Related papers: A Topological Representation Theorem for Tropical …
Tropical oriented matroids were defined by Ardila and Develin in 2007 in analogy to (classical) oriented matroids. In this paper we present one tropical analogue for the Topological Representation Theorem.
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid…
We present a comprehensive overview of Ardila and Develin's (generic) tropical oriented matroids, as well as many related objects and their axiomatics. Moreover, we use a unifying framework that elucidates the connections between these…
Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with…
We study a generalization of tropical oriented matroids by Ardila and Develin, and show that they are in bijection with subdivisions of root polytopes, which are sub-polytopes of a product of two simplices.
Ardila and Develin's paper on tropical oriented hyperplane arrangements and tropical oriented matroids defines tropical oriented matroids and conjectures a bijection between them and triangulations of products of simplices $\Delta_{n-1}…
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{\"o}nflies theorem.…
We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and…
Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its…
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…
Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the…
The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we…
We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of…
In this paper we extend the theory of oriented matroids to Lagrangian orthogonal matroids and their representations, and give a completely natural transformation from a representation of a classical oriented matroid to a representation of…
To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study…
Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that…
The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of "oriented arithmetic matroid" and prove some properties like the "uniqueness of orientation".
Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces…
We study representations of tropical linear spaces as intersections of tropical hyperplanes of circuits. For several classes of matroids, we describe minimal tropical bases. We also show that every realizable tropical linear space has a…