Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Totally bounded ultrametric spaces and locally finite trees

Oleksiy Dovgoshey

TL;DR

The paper advances the theory of totally bounded ultrametric spaces by extending the Gurvich–Vyalyi tree representation to this broader class and establishing that isometric completions correspond to isomorphic labeled representing trees. It develops a comprehensive framework linking metric, order, and ball-structure, including a cycle-based ultrametricity criterion, a graph-theoretic characterization via complete multipartite diametrical graphs, and a detailed description of distance sets and ultrametric-preserving functions. It also introduces and analyzes ultrametric balleans, showing how ball structures and completions influence order isomorphisms between balleans of dense subsets and the existence of isometric embeddings into compact models. Together, these results yield constructive classifications of representing trees, open-ball posets, and the interplay between weak similarities, distance sets, and ultrametric preservation, with implications for completions, embeddings, and ballean theory.

Abstract

We investigate the interrelations between the metric properties, order properties and combinatorial properties of the set of balls in totally bounded ultrametric space. In particular, the Gurvich-Vyalyi representation of finite, ultrametric spaces by monotone rooted trees is generalized to the case of totally bounded ultrametric spaces. It is shown that such spaces have isometric completions if and only if their labeled representing trees are isomorphic. We characterize up to isomorphism the representing trees of these spaces and, up to order isomorphism, the posets of open balls in such spaces.

Totally bounded ultrametric spaces and locally finite trees

TL;DR

The paper advances the theory of totally bounded ultrametric spaces by extending the Gurvich–Vyalyi tree representation to this broader class and establishing that isometric completions correspond to isomorphic labeled representing trees. It develops a comprehensive framework linking metric, order, and ball-structure, including a cycle-based ultrametricity criterion, a graph-theoretic characterization via complete multipartite diametrical graphs, and a detailed description of distance sets and ultrametric-preserving functions. It also introduces and analyzes ultrametric balleans, showing how ball structures and completions influence order isomorphisms between balleans of dense subsets and the existence of isometric embeddings into compact models. Together, these results yield constructive classifications of representing trees, open-ball posets, and the interplay between weak similarities, distance sets, and ultrametric preservation, with implications for completions, embeddings, and ballean theory.

Abstract

We investigate the interrelations between the metric properties, order properties and combinatorial properties of the set of balls in totally bounded ultrametric space. In particular, the Gurvich-Vyalyi representation of finite, ultrametric spaces by monotone rooted trees is generalized to the case of totally bounded ultrametric spaces. It is shown that such spaces have isometric completions if and only if their labeled representing trees are isomorphic. We characterize up to isomorphism the representing trees of these spaces and, up to order isomorphism, the posets of open balls in such spaces.

Paper Structure

This paper contains 22 sections, 127 theorems, 696 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Ultrametric balls
  3. Totally bounded metric spaces
  4. To ultrametricity via cycles and complete multipartite graphs
  5. Cycles in ultrametric spaces
  6. Complete multipartite graphs and ultrametric spaces
  7. The structure of distance sets and ultrametric preserving functions
  8. Basic facts about posets and isotone mappings
  9. Distance sets of ultrametric spaces
  10. Ultrametric preserving functions and weak similarities
  11. The first look on ultrametric balleans
  12. Metric properties of ultrametric balleans
  13. Free representing trees
  14. Rooted trees and related orderings
  15. Generating order by choosing root of the tree
  16. ...and 7 more sections

Key Result

Proposition 2.5

Let $(X, d)$ be ultrametric. Then the following statements hold:

Figures (15)

  • Figure 1: The graph of strictly increasing ultrametric preserving function $g \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ corresponding to the scaling function $\psi$ defined by rule \ref{['ex5.30:e5']}.
  • Figure 2: $(X, d)$ is an interval $I$ in $E^{1} = \mathbb{R}$ and $(Y, \rho)$ is a subspace of the Euclidean plane $E^{2}$ such that $Y \supset I$, all points $y \in Y \setminus I$ are isolated and the distances between them decrease as they approach to $I$.
  • Figure 3: The graph $T_X$ corresponding to $(X, d)$ is a tree.
  • Figure 4: The upper cover of $s_0$ is not unique in $(S, {\preccurlyeq_S})$.
  • Figure 5: The star $S$ and the path $P$ are not isomorphic as free trees, but the labelings $l_S$ and $l_P$ generate the same ultrametric on $V$.
  • ...and 10 more figures

Theorems & Definitions (323)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Remark 2.8
  • ...and 313 more