Oriented matroids 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. The creators of the theory of oriented matroids have, in fact, drawn their motivation from these diverse mathematical theories (see the historical sketch in Section 3.9), but they have nevertheless arrived at equivalent axiom systems – which manifests the fact that oriented matroids are “the right concept”.
We will start out by illustrating these different aspects of oriented matroids. Doing this, we will present a number of examples while at the same time introducing the main concepts and terminology of oriented matroids. This should assist the reader who wishes to access the later chapters in a non-linear order, or who first wants a quick idea of what is going on. It should also provide intuition and motivation both for the axiomatics and the further development of the theory.
Hence, our first two chapters will avoid an extensive discussion of the axiom systems for oriented matroids, which are treated in Chapter 3. We will also minimize dependence on background from ordinary matroids. Furthermore, extensive attributions will not be given in these introductory chapters; we refer to later chapters and the bibliography.
Oriented matroids from directed graphs
Let us consider the simple cycles of a directed graph D = (V,E) with arc set E, together with an orientation of each such cycle. Then every arc of a cycle is either a forward (positive) arc or a backward (negative) arc in the cycle. This allows us to consider the cycle as a signed subset of E, which consists of a positive and a negative part.