The Poset (O, ≼)

Abstract

In Chapter 6 we have analyzed the structure of an oriented matroid O, considered as a system of sign vectors. Here we will investigate its structure as a poset. These two points of view are strongly related, of course, though the relationship is not as clear as one might expect at the first glance. For example, if a set of sign vectors is given and we are to decide whether this is an OM, then we may simply check the axioms in order to find out the answer. On the other hand, if we are given a poset (P, ⪳), there is no obvious way to decide wether it is an OM-poset or not, except by “brute force”, i.e. by trying all possible OMs up to a certain size and each time comparing their posets to the given one. [21] and [48] provide a more clever way of computing a set O of sign vectors such that (O, ⪳) equals the given poset (P,⪳). But nonetheless, the final check whether or not (O, ⪳) is an OM poset has to be done by checking the sign vector axioms. The reason is that, so far, there is no neat characterization of OMs in terms of their posets.