Title:

The classification of ultrafilters on N

This thesis has as its aim the classification of ultrafilters on N, by use of partitions and collections of partitions of N, and the investigation of the operations under which each class is closed and the inclusion/exclusion relationships between them. Choquet (1) asked whether there was a n.p.u.f ? on N such that for no map 0 : N ? N is 0 absolute; Mathias answered that there was by constructing a n.p.u.f. ? on N with the stronger property that for no map 0 : N ? N is 0 a P point (of ?NN). In Chapter II we construct a P point ? such that for no map 0 : N ? N is 0 rare and, using this result, we construct a n.p.u.f. ? on N such that for no map 0 : N ? N is 0 a P point or a rare ultrafilter. We also show that if ? is a n.p.u.f. on N that cannot be mapped to a rare ultrafilter then ? cannot be mapped to a countable limit of absolute ultrafilters.;NOTATION: Let ? be a n.p.u.f. on N and s be a partition of N into finite sets (p.o.N.i.f.s.); we will write ? ? s whenever F ? ? implies F ? A ? 1 for each A ? s. Let S1,S2 be p.o.N.i.f.s.; we will write S1?S2 whenever A E S1 and B ? S2 imply A ? B ? 1.;DEFINITION: A n.p.u.f. ? on N is an n(a) point (point of degree of complexity n) if for every collection S = S1,S2,...,Sn+1 of p.o.N.i.f.s. satisfying Si?sj for 1 ? i < j ? n+1 there is a t, 1 ? t ? n+1 such that ??st, and this is the least n for which it is true. Chapter III is devoted to the investigation of this and allied notions. We extend the idea to allow infinite degrees of complexity (S. and c) and show that for any n.p.u.f. ? on N there is a n ? {0,1,.., s.,c} such that ? is a n(a) point. We also show that for any n,n(a) P points exist. Many of the theorems give bounds for the degree of complexity of ultrafilters of the form ? = ? lim ?i given the degree of complexity of ?, ?1, ?2,.. and given that ? has a certain property (e.g. ? is a P point). The first section of Chapter IV gives counterexamples to the following plausible hypotheses: 1) each 1(a) point is rapid; 2) each c(a ) point is not rapid. The final section of the thesis deals with a concept appearing in a letter from Professor G. Choquet to Dr. R.O.Davies.;DEFINITION: A n.p.u.f. ? on N has property c if for any pair of maps 0,? : N ? N, 0 = ? implies 0 and ? agree on some member of ?. We show that there is a n.p.u.f. ? on N that is neither a P point nor a rare ultrafilter with property c and a n.p.u.f. on N that is a P point without property c. We investigate the relationships between the class of n.p.u.f.'s with property c and the classes defined previously. 1) G. Choquet, Deux classes remarquables d'ultrafiltres sur N, Bull.Sci.Math.(2) 92 (1968), 143153.
