Combinatorial characterization of pseudometrics


Let X, Y be sets and let \(\Phi, \Psi\) be mappings with the domains X2 and Y2 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 \(d \colon X^{2} \to \mathbb{R}\).

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Dovgoshey, O., Luukkainen, J. Combinatorial characterization of pseudometrics. Acta Math. Hungar. 161, 257–291 (2020).

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Key words and phrases

  • pseudometric
  • strongly rigid metric
  • equivalence relation
  • semigroup of binary relations

Mathematics Subject Classification

  • primary 54E35
  • secondary 20M05