Combinatorial properties of ultrametrics and generalized ultrametrics (Q1999126)

Let \(\Phi\) and \(\Psi\) be mappings with \(X^2=X\times X\) and \(Y^2=Y\times Y\) as domains, respectively. The mappings \(\Phi\) and \(\Psi\) are said to be combinatorially similar if there exist bijections \(f : \Phi(X^2)\rightarrow \Psi(Y^2)\) and \(g: Y\rightarrow X\), such that \(\Psi(x, y)=f(\Phi(g(x), g(y)))\) holds for all \(x, y\in Y\). In this paper, the author proves some conditions for a given mapping to be combinatorially similar to an ultrametric or a pseudoultrametric. Some combinatorial characterizations for poset-valued ultrametric distances are also obtained. The author poses one problem at the end of the paper.