  We propose a definition of an "oriented interval greedoid" that
simultaneously generalizes the notion of an oriented matroid and the
construction on antimatroids introduced by L. J. Billera, S. K. Hsiao, and J.
S. Provan in "Enumeration in convex geometries and associated polytopal
subdivisions of spheres" [Discrete Comput. Geom. 39 (2008), no. 1-3, 123--137].
As for of oriented matroids, associated to each oriented interval greedoid is a
spherical simplicial complex whose face enumeration depends only on the
underlying interval greedoid.
