On the properties of well-graded partially union-closed families

摘要

In this paper we will study several properties of well-graded union-closed families that do not contain the empty set. Such union-closed families without the empty set are said to be partially union-closed. We will extend several results for well-graded union-closed families to the partially union-closed case, and we will also extend the concept of being intersection-closed to families without the empty set. (C) 2017 Elsevier Inc. All rights reserved.