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 $\{A_1,\dots ,A_r\}$, $\{B_1,\dots ,B_s\}$ of $ℱ$, $\bigcup _{i=1}^r{A}_i\ne \bigcup _{i=1}^s{B}_i$. (2) If $ℱ$ satisfies the condition of the title, then the number of subsets of $\bigcup _{A\in ℱ}A$ 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.