Ideals in $\mathcal{P} _G$ and $\beta G$

TL;DR

This paper explores the structure of ideals in the Stone-Čech compactification of a group, revealing differences based on group properties and characterizing minimal closed ideals containing idempotents.

Contribution

It introduces and characterizes new ideals in ech compactification , especially relating to countable, Abelian, and permutation groups, and analyzes their properties.

Findings

No closed maximal ideals in ech G for countable or Abelian groups.

Existence of closed maximal ideals in ech G for permutation groups of infinite cardinal.

Characterization of the minimal closed ideal containing all idempotents.

Abstract

For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $ \mathcal{P}_G$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $\beta G$ as a right topological semigroup to introduce and characterize some new ideals in $\beta G$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $\beta G$ maximal in $G^*$, $G^* = \beta G \setminus G$, but this statement does not hold for the group $S_\kappa$ of all permutations of an infinite cardinal $\kappa$. We characterize the minimal closed ideal in $\beta G$ containing all idempotents of $G^*$.

Equations

$${\mathcal{I}}^{\wedge }=\{p\in \beta G:p\in G\setminus A\text{ for each }A\in \mathcal{I}\},$$

$$K^{\vee }=\{G\setminus A:A\in \phi \}.$$

$${\Delta }_p(A)=A^{\ast }\cap Gp=\{gp:g\in G,A\in gp\}.$$

$$S{p}_G\subset S{c}_G\subset S{m}_G.$$

$$\{x\in T:gx\ne x,gx\in T,g\in G\}$$

$$\mathcal{J}=\{Z\subseteq G:Z\ne \varnothing ,Z\subset FB\text{ for some }F\in {\mathcal{F}}_G\}.$$

$$\{x_{0n+1},x_{1n+1},\dots ,x_{nn+1}\},\{x_{n+10},x_{n+11},\dots ,x_{n+1n+1}\}$$

$$g_i{X}_{n+1}{Y}_{n+1}\bigcap X_{n+1}{Y}_{n+1}=\varnothing ,g_i{X}_n{Y}_n\bigcap (X_0{Y}_0\cup \cdots \cup X_n{Y}_n)=\varnothing$$

$$\{g_{i_1}{g}_{i_2}\dots g_{i_n}:0\le i_1<i_2<\cdots <g_{i_n}<\omega \}$$

$$\{g_{i_1}{g}_{i_2}\dots g_{i_n}{b}_{i_n}:0\le i_1<i_2<\cdots <i_n<\omega \}$$

Full text

Ideals in $\mathcal{P}_{G}$ and $\beta G$

Igor Protasov and Ksenia Protasova

Abstract. For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $\mathcal{P}_{G}$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $\beta G$ as a right topological semigroup to introduce and characterize some new ideals in $\beta G$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $\beta G$ maximal in $G^{*}$, $G^{*}=\beta G\setminus G$, but this statement does not hold for the group $S_{\kappa}$ of all permutations of an infinite cardinal $\kappa$. We characterize the minimal closed ideal in $\beta G$ containing all idempotents of $G^{*}$.

2010 MSC: 22A15, 03E05

Keywords: Stone-$\check{C}$ech compactification, Boolean algebra, ideal, filter, ultrafilter.

1 Introduction

We recall that a family $\mathcal{I}$ of subsets of a set $X$ is an ideal in the Boolean algebra $\mathcal{P}_{G}$ of all subsets of $G$ if $\emptyset\notin\mathcal{I}$ and $A\in\mathcal{I}$, $B\in\mathcal{I}$, $C\subseteq A$ imply $A\cup B\in\mathcal{I}$, $C\in\mathcal{I}$. A family $\varphi$ of subsets of $G$ is a filter if the family $\{X\setminus A:A\in\varphi\}$ is an ideal. A filter maximal by the inclusion is called an ultrafilter.

For an infinite group $G$, an ideal $\mathcal{I}$ in $\mathcal{P}_{G}$ is called left (right) translation invariant if $gA\in\mathcal{I}$ ($Ag\in\mathcal{I}$) for all $g\in G$, $A\in\mathcal{I}$. If $\mathcal{I}$ is left and right translation invariant then $\mathcal{I}$ is called translation invariant. Clearly, each left (right) translation invariant ideal of $G$ contains the ideal $\mathcal{F}_{G}$ of all finite subsets of $G$. An ideal $\mathcal{I}$ in $\mathcal{P}_{G}$ is called a group ideal if $\mathcal{F}_{G}\subseteq\mathcal{I}$ and if $A\in\mathcal{I}$, $B\in\mathcal{I}$ then $AB^{-1}\in\mathcal{I}$.

Now we endow $G$ with the discrete topology and identify the Stone-$\check{C}$ech compactifi-cation of $G$ with the set of all ultrafilters on $G$ and denote $G^{*}=\beta G\setminus G$, so $G^{*}$ is the set of all free ultrafilters on $G$. Then the family $\{\overline{A}:A\subseteq G\}$, where $\overline{A}=\{p\in\beta G:A\in p\}$ forms the base for the topology of $\beta G$. Given a filter $\varphi$ on $G$, we denote $\overline{\varphi}=\cap\{\overline{A}:A\in\varphi\}$, so $\varphi$ defines the closed subset $\overline{\varphi}$ of $\beta G$, and each non-empty closed subset $K$ of $\beta G$ can be defined in this way: $K=\overline{\varphi}$ where $\overline{\varphi}=\{A\subseteq G:K\subseteq\overline{A}\}$.

We use the standard extension [4, Section 4.1] of the multiplication on $G$ to the semigroup multiplication on $\beta G$ such that, for each $p\in\beta G$, the mapping $x\longmapsto xp$, $x\in\beta G$ is continuous, and for each $g\in G$, the mapping, $x\longmapsto gx$, $x\in\beta G$ is continuous. Given two ultrafilters $p,q\in\beta G$, we choose $P\in p$ and, for each $x\in P$, pick $Q_{x}\in q$. Then $\bigcup_{x\in P}\ xQ_{x}\in pq$ and the family of all these subsets forms the base of the product $pq$.

It follows directly from the definition of the multiplication in $\beta G$ that $G^{*}$, $\overline{G^{*}G^{*}}$ are ideals in $\beta G$, and $G^{*}$ is the unique maximal closed ideal in $\beta G$. By Theorem 4.44 from [4], the closure $\overline{K(\beta G)}$ of the minimal ideal $K(G)$ of $\beta G$ is an ideal, so $\overline{K(\beta G)}$ is the smallest closed ideal in $\beta G$. For the structure of $\overline{K(\beta G)}$ and some other ideals in $\beta G$ see [4, Sections 4,6].

For an ideal $\mathcal{I}$ in $\mathcal{P}_{G}$, we put

[TABLE]

and use the following observations:

• •

$\mathcal{I}$ is left translation invariant if and only if $\mathcal{I}^{\wedge}$ is a left ideal of the semigroup $\beta G$ ;

• •

$\mathcal{I}$ is right translation invariant if and only if $(\mathcal{I}^{\wedge})G\subseteq\mathcal{I}^{\wedge}$.

We use also the inverse to ∧ mapping ∨. For a closed subset $K$ of $\beta G$, we take a filter $\varphi$ on $G$ such that $K=\overline{\varphi}$ and put

[TABLE]

In section 2, we use a classification of subsets of a group by their size to define some special ideals in $\mathcal{P}_{G}$. In section 3, we study ideals of $\beta G$ between $\overline{G^{*}G^{*}}$ and $G^{*}$. In section 4, we study ideals between $\overline{K(\beta G)}$ and $\overline{G^{*}G^{*}}$ and characterize the minimal closed ideal in $\beta G$ containing all idempotents of $G^{*}$.

2 Diversity of subsets of a group

In what follows, all group are supposed to be infinite. Let $G$ be a group with the identity $e$. We say that a subset $A$ of $G$ is

• •

large if $G=FA$ for some $F\in\mathcal{F}_{G}$;

• •

small if $L\setminus A$ is large for every large subset $L$;

• •

thin if $gA\cap A$ is finite for each $g\in G\setminus\{e\}$;

• •

$n$-thin, $n\in{\mathbb{N}}$ if, for every distinct elements $g_{0},\dots,g_{n}\in G$, the set $g_{0}A\cap\dots\cap g_{n}A$ is finite;

• •

sparse if, for every infinite subset $X$ of $G$, there exists a finite subset $F\subset X$ such that $\bigcap_{g\in F}gA$ is finite.

All above definitions can be unified with usage the following notion [16]. Given a subset $A$ of a group $G$ and an ultrafilter $p\in G^{*}$, we define a $p$-companion of $A$ by

[TABLE]

Then the following statement hold [16]:

• •

$A$ is large if and only if $\Delta_{p}(A)\neq\emptyset$ for each $p\in G^{*}$;

• •

$A$ is small if and only if, for every $p\in G^{*}$ and every $F\in\mathcal{F}_{G}$, we have $\Delta_{p}(FA)\neq Gp$;

• •

$A$ is thin if and only if, $\Delta_{p}(A)\leq 1$ for every $p\in G^{*}$;

• •

$A$ is $n$-thin if and only if, $\Delta_{p}(A)\leq n$ for every $p\in G^{*}$;

• •

$A$ is sparse if and only if, $\Delta_{p}(A)$ is finite for each $p\in G^{*}$.

Following [1], we say that a subset $A$ of $G$ is scattered if, for every infinite subset $X$ of $A$, there is $p\in X^{*}$ such that $\Delta_{p}(X)$ is finite. Equivalently [1, Theorem 1], $A$ is scattered if each subset $\Delta_{p}(A)$ is discrete in $G^{*}$.

We denote by $Sm_{G}$, $Sc_{G}$, $Sp_{G}$ the families of all small, scattered and sparse subsets of a group $G$. These families are translation invariant ideals in $\mathcal{P}_{G}$ (see [16, Proposition 1 ]), and for every group $G$, the following inclusions are strict [16, Proposition 12]

[TABLE]

We say that a subset $A$ of $G$ is finitely thin if $A$ is $n$-thin for some $n\in\mathbb{N}$. The family $FT_{G}$ of all finitely thin subsets of $G$ is a translation invariant ideal in $\mathcal{P}_{G}$ which contains the ideal $<T_{G}>$ generated by the family of all thin subsets of $G$. By [6, Theorem 1.2] and [14, Theorem 3], if $G$ is either countable or Abelian and $|G|<\aleph_{\omega}$ then $FT_{G}=<T_{G}>$. By [14, Example 3], there exists a group $G$ of cardinality $\aleph_{\omega}$ such that $<T_{G}>\subset FT_{G}$.

Clearly, $FT_{G}\subseteq Sp_{G}$. In the next section, we show that $FT_{G}\subset Sp_{G}$ for every group $G$

Theorem 2.1. For every group $G$, we have $Sm_{G}^{\wedge}=\overline{K(\beta G)}$.

This is Theorem 4.40 from [4] in the form given in [10, Theorem 12.5].

Theorem 2.2. For every group $G$, the following statements hold:

$(i)$* $Sp_{G}^{\wedge}=\overline{G^{*}G^{*}}$;*

$(ii)$* for a subset $A$ of $G$, $\overline{G^{*}G^{*}}\subset\overline{A}$ if and only if, for any infinite subsets $X,Y$ of $G$, there exist $x\in X$, $y\in Y$ such that $xy\in A$, $yx\in A$. *

The statement $(i)$ is Theorem 10 from [2], $(ii)$ is a recent result [11].

For more delicate classifications of subsets of groups and $G$-spaces see [5], [9], [15].

3 Between $\overline{G^{*}G^{*}}$ and $G^{*}$

Theorem 3.1. For every group $G$, the following statements hold:

$(i)$* if $\mathcal{I}$ is a left translation invariant ideal in $\mathcal{P}_{G}$ and $\mathcal{I}\neq\mathcal{F}_{G}$ then there exists a left translation invariant ideal $\mathcal{J}$ in $\mathcal{P}_{G}$ such that $\mathcal{F}_{G}\subset\mathcal{J}\subset\mathcal{I}$ and $\mathcal{J}\subset Sp_{G}$;*

$(ii)$* if $\mathcal{I}$ is a right translation invariant ideal in $\mathcal{P}_{G}$ and $\mathcal{I}\neq\mathcal{F}_{G}$ then there exists a right translation invariant $\mathcal{J}$ in $\mathcal{P}_{G}$ such that $\mathcal{F}_{G}\subset\mathcal{J}\subset\mathcal{I}$;*

$(iii)$* if $G$ is either countable or Abelian and $\mathcal{I}$ is a translation invariant ideal in $\mathcal{P}_{G}$ such that $\mathcal{I}\neq\mathcal{F}_{G}$ then there exists a translation invariant ideal $\mathcal{J}$ in $\mathcal{P}_{G}$ such that $\mathcal{F}_{G}\subset\mathcal{J}\subset\mathcal{I}$ and $\mathcal{J}\subset Sp_{G}$;*

Proof. We use the following auxiliary statement [8. Example 3]:

$(*)$ if a countable group $\Gamma$ acts on a set $X$ then, for every infinite subset $A$ of $X$, there exists a countable subset $T\subset A$ such that the set

[TABLE]

is finite.

$(i)$ We suppose that $G$ is countable, put $\Gamma=G$, $X=G$ and consider the action of $G$ on $G$ by the left shifts. We take an infinite subset $A\in\mathcal{I}$ and apply $(*)$ to choose a countable thin subset $T\subseteq A$. We partition $T$ into two infinite subsets $T=B\cup C$ and denote

[TABLE]

Clearly, $\mathcal{J}$ is a left translation invariant ideal and $\mathcal{J}\neq\mathcal{F}_{G}$. Since $gT\cap T$ is finite for every $g\in G\setminus\{e\}$, we have $C\notin\mathcal{I}$.

Hence, $\mathcal{F}_{G}\subset\mathcal{J}\subset\mathcal{I}$. By the choice of $T$, each subset $Y\in\mathcal{J}$ is a finite union of thin subsets, so $\mathcal{J}\subset Sp_{G}$.

If $G$ is an arbitrary infinite group then we take a countable subset $A\in\mathcal{I}$, consider the subgroup $H$ of $G$ generated by $A$ and denote by $\mathcal{I}_{H}$ the restriction of $\mathcal{I}$ to $H$, $\mathcal{I}_{H}=\{Y\cap H:Y\cap H\in\mathcal{I}\}$. By above paragraph, there exists a left invariant ideal $\mathcal{J}^{\prime}$ in $\mathcal{P}_{H}$ such that $\mathcal{F}_{H}\subset\mathcal{J}^{\prime}\subset\mathcal{I}_{H}$, $\mathcal{J}^{\prime}\subset Sp_{H}$. Then we put $\mathcal{J}=\{Y\subseteq G:Y\neq\emptyset,\ Y\subseteq FZ,\ F\in\mathcal{F}_{G},\ Z\in\mathcal{J}^{\prime}\}$.

$(ii)$ We repeat the proof of $(i)$ with the action of $G$ on $G$ by the right shifts.

$(iii)$ If $G$ is countable then we put $\Gamma=G\times G$, consider the action of $\Gamma$ on $G$ defined by $(g,h)x=g^{-1}xh$ and repeat the proof of $(i)$ in the countable case. If $G$ is Abelian then we apply $(i)$ directly. $\ \ \ \ \Box$

Theorem 3.2. For every group $G$, the following statements hold:

$(i)$* if $L$ is a closed left ideal in $\beta G$ such that $L\subset G^{*}$ then there exists a closed left ideal $L^{\prime}$ of $\beta G$ such that $L\subset L^{\prime}\subset G^{*}$, $\overline{G^{*}G^{*}}\subset L^{\prime}$;*

$(ii)$* if $R$ is a closed subset of $G^{*}$ such that $R\neq G^{*}$ and $RG\subseteq R$ then there exists a closed subset $R^{\prime}$ of $G^{*}$ such that $R\subset R^{\prime}\subset G^{*}$, $R^{\prime}G\subseteq R$;*

$(iii)$* if $G$ is either countable or Abelian and $I$ is a closed ideal in $\beta G$ such that $I\subset G^{*}$ then there exists a closed ideal $I^{\prime}$ in $\beta G$ such that $I\subset I^{\prime}\subset G^{*}$, $\overline{G^{*}G^{*}}\subset I$.*

Proof. $(i)$ We put $\mathcal{I}=L^{\vee}$, apply Theorem 3.2 (i) and set $L^{\prime}=\mathcal{J}^{\vee}$. Then $L^{\prime}$ is a left ideal in $\beta G$ and $L\subset L^{\prime}\subset G^{*}$. Since $\mathcal{J}\subset Sp_{G}$, by Theorem 2.2, we have $\overline{G^{*}G^{*}}\subset L^{\prime}$.

$(ii)$ We put $\mathcal{I}=R^{\vee}$, and note that $\mathcal{I}$ is right translation invariant. We apply Theorem 3.2(ii) and set $R^{\prime}=\mathcal{J}^{\wedge}$.

$(iii)$ We put $\mathcal{I}=I^{\vee}$, apply Theorem 3.2 (iii) and set $I^{\prime}=\mathcal{J}^{\wedge}$. Then $I^{\prime}$ is a left ideal in $\beta G$ and $I^{\prime}G\subseteq I^{\prime}$. Since $\mathcal{J}\subset Sp_{G}$, we have $I^{\prime}G\subseteq I^{\prime}$ so $I^{\prime}$ is a right ideal.

Remark 3.1. If $\mathcal{I}$ is a group ideal in $\mathcal{P}_{G}$ then, by [13], $\mathcal{I}^{\wedge}$ is an ideal in $\beta G$. By [12, Theorem 4], if $G$ is either contable or Abelian and $\mathcal{I}$ is a group ideal such that $\mathcal{I}\neq\mathcal{F}_{G}$ then there exists a group ideal $\mathcal{J}$ in $\mathcal{P}_{G}$ such that $\mathcal{F}_{G}\subset\mathcal{J}\subset\mathcal{I}$. If $A$ is an infinite subset of $G$ then the subset $AA$ is not sparse (put $X=A^{-1}$ in corresponding definition). It follows that if $\mathcal{I}$ is a group ideal and $\mathcal{I}\subseteq Sp_{G}$ then $\mathcal{I}=\mathcal{F}_{G}$ .

For a cardinal $\kappa$, $S_{\kappa}$ denotes the group of all permutations of $\kappa$.

Theorem 3.3. For every infinite cardinal $\kappa$, there exists a closed ideal $I$ in $\beta S_{\kappa}$ such that

$(i)$* $S_{\kappa}^{*}S_{\kappa^{*}}\subset I$;*

$(ii)$* if $M$ is a closed ideal in $\beta S_{\kappa}$ and $I\subseteq M\subseteq G^{*}$ then either $M=I$ or $M=S_{\kappa}^{*}$*

Proof. We take an arbitrary closed subset $X=\{x_{i}:i<\omega\}$ of $\kappa$ and define a permutation $f_{i}$ of $\kappa$ by $f_{i}(x_{2i})=x_{2i+1}$, $\ f_{i}(x_{2i+1})=x_{2i}$ and $f_{i}(x)=x$ for all $x\in\kappa\setminus\{x_{2i},x_{2i+1}\}$. We put $T=\{f_{i}:i<\omega\}$ and denote by $\mathcal{I}$ the smallest translation invariant ideal in $\mathcal{P}_{S_{\kappa}}$ containing $T$.

We note that $|gT\cap T|\leq 1$ for every $g\in G\setminus\{e\}$. Hence, $T$ is thin and $\mathcal{I}\subseteq Sp_{S_{\kappa}}$. To see that $\mathcal{I}\subset Sp_{S_{\kappa}}$, we observe that each element of $\mathcal{I}$ is a countable subset of $S_{\kappa}$, but there are uncountable thin subsets of $S_{\kappa}$.

We assume that there is a translation invariant ideal $\mathcal{J}$ in $\mathcal{P}_{S_{\kappa}}$ such that $\mathcal{F}_{G}\subset\mathcal{J}\subset\mathcal{I}$. Then there exists a countable subset $T_{1}$ of $T$ such that $T_{1}\in\mathcal{J}$, $T\setminus T_{1}$ is infinite and $T\setminus T_{1}\notin\mathcal{I}$ . We denote $T_{2}=T\setminus T_{1}$ and take a partition $\omega=W_{1}\cup W_{2}$ such that $T_{1}=\{f_{i}:i\in W_{1}\}$, $T_{2}=\{f_{i}:i\in W_{2}\}$. We fix an arbitrary bijection $\varphi:W_{1}\rightarrow W_{2}$ and define a permutation $h$ of $\kappa$ by the following rule.

If $x\in\kappa\setminus X$ then $f(x)=x$.

If $x\in X$ then we take $i<\omega$ such that $x\in\{x_{2i},x_{2i+1}\}$.

If $i\in W_{1}$ then we choose $j\in W_{2}$ such that $j=\varphi(i)$ and put h(x_{2i})=x_{2j},\ \ h(x_{2i+1})=x_{2j+1}.\

If $i\in W_{2}$ then we take k=\varphi^{-1}(i)\ and put $\ h(x_{2i})=x_{2k}$, $\ \ h(x_{2i+1})=x_{2k+1}$.

By the construction of $h$, we have $hT_{1}h=T_{2}$. Since $\mathcal{J}$ is translation invariant, we have $T_{2}\in\mathcal{J}$, $T\in\mathcal{J}$ so $\mathcal{J}=\mathcal{I}$ contradicting $\mathcal{J}\subset\mathcal{I}$.

To conclude the proof, we put $I=\mathcal{I}^{\wedge}$. By the construction of $\mathcal{I}$, $I$ is a closed ideal in $\beta S_{\kappa}$ satisfying $(i)$, $(ii)$. $\ \ \ \ \Box$

Remark 3.2. If $I$ is a subset of $\beta G$ such that $\overline{G^{*}G^{*}}\subseteq I$ then $I$ is an ideal in $G^{*}$. It follows that between $\overline{G^{*}G^{*}}$ and $G^{*}$ there are no maximal closed ideals in $G^{*}$.

Lemma 3.1. Let $\{A_{n}:n<\omega\}$ be a family of sparse subsets of a group $G$, $A=\cup_{n<\omega}A_{n}$. Then $A$ is sparse provided that the following two conditions are satisfied :

$(i)$* for every $F\in\mathcal{F}_{G}$ there exists $K\in\mathcal{F}_{G}$ such that $F(A_{i}\setminus K)\cap F(A_{j}\setminus K)=\emptyset$ for all $i<j<\omega$;*

$(ii)$* for every $g\in G\setminus\{e\}$, there exists $m\in\omega$ such that $gx\notin A$ for each $x\in\cup_{n>m}A_{n}$.*

Proof. We take an arbitrary ultrafilter $p\in G^{*}$ and prove that $\Delta_{p}(A)$ is finite. We split the proof in two cases.

Case $\Delta_{p}(A_{n})\neq\emptyset$ for some $n<\omega$. Since $A_{n}$ is sparse, we have $\Delta_{p}(A_{n})=T_{p}$ for some $F\in\mathcal{F}_{G}$ . We show that $\Delta_{p}(A)=\Delta_{p}(A_{n})$. Clearly, $\Delta_{p}(A_{n})\subseteq\Delta_{p}(A)$. We take an arbitrary $g\in G\setminus T$, put $F=T\cup\{g\}$ and choose $K$ satisfying $(i)$. Then $A_{n}\notin gp$ and $\cup\{A_{i}:i<\omega,\ i\neq n\}\in gp$ so $gp\notin\Delta_{p}(A)$ and $\Delta_{p}(A)\subseteq\Delta_{p}(A_{n})$.

Case $\Delta_{p}(A_{n})=\emptyset$ for each $n<\omega$. We show that $|\Delta_{p}(A)|\leq 1$. Assume the contrary : $A\in g_{1}p$, $A\in g_{2}p$ for distinct $g_{1}g_{2}\in G$. We denote $g=g_{1}g_{2}^{-1}$, $q=g_{1}p$. Then $\Delta_{p}(A)=\Delta_{q}(A)$, $\ A\in q$ and $\ g^{-1}A\in q$. We choose $m$ satisfying $(ii)$. Since $\Delta_{q}(A_{n})=\emptyset$ for each $n<\omega$, we have $\cup_{n>m}(A_{n})\in q$ but $A\notin gq$ and we get a contradiction with $\ g^{-1}A\in q$ . $\ \ \ \ \Box$

Theorem 3.4. For every group $G$, we have ${\it FT_{G}\subset Sp_{G}}$ so $\overline{G^{*}G^{*}}\subset{\it FT_{G}^{\wedge}}$.

*Proof. * Since ${\it FT_{G}\subseteq Sp_{G}}$, we should find a sparse subset $A$ of $G$ which is not $n$-thin for each $n\in\mathbb{N}$. Passing to a countable subgroup of $G$, we suppose that $G$ itself is countable.

We construct $A$ in the form $A=\cup_{n<\omega}A_{n}$ to satisfy the conditions $(i)$, $(ii)$ of Lemma 3.1 and such that $A_{n}$ is not $n$-thin for each $n>0$. For each $n<\omega$, we construct $An$ in the form $A_{n}=\cup_{i<\omega}K_{n}x_{n_{i}}$ for some finite $K_{n}$, $|K_{n}|=n+1$. $e\in K_{n}$ and some sequence $(x_{n_{i}})_{i<\omega}$ in $G$.

We enumerate $G=\{g_{n}:n<\omega\}$, $g_{0}=e$ and denote $F_{n}=\{g_{n},$ $\dots,g_{n}\}$. We put $K_{0}=\{e\}$, $g_{00}=e$. Assume that we have chosen $K_{0},\dots,K_{n}$ and $\{x_{00},x_{01},\dots x_{0n},$ \ \dots\, $x_{n0},x_{n1},\dots x_{nn}\},$ so that following conditions are satisfied:

(1) $\ \ \{g_{n},\ \dots,\ g_{n}\}\cap K_{n}K_{n}^{-1}=\emptyset$;

(2) $\ \ F_{n}K_{m}x_{mn}\cap F_{n}K_{m}$ \{x_{m0},\dots x_{m\ n-1}\}=\emptyset,\ $\ 0\leq m\leq n$;

(3) $\ \ F_{n}K_{n}\{x_{n0},\dots,x_{nn}\}\cap F_{n}K_{m}$ \{x_{m0},\dots,x_{mn}\}=\emptyset,\ $\ 0\leq m<n$;

(4) $\ \ F_{n}K_{n}x_{ni}\cap F_{n}K_{n}$ x_{nj}=\emptyset,\ $\ 0\leq i<j\leq n$.

Then we choose $K_{n+1}$ and

[TABLE]

to satisfy (1), (2), (3), (4) with $n+1$ in place of $n$. After $\omega$ steps, we get the family $\{A_{n}:n<\omega\}$.

We put $K=K_{0}\{x_{00},x_{01},\dots x_{0n}\}\bigcup\dots\bigcup K_{n}\{x_{n0},x_{n1},\dots,x_{nn}\}$ .

By (2), (3), (4), $F_{n}(A_{i}\setminus K)\cap F_{n}(A_{j}\setminus K)=\emptyset$ for all $i<j<\omega.$ Hence, the condition $(i)$ of Lemma 3.1 is satisfied.

By (1), (3), (4), $g_{n}(\cup_{i>n}A_{i})\cap A=\emptyset$, so the condition $(ii)$ is satisfied. By Lemma 3.1, $A$ is sparse. For every $n<\omega$, the subsets $\{g\{x_{n_{i}}:i<\omega\}:g\in K_{n}\}$ of $A_{n}$ are pairwise disjoint. Since $|K_{n}|=n+1$, $A_{n}$ is not $n$-thin. $\ \ \ \ \Box$

For subsets $X,Y$ of a group $G$, we say that the product $XY$ is an $n$-stripe if $|X|=n$, $n\in\mathbb{N}$ and $|X|=\omega$. It is easy to see that a subset $A$ of $G$ is $n$-thin if and only if $A$ has no $(n+1)$-stripes. Thus, $p\in FT_{G}^{\wedge}$ is and only if each member $P\in p$ has an $n$-stripe for every $n\in\mathbb{N}$.

We say that $XY$ is an $(n,m)$-rectangle if $|X|=n$, |Y|=m\, $\ n,m\in\mathbb{N}$. We say that a subset $A$ of $G$ has bounded rectangles if there is $n\in\mathbb{N}$ such that $A$ has no $(n,n)$-rectangles (and so $(n,m)$-rectangles for each $m>n$).

We denote by $BR_{G}$ the family of all subsets of $G$ with bounded rectangles.

Theorem 3.5. For a group $G$, the following statements hold:

$(i)$* $BR_{G}$ is a left translation invariant ideal in $\mathcal{P}_{G}$ ;*

$(ii)$* $BR_{G}^{\wedge}$ is a closed ideal in $\beta G$ and $p\in BR_{G}^{\wedge}$ if and only if each member $P\in p$ has an $(n,n)$-rectangle for every $n\in\mathbb{N}$;*

$(iii)$* $BP_{G}\subset FT_{G}$.*

*Proof. * (i)\ If $XY$ is an $(n,n)$-rectangle then $(gX)Y$ and $X(Yg)$ are $(n,n)$-rectangles, so the family $BP_{G}$ is translation invariant.

We take $AB\in BP_{G}$ and choose $n\in\mathbb{N}$ such that $A,B$ have no $(n,n)$-rectangles. By the bipartite Ramsey theorem [3, p. 95], there is a natural number $r$ such that, for every 2-coloring of edges of the complete biparte graph $K_{r,r}$, one can find a monochrome copy of $K_{n,n}$. We assume that $A\cup B$ contains an $(r,r)$-rectangle $XY$. We define a coloring $\chi:X\times Y\rightarrow\{0,1\}$ of the Cartesian product $X\times Y$ by the rule: $\chi((x,y))=1$ if and only if $xy\in A$. By the choice of $r$, there exist X^{\prime}\subset X\, $\ Y^{\prime}\subset Y$ such that |X^{\prime}|=|Y^{\prime}|=n\ and \ X^{\prime}\times Y^{\prime}\ is monochrome. Then either \ X^{\prime}Y^{\prime}\subset A\ or \ X^{\prime}Y^{\prime}\subset B\ and we get a contradiction with the choice of $A$ and $B$. Hence, $BP_{G}$ is an ideal in $\mathcal{P}_{G}$ .

$(ii\ )$ By $(i)$, \ BP_{G}^{\wedge}\ is a left ideal and $\ (BP_{G}^{\wedge})G\subseteq BP_{G}^{\wedge}$ . Since $\ BP_{G}\subseteq FT_{G}\subset Sp_{G}$ and $Sp_{G}^{\wedge}=\overline{G^{*}G^{*}}$, we have $\ (BP_{G}^{\wedge})G^{*}\subseteq BP_{G}^{\wedge}$ so $BP_{G}^{\wedge}$ is a right ideal. The second statement of $(ii)$ is evident.

$(iii)$ Passing to subgroups, we suppose that $G$ is countable and construct $A\in FT_{G}\setminus BP_{G}$ in the form $A=\bigcup_{n<\omega}X_{N}Y_{N}$, $\ \ |X_{n}|=|Y_{n}|=n+1$. We enumerate $G=\{g_{n}:n<\omega\}$, $g_{0}=e$ and put $X_{0}=Y_{0}=\{0\}$. Suppose that we have chosen $X_{0}Y_{0},\dots,X_{n}Y_{n}$. We choose $X_{n+1}Y_{n+1}$, $|X_{n+1}|=|Y_{n+1}|=n+2$ to satisfy the following conditions for each $i\in\{1,\dots.n+1\}$:

[TABLE]

After $\omega$ steps, we get the desired $A$. Indeed, $X_{n}Y_{n}\subset A$ so $A\notin BP_{G}$. By the construction, $gA\cap A$ is finite for each $g\in G\setminus\{e\}$, so $A$ is thin and $A\in FT_{G}$. $\ \ \ \ \Box$

4 Between $\overline{K(G)}$ and $\overline{G^{*}G^{*}}$

Let $(g_{n})_{n\in\omega}$ be an injective sequence in a group $G$. The set

[TABLE]

is called an $FP$*-set *.

Given a sequence $(b_{n})_{n\in\omega}$ in $G$, we say that the set

[TABLE]

is a *piecewise shifted $FP$-set *.

Theorem 4.1. For a group $G$, the following statements hold:

$(i)$* $Sc_{G}^{\wedge}=cl\{\epsilon p:\epsilon\in G^{*},\ p\in\beta G,\ \epsilon\epsilon=\epsilon\}$;*

$(ii)$* $Sc_{G}^{\wedge}$ is an ideal in $\beta G$ and $p\in Sc_{G}^{\wedge}$ if and only if each member of $p$ contains a pierwise shifted $FP$-set;*

$(iii)$* $Sc_{G}^{\wedge}$ is the minimal close ideal in $\beta G$ containing all idempotents of $G^{*}$.*

*Proof. * (i)\ We remind that a subset $A$ of $G$ is scattered if and only if, for each $p\in A^{*}$, the subset $Gp$ is discrete in $\beta G$. Hence, $A$ is not scattered if and only if, there is $p\in A^{*}$ such that $Gp$ is not discrete. On the other hand $Gp$ is not discrete if and only if $p=\epsilon p$ for some idempotent $\epsilon\in G^{*}$.

(ii)\ Since $Sc_{G}$ is a left translation invariant, $Sc_{G}^{\wedge}$ is a left ideal in $\beta G$. By $(i)$, $(Sc_{G}^{\wedge})q\subseteq Sc_{G}^{\wedge}$ for each $q\in\beta G$, so $Sc_{G}^{\wedge}$ is a right ideal .

By [1, Theorem 1], a subset $A$ is scattered if and only if $A$ contains no pierwise shifted $FP$-sets.

(iii)\ Let $\mathcal{M}$ denotes the minimal closed ideals of $\beta G$ containing all idempotents of $\beta G$. By $(i)$, $Sc_{G}^{\wedge}\subseteq\mathcal{M}$. Since $Sc_{G}^{\wedge}$ is a closed ideal, we have $\mathcal{M}=Sc_{G}^{\wedge}$ . $\ \ \ \ \ \ \Box$

Remark 4.1. If $\mathcal{I}$ is a group ideal in $\mathcal{P}_{G}$ and $\mathcal{I}\subseteq Sp_{G}$ then $\mathcal{I}=\mathcal{F}_{G}$ (see Remark 3.1). We can not state the same if $\mathcal{I}\subseteq Sc_{G}$.

Let $G$ be the direct $sum\ \oplus_{\omega}\mathbb{Z}_{2}$ of $\omega$ copies of $\mathbb{Z}_{2}=\{0,1\}$. For $g\in G$, we denote by $supt(g)$ the number of non-zero coordinates of $g$. We put $A=\{g\in G:supt(g)=1\}$ and consider the minimal group ideal $\mathcal{I}$ in $\mathcal{P}_{G}$ such that $A\in\mathcal{I}$ . If $S\in\mathcal{I}$ then there is $m\in\mathbb{N}$ such that $supt(g)\leq m$ for each $g\in S$. It follows that $S$ has no piecewise shifted $FP$-sets, so $S$ is scattered and $\mathcal{I}\subset Sc_{G}$.

The following observation follows directly from the basic properties of multiplication in $\beta G$: each right shift is continuous and each left shift on element of $g$ is continuous.

Lemma 3.1. If $L$ is a left ideal in $\beta G$ and $R$ is a right ideal in $\beta G$ then $\overline{LR}$ is an ideal in $\beta G$.

For a group $G$, we put $I_{G,0}=G^{*}$ and $I_{G,n+1}=\overline{G^{*}I_{G,n}}$. By Lemma 4.1, each $I_{G,n}$ is an ideal in $\beta G$.

Clearly, $I_{G,n+1}\subseteq I_{G,n}$ so $I_{G,n}\subseteq\overline{G^{*}G^{*}}$ for $n>0$.

Theorem 4.2. For every group $G$ and $n\in\omega$, we have

(i)\* $I_{G,n+1}\subset I_{G,n}$*

(ii)\* $Sc_{G}^{\wedge}\subset I_{G,n}$.*

*Proof. * (i)\ We note that $I_{G,n+1}^{\vee}=\{A\subseteq G:\Delta_{p}(A)$ is finite for each $p\in I_{G,n}\}$ and apply Theorem 4 from [7] stating that $I_{G,n}^{\vee}\subset I_{G,n+1}^{\vee}$.

(ii)\ For $n=0$, this is evident. We take an idempotent $\epsilon\in G^{*}$, $p\in\beta G$ and assume that $\epsilon p\in I_{G,n-1}$. Then $\epsilon\epsilon p\in G^{*}I_{G,n-1}$, so $\epsilon p\in I_{G,n}$. Applying Theorem 4.1, we conclude that $Se_{G}^{\wedge}\subseteq I_{G,n}$. The strict inclusion follows from $(i)$ . $\ \ \ \ \ \ \Box$

For a natural number $n$, we denote by $(G^{*})^{n}$ the product of $n$ copies of $n$. By Lemma 4.1, $(G^{*})^{n}$ is an ideal in $\beta G$.. Clearly, $\overline{(G^{*})^{n+1}}\subseteq\overline{(G^{*})^{n}}$. and $\overline{(G^{*})^{n}}\subseteq I_{G,n}$ .

By analogy with Theorem 4.2, we can prove

Theorem 4.3. For every group $G$ and $n\in\omega$, we have

(i)\* $\overline{(G^{*})^{n+1}}\subset\overline{(G^{*})^{n}}$;*

(ii)\* $Sc_{G}^{\wedge}\subset\overline{(G^{*})^{n}}$.*

References

[1] T. Banakh, I. Protasov, S. Slobodianiuk, Scattered subsets of groups, Ukr. Math. J. 67 (2015), 347-356.

[2] M. Filali, Ie. Lutsenko, I. Protasov, Boolean group ideal and the ideal structure of $\beta G$, Mat. Stud. 31 (2009), 19-28.

[3] R. Graham, B. Rotschild, J. Spencer, Ramsey Theory, Willey, New York, 1980.

[4] N. Hindman, D. Strauss Algebra in the Stone-$\check{C}$ech compactification, de Gruyter, Berlin, New York, 1998.

[5] Ie. Lutsenko, I. Protasov, Sparse, thin and other subsets of groups, Algebra Computa-tion 19 (2009), 491-510.

[6] Ie. Lutsenko, I. Protasov, Thin subsets of balleans, Appl. Gen. Topology 11 (2010), 89-93.

[7] Ie. Lutsenko, I. Protasov, Relatively thin and sparse subsets of groups, Ukr. Math. J. 63 (2011), 254-264.

[8] O. Petrenko, I. Protasov, Thin ultrafilters, Notre Dame J. Formal Logic 53 (2012), 79-88.

[9] I. Protasov, *Selective survey on subset combinatories of groups, * J. Math. Sciences 174 (2011), 486-514.

[10] I. Protasov, T. Banakh, Ball structures and colorings of groups and graphs, Math. Stud. Monogr. Ser., Vol. 11, VNTL, Lviv, 2003.

[11] I. Protasov, K. Protasova, *Ramsey-product subsets of a group, *, preprint (arxiv: 1703, 03874).

[12] I. Protasov, O. Protasova, *Sketch of group balleans, * Math. Stud. 22 (2004), 10-20.

[13] I. Protasov, O. Protasova, On closed ideals in $\beta G$, Semigroup Forum 75 (2007), 237-240.

[14] I. Protasov, S. Slobodianiuk, Thin subsets of groups, Ukr. Math. J. 65 (2013), 1384-1393.

[15] I. Protasov, S. Slobodianiuk, On the subset combinatorics of $G$-spaces, Algebra Discrete Math. 17 (2014), No 1, 98-109.

[16] I. Protasov, S. Slobodianiuk, *Ultracompanions of subsets of a group, * Comment. Math. Univ. Carolin. 55 (2014), 257-265.

CONTACT INFORMATION

I. Protasov:

Faculty of Computer Science and Cybernetics

Kyiv University

Academic Glushkov pr. 4d

03680 Kyiv, Ukraine

i.v.protasov@gmail.com

K. Protasova:

Faculty of Computer Science and Cybernetics

Kyiv University

Academic Glushkov pr. 4d

03680 Kyiv, Ukraine

ksuha@freenet.com.ua