Ultrafilter mappings and their Dedekind cuts

Abstract

Let D be an ultrafilter on the set N of natural numbers. To each function p : N → N p:N \to N and each ultrafilter E that is mapped to D by p, we associate a Dedekind cut in the ultrapower D-prod N. We characterize, in terms of rather simple closure conditions, the cuts obtainable in this manner when various restrictions are imposed on E and p. These results imply existence theorems, some known and some new, for various special kinds of ultrafilters and maps.