Ultrafilter with $\aleph _0$ predecessors in Rudin-Frolík order
Lev Bukovský, Eva Butkovičová · Czech digital mathematics library · 1981
We describe a construction of an ultrafilter p on the set |N of integers with countable set of types of predecessors of p in the Rudin-Frolik order.The relation ship between characters of ultrafilters and Rudin-Frolik or der is studied and the obtained result is used in the abovementioned construction.Key words; Ultrafilter, type of ultrafilter, Rudin-Fro lik order, character of a filter, P-point.Classification: 04A20 § 0. Introduction* The main result of this paper is a proof of the following theorem.Theorem A. There exists an ultrafilter p on the set \H such that the set of types (0.1) i*Cq); qi-pJ in the Rudin-Frolik order is isomorphic to the inverse order of the set of natural numbers.Assuming the continuum hypothesis, this theorem has been proved by A. Louveau 111 and R.C. Solomon LT21.Our proof does not need any set-theoretical assumption and works in any rea sonable set theory, e.g. in the Zermelo-Fraenkel set theory with the axiom of choice.By a slight modification we obtain also