Closures of Union-Closed Families

Dhruv Bhasin

Published 2020-03-20Version 1

Given a Union-Closed family, which is not equal to the power set of its universe, say $[n]$, one can always add a new set $A\subsetneq[n]$ to it, such that the new family remains Union-Closed. We construct a new family by collecting all such $A$'s and call this family the closure of $\mathcal F$. We study various properties of this closure. We characterize families whose closure becomes the power set of $[n]$ and give a checking criteria of closure roots of such families, i.e., existence of $\mathcal H$ such that closure of $\mathcal H=\mathcal F$.