Cardinal invariants of analytic P -ideals

Fernando Hernández-Hernández Michael Hrušák ^{∗}

Instituto de Matemáticas, UNAM, Unidad Morelia Apartado Postal 61-3 (Xangari) Morelia, Michoacán, México C.P. 58089

May 6, 2005

Abstract We study the cardinal invariants of analytic P -ideals, concentrating on the ideal Z of asymptotic density zero. Among other results we prove that min {𝔟, cov (N)} ≤ cov^{∗} (Z) ≤ max {𝔟, non (N)}.

0 Introduction

Analytic P -ideals and their quotients have been extensively studied in recent past. The first step to better understand the structure of the quotient forcings P (ω) /I is to understand the structure of the ideal itself. Significant progress in understanding the way in which the structure of an ideal affects the structure of its quotient has been done by I. Farah (see [Far00], [Far02a], [Far02b], [Far04a], [Far04b]).

Typically (but not always) the quotients P (ω) /I, where I is an analytic P -ideal, are proper and weakly distributive. For some special ideals these quotients have been identified: [Far04a] P (ω) /Z is as forcing notion equivalent to P (ω) /fin ∗ 𝔹 (2^{ω}), and [HZ04] P (ω) / tr (N) is as forcing notion equivalent to the iteration of 𝔹 (ω) followed by an ℵ_{0}-distributive forcing (see the definitions below).

∗2000 AMS Classification: 03E17, 03E40 The authors gratefully acknowledge support from PAPIIT grant IN106705. The second author’s research was also partially supported by GA ČR grant 201-03-0933 and CONA-CYT grant 40057-F

A secondary motivation comes from the problem of which ideals can be destroyed by a weakly distributive forcing. Even for the class of analytic P -ideals only partial results are known (see Section 3).

In this note we contribute to this line of research by investigating cardinal invariants of analytic P -ideals, comparing them to other, standard, cardinal invariants of the continuum.

In the first section we introduce cardinal invariants of ideals on ω, along the lines of the cardinal invariants contained in the Cichoń’s diagram. We also recall the definitions of standard orderings on ideals on ω (Rudin-Keisler, Tukey, Katětov) and their impact on the cardinal invariants of the ideals. Basic theory of analytic P -ideals on ω and examples are also reviewed here.

Known results on additivity and cofinality of analytic P -ideals are summarized in the second section.

The main part of the paper is contained in the third section. There we study the order of Katětov restricted to analytic P -ideals, giving a detailed description of how the summable and density ideals are placed in the Katětov order. For the rest of the section, we focus on the ideal of asymptotic density zero and compare its covering number to standard cardinal invariants of the continuum. We prove that min {𝔟, cov (N)} ≤ cov^{∗} (Z) ≤ max {𝔟, non (N)} and mention some consistency results. We introduce the notion of a totally bounded analytic P -ideal and show that all analytic P -ideals which are not totally bounded can be destroyed by a weakly distributive forcing.

In the last section we study the separating number of analytic P -ideals, an invariant closely related to the Laver and Mathias-Prikry type forcings associated with the ideal.

The two major problems that remain open here are: (1) Is add^{∗} (I) = add (N) for every tall analytic P -ideal I? and (2) Can every analytic P - ideal be destroyed by a weakly distributive forcing? What about Z?

We assume knowledge of the method of forcing as well as the basic theory of cardinal invariants of the continuum as covered in [BJ95]. Our notation is standard and follows [Kun83], [Jec97] and [BJ95]. In particular, c_{0}, ℓ_{1} and ℓ_{∞} denote the standard Banach spaces of sequences of reals. For A, B infinite subsets of ω, we say that A is almost contained in B (A ⊆^{∗} B) if A \ B is finite. The symbol A =^{∗} B means that A ⊆^{∗} B and B ⊆^{∗} A. For functions f, g ∈ ω^{ω} we write f ≤^{∗} g to mean that there is some m ∈ ω such that f (n) ≤ g (n) for all n ≥ m. The bounding number 𝔟 is the least cardinal of an ≤^{∗}-unbounded family of functions in ω^{ω}. The dominating number 𝔡 is the least cardinal of a ≤^{∗}-cofinal family of functions in ω^{ω}. Recall that a family of subsets of ω has the strong finite intersection property if any finite subfamily has infinite intersection. The pseudointersection number 𝔭 is the minimal size of a family of subsets of ω with the strong finite intersection property but without an infinite pseudointersection (i.e. without a common lower bound in the ⊆^{∗} order). A family S ⊆ P (ω) is a splitting family if for every infinite A ⊆ ω there is an S ∈ S such that S ∩ A and A \ S are infinite. The splitting number 𝔰 is the minimal size of a splitting family in P (ω).

The set 2^{ω} is equipped with the product topology, that is the topology with basic open sets of the form [s] = {x ∈ 2^{ω} : s ⊆ x}, where s ∈ 2^{<ω}. The topology of P (ω) is that obtained via the identification of each subset of ω with its characteristic function.

An ideal on X is a family of subsets of X closed under taking finite unions and subsets of its members. We assume throughout the paper that all ideals contain all singletons {x} for x ∈ X. An ideal I on ω is called P -ideal if for any sequence X_{n} ∈ I, n ∈ ω, there exists X ∈ I such that X_{n} ⊆^{∗} X for all n ∈ ω. An ideal I on ω is analytic if it is analytic as a subspace of P (ω) with the above topology. Recall that an ideal on ω is tall (or dense) if every infinite set of ω contains an infinite set from the ideal. If I is an ideal on ω and Y ⊆ ω is an infinite set, then we denote by I ↾ Y the ideal {I ∩ Y : I ∈ I}; note that the underlying set of the ideal I ↾ Y is not the underlying set of I but Y. For an ideal I on ω, we denote by I^{∗} the dual filter. M denotes the ideal of meager subsets of ℝ and N the ideal of Lebesgue null subsets of ℝ. Given an ideal I on a set X, the following are standard cardinal invariants associated with I :

add (I) = min {|A| : A ⊆ I∧⋃A ∉ I} ,
cov (I) = min{|A| : A ⊆ I ∧ ⋃A = X},
cof (I) = min {|A| : A ⊆ I ∧ (∀I ∈ I) (∃A ∈ A) (I ⊆ A)} ,
non (I) = min{|Y | : Y ⊆ X ∧ Y ∉ I}.

The provable relationships between the cardinal invariants of M and N are summed up in the following diagram:

\xymatrix{cov (N) \ar[r] & non (M) \ar[r] & cof (M) \ar[r] & cof (N) \\  & 𝔟 \ar[u] \ar[r] & 𝔡 \ar[u] \\ _{add (N)} \ar[uu] \ar[r] & _{add (M)} \ar[r] \ar[u] & _{cov (M)} \ar[r] \ar[u] & _{non (N)} \ar[uu]}

Cichoń’s Diagram

Regarding our forcing terminology, p ≤ q means that p is a stronger condition than q. By 𝔹 (κ) we denote the measure algebra of Maharam type κ (the algebra of Baire subsets of 2^{κ} modulo the ideal N_{κ} of Haar measure zero subsets of 2^{κ}). Recall that a forcing ℙ is weakly distributive (ω^{ω}-bounding) if every new function in ω^{ω} is pointwise dominated by a ground model function. A partial order ℙ satisfies Axiom A if there is a sequence ⟨≤_{n}: n ∈ ω⟩ of orderings on ℙ such that

1. p ≤_{0} q if p ≤ q for every p, q ∈ ℙ,

2. p ≤_{n+1} q ⇒ p ≤_{n} q for every p, q ∈ ℙ,

3. if {p_{n} : n ∈ ω} is such that if p_{n+1} ≤_{n} p_{n} for every n ∈ ω then there is a p ∈ ℙ such that p ≤_{n} p_{n} for every n ∈ ω,

4. for every maximal antichain A in ℙ, for every p ∈ ℙ and n ∈ ω there is a q ≤_{n} p such that {r ∈ A : r is compatible with q} is countable.

For more on forcing, e.g. the definition of proper forcing, consult [Kun83] and [BJ95]. Even though we do not define properness, we remind the reader that every Axiom A forcing is proper.

1 Analytic P -ideals and their cardinal invariants

Definition 1.1 Let I be a tall ideal on ω containing the ideal of finite sets. Define the following cardinals associated with I :

add^{∗} (I) = min {|A| : A ⊆ I∧ (∀X ∈ I) (∃A ∈ A) (A ⊈^{∗} X)} ,
cov^{∗} (I) = min{|A| : A ⊆ I ∧ (∀X ∈ [ω]^{ℵ_{0}}) (∃A ∈ A) (|A ∩ X| = ℵ_{0})},
cof^{∗} (I) = min {|A| : A ⊆ I ∧ (∀I ∈ I) (∃A ∈ A) (I ⊆^{∗} A)} ,
non^{∗} (I) = min{|A| : A ⊆ [ω]^{ℵ_{0}} ∧ (∀I ∈ I) (∃A ∈ A) (|A ∩ I| < ℵ_{0})}.

These cardinals have been studied, some of them not in the context of ideals on ω, but as cardinals associated to the dual filters.^{1} The notation for add^{∗} (I) is taken from [Bar99], other authors use 𝔭 (I^{∗}), 𝔟 (I, ⊆^{∗}) or 𝔟 (I). We justify our preference for the names chosen here by the next proposition.

For every tall ideal I on ω, there is a natural ideal of Borel subsets of P (ω) associated with I defined as

Î = {X ⊆ P (ω) : (∃I ∈ I) (X ⊆Î)} ,

where Î = {X ⊆ ω : |X ∩ I| = ℵ_{0}}. One can easily check that I ⊆^{∗} J if and only if Î ⊆ Ĵ. Hence, J ⊆ P (ω) is a P -ideal if and only if Ĵ is a σ-ideal.

Proposition 1.2 The following equalities hold:

add(Î) = add^{∗} (I) ,  cof(Î) = cof^{∗} (I) ,
non(Î) = non^{∗} (I) ,  cov(Î) = cov^{∗} (I) .

Proof. The facts that add(Î) = add^{∗} (I) and cof(Î) = cof^{∗} (I) follow directly from the observation that I ⊆^{∗} J if and only if Î ⊆ Ĵ. To see cov^{∗} (I) = cov(Î) observe that if {I_{α} : α < cov^{∗} (I)} is a family witnessing the definition of cov^{∗} (I) and X ∈ [ω]^{ω} then there is some α < cov^{∗} (I) such that |X ∩ I| = ℵ_{0}, so X ∈ Î_{α}. On the other hand, if κ < cov^{∗} (I) and {X_{α} : α < κ} ⊆ Î, for every α < κ choose I_{α} ∈ I so that X_{α} ⊆ Î_{α}. As κ < cov^{∗} (I), there is some X ∈ [ω]^{ω} such that |X ∩ I_{α}| < ℵ_{0} for all α < κ. Thus ⋃_{α<κ} X_{α} ≠ [ω]^{ω} .

1Brendle and Shelah in [BS99] introduced cardinal invariants 𝔭 (F) and π𝔭 (F) associated with an (ultra)filter F. For tall ideal I, add^{∗} (I) = 𝔭 (I^{∗}) , cov^{∗} (I) = π𝔭(I^{∗} ), non^{∗} (I) = πχ (I^{∗}) and cof^{∗} (I) = cof (I) = χ (I^{∗}) .

The inequalities holding among these cardinals are summarized in the following diagram:

\xymatrix{ &  & non^{∗} (I) \ar[dr] \\ ℵ_{0} \ar[r] & _{add_{∗} (I)} \ar[ur] \ar[dr] &  & _{cof_{∗} (I)} \ar[r] & 2^{ℵ_{0}} \\  &  & _{cov_{∗} (I)} \ar[ur]}

It follows directly from the definition that cov^{∗} (I) ≥ 𝔭 for any tall ideal I.

The main result about the structure of analytic P -ideals is due to Solecki [Far00, Theorem 1.2.5]. Recall that a function φ : P (ω) → [0, ∞] is a submeasure if φ (∅) = 0, φ (X) ≤ φ (Y) whenever X ⊆ Y, φ (X ∪ Y) ≤ φ (X) + φ (Y) for every X, Y ∈ P (ω) , and φ ({n}) < ∞ for every n ∈ ω. A submeasure on ω is called lower semicontinuous if φ (A) = lim_{n→∞} φ (A ∩ n) for every A ⊆ ω. There are two ideals associated with lower semicontinuous submeasures on ω:

Exh (φ) = {X ∈ P (ω) : lim_{m→∞} φ (X \ m) = 0} ,

and

Fin (φ) = {X ∈ P (ω) : φ (X) < ∞}.

Theorem 1.3 Let I be an ideal on ω. Then

(a) (Mazur) I is an F_{σ} ideal if and only if I = Fin (φ) for some lower semicontinuous submeasure φ.

(b) (Solecki) I is an analytic P -ideal if and only if I = Exh (φ) for some lower semicontinuous submeasure φ.

(c) (Solecki) I is an F_{σ} P -ideal if and only if I = Fin (φ) = Exh (φ) for some lower semicontinuous submeasure φ.

We will use the following simple fact several times in the paper.

Lemma 1.4 Let I = Exh (φ), where φ is lower semicontinuous submeasure. Then I is a tall ideal if and only if lim_{n→∞} φ ({n}) = 0.

Proof. If lim_{n→∞} φ ({n}) ≠ 0, then the set E = {n ∈ ω : φ ({n}) ≥ ε} is infinite for some ε > 0, and lim_{n→∞} φ (I \ n) ≠ 0 for all infinite I ⊆ E. Thus I is not a tall ideal.

On the other hand, if lim_{n→∞} φ ({n}) = 0 and X ⊆ ω is infinite, find an increasing sequence of k_{n} ∈ X such that φ (\{m\}) ≤ \frac{1}{2^{n}} for all m ≥ k_{n}. Then I = \{k_{n} : n ∈ ω\} ∈ I since φ (I \ k_{m}) ≤ ∑_{n=m}^{∞} \frac{1}{2^{n}} and ∑_{n=m}^{∞} \frac{1}{2^{n}} converges to zero.

Orderings on ideals. We shall take advantage of the (pre-)orderings on the family of ideals. Let I and J be ideals on ω. Recall:

1. (Rudin-Keisler order) I ≤_{RK}J if there exists a function f : ω → ω such that I ∈ I if and only if f^{−1} (I) ∈ J for every I ⊆ ω.

2. (Rudin-Blass order) I ≤_{RB}J if there exists a finite-to-one function f : ω → ω such that I ∈ I if and only if f^{−1} (I) ∈ J for every I ⊆ ω.

3. (Katětov order) I ≤_{K}J if there exists a function f : ω → ω such that f^{−1} (I) ∈ J for every I ∈ I.

4. (Tukey order^{2}) I ≤_{T}^{∗} J if there exists a function f : I → J such that for every ⊆^{∗}-bounded set X ⊆ J , f^{−1} (X) is ⊆^{∗}-bounded in I.

2The classical definition of Tukey order for partially ordered sets is as follows:

Definition 1.5 ([Fre91]) Let P and Q be partially ordered sets. Then:

(a) P ≤_{T} Q if there is a function f : P → Q such that {p ∈ P : f (p) ≤ q} is bounded for every q ∈ Q.

(b) P ≤_{ω} Q if there is a function f : P → Q such that {p ∈ P : f (p) ≤ q} is countably dominated for every q ∈ Q.

When restricted to ideals, most authors, e.g. [LV99], [Fre91], [Far00], define the Tukey order using (I, ⊆) rather than (I, ⊆^{∗}). Following them we denote by ≤_{T} the standard Tukey order on ideals. Even though there does not seem to be any relation between ≤_{T} and ≤_{T}^{∗} for ideals in general, for P -ideals ≤_{T} is finer than ≤_{T}^{∗}.

Proposition 1.6 Let I and J be P -ideals. Then:

These orderings have a deep impact on the cardinal invariants of related ideals. Note that if I ⊆ J then I ≤_{K}J, and that if X ∈ I^{+} then I ↾X ≤_{T}^{∗} I while I ↾ X ≥_{K} I. Notice also that I ≤_{RB}J ⇒ I ≤_{RK}J ⇒ I ≤_{K}J and for P -ideals I and J, I ≤_{RK}J ⇒ I ≤_{T}^{∗}J.

Any f : ω → ω witnessing I ≤_{K}J is called a Katětov reduction. Also, I and J are Katětov equivalent if I ≤_{K}J and J ≤_{K}I, which we denote by I ≅_{K} J. Similarly for the other orderings. We use I ≅ J to denote that there is a permutation ρ of ω such that I ∈ I if and only if ρ [I] ∈ J.^{3}

Some ideals on ω. All ideals we are interested in are tall, with one exception:

∅ × Fin = {A ⊆ ω × ω : (∀n ∈ ω) ({m ∈ ω : ⟨n, m⟩ ∈ A} is finite)} .

For A ⊆ 2^{<ω}, let π (A) = {f ∈ 2^{ω} : (∃^{∞}n ∈ ω) (f ↾ n ∈ A)}. The trace of the null ideal is

tr (N) = {A ⊆ 2^{<ω} : μ (π (A)) = 0} ,

where μ denotes the standard product measure on 2^{ω}. This ideal is, of course, very much related with the null ideal N on the reals.

(a) I ≤_{ω}J if and only if I ≤_{T}^{∗}J,

(b) I ≤_{T}J implies I ≤_{T}^{∗}J

Proof. (a) For the forward implication, first observe that I ≤_{T}^{∗}J if and only if there is a function g : I → J such that for every J ∈ J there is an I ∈ I such that A ⊆^{∗} I whenever A ∈ I and g (A) ⊆^{∗} J.

Take an f witnessing I ≤_{ω}J and fix J ∈ J. Consider the family {J_{n} : n ∈ ω} of all finite modifications of J. Then {A ∈ I : f (A) ⊆^{∗} J} = ⋃_{n∈ω} {A ∈ I : f (A) ⊆ J_{n}}. Since f witnesses I ≤_{ω}J, there are families {I_{n,m} : m ∈ ω} ⊆ I, for every n ∈ ω, such that whenever f (A) ⊆ J_{n} there is an m ∈ ω such that A ⊆ I_{n,m}. As I is a P -ideal there is an I ∈ I such that I_{n,m} ⊆^{∗} I for every m, n ∈ ω. Then A ⊆^{∗} I whenever f (A) ⊆^{∗} J; that is, f witnesses I ≤_{T}^{∗}J.

For the reverse implications let f : I → J be a witness to I ≤_{T}^{∗}J. That is, given J ∈ J, there is an I ∈ I such that f (A) ⊆^{∗} J implies A ⊆^{∗} I. Then all finite modifications of I countably dominate {A ∈ I : f (A) ⊆ J}.

(b) follows from (a) and the trivial fact that I ≤_{T}J implies I ≤_{ω}J.

3According to [Far00, 1.2.7] two ideals I and J on ω are isomorphic if there is a partial one-to-one function f : ω → ω such that ω \ ran (f) ∈ J , ω \ dom (f) ∈ I and A ∈ I ⇔f [A] ∈ J for all A ⊆ ω. As the two definitions are equivalent for tall ideals, we choose the simpler one.

The eventually different ideal is defined as

ED = {A ⊆ ω × ω : (∃m, n ∈ ω) (∀k > n) (|{l : ⟨k, l⟩ ∈ A}| ≤ m)} ,

i.e. it is the ideal generated by vertical sections and graphs of functions. Define ED_{fin} = ED ↾ Δ, where Δ = {⟨m, n⟩ ∈ ω × ω : n ≤ m}. The ideals ED and ED_{fin} are the only ideals mentioned in the paper which are not P -ideals. They are both F_{σ}.

Proposition 1.7 cov^{∗} (ED) = cov^{∗} (ED_{fin}) = non (M), non^{∗} (ED) = ω and non^{∗} (ED_{fin}) = cov (M).

Proof. It is enough to note that cov^{∗} (ED) ≥ cov^{∗} (ED_{fin}) ≥ non (M) ≥ cov^{∗} (ED). The first inequality follows from the fact that ED_{fin} = ED ↾ Δ, the second one follows from the definition and Theorem 2.4.7 in [BJ95]. The last one is essentially contained in Lemma 2.4.8 of [BJ95].

Take the columns in ω×ω to see that non^{∗} (ED) = ω. That non^{∗} (ED_{fin}) = cov (M) follows from Theorem 2.4.5 in [BJ95].

Given f : ℕ → ℝ^{+} such that ∑_{n∈ω} f (n) = ∞, the summable ideal corresponding to f is the ideal

I_{f} = {A ⊆ ω : ∑_{n∈A} f (n) < ∞} .

The ideal I_{f} is tall if and only if lim_{n→∞} f (n) = 0. The lower semicontinuous submeasure on ω corresponding to I_{f} is: φ_{f} (A) = ∑_{n∈A} f (n). By definition I_{f} = Fin (φ_{f}). So, summable ideals are F_{σ}. A typical example of a summable ideal is the ideal

I_{\frac{1}{n}} = \{A ⊆ ω : ∑_{n∈A} \frac{1}{n} < ∞\} .

An Erdős-Ulam ideal is an ideal associated to a function f : ℕ → ℝ. EU_{f} is the ideal of all subsets of f-density zero, i.e. sets A such that

lim_{n→∞} \frac{∑_{i∈A∩(n+1)} f (i)}{∑_{i=0}^{n} f (i)} = 0.

The lower semicontinuous submeasure corresponding to EU_{f} is given by

φ_{f} (A) = sup_{n∈ω} \frac{∑_{i∈A∩(n+1)} f (i)}{∑_{i=0}^{n} f (i)}.

A class of analytic P -ideals further extending the class of Erdős-Ulam ideals is the class of density ideals. For a submeasure φ let supp (φ) = {n ∈ ω : φ ({n}) ≠ 0}. Submeasures φ and ψ are orthogonal if they have disjoint supports.

Definition 1.8 For a sequence \overrightarrow{μ} = \{μ_{i}\}_{i∈ω} of orthogonal measures on ω each of which concentrates on some finite set define the submeasure φ\overrightarrow{μ} by

φ\overrightarrow{μ} = sup_{i∈ω} μ_{i}.

Then

Z\overrightarrow{μ} = Exh (φ\overrightarrow{μ}) = \{A ⊆ ω : lim_{n→∞} φ\overrightarrow{μ} (A \ n) = 0\}

is the density ideal generated by the sequence of measures.

The proof of the following result can be consulted in [Far00, 1.13.3].

Theorem 1.9 Every Erdős-Ulam ideal is equal to some density ideal Z\overrightarrow{μ}, where each μ_{n} is a probability measure.

The most common of the density ideals is the ideal Z of subsets of ω of asymptotic density zero, that is:

Z = \{A ⊆ ω : lim_{n→∞} \frac{|A∩n|}{n} = 0\} .

Equivalently, A ∈ Z if and only if

lim_{n→∞} \frac{|A ∩ [2^{n}, 2^{n+1})|}{2^{n}} = 0.

For more on analytic P -ideals and historical notes consult [Far00].

2 Additivity and cofinality

In this section we review some known results about additivity and cofinality of analytic P -ideals. The basic tool for studying these cardinal invariants is the Tukey ordering. This is largely due to the following observation. Although it is well known [Fre91, 1J(a)], we include its short proof for the sake of completeness.

Proposition 2.1 I ≤_{T}^{∗} J ⇒ add^{∗} (I) ≥ add^{∗} (J) and cof^{∗} (I) ≤ cof^{∗} (J) .

Proof. Suppose that f : I → J witnesses the definition of ≤_{T}^{∗}. Let A ⊆ I be a family with |A| < add^{∗} (J). There is a set B ∈ J such that f [A] ⊆^{∗} B for every A ∈ A. As f maps unbounded sets to unbounded sets, it follows that A must be bounded.

Let B ⊆ J be cofinal. For every B ∈ B, there is an A_{B} ∈ I such that if f [I] ⊆^{∗} B then I ⊆^{∗} A_{B}. It follows that the family {A_{B} : B ∈ B} is a cofinal subset of I.

The following theorem summarizes the known results.

Theorem 2.2 (a) add^{∗}(I_{\frac{1}{n}}) = add^{∗} (tr (N)) = add (N) .

(b) (Todorčević, [Tod96]) ∅ × Fin ≤_{T} I ≤_{T} I_{\frac{1}{n}} for every analytic P -ideal I. In particular, add (N) ≤ add^{∗} (I) ≤ 𝔟 for all analytic P -ideals I.

(c) (Fremlin, [Fre04, 526H]) add^{∗} (Z) = add (N) and cof^{∗} (Z) = cof (N) .

(d) (Farah, [Far00, 1.13.10]) Every tall Erdős-Ulam ideal is Rudin-Blass equivalent to Z.

(e) Every tall summable ideal is Tukey equivalent to I_{\frac{1}{n}}.

(f) add^{∗} (I) = add (N) and cof^{∗} (I) = cof (N), for every tall ideal I which is either summable or a density ideal.

Proof. For (e) by Claim 1 in [Far00, 1.12.14], given two summable ideals I_{f} and I_{g} there is an X ∈ I_{g}^{+} such that I_{f} ≤_{RB} I_{g} ↾ X. On the other hand, I_{g} ↾ X ≤_{T} I_{g}, hence I_{f} and I_{g} are Tukey equivalent. (f) for summable ideals follows directly from (a) and (e), and for tall density ideals it follows from (b), (c) and the fact that for every tall density ideal I there is an X ∈ I^{+} such that I ↾ X is Erdős-Ulam, [Far00, 1.13.10].

Louveau and Veličković in [LV99] showed that there are many ≤_{T} nonequivalent analytic P -ideals. On the other hand, assuming add (N) = cof (N) all tall analytic P -ideals are ≤_{T}^{∗}-equivalent. It is natural to ask:

Questions 2.3 (a) Are all tall analytic P -ideals ≤_{T}^{∗}-equivalent? (b) Is at least add^{∗} (I) = add (N) for every tall analytic P -ideal?

3 Covering and uniformity

The Katětov ordering relates to covering and uniformity of ideals in analogous way as the Tukey ordering does to additivity and cofinality.

Proposition 3.1 I ≤_{K}J ⇒ cov^{∗} (I) ≥ cov^{∗} (J) and non^{∗} (I) ≤ non^{∗} (J) .

Proof. Let A ⊆ I witness the definition of cov^{∗} (I), and let f : ω → ω be a Katětov reduction witnessing I ≤_{K}J. Then

B = {f^{−1} (A) : A ∈ A} ∪ {f^{−1} (F) : F ∈ [ω]^{<ℵ_{0}}}

witnesses the definition of cov^{∗} (J) . Indeed, if X ⊆ ω is infinite then either f [X] is infinite and hence, there is some A ∈ A such that for infinitely many n ∈ X, f (n) ∈ A; therefore |X ∩ f^{−1} (A)| = ℵ_{0}. Or else, f [X] is finite and hence X ⊆ f^{−1} (f [X]). In either case X has infinite intersection with some member of B.

If A ⊆ [ω]^{ℵ_{0}} witnesses non^{∗} (J), then {f [A] : A ∈ A} witnesses non^{∗} (I), for if I ∈ I, then f^{−1} (I) ∈ J and there is some A ∈ A such that A ∩ f^{−1} (I) is finite. Hence f [A] ∩ I must be finite.

First we investigate the behaviour of the Katětov order restricted to analytic P -ideals. It turns out that the ideal ED_{fin} is a lower bound for all analytic P -ideals in the Katětov order.

Proposition 3.2 ED_{fin} ≤_{K} I for every tall analytic P -ideal.

Proof. Let φ be a lower semicontinuous submeasure such that I = Exh (φ). By Lemma 1.4 there is a strictly increasing sequence ⟨a_{n} : n ∈ ω⟩ such that φ ({m}) < 2^{−n} for m ≥ a_{n}. Let g : ω → Δ ⊆ ω × ω be a one-to-one function mapping intervals [a_{n}, a_{n+1}) into distinct vertical sections of Δ, say {⟨b_{n}, i⟩ : i ≤ b_{n}}. This function g is a Katětov reduction. Indeed, let A ∈ ED_{fin}. Then there is a k ∈ ω such that |A ∩ {⟨b_{n}, i⟩ : i ≤ b_{n}}| ≤ k for all n ∈ ω. Now given ε > 0, φ (A ∩ [a_{n}, a_{n+1})) ≤ k2^{−n} and therefore φ (A \ n) ≤ ε for large enough n ∈ ω.

As ED_{fin} is not a P -ideal, the relation ED_{fin} ≤_{K} I can not be strengthened to ED_{fin} ≤_{RK} I. In the proof of the next result we use the following lemma which is probably folklore but we could not find any reference for it.

Lemma 3.3 Let ε > 0 and U be an infinite family of clopen subsets of 2^{ω}, each of measure at least ε. Then

μ ({x ∈ 2^{ω} : (∃^{∞}U ∈ U) (x ∈ U)}) ≥ ε.

Proof. Suppose not. Then there is a compact set K ⊆ 2^{ω} disjoint with {x ∈ 2^{ω} : (∃^{∞}U ∈ U) (x ∈ U)} such that μ (K) > 1 − ε. Let δ = ε + μ (K) − 1 > 0. Then μ (U ∩ K) ≥ δ for each U ∈ U. Write K = ⋃_{n∈ω} A_{n}, where

A_{n} = {x ∈ K : |{U ∈ U : x ∈ U}| = n} .

Then μ (K) = ∑_{n∈ω} μ (A_{n}), so there is an m ∈ ω such that ∑_{n=m}^{∞} μ (A_{n}) < \frac{δ}{2}. For each n < m let C_{n} ⊆ A_{n} be compact such that μ (A_{n} \ C_{n}) ≤ \frac{δ}{2m}. Putting C = ⋃_{n<m} C_{n}, μ (⋃_{n<m} A_{n} \ C) < \frac{δ}{2}.

Now it is not hard to see that {U ∈ U : U ∩ C ≠ ∅} is finite. Since μ (K \ C) < δ, U itself must be finite.

Theorem 3.4 I_{\frac{1}{n}} ≤_{K} tr (N) ≤_{K} Z.

Proof. It is easy to see that Z ≅ \{A ⊆ 2^{<ω} : lim_{n→∞} \frac{|A∩2^{n}|}{2^{n}} = 0\} and I_{\frac{1}{n}} ≅

\{A ⊆ 2^{<ω} : ∑_{n∈ω} \frac{|A∩2^{n}|}{2^{n}} < ∞\}. To see the latter first note that ∑_{n∈A} \frac{1}{n} =

∑_{n∈ω} ∑ \{\frac{1}{m} : m ∈ A ∩ [2^{n}, 2^{n+1})\} and then the isomorphism can be deduced observing that

\frac{1}{2} ∑_{n∈ω} \frac{|A ∩ [2^{n}, 2^{n+1})|}{2^{n}} ≤ ∑_{n∈ω} ∑_{m∈A∩[2^{n},2^{n+1})} \frac{1}{2^{n+1}} ≤ ∑_{n∈ω} ∑_{m∈A∩[2^{n},2^{n+1})} \frac{1}{m} ≤

≤ ∑_{n∈ω} ∑_{m∈A∩[2^{n},2^{n+1})} \frac{1}{2^{n}} ≤ ∑_{n∈ω} \frac{|A ∩ [2^{n}, 2^{n+1})|}{2^{n}}.

For I_{\frac{1}{n}} ≤_{K} tr (N), just note that if A ⊆ 2^{<ω} is such that ∑_{n∈ω} \frac{|A∩2^{n}|}{2^{n}} < ∞

then π (A) ∈ N. So, \{A ⊆ 2^{<ω} : ∑_{n∈ω} \frac{|A∩2^{n}|}{2^{n}} < ∞\} ⊆ tr (N).

To see that tr (N) ≤_{K} Z it suffices to prove that

A ∈ tr (N) ⇒ lim_{n→∞} \frac{|2^{n} ∩ A|}{2^{n}} = 0.

If it were not the case, then for some A ∈ tr (N) and for some ε > 0 the set J = \{n ∈ ω : \frac{|2^{n}∩A|}{2^{n}} ≥ ε\} is infinite. As A ∈ tr (N) , π (A) ∈ N, so there is a compact K ⊆ 2^{ω} such that μ (K) > 1 − \frac{ε}{3} and K ∩ π (A) = ∅. For n ∈ J, let U_{n} = ⋃_{s∈2^{n}∩A} [s] . Then μ (U_{n} ∩ K) ≥ \frac{2ε}{3}. By the previous lemma there is an x ∈ K ∩ π (A), contradicting the choice of K.

Proposition 3.5 I ≤_{K} tr (N) for all summable ideals I.

Proof. Let I be a summable ideal. By Lemma 1.12.4 in [Far00], it follows that there is an X ∈ I+_{\frac{1}{n}} such that I ≤_{K}I_{\frac{1}{n}} ↾ X. Fix such X ∈ I+_{\frac{1}{n}}. It suffices to show that I_{\frac{1}{n}} ↾ X ≤_{K} tr (N) .

Identify I_{\frac{1}{n}} with \{A ⊆ 2^{<ω} : ∑_{n∈ω} \frac{|A∩2^{n}|}{2^{n}} < ∞\}. So, ∑_{n∈ω} \frac{|X∩2^{n}|}{2^{n}} = ∞. It may happen that X ∈ tr (N). However, this problem is easily fixable. Recursively choose an increasing sequence of integers ⟨n_{i} : i ∈ ω⟩ such that ∑_{n∈[n_{i},n_{i+1})} \frac{|A∩2^{n}|}{2^{n}} ≥ 1. There is a bijection g : 2^{<ω} → 2^{<ω} such that

(1) g [2^{n}] = 2^{n}, for every n ∈ ω and

(2) for each s ∈ 2^{n_{i}+1} there is an m ∈ [n_{i}, n_{i+1}) such that s ↾ m ∈ g [X].

Note that for every I ⊆ X, I ∈ I_{\frac{1}{n}} if and only if g [I] ∈ I_{\frac{1}{n}}. So without loss of generality we can assume that g is the identity function, and hence π [X] = 2^{ω}. In particular, X ∉ tr (N) and I_{\frac{1}{n}} ↾ X ⊆ tr (N) ↾ X.

To prove that tr (N) ↾ X ≤_{K} tr (N) define a function h : 2^{<ω} → X as follows: For every s ∈ 2^{<ω} let m_{s} = max {k ∈ dom (s) : s ↾ k ∈ X}. This m_{s} is well defined for almost all s ∈ 2^{<ω} by (2). Fix t ∈ X and let

h (s) = \{ \begin{matrix}s ↾ m_{s} & if m_{s} is defined, \\ t & otherwise. \end{matrix}

To finish the proof, it suffices to notice that π [A] ⊆ π [h [A]] and hence h is a Katětov reduction witnessing tr (N) ↾ X ≤_{K} tr (N) .

Proposition 3.6 (a) (Farah) Every Erdős-Ulam ideal is Rudin-Blass equivalent to Z.

(b) If I is a density ideal then I ≤_{K}Z.

Proof. (a) follows directly from [Far00, p. 46] as any two Erdős-Ulam ideals EU_{f} and EU_{g} are even Rudin-Blass equivalent.

For (b) it suffices to note that for every tall density ideal I there is a positive set X such that I ↾X is Erdős-Ulam.

From the above propositions it follows that the density ideal Z is Katětov maximal among all ideals considered so far. On the other hand, the summable ideals are very low in the Katětov order. In particular, for any tall analytic P -ideal I there is an X ∈ I+_{\frac{1}{n}} such that I_{\frac{1}{n}} ↾ X ≤_{K} I. To see this, fix a lower semicontinuous submeasure φ such that I = Exh (φ) and note that I_{f} ⊆ I for f defined by f (n) = φ (\{n\}) as φ_{f} ≥ φ. By [Far00, p. 36] I_{f} ≤_{K} I_{\frac{1}{n}} ↾ X for some I_{\frac{1}{n}} -positive set X.

However, the summable ideals are highly non-homogeneous and we conjecture that they have consistently distinct covering numbers.

Using Proposition 3.1 we can summarize the impact of the Katětov order on analytic P -ideals as follows.

Theorem 3.7 (a) add (N) ≤ cov^{∗} (I) ≤ non (M) and cov (M) ≤ non^{∗} (I) ≤ cof (N) for every tall analytic P -ideal I.

(b) cov (N) ≤ cov^{∗} (I) and non^{∗} (I) ≤ non (N) for every tall summable ideal I.

(c) cov^{∗} (I) = cov^{∗} (Z) and non^{∗} (I) = non^{∗} (Z) for every Erdős-Ulam ideal I.

(d) cov^{∗} (Z) ≤ cov^{∗} (I) and non^{∗} (I) ≤ non^{∗} (Z) for every tall density or summable ideal I.

The next proposition resembles well known characterization of the splitting number (see [Vau90])

𝔰 = min {|S| : S ⊆ ℓ_{∞} ∧ (∀X ∈ [ω]^{ℵ_{0}}) (∃s ∈ S) (s ↾ X is not convergent)}

and may be of independent interest.

Proposition 3.8

𝔟 = min {|S| : S ⊆ c_{0} ∧ (∀X ∈ [ω]^{ℵ_{0}}) (∃s ∈ S) (s ↾ X ∉ ℓ_{1})} .

Proof. Denote by ν the cardinal on the right hand side of the equation. We first show that 𝔟 ≤ ν. In order to do this assume that κ < 𝔟 and show that κ < ν. Let S ∈ [c_{0}]^{κ}. For each s ∈ S define f_{s} : ω → ω by letting

f_{s} (n) = min \{m ∈ ω : (∀k ≥ m) (|s (k)| < \frac{1}{2^{k}} )\} .

There is a g ∈ ω^{ω} such that (∀s ∈ S) (f_{s} ≤^{∗} g). We can assume that g is strictly increasing. Put X = ran (g). If s ∈ S then there is an m ∈ ω such that f_{s} (k) ≤ g (k) for all k ≥ m and hence

∑_{x∈X\g(m)} |s (x)| = ∑_{n=m}^{∞} |s (g (n))| ≤ ∑_{n=m}^{∞} \frac{1}{2^{n}} < ∞.

Therefore, (∀s ∈ S) (s ↾ X ∈ ℓ_{1}) and hence κ < ν.

To see that ν ≤ 𝔟 proceed as follows. Consider a ≤^{∗}-unbounded family of strictly increasing functions, {f_{α} : α < 𝔟}, such that f_{α} (0) > 0, for each α < 𝔟. For all α < 𝔟 and n ∈ ω, let:

s_{α} (n) = \{ \begin{matrix}\frac{1}{m+1} & if f_{α} (m) ≤ n < f_{α} (m + 1) \\ 1 & if n < f_{α} (0) . \end{matrix}

We claim that S = {s_{α} : α < 𝔟} witnesses the definition of ν. To see this, let X ∈ [ω]^{ℵ_{0}} and let {x_{n} : n ∈ ω} be the increasing enumeration of X. There is an α < 𝔟 such that (∃^{∞}n ∈ ω) (x_{n} < f_{α} (n)). Define a sequence of natural numbers recursively by setting n_{k+1} = min {m > n_{k} : x_{m} < f_{α} (m)}. To see that s_{α} ↾ X ∉ ℓ_{1} note that

∑_{n=0}^{∞} s_{α} (x_{n}) ≥ n_{0} · \frac{1}{n_{0}} + (n_{1} − n_{0}) · \frac{1}{n_{1}} + ⋯ + (n_{k+1} − n_{k}) · \frac{1}{n_{k}} + ⋯

= 1 + ∑_{k=1}^{∞} (1 − \frac{n_{k}}{n_{k+1}} )

By the next lemma the last series never converges, which completes the proof of the proposition. We thank M. Garaev for supplying a short proof of the lemma.

Lemma 3.9 If ⟨x_{n} : n ∈ ω⟩ is a strictly increasing sequence of natural numbers then ∑_{n=0}^{∞} (1 − \frac{x_{n}}{x_{n+1}} ) diverges.

Proof. Without loss of generality x_{0} = 0. For every n ∈ ω, let y_{n} = x_{n+1} − x_{n}. Then y_{n} ≥ 1 and x_{n+1} = y_{0} + ⋯ + y_{n} for each n ∈ ω. So, 1 − \frac{x_{n}}{x_{n+1}} = \frac{y_{n}}{y_{0}+···+y_{n}}.

Assume that ∑_{n=0}^{∞} \frac{y_{n}}{y_{0}+···+y_{n}} converges. If ⟨y_{n} : n ∈ ω⟩ is bounded by some b then

∑_{n=0}^{m} \frac{y_{n}}{y_{0} + ⋯ + y_{n}} ≥ \frac{1}{b} ∑_{n=0}^{m} \frac{1}{n + 1},

which is a contradiction. So, assume that ⟨y_{n} : n ∈ ω⟩ is unbounded. There is a c ∈ ω such that ∑_{n=c+1}^{∞} \frac{y_{n}}{y_{0}+···+y_{n}} < \frac{1}{10}. Choose N ∈ ω so that y_{N} > y_{0} + ⋯ + y_{c}. Then

\frac{1}{10} > ∑_{n=c+1}^{N} \frac{y_{n}}{y_{0} + ⋯ + y_{n}} > \frac{\frac{1}{2} (y_{c+1} + ⋯ + y_{N}) + \frac{1}{2}y_{N}}{y_{0} + ⋯ + y_{N}} > \frac{1}{2},

which too is a contradiction.

For the rest of this section we will concentrate on cardinal invariants of the density ideal Z, trying to locate them among the standard cardinal invariants of the continuum.

Theorem 3.10 The following inequalities hold:

min {cov (N) , 𝔟} ≤ cov^{∗} (Z) ≤ non (M)  (1)

and

cov (M) ≤ non^{∗} (Z) ≤ max {𝔡, non (N)} .  (2)

Proof. The facts that cov^{∗} (Z) ≤ non (M) and cov (M) ≤ non^{∗} (Z) follow directly from Theorem 3.7(a).

To show that min {𝔟, cov (N)} ≤ cov^{∗} (Z), let κ < min {𝔟, cov (N)}. We show that κ < cov^{∗} (Z). Let {A_{α} : α < κ} ⊆ Z. To each A_{α} we associate a null set. To that end, we work on ∏_{n∈ω} 2^{n} endowed with μ^{∗}, the natural product of probability measures (each factor has the counting measure multiplied by \frac{1}{2^{n}}, so that μ (2^{n}) = 1). Let A_{α,n} = \{m < 2^{n} : 2^{n} + m ∈ A_{α}\}. Then lim_{n→∞} \frac{|A_{α,n}|}{2^{n}} = 0, for each α < κ, and since κ < 𝔟 and by Proposition 3.8 there is an X ∈ [ω]^{ℵ_{0}} such that

(∀α < κ) (∑_{n∈X} \frac{|A_{α,n}|}{2^{n}} < ε) .  (3)

Then let N_{α} = {h ∈ ∏_{n∈X} 2^{n} : (∃^{∞}n ∈ ω) (h (n) ∈ A_{α,n})}. The relation (3) shows that μ^{∗} (N_{α}) = 0, so N_{α} ∈ N, for every α < κ. Moreover, as κ < cov (N) there is an h ∈ ∏_{n∈X} 2^{n} \ ⋃ {N_{α} : α < κ} . Let Z be the set {2^{n} + h (n) : n ∈ X}. Then Z ∈ Z and Z ∩ A_{α} =^{∗} ∅, for every α < κ. So, κ < cov^{∗} (Z).

To conclude the proof, we show that non^{∗} (Z) ≤ max {𝔡, non (N)}. Fix a ≤^{∗}-dominating family {g_{α} : α < 𝔡} consisting of strictly increasing functions. For every α < 𝔡, choose non-null sets Y_{α} ⊆ ∏_{n∈ω} 2^{g_{α}(n)} of cardinality non (N) and enumerate each of them as {f_{α,β} : β < non (N)}. For α < 𝔡 and β < non (N), let

L_{α,β} = {2^{g_{α}(n)} + f_{α,β} (n) : n ∈ ω} ∈ Z.

We claim that L = {L_{α,β} : α < 𝔡 ∧ β < non (N)} witnesses the definition of non^{∗} (Z). To see this fix Z ∈ Z. There is a function f_{Z} : ω → ω such that

whenever g ≥^{∗} f_{Z} then ∑_{n=0}^{∞} \frac{|Z∩[2^{g(n)},2^{g(n+1)})|}{2^{g(n)}} < ∞. Let α < 𝔡 be such that f_{Z} ≤^{∗} g_{α}. Then the set

N_{α} = {f ∈ ∏_{n∈ω} 2^{g_{α}(n)} : (∃^{∞}m ∈ ω) (2^{g_{α}(m)} + f (m) ∈ Z)}

has measure zero, so there is a β < non (N) such that f_{α,β} ∉ N_{α}. Therefore

|L_{α,β} ∩ Z| < ℵ_{0}

as required.

For A ⊆ ω, the asymptotic density of A is defined as d (A) = lim_{n→∞} \frac{|A∩n|}{n}, whenever the limit exists. Define a probability measure on ω by μ ({n}) = \frac{1}{2^{n+1}} . Given an infinite set B consider the product measure μ_{B} on ω^{B}. For our next result we will use the following direct consequence of the Law of Large Numbers.

Proposition 3.11 Let B be any countable set. Then

μ_{B} (\{f ∈ ω^{B} : (∀n ∈ ω) (d (f^{−1} (n)) = \frac{1}{2^{n+1}} )\}) = 1.

Theorem 3.12 (a) cov^{∗} (Z) ≤ max {𝔟, non (N)}, ^{4} (b) non^{∗} (Z) ≤ min {𝔡, cov (N)} .

4Hrušák and Zapletal [HZ05] proved a general fact from which it follows that cov (N) ≤ cov^{∗} (tr (N)) ≤ max {cov (N) , 𝔡}. In particular, cov^{∗} (Z) ≤ max {cov (N) , 𝔡}.

Proof. For (a), let A = \{f ∈ ω^{ω} : (∀n ∈ ω) (d (f^{−1} (n)) = \frac{1}{2^{n+1}} )\}. By Proposition 3.11 μ_{ω} (A) = 1. Fix a μ_{ω}-non-null set X ⊆ A and an unbounded family D ⊆ ω^{ω}. For f ∈ D and g ∈ X let

I_{g,f} = ⋃_{n∈ω} g^{−1} (n) ∩ f (n) .

We claim that I_{g,f} ∈ Z. To see this note that {g^{−1} (n) : n ∈ ω} is a partition of ω such that d (g^{−1} (n)) = \frac{1}{2^{n+1}} for each n ∈ ω. The set I_{g,f} has only finite intersection with each element of the partition, hence I_{g,f} ∈ Z.

To finish the proof it suffices to verify that {I_{g,f} : x ∈ X, f ∈ D} is a covering family for Z. Indeed, if B ⊆ ω is an infinite set, put

F_{B} = {f ∈ A : (∃n ∈ ω) (f^{−1} (n) ∩ B) = ∅} .

We claim that μ_{ω} (F_{B}) = 0. By Proposition 3.11,

μ_{B} (\{f ∈ ω^{B} : (∀n ∈ ω)(d (f^{−1} (n)) = \frac{1}{2^{n+1}} )\}) = 1.

Let π : ω^{ω} → ω^{B} be the natural projection. Then

μ_{ω} (π^{−1}[\{f ∈ ω^{B} : (∀n ∈ ω)(d (f^{−1} (n)) = \frac{1}{2^{n+1}} )\}]) = 1

and F_{B} ∩ \{f ∈ ω^{B} : (∀n ∈ ω) (d (f^{−1} (n)) = \frac{1}{2^{n+1}} )\} = ∅.

Pick g ∈ X \F_{B}. Then B∩g^{−1} (n) ≠ ∅ for every n ∈ ω. Define f_{B} : ω → ω by

f_{B} (n) = min (B ∩ g^{−1} (n)) .

There is an f ∈ D such that f_{B} <^{∗} f. Then |I_{x,f} ∩ B| = ℵ_{0}, and the proof is done.

For (b) dualize the argument for (a). Let κ < min {𝔡, cov (N)} and fix {X_{α} : α < κ} ⊆ [ω]^{ℵ_{0}}. As each F_{X_{α}} is null there is a g ∈ A \ ⋃_{α<κ} F_{X_{α}}. That is, each X_{α} intersects g^{−1} (n) for all n ∈ ω. For α < κ, let

f_{α} (n) = min (g^{−1} (n) ∩ X_{α}) + 1.

As κ < 𝔡, there is a f ∈ ω^{ω} not dominated by any f_{α}. Let Z = ⋃_{n∈ω} g^{−1} (n)∩ f (n). Then Z ∈ Z and Z ∩ X_{α} is infinite for each α < κ. So, κ < non^{∗} (Z).

Next we mention some consistency results.

Theorem 3.13 The following are consistent with ZFC:

(a) cov (N) < cov^{∗} (I) for all tall analytic P -ideals,

(b) cov^{∗} (I) < add (M) for all tall analytic P -ideals.

Proof. (a) holds in any model where cov (N) < 𝔭 (see [BJ95]).

(b) holds in the Hechler model. To see this it suffices to note that under CH every tall P -ideal has a cofinal family which forms a strictly increasing tower. By a result of Baumgartner and Dordal [BD85], the iteration of Hechler forcing preserves towers and hence the cofinal family from V still witnesses cov^{∗} (Z) = ω_{1} in the extension. It is well known that add (M) = 𝔠 in the Hechler model (see [BJ95]).

A forcing ℙ destroys (diagonalizes) an ideal I if it introduces a set A ⊆ ω such that |A ∩ I| < ℵ_{0} for every I ∈ I ∩ V. The covering number of an ideal can be viewed as a measure of how difficult it is to diagonalize it. In particular, it is of interest to identify those ideals which can be diagonalized by an ω^{ω}-bounding forcing. The results of the previous section together with the following result of Brendle and Yatabe [BY05] tell us which (analytic P -)ideals are destroyed by the random forcing (see also [HZ04]).

Theorem 3.14 (Brendle-Yatabe) Let I be a tall ideal on ω. Then the random forcing destroys I if and only if I ≤_{K} tr (N).

Theorem 3.15 It is relatively consistent with ZFC that cov^{∗} (Z) < cov (N) = cov^{∗} (tr (N)) = cov (I) for all summable ideals I.

Proof. The model for this is the random real model. In this model, 𝔡 = non (N) = ω_{1} < cov (N) = 𝔠 (see [BJ95]). By Theorem 3.7 cov (N) = cov (I) for all summable ideals I and by Theorem 3.12 cov^{∗} (Z) = ω_{1}.

Definition 3.16 An analytic ideal I on ω is said to be totally bounded if whenever φ is a lower semicontinuous submeasure on P (ω) for which I = Exh (φ) then φ (ω) < ∞.

The following lemma provides an easy criterion for checking that an ideal is totally bounded. Recall that the splitting number 𝔰 (𝔹) of a Boolean algebra is defined as the minimal size of a family S of non-zero elements of 𝔹 such that for every non-zero b ∈ 𝔹 there is an s ∈ S such that b∧s ≠ 0 ≠ b−s.

Lemma 3.17 If I is an analytic P -ideal such than 𝔰 (P (ω) /I) = ω, then I is totally bounded.

Proof. Consider a submeasure φ on P (ω) such that I = Exh (φ) and φ (ω) = ∞. Fix any family {A_{n} : n ∈ ω} of I-positive sets. To show that this family is not a splitting family note that either A_{0} or else ω \ A_{0} has infinite submeasure. Without loss of generality assume A_{0} has. Choose then a finite F_{0} ⊆ A_{0} with φ (F_{0}) ≥ 1. Now, either (A_{0} \ F_{0})∩A_{1} or else (A_{0} \ F_{0})\ A_{1} has infinite submeasure. Once again, without loss of generality, assume (A_{0} \ F_{0})∩A_{1} has infinite submeasure and choose some finite F_{1} ⊆ (A_{0} \ F_{0})∩ A_{1} such that φ (F_{1}) ≥ 1. Now proceed alike with (A_{0} ∪ A_{1} \ (F_{0} ∪ F_{1})) ∩ A_{2} and (A_{0} ∪ A_{1} \ (F_{0} ∪ F_{1})) \ A_{2} and so on.

Let B = ⋃_{n∈ω} F_{n}. Then B ∉ I and it is clear that B is not split by any element of the family {A_{n}}_{n∈ω} .

Proposition 3.18 Z, tr (N) are totally bounded.

Proof. By previous lemma it suffices to show that there are countable splitting families in the respective algebras. For n > 0 and k < n, let S_{n,k} = {k + mn : m ∈ ω}. It is easily seen that S = {[S_{n,k}] : n, k ∈ ω} is a splitting family in P (ω) /Z. For s ∈ 2^{<ω}, let ⟨s⟩ = {t ∈ 2^{<ω} : s ⊆ t}. It is just as easy to see that S = {[⟨s⟩] : s ∈ 2^{<ω}} is a splitting family for P (2^{<ω}) / tr (N).

If an analytic P -ideal is not totally bounded then it is contained in a proper F_{σ}-ideal by Theorem 1.3. An ω^{ω}-bounding forcing for diagonalizing F_{σ}-ideals was discovered by C. Laflamme [Laf92]. We briefly review (an incarnation of) Laflamme’s forcing. Let I be an F_{σ}-ideal. I = Fin (φ) for some lower semicontinuous submeasure φ by Theorem 1.3. Define the poset ℙ_{φ} as the set of all trees T ∈ ω^{<ω} with stem s and such that:

1. succ_{T} (t) = {n ∈ ω : t^{⌢}n ∈ T } is finite for every t ∈ T, and

2. lim_{t∈T} φ (succ_{T} (t)) = ∞.

ordered by inclusion.

Call a tree T ∈ ℙ_{φ} a k-tree with stem s if 1. (∀t ∈ T) (t ⊆ s or t ⊇ s) and

2. (∀t ∈ T) (s ⊆ t ⇒ ϕ (succ_{T} (t)) ≥ k).

If n ∈ ω and T, T′ ∈ ℙ_{φ} we say that

T ≤_{n} T′ if (∀t ∈ T) φ (succ_{T′}(t)) ≤ n ⇒ succ_{T} (t) = succ_{T′} (t)
φ (succ_{T′}(t)) ≥ n ⇒ φ (succ_{T} (t)) ≥ n

Lemma 3.19 The forcing ℙ_{φ} is proper and ω^{ω}-bounding.

Proof. We actually show that ℙ_{φ} together with the sequence ⟨≤_{n}: n ∈ ω⟩ satisfies Axiom A. (1)-(3) from the definition of Axiom A are easy to verify. To show (4) and the fact that ℙ_{φ} is ω^{ω}-bounding it suffices to prove [BJ95, Lemma 7.2.10]:

Claim 3.20 If n ∈ ω, T ∈ ℙ_{φ} and ȧ is a ℙ_{φ}-name for an ordinal, then there exist S ≤_{n} T and a finite set of ordinals F such that S ⊩ ȧ ∈ F.

Fix T, n and ȧ. Define a rank function on the nodes of T ∈ ℙ_{φ} by

1. rk (t) = 0, if there is an n-tree T′ with stem t which decides ȧ, and

2. rk (t) ≤ i + 1, if rk (t) ≰ i and

φ (\{t′ ∈ succ_{T} (t) : rk (t′) ≤ i\}) ≥ \frac{1}{2}φ (succ_{T} (t)) .

Note that, for every t ∈ T, there is an i ∈ ω such that rk (t) = i. For if not, then there is a t ∈ T with undefined rank and one can recursively construct a tree T′ ⊆ T with stem t such that for every s ∈ T′, if s ≥ t then

succ_{T′}(s) = {s′ ∈ succ_{T} (s) : rk (s′) is undefined}

and

φ(succ_{T′}(s)) ≥ \frac{1}{2}φ (succ_{T} (s)) .

Then T′ ∈ ℙ_{φ} and no extension of T′ decides ȧ, which is a contradiction.

Now, E = {t ∈ T : rk (t) > 0} is finite as T is finitely branching and rk is strictly decreasing on T. So, there is a k ∈ ω such that E ⊆ {t ∈ T : |t| < k}.

In particular, for every t ∈ T with |t| = k there is an n-tree S_{t} with stem t and an ordinal α_{t} such that S_{t} ⊩ ȧ = α_{t}. Let

S = ⋃_{t∈ω^{k}∩T} S_{t} and F = {α_{t} : t ∈ ω^{k} ∩ T } .

Then S ≤_{n} T and S ⊩ ȧ ∈ F.

Lemma 3.21 ℙ_{φ} destroys I.

Proof. Let Ȧ_{gen} be the canonical name for the subset of ω coded by a generic filter in ℙ_{φ}. We show that if I ∈ I and T ∈ ℙ_{φ} there exist S ≤ T and m ∈ ω such that S ⊩ Ȧ_{gen} ∩ (I \ m) = ∅. Fix I ∈ I and T ∈ ℙ_{φ}. As I = Fin (φ), there is an n ∈ ω such that φ (I) < n. Let T′ ≤ T be a 2n-tree with stem s. Then, for every t ∈ T′ such that s ⊆ t, φ (succ_{T′} (t) \ I) ≥ φ (succ_{T′} (t)) − n. Define S ≤ T′ recursively by :

1. t ⊆ s ⇒ t ∈ S,

2. t ∈ S and t ⊇ s ⇒ succ_{S}(t) = succ_{T′} (t) \ I.

Let m = max (ran (s)) + 1. Then S ∈ ℙ_{φ} and S ≤ T and S ⊩ Ȧ_{gen} ∩ (I \ m) = ∅.

Theorem 3.22 It is relatively consistent with ZFC that ω_{1} = 𝔡 < cov^{∗} (I) for all analytic P -ideals which are not totally bounded.

Proof. Start with a model of CH and iterate forcings of the type ℙ_{φ} with countable support of length ω_{2}, where each ℙ_{φ} appears cofinally. Then cov^{∗} (I) = 𝔠 = ω_{2} as ℙ_{φ} destroys I = Fin (φ), and hence also Exh (φ). On the other hand, 𝔡 = ω_{1} in the resulting model as countable support iteration of ω^{ω}-bounding forcings is ω^{ω}-bounding [She98].

The advantage of our forcings over those of Laflamme is the simplicity of their definition. On the other hand, it seems to be more difficult to decide stronger properties of the forcings ℙ_{φ}. We do not even know whether a forcing ℙ_{φ} can add random reals. Forcings of the type ℙ_{φ} are still not quite well understood and deserve further investigation. For instance, it would be nice to know if/when they preserve P -points and if/when they preserve positive outer measure. As the forcings of the type ℙ_{φ} are in general not homogeneous, it is reasonable to expect that the answers to these questions may differ depending on the choice of the generic filter.

There are several natural open problems concerning cardinal invariants of analytic P -ideals. We explicitly mention at least four:

Questions 3.23 (a) Is cov^{∗} (Z) ≤ 𝔡 (or even cov^{∗} (Z) ≤ 𝔟)? ^{5}

(b) Is cov^{∗} (Z) minimal among the covering numbers of analytic P -ideals?

(c) Is cov^{∗} (I) = cov^{∗}(I_{\frac{1}{n}}) for all tall summable ideals I?

(d) Is cov^{∗} (I) = cov^{∗} (J) for all density ideals I, J which are not Erdős-Ulam?

4 The separating number

Let I be an ideal on ω and let G ⊆ I, H ⊆ I^{+} and A ⊆ ω. The set A separates G from H if

1. |A ∩ I| < ℵ_{0} for every I ∈ G, and

2. |A ∩ X| = ℵ_{0} for every X ∈ H.

A forcing ℙ separates an ideal I if it introduces a set A ⊆ ω such that A separates I ∩ V from I^{+} ∩ V. That is, I ∩ V is contained in {B ⊆ ω : A ∩ B is finite} while I^{+} ∩ V ∩ {B ⊆ ω : A ∩ B is finite} is empty. So, ℙ separates I and I^{+} (as subsets of P (ω)) by a very simple F_{σ}-set. In other words, separation is a strong kind of diagonalization (see [HZ04]).

Definition 4.1 For an ideal I on ω, the separating number of I is

sep (I) = min{|G| + |H| : G ⊆ I, H ⊆ I^{+} and

(∀A ⊆ ω) (A does not separate G from H)}.

5By Theorem 3.12 and the result of [HZ05] cov^{∗} (Z) ≤ max {𝔟, non (N)} as well as cov^{∗} (Z) ≤ max {𝔡, cov (N)}. Consequently, if Z can be destroyed by an ω^{ω}-bounding forcing ℙ, then ℙ must add random reals as well as make the ground model reals of measure zero.

Just like cov^{∗} (I) measures destructibility of the ideal I, the separating number measures how hard/easy it is to separate I. It is easily seen that

add^{∗} (I) ≤ sep (I) ≤ cov^{∗} (I)

for every tall ideal I. For maximal ideals I, sep (I) = cov^{∗} (I).

The Rudin-Keisler order relates to sep in the same way that the Tukey order relates to add^{∗} and the Katětov order relates to cov^{∗}.

Proposition 4.2 I ≤_{RK}J ⇒ sep (J) ≤ sep (I) .

Proof. Fix f : ω → ω witnessing that I ≤_{RK}J. Let G ⊆ I and H ⊆ I^{+}, each of cardinality less than κ ≤ sep (J). Without loss of generality assume that f^{−1} (G) ∈ G for every finite G ⊆ ω. Then for G^{←} = {f^{−1} (G) : G ∈ G} and H^{←} = {f^{−1} (H) : H ∈ H} there is an A ⊆ ω such that |A ∩ G| < ℵ_{0} for every G ∈ G^{←} and |A ∩ H| = ℵ_{0} for every H ∈ H^{←}. Then f [A] ∩ G is finite for every G ∈ G and f [A]∩H is infinite for every H ∈ H. Indeed, if f [A]∩H were finite for some H ∈ H, then f^{−1} (f [A] ∩ H) ∈ G^{−1} and thus A∩f^{−1} (H) is also finite contradicting the choice of A. Therefore κ ≤ sep (I).

To highlight the importance of the separating number we mention the following theorem. Recall that 𝕃_{I^{∗}} is the standard Laver type forcing for diagonalizing (in fact, separating) of the ideal I. 𝔪 (ℙ) denotes the minimum cardinal κ for which MA_{κ} (ℙ) fails. Recall also that a free ultrafilter U on ω is nowhere dense (nwd) if for every f : ω → ℝ there is U ∈ U such that f [U] is nowhere dense in ℝ.

Theorem 4.3 ([GHH04]) Let I be an ideal on ω. Then

𝔪 (𝕃_{I^{∗}}) = \{ \begin{matrix}min \{𝔟, sep (I)\} & if I^{∗} is an nwd-ultrafilter, \\ min \{add (M) , sep (I)\} & otherwise. \end{matrix}

The forcing 𝕃_{I^{∗}} always adds a dominating real. Another standard forcing for diagonalizing an ideal I is the Mathias-Prikry forcing 𝕄_{I^{∗}} defined as follows:

𝕄_{I^{∗}} = {⟨s, F ⟩ : s ∈ [ω]^{<ω} and F ∈ I^{∗}}

ordered by

⟨s, F ⟩ ≤ ⟨t, G⟩ if and only if s ⊇ t,
F ⊆ G and
s \ t ⊆ G.

A straightforward density argument shows that (just like 𝕃_{I^{∗}}) the forcing 𝕄_{I^{∗}} not only diagonalizes I, it separates I. It is not known in general when 𝕄_{I^{∗}} adds a dominating real, however, if I is F_{σ} then 𝕄_{I^{∗}} does not add dominating reals even when iterated (see [HZ04] or [Bre97, 3.2 Case 1] for a proof).

Proposition 4.4 (a) If I is a Erdős-Ulam ideal then sep (I) = sep (Z).

(b) If I is a tall density ideal then sep (I) ≤ sep (Z).

(c) If I_{f} and I_{g} are tall summable ideals then sep (I_{f}) = sep (I_{g}) .

Proof. First note that sep (I) ≤ sep (I ↾ X) for every ideal I and X ∈ I^{+}.

(a) follows directly from Proposition 4.2 and 1.13.10 in [Far00]. (b) follows from (a) and the fact for every tall density I there is a X ∈ I^{+} such that I ↾X is Erdős-Ulam.

Using the observation at the beginning of the proof, (c) follows from Proposition 4.2 and Claim 1 of Lemma 1.12.4 in [Far00].

Theorem 4.5 (a) sep (I) ≤ 𝔟 for all analytic P -ideals I which are not F_{σ}.

(b) It is consistent with ZFC that sep (I) > 𝔟 for all F_{σ} P -ideals I .

Proof. Let I be an analytic P -ideal which is not F_{σ}. By a result of Solecki, see [Sol99, Theorem 3.4] there is a partition ⟨A_{n} : n ∈ ω⟩ of ω into I-positive sets such that every I ⊆ ω intersecting each A_{n} in a finite set is in I. Let κ < sep (I) and fix a family of functions {f_{α} ∈ ω^{ω} : α < κ}. Enumerate each A_{n} as {a_{i}^{n} : i ∈ ω} and let

B_{α} = {a_{m}^{n} : m ≤ f_{α} (n)} .

As κ < sep (I), there is an X ⊆ ω which intersects every A_{n} in an infinite set and contains only finitely many elements of each B_{α}, α < κ. Define f : ω → ω by

f (n) = min {k ∈ A_{n} : (A_{n} \ k) ∩ X = ∅} .

Then f dominates f_{α} for every α < κ.

To prove (b) start with a model of CH and iterate forcings of the type 𝕄_{I^{∗}}, where I is an F_{σ} P -ideal, with finite support of length ω_{2}, where each 𝕄_{I^{∗}} appears cofinally. Then sep (I) = 𝔠 = ω_{2} as 𝕄_{I^{∗}} separates I. On the other hand, 𝔟 = ω_{1} in the resulting model as finite support iteration of forcings of type 𝕄_{I^{∗}}, for I F_{σ}, does not add dominating reals.

References

[Bar99] Tomek Bartoszyński, Invariants of Measure and Category, Available at arXiv:math.LO/9910015, 1999.

[BJ95] Tomek Bartoszyński and Haim Judah, Set theory, A K Peters Ltd., Wellesley, MA, 1995, On the structure of the real line. MR 96k:03002

[BD85] James E. Baumgartner and Peter Dordal, Adjoining dominating functions, J. Symbolic Logic 50 (1985), no. 1, 94–101. MR 86i:03064

[Bre97] Jörg Brendle, Mob families and mad families, Arch. Math. Logic 37 (1997), no. 3, 183–197. MR MR1619601 (99m:03098)

[BS99] Jörg Brendle and Saharon Shelah, Ultrafilters on ω—their ideals and their cardinal characteristics, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2643–2674. MR 2000m:03111

[BY05] Jörg Brendle and Shunsuke Yatabe, Forcing indestructibility of MAD families, Ann. Pure Appl. Logic 132 (2005), no. 2-3, 271– 312. MR MR2110827

[Far00] Ilijas Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, xvi+177. MR 2001c:03076

[Far02a] , Analytic Hausdorff gaps, Set theory (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 58, Amer. Math. Soc., Providence, RI, 2002, pp. 65–72. MR MR1903850 (2003e:03092)

[Far02b] , How many Boolean algebras P(ℕ)/I are there?, Illinois J. Math. 46 (2002), no. 4, 999–1033. MR MR1988247 (2004f:03091)

[Far04a] , Analytic Hausdorff gaps II: The density zero ideal, Preprint, 2004.

[Far04b] , Luzin gaps, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2197–2239 (electronic). MR 2 048 515

[Fre91] David H. Fremlin, The partially ordered sets of measure theory and Tukey’s ordering, Note Mat. 11 (1991), 177–214. MR 95e:06006

[Fre04] , Measure theory, Torres Fremlin, Colchester, England, 2004, Set theoretic measure theory.

[GHH04] R. A. González, F. Hernández, and M. Hrušák, Covering separable spaces by nowhere dense sets., In preparation, 2004.

[HZ04] Michael Hrušák and Jindřich Zapletal, Forcing with quotients, Preprint, 2004.

[HZ05] , Katětov order, In preparation, 2005.

[Jec97] Thomas Jech, Set theory, second ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1997. MR 99b:03061

[Kun83] Kenneth Kunen, Set theory. An introduction to independence proofs. Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983, Reprint of the 1980 original. MR 85e:03003

[Laf92] Claude Laflamme, Zapping small filters, Proc. Amer. Math. Soc. 114 (1992), no. 2, 535–544. MR 92e:03076

[LV99] Alain Louveau and Boban Veličković, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 (1999), no. 1-3, 171–195. MR MR1708151 (2000g:03111)

[She98] Saharon Shelah, Proper and improper forcing, second ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 98m:03002 [Sol99] Sławomir Solecki, Analytic ideals and their applications, Ann. Pure

Appl. Logic 99 (1999), no. 1-3, 51–72. MR 2000g:03112

[Tod96] Stevo Todorčević, Analytic gaps, Fund. Math. 150 (1996), no. 1, 55–66. MR MR1387957 (98j:03070)

[Vau90] Jerry E. Vaughan, Small uncountable cardinals and topology, Open problems in topology, North-Holland, Amsterdam, 1990, With an appendix by S. Shelah, pp. 195–218. MR 1 078 647