Combinatorial characterization of pseudometrics

@article{Dovgoshey2020CombinatorialCO,
  title={Combinatorial characterization of pseudometrics},
  author={Oleksiy Dovgoshey and Juoni Luukkainen},
  journal={Acta Mathematica Hungarica},
  year={2020},
  volume={161},
  pages={257-291}
}
Let X, Y be sets and let $$\Phi, \Psi$$ Φ , Ψ be mappings with the domains X 2 and Y 2 respectively. We say that $$\Phi$$ Φ is combinatorially similar to $$\Psi$$ Ψ if there are bijections $$f \colon \Phi(X^2) \to \Psi(Y^{2})$$ f : Φ ( X 2 ) → Ψ ( Y 2 ) and $$g \colon Y \to X$$ g : Y → X such that $$\Psi(x, y) = f(\Phi(g(x), g(y)))$$ Ψ ( x , y ) = f ( Φ ( g ( x ) , g ( y ) ) ) for all $$x, y \in Y$$ x , y ∈ Y . It is shown that the semigroups of binary relations generated by sets $$\{\Phi^{-1… 
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