Relatively thin and sparse subsets of groups
Let G be a group with identity e and let I be a left-invariant ideal in the Boolean algebra PG of all subsets of G. A subset A of G is called I-thin if gA⋂A∈I for every g∈G {e}. A subset A of G is called I-sparse if, for every infinite subset S of G, there exists a finite subset F⊂S such that ⋂g∈FgA...
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| Цитувати: | Relatively thin and sparse subsets of groups / Ie. Lutsenko, I.V. Protasov // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 216–225. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860007276148948992 |
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| author | Lutsenko, Ie. Protasov, I.V. |
| author_facet | Lutsenko, Ie. Protasov, I.V. |
| citation_txt | Relatively thin and sparse subsets of groups / Ie. Lutsenko, I.V. Protasov // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 216–225. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
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| description | Let G be a group with identity e and let I be a left-invariant ideal in the Boolean algebra PG of all subsets of G. A subset A of G is called I-thin if gA⋂A∈I for every g∈G {e}. A subset A of G is called I-sparse if, for every infinite subset S of G, there exists a finite subset F⊂S such that ⋂g∈FgA∈F. An ideal I is said to be thin-complete (sparse-complete) if every I-thin (I-sparse) subset of G belongs to I. We define and describe the thin-completion and the sparse-completion of an ideal in PG.
Припустимо, що G — група з одиницею e, I — iнварiантний злiва iдеал в булевiй алгебрi PG всiх пiдмножин групи G. Пiдмножина A групи G називається I-тонкою, якщо gA⋂A∈I для кожного g∈G {e}. Пiдмножина A групи G називається P-розрiдженою, якщо для кожної нескiнченної множини S групи G iснує скiнченна пiдмножина F⊂S така, що ⋂g∈FgA∈F. Говорять, що iдеал I тонко-повний (розрiджено-повний), якщо кожна I-тонка (I-розрiджена) множина групи G належить I. Визначено та описано тонке та розрiджене доповнення iдеалу в PG.
|
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UDC 517.5
Ie. Lutsenko, I. V. Protasov (Kyiv Nat. Taras Shevchenko Univ.)
RELATIVELY THIN AND SPARSE SUBSETS OF GROUPS
ВIДНОСНО ТОНКI ТА РОЗРIДЖЕНI ПIДМНОЖИНИ ГРУП
Let G be a group with the identity e, I be a left-invariant ideal in the Boolean algebra PG of all subsets of G.
A subset A of G is called I-thin if gA ∩ A ∈ I for every g ∈ G \ {e}. A subset A of G is called I-sparse
if, for every infinite subset S of G, there exists a finite subset F ⊂ S such that
⋂
g∈F gA ∈ F . An ideal I
is said to be thin-complete (sparse-complete) if every I-thin (I-sparse) subset of G belongs to I. We define
and describe the thin-completion and the sparse-completion of an ideal in PG.
Припустимо, що G — група з одиницею e, I — iнварiантний злiва iдеал в булевiй алгебрi PG всiх
пiдмножин групи G. Пiдмножина A групи G називається I-тонкою, якщо gA ∩ A ∈ I для кожного
g ∈ G \ {e}. Пiдмножина A групи G називається I-розрiдженою, якщо для кожної нескiнченної
множини S групи G iснує скiнченна пiдмножина F ⊂ S така, що
⋂
g∈F gA ∈ F . Говорять, що iдеал
I тонко-повний (розрiджено-повний), якщо кожна I-тонка (I-розрiджена) множина групи G належить
I. Визначено та описано тонке та розрiджене доповнення iдеалу в PG.
Let G be a group with the identity e, PG be the Boolean algebra of all subsets of G. A
family F of subsets of G is called
left-invariant if gF ∈ F for all g ∈ G and F ∈ F ;
downward closed if E ⊆ F and F ∈ F implies E ∈ F ;
additive if E ∪ F for all subsets E,F ∈ F ;
an ideal if F is downward closed and additive.
The family FG of all finite subsets of G is a left-invariant ideal of PG.
Given a left-invariant ideal I in PG, we classify the subsets of G by their size with
respect to I.
A subset A ⊆ G is said to be
I-large if there exist F ∈ FG and I ∈ I such that G = FA ∪ I;
I-small if L \A is I-large for every I-large subset L;
I-thick if L ∩A 6= ∅ for every I-large subset L;
I-thin if A ∩ gA ∈ I for every g ∈ G \ {e}.
I-sparse if each infinite set S ⊂ G contains a finite subset F ⊂ S with
⋂
g∈F gA ∈ I.
For the smallest ideal I∅ = {∅}, I∅-large, I∅-small and I∅-thick sets turn into
large, small and thick subsets which have been intensively studied last time (see the
survey [1]). On the other hand, FG-thin and FG-sparse subsets are known as thin and
sparse sets, see [2]. The I-large and I-small subsets appeared in [3]. For every left-
invariant ideal I, the family S(I) of all I-small subsets of G is a left-invariant ideal
containing I.
The paper consists of two sections. In the first section we study the thin-extension
τ(I) and the thin-completion τ∗(I) of the ideal I in PG. In the second section we study
the sparse-extension σ(I) and the sparse-completion σ∗(I) of I.
1. Relatively thin subsets of groups.
Proposition 1. For a subset A of a group G and I be a left-invariant ideal in
PG, the following statements hold:
(1) A is I-small if and only if G \ FA is I-large for every F ∈ FG;
(2) A is I-thick if and only if for any F ∈ FG and I ∈ I there exists x ∈ G such
that Fx ⊆ A \ I;
(3) A is not I-small if and only if there exists F ∈ FG such that FA is I-thick;
c© IE. LUTSENKO, I. V. PROTASOV, 2011
216 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
RELATIVELY THIN AND SPARSE SUBSETS OF GROUPS 217
(4) if A is I-thick, then for every F ∈ FG the set {g ∈ A : Fg ⊆ A} is I-thick.
Proof. 1. Theorem 2.1 from [3].
2. We suppose that A is I-thick and take F ∈ FG and I ∈ I. If Fx * A \ I for
every x ∈ G, then G = F−1(G \ (A \ I)) and thus the set L = G \ (A \ I) is I-large
and so is the set L \ I. Since A ∩ (L \ I) = L ∩ (A \ I) = ∅, the set A is not I-thick,
which is a contradiction.
If A is not I-thick, then L ∩ A = ∅ for some I-large subset L. Find I ∈ I and
F ∈ FG such that G = F (L ∪ I). Then for each x ∈ G the set F−1x meets L ∪ I and
hence cannot lie in A \ I ⊂ G \ (L ∪ I).
3. By (1), A is not I-small if and only if there exists F ∈ FG such that G \ FA is
not I-large. On the other hand, G \FA is not I-large if and only if for each I-large set
L ⊂ G we get L 6⊂ G \ FA, which is equivalent to L ∩ FA 6= ∅.
4. We fix F ∈ FG, e ∈ F and put B = {g ∈ A : Fg ⊆ A}. Then we take an arbitrary
H ∈ FG, e ∈ H. Given any I ∈ I, there exists a ∈ A \ I such that FHa ⊆ A \ I. By
the definition of B, Ha ⊆ B so Ha ⊆ B \ I and B is I-thick.
Proposition 1 is proved.
Let F be a left-invariant downward closed family of subsets of a group G. A subset
A ⊆ G is called F-thin if gA ∩A ∈ F for every g ∈ G \ {e}. The family of all F-thin
subsets of G is denoted by τ(F). The definition implies that τ(F) is left-invariant,
downward closed and F ⊆ τ(F). If F = τ(F), then the family F is called thin-
complete. The intersection τ∗(F) of all thin-complete families that contain F is called
the thin-completion of F . The thin-completion τ∗(F) contains the subfamilies τα(F)
defined by transfinite induction:
τ0(F) = F and τα(F) = τ(τ<α(F)), where τ<α(F) =
⋃
β<α
τβ(F)
for each ordinal α.
The families τn(F) for n ∈ ω admit a simple characterization:
Proposition 2. Let F ⊂ PG be a left-invariant downward closed family of subsets
of a group G and n ∈ ω. A subset A ⊂ G belongs to the family τn(F) if and only if⋂
i0,...,in∈{0,1}
gi00 . . . ginn A ∈ F
for any elements g0, . . . , gn ∈ G \ {e}.
Proof. For n = 0, the statement follows from the left-invariance of F . Assume that
the proposition is true for some n ∈ ω. Then A ∈ τn+1(F) if and only if A ∩ gn+1A ∈
∈ τn(F) for each gn+1 ∈ G, gn+1 6= e. By the inductive hypothesis
A ∩ gn+1A ∈ τn(F) ⇔
⋂
i0,...,in∈{0,1}
gi00 . . . ginn (A ∩ gn+1A) ∈ F
for all g0, . . . , gn ∈ G \ {e}, which is equivalent to⋂
i0,...,in+1∈{0,1}
gi00 . . . g
in+1
n+1A ∈ F .
Proposition 2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
218 IE. LUTSENKO, I. V. PROTASOV
Remark 1. In [4] T. Banakh and N. Lyaskovska (answering a problem posed in a
preliminary version of this paper) proved that a subset A of a group G belongs to the
family τ∗(F) if and only if for each sequence (gn)n∈ω ∈ (G \ {e})ω there is n ∈ ω
such that ⋂
i0,...,in∈{0,1}
gi00 . . . ginn A ∈ F .
Next, we describe the structure of the thin-completion τ∗(F).
Proposition 3. If G is a group of cardinality κ = |G| and F ⊂ PG is a left-
invariant downward closed family of subsets of G, then
τ∗(F) =
⋃
α<κ+
τα(F).
Proof. Clearly, τ<κ
+
(F) ⊆ τ∗(F). So, it suffices to show that each set A ∈ τκ+
(F)
belongs to τ<κ
+
(F).
First we consider the case of infinite cardinal κ = |G|. For any A ∈ τκ
+
(F)
and x ∈ G \ {e}, we get A ∩ xA ∈ τ<κ
+
(F) and hence A ∩ xA ∈ ταx(F) for
some ordinal αx < κ+. Let α = sup{αx : x ∈ G \ {e}} < κ+ and observe that
A ∩ xA ∈ ταx(F) ⊂ τα(F) for all x ∈ G \ {e} and thus A ∈ τα+1(F) ⊂ τ<κ+
(F).
Now consider the case of finite κ. In this case τ<κ
+
(F) = τκ(F). By Proposition 2,
the inclusion A ∈ τκ(F) will follow as soon as we check that for any elements
g0, . . . , gκ ∈ G \ {e} ⋂
i0,...,iκ∈{0,1}
gi00 . . . giκκ A ∈ F .
Define a sequence of subsets (Cn)κn=0 letting C0 = {e, g0} and Cn+1 = Cn · {e, gn+1}
for n < κ. Since Cκ \ C0 =
⋃
1≤n≤κ Cn \ Cn−1 and |Cκ \ C0| ≤ |G \ C0| = κ − 2,
there is a positive number n ≤ κ such that Cn−1 = Cn. For this number we get
Cn · {e, gn} = Cn−1 · {e, gn} = Cn = Cn−1.
Now consider the sequence h0, . . . , hκ, hκ+ defined by hi = gi if i ≤ n and hi = gi−1
if n < i ≤ κ+. This sequence induces a sequence of sets (Di)i≤κ+ defined inductively
by D0 = {e, h0} and Di = Di−1 · {e, hi} for 0 < i ≤ κ+. It follows that Di = Ci for
i ≤ n and Di = Ci−1 for n < i ≤ κ+. In particular, Dκ+ = Cκ. Since A ∈ τκ+
(F),
we get the required inclusion⋂
i0,...,iκ∈{0,1}
gi00 . . . giκκ A =
⋂
g∈Cκ
gA =
⋂
g∈Dκ+
gA =
⋂
i0,...,iκ+∈{0,1}
hi00 . . . h
iκ+
κ+ A ∈ F
implying A ∈ τκ(F) = τ<κ
+
(F).
Proposition 3 is proved.
Remark 2. In general the ordinal κ+ in Proposition 3 cannot be replaced by a
smaller ordinal: by [4], for a group G containing an element of infinite order, we get
τ∗(FG) 6= τα(FG) for each countable ordinal α.
In Boolean groups the situation is totally different. By a Boolean group we understand
a group G such that x2 = e for all x ∈ G. Let [G]≤n = {A ⊂ G : |A| ≤ n}.
Theorem 1. For a group G, the following statements hold:
(1) G is Boolean if and only if τ∗(I∅) = τ(I∅) = [G]≤1;
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
RELATIVELY THIN AND SPARSE SUBSETS OF GROUPS 219
(2) if G is Boolean, then τ∗(FG) = τ(FG);
(3) if G is infinite and τ∗(FG) = τ(FG), then G is Boolean;
(4) if I is a left invariant ideal, G has no elements of order 2 and T1, T2 ∈ τ(I),
then T1 ∪ T2 ∈ τ2(I).
Proof. 1. For every group G, τ(I∅) = [G]≤1. Let G be a Boolean group, A ∈
∈ P(G), |A| > 1, a, b ∈ A, a 6= b, g = ab−1. Then {a, b} ⊆ gA ∩ A so A /∈ τ([G]≤1)
and τ∗(I∅) = [G]≤1.
On the other hand, assume that G has an element a such that a2 6= e. We put
A = {e, a} and note that |gA ∩ A| 6 1 for every g ∈ G, g 6= e. It follows that
A ∈ τ([G]≤1) so τ∗(I∅) 6= [G]≤1.
2. We take an arbitrary subset A ∈ τ2(FG). By Proposition 2(2),
A ∩ gA ∩ fA ∩ fgA ∈ FG
for all f, g ∈ G \ {e}. We put f = g and get A ∩ gA ∈ FG so A ∈ τ(FG) and
τ∗(FG) = τ(FG).
3. We suppose the contrary, choose an element g ∈ G such that g2 6= e and construct
a subset A ∈ τ2(FG) \ τ(FG). Assume that G is countable, G = {gn : n < ω}, g0 = e
and put Gn = {gi : i 6 n}. We put x0 = e and choose inductively a sequence (xn)n∈ω
of elements of G such that, for every n < ω,
(Gn ∪ {g, g−1}){xn+1, gxn+1} ∩ (Gn ∪ {g, g−1}){x0, gx0, . . . , xn, gxn} = ∅.
We consider the set A = {xn, gxn : n ∈ ω} and observe that
gA ∩A = {gxn : n ∈ ω}, g−1A ∩A = {xn : n ∈ ω}.
By the choice of (xn)n∈ω, gA∩A ∈ τ(FG), g−1A∩A ∈ τ(FG). If f ∈ G\{g, g−1, e}
then fA ∩ A is finite. Hence A ∈ τ2(FG). Since gA ∩ A is infinite, A /∈ τ(FG) so
A ∈ τ2(FG) \ τ(FG).
If G is uncountable, we choose a countable subgroup G′ of G containing g and
repeat the construction of A inside G′.
4. Assuming the converse, we put X = T1 ∪ T2. By Proposition 2(2), there exist
g, f ∈ G \ {e} such that X ∩ gX ∩ fX ∩ gfX /∈ I. We observe that
X ∩ gX ∩ fX ∩ fgX =
⋃
i,j,k,l∈{1,2}
(Ti ∩ gTj ∩ fTk ∩ fgTl).
We choose i, j, k, l ∈ {1, 2} such that Ti ∩ gTj ∩ fTk ∩ fgTl /∈ I. Without loss of
generality, i = 1. Since T1 ∈ τ(I), we get j = k = 2. Since T2 ∈ τ(I), we get, g = f.
Since G has no elements of order 2, we have fg 6= e. Thus, T1 ∩ fgT1 ∈ I and we get
a contradiction.
Theorem 1 is proved.
Remark 3. Let G be an infinite Boolean group. By Theorem 1(2), τ∗(FG) =
= τ(FG). We take any infinite thin subset A and x ∈ G \ {e}. Then the union A ∪ xA
is not thin because (A ∪ xA) ∩ x(A ∪ xA) ⊇ xA is infinite. Consequently, the family
τ∗(FG) is not additive and τ∗(FG) is not an ideal.
In contrast, for every left-invariant ideal F in a torsion-free group G the family
τ<α(F) is a left-invariant ideal for each limit ordinal α, see [4]. In particular, the family
τ∗(F) is an ideal in PG.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
220 IE. LUTSENKO, I. V. PROTASOV
Theorem 2. Let G be an infinite group and I be a left-invariant ideal in PG.
Then τ(I) ⊆ S(I), where S(I) is the ideal of all I-small subsets of G.
Proof. We suppose the contrary and fix A ∈ τ(I) such that A /∈ S(I). Since A
is not I-small, by Proposition 1(3), there exists F ∈ FG such that FA is I-thick. Let
F = {f1, . . . , fn}. Since G is infinite, G \ F−1F 6= ∅. We choose h ∈ G \ F−1F and
put
Aij = A ∩ f−1j hfiA, i, j ∈ {1, . . . , n}.
Taking into account that A ∈ τ(I) and f−1j hfi 6= e, we conclude that Aij ∈ I. We
put B = {x ∈ FA : hx ∈ FA}. By Proposition 1(4), B is I-thick. Given any x ∈ B,
we choose fi, fj and a, b ∈ A such that x = fia, hx = fjb so fia = h−1fjb and
b = f−1j hfia. Hence, a ∈ Aij and x ∈ FAij . It follows that B ⊆
⋃n
i,j=1 FAij so
B ∈ I, which is impossible because B is I-thick.
Theorem 2 is proved.
Corollary 1. Let G be an infinite group, I be a left-invariant ideal in PG. Then
the ideal S(I) is thin-complete.
Proof. Applying Theorem 2 to S(I), we get S(I) ⊆ τ(S(I)) ⊆ S(S(I)). To verify
that S(S(I)) ⊆ S(I), we show that every S(I)-large subset L is I-large. We choose
F ∈ FG and S ∈ S(I) such that G = FL ∪ S. By Proposition 1(1), G \ S is I-large.
Hence, there exist H ∈ FG and I ∈ I such that G = H(G \ S) ∪ I = HFL ∪ I, so L
is I-large.
Corollary 1 is proved.
For every groupG, τ(I∅) coincides with the family [G]≤1 of all at most one-element
subsets. If G is finite, then S(I∅) = {∅}. Thus, Corollary 1 is not true for finite groups.
2. Relatively sparse subsets of groups. Let G be an infinite group and F be a
subfamily in PG. We say that a subset A ⊆ G is F-sparse if, for any infinite subset S
of G, there exists a finite subset F ⊂ S such that⋂
x∈F
xA ∈ F .
We denote by σ(F) the family of all F-sparse subsets of G. If F is left invariant
and downward closed then so is σ(F). Repeating the arguments from [2, p. 494, 495],
the reader can verify that σ(F) is a left invariant ideal provided that F is a left invariant
ideal. Alternatively, this statement can be derived from Theorem 3, see Corollary 2.
Now we need some information on the algebraic structure of the Stone – Čech
compactification βG of a discrete group G. We take βG to be the set of all ultrafilters
on G identifying G with the set of all principal ultrafilters. The topology of can be
described by stating that the sets {A : A ⊆ G} form a base for open sets in βG
where A = {p ∈ βG : A ∈ p}. The set G∗ = βG \ G of all free ultrafilters on G
is closed in βG and the family {A∗ : A ⊆ G} is a base for open sets in G∗ where
A∗ = {p ∈ G∗ : A ∈ p}.
Using the universal property of the Stone-Čech compactifications, the multiplication
on G can be extended to the semigroup operation on βG in such a way that all mappings
x → gx, g ∈ G and x → xp, p ∈ βG from βG to βG are continuous. Given any
q, p ∈ βG and A ⊆ G, the product qp is defined by the rule:
A ∈ qp⇔ {x ∈ G : x−1A ∈ q} ∈ p.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
RELATIVELY THIN AND SPARSE SUBSETS OF GROUPS 221
For the structure of the compact right-topological semigroup βG and its combinatorial
applications, see [5, 6].
For a family F of subsets of a group G, we put
F∧ =
{
q ∈ βG : G \A ∈ q for any A ∈ F
}
and note that ∧ is a bijection between the family of all left-invariant ideals of PG and
the family of all closed left ideals of βG. For more information on this correspondence,
see [2, 7].
Given an ultrafilter p ∈ G∗, we say that a subset A of G is (F , p)-sparse if for
any P ∈ p there exists a finite subset F ⊂ P such that
⋂
x∈F
x−1A ∈ F . We denote by
σ(F , p) the family of all (F , p)-sparse subsets of G. Clearly,
σ(F) =
⋂
p∈G∗
σ(F , p).
Theorem 3. Let F be a left-invariant ideal in PG, A ⊆ G, p ∈ G∗. Then
(1) A ∈ σ(F , p) if and only if A∗ ∩ pF∧ = ∅;
(2) (σ(F))∧ = cl(G∗F∧).
Proof. (1) A /∈ σ(F , p)⇔ ∃P ∈ p ∀F ⊂ P F ∈ FG :
⋂
x∈F x
−1A /∈ F ⇔
∃P ∈ p ∃q ∈ F∧ ∀x ∈ P : x−1A ∈ q ⇔
⇔ ∃q ∈ F∧ : A ∈ pq ⇔ A∗ ∩ pF∧ 6= ∅.
(2) q ∈ (σ(F))∧ ⇔ ∀Q ∈ q : G \ Q ∈ σ(F) ⇔ ∀Q ∈ q ∀p ∈ G∗ : G \ Q ∈
∈ σ(F , p) ⇔(1) ∀Q ∈ q ∀p ∈ G∗ : (G \Q)∗ ∩ pF∧ = ∅ ⇔ ∀Q ∈ q : (G \Q)∗ ∩
∩G∗F∧ = ∅⇔ ∀Q ∈ q : (G \Q)∗ ∩ cl(G∗F∧) = ∅ ⇔ q ∈ cl(G∗F∧).
Corollary 2. Let F be a left-invariant ideal in PG. Then
(1) σ(F) is a left-invariant ideal in PG;
(2) if F∧ is a right ideal in βG then (σ(F))∧ is a right ideal.
Proof. We note that a closure of an arbitrary left (right) ideal of βG is a left
(right) ideal, see Theorems 2.15 and 2.17 in [5]. Then both statements follow from
Theorem 3(2).
We say that a left-invariant ideal F in PG is sparse-complete if σ(F) = F (or
equivalently, by Theorem 3(2), F∧ = cl(G∗F∧)) and denote by σ∗(F) the intersection
of all sparse-complete ideals containing F . Clearly, the sparse-completion σ∗(F) is the
smallest sparse-complete ideal such that F ⊆ σ∗(F).
We define also a sequence (σn(F))n∈ω of ideals by recursion: σ0(F) = F ,
σn+1(F) = σ(σn(F)) for n ∈ ω.
Theorem 4. Let G be an infinite group, F be a left invariant ideal in PG. Then
(1) σ∗(F) =
⋃
n∈ω σ
n(F);
(2) σn+1(FG) 6= σn(FG) for every n ∈ ω.
Proof. 1. Clearly,
⋃
n∈ω σ
n(F) ⊆ σ∗(F). On the other hand,(
σ
( ⋃
n∈ω
σn(F)
))∧
= cl
(
G∗
(⋃
n∈ω
σn(F)
)∧)
=
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
222 IE. LUTSENKO, I. V. PROTASOV
= cl
(
G∗
⋂
n∈ω
(σn(F))∧
)
⊆ cl
(⋂
n∈ω
G∗(σn(F))∧
)
⊆
⊆ cl
(⋂
n∈ω
(σn+1(F))∧
)
⊆ cl
(⋂
n∈ω
(σn(F))∧
)
=
=
⋂
n∈ω
(σn(F))∧ =
(⋃
n∈ω
σn(F)
)∧
,
so
⋃
n∈ω σ
n(F) is sparse-complete and σ∗(F) ⊆
⋃
n∈ω σ
n(F).
2. We suppose that G is countable, G = {gn : n ∈ ω}, Fn = {gi : i 6 n}. For
n = 0, we take an arbitrary countable thin subset T and note that T ∈ σ(FG) \ FG.
For n > 0, it suffices to choose an injective sequence (xm)x∈ω in G and a decreasing
sequence (Xm)m∈ω of subsets of G such that
(1)n FmxmXm ∩ FkxkXk = ∅ for all m < k < ω;
(2)n gxmXm ∩ xmXm = ∅ for all g ∈ Fm \ {e} and m < ω;
(3)n Xm ∈ σn(FG) \ σn−1(FG) for each m < ω.
Indeed, we put Qn =
⋃
m∈ω xmXm. Let S = {x−1m : m ∈ ω}, F be a finite subset
of S. Then
⋂
z∈F zQn contains some subset Xm and, by (3)n, Qn /∈ σn(FG). On the
other hand, by (1)n, (2)n and (3)n, Qn ∈ σn+1(FG). Thus, Qn ∈ σn+1(FG)\σn(FG).
We show only how to construct Q1 and Q2.
To satisfy (1)1, (2)1, (3)1, we choose inductively two injective sequences (xm)m∈ω,
(ym)m∈ω in G (xm after ym) such that for every N ∈ ω and all m < k 6 N,
g ∈ Fm \ {e}:
Fmxm{yi : m < i 6 N} ∩ Fkxk{yi : k < i 6 N} = ∅,
gxm{yi : m < i 6 N} ∩ xm{yi : m < i 6 N} = ∅,
Fmym ∩ Fkyk = ∅.
Then we put Xm = {yi : m < i < ω}, and note that (xm)m∈ω, and (Xm)m∈ω
satisfy (1)1, (2)1, (3)1.
To satisfy (1)2, (2)2, (3)2,we choose inductively three injective sequences (xm)m∈ω,
(ym)m∈ω, (zm)m∈ω in G (ym after zm, xm after ym) such that for every N ∈ ω and all
m < k 6 N, g ∈ Fm \ {e}:
Fmxm{yizj : m < i < j 6 N} ∩ Fkxk{yizj : k < i < j 6 N} = ∅,
gxm{yizj : m < i < j 6 N} ∩ xm{yizj : m < i < j 6 N} = ∅,
Fmym{zi : m < i 6 N} ∩ Fkyk{zi : k < i 6 N} = ∅,
gym{zi : m < i 6 N} ∩ ym{zi : m < i 6 N} = ∅,
Fmzm ∩ Fkzk = ∅.
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RELATIVELY THIN AND SPARSE SUBSETS OF GROUPS 223
Then we put Xm = {yizj : m < i < j < ω}, and note that (xm)m∈ω and (Xm)m∈ω
satisfy (1)2, (2)2, (3)2.
If G is uncountable, we fix some countable subgroup of G and, for each n ∈ ω, pick
Q ⊆ H, Q ∈ (σn+1(FH)∩PH) \ (σn(FH)∩PH). Clearly, Q ∈ σn+1(FG) \ σn(FG).
Theorem 4 is proved.
Theorem 5. For a left-invariant ideal F of subsets of an infinite group G, we
have
(1) τ(F) ⊆ σ(F);
(2) if G is torsion-free, then σ(F) ⊆ τ∗(F);
(3) if G is Abelian and {g ∈ G : g2 = e} is finite, then τ2(FG) * σ(FG).
Proof. 1. The inclusion τ(F) ⊆ σ(F) follows from the definitions.
2. Assume that the group G is torsion-free and let A ⊆ G be F-sparse. According to
a characterization of τ∗(F) proved in [4] and mentioned in Remark 1, in order to prove
that A ∈ τ∗(F) we need to show that for each sequence (gn)n∈ω ∈ (G \ {e})ω there is
n ∈ ω such that
(4)
⋂
i0,...,in∈{0,1}
gi00 . . . ginn A ∈ F .
It follows from the torsion-free property of G that the set
C = {gi00 . . . ginn : n ∈ ω, i0, . . . , in ∈ {0, 1}}
is infinite. Since A is F-sparse, there is a finite subset F ⊂ C such that
⋂
x∈F xA ∈ F .
For this set F we can find n ∈ ω such that
F ⊂ {gi00 . . . ginn : i0, . . . , in ∈ {0, 1}}
and conclude that (4) holds.
3. First, we consider the case of countable groupG. Suppose that we have constructed
an infinite subset X of G such that
(5) ∀a, b, c ∈ X, a 6= b 6= c⇒ ab−1c /∈ X.
We choose a sequence (Fn)n∈ω of finite subsets of X such that each finite subset of
X appears in (Fn)n∈ω infinitely many times. We enumerate G = {gn : n < ω}, g0 = e,
and put Gn = {gi : i 6 n}. We put a0 = e and choose inductively a sequence (an)n∈ω
in G such that, for all n ∈ ω,
(6) Gn+1an+1Fn+1 ∩Gn+1(a0F0 ∪ . . . ∪ anFn) = ∅.
We claim that the set A =
⋃
n∈ω anFn belongs to τ2(FG) \ σ∗(FG).
Assuming that A ∈ σ∗(FG), we can find a finite subset F−1 of X−1 such that⋂
g∈F−1 gA = ∅. Since the set F appears in (Fn)n∈ω infinitely often, the intersection⋂
g∈F−1 gA contains infinitely many members of the injective sequence (an)n∈ω, so we
get a contradiction.
The inclusion A ∈ τ2(FG) will follow from Proposition 2(2) as soon as we show
that
|A ∩ gA ∩ fA ∩ gfA| <∞
for all g, f ∈ G\{e}. Suppose the contrary and choose corresponding g, f. By (6), there
exists n ∈ ω such that
anFn ∩ ganFn ∩ fanFn ∩ gfanFn 6= ∅.
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224 IE. LUTSENKO, I. V. PROTASOV
We pick t ∈ Fn such that
gt ∈ Fn, ft ∈ Fn, gft ∈ Fn,
so t, tg, tf, tfg ∈ X. But tf(t−1)tg = tfg and, by (5), tfg /∈ X.
If G is uncountable, we take a countable subgroup G′ and construct A inside G′.
To complete the proof, we construct X as a union X =
⋃
n∈ωXn of an increasing
sequence of finite subsets {Xn : n ∈ ω}, |Xn| = n. We put X0 = ∅ and assume that,
for some n ∈ ω, we have chosen a subset Xn satisfying (5) with Xn instead of X, and
Xn ∩X−1n = ∅. Since G has only finite number of elements of order 2, we can choose
an element xn+1 ∈ G such that
(7) (xn+1XnX
−1
n ∪ x−1n+1XnXn) ∩Xn = ∅;
(8) x2n+1 6= e;
(9) xn+1 /∈ X−1n ;
(10) xn+1X
−1
n xn+1 = ∅.
We put Xn+1 = Xn ∪{xn+1}. By (8) and (9), Xn+1 ∩X−1n+1 = ∅. By (7) and (10),
Xn+1 satisfies (5).
Theorem 5 is proved.
Corollary 3. Let G be an infinite Abelian group with finite number of elements
of order 2. Then τ2(FG) * τ(FG), τ(σ(FG)) * σ(FG), and σ(σ(FG)) * σ(FG). In
particular, the ideal σ(FG) of sparse sets is not thin-complete.
If the group G is torsion-free, then Theorem 5 guarantees that F ⊆ τ(F) ⊆ σ(F) ⊆
⊆ τ∗(F), which implies the equivalence of the equalities F = τ(F) = τ∗(F) (the thin-
completeness) and F = σ(F) (the sparse-completeness). Thus we obtain the following
surprising:
Corollary 4. Let F be a left-invariant ideal of subsets of an infinite group G. If G
is torsion-free, then F is thin-complete if and only if F is sparse-complete. Consequently,
τ∗(F) = σ∗(F).
We conclude this section discussing the intrinsic structure of σ(F).
Remark 4. Let G be an infinite group and F be a left-invariant ideal in PG. For
a subset A ⊂ G, we consider the set
ΣA =
{
F ∈ FG :
⋂
x∈F
xA /∈ F
}
,
partially ordered by the relation ⊂ . It follows from the definition that A is F-sparse if
and only if the ΣA is well-founded in the sense that it contains no infinite chains. In this
case we can assign to each set F ∈ ΣA the ordinal
rank(F ) = sup
{
rank(E) + 1: F ⊂ E ∈ ΣA, |E \ F | = 1
}
,
where sup(∅) = 0. So, the maximal elements of ΣA have rank 0, their immediate
predecessors have rank 1 and so on. Let
rank(ΣA) = sup
{
rank(F ) + 1: F ∈ ΣA
}
= rank(∅) + 1
be the rank of the family ΣA.
For an ordinal α let
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RELATIVELY THIN AND SPARSE SUBSETS OF GROUPS 225
σα(F) = {A ∈ σ(F) : rank(ΣA) 6 α+ 1} and σ<α(F) =
⋃
β<α
σβ(F).
Sets from the family σα(F) are called (α,F)-sparse. Observe that a set A ⊂ G is
(n,F)-sparse for a natural number n ∈ ω if and only if for each infinite set S ⊂ G there
is a set F ⊂ G of cardinality |F | ≤ n + 1 such that
⋂
x∈F xA ∈ F . Now we see that
(n,FG)-sparse sets coincide with (n + 1)-sparse sets studied in [2]. By Lemma 1.2 of
[2] the union A ∪B of an (n,FG)-space set A ⊂ G and an (m,FG)-sparse set B ⊂ G
is (m+ n,FG)-sparse. Consequently, the family σ<ω(FG) is an ideal in G.
Question 1. For which ordinals α the family σ<α(F) is an ideal in PG? Is it true
for each limit (additively indecomposable*) ordinal α?
Repeating the argument of Proposition 3, we can prove that
σ(F) =
⋃
α<|G|+
σα(F),
so σ<|G|+(F) = σ(F) is an ideal in PG according to Corollary 2.
We note that a similar construction using the rank function of well-founded trees has
been used in [4] for describing the intrinsic structure of the ideal τ∗(G).
Acknowledgment. We would like to thank Taras Banakh for many helpful conversat-
ions on the subject of the paper, and to a referee for valuable remarks that substantially
improved the presentation.
1. Protasov I. V. Selective survey on subset combinatorics of groups // Ukr. Math. Bull. – 2010. – 7. –
P. 220 – 257.
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– 19. – P. 491 – 510.
3. Banakh T., Lyaskovska N. Completeness of translation-invariant ideals in groups // Ukr. Math. J. – 2010.
– 62, № 7. – P. 1022 – 1031.
4. Banakh T., Lyaskovska N. On thin-complete ideals of subsets of groups. – Preprint
(http://arxiv.org/abs/1011.2585).
5. Hindman N., Strauss D. Algebra in the Stone – Čech compactification — theory and application. – Berlin,
New York: Walter de Grueter, 1998.
6. Protasov I. V. Combinatorics of numbers // Math. Stud. Monogr. Ser. – Lviv: VNTL Publ., 1997. – Vol.
2.
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2008. – 30. – P. 1 – 10.
Received 19.07.10
*An ordinal α is called additively indecomposable if for any ordinals β, γ < α we get β + γ < α.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
|
| id | nasplib_isofts_kiev_ua-123456789-166348 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | English |
| last_indexed | 2025-12-07T16:39:44Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Lutsenko, Ie. Protasov, I.V. 2020-02-19T04:48:02Z 2020-02-19T04:48:02Z 2011 Relatively thin and sparse subsets of groups / Ie. Lutsenko, I.V. Protasov // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 216–225. — Бібліогр.: 7 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166348 517.5 Let G be a group with identity e and let I be a left-invariant ideal in the Boolean algebra PG of all subsets of G. A subset A of G is called I-thin if gA⋂A∈I for every g∈G {e}. A subset A of G is called I-sparse if, for every infinite subset S of G, there exists a finite subset F⊂S such that ⋂g∈FgA∈F. An ideal I is said to be thin-complete (sparse-complete) if every I-thin (I-sparse) subset of G belongs to I. We define and describe the thin-completion and the sparse-completion of an ideal in PG. Припустимо, що G — група з одиницею e, I — iнварiантний злiва iдеал в булевiй алгебрi PG всiх пiдмножин групи G. Пiдмножина A групи G називається I-тонкою, якщо gA⋂A∈I для кожного g∈G {e}. Пiдмножина A групи G називається P-розрiдженою, якщо для кожної нескiнченної множини S групи G iснує скiнченна пiдмножина F⊂S така, що ⋂g∈FgA∈F. Говорять, що iдеал I тонко-повний (розрiджено-повний), якщо кожна I-тонка (I-розрiджена) множина групи G належить I. Визначено та описано тонке та розрiджене доповнення iдеалу в PG. en Інститут математики НАН України Український математичний журнал Статті Relatively thin and sparse subsets of groups Вiдносно тонкi та розрiдженi пiдмножини груп Article published earlier |
| spellingShingle | Relatively thin and sparse subsets of groups Lutsenko, Ie. Protasov, I.V. Статті |
| title | Relatively thin and sparse subsets of groups |
| title_alt | Вiдносно тонкi та розрiдженi пiдмножини груп |
| title_full | Relatively thin and sparse subsets of groups |
| title_fullStr | Relatively thin and sparse subsets of groups |
| title_full_unstemmed | Relatively thin and sparse subsets of groups |
| title_short | Relatively thin and sparse subsets of groups |
| title_sort | relatively thin and sparse subsets of groups |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/166348 |
| work_keys_str_mv | AT lutsenkoie relativelythinandsparsesubsetsofgroups AT protasoviv relativelythinandsparsesubsetsofgroups AT lutsenkoie vidnosnotonkitarozridženipidmnožinigrup AT protasoviv vidnosnotonkitarozridženipidmnožinigrup |