Simplicial Complexes Obtained from Qualitative Probability Orders

TL;DR

This paper explores the structure of simplicial complexes derived from qualitative probability orders, introducing new examples and conditions that distinguish them from known classes like threshold and shifted complexes.

Contribution

It constructs a novel qualitative probability order with an initial segment that is not a threshold complex and proposes necessary conditions and conjectures for their characterization.

Findings

Constructed a qualitative probability order on 26 atoms with a non-threshold initial segment

Provided evidence that smaller examples may not exist

Proposed necessary conditions and a conjecture for classifying these complexes

Abstract

In this paper we inititate the study of abstract simplicial complexes which are initial segments of qualitative probability orders. This is a natural class that contains the threshold complexes and is contained in the shifted complexes, but is equal to neither. In particular we construct a qualitative probability order on 26 atoms that has an initial segment which is not a threshold simplicial complex. Although 26 is probably not the minimal number for which such example exists we provide some evidence that it cannot be much smaller. We prove some necessary conditions for this class and make a conjecture as to a characterization of them. The conjectured characterization relies on some ideas from cooperative game theory.