Bulletin of the Belgian Mathematical Society - Simon Stevin

Combinatorial properties of ultrametrics and generalized ultrametrics

Oleksiy Dovgoshey

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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


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