Oriented Interval Greedoids

TL;DR

This paper introduces the concept of oriented interval greedoids, a new mathematical structure that generalizes oriented matroids and antimatroid constructions, linking combinatorial properties to geometric complexes.

Contribution

It defines oriented interval greedoids, extending the framework of oriented matroids and antimatroids, and explores their associated spherical simplicial complexes.

Findings

Defines oriented interval greedoids as a new combinatorial structure.

Shows the face enumeration depends only on the underlying interval greedoid.

Establishes a connection between these structures and geometric complexes.

Abstract

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.