Title: On the lattice formulation of the union-closed sets conjecture Show Chinese title

Title: 关于闭包集合猜想的格点表述

Abstract: The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice $L$ with more than one element contains a join-irreducible element that is less than or equal to at most half of the elements in $L$. In this work, we obtain several necessary conditions for any counterexample $\tilde{L}$ of minimum size. Show Chinese abstract

Abstract: 集合闭包猜想，也称为弗兰克尔猜想，是一个被广泛研究的问题，具有多种表述形式。 在格论方面，该猜想指出，每个包含多于一个元素的有限格 $L$ 都包含一个_join-不可约元，它小于或等于格 $L$ 中不超过一半的元素。 在本工作中，我们得到了最小反例 $\tilde{L}$ 的若干必要条件。