3 - Axiomatics

Summary

Introductory remarks

The first two chapters were devoted to an informal introduction to oriented matroids and an overview of their place in pure and applied mathematics. In this chapter we begin the systematic development of oriented matroid theory based on an axiomatic foundation.

A characteristic feature of oriented matroids is the variety of different axiom systems. The lack of one leading axiomatization is not a deficiency but adds richness and widens the applicability of the theory.

There are four basic axiom systems for oriented matroids:

1. (a) circuit axioms (Definition 3.2.1),

2. (b) orthogonality axioms (Theorem 3.4.3),

3. (c) chirotopes, or basis orientations (Definition 3.5.3), and

4. (d) vector axioms (Theorem 3.7.5).

The axioms abstract natural properties of the examples which motivate the theory. The four basic axiom systems arise this way from

1. (a) directed graphs,

2. (b) orthogonal pairs of real vector subspaces,

3. (c) point configurations and convex polytopes, and

4. (d) real hyperplane arrangements.

The various axiom systems look quite different at first glance, and the proofs of equivalence are far from trivial. This is in contrast to the theory of ordinary matroids, which is even richer in axiom systems, but for which the proofs of equivalence are considerably easier.

All the equivalences will be established in this chapter, and they provide the basic links between different aspects of the theory, and between different subsequent chapters of this book. One additional axiomatization of oriented matroids – equivalence classes of pseudosphere arrangements – will be deferred to Chapters 4 and 5, because of its different nature and because of the special tools its analysis and its equivalence proofs require.