Removing critical pairs
type: Item
instance of: scholarly article
title: Removing critical pairs
zbMATH Open document ID: 0693.06002
DOI: 10.1007/BF02034329
author: Klaus Reuter
published in: Order
publication date: 1989-01-01T00:00:00Z
review text: Let P be a poset. A pair (x,y) of incomparable elements of P is called critical if \(a<x\) implies \(a<y\), and \(b>y\) impliex \(b>x\) for all \(a,b\in P\). A pair (x,y) is said to be removable if its removal decreases the dimension by at most one. In the paper under review the author gives counterexamples to the following conjectures for posets with at least three elements. 1) If \(x\in P\) is contained in some critical pair, then there exists \(y\in P\setminus \{x\}\) such that (x,y) or (y,x) is a critical and removable pair. 2) Every critical pair is removable. The author believes that the well-known weaker conjecture is true: Every poset contains a removable pair. Here he describes how Ferrers dimension can be used in order to construct the counterexamples. In particular, removal theorems via Ferrers dimensions are proved.
reviewed by: Konrad Engel
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4134105
zbMATH Keywords: poset
zbMATH Keywords: order dimension
zbMATH Keywords: arrow relation
zbMATH Keywords: critical pair
zbMATH Keywords: removable pair
zbMATH Keywords: Ferrers dimension
zbMATH Keywords: removal theorems
MaRDI profile type: Publication
cites work: On realizable biorders and the biorder dimension of a relation
cites work: On finding the bidimension of a relation
cites work: A combinatorial problem involving graphs and matrices
cites work: Subdirect decomposition of concept lattices
description: scientific article; zbMATH DE number 4134105