标题： 有限集族，其中没有一个集合被其他集合的并所覆盖 显示英文标题

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

摘要： 设F为一个有限的非空集合族。 我们证明以下结论：(i) F满足标题中的条件当且仅当对于F的每一对不同的子族{A_1,...,A_r},{B_1,...,B_s}，A_i的并集与B_i的并集不同。 (ii) 如果F满足标题中的条件，那么包含至少一个F中的集合的并集的子集的数量是奇数。 我们给出这两个结果的两个应用，一个用于数论，另一个用于交换代数。 显示英文摘要

摘要： Let F be a finite nonempty family of finite nonempty sets. We prove the following: (i) F satisfies the condition of the title if and only if for every pair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union of the A_i is different from the union of the B_i. (ii) If F satisfies the condition of the title, then the number of subsets of the union of the members of F 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.