以集合交集表示的元素
Elements represented as intersections of sets
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中文总结 AI 辅助
该数学研究确定了[n]的表示族集的最小基数为Sperner函数的离散逆s(n)，并给出相关推广、应用及渐近结果。
中文摘要 AI 辅助
对于自然数n，令[n] = {1,2,…,n}。若族集𝒮⊆2^{[n]}中，[n]的每个单元素集都可表示为𝒮中若干集合的交集，则称𝒮为“表示族集”。本文证明，[n]的表示族集的最小可能基数为Sperner函数n↦C(n,⌊n/2⌋)的离散逆s(n)；根据Sperner定理，该函数是2^{[n]}作为子集（或布尔）格中反链的最大元素数。具体而言，s(n)是使2^{[n]}包含n元反链的最小正整数。本文还给出了相关推广、进一步应用，以及用Lambert W函数的第二实分支表示的渐近结果。
英文摘要
For a natural number $n$ let $[n] = \{1,\ldots,n\}$. We say that a family ${\cal{S}}\subseteq 2^{[n]}$ is \emph{representing} if every singleton set of $[n]$ is an intersection of some sets from ${\cal{S}}$. We show that the smallest possible cardinality of a representing set for $[n]$ is the discrete inverse $s(n)$ of the Sperner's function $n\mapsto \binom{n}{\lfloor n/2\rfloor}$, which by Sperner's Theorem is the maximum number of elements in an antichain in $2^{[n]}$ when viewed as subset (or boolean) lattice. Specifically, $s(n)$ is then the smallest positive integer such that $2^{[n]}$ contains an $n$-element antichain. Some generalization, further applications and asymptotics in terms of the second real branch of the Lambert $W$ function are presented.
