Continuous and other finitely generated canonical cofinal maps on ultrafilters

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.