On thin-complete ideals of subsets of groups
Let $F \subset \mathcal{P}_G$ be a left-invariant lower family of subsets of a group $G$. A subset $A \subset G$ is called $\mathcal{F}$-thin if $xA \bigcap yA \in \mathcal{F}$ for any distinct elements $x, y \in G$. The family of all $\mathcal{F}$-thin subsets of G is denoted by $\tau(\mathcal{F})$...
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| Дата: | 2011 |
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| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2011
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508723175227392 |
|---|---|
| author | Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. |
| author_facet | Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. |
| author_sort | Banakh, T. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:35:28Z |
| description | Let $F \subset \mathcal{P}_G$ be a left-invariant lower family of subsets of a group $G$. A subset $A \subset G$ is called $\mathcal{F}$-thin if
$xA \bigcap yA \in \mathcal{F}$ for any distinct elements $x, y \in G$. The family of all $\mathcal{F}$-thin subsets of G is denoted by $\tau(\mathcal{F})$.
If $\tau(\mathcal{F}) = \mathcal{F}$, then $\mathcal{F}$ is called thin-complete.
The thin-completion $\tau*(\mathcal{F})$ of $\mathcal{F}$ is the smallest thin-complete subfamily of $\mathcal{P}_G$ that contains $\mathcal{F}$.
Answering questions of Lutsenko and Protasov, we prove that a set $A \subset G$ belongs to $\tau*(G)$ if and only if for any sequence $(g_n)_{n\in \omega}$ of non-zero elements of G there is $n\in \omega$ such that
$$\bigcap_{i_0,...,i_n \in \{0, 1\}}g_0^{i_0}...g_n^{i_n} A \in \mathcal{F}.$$
Also we prove that for an additive family $\mathcal{F} \subset \mathcal{P}_G$ its thin-completion $\tau*(\mathcal{F})$ is additive. If the group $G$ is countable and torsion-free, then the completion $\tau*(\mathcal{F}_G)$ of the ideal $\mathcal{F}_G$ of finite subsets of $G$ is coanalytic and not Borel in the power-set $\mathcal{P}_G$ endowed with the natural compact metrizable topology. |
| first_indexed | 2026-03-24T02:29:44Z |
| format | Article |
| fulltext |
UDC 512.5
T. Banakh, N. Lyaskovska (Lviv Nat. Ivan Franko Univ.)
ON THIN-COMPLETE IDEALS OF SUBSETS OF GROUPS
ПРО ТОНКО-ПОВНI IДЕАЛИ НА ГРУПАХ
Let F ⊂ PG be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F -thin if
xA∩ yA ∈ F for any distinct elements x, y ∈ G. The family of all F -thin subsets of G is denoted by τ(F).
If τ(F) = F , then F is called thin-complete. The thin-completion τ∗(F) of F is the smallest thin-complete
subfamily of PG that contains F .
Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ∗(G) if and only
if for any sequence (gn)n∈ω of non-zero elements of G there is n ∈ ω such that⋂
i0,...,in∈{0,1}
gi00 . . . ginn A ∈ F .
Also we prove that for an additive family F ⊂ PG its thin-completion τ∗(F) is additive. If the group G is
countable and torsion-free, then the completion τ∗(FG) of the ideal FG of finite subsets of G is coanalytic
and not Borel in the power-set PG endowed with the natural compact metrizable topology.
Нехай F ⊂ PG — iнварiантна злiва нижня сiм’я пiдмножин групи G. Пiдмножина A ⊂ G називається
F -тонкою, якщо xA∩yA ∈ F для будь-яких рiзних елементiв x, y ∈ G. Сiм’я всiхF -тонких пiдмножин
G позначається як τ(F). Якщо τ(F) = F , тоF називається тонко-повною. Тонким поповненням τ∗(F)
сiм’ї F є найменша тонко-повна пiдсiм’я з PG, що мiстить F .
Як вiдповiдь на питання Луценка та Протасова доведено, що множинаA ⊂ G належить сiм’ї τ∗(G)
тодi i тiльки тодi, коли для будь-якої послiдовностi (gn)n∈ω ненульових елементiв G iснує n ∈ ω таке,
що ⋂
i0,...,in∈{0,1}
gi00 . . . ginn A ∈ F .
Також доведено, що для адитивної сiм’їF ⊂ PG її тонке поповнення τ∗(F) є адитивним. Якщо групаG
злiченна та без скруту, поповнення τ∗(FG) iдеалу FG скiнченних пiдмножин групи G є коаналiтичним
i не борелевим.
1. Introduction. This paper was motivated by problems posed by Ie. Lutsenko and
I. V. Protasov in a preliminary version of the paper [1] devoted to relatively thin sets in
groups.
Following [2], we say that a subset A of a group G is thin if for any distinct points
x, y ∈ G the intersection xA ∩ yA is finite. In [1] (following the approach of [3])
Lutsenko and Protasov generalized the notion of a thin set to that of a F-thin set where
F is a family of subsets of G. By PG we shall denote the Boolean algebra of all subsets
of the group G.
We shall say that a family F ⊂ PG is
left-invariant if xF ∈ F for all F ∈ F and x ∈ G;
additive if A ∪B ∈ F for all A,B ∈ F ;
lower if A ∈ F for any A ⊂ B ∈ F ;
an ideal if F is lower and additive.
Let F ⊂ PG be a left-invariant lower family of subsets of a group G. A subset
A ⊂ G is defined to be F-thin if for any distinct points x, y ∈ G we get xA ∩ yA ∈ F .
The family of all F-thin subsets of G will be denoted by τ(F). It is clear that τ(F)
is a left-invariant lower family of subsets of G and F ⊂ τ(F). If τ(F) = F , then the
family F will be called thin-complete.
Let τ∗(F) be the intersection of all thin-complete families F̃ ⊂ PG that contain F .
It is clear that τ∗(F) is the smallest thin-complete family containing F . This family is
called the thin-completion of F .
c© T. BANAKH, N. LYASKOVSKA, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6 741
742 T. BANAKH, N. LYASKOVSKA
The family τ∗(F) has an interesting hierarchic structure that can be described as
follows. Let τ0(F) = F and for each ordinal α put τα(F) be the family of all sets
A ⊂ G such that for any distinct points x, y ∈ G we get xA ∩ yA ∈
⋃
β<α τ
β(F). So,
τα(F) = τ(τ<α(F)), where τ<α(F) =
⋃
β<α
τβ(F).
By Proposition 3 of [1], τ∗(F) =
⋃
α<|G|+
τα(F).
The following theorem (that will be proved in Section 3) answers the problem of
combinatorial characterization of the family τ∗(F) posed by Ie. Lutsenko and I. V. Pro-
tasov. Below by e we denote the neutral element of the group G.
Theorem 1.1. Let F ⊂ PG be a left-invariant lower family of subsets of a group
G. A subset A ⊂ G belongs to the family τ∗(F) if and only if for any sequence
(gn)n∈ω ∈ (G \ {e})N there is a number n ∈ ω such that⋂
k0,...,kn∈{0,1}
gk00 . . . gknn A ∈ F .
We recall that a family F ⊂ PG is called additive if {A ∪ B : A,B ∈ F} ⊂ F . It
is clear that the family FG of finite subsets of a group G is additive. If G is an infinite
Boolean group, then the family τ∗(FG) = τ(FG) is not additive, see Remark 3 in [1].
For torsion-free groups the situation is totally diferent. Let us recall that a group G is
torsion-free if each non-zero element of G has infinite order.
Theorem 1.2. For a torsion-free group G and a left-invariant ideal F ⊂ PG the
family τ<α(F) is additive for any limit ordinal α. In particular, the thin-completion
τ∗(F) of F is an ideal in PG.
We define a subset of a group G to be ∗-thin if its belongs to the thin-completion
τ∗(FG) of the family FG of all finite subsets of the group G. By Proposition 3 of [1],
for each countable group G we get τ∗(FG) = τ<ω1(FG). It is natural to ask if the
equality τ∗(FG) = τ<α(FG) can happen for some cardinal α < ω1. If the group G is
Boolean, then the answer is affirmative: τ∗(F) = τ1(F) according to Theorem 1 of [1].
The situation is different for non-torsion groups:
Theorem 1.3. If an infinite group G contains an Abelian torsion-free subgroup
H of cardinality |H| = |G|, then τ∗(FG) 6= τα(FG) 6= τ<α(FG) for each ordinal
α < |G|+.
Theorems 1.2 and 1.3 will be proved in Sections 4 and 6, respectively. In Section 7
we shall study the Borel complexity of the family τ∗(FG) for a countable group G. In
this case the power-set PG carries a natural compact metrizable topology, so we can talk
about topological properties of subsets of PG.
Theorem 1.4. For a countable group G and a countable ordinal α the subset
τα(FG) of PG is Borel while τ∗(FG) = τ<ω1(FG) is coanalytic. If G contains an
element of infinite order, then the space τ∗(FG) is coanalytic but not analytic.
2. Preliminaries on well-founded posets and trees. In this section we collect the
neccessary information on well-founded posets and trees. A poset is an abbreviation
from a partially ordered set. A poset (X,≤) is well-founded if each subset A ⊂ X has
a maximal element a ∈ A (this means that each element x ∈ A with x ≥ a is equal
to a). In a well-founded poset (X,≤) to each point x ∈ X we can assign the ordinal
rankX(x) defined by the recursive formula
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON THIN-COMPLETE IDEALS OF SUBSETS OF GROUPS 743
rankX(x) = sup
{
rankX(y) + 1: y > x
}
,
where sup∅ = 0. Thus maximal elements of X have rank 0, their immediate predeces-
sors 1, and so on. If X is not empty, then the ordinal rank(X) = sup{rankX(x) + 1:
x ∈ X} is called the rank of the poset X. In particular, a one-element poset has rank 1.
If X is empty, then we put rank(X) = 0.
A tree is a poset (T,≤) with the smallest element ∅T such that for each t ∈ T the
lower set ↓t = {s ∈ T : s ≤ t} is well-ordered in the sense that each subset A ⊂ ↓t has
the smallest element. A branch of a tree T is any maximal linearly ordered subset of T.
If a tree is well-founded, then all its branches are finite.
A subset S ⊂ T of a tree is called a subtree if it is a tree with respect to the induced
partial order. A subtree S ⊂ T is lower if S = ↓S = {t ∈ T : ∃s ∈ S t ≤ s}.
All trees that appear in this paper are (lower) subtrees of the tree X<ω =
⋃
n∈ωX
n
of finite sequences of a set X. The tree X<ω carries the following partial order:
(x0, . . . , xn) ≤ (y0, . . . , ym) iff n ≤ m and xi = yi for all i ≤ n.
The empty sequence s∅ ∈ X0 is the smallest element (the root) of the tree X<ω.
For a finite sequence s = (x0, . . . , xn) ∈ X<ω and an element x ∈ X by ŝ x =
= (x0, . . . , xn, x) we denote the concatenation of s and x. So, ŝ x is one of |X| many
immediate successors of s. The set of all branches of X<ω can be naturally identified
with the countable power Xω. For each branch s = (sn)n∈ω ∈ Xω and n ∈ ω by
s|n = (s0, . . . , sn−1) we denote the initial interval of length n.
Let Tr ⊂ PX<ω denote the family of all lower subtrees of the tree X<ω and WF ⊂
⊂ Tr be the subset consisting of all well-founded lower subtrees of X<ω.
In Section 7 we shall exploit some deep facts about the descriptive properties of the
sets WF ⊂ Tr ⊂ PX<ω for a countable set X. In this case the tree X<ω is countable
and the power-set PX<ω carries a natural compact metrizable topology of the Tychonoff
power 2X
<ω
. So, we can speak about topological properties of the subsets WF and Tr
of the compact metrizable space PX<ω .
We recall that a topological space X is Polish if X is homeomorphic to a separable
complete metric space. A subset A of a Polish space X is called
Borel if A belongs to the smallest σ-algebra that contains all open subsets of X;
analytic if A is the image of a Polish space P under a continuous map f : P → A;
coanalytic if X \A is analytic.
By Souslin’s Theorem 14.11 [4], a subset of a Polish space is Borel if and only if it is
both analytic and coanalytic. By Σ1
1 and Π1
1 we denote the classes of spaces homeomor-
phic to analytic and coanalytic subsets of Polish spaces, respectively.
A coanalytic subset X of a compact metric space K is called Π1
1-complete if for
each coanalytic subset C of the Cantor cube 2ω there is a continuous map f : 2ω → K
such that f−1(X) = C. It follows from the existence of a coanalytic non-Borel set in
2ω that each Π1
1-complete set X ⊂ K is non-Borel.
The following deep theorem is classical and belongs to Lusin, see [4] (32.B and
35.23).
Theorem 2.1. Let X be a countable set.
(1) The subspace Tr is closed (and hence compact) in PX<ω .
(2) The set of well-founded trees WF is Π1
1-complete in Tr. In particular, WF is
coanalytic but not analytic (and hence not Borel).
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
744 T. BANAKH, N. LYASKOVSKA
(3) For each ordinal α < ω1 the subset WFα = {T ∈WF : rank(T ) ≤ α} is Borel
in Tr.
(4) Each analytic subspace of WF lies in WFα for some ordinal α < ω1.
3. Combinatorial characterization of ∗-thin subsets. In this section we prove
Theorem 1.1. Let F ⊂ PG be a left-invariant lower family of subsets of a group G.
Theorem 1.1 trivially holds if F = PG (which happens if and only if G ∈ F). So, it
remains to consider the case G /∈ F . Let e be the neutral element of G and G◦ = G\{e}.
We shall work with the tree G<ω◦ discussed in the preceding section.
Let A be any subset of G. To each finite sequence s ∈ G<ω◦ assign the set As ⊂ G,
defined by induction: A∅ = A and Aŝ x = As ∩ xAs for s ∈ G<ω◦ and x ∈ G◦.
Repeating the inductive argument of the proof of Proposition 2 [1], we can obtain the
following direct description of the sets As:
Claim 3.1. For every sequence s = (g0, . . . , gn) ∈ G<ω◦
As =
⋂
k0,...,kn∈{0,1}
gk00 . . . gknn A.
The set
TA = {s ∈ G<ω◦ : As /∈ F}
is a subtree of G<ω◦ called the τ -tree of the set A.
For a non-zero ordinal α let −1 +α be a unique ordinal β such that 1 + β = α. For
α = 0 we put −1 + α = −1. It follows that −1 + α = α for each infinite ordinal α.
Theorem 3.1. A set A ⊂ G belongs to the family τα(F) for some ordinal α if
and only if its τ -tree TA is well-founded and has rank(TA) ≤ −1 + α+ 1.
Proof. By induction on α. Observe that A ∈ τ0(F) = F if and only if TA = ∅ if
and only if rank(TA) = 0 = −1 + 0 + 1. So, Theorem holds for α = 0.
Assume that for some ordinal α > 0 and any ordinal β < α we know that a set
A ⊂ G belongs to τβ(G) if and only if TA is a well-founded tree with rank(TA) ≤
≤ −1 +β+ 1. Given a subset A ⊂ G we should check that that A ∈ τα(F) if and only
if its τ -tree TA is well-founded and has rank(TA) ≤ −1 + α+ 1.
First assume that A ∈ τα(F). Then for every x ∈ G◦ the set A ∩ xA belongs to
τβx(F) ⊂ τ<α(F) for some ordinal βx < α. By the inductive assumption, the τ -tree
TA∩xA is well-founded and has rank(TA∩xA) ≤ −1 + βx + 1.
If A ∈ τ(F), then TA ⊂ {s∅} and rank(TA) ≤ 1 ≤ −1 + α + 1. So, we can
assume that A /∈ τ(F). In this case each point x ∈ G◦ = G1
◦ considered as the sequence
(x) ∈ G1 of length 1 belongs to the τ -tree TA of the set A. So we can consider the upper
set TA(x) = {s ∈ TA : s ≥ x} and observe that the subtree TA(x) of TA is isomorphic
to the τ -tree TA∩xA of the set A ∩ xA and hence rank(TA(x)) = rank(TA∩xA) ≤
≤ −1 + βx + 1. It follows that
rank(TA) = rankTA(s∅) + 1 =
(
sup
x∈G◦
(rankTA(x) + 1)
)
+ 1 =
=
(
sup
x∈G◦
rankTA(x)
)
+ 1 ≤
(
sup
x∈G◦
(−1 + βx + 1
))
+ 1 ≤ −1 + α+ 1.
Now assume conversely that the τ -tree TA of A is well-founded and has rank(TA) ≤
≤ −1+α+1. For each x ∈ G◦, find a unique ordinal βx such that−1+βx = rankTA(x).
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON THIN-COMPLETE IDEALS OF SUBSETS OF GROUPS 745
It follows from
−1 + βx + 2 = rankTA(x) + 2 ≤ rankTA(s∅) + 1 = rank(TA) ≤ −1 + α+ 1
that βx < α. Since the subtree TA(x) = TA ∩ ↑x is isomorphic to the τ -tree TA∩xA
of the set A ∩ xA, we conclude that TA∩xA is well-founded and has rank(TA∩xA) =
= rank(TA(x)) = rankTA(x) + 1 = −1 + βx + 1. Then the inductive assumption
guarantees that A ∩ xA ∈ τβx(F) ⊂ τ<α(F) and hence A ∈ τα(F) by the definition
of the family τα(F).
Theorem 3.1 is proved.
As a corollary of Theorem 3.1, we obtain the following characterization proved
in [1]:
Corollary 3.1. A subset A ⊂ G belongs to the family τn(F) for some n ∈ ω if and
only if for each sequence (gi)
n
i=0 ∈ Gn+1
◦ we get⋂
k0,...,kn∈{0,1}
gk00 . . . gknn A ∈ F .
Theorem 3.1 also implies the following explicit description of the family τ∗(F),
which was announced in Theorem 1.1:
Corollary 3.2. For a subset A ⊂ G the following conditions are equivalent:
(1) A ∈ τ∗(F);
(2) the τ -tree TA of A is well-founded;
(3) for each sequence (gn)n∈ω ∈ Gω◦ there is n ∈ ω such that (g0, . . . , gn) /∈ TA;
(4) for each sequence (gn)n∈ω ∈ Gω◦ there is n ∈ ω such that⋂
k0,...,kn∈{0,1}
gk00 . . . gknn A ∈ F .
4. The additivity of the families τ<α(F). In this section we shall prove Theo-
rem 1.2. Let G be an infinite group and e be the neutral element of G.
For a natural number m let 2m denote the finite cube {0, 1}m. For vectors g =
= (g1, . . . , gm) ∈ (G \ {e})m and x = (x1, . . . , xm) ∈ 2m let
gx = gx1
1 . . . gxmm ∈ G.
A function f : 2m → G to a group G will be called cubic if there is a vector
g = (g1, . . . , gm) ∈ (G \ {e})m such that f(x) = gx for all x ∈ 2m.
Lemma 4.1. If the group G is torsion-free, then for every n ∈ N, m > (n− 1)2,
and a cubic function f : 2m → G we get |f(2m)| > n.
Proof. Assume conversely that |f(2m)| ≤ n. Consider the set B =
{
(k1, . . . , km) ∈
∈ 2m :
∑m
i=1
ki = 1
}
having cardinality |B| = m > (n − 1)2. Since e /∈ f(B), we
conclude that |f(B)| ≤ |f(2m)| − 1 ≤ n − 1 and hence |B ∩ f−1(y)| ≥ n for some
y ∈ f(B). Let By = B ∩ f−1(y) and observe that f(2m) ⊃
{
e, y, y2, . . . , y|By|
}
and
thus |f(2m)| ≥ |By|+ 1 ≥ n+ 1, which contradicts our assumption.
Lemma 4.1 is proved.
For every n ∈ N let c(n) be the smallest number m ∈ N such that for each cubic
function f : 2m → G we get |f(2m)| > n. It is easy to see that c(n) ≥ n. On the other
hand, Lemma 4.1 implies that c(n) ≤ (n− 1)2 + 1 if G is torsion-free.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
746 T. BANAKH, N. LYASKOVSKA
For a family F and a natural number n ∈ N, let∨
n
F = {∪A : A ⊂ F , |A| ≤ n}.
Lemma 4.2. Let F ⊂ PG be a left-invariant lower family of subsets in a torsion-
free group G. For every n ∈ N we get
∨
n
τ(F) ⊂ τ c(n)−1
(∨
m
F
)
,
where m = n2
c(n)
.
Proof. Fix any A ∈
∨
n
τ(F) and write it as the union A = A1 ∪ · · · ∪ An of sets
A1, . . . , An ∈ τ(F). The inclusion A ∈ τ c(n)−1
(∨
m
F
)
will follow from Corollary 3.1
as soon as we check that ⋂
x∈2c(n)
gxA ∈
∨
m
F
for each vector g ∈ (G \ {e})c(n). De Morgan’s law guarantees that
⋂
x∈2c(n)
gx ·
(
n⋃
i=1
Ai
)
=
⋃
f∈n2c(n)
⋂
x∈2c(n)
gxAf(x).
So, the proof will be complete as soon as we check that for every function f : 2c(n) → n
the set
⋂
x∈2c(n) gxAf(x) belongs to F . The vector g ∈ (G \ {e})c(n) induces the cubic
function g : 2c(n) → G, g : x 7→ gx. The definition of the function c(n) guarantees that
|g(2c(n))| > n. The function f : 2c(n) → n can be thought as a coloring of the cube
2c(n) into n colors. Since |g(2c(n))| > n, there are two points y, z ∈ 2c(n) colored by the
same color such that g(y) 6= g(z). Then gy = g(y) 6= g(z) = gz but f(y) = f(z) = k
for some k ≤ n. Consequently,⋂
x∈2c(n)
gxAf(x) ⊂ gyAk ∩ gzAk ∈ F
because the set Ak ∈ τ(F).
Lemma 4.2 is proved.
Now consider the function c : N × ω → ω defined recursively as c(n, 0) = 0 for
all n ∈ N and c(n, k + 1) = c(n) − 1 + c(n2
c(n)
, k) for (n, k) ∈ N × ω. Observe that
c(n, 1) = c(n)− 1 for all n ∈ N.
Lemma 4.3. If the group G is torsion-free and F ⊂ PG is a left-invariant ideal,
then ∨
n
τk(F) ⊂ τ c(n,k)(F)
for all pairs (n, k) ∈ N× ω.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON THIN-COMPLETE IDEALS OF SUBSETS OF GROUPS 747
Proof. By induction on k. For k = 0 the equality
∨
n τ
0(F) = F = τ c(n,0)(F)
holds because F is additive.
Assume that Lemma is true for some k ∈ ω. By Lemma 4.2 and by the inductive
assumption, for every n ∈ N we get
∨
n
τk+1(F) =
∨
n
τ(τk(F)) ⊂ τ c(n)−1
∨
n2c(n)
τk(F)
⊂
⊂ τ c(n)−1
(
τ c(n
2c(n)
,k)(F)
)
= τ c(n)−1+c(n
2c(n)
,k)(F) = τ c(n,k+1)(F).
Lemma 4.3 is proved.
Now we are able to present:
Proof of Theorem 1.2. Assume that G is a torsion-free group G and F ⊂ PG is a
left-invariant ideal. By transfinite induction we shall prove that for each limit ordinal α
the family τ<α(F) is additive. For the smallest limit ordinal α = 0 the additivity of the
family τ0(F) = F is included into the hypothesis. Assume that for some non-zero limit
ordinal α we have proved that the families τ<β(F) are additive for all limit ordinals
β < α. Two cases are possible:
1) α = β + ω for some limit ordinal β. By the inductive assumption, the family
τ<β(F) is additive. Then Lemma 4.3 implies that the family τ<α(F) = τ<ω(τ<β(F))
is additive.
2) α = supB for some family B 63 α of limit ordinals. By the inductive assumption
for each limit ordinal β ∈ B the family τ<β(F) is additive and then the union
τ<α(F) =
⋃
β∈B
τ<β(F)
is additive too.
This completes the proof of the additivity of the families τ<α(F) for all limit or-
dinals α. Since the torsion-free group G is infinite, the ordinal α = |G|+ is limit and
hence the family τ∗(F) = τ<α(F) is additive. Being left-invariant and lower, the family
τ∗(F) is a left-invariant ideal in PG.
Theorem 1.2 is proved.
Remark 4.1. Theorem 1.2 is not true for an infinite Boolean group G. In this case
Theorem 1(2) of [1] implies that τ∗(FG) = τ(FG). Then for any infinite thin subset
A ⊂ G and any x ∈ G \ {e} the union A ∪ xA is not thin as (A ∪ xA) ∩ x(A ∪ xA) =
= A ∪ xA is infinite. Consequently, the family τ∗(FG) = τ(FG) is not additive.
5. h-Invariant families of subsets in groups. Let G be a group and h : H → K
be an isomorphism between subgroups of G. A family F of subsets of G is called
h-invariant if a subset A ⊂ H belongs to F if and only if h(A) ∈ F .
Example 5.1. The ideal FZ of finite subsets of the group Z is h-invariant for each
isomorphism hk : Z→ kZ, h : x 7→ kx, where k ∈ N.
Proposition 5.1. Let h : H → K be an isomorphism between subgroups of a
group G. For any h-invariant family F ⊂ PG and any ordinal α the family τα(F) is
h-invariant.
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748 T. BANAKH, N. LYASKOVSKA
Proof. For α = 0 the h-invariance of τ0(F) = F follows from our assumption.
Assume that for some ordinal α we have established that the families τβ(F) are h-
invariant for all ordinals β < α. Then the union τ<α(F) =
⋃
β<α τ
β(F) is also
h-invariant.
We shall prove that the family τα(F) is h-invariant. Given a set A ⊂ H we need to
prove that A ∈ τα(F) if and only if h(A) ∈ τα(F).
Assume first that A ∈ τα(F). To show that h(A) ∈ τα(F), take any element
y ∈ G\{e}. If y /∈ K, then h(A)∩yh(A) = ∅ ∈ τ<α(F). If y ∈ K, then y = h(x) for
some x ∈ H and then h(A)∩ yh(A) = h(A∩ xA) ∈ τ<α(F) since A∩ xA ∈ τ<α(F)
and the family τ<α(F) is h-invariant.
Now assume that A /∈ τα(F). Then there is an element x ∈ G \ {e} such that
A∩xA /∈ τ<α(F). Since A ⊂ H, the element x must belong to H (otherwise A∩xA =
= ∅ ∈ τ<α(F)). Then for the element y = h(x) we get h(A) ∩ yh(A) /∈ τ<α(F) by
the h-invariance of the family τ<α(F). Consequently, h(A) /∈ τα(F).
Proposition 5.1 is proved.
Corollary 5.1. Let h : H → K be an isomorphism between subgroups of a group
G. For any h-invariant family F ⊂ PG the family τ∗(F) is h-invariant.
Definition 5.1. A left-invariant family F ⊂ PG of subsets of a group G is called
auto-invariant if F is h-invariant for each injective homomorphism h : G→ G;
sub-invariant if F is h-invariant for each isomorphism h : H → K between sub-
groups K ⊂ H of G.
strongly invariant if F is h-invariant for each isomorphism h : H → K between
subgroups of G.
It is clear that
strongly invariant ⇒ sub-invariant ⇒ auto-invariant.
Remark 5.1. Each auto-invariant family F ⊂ PG, being left-invariant is also right-
invariant.
Proposition 5.1 implies:
Corollary 5.2. If F ⊂ PG is an auto-invariant (sub-invariant, strongly invariant)
family of subsets of a group G, then so are the families τ∗(F) and τα(F) for all
ordinals α.
It is clear that the famly FG of finite subsets of a group G is strongly invariant.
Now we present some natural examples of families, which are not strongly invariant.
Following [5], we call a subset A of a group G
large if there is a finite subset F ⊂ G with G = FA;
small if for any large set L ⊂ G the set L \A remains large.
It follows that the family SG of small subsets of G is a left-invariant ideal in PG.
According to [5], a subset A ⊂ G is small if and only if for every finite subset F ⊂ G
the complement G \ FA is large. We shall need the following (probably known) fact.
Lemma 5.1. Let H be a subgroup of finite index in a group G. A subset A ⊂ H
is small in H if and only if A is small in G.
Proof. First assume that A is small in G. To show that A is small in H, take any
large subset L ⊂ H. Since H has finite index in G, the set L is large in G. Since
A is small in G, the complement L \ A is large in G. Consequently, there is a finite
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ON THIN-COMPLETE IDEALS OF SUBSETS OF GROUPS 749
subset F ⊂ G such that F (L \ A) = G. Then for the finite set FH = F ∩ H, we get
FH(L \A) = H, which means that L \A is large in H.
Now assume that A is small in H. To show that A is small in G, it suffices to show
that for every finite subset F ⊂ G the complement G \ FA is large in G. Observe
that (G \ FA) ∩ H = H \ FHA where FH = F ∩ H. Since A is small in H, the set
H \ FHA is large in H and hence large in G (as H has finite index in G). Then the set
G \ FA ⊃ H \ FHA is large in G too.
Lemma 5.1 is proved.
Proposition 5.2. Let G be an infinite Abelian group.
(1) If G is finitely generated, then the ideal SG is strongly invariant.
(2) If G is infinitely generated free Abelian group, then the ideal SG is not auto-
invariant.
Proof. 1. Assume that G is a finitely generated Abelian group. To show that SG
is strongly invariant, fix any isomorphism h : H → K between subgroups of G and
let A ⊂ H be any subset. The groups H,K are isomorphic and hence have the same
free rank r0(H) = r0(K). If r0(H) = r0(K) < r0(G), then the subgroups H,K
have infinite index in G and hence are small. In this case the inclusions A ∈ SG and
h(A) ∈ SG hold and so are equivalent.
If the free ranks r0(H) = r0(K) and r0(G) coincide, then H and K are subgroups
of finite index in the finitely generated group G. By Lemma 5.1, a subset A ⊂ H is
small in G if and only if A is small in H if and only if h(A) is small in the group
h(H) = K if and only if h(A) is small in G.
2. Now assume that G is an infinitely generated free Abelian group. Then G is
isomorphic to the direct sum ⊕κZ of κ = |G| ≥ ℵ0 many copies of the infinite cyclic
group Z. Take any subset λ ⊂ κ with infinite complement κ\λ and cardinality |λ| = |κ|
and fix an isomorphism h : G→ H of the group G = ⊕κZ onto its subgroup H = ⊕λZ.
The subgroup H has infinite index in G and hence is small in G. Yet h−1(H) = G is
not small in G, witnessing that the ideal SG of small subsets of G is not auto-invariant.
Proposition 5.2 is proved.
6. Thin-completeness of the families τα(F). In this section we shall prove that in
general the families τα(F) are not thin-complete. Our principal result is the following
theorem that implies Theorem 1.3 announced in the Introduction.
Theorem 6.1. Let G be a group containing a free Abelian subgroup H of cardi-
nality |H| = |G|. If F is a sub-invariant ideal of subsets of G such that τ(F)∩PH 6⊂ F ,
then τ∗(F) 6= τα(F) 6= τ<α(F) for all ordinals α < |G|+.
We divide the proof of this theorem in a series of lemmas.
Lemma 6.1. Let h : H → K be an isomorphism between subgroups of a group
G, F be an h-invariant left-invariant lower family of subsets of G. If a subset A ⊂ H
does not belong to τα(F) for some ordinal α, then for every point z ∈ G \ {e} the set
h(A) ∪ zh(A) /∈ τα+1(F).
Proof. Proposition 5.1 implies that h(A) /∈ τα(F). Since(
h(A) ∪ zh(A)
)
∩ z−1
(
h(A) ∪ zh(A)
)
⊃ h(A) /∈ τα(F),
the set h(A) ∪ zh(A) /∈ τα+1(F) by the definition of τα+1(F).
Lemma 6.1 is proved.
In the following lemma for a subgroup K of a group H by
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750 T. BANAKH, N. LYASKOVSKA
ZH(K) =
{
z ∈ H : ∀x ∈ K zx = xz
}
we denote the centralizer of K in H.
Lemma 6.2. Let h : H → K be an isomorphism between subgroups K ⊂ H of
a group G such that there is a point z ∈ ZH(K) with z2 /∈ K. Let F ⊂ PG be an
h-invariant left-invariant ideal. If a subset A ⊂ H belongs to the family τα(F) for some
ordinal α, then h(A) ∪ zh(A) ∈ τα+1(F).
Proof. By induction on α. For α = 0 and A ∈ F the inclusion h(A) ∪ zh(A) ∈
∈ F ⊂ τ(F) follows from the h-invariance and the additivity of F .
Now assume that for some ordinal α we have proved that for every β < α and
A ∈ PH ∩ τβ(F) the set h(A) ∪ zh(A) belongs to τβ+1(F). Given any set A ∈
∈ PH ∩ τα(F), we need to prove that h(A) ∪ zh(A) ∈ τα+1(F). This will follow as
soon as we check that (h(A)∪zh(A))∩y(h(A)∪zh(A) ∈ τα(F) for every y ∈ G\{e}.
If y /∈ K ∪ zK ∪ z−1K, then
(h(A) ∪ zh(A)) ∩ y(h(A) ∪ zh(A)) ⊂ (K ∪ zK) ∩ y(K ∪ zK) = ∅ ∈ τα+1(F).
So, it remains to consider the case y ∈ K ∪ zK ∪ z−1K ⊂ H. If y ∈ K, then
(h(A) ∪ zh(A)) ∩ y(h(A) ∪ zh(A)) = (h(A) ∩ yh(A)) ∪ z(h(A) ∩ y h(A)).
Since y ∈ K, there is an element x ∈ H with y = h(x). Since A ∈ τα(F), A ∩ xA ∈
∈ τβ(F) for some β < α and then
(h(A) ∪ zh(A)) ∩ y(h(A) ∪ zh(A)) =
= h(A ∩ xA) ∪ zh(A ∩ xA) ∈ τβ+1(F) ⊂ τα(F)
by the inductive assumption. If y ∈ zK, then z2 /∈ K implies that
(h(A) ∪ zh(A)) ∩ y(h(A) ∪ zh(A)) = zh(A) ∩ yh(A) ⊂ zh(A) ∈ τα(F)
by the h-invariance and the left-invariance of the family τα(F), see Proposition 5.1.
If y ∈ z−1K, then by the same reason,
(h(A) ∪ zh(A)) ∩ y(h(A) ∪ zh(A)) = h(A) ∩ yzh(A) ⊂ h(A) ∈ τα(F).
Lemma 6.2 is proved.
Given an isomorphism h : H → K between subgroups K ⊂ H of a group G, for
every n ∈ N define the iteration hn : H → K of the isomorphism h letting h1 =
= h : H → K and hn+1 = h ◦ hn for n ≥ 1.
The isomorphism h : H → K will be called expanding if
⋂
n∈N h
n(H) = {e}.
Example 6.1. For every integer k ≥ 2 the isomorphism
hk : Z→ kZ, hk : x 7→ kx,
is expanding.
Lemma 6.3. Let h : H → K be an expanding isomorphism between torsion-free
subgroups K ⊂ H of a group G and F ⊂ PG be an h-invariant left-invariant ideal of
subsets of G. For any limit ordinal α and a family {An}n∈ω ⊂ τ<α(F) of subsets of
the group H, the union A =
⋃
n∈ω h
n(An) belongs to the family τα(F).
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ON THIN-COMPLETE IDEALS OF SUBSETS OF GROUPS 751
Proof. First observe that {hn(An)}n∈ω ⊂ τ<α(F) by Proposition 5.1. To show
that A =
⋃
n∈ω h
n(An) ∈ τα(F) we need to check that A ∩ xA ∈ τ<α(F) for all
x ∈ G \ {e}. This is trivially true if x /∈ H as A ⊂ H. So, we assume that x ∈ H.
By the expanding property of the isomorphism h, there is a number m ∈ ω such that
x /∈ hm(H). Put B =
⋃m−1
n=0 h
n(An) and observe that A ∩ xA ⊂ B ∪ xB ∈ τ<α(F)
as τ<α(F) is additive according to Theorem 1.2.
Lemma 6.3 is proved.
Lemma 6.4. Assume that a left-invariant ideal F on a group G is h-invariant for
some expanding isomorphism h : H → K between torsion-free subgroups K ⊂ H of G
such that ZH(K) 6⊂ K. If τ(F) ∩ PH 6⊂ F , then τα(F) 6= τ<α(F) for all ordinals
α < ω1.
Proof. Fix any point z ∈ ZK(H) \K. Since H is torsion-free, z2 6= e. Since the
isomorphism h is expanding, z2 /∈ hm(H) for some m ∈ N. Replacing the isomorphism
h by its iterate hm, we lose no generality assuming that z2 /∈ h(H) = K.
By induction on α < ω1 we shall prove that τα(F) ∩ PH 6= τ<α(F) ∩ PH .
For α = 1 the non-equality τ(F)∩PH 6= τ0(F)∩PH is included into the hypothesis.
Assume that for some ordinal α < ω1 we proved that τβ(F)∩PH 6= τ<β(F)∩PH for
all ordinals β < α.
If α = β + 1 is a successor ordinal, then by the inductive assumption we can find
a set A ∈ τβ(F) \ τ<β(F) in the subgroup H. By Lemmas 6.1 and 6.2, A ∪ zA ∈
∈ τβ+1(F) \ τβ(F) = τα(F) \ τ<α(F) and we are done.
If α is a limit ordinal, then we can find an increasing sequence of ordinals (αn)n∈ω
with α = supn∈ω αn. By the inductive assumption, for every n ∈ ω there is a subset
An ⊂ H with An ∈ ταn+1(F) \ ταn(F). Then we can put A =
⋃
n∈ω h
n(An). By
Proposition 5.1, for every n ∈ ω, we get
hn(An) ∈ ταn+1(F) \ ταn(F)
and thus A /∈ ταn(F) for all n ∈ ω, which implies that A /∈ τ<α(F). On the other
hand, Lemma 6.3 guarantees that A ∈ τα(F).
Lemma 6.4 is proved.
Lemma 6.5. Assume that a left-invariant ideal F on a group G is h-invariant
for some isomorphism h : H → K between torsion-free subgroups K ⊂ H of G such
that z2 /∈ K for some z ∈ ZK(H). Assume that for an infinite cardinal κ there are
isomorphisms hn : H → Hn, n ∈ κ, onto subgroups Hn ⊂ H such that F is hn-
invariant and Hn · Hm ∩ Hk · Hl = {e} for all indices n,m, k, l ∈ κ with {n,m} ∩
∩ {k, l} = ∅.
If τ(F) ∩ PH 6⊂ F , then τα(F) 6= τ<α(F) for all ordinals α < κ+.
Proof. By induction on α < κ+ we shall prove that τα(F)∩PH 6= τ<α(F)∩PH .
For α = 1 the non-equality τ1(F) ∩ PH 6= τ0(F) ∩ PH is included into the
hypothesis. Assume that for some ordinal α < κ+ we proved that τβ(F) ∩ PH 6=
6= τ<β(F) ∩ PH for all ordinals β < α.
If α = β + 1 is a successor ordinal, then by the inductive assumption we can find a
set A ∈ τβ(F) \ τ<β(F) in the subgroup H. By Lemmas 6.1 and 6.2, h(A)∪ zh(A) ∈
∈ τβ+1(F) \ τβ(F) and we are done.
If α is a limit ordinal, then we can fix a family of ordinals (αn)n∈κ with α =
= supn∈κ(αn + 1). By the inductive assumption, for every n ∈ κ there is a subset
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752 T. BANAKH, N. LYASKOVSKA
An ⊂ H such that An ∈ ταn+1(F)\ ταn(F). After a suitable shift, we can assume that
e /∈ An. Since the ideal F is hn-invariant, hn(An) ∈ ταn+1(F) \ ταn(F) according to
Lemma 5.1.
Then the set A =
⋃
n∈ω hn(An) does not belong to τ<α(F). The inclusion A ∈
∈ τα(F) will follow as soon as we check that A ∩ xA ∈ τ<α(F) for all x ∈ G \ {e}.
This is clear if A∩xA is empty. If A∩xA is not empty, then x ∈ hn(An)hm(Am)−1 ⊂
⊂ HnHm for some n,m ∈ κ. Taking into account that HnHm ∩ HkHl = {e} for all
k, l ∈ κ \ {n,m} and e /∈ A, we conclude that
A ∩ xA ⊂ hn(An) ∪ hm(Am) ∪ xhn(An) ∪ xhm(Am) ∈ τ<α(F)
as τ<α(F) is additive according to Theorem 1.2.
Lemma 6.5 is proved.
Let us recall that a family F of subsets of a group G is called auto-invariant if for
any injective homomorphism h : G → G a subset A ⊂ G belongs to F if and only if
h(A) ∈ F .
Lemma 6.6. Let G be a free Abelian group G and F be an auto-invariant ideal
of subsets of G. If F is not thin-complete, then for each ordinal α < |G|+ the family
τα(F) is not thin-complete.
Proof. Being free Abelian, the group G is generated by some linearly independent
subset B ⊂ G. Consider the isomorphism h : G → 3G of G onto the subgroup 3G =
= {g3 : g ∈ G} and observe that h is expanding and for each z ∈ B we get z2 /∈ 3G.
The ideal F being auto-invariant, is h-invariant. Applying Lemma 6.4, we conclude that
τα(F) 6= τ<α(F) for all ordinals α < ω1. If the group G is countable, then this is
exactly what we need.
Now consider the case of uncountable κ = |G|. Being free Abelian, the group G
is isomorphic to the direct sum ⊕κZ of κ-many copies of the infinite cyclic group Z.
Write the cardinal κ as the disjoint union κ =
⋃
α∈κ κα of κ many subsets κα ⊂ κ of
cardinality |κα| = κ. For every α ∈ κ consider the free Abelian subgroup Gα = ⊕καZ
of G and fix any isomorphism hα : G→ Gα. It is clear that Gα⊕Gβ ∩Gγ ⊕Gδ = {0}
for all ordinals α, β, γ, δ ∈ κ with {α, β} ∩ {γ, δ} = ∅.
Being auto-invariant, the ideal F is hα-invariant for every α ∈ κ. Now it is legal to
apply Lemma 6.5 to conclude that τα(F) 6= τ<α(F) for all ordinals α < κ+.
Lemma 6.6 is proved.
Proof of Theorem 6.1. Let F be a sub-invariant ideal of subsets of a group G
and let H ⊂ G be a free Abelian subgroup of cardinality |H| = |G|. Assume that
τ(F) ∩ PH 6⊂ F .
Consider the ideal F ′ = PH ∩F of subsets of the group H. By transfinite induction
it can be shown that τα(F ′) = PH ∩ τα(F) for all ordinals α.
The sub-invariance of F implies the sub-invariance (and hence auto-invariance) of
F ′. By Lemma 6.6, we get τα(F ′) 6= τ<α(F ′) for each α < |H|+ = |G|+. Then also
τ∗(F) 6= τα(F) 6= τ<α(F) for all α < |G|+.
Theorem 6.1 is proved.
7. The descriptive complexity of the family τ∗(F). In this section given a count-
able group G and a left-invariant monotone subfamily F ⊂ PG, we study the de-
scriptive complexity of the family τ∗(F), considered as a subspace of the power-set
PG endowed with the compact metrizable topology of the Tychonoff product 2G (we
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ON THIN-COMPLETE IDEALS OF SUBSETS OF GROUPS 753
identify PG with 2G by identifying each subset A ⊂ G with its characteristic function
χA : G→ 2 = {0, 1}).
Theorem 7.1. Let G be a countable group and F ⊂ PG be a Borel left-invariant
lower family of subsets of G.
(1) For every ordinal α < ω1 the family τα(F) is Borel in PG.
(2) The family τ∗(F) = τ<ω1(F) is coanalytic.
(3) If τ∗(F) 6= τα(F) for all α < ω1, then τ∗(F) is not Borel in PG.
Proof. Let us recall that G◦ = G \ {e}.
In Section 3 to each subset A ⊂ G we assigned the τ -tree
TA = {s ∈ G<ω◦ : As /∈ F},
where for a finite sequence s = (g0, . . . , gn−1) ∈ Gn◦ ⊂ G<ω◦ we put
As =
⋂
x0,...,xn−1∈2n
gx0
0 . . . g
xn−1
n−1 A.
Consider the subspaces WF ⊂ Tr of PG<ω◦
, consisting of all (well-founded) lower
subtrees of the tree G<ω◦ .
Claim 7.1. The function
T∗ : PG → Tr, T∗ : A 7→ TA
is Borel measurable.
Proof. The Borel measurability of T∗ means that for each open subset U ⊂ Tr the
preimage T−1∗ (U) is a Borel subset of PG. Let us observe that the topology of the space
Tr is generated by the sub-base consisting of the sets
〈s〉+ = {T ∈ Tr : s ∈ T} and 〈s〉− = {T ∈ Tr : s /∈ T}, where s ∈ G<ω◦ .
Since 〈s〉− = Tr \ 〈s〉+, the Borel measurability of T∗ will follow as soon as we check
that for every s ∈ G<ω◦ the preimage T−1∗ (〈s〉+) = {A ∈ PG : s ∈ TA} is Borel.
For this observe that the function
f : PG ×G<ω◦ → PG, f : (A, s) 7→ As,
is continuous. Here the tree G<ω◦ is endowed with the discrete topology.
Since F is Borel in PG, the preimage E = f−1(PG \ F) is Borel in PG × G<ω◦ .
Now observe that for every s ∈ G<ω◦ the set
T−1∗ (〈s〉+) = {A ∈ PG : s ∈ TA} = {A ∈ PG : (A, s) ∈ E}
is Borel.
Theorem 7.1 is proved.
By Theorem 3.1, τ∗(F) = T−1∗ (WF) and τα(F) = T−1∗ (WF−1+α+1) for α < ω1.
Now Theorem 2.1 and the Borel measurablity of the function T∗ imply that the preimage
τ∗(F) = T−1∗ (WF) is coanalytic while τα(F) = T−1∗ (WF−1+α+1) is Borel for every
α < ω1, see [4] (14.4).
Now assuming that τα+1(F) 6= τα(F) for all α < ω1, we shall show that τ∗(F)
is not Borel. In the opposite case, τ∗(F) is analytic and then its image T∗(τ∗(F)) ⊂
⊂ WF under the Borel function T∗ is an analytic subspace of WF, see [4] (14.4).
By Theorem 2.1(4), T∗(τ∗(F)) ⊂ WFα+1 for some infinite ordinal α < ω1 and thus
τ∗(F) = T−1∗ (WFα+1) = τα(F), which is a contradiction.
Theorems 6.1 and 7.1 imply:
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754 T. BANAKH, N. LYASKOVSKA
Corollary 7.1. For any countable non-torsion group G the ideal τ∗(FG) ⊂ PG is
coanalytic but not analytic.
By [4] (26.4), the Σ1
1-Determinacy (i.e., the assumption of the determinacy of all
analytic games) implies that each coanalytic non-analytic space is Π1
1-complete. By
[6], the Σ1
1-Determinacy follows from the existence of a measurable cardinal. So, the
existence of a measurable cardinal implies that for each countable non-torsion group G
the subspace τ∗(FG) ⊂ PG, being coanalytic and non-analytic, is Π1
1-complete.
Question 7.1. Is the space τ∗(FZ) Π1
1-complete in ZFC?
1. Lutsenko Ie., Protasov I. V. Relatively thin and sparse subsets of groups // Ukr. Math. Zh. – 2011. – 63,
№ 2. – P. 216 – 225.
2. Lutsenko Ie., Protasov I. V. Sparse, thin and other subsets of groups // Int. J. Algebra Comput. – 2009.
– 19, №. 4. – P. 491 – 510.
3. Banakh T., Lyaskovska N. Completeness of translation-invariant ideals in groups // Ukr. Mat. Zh. – 2010.
– 62, №. 8. – P. 1022 – 1031.
4. Kechris A. Classical descriptive set theory. – Springer, 1995.
5. Bella A., Malykhin V. I. On certain subsets of a group // Questions Answers Gen. Top. – 1999. – 17,
№ 2. – P. 183 – 197.
6. Martin D. A. Measurable cardinals and analytic games // Fund. Math. – 1970. – 66. – P. 287 – 291.
Received 02.12.10
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| id | umjimathkievua-article-2759 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:29:44Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/839b410c2487a6dc742d4bfb8df44194.pdf |
| spelling | umjimathkievua-article-27592020-03-18T19:35:28Z On thin-complete ideals of subsets of groups Про тонко-повнi iдеали на групах Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. Let $F \subset \mathcal{P}_G$ be a left-invariant lower family of subsets of a group $G$. A subset $A \subset G$ is called $\mathcal{F}$-thin if $xA \bigcap yA \in \mathcal{F}$ for any distinct elements $x, y \in G$. The family of all $\mathcal{F}$-thin subsets of G is denoted by $\tau(\mathcal{F})$. If $\tau(\mathcal{F}) = \mathcal{F}$, then $\mathcal{F}$ is called thin-complete. The thin-completion $\tau*(\mathcal{F})$ of $\mathcal{F}$ is the smallest thin-complete subfamily of $\mathcal{P}_G$ that contains $\mathcal{F}$. Answering questions of Lutsenko and Protasov, we prove that a set $A \subset G$ belongs to $\tau*(G)$ if and only if for any sequence $(g_n)_{n\in \omega}$ of non-zero elements of G there is $n\in \omega$ such that $$\bigcap_{i_0,...,i_n \in \{0, 1\}}g_0^{i_0}...g_n^{i_n} A \in \mathcal{F}.$$ Also we prove that for an additive family $\mathcal{F} \subset \mathcal{P}_G$ its thin-completion $\tau*(\mathcal{F})$ is additive. If the group $G$ is countable and torsion-free, then the completion $\tau*(\mathcal{F}_G)$ of the ideal $\mathcal{F}_G$ of finite subsets of $G$ is coanalytic and not Borel in the power-set $\mathcal{P}_G$ endowed with the natural compact metrizable topology. Нехай $F \subset \mathcal{P}_G$ — iнварiантна злiва нижня сiм’я пiдмножин групи $G$. Пiдмножина $A \subset G$ називається $\mathcal{F}$-тонкою, якщо $xA \bigcap yA \in \mathcal{F}$ для будь-яких рiзних елементiв $x, y \in G$. Сiм’я всiх $\mathcal{F}$-тонких пiдмножин $G$ позначається як $\tau(\mathcal{F})$. Якщо $\tau(\mathcal{F}) = \mathcal{F}$, то $\mathcal{F}$ називається тонко-повною. Тонким поповненням $\tau*(\mathcal{F})$ сiм’ї $\mathcal{F}$ є найменша тонко-повна пiдсiм’я з $\mathcal{P}_G$, що мiстить $\mathcal{F}$. Як вiдповiдь на питання Луценка та Протасова доведено, що множина $A \subset G$ належить сiм’ї $\tau*(G)$ тодi i тiльки тодi, коли для будь-якої послiдовностi $(g_n)_{n\in \omega}$ ненульових елементiв $G$ iснує $n\in \omega$ таке, що $$\bigcap_{i_0,...,i_n \in \{0, 1\}}g_0^{i_0}...g_n^{i_n} A \in \mathcal{F}.$$ Також доведено, що для адитивної сiм’ї $\mathcal{F} \subset \mathcal{P}_G$ її тонке поповнення $\tau*(\mathcal{F})$ є адитивним. Якщо група $G$ злiченна та без скруту, поповнення $\tau*(\mathcal{F}_G)$ iдеалу $\mathcal{F}_G$ скiнченних пiдмножин групи $G$ є коаналiтичним i не борелевим. Institute of Mathematics, NAS of Ukraine 2011-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2759 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 6 (2011); 741-754 Український математичний журнал; Том 63 № 6 (2011); 741-754 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2759/2274 https://umj.imath.kiev.ua/index.php/umj/article/view/2759/2275 Copyright (c) 2011 Banakh T. O.; Lyaskovska N. |
| spellingShingle | Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. On thin-complete ideals of subsets of groups |
| title | On thin-complete ideals of subsets of groups |
| title_alt | Про тонко-повнi iдеали на групах |
| title_full | On thin-complete ideals of subsets of groups |
| title_fullStr | On thin-complete ideals of subsets of groups |
| title_full_unstemmed | On thin-complete ideals of subsets of groups |
| title_short | On thin-complete ideals of subsets of groups |
| title_sort | on thin-complete ideals of subsets of groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2759 |
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