Bulletin of the Belgian Mathematical Society - Simon Stevin

Combinatorial properties of ultrametrics and generalized ultrametrics

Oleksiy Dovgoshey

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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

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

First available in Project Euclid: 9 September 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 54E35: Metric spaces, metrizability
Secondary: 06A05: Total order 06A06: Partial order, general

ultrametric generalized ultrametric equivalence relation poset totally ordered set isotone mapping


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

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