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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

##### MSC:
 300000 Other combinatorial set theory
##### Keywords:
Rudin-Frolík order; types of ultrafilters
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