# Product of some large sets near idempotent

Surajit Biswas<EMAIL_ADDRESS>Sourav Kanti Patra
<EMAIL_ADDRESS>Sabyasachi Dey<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.  Department of Mathematics, Birla Institute of
Technology and Science, Pilani, Hyderabad Campus, Jawahar Nagar, Hyderabad,
India, 500078.

###### Abstract

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. Then, we
study the abundance of some large sets near idempotent. Also, we investigate
the effect of tensor product near idempotent. Finally, as an application we
provide the polynomial extension of Milliken-Taylor theorem near zero.

###### keywords:

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

###### MSC:

[2020] 03E05, 05D10.

##  1 Introduction

The notion of "central subset" was first introduced by Furstenberg [8] in the
semigroup $(\mathbb{N},+)$ of natural numbers. Later, Bergelson and Hindman
[5] 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 [10, Corollary 14.14.10]. A localized
notion of central set, called central set near idempotent was introduced by
Tootkaboni and Vahed in [15]. 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 [15, Theorem 4.3]. In [12], Hindman and Strauss have characterized
when the (finite and infinite) Cartesian product of sets is piecewise syndetic
or central. In [2], similar (partial) characterization has been provided for
some other class of large sets in an arbitrary semigroup. Here, in Theorem
2.14 and Theorem 2.15, 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 [16] 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 [15],
Tootkaboni and Vahed introduces the notion of some largeness near idempotent.
We study the abundance of some large sets near idempotent in Theorem 3.13 and
3.16. The matrix version of these results are deduced in [13] and [14]. In
Theorem 3.17 and 3.19, we provide the near idempotent 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 [6], 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 near idempotent. As an application, then in Theorem 4.10 and 4.11, and
in Corollary 4.12 we provide the near zero versions of polynomial Milliken-
Taylor theorem. Finally, in Theorem 4.13 and 4.14, 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 $S$ [10, 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 [10, 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 consider it with respect to the discrete topology on $S$, so
that it can be identified with the ultrafilters on $S$, and to reduce
notations we simply denote it by $\beta S$. Also, for $A\subseteq S$, by the
notation $\overline{A}$, we denote the closure of $A$ in $\beta S$. For other
details regarding Stone-Čech compactification, see [10].

##  2 Cartesian product near Idempotent

Let $T$ be a Hausdorff semitopological semigroup. Let $S$ be a dense
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 $U$. Let us
introduce the following terminology for ultrafilters on $S$ near a point [15,
Definition 2.1(a)].

######  Definition 2.1.

Let $S$ be a subsemigroup of a semitopological semigroup $T$. Given $x\in T$,
$x^{*}_{S}=\big{\\{}p\in\beta S:x\in\bigcap_{A\in
p}\text{{cl}}_{T}A\big{\\}}$.

By [15, Lemma 2.2], we have that for any $x\in T$, $x^{*}_{S}\neq\emptyset$ if
and only if $x\in\text{{cl}}_{T}S$. Moreover, if $e\in E(T)$, then $e^{*}_{S}$
is a compact subsemigroup of $\beta S$ [15, Lemma 2.3]. In the next
definition, first we recall the notion of central set near an idempotent [15,
Definition 3.9], and then introduce the notion of central∗ set near
idempotent.

######  Definition 2.2.

Let $S$ be a dense subsemigroup of $T$ and $e\in E(T)$. Let $K(e^{*}_{S})$ be
the smallest ideal of the semigroup $e^{*}$. 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 $e$.

The following terminologies [15, Definition 3.1] will be used in the
definition of $IP$ set and $IP^{*}$ set near idempotent.

######  Definition 2.3.

Let $(T,+)$ be a semitopological semigroup.

  1. (1)

Let $\mathcal{B}$ be a local base at the point $x\in T$. We say $\mathcal{B}$
has finite cover property if $\\{V\in\mathcal{B}:y\in V\\}$ is finite for each
$y\in T\setminus\\{x\\}$.

  2. (2)

Let $S$ be a dense subsemigroup of $T$ and $e\in E(T)$. 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 $FS\big{(}\langle
x_{n}\rangle_{n=k}^{l}\big{)}\subseteq U$ for each $l>k\geq m$.

  3. (3)

Let $\mathcal{B}=\\{U_{n}:n\in\mathbb{N}\\}$ be a countable local base at the
point $x\in T$ such that for each $n\in\mathbb{N}$, $U_{n+1}\subseteq U_{n}$
and $U_{n+1}+U_{n+1}\subseteq U_{n}$, and for each sequence $\langle
x_{n}\rangle_{n=1}^{\infty}$ if $x_{n}\in U_{n}$ for each $n\in\mathbb{N}$
then $\sum_{n=1}^{\infty}x_{n}$ converges near $x$. Then we say $\mathcal{B}$
is a countable local base for convergence at the point $x\in T$.

  4. (4)

Let $\mathcal{B}$ be a local base at the point $x\in T$. If $\mathcal{B}$
satisfies the conditions (1) and (3), then $\mathcal{B}$ is called a countable
local base that has the finite cover property for convergence at the point
$x$. For simplicity, we say $\mathcal{B}$ has the F property at the point $x$.

######  Definition 2.4.

Let $S$ be a dense subsemigroup of a semitopological semigroup $T$, and let
$e\in E(T)$. 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
$\lim_{n\rightarrow\infty}x_{n}=e,\,\sum_{n=1}^{\infty}x_{n}$ 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 $e$.

Note that, given a dense subsemigroup $S$ of a semitopological semigroup $T$
and given $e\in E(T)$, if there is a local base $\mathcal{B}$ at the $e$ with
F property, then $A\subseteq S$ is $IP$ near $e$ if and only if there exists
an idempotent $p\in e^{*}_{S}$ with $A\in p$ [15, Theorem 3.2]. For notational
simplicity, from the next lemma onwards, we use the notation $e^{*}_{S_{i}}$
instead of $(e_{i})^{*}_{S_{i}}$ for the corresponding subsemigroup $S_{i}$ of
$T_{i}$ with $e_{i}\in E(T_{i})$.

######  Lemma 2.5.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a dense subsemigroup of a
semitopological semigroup $T_{i}$ with $e_{i}\in E(T_{i})$. Let
$T=\bigtimes_{i\in I}T_{i}$, $S=\bigtimes_{i\in I}S_{i}$ and $e=\times_{i\in
I}e_{i}$. For each $i\in I$, let $\pi_{i}:S\rightarrow S_{i}$ be the
projection homomorphism from $S$ onto $S_{i}$. Then
$\tilde{\pi}_{i}[e^{*}_{S}]=e^{*}_{S_{i}}$.

###### Proof.

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

To prove the reverse inclusion, let $p_{i}\in e^{*}_{S_{i}}$, i.e.
$e_{i}\in\bigcap_{A_{i}\in p_{i}}\text{cl}_{T_{i}}(A_{i})$. Consider the
families $\mathcal{R}:=\\{A\subseteq S:e\in\text{cl}_{T}(A)\\}$ and
$\mathcal{A}=\\{\pi_{i}^{-1}(A_{i}):A_{i}\in p_{i}\\}$. Then
$e\in\text{cl}_{T}(\pi_{i}^{-1}(A_{i}))$ for each $A_{i}\in p_{i}$. So,
$\mathcal{A\subseteq\mathcal{R}}$ and by [10, 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. $A_{i}\in\tilde{\pi}_{i}(p)$. Thus,
$p_{i}\subseteq\tilde{\pi}_{i}(p)$, and both being ultrafilters, we have
$p_{i}=\tilde{\pi}_{i}(p)$. Therefore
$\tilde{\pi}_{i}[e^{*}_{S}]=e^{*}_{S_{i}}$. ∎

######  Lemma 2.6.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a dense subsemigroup of a
semitopological semigroup $T_{i}$ with $e_{i}\in E(T_{i})$. 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 $e=\times_{i\in I}e_{i}$. 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 $\tilde{\iota}[e^{*}_{S}]=\bigtimes_{i\in
I}e^{*}_{S_{i}}$.

###### Proof.

For each $i\in I$, if $\pi_{i}:S\rightarrow S_{i}$ be the natural projection
homomorphism, and if $\tilde{\pi}_{i}:\beta S\rightarrow\beta S_{i}$ be the
continuous extension of $\pi_{i}$, then we have $\tilde{\iota}=\bigtimes_{i\in
I}\tilde{\pi}_{i}$. Hence, by Lemma 2.5,
$\tilde{\iota}[e^{*}_{S}]\subseteq\bigtimes_{i\in I}e^{*}_{S_{i}}$.
Conversely, let $\times_{i\in I}p_{i}\in\bigtimes_{i\in I}e^{*}_{S_{i}}$, i.e.
$p_{i}\in e^{*}_{S_{i}}$ for each $i\in I$, i.e. $e_{i}\in\bigcap_{A_{i}\in
p_{i}}\text{cl}_{T_{i}}(A_{i})$. Consider the nonempty families
$\mathcal{R}=\\{A\subseteq S:e\in\text{cl}_{T}(A)\\}$ and
$\mathcal{A}=\\{\bigtimes_{i\in I}A_{i}:A_{i}\in p_{i}\text{ for each }i\in
I\\}$. Then, for $\bigtimes_{i\in I}A_{i}\in\mathcal{A}$, we have
$e_{i}\in\text{cl}_{T_{i}}(A_{i})$ for each $i\in I$, i.e.
$e\in\bigtimes_{i\in I}\text{cl}_{T_{i}}(A_{i})=\text{cl}_{T}(\bigtimes_{i\in
I}A_{i})$. Thus, $\bigtimes_{i\in I}A_{i}\in\mathcal{R}$, and we have
$\mathcal{A}\subseteq\mathcal{R}$. Hence, by [10, 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 $A_{i}\in\tilde{\pi}_{i}(p)$. Thus
$p_{i}\subseteq\tilde{\pi}_{i}(p)$, and both being ultrafilters, we have
$p_{i}=\tilde{\pi}_{i}(p)$. Therefore, we have
$\tilde{\iota}[e^{*}_{S}]=\bigtimes_{i\in I}e^{*}_{S_{i}}$. ∎

######  Theorem 2.7.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a dense subsemigroup of a
semitopological semigroup $T_{i}$ with $e_{i}\in E(T_{i})$ 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 $p=\times_{i\in I}p_{i}$. 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 [10, Corollary 4.22] and by
Lemma 2.6, $\tilde{\iota}[e^{*}_{S}]=\bigtimes_{i\in I}e^{*}_{S_{i}}$.
Consequently, $M$ is a compact subsemigroup of $e^{*}_{S}$. By [10, Exercise
1.7.3], $\tilde{\iota}[K(e^{*}_{S})]=K(\bigtimes_{i\in I}e^{*}_{S_{i}})$.
Also, by [10, Theorem 2.23], $K(\bigtimes_{i\in
I}e^{*}_{S_{i}})=\bigtimes_{i\in I}K(e^{*}_{S_{i}})$ and so $p\in
K(\bigtimes_{i\in I}e^{*}_{S_{i}})$. Consequently, $K(e^{*}_{S})\cap
M\neq\emptyset$ and therefore $K(e^{*}_{S})\cap M$ is an ideal of $M$ and so
$K(M)\subseteq K(e^{*}_{S})\cap M$. ∎

######  Corollary 2.8.

Let $S_{1}$ and $S_{2}$ be dense subsemigroups of semitopological semigroups
$T_{1}$ and $T_{2}$, respectively. Let $e_{1}\in E(T_{1})$ and $e_{2}\in
E(T_{2})$. Let $A_{1}\subseteq S_{1}$ be a central set near $e_{1}$ and let
$A_{2}\subseteq S_{2}$ be a central set near $e_{2}$. Then $A_{1}\times A_{2}$
is a central set near $(e_{1},e_{2})$ in $S_{1}\times S_{2}$.

###### Proof.

Pick an idempotent $p_{1}\in K(e^{*}_{S_{1}})$ and an idempotent $p_{2}\in
K(e^{*}_{S_{2}})$ such that $A_{1}\in p_{1}$ and $A_{2}\in p_{2}$. Let $M$ be
as in Theorem 2.7 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 $A_{1}\times A_{2}$ is
a central set near $(e_{1},e_{2})$ in $S_{1}\times S_{2}$. ∎

We will utilize the following notion of thick set in the final two results of
this section. Here, we also recall the definition of piecewise syndetic set
[15, Definition 3.5]. Note that, though it is called topologically piecewise
syndetic set in [15, Definition 3.5], but due to its natural algebraic
equivalence [15, Theorem 3.6] we will simply call it piecewise syndetic set.
(Given a set $X$ we write $\mathcal{P}_{f}(X)$ for the set of finite nonempty
subset of $X$.)

######  Definition 2.9.

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

  1. (1)

The set $A$ is thick near $e$ if there exists $U\in\tau_{e}$ such that for
each $F\in\mathcal{P}_{f}(S)$ and $V\in\tau_{e}$, there exists $x\in V\cap S$
such that $(F\cap U)+x\subseteq A$.

  2. (2)

The set $A$ is piecewise 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
for each $G\in\mathcal{P}_{f}(S)$ and $O\in\tau_{e}$ there exists $x\in S\cap
O$ such that $(G\cap V)+x\subseteq\bigcup_{t\in F}(-t+A)$.

  3. (3)

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

Note that if $A\subseteq S$ is thick near $e$, then $e\in\text{cl}_{T}(A)$.

######  Example 2.10.

For a given dense subsemigroup of a semitopological semigroup $T$, if $e\in
E(T)$ and $U\in\tau_{e}$, then $U\cap S$ is thick near $e$ in $S$.

The following lemma provides an algebraic correspondence of the above notions.

######  Lemma 2.11.

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

  1. (1)

If $A$ is thick near $e$, then there exists a left ideal $L$ of $e^{*}_{S}$
such that $L\subseteq\overline{A}$.

  2. (2)

The set $A$ is piecewise syndetic near $e$ if and only if $\overline{A}\cap
K(e^{*}_{S})\neq\emptyset$.

###### Proof.

  1. (1)

If $A\subseteq S$ is thick near $e$, then pick $U\in\tau_{e}$ such that
$\\{-s+A:s\in U\cap S\\}$ has finite intersection property and also
$e\in\text{cl}_{T}(-s+A)$. Consider the family $\mathcal{R}:=\\{B\subseteq
S:e\in\text{cl}_{T}(B)\\}$ and its subfamily $\mathcal{A}:=\\{-s+A:s\in U\cap
S\\}$. Then $\emptyset\notin\mathcal{R}$ and
$\mathcal{R}^{\uparrow}:=\\{C\subseteq S:B\subseteq C\text{ for some
}B\in\mathcal{R}\\}=\mathcal{R}$. Moreover, $\mathcal{R}$ is partition
regular. Therefore, by [10, Theorem 3.11] pick an ultrafilter $p\in\beta S$
such that $\mathcal{A}\subseteq p\subseteq\mathcal{R}$. Hence $p\in
e^{*}_{S}$. Now we will show that the left ideal $e^{*}_{S}+p$ of $e^{*}_{S}$
is contained in $\overline{A}$, 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 $U\cap S\subseteq\\{s\in S:-s+A\in p\\}$. Hence,
by [15, Lemma 2.2(d)] $U\cap S\in q$, and we have $\\{s\in S:-s+A\in p\\}\in
q$, i.e. $A\in q+p$.

  2. (2)

[15, Theorem 3.6].

∎

From

these algebraic correspondences it is immediate that for any idempotent $e$,
any thick set near $e$ is central near $e$, and any central set near $e$ is
piecewise syndetic near $e$.

######  Lemma 2.12.

Let $I$ be a set. For $i\in I$, let $S_{i}$ be a dense subsemigroup of a
semitopological semigroup $(T_{i},+)$, let $e_{i}\in E(T_{i})$, 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 $i\in I$, $A_{i}$ is thick near $e_{i}$, then $A$ is thick near
$e$.

  2. (2)

If $A$ is central near $e$, then each $A_{i}$ is central near $e_{i}$.

  3. (3)

If $A$ is piecewise syndetic near $e$, then for each $i\in I$, $A_{i}$ is
piecewise syndetic near $e_{i}$.

###### 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 given $F\in\mathcal{P}_{f}(S)$ and $V\in\tau_{e}$, pick
$V_{i}\in\tau_{e_{i}}$ such that $\bigtimes_{i\in I}V_{i}\subseteq V$. As each
$A_{i}$ is thick near $e_{i}$, so pick $U_{i}\in\tau_{e_{i}}$ and $x_{i}\in
V_{i}\cap S_{i}$ such that $(\pi_{i}(F)\cap U_{i})+x_{i}\subseteq A_{i}$.
Thus, we have $F\cap(\bigtimes_{i\in I}U_{i})\subseteq\big{(}\bigtimes_{i\in
I}(\pi_{i}(F)\cap U_{i})\big{)}+x\subseteq A$, i.e. we have $U=\bigtimes_{i\in
I}U_{i}\in\tau_{e}$, such that for each $F\in\mathcal{P}_{f}(S)$ and
$V\in\tau_{e}$, there exists $x\in V\cap S$ such that $(F\cap U)+x\subseteq
A$. Thus $A$ is thick near $e$.

  2. (2)

For each $i\in I$, let $\tilde{\pi}_{i}:\beta S\rightarrow\beta S_{i}$ be the
continuous extension of $\pi_{i}$. By Lemma 2.5, each
$\pi_{i}|_{e^{*}_{S}}:e^{*}_{S}\rightarrow e^{*}_{S_{i}}$ is a surjective
homomorphism. Hence, by [10, Exercise 1.7.3] we have
$\tilde{\pi}_{i}[K(e^{*}_{S})]=K(e^{*}_{S_{i}})$. Pick an idempotent $p\in
K(e^{*}_{S})$ such that $A\in p$. Then $\tilde{\pi}_{i}(p)$ is an idempotent
in $K(e^{*}_{S_{i}})$ and $A_{i}\in p$.

  3. (3)

For each $i\in I$ and $U_{i}\in\tau_{e_{i}}$, consider $\bigtimes_{\iota\in
I}U_{\iota}\in\tau_{e}$, where $U_{\iota}=T_{\iota}$ for each $\iota\neq i$.
Let $G_{i}\in\mathcal{P}_{f}(S_{i})$ and $O_{i}\in\tau_{e_{i}}$. Consider
$G=\bigtimes_{\iota\in I}G_{\iota}\in\mathcal{P}_{f}\big{(}\bigtimes_{\iota\in
I}S_{\iota}\big{)}$, where for each $\iota\neq i$, $G_{\iota}=\\{s_{\iota}\\}$
for some fixed $s_{\iota}\in S_{\iota}$. Also, consider $O=\bigtimes_{\iota\in
I}O_{\iota}\in\tau_{e}$, where $O_{\iota}=T_{\iota}$ for each $\iota\neq i$.
Now pick $F\in\mathcal{P}_{f}\big{(}\bigtimes_{\iota\in I}(U_{\iota}\cap
S_{\iota})\big{)}$, $\bigtimes_{\iota\in I}V_{\iota}\in\tau_{e}$ and
$x=\times_{\iota\in I}x_{\iota}\in S\cap O$ such that $(G\cap
V)+x\subseteq\bigcup_{t\in F}(-t+A)$, i.e. in particular $(G_{i}\cap
V_{i})+x_{i}\subseteq\bigcup_{t\in F_{i}}(-t+A_{i})$, and therefore each
$A_{i}$ is piecewise syndetic near $e_{i}$.

∎

The proof of the following lemma directly follows from Definition 2.9.

######  Lemma 2.13.

Let $S_{1}$ and $S_{2}$ be dense subsemigroups of semitopological semigroups
$T_{1}$ and $T_{2}$, respectively. Let $e_{1}\in E(T_{1})$ and $e_{2}\in
E(T_{2})$. Let $A_{1}\subseteq S_{1}$ be piecewise syndetic near $e_{1}$, and
let $A_{2}\subseteq S_{2}$ be piecewise syndetic near $e_{2}$. Then
$A_{1}\times A_{2}$ is piecewise syndetic near $(e_{1},e_{2})$ in $S_{1}\times
S_{2}$.

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 2.14.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a dense subsemigroup of a
semitopological semigroup $T_{i}$, let $e_{i}\in E(T_{i})$, 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 $e=\times_{i\in I}e_{i}$. Let
$J=\\{i\in I:\,A_{i}$ $\text{ is not thick near }e_{i}\text{ in }S_{i}\\}$ be
a finite set. Then $A$ is piecewise syndetic near $e$ in $S$ if and only if
for each $i\in I$, $A_{i}$ is piecewise syndetic near $e_{i}$ in $S_{i}$.

###### Proof.

The forward implication follows from Lemma 2.12. 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 2.12, $\bigtimes_{i\in I\setminus J}A_{i}$ 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 2.13, $\bigtimes_{i\in I}A_{j}$ is piecewise syndetic near $\times_{i\in
J}e_{j}$. Thus, $\big{(}\bigtimes_{i\in I\setminus
J}A_{i}\big{)}\times\big{(}\bigtimes_{i\in J}A_{i}\big{)}$ 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. $A$ is piecewise
syndetic near $e$ in $S$. ∎

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

######  Theorem 2.15.

Let $I$ be a set. For each $i\in I$, let $S_{i}$ be a dense subsemigroup of a
semitopological semigroup $T_{i}$, let $e_{i}\in E(T_{i})$, 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 $e=\bigtimes_{i\in I}e_{i}$. Let
$J=\\{i\in I:A_{i}\text{ is not thick near }e_{i}\\}$ be a finite set. Then
$A$ is central near $e$ in $S$ if and only if for each $i\in I$, $A_{i}$ is
central near $e_{i}$ in $S_{i}$.

###### Proof.

The forward implication follows from Lemma 2.12. 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 2.12, $\bigtimes_{i\in I\setminus J}A_{i}$ 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 2.8, $\bigtimes_{i\in J}A_{j}$ is central near $\times_{i\in
J}e_{j}$. Thus, $\big{(}\bigtimes_{i\in I\setminus
J}A_{i}\big{)}\times\big{(}\bigtimes_{i\in J}A_{i}\big{)}$ is central near
$\big{(}\times_{i\in I\setminus J}e_{i}\big{)}\times\big{(}\times_{i\in
J}e_{i}\big{)}$, i.e. $A$ is central near $e$ in $S$. ∎

##  3 Abundance of Large sets near idempotent

In [4], the authors have studied the abundance of large sets. Following them,
we deduce the corresponding results near idempotent.

######  Definition 3.1.

Let $S$ be a subsemigroup of a semitopological semigroup $T$, and let $e\in
E(T)$. $S$ is said to be dense near $e$ if for every neighborhood $U$ of $e$,
we have $U\cap S\neq\phi$.

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
$\pi_{i}$ we mean the projection onto the $i$-th coordinate.

######  Lemma 3.2.

Let $\varepsilon$ be a partition regular property which may be possessed by
subsets of a semigroup. Let $S$ be a semigroup, $l\in\mathbb{N}$, and
$I^{\Diamond}$ be a subsemigroup of $S^{l}$. Assume that for every
$\varepsilon$ set $A$ in $I^{\Diamond}$ and every
$i\in\\{1,2,\ldots,l\\},\,\pi_{i}[A]$ is an $\varepsilon$ set in $S$. For
every $\varepsilon$ set $A$ in $T^{\Diamond}$, every $i\in\\{1,2,\ldots,l\\}$
and every $\varepsilon^{*}$ set $B$ in $S$, there exists an $\varepsilon^{*}$
set $C$ in $I^{\Diamond}$ such that $C\subset A$ and $\pi_{i}[C]\subset B$.

###### Proof.

See [4, Lemma 2.1]. ∎

######  Lemma 3.3.

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 $S^{l}$. Then, statement $(1)$ implies
statement $(2)$. If every 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 a $\varepsilon$ set in $S$.

  2. (2)

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

###### Proof.

See [4, Lemma 2.2]. ∎

######  Corollary 3.4.

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

###### Proof.

It is immediate from the definition that whenever $A$ is an $IP$ set near
$e^{\Diamond}$ in $I^{\Diamond}$ and $i\in\\{1,2,\ldots,l\\}$, then
$\pi_{i}[A]$ is an IP set near $e$ in $S$. Now, the result follows by Lemma
3.3. ∎

######  Lemma 3.5.

Let $T_{1}$ and $T_{2}$ are semitopological semigroups with $e_{i}\in
E(T_{i})$ for $i=1,2$. Let $S_{1}$ and $S_{2}$ are dense near $e_{1}$ and
dense near $e_{2}$ semigroups of $T_{1}$ and $T_{2}$ respectively. Suppose
$\phi:T_{1}\rightarrow T_{2}$ be a continuous, open, surjective homomorphism
such that $\phi(e_{1})=e_{2}$. Let $A\subseteq S_{1}$.

  1. (1)

If $A$ is piecewise syndetic near $e_{1}$ in $S_{1}$, then $\phi[A]$ is
piecewise syndetic near $e_{2}$ in $S_{2}$.

  2. (2)

If $A$ is central near $e_{1}$ in $S_{1}$, then $\phi[A]$ is central near
$e_{2}$ in $S_{2}$.

###### Proof.

Let $\tilde{\phi}:\beta T_{1}\rightarrow\beta T_{2}$ be the continuous
extension of $\phi$ and note that by [10, Lemma 2.14], $\tilde{\phi}$ is a
homomorphism. Clearly, $\tilde{\phi}:e^{*}_{S_{1}}\rightarrow e^{*}_{S_{2}}$
be a surjective homomorphism. Therefore, we know by [10, Exercise 1.7.3] that
$\tilde{\phi}[K(e^{*}_{S_{1}})]=K(e^{*}_{S_{2}})$.

  1. (1)

By Lemma 2.11, pick $p\in\overline{A}\cup K(e^{*}_{S_{1}})$. Then
$\tilde{\phi}(p)=\overline{\phi[A]}\cup K(e^{*}_{S_{2}})$.

  2. (2)

By Definition 2.2, pick some idempotent $p\in\overline{A}\cup
K(e^{*}_{S_{1}})$. Then $\tilde{\phi}(p)$ is an idempotent in
$\overline{\phi[A]}\cup K(e^{*}_{S_{2}})$.

∎

######  Theorem 3.6.

Let $T$ be a semitopological semigroup and $e\in E(T)$. Suppose $S$ be a dense
near $e$ semigroup of $T$, $l\in\mathbb{N}$ and let $I^{\Diamond}\subseteq
S^{l}$ be a dense near $e^{\Diamond}$ subsemigroup of $T^{l}$. Assume that for
each $i\in\\{1,2,\ldots,l\\}$, $\pi_{i}[I^{\Diamond}]$ is piecewise syndetic
near $e$ in $S$. Let $B\subseteq S$.

  1. (1)

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

  2. (2)

If $B$ is $Central^{*}$ near $e$ in $S$, then $B^{l}\cup I^{\Diamond}$ is
$Central^{*}$ near $e^{\Diamond}$ in $I^{\Diamond}$.

###### Proof.

Let $i\in\\{1,2,\ldots,l\\}$. By Lemma 3.3, it suffices to show that whenever
$A$ is piecewise syndetic near $e^{\Diamond}$ in $I^{\Diamond}$, then
$\pi_{i}[A]$ is piecewise syndetic near $e$ in $S$ and whenever $A$ is central
near $e^{\Diamond}$ in $I^{\Diamond}$, then $\pi_{i}[A]$ is central near $e$
in $S$. Since $\pi_{i}[I^{\Diamond}]$ is piecewise syndetic near $e$ in $S$,
we have $\overline{\pi_{i}[I^{\Diamond}]}\cup K(e^{*}_{S})\neq\phi$. So, by
[10, Theorem 1.65],
$K\big{(}e^{*}_{\pi_{i}[I^{\Diamond}]}\big{)}=K(\overline{\pi_{i}[I^{\Diamond}]}\cup
e^{*}_{S})=\overline{\pi_{i}[I^{\Diamond}]}\cup K(e^{*}_{S})$. Now, assume
that $A$ is piecewise syndetic near $e^{\Diamond}$ in $I^{\Diamond}$. Then, by
Lemma 3.5, $\pi_{i}[A]$ is piecewise syndetic near $e$ in
$\pi_{i}[I^{\Diamond}]$. Thus, $\pi_{i}[A]\cup
K\big{(}e^{*}_{\pi_{i}[I^{\Diamond}]}\big{)}\neq\phi$ and consequently,
$\pi_{i}[A]\cup K(e^{*}_{S})\neq\phi$. Finally, assume that $A$ is central
near $e^{\Diamond}$ in $I^{\Diamond}$. Then, by Lemma 3.5, $\pi_{i}[A]$ is
central near $e$ in $\pi_{i}[I^{\Diamond}]$. Pick an idempotent
$p\in\pi_{i}[A]\cup K(e^{*}_{\pi[I^{\Diamond}]})$. Then $p\in\pi_{i}[A]\cup
K(e^{*}_{S})$.

∎

From now onwards, let $T$ be a Hausdorff semitopological semigroup with $e\in
E(T)$, let $S$ be a dense near $e$ subsemigroup of $T$, let $l\in\mathbb{N}$,
let $E^{\Diamond}\subseteq S^{l}$ be a dense near $e^{\Diamond}$ subsemigroup
of $T^{l}$ with $\\{(a,a,\ldots,a):a\in S\\}\subseteq E^{\Diamond}$, and let
$I^{\Diamond}$ be a near $e^{\Diamond}$ two-sided ideal of $E^{\Diamond}$.

######  Definition 3.7.

Let $X=(\beta T)^{l},Y=(e^{*}_{S})^{l}$ are with product topology and the
coordinate-wise operation. Then
$E=Cl_{X}E^{\Diamond},I=Cl_{X}I^{\Diamond},E_{0}=E\cap Y$ and $I_{0}=I\cap Y$.
Given $p\in\beta T,\,\overline{p}=(p,p,\ldots,p)\in X$.

######  Lemma 3.8.

X is a compact right topological semigroup, for each $\vec{x}\in
T^{l},\,\lambda_{\vec{x}}:X\rightarrow X$ 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.

The proof follows from [10, Theorem 2.22, 2.23 and 4.17]. ∎

######  Lemma 3.9.

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

###### Proof.

At first, we prove that for any $q\in e^{*}_{S}$, we have
$\overline{q}=(q,q,\ldots,q)\in E_{0}$. To this end, let $U$ be a neighborhood
of $\overline{q}$ in $Y$, and for each $i\in\\{1,2,\ldots,l\\}$, pick
$A_{i}\in q$ such that $\bigtimes_{i=1}^{l}\overline{A_{i}}\subseteq U$. Let,
$A=\bigcap_{i=1}^{l}A_{i}$, then $\tau_{e}\cup\\{A\\}$ has finite intersection
property. Now, choose $r\in e^{*}_{S}$ such that $\tau_{e}\cup\\{A\\}\subseteq
r$. Then $\overline{r}=(r,r,\ldots,r)\in E\cap U$. Thus, we have
$\overline{p}\in(K(e^{*}_{S}))^{l}\cap E$. Since by Lemma 3.8,
$K(Y)=(K(e^{*}_{S}))^{l}$, we thus have that $K(Y)\cap E_{0}\neq\phi$. Now, by
[10, Theorem 1.65], $K(E_{0})=K(Y)\cap E=(K(e^{*}_{S}))^{l}\cap E$. Since $I$
is an ideal of $E$, we get $K(E)\subseteq I$. Consequently, again by [10,
Theorem 1.65] we have that $K(I_{0})=(K(e^{*}_{S}))^{l}\cap E_{0}$. Thus,
$\overline{p}\in K(I_{0})$ as required. ∎

Now, $I^{\Diamond}\subseteq S^{l}$ itself is a dense near $e^{\Diamond}$
subsemigroup of $T^{l}$, and thus $\beta T^{\Diamond}$ is a compact right
topological semigroup. Let $e_{I^{\Diamond}}^{*}$ be the semigroup of
ultrafilters converging to $e^{\Diamond}$ in $\beta I^{\Diamond}$.

######  Definition 3.10.

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

Now, we recall [4, Lemma 3.6] which will be used subsequently.

######  Lemma 3.11.

If $r\in\beta I^{\Diamond}$ and $\tilde{\iota}(r)\in\overline{B}^{l}$. Then
$B^{l}\cup I^{\Diamond}\in r$.

######  Lemma 3.12.

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

###### Proof.

$\tilde{\iota}$ is a homomorphism follows from [10, Lemma 2.14]. Now we show
that $\tilde{\iota}[e^{*}_{I^{\Diamond}}]=I_{0}$. To see this let $r\in
e^{*}_{I^{\Diamond}}$, then $\tilde{\iota}(r)\in I$. So, it is enough to show
that $\tilde{\iota}(r)\in Y$. If $\tilde{\iota}(r)\in X\setminus Y$, then
there exists an open neighborhood $U$ of $\tilde{\iota}(r)$ such that $U\cap
Y=\emptyset$. We pick $D\subseteq S$ such that
$\tilde{\iota}(r)\in\overline{D}^{l}$ and $\overline{D}^{l}\subseteq U$. Then,
by Lemma 3.11, $D^{l}\cap I^{\Diamond}\in r$. Choose $p\in e^{*}_{S}$ such
that $D\in p$. Then $\overline{p}=(p,p,\ldots,p)\in Y\cap U$, a contradiction.
So, $\tilde{\iota}[e^{*}_{I^{\Diamond}}]\subseteq I_{0}$.
Conversely, let $\overline{q}=(q,q,\ldots,q)\in I_{0}$. Since $\tilde{\iota}$
is surjective, there exists $s\in\beta I^{\Diamond}$ such that
$\tilde{\iota}(s)=\overline{q}$. We now claim that $s\in
e^{*}_{I^{\Diamond}}$. If $s\in\beta I^{\Diamond}\setminus
e^{*}_{I^{\Diamond}}$, then there exists $Q\in s$ such that $\overline{Q}\cap
e^{*}_{I^{\Diamond}}=\emptyset$. Now, since
$\overline{q}\in\tilde{\iota}[\overline{Q}]$, we have $\pi_{i}[Q]\in q$ for
each $i=1,2,\ldots,l$, which is a contradiction to the fact that
$\overline{Q}\cap e^{*}_{I^{\Diamond}}=\emptyset$. Thus,
$I_{0}\subseteq\tilde{\iota}[e^{*}_{I^{\Diamond}}]$ and so
$I_{0}=\tilde{\iota}[e^{*}_{I^{\Diamond}}]$. Therefore,
$\tilde{\iota}|_{e^{*}_{I^{\Diamond}}}:{e^{*}_{I^{\Diamond}}}\rightarrow
I_{0}$ is a surjective homomorphism. Now by [10, Exercise 1.7.3], we have
$\tilde{\iota}[K(e^{*}_{I^{\Diamond}})]=K(I_{0})$. ∎

######  Theorem 3.13.

Let $(T,+)$ be a semitopological semigroup and $e\in E(T)$. Let $S$ be a near
$e$ subsemigroup of $T$, $l\in\mathbb{N}$ and $E^{\Diamond}\subseteq S^{l}$ be
a near $e^{\Diamond}$ subsemigroup of $T^{l}$ with $\\{(a,a,\ldots,a):a\in
S\\}\subseteq E^{\Diamond}$, and let $I^{\Diamond}$ be a near $e^{\Diamond}$
ideal of $E^{\Diamond}$. Let $B\subseteq S$.

  1. (1)

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

  2. (2)

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

  3. (3)

If $B$ is thick near $e$ in $S$, then $B^{l}\cap I^{\Diamond}$ is thick near
$e^{\Diamond}$ in $I^{\Diamond}$.

  4. (4)

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

  5. (5)

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

  6. (6)

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

###### Proof.

  1. (1)

By Lemma 2.11, pick some $p\in K(e^{*}_{S})$ such that $B\in p$. Let
$\overline{p}=(p,p,\ldots,p)$. Now, Lemma 3.9 provides that $\overline{p}\in
K(I_{0})$. Then, by Lemma 3.12, pick some $r\in K(e^{*}_{I^{\Diamond}})$ such
that $\tilde{\iota}(r)=\overline{p}$. Hence, by Lemma 3.11, we have $B^{l}\cap
I^{\Diamond}\in r$.

  2. (2)

Pick by Definition 2.2, some $p\in K(e^{*}_{S})$ such that $p+p=p$ and $B\in
p$. Let $\overline{p}=(p,p,\ldots,p)$. By Lemma 3.9, $\overline{p}\in
K(I_{0})$. By Lemma 3.12, pick $s\in K(e^{*}_{I^{\Diamond}})$ such that
$\tilde{\iota}(s)=\overline{p}$. Pick a minimal left ideal $L$ of
$e^{*}_{I^{\Diamond}}$ such that $s\in L$. Let $H=\\{r\in
L:\tilde{\iota}(r)=\overline{p}\\}$. Then $T$ is a compact subsemigroup of
$e^{*}_{I^{\Diamond}}$, so pick an idempotent $r\in H$. By Lemma 3.11, we have
$B^{l}\cap I^{\Diamond}\in r$.

  3. (3)

Pick a left ideal $L$ of $e^{*}_{S}$ such that $L\subseteq\overline{B}$. Since
each left ideal contains a minimal left ideal, and consequently $L\subseteq
K(e^{*}_{S})$. Pick $p\in L$ and let $\overline{p}=(p,p,\ldots,p)$. By Lemma
3.9, $\overline{p}\in K(I_{0})$. Pick by Lemma 3.12, some $r\in
K(e^{*}_{I^{\Diamond}})$ such that $\tilde{\iota}(r)=\overline{p}$. We claim
that $e^{*}_{I^{\Diamond}}+r\subseteq\overline{B^{l}\cap I^{\Diamond}}$ for
which it suffices by Lemma 3.11 to let $q\in e^{*}_{I^{\Diamond}}+r$ and show
that $\tilde{\iota}(q)\in\overline{B}^{l}$. Pick $v\in e^{*}_{I^{\Diamond}}$
such that $q=v+r$. Then for some $s_{1},s_{2},\ldots,s_{r}\in e^{*}_{S}$ we
have $\tilde{\iota}(v)=(s_{1},s_{2},\ldots,s_{r})$. Thus,

$\displaystyle\tilde{\iota}(q)$
$\displaystyle=\tilde{\iota}(v)+\tilde{\iota}(r)$
$\displaystyle=(s_{1},s_{2},\ldots,s_{r})+(p,p,\ldots,p)$
$\displaystyle=(s_{1}+p,s_{2}+p,\ldots,s_{r}+p)\in
L^{l}\subseteq\overline{B}^{l}.$

To establish statement $(4)$ and $(5)$, it suffices by Theorem 3.6 to let
$i\in\\{1,2,\ldots,l\\}$ and show that $\pi_{i}[I^{\Diamond}]$ is piecewise
syndetic near $e$ in $S$. Clearly
$\pi_{i}[e^{*}_{I^{\Diamond}}]\subseteq\pi_{i}[\beta
I^{\Diamond}]\subseteq\overline{\pi_{i}[I^{\Diamond}]}$. So, $\pi_{i}[\beta
I^{\Diamond}]$ is in fact thick near $e$. Statement $(6)$ follows immediately
from Corollary 3.4. ∎

######  Corollary 3.14.

Let $S$ be a dense near $0$ subsemigroup of $((0,\infty),+)$. Let $B\subseteq
S$, 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 "piecewise syndetic near zero",
"central near zero", "$central^{*}$ near zero", "thick near zero", "$PS^{*}$"
or "$IP^{*}$". If $B$ be large near zero in $(S,+)$, then $B^{l}\cap AP_{l}$
is large near zero in $AP_{l}$.

###### Proof.

Let $T=((0,\infty),+),\,I^{\Diamond}=AP_{l}$ and let
$E^{\Diamond}=I^{\Diamond}\cup\\{(a,a,\ldots,a):a\in S\\}$. Then apply Theorem
3.13. ∎

######  Theorem 3.15.

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

###### Proof.

Let $AP_{l}=\\{(a,a+d,a+2d,\ldots,a+(l-1)d):a,d\in S\\}$. Then the function
$\psi:S\times S\rightarrow AP_{l}$ given by
$\psi(a,d)=(a,a+d,\ldots,a+(l-1)d)$ is a homeomorphism and an isomorphism. So,
the conclusion follows from Corollary 3.4. ∎

######  Lemma 3.16.

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 $S$ be a dense
subsemigroup of $((0,\infty),+)$. Define ${\phi}:S^{v}\rightarrow S^{u}$ by
$\phi(\vec{x})=A\vec{x}$. Let $\tilde{\phi}:\beta(S^{v})\rightarrow(\beta
S)^{u}$ be its continuous extension. Let $p$ be a minimal idempotent of
$O^{+}(S)$ 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 $q\in
O^{+}(S^{v})$ such that
$\tilde{\phi}(q)=\overline{p}=(p,p,\ldots,p)\in(O^{+}(S))^{u}$.

###### Proof.

First, we claim that $\overline{p}\in\tilde{\phi}[O^{+}(S^{v})]$. Suppose that
$\overline{p}\in(\beta S)^{u}\setminus\tilde{\phi}[O^{+}(S^{v})]$. Since,
$\tilde{\phi}[O^{+}(S^{v})]$ is closed, so pick a neighborhood $U$ of
$\overline{p}$ such that $U\cap\tilde{\phi}[O^{+}(S^{v})]=\emptyset$. Pick
$D\in p$ such that $\overline{D}^{u}\subseteq U$. Now,
$\\{(S\cap(0,\eta))^{v}:\eta>0\\}\cup\\{\mathcal{Y}\\}$ has finite
intersection property, where $\mathcal{Y}=\\{\vec{y}\in S^{v}:A\vec{y}\in
D^{u}\\}$. So, there exists $r\in O^{+}(S^{v})$ such that
$\\{(S\cup(0,\eta))^{v}:\eta>0\\}\cup\\{\mathcal{Y}\\}\subseteq r$. Therefore,
$\tilde{\phi}(r)\in U\cap\tilde{\phi}[O^{+}(S^{v})]$, a contradiction. Let
$M=\\{q\in O^{+}(S^{v}):\tilde{\phi}(q)=\overline{p}\\}$. Then $M$ is a
compact subsemigroup of $O^{+}(S^{v})$, so pick an idempotent $w\in M$. Pick
by [10, Theorem 1.60] a minimal idempotent $q\in O^{+}(S^{v})$ such that
$q\leq w$. Since $\tilde{\phi}$ is a homomorphism, we have
$\tilde{\phi}(q)\leq\tilde{\phi}(w)=\overline{p}$. Now, by [10, Theorem 2.23],
$\overline{p}$ is minimal in $(O^{+}(S))^{u}$ and so
$\tilde{\phi}(q)=\overline{p}$. ∎

######  Theorem 3.17.

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

###### Proof.

Let $C$ be central near zero in $S$ and pick a minimal idempotent $q\in
O^{+}(S^{v})$ such that $\tilde{\phi}(q)=\overline{p}\in(O^{+}(S))^{u}$. Now,
$\bigtimes_{i=1}^{u}\overline{C}$ is a neighborhood of $\overline{p}$, so pick
$B\in q$ such that
$\tilde{\phi}[\overline{B}]\subseteq\bigtimes_{i=1}^{u}\overline{C}$. 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. ∎

######  Lemma 3.18.

Let $S_{0}\cup\\{S_{i}:i\in I\\}$ be a finite collection of semigroups with a
map $\phi:S_{0}\rightarrow\bigtimes_{i\in I}S_{i}$. Let $\tilde{\phi}:\beta
S_{0}\rightarrow\beta(\bigtimes_{i\in I}S_{i})$ be the continuous extension of
$\phi$ and let $\tilde{\iota}:\beta(\bigtimes_{i\in
I}S_{i})\rightarrow\bigtimes_{i\in I}\beta S_{i}$ be the continuous extension
of the inclusion map $\iota:(\bigtimes_{i\in
I}S_{i})\rightarrow\bigtimes_{i\in I}\beta S_{i}$. Let $p\in\beta S_{0}$. Then
$C\in\tilde{\iota}\circ\tilde{\phi}(p)$ if and only if $C=\bigtimes_{i\in
I}C_{i}$ and $\phi^{-1}(C)\in p$.

###### Proof.

For each $i_{0}\in I$, if $\tilde{\pi}_{i_{0}}:\beta(\bigtimes_{i\in
I}S_{i})\rightarrow\beta S_{i_{0}}$ be the continuous extension of the natural
projection $\pi_{i_{0}}:\bigtimes_{i\in I}S_{i}\rightarrow S_{i_{0}}$, then we
have $\tilde{\iota}=\bigtimes_{i\in I}\tilde{\pi}_{i}$. Hence,
$\tilde{\iota}\circ\tilde{\phi}(p)=\bigtimes_{i\in
I}\tilde{\pi}_{i}(\tilde{\phi}(p))$. Therefore,
$C\in\tilde{\iota}\circ\tilde{\phi}(p)$ if and only if $C=\bigtimes_{i\in
I}C_{i}$, where $C_{i}\in\tilde{\pi_{i}}\circ\tilde{\phi}(p)$, for each $i\in
I$. Now, by [10, Lemma 3.30], $C_{i}\in\tilde{\pi}_{i}\circ\tilde{\phi}(p)$ if
and only if $\phi^{-1}(\pi_{i}^{-1}(C_{i}))\in p$. But
$\phi^{-1}(\pi_{i}^{-1}(C_{i}))\in p$ for all $i\in I$, if and only if
$\bigcap_{i\in I}(\pi_{i}^{-1}(C_{i}))=\phi^{-1}(C)\in p$, and this completes
the proof. ∎

######  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 $S$ be
a dense subsemigroup of $((0,\infty),+)$. If

  1. (1)

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

  2. (2)

for every $s\in S$, there exists $\vec{x}\in(S\cap(0,s))^{v}$ such that
$A\vec{x}=\overline{s}$, where $\overline{s}=\begin{pmatrix}s\\\ s\\\
\vdots\\\ s\end{pmatrix}\in S^{v}$,

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

###### Proof.

Let $B$ be a piecewise syndetic near zero of $S$ and by [3, Theorem 2.5],
there exists $p\in E(K(O^{+}(S)))$ such that for every $\epsilon>0$ there
exists $s\in S\cap(0,\epsilon)$ for which $-s+B\in p$. Now, by Lemma 3.16
there exists $q\in E(K(O^{+}(S^{u})))$ such that
$\tilde{\phi}(q)=\overline{p}$. Let $F=\\{\vec{z}\in S^{v}:A\vec{z}\in
B^{u}\\}$. We show that $F$ is piecewise syndetic near zero in $S^{v}$. Let
$H=\\{\vec{x}\in S^{v}:-\vec{x}+F\in q\\}$. We now prove that for each
$\epsilon>0,\,(S\cap(0,\epsilon))^{v}\cap H\neq\emptyset$. Now, pick $s\in
S\cap(0,\epsilon)$ such that $-s+B\in p$ and let $C=-s+B$. Let
$D=\\{\vec{y}\in S^{v}:A\vec{y}\in C^{u}\\}$. By assumption (2), pick
$\vec{x}\in S\cap(0,\epsilon)^{u}$ such that
$A\vec{x}=\overline{s}\in(S\cap(0,F))^{u}$. Since $C^{u}\in\overline{p}$, so
by Lemma 3.17 we have $\phi^{-1}[C^{u}]\in q$, i.e. $D=\phi^{-1}[C^{u}]\in q$.
Let $\vec{y}\in D$, then we claim that $A(\vec{x}+\vec{y})\in B^{u}$. So, for
each $i\in\\{1,2,\ldots u\\}$ we have
$\sum_{j=1}^{v}a_{ij}(x_{i}+y_{j})=s+\sum_{j=1}^{v}a_{ij}y_{j}\in s+C\subseteq
B$, as required. ∎

##  4 Polynomial Milliken-Taylor theorem near idempotent

Let us begin by recalling the notion of tensor product [11, Definition 1.1].

######  Definition 4.1.

Let $X_{1}$ and $X_{2}$ be discrete spaces, let $p\in\beta X_{1}$, and
$q\in\beta X_{2}$. Then the tensor product of $p$ and $q$ is $p\otimes
q=\\{A\subseteq X_{1}\times X_{2}:\\{x_{1}\in X_{1}:\\{x_{2}\in
X_{2}:(x_{1},x_{2})\in A\\}\in q\\}\in p\\}$.

Let us consider a discrete space $X$. Then by Riesz representation theorem [7,
Theorem 7.17], the dual space $C(\beta X)^{*}$ of the Banach space $C(\beta
X):=\\{f:f:\beta X\rightarrow\mathbb{C}\text{ is continuous}\\}$ is
$\mathcal{M}(\beta X)$, the space of Radon measures with bounded variation,
and the identification $I:\mathcal{M}(\beta X)\rightarrow C(\beta X)^{*}$ is
given by $I_{\mu}(f):=\int_{\beta X}f\,d\mu$ for $\mu\in\mathcal{M}(\beta X)$
and $f\in C(\beta X)$.
For an ultrafilter $p$ in $\beta X$, in particular, if we consider the linear
functional $\varphi_{p}\in C(\beta X)^{*}$ defined by $\varphi_{p}(f)=f(p)$
for $f\in C(\beta X)$, then the corresponding measure in $\mathcal{M}(\beta
X)$ will be the Dirac measure $\delta_{p}$ on $\beta X$ defined by

$\delta_{p}(Z)=\begin{cases}1&\text{ if }p\in Z\\\ 0&\text{ if }p\notin
Z.\end{cases}$

By the definition of ultrafilter and [10, Theorem 3.6(e)], it is now very easy
to see that $\delta_{p}$ naturally corresponds to a finitely additive
$\\{0,1\\}$ valued measure $\mu_{p}$ on $X$ defined by

$\mu_{p}(A)=\delta_{p}(\text{cl}_{\beta X}A)=\begin{cases}1&\text{ if }A\in
p\\\ 0&\text{ if }A\notin p.\end{cases}$

Note that the final identification $p$ with $\mu_{p}$ was already mentioned in
[1, Section 3.1]. Now, the following theorem shows that the measure
corresponding to a tensor product of ultrafilters indeed behaves like a
product measure.

######  Proposition 4.2.

Let $X_{1}$ and $X_{2}$ be two discrete spaces. Let $\mu_{1}$ and $\mu_{2}$ be
the corresponding measures on $X_{1}$ and $X_{2}$, for $p\in\beta X_{1}$ and
$q\in\beta X_{2}$, respectively. Let $\mu$ be the corresponding measure on
$X_{1}\times X_{2}$ for the ultrafilter $p\otimes q\in\beta(X_{1}\times
X_{2})$. Then $\mu(A\times B)=\mu_{1}(A)\cdot\mu_{2}(B)$ for each $A\subseteq
X_{1}$ and $B\subseteq X_{2}$.

###### Proof.

By the definition of tensor product, it is very easy to see that $A\times B\in
p\otimes q$ if and only if $A\in p$ and $B\in q$, i.e. $\mu(A\times B)=1$ if
and only if $\mu_{1}(A)=1$ and $\mu_{2}(B)=1$. Hence, we have $\mu(A\times
B)=\mu_{1}(A)\cdot\mu_{2}(B)$ for all $A\subseteq X_{1}$ and $B\subseteq
X_{2}$. ∎

######  Theorem 4.3.

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 $\iota:S_{1}\times
S_{2}\rightarrow\beta S_{1}\times\beta S_{2}$. Then, $\tilde{\iota}(p\otimes
q)=(p,q)$ for $p\in\beta S_{1}$ and $q\in\beta S_{2}$.

###### Proof.

We have $\tilde{\iota}(p\otimes q)=(\tilde{\pi}_{1}(p\otimes
q),\tilde{\pi}_{2}(p\otimes q))$, where $\tilde{\pi}_{i}:\beta(S_{1}\times
S_{2})\rightarrow\beta S_{i}$ is the continuous extension of the coordinate
homomorphism $\pi_{i}:S_{1}\times S_{2}\rightarrow S_{i}$, $i=1,2$. Now,
$\tilde{\pi}_{1}(p\otimes q)=\\{A_{1}\subseteq S_{1}:\pi_{1}^{-1}(A_{1})\in
p\otimes q\\}=\\{A_{1}\subseteq S_{1}:A_{1}\times S_{2}\in p\otimes q\\}=p$.
The proof of $\tilde{\pi}_{2}(p\otimes q)=q$ is exactly similar. Thus, we have
$\tilde{\iota}(p\otimes q)=(p,q)$. ∎

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

######  Corollary 4.4.

Let $S_{1}$ and $S_{2}$ be dense subsemigroups of semitopological semigroups
$T_{1}$ and $T_{2}$, respectively, with $e_{1}\in E(T_{1})$ and $e_{2}\in
E(T_{2})$. Let $e^{*}_{S_{1}}\otimes e^{*}_{S_{2}}=\\{p\otimes q:p\in
e^{*}_{S_{1}},\,q\in e^{*}_{S_{2}}\\}$. 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, $\tilde{\iota}[e^{*}_{S_{1}}\otimes
e^{*}_{S_{2}}]=e^{*}_{S_{1}}\times e^{*}_{S_{2}}$.

We now recall the definition of higher dimensional tensor product of
ultrafilters from [6, definition 1.15].

######  Definition 4.5.

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

  1. (1)

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

  2. (2)

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

######  Lemma 4.6.

Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and $k\in\mathbb{N}$. Let
$p_{i}\in O^{+}(S)$ for $1\leq i\leq k$. Then $\otimes_{i=1}^{k}p_{i}\in
O^{+}\big{(}\bigtimes_{i=1}^{k}S_{i}\big{)}$.

###### Proof.

Follows immediately from the definition. ∎

Now, we recall [6, Corollary 2.8] and [6, Lemma 2.9]. From the next result
onwards, for $F,G\in\mathcal{P}_{f}(\mathbb{N})$, we denote $F<G$ if $\max
F<\min G$.

######  Lemma 4.7.

Let $m,k\in\mathbb{N}$. For each $i\in\\{1,2,\ldots,k\\}$, let $S_{i}$ be a
semigroup, $\langle x_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in $S_{i}$
and $p_{i}$ be an idempotent in $\bigcap_{r=1}^{\infty}\overline{FP(\langle
x_{i,n}\rangle_{n=r}^{\infty})}$. Let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots k\\}$ be a function and let
$A\in\otimes_{j=1}^{m}p_{f(j)}$. Then for each $i\in\\{1,2,\ldots,k\\}$ there
is a product subsystem $FP(\langle y_{i,n}\rangle_{n=1}^{\infty})$ of
$FP(\langle x_{i,n}\rangle_{n=1}^{\infty})$ such that $\\{(\prod_{t\in
F_{1}}y_{f(1),t},\prod_{t\in F_{2}}y_{f(2),t},\ldots,\prod_{t\in
F_{m}}y_{f(m),t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\\}\subseteq A$.

######  Lemma 4.8.

Let $m,k\in\mathbb{N}$. For each $i\in\\{1,2,\ldots,k\\}$, let $S_{i}$ be a
semigroup, let $\langle x_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in
$S_{i}$, and let $p_{i}\in\bigcap_{r=1}^{\infty}\overline{FP(\langle
x_{i,n}\rangle_{n=r}^{\infty})}$. Then, for any function
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,k\\}$, $\\{(\prod_{t\in
F_{1}}y_{f(1),t},\prod_{t\in F_{2}}y_{f(2),t},$ $\ldots,\prod_{t\in
F_{m}}y_{f(m),t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\\}\in\otimes_{j=1}^{m}p_{f(j)}$.

We now state the Milliken-Taylor Theorem near zero.

######  Theorem 4.9.

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

  1. (1)

There is an idempotent $p\in\bigcap_{m=1}^{\infty}\overline{FS(\langle
x_{n}\rangle_{n=m}^{\infty})}\cap O^{+}(S)$ 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 $\\{\sum_{j=1}^{k}a_{j}\sum_{t\in
F_{j}}y_{t}:F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq k\text{
with }F_{1}<F_{2}<\ldots<F_{k}\\}\subseteq A\cap(0,\epsilon)$.

Now, in the next two theorems along with Corollary 4.12 we generalize the
above theorem.

######  Theorem 4.10.

Let $S$ be a dense subsemigroup of $((0,\infty),+)$ such that $S\cap(0,1)$ is
a subsemigroup of $((0,1),\cdot)$ and assume that for each $y\in S\cap(0,1)$
and for each $x\in S$, $\frac{x}{y}\in S$ and $yx\in S$. Let
$m,k\in\mathbb{N}$. For each $i\in\\{1,2,\ldots,k\\}$, let $\langle
x_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in $S$ with
$\sum_{n=1}^{\infty}x_{i,n}<\infty$ and let
$p_{i}+p_{i}=p_{i}\in\bigcap_{r=1}^{\infty}\overline{FS(\langle
x_{i,n}\rangle_{n=r}^{\infty})}\cap O^{+}(S)$. Let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,k\\}$ be a function and
$g\in\mathbb{N}[x_{1},x_{2},\ldots,x_{m}]$ or $S[x_{1},x_{2},\ldots,x_{m}]$ be
a polynomial with $g(0,0,\ldots,0)=0$. Then for each
$\epsilon>0,\big{\\{}g(\sum_{t\in F_{1}}x_{f(1),t},\sum_{t\in
F_{2}}x_{f(2),t},\ldots,$ $\sum_{t\in
F_{m}}x_{f(m),t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\\}\cap(0,\epsilon)\subseteq
g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})$.

###### Proof.

By [6, Theorem 3.2],
$\tilde{g}(\otimes_{j=1}^{m}p_{f(j)})=g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})$.
Let $A=\\{g(\sum_{t\in F_{1}}x_{f(1),t},\sum_{t\in
F_{2}}x_{f(2),t},\ldots,\sum_{t\in
F_{m}}x_{f(m),t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\\}$. Since $g$ is continuous,
$g(0,0,\ldots,0)=0$, and $\otimes_{j=1}^{m}p_{f(j)}\in
O^{+}(\bigtimes_{i=1}^{m}S)$. Therefore,
$\tilde{g}\big{(}\otimes_{j=1}^{m}p_{f(j)}\big{)}=g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})\in
O^{+}(S)$. Thus by Lemma 4.8, $A\in\otimes_{j=1}^{m}p_{f(j)}$, and so by [10,
Lemma 3.30], $g[A]\in\tilde{g}(\otimes_{j=1}^{m}p_{f(j)})$. Hence
$g[A]\cap(0,\epsilon)\in g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})$. ∎

######  Theorem 4.11.

Let $m,k\in\mathbb{N}$ and let $S$ be a dense subsemigroup of $((0,\infty),+)$
such that $S\cap(0,1)$ is a subsemigroup of $((0,1),\cdot)$ and assume that
for each $y\in S\cap(0,1)$ and each $x\in S$, $\frac{x}{y}\in S$ and $yx\in
S$. For each $i\in\\{1,2,\ldots,k\\}$, let $\langle
x_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in $S$ with
$\sum_{n=1}^{\infty}x_{i,n}<\infty$ and let
$p_{i}+p_{i}=p_{i}\in\bigcap_{r=1}^{\infty}\overline{FS(\langle
x_{i,n}\rangle_{n=r}^{\infty}}\cap O^{+}(S)$. Let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,k\\}$ be a function and $g$ in
$\mathbb{N}[x_{1},x_{2},\ldots,x_{m}]$ or $S[x_{1},x_{2},\ldots,x_{m}]$ be a
polynomial with $g(0,0,\ldots,0)=0$ and let $A\in
g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})$. Then for each $\epsilon>0$, and for
each $i\in\\{1,2,\ldots,k\\}$ there is a sum subsystem $FS(\langle
y_{i,n}\rangle_{n=1}^{\infty})$ of $FS(\langle x_{i,n}\rangle_{n=1}^{\infty})$
such that $\Big{\\{}g(\sum_{t\in F_{1}}y_{f(1),t},\sum_{t\in
F_{2}}y_{f(2),t},\ldots,\sum_{t\in
F_{m}}y_{f(m),t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\Big{\\}}\subseteq A\cap(0,\epsilon)$.

###### Proof.

By [6, Theorem 3.2],
$\tilde{g}(\otimes_{j=1}^{m}p_{f(j)})=g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})$.
Now, $\tilde{g}(\otimes_{j=1}^{m}p_{f(j)})\in O^{+}(S).$ So, pick
$B\in\otimes_{j=1}^{m}p_{f(j)}$ such that
$\tilde{g}[\overline{B}]\subseteq\overline{A\cap(0,\epsilon)}$. By Lemma 4.7,
for each $i\in\\{1,2,\ldots,k\\}$ pick a sum subsystem $FS(\langle
y_{i,n}\rangle_{n=1}^{\infty})$ of $FS(\langle x_{i,n}\rangle_{n=1}^{\infty})$
such that $\Big{\\{}(\sum_{t\in F_{1}}y_{f(1),t},\sum_{t\in
F_{2}}y_{f(2),t},\ldots,\sum_{t\in
F_{m}}y_{f(m),t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\Big{\\}}\subseteq B$. Then
$\Big{\\{}g(\sum_{t\in F_{1}}y_{f(1),t},$ $\sum_{t\in
F_{2}}y_{f(2),t},\ldots,\sum_{t\in
F_{m}}y_{f(m),t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\Big{\\}}\subseteq A\cap(0,\epsilon)$. ∎

######  Corollary 4.12.

Let $S$ be a dense subsemigroup of $((0,\infty),+)$ such that $S\cap(0,1)$ is
a subsemigroup of $((0,1),\cdot)$ and assume that for each $y\in S\cap(0,1)$
and each $x\in S$, $\frac{x}{y}\in S$ and $yx\in S$. Let $m,k\in\mathbb{N}$.
For each $i\in\\{1,2,\ldots,k\\}$, let $\langle x_{i,n}\rangle_{n=1}^{\infty}$
be a sequence in $S$ with $\sum_{n=1}^{\infty}x_{i,n}<\infty$ and let
$p_{i}+p_{i}=p_{i}\in\bigcap_{r=1}^{\infty}\overline{FS(\langle
x_{i,n}\rangle_{n=r}^{\infty})}\cap O^{+}(S)$. Let
$f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,k\\}$ be a function and $g$ in
$\mathbb{N}[x_{1},x_{2},\ldots,x_{m}]$ or $S[x_{1},x_{2},\ldots,x_{m}]$ be a
polynomial with $g(0,0,\ldots,0)=0$. Then, the following statements are
equivalent.

  1. (1)

For each $\epsilon>0$ and for each $i\in\\{1,2,\ldots,k\\}$ there exists
$p_{i}=p_{i}+p_{i}\in\bigcap_{r=1}^{\infty}\overline{FS(\langle
x_{i,n}\rangle_{n=r}^{\infty})}\cap O^{+}(S)$ such that $A\cap(0,\epsilon)\in
g(p_{f(1)},p_{f(2)},\ldots,p_{f(m)})$.

  2. (2)

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

###### Proof.

$(1)$ implies $(2)$: The proof follows from Theorem 4.11.
$(2)$ implies $(1)$: By [9, Theorem 3.2], for each $i\in\\{1,2,\ldots,k\\}$
pick $p_{i}=p_{i}+p_{i}\in\bigcap_{r=1}^{\infty}FS(\langle
y_{i,n}\rangle_{n=1}^{\infty})$. Since for each $i\in\\{1,2,\ldots,k\\}$,

$\bigcap_{n=r}^{\infty}\overline{FS(\langle
y_{i,n}\rangle_{n=1}^{\infty})}\subseteq\bigcap_{n=r}^{\infty}\overline{FS(\langle
x_{i,n}\rangle_{n=1}^{\infty})},$

we have for each
$i\in\\{1,2,\ldots,k\\},\,p_{i}\in\cap_{r=1}^{\infty}\overline{FS(\langle
y_{i,n}\rangle_{n=1}^{\infty})}$. Now we use Theorem 4.11. ∎

######  Theorem 4.13.

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

  1. (1)

For each $i\in\\{1,2,\ldots,m\\}$ and for each $\epsilon>0$ there is a
sequence $\langle x_{i,n}\rangle_{n=1}^{\infty}$ with
$\sum_{n=1}^{\infty}x_{i,n}<\infty$ such that $\Big{\\{}(\sum_{t\in
F_{1}}x_{1,t},\sum_{t\in F_{2}}x_{2,t},\ldots,\sum_{t\in
F_{m}}x_{m,t}):F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\\}\subseteq A\cap(0,\epsilon)$.

  2. (2)

For each $i\in\\{1,2,\ldots,m\\}$, there is an idempotent $p_{i}\in O^{+}(S)$
such that $A\in\otimes_{i=1}^{m}p_{i}$.

###### Proof.

$(1)$ implies $(2)$: For each $i\in\\{1,2,\ldots,m\\}$, pick an idempotent
$p_{i}\in\bigcap_{r=1}^{\infty}\overline{FS(\langle
x_{i,n}\rangle_{n=r}^{\infty})}\cap O^{+}(S)$, by [9, Theorem 3.1]. Let $k=m$
and let $f:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,m\\}$ be the identity
function. After that, we apply Lemma 4.8.

$(2)$ implies $(1)$: By Lemma 4.6, $\otimes_{i=1}^{m}p_{i}\in
O^{+}(\bigtimes_{i=1}^{m}S)$ and apply [6, Theorem 1.16].

∎

######  Theorem 4.14.

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

  1. (1)

For each $\epsilon>0$ there is a sequence $\langle
x_{n}\rangle_{n=1}^{\infty}$ with $\sum_{n=1}^{\infty}x_{n}<\infty$ such that
$\Big{\\{}\big{(}\sum_{t\in F_{1}}x_{t},\sum_{t\in
F_{2}}x_{t},\ldots,\sum_{t\in
F_{m}}x_{t}\big{)}:F_{i}\in\mathcal{P}_{f}(\mathbb{N})\text{ for }1\leq i\leq
m\text{ with }F_{1}<F_{2}<\ldots<F_{m}\Big{\\}}\subseteq
A\cap(0,\epsilon)^{m}$.

  2. (2)

There is an idempotent $p\in O^{+}(S)$ such that $A\in\otimes_{i=1}^{m}p$.

###### Proof.

$(1)$ implies $(2)$: Pick an idempotent
$p\in\bigcap_{r=1}^{\infty}\overline{FS(\langle
x_{n}\rangle_{n=r}^{\infty})}$, by [9, Theorem 3.1]. Then, by Lemma 4.8 with
$k=1$, $A\in\otimes_{i=1}^{m}p$, as required.

$(2)$ implies $(1)$: By Lemma 4.6, $\otimes_{i=1}^{m}p\in
O^{+}(\bigtimes_{i=1}^{m}S)$. So for each $\epsilon>0$,
$A\cap(0,\epsilon)^{m}\in\otimes_{i=1}^{m}p$. Now use [6, Theorem 1.17]. ∎

## References

  * [1] V. Bergelson, _Combinatorial and Diophantine applications of ergodic theory_ , Appendix A by A. Leibman and Appendix B by Anthony Quas and Mátè Wierdl. Handbook of dynamical systems. Vol. 1B, 745 –869, Elsevier B. V., Amsterdam, 2006.
  * [2] S. Biswas, B. Bose, S.K. Patra, _D-sets in arbitrary semigroup_ Topology Appl. 282 (2020), 107329, 16 pp.
  * [3] T. Bhattacharya, S. Chakraborty, S. Patra, _Combined algebraic properties of C ∗-sets near zero_, Semigroup Forum 100 (2020), no. 3, 717 –731.
  * [4] V. Bergelson, N. Hindman, _Partition regular structures contained in large sets are abundant_ , J. Combin. Theory Ser. A 93 (2001), no. 1, 18 –36.
  * [5] V. Bergelson, N. Hindman, _Nonmetrizable topological dynamics and Ramsey theory_ , Trans. Amer. Math. Soc. 320 (1990), no. 1, 293 –320.
  * [6] V. Bergelson, N. Hindman, K. Williams, _Polynomial extensions of the Milliken-Taylor theorem_ , Trans. Amer. Math. Soc. 366 (2014), no. 11, 5727 –5748.
  * [7] G.B. Folland, _Real analysis. Modern techniques and their applications_ , Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. xvi+386 pp.
  * [8] H. Furstenberg, _Recurrence in ergodic theory and combinatorial number theory_ , M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981. xi+203 pp.
  * [9] N. Hindman, I. Leader, _The semigroup of ultrafilters near 0_ , Semigroup Forum 59 (1999), no. 1, 33–55.
  * [10] N. Hindman, D. Strauss, _Algebra in the Stone-Čech compactification. Theory and applications_ , Second revised and extended edition, De Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012.
  * [11] N. Hindman, D. Strauss, _Algebraic products of tensor products_ , Semigroup Forum 103 (2021), 888–898.
  * [12] N. Hindman, D. Strauss, _Cartesian products of sets satisfying the central sets theorem_ , Topology Proc. 35 (2010), 203 –223.
  * [13] N. Hindman, D. Strauss, _Image partition regular matrices and concepts of largeness_ New York J. Math. 26 (2020), 230 –260.
  * [14] N. Hindman, D. Strauss, _Image partition regular matrices and concepts of largeness II_, to appear in Topology Proceedings.
  * [15] M.A. Tootkaboni, T. Vahed, _The semigroup of ultrafilters near an idempotent of semitopological semigroup_ , Topology Appl. 159 (2012), 3494 –3503.
  * [16] B. van der Waerden, _Beweis einer Baudetschen Vermutung_ , Nieuw Arch. Wisk. 19 (1927), 212 –216.