Projective and External Saturation Problem for Posets

Order Pub Date : 2024-07-04 DOI:10.1007/s11083-024-09674-w
Dömötör Pálvölgyi, Balázs Patkós
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Abstract

We introduce two variants of the poset saturation problem. For a poset P and the Boolean lattice \(\mathcal {B}_n\), a family \(\mathcal {F}\) of finite subsets of \(\mathbb {N}\), not necessarily from \(\mathcal {B}_n\), is projective P-saturated if (i) it does not contain any strong copies of P, (ii) for any \(G\in \mathcal {B}_n\setminus \mathcal {F}\), the family \(\mathcal {F}\cup \{G\}\) contains a strong copy of P, and (iii) for any two different \(F,F'\in \mathcal {F}\) we have \(F\cap [n]\ne F'\cap [n]\). Ordinary strongly P-saturated families, i.e., families \(\mathcal {F}\subseteq \mathcal {B}_n\) satisfying (i) and (ii), automatically satisfy (iii) as they lie within \(\mathcal {B}_n\). We study what phenomena are valid both for the ordinary saturation number \(\text {sat}^{*}(n,P)\) and the projective saturation number \(\top \hspace{-10pt}\top \text {sat}(n,P)\), the size of the smallest projective P-saturated family. Note that the projective saturation number might differ for a poset and its dual. Motivated by this, we introduce an even more relaxed and symmetric version of poset saturation, external saturation. We conjecture that all finite posets have bounded external saturation number, and prove this in some special cases.

Posets 的投影和外部饱和问题
我们介绍正集饱和问题的两种变体。对于一个poset P和布尔网格\(\mathcal {B}_n\),\(\mathbb {N}\)的有限子集的族\(\mathcal {F}\),不一定来自\(\mathcal {B}_n\),如果(i) 它不包含P的任何强副本,那么它就是投影P饱和的、(ii) 对于任何一个 \(G\in \mathcal {B}_n\setminus \mathcal {F}/),族 \(\mathcal {F}\cup \{G/}/)包含一个 P 的强副本,并且 (iii) 对于任何两个不同的 \(F,F'\in \mathcal {F}/),我们有 \(F\cap [n]\ne F'\cap [n]\).普通的强 P 饱和族,即满足(i)和(ii)的族 \(\mathcal {F}\subseteq \mathcal {B}_n\) 自动满足(iii),因为它们位于 \(\mathcal {B}_n\) 内。我们将研究普通饱和数(\text {sat}^{*}(n,P)\) 和投影饱和数(\top \hspace{-10pt}\top \text {sat}(n,P)\) --最小投影 P 饱和族的大小--的有效现象。请注意,投影饱和数对于正集和它的对偶集可能是不同的。受此启发,我们引入了更宽松、更对称的正集饱和度版本,即外部饱和度。我们猜想所有有限正集都具有有界的外部饱和数,并在一些特殊情况下证明了这一点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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