# Large sets near idempotent and its product

Surajit Biswas<EMAIL_ADDRESS>Sourav Kanti Patra
<EMAIL_ADDRESS>Department of Mathematics, Ramakrishna Mission
Vidyamandira, Belur Math, Howrah, West Bengal, India, 711202.  Department of
Mathematics, Centre for Distance And Online Education, The University of
Burdwan, Golapbag, Purba Bardhaman, West Bengal, India, 713104.

###### Abstract

Tootkaboni and Vahed introduced the notion of some large sets near idempotent
along with some combinatorial properties. We characterize when the finite
Cartesian product of central sets near idempotent is central near idempotent.
Moreover, we provide a partial characterization for the infinite Cartesian
product of the same. We then study the abundance of some large sets near
idempotent. We also investigate the effect of tensor product near zero.
Finally, as an application we provide a characterization of members of
polynomials (with constant term 0) evaluated at idempotents in a near zero
semigroup.

###### keywords:

Central set near idempotent, Piecewise syndetic set near idempotent, Tensor
product, Milliken-Taylor System near zero.

###### MSC:

[2020] 03E05, 05D10.

##  1 Introduction

The notion of "central subset" was first introduced by Furstenberg [7] in the
semigroup $(\mathbb{N},+)$ of natural numbers. Later, Bergelson and Hindman
[4] defined the notion of a central set in an arbitrary semigroup. The class
of central sets is an important class of combinatorially rich large sets, as
they satisfy the Central Sets Theorem [9, Corollary 14.14.10]. A localized
notion of central set, called central set near idempotent was introduced by
Tootkaboni and Vahed in [14]. In a commutative semigroup, central sets near
idempotent is a class of localized sets around an idempotent, those are
combinatorially rich, because they satisfy the Central Sets Theorem near
idempotent [14, Theorem 4.3]. In [11], Hindman and Strauss have characterized
when the (finite and infinite) Cartesian product of sets is piecewise syndetic
or central. In [1], similar (partial) characterization has been provided for
some other class of large sets in an arbitrary semigroup. In Theorem 2.24 and
Theorem 2.25, we provide similar partial characterizations for piecewise
syndetic set near idempotent and central set near idempotent, respectively.

On the other hand, central sets can also be studied from a view point of
Ramsey theory. The main goal of Ramsey theory is to study the structures which
can not be broken under a finite partition of the system. Van der Waerden’s
theorem [15] is one such result, which says that, under any finite partition
of an infinite arithmetic progression, one of the cells always contains
arbitrary long (finite) arithmetic progressions. A deeper observation reveals
that for any large set $C$ in the semigroup $(\mathbb{N},+)$, the largeness of
the set
$R=\\{(a,d)\in\mathbb{N}\times\mathbb{N}:\\{a,a+d,\ldots,a+(k-1)d\\}\subseteq
C\\}$ in the semigroup $(\mathbb{N}\times\mathbb{N},+)$ are of special
interest, which leads to the study of abundance of large sets. In [14],
Tootkaboni and Vahed introduces the notion of some largeness near idempotent.
We study the abundance of some large sets near idempotent in Theorem 3.12.
Similar results for near zero semigroup is provided in Corollary 3.13 and
Theorem 3.14. The matrix version of these results are deduced in [12] and
[13]. In Theorem 3.18 and 3.19, we provide the near zero analogue of these
results.

Another important result in Ramsey theory is Milliken-Taylor theorem, which
ensures the partition regularity of a very general class of infinite
configurations called Milliken-Taylor systems. In [5], authors extended these
systems to images of very general extended polynomials using the tensor
product of ultrafilters. In Section 4, we begin with the study of tensor
product and Lemma 4.5 ensures that the near zero property gets preserved under
the tensor product. In Theorem 4.9, we provide a relationship between
Milliken-Taylor System and a linear form in one variable evaluated at an
ultrafilter in a near zero semigroup. In Theorem 4.11, we provide a
combinatorial characterization of members of polynomials (with constant term
0) evaluated at idempotents for a near zero subsemigroup of both
$([0,\infty),+)$ and $([0,\infty),\cdot)$. Finally, in Theorem 4.12 and 4.13,
we conclude with two multidimensional Ramsey theoretic results near zero.

The notion of central set near idempotent is algebraically defined in terms of
Stone-Čech compactification of the corresponding semigroup. For a given
semigroup $S$, the Stone-Čech compactification of $S_{d}$ (Given a topological
space $X$, the notation $X_{d}$ represents the set $X$ with the discrete
topology), i.e. $\beta S_{d}$ can be naturally identified with the collection
of all ultrafilters on $S$ [9, Theorem 3.27], and the semigroup structure on
$S$ induces a unique semigroup structure on $\beta S_{d}$ so that $\beta
S_{d}$ becomes a right topological semigroup with $S$ contained in its
topological center [9, Theorem 4.1]. This semigroup structure is explicitly
given by, for $p,q\in\beta S_{d}$, $p+q:=\\{A\subseteq S:\\{s\in S:-s+A\in
q\\}\in p\\}$. From now onwards, whenever we consider Stone-Čech
compactification of a semigroup $S$, we write $\beta S$ for $\beta S_{d}$ and
identify with the ultrafilters on $S$. For other details regarding Stone-Čech
compactification, see [9].

##  2 Cartesian product near Idempotent

Let $(T,+)$ be a semitopological semigroup. Throughout this article, we assume
all our semitopological semigroups to be Hausdorff. Note that the
semitopological structure on the semigroup $T$ is the topology on $T$ such
that for each $x\in T$, the left translation map $L_{x}$ and the right
translation map $R_{x}$ are continuous, where $L_{x}(y):=x+y$ and
$R_{x}(y):=y+x$, $y\in T$. Let $S$ be a subsemigroup of $T$. We denote the
collection of idempotents in $T$ by $E(T)$. Moreover, for every $x\in T$,
$\tau_{x}$ denotes the collection of all neighborhoods of $x$, where a set $U$
is called a neighborhood of $x\in T$ if $x\in\text{{int}}_{T}(U)$, i.e. if $x$
is an interior point of $U$. Following [14, Definition 2.1(a)], here we define
the collection of ultrafilters on $S$ near an idempotent.

######  Definition 2.1.

Let $S$ be a subsemigroup of a semitopological semigroup $T$. For $e\in
E(T)\setminus S$ we denote $e_{S}^{*}=\big{\\{}p\in\beta S:e\in\bigcap_{A\in
p}cl_{T}A\big{\\}}$.

Observe that $e_{S}^{*}=\\{p\in\beta S:U\cap S\in p\text{ for each
}U\in\tau_{e}\\}$.

######  Definition 2.2.

Let $S$ be a subsemigroup of a semitopological semigroup $T$, and let $e\in
E(T)\setminus S$. $S$ is said to be near $e$ subsemigroup of $T$ if $e\in
cl_{T}(S)$.

For example, $((0,\infty),+)$ is a near $0$ subsemigroup of $([0,\infty),+)$.
The semigroup $((0,1),\cdot)$ is both a near 0 subsemigroup and a near $1$
subsemigroup of $([0,1],\cdot)$. Note that here we consider the above
semigroups with the usual Euclidean topology. These examples are already
introduced and studied in [8]. For a noncommutative example, consider the
Heisenberg group $H:=\big{\\{}[x\,\,y\,\,z]:x,y,z\in\mathbb{R}\big{\\}}$
(where we use the notation $[x\,\,y\,\,z]$ to denote the upper triangular
matrix $\begin{pmatrix}1&x&z\\\ &1&y\\\ &&1\end{pmatrix}$) with the usual
matrix multiplication, and with the topology induced via its identification
with $\mathbb{R}^{3}$ given by $[x\,\,y\,\,z]\longleftrightarrow(x,y,z)$,
$(x,y,z)\in\mathbb{R}^{3}$. Then $H$ is a nilpotent real Lie group. Let
$I=[0\,\,0\,\,0]$. Then the subsemigroup
$H_{+}:=\big{\\{}[x\,\,y\,\,z]:x,y,z\in(0,\infty)\\}$ is a near $I$
subsemigroup of $H$.

######  Lemma 2.3.

If $S$ is a near $e$ subsemigroup of a semitopological semigroup $T$, then
$e_{S}^{*}$ is a compact subsemigroup of $\beta S$.

###### Proof.

If $e\in cl_{T}(S)$, pick by [9, Theorem 3.11] an ultrafilter $p\in\beta S$ so
that $\\{U\cap S:U\in\tau_{e}\\}\subseteq p\subseteq\\{A\subseteq S:e\in
cl_{T}(A)\\}$. Hence $p\in e_{S}^{*}$, i.e. $e_{S}^{*}$ is nonempty. The fact
that $e_{S}^{*}$ is a subsemigroup of $\beta S$ follows from [14, Lemma
2.3(i)].

Finally, observe that $p\in e_{S}^{*}$ if and only if $\\{U\cap
S:U\in\tau_{e}\\}\subseteq p$, i.e. if and only if
$p\in\bigcap_{U\in\tau_{e}}cl_{\beta S}(U\cap S)$. Thus
$e_{S}^{*}=\bigcap_{U\in\tau_{e}}cl_{\beta S}(U\cap S)$, and therefore
$e_{S}^{*}$ is compact. ∎

By [9, Theorem 2.8], any compact right topological semigroup $\mathcal{S}$ has
a smallest ideal $K(\mathcal{S})$ which is the union of all minimal left
ideals of $\mathcal{S}$ and also the union of all minimal right ideals of
$\mathcal{S}$. In the next definition, first we recall the notion of central
set near an idempotent [14, Definition 3.9], and then introduce the notion of
$central^{*}$ set near idempotent.

######  Definition 2.4.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $T$. Let
$K(e_{S}^{*})$ be the smallest ideal of the semigroup $e_{S}^{*}$. Let
$A\subseteq S$.

  1. (1)

The set $A$ is central near $e$ if there is some idempotent $p\in
K(e_{S}^{*})$ with $A\in p$.

  2. (2)

The set $A$ is $central^{*}$ near $e$ if $A$ intersects every central set near
$e$.

For a given sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in a noncommutative
semigroup, and for $F\in\mathcal{P}_{f}(\mathbb{N})$ we define $\sum_{n\in
F}x_{n}$ to be the sum in increasing order of indices, i.e. if
$F=\\{n_{1}<n_{2}<\cdots<n_{k}\\}\in\mathcal{P}_{f}(\mathbb{N})$, then
$\sum_{n\in F}x_{n}:=x_{n_{1}}+x_{n_{2}}+\cdots+x_{n_{k}}$. Moreover, for
$F,G\in\mathcal{P}_{f}(\mathbb{N})$, we denote $F<G$ if $\max F<\min G$. When
we say that a sequence $\langle H_{n}\rangle_{n=1}^{\infty}$ in
$\mathcal{P}_{f}(\mathbb{N})$ is an increasing sequence we mean that for each
$n\in\mathbb{N}$, $H_{n}<H_{n+1}$. We recall the following notion of finite
sum ("$FS$" in short) [9, Definition 5.1(b)] and sum subsystem [5, Definition
1.3].

######  Definition 2.5.

Let $(S,+)$ be a semigroup. Given a sequence $\langle
x_{n}\rangle_{n=1}^{\infty}$ in $S$, $FS\big{(}\langle
x_{n}\rangle_{n=1}^{\infty}\big{)}=\big{\\{}\sum_{n\in
F}x_{n}:F\in\mathcal{P}_{f}(\mathbb{N})\big{\\}}$.

######  Definition 2.6.

In a semigroup $(S,+)$, $FS\big{(}\langle y_{n}\rangle_{n=1}^{\infty}\big{)}$
is a sum subsystem of $FS\big{(}\langle x_{n}\rangle_{n=1}^{\infty}\big{)}$ if
there exists an increasing sequence $\langle H_{n}\rangle_{n=1}^{\infty}$ in
$\mathcal{P}_{f}(\mathbb{N})$ such that for each $n\in\mathbb{N}$,
$y_{n}=\sum_{t\in H_{n}}x_{t}$.

The following notion of convergence [14, Definition 3.1] will be used in the
definition of $IP$ set and $IP^{*}$ set near idempotent.

######  Definition 2.7.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$. Let
$\langle x_{n}\rangle_{n=1}^{\infty}$ be a sequence in $S$. We say
$\sum_{n=1}^{\infty}x_{n}$ converges near $e$ if for each $U\in\tau_{e}$ there
exists $m\in\mathbb{N}$ such that $FS(\langle
x_{n}\rangle_{n=m}^{\infty})\subseteq U$.

######  Definition 2.8.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$. Let
$A\subseteq S$.

  1. (1)

The set $A$ is $IP$ near $e$ if there is some sequence $\langle
x_{n}\rangle_{n=1}^{\infty}$ in $S$ such that $\sum_{n=1}^{\infty}x_{n}$
converges near $e$ and $FS\big{(}\langle
x_{n}\rangle_{n=1}^{\infty}\big{)}\subseteq A$.

  2. (2)

The set $A$ is $IP^{*}$ near $e$ if $A$ intersects all the $IP$ sets near $e$.

For a topological space $X$ with $x\in X$, we say that $x$ has a countable
local base in $X$ if $x$ has a countable neighborhood base $\langle
U_{n}\rangle_{n=1}^{\infty}$ in $X$. If $\langle U_{n}\rangle_{n=1}^{\infty}$
is a countable neighborhood base for $x$, one can let
$V_{n}=\bigcap_{k=1}^{n}U_{k}$ so that $V_{n+1}\subseteq V_{n}$ for each
$n\in\mathbb{N}$. Therefore, for a countable local base $\langle
U_{n}\rangle_{n=1}^{\infty}$ we will always assume that $U_{n+1}\subseteq
U_{n}$ for each $n\in\mathbb{N}$. The following theorem provides an algebraic
characterization of $IP$ sets near idempotent. (Given a set $Y$ we write
$\mathcal{P}_{f}(Y)$ for the set of finite nonempty subset of $Y$.)

######  Theorem 2.9.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$ and
let $A\subseteq S$.

  1. (1)

If there is a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ such that
$\sum_{n=1}^{\infty}x_{n}$ converges near $e$ and $FS(\langle
x_{n}\rangle_{n=1}^{\infty})\subseteq A$, then there is an idempotent $p\in
e_{S}^{*}$ such that $A\in p$.

  2. (2)

If $e$ has a countable local base in $T$ and there is an idempotent $p\in
e_{S}^{*}$ such that $A\in p$, then there is a sequence $\langle
x_{n}\rangle_{n=1}^{\infty}$ such that $\sum_{n=1}^{\infty}x_{n}$ converges
near $e$ and $FS(\langle x_{n}\rangle_{n=1}^{\infty})\subseteq A$.

###### Proof.

  1. (1)

Let $J=\bigcap_{m=1}^{\infty}cl_{\beta S}(FS(\langle
x_{n}\rangle_{n=m}^{\infty}))$. By [9, Lemma 5.11], $J$ is a closed
subsemigroup of $\beta S$ so there is an idempotent $p\in J$. It suffices to
show that $p\in e_{S}^{*}$ so suppose instead that $p\notin e_{S}^{*}$. Pick
$B\in p$ such that $e\notin cl_{T}(B)$, and pick $U\in\tau_{e}$ such that
$U\cap B=\emptyset$. Pick $m\in\mathbb{N}$ such that $FS(\langle
x_{n}\rangle_{n=m}^{\infty})\subseteq U$. This is a contradiction because
$FS(\langle x_{n}\rangle_{n=m}^{\infty})\in p$ and $B\in p$.

  2. (2)

Let $\langle W_{n}\rangle_{n=1}^{\infty}$ be a local base at $e$. Pick an
idempotent $p\in e_{S}^{*}$ such that $A\in p$. By [9, Lemma 4.14] if $B\in
p$, $B^{\star}=\\{x\in B:-x+B\in p\\}$, and $x\in B^{\star}$, then
$-x+B^{\star}\in p$. Note that for each $B\in p$ and each $U\in\tau_{e}$,
$B\cap U\in p$.

Let $B_{1}=A\cap W_{1}$. Pick $x_{1}\in B_{1}^{\star}$, let
$B_{2}=(-x_{1}+B_{1}^{\star})\cap(A\cap W_{2})$, and pick $x_{2}\in
B_{2}^{\star}$. Inductively let $n\geq 2$ and assume we have chosen $\langle
B_{t}\rangle_{t=1}^{n}$ and $\langle x_{t}\rangle_{t=1}^{n}$ such that if
$k\in\\{1,2,\ldots,n-1\\}$, then $B_{k+1}=(-x_{k}+B_{k}^{\star})\cap
B_{k}\cap(A\cap W_{k+1})$ and for $k\in\\{1,2,\ldots,n\\}$, $B_{k}\in p$ and
$x_{k}\in B_{k}^{\star}$. Let $B_{n+1}=(-x_{n}+B_{n}^{\star})\cap
B_{n}\cap(A\cap W_{n+1})$.

Having chosen $\langle B_{t}\rangle_{t=1}^{\infty}$ and $\langle
x_{t}\rangle_{t=1}^{\infty}$, we claim that if
$F\in\mathcal{P}_{f}(\mathbb{N})$ and $\min F=k$, then $\sum_{t\in F}x_{t}\in
B_{k}$. We establish this by induction on $|F|$. If $F=\\{k\\}$, we have
$x_{k}\in B_{k}$ as required. So assume that $|F|>1$, let
$G=F\setminus\\{k\\}$, and let $m=\min G$. Then $\sum_{t\in G}x_{t}\in
B_{m}\subseteq B_{k+1}\subseteq(-x_{k}+B_{k})$ so $\sum_{t\in F}x_{t}\in
B_{k}$.

To see that $\sum_{n=1}^{\infty}$ converges near $e$, let $U\in\tau_{e}$. Pick
$m$ such that $W_{m}\subseteq U$. Then $FS(\langle
x_{n}\rangle_{n=m}^{\infty})\subseteq B_{m}\subseteq W_{m}$.

∎

If for each $i\in I$, $X_{i}$ is a set and $x_{i}\in X_{i}$, we will write
$\times_{i\in I}x_{i}$ for the element of $\bigtimes_{i\in I}X_{i}$ whose
$i^{th}$ coordinate is $x_{i}$.

######  Lemma 2.10.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a near $e_{i}$
subsemigroup of a semitopological semigroup $T_{i}$. Let $T=\bigtimes_{i\in
I}T_{i}$, $S=\bigtimes_{i\in I}S_{i}$ and $e=\times_{i\in I}e_{i}$. If for
each $i\in I$, $\widetilde{\pi}_{i}:\beta S\rightarrow\beta S_{i}$ is the
continuous extension of the projection homomorphism $\pi_{i}:S\rightarrow
S_{i}$, then $\widetilde{\pi}_{i}[e_{S}^{*}]=(e_{i})_{S_{i}}^{*}$.

###### Proof.

For $p\in e_{S}^{*}$, we have $e\in\bigcap_{A\in p}cl_{T}(A)$. Also, by [9,
Lemma 3.30], $\widetilde{\pi}_{i}(p)=\\{A_{i}\subseteq
S_{i}:\pi_{i}^{-1}(A_{i})\in p\\}$ where $\pi_{i}^{-1}(A_{i})=\\{x\in
S:\pi_{i}(x)\in A_{i}\\}\subseteq S$. Thus, for each
$A_{i}\in\widetilde{\pi}_{i}(p)$, we have $\pi_{i}^{-1}(A_{i})\in p$, which
implies $e\in cl_{T}(\pi_{i}^{-1}(A_{i}))$, i.e. $e_{i}\in cl_{T_{i}}(A_{i})$.
Hence $e_{i}\in\bigcap_{A_{i}\in\widetilde{\pi}_{i}(p)}cl_{T_{i}}(A_{i})$,
i.e. $\widetilde{\pi}_{i}(p)\in(e_{i})_{S_{i}}^{*}$. Therefore
$\widetilde{\pi}_{i}[e_{S}^{*}]\subseteq(e_{i})_{S_{i}}^{*}$.

To prove the reverse inclusion, let $p_{i}\in(e_{i})_{S_{i}}^{*}$, i.e.
$e_{i}\in\bigcap_{A_{i}\in p_{i}}cl_{T_{i}}(A_{i})$. Consider the family
$\mathcal{A}:=\\{\pi_{i}^{-1}(A_{i}):A_{i}\in p_{i}\\}$ and the family
$\mathcal{R}:=\\{A\subseteq S:e\in cl_{T}(A)\\}$. Then $\mathcal{R}$ is
nonempty, $\emptyset\notin\mathcal{R}$, $\mathcal{R}^{\uparrow}:=\\{B\subseteq
S:A\subseteq B\text{ for some }A\in\mathcal{R}\\}=\mathcal{R}$ and
$\mathcal{R}$ is partition regular. Now, for any nonempty finite collection
$\mathcal{F}$ of elements from $p_{i}$ we have that
$\bigcap_{A_{i}\in\mathcal{F}}A_{i}\in p_{i}$, therefore $e_{i}\in
cl_{T_{i}}\big{(}\bigcap_{A_{i}\in\mathcal{F}}A_{i}\big{)}$, which implies
$e\in
cl_{T}\big{(}\pi_{i}^{-1}\big{(}\bigcap_{A_{i}\in\mathcal{F}}A_{i}\big{)}\big{)}=cl_{T}\big{(}\bigcap_{A_{i}\in\mathcal{F}}\pi_{i}^{-1}(A_{i})\big{)}$,
and hence $\bigcap_{A_{i}\in\mathcal{F}}\pi_{i}^{-1}(A_{i})\in\mathcal{R}$.
So, by [9, Theorem 3.11], pick $p\in\beta S$ such that
$\mathcal{A\subseteq}p\subseteq\mathcal{R}$. Hence $p\in e_{S}^{*}$. Moreover,
as $\mathcal{A}\subseteq p$, so for $A_{i}\in p_{i}$, $\pi_{i}^{-1}(A_{i})\in
p$, i.e. $A_{i}\in\widetilde{\pi}_{i}(p)$. Thus
$p_{i}\subseteq\widetilde{\pi}_{i}(p)$, and both being ultrafilters, we have
$p_{i}=\widetilde{\pi}_{i}(p)$. Therefore
$\widetilde{\pi}_{i}[e_{S}^{*}]=(e_{i})_{S_{i}}^{*}$. ∎

If for each $i\in I$, $X_{i}$ is a topological space, then recall that the box
topology on $\bigtimes_{i\in I}X_{i}$ is the topology generated by the sets
$\big{\\{}\bigtimes_{i\in I}U_{i}:U_{i}\text{ is open in }X_{i}\big{\\}}$.

######  Lemma 2.11.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a near $e_{i}$
subsemigroup of a semitopological semigroup $T_{i}$. Let us consider the
product semitopological semigroup $T=\bigtimes_{i\in I}T_{i}$ with box
topology, let $S=\bigtimes_{i\in I}S_{i}$, and $e=\times_{i\in I}e_{i}$.
Consider $\bigtimes_{i\in I}\beta S_{i}$ with product topology, and let
$\tilde{\iota}:\beta S\rightarrow\bigtimes_{i\in I}\beta S_{i}$ be the
continuous extension of the inclusion $\iota:S_{d}\rightarrow\bigtimes_{i\in
I}\beta S_{i}$. Then $\tilde{\iota}[e_{S}^{*}]=\bigtimes_{i\in
I}(e_{i})_{S_{i}}^{*}$.

###### Proof.

For each $i\in I$, if $\pi_{i}:S\rightarrow S_{i}$ is the natural projection
homomorphism onto the $i^{th}$ coordinate, and if $\widetilde{\pi}_{i}:\beta
S\rightarrow\beta S_{i}$ is the continuous extension of $\pi_{i}$, then for
each $p\in\beta S$ we have that $\tilde{\pi}_{i}(p)=\tilde{\iota}(p)_{i}$,
i.e. the $i^{th}$ coordinate of $\tilde{\iota}(p)$ is $\tilde{\pi}_{i}(p)$.
Hence we get $\tilde{\iota}[e_{S}^{*}]\subseteq\bigtimes_{i\in
I}\tilde{\iota}[e_{S}^{*}]_{i}=\bigtimes_{i\in
I}\tilde{\pi}_{i}[e_{S}^{*}]=\bigtimes_{i\in I}(e_{i})_{S_{i}}^{*}$, where the
last eqality follows from Lemma 2.10.

Conversely, let $\times_{i\in I}p_{i}\in\bigtimes_{i\in
I}(e_{i})_{S_{i}}^{*}$, i.e. for each $i\in I$ we have
$e_{i}\in\bigcap_{A_{i}\in p_{i}}cl_{T_{i}}(A_{i})$. Consider the family
$\mathcal{A}:=\\{\bigtimes_{i\in I}A_{i}:A_{i}\in p_{i}\text{ for each }i\in
I\\}$ and the family $\mathcal{R}:=\\{A\subseteq S:e\in cl_{T}(A)\\}$. Then
$\mathcal{R}$ is nonempty, $\emptyset\notin\mathcal{R}$,
$\mathcal{R}^{\uparrow}:=\\{B\subseteq S:A\subseteq B\text{ for some
}A\in\mathcal{R}\\}=\mathcal{R}$ and $\mathcal{R}$ is partition regular. Now,
for any nonempty finite collection $\big{\langle}\bigtimes_{i\in
I}A_{i}^{(n)}\big{\rangle}_{n=1}^{k}$ of elements from $\mathcal{A}$, we have
that for each $i\in I$, $\bigcap_{n=1}^{k}A_{i}^{(n)}\in p_{i}$, therefore
$e_{i}\in cl_{T_{i}}\big{(}\bigcap_{n=1}^{k}A_{i}^{(n)}\big{)}$, and this
implies $e\in cl_{T}\big{(}\bigtimes_{i\in
I}\big{(}\bigcap_{n=1}^{k}A_{i}^{(n)}\big{)}\big{)}=cl_{T}\big{(}\bigcap_{n=1}^{k}\big{(}\bigtimes_{i\in
I}A_{i}^{(n)}\big{)}\big{)}$, i.e. $\bigcap_{n=1}^{k}\big{(}\bigtimes_{i\in
I}A_{i}^{(n)}\big{)}\in\mathcal{R}$. Hence, by [9, Theorem 3.11], pick
$p\in\beta S$ such that $\mathcal{A}\subseteq p\subseteq\mathcal{R}$. Then
$p\in e_{S}^{*}$. Moreover, as $\mathcal{A}\subseteq p$, so for each $i\in I$,
if $A_{i}\in p_{i}$, then $\pi_{i}^{-1}(A_{i})\in\mathcal{A}\subseteq p$, and
this implies $A_{i}\in\widetilde{\pi}_{i}(p)$. Thus
$p_{i}\subseteq\widetilde{\pi}_{i}(p)$, and both being ultrafilters, we have
$p_{i}=\widetilde{\pi}_{i}(p)$. Therefore we have
$\tilde{\iota}[e_{S}^{*}]=\bigtimes_{i\in I}(e_{i})_{S_{i}}^{*}$. ∎

######  Theorem 2.12.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a near $e_{i}$
subsemigroup of a semitopological semigroup $T_{i}$ and let $p_{i}\in
E(K(e_{S_{i}}^{*}))$. Let us consider the product semitopological semigroup
$T=\bigtimes_{i\in I}T_{i}$ with box topology. Let $S=\bigtimes_{i\in
I}S_{i}$, $e=\times_{i\in I}e_{i}$ and $p=\times_{i\in I}p_{i}$. Consider
$\bigtimes_{i\in I}\beta S_{i}$ with product topology, and let
$\tilde{\iota}:\beta S\rightarrow\bigtimes_{i\in I}\beta S_{i}$ be the
continuous extension of the inclusion $\iota:S_{d}\rightarrow\bigtimes_{i\in
I}\beta S_{i}$. Let $M=\tilde{\iota}^{-1}\\{p\\}\cap e_{S}^{*}$. Then $M$ is a
compact subsemigroup of $e_{S}^{*}$ and $K(M)\subseteq K(e_{S}^{*})$.

###### Proof.

We have that $\tilde{\iota}$ is a homomorphism by [9, Corollary 4.22] and by
Lemma 2.11, $\tilde{\iota}[e_{S}^{*}]=\bigtimes_{i\in I}(e_{i})_{S_{i}}^{*}$.
Consequently, $M$ is a compact subsemigroup of $e_{S}^{*}$. By [9, Exercise
1.7.3], $\tilde{\iota}[K(e_{S}^{*})]=K(\bigtimes_{i\in
I}(e_{i})_{S_{i}}^{*})$. Also, by [9, Theorem 2.23], $K(\bigtimes_{i\in
I}e_{S_{i}}^{*})=\bigtimes_{i\in I}K((e_{i})_{S_{i}}^{*})$ and so $p\in
K(\bigtimes_{i\in I}(e_{i})_{S_{i}}^{*})$. Consequently, $K(e_{S}^{*})\cap
M\neq\emptyset$ and therefore $K(e_{S}^{*})\cap M$ is an ideal of $M$ and so
$K(M)\subseteq K(e_{S}^{*})\cap M$. ∎

From now onwards whenever we say that $A\subseteq S$ is a large set near an
idempotent $e$, we always mean that $A$ is a large set in $S$ near $e$, where
"large" can be any of central, $central^{*}$, $IP$, $IP^{*}$, thick, piecewise
syndetic and $PS^{*}$.

######  Corollary 2.13.

For each $i\in\\{1,2\\}$, let $S_{i}$ be a near $e_{i}$ subsemigroup of a
semitopological semigroups $T_{i}$. If for each $i\in\\{1,2\\}$,
$A_{i}\subseteq S_{i}$ is central near $e_{i}$, then $A_{1}\times
A_{2}\subseteq S_{1}\times S_{2}$ is a central set near $(e_{1},e_{2})$.

###### Proof.

For each $i\in\\{1,2\\}$, pick an idempotent $p_{i}\in K((e_{i})_{S_{i}}^{*})$
such that $A_{i}\in p_{i}$. Let $M$ be as in Theorem 2.12 and pick an
idempotent $r\in K(M)$. Then $\tilde{\iota}(r)=(p_{1},p_{2})$ and so
$A_{1}\times A_{2}\in r$. Since $r\in K((e_{1},e_{2})_{S_{1}\times
S_{2}}^{*})$, we have that $A_{1}\times A_{2}\subseteq S_{1}\times S_{2}$ is a
central set near $(e_{1},e_{2})$. ∎

The following notions will be used subsequently.

######  Definition 2.14.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$. Let
$A\subseteq S$. Then $A$ is thick near $e$ if there exists $U\in\tau_{e}$ such
that for each $F\in\mathcal{P}_{f}(U\cap S)$ and $V\in\tau_{e}$, there exists
$x\in V\cap S$ such that $F+x\subseteq A$.

######  Definition 2.15.

([14, Definition 3.3(b)]) Let $S$ be a near $e$ subsemigroup of a
semitopological semigroup $(T,+)$. A subset $B$ of $S$ is syndetic near $e$ if
for each $U\in\tau_{e}$, there exists some $F\in\mathcal{P}_{f}(U\cap S)$ and
some $V\in\tau_{e}$ such that $S\cap V\subseteq\bigcup_{t\in F}(-t+B)$.

######  Definition 2.16.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$
where $e$ has a countable local base $\langle W_{n}\rangle_{n=1}^{\infty}$ in
$T$, and let $A\subseteq S$.

  1. (1)

The set $A$ is piecewise syndetic near $e$ if there exist sequences $\langle
F_{n}\rangle_{n=1}^{\infty}$ and $\langle V_{n}\rangle_{n=1}^{\infty}$ such
that

    1. (a)

for each $n\in\mathbb{N}$, $F_{n}\in\mathcal{P}_{f}(W_{n}\cap S)$,
$V_{n}\in\tau_{e}$, and $V_{n}\subseteq W_{n}$ and

    2. (b)

for each $G\in\mathcal{P}_{f}(S)$ and for each $O\in\tau_{e}$, there exists
$x\in O\cap S$ such that for each $n\in\mathbb{N}$, $(G\cap
V_{n})+x\subseteq\bigcup_{t\in F_{n}}(-t+A)$.

  2. (2)

The set $A$ is $PS^{*}$ near $e$ if $A$ intersects all the piecewise syndetic
sets near $e$.

The following proposition shows that the definition of piecewise syndetic set
near idempotent is independent of the choice of countable local base.

######  Proposition 2.17.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$. Let
$\langle W_{n}^{(1)}\rangle_{n=1}^{\infty}$ and $\langle
W_{n}^{(2)}\rangle_{n=1}^{\infty}$ be two countable local bases for $e$ in
$T$. If $A\subseteq S$ is piecewise syndetic near $e$ with respect to $\langle
W_{n}^{(1)}\rangle_{n=1}^{\infty}$, then it is also piecewise syndetic near
$e$ with respect to $\langle W_{n}^{(2)}\rangle_{n=1}^{\infty}$.

###### Proof.

Pick a subsequence $\langle W_{n_{k}}^{(1)}\rangle_{k=1}^{\infty}$ of $\langle
W_{n}^{(1)}\rangle_{n=1}^{\infty}$ such that $W_{n_{k}}^{(1)}\subseteq
W_{k}^{(2)}$ for each $k\in\mathbb{N}$. Pick $\langle
F_{n}^{(1)}\rangle_{n=1}^{\infty}$ and $\langle
V_{n}^{(1)}\rangle_{n=1}^{\infty}$ such that

  1. (1)

for each $n\in\mathbb{N}$,
$F_{n}^{(1)}\in\mathcal{P}_{f}\big{(}W_{n}^{(1)}\cap S\big{)}$,
$V_{n}^{(1)}\in\tau_{e}$ and $V_{n}^{(1)}\subseteq W_{n}^{(1)}$, and

  2. (2)

for each $G\in\mathcal{P}_{f}(S)$ and for each $O\in\tau_{e}$, there exists
$x\in O\cap S$ such that for each $n\in\mathbb{N}$, $\big{(}G\cap
V_{n}^{(1)}\big{)}+x\subseteq\bigcup_{t\in F_{n}^{(1)}}(-t+A)$.

For each $k\in\mathbb{N}$, let $F_{k}^{(2)}=F_{n_{k}}^{(1)}$, and let
$V_{k}^{(2)}=V_{n_{k}}^{(1)}$. Then it is easy to see that

  1. (1)

for each $k\in\mathbb{N}$,
$F_{k}^{(2)}\in\mathcal{P}_{f}\big{(}W_{k}^{(2)}\cap S\big{)}$,
$V_{k}^{(2)}\in\tau_{e}$ and $V_{k}^{(2)}\subseteq W_{k}^{(2)}$, and

  2. (2)

for each $G\in\mathcal{P}_{f}(S)$ and for each $O\in\tau_{e}$, there exists
$x\in O\cap S$ such that for each $k\in\mathbb{N}$, $\big{(}G\cap
V_{k}^{(2)}\big{)}+x\subseteq\bigcup_{t\in F_{k}^{(2)}}(-t+A)$.

Hence $A\subseteq S$ is piecewise syndetic near $e$ with respect to $\langle
W_{n}^{(2)}\rangle_{n=1}^{\infty}$. ∎

######  Proposition 2.18.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$. If
$A\subseteq S$ is thick near $e$, then $e\in cl_{T}(A)$.

###### Proof.

Pick $U\in\tau_{e}$ as guaranteed for $A$ by the definition. Suppose that
$e\notin cl_{T}(A)$ and pick $W\in\tau_{e}$ such that $W\cap A=\emptyset$. We
may assume that $W$ is open. Now $e+e\in W$ and $T$ is a semitopological so
pick $X\in\tau_{e}$ such that $X+e\subseteq W$. Since $S$ is a near $e$
subsemigroup of $T$, pick $a\in S\cap U\cap X$. Pick $V\in\tau_{e}$ such that
$a+V\subseteq W$. Pick $x\in V\cap S$ such that $a+x\in A$. Then $a+x\in W\cap
A$, a contradiction. ∎

The following proposition provides a natural class of thick sets near
idempotent.

######  Proposition 2.19.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$. If
$U\in\tau_{e}$, then $U\cap S\subseteq S$ is a thick set near $e$.

###### Proof.

Without loss of generality we assume that $U$ is open. Now $e+e\in U$ and $T$
is semitopological so pick $W\in\tau_{e}$ such that $W+e\subseteq U$. For
given $F\in\mathcal{P}_{f}(W\cap S)$ and $V\in\tau_{e}$, we pick
$V_{w}\in\tau_{e}$ for each $w\in F\cap W$ such that $w+V_{w}\subseteq U$. For
any chosen $x\in\big{(}\bigcap_{w\in F\cap W}V_{w}\big{)}\cap S\cap V$ we then
have $F+x\subseteq U$, i.e. $F+x\subseteq U\cap S$. Hence $U\cap S\subseteq S$
is thick near $e$. ∎

The next two lemmas provide algebraic analogues of thick set near idempotent
and piecewise syndetic set near idempotent.

######  Lemma 2.20.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$. If
$A\subseteq S$ is thick near $e$, then there exists a left ideal $L$ of
$e_{S}^{*}$ such that $L\subseteq cl_{\beta S}A$.

###### Proof.

Pick $U\in\tau_{e}$ such that for every $F\in\mathcal{P}_{f}(U\cap S)$ and
every $V\in\tau_{e}$, there exists $x\in V\cap S$ such that $F+x\subseteq A$.
Consider the family $\mathcal{A}:=\\{-s+A:s\in U\cap S\\}$ and the family
$\mathcal{R}:=\\{B\subseteq S:e\in cl_{T}(B)\\}$. Then $\mathcal{R}$ is
nonempty, $\emptyset\notin\mathcal{R}$, $\mathcal{R}^{\uparrow}:=\\{C\subseteq
S:B\subseteq C\text{ for some }B\in\mathcal{R}\\}=\mathcal{R}$ and
$\mathcal{R}$ is partition regular. We claim that for every
$F\in\mathcal{P}_{f}(U\cap S)$, $\bigcap_{s\in
F}(-s+A)\in\mathcal{R}^{\uparrow}$. For $V\in\tau_{e}$ we pick $x\in V\cap S$
such that $F+x\subseteq A$, so we get $x\in\bigcap_{s\in F}(-s+A)$, and
therefore $\big{(}\bigcap_{s\in F}(-s+A)\big{)}\cap V\neq\emptyset$, i.e.
$e\in cl_{T}\big{(}\bigcap_{s\in F}(-s+A)\big{)}$. This completes the proof of
the above claim, and hence pick by [9, Theorem 3.11] an ultrafilter $p\in\beta
S$ such that $\mathcal{A}\subseteq p\subseteq\mathcal{R}$. Then $p\in
e_{S}^{*}$. Now we will show that the left ideal $e_{S}^{*}+p$ of $e_{S}^{*}$
is contained in $cl_{\beta S}A$, i.e. for each $q\in e_{S}^{*}$, $A\in q+p$.
We know that $A\in q+p$ if and only if $\\{s\in S:-s+A\in p\\}\in q$. Now, as
$\mathcal{A}\subseteq p$, so $U\cap S\subseteq\\{s\in S:-s+A\in p\\}$. But
$q\in e_{S}^{*}$ implies $U\cap S\in q$, and we have $\\{s\in S:-s+A\in
p\\}\in q$, i.e. $A\in q+p$. ∎

Note that from the above lemma it is immediate that any thick set near
idempotent is a central set near idempotent. The next lemma is generalization
of [8, Theorem 3.5].

######  Lemma 2.21.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$
where $e$ has a countable local base $\langle W_{n}\rangle_{n=1}^{\infty}$ in
$T$, and let $A\subseteq S$. Then $A\subseteq S$ is piecewise syndetic near
$e$ if and only if $K(e_{S}^{*})\cap cl_{\beta S}A\neq\emptyset$.

###### Proof.

Necessity. Pick $\langle F_{n}\rangle_{n=1}^{\infty}$ and $\langle
V_{n}\rangle_{n=1}^{\infty}$ satisfying conditions (a) and (b) of Definition
2.16(1). Given $G\in\mathcal{P}_{f}(S)$ and $O\in\tau_{e}$, let

$C(G,O)=\\{x\in O\cap S:\text{for all }n\in\mathbb{N},\,(G\cap
V_{n})+x\subseteq\bigcup_{t\in F_{n}}(-t+A)\\}.$

By assumption, each $C(G,O)\neq\emptyset$. Further, given $G_{1}$ and $G_{2}$
in $\mathcal{P}_{f}(S)$ and $O_{1},O_{2}\in\tau_{e}$ we have that $C(G_{1}\cup
G_{2},O_{1}\cap O_{2})\subseteq C(G_{1},O_{1})\cap C(G_{2},O_{2})$ so
$\\{C(G,O):G\in\mathcal{P}_{f}(S)\text{ and }O\in\tau_{e}\\}$ has the finite
intersection property so pick $p\in\beta S$ with
$\\{C(G,O):G\in\mathcal{P}_{f}(S)\text{ and }O\in\tau_{e}\\}\subseteq p$. Note
that since each $C(G,O)\subseteq O\cap S$, so $p\in e_{S}^{*}$.

Now we claim that for each $n\in\mathbb{N}$, $e_{S}^{*}+p\subseteq cl_{\beta
S}\big{(}\bigcup_{t\in F_{n}}(-t+A)\big{)}$, so let $n\in\mathbb{N}$, and let
$q\in e_{S}^{*}$. To show that $\bigcup_{t\in F_{n}}(-t+A)\in q+p$, we show
that

$V_{n}\cap S\subseteq\big{\\{}y\in S:-y+\bigcup_{t\in F_{n}}(-t+A)\in
p\big{\\}}.$

So let $y\in V_{n}\cap S$. Then $C(\\{y\\},V_{n})\in p$ and
$C(\\{y\\},V_{n})\subseteq-y+\bigcup_{t\in F_{n}}(-t+A)$.

Now pick $r\in(e_{S}^{*}+p)\cap K(e_{S}^{*})$ (since $e_{S}^{*}+p$ is a left
ideal of $e_{S}^{*}$). Given $n\in\mathbb{N}$, $\bigcup_{t\in F_{n}}(-t+A)\in
r$ so pick $t_{n}\in F_{n}$ such that $-t_{n}+A\in r$. Now each $t_{n}\in
F_{n}\subseteq W_{n}$ so $\lim_{n\to\infty}t_{n}=e$ so pick $q\in
e_{S}^{*}\cap cl_{\beta S}(\\{t_{n}:n\in\mathbb{N}\\})$. Then $q+r\in
K(e_{S}^{*})$ and $\\{t_{n}:n\in\mathbb{N}\\}\subseteq\\{t\in S:-t+A\in r\\}$
so $A\in q+r$.

Sufficiency. Pick $p\in K(e_{S}^{*})\cap cl_{\beta S}A$. Let $B=\\{x\in
S:-x+A\in p\\}$. By [14, Theorem 3.4], $B$ is syndetic near $e$. Inductively
for each $n\in\mathbb{N}$ pick $F_{n}\in\mathcal{P}_{f}(W_{n}\cap S)$ and
$V_{n}\in\tau_{e}$ (with $V_{n}\subseteq W_{n}$ and $V_{n+1}\subseteq V_{n}$)
such that $S\cap V_{n}\subseteq\bigcup_{t\in F_{n}}(-t+B)$.

Let $G\in\mathcal{P}_{f}(S)$ be given. If $G\cap V_{1}=\emptyset$, the
conclusion is trivial, so assume $G\cap V_{1}\neq\emptyset$ and let $H=G\cap
V_{1}$. For each $y\in H$, let $m(y)=\max\\{n\in\mathbb{N}:y\in V_{n}\\}$. For
each $y\in H$ and each $n\in\\{1,2,\ldots,m(y)\\}$, we have $y\in\bigcup_{t\in
F}(-t+B)$ so pick $t(y,n)\in F_{n}$ such that $y\in-t(y,n)+B$. Then given
$y\in H$ and $n\in\\{1,2,\ldots,m(y)\\}$, we have that $-(t(y,n)+y)+A\in p$.

Now let $O\in\tau_{e}$ be given. Then $O\in p$ so pick

$x\in O\cap\bigcap_{y\in H}\bigcap_{n=1}^{m(y)}(-(t(y,n)+y)+A).$

Then given $n\in\mathbb{N}$ and $y\in G\cap V_{n}$, we have that $y\in H$ and
$n\leq m(y)$, so $t(y,n)+y+x\in A$, so $y+x\in-t(y,n)+A\subseteq\bigcup_{t\in
F_{n}}(-t+A)$. ∎

If $S$ is a near $e$ subsemigroup of a semitopological semigroup $T$ where $e$
has a countable local base in $T$, then the above lemma provides that any
central set in $S$ near $e$ is a piecewise syndetic set in $S$ near $e$.
Moreover, Theorem 2.9 provides that any central set in $S$ near $e$ is also an
$IP$ set in $S$ near $e$.

######  Lemma 2.22.

Let $I$ be a set. For $i\in I$, let $S_{i}$ be a near $e_{i}$ subsemigroup of
a semitopological semigroup $(T_{i},+)$ and let $A_{i}\subseteq S_{i}$. Let us
consider product semitopological semigroup $T=\bigtimes_{i\in I}T_{i}$ with
box topology. Let $S=\bigtimes_{i\in I}S_{i},\,A=\bigtimes_{i\in I}A_{i}$, and
$e=\times_{i\in I}e_{i}$.

  1. (1)

If for each $i\in I$, $A_{i}\subseteq S_{i}$ is thick near $e_{i}$, then
$A\subseteq S$ is thick near $e$.

  2. (2)

If $A\subseteq S$ is central near $e$, then $A_{i}\subseteq S_{i}$ is central
near $e_{i}$ for each $i\in I$.

  3. (3)

If $e$ has a countable local base in $T$ and if $A\subseteq S$ is piecewise
syndetic near $e$, then $A_{i}\subseteq S_{i}$ is piecewise syndetic near
$e_{i}$ for each $i\in I$.

###### Proof.

For each $i\in I$, let $\pi_{i}:S\rightarrow S_{i}$ be the projection
homomorphism from $S$ onto $S_{i}$.

  1. (1)

For each $i\in I$, pick $U_{i}\in\tau_{e_{i}}$ such that for each
$F_{i}\in\mathcal{P}_{f}(U_{i}\cap S_{i})$ and $V_{i}\in\tau_{e_{i}}$, there
exists $x_{i}\in V_{i}\cap S_{i}$ such that $F_{i}+x_{i}\subseteq A_{i}$. Let
$U=\bigtimes_{i\in I}U_{i}$. Let $F\in\mathcal{P}_{f}(U\cap S)$ and
$V\in\tau_{e}$, and pick $V_{i}\in\tau_{e_{i}}$ for each $i\in I$ such that
$\bigtimes_{i\in I}V_{i}\subseteq V$.

Now, for $\pi_{i}[F]\in\mathcal{P}_{f}(S_{i})$ and $V_{i}\in\tau_{e_{i}}$ pick
$x_{i}\in V_{i}\cap S_{i}$ such that $\pi_{i}[F]+x_{i}\subseteq A_{i}$. Let
$x=\times_{i\in I}x_{i}$. Then

$F+x\subseteq\bigtimes_{i\in I}(\pi_{i}[F]+x_{i})\subseteq\bigtimes_{i\in
I}A_{i}=A,$

and therefore $A\subseteq S$ is thick near $e$.

  2. (2)

For each $i\in I$, let $\widetilde{\pi}_{i}:\beta S\rightarrow\beta S_{i}$ be
the continuous extension of $\pi_{i}$. By Lemma 2.10, each
$\widetilde{\pi}_{i}|_{e_{S}^{*}}:e_{S}^{*}\rightarrow(e_{i})_{S_{i}}^{*}$ is
a surjective homomorphism. Hence, by [9, Exercise 1.7.3] we have
$\widetilde{\pi}_{i}[K(e_{S}^{*})]=K((e_{i})_{S_{i}}^{*})$. Pick an idempotent
$p\in K(e_{S}^{*})$ such that $A\in p$. Then $\widetilde{\pi}_{i}(p)$ is an
idempotent in $K((e_{i})_{S_{i}}^{*})$ and $A_{i}\in\widetilde{\pi}_{i}(p)$.

  3. (3)

Let $\langle W_{n}\rangle_{n=1}^{\infty}$ be a countable local base for $e$ in
$T$. For each $n\in\mathbb{N}$ and for each $i\in I$ pick a neighborhood
$W_{i}^{(n)}$ of $e_{i}$ in $T_{i}$ such that $e\in\bigtimes_{i\in
I}W_{i}^{(n)}\subseteq W_{n}$. Then $\big{\langle}\bigtimes_{i\in
I}W_{i}^{(n)}\big{\rangle}_{n=1}^{\infty}$ is a countable local base for $e$
in $T$ and for each $i\in I$,
$\big{\langle}W_{i}^{(n)}\big{\rangle}_{n=1}^{\infty}$ is a countable local
base for $e_{i}$ in $T_{i}$. With respect to the countable local base
$\big{\langle}\bigtimes_{i\in I}W_{i}^{(n)}\big{\rangle}_{n=1}^{\infty}$ for
$e$ in $T$, pick $\langle F_{n}\rangle_{n=1}^{\infty}$ and $\langle
V_{n}\rangle_{n=1}^{\infty}$ as guaranteed for $A$ by Definition 2.16(1). But,
then the conditions $(a)$ and $(b)$ of Definition 2.16(1) also hold if we
replace $V_{n}$ by any element of $\tau_{e}$ contained in $V_{n}$ for all
$n\in\mathbb{N}$. So without loss of generality we can assume that for all
$n\in\mathbb{N}$, $V_{n}=\bigtimes_{i\in I}V_{i}^{(n)}$.

Fix $i\in I$. Then for each $n\in\mathbb{N}$ we have that
$\pi_{i}[F_{n}]\in\mathcal{P}_{f}\big{(}W_{i}^{(n)}\cap S_{i}\big{)}$,
$V_{i}^{(n)}\in\tau_{e_{i}}$ and $V_{i}^{(n)}\subseteq W_{i}^{(n)}$. Now let
$G_{i}\in\mathcal{P}_{f}(S_{i})$ and let $O_{i}\in\tau_{e_{i}}$. Then fix some
element $s_{j}\in S_{j}$ for each $j\neq i$ and define $G=\bigtimes_{j\in
I}G_{j}$, where $G_{j}:=\\{s_{j}\\}$ for $j\neq i$. Also, define
$O=\bigtimes_{j\in I}O_{j}$, where $O_{j}:=T_{j}$ for each $j\neq i$. Pick
$x=\times_{j\in I}x_{j}\in O\cap S$ such that for each $n\in\mathbb{N}$,
$(G\cap V_{n})+x\subseteq\bigcup_{t\in F_{n}}(-t+A)$, which implies

$\bigtimes_{j\in I}\big{(}\big{(}G_{j}\cap
V_{j}^{(n)}\big{)}+x_{j}\big{)}\subseteq\bigcup_{t\in
F_{n}}(-t+A)\subseteq\bigtimes_{j\in
I}\bigcup_{t\in\pi_{j}[F_{n}]}(-t+A_{j}),$

and in particular we have for each $n\in\mathbb{N}$ that $\big{(}G_{i}\cap
V_{i}^{(n)}\big{)}+x_{i}\subseteq\bigcup_{t\in\pi_{i}[F_{n}]}(-t+A_{i})$ where
$x_{i}\in O_{i}\cap S_{i}$. Hence $A_{i}\subseteq S_{i}$ is piecewise syndetic
near $e_{i}$.

∎

The proof of the following lemma directly follows from Definition 2.16(1).

######  Lemma 2.23.

For each $i\in\\{1,2\\}$, let $S_{i}$ be a near $e_{i}$ subsemigroup of a
semitopological semigroup $T_{i}$ where $e_{i}$ has a countable local base in
$T_{i}$. If for each $i\in\\{1,2\\}$, $A_{i}\subseteq S_{i}$ is piecewise
syndetic near $e_{i}$, then $A_{1}\times A_{2}\subseteq S_{1}\times S_{2}$ is
piecewise syndetic near $(e_{1},e_{2})$.

Now we have the following partial characterization of when the Cartesian
product of piecewise syndetic sets near idempotent is a piecewise syndetic set
near idempotent.

######  Theorem 2.24.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a near $e_{i}$
subsemigroup of a semitopological semigroup $T_{i}$ and let $A_{i}\subseteq
S_{i}$. Let us consider the product semitopological semigroup
$T=\bigtimes_{i\in I}T_{i}$ with box topology. Let $S=\bigtimes_{i\in
I}S_{i}$, $A=\bigtimes_{i\in I}A_{i}$ and $e=\times_{i\in I}e_{i}$. Let $e$
have a countable local base in $T$. Let $J=\\{i\in I:A_{i}\subseteq
S_{i}\text{ is not thick near }e_{i}\\}$ be a finite set. Then $A\subseteq S$
is piecewise syndetic near $e$ if and only if for each $i\in I$,
$A_{i}\subseteq S_{i}$ is piecewise syndetic near $e_{i}$.

###### Proof.

The forward implication follows from Lemma 2.22. For the converse part, if
$J=I$, then delete the references to $I\setminus J$ in the part that follows.
Similarly, if $J=\emptyset$, then delete the references to $J$ in the part
that follows. By Lemma 2.22, $\bigtimes_{i\in I\setminus
J}A_{i}\subseteq\bigtimes_{i\in I\setminus J}S_{i}$ is thick near
$\times_{i\in I\setminus J}e_{i}$, and therefore it is piecewise syndetic near
$\times_{i\in I\setminus J}e_{i}$. Moreover, by inductive application of Lemma
2.23, $\bigtimes_{i\in J}A_{i}\subseteq\bigtimes_{i\in J}S_{i}$ is piecewise
syndetic near $\times_{i\in J}e_{i}$. Thus $\big{(}\bigtimes_{i\in I\setminus
J}A_{i}\big{)}\times\big{(}\bigtimes_{i\in
J}A_{i}\big{)}\subseteq\big{(}\bigtimes_{i\in I\setminus
J}S_{i}\big{)}\times\big{(}\bigtimes_{i\in J}S_{i}\big{)}$ is piecewise
syndetic near $\big{(}\times_{i\in I\setminus
J}e_{i}\big{)}\times\big{(}\times_{i\in J}e_{i}\big{)}$, i.e. $A\subseteq S$
is piecewise syndetic near $e$. ∎

Regarding the Cartesian product of Central sets near idempotent, we have the
following theorem.

######  Theorem 2.25.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a near $e_{i}$
subsemigroup of a semitopological semigroup $T_{i}$ and let $A_{i}\subseteq
S_{i}$. Let us consider the product semitopological semigroup
$T=\bigtimes_{i\in I}T_{i}$ with box topology. Let $S=\bigtimes_{i\in
I}S_{i}$, $A=\bigtimes_{i\in I}A_{i}$ and $e=\times_{i\in I}e_{i}$. Let
$J=\\{i\in I:A_{i}\subseteq S_{i}\text{ is not thick near }e_{i}\\}$ be a
finite set. Then $A\subseteq S$ is central near $e$ if and only if for each
$i\in I$, $A_{i}\subseteq S_{i}$ is central near $e_{i}$.

###### Proof.

The forward implication follows from Lemma 2.22. For the converse part, if
$J=I$, then delete the references to $I\setminus J$ in the part that follows.
Similarly, if $J=\emptyset$, then delete the references to $J$ in the part
that follows. By Lemma 2.22, $\bigtimes_{i\in I\setminus
J}A_{i}\subseteq\bigtimes_{i\in I\setminus J}S_{i}$ is thick near
$\times_{i\in I\setminus J}e_{i}$, and therefore it is central near
$\times_{i\in I\setminus J}e_{i}$. Moreover, by inductive application of
Corollary 2.13, $\bigtimes_{i\in J}A_{i}\subseteq\bigtimes_{i\in J}S_{i}$ is
central near $\times_{i\in J}e_{i}$. Thus $\big{(}\bigtimes_{i\in I\setminus
J}A_{i}\big{)}\times\big{(}\bigtimes_{i\in
J}A_{i}\big{)}\subseteq\big{(}\bigtimes_{i\in I\setminus
J}S_{i}\big{)}\times\big{(}\bigtimes_{i\in J}S_{i}\big{)}$ is central near
$\big{(}\times_{i\in I\setminus J}e_{i}\big{)}\times\big{(}\times_{i\in
J}e_{i}\big{)}$, i.e. $A\subseteq S$ is central near $e$. ∎

##  3 Abundance of Large sets near idempotent

In [3], the authors have studied the abundance of large sets. Following them,
we deduce the corresponding results near idempotent. In the following, we
write $I^{\Diamond}$ for a subsemigroup of $S^{l}$. When we say that
$\varepsilon$ is a property which may be possessed by subsets of a semigroup,
we mean properties such as those we have been considering, whose definition
depends on the particular semigroup in which the sets reside. By
$\varepsilon^{*}$ set in a semigroup, we mean a subset in the semigroup which
intersects all the $\varepsilon$ sets in that semigroup. By $\pi_{i}$ we mean
the projection onto the $i^{th}$ coordinate.

######  Lemma 3.1.

Let $\varepsilon$ be a partition regular property which may be possessed by
subsets of a semigroup. Let $S$ be a semigroup, let $l\in\mathbb{N}$, and let
$I^{\Diamond}$ be a subsemigroup of $S^{l}$. Statement $(1)$ implies statement
$(2)$. If each superset of an $\varepsilon$ set in $S$ is an $\varepsilon$
set, then statements $(1)$ and $(2)$ are equivalent.

  1. (1)

For every $\varepsilon$ set $A$ in $I^{\Diamond}$ and every
$i\in\\{1,2,\ldots,l\\}$, $\pi_{i}[A]$ is an $\varepsilon$ set in $S$.

  2. (2)

Whenever $B$ is an $\varepsilon^{*}$ set in $S$, $B^{l}\cap I^{\Diamond}$ is
an $\varepsilon^{*}$ set in $I^{\Diamond}$.

###### Proof.

[3, Theorem 2.2]. ∎

Note that Definition 2.4, Theorem 2.9 and Lemma 2.21 ensures the partition
regular property of central sets near idempotent, IP sets near idempotent and
piecewise syndetic sets near idempotent, respectively. If $X$ is a set, $x\in
X$ and $l\in\mathbb{N}$, we will write $\overline{x}$ for the element in
$X^{l}$ each of whose coordinates is $x$.

######  Corollary 3.2.

Let $S$ be a dense near $e$ subsemigroup of a semitopological semigroup $T$.
Let $l\in\mathbb{N}$ and let $I^{\Diamond}\subseteq S^{l}$ be a near
$\overline{e}$ semigroup of $T^{l}$. If $B\subseteq S$ be an $IP^{*}$ set near
$e$, then $B^{l}\cap I^{\Diamond}\subseteq I^{\Diamond}$ is an $IP^{*}$ set
near $\overline{e}$.

###### Proof.

It is immediate from the definition that whenever $A\subseteq I^{\Diamond}$ is
an $IP$ set near $\overline{e}$ and $i\in\\{1,2,\ldots,l\\}$, then
$\pi_{i}[A]\subseteq S$ is an IP set near $e$. Now the result follows from
Lemma 3.1. ∎

######  Lemma 3.3.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $T$. Let
$l\in\mathbb{N}$. Let $I^{\Diamond}\subseteq S^{l}$ be a near $\overline{e}$
subsemigroup of $T^{l}$ such that for each $i\in\\{1,2,\ldots,l\\}$ and for
$\mathcal{E}_{i}\subseteq\pi_{i}[I^{\Diamond}]$, if $e\in
cl_{T}(\mathcal{E}_{i})$ then $\overline{e}\in
cl_{T^{l}}\big{(}\pi_{i}^{-1}(\mathcal{E}_{i})\big{)}$ where
$\pi_{i}:I^{\Diamond}\rightarrow\pi_{i}[I^{\Diamond}]$ is the natural
projection homomorphism onto the $i^{th}$ coordinate. If for each
$i\in\\{1,2,\ldots,l\\}$, $\widetilde{\pi}_{i}:\beta
I^{\Diamond}\rightarrow\beta(\pi_{i}[I^{\Diamond}])$ is the continuous
extension of $\pi_{i}$, then
$\widetilde{\pi}_{i}\big{[}\overline{e}_{I^{\Diamond}}^{*}\big{]}=e_{\pi_{i}[I^{\Diamond}]}^{*}$.

###### Proof.

Let $i\in\\{1,2,\ldots,l\\}$. For $p\in\overline{e}_{I^{\Diamond}}^{*}$, we
have $\overline{e}\in\bigcap_{A\in p}cl_{T^{l}}(A)$. Also, by [9, Lemma 3.30],
$\widetilde{\pi}_{i}(p)=\\{A_{i}\subseteq\pi_{i}[I^{\Diamond}]:\pi_{i}^{-1}(A_{i})\in
p\\}$. Thus, for each $A_{i}\in\widetilde{\pi}_{i}(p)$ we have
$\pi_{i}^{-1}(A_{i})\in p$, which implies $\overline{e}\in
cl_{T^{l}}\big{(}\pi_{i}^{-1}(A_{i})\big{)}$, i.e. $e\in cl_{T}(A_{i})$. Hence
$e\in\bigcap_{A_{i}\in\widetilde{\pi}_{i}(p)}cl_{T}(A_{i})$, i.e.
$\widetilde{\pi}_{i}(p)\in e_{\pi_{i}[I^{\Diamond}]}^{*}$.

To prove the reverse inclusion, let $p_{i}\in e_{\pi_{i}[I^{\Diamond}]}^{*}$.
Consider the family $\mathcal{A}:=\\{\pi_{i}^{-1}(A_{i}):A_{i}\in p_{i}\\}$
and the family $\mathcal{R}:=\\{A\subseteq I^{\Diamond}:\overline{e}\in
cl_{T^{l}}(A)\\}$. Then $\mathcal{A}$ is closed under finite intersection,
$\mathcal{R}$ is nonempty, $\emptyset\notin\mathcal{R}$,
$\mathcal{R}^{\uparrow}:=\\{B\subseteq I^{\Diamond}:A\subseteq B\text{ for
some }A\in\mathcal{R}\\}=\mathcal{R}$ and $\mathcal{R}$ is partition regular.
Moreover, if $A_{i}\in p_{i}$, then $A_{i}\subseteq\pi_{i}[I^{\Diamond}]$ and
$e\in cl_{T}(A_{i})$ implies $\overline{e}\in
cl_{T^{l}}\big{(}\pi_{i}^{-1}(A_{i})\big{)}$. Thus
$\mathcal{A}\subseteq\mathcal{R}$. Hence, by [9, Theorem 3.11] pick $p\in\beta
I^{\Diamond}$ such that $\mathcal{A}\subseteq p\subseteq\mathcal{R}$. Then
$p\in\overline{e}_{I^{\Diamond}}^{*}$. Also, if $A_{i}\in p_{i}$, then
$\pi_{i}^{-1}(A_{i})\in p$, i.e. $A_{i}\in\widetilde{\pi}_{i}(p)$. Thus
$p_{i}\subseteq\widetilde{\pi}_{i}(p)$, and both being ultrafilters, we have
that $p_{i}=\widetilde{\pi}_{i}(p)$. Therefore
$\widetilde{\pi}_{i}\big{[}\overline{e}_{I^{\Diamond}}^{*}\big{]}=e_{\pi_{i}[I^{\Diamond}]}^{*}$.
∎

######  Lemma 3.4.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $T$. Let
$l\in\mathbb{N}$. Let $I^{\Diamond}\subseteq S^{l}$ be a near $\overline{e}$
subsemigroup of $T^{l}$ such that for each $i\in\\{1,2,\ldots,l\\}$ and for
$\mathcal{E}_{i}\subseteq\pi_{i}[I^{\Diamond}]$, if $e\in
cl_{T}(\mathcal{E}_{i})$, then $\overline{e}\in
cl_{T^{l}}\big{(}\pi_{i}^{-1}(\mathcal{E}_{i})\big{)}$ where
$\pi_{i}:I^{\Diamond}\rightarrow\pi_{i}[I^{\Diamond}]$ is the natural
projection homomorphism onto the $i^{th}$ coordinate. Let $A\subseteq
I^{\Diamond}$.

  1. (1)

If $A\subseteq I^{\Diamond}$ is central near $\overline{e}$, then
$\pi_{i}[A]\subseteq\pi_{i}[I^{\Diamond}]$ is central near $e$ for each
$i\in\\{1,2,\ldots,l\\}$.

  2. (2)

If $e$ has a countable local base in $T$ and if $A\subseteq I^{\Diamond}$ is
piecewise syndetic near $\overline{e}$, then
$\pi_{i}[A]\subseteq\pi_{i}[I^{\Diamond}]$ is piecewise syndetic near $e$ for
each $i\in\\{1,2,\ldots,l\\}$.

###### Proof.

For each $i\in\\{1,2,\ldots,l\\}$, let $\widetilde{\pi}_{i}:\beta
I^{\Diamond}\rightarrow\beta\big{(}\pi_{i}[I^{\Diamond}]\big{)}$ be the
continuous extension of $\pi_{i}$. By Lemma 3.3 and [9, Lemma 2.14],
$\widetilde{\pi}_{i}|_{\overline{e}_{I^{\Diamond}}^{*}}:\overline{e}_{I^{\Diamond}}^{*}\rightarrow
e_{\pi_{i}[I^{\Diamond}]}^{*}$ is a surjective homomorphism. Hence, by [9,
Exercise 1.7.3] we have that
$\widetilde{\pi}_{i}[K(\overline{e}_{I^{\Diamond}}^{*})]=K\big{(}e_{\pi_{i}[I^{\Diamond}]}^{*}\big{)}$.

  1. (1)

Pick an idempotent $p\in K(\overline{e}_{I^{\Diamond}}^{*})$ such that $A\in
p$. Then $\widetilde{\pi}_{i}(p)$ is an idempotent in
$K\big{(}e_{\pi_{i}[I^{\Diamond}]}^{*}\big{)}$ and
$\pi_{i}[A]\in\widetilde{\pi}_{i}(p)$.

  2. (2)

Using Lemma 2.21 pick $p\in K(\overline{e}_{I^{\Diamond}}^{*})$ such that
$A\in p$. Then $\widetilde{\pi}_{i}(p)\in
K\big{(}e_{\pi_{i}[I^{\Diamond}]}^{*}\big{)}$ and
$\pi_{i}[A]\in\widetilde{\pi}_{i}(p)$.

∎

######  Theorem 3.5.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $T$ where
$e$ has a countable local base in $T$. Let $l\in\mathbb{N}$. Let
$I^{\Diamond}\subseteq S^{l}$ be a near $\overline{e}$ subsemigroup of $T^{l}$
such that for each $i\in\\{1,2,\ldots,l\\}$, $\pi_{i}[I^{\Diamond}]\subseteq
S$ is piecewise syndetic near $e$, and for
$\mathcal{E}_{i}\subseteq\pi_{i}[I^{\Diamond}]$ if $e\in
cl_{T}(\mathcal{E}_{i})$, then $\overline{e}\in
cl_{T^{l}}\big{(}\pi_{i}^{-1}(\mathcal{E}_{i})\big{)}$ where
$\pi_{i}:I^{\Diamond}\rightarrow\pi_{i}[I^{\Diamond}]$ is the natural
projection homomorphism onto the $i^{th}$ coordinate. Let $B\subseteq S$.

  1. (1)

If $B\subseteq S$ is $central^{*}$ near $e$, then $B^{l}\cap
I^{\Diamond}\subseteq I^{\Diamond}$ is $central^{*}$ near $\overline{e}$.

  2. (2)

If $B\subseteq S$ is $PS^{*}$ near $e$, then $B^{l}\cap I^{\Diamond}\subseteq
I^{\Diamond}$ is piecewise syndetic near $\overline{e}$.

###### Proof.

Let $i\in\\{1,2,\ldots,l\\}$ and pick $p\in K(e_{S}^{*})\cap cl_{\beta
S}(\pi_{i}[I^{\Diamond}])$. Then for each $U\in\tau_{e}$,
$U\cap\pi_{i}[I^{\Diamond}]=(U\cap S)\cap\pi_{i}[I^{\Diamond}]\in p$, and
therefore using the natural inclusion of $e_{\pi_{i}[I^{\Diamond}]}^{*}$
inside $e_{S}^{*}$ we have that $p\in e_{\pi_{i}[I^{\Diamond}]}^{*}$. Thus
$e_{\pi_{i}[I^{\Diamond}]}^{*}\cap K(e_{S}^{*})\neq\emptyset$ and therefore by
[9, Theorem 1.65], we have that
$K\big{(}e_{\pi_{i}[I^{\Diamond}]}^{*}\big{)}\subseteq K(e_{S}^{*})$.

  1. (1)

By Lemma 3.1 it suffices to show that whenever $A\subseteq I^{\Diamond}$ is
central near $\overline{e}$, then $\pi_{i}[A]\subseteq S$ is central near $e$
for each $i\in\\{1,2,\ldots,l\\}$. Fix $i\in\\{1,2,\ldots,l\\}$ and let
$A\subseteq I^{\Diamond}$ be central near $\overline{e}$. Then by Lemma 3.4,
$\pi_{i}[A]\subseteq\pi_{i}[I^{\Diamond}]$ is central near $e$. So pick an
idempotent $p\in K\big{(}e_{\pi_{i}[I^{\Diamond}]}^{*}\big{)}\cap
cl_{\beta(\pi_{i}[I^{\Diamond}])}(\pi_{i}[A])$ and consequently we have that
$p\in K(e_{S}^{*})\cap cl_{\beta S}(\pi_{i}[A])$.

  2. (2)

By Lemma 3.1 it suffices to show that whenever $A\subseteq I^{\Diamond}$ is
piecewise syndetic near $\overline{e}$, then $\pi_{i}[A]\subseteq S$ is
piecewise syndetic near $e$ for each $i\in\\{1,2,\ldots,l\\}$. Fix
$i\in\\{1,2,\ldots,l\\}$ and let $A\subseteq I^{\Diamond}$ be piecewise
syndetic near $\overline{e}$. Then by Lemma 3.4,
$\pi_{i}[A]\subseteq\pi_{i}[I^{\Diamond}]$ is piecewise syndetic near $e$. So
$K\big{(}e_{\pi_{i}[I^{\Diamond}]}^{*}\big{)}\cap
cl_{\beta(\pi_{i}[I^{\Diamond}])}(\pi_{i}[A])\neq\emptyset$ and consequently
we have that $K(e_{S}^{*})\cap cl_{\beta S}(\pi_{i}[A])\neq\emptyset$.

∎

From now onwards, let $S$ be a near $e$ subsemigroup of a semitopological
semigroup $(T,+)$, let $l\in\mathbb{N}$ and let $E^{\Diamond}\subseteq S^{l}$
be a near $\overline{e}$ subsemigroup of $T^{l}$ such that
$\\{\overline{a}:a\in S\\}\subseteq E^{\Diamond}$. Moreover, let
$I^{\Diamond}$ be a near $\overline{e}$ ideal of $E^{\Diamond}$ such that for
each $i\in\\{1,2,\ldots,l\\}$ and for
$\mathcal{E}_{i}\subseteq\pi_{i}[I^{\Diamond}]$, if $e\in
cl_{T}(\mathcal{E}_{i})$, then $\overline{e}\in
cl_{T^{l}}\big{(}\pi_{i}^{-1}(\mathcal{E}_{i})\big{)}$ where
$\pi_{i}:I^{\Diamond}\rightarrow\pi_{i}[I^{\Diamond}]$ is the natural
projection homomorphism onto the $i^{th}$ coordinate. Note that when we say
$I^{\Diamond}$ is a near $\overline{e}$ ideal of $E^{\Diamond}$, we mean that
$I^{\Diamond}$ is an ideal of $E^{\Diamond}$ which is also a near
$\overline{e}$ subsemigroup of $T^{l}$.

######  Definition 3.6.

Let $X=(\beta S)^{l},Y=(e_{S}^{*})^{l}$ are with product topology and the
coordinatewise operation. Then
$E=cl_{X}E^{\Diamond},I=cl_{X}I^{\Diamond},E_{0}=E\cap Y$ and $I_{0}=I\cap Y$.

######  Lemma 3.7.

X is a compact right topological semigroup, for each $\vec{x}\in S^{l}$,
$\lambda_{\vec{x}}:X\rightarrow X$ given by
$\lambda_{\vec{x}}(\vec{p})=\vec{x}+\vec{p},\,\vec{p}\in X$ is continuous, $Y$
is a subsemigroup of $X$, $E_{0}$ is a subsemigroup of $Y$, $I_{0}$ is an
ideal of $E_{0}$, and $K(Y)=(K(e_{S}^{*}))^{l}$.

###### Proof.

It suffices to show that $E_{0}\neq\emptyset$ and $I_{0}\neq\emptyset$.
Remaining of the proof follows from [9, Theorem 2.22, 2.23 and 4.17]. Let
$\tilde{\iota}:\beta E^{\Diamond}\rightarrow X$ be the continuous extension
the natural inclusion map $\iota:E^{\Diamond}\rightarrow X$. Now we prove that
$E_{0}\neq\emptyset$ and the proof for $I_{0}$ is same as of $E_{0}$. Pick
$p\in\overline{e}_{E^{\Diamond}}^{*}$. Following the proof of [9, Theorem
3.27] we then have that $\tilde{\iota}(p)\in\bigcap_{A\in p}cl_{X}(\iota[A])$.
In particular, $\tilde{\iota}(p)\in cl_{X}(\iota[E^{\Diamond}])=E$. Let
$i\in\\{1,2,\ldots,l\\}$ and let
$\pi_{i}:E^{\Diamond}\rightarrow\pi_{i}[E^{\Diamond}]$ and
$proj_{i}:X\rightarrow\beta S$ be the natural projection maps onto the
$i^{th}$ coordinates of $E^{\Diamond}$ and $X$, respectively. If
$\widetilde{\pi}_{i}:\beta(E^{\Diamond})\rightarrow\beta(\pi_{i}[E^{\Diamond}])\subseteq\beta
S$ is the continuous extension of $\pi_{i}$, then
$proj_{i}\circ\tilde{\iota}=\widetilde{\pi}_{i}$ as both the sides agree on a
dense set $E^{\Diamond}$ of $\beta E^{\Diamond}$. Now, for each $U\in\tau_{e}$
we have that $U^{l}\cap E^{\Diamond}\in p$, which implies $\pi_{i}[U^{l}\cap
E^{\Diamond}]\in\widetilde{\pi}_{i}(p)=proj_{i}(\tilde{\iota}(p))$, i.e.
$U\cap\pi_{i}[E^{\Diamond}]\in proj_{i}(\tilde{\iota}(p))$, and therefore
$U\cap S\in proj_{i}(\tilde{\iota}(p))$. Hence $proj_{i}(\tilde{\iota}(p))\in
e_{S}^{*}$ and thus $\tilde{\iota}(p)\in E\cap Y=E_{0}$. ∎

######  Lemma 3.8.

Let $p\in K(e_{S}^{*})$. Then $\overline{p}\in
K(I_{0})=(K(e_{S}^{*}))^{l}\cap{E_{0}}$.

###### Proof.

Observe first that $\overline{p}\in E$. To see this let $U$ be a neighborhood
of $\overline{p}$ in $X$ and for each $i\in\\{1,2,\ldots,l\\}$, pick $A_{i}\in
p$ such that $\bigtimes_{i=1}^{l}cl_{\beta S}(A_{i})\subseteq U$. Then
$\bigcap_{i=1}^{l}A_{i}\in q$ so pick $a\in\bigcap_{i=1}^{l}A_{i}$. Then
$\overline{a}\in E^{\Diamond}\cap U$.

Thus $\overline{p}\in(K(e_{S}^{*}))^{l}\cap E=(K(e_{S}^{*}))^{l}\cap E_{0}$.
Since, by Lemma 3.7, $K(Y)=(K(e_{S}^{*}))^{l}$ we thus have that $K(Y)\cap
E_{0}\neq\emptyset$. Thus, by [9, Theorem 1.65] $K(E_{0})=K(Y)\cap
E_{0}=(K(e_{S}^{*}))^{l}\cap E_{0}$.

Since $I_{0}$ is an ideal of $E_{0}$, $K(E_{0})\subseteq I_{0}$. Consequently,
again by [9, Theorem 1.65], $K(I_{0})=K(E_{0})=(K(e_{S}^{*}))^{l}\cap E_{0}$.
Thus $\overline{p}\in K(I_{0})$. ∎

Now $I^{\Diamond}$ itself is a near $\overline{e}$ subsemigroup of $T^{l}$,
and thus $\beta I^{\Diamond}$ is a compact right topological semigroup.

######  Definition 3.9.

$\iota:I^{\Diamond}\rightarrow I^{\Diamond}\subseteq I$ is the identity
function and $\tilde{\iota}:\beta I^{\Diamond}\rightarrow I$ is its continuous
extension.

######  Lemma 3.10.

The function $\tilde{\iota}$ is a homomorphism and
$\tilde{\iota}[K(\overline{e}_{I^{\Diamond}}^{*})]=K(I_{0})$.

###### Proof.

Let $i\in\\{1,2,\ldots,l\\}$. Let $proj_{i}:X\rightarrow\beta S$ be the
natural projection map onto the $i^{th}$ coordinates of $X$. If
$\widetilde{\pi}_{i}:\beta
I^{\Diamond}\rightarrow\beta(\pi_{i}[I^{\Diamond}])\subseteq\beta S$ is the
continuous extension of $\pi_{i}$, then
$\widetilde{\pi}_{i}=proj_{i}\circ\tilde{\iota}$ as both sides agree on a
dense set $I^{\Diamond}$ of $\beta I^{\Diamond}$.

By [9, Lemma 2.14], $\tilde{\iota}$ is a homomorphism. Now we show that
$\tilde{\iota}[\overline{e}_{I^{\Diamond}}^{*}]=I_{0}$. To see this, first let
$p\in\overline{e}_{I^{\Diamond}}^{*}$. Then $\tilde{\iota}(p)\in I$,
$proj_{i}(\tilde{\iota}(p))=\widetilde{\pi}_{i}(p)$ and by Lemma 3.3 we have
that $\widetilde{\pi}_{i}(p)\in e_{\pi_{i}[I^{\Diamond}]}^{*}\subseteq
e_{S}^{*}$. Hence $proj_{i}(\tilde{\iota}(p))\in e_{S}^{*}$ and thus
$\tilde{\iota}(p)\in Y$.

Conversely, let $(p_{1},p_{2},\ldots,p_{l})\in I_{0}$. For each
$j\in\\{1,2,\ldots,l\\}$, if $A_{j}\in p_{j}$, then
$\bigtimes_{j=1}^{l}cl_{\beta S}(A_{j})$ is a neighborhood of
$(p_{1},p_{2},\ldots,p_{l})$ in $X$ and therefore
$\big{(}\bigtimes_{j=1}^{l}A_{j}\big{)}\cap
I^{\Diamond}=\big{(}\bigtimes_{j=1}^{l}cl_{\beta S}(A_{j})\big{)}\cap
I^{\Diamond}\neq\emptyset$. Moreover, the family
$\big{\\{}\big{(}\bigtimes_{j=1}^{l}A_{j}\big{)}\cap I^{\Diamond}:A_{j}\in
p_{j}\text{ for each }j\in\\{1,2,\ldots,l\\}\big{\\}}$ has finite intersection
property, hence pick $p\in\beta I^{\Diamond}$ such that
$\big{\\{}\big{(}\bigtimes_{j=1}^{l}A_{j}\big{)}\cap I^{\Diamond}:A_{j}\in
p_{j}\text{ for each }j\in\\{1,2,\ldots,l\\}\big{\\}}\subseteq p$. We claim
that $\tilde{\iota}(p)=(p_{1},p_{2},\ldots,p_{l})$ for which it suffices to
show that $proj_{i}\circ\tilde{\iota}(p)=p_{i}$, i.e.
$\widetilde{\pi}_{i}(p)=p_{i}$. To prove the claim, let $B_{i}\in p_{i}$. Then
$\pi_{i}^{-1}(B_{i})=\big{(}\bigtimes_{j=1}^{l}B_{j}\big{)}\cap
I^{\Diamond}\in p$, where $B_{j}:=S$ for each $j\neq i$, and therefore
$B_{i}\in\widetilde{\pi}_{i}(p)$. Hence
$p_{i}\subseteq\widetilde{\pi}_{i}(p)$, and both being ultrafilters, we have
that $p_{i}=\widetilde{\pi}_{i}(p)$. Now we show that
$p\in\overline{e}_{I^{\Diamond}}^{*}$. For any $U\in\tau_{\overline{e}}$ pick
$V\in\tau_{e}$ such that $V^{l}\subseteq U$. Then $V\cap S\in p_{j}$ for each
$j\in\\{1,2,\ldots,l\\}$, so $V^{l}\cap I^{\Diamond}=(V\cap S)^{l}\cap
I^{\Diamond}\in p$ and therefore $U\cap I^{\Diamond}\in p$. Thus
$p\in\overline{e}_{I^{\Diamond}}^{*}$.

Therefore
$\tilde{\iota}|_{\overline{e}_{I^{\Diamond}}^{*}}:{\overline{e}_{I^{\Diamond}}^{*}}\rightarrow
I_{0}$ is a surjective homomorphism. Now, by [9, Exercise 1.7.3], we have that
$\tilde{\iota}[K(\overline{e}_{I^{\Diamond}}^{*})]=K(I_{0})$. ∎

Now we recall [3, Lemma 3.6] which will be used subsequently.

######  Lemma 3.11.

Let $B\subseteq S$. If $r\in\beta I^{\Diamond}$ and
$\tilde{\iota}(r)\in(cl_{\beta S}B)^{l}$. Then $B^{l}\cap I^{\Diamond}\in r$.

######  Theorem 3.12.

Let $S$ be a near $e$ subsemigroup of a semitopological semigroup $(T,+)$, let
$l\in\mathbb{N}$ and let $E^{\Diamond}\subseteq S^{l}$ be a near
$\overline{e}$ subsemigroup of $T^{l}$ such that $\\{\overline{a}:a\in
S\\}\subseteq E^{\Diamond}$. Let $I^{\Diamond}$ be a near $\overline{e}$ ideal
of $E^{\Diamond}$ such that for each $i\in\\{1,2,\ldots,l\\}$ and for
$\mathcal{E}_{i}\subseteq\pi_{i}[I^{\Diamond}]$, if $e\in
cl_{T}(\mathcal{E}_{i})$, then $\overline{e}\in
cl_{T^{l}}\big{(}\pi_{i}^{-1}(\mathcal{E}_{i})\big{)}$ where
$\pi_{i}:I^{\Diamond}\rightarrow\pi_{i}[I^{\Diamond}]$ is the natural
projection homomorphism onto the $i^{th}$ coordinate. Let $B\subseteq S$.

  1. (1)

If $B\subseteq S$ is central near $e$, then $B^{l}\cap I^{\Diamond}\subseteq
I^{\Diamond}$ is central near $\overline{e}$.

  2. (2)

Let $e$ have a countable local base in $T$. If $B\subseteq S$ is piecewise
syndetic near $e$, then $B^{l}\cap I^{\Diamond}\subseteq I^{\Diamond}$ is
piecewise syndetic near $\overline{e}$.

  3. (3)

Let $e$ have a countable local base in $T$. If $B\subseteq S$ is $central^{*}$
near $e$, then $B^{l}\cap I^{\Diamond}\subseteq I^{\Diamond}$ is $central^{*}$
near $\overline{e}$.

  4. (4)

Let $e$ have a countable local base in $T$. If $B\subseteq S$ is $PS^{*}$ near
$e$, then $B^{l}\cap I^{\Diamond}\subseteq I^{\Diamond}$ is $PS^{*}$ near
$\overline{e}$.

  5. (5)

If $B\subseteq S$ is $IP^{*}$ near $e$, then $B^{l}\cap I^{\Diamond}\subseteq
I^{\Diamond}$ is $IP^{*}$ near $\overline{e}$.

###### Proof.

  1. (1)

Pick $p\in K(e_{S}^{*})$ such that $p+p=p$ and $B\in p$. By Lemma 3.8,
$\overline{p}\in K(I_{0})$. Pick by Lemma 3.10 $q\in
K(\overline{e}_{I^{\Diamond}}^{*})$ such that $\tilde{\iota}(q)=\overline{p}$.
Pick a minimal left ideal $L$ of $\overline{e}_{I^{\Diamond}}^{*}$ such that
$q\in L$. Let $M=\\{r\in L:\tilde{\iota}(r)=\overline{p}\\}$. Then $M$ is a
compact subsemigroup of $\overline{e}_{I^{\Diamond}}^{*}$, so pick an
idempotent $r\in M$. By Lemma 3.11, $B^{l}\cap I^{\Diamond}\in r$.

  2. (2)

Pick by Lemma 2.21 $p\in K(e_{S}^{*})$ such that $B\in p$. Now Lemma 3.8
provides that $\overline{p}\in K(I_{0})$. Pick by Lemma 3.10 $q\in
K(\overline{e}_{I^{\Diamond}}^{*})$ such that $\tilde{\iota}(q)=\overline{p}$.
Hence, by Lemma 3.11, we have $B^{l}\cap I^{\Diamond}\in r$.

To establish statements (3) and (4), it suffices by Theorem 3.5 to let
$i\in\\{1,2,\ldots,l\\}$ and show that $\pi_{i}[I^{\Diamond}]\subseteq S$ is
piecewise syndetic near $e$. Since $I^{\Diamond}$ is a near $\overline{e}$
ideal of $E^{\Diamond}$, so in particular $\pi_{i}[I^{\Diamond}]$ is a left
ideal of $\pi_{i}[E^{\Diamond}]=S$ and therefore $cl_{\beta
S}(\pi_{i}[I^{\Diamond}])$ is a left ideal of $\beta S$. Moreover, as $e\in
cl_{T}(\pi_{i}[I^{\Diamond}])$, so $cl_{\beta S}(\pi_{i}[I^{\Diamond}])\cap
e_{S}^{*}\neq\emptyset$. Hence $cl_{\beta S}(\pi_{i}[I^{\Diamond}])\cap
e_{S}^{*}$ is a left ideal of $e_{S}^{*}$. Pick a minimal left ideal $L$ of
$e_{S}^{*}$ contained in $cl_{\beta S}(\pi_{i}[I^{\Diamond}])\cap e_{S}^{*}$.
Then $L\cap cl_{\beta S}(\pi_{i}[I^{\Diamond}])=L\neq\emptyset$ and therefore
$K(e_{S}^{*})\cap cl_{\beta S}(\pi_{i}[I^{\Diamond}])\neq\emptyset$. Thus, by
Lemma 2.21, $\pi_{i}[I^{\Diamond}]\subseteq S$ is piecewise syndetic near $e$.
Statement (5) follows immediately from Corollary 3.2. ∎

######  Corollary 3.13.

Let $S\subseteq(0,\infty)$ be a near zero subsemigroup of $([0,\infty),+)$.
Let $l\in\mathbb{N}$ and let $AP_{l}=\\{(a,a+d,\ldots,a+(l-1)d):a,d\in S\\}$.
Let "large near zero" be any of "central near zero", "piecewise syndetic near
zero", "$central^{*}$ near zero", "$PS^{*}$ near zero" or "$IP^{*}$ near
zero". Let $B\subseteq S$. If $B\subseteq S$ is large near zero, then
$B^{l}\cap AP_{l}\subseteq AP_{l}$ is large near zero.

###### Proof.

Let $T=([0,\infty),+),\,I^{\Diamond}=AP_{l}$ and let
$E^{\Diamond}=I^{\Diamond}\cup\\{\overline{a}:a\in S\\}$. It suffices by
Theorem 3.12 to let $i\in\\{1,2,\ldots,l\\}$ and show that if
$\mathcal{E}_{i}\subseteq\pi_{i}[AP_{l}]$ and if $0\in
cl_{[0,\infty)}(\mathcal{E}_{i})$, then $\overline{0}\in
cl_{[0,\infty)^{l}}\big{(}\pi_{i}^{-1}(\mathcal{E}_{i})\big{)}$. Note that
$\pi_{i}[AP_{l}]=\\{a+(i-1)d:a,d\in S\\}$ and
$\pi_{i}^{-1}(\mathcal{E}_{i})=\\{(a,a+d,\ldots,a+(l-1)d)\in AP_{l}:a,d\in
S\text{ and }a+(i-1)d\in\mathcal{E}_{i}\\}$. To show $\overline{0}\in
cl_{[0,\infty)^{l}}\big{(}\pi_{i}^{-1}(\mathcal{E}_{i})\big{)}$, it suffices
to let $\epsilon>0$ and show that
$[0,\epsilon)^{l}\cap\pi_{i}^{-1}(\mathcal{E}_{i})\neq\emptyset$. Since $0\in
cl_{[0,\infty)}(\mathcal{E}_{i})$, so pick $a,d\in S$ such that
$0<a+(i-1)d<\epsilon/l$. Then for each $j\in\\{1,2,\ldots,l\\}$,
$0<a+(j-1)d<\epsilon$, and therefore
$(a,a+d,\ldots,a+(l-1)d)\in[0,\epsilon)^{l}\cap\pi_{i}^{-1}(\mathcal{E}_{i})$
∎

######  Theorem 3.14.

Let $S\subseteq(0,\infty)$ be a near zero subsemigroup of $([0,\infty),+)$.
Let $l\in\mathbb{N}$ and let $AP_{l}=\\{(a,a+d,\ldots,a+(l-1)d):a,d\in S\\}$.
Let "large near zero" be any of "central near zero", "piecewise syndetic near
zero", "$central^{*}$ near zero", "$PS^{*}$ near zero" or "$IP^{*}$ near
zero". Let $B\subseteq S$. If $B\subseteq S$ be large near zero, then
$\\{(a,d)\in S\times S:\\{a,a+d,\ldots,a+(l-1)d\\}\subseteq B\\}\subseteq
S\times S$ is large near zero.

###### Proof.

Let $AP_{l}=\\{(a,a+d,a+2d,\ldots,a+(l-1)d):a,d\in S\\}$. Then the function
$\psi:(S\times S)\cup\\{(0,0)\\}\rightarrow AP_{l}\cup\\{\overline{0}\\}$
defined by $\psi(0,0)=\overline{0}$ and $\psi(a,d)=(a,a+d,\ldots,a+(l-1)d)$,
$a,d\in S$ is an isomorphism. If $B\subseteq S$ is large near zero, then by
Corollary 3.13, $B^{l}\cap AP_{l}\subseteq AP_{l}$ is large near zero, hence
$\psi^{-1}(B^{l}\cap AP_{l})\subseteq\psi^{-1}(AP_{l})$is large near zero,
i.e. $\\{(a,d)\in S\times S:\\{a,a+d,\ldots,a+(l-1)d\\}\subseteq B\\}\subseteq
S\times S$ is large near zero. ∎

Now we recall the notion of minimal idempotent [9, Definition 1.37] which will
be used subsequently.

######  Definition 3.15.

Let $(\mathcal{S},+)$ be a semigroup and let $e,f\in E(\mathcal{S})$. Then
$e\leq f$ if $e=e+f=f+e$.

######  Definition 3.16.

Let $\mathcal{S}$ be a semigroup. Let $e\in E(\mathcal{S})$. Then $e$ is a
minimal idempotent if $f\in E(\mathcal{S})$ and $f\leq e$ implies $f=e$.

######  Lemma 3.17.

Let $u,v\in\mathbb{N}$ and let $A$ be an $u\times v$ matrix with entries from
$\mathbb{N}\cup\\{0\\}$ such that no row of $A$ is zero. Let
$S\subseteq(0,\infty)$ be a dense subsemigroup of $([0,\infty),+)$. Let
$\widetilde{\phi}:\beta(S^{v})\rightarrow(\beta S)^{u}$ be the continuous
extension of the map $\phi:S^{v}\rightarrow S^{u}\subseteq(\beta S)^{u}$
defined by $\phi(\vec{x})=A\vec{x}$, $\vec{x}\in S^{v}$. Let $p$ be a minimal
idempotent of $0_{S}^{*}$ and assume that for every $C\in p$ there exists
$\vec{x}\in S^{v}$ such that $A\vec{x}\in C^{u}$. Then there is a minimal
idempotent $q\in\overline{0}_{S^{v}}^{*}$ such that
$\widetilde{\phi}(q)=\overline{p}$.

###### Proof.

First, we claim that
$\overline{p}\in\widetilde{\phi}\big{[}\overline{0}_{S^{v}}^{*}\big{]}$.
Suppose that $\overline{p}\in(\beta
S)^{u}\setminus\widetilde{\phi}\big{[}\overline{0}_{S^{v}}^{*}\big{]}$. Since
$\widetilde{\phi}\big{[}0_{S^{v}}^{*}\big{]}$ is closed, so pick a
neighborhood $U$ of $\overline{p}$ such that
$U\cap\widetilde{\phi}\big{[}\overline{0}_{S^{v}}^{*}\big{]}=\emptyset$. Pick
$C\in p$ such that $(cl_{\beta S}C)^{u}\subseteq U$. Let
$\mathcal{X}=\\{\vec{x}\in S^{v}:A\vec{x}\in C^{u}\\}$. Then
$\\{(S\cap(0,\epsilon))^{v}:\epsilon>0\\}\cup\\{\mathcal{X}\\}$ has finite
intersection property, and so pick $r\in\overline{0}_{S^{v}}^{*}$ such that
$\\{(S\cup(0,\epsilon))^{v}:\epsilon>0\\}\cup\\{\mathcal{X}\\}\subseteq r$.
Then $\widetilde{\phi}(r)\in\widetilde{\phi}[0_{S^{v}}^{*}]$. Moreover,
$\phi[\mathcal{X}]\in\widetilde{\phi}(r)$ and $\phi[\mathcal{X}]\subseteq
C^{u}$ implies $C^{u}\in\widetilde{\phi}(r)$, i.e.
$\widetilde{\phi}(r)\in(cl_{\beta S}C)^{u}$ and thus $\widetilde{\phi}(r)\in
U\cap\widetilde{\phi}\big{[}\overline{0}_{S^{v}}^{*}\big{]}$, a contradiction.

Let $M=\\{q\in\overline{0}_{S^{v}}^{*}:\widetilde{\phi}(q)=\overline{p}\\}$.
Then $M$ is a compact subsemigroup of $0_{S^{v}}^{*}$, so pick an idempotent
$w\in M$. Pick by [9, Theorem 1.60] a minimal idempotent
$q\in\overline{0}_{S^{v}}^{*}$ such that $q\leq w$. Since $\widetilde{\phi}$
is a homomorphism, we have
$\widetilde{\phi}(q)\leq\widetilde{\phi}(w)=\overline{p}$ so, since
$\overline{p}$ is minimal in $(0_{S}^{*})^{u}$, we have that
$\widetilde{\phi}(q)=\overline{p}$. ∎

######  Theorem 3.18.

Let $u,v\in\mathbb{N}$ and let $A$ be an $u\times v$ matrix with entries from
$\mathbb{N}\cup\\{0\\}$ such that no row of $A$ is zero. Let
$S\subseteq(0,\infty)$ be a dense subsemigroup of $([0,\infty),+)$. Assume
that for every central set near zero $C$ in $S$, there exists $\vec{x}\in
S^{v}$ such that $A\vec{x}\in C^{u}$. Then for every central set near zero $C$
in $S$, $\\{\vec{x}\in S^{v}:A\vec{x}\in C^{u}\\}$ is central near zero in
$S^{v}$.

###### Proof.

Let $C\subseteq S$ be a central set near zero. Pick an idempotent $p\in
K(0_{S}^{*})$ such that $C\in p$. By [9, Theorem 1.59], $p$ is a minimal
idempotent in $0_{S}^{*}$. Let $\widetilde{\phi}:\beta(S^{v})\rightarrow(\beta
S)^{u}$ be the continuous extension of the map $\phi:S^{v}\rightarrow
S^{u}\subseteq(\beta S)^{u}$ defined by $\phi(\vec{x})=A\vec{x}$, $\vec{x}\in
S^{v}$. Pick by Lemma 3.17 a minimal idempotent $q\in\overline{0}_{S^{v}}^{*}$
such that $\widetilde{\phi}(q)=\overline{p}$. By [9, Theorem 1.59], $q$ is an
idempotent in $K\big{(}\overline{0}_{S^{v}}^{*}\big{)}$. Since
$\bigtimes_{i=1}^{u}cl_{\beta S}C$ is a neighborhood of $\overline{p}$, so
pick $B\in q$ such that
$\widetilde{\phi}\big{[}cl_{\beta(S^{v})}B\big{]}\subseteq\bigtimes_{i=1}^{u}cl_{\beta
S}C$. Then $B\subseteq\\{\vec{x}\in S^{v}:A\vec{x}\in C^{u}\\}$, so
$\\{\vec{x}\in S^{v}:A\vec{x}\in C^{u}\\}\in q$ as required. ∎

######  Theorem 3.19.

Let $u,v\in\mathbb{N}$ and $A$ be an $u\times v$ matrix with entries from
$\mathbb{N}\cup\\{0\\}$ such that none of the rows of $A$ is zero. Let
$S\subseteq(0,\infty)$ be a dense subsemigroup of $([0,\infty),+)$. If

  1. (1)

for each central set near zero $C$ in $S$, there exists $\vec{x}\in S^{v}$
such that $A\vec{x}\in C^{u}$ and

  2. (2)

for each $\epsilon>0$ and for each $s\in S\cap(0,\epsilon)$, there exists
$\vec{x}\in(S\cap(0,\epsilon))^{v}$ such that $A\vec{x}=\overline{s}$,

then for every piecewise syndetic set near zero $B$ in $S,\,\\{\vec{x}\in
S^{v}:A\vec{x}\in B^{u}\\}\subseteq S^{v}$ is piecewise syndetic near zero.

###### Proof.

Let $B\subseteq S$ be piecewise syndetic near zero. Pick by [2, Theorem 2.5]
$p\in E(K(0_{S}^{*}))$ such that for each $\epsilon>0$ there exists $s\in
S\cap(0,\epsilon)$ for which $-s+B\in p$. Let
$\widetilde{\phi}:\beta(S^{v})\rightarrow(\beta S)^{u}$ be the continuous
extension of the map $\phi:S^{v}\rightarrow S^{u}\subseteq(\beta S)^{u}$
defined by $\phi(\vec{x})=A\vec{x}$, $\vec{x}\in S^{v}$. Since $p$ is a
minimal idempotent in $0_{S}^{*}$ [9, Theorem 1.59], pick by Lemma 3.17 $q\in
E\big{(}K\big{(}\overline{0}_{S^{v}}^{*}\big{)}\big{)}$ such that
$\widetilde{\phi}(q)=\overline{p}$. To show that $\\{\vec{x}\in
S^{v}:A\vec{x}\in B^{u}\\}=\phi^{-1}[B^{u}]\subseteq S^{v}$ is piecewise
syndetic near zero, it suffices by [2, Theorem 2.5] to let $\epsilon>0$ and
find $\vec{x}\in(S\cap(0,\epsilon))^{v}$ such that
$-\vec{x}+\phi^{-1}[B^{u}]\in q$. Pick $s\in S\cap(0,\epsilon)$ such that
$-s+B\in p$. Then $(-s+B)^{u}\in\overline{p}=\widetilde{\phi}(q)$, i.e.
$\phi^{-1}[(-s+B)^{u}]\in q$. Pick by Assumption (2)
$\vec{x}\in(S\cap(0,\epsilon))^{v}$ such that $A\vec{x}=\overline{s}$. Then

$\displaystyle\phi^{-1}[(-s+B)^{u}]$
$\displaystyle\subseteq\phi^{-1}[-\overline{s}+B^{u}]$
$\displaystyle\subseteq\\{\vec{y}\in S^{v}:\overline{s}+A\vec{y}\in B^{u}\\}$
$\displaystyle=\\{\vec{y}\in S^{v}:\phi(\vec{x}+\vec{y})\in B^{u}\\}$
$\displaystyle\subseteq\\{\vec{y}\in
S^{v}:\vec{x}+\vec{y}\in\phi^{-1}[B^{u}]\\}=-\vec{x}+\phi^{-1}[B^{u}],$

and therefore $-\vec{x}+\phi^{-1}[B^{u}]\in q$. ∎

##  4 Tensor product near zero

Let us begin by recalling the notion of tensor product [10, Definition 1.1].

######  Definition 4.1.

Let $X$ and $Y$ be discrete spaces, let $p\in\beta X$, and $q\in\beta Y$. Then
the tensor product of $p$ and $q$ is $p\otimes q=\\{A\subseteq X\times
Y:\\{x\in X:\\{y\in Y:(x,y)\in A\\}\in q\\}\in p\\}$.

It is easy to see that $A\times B\in p\otimes q$ if and only if $A\in p$ and
$B\in q$.

######  Theorem 4.2.

Let $S_{1}$ and $S_{2}$ be two discrete subsemigroups, and let
$\tilde{\iota}:\beta(S_{1}\times S_{2})\rightarrow\beta S_{1}\times\beta
S_{2}$ be the continuous extension of the inclusion map $\iota:(S_{1}\times
S_{2})_{d}\rightarrow\beta S_{1}\times\beta S_{2}$. Let $p_{1}\in\beta S_{1}$
and $p_{2}\in\beta S_{2}$. Then $\tilde{\iota}(p_{1}\otimes
p_{2})=(p_{1},p_{2})$.

###### Proof.

For each $i\in\\{1,2\\}$, if $\pi_{i}:S_{1}\times S_{2}\rightarrow S_{i}$ is
the natural projection homomorphism onto the $i^{th}$ coordinate and if
$\widetilde{\pi}_{i}:\beta(S_{1}\times S_{2})\rightarrow\beta S_{i}$ is the
continuous extension of $\pi_{i}$, then $\tilde{\iota}(p_{1}\otimes
p_{2})=(\widetilde{\pi}_{1}(p_{1}\otimes
p_{2}),\widetilde{\pi}_{2}(p_{1}\otimes p_{2}))$. Now
$\widetilde{\pi}_{1}(p_{1}\otimes p_{2})=\\{A_{1}\subseteq
S_{1}:\pi_{1}^{-1}(A_{1})\in p_{1}\otimes p_{2}\\}=\\{A_{1}\subseteq
S_{1}:A_{1}\times S_{2}\in p_{1}\otimes p_{2}\\}=p_{1}$. The proof of
$\widetilde{\pi}_{2}(p_{1}\otimes p_{2})=p_{2}$ is exactly similar. Thus
$\tilde{\iota}(p_{1}\otimes p_{2})=(p_{1},p_{2})$. ∎

The above theorem now provides us the following version near idempotent.

######  Corollary 4.3.

For each $i\in\\{1,2\\}$, let $S_{i}$ be a near $e_{i}$ subsemigroup of a
semitopological semigroup $T_{i}$. Let
$(e_{1})_{S_{1}}^{*}\otimes(e_{2})_{S_{2}}^{*}=\\{p\otimes
q:p\in(e_{1})_{S_{1}}^{*},\,q\in(e_{2})_{S_{2}}^{*}\\}$. Let
$\tilde{\iota}:\beta(S_{1}\times S_{2})\rightarrow\beta S_{1}\times\beta
S_{2}$ be the continuous extension of the inclusion map $\iota:(S_{1}\times
S_{2})_{d}\rightarrow\beta S_{1}\times\beta S_{2}$. Then
$\tilde{\iota}[(e_{1})_{S_{1}}^{*}\otimes(e_{2})_{S_{2}}^{*}]=(e_{1})_{S_{1}}^{*}\times(e_{2})_{S_{2}}^{*}$.

We now recall the definition of higher dimensional tensor product of
ultrafilters [5, definition 1.15].

######  Definition 4.4.

Let $l\in\mathbb{N}$, and for each $i\in\\{1,2,\ldots,l\\}$ let $S_{i}$ be a
semigroup and let $p_{i}\in\beta S_{i}$. We define
$\otimes_{i=1}^{l}p_{i}\in\beta\big{(}\bigtimes_{i=1}^{l}S_{i}\big{)}$
inductively as follows.

  1. (1)

$\otimes_{i=1}^{1}p_{i}=p_{1}$,

  2. (2)

Given $l\in\mathbb{N}$ and $A\subseteq\bigtimes_{i=1}^{l+1}S_{i}$,
$A\in\otimes_{i=1}^{l+1}p_{i}$ if
$\\{(x_{1},x_{2},\ldots,x_{l})\in\bigtimes_{i=1}^{l}S_{i}:\\{x_{l+1}\in
S_{l+1}:(x_{1},x_{2},\ldots,x_{l+1})\in A\\}\in
p_{l+1}\\}\in\otimes_{i=1}^{l}p_{i}$.

######  Lemma 4.5.

Let $l\in\mathbb{N}$. For each $i\in\\{1,2,\ldots,l\\}$ let
$S_{i}\subseteq(0,\infty)$ be a dense subsemigroup of $([0,\infty),+)$ and let
$p_{i}\in 0_{S_{i}}^{*}$. Let us consider the product semigroup
$S=\bigtimes_{i=1}^{l}S_{i}$. Then
$\otimes_{i=1}^{l}p_{i}\in\overline{0}_{S}^{*}$.

###### Proof.

It suffices to show that for each $U\in\tau_{0}$, $U^{l}\cap
S\in\otimes_{i=1}^{l}p_{i}$ which follows directly from Definition 4.4. ∎

We now recall [5, Corollary 2.8 and Lemma 2.9].

######  Lemma 4.6.

Let $m,l\in\mathbb{N}$. For each $i\in\\{1,2,\ldots,l\\}$, let $(S_{i},+)$ be
a semigroup, $\langle x_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in $S_{i}$
and let $p_{i}$ be an idempotent in $\bigcap_{r=1}^{\infty}cl_{\beta
S_{i}}\big{(}FS(\langle x_{i,n}\rangle_{n=r}^{\infty})\big{)}$. Let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots l\\}$ be a function and let
$A\in\otimes_{j=1}^{m}p_{f(j)}$. Then for each $i\in\\{1,2,\ldots,l\\}$ there
is a sum subsystem $FS\big{(}\langle y_{i,n}\rangle_{n=1}^{\infty}\big{)}$ of
$FS\big{(}\langle x_{i,n}\rangle_{n=1}^{\infty}\big{)}$ such that
$\big{\\{}\big{(}\sum_{n\in F_{1}}y_{f(1),n},\sum_{n\in F_{2}}y_{f(2),n},$
$\ldots,\sum_{n\in F_{m}}y_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\ldots<F_{m}\big{\\}}\subseteq A$.

######  Lemma 4.7.

Let $m,l\in\mathbb{N}$. For each $i\in\\{1,2,\ldots,l\\}$, let $(S_{i},+)$ be
a semigroup, let $\langle x_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in
$S_{i}$ and let $p_{i}\in\bigcap_{r=1}^{\infty}cl_{\beta
S_{i}}\big{(}FS(\langle x_{i,n}\rangle_{n=r}^{\infty})\big{)}$. Then for any
function $f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,l\\}$,
$\big{\\{}\big{(}\sum_{n\in F_{1}}x_{f(1),n},$ $\sum_{n\in
F_{2}}x_{f(2),n},\ldots,\sum_{n\in F_{m}}x_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\ldots<F_{m}\big{\\}}\in\otimes_{j=1}^{m}p_{f(j)}$.

######  Definition 4.8.

Let $m\in\mathbb{N}$ and let $(S,+)$ be a semigroup. Let $\langle
a_{j}\rangle_{j=1}^{m}$ and $\langle x_{n}\rangle_{n=1}^{\infty}$ be sequences
in $\mathbb{N}$ and $S$, respectively. The Milliken-Taylor System determined
by $\langle a_{j}\rangle_{j=1}^{m}$ and $\langle x_{n}\rangle_{n=1}^{\infty}$
is $MT\big{(}\langle a_{j}\rangle_{j=1}^{m},\langle
x_{n}\rangle_{n=1}^{\infty}\big{)}=\big{\\{}\sum_{j=1}^{m}a_{j}\sum_{n\in
F_{j}}x_{n}:\text{each }F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}$.

Regarding the following discussion, note that if $p\in\beta([0,\infty))$ and
$a\in\mathbb{N}$, then $ap$ is the product in $\beta([0,\infty))$ and not the
sum of $p$ with itself $a$ times. It is not true in general that
$a_{1}p+a_{2}p=(a_{1}+a_{2})p$.

######  Theorem 4.9.

Let $m\in\mathbb{N}$ and let $S\subseteq(0,\infty)$ be a dense subsemigroup of
$([0,\infty),+)$. Let $\langle a_{j}\rangle_{j=1}^{m}$ be a sequence in
$\mathbb{N}$ and let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a sequence in
$S$ such that $\lim_{n\rightarrow\infty}x_{n}=0$. Let
$g(z)=\sum_{j=1}^{m}a_{j}z$ and let $A\subseteq S$. The following statements
are equivalent.

  1. (1)

There is an idempotent $p\in\bigcap_{r=1}^{\infty}cl_{\beta
S}\big{(}FS(\langle x_{n}\rangle_{n=r}^{\infty})\big{)}$ such that $A\in
g(p)$.

  2. (2)

For each $\epsilon>0$, there is a sum subsystem $FS(\langle
y_{n}\rangle_{n=1}^{\infty})$ of $FS(\langle x_{n}\rangle_{n=1}^{\infty})$
such that $MT\big{(}\langle a_{j}\rangle_{j=1}^{m},\langle
y_{n}\rangle_{n=1}^{\infty}\big{)}\subseteq A\cap(0,\epsilon)$.

###### Proof.

(1) implies (2). Let $p$ be an idempotent in $\bigcap_{r=1}^{\infty}cl_{\beta
S}\big{(}FS(\langle x_{n}\rangle_{n=r}^{\infty})\big{)}$ such that $A\in
g(p)$. By passing to a subsequence, we may presume that
$\sum_{n=1}^{\infty}x_{n}$ converges. Note that since
$\sum_{n=1}^{\infty}x_{n}$ converges, $p\in 0_{S}^{*}$. Moreover, by using the
definition of product it is very easy to see that for each
$j\in\\{1,2,\ldots,m\\}$ and for each $\epsilon>0$ we have that
$(0,\epsilon)\cap S\in a_{j}p$, i.e. $a_{j}p\in 0_{S}^{*}$. Thus $g(p)\in
0_{S}^{*}$. So it suffices to show that whenever $Q\in g(p)$, there is a sum
subsystem $FS\big{(}\langle y_{n}\rangle_{n=1}^{\infty}\big{)}$ of
$FS\big{(}\langle x_{n}\rangle_{n=1}^{\infty}\big{)}$ such that
$MT\big{(}\langle a_{j}\rangle_{j=1}^{m},\langle
y_{n}\rangle_{n=1}^{\infty}\big{)}\subseteq Q$. The rest of the part now
follows from the proof of [6, Theorem 5.7].

(2) implies (1). Fix $\epsilon>0$ and pick a sum subsystem $FS\big{(}\langle
y_{n}\rangle_{n=1}^{\infty}\big{)}$ of $FS\big{(}\langle
x_{n}\rangle_{n=1}^{\infty}\big{)}$ such that $MT\big{(}\langle
a_{j}\rangle_{j=1}^{m},\langle y_{n}\rangle_{n=1}^{\infty}\big{)}\subseteq
A\cap(0,\epsilon)$. Pick by [9, Lemma 5.11] an idempotent
$p\in\bigcap_{r=1}^{\infty}cl_{\beta S}\big{(}FS\big{(}\langle
y_{n}\rangle_{n=r}^{\infty}\big{)}\big{)}\subseteq\bigcap_{r=1}^{\infty}cl_{\beta
S}\big{(}FS\big{(}\langle x_{n}\rangle_{n=r}^{\infty})\big{)}$. We claim that
$MT\big{(}\langle a_{j}\rangle_{j=1}^{m},\langle
y_{n}\rangle_{n=1}^{\infty}\big{)}\in g(p)$, for which it suffices to show by
downward induction on $k\in\\{1,2,\ldots,m\\}$ that for each $r\in\mathbb{N}$,
$\big{\\{}\sum_{j=k}^{m}a_{j}\sum_{n\in F_{j}}y_{t}:\text{each
}F_{j}\in\mathcal{P}_{f}(\\{r,r+1,\ldots\\})\text{ and
}F_{k}<F_{k+1}<\cdots<F_{m}\big{\\}}\in a_{k}p+a_{k+1}p+\cdots+a_{m}p$.

For $k=m$, we have $FS\big{(}\langle y_{n}\rangle_{n=r}^{\infty}\big{)}\in p$
so that $FS\big{(}\langle a_{m}y_{n}\rangle_{n=r}^{\infty}\big{)}\in a_{m}p$.
So let $k\in\\{1,2,\ldots,m-1\\}$ and assume that the assertion is true for
$k+1$. Let $r\in\mathbb{N}$ and let $B=\big{\\{}\sum_{j=k}^{m}a_{j}\sum_{n\in
F_{j}}y_{n}:\text{each }F_{j}\in\mathcal{P}_{f}(\\{r,r+1,\ldots\\})\text{ and
}F_{k}<F_{k+1}<\cdots<F_{m}\big{\\}}$. We show that $FS\big{(}\langle
a_{k}y_{n}\rangle_{n=r}^{\infty}\big{)}\subseteq\\{y\in S:-y+B\in
a_{k+1}p+a_{k+2}p+\cdots+a_{m}p\\}$.

So let $b\in FS\big{(}\langle a_{k}y_{n}\rangle_{n=r}^{\infty}\big{)}$ and
pick $F_{k}\in\mathcal{P}_{f}(\\{r,r+1,\ldots\\})$ such that
$b=a_{k}\sum_{n\in F_{k}}y_{n}$. Let $t=\max F_{k}+1$. Then
$\big{\\{}\sum_{j=k+1}^{m}a_{j}\sum_{n\in F_{j}}y_{n}:\text{each
}F_{j}\in\mathcal{P}_{f}(\\{t,t+1,\ldots\\})\text{ and
}F_{k+1}<F_{k+2}<\cdots<F_{m}\big{\\}}\subseteq-b+B$, so $-b+B\in
a_{k+1}p+a_{k+2}p+\cdots+a_{m}p$ as required. Since $MT\big{(}\langle
a_{j}\rangle_{j=1}^{m},\langle y_{n}\rangle_{n=1}^{\infty}\big{)}\subseteq
A\cap(0,\epsilon)\subseteq A$ and $MT\big{(}\langle
a_{j}\rangle_{j=1}^{m},\langle y_{n}\rangle_{n=1}^{\infty}\big{)}\in g(p)$, we
have $A\in g(p)$. ∎

######  Theorem 4.10.

Let $S\subseteq(0,\infty)$ be a dense subsemigroup of both $([0,\infty),+)$
and $([0,\infty),\cdot)$. Let $m,l\in\mathbb{N}$. For each
$i\in\\{1,2,\ldots,l\\}$, let $\langle x_{i,n}\rangle_{n=1}^{\infty}$ be a
sequence in $S$ such that $\lim_{n\rightarrow\infty}x_{i,n}=0$ and let
$p_{i}\in\bigcap_{r=1}^{\infty}cl_{\beta S}\big{(}FS\big{(}\langle
x_{i,n}\rangle_{n=r}^{\infty}\big{)}\big{)}$. Let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,l\\}$ be a function and let
$g\in\mathbb{N}[x_{1},x_{2},\ldots,x_{m}]$ be a polynomial with
$g\big{(}\overline{0}\big{)}=0$. Then for each $\epsilon>0$,
$\big{\\{}g\big{(}\sum_{n\in F_{1}}x_{f(1),n},$ $\sum_{n\in
F_{2}}x_{f(2),n},\ldots,\sum_{n\in F_{m}}x_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\cap(0,\epsilon)\in
g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}$.

###### Proof.

Let $\widetilde{g}:\beta\big{(}\bigtimes_{j=1}^{m}S\big{)}\rightarrow\beta S$
be the continuous extension of $g$. Then by [5, Theorem 3.2],
$\widetilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}=g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}$.
Let $A=\big{\\{}\big{(}\sum_{n\in F_{1}}x_{f(1),n},\sum_{n\in
F_{2}}x_{f(2),n},\ldots,\sum_{n\in F_{m}}x_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}$.

By passing to a subsequence, we may presume that for each
$i\in\\{1,2,\ldots,l\\}$, $\sum_{n=1}^{\infty}x_{i,n}$ converges, and then we
have that $p_{i}\in 0_{S}^{*}$. By Lemma 4.5, $\otimes_{j=1}^{m}p_{f(j)}\in
0_{S^{m}}^{*}$. Since $g$ is continuous and $g\big{(}\overline{0}\big{)}=0$,
so for each $\delta>0$, it follows easily that $[0,\epsilon)\cap
S\in\widetilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}$, and thus
$\widetilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}=g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}\in
0_{S}^{*}$. Then, by Lemma 4.7, $A\in\otimes_{j=1}^{m}p_{f(j)}$, so by [9,
Lemma 3.30], $g[A]\in\widetilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}$.
Hence $g[A]\cap(0,\epsilon)\in
g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}$. ∎

######  Theorem 4.11.

Let $S\subseteq(0,\infty)$ be a dense subsemigroup of both $([0,\infty),+)$
and $([0,\infty),\cdot)$. Let $m,l\in\mathbb{N}$. For each
$i\in\\{1,2,\ldots,l\\}$, let $\langle x_{i,n}\rangle_{n=1}^{\infty}$ be a
sequence in $S$ such that $\lim_{n\rightarrow\infty}x_{i,n}=0$. Let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,l\\}$ be a function, let
$g\in\mathbb{N}[x_{1},x_{2},\ldots,x_{m}]$ be a polynomial with
$g\big{(}\overline{0}\big{)}=0$ and let $A\subseteq S$. The following
statements are equivalent.

  1. (1)

For each $i\in\\{1,2,\ldots,l\\}$ there exists
$p_{i}=p_{i}+p_{i}\in\bigcap_{r=1}^{\infty}cl_{\beta S}\big{(}FS(\langle
x_{i,n}\rangle_{n=r}^{\infty}\big{)}\big{)}$ such that $A\in
g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})$.

  2. (2)

For each $\epsilon>0$ and for each $i\in\\{1,2,\ldots,l\\}$ there is a sum
subsystem $FS\big{(}\langle y_{i,n}\rangle_{n=1}^{\infty}\big{)}$ of
$FS\big{(}\langle x_{i,n}\rangle_{n=1}^{\infty}\big{)}$ such that
$\Big{\\{}g\big{(}\sum_{n\in F_{1}}y_{f(1),n},\sum_{n\in F_{2}}y_{f(2),n},$
$\ldots,\sum_{n\in F_{m}}y_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\ldots<F_{m}\Big{\\}}\subseteq A\cap(0,\epsilon)$.

###### Proof.

(1) implies (2). By [5, Theorem 3.2]
$g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}=\widetilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}$
and from the proof of Theorem 4.10, we have that
$\widetilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}\in 0_{S}^{*}$. Let
$\epsilon>0$. Then
$A\cap(0,\epsilon)\in\widetilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}$.
Pick $B\in\otimes_{j=1}^{m}p_{f(j)}$ such that
$\widetilde{g}\big{[}cl_{\beta(S^{m})}B\big{]}\subseteq cl_{\beta
S}(A\cap(0,\epsilon))$. By Lemma 4.6, for each $i\in\\{1,2,\ldots,l\\}$ pick a
sum subsystem $FS\big{(}\langle y_{i,n}\rangle_{n=1}^{\infty}\big{)}$ of
$FS\big{(}\langle x_{i,n}\rangle_{n=1}^{\infty}\big{)}$ such that
$\big{\\{}\big{(}\sum_{n\in F_{1}}y_{f(1),n},\sum_{n\in
F_{2}}y_{f(2),n},\ldots,$ $\sum_{n\in F_{m}}y_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\subseteq B$. Then
$\big{\\{}g\big{(}\sum_{n\in F_{1}}y_{f(1),n},\sum_{n\in
F_{2}}y_{f(2),n},\ldots,\sum_{n\in F_{m}}y_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and }$
$F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\subseteq g[B]\subseteq A\cap(0,\epsilon)$.

(2) implies (1). Fix $\epsilon>0$. For each $i\in\\{1,2,\ldots,l\\}$ pick a
sum subsystem $FS\big{(}\langle y_{i,n}\rangle_{n=1}^{\infty}\big{)}$ of
$FS\big{(}\langle x_{i,n}\rangle_{n=1}^{\infty}\big{)}$ as guaranteed in (2),
and pick by [9, Lemma 5.11]
$p_{i}=p_{i}+p_{i}\in\bigcap_{r=1}^{\infty}cl_{\beta S}\big{(}FS\big{(}\langle
y_{i,n}\rangle_{n=r}^{\infty}\big{)}\big{)}\subseteq\bigcap_{r=1}^{\infty}cl_{\beta
S}\big{(}FS\big{(}\langle x_{i,n}\rangle_{n=r}^{\infty}\big{)}\big{)}$. By
Theorem 4.10, $\big{\\{}g\big{(}\sum_{n\in F_{1}}y_{f(1),n},\sum_{n\in
F_{2}}y_{f(2),n},\ldots,\sum_{n\in F_{m}}y_{f(m),n}\big{)}:\text{each
}F_{j}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\cap(0,\epsilon)\in
g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}$, so $A\cap(0,\epsilon)\in
g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}$ and thus $A\in
g\big{(}p_{f(1)},p_{f(2)},\ldots,p_{f(m)}\big{)}$. ∎

The following two theorems are generalizations of Theorem 2.9 for near zero
semigroups.

######  Theorem 4.12.

Let $m\in\mathbb{N}$, and for $i\in\\{1,2,\ldots,m\\}$ let
$S_{i}\subseteq(0,\infty)$ be a dense subsemigroup of $([0,\infty),+)$. Let
$S=\bigtimes_{i=1}^{m}S_{i}$ and let $A\subseteq S$. The following statements
are equivalent.

  1. (1)

For each $i\in\\{1,2,\ldots,m\\}$, there is an idempotent $p_{i}\in
0_{S_{i}}^{*}$ such that $A\in\otimes_{i=1}^{m}p_{i}$.

  2. (2)

For each $i\in\\{1,2,\ldots,m\\}$, there is a sequence $\langle
x_{i,n}\rangle_{n=1}^{\infty}$ in $S_{i}$ such that
$\sum_{n=1}^{\infty}x_{i,n}$ converges and $\big{\\{}\big{(}\sum_{n\in
F_{1}}x_{1,n},\sum_{n\in F_{2}}x_{2,n},\ldots,\sum_{n\in
F_{m}}x_{m,n}\big{)}:\text{each }F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\subseteq A$.

###### Proof.

(1) implies (2). For $i\in\\{1,2,\ldots,m\\}$ pick an idempotent $p_{i}\in
0_{S_{i}}^{*}$ such that $A\in\otimes_{i=1}^{m}p_{i}$. By Lemma 4.5,
$\otimes_{i=1}^{m}p_{i}\in\overline{0}_{S}^{*}$ so
$\bigtimes_{i=1}^{m}\big{(}\big{(}0,\frac{1}{2}\big{)}\cap
S_{i}\big{)}\in\otimes_{i=1}^{m}p_{i}$. Therefore
$A\cap\bigtimes_{i=1}^{m}\big{(}0,\frac{1}{2}\big{)}\in\otimes_{i=1}^{m}p_{i}$.
By [5, Theorem 1.16], for each $i\in\\{1,2,\ldots,m\\}$ pick a sequence
$\langle x_{i,n}\rangle_{n=1}^{\infty}$ in $S_{i}$ such that
$\big{\\{}\big{(}\sum_{n\in F_{1}}x_{1,n},\sum_{n\in F_{2}}x_{2,n},$
$\ldots,\sum_{n\in F_{m}}x_{m,n}\big{)}:\text{each
}F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\subseteq
A\cap\bigtimes_{i=1}^{m}\big{(}0,\frac{1}{2}\big{)}\subseteq A$. Then for each
$i\in\\{1,2,\ldots,m\\}$ and each $F\in\mathcal{P}_{f}(\mathbb{N})$ with $\min
F\geq i$, $\sum_{n\in F}x_{i,n}<\frac{1}{2}$ so $\sum_{n=1}^{\infty}x_{i,n}$
converges.

(2) implies (1). For each $i\in\\{1,2,\ldots,m\\}$, pick by [9, Lemma 5.11] an
idempotent $p_{i}\in\bigcap_{r=1}^{\infty}cl_{\beta
S_{i}}\big{(}FS\big{\langle}x_{i,n}\rangle_{n=r}^{\infty}\big{)}\big{)}$. For
each $i\in\\{1,2,\ldots,m\\}$, since $\sum_{n=1}^{\infty}x_{i,n}$ converges,
so $p_{i}\in 0_{S_{i}}^{*}$. Let $l=m$ and let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,l\\}$ be the identity function.
Apply Lemma 4.7. ∎

######  Theorem 4.13.

Let $m\in\mathbb{N}$, let $S\subseteq(0,\infty)$ be a dense subsemigroup of
$([0,\infty),+)$ and let $A\subseteq S^{m}$. The following statements are
equivalent.

  1. (1)

There is an idempotent $p\in 0_{S}^{*}$ such that $A\in\otimes_{i=1}^{m}p$.

  2. (2)

There is a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $S$ such that
$\sum_{n=1}^{\infty}x_{n}$ converges and $\big{\\{}\big{(}\sum_{n\in
F_{1}}x_{n},\sum_{n\in F_{2}}x_{n},\ldots,\sum_{n\in
F_{m}}x_{n}\big{)}:\text{each }F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\subseteq A$.

###### Proof.

(1) implies (2). Pick an idempotent $p\in 0_{S}^{*}$ such that
$A\in\otimes_{i=1}^{m}p$. By Lemma 4.5,
$\otimes_{i=1}^{m}p\in\overline{0}_{S^{m}}^{*}$ so
$\bigtimes_{i=1}^{m}\big{(}\big{(}0,\frac{1}{2}\big{)}\cap
S\big{)}\in\otimes_{i=1}^{m}p$. Therefore
$A\cap\bigtimes_{i=1}^{m}\big{(}0,\frac{1}{2}\big{)}\in\otimes_{i=1}^{m}p$.
Pick by [5, Theorem 1.17] a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in
$S$ such that $\big{\\{}\big{(}\sum_{n\in F_{1}}x_{n},\sum_{n\in
F_{2}}x_{n},\ldots,\sum_{n\in F_{m}}x_{n}\big{)}:\text{each
}F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ and
}F_{1}<F_{2}<\cdots<F_{m}\big{\\}}\subseteq
A\cap\bigtimes_{i=1}^{m}\big{(}0,\frac{1}{2}\big{)}\subseteq A$. Then for each
$F\in\mathcal{P}_{f}(\mathbb{N})$ with $\min F\geq i$, $\sum_{n\in
F}x_{n}<\frac{1}{2}$ so $\sum_{n=1}^{\infty}x_{n}$ converges.

(2) implies (1). Pick by [9, Lemma 5.11] an idempotent
$p\in\bigcap_{r=1}^{\infty}cl_{\beta S}\big{(}FS\big{(}\langle
x_{n}\rangle_{n=r}^{\infty}\big{)}\big{)}$. Since $\sum_{n=1}^{\infty}x_{n}$
converges, so $p\in 0_{S}^{*}$. By Lemma 4.7 with $l=1$,
$A\in\otimes_{i=1}^{m}p$. ∎

## Acknowledgments

The authors are indebted to the anonymous reviewers for their generous
comments and suggestions on the previous manuscripts. The first author
gratefully acknowledges the grant UGC-NET SRF fellowship with Ref. No.
20/12/2015(ii)EU-V of CSIR-UGC NET December 2015.

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