TRANSITIVE PROPERTIES OF IDEALS ON GENERALIZED CANTOR SPACES.

JAN KRASZEWSKI

Abstract. In this paper we compute transitive cardinal coefficients of ideals on generalized Cantor spaces. In particular, we observe that there exists a null set A ⊆ 2^{ω_{1}} such that for every null set B ⊆ 2^{ω_{1}} we can find x ∈ 2^{ω_{1}} such that the set A ∪ (A + x) cannot be covered by any translation of the set B.

1. Introduction, definitions and basic properties

In 2001 Kraszewski in [5] defined a class of productive σ-ideals of subsets of the Cantor space 2^{ω} and observed that both σ-ideals of meagre sets and of null sets are in this class. Next, from every productive σ-ideal J one can produce a σ-ideal J_{κ} of subsets of the generalized Cantor space 2^{κ}. In particular, starting from meagre sets and null sets in 2^{ω} we obtain meagre sets and null sets in 2^{κ}, respectively. This description gives us a powerful tool for investigating combinatorial properties of ideals on 2^{κ}, which was done in [5]. In this paper we continue our research, focusing on transitive cardinal coefficients of ideals of subsets of 2^{κ}.

We use standard set-theoretical notation and terminology from [2]. Let (G, +) be an infinite abelian group. We consider a σ-ideal J of subsets of G which is proper, contains all singletons and is invariant (under group operations).

For an ideal J we consider the following transitive 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 ∧ (∀A ∈ J)(∃B ∈ B)(∃g ∈ G) A ⊆ B + g}.

First two ones are both called a transitive additivity. The remaining two ones are called a transitive covering number and a transitive cofinality, respectively.

We say that an ideal J is κ-translatable if

(∀A ∈ J)(∃B_{A} ∈ J)(∀S ∈ [G]^{κ})(∃t_{S} ∈ G) A + S ⊆ B_{A} + t_{S}.

We define a translatability number of J as follows

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

2000 Mathematics Subject Classification. 03E05, 03E17.

Key words and phrases. generalized Cantor spaces, transitive cardinal coefficients.

For more information about relations between classical and transitive cardinal coefficients of ideals – see [2]. For more about translatability – see [1], [3] and [6].

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 space is endowed with the standard product topology. Moreover, we consider the standard product measure and the standard product group structure on 2^{κ}.

We introduce some extra notation in order to simplify further considerations. Let κ be an infinite cardinal number. We put Inj(ω, κ) = {φ ∈ κ^{ω} : φ is an injection}. For A ⊆ 2^{κ}, B ⊆ 2^{ω} and φ ∈ Inj(ω, κ) we put φ ∗ A = {x ∘ φ : x ∈ A} and B_{φ} = {x ∈ 2^{κ} : x ∘ φ ∈ B}.

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

Let J be a σ-ideal of subsets of 2^{ω}. We say that J is productive if

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

It is easy to show that J is productive if and only if for every A ⊆ 2^{ω} and φ ∈ Inj(ω, ω) if A ∈ J then 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, the ideal 𝕊_{2} investigated in [4] is the least non-trivial productive σ-ideal of subsets of the Cantor space.

For any productive σ-ideal J we define

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

A standard consideration shows that J_{κ} is a σ-ideal of subsets of 2^{κ}. If J is invariant then so is J_{κ}. If A ∈ J_{κ} then any φ ∈ Inj(ω, κ) such that φ ∗ A ∈ J is called a witness for A.

Let us also recall one useful definition used in [5]. We say that an 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 of subsets of 2^{ω} mentioned above obviously have WFP. We will need the following technical lemma proved in [5].

Lemma 1.1. 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. □

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

From now on we assume that J is a proper, invariant and productive σ-ideal of subsets of 2^{ω} containing all singletons and that κ ≥ ω_{1}. We investigate relations between transitive cardinal coefficients of J and those of J_{κ}. Some of them are similar to relations between standard cardinal coefficients of J and J_{κ} proved in [5]. We omit the proofs, as they are also analogous.

Theorem 2.1. add_{t}(J_{κ}) = ω_{1}. □

Theorem 2.2. cof_{t}(J_{κ}) ≤ max{cof([κ]^{≤ω}), cof_{t}(J)}. Moreover, if J has WFP then cof_{t}(J_{κ}) ≥ cof_{t}(J). □

However, other transitive cardinal coefficients behave in a radically different way.

Theorem 2.3. If J has WFP then add_{t}^{∗}(J_{κ}) = add_{t}^{∗}(J).

Proof. Let T ⊆ 2^{κ} be such that A + T ∉ J_{κ} for some A ∈ J_{κ} and let φ be a witness for A. Then φ ∗ A ∈ J and φ ∗ A + φ ∗ T = φ ∗ (A + T) ∉ J. Hence add_{t}^{∗}(J_{κ}) ≥ add_{t}^{∗}(J).

To show the other inequality, let us fix T ⊆ 2^{ω} such that A + T ∉ J for some A ∈ J. We have A_{id_{ω}} ∈ J_{κ} (because id_{ω} ∈ Inj(ω, κ) and J is productive). We define T′ = {t ∈ 2^{κ} : t ↾ ω ∈ T ∧ t ↾ (κ \ ω) ≡ 0}. Then A_{id_{ω}} + T′ = (A + T)_{id_{ω}} and from Lemma 1.1 we get (A + T)_{id_{ω}} ∉ J_{κ}, which ends the proof. □

Theorem 2.4. cov_{t}(J_{κ}) = cov_{t}(J).

Proof. Similar to the proof of Theorem 2.3. □

Theorem 2.5. If J has WFP then τ(J_{κ}) = τ(J).

Proof. Suppose that J is ξ-translatable. We consider any A ∈ J_{κ} and φ ∈ Inj(ω, κ) being 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}. Then

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

for some t ∈ 2^{κ}. Hence J_{κ} is ξ-translatable.

On the other hand, let us assume that J_{κ} is ξ-translatable and 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}]^{ξ} like in the proof of Theorem 2.3. There exists an appropriate t_{T′} ∈ 2^{κ} such that A′ + T′ ⊆ B_{A′} + t_{T′}. But A′ + T′ = (A + T)_{id_{ω}} and

(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 1.1 we know that C ∈ J.

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

A + S = A + S + t_{S} + t_{S} ⊆ C + t_{S}

and we are done. □ As an immediate corollary we obtain the following interesting result.

Corollary 2.6. There exists a null set A ⊆ 2^{ω_{1}} such that for every null set B ⊆ 2^{ω_{1}} we can find x ∈ 2^{ω_{1}} such that the set A ∪ (A + x) cannot be covered by any translation of the set B.

Proof. From [1] we know that τ(N) = 2, where N states for the ideal of null subsets of 2^{ω}. In [5] is shown that N_{ω_{1}} is exactly the ideal of null subsets of 2^{ω_{1}}. But from Theorem 2.5 we know that τ(N_{ω_{1}}) = 2 and this is what we have been supposed to show. □

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

kraszew@math.uni.wroc.pl

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