# Essential CR-sets near zero

Sujan Pal  Department of Mathematics, University of Kalyani, Kalyani,
Nadia-741235, West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

Hindman and Leader were the first to introduce the concept of central sets
near zero for dense semigroups of $\left(\left(0,\infty\right),+\right)$ and
demonstrated an important combinatorial theorem regarding these sets. Using
the algebraic structure of the Stone-Čech compactification, Bayatmanesh and
Tootkabani generalised and extended this combinatorial theorem to the central
theorem near zero. We introduce the CR-sets near zero and the essential CR-
sets near zero in this article. Then dynamically and combinatorially
characterize essential CR-sets near zero. Thereafter, we explore the cartesian
product of these sets.

##  1\. Introduction

There are many notions of largeness in a semigroup $\left(S,+\right)$ that are
related to the algebraic structure of the Stone-Čech compactification of the
discrete set $S$.

In our work we will use the structure of Stone-Čech compactification of
discrete semigroup. Let $\left(S,\cdot\right)$ be a discrete semigroup and
$\beta S$ be the Stone-Čech compactification of the discrete semigroup $S$ and
$\cdot$ on $\beta S$ is the extension of ${}^{\prime}\cdot^{\prime}$ on $S$.
The points of $\beta S$ are ultrafilters and principal ultrafilters are
identified by the points of $S$. The extension is unique extension for which
$\left(\beta S,\cdot\right)$ is compact, right topological semigroup with $S$
contained in its topological center. That is, for all $p\in\beta S$ the
function $\rho_{p}:\beta S\to\beta S$ is continuous, where $\rho_{p}(q)=q\cdot
p$ and for all $x\in S$, the function $\lambda_{x}:\beta S\to\beta S$ is
continuous, where $\lambda_{x}(q)=x\cdot q$. For $p,q\in\beta S$, $p\cdot
q=\left\\{A\subseteq S:\left\\{x\in S:x^{-1}A\in q\right\\}\in p\right\\}$,
where $x^{-1}A=\left\\{y\in S:x\cdot y\in A\right\\}$.

Here we work for those subsemigroup $S$, which are dense subsemigroup of
$\left(\left(0,\infty\right),+\right)$, then one can define
$0^{+}\left(S\right)=\left\\{p\in\beta S_{d}:\text{ for all
}\left(\delta>0\right)\left(\left(0,\delta\right)\in p\right)\right\\}$,
($S_{d}$ is the set $S$ with the discrete topology). In this paper the set
$A^{B}$ is the collection of all functions from the set $B$ to the set $A$.

To proceed with the main work on this paper, we require some definitions and
lemmas.

######  Definition 1.

Let $\left(S,+\right)$ be a commutative semigroup and let $A\subseteq S$.

  1. (1)

$A$ is said to be a J-set if and only if whenever
$F\in\mathcal{P}_{f}\left(S^{\mathbb{N}}\right)$, there exist $a\in S$ and
$H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ such that for each $f\in F$,

$a+\sum_{t\in H}f\left(t\right)\in A.$

  2. (2)

$A$ is said to be a CR-set if for each $k\in\mathbb{N}$ there exist
$r\in\mathbb{N}$ and whenever $F\in\mathcal{P}_{f}\left(S^{\mathbb{N}}\right)$
with $\mid F\mid\leq k$ , $a\in S$ and
$H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ with $maxH\leq r$ such that for
each $f\in F$,

$a+\sum_{t\in H}f\left(t\right)\in A.$

  3. (3)

$A$ is said to be a k-CR-set if there exist $r\in\mathbb{N}$ and whenever
$F\in\mathcal{P}_{f}\left(S^{\mathbb{N}}\right)$ with $\mid F\mid\leq k$ ,
$a\in S$ and $H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ with $maxH\leq r$
such that for each $f\in F$,

$a+\sum_{t\in H}f\left(t\right)\in A.$

Then clearly CR-sets are J-sets.

######  Definition 2.

Let $\left(S,+\right)$ be a commutative semigroup and let $A\subseteq S$.

  1. (1)

$J\left(S\right)=\left\\{p\in\beta S:\text{ for all }A\in p,A\text{ is a
J-set}\right\\}$

  2. (2)

$CR\left(S\right)=\left\\{p\in\beta S:\text{ for all }A\in p,A\text{ is a CR-
set}\right\\}$

  3. (3)

For $k\in\mathbb{N}$, $k-CR\left(S\right)=\left\\{p\in\beta S:\text{ for all
}A\in p,A\text{ is a k-CR-set}\right\\}$.

In the paper [P, Def 4.3] Patra introduced the J-sets near zero.

Here $\mathcal{T}_{0}$ be the set of all sequences in $S$ which converge to
zero.

######  Definition 3.

Let $\left(S,+\right)$ be a dense subsemigroup of
$\left(\left(0,\infty\right),+\right)$ and let $A\subseteq S$.

  1. (1)

$A$ is said to be J-set near zero if and only if whenever $F\in
P_{f}\left(\mathcal{T}_{0}\right)$ and $\delta>0$, there exists $a\in
S\cap\left(0,\delta\right)$ and $H\in P_{f}\left(\mathbb{N}\right)$ such that
for each $f\in F$,

$a+\sum_{t\in H}f\left(t\right)\in A.$

  2. (2)

$J^{0}\left(S\right)=\left\\{p\in 0^{+}\left(S\right):\forall A\in p,A\text{
is J-set near zero}\right\\}$.

######  Lemma 4.

Let $A$ be a finite nonempty alphabet and let $r\in\mathbb{N}$. There is some
$m\in\mathbb{N}$ such that whenever the length $m$ words over $A$ are
r-colored, there is a variable word $w\left(v\right)$ such that
$w\left(v\right)$ begins and ends with a constant, $w\left(v\right)$ has no
successive occurrences of $v$, and $\left\\{w\left(a\right):a\in A\right\\}$
is monochromatic.

###### Proof.

[HS, Lemma 14.8.1, page-335] ∎

##  2\. The notion of combinatorially rich sets near zero

The concept of CR-set near zero is closely associated with the concept of
J-set near zero, as is the case with CR-set and J-set.

######  Definition 5.

Let $\left(S,+\right)$ be a dense subsemigroup of
$\left(\left(0,\infty\right),+\right)$.

  1. (1)

Then $A$ is a CR-set near zero if for each $k\in\mathbb{N}$ and $\delta>0$
there exists $r\in\mathbb{N}$ whenever $F\in
P_{f}\left(\mathcal{T}_{0}\right)$ with $\mid F\mid<k$ , there exist $a\in
S\cap\left(0,\delta\right)$ and $H\subseteq\left\\{1,2,\ldots,r\right\\}$ such
that for all $f\in F$,

$a+\sum_{t\in H}f\left(t\right)\in A.$

  2. (2)

Then $A$ is a k-CR-set near zero if for each $\delta>0$ there exists
$r\in\mathbb{N}$ whenever $F\in P_{f}\left(\mathcal{T}_{0}\right)$ with $\mid
F\mid<k$ , there exist $a\in S\cap\left(0,\delta\right)$ and
$H\subseteq\left\\{1,2,\ldots,r\right\\}$ such that for all $f\in F$

$a+\sum_{t\in H}f\left(t\right)\in A.$

Now, we show that the notion of combinatorially rich sets near zero is
partition regular.

######  Lemma 6.

Let $S$ be a dense subsemigroup of $\left(\left(0,\infty\right),+\right)$ and
$A_{1},A_{2}\subseteq S$. If $A_{1}\cup A_{2}$ is a CR-set near zero, then
either $A_{1}$ or $A_{2}$ is a CR-set near zero.

###### Proof.

If possible let $A_{1}\cup A_{2}$ is a CR-set near zero in $S$ and neither
$A_{1}$ nor $A_{2}$ is a CR-set near zero in $S$.

For $i\in\left\\{1,2\right\\}$ pick $k_{i}\in\mathbb{N}$ and $\delta>0$ such
that for every $r\in\mathbb{N}$ there exists
$F\in\mathcal{P}_{f}\left(\mathcal{T}_{0}\right)$ such that $\mid F\mid\leq
k_{i}$, for all $a\in S\cap\left(0,\delta\right)$ and for all
$H\subseteq\left\\{1,2,\ldots,r\right\\}$, there exists $f\in F$

$a+\sum_{t\in H}f(t)\notin A_{i}.$

Let $k=k_{1}+k_{2}$ and pick by lemma 4, some $n\in\mathbb{N}$ such that
whenever the length $n$ words over $\left\\{1,2,\ldots,k\right\\}$ are
2-colored, there is a variable word $w\left(v\right)$ beginning and ending
with a constant and having no adjacent occurrences of $v$ such that
$\left\\{w\left(l\right):l\in\left\\{1,2,\ldots,k\right\\}\right\\}$ is
monochromatic.

Let $h\in\mathcal{T}_{0}$. For each $t\in\mathbb{N}$, we can choose
$s_{t}\in\mathbb{N}$ such that $h\left(s_{t}\right)<\frac{1}{2^{t+1}}\delta$.
Since we could replace each $f_{i}$ by the sequence $f_{i}^{{}^{\prime}}$,
defined by $f_{i}^{\prime}\left(t\right)=f_{i}\left(s_{t}\right)$, we may
suppose that $\sum_{t=1}^{\infty}h\left(t\right)<\frac{1}{2}\delta$.

Let $W$ be the set of length $n$ words over $\left\\{1,2,\ldots,k\right\\}$.
For $w=b_{1}b_{2}\ldots b_{n}$ where each
$b_{i}\in\left\\{1,2,...,k\right\\}$. Define $g_{w}:\mathbb{N}\rightarrow S$
for $y\in\mathbb{N}$ by, $g_{w}(y)=\sum_{i=1}^{n}f_{b_{i}}\left(ny+i\right)$.
Since $\left\\{g_{w}:w\in W\right\\}$ is finite and $A_{1}\cup A_{2}$ is a CR-
set near zero in $S$, there exist $c\in\left(0,\frac{1}{2}\delta\right)$ and
$r_{1}\in\mathbb{N}$ with $K\subseteq\left\\{1,2,\ldots,r_{1}\right\\}$ such
that for all $w\in W$

$c+\sum_{y\in K}g_{w}\left(y\right)\in A_{1}\cup A_{2}.$

Define $\varphi:W\to\left\\{1,2\right\\}$ by $\varphi\left(w\right)=1$ if and
only if $c+\sum_{y\in K}g_{w}\left(y\right)\in A_{1}$. Pick a variable word
$w\left(v\right)$, beginning and ending with a constant and without successive
occurrences of $v$ such that $\varphi$ is constant on
$\left\\{w\left(l\right):l\in\left\\{1,2,\ldots,k\right\\}\right\\}.$ Assume
without loss of generality that $\varphi\left(w\left(l\right)\right)=1$ for
all $l\in\left\\{1,2,\cdots,k\right\\},$then for all
$l\in\left\\{1,2,\ldots,k\right\\}$

$c+\sum_{t\in K}g_{w\left(l\right)}\left(t\right)\in A_{1}.$

Let $w\left(v\right)=b_{1}b_{2}\ldots b_{n}$ where each
$b_{i}\in\left\\{1,2,\ldots,k\right\\}\cup\left\\{v\right\\}$, some
$b_{i}=v,b_{1},b_{n}\neq v$, and if $b_{i}=v$, then $b_{i+1}\neq v$.

Let
$C=\left\\{i\in\left\\{1,2,\ldots,n\right\\}:b_{i}\in\left\\{1,2,\ldots,k\right\\}\right\\}$
and $I=\left\\{i\in\left\\{1,2,\ldots,n\right\\}:b_{i}=v\right\\}$. Then, for
each $l\in\left\\{1,2,\ldots,k\right\\}$we have

$c+\sum_{t\in K}g_{w\left(l\right)}\left(t\right)=c+\sum_{y\in K}\sum_{i\in
C}f_{b_{i}}\left(ny+i\right)+\sum_{y\in K}\sum_{i\in
I}f_{l}\left(ny+i\right).$

Let $a=c+\sum_{y\in K}\sum_{i\in C}f_{b_{i}}\left(ny+i\right)$. Then $a\in
S\cap\left(0,\delta\right)$ and $a+\sum_{y\in K}\sum_{i\in
I}f_{l}\left(ny+i\right)\in A_{1}$ for every
$l\in\left\\{1,2,\ldots,k\right\\}$.

Putting $H=nK+I$, then $H\subseteq\left\\{1,2,\ldots,nr_{1}+n\right\\}$, we
have

$\sum_{y\in K}\sum_{i\in I}f_{l}\left(ny+i\right)=\sum_{y\in
K}f_{l}\left(y\right)$

because the sets of the form $nK+i$, with $i\in I$, are pairwise disjoint. It
follows that $a+\sum_{t\in H}f_{l}(t)\notin A_{i}$ for every
$l\in\left\\{1,2,\ldots,k\right\\}$ and hence a contradiction arise. ∎

So CR-set near zero satify partition regularity.

Now $k\text{-}CR^{0}\left(S\right)=\left\\{p\in 0^{+}\left(S\right):\forall
A\in p,A\text{ is k-CR-\text{set near zero}}\right\\}$ and

$CR^{0}\left(S\right)=\left\\{p\in 0^{+}\left(S\right):\forall A\in p,A\text{
is a CR-set near zero}\right\\}$

Then
$CR^{0}\left(S\right)=\bigcup_{k\in\mathbb{N}}k\text{-}CR^{0}\left(S\right)$.

######  Lemma 7.

Let $S$ be a dense subsemigroup of $\left(\left(0,\infty\right),+\right)$.
Then $k\text{-}CR^{0}\left(S\right)$ is two sided ideal of $\beta S$ for all
$k\in\mathbb{N}$.

###### Proof.

Let us first fixed $k$ and $p\in k\text{-}CR^{0}\left(S\right)$ and $q\in\beta
S$. We have to show $p+q,q+p\in k\text{-}CR^{0}\left(S\right)$.

Let $A\in p+q$ , So $B=\left\\{x\in S:-x+A\in q\right\\}\in p\in
k\text{-}CR^{0}\left(S\right)$. So by definition of
$k\text{-}CR^{0}\left(S\right)$, $B$ is a k-CR-set near zero.

So there exists $r\in\mathbb{N}$ whenever $F\in
P_{f}\left(\mathcal{T}_{0}\right)$ with $\mid F\mid<k$ and $\delta>0$, there
exist $a\in S\cap\left(0,\delta\right)$, and
$H\subseteq\left\\{1,2,\ldots,r\right\\}$ such that for all $f\in F$

$a+\sum_{t\in H}f\left(t\right)\in B$

Then for all $f\in F$,

$-\left(a+\sum_{t\in H}f\left(t\right)\right)+A\in q$

i,e,

$\bigcap_{f\in F}\left(-\left(a+\sum_{t\in
H}f\left(t\right)\right)+A\right)\in q$

Let $y\in\bigcap_{f\in F}\left(-\left(a+\sum_{t\in
H}f\left(t\right)\right)+A\right)$, So $b+\sum_{t\in H}f\left(t\right)\in A$
for all $f\in F$, where $b=y+a\in S$. So $A$ is a k-CR-set near zero. Since
$A$ is arbitrary element of $p+q$, Then $p+q\in
k\text{-}CR^{0}\left(S\right)$.

Let $A\in q+p$ , Assumse $B=\left\\{x\in S:-x+A\in p\right\\}\in q$ , $B$ is
nonempty. Let $x\in B$, then $-x+A\in p\in k\text{-}CR^{0}\left(S\right)$. By
definition of $k\text{-}CR^{0}\left(S\right)$, $-x+A$ is a $k\text{-}CR$-set
near zero. So there exists $r\in\mathbb{N}$ whenever $F\in
P_{f}\left(\mathcal{T}_{0}\right)$ with $\mid F\mid<k$ and $\delta>0$, there
exist $a\in S\cap\left(0,\delta\right)$, and
$H\subseteq\left\\{1,2,\ldots,r\right\\}$ such that for all $f\in F$

$a+\sum_{t\in H}f\left(t\right)\in-x+A.$

i.e, For all $f\in F$,

$b+\sum_{t\in H}f\left(t\right)\in A$

where $b=x+a\in S$ . So $A$ is a k-CR-set near zero. Since $A$ is arbitrary
element of $p+q$, Then $p+q\in k\text{-}CR^{0}\left(S\right)$.

So $k\text{-}CR^{0}\left(S\right)$ is two sided ideal of $\beta S$ for all
$k\in\mathbb{N}$. ∎

##  3\. Characterisation of essential CR-sets

There are several notions of largeness in topological dynamics, and most of
them play a significant role in Ramsey Theory. One such notion is the central
set. The concept of central set was first introduced by H. Frustenberg using
the notions of proximality and uniformly recurrent from topological dynamics
in [F81,Def 8.3]. Later, Patra characterised central sets near zero
dynamically in [P].

We present a dynamic and basic characterization of essential CR sets in this
section. Let’s give a brief description of the well-known sets. We can begin
by stating the well-known definition of a topological dynamical system.

######  Definition 8.

A dynamical system is a pair $\left(X,\left\langle T_{s}\right\rangle_{s\in
S}\right)$ such that

  1. (1)

$X$ is compact Hausdorff space,

  2. (2)

$S$ is a semigroup,

  3. (3)

For each $s\in S$, $T_{s}:X\to X$ and $T_{s}$ is continuous, and

  4. (4)

For all $s,t\in S$, $T_{s}\circ T_{t}=T_{st}$.

.

######  Definition 9.

Let $S$ be a nonempty discrete space and $\mathcal{K}$ be a filter on $S$.

  1. (1)

$\overline{\mathcal{K}}=\left\\{p\in\beta S:\mathcal{K}\subseteq p\right\\}$

  2. (2)

$L\left(\mathcal{K}\right)=\left\\{A\subseteq S:S\setminus
A\notin\mathcal{K}\right\\}$

######  Theorem 10.

Let $S$ be a nonempty discrete space and $\mathcal{K}$ be a filter on $S$.

  1. (1)

$\overline{\mathcal{K}}=\left\\{p\in\beta S:A\in
L\left(\mathcal{K}\right)\text{ for all }A\in p\right\\}$

  2. (2)

Let $\beta\subseteq L\left(\mathcal{K}\right)$ be closed under finite
intersections then there exists a $p\in\beta S$ with $\beta\subseteq
p\subseteq L\left(\mathcal{K}\right)$.

######  Definition 11.

Let $\left(X,\left\langle T_{s}\right\rangle_{s\in S}\right)$ be a dynamical
system, $x$ and $y$ in $X$ and $\mathcal{K}$ be a filter on $S$. The pair
$\left(x,y\right)$ is called jointly $\mathcal{K}$-recurrence if and only if
for every neighborhood $U$ of $y$ we have $\left\\{s\in
S:T_{s}\left(x\right),T_{s}\left(y\right)\in U\right\\}\in
L\left(\mathcal{K}\right)$.

######  Theorem 12.

Let $S$ be a nonempty discrete space and $\mathcal{K}$ be a filter on $S$ such
that $\overline{\mathcal{K}}$ is a compact subsemigroup of $\beta S$, and let
$A\subseteq S$. Then $A$ is a member of an idempotent in
$\overline{\mathcal{K}}$ if and only if there exists a dynamics system
$\left(X,\left\langle T_{s}\right\rangle_{s\in S}\right)$ with points $x$ and
$y$ in $X$ and there exists a neighborhood $U$ of $y$ such that the pair
$\left(x,y\right)$ is jointly $\mathcal{K}$-recurrence and $A=\left\\{s\in
S:T_{s}\left(x\right)\in U\right\\}$.

######  Definition 13.

Let $S$ be a dense subsemigroup of $\left(\left(0,\infty\right),+\right)$ and
$A\subseteq S$. Then $A$ is an essential CR-set near zero if there exists an
idempotent in $p\in CR^{0}\left(S\right)$ such that $A\in p$.

######  Theorem 14.

Let $S$ be a dense subsemigroup of $\left(\left(0,\infty\right),+\right)$ and
$A\subseteq S$. Then $A$ is an essential CR-set near zero if and only if there
exists a dynamical system $\left(X,\left\langle T_{s}\right\rangle_{s\in
S}\right)$ with points $x$ and $y$ in $X$ such that for every neighbourhood
$U$ of $y$, $\left\\{s\in S:T_{s}\left(x\right),T_{s}\left(y\right)\in
U\right\\}$ is a CR-set near zero and a neighbourhood $V$ of $y$ such that
$A=\left\\{s\in S:T_{s}\left(x\right)\in V\right\\}$.

To prove this theorem, we need some results.

######  Lemma 15.

Let $S$ be a dense subsemigroup of $\left(\left(0,\infty\right),+\right)$ and

$\mathcal{K}=\left\\{A\subseteq S:S\setminus A\text{ is not a CR-set near
zero}\right\\}.$

Then $K$ is a filter on $S$ with $CR^{0}\left(S\right)=\overline{\mathcal{K}}$
and $CR^{0}\left(S\right)$ is a compact subsemigroup of $\beta S$.

###### Proof.

Clearly $\emptyset\notin\mathcal{K}$, So $\mathcal{K}$ is nonempty. If
$A\in\mathcal{K}$ and $A\subseteq B$, then $B\in\mathcal{K}$. So $\mathcal{K}$
is closed under super sets. So $\mathcal{K}$ is a filter. From theorem 10 ,
$CR^{0}\left(S\right)=\overline{\mathcal{K}}$. By lemma 6, lemma7 and
definition 13 we get that $CR^{0}\left(S\right)$ is a compact subsemigroup of
$\beta S$. ∎

Now we ready to prove the theorem 14.

###### Proof.

Let $\mathcal{K}=\left\\{B\subseteq S:S\setminus B\text{ is not a CR-set near
zero}\right\\}$. Since definition13, characterizes essential CR-set near zero
in terms of idempotents in $\overline{\mathcal{K}}$, by theorem 12 the result
follow. ∎

We are now like to elementary characterise the essential CR-set near zero.

######  Theorem 16.

Let $S$ be a dense subsemigroup of $\left(\left(0,\infty\right),+\right)$and
$A\subseteq S$. Statements (1), (2) and (3) are equivalent.

  1. (1)

$A$ is an essential CR-set near zero.

  2. (2)

There is a non empty set $T$ of functions such that

    1. (a)

For all $f\in T$, $domain\left(f\right)\in\omega$ and
$rang\left(f\right)\subseteq A$.

    2. (b)

For all $f\in T$ and all $x\in B_{f}\left(T\right)$, $B_{f^{\frown}x}\subseteq
x^{-1}B_{f}$.

    3. (c)

For each $F\in P_{f}\left(T\right)$, $\bigcap_{f\in F}B_{f}\left(T\right)$ is
a CR-set near zero.

  3. (3)

There is a downward directed family $\left\langle C_{F}\right\rangle_{F\in I}$
of subsets of $A$ such that

    1. (a)

For each $F\in I$ and each $x\in C_{F}$ there exists $G\in I$ with
$C_{G}\subseteq x^{-1}C_{F}$ and

    2. (b)

For all $\mathcal{F}\in P_{f}\left(I\right)$, $\bigcap_{F\in\mathcal{F}}C_{F}$
is a CR-set near zero.

###### Proof.

(1) $\Rightarrow$ (2) As $A$ be an essential CR-set near zero, then there
exists an idempotent $p\in CR^{0}\left(S\right)$ such that $A\in p$. Pick a
set $T$ of functions as guaranteed by Lemma 2.2. Conclusions (a) and (b) hold
directly. Given $F\in P_{f}\left(T\right)$, $B_{f}\in p$ for all $f\in F$,
hence $\bigcap_{f\in F}B_{f}\left(T\right)\in p$ and so $\bigcap_{f\in
F}B_{f}\left(T\right)$ is a CR-set near zero.

(2) $\Rightarrow$ (3) Let $T$ be guaranteed by (2). Let
$I=P_{f}\left(T\right)$ For each $F\in I$, let $C_{F}=\bigcap_{f\in
F}B_{f}\left(T\right)$ . Then directly each $C_{F}$ is a CR-set near zero.
Given $\mathcal{F}\in P_{f}\left(I\right)$, if $G=\cup\mathcal{F}$, then
$\bigcap_{F\in\mathcal{F}}C_{F}=C_{G}$ and is therefore a CR-set near zero. To
verify (a), let $F\in I$ and let $x\in C_{F}$ . Let
$G=\left\\{f^{\frown}x:f\in F\right\\}$. For each $f\in F$,
$B_{f^{\frown}x}\subseteq x^{-1}B_{f}$ and so $C_{G}\subseteq x^{-1}C_{F}$.

(3) $\Rightarrow$ (1) Let $\left\langle C_{F}\right\rangle_{F\in I}$ i is
guaranteed by (3). Let$M=\bigcap_{F\in I}\overline{C_{F}}$ . By [12, Theorem
4.20], $M$ is a subsemigroup of $\beta S$. By [12, Theorem 3.11] there is some
$p\in\beta S$ such that $\left\\{C_{F}:F\in I\right\\}\subseteq
p\subseteq\mathcal{F}$. Therefore $M\cap CR^{0}\left(S\right)\neq\emptyset$;
and so $M\cap CR^{0}\left(S\right)$ is a compact subsemigroup of $\beta S$.
Thus there is an idempotent $p\in M\cap CR^{0}\left(S\right)$, and so $A$ is
an essential CR-set near zero. ∎

##  4\. Cartetian product of essential cr-sets

The cartesian product of two central sets is a central set; the cartesian
product of two J-sets is a J-set; and the cartesian product of two C-sets is a
C-set. We are looking into whether the same result is true for CR-sets near
zero and essential CR-sets near zero.

Here
$\mathcal{D}_{0}=\left\\{\left(z_{n}=\left(x_{n},y_{n}\right)\right)_{n\in\mathbb{N}}\in\left(S\times
T\right)^{\mathbb{N}}:\text{ both }x_{n},y_{n}\text{ converges to
zero}\right\\}$

If $\left(x,y\right)$ be a point in $S\times T$, Then projection on $S$,
$P_{S}\left(x,y\right)=x$ and projection on $T$, $P_{T}\left(x,y\right)=y$.

Now we define $J$-sets near zero and CR-sets near zero for the product space.

######  Definition 17.

Let $S$ and $T$ be a dense subsemigroup of
$\left(\left(0,\infty\right),+\right)$ and $D\subseteq S\times T$ is called
J-set near zero in $S\times T$ if for $\delta>0$ and
$F\in\mathcal{P}_{f}\left(\mathcal{D}_{0}\right)$ there exist
$\overline{a}=\left(a_{1},a_{2}\right)\in\left(S\times
T\right)\bigcap\left(0,\delta\right)^{2}$ and
$H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ such that for all $f\in F$

$\overline{a}+\sum_{t\in H}f\left(t\right)\in D.$

.

######  Definition 18.

Let $S$ and $T$ be a dense subsemigroup of
$\left(\left(0,\infty\right),+\right)$ and $D\subseteq S\times T$ is called
CR-set near zero in $S\times T$ if for $k\in\mathbb{N}$, $\delta>0$ for every
$F\in\mathcal{P}_{f}\left(\mathcal{D}_{0}\right)$ with $\mid F\mid\leq k$
there exist $r\in\mathbb{N}$,
$\overline{a}=\left(a_{1},a_{2}\right)\in\left(S\times
T\right)\bigcap\left(0,\delta\right)^{2}$ and
$H\subseteq\left\\{1,2,\ldots,r\right\\}$ such that for all $f\in F$

$\overline{a}+\sum_{t\in H}f\left(t\right)\in D.$

######  Lemma 19.

Let $S$ and $T$ be a dense subsemigroups of
$\left(\left(0,\infty\right),+\right)$. Let $A$ and $B$ be an CR-set near zero
in $S$ and $T$ respectively. Then $A\times B$ is essential CR-set near zero in
$S\times T$.

###### Proof.

Let $k\in\mathbb{N}$, $\delta>0$ and
$F\in\mathcal{P}_{f}\left(\mathcal{D}_{0}\right)$ with $\mid F\mid\leq k$.

Let $F_{1}=\left\\{x=P_{S}\left(z\right):z\in F\right\\}$ and
$F_{2}=\left\\{y=P_{T}\left(z\right):z\in F\right\\}$. Then
$F_{1},F_{2}\in\mathcal{P}_{f}\left(\mathcal{T}_{0}\right)$ and $\mid
F_{1}\mid,\mid F_{1}\mid\leq k$.

Since $A$ is an CR-set near zero in $S$ for this $k\in\mathbb{N}$, $\delta>0$
and $F_{1}\in\mathcal{P}_{f}\left(\mathcal{T}_{0}\right)$ with $\mid
F_{1}\mid\leq k$, there exist $r_{1}\in\mathbb{N}$, $a_{1}\in
S\cap\left(0,\delta\right)$, and $H\subseteq\left\\{1,2,\ldots,r_{1}\right\\}$
such that for all $f_{1}\in F_{1}$

$a_{1}+\sum_{t\in H}f_{1}\left(t\right)\in A.$

And since $B$ is an CR-set near zero in $T$ for this $k\in\mathbb{N}$ ,
$\delta>0$ and $F_{2}\in\mathcal{P}_{f}\left(\mathcal{T}_{0}\right)$ with
$\mid F_{2}\mid\leq k$ there exist $r_{2}\in\mathbb{N}$, $a_{2}\in
T\cap\left(0,\delta\right)$ and $H\subseteq\left\\{1,2,\ldots,r_{2}\right\\}$
such that for all $f_{2}\in F_{1}$

$a_{2}+\sum_{t\in H}f_{2}\left(t\right)\in B.$

Let $r=min\left\\{r_{1},r_{2}\right\\}$, then there exist $r\in\mathbb{N}$,
$\overline{a}=\left(a_{1},a_{2}\right)\in\left(S\times
T\right)\bigcap\left(0,\delta\right)^{2}$ and
$H\subseteq\left\\{1,2,\ldots,r\right\\}$ such that for all $f\in F$

$\overline{a}+\sum_{t\in H}f\left(t\right)\in A\times B.$

∎

> By proceeding in the same way, the above result holds true for J-sets near
> zero.

######  Theorem 20.

Let $S$ and $T$ be a dense subsemigroups of
$\left(\left(0,\infty\right),+\right)$. Let $A$ be an essential CR-set near
zero in $S$ and $B$ be an essential CR-set near zero in $T$. Then $A\times B$
is essential CR-set near zero in $S\times T$.

###### Proof.

Given that $A$ be an essential CR-set near zero in $S$ , then $\left\langle
C_{F}\right\rangle_{F\in I}$ be as guaranteed by theorem 16 for $A$. It is
also given that $B$ be an essential CR-set near zero in $T$, then
$\left\langle D_{G}\right\rangle_{G\in J}$ be as guaranteed by theorem 18(c)
for $B$. Direct $I\times J$ by agreeing that
$\left(F,G\right)\geq\left(F^{\prime},G^{\prime}\right)$ if and only if $F\geq
F^{\prime}$ and $G\geq G^{\prime}$. We claim that $\left\langle C_{F}\times
D_{G}\right\rangle_{F\times G\in I\times J}$ is as required by theorem 16 to
show that $A\times B$ is an essential CR-set near zero in $S\times T$. Let
$F\times G\in I\times J$ and let $\left(x,y\right)\in C_{F}\times D_{G}$. Pick
$H\in I$ and $K\in J$ such that $C_{H}\subseteq x^{-1}C_{F}$ and
$D_{K}\subseteq x^{-1}D_{G}$ . Then $\left(F,G\right)\in I\times J$ ,
$\left(C_{H},D_{K}\right)\subseteq\left(x,y\right)^{-1}\left(C_{F},D_{G}\right)$
and $C_{H}\times D_{K}$ and $C_{F}\times D_{G}$ are CR-set near zero in
$S\times T$ by lemma 19. ∎

Acknowledgment: The second author acknowledges the Grant CSIR-UGC NET
fellowship with file No. 09/106(0199)/2019-EMR-I.

## References

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