Combinatorial properties of ultrametrics and generalized ultrametrics

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 $f colon Phi(X^2) to Psi(Y^{2})$ 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.