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 .

中文翻译：

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伪度量的组合表征

设 X, Y 为集合，并设 $$\Phi, \Psi$$ Φ , Ψ 分别为域 X 2 和 Y 2 的映射。如果存在双射 $$f \colon \Phi(X^2) \to \Psi(Y^{2})$$，我们说 $$\Phi$$ Φ 在组合上类似于 $$\Psi$$ Ψ f : Φ ( X 2 ) → Ψ ( Y 2 ) 和 $$g \colon Y \to X$$ g : Y → X 使得 $$\Psi(x, y) = f(\Phi(g(x) ), g(y)))$$ Ψ ( x , y ) = f ( Φ ( g ( x ) , g ( y ) ) ) 对于所有 $$x, y \in Y$$ x , y ∈ Y 。表明由集合 $$\{\Phi^{-1}(a) \colon a \in \Phi(X^{2})\}$$ { Φ - 1 ( a ) : a ∈ Φ ( X 2 ) } 和 $$\{\Psi^{-1}(b) \colon b \in \Psi(Y^{2})\}$$ { Ψ - 1 ( b ) : b ∈ Ψ ( Y 2 ) } 对于组合相似的 $$\Phi$$ Φ 和 $$\Psi$$ Ψ 是同构的。给定映射在组合上与伪度量相似的充分必要条件，或强刚性伪度量，或离散伪度量。$$\{d^{-1}(r) \colon r \in d(X^{2})\}$$ { d - 1 ( r ) : r ∈ d ( X 2 ) } 完全描述了非离散的、强刚性的伪度量，以及离散伪度量 $$d \colon X^{2} \to \mathbb{R}$$ d : X 2 → R 。