Is there a characterization of oriented matroids in terms of order theory, similar to that of matroids as geometric lattices?
Does this question make sense at all? I have seen (for instance in Ziegler's "Lectures on Polytopes") that the signed (co-)vectors of an oriented matroid can be given a partial order, so I was wondering whether this can be pushed further and perhaps be of use.
Would such an identification allow for employing constructions for posets, like Moebius inversion, incidence algebras, etc. with oriented matroids?