Orderings of ultrafilters on Boolean algebras

TL;DR

This paper explores two generalizations of the Rudin-Keisler ordering for ultrafilters on Boolean algebras, introduces new construction techniques, and examines their relation to Tukey reducibility, revealing complex interactions under the Continuum Hypothesis.

Contribution

It develops new methods to construct incomparable ultrafilters in Boolean algebras and analyzes their relationships with Tukey reducibility, highlighting differences from classical cases.

Findings

Existence of ultrafilters on Cohen algebra that are RK-equivalent but Tukey-incomparable under CH

New techniques for constructing incomparable ultrafilters on Boolean algebras

Distinct behaviors of ultrafilter orderings in Boolean algebras compared to classical settings

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 Tukey-incomparable, in stark contrast with the classical setting.