Closures of Union-Closed Families

Abstract

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[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 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 H such that closure of H = F.