A Cubical Perspective on Complements of Union-Closed Families of Sets

TL;DR

This paper explores the topological structure of complements of union-closed set families using cubical sets, revealing their acyclicity and deriving related combinatorial formulas.

Contribution

It introduces a novel cubical set construction for simply rooted families of sets and proves their acyclicity, providing new insights into their topological and combinatorial properties.

Findings

Cubical sets associated with simply rooted families are always acyclic.

A new combinatorial formula is derived for families containing the empty set.

Elementary proof provided for the acyclicity and formula.

Abstract

Complements of union-closed families of sets, over a finite ground set, are known as simply rooted families of sets. Cubical sets are widely studied topological objects having applications in computational homology. In this paper, we look at simply rooted families of sets from the perspective of cubical sets. That is, for every family $\mathcal F$ of subsets of a finite set, we construct a natural cubical set $X(\mathcal F)$ (corresponding to it). We show that for every simply rooted family $\mathcal F$, containing the empty set, the cubical set $X(\mathcal F)$ is always acyclic (that is, it has trivial reduced cubical homology). As a consequence of this, using the Euler-Poincar\`{e} formula, we obtain a formula satisfied by all simply rooted families of sets which contain the empty set. We also provide an elementary proof of this formula.