated matrix over the tropical semiring. Recent research has shed new light on the structure (algebraic, combinatorial and geometric) of tropical idempotents, and in this paper we explore the consequences of this for the metric geometry of tropical polytopes. We prove, for example, that every n-point metric space is realised by the Hilbert projective metric on the vertices of a pure n-dimensional tropical polytope in tropical n-space. More generally, every n-point asymmetric distance function is realised by a residuation operator on the vertices of such a polytope. In the symmetric case, we show that the maximal group of tropical matrices containing the idempotent associated to a metric space is a direct product of the real numbers with the isometry group of the space; it follows that every direct product of a finite group with the real numbers arises as a maximal subgroup of a sufficiently large finitary full tropical matrix semigroup. In the process we also prove some new results about tropical idempotent matrices, and note some semigroup-theoretic consequences which may be of independent interest.