In this note, we present a conjecture on intersections of set families, and a rephrasing of the conjecture in terms of principal downsets of Boolean lattices. The conjecture informally states that, whenever we can express the measure of a union of sets in terms of the measure of some of their intersections using the inclusion-exclusion formula, then we can express the union as a set from these same intersections via the set operations of disjoint union and subset complement. We also present a partial result towards establishing the conjecture.

翻译：本文提出一个关于集族交集的猜想，并将其重新表述为布尔格主下集形式。该猜想非正式地指出：当我们能通过容斥原理用某些交集测度来表达集族并集的测度时，那么通过互不相交并集和子集补集的集合运算，总能将这些并集表达为相同交集构成的集合。本文还给出了该猜想的局部结果。