Weakly Normal Filters and Irregular Ultrafilters

For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of least functions and irregularity of ultrafilters, are then proved: Let U be a uniform ultrafilter over a regular cardinal k. (a) If k = \+, then U is not (\, \ )-regular iff V has a least function / such that {£ < \+| cf(/(£)) = \} G U. (b) If w < u < k and U is not (u>, u)-regular, then U has a least function. In this paper is considered a relatively new class of filters, those which satisfy a property abstracted from normal ultrafilters over a measurable cardinal. The first section discusses these filters in a general context, and the second shows their relevance to the study of the regularity of ultrafilters. The set theoretical notation and terminology is standard, in particular a,ß,y, . . . are variables for ordinals while k, X, p,... are reserved for cardinals. In fact, k will denote an arbitrary but fixed regular cardinal throughout the discussion. It is always assumed that a filter over K is proper and contains the sets {£|a < % < k} for every a < K, so that ultrafilters are always uniform. This material forms part of the third chapter of the author's doctoral dissertation [4], but the results (except 2.5) were obtained some time ago in 1974. 1. Weakly normal filters. A series of easy definitions culminate in the main concept; recall that a set X has positive measure with respect to a filter F iff X meets every element of F. 1.1. Definitions. Let F be a filter over k. (i) fEKK is unbounded (mod F) iff{% < n\a </(£)} has positive measure for every a<n. (ii) /£ kk is almost 1-1 iff foi every a<K, l/_1({a})| < k. /E kk is almost 1-1 (mod F) iff there is a set X of positive measure so that f\X is almost 1-1, i.e. for every a < k, l/_1({a}) n X\ <K. (iii) F is a p-point filter iff every function unbounded (mod F) is almost 1-1 (mod F). (iv) fEKK is a least function (mod F) ifff is unbounded (mod F) yet Received by the editors May 7, 1975 and, in revised form, June 2, 1975. AMS (MOS) subject classifications (1970). Primary 04A10, 02K35.