Generalized ultrametric spaces. I
type: Item
instance of: scholarly article
title: Generalized ultrametric spaces. I
zbMATH Open document ID: 0922.54028
DOI: 10.1007/BF02940794
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1999-09-17T00:00:00Z
review text: The authors study ultrametric spaces \((X, d, \Gamma)\), where \(X\) is a nonempty set, \(\Gamma\) a partially ordered set with smallest element \(0\), and \(d: X\times X \rightarrow\Gamma\) a (surjective) ultrametric distance function (i.e., \(d(x,y)=0\) if and only if \(x=y\), \(d(x,y)=d(y,x)\), if \(d(x,y) \leq \gamma\) and \(d(y,z) \leq \gamma\) then \(d(x,z) \leq \gamma\)). A description of all ultrametrics on a given nonempty set \(X\) is provided. This is done as follows. Let \(\mathcal D\) be a set of equivalence relations on \(X\) such that equality is in \(\mathcal D\) and for all \(x,y \in X\), there exists the smallest \(\alpha \in \mathcal D\) for which \(x\alpha y\). Let \(\mathbb U\) be the set of all such \(\mathcal D\). Set \(d_{\mathcal D}(x,y)=\alpha\) and \(\text{Prin}(\mathcal D)\) the image of \(X\times X\) under \(d_{\mathcal D}\). Then \((X, d_{\mathcal D}, \text{Prin}(\mathcal D))\) is an ultrametric space. On the other hand, given an ultrametric space \((X, d, \Gamma)\), an equivalence relation \(\alpha\) is said to be \(d\)-compatible if \(x\alpha y\) and \(d(x', y') \leq d(x, y)\) then \(x'\alpha y'\). Call \(\mathcal D \in \mathbb U\) saturated if it coincides with the set of all \(d_{\mathcal D}\)-compatible equivalence relations. Let \(\mathbb U_{\text{sat}}\) be the set of all such saturated \(\mathcal D\), and \({\mathcal U}\) the set of isomorphism classes of ultrametrics on \(X\). Then \[ {\mathcal D} \mapsto \text{ the isomorphism class of }(X, d_{\mathcal D}, \text{Prin}(\mathcal D)) \] defines a bijection from \(\mathbb U_{\text{sat}}\) onto \({\mathcal U}\). The inverse mapping is defined as follows: given \((X, d, \Gamma)\), we let \({\mathcal D}\) be the set of \(d\)-compatible equivalence relations. Then \({\mathcal D} \in \mathbb U_{\text{sat}}\) and \((X, d_{\mathcal D}, \text{Prin}(\mathcal D))\), \((X, d, \Gamma)\) are in the same isomorphism class. An ultrametric space is called spherically complete when any nonempty ch…
reviewed by: Salma Kuhlmann
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 12J99
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 13J99
Mathematics Subject Classification ID: 16W80
zbMATH DE Number: 977856
zbMATH Keywords: ultrametric spaces
zbMATH Keywords: compatible equivalence relations
zbMATH Keywords: spherically complete spaces
author: Sibylla Priess-Crampe
author: Paulo Ribenboim
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02940794
OpenAlex ID: W4250324347
description: scientific article; zbMATH DE number 977856