On trees and tree dimension of ordered sets
type: Item
instance of: scholarly article
title: On trees and tree dimension of ordered sets
zbMATH Open document ID: 0796.06001
DOI: 10.1007/BF01111300
author: Gerhard Behrendt
published in: Order
publication date: 1994-03-27T00:00:00Z
review text: We call an ordered set \((X,\leq)\) a tree if no pair of incomparable elements of \(X\) has an upper bound. We denote by \(\text{St}(X)\) the set of stalagmites of \((X,\leq)\) which have a greatest element. Theorem. Let \((X,\leq)\) be an ordered set, \((Z,\leq)\) a tree and \(\tau: Z\to X\) an order-preserving surjection such that for every tree \((Y,\leq)\) and every order-preserving surjection \(f: Y\to X\) there exists an order- preserving mapping \(g: Y\to Z\) such that \(f= g\tau\). Then \((Z,\leq)\) contains a subset which, with respect to the induced order, is isomorphic to \((\text{St}(X),\leq)\). We define the tree dimension \(\text{td}(X,\leq)\) of an ordered set as the minimal number of extensions of \((X,\leq)\) which are trees such that the given order is the intersection of those tree orders. The author characterizes the tree dimension, describes relations between dimension and tree dimension and gives removal theorems.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 540217
zbMATH Keywords: stalagmites
zbMATH Keywords: tree dimension
zbMATH Keywords: extensions
reviewed by: Bohumil František Šmarda
MaRDI profile type: Publication
cites work: On the Number of Normal Subgroups of an Infinite Group
cites work: On the dimension of the Cartesian product of relations and orders
cites work: Dimension of the crown \(S^k_n\)
cites work: The Comparability Graph of a Tree
cites work: A Note on "The Comparability Graph of a Tree"
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description: scientific article; zbMATH DE number 540217