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 combinatorially similar if there are bijections $f:\Phi (X^2)\to \Psi (Y^2)$ and $g:Y\to X$ 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.

41 pages, 2 figures

References in corpus (3)

• Products of metric spaces, covering numbers, packing numbers and characterizations of ultrametric spaces
• Weak similarities of finite ultrametric and semimetric spaces
• Combinatorial characterization of pseudometrics

Cited by in corpus (1)

• On characterization of functions preserving metric-type conditions via triangular and polygonal structures