March 6
Tom Benhamou,
Rutgers University
A characterization of ultrafilters coding dominating reals
Canjar and Laflamme studied the equivalent notions of Canjar ultrafilters and Strong P-points (resp.). These are ultrafilters with the property that the respective Mathias forcing does not add a dominating real. They showed that such ultrafilters must be P-points without rapid RK-predecessors, and conjectured that these two properties in fact characterize Canjar ultrafilters. While the Canjar-Laflamme conjecture was proven to be false by Blass-Hrusak-Verner, we present here a characterization of Canjar ultrafilters that catches the underlying intuition of the Canjar-Laflamme conjecture using the cofinal type of $\omega^{\omega}$ with the everywhere domination order. After proving this theorem, we will present several applications of our characterization, including a classification of the class of Tukey-idempotent ultrafilters and an answer to a question of Hrusak-Verner about the possibility of $P(\omega)/I$ adding a Canjar ultrafilter, where $I$ is analytic. This is joint work with Natasha Dobrinen and Tan Ozlap.