The comb-like representations of cellular ordinal balleans
Given two ordinal λ and γ, let f:[0,λ)→[0,γ) be a function such that, for each α<γ, sup{f(t):t∈[0,α]}<γ. We define a mapping df:[0,λ)×[0,λ)⟶[0,γ) by the rule: if x<y then df(x,y)=df(y,x)=sup{f(t):t∈(x,y]}, d(x,x)=0. The pair ([0,λ),df) is called a γ−comb defined by f. We show that each cel...
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Інститут прикладної математики і механіки НАН України
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| Cite this: | The comb-like representations of cellular ordinal balleans / I. Protasov, K. Protasova // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 282-286. — Бібліогр.: 8 назв. — англ. |
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| citation_txt | The comb-like representations of cellular ordinal balleans / I. Protasov, K. Protasova // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 282-286. — Бібліогр.: 8 назв. — англ. |
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| description | Given two ordinal λ and γ, let f:[0,λ)→[0,γ) be a function such that, for each α<γ, sup{f(t):t∈[0,α]}<γ. We define a mapping df:[0,λ)×[0,λ)⟶[0,γ) by the rule: if x<y then df(x,y)=df(y,x)=sup{f(t):t∈(x,y]}, d(x,x)=0. The pair ([0,λ),df) is called a γ−comb defined by f. We show that each cellular ordinal ballean can be represented as a γ−comb. In General Asymptology, cellular ordinal balleans play a part of ultrametric spaces.
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 2, pp. 282–286
© Journal “Algebra and Discrete Mathematics”
The comb-like representations of cellular ordinal
balleans
Igor Protasov and Ksenia Protasova
Abstract. Given two ordinal λ and γ, let f : [0, λ) → [0, γ)
be a function such that, for each α < γ, sup{f(t) : t ∈ [0, α]} < γ.
We define a mapping df : [0, λ) × [0, λ) −→ [0, γ) by the rule: if
x < y then df (x, y) = df (y, x) = sup{f(t) : t ∈ (x, y]}, d(x, x) = 0.
The pair ([0, λ), df ) is called a γ−comb defined by f . We show
that each cellular ordinal ballean can be represented as a γ−comb.
In General Asymptology, cellular ordinal balleans play a part of
ultrametric spaces.
Introduction
In [3], a function f : [0, 1] → [0, ∞) is called a comb if, for every
ε > 0, the set {t ∈ [0, 1] : f(t) > ε} is finite. Each comb f defines a
pseudo-metric df on the set If = {t ∈ [0, 1] : f(t) = 0} by the rule: if
x < y then
df (x, y) = max{f(t) : t ∈ (x, y)},
df (y, x) = df (x, y), d(x, x) = 0.
After some reduced completion of (If , df ), the authors get a compact
ultrametric space and show that each compact ultrametric space with no
isolated points can be obtained in this way.
In this note, we modify the basic construction from [3] to get the
comb-like representations of cellular ordinal balleans which, in General
Asymptology [7], play a part of ultrametric spaces.
2010 MSC: 54A05, 54E15, 54E30.
Key words and phrases: ultrametric space, cellular ballean, ordinal ballean,
(λ, γ)−comb.
I . V. Protasov, K. D. Protasova 283
1. Balleans
Following [5], [7], we say that a ball structure is a triple B = (X, P, B),
where X, P are non-empty sets, and for all x ∈ X and α ∈ P , B(x, α) is
a subset of X which is called a ball of radius α around x. It is supposed
that x ∈ B(x, α) for all x ∈ X, α ∈ P . The set X is called the support
of B, P is called the set of radii.
Given any x ∈ X, A ⊆ X, α ∈ P , we set
B∗(x, α) = {y ∈ X : x ∈ B(y, α)},
B(A, α) =
⋃
a∈A
B(a, α) and B∗(A, α) =
⋃
a∈A
B∗(a, α).
A ball structure B = (X, P, B) is called a ballean if
• for any α, β ∈ P , there exist α′, β′ ∈ P such that, for every x ∈ X,
B(x, α) ⊆ B∗(x, α′) and B∗(x, β) ⊆ B(x, β′);
• for any α, β ∈ P , there exists γ ∈ P such that, for every x ∈ X,
B(B(x, α), β) ⊆ B(x, γ);
• for any x, y ∈ X, there exists α ∈ P such that y ∈ B(x, α).
We note that a ballean can be considered as an asymptotic counterpart
of a uniform space, and could be defined [8] in terms of the entourages of
the diagonal ∆X = {(x, x) : x ∈ X} in X × X. In this case a ballean is
called a coarse structure.
For categorical look at the balleans and coarse structures as “two faces
of the same coin” see [2].
Let B =(X, P, B), B′ =(X ′, P ′, B′) be balleans. A mapping f : X → X ′
is called a ≺-mapping if, for every α ∈ P , there exists α′ ∈ P ′ such that,
for every x ∈ X, f(B(x, α)) ⊆ B′(f(x), α′).
A bijection f : X → X ′ is called an asymorphism between B and B′ if
f and f−1 are ≺-mappings. In this case B and B′ are called asymorphic.
Given a ballean B = (X, P, B), we define a preodering < on P by the
rule: α < β if and only if B(x, α) ⊆ B(x, β) for each x ∈ X. A subset P ′
of P is called cofinal if, for every α ∈ P , there exists α′ ∈ P ′ such that
α < α′. A ballean B is called ordinal if there exists a cofinal well-ordered
(by <) subset P ′ of P .
For a ballean B = (X, P, B), x, y ∈ X and α ∈ P , we say that x and
y are α-path connected if there exists a finite sequence x0, . . . , xn, x0 = x,
284 The comb-like representations of balleans
xn = y such that xi+1 ∈ B(xi, α) for each i ∈ {0, . . . , n − 1}. For any
x ∈ X and α ∈ P , we set
B⋄(x, α) = {y ∈ X : x, y are α-path connected},
and say that the ballean B⋄ = (X, P, B⋄) is a cellularization of B. A
ballean B is called cellular if the identity id : X → X is an asymorphism
between B and B⋄.
Each metric space (X, d) defines a metric ballean
B(X, d) = (X,R+, Bα),
where Bd(x, r) = {y ∈ X : d(x, y) 6 r}. Clearly, B(X, d) is ordinal and, if
d is an ultrametric then B(X, d) is cellular.
For examples, decompositions and classification of cellular ordinal
balleans see [1], [2], [4], [6].
2. Representations
For ordinals α, β, α < β, we use the standard notations
[α, β] = {t : α 6 t 6 β}, [α, β) = {t : α 6 t < β},
(α, β] = {t : α < t 6 β}.
Let X be a set and γ be an ordinal. We say that a mapping d : X×X →
[0, γ) is a γ-ultrametric if d(x, x) = 0, d(x, y) = d(y, x) and
d(x, y) 6 max{d(x, z), d(z, y)}.
Clearly, each ultrametric space with integer valued metric is an ω-
ultrametric space. By [7, Theorem 3.1.1], every cellular metrizable ballean
is asymorphic to some ω-ultrametric space.
Given two γ-ultrametric spaces (X, d), (X ′, d′), a bijection h : X → X ′
is called an isometry if, for any x, y ∈ X, we have
d(x, y) = d′(h(x), h(y)).
Now let λ, γ be ordinal and f : [0, λ) → [0, γ) be a function such
that, for each α < λ, sup{f(t) : t ∈ [0, α]} < γ. We define a mapping
df : [0, λ) × [0, λ) → [0, γ) by the rule: if x < y then
df (x, y) = df (y, x) = sup{f(t) : t ∈ (x, y]}, d(x, x) = 0,
and say that ([0, λ), df ) is a γ-comb determined by f . Evidently, each
γ-comb is a γ-ultrametric space.
I . V. Protasov, K. D. Protasova 285
Theorem. Every γ-ultrametric space (X, d) is isometric to some γ-comb
([0, λ), df ).
Proof. We proceed on induction by γ. For γ = 1, we just enumerate X
as [0, λ) and take f ≡ 0. Assume that we have proved the statement for
all ordinals less than γ and consider two cases.
Case 1. Let γ is not a limit ordinal, so γ = γ′ + 1. We partition X =⋃
{Xδ : δ ∈ [0, ν)} into classes of the equivalence ∼ defined by x ∼ y if and
only if d(x, y) < γ′. If δ < δ′ < ν and x ∈ Xδ, y ∈ Xδ′ then d(x, y) = γ′.
By the inductive hypothesis, each Xδ is isometric to some γ′-comb
([0, λδ), dfδ
). We replace inductively each δ ∈ [0, ν) with consecutive
intervals {[lδ, lδ +λδ) : δ ∈ [0, ν)}, l0 = 0 and define a function f : [0, λ) →
[0, γ), [0, λ) =
⋃
{[lδ, lδ + λδ) : δ ∈ [0, ν)} as follows. We put f = f0
on [0, λ0). If δ > 0 then we put f(lδ) = γ′ and f(lδ + x) = fδ(x) for
x ∈ (0, λδ).
After |ν| steps, we get the desired γ-comb ([0, λ), df ).
Case 2. γ is a limit ordinal. We fix some x0 ∈ X and, for each δ < γ,
denote Xδ = {x ∈ X : d(x0, x) < δ}. By the inductive hypothesis, there
is an isometry hδ : Xδ → ([0, λδ), dfδ
). Moreover, in view of Case 1, fδ+1
and hδ+1 can be chosen as the extensions of fδ and hδ. Hence, we can use
induction by δ to get the desired γ-comb and isometry.
Every γ-ultrametric space (X, d) can be considered as the ballean
(X, [0, γ), Bd), where Bd(x, α) = {y ∈ X : d(x, y) 6 α}.
On the other hand, let (X, P, B) be a cellular ordinal ballean. We may
suppose that P = [0, γ) and B(x, α) = B⋄(x, α) for all x ∈ X, α ∈ [0, γ).
We define a γ-ultrametric d on X by d(x, y) = min{α ∈ [0, γ) : y ∈
B(x, α)}. Then (X, P, B) is asymorphic to (X, d).
Corollary. Every cellular ordinal ballean is asymorphic to some γ-comb.
References
[1] I. Protasov, T. Banakh, D. Repoš, S. Slobodianiuk Classifying homogeneous cellular
ordinal balleans up to course equivalence, preprint (arXiv:1409.3910).
[2] T. Banakh, D. Repovš, Classifying homogeneous ultrametric spaces up to coarse
equivalence, preprint (arXiv: 1408.4818).
[3] A. Lambert, G. Uribe Bravo, The comb representation of compact ultrametric spaces,
preprint (arXiv: 1602.08246).
[4] I. Protasov, O. Petrenko, S. Slobodianiuk Asymptotic structures of cardinals, Appl.
Gen.Topology 15, N2 (2014), pp.137-146.
286 The comb-like representations of balleans
[5] I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Math.
Stud. Monogr. Ser., Vol.11, VNTL, Lviv, 2003.
[6] I.V. Protasov, A. Tsvietkova, Decomposition of cellular balleans, Topology Proc.
36 (2010), pp.77-83.
[7] I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol.
12, VNTL, Lviv, 2007.
[8] Roe J., Lectures on Coarse Geometry, Univ. Lecture Series, Vol.31, Amer. Math.
Soc, Providence, RI, 2003.
Contact information
I. V. Protasov,
K. D. Protasova
Taras Shevchenko National University of Kyiv,
Department of Cybernetics, Volodymyrska 64,
01033, Kyiv Ukraine
E-Mail(s): i.v.protasov@gmail.com,
k.d.ushakova@gmail.com
Web-page(s): do.unicyb.kiev.ua/index.php
/uk/2014-08-31-19-03-19/38,
is.unicyb.kiev.ua
/uk/staff.protasova.html
Received by the editors: 29.01.2016.
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| id | nasplib_isofts_kiev_ua-123456789-155256 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-24T09:18:50Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Protasov, I. Protasova, K. 2019-06-16T14:51:28Z 2019-06-16T14:51:28Z 2016 The comb-like representations of cellular ordinal balleans / I. Protasov, K. Protasova // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 282-286. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC:54A05, 54E15, 54E30. https://nasplib.isofts.kiev.ua/handle/123456789/155256 Given two ordinal λ and γ, let f:[0,λ)→[0,γ) be a function such that, for each α<γ, sup{f(t):t∈[0,α]}<γ. We define a mapping df:[0,λ)×[0,λ)⟶[0,γ) by the rule: if x<y then df(x,y)=df(y,x)=sup{f(t):t∈(x,y]}, d(x,x)=0. The pair ([0,λ),df) is called a γ−comb defined by f. We show that each cellular ordinal ballean can be represented as a γ−comb. In General Asymptology, cellular ordinal balleans play a part of ultrametric spaces. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics The comb-like representations of cellular ordinal balleans Article published earlier |
| spellingShingle | The comb-like representations of cellular ordinal balleans Protasov, I. Protasova, K. |
| title | The comb-like representations of cellular ordinal balleans |
| title_full | The comb-like representations of cellular ordinal balleans |
| title_fullStr | The comb-like representations of cellular ordinal balleans |
| title_full_unstemmed | The comb-like representations of cellular ordinal balleans |
| title_short | The comb-like representations of cellular ordinal balleans |
| title_sort | comb-like representations of cellular ordinal balleans |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155256 |
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