# Elementary characterization of essential $\mathcal{F}$-sets and its
combinatorial consequences

Dibyendu De
<EMAIL_ADDRESS>111 Department of Mathematics, University of Kalyani,
Kalyani-741235, Nadia, West Bengal, India.  Pintu Debnath
<EMAIL_ADDRESS>222Department of Mathematics, Basirhat College,
Basirhat -743412, North 24th parganas, West Bengal, India.  Sayan Goswami
<EMAIL_ADDRESS>333Department of Mathematics, University of Kalyani,
Kalyani-741235, Nadia, West Bengal, India  444The author acknowledges the
grant of UGC NET SRF fellowship.

###### Abstract

There is a long history of studying Ramsey theory using the algebraic
structure of the Stone-Čech compactification of discrete semigroup. It has
been shown that various Ramsey theoretic structures are contained in different
algebraic large sets. In this article we will deduce the combinatorial
characterization of certain sets, that are the member of the idempotent
ultrafilters of the closed subsemigroup of $\beta S$, arising from certain
Ramsey family. In a special case when $S=\mathbb{N}$, we will deduce that sets
which are the members of all idempotent ultrafilters of those semigroups
contain certain additive-multiplicative structures. Later we will generalize
this result for weak rings, where we will show a non-commutative version of
the additive-multiplicative structure.

Mathematics subject classification 2020: 05D10

Keywords: Stone-Čech compactification, closed subsets of $\beta\mathbb{N}$,
Ramsay families

## 1 Introduction

For a set $S$, let $\mathcal{P}\left(S\right)$ be the collection of all
subsets of $S$ and $\mathcal{P}_{f}\left(S\right)$ be the set of all finite
subsets of $S$. A collection $\mathcal{F\subseteq
P}\left(S\right)\setminus\left\\{\emptyset\right\\}$ is called upward
hereditary if whenever $A\in\mathcal{F}$ and $A\subseteq B\subseteq S,$ then
it follows that $B\in\mathcal{F}$. A non-empty and upward hereditary
collection $\mathcal{F\subseteq
P}\left(S\right)\setminus\left\\{\emptyset\right\\}$ is called a family. If
$\mathcal{F}$ is a family, the dual family $\mathcal{F}^{*}$ is given by,

$\mathcal{F}^{*}=\\{E\subseteq S:\forall A\in\mathcal{F},E\cap
A\neq\emptyset.\\}$

A family $\mathcal{F}$ possesses the Ramsey property if whenever
$A\in\mathcal{F}$ and $A=A_{1}\cup A_{2}$ there is some
$i\in\left\\{1,2\right\\}$ such that $A_{i}\in\mathcal{F}$.

Throughout the article we will need the following definitions.

######  Definition 1.1.

[12] Let $\left(S,\cdot\right)$ be a discrete semigroup.

  1. 1.

The set $A$ is thick if and only if for any finite subset $F$ of $S$, there
exists an element $x\in S$ such that $F\cdot x\subset A$. This means the sets
which contains a translation of any finite subset. For example, one can see
$\cup_{n\in\mathbb{N}}\left[2^{n},2^{n}+n\right]$ is a thick set in
$\mathbb{N}$.

  2. 2.

The set $A$ is syndetic if and only if there exists a finite subset $G$ of $S$
such that $\bigcup_{t\in G}t^{-1}A=S$. That is, with a finite translation if,
the set which covers the entire semigroup, then it will be called a Syndetic
set. For example, the set of even and odd numbers are both syndetic in
$\mathbb{N}$.

  3. 3.

A set $A\subseteq S$ is $IP$ set if and only if there exists a sequence
$\left\\{x_{n}\right\\}_{n=1}^{\infty}$ in $S$ such that
$FP\left(\left\\{x_{n}\right\\}_{n=1}^{\infty}\right)\subseteq A$. Where

$FP\left(\left\\{x_{n}\right\\}_{n=1}^{\infty}\right)=\left\\{\prod_{n\in
F}x_{n}:F\in\mathcal{P}_{f}\left(\mathbb{N}\right)\right\\}$

and $\prod_{n\in F}x_{n}$ to be the product in increasing order.

  4. 4.

The sets which can be written as an intersection of a syndetic and a thick set
are called $\mathit{Piecewise}$ $\mathit{syndetic}$ sets. More formally a set
$A$ is $\mathit{Piecewise}$ $\mathit{syndetic}$ if and only if there exists
$G\in\mathcal{P}_{f}\left(S\right)$ such that for every
$F\in\mathcal{P}_{f}\left(S\right)$, there exists $x\in S$ such that $F\cdot
x\subseteq\bigcup_{t\in G}t^{-1}A$. Clearly the thick sets and syndetic sets
are natural examples of $\mathit{Piecewise}$ $\mathit{syndetic}$ sets. From
definition one can immediately see that
$2\mathbb{N}\cap\bigcup_{n\in\mathbb{N}}\left[2^{n},2^{n}+n\right]$ is a non-
trivial example of $\mathit{Piecewise}$ $\mathit{syndetic}$ sets in
$\mathbb{N}$.

  5. 5.

$\mathcal{T}=\,^{\mathbb{N}}S$.

  6. 6.

For $m\in\mathbb{N}$,
$\mathcal{J}_{m}=\left\\{\left(t\left(1\right),\ldots,t\left(m\right)\right)\in\mathbb{N}^{m}:t\left(1\right)<\ldots<t\left(m\right)\right\\}.$

  7. 7.

Given $m\in\mathbb{N}$, $a\in S^{m+1}$, $t\in\mathcal{J}_{m}$ and $f\in F$,

$x\left(m,a,t,f\right)=\left(\prod_{j=1}^{m}\left(a\left(j\right)\cdot
f\left(t\left(j\right)\right)\right)\right)\cdot a\left(m+1\right)$

where the terms in the product $\prod$ are arranged in increasing order.

  8. 8.

$A\subseteq S$ is called a $J$-set iff for each
$F\in\mathcal{P}_{f}\left(\mathcal{T}\right)$, there exists $m\in\mathbb{N}$,
$a\in S^{m+1}$, $t\in\mathcal{J}_{m}$ such that, for each $f\in\mathcal{T}$,

$x\left(m,a,t,f\right)\in A.$

  9. 9.

If the semigroup $S$ is commutative, the definition is rather simple. In that
case, a set $A\subseteq S$ is a $J$-set if and only if whenever
$F\in\mathcal{P}_{f}\left({}^{\mathbb{N}}S\right)$, there exist $a\in S$ and
$H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$, such that for each $f\in F$,
$a+\sum_{t\in H}f(t)\in A$.

Let us talk about positive density sets of $\mathbb{N}$. Unmodified positive
density means positive asymptotic density, and set of subsets of $\mathbb{N}$
with positive asymptotic density is not a family since it is not closed under
passage to supersets. (The asymptotic density of $A\subseteq\mathbb{N}$ is
$d\left(A\right)=lim_{n\rightarrow\infty}\frac{\mid
A\cap\left\\{1,2,\ldots,n\right\\}\mid}{n}$ provided that limit exists and
undefined otherwise.). But $\mathcal{F}$ is the family of subsets of
$\mathbb{N}$ with positive upper asymptotic density. The upper asymptotic
density of $A\subseteq\mathbb{N}$ is
$d\left(A\right)=limsup_{n\rightarrow\infty}\frac{\mid
A\cap\left\\{1,2,\ldots,n\right\\}\mid}{n}$.

There are many families $\mathcal{F}$ with Ramsay property, where for
uniformity we consider following families for $\mathbb{\mathbb{N}}$.

  * •

The infinite sets,

  * •

The piecewise syndetic sets,

  * •

The sets of positive upper asymptotic density,

  * •

The set containing arbitrary large arithmetic progression,

  * •

The set with property that $\sum_{n\in A}\frac{1}{n}=\infty$,

  * •

The $J$-sets,

  * •

The $IP$-sets.

###  A brief review of Topological algebra:

Let us recall some basic algebraic structure of the Stone-Čech
compactification. The set $\\{\overline{A}:A\subset S\\}$ is a basis for the
closed sets of $\beta S$. The operation ‘$\cdot$’ on $S$ can be extended to
the Stone-Čech compactification $\beta S$ of $S$ so that$(\beta S,\cdot)$ is a
compact right topological semigroup (meaning that for any is continuous) with
$S$ contained in its topological center (meaning that for any $x\in S$, the
function $\lambda_{x}:\beta S\rightarrow\beta S$ defined by
$\lambda_{x}(q)=x\cdot q$ is continuous). This is a famous Theorem due to
Ellis that if $S$ is a compact right topological semigroup then the set of
idempotents $E\left(S\right)\neq\emptyset$. A non-empty subset $I$ of a
semigroup $T$ is called a left ideal of $S$ if $TI\subset I$, a right ideal if
$IT\subset I$, and a two sided ideal (or simply an ideal) if it is both a left
and right ideal. A minimal left ideal is the left ideal that does not contain
any proper left ideal. Similarly, we can define minimal right ideal and
smallest ideal.

Any compact Hausdorff right topological semigroup $T$ has the smallest two
sided ideal

$\begin{array}[]{ccc}K(T)&=&\bigcup\\{L:L\text{ is a minimal left ideal of
}T\\}\\\ &=&\,\,\,\,\,\bigcup\\{R:R\text{ is a minimal right ideal of
}T\\}.\end{array}$

Given a minimal left ideal $L$ and a minimal right ideal $R$, $L\cap R$ is a
group, and in particular contains an idempotent. If $p$ and $q$ are
idempotents in $T$ we write $p\leq q$ if and only if $pq=qp=p$. An idempotent
is minimal with respect to this relation if and only if it is a member of the
smallest ideal $K(T)$ of $T$. Given $p,q\in\beta S$ and $A\subseteq S$, $A\in
p\cdot q$ if and only if the set $\\{x\in S:x^{-1}A\in q\\}\in p$, where
$x^{-1}A=\\{y\in S:x\cdot y\in A\\}$. See [12] for an elementary introduction
to the algebra of $\beta S$ and for any unfamiliar details.

It will be easy to check that the family $\mathcal{F}$ has the Ramsey property
if and only if the family $\mathcal{F}^{*}$ is a filter. For a family
$\mathcal{F}$ with the Ramsey property, let $\beta(\mathcal{F})=\\{p\in\beta
S:p\subseteq\mathcal{F}\\}$. Then the following from [10, Theorem 5.1.1].

######  Theorem 1.2.

Let $S$ be a discrete set. For every family $\mathcal{F\subseteq
P}\left(S\right)$ with the Ramsay property,
$\beta\left(\mathcal{F}\right)\subseteq\beta S$ is closed. Furthermore,
$\mathcal{F}=\cup\beta\left(\mathcal{F}\right)$. Also if $K\subseteq\beta S$
is closed, $\mathcal{F}_{K}=\left\\{E\subseteq S:\overline{E}\cap
K\neq\emptyset\right\\}$ is a family with the Ramsay property and
$\overline{K}=\beta\left(\mathcal{F}_{K}\right)$.

Let $S$ be a discrete semigroup, then for every family $\mathcal{F\subseteq
P}\left(S\right)$ with the Ramsay property,
$\beta\left(\mathcal{F}\right)\subseteq\beta S$ is closed. If
$\beta\left(\mathcal{F}\right)$ be a subsemigroup of $\beta S$, then
$E\left(\beta\mathcal{F}\right)\neq\emptyset$. But may not be subsemigroup.
For example, let $\mathcal{F}=\mathcal{IP}$, the family of IP- sets. It is
easy to show that
$\beta\left(\mathcal{F}\right)=\beta\left(\mathcal{IP}\right)=E\left(\beta
S\right)$. But $E\left(\beta S\right)$ is not a subsemigroup of $\beta S$.

######  Definition 1.3.

Let $\mathcal{F}$ be a family with Ramsay property such that
$\beta(\mathcal{F})$ is a subsemigroup of $\beta S$ and $p$ be an idempotent
in $\beta(\mathcal{F})$, then each member of $p$ is called essential
$\mathcal{F}$-set. And $A\subset S$ is called essential
$\mathcal{F}^{\star}$-set if $A$ intersects with all essential
$\mathcal{F}$-sets. That is a set is essential $\mathcal{F}^{\star}$ if and
only if $A\in p$, for every $p\in E\left(\beta(\mathcal{F})\right).$

The family $\mathcal{F}$ is called left (right) shift-invariant if for all
$s\in S$ and all $E\in\mathcal{F}$ one has
$sE\in\mathcal{F}(Es\in\mathcal{F})$. The family $\mathcal{F}$ is called left
(right) inverse shift-invariant if for all $s\in S$ and all $E\in\mathcal{F}$
one has $s^{-1}E\in\mathcal{F}(Es^{-1}\in\mathcal{F})$. From[10, Theorem
5.1.2] we have the following one:

######  Theorem 1.4.

If $\mathcal{F}$ is a family having the Ramsey property then
$\beta\mathcal{F}\subseteq\beta S$ is a left ideal if and only if
$\mathcal{F}$ is left shift-invariant. Similarly,
$\beta\mathcal{F}\subseteq\beta S$ is a right ideal if and only if
$\mathcal{F}$ is right shift-invariant.

From [10, Theorem 5.1.10], we can identify those families $\mathcal{F}$ with
Ramsey property for which $\beta\left(\mathcal{F}\right)$ is a subsemigroup of
$\beta S$ . The condition is a rather technical weakening of left shift-
invariance.

######  Theorem 1.5.

Let $S$ be any semigroup, and let $\mathcal{F}$ be a family of subsets of $S$
having the Ramsey property. Then the following are equivalent:

  1. 1.

$\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$.

  2. 2.

$\mathcal{F}$ has the following property: If $E\subseteq S$ is any set, and if
there is $A\in\mathcal{F}$ such that for all finite $H\subseteq A$ one has
$\left(\cap_{q\in H}x^{-1}E\right)\in\mathcal{F}$, then $E\in\mathcal{F}$.

Let us abbreviate the family of infinite sets as $\mathcal{IF}$, the family of
piecewise syndetic sets as $\mathcal{PS}$, the family of positive upper
asymptotic density as $\Delta$, the family sets containing arithmetic
progression of arbitrary length as $\mathcal{AP}$, the family of sets with the
property $\sum_{n\in A}\frac{1}{n}=\infty$ as $\mathcal{HSD}$ and the family
of $J$-sets as $\mathcal{J}$.

From the above definition together with the abbreviations, we get quasi
central set is an essential $\mathcal{PS}$-set, $D$-set is an essential
$\Delta$-set and $C$-set is an essential $\mathcal{J}$-set.

## 2 Elementary characterization of essential $\mathcal{F}$-sets

In [10, Theorem 5.2.3], the author has established dynamical characterization
of essential $\mathcal{F}$-sets. But the elementary characterization of
essential $\mathcal{F}$-sets are still unknown. Although elementary
characterization of quasi central-sets and $C$-sets are known from [10,
Theorem 3.7] and [11, Theorem 2.7] respectively. Since quasi central sets and
$C$ sets comes from the settings of essential $\mathcal{F}$-set and this fact
confines the fact that essential $\mathcal{F}$-sets might have elementary
characterization. In this section we will prove the supposition that
elementary characterization of essential $\mathcal{F}$-sets could be found
exactly the same way what the authors did in [11] for $C$-sets.

Let $\omega$ be the first infinite ordinal and each ordinal indicates the set
of all it’s predecessor. In particular, $0=\emptyset,$ for each
$n\in\mathbb{N},\>n=\left\\{0,1,...,n-1\right\\}$.

######  Definition 2.1.

Let us recall the following definitions from [11, Definition 2.5].

  1. 1.

If $f$ is a function and $dom\left(f\right)=n\in\omega$, then for all $x$,
$f^{\frown}x=f\cup\left\\{\left(n,x\right)\right\\}$.

  2. 2.

Let $T$ be a set functions whose domains are members of $\omega$. For each
$f\in T$, $B_{f}\left(T\right)=\left\\{x:f^{\frown}x\in T\right\\}.$

We recall the following lemma from [11, Lemma 2.6] which is key to our
characterization of essential $\mathcal{F}$-sets.

######  Lemma 2.2.

Let $p\in\beta S$. Then $p$ is an idempotent if and only if for each $A\in p$
there is a non-empty set $T$ of functions such that

  1. 1.

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

  2. 2.

For all $f\in T$, $B_{f}\left(T\right)\in p$.

  3. 3.

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

The following theorem is the characterization of essential $\mathcal{F}$\-
set.

######  Theorem 2.3.

Let $\left(S,\cdot\right)$ be a semigroup, and assume that $\mathcal{F}$ is a
family of subsets of $S$ with the Ramsay property such that
$\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$. Let
$A\subseteq S$. Statements $(1)$, $(2)$ and $(3)$ are equivalent and are
implied by statement $(4)$. If $S$ is countable, then all the five statements
are equivalent.

  1. 1.

$A$ is an essential $\mathcal{F}$-set.

  2. 2.

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

    1. (a)

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

    2. (b)

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

    3. (c)

For all $F\in\mathcal{P}_{f}\left(T\right)$, $\cap_{f\in F}B_{f}(T)$ is a
$\mathcal{F}$-set.

  3. 3.

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

    1. (a)

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

    2. (b)

For each
$\mathcal{F}\in\mathcal{P}_{f}\left(I\right),\,\bigcap_{F\in\mathcal{F}}C_{F}$
is a $\mathcal{F}$-set.

  4. 4.

There is a decreasing sequence $\left\langle
C_{n}\right\rangle_{n=1}^{\infty}$ of subsets of $A$ such that

    1. (a)

For each $n\in\mathbb{N}$ and each $x\in C_{n}$, there exists $m\in\mathbb{N}$
with $C_{m}\subseteq x^{-1}C_{n}$.

    2. (b)

For each $n\in\mathbb{N}$, $C_{n}$ is a $\mathcal{F}$-set.

###### Proof.

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

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

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

It is trivial that $(4)\Rightarrow(3)$. Assume now that $S$ is countable. We
shall show that $(2)\Rightarrow(4)$. So let $T$ be as guaranteed by $(2)$.
Then $T$ is countable so enumerate $T$ as
$\left\\{f_{n}:n\in\mathbb{N}\right\\}$. For $n\in\mathbb{N}$, let
$C_{n}=\bigcap_{k=1}^{n}B_{f_{k}}$. Then each $C_{n}$ is a $\mathcal{F}$-set.
Let $n\in\mathbb{N}$ and let $x\in C_{n}$. Pick $m\in\mathbb{N}$ such that

$\left\\{f_{k}^{\frown}x:k\in\left\\{1,2,\ldots,n\right\\}\right\\}\subseteq\left\\{f_{1},f_{2},\ldots,f_{m}\right\\}.$

Then $C_{m}\subseteq x^{-1}C_{n}$. ∎

Now, we will conclude this section with the following fascinating results:

######  Corolarry 2.4.

Let $\left(S,\cdot\right)$ be a countable discrete semigroup and let
$\mathcal{F}$ be a left inverse shift invariant family with Ramsay property
such that $\beta\left(\mathcal{F}\right)$ is subsemigroup of $\beta S$. If
there exists a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $S$ such that
$FP\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)$ is $\mathcal{F}$-set,
then $FP\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)$ is essential
$\mathcal{F}$-set.

###### Proof.

Now choose arbitrarily $m\in\mathbb{N}$. Then

$FP\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)=FP\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\cup FP\left(\langle
x_{n}\rangle_{n=1}^{m-1}\right)$
$\qquad\qquad\qquad\qquad\qquad\;\;\;\cup\left\\{tFP\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right):t\in FP\left(\langle
x_{n}\rangle_{n=1}^{m-1}\right)\right\\}$

As $\mathcal{F}$ is a Ramsay family, $FP\left(\langle
x_{n}\rangle_{n=1}^{\infty}\right)$ is $\mathcal{F}$-set, and $FP\left(\langle
x_{n}\rangle_{n=1}^{m-1}\right)$ is a finite set, we have either
$FP\left(\langle x_{n}\rangle_{n=m}^{\infty}\right)\in\mathcal{F}$ or
$tFP\left(\langle x_{n}\rangle_{n=m}^{\infty}\right)\in\mathcal{F}$ for some
$t\in FP\left(\langle x_{n}\rangle_{n=1}^{m-1}\right)$. Now $\mathcal{F}$
being left inverse shift invariant, in either cases $FP\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)$ is $\mathcal{F}$\- set. Let us consider
the sequence

$FP\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)\supseteq FP\left(\langle
x_{n}\rangle_{n=2}^{\infty}\right)\supseteq\cdots\supseteq FP\left(\langle
x_{n}\rangle_{n=k}^{\infty}\right)\supseteq\cdots.$

Let $C_{i}=FP\left(\langle x_{n}\rangle_{n=i+1}^{\infty}\right)$ for all
$i\in\mathbb{N}$. Then for any $n\in\mathbb{N}$ , any $x\in C_{n}$ , we have
$l\in\mathbb{N}$ such that $C_{l}\subseteq x^{-1}C_{n}$. And this concludes
that $FP\left(\left\\{x_{n}\right\\}_{n=1}^{\infty}\right)$ is an essential
$\mathcal{F}$-set from the previous theorem. ∎

######  Corolarry 2.5.

If $S$ is a group and $G$ be a subgroup of $S$ which is a $\mathcal{F}$\- set,
then it is an essential $\mathcal{F}$\- set.

###### Proof.

As

$G\supseteq G\supseteq\cdots\supseteq G\supseteq\cdots$

is the necessary chain of condition 3 of theorem 2.3, it follows. ∎

## 3 Combined additive and multiplicative structure

Given a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$, we say
that $\langle y_{n}\rangle_{n=1}^{\infty}$ is a sum subsystem of $\langle
x_{n}\rangle_{n=1}^{\infty}$ provided there exists a sequence $\langle
H_{n}\rangle_{n=1}^{\infty}$ of non-empty finite subset such that
$\text{max}H_{n}<\text{min}H_{n+1}$ and $y_{n}=\sum_{t\in H_{n}}x_{t}$ for
each $n\in\mathbb{N}$. In [7] N. Hindman and V. Bergelson proved the following
theorem.

######  Theorem 3.1.

Let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a sequence in $\mathbb{N}$ and
$A$ be IP⋆-set in $\left(\mathbb{N},+\right)$. Then there exists a subsystem
$\langle y_{n}\rangle_{n=1}^{\infty}$ of $\langle x_{n}\rangle_{n=1}^{\infty}$
such that $FS\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\cup
FP\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\subseteq A$.

In [6, Theorem 2.4], it was proved that central⋆ sets also possess some
IP⋆-set-like properties for some specified sequences called minimal
sequence[6, Definition 2.4]:

######  Definition 3.2.

A sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ is minimal
sequence if

$\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)\cap
K\left(\beta\mathbb{N}\right)\neq\emptyset.$

It is known that $\langle 2^{n}\rangle_{n=1}^{\infty}$ is a minimal sequence
while the sequence $\langle 2^{2n}\rangle_{n=1}^{\infty}$ is not a minimal
sequence. And in[6, Theorem 2.4], it was proved the following substantial
multiplicative result of central⋆ sets.

######  Theorem 3.3.

Let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a minimal sequence in
$\mathbb{N}$ and $A$ be central⋆ set in $\left(\mathbb{N},+\right)$. Then
there exists a subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ of $\langle
x_{n}\rangle_{n=1}^{\infty}$ such that $FS\left(\langle
y_{n}\rangle_{n=1}^{\infty}\right)\cup FP\left(\langle
y_{n}\rangle_{n=1}^{\infty}\right)\subseteq A$.

In [5, Theorem 2.10], it was established an analogue version of the above
theorem in case of $C^{\star}$ sets for some specific type of sequences called
almost minimal sequence [5, Definition 2.3]:

######  Definition 3.4.

A sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ is almost
minimal sequence if

$\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)\cap
J\left(\mathbb{N}\right)\neq\emptyset$

.

In [5, Theorem 2.7], it has been characterized the almost minimal sequences by
the following theorem.

######  Theorem 3.5.

In $(\mathbb{N},+)$ the following conditions are equivalent:

  1. 1.

$\langle x_{n}\rangle_{n=1}^{\infty}$ is almost minimal sequence.

  2. 2.

$FS(\langle x_{n}\rangle_{n=1}^{\infty})$ is a $J$-set.

  3. 3.

There is an idempotent in $\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)\cap J\left(\mathbb{N}\right)$.

Now we are in position to state the main theorem of [5, Theorem 2.10]:

######  Theorem 3.6.

Let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a minimal sequence in
$\mathbb{N}$ and $A$ be $C^{\star}$ set in $\left(\mathbb{N},+\right)$. Then
there exists a subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ of $\langle
x_{n}\rangle_{n=1}^{\infty}$ such that

$FS\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\cup FP\left(\langle
y_{n}\rangle_{n=1}^{\infty}\right)\subseteq A$

.

As we know that $C$-sets are essential $\mathcal{J}$-sets, the above theorem
motives us to think some analogue result for essential $\mathcal{F}$-sets.
First let us define , $\mathcal{F}$-minimal sequence.

######  Definition 3.7.

A sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ is
$\mathcal{F}$-minimal sequence if

$\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)\cap\beta\left(F\right)\neq\emptyset.$

We can characterize $\mathcal{F}$\- minimal sequences as like as almost
minimal sequence given below and can be proved in the same way as the author
did in [5, Theorem 2.7] for almost minimal sequences:

######  Theorem 3.8.

For an inverse shift invariant family $\mathcal{F}$ in $(\mathbb{N},+)$ with
Ramsay property such that $\beta\left(\mathcal{F}\right)$ is a subsemigroup of
$\beta S$, the following conditions are equivalent:

  1. 1.

$\langle x_{n}\rangle_{n=1}^{\infty}$ is almost $\mathcal{F}$\- minimal
sequence.

  2. 2.

$FS(\langle x_{n}\rangle_{n=1}^{\infty})\in q$, for some
$q\in\beta(\mathcal{F})$.

  3. 3.

There is an idempotent in $\cap_{m=1}^{\infty}cl(FS(\langle
x_{n}\rangle_{n=m}^{\infty}))\cap\beta(\mathcal{F})$.

###### Proof.

$(1)\implies(2)$ follows from definition.

$(2)\implies(3)$ Since $FS\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)\in
q\in\beta(\mathcal{F})$ we get $cl\left(FS\left(\langle
x_{n}\rangle_{n=1}^{\infty}\right)\right)\cap\beta\left(\mathcal{F}\right)\neq\emptyset$.
From [12, Lemma 5.11], choose $\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)$. It will easy to see that
$\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=1}^{\infty}\right)\right)$ is a closed subsemigroup of
$\beta\mathbb{N}$ and as well as $\beta(\mathcal{F})$ is also closed
subsemigroup $\beta\mathbb{N}$. Hence
$\,\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)\cap\beta\left(\mathcal{F}\right)$ is
a compact subsemigroup of $\left(\beta\mathbb{N},+\right)$. So it will be
sufficient to check that $\cap_{m=1}^{\infty}cl\left(FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)\cap\beta\left(\mathcal{F}\right)\neq\emptyset$.

Now choose arbitrarily $m\in\mathbb{N}$ and then $FS\left(\langle
x_{n}\rangle_{n=1}^{\infty}\right)=FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\cup FS\left(\langle
x_{n}\rangle_{n=1}^{m-1}\right)\cup\left\\{t+FS\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right):t\in FS\left(\langle
x_{n}\rangle_{n=1}^{m-1}\right)\right\\}$ and so we have one of the
followings:

  1. 1.

$FS\left(\langle x_{n}\rangle_{n=m}^{\infty}\right)\in p$

  2. 2.

$FS\left(\langle x_{n}\rangle_{n=1}^{m-1}\right)\in p$

  3. 3.

$t+FS\left(\langle x_{n}\rangle_{n=m}^{\infty}\right)\in p$ for some $t\in
FS\left(\langle x_{n}\rangle_{n=1}^{m-1}\right)$.

Now $(2)$ is not possible as in that case $p$ will be a member of principle
ultrafilter. If $(1)$ holds then we have done. Now if we assume $(3)$ holds
then for some $t\in FS\left(\langle x_{n}\rangle_{n=1}^{m-1}\right)$, we have
$t+FS\left(\langle x_{n}\rangle_{n=m}^{\infty}\right)\in p$. Choose $q\in
cl\left(FS\left(\langle x_{n}\rangle_{n=m}^{\infty}\right)\right)$ so that
$t+q=p$. Now for every $F\in q$,
$t\in\left\\{n\in\mathbb{N}:-n+\left(t+F\right)\in q\right\\}$ so that $t+F\in
p$. Since $\mathcal{F}$-sets are inverse shift invariant invariant, $F$ is a
$\mathcal{F}$-sets. We have $q\in\beta\left(\mathcal{F}\right)\cap
cl\left(FS\left(\langle x_{n}\rangle_{n=m}^{\infty}\right)\right)$.

$(3)\implies(1)$ follows from definition of $\mathcal{F}$\- minimal sequence
and condition $(3)$. ∎

To prove the main theorem, we need the following two lemmas are essential.

######  Lemma 3.9.

Let $\mathcal{F}$ be a dilation invariant family (i.e. the family is invariant
under taking product by any element of $\mathbb{N}$) with Ramsay property such
that $\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$. If $A$ be
an essential $\mathcal{F}$-set in $(\mathbb{N},+)$ then $nA$ is also an
essential $\mathcal{F}$-set in $(\mathbb{N},+)$ for any $n\in\mathbb{N}$.

###### Proof.

If $A$ be an essential $\mathcal{F}$-set, then by elementary characterization
of essential $\mathcal{F}$-set, we get a sequence of $\mathcal{F}$-sets
$\left\langle C_{k}\right\rangle_{k=1}^{\infty}$ with

$A\supseteq C_{1}\supseteq C_{2}\supseteq\cdots$

satisfying property 4 of theorem 2.3. Now consider the sequence $\left\langle
nC_{k}\right\rangle_{k=1}^{\infty}$ of $\mathcal{F}$-sets which satisfies

$nA\supseteq nC_{1}\supseteq nC_{2}\supseteq\cdots$

and for each $k\in\mathbb{N}$ and each $t\in nC_{k}$, there exists
$p\in\mathbb{N}$ with $nC_{p}\subseteq-t+nC_{k}$. This proves that $nA$ is an
essential $\mathcal{F}$-set in $(\mathbb{N},+)$ for any $n\in\mathbb{N}$. ∎

We get another lemma given below.

######  Lemma 3.10.

Let $\mathcal{F}$ be a dilation invariant family with Ramsay property such
that $\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$. If $A$ be
an essential $\mathcal{F}^{\star}$-set in $(\mathbb{N},+)$ then $n^{-1}A$ is
also a essential $\mathcal{F^{\star}}$-set in $(\mathbb{N},+)$ for any
$n\in\mathbb{N}$.

###### Proof.

It is sufficient to show that for any essential $\mathcal{F}$-set $B$, $B\cap
n^{-1}A\neq\emptyset$. Since $B$ is essential $\mathcal{F}$-set , $nB$ is
essential $\mathcal{F}$-set and $A\cap nB\neq\emptyset$. Choose $m\in A\cap
nB$ and $k\in B$ such that $m=nk$. Therefore $k=m/n\in n^{-1}A$ so $B\cap
n^{-1}A\neq\emptyset$. ∎

Now we will show that all $\mathcal{F}^{\star}$-set have a substantial
multiplicative property.

######  Theorem 3.11.

Let $\mathcal{F}$ be an inverse shift invariant and dilation invariant family
with Ramsay property such that $\beta\left(\mathcal{F}\right)$ is a
subsemigroup of $\beta S$. Let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a
$\mathcal{F}$-minimal sequence and $A$ be a an essential $\mathcal{F}$-set in
$(\mathbb{N},+)$. Then there exists a sum subsystem $\langle
y_{n}\rangle_{n=1}^{\infty}$ of $\langle x_{n}\rangle_{n=1}^{\infty}$ such
that $FS(\langle y_{n}\rangle_{n=1}^{\infty})\cup FP(\langle
y_{n}\rangle_{n=1}^{\infty})\subseteq A$.

###### Proof.

Since $\langle x_{n}\rangle_{n=1}^{\infty}$ is a $\mathcal{F}$-minimal
sequence in $\mathbb{N}$, we can find some essential idempotent $p\in
E\left(\beta\left(\mathcal{\mathcal{F}}\right)\right)$ for which $FS(\langle
x_{n}\rangle_{n=m}^{\infty})\in p$ for each $m\in\mathbb{N}$. Since A be an
essential $\mathcal{F}^{\star}$-set for every $n\in\mathbb{N}$, $n^{-1}A\in
p$. Let $A^{\star}=\\{n\in A:-n+A\in p\\}$, then $A^{\star}\in p$. We can
choose $y_{1}\in A^{\star}\cap FS(\langle x_{n}\rangle_{n=1}^{\infty})$.
Inductively, let $m\in\mathbb{N}$ and $\langle y_{i}\rangle_{i=1}^{m}$,
$\langle H_{i}\rangle_{i=1}^{m}$ in $\mathcal{P}_{f}(\mathbb{N})$ be chosen
with the following property:

  1. 1.

$i\in\\{1,2,\ldots\,,m-1\\},\,\text{max}H_{i}<\text{min}H_{i+1}.$

  2. 2.

$\text{If}\,y_{i}={\displaystyle\sum_{t\in
H_{i}}x_{t},\,\text{then}\,\sum_{t\in H_{i}}x_{t}\in
A^{\star},\,\text{and}\,FP(\langle y_{i}\rangle_{i=1}^{m})\subseteq
A^{\star}}.$

We observe that $\\{\,\sum_{t\in
H}x_{t}:\,H\in\mathcal{P}_{f}(\mathbb{N}),\,\text{min}H>\text{max}H_{m}\\}\in
p$. Let us set $B=\\{\,\sum_{t\in
H}x_{t}:\,H\in\mathcal{P}_{f}(\mathbb{N}),\,\text{min}H>\text{max}H_{m}\\}$,
$E_{1}=FS(\langle y_{t}\rangle_{n=1}^{m})$ and $E_{2}=FP(\langle
y_{t}\rangle_{n=1}^{m})$. Now consider $D=B\cap A^{\star}\cap\bigcap_{s\in
E_{1}}\left(-s+A^{\star}\right)\cap\bigcap_{s\in
E_{2}}\left(s^{-1}A^{\star}\right)$. Then $D\in p$. Choose $y_{m+1}\in D$ and
$H_{m+1}\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ such that
$\text{min}H_{m+1}>\text{max}H_{m}$. Putting $y_{m+1}=\sum_{t\in
H_{m+1}}x_{t}$, it shows that the induction can be continued and proves the
theorem. ∎

## 4 Essential $\mathcal{F}^{\star}$-sets in weak rings

In this section we extend the previous theorem to a much wider class, called
“weak rings” and start with the following definition:

######  Definition 4.1.

[7, Definition 16.33, Page 419]

  1. 1.

A left weak ring is a triple $\left(S,+,\cdot\right)$ such that
$\left(S,+\right)$ and $\left(S,.\right)$ are semigroups and the left
distributive law holds. That is, for all $x,y,z\in S$ one
has$x\cdot\left(y+z\right)=x\cdot y+x\cdot z$.

  2. 2.

A right weak ring is a triple $\left(S,+,\cdot\right)$ such that
$\left(S,+\right)$ and $\left(S,.\right)$ are semigroups and the right
distributive law holds. That is, for all $x,y,z\in S$ one
has$\left(x+y\right)\cdot z=x\cdot z+y\cdot z$.

  3. 3.

A weak ring is a triple $\left(S,+,\cdot\right)$ which is both a left weak
ring and a right weak ring.

Dilation invariance of a family is defined as:

######  Definition 4.2.

This is similar as for dilation invariance of $\mathbb{N}$.

  1. 1.

Let $\left(S,+,\cdot\right)$ be a left weak ring. A family $\mathcal{F}$ is
called left dilation invariant if for any $s\in S$ and $A\in\mathcal{F}$,
$sA\in\mathcal{F}$.

  2. 2.

Let $\left(S,+,\cdot\right)$ be a right weak ring. A family $\mathcal{F}$ is
called right dilation invariant if for any $s\in S$ and $A\in\mathcal{F}$,
$As\in\mathcal{F}$.

  3. 3.

Let $\left(S,+,\cdot\right)$ be a weak ring. A family $\mathcal{F}$ is called
dilation invariant if it is both left and right dilation invariant.

Recall that in $FP\left(\langle x_{n}\rangle_{n=1}^{\infty}\right)$ the
products are taken in increasing order of indices and the following definition
is taken from [12, Definition 16.36]:

######  Definition 4.3.

Let $\left(S,\cdot\right)$ be a semigroup, let $\langle
x_{n}\rangle_{n=1}^{\infty}$ be a sequence in $S$, and let $k\in\mathbb{N}$.
Then $AP\left(\langle x_{n}\rangle_{n=1}^{k}\right)$ is the set of all
products of terms of $\langle x_{n}\rangle_{n=1}^{k}$in any order with no
repetitions. Similarly $AP\langle x_{n}\rangle_{n=1}^{\infty}$is the set of
all products of terms of $\langle x_{n}\rangle_{n=1}^{\infty}$ in any order
with no repetitions.

For example, for $k=3$, we obtain the following:

$AP\left(\langle
x_{n}\rangle_{n=1}^{3}\right)=\left\\{x_{1},x_{2},x_{3},x_{1}x_{2},x_{1}x_{2},x_{1}x_{3},x_{2}x_{3},x_{2}x_{1},x_{3}x_{2},\right.$
$\left.\qquad\qquad\qquad\qquad
x_{1}x_{2}x_{3},x_{1}x_{3}x_{2},x_{2}x_{1}x_{3},x_{2}x_{1}x_{3},x_{2}x_{3}x_{1},x_{3}x_{1}x_{2},x_{3}x_{2}x_{1}\right\\}$

From [12, Theorem 16.38], we get the following theorem for IP⋆-sets which is
our main aim in this section to prove analogous result for essential
$\mathcal{F}^{\star}$-sets.

######  Theorem 4.4.

Let $\left(S,+,\cdot\right)$ be a weak ring, let $A$ be an IP⋆ set in
$\left(S,+\right)$, and let $\langle x_{n}\rangle_{n=1}^{\infty}$ be any
sequence in $S$. Then there exists a sum subsystem $\langle
y_{n}\rangle_{n=1}^{\infty}$ of $\langle x_{n}\rangle_{n=1}^{\infty}$ in S
such that $FS(\langle y_{n}\rangle_{n=1}^{\infty})\cup FP(\langle
y_{n}\rangle_{n=1}^{\infty})\subseteq A$.

The above theorem is true for any sequence in $S$. But we show that the above
result is true for essential $\mathcal{F}^{\star}$-set for
$\mathcal{F}$-minimal sequences.

######  Lemma 4.5.

Let $S$ be a set, let $A\subseteq S$.

  1. 1.

If $\left(S,+,\cdot\right)$ is a left weak ring and $\mathcal{F}$ is left
dilation invariant family with Ramsay property and $A$ is an essential
$\mathcal{F}^{\star}$-set in $\left(S,+\right)$, then $sA$ is an essential
$\mathcal{F^{\star}}$-set in $(S,+)$.

  2. 2.

If $\left(S,+,\cdot\right)$ is a right weak ring and $\mathcal{F}$ is right
dilation invariant family with Ramsay property and $A$ is an essential
$\mathcal{F}^{\star}$-set in $\left(S,+\right)$, then $As$ is an essential
$\mathcal{F^{\star}}$-set in $(S,+)$.

  3. 3.

If $\left(S,+,\cdot\right)$ is a weak ring and $\mathcal{F}$ is dilation
invariant family with Ramsay property and $A$ is an essential
$\mathcal{F}^{\star}$-set in $\left(S,+\right)$, then $sAt$ is an essential
$\mathcal{F^{\star}}$-set in $(S,+)$.

###### Proof.

It suffices to establish $(1)$ since then $(2)$ follows from a left-right
switch and $(3)$ follows from $(1)$ and $(2)$. If $A$ be an essential
$\mathcal{F}$-set, then by elementary characterization of essential
$\mathcal{F}$-set, we get a downward directed family of $\mathcal{F}$-sets
$\left\langle C_{F}\right\rangle_{F\in I}$ such that for each $F\in I$ and
each $u\in C_{F}$, there exists $G\in I$ with $C_{G}\subseteq-u+C_{F}$. Now
consider the downward directed family $\left\langle sC_{F}\right\rangle_{F}\in
I$ of $\mathcal{F}$-sets. Now for each $F\in I$ and each $su\in sC_{F}$, there
exists $G\in I$ with $sC_{G}\subseteq-su+sC_{F}$(using the left distributive
law of left weak ring). This proves that $sA$ is an essential
$\mathcal{F}$-set in $(S,+)$ for any $s\in S$. ∎

Now we are in position two prove the following two lemmas:

######  Lemma 4.6.

Let $S$ be a set, let $A\subseteq S$.

  1. 1.

If $\left(S,+,\cdot\right)$ is a left weak ring and $\mathcal{F}$ is left
dilation invariant family with Ramsay property such that
$\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$ and $A$ is an
essential $\mathcal{F}^{\star}$-set in $\left(S,+\right)$, then $s^{-1}A$ is
an essential $\mathcal{F^{\star}}$-set in $(S,+)$.

  2. 2.

If $\left(S,+,\cdot\right)$ is a right weak ring and $\mathcal{F}$ is right
dilation invariant family with Ramsay property such that
$\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$ and $A$ is an
essential $\mathcal{F}^{\star}$-set in $\left(S,+\right)$, then $As^{-1}$ is
an essential $\mathcal{F^{\star}}$-set in $(S,+)$.

  3. 3.

If $\left(S,+,\cdot\right)$ is a weak ring and $\mathcal{F}$ is dilation
invariant family with Ramsay property such that
$\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$ and $A$ is an
essential $\mathcal{F}^{\star}$-set in $\left(S,+\right)$, then
$s^{-1}At^{-1}$ is an essential $\mathcal{F^{\star}}$-set in $(S,+)$.

###### Proof.

It suffices to establish $(1)$ since then $(1)$ follows from a left-right
switch and $(3)$ follows from $(1)$ and $(2)$. It is sufficient to show that
for any essential $\mathcal{F}$-set $B$, $B\cap s^{-1}A\neq\emptyset$. Since
$B$ is essential $\mathcal{F}$-set , $sB$ is essential $\mathcal{F}$-set and
$A\cap sB\neq\emptyset$. Choose $u\in A\cap sB$ and $v\in B$ such that $u=sv$.
Therefore $v\in s^{-1}A$ so $B\cap s^{-1}A\neq\emptyset$. ∎

######  Definition 4.7.

A sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $(S,\cdot)$ is
$\mathcal{F}$-minimal sequence if

$\cap_{m=1}^{\infty}cl\left(FP\left(\langle
x_{n}\rangle_{n=m}^{\infty}\right)\right)\cap\beta\left(F\right)\neq\emptyset.$

The following is the characterization of $\mathcal{F}$\- minimal sequences for
arbitrary semigroup.

######  Theorem 4.8.

For a left inverse shift invariant family $\mathcal{F}$ in a semigroup
$(S,\cdot)$ with Ramsay property such that $\beta\left(\mathcal{F}\right)$ is
a subsemigroup of $\beta S$, the following conditions are equivalent:

  1. 1.

$\langle x_{n}\rangle_{n=1}^{\infty}$ is almost $\mathcal{F}$\- minimal
sequence.

  2. 2.

$FP(\langle x_{n}\rangle_{n=1}^{\infty})\in q$, for some
$q\in\beta(\mathcal{F})$.

  3. 3.

There is an idempotent in $\cap_{m=1}^{\infty}cl(FP(\langle
x_{n}\rangle_{n=m}^{\infty}))\cap\beta(\mathcal{F})$.

###### Proof.

The proof is same as the proof of Theorem 3.8, and so we omit the proof. ∎

We now show that all $\mathcal{F}^{\star}$-set have a substantial
multiplicative property.

######  Theorem 4.9.

Let be a $\left(S,+,\cdot\right)$ be a weak ring. Let $\mathcal{F}$ be a left
inverse shift invariant and dilation invariant family with Ramsay property
such that $\beta\left(\mathcal{F}\right)$ is a subsemigroup of $\beta S$. Let
$\langle x_{n}\rangle_{n=1}^{\infty}$ be a $\mathcal{F}$-minimal sequence and
$A$ be an essential $\mathcal{F^{\star}}$-set in $(S,+)$. Then there exists a
sum subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ of $\langle
x_{n}\rangle_{n=1}^{\infty}$ such that $FS(\langle
y_{n}\rangle_{n=1}^{\infty})\cup AP(\langle
y_{n}\rangle_{n=1}^{\infty})\subseteq A$.

###### Proof.

Since $\langle x_{n}\rangle_{n=1}^{\infty}$ is a $\mathcal{F}$-minimal
sequence in $(S,+)$, we can find some essential idempotent $p\in
E\left(\beta\left(\mathcal{\mathcal{F}}\right)\right)$ for which $FS(\langle
x_{n}\rangle_{n=m}^{\infty})\in p$ for each $m\in p$. Since $A$ is an
essential $\mathcal{F}^{\star}$-set for every $s,t\in S$,
$s^{-1}A,t^{-1}A,s^{-1}At^{-1}\in p$. Let $A^{\star}=\\{s\in A:-s+A\in p\\}$,
then $A^{\star}\in p$. We can choose $y_{1}\in A^{\star}\cap FS(\langle
x_{n}\rangle_{n=1}^{\infty})$. Inductively, let $m\in\mathbb{N}$ and $\langle
y_{i}\rangle_{i=1}^{m}$, $\langle H_{i}\rangle_{i=1}^{m}$ in
$\mathcal{P}_{f}(\mathbb{N})$ be chosen with the following properties:

  1. 1.

For $i\in\\{1,2,\ldots\,,m-1\\},\,\text{max}H_{i}<\text{min}H_{i+1}$

  2. 2.

$\text{If}\,y_{i}={\displaystyle\sum_{t\in
H_{i}}x_{t},\,\text{then}\,\sum_{t\in H_{i}}x_{t}\in
A^{\star},\,\text{and}\,AP(\langle y_{i}\rangle_{i=1}^{m})\subseteq
A^{\star}}.$

We observe that $\\{\,\sum_{t\in
H}x_{t}:\,H\in\mathcal{P}_{f}(\mathbb{N}),\,\text{min}H>\text{max}H_{m}\\}\in
p$. Let us set $B=\\{\,\sum_{t\in
H}x_{t}:\,H\in\mathcal{P}_{f}(\mathbb{N}),\,\text{min}H>\text{max}H_{m}\\}$,
$E_{1}=FS(\langle y_{t}\rangle_{n=1}^{m})$, and $E_{2}=AP(\langle
y_{t}\rangle_{n=1}^{m})$. Now consider

$D=B\cap A^{\star}\cap\bigcap_{s\in
E_{1}}\left(-s+A^{\star}\right)\cap\bigcap_{s\in
E_{2}}\left(s^{-1}A^{\star}\right)\cap\bigcap_{s\in
E_{2}}\left(A^{\star}s^{-1}\right)\cap\bigcap_{s,t\in
E_{2}}\left(s^{-1}A^{\star}t^{-1}\right).$

Then $D\in p$. Choose $y_{m+1}\in D$, and
$H_{m+1}\in\mathcal{P}_{f}\left(\mathbb{N}\right)$, such that
$\text{min}H_{m+1}>\text{max}H_{m}$. Putting $y_{m+1}=\sum_{t\in
H_{m+1}}x_{t}$, it shows that the induction can be continued and proves the
theorem. ∎

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# DialPort: Connecting the Spoken Dialog Research Community to Real User Data

Tiancheng Zhao 1 Kyusong Lee∗1,2 Maxine Eskenazi1

1Language Technologies Institute, Carnegie Mellon University
2Pohang University of Science and Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS>Both authors equally
contributed to this work

###### Abstract

This paper describes a new spoken dialog portal that connects systems produced
by the spoken dialog academic research community and gives them access to real
users. We introduce a distributed, multi-modal, multi-agent prototype dialog
framework that affords easy integration with various remote resources, ranging
from end-to-end dialog systems to external knowledge APIs. To date, the
DialPort portal has successfully connected to the multi-domain spoken dialog
system at Cambridge University, the NOAA (National Oceanic and Atmospheric
Administration) weather API and the Yelp API.

##  1 Introduction

The advent of Siri, Cortana and other agents has generated interest in spoken
dialog research. These applications have sparked the imagination of many and
led them to believe that speaking to intelligent agents is useful. The
research community needs to profit from this interest by creating a service
for the general public that can gather real user data that can be used to make
dialog systems more robust. It can also be used to carry out experiments on
comparative studies. Industry already has access to large data sets and
sometimes to pools of real users that are viewed as strategic competitive
resources and so not shared with the research community. Yet much fundamental
research remains to be done, such as signal processing in noisy conditions,
recognition of groups of difficult users (like the elderly and non-natives),
management of complex dialogs (such as multi party meetings, negotiations, and
multimodal interaction), and the automatic use of meta linguistic information
such as prosody. The academic community needs to unite through one common
portal to be able to gather enough real user data to make significant impact.
It is extremely difficult for any one group to devote time to collecting a
significant amount of real user data. The users must be found and kept
interested and the interface must be created and maintained. One data
gathering portal that all dialog systems can be connected to gives potential
users a variety of interesting applications, much in the way that virtual
assistants do not only provide information about scheduling. The DialPort
portal was created for this purpose.
The Dialog Research Center at Carnegie Mellon (DialRC) has given the community
real user data from the Lets Go System [Raux et al., 2005] as well as access
to the system to run studies. But research is carried out in other areas,
beyond simple form filling. Just as one research group cannot attract a
diverse pool of regular users, one group cannot cover all of the possible
applications, such as virtual humans and robots with multimodal communication.
Thus the goal of DialPort is to attract and maintain a pool of real users to a
group of spoken dialog applications.
The first year goal is to create the portal and link it to other systems. Once
the working portal can give a variety of useful information, a service such as
Prefinery111https://www.prefinery.com/ will be used to attract the real users.
These services solicit potential users, giving bonuses for signup and usage as
well as for getting friends to sign up. In this paper, we present the DialPort
portal that will link many different research applications and will provide
real user data. Section 2 discusses related work; Section 3 describes the
overall architecture; Section 4, 5 and 6 discuss the core modules; Section 7
explains the integration protocol; Section 8 reviews the current progress and
Section 9 concludes.

##  2 Related Work

Several SDS-building industrial platforms have recently become available to
non-expert developers. Microsoft Research released Language Understanding
Intelligent Service (LUIS) [Williams et al., 2015] which helps software
developers create cloud-based, machine-learning powered, language
understanding models for specific application domains. Facebook is building an
AI platform that helps developers create chat bots that can converse in
natural language222https://wit.ai. The HALEF (Help Assistant–Language-Enabled
and Free) framework from ETS leverages different open-source components to
form an SDS framework that is modular and industry-standard-compliant
[Suendermann-Oeft et al., 2015]. Although these platforms have helped
researchers build robust SDSs more efficiently, the data that they collected
has not been shared with the academic research community.
Most early SDS work focused on single-domain SDSs, such as bus schedules,
restaurant, or museum information, etc. A single-domain dialog system has
limited semantic coverage. Thus, multi-domain dialog systems have appeared,
enabling one system to handle several domains. Past approaches usually
followed a two-stage framework [Komatani et al., 2009, Nakano et al., 2011],
in which the first stage classifies the domain and the second stage forwards
the user’s request to the relevant single-domain dialog manager. This method
has shown promising results for scaling up dialog systems to handle multiple
domains. DialPort differs from previous frameworks in this area by proposing
the concept of a multi-agent dialog system. This system combines both goal-
driven and non-goal-driven dialog agents that have been independently
developed by different research teams. The task of DialPort is to judiciously
assign the user’s utterance to the most relevant dialog agent and to carry out
complex nested conversations with real users. The long term goal is to enable
any research group to connect their SDS to DialPort using a lightweight
integration protocol. DialPort makes it easy for real users to access many
state-of-the-art dialog system services all in one place through a universal
web-based entry point.

##  3 Architecture

Figure 1 presents the system architecture, comprised of three sections: User
Interface, DialPort and Remote Agents.

###  3.1 User Interface

The user Interface is the publicly available front end for real
users333https://skylar.speech.cs.cmu.edu. It is in charge of both the visual
and audio agents representing each dialog system. The visual representation
uses WebGL Unity 3D, which will be discussed in Section 4. The audio side uses
the Google Chrome Speech ASR API to transform the user’s speech into text and
the Google Chrome TTS API to convert DialPort’s text output into English
speech.

Figure 1: The overall architecture of DialPort.

###  3.2 DialPort

DialPort is scalable and distributed. Its central message broker is ActiveMQ,
a well-known open source message broker. ActiveMQ allows us to easily connect
multiple components in order to create a larger system. Building on ActiveMQ,
DialPort has four main modules: the HTTP API Server, the Natural Language
Understanding (NLU), the ReinForest Dialog Manager (DM) and the Natural
Language Generation (NLG). With the exception of the ReinForest DM, the
modules are RESTful (Representational state transfer) web services: they do
not consider any state information when handling requests. All contextual
information about a dialog is maintained by the ReinForest DM. The HTTP API
Server is the front gate of DialPort. It converts the incoming HTTP messages
into proper ActiveMQ messages and sends them to the NLU. The NLU outputs a
semantic frame that contains the original utterance along with: entities, an
intent and a domain. Assuming the user utterance is: ”Recommend a restaurant
in Pittsburgh”, an example output from the NLU is given in Table 1. Given the
user input annotated by the NLU, ReinForest updates its internal dialog state
and generates the next system response. The response is in the format of a
list of dialog acts and content value tuples,
$a_{sys}=[(DA_{0},v_{0}),...(DA_{k},v_{k})]$. The NLG is responsible for
transforming $a_{sys}$ to the natural language surface form. Given the
utterance in the previous example, an example ReinForest response is:
$a_{sys}$ = [(CONFIRM, value=Pittsburgh), (ASK, value=food_type)]. And the
final system output is ”I believe you said Pittsburgh. What kind of food do
you want?”

Domain |  {Restaurant: 0.95; Hotel: 0.05}
---|---
Intent |  {Request: 0.9; Inform: 0.1}
Entities |  {Type: Location; Value: Pittsburgh}
Table 1: Example Semantic Frame Output from the NLU. Figure 2: Non-verbal
Generation of each agent - some positioning possibilities

###  3.3 Remote Agents

Easy integration with remote Agents is a major contribution of the proposed
architecture. We define a remote agent as any external resource that can be
integrated into the DialPort ecosystem. Generally there are three types of
remote agents: audio remote agent, text remote agent and knowledge remote
agent.

Audio Remote Agent: this is a self-sustaining spoken dialog system that only
has a public audio API. Therefore, an audio remote agent expects audio
streaming input and returns an audio clip that contains the system’s spoken
response. DialPort does not presently support this type of remote agent due to
the difficulty of dealing with real-time audio streaming amongst remote
servers. This will be dealt with when connection to a system of this type is
proposed.

Text Remote Agent: this type of agent provides text API, which inputs the
latest ASR text output and returns the next system response in text form. It
should be noted that even end-to-end spoken dialog systems can belong to text
remote agents. They only need to provide a text-based API. The Cambridge
Multi-domain spoken dialog system in Figure 1 [Gasic et al., 2015] is one
example. It has its own VoIP audio server and also provides a text-based API.
Therefore, when the Cambridge system connects with DialPort, the latter sends
the transcribed speech to Cambridge’s text-based API and bypasses its VoIP
server.

Knowledge Remote Agent: the third type of remote agent is an external
knowledge base, ranging from a web API (e.g. Yelp API) to an in-house
relational database (e.g Pittsburgh bus schedule). DialPort is in charge of
all of the dialog processing and uses the knowledge remote agent as the back-
end.

##  4 Virtual Agent

This section introduces Skylar and Jasmin. Skylar is the virtual agent for
DialPort. Jasmin is the Cambridge University dialog system agent. Both agents
interact with users via a web speech interface and have 3D animated
embodiments powered by the Unity 3D Engine444http://unity3d.com/.
The way in which non-verbal expression (agent behavior) is handled is
important. Users need to be able to easily interpret what the agents’ current
and next status are. DialPort uses a variety of character animations such as
(a) standing, (b) listening, (c) understanding, (d) thinking, and (e) talking
(Figure 2), so that users implicitly know when they should talk or should wait
for the ASR results. Moreover, the non-verbal expressions of a virtual agent
indicate which agent the system thinks the user is talking to. Generating
natural non-verbal expression is an open research problem. At present, non-
verbal expressions are manually encoded using the Mixamo
engine555https://www.mixamo.com as the following:

  * •

When nothing is happening, he/she is standing with his/her arms lowered to
his/her sides. (Figure 2-a).

  * •

When a user starts to talk to Skylar, he puts his hands/arms behind his back
and inclines his head slightly forward and down. Jasmin puts a hand to her
ear, indicating that she is listening (Figure 2-b).

  * •

When he/she has finished listening, he/she quickly nods twice to indicate that
he/she has understood and is starting to process the turn (Figure 2-c)

  * •

After he/she nods, he/she brings his/her arms back down to his/her sides and
raises his/her head, looking up at the ceiling - this indicates that the
agents is thinking of a response (Figure 2-d).

  * •

when he/she is speaking, he/she lowers his/her head and his/her gaze to look
at the user and uses various hand gestures. (Figure 2-e)

##  5 The ReinForest Dialog Manager

The challenges facing the DialPort dialog manager are that it must 1) support
easy extension to a large variety of domains and 2) support mixed initiative
and mixed (non)-goal driven dialogs. Specifically to deal with these two
challenges, we have developed a new dialog manager based on RavenClaw [Bohus
and Rudnicky, 2003]. The overview of the ReinForest DM is shown in Figure3.

Figure 3: Overview of Reinforest Dialog Manager.

The core of ReinForest has two parts: the knowledge ontology and the dialog
engine. The knowledge ontology is a domain-dependent knowledge graph that
developers can use to quickly encode domain knowledge and relations between
various concepts. The dialog engine is a domain-independent execution
mechanism that generates the next system response given the current dialog
state. The following sections formally define these two components.

###  5.1 Knowledge Ontology

The Knowledge Ontology can be thought of as fuel for ReinForest. The basic
unit of the ontology is a concept, which is an abstraction of knowledge. For
example, knowledge about the weather is a concept. A concept can have a number
of dependent concepts that encode the causal relationship. In the weather
domain for example, the weather concepts depend on the location and date time
concepts. Therefore, in order to give weather information, the system has to
have already acquired the values of date time and location. More importantly,
each remote agent is also represented as a concept. For example, the Cambridge
Multi-Domain Dialog System is represented as a single concept that contains
information about all of the domains that it covers.

Concept Pool: Given the definition of a concept, ReinForest enables developers
to construct domain knowledge in the form of a directed acyclic graph (DAG).
ReinForest also introduces the idea of a concept pool to create groups of
concepts. Figure 4 gives a simple example of a concept pool for a slot-filling
dialog manager. Essentially a concept pool is a collection of concepts that
share some common properties. There are three concept pools in ReinForest: the
agent concept pool, the user concept pool and the remote concept pool. The
agent concept pool consists of concepts that are powered by knowledge remote
agents, such as the agent’s name and the weather information. The user concept
pool contains concepts that only users know, such as the date that the user is
asking about, the user’s names etc. Finally the remote concept pool contains
all the text remote agents, where each is considered to be a black box that
can generate the next system response given the context of certain domains.

Figure 4: Example concept pools.

###  5.2 Dialog Engine

As illustrated in Figure 3, the dialog engine of ReinForest is a domain-
independent execution mechanism that consists of 4 modules: hierarchical
policy execution, belief update, tree transformation and error handling.
Algorithm 1 shows the pseudo code of the main execution loop inside
ReinForest. The dialog engine establishes its connection to the knowledge
ontology via the dialog state, $s$, which captures all the information about
the ongoing dialogs. Next we formally define these four modules.

Algorithm 1 ReinForest Main Loop

while $dialog.end()\neq True$ do

while $dialog\\_stack.top()\in O$ do

$execute(dialog\\_stack.pop())$

end while

if user has input then

$belief\\_update()$

$tree\\_transformation()$

$error\\_handle()$

end if

end while

###  5.3 Hierarchical Policy

Hierarchical policy has been studied extensively in the literature of both
hierarchical reinforcement learning (HRL) [Dietterich, 2000, Parr and Russell,
1998, Sutton et al., 1999] and plan-based dialog management [Bohus and
Rudnicky, 2003, Bohus and Rudnicky, 2009]. The contribution of ReinForest is
that it formalizes plan-based dialog management in the language of HRL, which
opens up the possibility of applying well-established HRL algorithms to
optimize the operations of the plan-based dialog manager. We first introduce
the notations of HRL and then define the dialog task tree using that language.

Figure 5: Example dialog with Skylar and the corresponding dialog tree. S:
Start, R: Root, DW: Domain weather, RR: Request root, IR: Inform root, ME:
Misunderstanding error, CAM: Cambridge Remote Agent. The red text is from
Skylar and the green text is from the Cambridge Remote Agent.

Hierarchical Reinforcement Learning: The mathematical framework of HRL is the
Markov Decision Process (MDP). An MDP is a tuple $(S,A,P,\gamma,R)$, where $S$
is a set of states; $A$ is a set of actions; $P$ defines the transition
probability $P(s^{\prime}|s,a)$; $R$ defines the expected immediate reward
$R(s,a)$; and $\gamma\in[0,1)$ is the discounting factor. Furthermore, an MDP
$M$ can be decomposed into a set of subtasks, $O=\\{O_{1},O_{2},..O_{k}\\}$,
where $O_{0}$, by convention, is the root and solving $O_{0}$ solves $M$. A
subtask is a Semi-Markov Decision Process (SMDP), which is characterized by
two additional variables compared to an MDP:

  1. 1.

$\beta_{i}(s)$ is the termination predicate of subtask $O_{i}$ that partitions
$S$ into a set of active states, $S_{i}$ and a set of terminal states $T_{i}$.
If $O_{i}$ enters a state in $T_{i}$, $O_{i}$ and its descendants exit
immediately, i.e. $\beta_{i}(s)=1$ if $s\in T_{i}$, otherwise
$\beta_{i}(s)=0$.

  2. 2.

$U_{i}$ is a nonempty set of actions that can be performed by $O_{i}$. The
actions can be either primitive actions from $A$ or other subtasks, $O_{j}$,
where $i\neq j$. We will refer to $U_{i}$ as the children of subtask $O_{i}$.

Eventually a hierarchical policy for $M$ is $\pi$ and this is simply a set of
policies for each subtask, i.e. $\pi=\\{\pi_{0},\pi_{1}...\pi_{n}\\}$. It is
evident that a valid hierarchical policy forms a DAG, whose roots come from
$O_{0}$ and whose terminal leaves are primitive actions belonging to $A$.

ReinForest Dialog Policy: ReinForest Dialog Policy forms a dialog task tree.
Each node is a subtask belonging to one of three types: dialog agency, dialog
choice agency and dialog agent.

  * •

Dialog Agency: a subtask $O_{i}\in O$ with a fixed policy that will execute
its children from left to right.

  * •

Dialog Choice Agency: a subtask $O_{i}\in O$ with a learned policy that
chooses the next executed child based on the context.

  * •

Dialog Agent: a primitive action $a\in A$ that actually delivers the action to
users.

###  5.4 Belief Update

Belief update takes place when there is new input from the user. This
component will first update the generic dialog states, such as incrementing
the dialog turn count. Then it loops through all the concepts in the knowledge
ontology and checks if the annotations in the new input match with any
subscribed domain/intent/entities in each concept. If a match is found, the
new values are stored in the concept’s attribute map.

###  5.5 Tree Transformation

Tree transformation is key in ReinForest in order to support mixed-initiative
and multi-domain dialogs. The transformation has two steps: candidate tree
generation and candidate tree selection.

Candidate tree generation: scans through the updates made by belief update and
generates a list of candidate trees that can be pushed to the dialog stack
(the list can be $\emptyset$). Usually a candidate is generated when the user
explicitly requests the type of information that is handled by a different
domain.

Candidate selection: the selected candidate trees are appended under a dialog
choice agency. The dialog choice agency selects one of the trees and pushes it
to the dialog stack.

Table 2: Characteristics of systems that handle non-goal driven utterances

|  Characteristics |  Main Technique |  Representative Systems
---|---|---|---
(a) Pattern Matching |  high precision, fast response time, time consuming to make patterns |  Artificial Intelligence Markup Language [Wallace and others, 2005] |  Alice [Wallace, 2009]
(b) Example-based |  high precision by threshold, slow response time if the data size is large |  vector space model [Turney et al., 2010] |  IRIS [Banchs and Li, 2012]
(c) Neural Chatbot |  open domain, sometimes inconsistent and ungrammatical, require large corpora |  sequence-to-sequence learning [Sutskever et al., 2014] |  CleverBot [Vinyals and Le, 2015]

###  5.6 Error Handling

There are two types of error handling: misunderstand error handling and non-
understand error handling [Bohus and Rudnicky, 2003]. Misunderstand handling
is used to conduct explicit or implicit confirms about concepts in the
knowledge ontology. Specifically, the dialog engine will loop through all
concepts in the user concept pool and select concepts that are updated but not
yet grounded for misunderstanding error handling. A misunderstanding subtask
will then be pushed to the stack. The misunderstanding subtask will choose a
built-in misunderstanding error handling strategy to confirm each concept. The
current implementation supports two types of strategies: implicit and explicit
confirm.
On the other hand, non-understanding handling is activated when there is user
input, but no update or tree-transformation is able to succeed. ReinForest
implements a wide range of non-understanding handling strategies, ranging from
the simple ”can you repeat that?” to a response from an external chat-bot.

###  5.7 Execution Demonstration

Figure 5 shows an example dialog with ReinForest. The dialog engine first
pushes the root on to the stack and asks what the user needs. After
recognizing that the user is looking for weather information, it pushes the
weather domain tree on to the stack. After acquiring all of the dependent
concepts, Skylar informs the the user of the weather information. The user
then decides to request restaurant information which is covered by the
Cambridge Remote Agent so it is pushed to the stack. At this time, ReinForest
transfers the control to Cambridge and simply calls next to the remote agent
in order to obtain the next response. When the remote agent terminates it’s
own session, Skylar takes back control and continues the conversation.

##  6 Non-goal-driven Dialog Manager

When a user’s input cannot be handled by ReinForest, such as out-of-domain
utterances (e.g., ”you are smart”), specific questions (e.g., “who founded
Microsoft?”, “how much is an iPhone?”), the non-understanding error handling
policy triggers the non-goal driven DM to generate a system response. Goal-
driven dialog systems focus on a set of predefined in-domain requests, non-
goal driven dialog systems must handle open domain utterances which have in
principle unlimited user intents. Most previous non-goal-driven approaches
either used handcrafted pattern matching rules or example-based approaches
that used a database manually designed by human experts. Recently, recurrent
neural network-based data-driven approaches have been proposed that train on
large movie transcription corpora (Table 2). Most of the past approaches focus
on building chat bots and do not provide a direct solution for DialPort,
because we use the non-goal driven dialog manager together with goal driven
dialog processing for both entertainment and error handling. Pattern-based
approaches are expensive and time consuming (Table 2-a). Moreover, existing
publicly available patterns were designed to maintain a conversation with a
chat bot for entertainment only. The Neural Network chat bots need a large
amount of data to achieve good performance and their dialogs are biased
towards the training data (e.g movie scripts). (Table 2-c). Thus the learned
dialog strategy is different from the one we want, which is used to recover
non-understanding errors and to encourage users to speak to remote agents in
other domains.

The initial prototype of DialPort uses an example-based chat bot for the
initial prototype, because the precision of the response can be controlled by
a similarity threshold (Table 2-b). We use the publicly available large
knowledge base, Freebase666http://freebase.com created by Google, to extend
coverage. For example, if a user asks about a person, a location or the
definition of a word, by using the Freebase ID extracted from the DBpedia
spotlight and the Freebase property ”common.topic.description”, the system can
find the requested information. Therefore, the non-understanding error
handling policy queries the chat bot agent with the out-of-domain user input
and the example-based chat bot calculates the similarity scores using sent2vec
[Shen et al., 2014] (rather than a traditional vector space model). If the
similarity score is over 80%, the system response is selected from the chat
bot agent. Otherwise ReinForest follows a deterministic error handling
strategy which first asks users to ”rephrase their request” and then provides
more instructions if the error cannot be recovered.

##  7 Integration Protocol

This section describes the integration protocol given to participants who are
joining DialPort. This simple integration protocol concerns the text remote
agent and the knowledge remote agent.

Text Remote Agent: a text remote agent is a dialog agency defined in Section 5
and it only needs to implement two APIs:

  * •

$NewCall(id,s_{0})$: The input parameters include the user $id$ and current
dialog state $s_{0}$. The output is the first system response. The initial
state $s_{0}$ enables the remote agent to skip redundant questions that were
already asked in the previous conversation. DialPort calls this function to
initialize a new session with the remote agent. Also, it is up to the remote
research group how they use $s_{0}$, so the remote agent can operate totally
independently. The exact format of $s_{0}$ can be customized if needed.

  * •

$Next(utt)$: The input is the users’ utterance and the output is the system’s
response and an end-of-session flag. After $NewCall$, DialPort continues to
call $Next$ to obtain the next system response until the end-of-session flag
is $true$. Thus, the remote agent has complete autonomy during its session. We
also note that the terminal flag is equivalent to the termination predicate
$\beta_{i}(s)$ in the definition of an SMDP.

The purpose of DialPort is to collect and share real user data. So when a text
remote agent finishes its session, it should be responsible for sending a
dialog report along with the response to the last $Next$ call. The report
should contain all the essential information about the conversation, such as
the utterance at each turn. The final report format will be found on the
DialPort website. Speech data that is collected will be made publicly
available by the group who collected the data.

Knowledge Remote Agent: a knowledge agent is simply a function that outputs a
list of matched entities, given a list of input constraints. Therefore, any
common database format (e.g. SQL) or service API (e.g. Yelp API) can be a
knowledge remote agent.

##  8 Current State of DialPort

The first academic system that was connected to DialPort is from Cambridge
University. In the near future, any academic system that can be connected will
be welcome to join the portal. Figure 6 shows the two current agents: a butler
named Skylar from CMU and a librarian named Jasmin from Cambridge. Their
appearance may change at a later time. DialPort will start to attract users to
these systems as soon as it passes a series of stability tests and when
several other remote agents, such as Yelp are added in order to broaden
interest. CMU’s Skylar will give information about the weather and restaurants
other than in Cambridge and San Francisco. Its job is also to ”sell” the other
systems, getting the user to want to try them. When the domain changes, a new
avatar appears and handles the conversation. When the dialog on the connected
system’s topic is over, the user is handed back to Skylar. The transition is
seamless from the users point of view. Jasmin from Cambridge gives information
on hotels and restaurants in San Francisco. When the user is speaking with
her, the logo in the background changes to the Cambridge log to indicate which
system the user is speaking to (Figure 6).

Figure 6: The present appearance of CMU-Skylar and Cambridge-Jasmin.

##  9 Conclusions

We propose a novel shared platform, DialPort, which can connect to many
research systems enabling them to test new application ideas and gather real
user data. In this paper, we have described the architecture of the user
interface, DialPort, the virtual agents, and the (non)goal driven dialog
managers and we have reported the current progress of the DialPort project. An
important purpose of this paper is to encourage our colleagues to link their
systems to DialPort so that we can help them to collect real user data.

##  10 Acknowledgements

This work was funded by NSF grant CNS-1512973. The opinions expressed in this
paper do not necessarily reflect those of NSF. We would also like to thank
Dialogue Systems Group at Cambridge for their hard work in making the
connection with DialPort.

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