Continuous and other finitely generated canonical cofinal maps on ultrafilters

TL;DR

This paper explores conditions under which canonical cofinal maps on ultrafilters are continuous or finitely generated, revealing their structure and connections to various reducibilities like Tukey, Rudin-Keisler, and Rudin-Blass.

Contribution

It establishes that monotone cofinal maps on certain ultrafilters are canonical on cofinal subsets, linking these maps to finitary and continuous structures, and explores their implications for reducibility hierarchies.

Findings

Every ultrafilter Tukey reducible to a p-point has continuous Tukey reductions.

Monotone cofinal maps from Fubini iterates of p-points are generated by finitary, end-extension preserving maps.

Ultrafilters Tukey reducible to Fubini iterates of p-points have finitely generated cofinal maps.

Abstract

This paper investigates conditions under which canonical cofinal maps of the following three types exist: continuous, generated by finitary end-extension preserving maps, and generated by finitary maps. The main theorems prove that every monotone cofinal map on an ultrafilter from a certain class of ultrafilters is actually canonical when restricted to some cofinal subset. These theorems are then applied to find connections between Tukey, Rudin-Keisler, and Rudin-Blass reducibilities on large classes of ultrafilters. The main theorems on canonical cofinal maps are the following. Under a mild assumption, basic Tukey reductions are inherited under Tukey reduction. In particular, every ultrafilter Tukey reducible to a p-point has continuous Tukey reductions. If $\mathcal{U}$ is a Fubini iterate of p-points, then each monotone cofinal map from $\mathcal{U}$ to some other ultrafilter is generated (on a cofinal subset of $\mathcal{U}$) by a finitary map on the base tree for $\mathcal{U}$ which is monotone and end-extension preserving - the analogue of continuous in this context. Further, every ultrafilter which is Tukey reducible to some Fubini iterate of p-points has finitely generated cofinal maps. Similar theorems also hold for some other classes of ultrafilters.