Oriented Matroids and Hyperplane Transversals

Tropical Hyperplane Arrangements and Oriented Matroids

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 determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.

Fully Optimal Bases and the Active Bijection in Graphs, Hyperplane Arrangements, and Oriented Matroids

In this note, we present the main results of a series of forthcoming papers, dealing with bijective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce fully optimal bases, which have a simple combinatorial characterizatio...

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids

Local equivalence of transversals in matroids

Given any system of n subsets in a matroid M , a transversal of this system is an n-tuple of elements of M , one from each set, which is independent. Two transversals differing by exactly one element are adjacent, and two transversals connected by a sequence of adjacencies are locally equivalent, the distance between them being the minimum number of adjacencies in such a sequence. We give two s...

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...