Families of finite sets in which no set is covered by the union of the others

Abstract

Let ℱ be a finite nonempty family of finite nonempty sets. We prove the following: (1) ℱ satisfies the condition of the title if and only if for every pair of distinct subfamilies {A1,…,Ar}, {B1,…,Bs} of ℱ, ⋃i=1rAi≠⋃i=1sBi. (2) If ℱ satisfies the condition of the title, then the number of subsets of ⋃A∈ℱA containing at least one set of ℱ is odd. We give two applications of these results, one to number theory and one to commutative algebra.