Arch. Math. Logic 41, 743–764 (2002) Digital Object Identifier (DOI): 10.1007/s001530200140
Mathematical Logic
Pierre Matet · C´edric P´ean · Stevo Todorcevic
Prime ideals on Pω (λ) with the partition property Received: 6 May 1999 / Revised version: 15 October 2001 Published online: 2 September 2002 – © Springer-Verlag 2002 Abstract. We use ideas of Fred Galvin to show that under Martin’s axiom, there is a prime ideal on Pω (λ) with the partition property for every λ < 2ℵ0 .
0. Introduction For an infinite cardinal λ, the collection of all finite subsets of λ is denoted by Pω (λ). We let Iω,λ denote the collection of all noncofinal subsets of Pω (λ). As a convenient convention the phrase “ideal on Pω (λ)” will mean “proper ideal on Pω (λ) extending Iω,λ ”. Now Ramsey’s theorem can be easily reformulated as + −→ (I + )n ). Theorem a. Iω,ω has the partition property (i.e. ∀ n Iω,ω ω,ω
Fred Galvin (see [1], [11] and [13]) proved the following, which can be seen (since MAℵ0 is always true) as a generalization of Theorem a. Theorem A. Assume MAκ . Then Iω,λ has the partition property for every λ ≤ κ. Let us observe that Theorem a can be restated as follows. Theorem a . There exists an ideal on Pω (ω) with the partition property. Theorem a should be compared with the following, which is an immediate consequence of the fact that Martin’s axiom implies the existence of Ramsey ultrafilters. Theorem b. Assume MA. Then there exists a prime ideal on Pω (ω) with the partition property. The main result of this paper is the following generalization of Theorem b. Theorem B. Assume MA. Then for every λ < 2ℵ0 , there exists a prime ideal on Pω (λ) with the partition property. P. Matet, C. P´ean: Universit´e de Caen-CNRS, ESA 6081, Laboratoire SDAD, Campus 2, 14032 Caen Cedex, France. e-mail:
[email protected] S. Todorcevic: Universit´e Paris 7-CNRS, UPRESA 7056, Equipe de Logique Math´ematique, T45/55 5`eme e´ tage, 2, place Jussieu - Case 7012, 75251 Paris Cedex 05, France e-mail:
[email protected]
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The original proof (by Booth [6]) of the existence of Ramsey ultrafilters under MA was based on the fact that assuming Martin’s axiom, the pseudo-intersection number ᒍ is equal to 2ℵ0 . Thus Theorem b can be refined as follows. Theorem b . Assume ᒍ = 2ℵ0 . Then there exists a prime ideal on Pω (ω) with the partition property. We do not know whether MA implies that the two-cardinal version ᒍω,λ of ᒍ is equal to 2ℵ0 for every λ < 2ℵ0 . (We tend to think that ᒍω,λ is indeed small, i.e. ᒍω,λ ≤ 2ℵ0 for all λ, but do not rule out the possibility that ᒍω,λ > λ+ is consistent for some λ with ω < λ < 2ℵ0 ). It is however possible to generalize Theorem b as follows. Theorem B . B . Assume ᒍ = 2ℵ0 . Then for every λ < 2ℵ0 , there exists a prime ideal on Pω (λ) with the partition property. Ketonen showed in [15] that if cov(M) (the covering number for category) is equal to 2ℵ0 , then every filter on ω with < 2ℵ0 many generators can be extended to a Ramsey ultrafilter. (Canjar [10] and Bartoszy´nski and Judah [3] have shown that the converse holds). This yields the following further refinement of Theorem b. Theorem b . Assume cov(M) = 2ℵ0 . Then there exists a prime ideal on Pω (ω) with the partition property. Theorem b can be generalized by substituting the Tychonoff product 2λ for the Cantor space 2ω . Theorem B . Assume cov(Mω,λ ) = 2λ . Then there exists a prime ideal on Pω (λ) with the partition property. Theorem B is not the end of the story. Ramsey ultrafilters on ω can be characterized in a number of ways, and some of these equivalences do not seem to remain true for Pω (λ). Given a prime ideal J on Pω (λ), let us consider the following list of properties that might be verified by J : (i) J has the partition property. (ii) J is weakly selective. (iii) J has the partition property, and so does every ideal on Pω (λ) that is RudinKeisler equivalent to it. (iv) J is minimal with respect to the Rudin-Keisler order. It is well-known (see Proposition 2.3) that (i) and (ii) are equivalent. What we prove is the following improved version of Theorem B . Theorem B . Assume cov(Mω,λ ) = 2λ . Then there is a prime ideal on Pω (λ) that satisfies (iii). We do not know how to obtain minimality from our assumption, and so use forcing in order to produce a prime ideal that verifies (iii) and (iv).
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In contrast to Pκ (λ) (for uncountable κ), which has been extensively studied in the last twenty-five years, the theory of Pω (λ) has been the object of very few published articles. Obviously, Pω (λ) and Pκ (λ) have much in common, and lots of results on Pκ (λ) are in fact valid for Pω (λ), one important difference being of course that there are no normal ideals on Pω (λ). We will take advantage of this similarity, which will guide us in our choice of notation and terminology and in the formulation of open problems. In Sections 1–3 we review some basic material. Section 1 deals with the notion of ideal on Pω (λ), while Section 2 is devoted to partition properties and associated combinatorial properties. Section 3 is concerned with the Rudin-Keisler order and some related properties of ideals, which are variants of properties considered in [19]. In Section 4 we show that small ideals have nice properties, and we adapt the traditional argument to show the existence of big ideals. Section 5 is concerned with the notion of pseudo-intersection number of an ideal. Results in this section are supposed to shed some light on the results of Sections 8 and 9. Take for instance Proposition 9.2. Its proof uses Corollary 8.3 which shows that if cov(Mω,λ ) > λ, then all ideals on Pω (λ) of cofinality λ have the partition property. Corollary 8.3 seems to give more, namely that if cov(Mω,λ ) > λ, then all ideals on Pω (λ) of cofinality λ have high pseudo-intersection number. It is however shown in Proposition 5.5 that if all ideals on Pω (λ) of cofinality λ have the partition property, then they all have high pseudo-intersection number. Section 6 deals with cov(Mω,λ ), the two-cardinal version of cov(M). In Section 7 we prove a result which will be used in Section 8 to show that the forcing notions used there are equivalent to adding a number of Cohen reals. In this we follow [13] (where the proof of Theorem 4 uses a special case of our result). We show in Section 8 that all ideals on Pω (λ) of cofinality < cov(Mω,λ ) have the partition property. In Section 9 we use the results of the preceding sections to show that cov(Mω,λ ) = 2λ implies the existence of (many) prime ideals on Pω (λ) such that all their RK-equivalents have the partition property. We also force over a model of cov(Mω,λ ) > λ to obtain such ideals with the additional property of being minimal with respect to the Rudin-Keisler order. 1. Ideals on Pω (λ) This section is devoted to the notion of ideal on Pω (λ). We recall basic definitions and review some elementary facts that will be needed later. For each set S, we let Pω (S) denote the collection of all finite subsets of S. Throughout the paper λ will denote an infinite cardinal. For each a ∈ Pω (λ), we set a = {b ∈ Pω (λ) : a ⊆ b} and a = a − {a}. We set Iω,λ = {B ⊆ Pω (λ) : ∃ a ∈ a = ∅}. An ideal on Pω (λ) is a collection J of subsets of Pω (λ) such Pω (λ) B ∩ that (i) ∀ A ∈ J P (A) ⊆ J ; (ii)∀ A, B ∈ J A ∪ B ∈ J ; (iii) Iω,λ ⊆ J ; and (iv) Pω (λ) ∈ J . The following is readily checked. Proposition 1.1. Iω,λ is an ideal on Pω (λ).
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Throughout the paper H will denote an ideal on Pω (λ). We put H + = P (Pω (λ)) − H and H ∗ = {Pω (λ) − B : B ∈ H }. H is prime if H + = H ∗ . We let non(H ) denote the smallest cardinality of any A ⊆ Pω (λ) with A ∈ H + . X ⊆ H generates P (B). We let cof (H ) denote the smallest cardinality of any set H if H = B∈X
X ⊆ H that generates H . The following is well-known. Proposition 1.2. non(H ) = λ ≤ cof (H ). Proof. Let us first show that non(H ) ≤ cof (H ). Thus let X ⊆ H generate H . (Pω (λ) − B). Then ran(f ) ∈ H + . Pick f ∈ B∈X
Let us next show that non(Iω,λ ) ≥ λ. Thus let C ⊆ Pω (λ) be such that |C| < λ. } = ∅. Select γ ∈ λ − C. Then C ∩ {γ It remains to observe that non(Iω,λ ) ≤ non(H ) ≤ λ. For each A ∈ H + , we set H |A = {B ⊆ Pω (λ) : B ∩ A ∈ H }. The easy proof of the following is left to the reader. Proposition 1.3. Let A ∈ H + . Then H |A is an ideal on Pω (λ). Moreover, H ⊆ H |A and cof (H |A) ≤ cof (H ). By Proposition 1.2, an ideal on Pω (λ) cannot be generated by a set of size < λ. There are, as is shown by the next proposition, ideals on Pω (λ) that are generated by a size λ set. Such ideals are of course easier to work with, and we will show in Section 4 that they have some nice properties. + . Proposition 1.4. cof (Iω,λ |A) = λ for every A ∈ Iω,λ
Proof. Use Proposition 1.2 and Proposition 1.3. H is tall if for every A ∈ H + , H |A = Iω,λ |A. The following is immediate.
Proposition 1.5. Let µ be a cardinal > 0, and let Hα for α < µ be ideals on Pω (λ) such that Hβ ⊆ Hα whenever β < α. Then Hα is an ideal on Pω (λ), and cof (
Hα ) ≤ µ ·
α<µ
α<µ
cof (Hα ).
α<µ
2. Partition properties We now consider partition properties of ideals on Pω (λ). Johnson [14] showed that the partition property may be characterized in terms of distributivity and selectivity, and the section ends with the statement of his result. Concerning selectivity (and the P -point property), it should be pointed out that the terminology is not standard. We follow [14] (rather than [23]). H is a weak P-point if given A ∈ H + and Ba ∈ H for a ∈ A, there is a D ∈ H + ∩ P (A) such that D ∩ Ba ∈ Iω,λ for all a ∈ D.
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Proposition 2.1. Assume that H is a weak P -point and cof (H ) = λ. Then H is not tall. P (Bα ). Pick C ∈ H + so that Proof. Let Bα ∈ H for α < λ be such that H = C ∩(
α<λ + Iω,λ
Bα ) ∈ Iω,λ for all a ∈ C. Now let D ∈ ∩ P (C). Given α < λ, select a ∈ C with α ∈ a. Then D ∩ ( Bα ) ∈ Iω,λ , and therefore D − Bα = ∅. Hence α∈a
D ∈ H +.
α∈a
Given A ⊆ Pω (λ) and n ∈ ω with n > 0, we let [A]n = {(a0 , a1 , . . . , an−1 ) ∈ A × A × . . . × A : a0 ⊂ a1 ⊂ . . . ⊂ an−1 }. H is weakly selective if given A ∈ H + and Ba ∈ H for a ∈ A, there is a D ∈ H + ∩ P (A) such that b ∈ Ba for all (a, b) ∈ [D]2 . It is shown in [18] that if the Generalized Continuum Hypothesis holds and λ has cofinality ω, then Iω,λ is not weakly selective. + + H is (λ, 2)-distributive if given C ∈ H and Aα ∈ H for α < λ, there are D ∈ H + ∩ P (C) and f ∈ {Aα , Pω (λ) − Aα } such that for all α < λ, α<λ
D − f (α) ∈ H . Given G ⊆ P (Pω (λ)) and n ∈ ω with n > 0, G −→ (G)n means that for all A ∈ G and F : [A]n −→ 2, there is a C ∈ G ∩ P (A) such that F is constant on [C]n . The following is trivial. Proposition 2.2. H is prime if and only if H ∗ −→ (H ∗ )1 . The following is due to Johnson [14]. Proposition 2.3. H + −→ (H + )n for every n ∈ ω with n > 0 if and only if H is weakly selective and (λ, 2)-distributive. It is not known whether it is consistent that there exists an ideal J on Pω (λ) such that J + −→ (J + )2 but J + −→ (J + )3 . 3. The Rudin-Keisler order This section is devoted to the Rudin-Keisler ordering of ideals on Pω (λ). Assuming λ > ω, the situation is more complicated than in the case of ideals over ω, since for example the image of a noncofinal set under a function from Pω (λ) to Pω (λ) may very well be cofinal, and there are functions f : Pω (λ) −→ Pω (λ) such that ran(f ) is cofinal but there does not exist any cofinal A for which (A, ⊂) is isomorphic to (f [A], ⊂). We are thus led, as e.g. in [20] and [23], to focus on ideals that satisfy various additional properties. f : Pω (λ) −→ Pω (λ) is cofinal mod H (respectively strongly cofinal mod H ) + + if {f [E] : E ∈ H ∗ } ⊆ Iω,λ (resp. {f [E] : E ∈ H + } ⊆ Iω,λ ). Given f : Pω (λ) −→ Pω (λ), we put f∗ (H ) = {B ⊆ Pω (λ) : f −1 (B) ∈ H }. The following is easily checked.
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Proposition 3.1. Let f : Pω (λ) −→ Pω (λ) be strongly cofinal mod H . Then the following hold: (i) f∗ (H ) is an ideal on Pω (λ). (ii) If H is prime, then so is f∗ (H ). (iii) If h : Pω (λ) −→ Pω (λ) is such that {a ∈ Pω (λ) : f (a) = h(a)} ∈ H ∗ , then f∗ (H ) = h∗ (H ). The following well-known fact will be used in Section 9. Lemma 3.2. Let I , J be two ideals on Pω (λ). Assume that J is not prime, and let f : Pω (λ) −→ Pω (λ). Then there are C ∈ I + and D ∈ J + with the property that J = f∗ (I ) for all ideals I , J on Pω (λ) such that I |C ⊆ I and J |D ⊆ J . Proof. Select A0 , A1 ∈ J + so that A0 ∩ A1 = ∅ and A0 ∪ A1 = Pω (λ). In case there is an i < 2 such that f −1 (Ai ) ∈ I , put C = Pω (λ) and D = Ai . Otherwise set C = f −1 (A0 ) and D = A1 . We define a transitive binary relation ≤RK over the set of all ideals on Pω (λ) by letting J ≤RK I if and only if there is an f : Pω (λ) −→ Pω (λ) such that J = f∗ (I ). The associated equivalence relation ≡RK is defined by letting J ≡RK I if and only if J ≤RK I and I ≤RK J . The following is readily checked. Proposition 3.3. Let f : Pω (λ) −→ Pω (λ) be one-to-one and such that f∗ (H ) is an ideal on Pω (λ). Then f is strongly cofinal mod H . Moreover, H ≡RK f∗ (H ) and cof (H ) = cof (f∗ (H )). Lemma 3.4. Assume H is prime, and let f : Pω (λ) −→ Pω (λ) be such that H = f∗ (H ). Then {a ∈ Pω (λ) : f (a) = a} ∈ H ∗ . Proof. Fix a bijection i : Pω (λ) −→ λ, and let R = {a : i(f (a)) > i(a)}, T = {a : i(f (a)) < i(a)}. Then follow the proof of Theorem 3.3 in [6] to show that R, T ∈ H . Proposition 3.5. Let I , J be two prime ideals on Pω (λ). Then I ≡RK J if and only if there is a bijection h : Pω (λ) −→ Pω (λ) such that J = h∗ (I ). Proof. The right-to-left implication is immediate from Proposition 3.3. For the other direction, assume that I ≡RK J . Let f : Pω (λ) −→ Pω (λ) and g : Pω (λ) −→ Pω (λ) be such that J = f∗ (I ) and I = g∗ (J ). Put C = {a ∈ Pω (λ) : (g ◦ f )(a) = a}. As I = (g ◦ f )∗ (I ), we have that C ∈ I ∗ by Lemma 3.4. Moreover, f is one-to-one on C. Now pick C0 , C1 ⊆ C so that C0 ∪ C1 = C, C0 ∩ C1 = ∅ and |C0 | = |C1 | = λ. Let i < 2 be such that Ci ∈ I ∗ . Select a bijection j : Pω (λ) − Ci −→ Pω (λ) − f [Ci ]. Now define h : Pω (λ) −→ Pω (λ) by letting h(a) = f (a) in case a ∈ Ci , and h(a) = j (a) otherwise. Clearly, h is bijective. Moreover, h∗ (I ) = f∗ (I ) by Proposition 3.1 (iii).
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Lemma 3.6. Let a ∈ Pω (λ). Then there is a bijection θ : P (a) −→ |P (a)| such that θ(b) < θ (c) for all (b, c) ∈ [P (a)]2 . Proof. Define θn : P (n) −→ 2n for n ∈ ω so that θn+1 (d) = θn (d) in case d ∈ P (n), and θn+1 (d) = 2n +θn (d −{n}) otherwise. Pick a bijection j : a −→ |a|. Then define θ by letting θ (b) = θ|a| (j [b]). The following was suggested by Theorem 5.10 of [23]. Lemma 3.7. Let f : Pω (λ) −→ Pω (λ). Then there is a one-to-one h : Pω (λ) −→ Pω (λ) such that a ∪ f (a) ⊆ h(a) for all a ∈ Pω (λ), and h(a) ⊂ h(b) for all (a, b) ∈ [Pω (λ)]2 . Proof. Fix a bijection j : λ −→ Pω (λ). The definition of h proceeds by induction on α ∈ λ. Thus let α ∈ λ be such that h has already been defined on P (j (β)). Set B =
β<α
P (j (β)) and C = P (j (α)) − B. By Lemma 3.6, there is a bijection
β<α
θ : P (j (α)) −→ |P (j (α))| such that θ (b) < θ (c) for all (b, c) ∈ [P (j (α))]2 . Suppose c ∈ C is such that h(b) has been defined for all b ∈ C with θ (b) < θ(c). By Proposition 1.4, there is a d ∈ Pω (λ) such that for all e ∈ B ∪ {b ∈ P (j (α)) : θ(b) < θ(c)}, d ⊆ h(e). Now put h(c) = c ∪ d ∪ f (c) ∪ ( h(b)). b⊂c
H is order-reflecting (respectively order-preserving) if for every one-to-one h : Pω (λ) −→ Pω (λ) such that h is cofinal mod H , there is an A ∈ H + with the property that ∀ a, b ∈ A (h(a) ⊂ h(b) −→ a ⊂ b) (resp. ∀ a, b ∈ A (a ⊂ b ←→ h(a) ⊂ h(b))). Proposition 3.8. Assume H is order-reflecting, and let f : Pω (λ) −→ Pω (λ). Then there are A ∈ H + and a one-to-one h : A −→ Pω (λ) such that (∀ a ∈ A f (a) ⊆ h(a)) and (∀ a, b ∈ A (h(a) ⊂ h(b) −→ a ⊂ b)). Proof. By Lemma 3.7 there is a one-to-one h : Pω (λ) −→ Pω (λ) such that for a ⊆ h−1 ( a ). Clearly, h is cofinal mod H , and all a ∈ Pω (λ), f (a) ⊆ h(a) and + therefore there is an A ∈ H such that ∀ a, b ∈ A (h(a) ⊂ h(b) −→ a ⊂ b). In [23] Zwicker introduces two properties of ideals, which he calls ‘the backwards order-preserving property’ and ‘the property (*)’. Proposition 3.8 shows that every order-reflecting prime ideal on Pω (λ) has the backwards order-preserving property. The proof of the following follows the proof of Observation 5.11 in [23], which states that the backwards order-preserving property implies the property (*). Corollary 3.9. Assume H is order-reflecting, and let f : Pω (λ) −→ Pω (λ). Then there is an A ∈ H + such that f [B] ∈ Iω,λ for every B ∈ Iω,λ ∩ P (A). Proof. By Proposition 3.8, there are A ∈ H + and h : A −→ Pω (λ) such that (∀ a ∈ A f (a) ⊆ h(a)) and (∀ a, b ∈ A (h(a) ⊆ h(b) −→ a ⊆ b)). Given = ∅, and consequently h[A − a ∈ A, we have h[A − a ] ∩ h(a) a ] ∈ Iω,λ . Hence for every B ∈ Iω,λ ∩ P (A), h[B] ∈ Iω,λ , and therefore f [B] ∈ Iω,λ .
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The following is essentially due to Menas (see Proposition 10 in [20]). Proposition 3.10. Assume H is prime and weakly selective. Then the following are equivalent: (i) H is order-preserving. (ii) H is order-reflecting. (iii) J is weakly selective for every ideal J on Pω (λ) with H ≡RK J . Proof. (i) → (ii): Trivial. (ii) → (iii): Assume (ii), and let J be a prime ideal on Pω (λ) with H ≡RK J . By Proposition 3.1 (ii) and Proposition 3.5, there is a one-to-one h : Pω (λ) −→ Pω (λ) such that J = h∗ (H ). Select A ∈ H ∗ so that ∀ a, b ∈ A (h(a) ⊂ h(b) −→ a ⊂ b). Given G : [Pω (λ)]2 −→ 2, define F : [A]2 −→ 2 by letting F (a, b) = 0 if and only if h(a) ⊂ h(b) and G(h(a), h(b)) = 0. By Proposition 2.3, there is a C ∈ H ∗ ∩ P (A) such that F is constant on [C]2 . Then clearly h[C] ∈ J ∗ and G is constant on [h[C]]2 . Hence by Proposition 2.3, J is weakly selective. (iii) → (i): Assume (iii), and let h : Pω (λ) −→ Pω (λ) be one-to-one and cofinal mod H . Then by Proposition 3.3, H ≡RK h∗ (H ). Define F : [ran(h)]2 −→ 2 by letting F (h(a), h(b)) = 0 if and only if a ⊂ b. By Proposition 2.3, there are C ∈ (h∗ (H ))∗ ∩ P (ran(h)) and i < 2 such that F is identically i on [C]2 . Then = ∅ for any a ∈ h−1 (C), we h−1 (C) ∈ H ∗ and as ( a ∩ h−1 (C)) ∩ h−1 (h(a)) −1 have that i = 0. Thus ∀ a, b ∈ h (C) (h(a) ⊂ h(b) −→ a ⊂ b). Now define G : [h−1 (C)]2 −→ 2 by letting G(a, b) = 0 if and only if h(a) ⊂ h(b). By Proposition 2.3, there are A ∈ H ∗ ∩ h−1 (C) and k < 2 such that G is identically k = ∅ for any a ∈ A, we have that k = 0. Clearly, on [A]2 . As ( a ∩ A) ∩ h−1 (h(a)) for all a, b ∈ A, a ⊂ b if and only if h(a) ⊂ h(b). H is minimal if H ≡RK J for every ideal J on Pω (λ) such that J ≤RK H . Proposition 3.11. Assume H is prime. Then H is minimal if and only if for every cofinal mod H f : Pω (λ) −→ Pω (λ), there is an A ∈ H ∗ such that f is one-to-one on A. Proof. The proof is the same as that of Exercise 38.4 in [12].
4. Small ideals By Proposition 1.2, the cofinality of an ideal on Pω (λ) is a cardinal that is not less than λ and not greater than 2λ . We will now see that ideals with cofinality λ satisfy various elementary properties. We end the section by showing that there are always prime ideals with cofinality equal to 2λ . Proposition 4.1. Assume cof (H ) = λ, and let f : Pω (λ) −→ Pω (λ) be cofinal mod H . Then there is an A ∈ H + such that f is one-to-one on A. Proof. Let Bα ∈ H for α < λ be such that H = P (Bα ). Using induction, we α<λ
define aα ∈ Pω (λ) for α < λ as follows. We start by selecting a0 ∈ Pω (λ) − B0 .
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Now suppose α > 0 is such that aβ has already been defined for each β < α. By Proposition 1.2, there is a b ∈ Pω (λ) such that for all β < α, b ⊆ f (aβ ). We select aα so that aα ∈ Pω (λ) − Bα and b ⊆ f (aα ). We finally set A = {aα : α < λ}. Clearly, A is as desired. Proposition 4.2. Assume cof (H ) = λ. Then there is a partition of Pω (λ) into λ sets in H + . Proof. Let Bβ for β < λ be such that H = P (Bβ ), and let Dξ : ξ < λ be an β<λ
enumeration of the set {Bβ : β < λ} which lists each element λ times. Construct ξ ξ ξ aη for η ≤ ξ < λ so that (i) all the aη are distinct, and (ii) aη ∈ Hξ ∪ Bη . Now ξ set Aη = {aη : η ≤ ξ < λ} for each η < λ. Then Aη ∈ H + , Aη ∩ Bη = ∅ and for all η < η, Aη ∩ Aη = ∅. Given two cardinals µ, ν with µ ≥ ν ≥ ω, H is (µ, ν)-feeble if there is a k : Pω (λ) −→ µ such that k −1 (E) ∈ Iω,λ for all E ⊆ µ with |E| < ν, and k −1 (E) ∈ H + for all E ⊆ µ with |E| = ν. Proposition 4.3. Assume cof (H ) = λ. Then H is (λ, cof (λ))-feeble. Proof. Let Aη for η < λ be as in the proof of Proposition 4.2. For each η < λ, pick a bijection jη : λ −→ Aη . Now define k : Pω (λ) −→ λ by letting k −1 ({α}) = {jβ (α) : β < α} ∪ {jα (γ ) : γ ≤ α}. Then by Proposition 1.2, k −1 (E) ∈ Iω,λ for all E ⊆ λ with |E| < cof (λ). Clearly, k −1 (E) ∈ H + for all E ⊆ λ with |E| = cof (λ). We recall that given a regular infinite cardinal µ, ᑿµ denotes the least cardinality of any collection F of functions from µ to µ with the following property: given g : µ −→ µ, |{α ∈ µ : g(α) ≤ f (α)}| = µ for some f ∈ F . ᑿω is usually denoted by ᑿ. Proposition 4.4. Let µ be a regular infinite cardinal ≤ λ such that cof (H ) < ᑿµ . Then H is (µ, µ)-feeble. Proof. Pick X ⊆ H such that |X| < ᑿµ and X generates H . For each B ∈ X, define fB : µ −→ µ so that (i) fB (α) < fB (β) for all α, β ∈ µ with α < β, and (ii) for every α ∈ µ, there is a c ∈ Pω (λ)−B with fB (α) = max(c∩µ). Pick g : µ −→ µ so that for every B ∈ X, |{α ∈ µ : fB (α) > g(α)}| < µ. Define h : µ −→ µ by letting h(α) = (α ∪g(α))+1. Then define ηα for α < µ so that η0 = 0, ηα+1 = h(ηα ) and in case α is a limit > 0, ηα = ηβ . Finally define k : Pω (λ) −→ µ by letting β<α
k −1 ({α}) = {c ∈ Pω (λ) : ηα ≤ max(c ∩ µ) < ηα+1 }. Clearly, k −1 (E) ∈ Iω,λ for all E ⊆ µ with |E| < µ. Given B ∈ X and α ∈ µ, we have ηα ≤ fB (ηα ) and g(ηα ) < h(ηα ) = ηα+1 . It clearly follows that k −1 (E) ∈ H + for all E ⊆ µ with |E| = µ. The following is immediate. Corollary 4.5. Assume H is prime. Then cof (H ) ≥ ᑿµ for every regular infinite cardinal µ ≤ λ.
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G ⊆ P (Pω (λ)) is independent if and f ∈
E∈X
+ f (E) ∈ Iω,λ whenever X ∈ Pω (G)−{∅}
{E, Pω (λ) − E}.
E∈X
Proposition 4.6. There is an independent G ⊆ P (Pω (λ)) with |G| = 2λ . Proof. Fix a bijection j : λ −→ Pω (λ). Let Q denote the collection of all infinite proper subsets A of λ. For each A ∈ Q, put SA = {a ∈ Pκ (λ) : ∃ α ∈ a ( j (β) ⊆ j (α) and j −1 (A ∩ j (α)) ∈ a)}. β∈a
g : Y −→ Now fix X, Y ∈ Pω (Q) with X ∩ Y = ∅. Define f : Y −→ λ, λ and h : X × Y −→ λ so that f (B) ∈ λ − B, g(B) ∈ B − j (β) and h(A, B) ∈ (A−B)∪(B −A). Then set c = (
β∈b
j (β))∪ran(f )∪ran(g)∪ran(h)
β∈b
and a = {j −1 (c)} ∪ b ∪ {j −1 (A ∩ c) : A ∈ X}. It is readily checked that a ∈ b ∩ (( SA ) − ( SB )). A∈X
B∈Y
A well-known argument (see Exercise A11 on page 289 of [16]) yields the following corollary. Corollary 4.7. There is a prime ideal K on Pω (λ) such that cof (K) = 2λ . Proof. By Proposition 4.6, there is an independent G ⊆ P (Pω (λ)) with |G| = 2λ . ∈ Z if and only if there is a one-to-one Define Z ⊆ P (Pω (λ)) by letting E g : ω −→ G such that E =Pω (λ) − ran(g). Then define J ⊆ P (Pω (λ)) by ∗ ∪ Z) − {∅}. letting B ∈ J if and only if ( X) ∩ B = ∅ for some X ∈ Pω (G ∪ Iω,λ Now let I be any ideal on Pω (λ) with J ⊆ I . Let Y ⊆ I generate I . Define f : G −→ Y so that f (A) ⊇ Pω (λ) − A. As f −1 ({B}) is finite for every B ∈ Y , we must have |Y | ≥ 2λ . 5. ᒍH With each ideal J on Pω (λ) we now associate its pseudo-intersection number ᒍJ . We will see that for J = Iω,λ , ᒍJ can be seen as a two cardinal version of the wellknown cardinal invariant ᒍ. We will show in the next section that under Martin’s axiom, ᒍJ > λ for all J with cofinality < 2ℵ0 , and so we are mostly interested now in the consequences of having a large ᒍJ . + We define the infinite cardinal ᒍH as follows.+Let W consist of all Z ⊆ H+ such that (i) for all S ∈ Pω (Z) − {∅}, S ∈ H , and (ii) for every D ∈ H , there is an A ∈ Z with D − A ∈ H + . We let ᒍH be the least cardinality of any Z ∈ W in case W = ∅, and set ᒍH = (2λ )+ otherwise. Proposition 5.1. Assume ᒍH > λ. Then H is (λ, 2)-distributive. Proof. Given C ∈ H + and Aα ∈ H + for α < λ, construct by induction Dβ ∈ H + for β ≤ λ so that
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(i) ∀ α < β Dβ − Dα ∈ H ; (ii) D0 ⊆ C; (iii) ∀ α < λ (Dα ⊆ Aα or Dα ∩ Aα = ∅).
We put ᒍω,λ = ᒍIω,λ . We recall that ᒍ denotes the least cardinality of any subset Y of [ω]ω(the collection of all infinite subsets of ω) such that (i) for all S ∈ Pω (Y ) − {∅}, S ∈ [ω]ω ; and (ii) for every D ∈ [ω]ω , there is an A ∈ Y with D − A ∈ [ω]ω . Proposition 5.2.
ᒍω,ω = ᒍ.
ᒍ ≥ ᒍω,ω . Thus let Y ⊆ [ω]ω be such that (∀ S ∈ Proof. Let us first show that ω Pω (Y ) − {∅} S ∈ [ω] (∀ D ∈ [ω]ω ∃ A ∈ Y D − A ∈ [ω]ω ). Then ) and + + clearly Y ⊆ Iω,ω , and S ∈ Iω,ω for all S ∈ Pω (Y ) − {∅}. Now suppose there is + with the property that D − A ∈ I + a D ∈ Iω,ω ω,ω for all A ∈ Y . Then D ∩ A ∈ Iω,ω ω ω for any A ∈ Y , and therefore D ∩ ω ∈ [ω] . Moreover, (D ∩ ω) − A ∈ [ω] for all A ∈ Y , a contradiction. + be such that Z ⊆ Iω,ω Conversely, let us show that ᒍ+ ≤ ᒍω,ω . Thus let + (∀ S ∈ Pω (Z) − {∅} S ∈ Iω,ω ) and (∀ D ∈ Iω,ω ∃ A ∈ Z D − A ∈ + ). Pick a bijection j : ω −→ P (ω) and a one-to-one f : ω −→ Z. Set Iω,ω ω Z = (X − ran(f )) ∪ {f (n) ∩ j (n) : n ∈ ω}. Clearly, we have that (∀ S ∈ + ) and (∀ D ∈ I + ∃ A ∈ Z D − A ∈ I + ). Pω (Z ) − {∅} S ∈ Iω,ω ω,ω ω,ω Put Y = {j −1 (A) : A ∈ Z }. Then T ∈ [ω]ω for all T ∈ Pω (Y ) − {∅}. Now + , pick A ∈ Z so that j [C] − A ∈ I + . let C ∈ [ω]ω . In case j [C] ∈ Iω,ω ω,ω Then C − j −1 (A) ∈ [ω]ω . Next suppose that j [C] ∈ Iω,ω . Select n ∈ ω so that j [C] ∩ j (n) = ∅. Then (C =) C − j −1 (f (n) ∩ j (n)) ∈ [ω]ω .
Proposition 5.3. The product λ · ᒍω,λ is the least cardinal ρ such that there is a tall ideal K on Pω (λ) with cof (K) = ρ. Proof. Let J be any tall ideal on Pω (λ). We have that cof (J ) ≥ λ by Proposition 1.2. We claim that cof (J ) ≥ ᒍω,λ . Suppose otherwise, and let X ⊆ J be such that + X generates J and |X| < ᒍω,λ . Select D ∈ Iω,λ so that D ∩ B ∈ Iω,λ for all B ∈ X. + + Then P (D) ∩ Iω,λ ⊆ J , a contradiction. Thus 2λ ≥ cof (J ) ≥ ᒍω,λ for any prime ideal J on Pω (λ), and therefore + λ 2 ≥ ᒍω,λ . So there is a Z ⊆ Iω,λ such that (i) |Z| = ᒍω,λ ; (ii) ∀ S ∈ + + + Pω (Z) − {∅} S ∈ Iω,λ ; and (iii) ∀ D ∈ Iω,λ ∃ A ∈ Z D − A ∈ Iω,λ . We define an ideal K on Pω (λ) by letting K = {B ⊆ Pω (λ) : ∃ a ∈ Pω (λ) ∃ S ∈ Pω (Z) B ∩ ( a∩ A) = ∅}. A∈S
K +,
We have that cof (K) ≤ λ · ᒍω,λ . Given D ∈ there is an A ∈ Z such that + D − A ∈ Iω,λ . Clearly, D − A ∈ K. Hence K is tall. Lemma 5.4. Assume that cof (H ) < ᒍω,λ . Then ᒍH ≥ ᒍω,λ . Proof. Let Z ⊆ H + be such that |Z| < ᒍω,λ and for all S ∈ Pω (Z) − {∅}, S ∈ H + . Let X ⊆ H be such that X generates H and |X| < ᒍω,λ . Then there is
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+ a D ∈ Iω,λ such that D − A ∈ Iω,λ for all A ∈ Z ∪ {Pω (λ) − B : B ∈ X}. Clearly, + D∈H .
Since Iω,λ is (trivially) a weak P -point, one might naively expect all ideals on Pω (λ) of cofinality λ to have the same property. However for λ > ω, this last statement does not necessarily hold. In fact, it implies, as will be seen now, that ᒍω,λ is large, whereas it can be shown (see [19]) that assuming, for instance, the Generalized Continuum Hypothesis, ᒍω,λ ≤ ω1 . Proposition 5.5. The following are equivalent: (i) J is a weak P -point whenever J is an ideal on Pω (λ) with cof (J ) = λ. (ii) ᒍω,λ > λ. (iii) ᒍJ > λ for all ideals J on Pω (λ) with cof (J ) = λ. Proof. (i) → (ii): By Proposition 5.3 and Proposition 2.1. (ii) → (iii): By Lemma 5.4. (iii) → (i): Assume (iii) holds. Then ᒍω,λ > λ by Proposition 1.4. Now let J be an ideal on Pω (λ) with cof (J ) = λ. Fix A ∈ J + . Then J |A is not tall by Proposition 1.3 and Proposition 5.3, and consequently there is a C ∈ J + ∩ P (A) + such that Iω,λ ∩ P (C) ⊆ J + . Clearly, C ∩ B ∈ Iω,λ for all B ∈ J . Hence J is a weak P -point. 6. cov(Mω,λ ) cov(Mω,λ ) is a two-cardinal version of the cardinal invariant cov(M). In this section we review what is known concerning its value. We endow the set 2λ of all functions from λ to 2 with the product topology, where 2 is given the discrete topology. Mω,λ denotes the ideal of meager subsets of 2λ . We put M = Mω,ω . cov(Mω,λ ) denotes the least cardinality of any F ⊆ Mω,λ such that 2λ = F. For any set A, we let F n(A, 2) denote the set of all functions s with dom(s) ∈ Pω (A) and ran(s) ⊆ 2, ordered by reverse inclusion. Thus F n(λ, 2) is the notion of forcing which adds λ Cohen reals. For each s ∈ F n(λ, 2), we put Osλ = {f ∈ 2λ : s ⊂ f }. Since 2λ is a compact Hausdorff space, it satisfies the Baire Category Theorem, therefore cov(Mω,λ ) ≥ ω1 . We are going to use the following obvious reformulation of the cardinal cov (Mω,λ ). least cardinality of any nonempty family F of Proposition 6.1. cov(Mω,λ ) is the dense open subsets of 2λ such that F = ∅. The following is due to Miller [21]. Proposition 6.2. cov(Mω,λ ) ≤ cov(Mω,µ ) whenever ω ≤ µ ≤ λ.
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Proof. Fix an infinite cardinal µ ≤ λ, and let O α for α < cov(Mω,µ ) be dense µ open subsets of 2 with the property that Oα = ∅. For each α, set α
λ ⊆ O λ ∩ U . Hence U is dense in 2λ . We that tµ ⊆ s. Then Os∪t an s ∈ Sα such α α t have that Uα = ∅, as clearly α
Thus cov(Mω,λ ) ≤ cov(M). Let Cω,λ denote the least cardinality of any family D of dense subsets of F n(λ, 2) such that the following holds: for every filter G ⊆ F n(λ, 2), there is a D ∈ D with G ∩ D = ∅. The following is due to Miller [21]. Proposition 6.3. Cω,λ = cov(Mω,λ ). Proof. Let us first show that Cω,λ ≥ cov(Mω,λ ). Thus let D be a family of dense subsets of F n(λ, 2) such that |D| < cov(Mω,λ ). For each D ∈ D, set OD = Otλ . Given s ∈ F n(λ, 2), there is a t such that t ∈ D and s ⊆ t, and therefore t∈D Otλ
⊆ OD ∩ Osλ . Thus OD is dense in 2λ . Now select f ∈
OD , and put
D∈D
G = {u ∈ F n(λ, 2) : u ⊂ f }. Then G is a filter with the property that G ∩ D = ∅ for all D ∈ D. Let us now show that Cω,λ ≤ cov(Mω,λ ). Thus let ρ be a cardinal with 0 < ρ < Cω,λ , and let Oα be a dense open subset of 2λ for each α < ρ. Set Dα = {t ∈ F n(λ, 2) : Otλ ⊆ Oα } for every α < ρ. As each Dα is clearly dense in F n(λ, 2), there is a filter G ⊆ F n(λ, 2) such that G ∩ Dα = ∅ for all α < ρ. Pick f ∈ 2λ with G ⊆ f . Then f ∈ Oα . α<ρ
Landver [17] established the following. Proposition 6.4. Suppose λ ≥ cov (Mω,2ℵ0 ). Then cov(Mω,λ ) = cov (Mω,2ℵ0 ). The proof of Proposition 6.4 given below is slightly different from that of Landver. It is due to Brendle [8]. Lemma 6.5. Suppose D is an infinite collection of dense subsets of F n(λ, 2) such that |D| < λ. Then there is a subset A of λ of size |D| such that D ∩ F n(A, 2) is a dense subset of F n(A, 2) for every D ∈ D. Proof. For D ∈ D, select ϕD : F n(λ, 2) −→ D so that s ⊆ ϕD (s) for every s ∈ F n(λ, 2). Define Am for m < ω by letting A0 = |D| and
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Am+1 = {dom(ϕD (s)) : D ∈ D Then A = Am is as desired.
and
s ∈ F n(Am , 2)}.
m<ω
Lemma 6.6. Let ν ≤ τ be two infinite cardinals. Then cov(Mω,τ ) ≥ min (ν + , cov (Mω,ν )). Proof. Let D be an infinite collection of dense subsets of F n(τ, 2) such that |D| ≤ ν and |D| < cov(Mω,ν) . By Lemma 6.5 there is an A ⊆ λ such that |A| = |D| and D∩ F n(A, 2) is a dense subset of F n(A, 2) for every D ∈ D. Since |D| < cov(Mω,|A| ) by Proposition 6.2, there exists a filter G ⊆ F n(A, 2) such that G ∩ D = ∅ for every D ∈ D. Proof of Proposition 6.4. Set µ = cov(Mω,2ℵ0 ). By Proposition 6.2 and Lemma 6.6, we have cov(Mω,µ ) ≥ cov(Mω,2ℵ0 ) ≥ min(µ+ , cov(Mω,µ )) and
cov(Mω,µ ) ≥ cov(Mω,λ ) ≥ min(µ+ , cov(Mω,µ )).
Hence, cov(Mω,µ ) = µ = cov(Mω,λ ).
Thus cov(Mω,λ ) > λ iff λ < cov(Mω,2ℵ0 ). θλ denotes the least cardinality of any collection X of infinite subsets of λ such that for every B ⊆ λ with |B| = λ, there is an A ∈ X with A ⊆ B. It is well-known that θω = 2ℵ0 (since there exists a pairwise almost disjoint family F of infinite subsets of ω with |F | = 2ℵ0 ). The following is essentially due to Miller [21]. Proposition 6.7.
λ · cov(Mω,λ ) ≤ θλ ≤ λℵ0 .
Proof. It is immediate that λ ≤ θλ ≤ λℵ0 . Now suppose λ < cov (Mω,λ ). We have to show that cov (Mω,λ ) ≤ θλ . Thus let X be an infinite collection of infinite subsets of λ such that |X| < cov (Mω,λ ). Fix a pairwise disjoint collection Y of infinite subsets of λ with |Y | = λ. Set DA = {f ∈ 2λ : ∃α ∈ A
f (α) = 0}
for A ∈ X, and
EC = {f ∈ 2λ : ∃α ∈ C f (α) = 1} for C ∈ Y . Now let g ∈ ( DA ) ∩ ( EC ). Then clearly | g −1 ({1}) |= λ. A∈X
C∈Y
Moreover, A − g −1 ({1}) = ∅ for all A ∈ X.
The following is well-known (see Lemma 5.13 in chapter VII of [16]). Lemma 6.8. Suppose κ is an uncountable cardinal in V and V [G] is the generic extension of V obtained by adding κ Cohen reals. Then for every infinite cardinal τ , (2τ )V [G] ≤ (κ τ )V .
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Thus if κ τ = κ in V , then in V [G] 2τ = κ. The following is due to Miller [21]. Proposition 6.9. Suppose κ is an uncountable cardinal in V and V [G] is the generic extension of V obtained by adding κ Cohen reals. Then the following hold : (i) In V [G] cov(Mω,κ ) ≥ κ. (ii) If κ ℵ0 = κ in V , then in V [G] cov(Mω,κ ) = κ = 2ℵ0 . (iii) If in V κ has cofinality ω and ν ℵ1 < κ for every cardinal ν < κ, then in V [G] θω1 = κ. Proof. (i) : By Proposition 6.4 it suffices to prove that in V [G] cov(Mω,τ ) ≥ κ for every infinite cardinal τ < κ. So let such a τ be fixed. In V [G], let ρ be a cardinal with 0 < ρ < κ, and let Oα be a dense open subset of 2τ for each α < ρ. Define E ⊆ ρ × F n(τ, 2) by letting (α, s) ∈ E if and only if Osτ ⊆ Oα . Then E ∈ V [G ∩ F n(A, 2)] for some A ⊆ κ such that A lies in V and |A| ≤ ρ · τ . In V , select a bijection j : κ × τ −→ κ. Pick δ ∈ κ so that A ∩ j [{δ} × τ ] = ∅. Then G ∩ F n(j [{δ} × τ ], 2) is generic overτ V [G ∩ F n(A, 2)]. Now set g = (G ∩ F n(j [{δ} × τ ], 2)), and define f ∈ 2 by letting f (β) = g(j (δ, β)). It is readily checked that f ∈ Oα . α<ρ
(ii) : By (i) and Lemma 6.8. (iii) : We have θω1 ≥ κ by (i) and Proposition 6.7. Let us now establish the reverse inequality. In V let < κm : m < ω > be a sequence of infinite cardinals cofinal in κ. For m < ω, set Xm = P (ω1 ) ∩ V [G ∩ F n(κm , 2)]. Then | Xm | ≤ κ m<ω
since by Lemma 6.8 |Xm | < κ for every m < ω. Now fix B ∈ V [G] such that B ⊆ ω1 and |B| = ℵ1 . For m < ω, let Am be the set of all γ < ω1 such that some s ∈ G ∩ F n(κ m , 2) forces that γ ∈ B. Note that Am ∈ Xm for every m < ω. Moreover, B = Am . Since B is uncountable, there must be a k < ω such that Ak is infinite.
m<ω
Miller [21] also showed that it is consistent that cov(M) = ℵ2 and cov(Mω,ω1 ) = ℵ1 . Let us now see how cov(Mω,2ℵ0 ) compares with some other cardinal invariants of the continuum. ω ᒂA denotes the least cardinality of any F ⊆ ω with the following property : given an infinite subset A of ω and ϕ ∈ {x ⊂ ω : |x| = n}, there is an f ∈ F n∈A
such that the set {n ∈ A : f (n) ∈ / ϕ(n)} is infinite. Brendle [7] established the following. Proposition 6.10.
ᒂA ≤ cov(Mω,2ℵ0 ).
add(N) denotes the least cardinality of any family F of measure zero subsets of the real line whose union is not of measure zero. The following is due to Blass [4] and Brendle and Shelah [9].
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Proposition 6.11.
ᒍ · add(N) ≤ ᒂA ≤ ᑿ.
Using a result of Shelah, Brendle [7] established the following. Proposition 6.12. It is consistent that the following three statements hold simultaneously : (a) ᑿ = cov(M) = 2ℵ0 = ℵ3 . (b) ᒍ = cov(Mω,ω2 ) = ℵ2 . (c) add(N) = ℵ1 . / M. non(M) denotes the least cardinality of any X ⊆ 2ω such that X ∈ The following is well-known (see [5]). Proposition 6.13.
ᑿ ≤ non(M).
Blass [5, 11.3] observed the following. Proposition 6.14. Every generic extension of V obtained by adding uncountably many Cohen reals satisfies non(M)= ω1 . Since ᒍ ≤ non(M) by Propositions 6.11 and 6.13, Proposition 6.14 can be combined with Proposition 8.2 to obtain the consistency of ᒍ < ᒍω,ω1 . 7. Cohen reals We present in this section the result which is the key to the results of the next two sections. Let us first recall the following definitions. A partially ordered set (P , ≺) is directed if for all p, p ∈ P , there is a p ∈ P such that p ! p and p ! p . (P , ≺) has finite character if for each p ∈ P , there are only finitely many p ∈ P such that p ≺ p. Until the end of this section, assume that (P , ≺) is a nonempty partially ordered set such that (0) (P , ≺) is directed, (1) (P , ≺) has finite character, (2) (P , ≺) has no maximal elements, and (3) every cofinal subset of P has size |P |. Further assume that Sp ⊆ P for p ∈ P are such that Sp is cofinal in P for every finite, p∈a
nonempty a ⊆ P . We let Q consist of all finite q ⊆ P such that p ∈ Sp for all p, p ∈ Q with p ≺ p . Given q, q ∈ Q, we let q ≤ q just in case (a)q ⊆ q , and (b) p ≺ p whenever p ∈ q and p ∈ q − q. Lemma 7.1. (i) |Q| = |P |. (ii) (Q, ≤) is separative (i.e. given q, q ∈ Q with q ≤ q, there is a q ≤ q that is incompatible with q). (iii) If q ∈ Q and Y is a dense subset of {q ∈ Q : q ≤ q}, then Y has size |P |. Proof. (i): Clear.
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(ii): Fix q, q ∈ Q with q ≤ q. If q ⊆ q , then q and q are clearly incompatible. Now assume that q ⊆ q , and let p ∈ q − q . Pick p ∈ P so that p ≺ p and for all p ∈ q , p ! p and p ∈ Sp . Then q and q ∪ {p } are incompatible. (iii): Fix q ∈ Q and a dense subset Y of {q ∈ Q : q ≤ p}. Given p ∈ P , select p ∈ P so that p ! p , q ∪ {p } ∈ Q and q ∪ {p } ≤ q. Then p ∈ q for every q ∈ Y such that q ≤ q ∪ {p }. Hence q is cofinal in P , and therefore q ∈Y
|Y | ≥ |P |.
We set Sp∗ = {Sp : p ∈ P and p ! p} for every p ∈ P . We let C denote the set of all countable X ⊆ P such that (a) {p ∈ P : p ≺ p} ⊆ X for every p ∈ X, and (b) for every p ∈ X, there is a p ∈ X such that p ≺ p and p ∈ Sp∗ .
Lemma 7.2. (i) P = C. (ii) C is closed under countable unions. Proof. (i): Given p ∈ P , define pn ∈ P for n ∈ ω so that p0 = p and for every n ∈ ω, pn+1 " pn and pn+1 ∈ Sp∗n . Set X = {p ∈ P : p ! pn }. Then X ∈ C. n∈ω
(ii): Clear.
X denote the set of Given X ∈ C, a finite subset F of X and x ∈ F , we let RF,x all p ∈ X such that x ≺ p and p ∈ Sy for all y ∈ F with y ≺ p. X = ∅ for every X ∈ C, every finite F ⊆ X and every x ∈ F . Lemma 7.3. RF,x
Proof. Given X, F and x, define xi ∈ X for i ∈ ω so that x0 = x and for all i, ∗ xi+1 " xi and xi+1 ∈ Sxi . Set G = {y ∈ F : y ! xi }. Since G is finite, there is i∈ω
X . a j ∈ ω such that y ! xj for all y ∈ G. Then xj +1 ∈ RF,x
We set QX = {q ∈ Q : q ⊆ X} for every X ∈ C. Notice that by Lemma 7.2, Q = QX . X∈C
Lemma 7.4. Let X ∈ C, and let A be a maximal antichain of QX . Then A is a maximal antichain of Q. Proof. Fix q ∈ Q with q = ∅. Set F = {x ∈ X : x ! p}. Notice that p∈q
q ∩ F = q ∩ X. Using Lemma 7.3, we define p0 , p1 , . . . , pn ∈ X and F0 ⊃ F1 ⊃ . . . ⊃ Fn ⊃ Fn+1 for some n ∈ ω so that (0) F0 = F ; X X (1) Given i ∈ {0, . . . , n}, pi ∈ RF,x and for all p ∈ RF,x , |{x ∈ F : x ≺ p}| ≤ |{x ∈ F : x ≺ pi }|;
x∈Fi
x∈Fi
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(2) Fi+1 = {x ∈ Fi : x ≺ pi } for i = 0, . . . , n; (3) Fn+1 = ∅. We claim that p0 , p1 , . . . , pn are pairwise incomparable. Thus fix i, j with X for some y ∈ F . Then y ≺ p and as i < j ≤ n. We have that pj ∈ RF,y j j X y ∈ Fi+1 , y ≺ pi . Hence pj ≺ pi . Moreover pj ∈ RF,x , and therefore x∈Fi
pi ≺ pj since otherwise |{x ∈ F : x ≺ pj }| > |{x ∈ F : x ≺ pi }|. Notice that F ⊆ {x ∈ X : x ≺ pi } and therefore by the claim above, F ⊆
i≤n
{x ∈ X : pi ! x}.
i≤n
Now set q = (q ∩ X) ∪ {pi : i ≤ n}. As q ∈ QX , there are s ∈ A and t ∈ QX such that t ≤ s and t ≤ q . Notice that t ∩ F = q ∩ F = q ∩ F . Put w = t ∪ (q − X). It is readily verified that w ∈ Q and w ≤ q. Moreover w ≤ t, and consequently w ≤ s. The following now follows from Theorem 3.2 in [2]. Proposition 7.5. Forcing with Q is equivalent to adding |P | many Cohen reals (i.e. r.o.(Q) and r.o.(F n(|P |, 2)) are isomorphic). 8. Ideals with the partition property We will now show that all ideals on Pω (λ) of cofinality < cov(Mω,λ ) have the partition property and are order-reflecting. All results are based on Proposition 7.5. Lemma 8.1. Assume that cof (H ) < cov(Mω,λ ) and let Sa ∈ H + for a ∈ Pω (λ) be such that Sa ∈ H + for every y ∈ Pω (Pω (λ)) − {∅}. Then there is a D ∈ H+ ∩ P(
a∈y
Sa ) such that b ∈ Sa for every (a, b) ∈ [D]2 .
a∈Pω (λ)
Proof. Set P =
Sa . Then clearly (P , ⊂) is directed, has finite character and
a∈Pω (λ)
has no maximal elements. Moreover by Proposition 1.2, every cofinal subset of P has size |P |. Put Q = {q ∈ Pω (P ) : ∀ (a, b) ∈ [q]2 b ∈ Sa }. Given q, q ∈ Q, we a ∈ q and b ∈ q − q. let q ≤ q just in case (0) q ⊆ q , and (1) b ⊂ a whenever P (Xα ). For each Now let Xα ∈ H for α < cof (H ) be such that H = α
α, set Dα = {q ∈ Q : ∃ b ∈ q b ∈ Xα }. Let us show that Dα is dense in Q. Thus let q ∈ Q − {∅}. Since ( x. Sx ) − Xα ∈ H + , we can pick b ∈ (( Sx ) − Xα ) ∩ x∈q
x∈q
x∈q
Then q ∪ {b} ∈ Dα . Moreover q ∪ {b} < q. By Propositions 6.3 and 7.5 there is a filter G ⊆ Q such that G ∩ Dα = ∅ for all α < cof (H ). Set D = G. Then D is as desired.
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Proposition 8.2. Assume that cof (H ) < cov(Mω,λ ). Then H is weakly selective, and ᒍH > λ. Proof. Given A ∈ H + and Ba ∈ H for a ∈ A, there is by Lemma 8.1 a D ∈ H + ∩ P (A) such that b ∈ A − Ba for all (a, b) ∈ [D]2 . Thus H is weakly selective. Now let Eα ∈ H + for α < λ be such that Eα ∈ H + for all y ∈ Pω (λ)−{∅}. α∈y
By Lemma 8.1, there is a D ∈ H + such that b ∈
Eα for all (a, b) ∈ [D]2 .
α∈a
Given α < λ, pick a ∈ D with α ∈ a. Then D ∩ a ⊆ Eα , and consequently D − Eα ∈ Iω,λ . It is shown in [22] that cov(M) is the least cardinal κ such that there is an ideal J on ω of cofinality κ which does not have the partition property. The following generalizes one half of this result. Corollary 8.3. Assume that cof (H ) < cov(Mω,λ ). Then H + −→ (H + )n for all n ∈ ω with n > 0.
Proof. By Propositions 2.3, 5.1 and 8.2.
Proposition 8.4. Assume that cof (H ) < cov(Mω,λ ). Then H is order-reflecting. Proof. Let D ∈ H ∗ and h : Pω (λ) −→ Pω (λ) be such that h is one-to-one on D and cofinal mod H . Set P = h[D]. Then (P , ⊂) is directed, has finite character and has no maximal elements. Moreover by Proposition 1.2 every cofinal subset of P has size |P |. We define a bijection g by letting g = h ∩ (D × P ). For each x ∈ P , set Sx = {y ∈ P ∩ x : g −1 (x) ⊂ g −1 (y)}. Let e be a finite, nonempty subset of P . Given z ∈ P , set a = g −1 [e] and b = z ∪ (∪e). As D ∩ a ∈ H ∗ , there is a d ∈ D ∩ a with b ⊂ h(d). Then h(d) ∈ z ∩ ( Sx ). Thus Sx is cofinal in P . x∈e
x∈e
We let Q consist of all finite q ⊆ P such that y ∈ Sx for all x, y ∈ Q with x ⊂ y. Given q, q ∈ Q, we let q ≤ q just in case (0) q ⊆ q , and (1) y ⊂ x whenever x ∈ q and y ∈ q − q. Let Bα ∈ H for α < cof (H ) be such that H = P (Bα ). For each α
α < cof (H ), set Dα = {q ∈ Q : q ∩ g[D (H ) − Bα ] = ∅}. Given α < cof and q ∈ Q, set r = g −1 [q] and s = q. As D ∩ ( r − Bα ) ∈ H ∗ , there is a c ∈ D ∩ ( r − Bα ) with s ⊂ h(c). Then q ∪ {h(c)} ∈ Q ∩ Dα . Moreover q ∪ {h(c)} ≤ q. Thus each Dα is dense in Q. By Propositions 6.3 and 7.5 there is a filter G ⊆ Q such that G ∩ Dα = ∅ for all α < cof (H ). Put A = g −1 [ G]. Then A ∈ H + ∩ P (D). Moreover a0 ⊂ a1 for all a0 , a1 ∈ A with h(a0 ) ⊂ h(a1 ). We conclude this section by showing that if cof (H ) < cov(Mω,λ ), then H is not prime.
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+ Proposition 8.5. Let µ be an infinite cardinal such that cov(M ω,µ ) ≥ µ ·cof (H ). Then there is a Q ⊆ P (Pω (λ)) such that |Q| = µ and f (E) ∈ H + whenever
X ∈ Pω (Q) − {∅} and f ∈
E∈X
{E, Pω (λ) − E}.
E∈X
Proof. Let Bα ∈ H for α < cof (H ) be such that H =
P (Bα ). Set
α
P = F n(µ × Pω (λ), 2). For all s ∈ F n(µ, 2) and α < cof (H ), set
Ds,α = {u ∈ P : ∃ a ∈ Pω (λ) − Bα ∀ γ ∈ dom(s) u(γ , a) = s(γ )}. Each Ds,α is clearly dense in P . By Proposition 6.4 there is a filter G ⊆ P such that G ∩ Ds,α = ∅ for all s ∈ F n(µ, 2) and α < cof (H ). Set Q = {{a ∈ Pω (λ) : ( G)(γ , a) = 0} : γ ∈ µ}. Then Q is as desired.
9. Prime ideals We will now show that if cov(Mω,λ ) = 2λ , then there is a prime ideal on Pω (λ) such that all its RK-equivalents have the partition property. Proposition 9.1. Assume that cov(Mω,λ ) = 2λ and cof (H ) < 2ℵ0 . Then there are 2ℵ0 pairwise incomparable prime ideals on Pω (λ) such that every one of them includes H and is weakly selective and order-preserving. Proof. We have that 2λ = 2ℵ0 by Proposition 6.2. Now let Fα for α < 2ℵ0 be an enumeration of all F : [Pω (λ)]2 −→ 2, and let hα for α < 2ℵ0 be an enumeration of all functions from Pω (λ) to Pω (λ). Fix a bijection i : 2ℵ0 × 2ℵ0 × 2ℵ0 −→ 2ℵ0 . Using Propositions 1.3, 1.5, 8.4 and 8.5, Lemma 3.2 and Corollary 8.3, we define for each (α, β) ∈ 2ℵ0 × 2ℵ0 an ideal Jα,β on Pω (λ) with cof (Jα,β ) ≤ |1 + α| · cof (H ), Aα,β ∈ Jα,β + , Cα,β ∈ Jα,β + ∩ P (Aα,β ) and Dα,β ∈ Jα,β + ∩ P (Cα,β ) so that (0) J0,β = H . (1) If hα is not one-to-one, then Aα,β = Pω (λ). (2) In case hα is one-to-one, set Zα = {Pω (λ) − h−1 a ) : a ∈ Pω (λ)}. Then α ( Aα,β ∈ Zα in case Zα ∩ Jα,β + = ∅. Otherwise, Aα,β is such that a ⊂ b for all a, b ∈ Aα,β with hα (a) ⊂ hα (b). (3) Fα is constant on Cα,β . (4) Suppose α = i(γ , δ, η). Then Dα,β = Cα,β in case γ = δ or β ∈ {γ , δ}. If γ = δ, then Dα,γ and Dα,δ are such that K = (hη )∗ (I ) for all ideals I, K on Pω (λ) such that Jα,γ |Dα,γ ⊆ I and Jα,δ |Dα,δ ⊆ K. (5) Jα+1,β = Jα,β |Dα,β . (6) Jα,β = Jγ ,β in case α is a limit > 0. γ <α
Prime ideals on Pω (λ) with the partition property
For each β < 2ℵ0 , set Hβ =
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Jα,β . Then Hβ is an ideal on Pω (λ) by Prop-
α<2ℵ0
osition 1.5. Moreover, we clearly have that Hβ∗ −→ (Hβ∗ )2 . Hence Hβ is prime by Proposition 2.2. Let us show that Hβ is weakly selective. Given Ba ∈ Hβ for a ∈ Pω (λ), define F : [Pω (λ)]2 −→ 2 by letting F (a, b) = 0 if and only if a ) − Ba = ∅ b ∈ Ba . Pick E ∈ Hβ∗ so that F is constant on [E]2 . We have that (E ∩ 2 for every a ∈ E, and therefore F is identically 1 on [E] . Hβ is order-reflecting, since given α < 2ℵ0 such that hα is one-to-one, either there is an a ∈ Pω (λ) with h−1 a ) ∈ Hβ , or else there is an A ∈ Hβ∗ such that a ⊂ b for all a, b ∈ A with α ( hα (a) ⊂ hα (b). Hence by Proposition 3.10, Hβ is order-preserving. Finally, given γ , δ, η ∈ 2ℵ0 with γ = δ, we have Hδ = (hη )∗ (Hγ ) as J(i(γ ,δ,η))+1,γ ⊆ Hγ and J(i(γ ,δ,η))+1,δ ⊆ Hδ . Is it true that if cof (H ) < cov(Mω,λ ), then for every f : Pω (λ) −→ Pω (λ) that is cofinal mod H , there is an A ∈ H + with f being one-to-one on A ? A positive answer would enable us to show that the prime ideals constructed in the proof of Proposition 9.1 are minimal. We are however unable to answer positively the question above, and so we will take another road to the construction of minimal ideals. It is well-known (see Exercise 24.16 in [12]) that forcing with P (ω)/f in adds a prime ideal on ω that is minimal. We will obtain minimal ideals on Pω (λ) by forcing with two-cardinal versions of P (ω)/f in. For each A ⊆ Pω (λ), we set [A]H = {B ⊆ Pω (λ) : (A − B) ∪ (B − A) ∈ H }. We put P (Pω (λ))/H = {[A]H : A ⊆ Pω (λ)}. Given A, B ⊆ Pω (λ), we let [A]H ≤ [B]H just in case A − B ∈ H . Proposition 9.2. Assume that cof (H ) = λ and λ < cov(Mω,λ ). Let G be generic for ((P (Pω (λ))/H − {[∅]H }, ≤) over V . Then in V [G], there is a prime ideal K on Pω (λ) such that H ⊆ K and K is weakly selective, order-preserving and minimal. Proof. By Lemma 19.6 in [12] and Proposition 8.2, we have that if f ∈ V [G] is a function from λ to V , then f ∈ V . Now in V [G], set K = {B ⊆ Pω (λ) : [Pω (λ) − B]H ∈ G}. It is easy to check that K is a prime ideal on Pω (λ) and H ⊆ K. Let f : Pω (λ) −→ Pω (λ) be given. Suppose A ∈ H + is such that f [C] ∈ Iω,λ for every C ∈ H + ∩ P (A). By Propositions 1.2, 1.3 and 4.1, there is a D ∈ H + ∩ P (A) such that f is one-to-one on D. Then by Proposition 1.2, Proposition 1.3 and the proof of Proposition 8.4, there is an E ∈ H + ∩ P (D) such that a ⊂ b for all a, b ∈ E with f (a) ⊂ f (b). Hence, either there is an S ∈ K ∗ such that f [S] ∈ Iω,λ , or else there is a T ∈ K ∗ such that f is one-to-one on T and a ⊂ b for all a, b ∈ T with f (a) ⊂ f (b). Thus K is order-reflecting and, by Proposition 3.11, minimal. Using Corollary 8.3, it is easy to see that K ∗ −→ (K ∗ )n for all n ∈ ω with n > 0. Hence by Proposition 2.3, K is weakly selective. Finally, K is order-preserving by Proposition 3.10. References [1] Alves, C.: Partitions of Finite Substructures, Ph. D. Thesis, Pennsylvania State University, 1985
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[2] Balcar, B., Jech, T., Zapletal, J.: Semi-Cohen Boolean algebras, Annals of Pure and Applied Logic 87, 187–208 (1997) [3] Bartoszy´nski, T., Judah, H.: Measure and category - filters on ω, in : Set Theory of the Continuum (H. Judah, W. Just and H. Woodin, eds.), Mathematical Sciences Research Institute Publications 26, Springer, New York, 175–201 (1992) [4] Blass, A.: Cardinal characteristics and the product of countably many infinite cyclic groups, Journal of Algebra 169, 512–540 (1994) [5] Blass, A.: Combinatorial cardinal characteristics of the continuum, in : Handbook of Set Theory (M. Foreman, A. Kanamori and M. Magidor, eds.), Kluwer, Dordrecht, to appear [6] Booth, D.: Ultrafilters on a countable set, Annals of Mathematical Logic 2, 1–24 (1970) [7] Brendle, J.: Cardinal invariants of the continuum and combinatorics on uncountable cardinals, preprint [8] Brendle, J.: personal communication [9] Brendle, J., Shelah, S.: Evasion and prediction II, Journal of the London Mathematical Society 53(2), 19–27 (1996) [10] Canjar, R.M.: On the generic existence of special ultrafilters, Proceedings of the American Mathematical Society 110, 233–241 (1990) [11] Jech, T.: Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic 5, 165–198 (1973) [12] Jech, T.: Set Theory, Academic Press, New York, 1978 [13] Jech, T., Shelah, S.: A partition theorem for pairs of finite sets, Journal of the American Mathematical Society 4, 647–656 (1991) [14] Johnson, C.A.: Some partition relations for ideals on Pκ (λ), Acta Mathematica Hungarica 56, 269–282 (1990) ˇ [15] Ketonen, J.: On the existence of P -points in the Stone-Cech compactification of integers, Fundamenta Mathematicae 92, 91–94 (1976) [16] Kunen, K.: Set Theory, North-Holland, Amsterdam, 1980 [17] Landver, A.: Baire numbers, uncountable Cohen sets and perfect-set forcing, Journal of Symbolic Logic 57, 1086–1107 (1992) [18] Matet, P.: Negative partition relations for uncountable cardinals of cofinality ω, Studia Scientiarum Mathematicarum Hungarica 37, 233–236 (2001) [19] Matet, P., Pean, C.: Distributivity properties on Pω (λ), preprint [20] Menas, T.K.: A combinatorial property of Pκ (λ), Journal of Symbolic Logic 41, 225– 234 (1976) [21] Miller,A.W.: The Baire category theorem and cardinals of countable cofinality, Journal of Symbolic Logic 47, 275–288 (1982) [22] Scheepers, M.: The least cardinal for which the Baire category theorem fails, Proceedings of the American Mathematical Society 125, 579–585 (1997) [23] Zwicker, W.: A beginning for structural properties of ideals on Pκ (λ), in : Set Theory and its Applications (J. Steprans and S. Watson, eds.), Springer, Berlin, 201–217 (1989)
Note added in proof. Kraszewskip [Properties of ideals on the generalized Cantor spaces, Journal of Symbolic Logic 66, 1303–1320 (2001)] showed that if cov(Mω,λ ) < min (ᑿ, cov(M)), then cov(Mω,λ ) ≤ u(ω1 , λ), where u(ω1 , λ) denotes the smallest cardinality of any family A of countable subsets of λ such that every countable subset of λ is included in some member of A. It follows that if (a) and (b) of Proposition 6.12 both hold, then cov(Mω,ω1 ) = ℵ3 .
