Simplicial Complexes Obtained from Qualitative Probability Orders

Abstract

Abstract. The goal of this paper is to introduce a new class of simplicial complexes that naturally generalize the threshold complexes. These will be derived from qualitative probability orders on subsets of a finite set that generalize subset orders induced by probability measures. We show that this new class strictly contains the threshold complexes and is strictly contained in the shifted complexes. We conjecture that this class of complexes is exactly the set of strongly acyclic complexes, a class that has previously appeared in the context of cooperative games. Beyond the results themselves, this new class of complexes allows us to refine our understanding of one-point extensions of a particular oriented matroid.

Minkowski complexes and convex threshold dimension

“…Hence, threshold complexes are shifted in the sense of Kalai [6] and topologically wedges of (not necessarily equidimensional) spheres. See [7] and [3] for more information regarding the combinatorics and topology of threshold and shifted complexes.…”