## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 27, Number 3 (2020), 379-417.

### Combinatorial properties of ultrametrics and generalized ultrametrics

#### Abstract

Let \(X\), \(Y\) be sets and let \(\Phi\), \(\Psi\) be mappings with domains \(X^{2}\) and \(Y^{2}\), respectively. We say that \(\Phi\) and \(\Psi\) are \emph{combinatorially similar} if there are bijections \(f \colon \Phi(X^2) \to \Psi(Y^{2})\) and \(g \colon Y \to X\) such that \(\Psi(x, y) = f(\Phi(g(x), g(y)))\) for all \(x\), \(y \in Y\). Conditions under which a given mapping is combinatorially similar to an ultrametric or a pseudoultrametric are found. Combinatorial characterizations are also obtained for poset-valued ultrametric distances recently defined by Priess-Crampe and Ribenboim.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 27, Number 3 (2020), 379-417.

**Dates**

First available in Project Euclid: 9 September 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1599616821

**Digital Object Identifier**

doi:10.36045/bbms/1599616821

**Mathematical Reviews number (MathSciNet)**

MR4146738

**Subjects**

Primary: 54E35: Metric spaces, metrizability

Secondary: 06A05: Total order 06A06: Partial order, general

**Keywords**

ultrametric generalized ultrametric equivalence relation poset totally ordered set isotone mapping

#### Citation

Dovgoshey, Oleksiy. Combinatorial properties of ultrametrics and generalized ultrametrics. Bull. Belg. Math. Soc. Simon Stevin 27 (2020), no. 3, 379--417. doi:10.36045/bbms/1599616821. https://projecteuclid.org/euclid.bbms/1599616821