Ultrafilters without immediate predecessors in Rudin-Frolik order for regulars

The aim of this paper is to construct ultrafilters without immediate predecessors in the Rudin-Frolik order in $\beta \kappa\setminus \kappa$, where $\kappa$ is a regular cardinal. This generalizes the problem posed by Peter Simon more than 40 years ago.


Introduction
The Rudin-Frolik ordering of ultrafilters has a long history. It was defined by Z. Frolik in [9] who used it to prove that βω \ω is not homogeneous. M.E. Rudin, who nearly defined this ordering in cite [16], is credited with being the first to notice that the relationship between filters she used is actually ordering. D. Booth in [2] showed that this relation is a partial ordering of the equivalence classes, that is a tree, and that it is not well-founded.
The author defined and studied the partial orders on the type of points in βω and in βω \ ω in [17]. These definitions were used later in [3] and [4].
E. Butkovičová between 1981 and1990, published a number of papers concerning ultrafilters in the Rudin-Frolik order in βω \ ω. In [3] with L. Bukovský and [4] she constructed an ultrafilter on ω with a countable set of its predecessors. In [5] she constructed ultrafilters without immediate predecessors. In [6] Butkovičová showed that there exists in Rudin-Frolik order an unbounded chain order-isomorphic in ω 1 . In [7] she proved that there is a set of 2 2 ℵ 0 ultrafilters incomparable in Rudin-Frolik order which is bounded from below and no its subset of cardinality more than one has an infimum. In [8] Butkovičová proved that for every cardinal between ω and c there is a strictly decreasing chain without a lower bound. In most of these papers, the method presented in [15].
In 1976, A. Kanamori published a paper [14] in which, among other things, it showed that the Rudin-Frolik tree cannot be very high if one considers it over a measurable cardinal. Moreover, in the same paper he left a number of open problems regarding the Rudin-Frolik order. Recently, M. Gitik in [10] answered some of them by using meta-mathematical methods. The solutions to some of the problems from [14] presented in combinatorial methods are in preparation, ( [13]).
Since the Rudin-Frolik order is still not well researched, especially for ultrafilters over cardinals greater than ω, it seems worthwhile to look at them for such cardinals. For this purpose, special techniques are needed.
In 2001, Baker and Kunen published in [1] a very useful method which can be recognized as a generalization of the method presented in [15]. It is worth emphasizing that both methods, (from [15] and [1]), provide useful "technology" for keeping the transfinite construction of an ultrafilter not finished before c steps, (see [15]), and 2 κ , for κ being infinite cardinal, (see [1]), but the second method has some limitations, among others κ must be regular. As it turned out, the method from [1] can be useful in keeping the results for the Rudin-Frolik order but for κ-regular. Due to the lack of adequate useful method for κ-singular, the similar results but for singulars are still open questions. Based on [1] and papers by Butkovičová, the investigations into the Rudin-Frolik order for regulars have further consequences, which will be presented in [11,12]. The starting point of this paper, based on [5], is the following problem posed by Peter Simon, known as Simon's problem: Does there exist a non-minimal ultrafilter in the Rudin-Frolik order of βω \ω without immediate predecessor?
One does not know how old the question is, but it was mentioned in [3] in 1981 and solved in [5] in 1982. As was mentioned above, the main method used in [5] is the method introduced by K. Kunen in [15] for keeping the transfinite construction of an ultrafilter from finishing before continuum steps.
As far as one knows, nobody considers the Simon's problem in the general case, i.e. for βκ, for arbitrary cardinal κ. This is the main idea for checking whether such a generalization is possible. The answer is yes, but for κregular. It is dictated by an adequate method based on the one presented in [1], which just works for regulars.
The paper is organized as follows: in Section 2, there are presented definitions and previous facts needed for proving the results in further parts of this paper. Section 3 contains main results preceded by auxiliary results. The paper is finished with the open problem for singulars.

Definitions and previous results
2.1. In the whole paper, we assume that κ is an infinite cardinal. Then βκ means theČech-Stone compactification, where κ has the discrete topology. Hence, βκ is the space of ultrafilters on κ and βκ \ κ is the space of nonprincipal ultrafilters on κ. We will write u(κ) to denote the space of uniform The following fact, taken from [17], is easily generalized Fact 1. Let κ be a regular infinite cardinal. If X, Y are κ-discrete sets of ultrafilters on κ and G ∈ X ∩ Y , then
An ultrafilter G ∈ βκ\κ has an immediate predecessor F in Rudin-Frolik order iff G = Σ(X, F) for some κ-discrete set X of minimal ultrafilters in Rudin-Frolik order.
A Simon point in βκ \ κ is an ultrafilter G ∈ βκ \ κ which is non-minimal and has no immediate predecessors.
2.6. Let X = {F ξ,ζ : ξ < τ, ζ < κ} be a stratified set of filters and let W be its subset. We define Intuitively, the above construction is used to select only certain filters from X with the desired property (see e.g. property (P) in 2.8), and then add (inductively) to the set W only those filters outside W which satisfy the condition (2) in the definition in 2.5. This construction will be used to define property (P), (see 2.8), the formulation of which would not be possible by taking the entire ultrafiter family into account.
A step family (over κ, with respect toφ) is a family of subsets of κ, satisfying the following conditions: Let I be an index set and F be a filter on κ. The family is an independent matrix of |I| step-families (over κ) with respect to (shortly w.r.t.) F,φ iff ). If κ is a regular cardinal andφ is a κ-shrinking function, then there exists an independent matrix of 2 κ step-families over κ with respect to the filter {D ⊆ κ: |κ \ D| < κ},φ.

Auxiliary Results
Lemma 1 Let κ be a regular cardinal and letφ be a κ-shrinking function.
By Fact 2, fix a matrix of 2 κ step-families over κ independent with respect to the filter FR(κ).
Theorem 1 Let κ be a regular cardinal and letφ be a κ-shrinking function. There exists a Simon point in βκ \ κ.
Proof. The theorem implies directly from Lemma 1 and Lemma 2.
Open problem Is there a Simon point in βκ \ κ for κ-singular?