A simplicial poset P (also called a boolean poset and a poset of boolean type) is a finite poset with a smallest element 0̂ such that every interval [0̂, y] for y ∈ P is a boolean algebra, i.e., [0̂, y] is isomorphic to the set of all subsets of a finite set, ordered by inclusion. The set of all faces of a (finite) simplicial complex with empty set added forms a simplicial poset ordered by inclusi...

We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in polynomial time whether a simplicial compl...

Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex Mn, which is the simplicial complex of matchings in the complete graph Kn. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of Mn. First, we demonstrate that there is nonvanish...

Edelman & Fishburn (2006) initiated the study of abstract simplicial complexes which are initial segments of qualitative probability orders. By their nature they are combinatorial generalizations of threshold complexes, however they have not been studied before. In this paper we prove that the class of initial segments of qualitative probabilities is a new class, different from all the classic ...

We provide a basic formal framework for the theory of chamber complexes and Coxeter systems, and for buildings as thick chamber complexes endowed with a system of apartments. Along the way, we develop some of the general theory of abstract simplicial complexes and of groups (relying on the group add class for the basics), including free groups and group presentations, and their universal proper...

In this paper we show results on the combinatorial properties of shifted simplicial complexes. We prove two intrinsic characterization theorems for this class. The first theorem is in terms of a generalized vicinal preorder. It is shown that a complex is shifted if and only if the preorder is total. Building on this we characterize obstructions to shiftedness and prove there are finitely many i...

We call a set of positive integers closed under taking unitary divisors an unitary ideal. It can be regarded as a simplicial complex. Moreover, a multiplicative arithmetical function on such a set corresponds to a function on the simplicial complex with the property that the value on a face is the product of the values at the vertices of that face. We use this observation to solve the following...

In the theory of imprecise probability it is often of interest to find the range of the expectation of some function over a convex family of probability measures. Here we show how to find the joint range of the expectations of a finite set of functions when the underlying space is finite and the family of probability distributions is defined by finitely many linear constraints.

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