Weakly normal filters and irregular ultrafilters

Abstract

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 ultrafi'lters, are then proved: Let U be a uniform ultrafilter over a regular cardinal K. (a) If K = X+, then U is not (X, A +)-regular iff U has a least function f such that {t < X+1 cf(f(0)) = X} E U. (b) If w S A < K and U is not (w, ,)-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, j, y, . . . are variables for ordinals while K, X, t, .... 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 {tl x < t < 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 [41, 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) f E KK is unbounded (mod F) iff{t < KIt <f(Q)} has positive measure for every a < K. (ii) f E KK is almost 1-1 iff for every a < K, If ({a})I < K. f E KK iS almost 1-1 (mod F) iff there is a set X of positive measure so that fIX is almost l-1,i.e. for every a< K, if-I({a}) fnXI < K. (iii) F is a p-point filter iff every function unbounded (mod F) is almost 1-1 (mod F). (iv) f E KK is a least function (mod F) iff f 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.