2015/05/02 by Natasha Dobrinen, Dobrinen, Natasha · 1 citation
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1505.00368
openalex publication_date 2015/05/02 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
This paper investigates conditions under which canonical cofinal maps of the\nfollowing three types exist: continuous, generated by finitary end-extension\npreserving maps, and generated by finitary maps. The main theorems prove that\nevery monotone cofinal map on an ultrafilter from a certain class of\nultrafilters is actually canonical when restricted to some cofinal subset.\nThese theorems are then applied to find connections between Tukey,\nRudin-Keisler, and Rudin-Blass reducibilities on large classes of ultrafilters.\n The main theorems on canonical cofinal maps are the following. Under a mild\nassumption, basic Tukey reductions are inherited under Tukey reduction. In\nparticular, every ultrafilter Tukey reducible to a p-point has continuous Tukey\nreductions. If \U is a Fubini iterate of p-points, then each\nmonotone cofinal map from \U to some other ultrafilter is generated\n(on a cofinal subset of \U) by a finitary map on the base tree for\n\U which is monotone and end-extension preserving - the analogue of\ncontinuous in this context. Further, every ultrafilter which is Tukey reducible\nto some Fubini iterate of p-points has finitely generated cofinal maps. Similar\ntheorems also hold for some other classes of ultrafilters.\n