| Résume | In this talk, I shall report on joint work with Jörg Brendle, focusing on the combinatorial properties of ultrafilters on Boolean algebras in relation to the Tukey and Rudin-Keisler orderings. First, I aim to introduce the framework of Tukey reducibility and discuss the existence of non-maximal ultrafilters. Furthermore, I shall connect this discussion with a cardinal invariant of Boolean algebras, the ultrafilter number, and sketch consistency results (and open questions) concerning its possible values on Cohen and random algebras. Finally, I will analyse and compare two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras, introducing new techniques to construct incomparable ultrafilters in this setting. |
