Two constructions of oriented matroids with disconnected extension space

New Examples of Oriented Matroids with Disconnected Realization Spaces

We construct oriented matroids of rank 3 on 13 points whose realization spaces are disconnected. They are defined on smaller points than the known examples with this property. Moreover, we construct the one on 13 points whose realization space is a connected and non-irreducible semialgebraic variety. 1 Oriented Matroids and Matrices Throughout this section, we fix positive integers r and n. Let...

Extension Spaces of Oriented Matroids

We study the space of all extensions of a real hyperplane arrangement by a new pseudo-hyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov & Sturmfels, via the Bohne-Dress Theorem on zonotopal tilings. We prove that the extension space ...

Disconnected coverings for oriented matroids via simultaneous mutations

Let Un; r be a uniform oriented matroid having as bases, B, all r-subsets (resp. as circuits, C, all (r + 1)-subsets) of {1; : : : ; n}. We say that C1 ⊆ C is a covering, of Un; r , if for any base B∈B there is a circuit C ∈C1 such that B ⊂ C. Let G(C1) be the graph having as set of vertices the elements of C1 and where two vertices are joined if they have one base in common. We say that C1 ⊆ C...

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids

Oriented Matroids

The theory of oriented matroids provides a broad setting in which to model, describe, and analyze combinatorial properties of geometric configurations. Apparently totally different mathematical objects such as point and vector configurations, arrangements of hyperplanes, convex polytopes, directed graphs, and linear programs find a common generalization in the language of oriented matroids. The...