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تسجيل مستخدم جديدCombinatorial properties of ultrametrics and generalized ultrametrics
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تأليف
O. Dovgoshey
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Let $X$, $Y$ be sets and let $Phi$, $Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $Phi$ and $Psi$ are combinatorially similar 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 $x$, $y in Y$. Conditions under which a given mapping is combinatorially similar to an ultrametric or a pseudoultrametric are found. Combinatorial characterizations are also obtained for poset-valued ultrametric distances recently defined by Priess-Crampe and Ribenboim.
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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 $Ps
i(x, y) = f(Phi(g(x), g(y)))$ 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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