Tukey-idempotency and strong p-points
Tom Benhamou, Natasha Dobrinen, Tan Özalp
TL;DR
This work characterizes strong $p$-points of $ olinebreak[4] ω$ in terms of Tukey order, showing an ultrafilter is a strong $p$-point iff it is not Tukey above $( olinebreak[4] ω^ω, ≤)$. The authors establish a web of equivalences linking strong $p$-points, Canjar ultrafilters, and $P^+$-filters via the relation $Uullet U>_T U$, and connect these to forcing notions and $I$-p.i.p. frameworks. They prove that no analytic ideal $I$ with $P( olinebreak[4] ω)/I$ not adding reals can yield a Canjar ultrafilter, and they show there are no Canjar ultrafilters on measurable cardinals. Extending to topological Ramsey spaces, they show that, under mild hypotheses, TRS-derived ultrafilters are Tukey-idempotent, with broad consequences for the structure and commutativity of Fubini products of such ultrafilters. The results resolve several open questions and illuminate the landscape of Tukey-types among ultrafilters, including applications to forcing and questions about definable ideals and large cardinals.
Abstract
We characterize strong $p$-point ultrafilters by showing that they are exactly those $p$-points that are not Tukey above $(ω^ω,\leq)$; or equivalently, those $p$-points that are not Tukey-idempotent. Moreover, we show that there are no Canjar ultrafilters on measurable cardinals. We make use of tools which were motivated by topological Ramsey spaces, developed in \cite{Benhamou/Dobrinen24}, and furthermore, show that ultrafilters arising from most of the known topological Ramsey spaces are Tukey-idempotent. Our results answer questions of Hrušák and Verner \cite[Question 5.7]{Hrusak/Verner11}, Brook-Taylor \cite[Question 3.6]{QuestionGeneralized}, and partially Benhamou and Dobrinen \cite[Question 5.6]{Benhamou/Dobrinen24}.