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 oriented matroid of a finite set of points P extracts “relative position” and “orientation” information from the configuration; for example, it can be given by a list of signs that encodes the orientations of all the bases of P . In the passage from a concrete point configuration to its oriented matroid metrical information is lost, but many structural properties of P have their counterparts at the — purely combinatorial — level of the oriented matroid. We first introduce oriented matroids in the context of several models and motivations (Section 7.1). Then we present some equivalent axiomatizations (Section 7.2). Finally, we discuss concepts that play central roles in the theory of oriented matroids (Section 7.3), among them duality, realizability, the study of simplicial cells, and the treatment of convexity.