Transitive properties of the ideal 𝕊_{2}.

Jan Kraszewski Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-156 Wrocław, Poland (e-mail: kraszew@math.uni.wroc.pl)

Abstract

In this paper we compute transitive cardinal coefficients of the σ–ideal 𝕊_{2}, the least nontrivial productive σ–ideal of subsets of the Cantor space 2^{ω}. We also apply transitive operations to 𝕊_{2}. In particular, we show that σ–ideal of strongly 𝕊_{2} sets is equal to 𝔹_{2}, one of Mycielski ideals.

0. Introduction. In this paper we investigate transitive properties of the σ–ideal 𝕊_{2}. This ideal appeared for the first time in [10], but only incidentally. It was thoroughly investigated by Cichoń and Kraszewski in [5]. It turned out that cardinal characteristics of 𝕊_{2} are strongly connected with some intensively studied combinatorial properties of subsets of natural numbers (the splitting and reaping numbers). Namely,

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

Moreover, 𝕊_{2} is the least nontrivial productive σ–ideal of subsets of the Cantor space 2^{ω}. The notion of productivity is a powerful tool for investigating properties of ideals on generalized Cantor spaces 2^{κ}. For more details see [9].

In the first part of this paper we completely describe all well-known transitive cardinal characteristics of 𝕊_{2}. In the second part we apply transitive operations to 𝕊_{2}. In particular, we show that the σ–ideal of strongly 𝕊_{2} sets is exactly 𝔹_{2}, one of Mycielski ideals.

1. Definitions and basic properties. We use standard set-theoretical notation and terminology from [2]. Recall that the cardinality of the set of all real numbers is denoted by 𝔠. The cardinality of a set X is denoted by |X|. The power set of a set X is denoted by P(X). If κ is a cardinal number then [X]^{κ} denotes the family of all subsets of the set X of cardinality κ. 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.

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

2000 Mathematics Subject Classification: 03E17, 03E02.

Key words and phrases: Cantor space, ideals, transitive cardinal coefficients, partitions.

Let J be an ideal of subsets of an abelian group G. We say that J is translation invariant if A + g = {x + g : x ∈ A} ∈ J for each A ∈ J and g ∈ G and that J is symmetric if −A = {−x : x ∈ A} ∈ J for each A ∈ J.

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 },

where a family B ⊆ J is called a transitive base if for each A ∈ J there exists B ∈ B and g ∈ G such that A ⊆ B + g. The first two of these are both called transitive additivity and the last two are called transitive covering number and transitive cofinality, respectively. Let us notice that all definitions of cardinal coefficients mentioned above are valid also for an arbitrary family A ⊆ P(G).

Let us also recall that by the uniformity of J we mean the following cardinal number

non(J) = min{|A| : A ⊆ G ∧ A ∉ J }.

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 translatability number of J as follows

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

In this paper we deal with the Cantor space 2^{ω} interpreted as the set of all functions from ω into the set {0, 1}. This space is endowed with the standard product topology. Moreover, we consider the standard product group structure on 2^{ω}.

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 the family {[f] : f ∈ P if}. We recall some properties of 𝕊_{2}, which were proved in [5].

Fact 1.1 (a) 𝕊_{2} is a proper σ-ideal, containing singletons, with a base consisting of Borel sets. Every A ∈ 𝕊_{2} is both meagre and null.

(b) 𝕊_{2} is translation invariant.

(c) There exists 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 [5] the following useful lemma was proved.

Lemma 1.2 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 [13], 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 [8]. It was proved in [5] that non(𝕊_{2}) = ℵ_{0}-𝔰.

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. [7], [15], [17]).

2. Transitive cardinal coefficients of 𝕊_{2}. Let J be an ideal of subsets of an abelian group G. The first cardinal coefficient on the stage was the transitive covering number of J that appeared implicitly in 1938 in the famous Rothberger theorem, which was originally formulated for classical ideals of meagre and null subsets of the real line (cf. [18]). In his general version it says that if J and I are translation invariant ideals of subsets of 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). It is worth observing that the transitive covering number may be different from the covering number and 𝕊_{2} is an example.

Theorem 2.1 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}. Without loss of generality we can assume that A = ⋃_{i<ω}[f_{i}], where the family {f_{i} : i < ω} ⊆ P if is normal. 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. □

In 1985 Pawlikowski in [16] introduced the transitive cofinality and gave the complete description of transitive cofinalities of ideals of meagre and null subsets of the real line. He also mentioned a dual coefficient to the transitive cofinality. Following the way of describing cardinal characteristics of the continuum presented by Blass in [3] 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 yet another coefficient, introduced in [16]. In order not to make a mess we will call the latter coefficient the starred transitive additivity and denote by add_{t}^{∗}(J).

Now we calculate these coefficients for 𝕊_{2}. To begin with, we observe the following general property concerning starred transitive additivity.

Proposition 2.2 Let J be a proper and translation invariant σ– ideal of subsets of a group G containing all singletons. Then add_{t}^{∗}(J) ≤ non(J).

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

Theorem 2.3 add_{t}^{∗}(𝕊_{2}) = ℵ_{0}-𝔰.

Proof. As non(𝕊_{2}) = ℵ_{0}-𝔰 then thanks to Proposition 2.2 it is enough to show that add_{t}^{∗}(𝕊_{2}) ≥ non(𝕊_{2}).

Suppose now that T ⊆ 2^{ω} and A ∈ 𝕊_{2}. To finish the proof we show that if |T | < non(𝕊_{2}) then A + T ∈ 𝕊_{2}. As in the proof of Theorem 2.1 we can assume that A = ⋃_{i<ω}[f_{i}], where f_{i} ∈ P if form a normal family. 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 a bijection. It induces a bijection 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. □

In order to prove results about add_{t}(𝕊_{2}) and cof_{t}(𝕊_{2}) we introduce some extra notation. For a set X ∈ [ω]^{ω} let (X)_{ω}^{ω} denote the family of all infinite partitions of X into infinite parts. For P_{1}, P_{2} ∈ (ω)_{ω}^{ω} we put P_{1} ⪯ P_{2} if for every p_{1} ∈ P_{1} there exists p_{2} ∈ P_{2} such that p_{2} ⊆ p_{1} (we say that P_{2} dominates P_{1}). It is not difficult to observe that ⪯ is a partial ordering on (ω)_{ω}^{ω}. Let us notice that if we consider ⪯ on the family (ω) of all partitions of ω (which is more common) then {ω} (one-element partition) is the smallest element of this ordering while the partition into singletons is the greatest one. Properties of relations on partitions of ω have been intensively studied lately by Matet, Majcher-Iwanow and others; for more information cf. [14], [6] or [12].

We define an unboundedness and dominating numbers 𝔟_{⪯} and 𝔡_{⪯} in a standard way.

\begin{matrix}𝔟_{⪯} = min\{|R| : R ⊆ (ω)_{ω}^{ω} ∧ (∀P ∈ (ω)_{ω}^{ω})(∃R ∈ R)R ⪯̸ P \}, \\ 𝔡_{⪯} = min\{|R| : R ⊆ (ω)_{ω}^{ω} ∧ (∀P ∈ (ω)_{ω}^{ω})(∃R ∈ R)P ⪯ R\}. \end{matrix}

We have the following well-known lemma.

Lemma 2.4 𝔟_{⪯} = ω_{1}, 𝔡_{⪯} = 𝔠.

Proof. Inequalities 𝔟_{⪯} ≥ ω_{1} and 𝔡_{⪯} ≤ 𝔠 are obvious. To show the other inequalities we first construct a family P ⊆ (ω)_{ω}^{ω} of cardinality 𝔠 such that for every two partitions P_{1}, P_{2} ∈ P if p_{1} ∈ P_{1} and p_{2} ∈ P_{2} then p_{1} ∩ p_{2} is finite.

We deal with partitions of ℤ × ℤ instead of 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.

Now, if R ⊆ (ω)_{ω}^{ω} is any subfamily of P of size ω_{1} then R cannot be dominated by one partition. Indeed, 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 is simultaneously contained in r_{1} and r_{2}.

On the other hand, let us consider a family R such that every partition from (ω)_{ω}^{ω} is dominated by a partition from R. 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|

and we are done. □

Theorem 2.5 add_{t}(𝕊_{2}) = ω_{1}, cof_{t}(𝕊_{2}) = 𝔠.

Proof. As ω_{1} ≤ add_{t}(𝕊_{2}) and cof_{t}(𝕊_{2}) ≤ 𝔠 then thanks to Lemma 2.4 we have to prove only add_{t}(𝕊_{2}) ≤ 𝔟_{⪯} and cof_{t}(𝕊_{2}) ≥ 𝔡_{⪯}.

We observe the following useful fact. Let P ⊆ (ω)_{ω}^{ω} be a family of partitions and A ⊆ 𝕊_{2}. Let us assume that for every partition P ∈ P there exist A_{P} ∈ A and x_{P} ∈ 2^{ω} such that ⋃_{p∈P} [0_{p}] ⊆ A_{P} + x_{P}, where 0_{p} denotes a function constantly equal to 0 on its domain, which is the set p. Then there exists a family R ⊆ (ω)_{ω}^{ω} of size |A| such that for every P ∈ P there exists R ∈ R such that P ⪯ R. Indeed, without loss of generality we can assume that A_{P} = ⋃_{i<ω}[f_{i}^{P}], where {f_{i}^{P} : i < ω} ⊆ P if and {dom(f_{i}^{P}) : i < ω} ∈ (ω)_{ω}^{ω} and by Lemma 1.2 we get that for every p ∈ P there exists a natural number i_{p} such that [0_{p}] ⊆ [f_{i_{p}}^{P} + x_{P} ↾ dom(f_{i_{p}}^{P})]. Thus dom(f_{i_{p}}^{P}) ⊆ p and, consequently, P ⪯ {dom(f_{i}^{P}) : i < ω}. Hence R = {{dom(f_{i}^{P}) : i < ω} : P ∈ P} is a family of the sort we are looking for.

Now, let P ⊆ (ω)_{ω}^{ω} be an arbitrary family of partitions of size less than add_{t}(𝕊_{2}). From the definition of 𝕊_{2} we obtain that our assumption is fulfilled for a family A having one element. Thus P is bounded by one partition and we get add_{t}(𝕊_{2}) ≤ 𝔟_{⪯}.

On the other hand, our assumption is fulfilled also for P = (ω)_{ω}^{ω} and A ⊆ 𝕊_{2} being a transitive base for 𝕊_{2}. In this situation, the family R obtained from the fact mentioned above is a dominating family of partitions, so we have cof_{t}(𝕊_{2}) ≥ 𝔡_{⪯}, which ends the proof. □

The last transitive property we deal with is translatability. In 1993 Carlson in [4] introduced the notion of κ–translatability and proved that the σ–ideal of meagre subsets 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 subsets of the Cantor space is not 2–translatable. Kysiak in [11] introduced a natural notion of a translatability number.

As far as 𝕊_{2} is concerned, its translatability number can be computed precisely.

Theorem 2.6 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 ∈ [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 A = {0_{ω}}. We claim that this set witnesses that 𝕊_{2} is not ω_{1}–translatable. So suppose B = ⋃_{i<ω}[h_{i}] where {h_{i} : i < ω} ⊆ P if and {dom(h_{i}) : i < ω} = P ∈ (ω)_{ω}^{ω}. Consider the set T defined as above. Then no translation of B covers T = A + T. □

3. Transitive operations on 𝕊_{2}. In this paragraph we apply transitive operations to the ideal 𝕊_{2}. To begin with, let us recall some definitions.

Let us assume that J is a σ–ideal of subsets of an abelian group G which is proper, translation invariant, symmetric and contains all singletons. We define (cf. [19])

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

(Seredyński used J^{∗} instead of s(J)). In [19] many basic properties of operations s and g can be found. If we apply these operations to the σ–ideal s of meagre sets M and of null sets N we obtain strongly null sets s(M), strongly meagre sets s(N), meagre-additive sets g(M) and null-additive sets g(N) (see [2] for more information).

The following are well-known.

Fact 3.1 non(s(J)) = cov_{t}(J), non(g(J)) = add_{t}^{∗}(J). □

We can also observe other basic relations.

Proposition 3.2 cov_{t}(s(J)) ≥ non(J), add_{t}^{∗}(g(J)) = non(g(J)).

Proof. Straightforward from definitions. □

We prove now that σ–ideal s 𝕊_{2} and 𝔹_{2} are closely related to each other.

Theorem 3.3 s(𝕊_{2}) = 𝔹_{2}.

Proof. Let us consider any A ⊆ 2^{ω}. A standard calculation 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 2, 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}. Then there exists i < ω such that C ↾ dom(f_{i}) = 2^{dom(f_{i})}. Indeed, 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 [7] 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 3.4 If Martin’s Axiom holds then cov_{t}(𝔹_{2}) = 𝔠.

Proof. From Theorem 3.3 and Proposition 3.2 we obtain that cov_{t}(𝔹_{2}) ≥ non(𝕊_{2}). It was proved in [5] that non(𝕊_{2}) = ℵ_{0}-𝔰 and it is well-known that under Martin’s Axiom we have ℵ_{0}-𝔰 = 𝔠. □

In order to describe g(𝕊_{2}) we need to introduce more definitions. By Inj we denote the set of all injections from ω into ω. For A ⊆ 2^{ω} and φ ∈ Inj we put φ∗A = {x∘φ : x ∈ A} and A_{φ} = {x ∈ 2^{ω} : x ∘ φ ∈ A}. It is easy to observe that we have φ ∗ A_{φ} = A and A ⊆ (φ ∗ A)_{φ}. Let J be a σ–ideal of subsets of 2^{ω}. We say that J is productive if for every A ⊆ 2^{ω} and φ ∈ Inj if φ ∗ A is in J then so is A. We say that J has WFP (Weak Fubini Property) if for every A ⊆ 2^{ω} and φ ∈ Inj if A_{φ} is in J then so is A. Straight from the definitions we obtain that 𝕊_{2}, σ–ideal s of meagre and null sets are productive and have WFP. For more discussion on these properties cf. [9].

We put

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

In other words, A ∈ p(J) if for every T ∈ [ω]^{ω} the set A ↾ T is in J (T), where J (T) denotes a version of J defined on 2^{T} instead of 2^{ω}.

Theorem 3.5 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 then [0_{rng(φ)}] + A = (φ ∗ A)_{φ}. Hence, reformulating our condition we obtain (∀φ ∈ Inj) (φ ∗ A)_{φ} ∈ 𝕊_{2}. Thus, as 𝕊_{2} is productive and has WFP, we show that this fact is equivalent to (∀φ ∈ Inj) 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 [19].

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}.

Then we have the following results.

Proposition 3.6 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 were observed in [19]. To prove (c) it is enough to repeat carefully the proof of Theorem 3.5. □

Acknowledgements The author would like to thank the referee for improving the language of the paper.

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