Oriented matroid

Summary

An oriented matroid is a matroid in which each minimal dependence also carries signs, recording which elements lie on which side. It abstracts the combinatorics of real vector configurations, of point configurations, of hyperplane arrangements and of directed graphs. Faces of convex polytopes, regions of real arrangements and the results of linear programming can all be described in terms of oriented matroids. Not every oriented matroid comes from vectors; by the topological representation theorem, oriented matroids correspond to arrangements of pseudospheres.