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. Not every oriented matroid can be realised by an arrangement of linear hyperplanes though. The famous Topological Representation Theorem by Folkman and Lawrence, however, states that every oriented matroid can be represented as an arrangement of $\textit{pseudo}$hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper I prove that this correspondence is a bijection. Moreover, I present a tropical analogue for the Topological Representation Theorem.

Source : oai:HAL:hal-01283141v1

Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)

Section: Proceedings

Published on: January 1, 2012

Submitted on: January 31, 2017

Keywords: combinatorics, oriented matroids, discrete topology, tropical geometry,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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