Flattening antichains with respect to the volume
type: Item
instance of: scholarly article
title: Flattening antichains with respect to the volume
zbMATH Open document ID: 0913.05092
author: Ljiljana Brankovic
author: Paulette Lieby
author: Mirka Miller
published in: The Electronic Journal of Combinatorics
publication date: 1999-02-02T00:00:00Z
full work available at URL: https://eudml.org/doc/119761
full work available at URL: v6i1toc.html
review text: Given a (finite) poset \((P,\leq)\) with a strictly increasing function \(h:P\to N=\{0,1,2, \dots\}\), define the volume of a subset \(S\) of \(P\) (with respect to \(h\)) to be \(\sum_{x\in S} h(x)\). A subset \(S\) of \(P\) is flat (with respect to \(h)\) if \(x,y\in S\) with \(h(x)<h(y)\) implies there is no \(z\in P\) such that \(h(x)< h(z)<h(y)\). Thus, if \(P=B_n\), the lattice of subsets of \([n]=\{1, \dots, n\}\), and \(h(x)= | x|\), then \(S\) is flat if there exists a \(k\) such that every element of \(S\) has cardinality \(k\) or \(k+1\). There are many posets which are themselves flat with respect to some function \(h\). Many more (e.g., chains) have flat antichains only. It has been conjectured that \(B_n\) is among the posets which have the property that for every antichain there is a flat antichain with the same volume \((h(x)= | x|)\). Special cases occur in the literature. In this paper the conjecture is settled positively via a sequence of quite elegant technical lemmas based on results by Sperner, Clements and Daykin, among others. Rather than closing a area of inquiry it may also be the case that this paper can be adapted to a more general situation and lead to other interesting conjectures.
reviewed by: J. Neggers
Mathematics Subject Classification ID: 05D99
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 06E99
zbMATH DE Number: 1247855
zbMATH Keywords: poset
zbMATH Keywords: volume
zbMATH Keywords: flat
zbMATH Keywords: antichain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1247855