A Remark on the Rudin-keisler Order of Ultrafilters

Define a (pre)order on /3co as follows: p • q iff there is a finite to one function f c woo such that/3f(cl) = p. Let ,•{ denote he Rudin-Keisler order on Ow. We show that here are points p,qCw* such that Px•q, but p and q are •<-incomparable. 0. Introduction. In [5], M. E. Rudin showed that if there are points p,q G co* Such that for any finite to one function f G cow we have that 13f(p) 4:13f(q), then the indecomposable continuum H*, where H denotes the half-line [0,oo), has at least two composants. This result suggests to define a (pre)order •< on 13co as follows: p •< q iff there is a finite to one f C cow such that 13f(q) = p. Observe that this order is quite similar to the Rudin-Keisler order on 13co, since this order is defined by p q• q iff there is an f C cow such that 13f(q) = p. Intuitively, •{ is finer than • and the aim of this note is to make this precise. First observe that if p •< q then p • q, but not conversely. For if p = 0 and q • co*, then p and q are •<.-incomparable andp x{ q. This shows that •< is only interesting on co*. We will show that there are points p,q Gco* such that p •{ q but p and q are •<.-incomparable. 1. Independent matrices and R-points. An indexed family {A•: i • I, j • J) of clopen subsets of co* is called a J by I independent matrix if i for all distinct j 0,j 1 c J and i • I we have that A•O 1 for each finite F C I and •o: F -} J it is true that iX . Ch (A•iX) iX • F ) 4: 1⁄2. This concept is due to K. Kunen. The following Lemma follows immediately from [3, 2.21.