Ultrafilters on G-spaces
For a discrete group G and a discrete G-space X, we identify the Stone-Cech compactifications βG and βX with the sets of all ultrafilters on G and X, and apply the natural action of βG on βX to characterize large, thick, thin, sparse and scattered subsets of X. We use G-invariant partitions and colo...
Gespeichert in:
| Datum: | 2015 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2015
|
| Schriftenreihe: | Algebra and Discrete Mathematics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/154258 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Ultrafilters on G-spaces / O.V. Petrenko, I.V. Protasov // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 254–269. — Бібліогр.: 28 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-154258 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1542582019-06-16T01:27:15Z Ultrafilters on G-spaces Petrenko, O.V. Protasov, I.V. For a discrete group G and a discrete G-space X, we identify the Stone-Cech compactifications βG and βX with the sets of all ultrafilters on G and X, and apply the natural action of βG on βX to characterize large, thick, thin, sparse and scattered subsets of X. We use G-invariant partitions and colorings to define G-selective and G-Ramsey ultrafilters on X. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on ω, the T-points, and study interrelations between these ultrafilters and some classical ultrafilters on ω. 2015 Article Ultrafilters on G-spaces / O.V. Petrenko, I.V. Protasov // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 254–269. — Бібліогр.: 28 назв. — англ. 1726-3255 2010 MSC:05D10, 22A15, 54H20 http://dspace.nbuv.gov.ua/handle/123456789/154258 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
For a discrete group G and a discrete G-space X, we identify the Stone-Cech compactifications βG and βX with the sets of all ultrafilters on G and X, and apply the natural action of βG on βX to characterize large, thick, thin, sparse and scattered subsets of X. We use G-invariant partitions and colorings to define G-selective and G-Ramsey ultrafilters on X. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on ω, the T-points, and study interrelations between these ultrafilters and some classical ultrafilters on ω. |
| format |
Article |
| author |
Petrenko, O.V. Protasov, I.V. |
| spellingShingle |
Petrenko, O.V. Protasov, I.V. Ultrafilters on G-spaces Algebra and Discrete Mathematics |
| author_facet |
Petrenko, O.V. Protasov, I.V. |
| author_sort |
Petrenko, O.V. |
| title |
Ultrafilters on G-spaces |
| title_short |
Ultrafilters on G-spaces |
| title_full |
Ultrafilters on G-spaces |
| title_fullStr |
Ultrafilters on G-spaces |
| title_full_unstemmed |
Ultrafilters on G-spaces |
| title_sort |
ultrafilters on g-spaces |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/154258 |
| citation_txt |
Ultrafilters on G-spaces / O.V. Petrenko, I.V. Protasov // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 254–269. — Бібліогр.: 28 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT petrenkoov ultrafiltersongspaces AT protasoviv ultrafiltersongspaces |
| first_indexed |
2025-07-14T05:54:56Z |
| last_indexed |
2025-07-14T05:54:56Z |
| _version_ |
1837600600688164864 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 2, pp. 254–269
© Journal “Algebra and Discrete Mathematics”
Ultrafilters on G-spaces
O. V. Petrenko, I. V. Protasov
Abstract. For a discrete group G and a discrete G-space
X, we identify the Stone-Čech compactifications βG and βX with
the sets of all ultrafilters on G and X, and apply the natural action
of βG on βX to characterize large, thick, thin, sparse and scattered
subsets of X. We use G-invariant partitions and colorings to define
G-selective and G-Ramsey ultrafilters on X. We show that, in
contrast to the set-theoretical case, these two classes of ultrafilters
are distinct. We consider also universally thin ultrafilters on ω, the
T -points, and study interrelations between these ultrafilters and
some classical ultrafilters on ω.
By a G-space, we mean a set X endowed with the action G × X →
X : (g, x) 7→ gx of a group G. All G-spaces are supposed to be transitive:
for any x, y ∈ X, there exists g ∈ G such that gx = y. If X = G and the
action is the group multiplication, we say that X is a regular G-space.
Several intersting and deep results in combinatorics, topological dy-
namics and topological algebra, functional analysis were obtained by
means of ultrafilters on groups (see [5–7,12,27,28]).
The goal of this paper is to systematize some recent and prove some
new results concerning ultrafilters on G-spaces, and point out the key
open problems.
In sections 1,2 and 3, we keep together all necessary definitions of
filters, ultrafilters and the Stone-Čech compactification βX of the discrete
space X. We extend the action of G on X to the action of βG on βX,
characterize the minimal invariant subsets of βX, define the corona X̌ of
X and the ultracompanions of subsets of X.
2010 MSC: 05D10, 22A15, 54H20.
Key words and phrases: G-space, ultrafilters, ultracompanion, G-selective ul-
trafilter, G-Ramsey ultrafilter, T -point, ballean, asymorphism.
O. V. Petrenko, I. V. Protasov 255
In section 4, we give ultrafilter charecterizations of large, thick, thin,
sparse and scattered subsets of X.
In section 5, we use G-invariant partitions and colorings to define
G-selective and G-Ramsey ultrafilters on X, and show that, in contrast
to the set-theoretical case, these two classes are essentially different.
In section 6, we use countable group of permutatious of ω = {0, 1, . . .}
to define thin ultrafilters on ω. We prove that some classical ultrafilters
on ω (for example, P - and Q-points) are thin ultrafilters.
We conclude the paper, showing in section 7, how all above results
can be considered and interpreted in the frames of general asymptology.
1. Filters and ultrafilters
A family F of subsets of a set X is called a filter if X ∈ F ,∅ /∈ F and
A, B ∈ F , A ⊆ C ⇒ A ∩ B ∈ F , C ∈ F
The family of all fillters on X is partially ordered by inclusion ⊆. A filter
U that is maximal in this ordering is called an ultrafilter. Equivalentely, U
is ultrafilter if A ∪ B ∈ U implies A ∈ U or B ∈ U . This characteristic of
ultrafilters plays the key role in the Ramsey Theory: to prove that, under
any finite partition of X, at least one cell of the partition has a given
property, it suffices to construct an ultrafilter U such that each member
of U has this property.
An ultrafilter U is called principal if {x} ∈ U for some x ∈ X. Non-
principal ultrafilters are called free and the set of all free ultrafilters on
X is denoted by X∗.
We endow a set X with the discrete topology. The Stone-Čech com-
pactification βX of X is a compact Hausdorff space such that X is a
subspace of βX and any mapping f : X → Y to a compact Hausdorff
space Y can be extended to the continuous mapping fβ : βX → Y . To
work with βX, we take the points of βX to be the ultrafilters on X, with
the points of X identified with the principal ultrafilters, so X∗ = βX \ X.
The topology of βX can be defined by stating that the sets of the form
A = {p ∈ βX : A ∈ p}, where A is a subset of X, are base for the open
sets. For a filter ϕ on X, the set ϕ = {A : A ∈ ϕ} is closed in βX, and
each non-empty closed subset of βX is of the form ϕ for an appropriate
filter ϕ on X.
256 Ultrafilters on G-spaces
2. The action of βG on βX
Given a G-space X, we endow G and X with the discrete topologies
and use the universal property of the Stone-Čech compactification to
define the action of βG on βX.
Given g ∈ G, the mapping x 7→ gx : X → βX extends to the
continuous mapping
p 7→ gp : βX → βX.
We note that gp = {gP : P ∈ p}, where gP = {gx : x ∈ P}.
Then, for each p ∈ βX, we extend the mapping g 7→ gp : G → βX to
the continuous mapping
q 7→ qp : βG → βX.
Let q ∈ βG and p ∈ βX. To describe a base for the ultrafilter qp ∈ βX,
we take any element Q ∈ q and, for every g ∈ Q, choose some element
Pg ∈ p. Then
⋃
g∈Q gPg ∈ qp and the family of subsets of this form is a
base for qp.
By the construction, for every g ∈ G, the mapping p 7→ gp : βX → βX
is continuous and, for every p ∈ βX, the mapping q 7→ qp : βG → βX
is continuous. In the case of the regular G-space X, X = G, we get well
known (see [7]) extention of multiplication from G to βG making βG
a compact right topological semigroup. For plenty applications of the
semigroup βG to combinatorics and topological algebra see [6,7, 12,28].
It should be marked that, for any q, r ∈ βG, and p ∈ βX, we have
(qr)p = q(rp) so semigroup βG acts on βX.
Now we define the main technical tool for study of subsets of X by
means of ultrafilters.
Given a subset A of X and an ultrafilter p ∈ βX we define the
p-companion of A by
Ap = {A ∩ Gp} = {gp : g ∈ G, A ∈ gp}.
Systematically, p-companions will be used in section 4. Here we demon-
strate only one appication of p-companion to characterize minimal invari-
ant subsets of βX. A closed subset S of βX is called invariant if g ∈ G
and p ∈ S imply gp ∈ S. Clearly, S is invariant if and only if (βG)p ⊆ S
for each p ∈ S. Every invariant subset S of βX contains minimal by
inclusion invariant subset. A subset M is minimal invariant if and only
if M = (βG)p for each p ∈ S. In the case of the regular G-space, the
minimal invariant subsets coincide with minimal left ideals of βG so the
following theorem generalizes Theorem 4.39 from [7].
O. V. Petrenko, I. V. Protasov 257
Theorem 2.1. Let X be a G-space and let p ∈ βX. Then (βG)p is
minimal invariant if and only if, for every A ∈ p, there exists a finite
subset F of G such that G = FAp.
Proof. We suppose that (βG)p is a minimal invariant subset and take
an arbitary r ∈ βG. Since (βG)rp = (βG)p and p ∈ (βG)p, there exists
qr ∈ βG such that qrrp = p. Since A ∈ qrrp, there exists xr ∈ G such that
A ∈ xrrp so x−1
r A ∈ rp. Then we choose Br ∈ r such that x−1
r A ⊇ Brp
and consider the open cover {Br : r ∈ βG} of βG. By compactness of
βG, there is its finite subcover {Br1
, . . . , Brn
}, so G = Br1
∪ . . . ∪ Brn
.
We put F −1 = {xr1
, . . . , xrn
}. Then G = (FA)p and it suffices to observe
that (FA)p = FAp.
To prove the converse statement, we suppose on the contrary that
(βG)p is not minimal and choose r ∈ βG such that p /∈ (βG)rp. Since
(βG)rp is closed in βX, there exists A ∈ p such that A ∩ (βG)rp = ∅.
It follows that A /∈ qrp for every q ∈ βG. Hence, G \ A ∈ qrp for
each q ∈ βG and, in particular, x(G \ A) ∈ rp for each x ∈ G. By the
assumption, gAp ∈ r for some g ∈ G so A ∈ g−1rp, gA ∈ rp and we get a
contradiction.
3. Dynamical equivalences and coronas
For an infinite discrete G-space, we define two basic equivalences on
the space X∗ of all free ultrafilter on X.
Given any r, q ∈ X∗, we say that r, q are parallel (and write r ‖ q) if
there exists g ∈ G such that q = gr. We denote by ∼ the minimal (by
inclusion) closed in X∗ × X∗ equivalences on X∗ such that ‖⊆∼. The
quotient X∗/ ∼ is a compact Hausdorff space. It is called the corona of
X and is denoted by X̌.
For every p ∈ X∗, we denote by p̌ the class of the equivalence ∼
containing p, and say that p, q ∈ X∗ are corona equivalent if p̌ = q̌. To
detect whether two ultrafilters p, q ∈ X∗ are corona equivalent, we use
G-slowly oscillating functions on X.
A function h : X → [0, 1] is called G-slowly oscillating if, for any ε > 0
and finite subset K ⊂ G, there exists a finite subset F of X such that
diam h(Kx) < ε,
for each x ∈ X \ F , where diam h(Kx) = sup{|h(y) − h(z)| : y, z ∈ Kx}.
Theorem 3.1. Let q, r ∈ X∗. Then q̌ = ř if and only if hβ(r) = hβ(q)
for every G-slowly oscillating function h : X → [0, 1].
258 Ultrafilters on G-spaces
For more detailed information on dynamical equivalences and topolo-
gies of coronas see [14] and [1, 13,17,19].
In the next section, for a subset A of X and p ∈ X∗, we use the corona
p-companion of A
Ap̌ = A∗ ∩ p̌.
4. Diversity of subsets of G-spaces
For a set S, we use the standard notation [S]<ω for the family of all
finite subsets of S.
Let X be a G-space, x ∈ X, A ⊆ X, K ∈ [G]<ω. We set
B(x, K) = Kx ∪ {x}, B(A, K) =
⋃
a∈A
B(a, K),
and say that B(x, K) is a ball of radius K around x. For motivation of
this notation, see the section 7.
Our first portion of definitions concerns the upward directed properties:
A ∈ P and A ⊆ B imply B ∈ P.
A subset A of a G-space X is called
• large if there exists K ∈ [G]<ω such that X = KA;
• thick if, for every K ∈ [G]<ω, there exists a ∈ A such, that Ka ⊆ A;
• prethick if there exists F ∈ [G]<ω such that FA is thick.
In the dynamical terminology [7], large and prethick subsets are known
as syndedic and piecewise syndedic subsets.
Theorem 4.1. For a subset A of an infinite G-space X, the following
statements hold:
(i) A is large if and only if Ap 6= ∅ for each p ∈ X∗;
(ii) A is thick if and only if, there exists p ∈ X∗ such that Ap = Gp.
Proof. (i) We suppose that A is large and choose F ∈ [G]<ω such that
X = FA. Given any p ∈ X∗, we choose g ∈ F such that gA ∈ p. Then
A ∈ g−1p and Ap 6= ∅.
To prove the converse statement, for every p ∈ X∗, we choose gp ∈ G
such that A ∈ gpp so g−1
p A ∈ p. We consider an open covering of X∗
by the subsets {g−1
p A∗ : p ∈ X∗} and choose its finite subcovering
g−1
p1
A∗, . . . , g−1
pn
A∗. Then the set H = X \ (g−1
p1
A∗ ∪ . . . ∪ g−1
pn
A∗) is finite.
O. V. Petrenko, I. V. Protasov 259
We choose F ∈ [G]<ω such that H ⊂ FA and {g−1
p1
, . . . , g−1
pn
} ⊂ F . Then
X = FA so A is large.
(ii) We note that A is thick if and only if X \ A is not large and
apply (i).
Theorem 4.2. A subset A of an infinite G-space X is prethick if and
only if there exists p ∈ X∗ such that A ∈ p and (βG)p is a minimal
invariant subsets of βX.
Proof. The theorem was proved for regular G-spaces in [7, Theorem 4.40].
This proof can be easily adopted to the general case if we use Theorem 2.1
in place of Theorem 4.39 from [7].
Corollary 4.1. For every finite partition of a G-space X, at least one
cell of the partition is prethick.
Remark 4.1. For a subset A of an infinite G-space X, we set
∆(A) = {g ∈ G : g−1A ∩ A is infinite}.
Let P be a finite partition of X. We take p ∈ X∗ such that the set (βG)p
is minimal invariant and choose A ∈ P such that A ∈ p. By Theorem 2.1,
Ap is large in G. If g ∈ Ap then g−1A ∈ p and A ∈ p. Hence, g−1A ∩ A is
infinite, so Ap ⊆ ∆(A) and ∆(A) is large.
In fact, this statement can be essentially strengthened: there is a
function f : N → N such that, for every n-partition P of a G-space X,
there are A ∈ P and F ⊂ G such that G = F∆(A) and |F | 6 f(n).
This is an old open problem (see the surveys [2, 22] whether the above
statement is true with f(n) = n).
In the second part of the section, we consider the downward directed
properties A ∈ P, B ⊆ A imply B ∈ P ) and present some results
from [3,23] A subset A of a G-space X is called
• thin if, for every F ∈ [G]<ω, there exists K ∈ [X]<ω, such that
BA(a, F ) = {a} for each a ∈ A \ K, where BA(a, F ) = B(a, F ) ∩ A;
• sparse if, for every infinite subset Y of X, there exists H ∈ [G]<ω
such that, for every F ∈ [G]<ω, there is y ∈ Y such that BA(y, F ) \
BA(y, H) = ∅;
• scattered if, for every infinite subset Y of X, there exists H ∈
[G]<ω, such that, for every F ∈ [G]<ω, there is y ∈ Y such that
BY (a, F ) \ BY (a, H) = ∅.
260 Ultrafilters on G-spaces
Theorem 4.3. For a subset A of a G-space X, the following statements
hold:
(i) A is thin if and only if |Ap| 6 1 for each p ∈ X∗;
(ii) A is sparse if and only if Ap is finite for every p ∈ X∗;
Let (gn)n∈ω be a sequence in G and let (xn)n∈ω be a sequence in X such
that
(1) {gε0
0 . . . gεn
n xn : εi ∈ {0, 1}} ∩ {gε0
0 . . . gεm
m xm : εi ∈ {0, 1}} = ∅ for
all distinct m, n ∈ ω;
(2) |{gε0
0 . . . gεn
n xn : εi ∈ {0, 1}}| = 2n+1 for every n ∈ ω.
We say that a subset Y of X is a piecewise shifted FP -set if there exist
(gn)n∈ω, (xn)n∈ω satisfying (1) and (2) such that
Y = {gε0
0 . . . gεn
n xn : εn ∈ {0, 1}, n ∈ ω}.
For definition of an FP -set in a group see [7].
Theorem 4.4. For a subset A of a G-space X, the following statements
are equivalent:
(i) A is scattered;
(ii) for every infinite subset Y of A, there exists p ∈ Y ∗ such that Yp is
finite;
(iii) App is discrete in X∗ for every p ∈ X∗;
(iv) A contains no piecewise shifted FP -sets.
Theorem 4.5. Let G be a countable group and let X be a G-space. For
a subset A of X, the following statements hold:
(i) A is large if and only if Ap̌ 6= ∅ for each p ∈ X∗;
(ii) A is thick if and only if p̌ ⊆ A∗ for some p ∈ X∗;
(iii) A is thin if and only if |Ap̌| 6 1 for each p ∈ X∗;
(iv) if Ap̌ is finite for each p ∈ X∗ then A is sparse;
(v) if, for every infinite subset Y of A, there is p ∈ Y ∗ such that Yp̌ is
finite then A is scattered.
Question 4.1. Does the conversion of Theorem 4.5 (iv) hold?
Question 4.2. Does the conversion of Theorem 4.5 (v) hold?
Remark 4.2. If G is an uncountable Abelian group then the corona Ǧ
is a singleton [13]. Thus, Theorem 4.5 does not hold (with X = G) for
uncountable Abelian groups.
O. V. Petrenko, I. V. Protasov 261
5. Selective and Ramsey ultrafilters on G-spaces
We recall (see [4]) that a free ultrafilter U on an infinite set X is
said to be selective if, for any partition P of X, either one cell of P is
a member of U , or some member of U meets each cell of P in at most
one point. Selective ultrafilters on ω are also known under the name
Ramsey ultrafilters because U is selective if and only if, for each colorings
χ : [ω]2 → {0, 1} of 2-element subsets of ω, there exists U ∈ U such that
the restriction χ|[U ]2 ≡ const.
Let G be a group, X be a G-space with the action G×X → X, (g, x) 7→
gx. All G-spaces under consideration are supposed to be transitive: for
any x, y ∈ X, there exists g ∈ G such that gx = y. If G = X and gx is
the product of g and x in G, X is called a regular G-space. A partition P
of a G-space X is G-invariant if gP ∈ P for all g ∈ G, P ∈ P.
Let X be an infinite G-space. We say that a free ultrafilter U on X is
G-selective if, for any G-invariant partition P of X, either some cell of P
is a member of U , or there exists U ∈ U such that |P ∩ U | 6 1 for each
P ∈ P.
Clearly, each selective ultrafilter on X is G-selective. Selective ultrafil-
ters on ω exist under some additional to ZFC set-theoretical assumptions
(say, CH), but there are models of ZFC with no selective ultrafilters on ω.
Let X be a G-space, x0 ∈ X. We put St(x0) = {g ∈ G : gx0 = x0}
and identify X with the left coset space G/St(x0). If P is a G-invariant
partition of X = G/S, S = St(x0), we take P0 ∈ P such that x0 ∈ P0,
put H = {g ∈ G : gS ∈ P0} and note that the subgroup H completely
determines P : xS and yS are in the same cell of P if and only if y−1x ∈ H.
Thus, P = {x(H/S) : x ∈ L} where L is a set of representatives of the
left cosets of G by H.
Theorem 5.1. For every infinite G-space X, there exists a G-selective
ultrafilter U on X in ZFC.
Proof. We take x0 ∈ X, put S = St(x0) and identify X with G/S. We
choose a maximal filter F on G/S having a base consisting of subsets
of the form A/S where A is a subgroup of G such that S ⊂ A and
|A : S| = ∞. Then we take an arbitrary ultrafilter U on G/S such that
F ⊆ U .
To show that U is G-selective, we take an arbitrary subgroup H of G
such that S ⊆ H and consider a partition PH of G/S determined by H.
If |H ∩ A : S| = ∞ for each subgroup A of G such that A/S ∈ F then,
by the maximality of F , we have H/S ∈ F . Hence, H/S ∈ U .
262 Ultrafilters on G-spaces
Otherwise, there exists a subgroup A of G such that A/S ∈ F and
|H ∩ A : S| is finite, |H ∩ A : S| = n. We take an arbitrary g ∈ G and
denote Tg = gH ∩ A. If a ∈ Tg then a−1Tg ⊆ A and a−1Tg ⊆ H. Hence,
a−1Tg/S ⊆ A ∩ H/S so |Tg/S| 6 n. If x and y determine the same coset
by H, then they determine the same set Tg. Then we choose U ∈ U such
that |U ∩ x(H ∩ A/S)| 6 1 for each x ∈ G. Thus, |U ∩ P | 6 1 for each
cell P of the partition PH .
The next theorem characterizes all G-spaces X such that each free
ultrafilter on X is G-selective.
Theorem 5.2. Let G be a group, S be a subgroup of G such that |G :
S| = ∞, X = G/S. Each free ultrafilter on X is G-selective if and only
if, for each subgroup T of G such that S ⊂ T ⊂ G, either |T : S| is finite
or |G : T | is finite.
Applying Theorem 2, we conclude that each free ultrafilter on an
infinite Abelian group G (as a regular G-space) is selective if and only if
G = Z⊕ F or G = Zp∞ × F , where F is finite, Zp∞ is the Prüffer p-group.
In particular, each free ultrafilter on Z is Z-selective.
For a G-space X and n > 2, a coloring χ : [X]n → {0, 1} is said to be
G-invariant if, for any {x1, . . . , xn} ∈ [X]n and g ∈ G, χ({x1, . . . , xn}) =
χ({gx1, . . . , gxn}). We say that a free ultrafilter U on X is (G, n)-Ramsey
if, for every G-invariant coloring χ : [X]n → {0, 1}, there exists U ∈ U
such that χ|[U ]n ≡ const. In the case n = 2, we write “G-Ramsey” instead
of (G, 2)-Ramsey.
Theorem 5.3. For any G-space X, each G-Ramsey ultrafilter on X is
G-selective.
The following three theorems show that the conversion of Theorem 5.3
is very far from truth. Let G be a discrete group, βG is the Stone-Čech
compactification of G as a left topological semigroup, K(βG) is the
minimal ideal of βG.
Theorem 5.4. Each ultrafilter from the closure cl K(βZ) is not Z-
Ramsey.
A free ultrafilter U on an Abelian group G is said to be a Schur
ultrafilter if, for any U ∈ U , there are distinct x, y ∈ U such that x+y ∈ U .
Theorem 5.5. Each Schur ultrafilter U on Z is not Z-Ramsey.
O. V. Petrenko, I. V. Protasov 263
A free ultrafilter U on Z is called prime if U cannot be represented as
a sum of two free ultrafilters.
Theorem 5.6. Every Z-Ramsey ultrafilter on Z is prime.
Question 5.1. Is each Z-Ramsey ultrafilter on Z selective?
Some partial positive answers to this question are in the following two
theorems.
Theorem 5.7. Assume that a free ultrafilter U on Z has a member A
such that |g + A ∩ A| 6 1 for each g ∈ Z \ {0}. If U is Z-Ramsey then U
is selective.
Theorem 5.8. Every (Z, 4)-Ramsey ultrafilter on Z is selective.
All above results are from [9].
Remark 5.1. Let G be an Abelian group. A coloring χ : [G]2 → {0, 1}
is called a PS-coloring if χ({a, b}) = χ({a − g, b + g}) for all {a, b} ∈ [G]2,
equivalently, a + b = c + d implies χ({a, b}) = χ({c, d}). A free ultrafilter
U on G is called a PS-ultrafilter if, for any PS-coloring χ of [G]2, there is
U ∈ U such that χ|[U ]2 ≡ const. The following statements were proved
in [18], see also [6, Chapter 10].
If G has no elements of order 2 then each PS-ultrafilter on G is selective.
A strongly summable ultrafilter on the countable Boolean group B is a
PS-ultrafilter but not selective. If there exists a PS-ultrafilter on some
countable Abelian group then there is a P -point in ω∗.
Clearly, an ultrafilter U on B is a PS-ultrafilter if and only if U is
B-Ramsey. Thus, a B-Ramsey ultrafilter needs not to be selective, but
such an ultrafilter cannot be constructed in ZFC with no additional
assumptions.
6. Thin ultrafilters
A free ultrafilter U on ω is said to be
• P -point if, for any partition P of ω, either A ∈ U for some cell A of
P or there exists U ∈ U such that U ∩ A is finite for each A ∈ P;
• Q-point if, for any partition P of ω into finite subsets, there exists
U ∈ U such that |U ∩ A| 6 1 for each A ∈ P.
264 Ultrafilters on G-spaces
Clearly, U is selective if and only if U is a P -point and a Q-point. It
is well known that the existence of P - or Q-points is independent of the
system of axioms ZFC.
We say that a free ultrafilter U on ω is a T -point if, for every countable
group G of permutations of ω, there is a thin subset U ∈ U in the G-space
ω.
To give a combinatorical characterization of T -points (see [8, 9]), we
need some definitions.
A covering F of a set X is called uniformly bounded if there exists
n ∈ N such that | ∪ {F ∈ F : x ∈ F}| 6 n for each x ∈ X.
For a metric space (X, d) and n ∈ N, we denote Bd(x, n) = {y ∈ X :
d(x, y) 6 n}. A metric d is called locally finite (uniformly locally finite) if,
for every n ∈ N, Bd(x, n) is finite for each x ∈ X (there exists c(n) ∈ N
such that |Bd(x, n)| 6 c(n) for each x ∈ X).
A subset A of (X, d) is called d-thin if, for every n ∈ N there exists a
bounded subset B of X such that Bd(a, n) ∩ A = {a} for each a ∈ A \ B.
Theorem 6.1. For a free ultrafilter U on ω, the following statement are
equivalent:
(i) U is a T -point;
(ii) for any sequence (Fn)n∈ω of uniformly bounded coverings of ω, there
exists U ∈ U such that, for each n ∈ ω, |F ∩ U | 6 1 for all but
finitely many F ∈ Fn;
(iii) for each uniformly locally finite metric d on ω, there is a d-thin
member U ∈ U .
We do not know if a sequence of coverings in (ii) can be replaced to a
sequence of partitions.
Remark 6.1. By [10, Theorem 3], a free ultrafilter U on ω in selective if
and only if, for every metric d on ω, there is a d-thin member of U .
Remark 6.2. By [10, Theorem 8], a free ultrafilter U on ω is a Q-point if
and only if, for every locally finite metric d on ω, there is a d-thin member
of U .
Remark 6.3. It is worth to be mentioned the following metric charac-
terization of P -points: a free ultrafilter U on ω is a P -point if and only if,
for every metric d on ω, either some member of U is bounded or there is
U ∈ U such that (U, d) is locally finite.
O. V. Petrenko, I. V. Protasov 265
A free ultrafilter U on ω is said to be a weak P -point (a NWD-point)
if U is not a limit point of a countable subset in ω∗ (for every injective
mapping f : ω → R, there is U ∈ U such that f(U) is nowhere dense in
R). We note that a weak P -point exists in ZFC.
In the next theorem, we summarize the main results from [8].
Theorem 6.2. Every P -point and every Q-point is a T -point. Under
CH, there exists a T -point which is neither P -point, nor NWD-point,
nor Q-point. For every ultrafilter V on ω, there exist a T -point U and a
mapping f : ω → ω such that V = fβ(U).
Question 6.1. Does there exist a T -point in ZFC?
Question 6.2. Is every weak P -point a T -point?
Question 6.3. (T. Banakh). Let U be a free ultrafilter on ω such that,
for any metric d on ω, some member of U is discrete in (X, d). Is U a
T -point?
A free ultrafilter U on ω is called a Tℵ0
-point if, for each minimal well
ordering < of ω, there is a d<-thin member of U , where d< is the natural
metric on ω defined by <. By Theorem 6.1, each T -point is Tℵ0
-point.
Question 6.4. Is every Tℵ0
-point a T -point? Does there exist a Tℵ0
-point
in ZFC?
Remark 6.4. An ultrafilter U on ω is called rapid if, for any partition
{Pn : n ∈ ω} of ω into finite subsets, there exists U ∈ U such that
|U ∩ Pn| 6 n for every n ∈ ω. Jana Flašková (see [10, p.350]) noticed that,
in contrast to Q-points, a rapid ultrafilter needs not to be a T -point.
Remark 6.5. A family F of infinite subsets of ω is coideal if M ⊆ N, M ∈
F ⇒ N ∈ F and M = N0 ∪ N1, M ∈ F ⇒ N0 ∈ F ∨ N1 ∈ F . Clearly,
the family of all infinite subsets of ω is a coideal.
Following [27], we say that a coideal F is
• P -coideal if, for every decreasing sequence (An)n∈ω in F there is
B ∈ F such that B \ An is finite for each n ∈ ω;
• Q-coideal if, for every A ∈ F and every partition A = ∪n∈ωFn with
Fn finite, there is B ∈ F such that B ⊆ A and |B ∩ Fn| 6 1 for
each n ∈ ω.
266 Ultrafilters on G-spaces
We say that a coideal F is a T -coideal if, for every countable group
G of permutations of ω and every M ∈ F there exists a G-thin subset
N ∈ F such that N ⊆ M .
Generalizing the first statement in Theorem 6.2, we get: every P -
coideal and every Q-coideal is a T -coideal.
Remark 6.6. We say that U ∈ ω∗ is sparse (scattered) if, for every
countable group G of permutations of ω, there is sparse (scattered) in
(G, w) member of U . Clearly, T -point ⇒ sparse ultrafilter ⇒ scattered
ultrafilter.
Question 6.5. Does there exist sparse (scattcred) ultrafilter in ZFC? Is
every weak P -point scattered ultrafilter?
Question 6.6. Let U be a free ultrafilter on ω such that, for every count-
able group G of permutations of ω, the orbit {gU : g ∈ G} is discrete in
ω∗. Is U a weak P -point?
7. The ballean context
Following [21,25], we say that a ball structure is a triple B = (X, P, B),
where X, P are non-empty sets and, for every x ∈ X and α ∈ P , B(x, α)
is a subset of X which is called a ball of radius α around x. It is supposed
that x ∈ B(x, α) for all x ∈ X and α ∈ P . The set X is called the support
of B, P is called the set of radii.
Given any x ∈ X, A ⊆ X and α ∈ P we set
B∗(x, α) = {y ∈ X : x ∈ B(y, α)}, B(A, α) =
⋃
a∈A
B(a, α)
A ball structure B = (X, P, B) is called a ballean if
• for any α, β ∈ P , there exist α′, β′ such that, for every x ∈ X,
B(x, α) ⊆ B∗(x, α′), B∗(x, β) ⊆ B(x, β′);
• for any α, β ∈ P , there exists γ ∈ P such that, for every x ∈ X,
B(B(x, α), β) ⊆ B(x, γ);
A ballean B on X can also be determined in terms of entourages of
the diagonal of X ×X ( in this case it is called a coarse structure [26]) and
O. V. Petrenko, I. V. Protasov 267
can be considered as an asymptotic counterpart of a uniform topological
space.
Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be balleans. A mapping
f : X1 → X2 is called a ≺-mapping if, for every α ∈ P1, there exists
β ∈ P2 such that, for every x ∈ X1, f(B1(x, α)) ⊆ B2(f(x), β). A bijection
f : X1 → X2 is called an asymorphism if f and f−1 are ≺-mappings.
Every metric space (X, d) defines the metric ballean (X,R+, Bd),
where Bd(x, r) = {y ∈ X : d(x, y) 6 r}. By [25, Theorem 2.1.1], a ballean
(X, P, B) is metrizable (i.e. asymorphic to some metric ballean) if and
only if there exists a sequence (αn)n∈ω in P such that, for every α ∈ P ,
one can find n ∈ ω such that B(x, α) ⊆ B(x, αn) for each x ∈ X.
Let G be a group, I be an ideal in the Boolean algebra PG of all
subsets of G, i.e. ∅ ∈ I and if A, B ∈ I and A′ ⊆ A then A ∪ B ∈ I and
A′ ∈ I. An ideal I is called a group ideal if, for all A, B ∈ I, we have
AB ∈ I and A−1 ∈ I. For construction of group ideals see [16].
For a G-space X and a group ideal I on G, we define the ballean
B(G, X, I) as the triple (X, I, B) where B(x, A) = Ax ∪ {x}. In the case
where I is the ideal of all finite subsets of G, we omit I and return to
the notation B(x, A) used from the very beginning of the paper.
The following couple of theorems from [10,15] demonstrate the tight
interrelations between balleans and G-spaces.
Theorem 7.1. Every ballean B with the support X is asymorphic to the
ballean B(G, X, I) for some subgroup G of the group SX of all permutations
of X and some group ideal I on G.
Theorem 7.2. Every metrizable ballean B with the support X is asy-
morphic to the ballean B(G, X, I) for some subgroup G of SX and some
group ideal I on G with countable base such that, for all x, y ∈ X, there
is A ∈ I such that y ∈ Ax.
A ballean B = (X, P, B) is called locally finite (uniformly locally finite)
if each ball B(x, α) is finite (for each α ∈ P , there exists n ∈ N such that
|B(x, α)| 6 n for every x ∈ X.
Theorem 7.3. Every locally finite ballean B with the support X is asy-
morphic to the ballean B(G, X, I) for some subgroup G of SX and some
group ideal I on G with a base consisting of subsets compact in the topology
of pointwise convergence on SX .
Theorem 7.4. Every uniformly locally finite ballean B with the support
X is asymorphic to the ballean B(G, X, [G]<ω) for some subgroup G of
SX .
268 Ultrafilters on G-spaces
We note that Theorem 7.4 plays the key part in the proof of Theo-
rem 6.1.
For ultrafilters on metric spaces and balleans we address the reader
to [12,20,24].
References
[1] T. Banakh, O. Chervak, L. Zdomskyy, On character of points in the Higson corona
of a metric space, Comment. Math. Univ. Carolin. 54 (2013), no 2, 159–178.
[2] T. Banakh, I. Protasov, S. Slobodianiuk, Densities, submeasures and partitions of
groups, Algebra Discrete Math. 17 (2014), no 2, 193–221.
[3] T. Banakh, I. Protasov, S. Slobodianiuk, Scattered subsets of groups, Ukr. Math.
J. 67 (2015), 304–312.
[4] W.W. Comfort, Ultrafilters: some old and some new results, Bull. Amer. Math.
Soc. 83(1977), 417-455.
[5] H. Dales, A. Lau, D. Strauss, Banach Algebras on a Semigroups and Their Com-
pactifications, Mem. Amer. Math. Soc., vol. 2005, 2010.
[6] M. Filali, I. Protasov, Ultrafilters and Topologies on Groups, Math. Stud. Monogr.
Ser., Vol. 13, VNTL Publishers, Lviv, 2011.
[7] N. Hindman, D. Strauss, Algebra in the Stone-Čech compactification: Theory and
Applications, Walter de Gruyter, Berlin, New York, 1998.
[8] O. Petrenko, I. Protasov, Thin ultrafilters, Notre Dame J. Formal Logic 53 (2012),
79–88.
[9] O. Petrenko, I. Protasov, Selective and Ramsey ultrafilters on G-spaces, Notre
Dame J. Formal Logic (to appear).
[10] O.V. Petrenko, I.V. Protasov, Balleans and G-spaces, Ukr. Math. J. 64 (2012),
344–350.
[11] O.V. Petrenko, I.V. Protasov, Balleans and filters, Matem. Stud. 38 (2012), 3–11.
[12] I. Protasov, Combinatorics of Numbers, Math. Stud. Monogr. Ser, Vol. 2, VNTL
Publisher, Lviv, 1997.
[13] I. Protasov, Coronas of balleans, Topology Appl. 149 (2005), 149–160.
[14] I. Protasov, Dynamical equivalences on G
∗, Topology Appl. 155 (2008), 1394–1402.
[15] I.V. Protasov, Balleans of bounded geometry and G-spaces, Mat. Stud. 30 (2008),
61–66.
[16] I. Protasov, Counting Ω-ideals, Algebra Universalis 62 (2010), 339–343.
[17] I. Protasov, Coronas of ultrametric spaces, Comment. Math. Univ. Carolin. 52
(2011), no 2, 303–307.
[18] I. Protasov, Ultrafilters and partitions of Abelian groups, Ukr. Math. J. 53 (2011),
99–107.
[19] I. Protasov, Weak P -points in coronas of G-spaces, Topology Appl. 159 (2012),
587–592.
[20] I. Protasov, Ultrafilters on metric spaces, Topology Appl. 164 (2014), 207–214.
O. V. Petrenko, I. V. Protasov 269
[21] I. Protasov, T. Banakh, Ball Structures and Colorings of Graphs and Groups,
Math. Stud. Monogr. Ser, Vol. 11, VNTL Publisher, Lviv, 2003.
[22] I. Protasov, S. Slobodianiuk, Partitions of groups, Math. Stud., 42 (2014), 115–128.
[23] I. Protasov, S. Slobodianiuk, On the subset combinatorics of G-spaces, Algebra
Discrete Math. 17 (2014), 98–109.
[24] I. Protasov, S. Slobodianiuk, Ultrafilters on balleans, Ukr. Math. J. (to appear).
[25] I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol.
12, VNTL Publishers, Lviv, 2007.
[26] J. Roe, Lectures on Coarse Geometry, Amer. Math. Soc., Providence, R.I. 2003.
[27] S. Todorcevic, Introduction to Ramsey Spaces, Princeton Univ. Press, 2010.
[28] Y. Zelenyuk, Ultrafilters and Topologies on Groups, de Grueter, 2012.
Contact information
O. V. Petrenko,
I. V. Protasov
Department of Cybernetics,
Taras Shevchenko National University,
Volodymyrs’ka St., 64, 01601 Kyiv, Ukraine
E-Mail(s): opetrenko72@gmail.com,
i.v.protasov@gmail.com
Received by the editors: 26.06.2015
and in final form 26.06.2015.
|