Partial orders on the types in $\beta N$

Abstract

Three partial orders on the types of points in fN are defined and studied in this paper. Their relation to the types of points in fNN is also described. Several natural partial orders can be given to the types of points in ,BN. The purpose of this paper is to give some of these orders wider publicity. I feel these orders are fundamental in the study of ultrafilters on the integers. I had hoped these orders would lead to a classification of the types of points in N*. I no longer feel this is true, but connections with this important unsolved problem are discussed. I. Let N denote the set of all positive integers and S the set of all subsets of N. Let ,BN denote the set of all ultrafilters on N and N* the set of all free ultrafilters on N. For Mc N let W(M) be the set of all terms of fN to which M belongs. Then the set of all W(M) for Mc N forms a basis for a topology on ,BN and the resulting space is topologically the Cech compactification of the integers and N* is topologically ,BNN. To avoid ambiguity let n' be the ultrafilter to which the integer n belongs and N' the set of all fixed ultrafilters; thus N* = PN-N'. If p and q are points of a topological space X, p and q are of the same type in X provided there is a homeomorphism of X onto itself taking p into q. It is easy to see [1] that two ultrafilters on N are of the same type in ,BN if and only if there is a permutation of N which takes the members of orne onto the members of the other. That Q and 0 are of the same type in ,BN will be denoted by Q 0 and [Q] will denote the set of all ultafilters on N which are of the same type as Q. Clearly is. an equivalence relation and [Q] has c members. The problem of characterizing the types of points in N* is the problem of finding reasonable necessary and sufficient conditions on terms Q and 0 of N* so that one can construct a permutation of S which preserves infinite intersections and takes the members of Q onto the members of 0. A term Q of N* is called a P-point provided, for every countable subcollection {En}nEN of Q, there is a term E of Q such that E-En is finite for all n. In [1] Walter Rudin proves that the continuum hypothesis [CH] implies the existence of P-points in N* and that all P-points are of the same type in N*. Booth [3] has shown, using Martin's axiom rather than [CH], that there are P-points in N* without an 9, base. Received by the editors March 13, 1970. AMS 1969 subject classifications. Primary 5453, 0415.