On free vector balleans
A vector balleans is a vector space over R endowed with a coarse structure in such a way that the vector operations are coarse mappings. We prove that, for every ballean (X, E), there exists the unique free vector ballean V(X, E) and describe the coarse structure of V(X, E). It is shown that normali...
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irk-123456789-1884232023-03-01T01:27:01Z On free vector balleans Protasov, I. Protasova, K. A vector balleans is a vector space over R endowed with a coarse structure in such a way that the vector operations are coarse mappings. We prove that, for every ballean (X, E), there exists the unique free vector ballean V(X, E) and describe the coarse structure of V(X, E). It is shown that normality of V(X, E) is equivalent to metrizability of (X, E). 2019 Article On free vector balleans / I. Protasov, K. Protasova // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 70–74. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC: 46A17, 54E35 http://dspace.nbuv.gov.ua/handle/123456789/188423 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A vector balleans is a vector space over R endowed with a coarse structure in such a way that the vector operations are coarse mappings. We prove that, for every ballean (X, E), there exists the unique free vector ballean V(X, E) and describe the coarse structure of V(X, E). It is shown that normality of V(X, E) is equivalent to metrizability of (X, E). |
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Protasov, I. Protasova, K. |
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Protasov, I. Protasova, K. On free vector balleans Algebra and Discrete Mathematics |
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Protasov, I. Protasova, K. |
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Protasov, I. |
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On free vector balleans |
| title_short |
On free vector balleans |
| title_full |
On free vector balleans |
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On free vector balleans |
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On free vector balleans |
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on free vector balleans |
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Інститут прикладної математики і механіки НАН України |
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2019 |
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On free vector balleans / I. Protasov, K. Protasova // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 70–74. — Бібліогр.: 11 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT protasovi onfreevectorballeans AT protasovak onfreevectorballeans |
| first_indexed |
2025-07-16T10:27:29Z |
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2025-07-16T10:27:29Z |
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1837798942764433408 |
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“adm-n1” — 2019/3/22 — 12:03 — page 70 — #78
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 1, pp. 70–74
c© Journal “Algebra and Discrete Mathematics”
On free vector balleans
Igor Protasov and Ksenia Protasova
Abstract. A vector balleans is a vector space over R en-
dowed with a coarse structure in such a way that the vector opera-
tions are coarse mappings. We prove that, for every ballean (X, E),
there exists the unique free vector ballean V(X, E) and describe the
coarse structure of V(X, E). It is shown that normality of V(X, E)
is equivalent to metrizability of (X, E).
1. Introduction
Let X be a set. A family E of subsets of X × X is called a coarse
structure if
• each ε ∈ E contains the diagonal △X , △X = {(x, x) : x ∈ X};
• if ε, δ ∈ E then ε◦δ∈E and ε−1∈E , where ε◦δ={(x, y) : ∃z((x, z)∈ε,
(z, y) ∈ δ)}, ε−1 = {(y, x) : (x, y) ∈ ε};
• if ε ∈ E and △X ⊆ ε′ ⊆ ε then ε′ ∈ E ;
• for any x, y ∈ X, there exists ε ∈ E such that (x, y) ∈ ε.
A subset E ′ ⊆ E is called a base for E if, for every ε ∈ E , there exists
ε′ ∈ E ′ such that ε ⊆ ε′. For x ∈ X, A ⊆ X and ε ∈ E , we denote
ε[x] = {y ∈ X : (x, y) ∈ ε}, ε[A] = ∪a∈A ε[a] and say that ε[x] and ε[A]
are balls of radius ε around x and A.
The pair (X, E) is called a coarse space [11] or a ballean [8], [10].
Each subset Y ⊆ X defines the subballean (Y, EY ), where EY is the
restriction of E to Y × Y . A subset Y is called bounded if Y ⊆ ε[x] for
some x ∈ X and ε ∈ E .
2010 MSC: 46A17, 54E35.
Key words and phrases: coarse structure, ballean, vector ballean, free vector
ballean.
“adm-n1” — 2019/3/22 — 12:03 — page 71 — #79
I . Protasov, K. Protasova 71
Let (X, E), (X ′, E ′) be balleans. A mapping f : X → X ′ is called
coarse or macro-uniform if, for every ε ∈ E , there exists ε′ ∈ E ′ such that
f(ε[x]) ⊆ ε′[f(x)] for each x ∈ X. If f is a bijection such that f and f−1
are coarse then f is called an asymorphism.
Every metric d on a set X defines the metric ballean (X, Ed), where Ed
has the base {{(x, y) : d(x, y) < r} : r ∈ R
+}. We say that a ballean (X, E)
is metrizable if there exists a metric d on X such that E = Ed. In what
follows, we consider R as a ballean defined by the metric d(x, y) = |x− y|.
Given two balleans (X1, E1), (X2, E2), we define the product
(X1 ×X2, E), where E has the base E1 × E2.
Let V be a vector space over R and let E be a coarse structure on V.
Following [5], we say that (V, E) is a vector ballean if the operations
V × V → V , (x, y) 7→ x+ y and R× V , (λ, x) 7→ λx
are coarse.
A family I of subsets of V is called a vector ideal if
(1) if U,U ′ ∈ I and W ⊆ U then U ∪ U ′ ∈ I and W ∈ I;
(2) for every U ∈ I, U + U ∈ I;
(3) for any U ∈ I and λ ∈ R
+, [−λ, λ]U ∈ I, where [−λ, λ]U =⋃
{λ′U : λ′ ∈ [−λ, λ]};
(4)
⋃
I = V .
A family I ′ ⊆ I is called a base for I if, for each U ∈ I, there is
U ′ ∈ I ′ such that U ⊆ U ′.
If (V, E) is a vector ballean then the family I of all bounded subsets
of V is a vector ideal. On the other hand, every vector ideal I on V
defines the vector ballean (V, E), where E is a coarse structure with
the base {{(x, y) : x − y ∈ U} : U ∈ I}. Thus, we have got a bijective
correspondence between vector balleans on V and vector ideas. Following
this correspondence, we write (V, I) in place of (V, E).
Let (V, I), (V ′, I ′) be vector balleans. We note that a linear mapping
f : V → V ′ is coarse if and only if f(U) ∈ I ′ for each U ∈ I.
In Section 2, we show that, for every ballean (X, E), there exists the
unique vector ideal I(X,E) on the vector space V(X) with the basis X such
that
• (X, E) is a subballean of (V(X), I(X,E));
• for every vector ballean (V, I), every coarse mapping (X, E) → (V, I)
gives rise to the unique coarse linear mapping (V(X), I(X,E)) →
(V, I).
We denote V(X, E) = (V(X), I(X,E)) and say that I(X,E) and V(X, E)
are free vector ideal and free vector ballean over (X, E).
“adm-n1” — 2019/3/22 — 12:03 — page 72 — #80
72 On free vector balleans
Free vector balleans can be considered as counterparts of free vector
spaces studied in many papers, for examples, [1], [3], [4]. It should be
mentioned that the free activity in topological algebra was initiated by
the famous paper of Markov on free topological groups [6]. For free coarse
groups see [9].
2. Construction
Given a ballean (X, E), we consider X as the basis of the vector space
V(X). For each ε ∈ E and n ∈ N, we set Dε = {x − y : (x, y) ∈ ε} and
denote by Sn,ε the sum of n copies of [−n, n]Dε.
Theorem 1. Let (X, E) be a ballean and let z ∈ X. Then the family
{Sn,ε + [−n, n]z : ε ∈ E , n ∈ N}
is a base of the free vector ideal I(X,E).
Proof. We denote by I the family of all subsets U of V(X) such that U is
contained in some Sn,ε+[−n, n]z. Clearly, I satisfies (1), (2), (3) from the
definition of a vector ideal. To see that
⋃
I = V(X), we take an arbitrary
y ∈ X and choose ε ∈ E so that (y, z) ∈ ε. Then y = (y− z) + z ∈ DE + z.
In view of (2), (3), we conclude that
⋃
I = V(X).
To show that (X, E) is a subballean of (V(X), I), we denote by E ′ the
coarse structure of the ballean (V(X), I). Since DE ∈ I for each ε ∈ E ,
E = E−1 we have E ⊆ E ′|X . To verify the inclusion E ′ |X⊆ E we take
x, y ∈ X, assume that x− y ∈ Sn,E + [−n, n]z and show that (x, y) ∈ εn.
We write x− y = λ1(x1 − y1) + . . .+ λn(xn − yn) + λn+1z, (xi, yi) ∈ ε,
λi ∈ [−n, n]. Since (λ1−λ1)+. . .+(λn−λn)+λn+1 = 0, we have λn+1 = 0
so x− y = λ1(x1− y1)+ . . .+λn(xn− yn). If (x1, y1), . . . , xn, yn) ∈ {x, y}
then the statement is evident because (xi, yi) ∈ ε. Assume that there
exists a ∈ {x1, y1, . . . , xn, yn} such that a /∈ {x, y}. We take all items
λi(xi − yi), i ∈ I such that a ∈ {xi, yi}, and denote by s the sum of
all these items. The coefficient before a in the canonical decomposition
of s by the basis X must be 0. We take λk(xk − yk), k ∈ I. If xk = a
or yk = a then we replace each a in λi(xi − yi), i ∈ I, to yk or xk
respectively. Then we get x − y = λ1(x
′
1 − y′1) + . . . + λn(x
′
n − y′n),
a /∈ {x′1, y
′
1, . . . , x
′
n − y′n} and (x′i, y
′
i) ∈ ε2. Repeating this trick, we run
into the case x1, y1, . . . , xn, yn ∈ {x, y} and (xi, yi) ∈ εn.
To conclude the proof, we observe that I is the minimal vector ideal on
V(X) such that (X, E) is a subballean of (V(X), I). If (V, I ′) is a ballean
“adm-n1” — 2019/3/22 — 12:03 — page 73 — #81
I . Protasov, K. Protasova 73
and f : (X, E) → (V, I ′) is a coarse mapping then the linear extension
h : (V(X), I) → (V, I ′) of f is coarse because h−1(I ′) is a vector ideal on
V(X) and I ⊆ h−1(I ′).
3. Metrizability and normality
Theorem 2. A ballean V(X, E) is metrizable if and only if (X, E) is
metrizable.
Proof. By [10, Theorem 2.1.1], (X, E) is metrizable if and only if E has a
countable base. Apply Theorem 1.
Let (X, E) be a ballean,A ⊆ X. A subset U of X is called an asymptotic
neighbourhood of A if, for every ε ∈ E , the set ε[A]\U is bounded.
Two subsets A,B of X are called asymptotically disjoint (asymptotically
separated) if, for every ε ∈ E , the intersection ε[A]∩ ε[B] is bounded (A,B
have disjoint asymptotic neighbourhoods).
A ballean (X, E) is called normal [7] if any two asymptotically disjoint
subsets of X are asymptotically separated. Every metrizable ballean is
normal.
Given an arbitrary ballean (X, E), the family BX of all bounded subsets
of X is called a bornology of (X, E). A subfamily B′ ⊆ BX is called a base
of BX if, for every B ∈ BX , there exists B′ ∈ BX such that B ⊆ B′. The
minimal cardinality of bases of BX is denoted by cof BX .
Theorem 3. For every ballean (X, E), the free vector ballean V(X, E) is
normal if and only if V(X, E) is metrizable.
Proof. For |X| = 1, the statement is evident. Let |X| > 1, a ∈ X, L = Ra,
Y = X \ {a}. Applying Theorem 1, we conclude that the canonical
isomorphism between V(X, E) and L × V(Y, EY ) is an asymorphism. If
V(X, E) is normal then, by Theorem 1.4 from [2], cof BL = cof BZ , where
Z = V(Y, EY ). Since cof BL = ℵ0, BZ has a countable base. To conclude
the proof, it suffices to note that BZ is the vector ideal such that V(Y, EY ) =
(V(Y ), I). Hence I has a countable base and V(Y, EY ) is metrizable by
Theorem 2.1.1. from [10].
References
[1] T. Banakh, A. Leiderman, ωω-dominated function spaces and ω
ω-bases in free
objects in topological algebra, Topology Appl. 241 (2018), 203-241.
[2] T. Banakh, I. Protasov, The normality and bounded growth of balleans, arXiv:
1810.07979.
“adm-n1” — 2019/3/22 — 12:03 — page 74 — #82
74 On free vector balleans
[3] S. S. Gabriyelyan, A description of the topology of free vector spaces, arXiv: 1804.
05199.
[4] S. S. Gabriyelyan, S. A. Morris, Free topological vector space, Topology Appl. 223
(2017), 30-49.
[5] Ie. Lutsenko, I.V. Protasov, Sketch of vector balleans, Math. Stud. 31 (2009),
219-224.
[6] A. A. Markov, On free topological groups, Izv. Acad. Nauk SSSR 9 (1945), 3-64.
English translation: Amer. Math. Soc. Transl. 30 (1950), 11-88.
[7] I.V. Protasov, Normal ball structures, Math. Stud. 20 (2003), 3-16.
[8] I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs,
Math. Stud. Monogr. Ser., Vol. 11, VNTL, Lviv, 2003.
[9] I. Protasov, K. Protasova Free coarse groups, J. Group Theory, DOI:
https://doi.org/10.1515/jgth-2019-2048; arXiv:1803.10504.
[10] I. Protasov, M. Zarichnyi, General Asymptopogy, Math. Stud. Monogr. Vol. 12,
VNTL, Lviv, 2007.
[11] J. Roe, Lectures on Coarse Geometry, AMS University Lecture Ser. 31, Providence,
RI, 2003.
Contact information
I. Protasov,
K. Protasova
Department of Computer Science and
Cybernetics, Kyiv University, Volodymyrska 64,
01033, Kyiv, Ukraine
E-Mail(s): i.v.protasov@gmail.com,
ksuha@freenet.com.ua
Received by the editors: 10.03.2019.
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