Ultrafilters over a Measurable Cardinal

The extensive theory that exists on ~ca, the set of uitrafilters ov-.r the integers, suggests an analogous study of the family of g-complete ultrafih~rs over a measurable cardinal g > w. This paper is devoted to such a study, with emphasis on those aspects which make the uncountable case interesting and distinctive. Section 1 is a preliminary section, recapitulating some knov, n concepts and results in the theory of ultrafilters, while Section 2 introduces the convenient frameworR of Puritz for discussing elementary embeddings of totally ordered structures. Section 3 then begins the study in earnest, and introduces a function r on ultrafilters which is a measure of complexi*7. Section 4 is devoted to p-points; partition properties akin to the familiar Ramsey property of normal ultrafilters are shown to yield non-trivial p-points, and examples are constructed. In Section 5 sum and limit constructions are considered; a new proof of a theorem of $oiovay and a generalization are given, and R is shown that the Rudin-Frol ik tree ca ,not have much height. Finally, Section 6 discusses filter related formulations of the well-known Jonsson and Rowbottom properties of cardinals. The notation used in this paper is much as in the most recent set theoretical literature, but the following are specified: The letters a,/3, % 6 ... denote ordinals whereas g, ~,,/~, ;, ... are reserved for cardinals. I f x and y are sets, zy denotes the set of functions from x to y, 'so that gx is the cardinality of xg. i f x is a set, ~ (x ) denotes Rs power set. id