arXiv:1503.06092v1 [math.LO] 20 Mar 2015

arXiv:1503.06092v1 [math.LO] 20 Mar 2015

Gδ SEMIFILTERS AND ω ∗ WILL BRIAN AND JONATHAN VERNER

Abstract. The ultrafilters on the partial order ([ω]ω , ⊆∗ ) are the free ultrafilters on ω, which constitute the space ω ∗ , the Stoneˇ Cech remainder of ω. If U is an upperset of this partial order (i.e., a semifilter ), then the ultrafilters on U correspond to closed subsets of ω ∗ via Stone duality. If, in addition, U is sufficiently “simple” (more precisely, Gδ as a subset of 2ω ), we show that U is similar to [ω]ω in several ways. First, pU = tU = p (this extends a result of Malliaris and Shelah). Second, if d = c then there are ultrafilters on U that are also P -filters (this extends a result of Ketonen). Third, there are ultrafilters on U that are weak P -filters (this extends a result of Kunen). By choosing appropriate U , these similarity theorems find applications in dynamics, algebra, and combinatorics. Most notably, we will answer two open questions of Hindman and Strauss by proving that there is an idempotent of ω ∗ that is both minimal and maximal.

1. Introduction The main theme of this paper is that there is a class of “simple” semifilters whose members all look essentially like [ω]ω , the set of infinite subsets of ω. We will prove several theorems along these lines, and also find applications of these theorems. Perhaps the most interesting of these applications is that we resolve two open questions of Hindman and Strauss (Questions 9.25 and 9.26 in [13]) by proving that there is an idempotent of ω ∗ that is both minimal and maximal. Recall that any semifilter is naturally identified with a subset of 2ω via characteristic functions. The “simple” class of semifilters we are interested in are those that are Gδ in 2ω (where 2ω has its standard topology as the Cantor set). Some of our proofs will also work for 2010 Mathematics Subject Classification. Primary: 03E17, 54D35. Secondary: 22A15, 03E35, 06A07. Key words and phrases. semifilter, ω ∗ , (weak) P -filter, (weak) P -set, minimal ideal, minimal/maximal idempotent. 1

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co-meager semifilters. Clearly [ω]ω is in this class because it is cocountable. We will show that many of its properties, including some that correspond to interesting properties of ω ∗, can be proved for any other semifilter in this class as well. For example, in Section 3, we show that Gδ semifilters all satisfy the Malliaris-Shelah equality p = t (see [19]). That is, defining pG and tG appropriately, we show that for a Gδ semifilter G, pG = tG = p. In Section 4, we show that if d = c then every Gδ semifilter admits an ultrafilter that is also a P -filter. This generalizes a result of Kentonen from [17], which says the same thing for [ω]ω . In Section 5, we show that every Gδ semifilter admits an ultrafilter that is a weak P -filter. This generalizes a result of Kunen from [18], which says the same thing for [ω]ω . In Section 6 we have collected a few applications of these results. As mentioned above, our primary application is to show there is an idempotent in ω ∗ that is both minimal and maximal. Hitherto, this was known to follow from p = c (see [26] or, later but independently, [8]), but it was not known whether ZFC could prove even the existence of maximal idempotents. 2. Preliminaries A semifilter is a subset S of P(ω) such that ∅ = 6 S = 6 P(ω) and S is closed upwards in ⊆∗ (as usual, A ⊆∗ B means A \ B is finite). We think of semifilters as partial orders, naturally ordered by ⊆∗ . The largest possible semifilter is [ω]ω , the set of all infinite subsets of ω. The separative quotient of [ω]ω is P(ω)/fin. While many authors prefer to speak of P(ω)/fin rather than [ω]ω when speaking of ultrafilters, it is less convenient for our purposes. We have no need (and no desire) to work with equivalence classes, and will not need to use the separativity or Boolean structure of P(ω)/fin. If S is a semifilter, then a filter on S is a filter on the partial order (S, ⊆∗ ). Specifically, F ⊆ S is a filter on S whenever • F= 6 ∅. • A ∈ F and A ⊆∗ B implies B ∈ F . • A, B ∈ F implies A ∩ B ∈ F . An ultrafilter on S is a maximal filter on S. B ⊆ S is a filter base on S if {A ⊆ N : B ⊆∗ A for some B ∈ B} is a filter on S. A set is centered in S if it is contained in some filter base. The collection of all ultrafilters on [ω]ω is denoted ω ∗ . This set has a ˇ natural topology as the Stone-Cech remainder of ω, with basic open sets ∗ ∗ of the form A = {F ∈ ω : A ∈ F }. Every filter F on [ω]ω corresponds

Gδ SEMIFILTERS AND ω ∗

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T to a closed subset of ω ∗ , namely Fˆ = A∈F A∗ . Fˆ is called the Stone dual of F . For more on Stone duality and the topology of ω ∗, we refer the reader to [22]. If F is a filter on some semifilter S, then F is also a filter on [ω]ω , although an ultrafilter on S may not be an ultrafilter on [ω]ω . Thus the (ultra)filters on some semifilter S correspond to closed subsets of ω ∗. For certain choices of S, these closed sets may have interesting algebraic/dynamic/combinatorial properties, and for certain choices of the ultrafilter they may also have interesting topological properties. The interplay between these two choices will give rise to our applications in Section 6. A subset X of ω ∗ is a P -set if, whenever hUn : n < ωi is aTsequence of open sets each of which contains X, X is in the interior of n