Ultrafilter Mappings and Their Dedekind Cuts

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 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]). A function p: N —» /V maps E to the ultrafilter p(E) = {A C ,V| p~ HA) e E\; p(E) depends only on E and the E-germ of p. If p maps E to D, we write p: E —* D. If, in addition, p is one-to-one on some set in E, then we call p an isomorphism from E to D and write p: E Sf D. In this case, there is a permutation of /V equal mod E to /, and there is an isomorphism g:D«E, unique mod D, such that f ° g and g °f are equal, mod D and E respectively, to the identity. It is easy to check that the composite of two isomorphisms is again an isomorphism. If we replace a map p: E —» D by another with the same germ, or if we compose p with an isomorphism E' = E, all the properties of p in which we shall be Received by the editors July 31, 1972. AMS (MOS) subject classifications (1970). Primary 04A20; Secondary 02H13, 02H20, 04A30.