On Milliken-Taylor Ultrafilters

We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k-coloured Milliken-Taylor ultrafilters have at least k + 1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model. 1. Milliken-Taylor ultrafilters and their projections We answer a question of López-Abad whether there can be more than two near coherence classes of ultrafilters in the core of a Milliken-Taylor ultrafilter. Then we show that in Milliken Taylor ultrafilter with k colours there are k + 1 near coherence classes in its projection to ω, generalising a result of Blass [7]. Then we investigate whether a Milliken-Taylor ultrafilter is preserved by forcing with another Milliken-Taylor ultrafilter. The somewhat surprising answer is no, independently of the relationship of the two ultrafilters. In the rest of this introductory section we review part of the relevant background. Our nomenclature follows [10] and [5]. We let F be the collection of all non-empty finite subsets of ω. For a, b ∈ F we write a < b if (∀n ∈ a)(∀m ∈ b)(n < m). We will work with proper filters on F, i.e. non-empty subsets of P(F) that are closed under binary intersections and supersets and do not contain the empty set. A filter on F is called non-principal if it does contain all sets of the form FrE, E finite. A sequence c̄ = 〈cn : n ∈ ω〉 of members of F is called unmeshed if for all n, cn < cn+1. Henceforth, barred lower case variables stand for such sequences. For n ≤ ω, the set (F)n denotes the collection of all unmeshed sequences in F of length n. If c̄ is a sequence in (F)ω, we write (FU)ω(c̄) for the set of all unmeshed sequences whose members are finite unions of some of the cn’s and we write FU(c̄) for the set of all finite unions of members of c̄. Date: June 4, 2010. 2010 Mathematics Subject Classification: 03E05, 03E17, 03E35.