Subgroups of $SF(\omega)$ and the relation of almost containedness

The relations of almost containedness and orthogonality in the lattice of groups of finitary permutations are studied in the paper. We define six cardinal numbers naturally corresponding to to these relations by the standard scheme of $P(\omega)$. We obtain some consistency results concerning these numbers and some versions of the Ramsey theorem.


Introduction
The paper is motivated by investigations of various versions of van Douwen's diagram, i.e. the set of relations between six cardinals related to simple properties of almost disjointness and almost containedness, for example see [2], [3], [4], [6], [7], [14], [17]. The following theorem proved by P.Matet in [13] became one of the motivating results in this direction: Let (ω) ω be the set of all partitions of ω having infinitely many classes. Let ≤ be the order on (ω) ω defined by: E 1 ≤ E 2 if E 2 is finer than E 1 . Then assuming the continuum hypothesis there is a filter F ⊂ (ω) ω such that for every (Σ 1 1 ∪ Π 1 1 )-coloring δ : (ω) ω → 2 there is a partition E ∈ F such that δ is constant on the set of all infinite partitions coarser than E.
The statement is a variant (and a consequence) of the dualized version of Ramsey's theorem proved by T.Carlson and S.Simpson in [5]. The argument of P.Matet uses the observation that the tower cardinal (we denote it by t d ) for the ordering of infinite partitions is uncountable. Here t d is defined by the same scheme as t for the lattice P (ω) of all subsets of ω (see [7]). Moreover it is proved in [13] that the tower cardinal for partitions is ω 1 in ZFC and it is proved in [6] that the size of a maximal almost orthogonal family of partitions must be 2 ω .
The lattice of partitions under the order reversing ≤ was studied in [12] and [3]. It is shown there that the corresponding cardinal invariants look differently. For example p and t defined in this case coincide with classical p.
Note that this lattice can be also defined to be the lattice of 1-closed subgroups of Sym(ω) (i.e. the automorphism groups of structures with only unary predicates). Indeed, the corresponding isomorphism of these lattices maps a partition E to the group G E of all permutations preserving the E-classes. On the other hand for any E the group G E is uniquely determined by the subgroup G E ∩ SF (ω) of the group SF (ω) of all finitary permutations of ω. The embedding obtained E → G E ∩ SF (ω) maps almost trivial equivalence relations into the ideal IF of all finite subgroups of SF (ω).
This motivates further questions. For example it is interesting to find a variant of the result of P.Matet in the lattice of all subgroup of the group SF (ω). Since the corresponding tower cardinal is involved in this question, a general problem of description the corresponding van Douwen's diagram for this lattice seems relevant. The paper is devoted to these questions.
We use standard set theoretic conventions and notation.
[ω] ω and [ω] <ω stand for all infinite and all finite subsets of ω respectively. For k ∈ ω \ {0}, let [ω] k be the set of all k-element subsets of ω. By (ω) we denote the set of all partitions of ω. A partition is finite if it has finitely many pieces (classes). The set of all finite partitions will be denoted by (ω) <ω and the set of all infinite partitions will be denoted by (ω) ω . In our paper we often identify n ∈ ω with {0, ..., n − 1}.

Almost containedness
Van Douwen's diagrams are due to [7] and [17], where the case of P (ω) was considered. The term was used in [6] where the case of partitions was studied. The general idea can be described as follows.
Let L be a lattice with 0 and 1, and let I be an ideal of L. We say that a, b For any a ∈ L we put a I = {b : b = a a}. It is clear that the relation ≤ a becomes the usual almost containedness if we consider the lattice (P (ω), ⊆) with respect to the ideal of finite subsets of ω.
In general, to characterize a lattice L under these relations we need some further notions. We say that a splits b if there are c, d ≤ b not in I such that c ≤ a and d, a are orthogonal. A family Γ ⊂ L \ I is a splitting family if for every b ∈ L \ I there exists a ∈ Γ that splits b. We say that Γ is a reaping family if for each a ∈ L \ I there is some b ∈ Γ such that b ≤ a a or a, b are orthogonal. We also define a family Γ ⊂ L \ I to be ≤-centered if any finite intersection of its elements is not in I.
We can now associate to L the following cardinals. Define a I to be the least cardinality of a maximal family of pairwise orthogonal elements from L \ 1 I . Let p I be the least cardinality of a ≤-centered family Γ such that there is no b ∈ L\I such that b is a lower bound of Γ under ≤ a and the family Γ ∪ {b} is still ≤-centered. Similarly, define t I (the tower cardinal) as the least cardinality of a ≤ a -decreasing ≤-centered chain without lower ≤ a -bound consistent (in the sense of ≤-centeredness) with the family. The cardinals s I , r I are the corresponding (least) cardinals for splitting families and reaping families respectively. It is worth noting that p I and t I can be undefined (for example if for any a ∈ L the set {b : b ≤ a} is finite). Also, (L, I) does not necessarily have a splitting family (for example if L is an atomic boolean algebra and I is trivial). So s I can be undefined too. On the other hand, it is clear that p I ≤ t I if they are defined.
The last cardinal h I is defined as follows. A family Σ of maximal families of pairwise orthogonal elements in L \ 1 I is shattering if for every a ∈ L \ I there are Γ ∈ Σ and distinct b, c ∈ Γ which are not orthogonal to a. Let h I be the least cardinality of a shattering family in L.
The following lemma seems to be folklore.
Proof. Take a splitting family Γ = {c ν : ν < s}. For each ν < s choose Ψ ν to be a maximal family of pairwise orthogonal elements such that c ν ∈ Ψ ν . Let us check that the set of these families is shattering. Let c ∈ L \ I. Since Γ is a splitting family there is ν and a, b ≤ c such that a ≤ c ν and b is orthogonal to c ν . By our construction there is d ∈ Ψ ν not orthogonal to b. So Ψ ν shatters c by c ν and d.
Remark. In the case of the lattice (P (ω), ∪, ∩) and the ideal [ω] <ω of all finite subsets of ω the introduced numbers are exactly the classical cardinals a, h, p, r, s, t (all of them occur in [17]). Indeed, our definitions of a I , r I , s I , h I are formulated as the corresponding classical ones in [7] and in [17]. The classical t is the least cardinality of a ≤ a -decreasing chain in P (ω) without ≤ a -bound. The classical p is defined as follows. We say that a family Γ ⊆ [ω] ω is ≤ a -centered if every its finite subset Γ ′ has an infinite pseudointersection: a set X ∈ [ω] ω almost contained in each element of Γ ′ . Then the classical p is the least cardinality of a ≤ a -centered family from P (ω) without lower ≤ a -bound. So, there is no assumption on ⊆-centeredness as in the definitions of p I and t I . On the other hand we do not need such assumptions because any ≤ a -centered family from P (ω) is centered. So p = p [ω] <ω and t = t [ω] <ω .
Note that the definitions of the above cardinals make sense if we consider L/ = a under the reverse order ≥ a replacing the ideal I by 1 I . In the case of P (ω) the converse cardinals are equal to the corresponding cardinals for ⊂ because P (ω) is a Boolean algebra. The fact that this is not true in general is quite important for the lattice of subgroups of SF (ω) and for partitions.
In the latter case we can consider partitions as subsets of ω 2 under the inclusion (denoted by ⊂ pairs ). The lattice that we get (with operations ∨ pairs and ∧ pairs ) is converse to the lattice ((ω), ≤). Let IF be the ideal of partitions (in ((ω), < pairs )) obtained from id ω by adding a finite set of pairs. Then the class 1 IF is exactly (ω) <ω . Note that papers [12] and [3] study cardinal invariants of this lattice. On the other hand the relation of almost containedness of partitions studied in [13] and [6] can be defined as follows: As a result we have that the cardinal a d , p d , t d , h d , s d , r d studied in [13] and [6] are the converse cardinals for the pair (((ω), < pairs ), IF ).

The lattice of subgroups of SF (ω)
Let SF (ω) be the group of all finitary permutations of ω. This means that the elements of SF (ω) are exactly the permutations g with finite support, where supp(g) = {x : g(x) = x}. The algebraic structure of subgroups of SF (ω) is described in [15], [16]. The aim of out paper is to study the van Douwen's invariants of the lattice of subgroups of SF (ω).
Throughout the paper LF is the lattice of all subgroups of SF (ω) and IF is the ideal of all finite subgroups. We say that G 1 and is a subgroup of a group finitely generated over G 2 by elements of SF (ω). Let SF (ω) IF = {G ≤ SF (ω) : SF (ω) is finitely generated over G}. As in Section 1.2 we define the cardinal numbers a SF , p SF , t SF , r SF , h SF and s SF . For example, a SF is the least cardinality of a maximal family of pairwise orthogonal elements from LF \ SF (ω) IF and p SF is the least cardinality of a ≤-centered family of elements in LF \ IF with no lower ≤ a -bound ≤-consistent (in the sense of ≤-centeredness) with the family.
We put a topology on LF in the following way. Let H ≤ SF (ω) be finite and A ⊂ ω is a finite set containing the union of the supports of the elements of H. Let [H, A] be the set of all subgroups of SF (ω) such that the groups that they induce on A are equal to H (we think of H as a permutation group on A). The topology that we consider is defined by the base consisting of all sets [H, A]. This topology is metrizable: fix an enumeration A 0 , A 1 , ... of all finite subsets of ω and define d(G 1 , G 2 ) = {2 −n : the groups induced by G 1 and G 2 on A n are not the same }.
Note that the space LF is complete. A function δ : LF → n, n ∈ ω, is then called a Borel (respectively Σ 1 1 ∪ Π 1 1 ) coloring if δ −1 (i) is Borel (respectively analytic or coanalytic) for every i < n (where n ∈ ω is viewed as {0, ..., n − 1}.) Consider the set LF 1 of all groups of the form SF (ω) ∩ G where G is 1-closed. We identify elements of LF 1 with elements of 2 ω×ω (the corresponding partitions). Then it is easily seen that the topology on LF 1 induced by the topology above becomes the restriction of the product topology on 2 ω×ω where 2 considered discrete. A theorem of T.Carlson and S.Simpson from [5] can be restated as follows: for every (Σ 1 The corresponding theorem for P (ω) proved by F.Galvin and K.Prikry in [8] is stated as follows: for every (Σ 1 1 ∪ Π 1 1 )coloring (originally: Borel coloring; see Remark 2.6 in [5]) δ : P (ω) → 2 there exists an infinite A ∈ P (ω) such that δ is constant on the set of all infinite subsets of A. It is shown in [5] that this theorem is a consequence of the Carlson-Simpson theorem.

A variant of Matet's theorem
The proof of the theorem of Matet stated in Section 1.1 (this is Proposition 8.1 from [13]) uses the Calson-Simpson's theorem and Proposition 4.2 from [13] asserting that t d is uncountable. We will use the same strategy. Our version of this theorem will be given in Section 2.2.

Comparing
We begin with the following useful lemma.
(ii) moreover, for any H ⊂ Sym(m) and any sequence G 0 , G 1 , ..., G n ∈ LF \ IF of groups orthogonal to G the above ρ can be chosen such that additionally Proof. (i). Suppose that the lemma is not true. Choose a minimal A = {a 0 , ..., a k } ⊂ m such that there are infinitely many g ∈ G such that m ∩ supp(g) ⊂ A. Then A = ∅. We fix some non-trivial g 0 with that property and consider all tuples g(ā) = (g(a 0 ), ..., g(a k )) for the above g's. If everyone of these tuples has non-empty intersection with supp(g 0 ) then there is i ≤ k such that g(a i ) is the same for infinitely many g's. Clearly, for such g and g ′ the set m ∩ supp( This contradicts the minimality of A. Choose g as above with g(ā) ∩ supp(g 0 ) = ∅ additionally. It is easily seen that g −1 · g 0 · g fixes m pointwise. This contradicts our assumption.
(ii). Suppose the contrary. By (i) we can find i ≤ n such that for infinitely many ρ ∈ G with supp(ρ) ∩ m = ∅ there is g ∈ H satisfying g · ρ ∈ G i . Since H is finite, there is g 0 ∈ H such that g 0 · ρ ∈ G i for infinitely many ρ ∈ G. Hence, for infinitely many ρ, ρ ′ ∈ G, ρ −1 · ρ ′ ∈ G i , which contradicts orthogonality.
This lemma alows us to imitate some arguments from [3] and [12]. Lemma 2.2 For any G ∈ LF \ IF there is H ∈ LF \ IF such that H ≤ G and the sublattice of (LF/ = a , ≤ a ) of all elements below the = a -class of H is isomorphic to In particular h ≤ h SF , s ≤ s SF , r SF ≤ r, t SF ≤ t and p SF ≤ p.
Proof. Given G we construct an infinite sequence σ i ∈ G, i ∈ ω, so that the supports of these σ i are pairwise disjoint. It is easy to arrange that all σ i are of prime orders, say p m i . Let H the group generated by all σ i , i ∈ ω. Then H is isomorphic to the direct sum of all Z/p m i Z.
To each A ⊂ ω we associate the subgroup of H generated by all σ i with i ∈ A. This maps (P(ω)/F in, ⊆ a ) to the sublattice of (LF/ = a , ≤ a ) of all elements below the = a -class of H. It is clear that this map is an isomorphism. The rest is easy.
The following easy statement is based on an argument which will be typical below.

Lemma 2.3 For any countable sequence
Proof. Assume we have a decreasing sequence G 0 > G 1 > ... in LF \ IF . For every i ∈ ω choose non-trivial g i ∈ G i such that supp(g i ) is disjoint from the supports of the previous elements. We can do this by Lemma 2.1(i). Let G be the group generated by all these g i . Then G ∈ LF \ IF , the family {G} ∪ {G i : i ∈ ω} is centered and G ≤ a G i for every i ∈ ω.
The following lemma uses another type of aruments.
Proof. Let g be a finitary permutation such that g(a) = b ∈ P j for a ∈ P i , i = j. Let a ′ ∈ P i \ supp(g) and b ′ ∈ P j \ supp(g). Below we denote the transposition of x and y by (x, y). It is clear that the element (a, a ′ ) · g −1 · (b, b ′ ) · g · (a, a ′ ) (which belongs to G 0 , g ) is the transposition (a ′ , b ′ ). This yields that the group inducing SF (P i ∪ P j ) and acting trivially on ω \ (P i ∪ P j ), is a subgroup of G 0 , g . The rest is clear.
Proof. p ≤ p SF . Take an ≤-centered family Γ ⊂ LF \ IF of cardinality p SF without bounds in LF , as in the definition of p SF . Consider the set P p of all pairs (H, F ) where H ⊂ SF (ω) is a finite set of permutations with pairwise disjoint supports and F is a finite subfamily of Γ. We define a forcing relation on P p as follows: Clearly P p is σ-centered and satisfies the ccc. For any k ∈ ω and G ∈ Γ the family {(H, F ) ∈ P p : k < |H|, G ∈ F )} is dense in P p by centeredness of Γ and Lemma 2.1(i). For a generic Φ define It is easily seen that for any G ∈ Γ, the group G 0 is almost contained in G. Since G 0 is generated by elements with pairwise disjoint supports, the family Γ ∪ {G 0 } is ≤-centered. This is a contradiction. By a theorem of M.Bell [1] MA κ (σ-centered) is equivalent to κ < p. Therefore, p ≤ p SF .
Let us prove t SF ≤ p. Since p SF ≤ t SF , we will have the equality of these three invariants.
We use the method of Theorem 3.2 of [3]. By Lemma 3.1 of this paper: p is the minimal cardinal of form |A| + |T | where A ∪ T is a family of infinite subsets of ω with the finite intersection property so that T is wellordered by ⊇ * and there is no infinite T so that T is almost contained in each member of T and A ∪ {T } has the finite intersection property.
Now it is easy to see that the sequence G α , α < λ + µ, is ≤ a -decreasing. If F ⊂ λ + µ is finite let The group generated by all transpositions (4i, 4i + 1), i ∈ B, is an infinite subgroup of any G γ with γ ∈ F . This shows the family G α , α < λ + µ, is ≤-centered.
If H ≤ a G α for all α < λ + µ, then H is finitely generated over some family of transpositions (4i, 4i + 1), i ∈ T . Since T is almost contained in all T β , β < µ, the assumptions above imply that A ∪ {T } does not have the finite intersection property. The latter easily implies that {G α : α < λ + µ} ∪ {H} is not ≤-centered. Corollary 2.6 (i) The following inequalities are true in (LF, IF ): (ii) p SF and t SF are equal to continuum under Martin's Axiom.

A version of Matet's theorem
We now prove the main result of the section.

Theorem 2.7
Assuming MA there is a filter F ⊂ LF \ IF such that for every (Σ 1 1 ∪ Π 1 1 )-coloring δ : LF → 2 there is G ∈ F such that δ is constant on the set of all infinite subgroups of G.
Proof. Let G 0 < SF (ω) be generated by an infinite sequence {g 1 , ..., g i , ...} with pairwise disjoint supports where each g i is a cycle of the prime length p i . Then G 0 is isomorphic to Σ i∈ω Z/p i Z. It is easy to see that if G ≤ G 0 then G is generated by a subset of the set {g i : i ∈ ω}. Then the lattice of all subgroups of G 0 is isomorphic to (P (ω), ⊆). We identify G ≤ G 0 with the corresponding subset of ω. Notice that then the topology defined in Section 1.3, on {G : G ≤ G 0 } becomes the product topology on 2 ω . Also, {G : G ≤ G 0 } is a closed subset of LF .
We now use the strategy of Proposition 8.1 from [13]. Let δ α , α < 2 ω , be an enumeration of all (Σ 1 1 ∪ Π 1 1 )-colorings δ : LF → 2. We construct a descending tower of subgroups of G 0 . Supposing that G β , β < α, have already been selected, use Corollary 2.6 (ii) to find G α ≤ G 0 such that the family {G γ : γ ≤ α} is ≤-centered and G α ≤ a G γ for all γ < α. By the Galvin-Prikry theorem ( [8]) there is an infinite subset of the set of generators of G α such that all its infinite subsets have the same color with respect to the coloring induced by δ α . This shows that G α can be chosen such that all its infinite subgroups have the same color with respect to δ α .
Let F be the filter generated by the tower obtained. It follows from the construction that F satisfies the conditions of the theorem.

The diagram in (LF, IF )
We begin this subsection with the following algebraic lemma.
Lemma 3.1 Let G 0 , ..., G n−1 be a sequence of infinite groups from LF not a-equivalent to SF (ω). Then for any k, m ∈ ω, k > 0, and H ⊂ Sym(m) there is a non-trivial finitary permutation ρ consisting of (k + 1)-cycles such that supp(ρ) ⊂ ω \ m and for every i < n, Proof. For each i < n set It is easily seen that each S i is a group. Choose a family {D j,0 : 0 ≤ j < n} of pairwise disjoint finite sets such that for every j, D j,0 ⊂ ω \ m and S j does not induce Sym(D j,0 ). Let D j,1 , ..., D j,k be sets from ω \ m of the same size as D j,0 . We may assume that every pair from {D j,i : 0 ≤ i ≤ k; j < n} has empty intersection. For every 0 < i ≤ k and j < n we choose a bijection f j,i from D j,0 onto D j,i such that it is not induced by any element of S j . The existence of such f j,i is a consequence of the fact that for any bijections f, g : D j,0 → D j,i induced by S j , the bijection g −1 · f defines a permutation on D j,0 induced by S j . We now define a permutation ρ with the support {D j,i : . Let us check that ρ satisfies the conclusion of the lemma. It is clear that ρ consists of cycles of length k + 1. Suppose, that for some g 0 ∈ H the element g = g 0 · ρ l , 0 < l ≤ k, is contained in some G j . Thus g ∈ S j and by our construction g maps D j,0 onto D j,l by f j,l . Since S j does not induce f j,l , we have a contradiction.
The van Douwen cardinals for (LF, IF ) are described in the following theorem.
(ii) All the coefficients are equal to continuum under Martin's Axiom; (iii) Each of the following equalities is consistent with {ZFC + ω 1 < 2 ω }: Proof. (i). The inequality h SF ≤ s SF is shown in Lemma 1.1. The inequality ω 1 ≤ p SF follows from Theorem 2.5.
To show that t SF ≤ h SF , suppose µ < t SF and Ψ = {Ψ ν : ν < µ} is a shattering family of maximal families of pairwise orthogonal elements in L \ SF (ω) IF . Choose an infinite G 0 < SF (ω) generated by an infinite sequence of elements with pairwise disjoint supports and of prime orders. Taking intersections with appropriate elements of Ψ ν 's we construct a ≤ a -decreasing ≤-centered sequence of infinite groups {G ν : ν < µ} such that G ν is not shattered by Ψ γ for all γ < ν. We can arrange that the groups are = a -distinct. Since µ < t SF , there is G ∈ LF \ IF such that G < a G ν for all ν < µ. We may assume that G < G 0 . If G is shattered by a pair H 1 , H 2 from some family from Ψ then H 1 and H 2 provide two disjoint infinite sets S 1 ⊂ G ∩ H 1 and S 2 ⊂ G ∩ H 2 such that each S i consists of elements with pairwise disjoint supports. Since G < a G ν , every S i has an infinite intersection with G ν . This shows that every G ν is shattered by H 1 and H 2 . This is a contradiction.
To prove ω 1 ≤ r SF it suffices to show that if a family Ψ ⊆ LF \ IF is countable then there exists G ∈ LF \ IF such that for every G ′ ∈ Ψ the groups G, G ′ are not orthogonal and G ′ ≤ a G. Let {G 0 , G 1 , ...} be an enumeration of Ψ. Assume that each member of Ψ occurs infinitely often. We construct two sequences g 0 , g 1 , ... and h 0 , h 1 , ... of finitary permutations with pairwise disjoint supports such that for all i, j ∈ ω we have supp(g i ) ∩ supp(h j ) = ∅ and g i , h i ∈ G i . It is easily seen that Lemma 2.1(i) implies the existence of such sequences. LetĜ 1 = {g i : i ∈ ω} and G 2 = {h i : i ∈ ω} . Clearly,Ĝ 1 andĜ 2 are orthogonal but they are not orthogonal to any G i (since each member of Ψ is enumerated infinitely often). Now it is easy to see that G =Ĝ 1 satisfies the conditions that we need.
To prove the inequality ω 1 ≤ a SF take a countable Ψ ⊂ LF \ SF (ω) IF . We construct a group G by induction. Fix an enumeration of Ψ : G 0 , G 1 , ... . Let H be the set of the elements which have been constructed at the first n − 1 steps. At the n-th step we choose a permutation ρ as in Lemma 3.1 with respect to G 0 , ..., G n and m large enough. It is easily seen that that the group generated by this sequence is orthogonal to any group from Ψ. For any k ∈ ω and G ∈ Ψ the family {(H, H ′ ) ∈ P r : k < |H ′ ∩ G|, k < |H ∩ G|} is dense in P r by Lemma 2.1(i) (see also the previous part of the proof). For a generic Φ define G 0 = {H : (H, H ′ ) ∈ Φ} . It is easy to see that for any G ∈ Ψ, the groups G and G 0 are not orthogonal and G is not contained in G 0 under ≤ a . Thus Ψ is not reaping.
To show a SF = 2 ω , given an infinite family Γ ⊂ LF of infinite groups define a forcing notion P a as follows. Let P a be the set of all pairs (H, F ) where F is a finite subset of Γ and H is a finite set of permutations such that their supports are pairwise disjoint. We define (H, F ) ≤ (H ′ , F ′ ) iff H ′ ⊂ H, F ′ ⊂ F and each h ∈ H \ H ′ is not contained in any G ∈ F ′ . It is easily verified that P a is a ccc forcing notion.
Consider P a with respect to Ψ ⊂ LF \ SF (ω) IF of cardinality < 2 ω . Clearly, the following sets are dense in P a (apply By MA we have a filter Φ ⊂ P a meeting all these Σ's. It is easy to see that the group G 0 = {H : (H, F ) ∈ Φ} is orthogonal to any group from Ψ.
(iii). Con(ZFC + a SF = ω 1 < 2 ω ). We start with an arbitrary countable family Ψ 0 ⊂ LF \ SF (ω) IF of parwise orthogonal groups. Take a sequence by a finite support iteration (P γ , Q γ : γ < ω 1 ) of the forcing P a (from the previous part of the proof) applied to potential Ψ γ 's. The canonical name for P γ of Ψ γ+1 is obtained from the canonical name of Ψ γ by adding the canonical name of the group G γ defined by Q γ as G 0 by P a above. Let Φ be generic for P ω 1 and Φ γ be the corresponding restriction to P γ . It is easily seen that P ω 1 fulfils the ccc. Since ω 1 is regular and each group G in LF [Φ] is defined by a countable set of finitary permutations, it is contained in some LF [Φ γ ]. Suppose that some G is orthogonal to each group from Ψ γ . Thus by Lemma 2.1(ii) every set is not orthogonal to G. This shows that the set {Ψ γ : γ < ω 1 } is a maximal family of pairwise orthogonal groups in LF [Φ]. The cases Con(ZFC + s SF = ω 1 < 2 ω ) and Con(ZFC + r SF = ω 1 < 2 ω ) can be handled in a similar way. In the first case constructing G γ we apply the forcing P r . In the second one at every step we should apply P a to the whole family LF \ (IF ∪ SF (ω) IF ). So every G ∈ LF [Φ] \ (IF ∪ SF (ω) IF ) is orthogonal to some G γ .
We conjecture that h SF = h, s SF = s and r SF = r. Note that the corresponding equalities hold for the lattice of partiotions under ≤ pairs [3]. At the moment we cannot adapt the arguments of [3] to our case. The case of a is also open.

Another version of Matet's theorem
As we noted in Introduction the lattice of partitions under the reverse order is a sublattice of LF . This suggests that in the lattice LF the most natural variant of the theorem of P.Matet cited there (which is Proposition 8.1 from [13]) is the folowing one.
Theorem 4.1 Assuming the continnuum hypothesis there is an ideal I ⊂ LF \ SF (ω) IF such that for every (Σ 1 1 ∪ Π 1 1 )-coloring δ : LF → 2 there is G ∈ I such that δ is constant on the set of all supergroups of G which do not belong to SF (ω) IF .
We may now consider the set L 0 = {G : G 0 ≤ G ≤ SF (ω)} as a sublattice of partitions coarser than E 0 . Notice that then the topology defined in Section 1.3, on L 0 becomes the product topology on 2 ω×ω . This follows from the fact that any finite permutation group (on a finite subset of ω) induced by a group G from L 0 is a finite 1-closed permutation group and can be identified with a partition induced by the partition corresponding to G. Moreover, it is easy to see that L 0 is closed in LF .
We now use the Matet's theorem. Take an ideal I 0 of L 0 provided by this theorem. Then I 0 generates an ideal of LF . This ideal works as I in the statement.
Since p d = t d SF = ω 1 (see [13]), we have the statemet of the lemma.
Using this proposition and the material of papers [13], [6], [2], [14] we obtain the following relations: Moreover the following relations are consistent with ZFC: We mention the following questions:

Remarks
Our results suggest the investigation of van Douwen's cardinals for the lattice of all closed subgroups of Sym(ω). The definition of the a-order in this case must be as follows: G ≤ a G ′ iff there exists a finite set X of finitary permutations such that G is a subgroup of the closed group generated by G ′ and X. It is worth noting that some results of [10] can be interpreted in this vein for some converse coefficients (for example, see Observation 3.3 in [10]). However, one can notice that the lattice of all closed subgroups admits several constructions which make some of the van Douwen's cardinals trivial. For example, the group Z with the natural action on itself can be considered as a closed subgroup of Sym(ω). It is clear that for every n ∈ ω no closed subgroups split any nZ. So, s I is undefined. On the other hand it is worth noting that for every n ∈ ω, nZ = a Sym(ω). Indeed, fixing some representatives a i of all the orbits, add the transpositions of the pairs a i , a i + n. This induces all permutations on every orbit. Adding transpositions of some pairs from distinct orbits we get Sym(ω).
Another easy observation is that r I = 1 in this case. Indeed, the Prüfer group Z(p ∞ ) with the natural action on itself forms a reaping family.
It is interesting to compare the lattices that we consider here with the lattice P (ω) of all subsets of ω and the ideal of finite subsets. Since = a is a congruence of P (ω), the orthogonality of infinite a and b means the absence of c such that c ≤ a a and c ≤ a b. So the van Douwen's cardinals can be defined only in terms of ≤ a (and originally it was so). On the other hand, this does not hold in lattices of subgroups of Sym(ω). Indeed, let σ be a transposition of some pair in ω. Then Z σ induces a closed subgroup of Sym(ω) which is a-equivalent to Z with the above action. Clearly, the intersection of these groups is trivial.
In the case of (LF, IF ) the corresponding example is as follows. Let infinite A, B, C ⊂ ω define a partition of ω and R be a bijection between A and B. Let E 0 = A 2 ∪ B 2 ∪ id C×C and E 1 = R ∪ id C×C . It is easily seen that E 0 and E 1 are orthogonal equivalence relations, but E 1 ≤ a E 0 . The groups G E 0 ∩ SF (ω) and G E 1 ∩ SF (ω) have the same properties.