Matroids, antimatroids, and the generalized external order

Las Vergnas's active orders are a collection of partial orders on the bases of a matroid which are derived from the classical notion of matroid activity. In this paper, we construct a generalization of Las Vergnas's external order which is defined on the independence complex of a matroid. We show that this poset is a refinement of the geometric lattice of flats of the matroid, and has the structure of a supersolvable join-distributive lattice. We uniquely characterize the lattices which are isomorphic to the external order of a matroid, and we explore a correspondence between matroid and antimatroid minors which arises from the poset construction.