Splitting property in infinite posets
type: Item
instance of: scholarly article
title: Splitting property in infinite posets
zbMATH Open document ID: 0872.05053
DOI: 10.1016/S0012-365X(96)00010-6
published in: Discrete Mathematics
publication date: 1997-10-06T00:00:00Z
review text: Given a poset \(P\), a subset \(H\subset P\) is a generator if \(D(H)\cup U(H)=P\), where \(D(H)\) is the ideal (downset) generated by \(H\) and \(U(H)\) is the filter (upset) generated by \(H\). Thus if \(S\) is an antichain which is a generator it is maximal and conversely a maximal antichain is a generator. \(H\) has the splitting property if \(H\) can be partitioned \(H=(H_1,H_2)\) such that \(U(H_1)\cup D(H_2)=P\). Thus, if \(P\) is finite, the collection of all minimal elements yields \(U(H)\cup D(\varnothing)=P\), while the collection of all maximal elements yields \(U(\varnothing)\cup D(H)=P\). If \(P\) is infinite there may be complete failure in the observations above and thus it becomes an interesting and more subtle problem to deal with even the existence of generators \(H\) with special splitting properties in general and more special cases as the author has done, basing his further observation on work by R. Ahlswede and L. H. Khachatrian among others.
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1018637
zbMATH Keywords: poset
zbMATH Keywords: antichain
zbMATH Keywords: generator
zbMATH Keywords: splitting property
author: Péter L. Erdős
reviewed by: J. Neggers
MaRDI profile type: Publication
cites work: A splitting property of maximal antichains
cites work: On the minimal covering of infinite sets
cites work: Q3880849
full work available at URL: s0012-365x(96)00010-6
OpenAlex ID: W2142262870
Wikidata QID: Q127473198
description: scientific article; zbMATH DE number 1018637