## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Combinatorial properties of ultrametrics and generalized ultrametrics

Oleksiy Dovgoshey

#### 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

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

#### 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