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

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.