Decomposition of Cellular Balleans

Given any x ∈ X, A ⊆ X, α ∈ P , we put

A ball structure B = (X, P, B) is called a ballean (or a a coarse structure) if

• ∀α, β ∈ P ∃α ′ , β ′ ∈ P such that ∀x ∈ X B(x, α) ⊆ B * (x, α ′ ), B * (x, β) ⊆ B(x, β ′ );

• ∀α, β ∈ P ∃γ ∈ P such that ∀x ∈ X B(B(x, α), β) ⊆ B(x, γ); Let B 1 = (X 1 , P 1 , B 1 ) and B 2 = (X 2 , P 2 , B 2 ) be balleans.

A mapping f : X 1 → X 2 is called a ≺-mapping if ∀α ∈ P 1 ∃β ∈ P 2 such that:

A bijection f : X 1 → X 2 is called an asymorphism between B 1 and B 2 if f and f -1 are ≺-mappings. In this case B 1 and B 2 are called asymorphic.

If X 1 = X 2 and the identity mapping id: X 1 → X 2 is an asymorphism, we identify B 1 and B 2 and write

For motivation to study balleans, see [1], [2], [3], [4].

Every metric space (X, d) determines the metric ballean B(X, d) = (X, R + , B d ), where R + is the set of non-negative real numbers, To define cf(B), we use the natural preordering on P : α ≤ β if and only if B(x, α) ⊆ B(x, β) for every x ∈ X. A subset P ′ is cofinal in P if, for every α ∈ P , there exists α ′ ∈ P ′ such that α ≤ α ′ , so cf(B) is the minimal cardinality of cofinal subsets of P . Given an arbitrary ballean B = (X, P, B), x, y ∈ X and α ∈ P , we say that x, y are α-path connected if there exists a finite sequence x 0 , x 1 , …, x n , x 0 = x, x n = y such that x i+1 ∈ B(x i , α), for every i ∈ {0, 1, …, n -1}. For any x ∈ X and α ∈ P , we put X,d) is an ultrametric space then the ballean B(X, d) is cellular. Moreover, by [3, Theorem 3.1], a ballean B is metrizable and cellular if and only if B is asymorphic to the metric ballean B(X, d) of some ultrametric space (X, d).

Example 2. Let G be an infinite group with the identity e, κ be an infinite cardinal such that κ ≤ |G|, F(G, κ) = {A ⊆ G : e ∈ A, |A| < κ}.

Given any g ∈ G and A ∈ F (G, κ), we put B(g, A) = gA and get the ballean B(G, κ) = (G, F (G, κ), B). In the case κ = |G|, we write B(G) instead of B(G, κ). A ballean B(G, κ) is cellular if and only if either κ > ℵ 0 or κ = ℵ 0 and G is locally finite (i.e. every finite subset of G is contained in some finite subgroup).

Every Boolean group ideal ℑ on G determines the ballean B(G, ℑ) = (G, ℑ, B), where B(g, A) = gA for all g ∈ G, A ∈ ℑ. The balleans on groups determined by the Boolean group ideals can be considered (see [3,ter 6]) as the asymptotic counterparts of the group topologies. A ballean B(G, ℑ) is cellular if and only if ℑ has a base consisting of the subgroups of G.

A connected ballean B = (X, P, B) is called ordinal if there exists a cofinal well-ordered (by ≤) subset of P . Clearly, every metrizable ballean is ordinal.

Theorem 1. Let B = (X, P, B) be an ordinal ballean. Then B is either metrizable or cellular.

Proof. If cf(B) ≤ ℵ 0 then B is metrizable by theorem 2.1 from [3]. Assume that cf(B) > ℵ 0 . Given an arbitrary α ∈ P , we choose inductively a sequence (α n ) n∈ω in P such that α 0 = α and B(B(x, α n ), α) ⊆ B(x, α n+1 ) for every x ∈ X. Since cf(B) > ℵ 0 , we can pick

Let γ be an ordinal, {Z λ : λ < γ} be a family of non-empty sets. For every λ < γ we fix some element e λ ∈ Z λ and say that the family {(Z λ , e λ ) :

is the set of all functions f : {λ : λ < γ} → ∪ λ<γ Z λ such that f (λ) ∈ Z λ and f (λ) = e λ for all but finitely many λ < γ. We consider the ball structure B(Z) = (Z, {λ : λ < γ}, B), where B(f, λ) = {g ∈ Z : f (λ ′ ) = g(λ ′ ) for all λ ′ ≥ λ} It is easy to verify that B(Z) is a cellular ballean.

We say that a ballean B is decomposable in a direct product if B is asymorphic to B(Z) for some direct product Z. Theorem 2. Let γ be a limit ordinal, B = (Z, {λ : λ < γ}, B) be a ballean such that:

(iii) if β is a limit ordinal and β < γ then B(x, β) = ∪ α<β B(x, α) for each

x ∈ X;

(iv) there exists a cardinal κ 0 such that B(x, 0) = κ 0 for each x ∈ X;

(v) for every α < γ there exists a cardinal κ α such that every ball of radius α + 1 is a disjoint union of κ α -many balls of radius α.

Then B is decomposable in a direct product.

Proof. We fix some set Z 0 of cardinality κ 0 and define inductively a family of sets {Z α , α < γ}. If α is a limit ordinal, we take Z α to be a singleton. If α = β + 1 we take a set Z α of cardinality κ β . For every α < γ, we choose some element e α ∈ Z α , put Z = ⊗ λ<γ (Z λ , e λ ) and show that B is asymorphic to B(Z). To this end we fix some element x 0 ∈ X and, for every α < γ, define a mapping f α : B(x 0 , α) → ⊗ β≤α (Z β , e β ) such that, for all β < α < γ, f α | B(x 0 ,β) = f β and the inductive limit f of the family {f α : α < γ} is an asymorphism between B and B(Z). Here we identify ⊗ β≤α (Z β , e β ) with the corresponding subset of ⊗ β<γ (Z β , e β ).

At the first step we fi