Completeness of invariant ideals in groups
We introduce and study various notions of completeness of translation-invariant ideals in groups.
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| author | Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. |
| author_facet | Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. |
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| description | We introduce and study various notions of completeness of translation-invariant ideals in groups. |
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UDC 512
T. Banakh (Ivan Franko Nat. Univ. Lviv, Ukraine; Inst. Mat., Univ. Humanist.-Przyrod. Jana Kochanow-
skiego Kielcach, Poland),
N. Lyaskovska (Ivan Franko Nat. Univ. Lviv, Ukraine)
COMPLETENESS OF INVARIANT IDEALS IN GROUPS
ПОВНОТА IНВАРIАНТНИХ IДЕАЛIВ У ГРУПАХ
We introduce and study various notions of completeness of translation-invariant ideals in groups.
Введено та дослiджено рiзнi поняття повноти iнварiантних iдеалiв у групах.
In this paper we introduce and study various notions of completeness of translation-
invariant ideals in groups. Those completeness notions have topological, measure-theoretic,
or packing nature.
1. The ideal of small subsets in a group. A proper family I ( P(G) of subsets of
a group G is an ideal if I is closed under taking subsets and unions. An ideal I on G is
translation-invariant (briefly, invariant) if xA ∈ I for each x ∈ G and A ∈ I.
Each group G possesses the trivial ideal I0 = {∅} and this is a unique invariant ideal
on a finite group. Each infinite group G possesses the invariant ideal F consisting of all
finite subsets of G and this is the smallest invariant ideal containing a non-empty set.
A less trivial example of an invariant ideal is the ideal of small sets. A subset A of a
group G is called
(i) large if FA = G for some finite set F ⊂ G;
(ii) small if for each large set B ⊂ G the complement B \A is large.
It follows from the definition that the family S of small subsets of G is an invariant
ideal in G.
To give a characterization of small subsets let us introduce the following combinatorial
notion that has a topological flavour. For two subsets B,U ⊂ G the set
IntBU = {x ∈ G : Bx ⊂ U} = G \ (B−1(G \ U))
is called the B-interior of U in G.
Small sets admit a topological characterization.
Theorem 1.1. For a subset A of a group G the following conditions are equiva-
lent:
(1) A is small;
(2) for any finite set F ⊂ G the complement G \ FA is large;
(3) for any finite set F ⊂ G there is a finite set B ⊂ G such that IntB(FA) = ∅;
(4) A is nowhere dense with respect to some left-invariant totally bounded topology
on G.
A topology τ on a group G is totally bounded if each non-empty open set U ∈ τ is
large.
Proof. The equivalence (1)⇔ (2)⇔ (3) was proved in [1, 2], see also Theorem 2.1.
(2)⇒ (4) Assuming that A ⊂ G satisfies (2), observe that the topology
τ = {∅} ∪
{
U ⊂ G : U ⊃ G \ FA for some finite F ⊂ G
}
c© T. BANAKH, N. LYASKOVSKA, 2010
1022 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
COMPLETENESS OF INVARIANT IDEALS IN GROUPS 1023
on G is left-invariant and totally bounded. The set A is closed in (G, τ) and has empty
interior. Otherwise we could find a finite subset F ⊂ G with G \ FA ⊂ A. Then for the
finite set Fe = F ∪ {e} we get FeA = FA ∪A = G, which contradicts (2).
(4) ⇒ (2) Assume that a subset A ⊂ G is nowhere dense in some totally bounded
left-invariant topology τ. Then for every finite F ⊂ G the set FA is nowhere dense in
(G, τ) and hence the complement G \ FA contains some open set U ∈ τ. Since τ is
totally bounded, U is large in G. Consequently, there is a finite subset B ⊂ G such that
G = BU ⊂ B(G \ FA) ⊂ G, witnessing that (2) holds.
By [3] or [4] (1.3, 9.4), on any group G there exists a totally bounded left-invariant
topology τ, which is Hausdorff and extremally disconnected. The latter means that the
closure of each open subset of (G, τ) is open. This result of I. Protasov implies the fol-
lowing description of the ideal of small sets.
Theorem 1.2. The ideal S of small subsets of any groupG coincides with the ideal
of nowhere dense subsets of G endowed with some Hausdorff extremally disconnected
left-invariant topology τs.
Proof. According to [3] or [4] (1.3, 9.4), the group G admits a Hausdorff extremally
disconnected totally bounded left-invariant topology τ. Let τs be the topology on G gen-
erated by the base
B =
{
U \A : U ∈ τ, A ∈ S
}
.
Let us show that the topology τs is totally bounded. Given any basic set U \ A ∈ B,
use the total boundedness of the topology τ in order to find a finite subset F ⊂ G such
that G = FU. Since the set A is small, for the finite set F there is a finite set E ⊂ G such
that E(G \ FA) = G. Now observe that
EF (U \A) ⊃ E(FU \ FA) = E(G \ FA) = G,
witnessing that the set U \A is large.
Applying Theorem 1.1, we conclude that the idealM of nowhere dense subset of the
totally bounded left-topological group (G, τs) lies in the ideal S of small sets.
In order to prove the reverse inclusion S ⊂ M, fix any small set S ⊂ G. It follows
from the definition of the topology τs that S is closed in (G, τs). We claim that S is
nowhere dense. Assuming the converse, we would find a non-empty basic set U \A ⊂ S
with U ∈ τ and A ∈ S. Then the set U ⊂ A ∪ S is small, which contradicts the total
boundedness of the topology τ. This completes the proof of the equality S =M.
Now we check that the topology τs is Hausdorff and extremally disconnected. Since
the topology τ is Hausdorff, so is the topology τs ⊃ τ.
In order to prove the extremal disconnectedness of the topology τs, take any open
subset W ∈ τs and write it as the union W =
⋃
i∈I(Ui \ Ai) of basic sets Ui \ Ai ∈ B,
i ∈ I. Consider the open set U =
⋃
i∈I Ui ∈ τ and its closure U in (G, τ). The extremal
disconnectedness of the topology τ implies that U ∈ τ ⊂ τs. It remains to check that W
is dense in U in the topology τs. In the opposite case, we would find a non-empty basic
set V \ A ∈ B that meets U but is disjoint with W. The set V is open in τ and meets the
closure U of the open set U ∈ τ. Consequently, V ∩ U 6= ∅ and there is an index i ∈ I
such that V ∩ Ui 6= ∅. Then
(Ui ∩ V \A) ∩ (Ui \Ai) ⊂ (Ui ∩ V \A) ∩W = ∅
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
1024 T. BANAKH, N. LYASKOVSKA
implies that V ∩ Ui ⊂ A ∪ Ai. But this is impossible as V ∩ Ui ∈ τ is large while the
set A ∪ Ai is small. This contradiction shows that the set W is dense in U ∈ τs in the
topology τs. Consequently, the closure of the open set W in the topology τs in open and
the left-topological group (G, τs) is extremally disconnected.
Theorem 1.2 is proved.
Problem 1.1. Is the ideal S of small subsets of the group Z equal to the ideal of
nowhere dense subsets of Z with respect to some regular totally bounded left-invariant
topology τ on Z?
In light of this problem, let us remark that each countable Abelian group G contains
a small subset which is dense with respect to any totally bounded group topology on G,
see [2].
2. S-complete ideals. In this section, given an invariant ideal I in a group G, we
introduce an invariant ideal SI called the S-completion of I. For two subsets A,B ⊂ G
we write A ⊂I B if A \B ∈ I and A =I B if A ⊂I B and B ⊂I A.
We define a subset A of a group G to be
(i) I-large if FA =I G for some finite subset F ⊂ G;
(ii) I-small if for each I-large subset L ⊂ G the complement L \A is I-large.
I-Small subsets admit the following characterization.
Theorem 2.1. For a subset A ⊂ G the following conditions are equivalent:
(1) A is I-small;
(2) for every finite subset F ⊂ G the set G \ FA is I-large;
(3) for every finite subset F ⊂ G there is a finite subset B ⊂ G such that
IntB(FA) ∈ I.
Proof. (1) ⇒ (2) Assume that A is I-small. Then for every finite set F ⊂ G the
set FA is I-small (because the ideal SI is invariant) and then its complement G \ FA is
I-large.
(2) ⇒ (3) By (2), for every finite subset F ⊂ G the set G \ FA is I-large. Conse-
quently, B(G \ FA) =I G for some finite set B ⊂ G. We can assume that B = B−1.
It follows that the set I = G \ B(G \ FA) belongs to the ideal I. We claim that
IntB(FA) ⊂ I. Indeed, for every x ∈ IntB(FA), we get Bx ⊂ FA and thus Bx∩ (G \
FA) = ∅. Then x ∈ G \B−1(G \ FA) = I.
(3)⇒ (2) By (3), for every finite set F ⊂ G there is a finite set B = B−1 ⊂ G such
that I = IntB(FA) ∈ I. Then for every x ∈ G \ I the set Bx meets G \ FA and hence
G \ I ⊂ B(G \ FA), which means that B(G \ FA) is I-large.
(2)⇒ (1) Given an I-large subset L ⊂ G we need to show that L\A is I-large. Find
a finite subset F ⊂ G such that FL =I G. By (2), the set G \ FA is I-large and hence
B(G\FA) =I G for some finite set B ⊂ G. Consider the finite set BF and observe that
BF (L \A) ⊃ B(FL \ FA) =I B(G \ FA) =I G
and hence BF (L \A) =I G witnessing that the set L \A is I-large.
Theorem 2.1 is proved.
It follows from the definition that the family SI of all I-small subsets is an invariant
ideal that contains the ideal I. This ideal SI is called the S-completion of the ideal I. An
invariant ideal I in a group G is called S-complete if SI = I.
It is clear that for the smallest ideal I0 = {∅} its S-completion SI0 coincides with
the ideal S of all small subsets of G.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
COMPLETENESS OF INVARIANT IDEALS IN GROUPS 1025
Theorem 2.2. For any invariant ideal I the ideal SI is S-complete. In particular,
the ideal S = SI0 is S-complete.
Proof. The S-completeness of the ideal SI will follow as soon as we check that each
SI-small subset A ⊂ G is I-small. Given a finite subset F ⊂ G, we need to find a finite
subset B ⊂ G such that IntB(FA) ∈ I. Since A is SI-small, we can apply Theorem 2.1
and find a finite subset B1 ⊂ G such that IntB1
(FA) ∈ SI . Applying this theorem once
more, find a finite subset B2 ⊂ G such that IntB2IntB1(FA) ∈ I. Then for the finite
set B = B1B2 ⊂ G we get IntB(FA) ⊂ IntB2
IntB1
FA ∈ I, which means that A is
I-small by Theorem 2.1.
Theorem 2.2 is proved.
Corollary 2.1. For any invariant ideal I ⊂ S we get SI = SS = S.
The definition of the I-small set implies that the S-completeness is preserved by
arbitrary intersections.
Proposition 2.1. For any S-complete ideals Iα, α ∈ A, in a group G the inter-
section I =
⋂
α∈A Iα is an S-complete ideal in G.
The following example shows that in general, the S-completion SI of an invariant
ideal I is strictly larger that the smallest ideal generated by the union S ∪ I.
Example 2.1. There is a set I ⊂ Z that generates an invariant ideal I such that
the ideal SI contains an I-small subset A ⊂ Z that cannot be written as the union
A = As ∪Ai of two subsets As ∈ S and Ai ∈ I.
Proof. Choose a double sequence (In,m)n,m∈ω of pairwise disjoint intervals in R
such that diamIn,m →∞ as n+m→∞. For every n ∈ ω let In =
⋃
m∈ω In,m. Let
I =
⋃
n∈ω
In ∩ n2Z and A =
⋃
n∈ω
In ∩ n(2Z+ 1).
The set I generates the invariant ideal
I = {J ⊂ Z : ∃F ∈ F J ⊂ FI}.
It is easy to check that the set A is I-small but for every J ∈ I the complement A \ J is
not small.
3. The N -completion of an invariant ideal in an amenable group. It turns out
that the S-completion SI of an invariant ideal I in an amenable groupG contains another
interesting ideal NI called the N -completion of I.
Let us recall that a group G is called amenable if G has a Banach measure, which
is a left-invariant probability measure µ : P(G) → [0, 1] defined on the family of all
subsets of G, see [5]. By the Følner condition [5] (0.7), a group G is amenable if and
only if for every finite set F ⊂ G and every ε > 0 there is a finite set E ⊂ G such that
|E4xE| < ε|E| for all x ∈ F. Here A4B = (A \ B) ∪ (B \ A) is the symmetric
difference of two sets A,B. The class of amenable groups contains all locally finite and
all abelian groups and is closed under many operations over groups, see [5] (0.16).
It is clear that for each Banach measure µ on an amenable group G the family of
µ-null sets
Nµ = {A ⊂ G : µ(A) = 0}
is an invariant ideal in G.
The following theorem suggested to the authors by I. V. Protasov shows that the ideals
Nµ form a cofinal subset in the family of all invariant ideals on an amenable group.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
1026 T. BANAKH, N. LYASKOVSKA
Theorem 3.1. Each invariant ideal I in an amenable group G lies in the ideal
Nµ for a suitable Banach measure µ on G.
Proof. The Banach measure µ with Nµ ⊃ I will be constructed as a limit point
of a net of probability measures (µd)d∈D indexed by elements of the directed set D =
= F × N× I endowed with the partial order (F, n,A) ≤ (E,m,B) iff F ⊂ E, n ≤ m,
and A ⊂ B. Here F is the family of finite subsets of the group G.
To each triple d = (F, n,A) ∈ D we assign a probability measure µd on G in the
following manner. Using the Følner criterion of amenability [5] (0.7), find a finite subset
Fd ⊂ G such that
|Fd4xFd|
|Fd|
<
1
n
for all x ∈ F.
Since A belongs to an invariant ideal, we can find a point yd ∈ G \ F−1d A. Now consider
the probability measure µd : P(G)→ [0, 1] defined by
µd(B) =
|B ∩ Fdyd|
|Fd|
for B ⊂ G
and observe that µd(A) = 0.
Each measure µd, d ∈ D, being a function on the family P(G) of all subsets of the
group G, is a point of the Tychonov cube [0, 1]P(G).
By the compactness of [0, 1]P(G), the net of measures (µd)d∈D has a limit point
µ ∈ [0, 1]P(G), see [6] (3.1.23). This is a function µ : P(G) → [0, 1] such that for each
neighborhood O(µ) of µ in [0, 1]P(G) and every d0 ∈ D there is d ≥ d0 in D with
µd ∈ O(µ).
We claim that µ is a Banach measure on G with I ⊂ Nµ. We need to check the
following conditions:
(1) µ(G) = 1;
(2) µ(A ∪B) = µ(A) + µ(B) for any disjoint sets A,B ⊂ G;
(3) µ(xB) = µ(B) for every x ∈ G and B ⊂ G;
(4) µ(B) = 0 for each B ∈ I.
1. Assuming that µ(G) < 1, consider the neighborhood O1(µ) = {η ∈ [0, 1]P(G) :
η(G) < 1} and note that in contains no measure µd, d ∈ D.
2. Assuming that µ(A∪B) 6= µ(A)+µ(B) for some disjoint setsA,B ⊂ G, observe
that O2(µ) =
{
η ∈ [0, 1]P(G) : η(A ∪ B) 6= η(A) + η(B)
}
is an open neighborhood of
µ in [0, 1]P(G) containing no measure µd, d ∈ D.
3. Assume that µ(xB) 6= µ(B) for some x ∈ G and B ⊂ G. Find n ∈ N such that
3
n
< |µ(xB) − µ(B)| and consider the element d0 = ({x−1}, n,∅) ∈ D. Since µ is a
limit point of the net (µd)d∈D, the neighborhood
O3(µ) =
{
η ∈ [0, 1]P(G) : max{|η(A)− µ(A)|, |η(xA)− µ(xA)|} < 1
n
}
of µ contains the measure µd for some d = (F,m,B) ∈ D with d ≥ d0.
Since {x−1} ⊂ F, the definition of the set Fd guarantees that
|Fd4x−1Fd|
|Fd|
<
1
m
≤
≤ 1
n
. Then
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
COMPLETENESS OF INVARIANT IDEALS IN GROUPS 1027
|µd(B)− µd(xB)| =
∣∣|B ∩ Fdyd| − |B ∩ x−1Fdyd|∣∣
|Fd|
≤
≤ |Fdyd4x
−1Fdyd|
|Fd|
=
|Fd4x−1Fd|
|Fd|
<
1
m
≤ 1
n
.
On the other hand, µd ∈ O(µ) implies
|µd(B)− µd(xB)| ≥
≥ |µ(B)− µ(xB)| − |µ(B)− µd(B)| − |µ(xB)− µd(xB)| > 1
n
and this is a contradiction.
4. Take any A0 ∈ I and assume that µ(A0) 6= 0. Find n ∈ N such that µ(A0) >
1
n
and consider the element d0 = (∅, n,A0) ∈ D. Since µ is a limit of the net (µd)d∈D,
the neighborhood O(µ) =
{
η ∈ [0, 1]P(G) : η(A0) >
1
n
}
contains the measure µd for
some d = (F,m,A) ≥ d0 in D. The choice of the point yd guarantees that Fdyd ∩
∩A0 ⊂ Fdyd ∩A = ∅ and hence µd(A0) =
|A0 ∩ Fdyd|
|Fd|
= 0 6> 1
n
.
Theorem 3.1 is proved.
Theorem 3.1 implies that for an invariant ideal I in an amenable group G the inter-
section
NI =
⋂
Nµ⊃I
Nµ
is a well-defined ideal that contains I. In this definition µ runs over all Banach measures
on G such that I ⊂ Nµ. The idealNI will be called theN -completion of the ideal I. An
invariant ideal I is defined to be N -complete if I coincides with its N -completion NI .
The N -completion N{∅} =
⋂
µNµ of the smallest ideal I = {∅} is denoted by N
and called the ideal of absolute null sets. The ideal N is well-defined for each amenable
group G.
The following properties of N -complete ideals follows immediately from the defini-
tion.
Proposition 3.1. Let G be an amenable group.
(1) For every Banach measure µ the ideal Nµ is N -complete.
(2) For any N -complete ideals Iα, α ∈ A, in G the intersection
⋂
α∈A Iα is an
N -complete ideal.
(3) For each invariant ideal I in G the ideal NI is N -complete.
(4) NI = N for each invariant ideal I ⊂ N .
The S- and N -completions relate as follows.
Theorem 3.2. For any invariant ideal I in an amenable group G we get I ⊂
⊂ NI ⊂ SI . In particular, N ⊂ S.
Proof. Assume conversely that some setA ∈ NI is not I-small. This means that there
is a finite set F ⊂ G such that the complement G \ FA is not I-large. Consequently, the
family I ∪ (G \ FA) generates an invariant ideal
J =
{
J ⊂ G : J ⊂ I ∪ E(G \ FA) for some I ∈ I and E ∈ F
}
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
1028 T. BANAKH, N. LYASKOVSKA
By Theorem 3.1, there is a Banach measure µ onG such thatJ ⊂ Nµ. Then µ(G\FA) =
= 0 and hence µ(FA) = 1 and µ(A) > 0. Now we see that I ⊂ NI ⊂ Nµ but A /∈ Nµ,
which contradicts the choice of A ∈ NI .
Theorem 3.2 is proved.
Corollary 3.1. Each S-complete ideal in an amenable group in N -complete.
4. Packing indices. Let I be an invariant ideal of subsets of a group G. To each
subset A of the group G we can assign the I-packing index
I-pack(A) = sup
{
|B| : B ⊂ G is such that {bA}b∈B is disjoint modulo I
}
.
An indexed family {Ab}b∈B of subsets of G is called disjoint modulo the ideal I if
Ab ∩Aβ ∈ I for any distinct indices b, β ∈ B.
If I = {∅} is the trivial ideal on G, then we write pack(A) instead of {∅}-pack(A).
For example, the packing index pack(2Z) of the set A = 2Z of even numbers in the
group G = Z is equal to 2. The same equality I-pack(2Z) = 2 holds for any ideal I
on Z.
It should be mentioned that in the definition of the I-packing index, the supremum
cannot be replaced by the maximum: by [7], each infinite group G contains a subset
A ⊂ G such that pack(A) ≥ ℵ0 but no infinite set B ⊂ G with disjoint {bA}b∈B exists.
To catch the difference between sup and max, for a subset A ⊂ G let us consider a
more informative cardinal characteristic
I-Pack(A) = sup
{
|B|+ : B ⊂ G such that {bA}b∈B is disjoint modulo I
}
.
It is clear that I-pack(A) ≤ I-Pack(A) and
I-pack(A) = sup{κ : κ < I-Pack(A)},
so the value of I-pack(A) can be recovered from that of I-Pack(A). The packing indices
{∅}-pack and {∅}-Pack were intensively studied in [7 – 10].
In fact, the I-packing indices I-pack(A) and I-Pack(A) are partial cases of the pack-
ing indices I-packn(A) and I-Packn(A) defined for every cardinal number n ≥ 2 by the
formulas:
I-packn(A) = sup
{
|B| : B ⊂ G such that
{ ⋂
c∈C
cA : C ∈ [B]n
}
⊂ I
}
and
I-Packn(A) = sup
{
|B|+ : B ⊂ G such that
{ ⋂
c∈C
cA : C ∈ [B]n
}
⊂ I
}
=
= min
{
κ : ∀B ⊂ G
(
|B| ≥ κ ⇒
(
∃C ∈ [B]n with
⋂
c∈C
cA /∈ I
))}
,
where [B]n stands for the family of all n-element subsets of B.
It is clear that I-pack(A) = I-pack2(A) and I-Pack(A) = I-Pack2(A). Also,
I-packn(A) ≤ I-packn+1(A) and I-Packn(A) ≤ I-Packn+1(A) for any finite n ≥ 2.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
COMPLETENESS OF INVARIANT IDEALS IN GROUPS 1029
If I = {∅}, then we shall write packn(A) and Packn(A) instead of {∅}-packn(A)
and {∅}-Packn(A).
The following example show that the difference between the packing indices pack2(A)
and pack3(A) can be infinite.
Example 4.1. The subset A = {n(n − 1)/2 : n ∈ N} of the group Z has
pack2(A) = 1 and pack3(A) = ℵ0. The latter equality follows from the observation that
the family {2m+A}m∈N is 3-disjoint in the sense that (2n+A)∩(2m+A)∩(2k+A) = ∅
for any pairwise distinct numbers n,m, k ∈ N.
5. Packing-complete ideals. It is clear that for each set A ∈ I and finite n ≥ 2
we get I-packn(A) = |G| and I-Packn(A) = |G|+. We shall be interested in ideals for
which the converse implication holds.
Definition 5.1. An invariant ideal I on a group G is called Packn-complete
(resp. packn-complete) if I contains each set A ⊂ G with I-Packn(A) ≥ ℵ0 (resp.
I-packn(A) ≥ ℵ0).
Since Packn(A) ≤ Packn+1(A), each Packn+1-complete ideal is Packn-complete.
Definition 5.2. An invariant ideal I on a group G is called Pack<ω-complete if
I is Packn-complete for every n ≥ 2.
For each ideal I in a group G and every n ≥ 2 we get the implications:
Pack<ω-complete⇒ Packn-complete⇒ Pack2-complete⇒ pack2-complete.
The simplest example of a Pack<ω-complete ideal is the ideal Nµ.
Theorem 5.1. For any Banach measure µ on a group G the ideal Nµ = {A ⊂
⊂ G : µ(A) = 0} is Pack<ω-complete.
Proof. We need to show that the ideal Nµ is Packn-complete for every n ≥ 2. Take
any subset A ⊂ G with Nµ-Packn(A) ≥ ℵ0.
This means that for any natural number m there is a subset Bm of size m such that
µ(b1A ∩ · · · ∩ bnA) = 0 for any distinct b1, . . . bn ∈ Bm. It follows that
1 ≥ µ
( ⋃
b∈Bm
bA
)
≥ 1
n
∑
b∈Bm
µ(bA) =
1
n
∑
b∈Bm
µ(A)
and hence µ(A) ≤ n
|Bm|
=
n
m
. Since this equality holds for any m we conclude that
µ(A) = 0 and hence A ∈ Nµ.
Theorem 5.1 is proved.
Since the intersection of Pack<ω-complete ideals is Pack<ω-complete, Theorems 5.1
and 3.2 imply
Corollary 5.1. An invariant ideal I in an amenable group G is Pack<ω-complete
provided I is N -complete or S-complete.
Thus for each ideal I in an amenable group G and every n ≥ 2 we get the implica-
tions:
S-complete ⇒ N -complete ⇒ Pack<ω-complete ⇒ pack2-complete.
The amenability assumption is essential in Corollary 5.1.
Proposition 5.1. If a group G contains an isomorphic copy of the free group F2
with two generators, then G = A ∪ B is the union of two subsets with infinite pack2-
index. Consequently, no ideal of G is pack2-complete. In particular, the ideal S of small
subsets of G is not pack2-complete.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
1030 T. BANAKH, N. LYASKOVSKA
Proof. Let a, b be the generators of the free subgroup F2 ⊂ G. Choose a subset
S ⊂ G that meets each coset F2 ·g, g ∈ G, at a single point. We shall additionally assume
that the singleton S ∩ F2 = {e} contains the neutral element of G. Each element of F2
can be uniquely written as an irreducible word in the alphabet {a, a−1, b, b−1}. Let Fa be
the set of irreducible words that start with a or a−1. It is clear that F2 = Fa ∪ (F2 \ Fa)
and thusG = A∪B whereA = FaS andB = G\A = (F2 \Fa)S. It remains to observe
that the sets A, B have infinite pack2-index.
Assuming that G contains a pack2-complete ideal I, we conclude that A,B ∈ I and
hence G = A ∪B ∈ I, which is a contradiction.
Proposition 5.1 is proved.
6. Pack<ω-completion of ideals. For an ideal I in a group G let Pack<ω(I) be
the intersection of all invariant Pack<ω-complete ideals J ⊂ P(G) that contain I. If
no such an ideal J exists, then we put Pack<ω(I) = P(G). The family Pack<ω(I)
is called the Pack<ω-completion on I. If Pack<ω(I) 6= P(G), then Pack<ω(I) is a
Pack<ω-complete ideal in G.
By analogy, for every n ≥ 2 we can define the Packn-completion Packn(I) of I.
Corollary 5.1 guarantees that for an invariant ideal I in an amenable group G its packing
completions Pack<ω(I) and Packn(I) are ideals lying in the Pack<ω-complete ideal
NI . On the other hand, for a group G containing a copy of the free group F2, the packing
completions Packn(I) and Pack<ω(I) coincide with P(G).
The following theorem describes the inner structure of the Pack<ω-completion. A
subset A ⊂ P(G) is called additive if A ∪B for any sets A,B ∈ A.
Proposition 6.1. The Pack<ω-completion Pack<ω(I) of an invariant ideal I on
a group G is equal to the union
I<ω1 =
⋃
α<ω1
Iα
where I0 = I and Iα is the smallest additive family containing all subsets A ⊂ G with
infinite index Iβ-Packn(A) for some β < α and n < ω.
Proof. The inclusion Pack<ω(I) ⊃ I<ω1 follows from the fact that each Pack<ω-
complete ideal which contains Iβ for all β < α also contains Iα. To show the equality
we need to prove that I<ω1
is a Pack<ω-complete ideal if I<ω1
6= P(G).
First we show that I<ω1 is Packn-complete for each n ≥ 2. Let A be subset of
G with I<ω1
-Packn(A) ≥ ℵ0. It means that there is countable sequence (Bm)m∈ω of
subsets Bm ∈ [G]m such that for any subset C ∈ [Bm]n the intersection
⋂
c∈C cA ∈
∈ I<ω1 and thus
⋂
c∈C cA ∈ Iα(m,C) for some countable ordinal α(m,C) < ω1. Since
the sequence (Bm) is countable and the sets Bm and C ∈ [Bm]n are finite, there is a
countable ordinal α such that α > α(m,C) for each m and each C. For this ordinal α we
get Iα-Packn(A) ≥ ℵ0. According to the definition of Iα+1 the set A ∈ Iα+1 ⊂ I<ω
and thus I<ω1 is Packn-complete for each n ≥ 2.
Proposition 6.1 is proved.
By analogy we can describe the Packn-completion Packn(I) of I.
Proposition 6.2. The Packn-completion Packn of an invariant ideal I on a group
G is equal to the union
I<ω1 =
⋃
α<ω1
Iα
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
COMPLETENESS OF INVARIANT IDEALS IN GROUPS 1031
where I0 = I and Iα is the smallest additive family that contains all subsets A ⊂ G
with infinite index Iβ-Packn(A) for some β < α.
7. Some open problems. For an invariant ideal I in an amenable group G and every
n ≥ 2 we get the following chain of ideals:
I ⊂ Pack2(I) ⊂ · · · ⊂ Packn(I) ⊂ Packn+1(I) ⊂ · · · ⊂ Pack<ω(I) ⊂ NI ⊂ SI .
For the ideal I = S all these ideals coincide. What happens for the smallest ideal I0 =
= {∅}?
Problem 7.1. Are the ideals Packn(I0), n ≥ 2, in the group Z pairwise distinct?
Do they differ from the ideal Pack<ω(I0)? Is Pack<ω(I0) 6= N?
It should be mentioned thatN 6= S for each countable amenable group, see [11] (5.3).
Problem 7.2. Give a combinatorial description of subsets lying in the ideals
Pack<ω(I0) and Packn(I0), n ≥ 2.
Each ideal of subsets of the group Z can be considered as a subspace of the Cantor
cube {0, 1}Z. So, we can speak about topological properties of ideals.
Problem 7.3. Describe the Borel complexity of the ideals Pack<ω(I0) and
Packn(I0) for n ≥ 2. Are they coanalytic? Is the ideal N of absolute null sets co-
analytic? Borel?
Let us recall that a subspace C of a Polish space X is coanalytic if its complement
X \C is analytic. The latter means that X \C is the continuous image of a Polish space,
see [12]. It easy to show that the ideal S of small subsets of a countable group G is an
Fσδ-subset of P(G).
By Corollary 5.1, for the smallest ideal I0 = {∅} in an amenable group G its
packing completion Pack<ω(I0) ⊂ S. On the other hand, Proposition 5.1 implies that
Pack<ω(I0) = P(G) if G contains a copy of the free group F2.
Problem 7.4 (I. Protasov). Is a group G amenable if Pack<ω(I0) ⊂ S?
8. Acknowledgment. The authors would like to thank Igor Protasov for careful and
creative reading the manuscript and many valuable remarks and suggestions that substan-
tially improved both the results and presentation of the paper.
1. Bella A., Malykhin V. I. On certain subsets of a group // Quest. Answers Gen. Topology. – 1999. – 17,
№ 2. – P. 183 – 197.
2. Bella A., Malykhin V. I. On certain subsets of a group II // Ibid. – 2001. – 19, № 1. – P. 81 – 94.
3. Protasov I. V. Maximal topologies on groups // Sib. Mat. Zh. – 1998. – 39, № 6. – P. 1368 – 1381.
4. Protasov I. V. Algebra in the Stone-Cech compactification: applications to topologies on groups // Algebra
Discrete Math. – 2009. – № 1. – P. 83 – 110.
5. Paterson A. Amenability. – Providence, RI: Amer. Math. Soc., 1988.
6. Engelking R. General topology // Sigma Ser. Pure Math. – Berlin: Heldermann, 1989. – 6.
7. Banakh T., Lyaskovska N. Weakly P -small not P -small subsets in groups // Int. J. Algebra Comput. –
2008. – 18, № 1. – P. 1 – 6.
8. Banakh T., Lyaskovska N. Weakly P -small not P -small subsets in Abelian groups // Algebra Discrete
Math. – 2006. – № 3. – P. 29 – 34.
9. Banakh T., Lyaskovska N., Repovs D. Packing index of subsets in Polish groups // Notre Dame J. Formal
Logic. – 2009. – 50, № 4. – P. 453 – 468.
10. Lyaskovska N. Constructing subsets of a given packing index in Abelian groups // Acta Univ. carol.
Math. et phys. – 2007. – 48, № 2. – P. 69 – 80 (ArXiv:0901.1151).
11. Lutsenko Ie., Protasov I. V. Sparse, thin and other subsets of groups // Int. J. Algebra Comput. – 2009.
– 19, № 4. – P. 491 – 510.
12. Kechris A. Classical descriptive set theory. – Berlin: Springer, 1995.
Received 09.03.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 8
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| id | umjimathkievua-article-2934 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:33:06Z |
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| spelling | umjimathkievua-article-29342020-03-18T19:40:46Z Completeness of invariant ideals in groups Повнота інваріантних ідеалів у групах Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. We introduce and study various notions of completeness of translation-invariant ideals in groups. Введено та досліджено різні поняття повноти інваріантних ідеалів у групах. Institute of Mathematics, NAS of Ukraine 2010-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2934 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 8 (2010); 1022–1031 Український математичний журнал; Том 62 № 8 (2010); 1022–1031 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2934/2618 https://umj.imath.kiev.ua/index.php/umj/article/view/2934/2619 Copyright (c) 2010 Banakh T. O.; Lyaskovska N. |
| spellingShingle | Banakh, T. O. Lyaskovska, N. Банах, Т. О. Лясковська, Н. Completeness of invariant ideals in groups |
| title | Completeness of invariant ideals in groups |
| title_alt | Повнота інваріантних ідеалів у групах |
| title_full | Completeness of invariant ideals in groups |
| title_fullStr | Completeness of invariant ideals in groups |
| title_full_unstemmed | Completeness of invariant ideals in groups |
| title_short | Completeness of invariant ideals in groups |
| title_sort | completeness of invariant ideals in groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2934 |
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