Combinatorics of Ultrafilters on Cohen and Random Algebras

Abstract

We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukeyincomparable, in stark contrast with the classical setting.Proposition 2.6. Let U and V be ultrafilters on a complete Boolean algebra B.Let us now introduce the notion of Tukey reducibility [17], which will play a role in Sections 3 and 5.Definition 2.7. Let U and V be ultrafilters on a Boolean algebra B. We define U, ≥ ≤ T V, ≥ if and only if there exist functions f : U → V and g :As usual, we let cof(U ) be the minimum cardinality of a cofinal subset of U .Remark 2.8. If a pair of functions f, g witnesses Tukey reducibility as above, then it is easy to verify that, whenever C ⊆ V is cofinal, its pointwise image g [C] is cofinal in U . Furthermore, as observed by Isbell [8], the function g may be taken to be monotonic, in the sense that ifFor ultrafilters over ω, if U ≤ RK V then U, ⊇ ≤ T V, ⊇ , which is a well-known fact proved in Dobrinen and Todorčević [6, Fact 1]. Indeed, the same is true in the general context for the M-ordering.