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

TL;DR

This paper characterizes families of finite sets where no set is covered by others and explores their properties, including a unique union condition and an odd count of certain subsets, with applications to number theory and algebra.

Contribution

It provides a characterization of such families via union uniqueness and establishes an oddness property of subset counts, with applications in number theory and algebra.

Findings

Families satisfy the union-uniqueness condition if and only if unions of distinct subfamilies differ.

If the family satisfies the condition, the count of certain subsets is always odd.

Applications demonstrate relevance to number theory and commutative algebra.

Abstract

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.