Pattern Recognition on Oriented Matroids

Notes

There is a surprising multitude of definitions for oriented matroids [30, p. 1]. They are defined by various equivalent axiom systems, and they can be thought of as a combinatorial abstraction of point configurations over the reals, of real hyperplane arrangements, of convex polytopes, and of directed graphs [27, p. 1].

Oriented matroids are surveyed, for example, in Refs. [25, 31, 78, 166]; they are extensively studied in the books [13, 27, 30, 34, 57, 59, 170, 186] and in numerous research articles.

In the very brief survey of oriented matroids given in Section 1.1, we mainly adopt the terminology of Ref. [27, Ch. 3, 4, 7].

Recall that the standard definition of a simple oriented matroid is that it has no loops and parallel elements, ...