TRANSITIVE PROPERTIES OF IDEALS

Jan Kraszewski

Abstract. In this paper we present a wide range of results connected with transitive properties of ideals. In particular, we present relations between standard and transitive coefficients of ideals and compute transitive cardinal coefficients of ideals on generalized Cantor spaces.

0. Introduction

In this paper we would like to give a description of cardinal characteristics connected with certain transitive properties of ideals. We begin our presentation from some history.

Let 𝔾 be any abelian group and let J be any proper ideal of subsets of 𝔾 which is translation invariant (that is I + g ∈ J for each I ∈ J and g ∈ 𝔾). The first cardinal coefficient on the stage was a transitive covering number of J (denoted by cov_{t}(J)) that appeared implicitely in 1938 in the famous Rothberger theorem, which was originally formulated for classical ideals of meagre and null subsets of the real line (cf. [21]). We can formulate this theorem more generally as follows.

Theorem 0.1. Let J and I be translation invariant ideals of subsets of a group 𝔾, orthogonal to each other (that is there exist A ∈ J and B ∈ I such that A∪B = 𝔾). Then

cov_{t}(J) ≤ non(I),

where non(I) is the minimal cardinality of the subset of 𝔾 that do not belong to I.

Proof. We fix A ∈ J and B ∈ I such that A ∪ B = 𝔾. Let T ⊆ 𝔾 be the set of cardinality non(I), which is not in I. One can notice that A − T = 𝔾, which ends the proof. □

In 1981 Carlson asked if it was possible to find a null subset B of the real line with a property that for every null subset A of the real line there exists a real number r such that A ⊆ B + r. We can reformulate this problem in the following way.

We call a family B ⊆ J a transitive base of J if for each A ∈ J there exists B ∈ B and g ∈ 𝔾 such that A ⊆ B +g. The minimal cardinality of a transitive base of J we call the transitive cofinality and denote by cof_{t}(J). Thus, the question was whether cof_{t}(N) = 1, where N denotes the σ−ideal of null subsets of the real line. The most general answer was obtained by Galvin in 1981 and independently by Brzuchowski, Cichoń and We,glorz in 1982 (cf. [7], p.166 or [22]).

Typeset by AMS-TEX

Key words and phrases. Set Theory, σ-ideals, groups, translation invariance, cardinal functions.

The author is supported by a KBN grant 5 P03A 051 20.

Theorem 0.2. Let J be a translation invariant ideal of subsets of a group 𝔾. If J is symmetric (that is −A ∈ J for each A ∈ J), then cof_{t}(J) > 1.

Proof. Suppose that there exists a set B ∈ J such that for every A ∈ J there exists g ∈ 𝔾 such that A ⊆ B + g. If b ∉ B then B ⊆_{}B̸ ∪ {b} ⊆ B + g for some g ∈ 𝔾. Thus −B ⊆_{}−̸B − g. But we can assume that B = −B. Hence B ⊂ B + g ⊆_{}(B − g) + g = B which is a contradiction. □

The complete description of transitive cofinalities of ideals of meagre and null subsets of the real line was presented by Pawlikowski in [19] in 1984. He also mentioned a dual coefficient to a transitive cofinality. Following the way of describing cardinal caracteristics of the continuum presented by Blass in [4] we will call it a transitive additivity and denote by add_{t}(J). Unfortunately, Pawlikowski (and then Bartoszyński and Judah in [2]) used this name and notation for another coefficient. In order not to make a mess we will call it a starred transitive additivity and denote by add_{t}^{∗}(J).

In 1989 Seredyński in [22] investigated properties of some transitive operations on ideals.

In 1993 Carlsson in [6] introduced the notion of κ-translatibility and proved that the σ−ideal of meagre substes of the real line and the σ−ideal generated by closed null subsets of the real line are ω-translatable. Bartoszyński in [1] proved that the σ −ideal of null substes of the Cantor space is not 2-translatable. Kysiak in [16] introduced a natural notion of a translatibility number.

In the second paragraph of this paper we present relations between standard and transitive coefficients of an ideal. In the fifth paragraph we discuss possibility of existence other relations. In the third paragraph we compute these characteristics for the σ−ideal 𝕊_{2}. In the fourth paragraph we show that the transitive covering number can be totally different from the standard cofinality. The sixth paragraph is devoted to transitive operations. Finally, we focus our attention on transitive cardinal coefficients of ideals of subsets of generalized Cantor spaces.

1. Definitions and basic properties

We use standard set-theoretical notation and terminology derived from [15]. Let us remind that the cardinality of the set of all real numbers is denoted by 𝔠. The cardinality of a set X is denoted by |X|. A power set of a set X is denoted by P(X). If κ is a cardinal number then [X]^{κ} ([X]^{≤κ}) denotes the family of all subsets of the set X of cardinality κ (not greater than κ, respectively). X^{<ω} denotes the set of all finite sequences of elements of the set X. If φ : X → Y is a function then rng(φ) denotes the range of φ. If A ⊆ Y then φ^{−1}[A] denotes the pre-image of A.

Let (G, +) be an infinite abelian group. We consider a σ−ideal J of subsets of G which is proper and contains all singletons (i.e. ⋃ J = G). Moreover, we assume that J is translation invariant (i.e. (∀A ∈ J)(∀g ∈ G) A+g = {a+g : a ∈ A} ∈ J) and symmetric (i.e. (∀A ∈ J) − A = {−a : a ∈ A} ∈ J).

We say that a family B ⊆ J is cofinal with J if for each A ∈ J there exists such B ∈ B that A ⊆ B. We also call such a family B a base of J.

For an ideal J we consider the following cardinal numbers

add(J) = min{|A| : A ⊆ J & ⋃ A ∉ J },
cov(J) = min{|A| : A ⊆ J & ⋃ A = G},
non(J) = min{|B| : B ⊆ G & B ∉ J },
cof(J) = min{|B| : B ⊆ J & B is a base of J }.

They are called the additivity, the covering number, the uniformity and the cofinality of J, respectively. Note that the following relations hold:

add(J) ≤ cov(J), add(J) ≤ non(J), cov(J) ≤ cof(J), non(J) ≤ cof(J).

Moreover, add(J) is regular and add(J) ≤ min{cf(non(J)), cf(cof(J))}.

We call a family B ⊆ J a transitive base if for each A ∈ J there exists B ∈ B and g ∈ G such that A ⊆ B + g.

For an ideal J we consider the following cardinal numbers

add_{t}(J) = min{|A| : A ⊆ J & ¬(∃B ∈ J)(∀A ∈ A)(∃g ∈ G) A ⊆ B + g},
add_{t}^{∗}(J) = min{|T | : T ⊆ G & (∃A ∈ J) A + T ∉ J },
cov_{t}(J) = min{|T | : T ⊆ G & (∃A ∈ J) A + T = G},
cof_{t}(J) = min{|B| : B ⊆ J & B is a transitive base of J }.

First two ones are both called transitive additivity. The latter two ones are called transitive covering number and transitive cofinality, respectively. Let us notice that all definitions of cardinal coefficients mentioned above (both normal and transitive) are valid also for an arbitrary family A ⊆ P (G).

We say that an ideal J is κ − translatable if

(∀A ∈ J)(∃B ∈ J)(∀T ∈ [G]^{κ})(∃g ∈ G) A + T ⊆ B + g.

We define a translatibility number of J as follows

τ(J) = min{κ : J is not κ − translatable}.

For a σ−ideal J of subsets of G we define the following families of subsets of G

\begin{matrix}s(J) = \{A ⊆ G : (∀B ∈ J) A + G ≠ G\}, \\ g(J) = \{A ⊆ G : (∀B ∈ J) A + G ∈ J \}, \end{matrix}

The following basic properties of operations s and g can be found e.g. in [22].

Proposition 1.1. Let us assume that J is a proper, translation invariant, symmetric σ−ideal which contains singletons. Then (a) g(J) is a proper, translation invariant, symmetric σ−ideal which contains singletons;

(b) s(J) is a proper, translation invariant, symmetric family of subsets of G which contains singletons; (c) g(J) ⊆ J ∩ s(J);

(d) s(s(s(J))) = s(J);

(e) g(g(J)) = g(J). □

If G is a fixed Polish locally compact group equipped with Haar measure then the σ−ideals of meagre subsets and of null subsets of G are denoted by M(G) and N (G), respectively. We will write M and N if it is clear which Polish group we consider or if the specification of a group is not necessary.

The Galvin-Mycielski-Solovay theorem [10] shows that s(M) is a σ − ideal of strongly null sets (this theorem is used to be formulated for G = 2^{ω} or G = ℝ but is true for every locally Polish group – see [16]). The family s(N) is called strongly meager sets. Recently Bartoszyński and Shelah show [3] that under CH strongly meager sets do not form an ideal.

Sets from σ−ideals g(M) and g(N) are called meager-additive and null-additive, respectively (see [2] for more information).

From now on we deal with the generalized Cantor space 2^{κ} interpreted as the set of all functions from an infinite cardinal number κ into the set {0, 1}. This spaces are endowed with the standard product topology. Moreover, we consider the standard product measure on 2^{κ}.

We define

P if = {f : f is a function & dom(f) ∈ [ω]^{ω} & rng(f) ⊆ 2}.

If f ∈ P if then we put

[f] = {x ∈ 2^{ω} : f ⊆ x}.

Let 𝕊_{2} denotes the σ-ideal of subsets of 2^{ω}, which is generated by a family {[f] : f ∈ P if}. We recall some properties of 𝕊_{2}, which were proved in [8].

Fact 1.2. (a) 𝕊_{2} is a proper σ-ideal, containig singletons, with a base consisting of Borel sets. Every A ∈ 𝕊_{2} is both meager and null.

(b) 𝕊_{2} is translation invariant ans symmetric.

(c) There exist a family of size 𝔠 of pairwise disjoint Borel subsets of 2^{ω} that do not belong to 𝕊_{2}. □

We call a family F ⊆ P if normal if for each two different f_{1}, f_{2} ∈ F we have dom(f_{1}) ∩ dom(f_{2}) = ∅. Directly from the definition of 𝕊_{2} we can deduce that

A ∈ 𝕊_{2} ⟺ A ⊆ ⋃_{f∈F}[f],

for some countable normal family F ⊆ P if. In [8] the following useful lemma was proved.

Lemma 1.3. Suppose that {f_{i} : i ∈ I} is a normal family of functions from P if, f ∈ P if and [f] ⊆ ⋃_{i∈I}[f_{i}]. Then [f] ⊆ [f_{i}] for some i ∈ I. □

Let A, S be two infinite subsets of ω. We say that S splits A if |A∩S| = |A\S| = ω. Let us recall a cardinal number related with a notion of splitting, introduced by Malychin in [17], namely

ℵ_{0}-𝔰 = min{|S| : S ⊆ [ω]^{ω} & (∀A ∈ [[ω]^{ω}]^{ω})(∃S ∈ S)(∀A ∈ A)(S splits A)}.

More about cardinal numbers connected with the relation of splitting can be found in [12].

We define also a reaping number

𝔯 = min{|R| : R ⊆ [ω]^{ω} & (∀A ∈ [ω]^{ω})(∃R ∈ R)(A does not splitR)}.

(cf. [23] for more details). In [8] the following fact was proved.

Fact 1.4. add(𝕊_{2}) = ω_{1}, non(𝕊_{2}) = ℵ_{0}-𝔰, cov(𝕊_{2}) = 𝔯, cof(𝕊_{2}) = 𝔠. □

We introduce some extra notation in order to simplify further considerations. Let λ and κ be any infinite cardinal numbers. We put Inj(λ, κ) = {φ ∈ κ^{λ} : φ is an injection}.

Definition. For A ⊆ 2^{κ} and φ ∈ Inj(λ, κ) we put φ ∗ A = {x ∘ φ : x ∈ A}. For B ⊆ 2^{ω} and φ ∈ Inj(ω, κ) we put B_{φ} = {x ∈ 2^{κ} : x ∘ φ ∈ B}.

Obviously, φ ∗ A ⊆ 2^{λ} and B_{φ} ⊆ 2^{κ}. Another simple observation is that for B ⊆ 2^{ω} and φ ∈ Inj(ω, κ) we have φ ∗ B_{φ} = B. Similarly, for A ⊆ 2^{κ} and φ ∈ Inj(ω, κ) we have A ⊆ (φ ∗ A)_{φ}.

From now on let J be a σ−ideal of subsets of 2^{ω} and κ be any infinite cardinal number. We define

κ(J) = {A ⊆ 2^{κ} : (∃φ ∈ Inj(ω, κ)) φ ∗ A ∈ J }.

If A ⊆ 2^{κ} then any φ ∈ Inj(ω, κ) such that φ ∗ A ∈ J we called a witness for A. If I ⊆ J then κ(I) ⊆ κ(J). Moreover, we have J ⊆ ω(J), because for every A ∈ J the identity on ω is a witness for A. The σ−ideal generated by the family κ(J) we denote by J_{κ}.

Definition. The σ−ideal J is productive if ω(J) ⊆ J.

We formulate some equivalent versions of this property now, proved in [13].

Fact 1.5. For a σ−ideal J of subsets of 2^{ω} the following conditions are equivalent:

(a) J is productive,

(b) (∀A ⊆ 2^{ω})(∀φ ∈ Inj(ω, ω))(φ ∗ A ∈ J ⇒ A ∈ J),

(c) (∀A ⊆ 2^{ω})(∀φ ∈ Inj(ω, ω))(A ∈ J ⇒ A_{φ} ∈ J). □

Directly from their definitions we deduce that the σ−ideals of meagre subsets and of null subsets of 2^{ω} are productive. Also the σ −ideal generated by closed null subsets of 2^{ω} is productive. Moreover, 𝕊_{2} is the least non-trivial productive σ−ideal of subsets of the Cantor space.

If J is productive then κ(J) = J_{κ} for any infinite cardinal number κ. σ−ideals J_{κ} for a certain productive σ−ideal J of subsets of 2^{ω} were intensively studied in [13].

We shall use in our further considerations the following simple lemma.

Lemma 1.6. If A, B ⊆ 2^{ω}, φ ∈ Inj(ω, κ), s ∈ 2^{ω} and t ∈ 2^{κ} then

(a) A_{φ} + t = (A + t ∘ φ)_{φ};

(b) (B + s)_{φ} = B_{φ} + s′ for some s′ ∈ 2^{κ} such that s′ ∘ φ = s;

(c) (A + B)_{φ} = A_{φ} + B_{φ}.

If A, B ⊆ 2^{κ}, φ ∈ Inj(λ, κ) then

(d) φ ∗ (A + B) = φ ∗ A + φ ∗ B.

Proof. Straightforward from the definitions. □

Let us also recall one useful definition used in [13].

The ideal J of subsets of 2^{ω} has WFP (Weak Fubini Property) if for every φ ∈ Inj(ω, ω) and every A ⊆ 2^{ω} if A_{φ} is in J then so is A.

The σ−ideals mentioned previously, i.e. σ−ideals of meagre sets and of null sets of 2^{ω}, the σ−ideal generated by closed null subsets of 2^{ω} and 𝕊_{2} obviously have WFP.

We will need the following technical lemma proved in [13].

Lemma 1.7. If J is a productive ideal of subsets of 2^{ω} having WFP then for every φ ∈ Inj(ω, κ) and every A ⊆ 2^{ω} if A_{φ} ∈ J_{κ} then A ∈ J. □

We introduce a notion which is in a sense dual to the notion of productivity. Let J be a σ−ideal of subsets of 2^{ω}. We put

p(J) = {A ⊆ 2^{κ} : (∀φ ∈ Inj(ω, ω)) φ ∗ A ∈ J }.

The following fact holds.

Fact 1.8. (a) p(J) ⊆ J.

(b) If J is proper, translation invariant, symmetric and contains singletons then so is p(J).

Proof. (a) is obvious as id_{ω} ∈ Inj(ω, ω). Let J be a proper, translation invariant and symmetric σ − ideal of subsets of 2^{ω}. Then p(J) is a σ −ideal because of the fact that for φ ∈ Inj(ω, ω) and sets A_{i} ⊆ 2^{ω} we have φ ∗ ⋃_{i<ω} A_{i} = ⋃_{i<ω} φ ∗ A_{i}. Properness is straight from (a) and containing singletons is straight from the definition and the assumption. To get translation invariance and symmetry it is enough to notice that for φ ∈ Inj(ω, ω) and A ⊆ 2^{ω}, x ∈ 2^{ω} we have φ ∗ (−A) = −φ ∗ A and φ ∗ (A + x) = φ ∗ A + x ∘ φ). □

The next theorem shows the duality mentioned above.

Theorem 1.9. Let us consider functions ω, p : P (P (2^{ω})) → P (P (2^{ω})) defined as follows

ω(A) = {A ⊆ 2^{κ} : (∃φ ∈ Inj(ω, ω)) φ ∗ A ∈ A}
p(A) = {A ⊆ 2^{κ} : (∀φ ∈ Inj(ω, ω)) φ ∗ A ∈ A}.

Then ω is a topological closure operator (in a sense of Kuratowski), p is a topological interior operator and they determine the same topology.

Proof. As far as an operation ω is concerned, we obtain straight from the definition that ω(∅) = ∅, A ⊆ ω(A) and ω(A ∪ B) = ω(A) ∪ ω(B). Furthemore, using the fact that for φ, ψ ∈ Inj(ω, ω) and A ⊆ 2^{ω} we have φ ∘ ψ ∈ Inj(ω, ω) and ψ ∗ (φ ∗ A) = (φ ∘ ψ) ∗ A we get by simple calculations ω(ω(A)) = ω(A), which implies that ω is a topological closure operation.

Similarly, we obtain from the definition that p(2^{ω}) = 2^{ω}, p(A) ⊆ A) and p(A ∩ B) = p(A)∩p(B). Also the same argument as above helps us to show that p(p(A)) = p(A). Consequently, p is a topological interior operation.

Let T_{ω} and T_{p} denote topologies on P (P (2^{ω})) determined by operations ω and p, respectively. We prove that T_{ω} = T_{p}. First, let us observe that for every family A we have p(A)^{c} = ω(A^{c}). We know that A ∈ T_{ω} ⟺ A^{c} = ω(A^{c}) and A ∈ T_{p} ⟺ A = p(A). To finish the proof it is enough to notice that A ∈ T_{p} implies ω(A^{c}) = p(A)^{c} = A^{c} and A ∈ T_{ω} implies A = (A^{c})^{c} = (ω(A^{c}))^{c} = (p(A)^{c})^{c} = p(A). □

We will need one more σ−ideal. Let us define

𝔹_{2} = {A ⊆ 2^{ω} : (∀X ∈ [ω]^{ω}) A↾X ≠ 2^{X}},

where A ↾ X = {x ↾ X : x ∈ A}. This is one of the Mycielski ideals and was intensively studied by many authors (cf. [9], [18], [20]). It is an easy observation that 𝔹_{2} = {A ⊆ 2^{ω} : (∀φ ∈ Inj(ω, ω)) φ ∗ A ≠ 2^{ω}} and, consequently, 𝔹_{2} = p(P (2^{ω}) \ {2^{ω}}).

2. Transitive cardinal coefficients of ideals

Let (G, +) be an infinite abelian group. We consider a σ −ideal J of subsets of G which is proper and contains all singletons. Moreover, we assume that J is translation invariant and symmetric.

In this section we present relations between standard and transitive cardinal characteristics of J. First of all, we have the following diagram.

Theorem 2.1.

add_{t}(J) → cof_{t}(J)
↗ ↘
add(J) → cov(J) ⟶ cof(J)
↘ ↘
add_{t}^{∗}(J) → cov_{t}(J) ↑
↘  |
non(J) ⌋

where κ → λ means κ ≤ λ. Moreover, every inequality may be strict.

Proof. Left to the reader. For possibility of strict inequalities – cf. Paragraph 5. □

There are also some extra connections between these coefficients.

Proposition 2.2. add(J) = min{add_{t}(J), add_{t}^{∗}(J)}.

Proof. Let us cosider A ⊆ J such that |A| < min{add_{t}(J), add_{t}^{∗}(J)}. Then there exists B ∈ J such that for every A ∈ A there exists g_{A} ∈ G such that A ⊆ B + g_{A}. Let T = {g_{A} : A ∈ A}. Then |T | < add_{t}^{∗}(J) so B + T ∈ J. Furthemore, ⋃ A ⊆ B + T, which ends the proof. □

8 JAN KRASZEWSKI

Proposition 2.3. cov(J) ≥ min{add_{t}(J), cov_{t}(J)}. Proof. Let us cosider A ⊆ J such that |A| < min{add_{t}(J), cov_{t}(J)}. Then there exists B ∈ J and T ∈ [G]^{|A|} such that ⋃ A ⊆ B + T. But B + T ≠ G and, consequently, ⋃ A ≠ G and we are done. □

This result gives an interesting corollary concerning an ideal of meagre sets. Corollary 2.4. If 𝔟 = 𝔠 then cov(M) = cov_{t}(M). Proof. Pawlikowski in [19] proved that add_{t}(M) = 𝔟. So the assumption cov(M) < cov_{t}(M) ≤ 𝔠 leads to a contradiction. □ Fact 2.5. If cof(J) > |G| then cof_{t}(J) = cof(J). Proof. Straightforward from the definitions. □

Our next observation is connected with cofinalities of transitive coefficients. Proposition 2.6. (a) cf(add_{t}(J)) ≥ add(J),

(b) cf(add_{t}^{∗}(J)) ≥ add(J),

(c) cf(cof_{t}(J)) ≥ add(J). Proof. To prove (a) let us consider A ⊆ J such that for every B ∈ J there exists A ∈ A which cannot be covered by any translation of the set B. Let us assume that λ = cf(add_{t}(J)) < add(J). Then there exist A_{ξ} ⊆ J for ξ < λ such that

A = ⋃_{ξ<λ} A_{ξ} and |A_{ξ}| < add_{t}(J).

Thus for every ξ < λ there exists B_{ξ} such that (∀A ∈ A_{ξ})(∃t ∈ G) A ⊆ B_{ξ} + t. But then every set A ∈ A can be covered by some translation of a set B = ⋃_{ξ<λ} B_{ξ} ∈ J, which leads to a contradiction.

Proofs of (b) and (c) are analogous. □ Finally, we show some interactions between a translatibility number and the

diagram from Theorem 2.1. Proposition 2.7. τ(J) ≤ add_{t}^{∗}(J) ≤ max{τ(J), cof_{t}(J)}. Proof. First inequality is an immediate consequence of the definitons. To prove the other one, let us assume that B is a transitive base and τ(J) = κ, i.e.

(∃A ∈ J)(∀B ∈ J)(∃T_{B} ∈ [G]^{κ})(∀g ∈ G) A + T ⊈ B + g.

We define T = ⋃_{B∈B} T_{B}. It is not difficult to show that A + T ∉ J, which ends the proof. □ Corollary 2.8. add_{t}^{∗}(J) = τ(J) or add_{t}^{∗}(J) ≤ cof_{t}(J). □

3. Transitive cardinal coefficients of 𝕊_{2}

In this section we compute transitive cardinal coefficients for the σ−ideal 𝕊_{2}.

Theorem 3.1. add_{t}^{∗}(𝕊_{2}) = non(𝕊_{2})

Proof. To prove that add_{t}^{∗}(𝕊_{2}) ≤ non(𝕊_{2}) it is enough to observe that for every set T ⊆ 2^{ω} such that T ∉ 𝕊_{2} we have |T | ≥ add_{t}^{∗}(𝕊_{2}) because {0} + T = T ∉ 𝕊_{2} and, of course, {0} ∈ 𝕊_{2}.

Suppose now that T ⊆ 2^{ω} and A ∈ 𝕊_{2}. To finish the proof we show that if T ∈ 𝕊_{2} then A + T ∈ 𝕊_{2}. Without loss of generality we can assume that A = ⋃_{i<ω}[f_{i}], where the family {f_{i} : i < ω} ⊆ P if is normal. Thus

A + T = ⋃_{t∈T} A + t = ⋃_{t∈T} ⋃_{i<ω}([f_{i}] + t) = ⋃_{i<ω} ⋃_{t∈T}[f_{i} + t↾dom(f_{i})]

Fix i < ω. Let ι : dom(f_{i}) → ω be an isomorphism. It induces an isomorphism ι̂: 2^{dom(f_{i})} → 2^{ω}. The image of the set {f_{i} + t ↾ dom(f_{i}) : t ∈ T } ⊆ 2^{dom(f_{i})} by ι̂ has cardinality strictly smaller than non(𝕊_{2}). Consequently, it can be covered by a set ⋃_{j<ω}[g_{j}], for some {g_{j} : j < ω} ⊆ P if. Hence

⋃_{t∈T}[f_{i} + t↾dom(f_{i})] ⊆ ⋃_{j<ω}[ι̂^{−1}(g_{j})] ∈ 𝕊_{2},

which ends the proof. □

Theorem 3.2. cov_{t}(𝕊_{2}) = 𝔠

Proof. It is obvious that cov_{t}(𝕊_{2}) ≤ 𝔠, so it is enough to show the other inequality. Let T ⊆ 2^{ω} and A ∈ 𝕊_{2}. We can assume as in the proof of Theorem 3.1 that

A + T = ⋃_{i<ω} ⋃_{t∈T}[f_{i} + t↾dom(f_{i})],

where f_{i} ∈ P if form a normal family. If |T | < 𝔠 then for every i < ω there exist a function g_{i} : dom(f_{i}) → 2 which is different from every function f_{i} + t ↾ dom(f_{i}), where t ∈ T. Because the family {f_{i} : i < ω} is normal then there exists a function x ∈ 2^{ω} such that ⋃_{i<ω} g_{i} ⊆ x and we have x ∉ (A + T) which ends the proof. □

For a set X ∈ [ω]^{ω} let (X)_{ω}^{ω} denotes the family of all infinite partitions of X into infinite parts. Let R be a family of partitions from (ω)_{ω}^{ω}. We say that R has a property (∗)_{1} if

(∀P ∈ (ω)_{ω}^{ω})(∃R ∈ R)(∀p ∈ P)(∃r ∈ R) r ⊆ p}

and a property (∗)_{2} if

(∀P ∈ (ω)_{ω}^{ω})(∃R ∈ R)(∃r ∈ R)(∀p ∈ P) p ⊈ r}.

We introduce new cardinal numbers connected with these properties:

\begin{matrix}λ_{1} = min\{|R| : R ⊆ (ω)_{ω}^{ω} & R has (∗)_{1}\} \\ λ_{2} = min\{|R| : R ⊆ (ω)_{ω}^{ω} & R has (∗)_{2}\}. \end{matrix}

The following theorem justifies the introduction of λ_{1} and λ_{2}.

Theorem 3.3. cof_{t}(𝕊_{2}) = λ_{1}, add_{t}(𝕊_{2}) = λ_{2}.

Proof. We prove only the first part of this theorem. The proof of the second part is analogous.

Let R ⊆ (ω)_{ω}^{ω} be a family of partitions having (∗)_{1}. Let

R^{∗} = { ⋃_{r∈R}[0_{r}] : R ∈ R} ⊆ 𝕊_{2}

where 0_{r} denotes a function constantly equal to 0 on its domain, which is the set r. Let A ∈ 𝕊_{2}. We can assume that A = ⋃_{i<ω}[f_{i}], where {f_{i} : i < ω} ⊆ P if and {dom(f_{i}) : i < ω} ∈ (ω)_{ω}^{ω}. From the definition of R we know that there exist R ∈ R such that if i is a natural number then r_{i} ⊆ dom(f_{i}) for some r_{i} ∈ R. We define a function x ∈ 2^{ω} in the following way:

x(n) = f_{i}(n) if n ∈ r_{i} for some i < ω

x(n) = 0 elsewhere It is a routine to check that

A ⊆ ⋃_{r∈R}[0_{r}] + x.

Hence the family R^{∗} is a transitive base for the ideal 𝕊_{2} and, consequently, cof_{t}(𝕊_{2}) ≤ λ_{1}.

Now let B ⊆ 𝕊_{2} be a transitive base for 𝕊_{2}. We can assume that if B ∈ B then B = ⋃_{i<ω}[f_{i}] where {f_{i} : i < ω} is a normal family from P if and {dom(f_{i}) : i < ω} ∈ (ω)_{ω}^{ω}. Let P ∈ (ω)_{ω}^{ω} and let A = ⋃_{p∈P} [0_{p}] ∈ 𝕊_{2}. Then

A ⊆ ⋃_{i<ω}[f_{i}] + f = ⋃_{i<ω}[f_{i} + f ↾dom(f_{i})]

for some ⋃_{i<ω}[f_{i}] ∈ B and f ∈ 2^{ω}. By the Lemma 1.2 we obtain that for every p ∈ P there exists a natural number i_{p} such that [0_{p}] ⊆ [f_{i_{p}} + f ↾ dom(f_{i_{p}})]. Thus dom(f_{i_{p}}) ⊆ p. Hence the family

R = {{dom(f_{i}) : i < ω} : ⋃_{i<ω}[f_{i}] ∈ B} ⊆ (ω)_{ω}^{ω}

has the property (∗)_{1}. It is easy to check that λ_{1} ≤ |R| ≤ |B| which ends the proof. □

To reach the final result we need a simple lemma:

Lemma 3.4. There exists a family P ⊆ (ω)_{ω}^{ω} of cardinality 𝔠 such that for every two partitions P_{1}, P_{2} ∈ R if p_{1} ∈ P_{1} and p_{2} ∈ P_{2} then p_{1} ∩ p_{2} is finite.

Proof. We deal with partitions of ℤ × ℤ except for partitions of ω. Let p_{i}^{α} = {(z_{1}, z_{2}) ∈ ℤ × ℤ : i ≤ z_{2} − αz_{1} < i + 1} for i ∈ ℤ and α ≥ 0. Then P^{α} = {p_{i}^{α} : i ∈ ℤ} is a partition from (ℤ × ℤ)_{ω}^{ω}. It is not difficult to check that a family P = {P^{α} : α ≥ 0} has a needed property. □

Theorem 3.5. cof_{t}(𝕊_{2}) = 𝔠, add_{t}(𝕊_{2}) = ω_{1}.

Proof. Directly from the definition and Proposition 1.2(a) we obtain cof_{t}(𝕊_{2}) ≤ cof(𝕊_{2}) ≤ 𝔠 and ω_{1} ≤ add(𝕊_{2}) add_{t}(𝕊_{2}), so it is enough to show that λ_{1} ≥ 𝔠 and λ_{2} ≤ ω_{1} .

Let P ⊆ (ω)_{ω}^{ω} be a family the existence of which we proved in the previous lemma. Let R ⊆ (ω)_{ω}^{ω} be a family with the (∗)_{1} property. For a given R ∈ R we define P_{R} = {P ∈ P : (∀p ∈ P)(∃r ∈ R)r ⊆ p}. Obviously P = ⋃_{R∈R} P_{R}. Moreover, every family P_{R} is at most countable because any element of R cannot be contained in elements of different partitions from P_{R}. Therefore

𝔠 ≤ |P| ≤ ω · |R|.

On the other hand, if we consider a family R to be any subfamily of P of size ω_{1} then R has the property (∗)_{2}. Actually, if there exists a partition P ∈ (ω)_{ω}^{ω} such that for every R ∈ R and every r ∈ R we have an element p ∈ P such that p ⊆ r then we get a contradiction as for different R_{1}, R_{2} ∈ R and r_{1} ∈ R_{1}, r_{2} ∈ R_{2} there is no p ∈ P which simultaneously contained in r_{1} and r_{2}. □

As a matter of fact, the proof of Theorem 3.5 could be shortened if we observe that the family P defined in Lemma 3.4 is a (𝔠, ω_{1})-Lusin set for a certain relation (see [14] for more disscusion).

Finally, we compute the transatibility number of 𝕊_{2}.

Theorem 3.6. τ(𝕊_{2}) = ω_{1}.

Proof. To begin with, we show that 𝕊_{2} is ω–translatable. Let A ∈ 𝕊_{2} be arbitrary. As usual, without loss of generality we can assume that A = ⋃_{i<ω}[f_{i}], where {f_{i} : i < ω} ⊆ P if and {dom(f_{i}) : i < ω} ∈ (ω)_{ω}^{ω}. For every i < ω let us fix a partition P_{i} = {p_{ij} : j < ω} ∈ (dom(f_{i}))_{ω}^{ω}. Then {p_{ij} : i, j < ω} ∈ (ω)_{ω}^{ω}. We define

B = ⋃_{i<ω} ⋃_{j<ω}[0_{p_{ij}}].

Obviously, B ∈ 𝕊_{2}. For every T = {t_{j} : j < ω} ∈ [2^{ω}]^{ω} we define g ∈ 2^{ω} as follows:

(∀i, j < ω) g ↾p_{ij} = (f_{i} + t_{j})↾p_{ij}.

It is a routine calculation to show that A + T ⊆ B + g.

To show the other inequality, let us consider first a partition P of ω into infinite parts. We can observe that there exists a set T_{P} ∈ [2^{ω}]^{ω_{1}} such that for every family {h_{i} : i < ω} ⊆ P if if {dom(h_{i}) : i < ω} = P then T ⊈ ⋃_{i<ω}[h_{i}]. Namely, it is enough to take T such that (∀p ∈ P)(∀x, y ∈ T)(x ≠ y ⇒ x↾p ≠ y ↾p).

Let us assume that 𝕊_{2} is ω_{1}–translatable and let us fix any A ∈ 𝕊_{2} of the form as in the first part of the proof. Hence there exists B ∈ 𝕊_{2} (we can assume again that B = ⋃_{j<ω}[h_{j}], where {h_{j} : j < ω} ⊆ P if and {dom(h_{j}) : j < ω} ∈ (ω)_{ω}^{ω}) such that for every T ∈ [2^{ω}]^{ω_{1}} there exists g ∈ 2^{ω} such that A + T ⊆ B + g. But then from Lemma 1.3 we obtain that (∀j < ω)(∃i < ω) dom(h_{j}) ⊆ dom(f_{i}) and, consequently, for every i < ω we have Z_{i} ∈ [ω]^{ω} such that {dom(h_{j}) : j ∈ Z_{i}} is a partition of dom(f_{i}). Thus for every i < ω we have

[f_{i}] + T ⊆ ⋃_{j∈Z_{i}}[h_{j}] + g. (∗)

Now, identyfying 2^{ω} with 2^{dom(f_{i})} as in the proof of Theorem 3.1, we obtain for every i < ω a set T_{i} = T_{{dom(h_{j}):j∈Z_{i}}}. Let T_{i} = {t_{α}^{i} : α < ω_{1}}. Then the set

T = { ⋃_{i<ω} t_{α}^{i} : α < ω_{1}}

contradicts the condition (∗), which ends the proof. □

4. Cofinality versus transitive covering

In this section we show that transitive covering of an ideal may be totally different from its cofinality.

Theorem 4.1. Let λ be a cardinal number of an uncountable cofinality and let ⟨G_{α} : α < λ⟩ be an increasing sequence of subgroups of a group G such that G = ⋃_{α<λ} G_{α}. If J is a σ−ideal of subsets of G generated by a family {G_{α} : α < λ} then cof(J) = cf(λ) and

cov_{t}(J) = inf{|^{G}/_{G_{α}}| : α < λ}.

Proof. Straight from the fact, that the sequence ⟨G_{α} : α < λ⟩ is increasing we can deduce that

J = {A ⊆ G : (∃ξ < λ) A ⊆ G_{ξ}}.

It is a simple observation that J is a translation invariant, symmetric σ −ideal containing sigletons. It is also proper as the cofinality of λ is uncountable.

Let us fix a given sequence of cardinal numbers ⟨ξ_{α} : α < cf(λ)⟩, cofinal in λ. Then the family {G_{ξ_{α}} : α < cf(λ)} is a base for J. Moreover, no family of elements of J of cardinality strictly less than cf(λ) can be a base of J as all elements of such a family are contained in G_{ξ} for some ξ < λ. Hence cof(J) = cf(λ).

Let us observe that a sequence of cardinal numbers ⟨|^{G}/_{G_{α}}| : α < λ⟩ is decreasing so there exists ζ < λ such that |^{G}/_{G_{α}}| = |^{G}/_{G_{ζ}}| for α ≥ ζ. Let us consider now a set T ⊆ G such that |T | = cov_{t}(J) and there exists A ∈ J such that A + T = G. Without loss of generality we may assume that A = G_{ξ} for some ζ ≤ ξ < λ. Then we may get T′ ⊆ T such that (∀t ∈ T′) T′ ∩ (G_{ξ} + t) = {t} and G_{ξ} + T′ = G. Thus cov_{t}(J) = |T′| = |^{G}/_{G_{ξ}}| and, consequently,

cov_{t}(J) = |^{G}/_{G_{ξ}}| = |^{G}/_{G_{ζ}}| = inf{|^{G}/_{G_{α}}| : α < λ},

which ends the proof. □

As an application of Theorem 4.1 we construct a σ−ideal, the transitive covering of which is in general radically bigger than its cofinality. First, we introduce some necessary notation.

A set H ⊆ ℝ is called a Hamel basis if it is a basis of (ℝ, +) treated as a linear space over a field ℚ of rational numbers.

From now on let us fix a Hamel basis H and its enumeration H = {h_{α} : α < 𝔠}. Then every real number x has the unique representation in this basis, i.e.

(∀x ∈ ℝ)(∃! r_{x} ∈ ℚ^{𝔠})(| supp(r_{x})| < ω & x = ∑_{α<𝔠} r_{x}(α)h_{α}),

where supp(r_{x}) = {α : r_{x}(α) ≠ 0}. In order to simplify the notation we replace supp(r_{x}) by supp(x).

Definition. Let {P_{ξ} : ξ < ω_{1}} be a fixed partition of 𝔠 into parts of cardinality 𝔠. Let A be any set. We say that a function f ∈ ℝ^{A} is Hamel-bounded if

(∃ξ < ω_{1})(∀a ∈ A)(supp(f(a)) ⊆ ⋃_{β<ξ} P_{β}).

Then we put HB(A) = {f ∈ ℝ^{A} : f is Hamel − bounded}. One can check that HB(A) is a subgroup of ℝ^{A} with a standard addition of functions. For any function f ∈ HB(A) its Hamel-bound hb (f) is defined as follows:

hb (f) = min{ξ < ω_{1} : (∀a ∈ A)(supp(f(a)) ⊆ ⋃_{β<ξ} P_{β})}.

Let κ be an infinite cardinal number. Let B_{ξ} = {f ∈ HB(κ) : hb (f) ≤ ξ}. Of course, ⟨B_{ξ} : ξ < ω_{1}⟩ is an increasing sequence of subgroups of a group HB(κ) and HB(κ) = ⋃_{ξ<ω_{1}} B_{ξ}. we define T (κ) as a σ −ideal generated by the family {B_{ξ} : ξ < ω_{1}}.

Lemma 4.2. |^{HB(κ)}/_{B_{ξ}} | = 2^{κ} for every ξ < ω_{1}.

Proof. Let us fix B_{ξ} for some ξ < ω_{1}. We consider a set T ⊆ HB(κ) such that (∀t ∈ T) T ∩ (B_{ξ} + t) = {t} and B_{ξ} + T = HB(κ).

Let us fix P ⊆ κ and a real number x such that x ∈ H \ {h_{α} : α ∈ ⋃_{β<ξ} P_{β}}. We define a function f_{P} ∈ HB(κ) as follows:

f_{P} (α) = χ_{P} (α) · x

where χ_{P} denotes the characteristic function of a set P. Then there exists t_{P} ∈ T and g ∈ b_{ξ} such that f_{P} = g + t_{P}. In particular, for each α ∈ P we have

x = f_{P} (α) = g(α) + t_{P} (α).

But we know from the assumption that supp(x) ⊈ ⋃_{β<ξ} P_{β}, so we have supp(x) ⊆ supp(t_{P} (α)) for each α ∈ P. On the other hand, if α ∉ P then f_{P} (α) = 0 and, consequently, supp(t_{P} (α)) = supp(g(α)) ⊆ ⋃_{β<ξ} P_{β} for such α’s.

Let P_{1} and P_{2} be two different subsets of κ and α ∈ P_{1} △ P_{2}. Suppose that t_{P_{1}} = t_{P_{2}} = t. Then

supp(x) ⊆ supp(t(α)) ⊆ ⋃_{β<ξ} P_{β},

which is a contradition. Hence t_{P_{1}} ≠ t_{P_{2}} and, consequently,

|^{HB(κ)}/_{B_{ξ}} | = |T | ≥ |P(κ)| = 2^{κ},

which ends the proof, as |HB(κ)| = 2^{κ}. □

Corollary 4.3. For every infinite cardinal number κ we have cof(T (κ)) = ω_{1} and cov_{t}(T (κ)) = 2^{κ}.

Proof. It is enough to apply Theorem 4.1 for λ = ω_{1}, G = HB(κ), G_{ξ} = B_{ξ} and J = T (κ). □

5. Possible diagrams

After reading Paragraph 2 it occurs a natural question whether there are any other relations between coefficients in the diagram form Theorem 2.1. One of possible ways of solving this problem (negatively) is to put ω_{1} and ω_{2} = 𝔠 in nodes of the diagram and try to find a σ−ideal J and a model for every legal configuration.

There are 23 legal configurations (taking under consideration Theorem 2.1 and Propositions 2.2 and 2.3). In this section we shall present models for 12 of them. From now on, ∘ will stand for ω_{1} and • will stand for ω_{2} = 𝔠.

Theorem 5.1. If G = 2^{ω} and J = 𝕊_{2} then there are the following possibilities for the diagram from Theorem 2.1.

(a) (b) ∘ → •

↗ ↘ ∘ → • ⟶ •

↘ ↘
• → • ↑
↘  |
• ⌋

∘ → •
↗ ↘
∘ → • ⟶ •
↘ ↘
∘ → • ↑
↘  |
∘ ⌋

(c) (d) ∘ → •

↗ ↘ ∘ → ∘ ⟶ •

↘ ↘
• → • ↑
↘  |
• ⌋

∘ → •
↗ ↘
∘ → ∘ ⟶ •
↘ ↘
∘ → • ↑
↘  |
∘ ⌋

Proof. Fact 1.4 and Theorems 3.1, 3.2 and 3.5 give us a complete description of a diagram from Theorem 2.1 for J = 𝕊_{2}.

To obtain the diagram (a) it is enough to assume Martin’s Axiom and 𝔠 = ω_{2}, as it well-known that we have then ℵ_{0}-𝔰 = 𝔯 = 𝔠. For the diagram (b) we need ℵ_{0}-𝔰 = ω_{1} and 𝔯 = ω_{2} = 𝔠. This sitiuation takes place in a model obtained by adding ω_{2} random reals to a model of CH, as ℵ_{0}-𝔰 ≤ max{𝔟, 𝔰} (cf. [12]) and cov(N) ≤ 𝔯 (cf. [23] or [8]). To obtain the diagram (c) we need 𝔯 = ω_{1} and ℵ_{0}-𝔰 = ω_{2} = 𝔠. To get such a model we use a countable support iteration of length ω_{2} (over CH) of the Blass-Shelah forcing introduced in [5] (see also [2]). This iteration preserves p-points, so a p-point from the ground model causes 𝔯 = ω_{1}. On the other hand on every step we add a subset of ω which is either almost contained or almost disjoint with every subset of ω from the ground model. A standard argument shows that 𝔰 = ω_{2} where 𝔰 is a classical splitting number (cf. [23]). But 𝔰 ≤ ℵ_{0}-𝔰 which ends the proof. Finally, the diagram (d) holds in the iterated Sacks model, because we have then ℵ_{0}-𝔰 = 𝔯 = ω_{1} < 𝔠 = ω_{2}. □

Theorem 5.2. If G = ℝ and J = M then there are the following possibilities for the diagram from Theorem 2.1.

(a) (b) • → •

↗ ↘ • → • ⟶ •

↘ ↘
• → • ↑
↘  |
• ⌋

• → •
↗ ↘
∘ → ∘ ⟶ •
↘ ↘
∘ → ∘ ↑
↘  |
• ⌋

(c) (d) ∘ → •

↗ ↘ ∘ → ∘ ⟶ •

↘ ↘
∘ → ∘ ↑
↘  |
∘ ⌋

∘ → ∘
↗ ↘
∘ → ∘ ⟶ •
↘ ↘
∘ → ∘ ↑
↘  |
• ⌋

Proof. We will use well-known Cichoń’s diagram (cf. [2]), Theorem 0.1 and Pawlikowski’s results: add_{t}(M) = 𝔟 and cof_{t}(M) = 𝔡 (cf. [19]).

The diagram (a) holds assuming Martin’s Axiom and 𝔠 = ω_{2}. For the diagram (b) we need non(N) = ω_{1} and 𝔟 = ω_{2} = 𝔠 which is true after adding ω_{1} random reals to the model of MA + 𝔠 = ω_{2}. To obtain the diagram (c) we need non(M) = non(N)ω_{1} and 𝔡 = ω_{2} = 𝔠 so it is enough to iterate ω_{2} times Miller forcing, with countable support (over CH). Finally, the diagram (d) requires non(N) = 𝔡 = ω_{1} and non(M) = ω_{2} = 𝔠 and adding ω_{2} random reals to the model of CH will do. □

Theorem 5.3. If G = ℝ and J = N then there are the following possibilities for the diagram from Theorem 2.1.

(a) (b) ∘ → •

↗ ↘ ∘ → • ⟶ •

↘ ↘
∘ → • ↑
↘  |
• ⌋

∘ → •
↗ ↘
∘ → ∘ ⟶ •
↘ ↘
∘ → ∘ ↑
↘  |
• ⌋

(c) ∘ → ∘

↗ ↘

∘ → ∘ ⟶ ∘

↘ ↘
∘ → ∘ ↑
↘  |
∘ ⌋

Proof. Again, we will use Cichoń’s diagram, Theorem 0.1 and Pawlikowski’s results: add_{t}(N) = add(N), cof_{t}(N) = cof(N) and add(N) = min{𝔟, add_{t}^{∗}(N)}.

The diagram (a) requires add(N) = ω_{1} and 𝔟 = non(N) = cov(N) = ω_{2} = 𝔠. A suitable model can be found in [2] or in [11]. For the diagram (b) we need non(M) = ω_{1} and non(N) = ω_{2} = 𝔠 so adding ω_{2} Cohen reals to a model of CH will do. Finally, the diagram (c) holds in iterated Sacks model, as we have cof(N) = ω_{1} < 𝔠 = ω_{2}. □

Theorem 5.4. If G = HB(ω) and J = T (ω) then there is the following possibility for the diagram from Theorem 2.1.

∘ → ∘
↗ ↘
∘ → ∘ ⟶ ∘
↘ ↘
∘ → • ↑
↘  |
∘ ⌋

Proof. It is enough to take any model of 𝔠 = ω_{2} and apply Corollary 4.3 □

Remark. Of course, some of diagrams mentioned above can be also obtained for other ideals, e.g. diagrams 5.1(b), 5.2(a) or 5.2(c) can be obtained for σ−ideal of null sets as well.

6. Transitive operations on ideals

In this paragraph we prove some results connected with operations s and g. As we have assumed at the beginning of this paper, J is a σ −ideal of subsets of G which is proper, translation invariant, symmetric and contains all singletons.

Fact 6.1. non(s(J)) = cov_{t}(J), non(g(J)) = add_{t}^{∗}(J).

Proof. Straightforward from definitions. □

Proposition 6.2. cov_{t}(s(J)) ≥ non(J).

Proof. Let us consider T ⊆ G such that A + T = G for some A ∈ s(J). But we know that for every B ∈ J we have A + B ≠ G. Hence T ∉ J. □

Proposition 6.3. add_{t}^{∗}(g(J)) = non(g(J)).

Proof. It occurs from Theorem 2.1 and Fact 7.1 that it is enough to show that add_{t}^{∗}(g(J)) ≥ add_{t}^{∗}(J). Let us consider T ⊆ G such that there exists A ∈ g(J) such that A + T ∉ g(J). This means that for some B ∈ J we have (A + T) + B = (A + B) + T ∉ J. But A + B ∈ J which ends the proof. □

Corollary 6.4. cov_{t}(g(J)) ≥ max{non(J), cov_{t}(J)}.

Proof. Straightforward from definitions, Proposition 1.1(c) and Proposition 6.2. □

In the next part of the paragraph we show that σ−ideals 𝕊_{2} and 𝔹_{2} are closely related to each other.

Theorem 6.5. s(𝕊_{2}) = 𝔹_{2}.

Proof. Let us consider any A ⊆ 2^{ω}. It is a standard calculation which shows that if for some X ∈ [ω]^{ω} we have A ↾ X = 2^{X} then A + [0_{X}] = 2^{ω}. Hence if A ∉ 𝔹_{2} then A ∉ s(𝕊_{2}).

On the other hand, let us consider any C ⊆ 2^{ω} such that B + C = 2^{ω} for some B ∈ 𝕊_{2}. As in proofs in Paragraph 3, without loss of generality we can assume that B = ⋃_{i<ω}[f_{i}], where {f_{i} : i < ω} ⊆ P if and {dom(f_{i}) : i < ω} ∈ (ω)_{ω}^{ω}. It occurs that there exists i < ω such that C ↾ dom(f_{i}) = 2^{dom(f_{i})}. Actually, if we suppose that for all i < ω there exists g_{i} ∈ 2^{dom(f_{i})} \ C ↾ dom(f_{i}) then we have ⋃_{i<ω}(f_{i} + g_{i}) ∈ 2^{ω} \ B + C. Thus if C ∉ s(𝕊_{2}) then C ∉ 𝔹_{2} which completes the proof. □

In [9] the authors showed that the covering number of 𝔹_{2} is a weird object and it is difficult to find reasonable estimations for it. In particular, it is relatively consistent that Martin’s Axiom holds, 𝔠 = ω_{2} and cov(𝔹_{2}) = ω_{1}. The following corollary shows that the situation for the transitive covering number of 𝔹_{2} is different.

Corollary 6.6. If Martin’s Axiom holds then cov_{t}(𝔹_{2}) = 𝔠.

Proof. From Theorem 6.5 and Proposition 6.2 we obtain that cov_{t}(𝔹_{2}) = non(𝕊_{2}). It was proved in [8] that non(𝕊_{2}) = ℵ_{0}-𝔰 and it is well-known that under Martin’s Axiom we have ℵ_{0}-𝔰 = 𝔠. □

It is a natural question to ask what we know about g(𝕊_{2}). The next theorem partially answers it.

Theorem 6.7. g(𝕊_{2}) = p(𝕊_{2}).

Proof. Let us assume that A ∈ g(𝕊_{2}) that is (∀B ∈ 𝕊_{2})A+B ∈ 𝕊_{2}. It is not difficult to observe that this condition is equivalent to (∀T ∈ [ω]^{ω})[0_{T}] + A ∈ 𝕊_{2}. But we can prove that if φ ∈ Inj(ω, ω) then [0_{dom(φ)}]+A = (φ∗A)_{φ}. Hence, reformulating our condition we obtain (∀φ ∈ Inj(ω, ω)) (φ ∗ A)_{φ} ∈ 𝕊_{2}. Thus, as 𝕊_{2} is productive and has WFP, we show that this fact is equivalent to (∀φ ∈ Inj(ω, ω)) φ ∗ A ∈ 𝕊_{2} and, consequently, to A ∈ p(𝕊_{2}). □

Finally, we will show that all operations that appeared in this paragraph are versions of one operation, defined in [22].

Let A, B be translation invariant families of subsets of a group G. We put

G_{t}(A, B) = {A ⊆ G : (∀B ∈ B) A + B ∈ A}.

The following proposition takes place.

Proposition 6.8. Let J be a translation invariant, symmetric σ−ideal of subsets of a group G. Then

(a) s(J) = G_{t}(P(G) \ {G}, J),

(b) g(J) = G_{t}(J , J). If G = 2^{ω} and J is productive and has WFP then

(c) p(J) = G_{t}(J , 𝕊_{2}).

Proof. (a) and (b) are reformulations of definitions and was observed in [22]. To prove (c) it is enough to repeat carefully the proof of Theorem 6.7 □

7. Transitive cardinal coefficients of ideals on 2^{κ}

From now on we assume that J is a proper and productive σ−ideal of subsets of 2^{ω} containing all singletons (i.e. ⋃ J = 2^{ω}) and that κ ≥ ω_{1}. We investigate relations between transitive cardinal coefficients of J and those of J_{κ}. Some of them are parallel to relations between standard cardinal coefficients of J and J_{κ} proved in [13].

Theorem 7.1. add_{t}(J_{κ}) = ω_{1}.

Proof. The proof is actually the same as the one of Theorem 2.1 from [13] which states that add(J) = ω_{1}. □

Theorem 7.2. cof_{t}(J_{κ}) ≤ max{cof([κ]^{≤ω}), cof_{t}(J)}. Moreover, if J has WFP then cof_{t}(J_{κ}) ≥ cof_{t}(J).

Proof. Similarly as above, constructions are analogous to those from Theorems 2.2 and 2.5 from [13] which state that cof(J_{κ}) ≤ max{cof([κ]^{≤ω}), cof(J)} and if J has WFP then cof(J_{κ}) ≥ cof(J). The only addition is making use of Lemma 1.6(a) and (b). □

Theorem 7.3. If ω_{1} ≤ λ ≤ κ then add_{t}^{∗}(J_{λ}) ≤ add_{t}^{∗}(J_{κ}).

Proof. Let us fix T ⊆ 2^{κ} such that A + T ∉ J_{κ} for some A ∈ J_{κ}. We take P hi ∈ Inj(λ, κ) such that rng(φ) ⊆ rng(Φ), where φ is a witness for A. Let us consider Φ∗A, Φ∗T ⊆ 2^{λ}. Obviously, |φ∗T | ≤ |T |. Furthemore ψ = Φ^{−1} ∘φ ∈ Inj(ω, λ) is a witness for Φ∗A ∈ J_{λ}. From Lemma 1.6(d) we know that Φ∗(A+T) = Φ∗A+Φ∗T.

Now, if χ ∈ Inj(ω, λ) was a witness for Φ∗A+Φ∗T ∈ J_{λ} then Φ∘χ ∈ Inj(ω, κ) would be a witness for A + T ∈ J_{κ} which contradicts the assumption. Hence Φ ∗ A + Φ ∗ T ∉ J_{λ} which ends the proof. □

Theorem 7.4. If J has WFP then add_{t}^{∗}(J_{κ}) = add_{t}^{∗}(J).

Proof. It is enough to prove that add_{t}^{∗}(J_{κ}) ≤ add_{t}^{∗}(J) and apply Theorem 7.3.

Let us fix T ⊆ 2^{ω} such that A + T ∉ J for some A ∈ J. We define T′ = {t ∈ 2^{κ} : t ↾ ω ∈ T & t ↾ (κ \ ω) ≡ 0}. Obviously, |T′| = |T |. From Lemma 1.6(a) we have

A_{id_{ω}} + T′ = ⋃_{t′∈T′}(A + t′ ↾ω)_{id_{ω}} = ⋃_{t∈T}(A + t)_{id_{ω}} = ( ⋃_{t∈T} A + t)_{id_{ω}} = (A + T)_{id_{ω}}

and A_{id_{ω}} ∈ J_{κ} as J is productive. But id_{ω} ∈ Inj(ω, κ) and J has WFP and from Lemma 1.7 we get (A + T)_{id_{ω}} ∉ J_{κ}, which ends the proof. □

Theorem 7.5. cov_{t}(J_{κ}) = cov_{t}(J).

Proof. Let us fix T ⊆ 2^{ω} such that A + T = 2^{ω} for some A ∈ J. We define T′ in the same way as in the proof of Theorem 7.4. Similarly, we get A_{id_{ω}} + T′ = (A + T)_{id_{ω}} = (2^{ω})_{id_{ω}} = 2^{κ} and, consequently, cov_{t}(J_{κ}) ≤ cov_{t}(J).

To show the other inequality, let us fix S ⊆ 2^{κ} such that A + S = 2^{κ} for some A ∈ J_{κ}. Let φ ∈ Inj(ω, κ) be a witness of A. We consider φ ∗ S ⊆ 2^{ω}. Then |φ ∗ S| ≤ |S| and from Lemma 1.6(d)

φ ∗ A + φ ∗ S = φ ∗ (A + S) = φ ∗ 2^{κ} = 2^{ω}.

But φ ∗ A ∈ J, which ends the proof. □

Theorem 7.6. τ(J) ≤ τ(J_{κ}).

Proof. Let us assume that J is ξ-transitive i.e. for every A ∈ J there exists B_{A} ∈ J such that for every S ∈ [2^{ω}]^{ξ} there exists t_{S} ∈ 2^{ω} such that A + S ⊆ B_{A} + t_{S}. We show that J_{κ} is also ξ-transitive.

Let us consider any A ∈ J_{κ} and let φ ∈ Inj(ω, κ) be its witness. Then φ∗A ∈ J and we fix B_{φ∗A} ∈ J. If S ∈ [2^{κ}]^{ξ} then without loss of generality we can assume that φ ∗ S ∈ [2^{ω}]^{ξ} and thus there exists t_{φ∗S} ∈ 2^{ω} such that φ ∗ A + φ ∗ S ⊆ B_{φ∗A} + t_{φ∗S}. Using Lemma 1.6(b) and (d) we obtain

A + T ⊆ (φ ∗ (A + T))_{φ} ⊆ (B_{φ∗A} + t_{φ∗S})_{φ} = (B_{φ∗A})_{φ} + t

for some t ∈ 2^{κ}. Hence J_{κ} is ξ-transitive as we can take (B_{φ∗A})_{φ} for B_{A} and t for t_{S} from the definition of ξ-transitivity of J_{κ}. □

Theorem 7.7. If J has WFP then τ(J_{κ}) = τ(J).

Proof. It is enough to prove that τ(J_{κ}) ≤ τ(J) and apply Theorem 7.6. As in the proof of Theorem 7.6, we assume that J_{κ} is ξ-transitive and show that J is also ξ-transitive.

Let us consider any A ∈ J. Then A′ = A_{id_{ω}} ∈ J_{κ} and we fix B_{A′} ∈ J_{κ}. If T ∈ [2^{ω}]^{ξ} then we define T′ ⊆ 2^{κ} like in the proof of Theorem 7.4. Obviously, T′ ∈ [2^{κ}]^{ξ} and there exists appropriate t_{T′} ∈ 2^{κ} such that A′ + T′ ⊆ B_{A′} + t_{T′}. But A′ + T′ = (A + T)_{id_{ω}}. From Lemma 1.6(b) we obtain

(A + T + t_{T′} ↾ω)_{id_{ω}} = (A + T)_{id_{ω}} + t_{T′} ⊆ B.

Let us define

C = ⋃_{T ∈[2^{ω}]^{ξ}}(A + T + t_{T′} ↾ω).

Then C ⊆ 2^{ω} and

C_{id_{ω}} = ⋃_{T ∈[2^{ω}]^{ξ}}(A + T + t_{T′} ↾ω)_{id_{ω}} ⊆ B ∈ J_{κ}.

Thus C_{id_{ω}} ∈ J_{κ} and from Lemma 1.7 we know that C ∈ J. To finish the proof it is enough to show that from every S ∈ [2^{ω}]^{ξ} there exists t_{S} ∈ 2^{ω} such that A + S ⊆ C + t_{S}.

Let us consider any S ∈ [2^{ω}]^{ξ} and put t_{S} = t_{T′} ↾ω. Then

A + S = A + S + t_{T′} ↾ω + t_{T′} ↾ω ⊆ C + t_{T′} ↾ω = C + t_{S}

and we are done. □

Acknowledgements. The author would like to thank Professor Cichoń for many fruitful disscusions.

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Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-156 Wrocław, Poland

E-mail address: kraszew@math.uni.wroc.pl