论文标题
关闭联盟封锁的家庭
Closures of Union-Closed Families
论文作者
论文摘要
给定一个宇宙$ [n] $子集的联合封闭的家庭$ \ MATHCAL {f} $，带有$ \ Mathcal {f} $不等于$ [n] $的功率集，可以将新的子集$ a $添加到它上，以便由此产生的家庭保留联盟。我们通过添加到$ \ Mathcal {f} $所有这样的$ a $'s构建一个新的family $ \ overline {\ mathcal {f}} $，并将其称为$ \ mathcal f $的关闭。本文致力于研究此类封闭的各种属性，包括表征封闭的家庭等于$ [n] $的功率集，为存在此类家庭的封闭根的存在提供了标准。
Given a union-closed family $\mathcal{F}$ of subsets of the universe $[n]$, with $\mathcal{F}$ not equal to the power set of $[n]$, a new subset $A$ can be added to it such that the resulting family remains union-closed. We construct a new family $\overline{\mathcal{F}}$ by adding to $\mathcal{F}$ all such $A$'s, and call this the closure of $\mathcal F$. This paper is dedicated to the study of various properties of such closures, including characterizing families whose closures equal the power set of $[n]$, providing a criterion for the existence of closure roots of such families etc.
