A family $\mathcal{F}$ of sets is union-closed if the union of any two sets from $\mathcal{F}$ belongs to $\mathcal{F}$. The union-closed sets conjecture states that if $\mathcal{F}$ is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in $\mathcal{F}$. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions.

一个家庭$\mathcal{F}$如果任意两个集合的并集，则集合是并闭的$\mathcal{F}$属于$\mathcal{F}$。并闭集猜想指出，如果$\mathcal{F}$是有限集的有限并闭族，则存在一个元素属于至少一半的集合$\mathcal{F}$。该猜想在其他组合结构（例如格和图）方面有几个等价的公式。就其整体普遍性而言，该猜想仍然是开放的，但它已在各种重要的格类（例如下半模格）和图（例如弦二分图）中得到验证。在本文中，我们开发了一种布尔方法来解决该猜想，并针对几类布尔函数（例如子模函数和双 Horn 函数）验证了它。