Continuous and other finitely generated canonical cofinal maps on ultrafilters
Volume 249 / 2020
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.
