Combinatorial characterization of pseudometrics

Abstract

Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ is combinatorially similar to $\Psi$ 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 $x$, $y \in Y$. It is shown that the semigroups of binary relations generated by sets $\{\Phi^{-1}(a) \colon a \in \Phi(X^{2})\}$ and $\{\Psi^{-1}(b) \colon b \in \Psi(Y^{2})\}$ are isomorphic for combinatorially similar $\Phi$ and $\Psi$. The necessary and sufficient conditions under which a given mapping is combinatorially similar to a pseudometric, or strongly rigid pseudometric, or discrete pseudometric are found. The algebraic structure of semigroups generated by $\{d^{-1}(r) \colon r \in d(X^{2})\}$ is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics $d \colon X^{2} \to \mathbb{R}$.