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Long chains in Rudin-Frolík order. (English) Zbl 0564.04005
From the result of Z. Frolík the possibilities for the cardinality of branches in Rudin-Frolík order of types of ultrafilters in \(\beta\) N are \(2^{\aleph_ 0}\) or \((2^{\aleph_ 0})^+\). In this paper the author shows that there is a set of ultrafilters \(\{U^{\alpha}_{\beta,n}|\) \(n\in \omega\), \(\alpha <(2^{\aleph_ 0})^+\), \(\beta <\alpha \}\subseteq \beta N\) such that (1) for each \(\alpha <(2^{\aleph_ 0})^+\), \(\beta <\alpha\), \(\{U^{\alpha}_{\beta,N}|\) \(n<\omega \}\) satisfies that there is \(\{D^{\alpha}_{\beta,N}|\) \(n<\omega \}\) such that \(D^{\alpha}_{\beta,k}\in U^{\alpha}_{\beta,k}\) for \(k<\omega\) and \(D^{\alpha}_{\beta,k}\cap D^{\alpha}_{\beta,l}=\emptyset\) for \(k\neq l\), (2) if \(\beta <\delta <\alpha\), \(k\in \omega\) then \(U^{\alpha}_{\beta,k}=\sum (\{U^{\alpha}_{\delta,l}|\) \(l<\omega \},U^{\delta}_{\beta,k})\). Hence there is a chain order- isomorphic to \((2^{\aleph_ 0})^+\) in Rudin-Frolík order of \(\beta\) N.
Reviewer: N.Kubota

03E05 Other combinatorial set theory
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