Ultrafilter Mappings and Their Dedekind Cuts

Andreas Blass
1974 Transactions of the American Mathematical Society  
Let D be an ultrafilter on the set N of natural numbers. To each function p: N -N and each ultrafilter E that is mapped to D by p, we associate a Dedekind cut in the ultrapower ö-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. Although some of what we say can be generalized
more » ... can be generalized to larger cardinals, we shall confine our attention to ultrafilters on a countable set, which we may take to be the set /V of natural numbers. It will be convenient to identify N xN with N by means of one of the standard pairing functions; thus, the projections, and ff2» ft°m 'V x N to N may be viewed as maps from iV to Af. If E is an ultrafilter on iV, we define an equivalence relation on the functions from /V to N by declaring two functions to be equal mod E iff their restrictions to some set in E are the same. The equivalence class [/] of a function / is called its germ (more precisely, its E-germ [/lg), and the set of all germs is the ultrapower E-prod N. All relations and operations defined on N have natural extensions making E-prod iV an elementary extension of N, provided we identify the germs of constant functions with the values of these functions (see [8] ).
doi:10.2307/1996783 fatcat:b5ifrb4kpfh2tp3vdu3cd3aouu