A central concept for the combinatorial study of convex polytopes and hyper-plane arrangements in Rd is that of a face lattice: the cells on the boundary of the polytope or the cells in the induced decomposition of Rd ordered by inclusion. These are precisely the kind of objects that will be studied in this chapter, except in a more general axiomatized version. A surprising amount of detailed information, much of it highly non-trivial also in the realizable case, can be obtained this way.

The lattices with which we will be concerned are formed by the covectors of an oriented matroid under a natural partial ordering. In this chapter oriented matroids will primarily be viewed as such ordered structures. The first two sections develop the elementary combinatorial properties, which are basic for an understanding of oriented matroid theory. The step to topology is taken in Section 4.3. There it is shown that the covector lattice of an oriented matroid uniquely determines a regular cell decomposition of a sphere.

The details of the constructions in Section 4.3 add up to a proof for one direction of the Topological Representation Theorem for oriented matroids (the more difficult one). This proof differs from the two previously known ones by Folkman and Lawrence (1978) and by Edmonds and Mandel (1982), although the mathematical underpinnings are similar to those of the Edmonds-Mandel proof. It makes systematic use of a general technique for poset shellability developed in Björner (1980, 1984a) and Björner and Wachs (1983), which has been applied to several classes of finite posets in connection with certain algebraic and topological questions.