• DOI:10.1007/s10474-020-01020-x
• Corpus ID: 189999792

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}
}

• O. Dovgoshey, J. Luukkainen
• Published 18 June 2019
• Mathematics
• Acta Mathematica Hungarica

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 $$\{\Phi^{-1}(a) \colon a \in \Phi(X^{2})\}$$ { Φ - 1 ( a ) : a ∈ Φ ( X 2 ) } and $$\{\Psi^{-1}(b) \colon b \in \Psi(Y^{2})\}$$ { Ψ - 1 ( b ) : b ∈ Ψ ( Y 2 ) } are isomorphic for combinatorially similar $$\Phi$$ Φ and $$\Psi$$ Ψ . 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 $$\{d^{-1}(r) \colon r \in d(X^{2})\}$$ { d - 1 ( r ) : r ∈ d ( X 2 ) } is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics $$d \colon X^{2} \to \mathbb{R}$$ d : X 2 → R .