A Topological Representation Theorem for Tropical Oriented Matroids: Part II

TL;DR

This paper establishes a bijective correspondence between tropical oriented matroids and mixed subdivisions of dilated simplices, and presents a tropical analogue of the classical Topological Representation Theorem.

Contribution

It proves the bijection between tropical oriented matroids and mixed subdivisions, and introduces a tropical version of the Topological Representation Theorem.

Findings

Proved the bijection between tropical oriented matroids and mixed subdivisions.

Established a tropical analogue of the classical Topological Representation Theorem.

Abstract

Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes -- in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper we show that this correspondence is a bijection. Moreover, a tropical analogue for the Topological Representation Theorem for (classical) oriented matroids by Folkman and Lawrence is presented.