TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 179, May 1973 THE RUDIN-KEISLER ORDERING OF F-POINTS BY ANDREAS BLASS ABSTRACT. The Stone-Cech compactification ßu of the discrete space u of natural numbers is weakly ordered by the relation "D is the image of E under the canonical extension ßf : ßu -> ßu of some map / : u -> u." We shall investigate the structure, with respect to this ordering, of the set of P-points of ßu - a. 1. Introduction. We shall be concerned with ultrafilters on co. These can be considered from several different viewpoints. In the above abstract, we consid- ered them topologically as points of ßu. From the viewpoint of model theory, we may think of ultrafilters as "the things you use to construct ultraproducts." The abstract would then read as follows. Let JVbe the model whose universe is w and whose relations and functions are all the relations and functions on u. The set of ultrafilters on « is weakly ordered by the relation "Z)-prod ^/Vcan be elementarily embedded into F-prod JV." We shall investigate the structure, with respect to this ordering, of the set of those ultrafilters D such that every nonstandard elementary submodel of D-prod JVis cofinal with Z)-prod JV. The two versions of the abstract describe the same ordering (called the Rudin- Keisler ordering) and the same set of ultrafilters (called the set of F-points). For the most part, we shall consider ultrafilters directly as families of subsets of w, but the model-theoretic interpretation will be used occasionally. Conversely, our theorems can be interpreted as asserting the existence of nonstandard models of arithmetic (i.e. elementary extensions of JV) with certain additional properties. f-points were studied by W. Rudin [9], who proved, using the continuum hypothesis, that they exist and that any one of them can be mapped to any other by a homeomorphism of ßu — w onto itself. Further work on F-points was done by Choquet [4], [5] (who called them ultrafiltres absolument ï-simples in [4] and ultrafiltres 8-stables in [5]), by M. E. Rudin [8] and by Booth [3]. Booth considered two orderings of the set of ultrafilters, the Rudin-Keisler ordering and a stronger one called the Rudin-Frolik ordering. In the latter ordering, to which most of [3] is devoted, all P-points are minimal. Booth did not explicitly consider the Rudin- Keisler ordering of f-points, but, as we shall see, his Theorem 4.12 implies that not all f-points are minimal and their RK ordering is nontrivial (if Martin's Received by the editors March 10, 1971 and, in revised form, March 6, 1972. A MS (A/05) subject classifications (1970). Primary 02H20, 54D35,04-00; Secondary 04A20, 04A25, 04A30,54G10. Key words and phrases. Ultrafilter, P-point, Rudin-Keisler ordering, nonstandard model, ultra- power, elementary embedding, cofinal submodel, Martin axiom. Copyright © 1973, American Mathematical Society 145