Generalized ultrametric spaces. I
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 chain of balls has a nonempty intersection. In view of the basic fixed point theorem [ibid. 63, 227-244 (1993; Zbl 0788.06004)], it is important to provide examples of such spaces. This is done in the last paragraph. In particular, given any ultrametric space, the authors associate canonically a spherically complete space and an injective ``expanding transformation'' from the given space to the enlarged space. Also, complete Boolean algebras provide natural examples of spherically complete spaces (when endowed with the distance \(d(a,b)=a+b\)). A nice characterization of completeness and spherical completeness for a Boolean algebra \((A, d, A)\) is given in terms of ``Hausdorff gaps''. For Part II, cf. ibid. 67, 19-31 (1997; Zbl 0887.54029).
scientific article; zbMATH DE number 977856