# On the Ranks of Semigroup of Order-preserving or Order-reversing Partial
Contraction Mappings on a Finite Chain

B. Ali, M. A. Jada and M. M. Zubairu111Corresponding Author. Email:
<EMAIL_ADDRESS>
Department of Mathematical Sciences, Nigerian Defence Academy, P.M.B. 2109,
Kaduna, Nigeria
<EMAIL_ADDRESS>
Department of Mathematics, Sule Lamido University, P.M.B. 048, Kafin Hausa,
Nigeria
<EMAIL_ADDRESS>
Department of Mathematics, Bayero University Kano, P.M.B. 3011, Kano, Nigeria
<EMAIL_ADDRESS>

###### Abstract

Let $\mathcal{CP}_{n}$ be the semigroup of partial contraction mappings on
$[n]=\\{1,2,\ldots,n\\}$ and let $\mathcal{OCP}_{n}$ and $\mathcal{ORCP}_{n}$
be its subsemigroups consisting of all order-preserving and of all order-
preserving or order-reversing, partial contraction mappings, respectively. In
this paper we obtain the rank of the two semigroups, $\mathcal{OCP}_{n}$ and
$\mathcal{ORCP}_{n}$, respectively.

_2010 Mathematics Subject Classification. 20M20._

Keywords: Transformation semigroup, Contraction mappings, Order-preserving,
Order-reversing, Generating sets, Ranks

##  1 Introduction

Let $[n]$ denote a finite chain, $\\{1,2,\ldots,n\\}$. A _partial
transformation_ is a map $\alpha:\textrm{dom
}\alpha\subseteq[n]\rightarrow\textrm{im }\alpha\subseteq[n]$. Denote
$\mathcal{P}_{n}$ to be the _semigroup of all partial transformation_ on
$[n]$. A map $\alpha\in\mathcal{P}_{n}$ is called a _total_ or _full_
transformation if $\textrm{dom }\alpha=[n]$. The collection of all full
transformations on $[n]$ is known to be the _semigroup of full transformation_
denoted by $\mathcal{T}_{n}$. For $\alpha\in\mathcal{P}_{n}$, $\alpha$ is said
to be _order-preserving_ (respectively, _order reversing_) if (for all
$x,y\in\textrm{dom }\alpha$) $x\leq y$ implies $x\alpha\leq y\alpha$
(respectively, $x\alpha\geq y\alpha$); an _isometry_ (i. e., _distance
preserving_) if (for all $x,y\in\textrm{dom }\alpha$)
$|x\alpha-y\alpha|=|x-y|$; and a _contraction_ if (for all $x,y\in\textrm{dom
}\alpha$) $|x\alpha-y\alpha|\leq|x-y|$. The collection of all contraction
mappings on $[n]$ is called the _semigroup of partial contraction mappings_
denoted by $\mathcal{CP}_{n}$.

Let

$\mathcal{OCP}_{n}=\\{\alpha\in\mathcal{CP}_{n}:(\textnormal{for all
}x,y\in\textnormal{dom }\alpha)~{}x\leq y~{}\Rightarrow~{}x\alpha\leq
y\alpha\\}$ (1)

and

$\mathcal{ORCP}_{n}=\mathcal{OCP}_{n}\cup\\{\alpha\in\mathcal{CP}_{n}:(\textnormal{for
all }x,y\in\textnormal{dom }\alpha)~{}x\leq y~{}\Rightarrow~{}x\alpha\geq
y\alpha\\}.$ (2)

Then, $\mathcal{OCP}_{n}$ and $\mathcal{ORCP}_{n}$ are subsemigroups of
$\mathcal{CP}_{n}$ known as the _semigroup of order-preserving partial
contraction mappings_ and the _semigroup of order-preserving or order-
reversing partial contraction mappings_ , respectively.

A non-empty subset $A$ of a semigroup $S$ is said to be a _generating set_ of
$S$ (denoted as $\langle A\rangle=S$) if for every $x\in S$, $x$ can be
written as finite product of some elements in $A$. If $S$ is generated by a
finite subset, then, the smallest subset that generate $S$ is called the
_minimal generating set_ of $S$ while its cardinality is called the _rank_ of
$S$, denoted by rank($S$). That is, rank$(S)=\min\\{|A|:A\subseteq
S\textnormal{ and }\langle A\rangle=S\\}$.

The importance of the study of rank properties in semigroup theory cannot be
over emphasized. Various scholars study rank properties on the semigroup
$\mathcal{P}_{n}$ and many of its subsemigroups over the years, and fruitful
results were recorded, see for example, [8, 11, 13, 14, 18].

The semigroup of partial contraction mappings being a relatively new semigroup
with promising research problems attracts the attention of many researchers in
the area. For example, Adeshola in [1], investigated some algebraic and
combinatorial properties of some subsemigroups of $\mathcal{CT}_{n}$ (the
semigroup of full contraction mappings) and obtain the cardinalities of some
of its subsemigroups. The study of Green’s and starred Green’s relations of
$\mathcal{CT}_{n}$ was presented in Garba _et al.,_ [12]. The results in [12]
were extended in [21] to give complete characterization of the Green’s
relations on $\mathcal{CT}_{n}$. Many more algebraic and combinatorial studies
on $\mathcal{CP}_{n}$ and some of its subsemigroup were conducted see, [4, 20,
22] which by no means a complete list.

However, a lot of questions regarding the rank properties of
$\mathcal{CP}_{n}$ and many of its subsemigroups are still unanswered.
Recently, Toker examined the ranks of the semigroup of _order-preserving or
order-reversing full contractions_ , $\mathcal{ORCT}_{n}$ and that of its
subsemigroup $\mathcal{OCT}_{n}$, consisting of all order-preserving elements
in [19]. Bugay [10] extended the work in [19] by considering the ideals of the
semigroups $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$ respectively. In [3],
the study of nilpotent elements in $\mathcal{OCP}_{n}$ and its subsemigroup
$\mathcal{OCI}_{n}$, consisting of all partial one-to-one transformations was
presented and the ranks of the subsemigroups generated by the nilpotents was
obtained. Also, in [7] the study of algebraic and rank properties of the
semigroup $\mathcal{OCI}_{n}$ was conducted. Umar and Alkharousi study the
rank and combinatorial properties of the semigroups of partial isometries in
[5, 6]. The study of the rank properties of isometries were further exploited
in [2, 9].

Having studied (by Toker [19] and Bugay [10]) the rank properties of the
semigroups, $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$, in this paper we
consider a larger semigroup, $\mathcal{ORCP}_{n}$ and its subsemigroup,
$\mathcal{OCP}_{n}$, and investigated their ranks. The structure of the paper
is as follows. After this introductory section, followed is the preliminary
section. In section $3$, we present the rank of $\mathcal{OCP}_{n}$, and show
that the semigroup $\mathcal{OCP}_{n}$ has rank $2n$ for all $n\geq 3$. The
results in section $3$ were extended in section $4$ and shown that, despite
the semigroup $\mathcal{ORCP}_{n}$ is larger, it also has the same rank as
$\mathcal{OCP}_{n}$.

##  2 Preliminaries

In this section, we give definition of some basic terms. We also quote some
results from related literature that will be needed in proving some results in
the paper.

Let $\alpha$ be an element of $\mathcal{CP}_{n}$. Denote $\textrm{dom
}\alpha$, $\textrm{im }\alpha$, and $h(\alpha)$ to be the domain of $\alpha$,
image of $\alpha$, and $|\textrm{im }\alpha|$, respectively. The kernel of
$\alpha$ is denoted by ker $\alpha$, and defined as ker
$\alpha=\\{(x,y)\in\textnormal{dom }\alpha\times\textnormal{dom
}\alpha:x\alpha=y\alpha\\}$. Furthermore, for
$\alpha,\beta\in\mathcal{CP}_{n}$, the composition of $\alpha$ and $\beta$ is
defined as $x(\alpha\circ\beta)=((x)\alpha)\beta$ for all $x$ in dom $\alpha$.
Without ambiguity, we shall be using the notation $\alpha\beta$ to denote
$\alpha\circ\beta$ and $x\alpha$ to serve as the image of $x$ under $\alpha$
instead of the notation $\alpha(x)$ so that our composition of maps reads from
left to right, i. e., $x\alpha\beta$ to serve as $\beta(\alpha(x))$. Also, a
non-empty subset $A$ of $[n]$ is called convex if for every $x,y\in A$,
$x<z<y$ implies $z\in A$. We shall write $1_{A}$ for any subset
$A\subseteq[n]$ to denote partial identity map on $A$.

Next, given any transformation $\alpha$ in $\mathcal{ORCP}_{n}$, $\alpha$ can
be expressed as

$\alpha=\left(\begin{array}[]{cccc}A_{1}&A_{2}&\ldots&A_{p}\\\
b_{1}&b_{2}&\ldots&b_{p}\end{array}\right)\,(1\leq p\leq n).$ (3)

The blocks $A_{i}(1\leq i\leq p)$ are equivalence classes under the relation
ker $\alpha$, and the collection of all the equivalence classes of the
relation ker $\alpha$ partitioned the domain of $\alpha$ called the _kernel
partition_ of $\alpha$, denoted by kp$(\alpha)$, i. e.,
$[n]\backslash\textnormal{ker }\alpha=\textnormal{kp}(\alpha)$. Moreover,
kp$(\alpha)$ is ordered under the usual ordering, that is,
$A_{1}<A_{2}<\cdots<A_{p}$, where $A_{i}<A_{j}$ if and only if $a<b$ for all
$a\in A_{i}$ and $b\in A_{j}$. Further, for each block $A_{i}$, if
$|A_{i}|\geq 2$ then $a_{i}<a_{i+1}\in A_{i}$. In addition, if $\alpha$ is
order-preserving then $b_{1}<b_{2}<\ldots<b_{p}$, and if $\alpha$ is order-
reversing then $b_{p}<b_{p-1}<\ldots<b_{1}$. Thus, for the rest of the content
of the work we shall consider $\alpha\in\mathcal{ORCP}_{n}$ to be as expressed
in equation (3) unless otherwise specified.

Let $S$ be a semigroup and $x\in S$. Then, $Sx$, $xS$, and $SxS$ are principal
ideals generated by $x$ in $S$. They are known as the _principal left,
principal right_ , and _principal two sided_ ideal, respectively generated by
$x$. Green’s relations are equivalence relations defined on a semigroup based
on the principal ideals its elements generate. These relations are vital in
understanding the algebraic structure of any semigroup more especially regular
semigroups. There are five of these equivalences, namely, $\mathcal{L}$,
$\mathcal{R}$, $\mathcal{J}$, $\mathcal{D}$ and $\mathcal{H}$ relation defined
as follows: For all $a,b\in S$,

$\displaystyle(a,b)\in\mathcal{L}$ $\displaystyle\Longleftrightarrow$
$\displaystyle S^{1}a=S^{1}b;$ $\displaystyle(a,b)\in\mathcal{R}$
$\displaystyle\Longleftrightarrow$ $\displaystyle aS^{1}=bS^{1};$
$\displaystyle(a,b)\in\mathcal{J}$ $\displaystyle\Longleftrightarrow$
$\displaystyle S^{1}aS^{1}=S^{1}bS^{1}.$

The symbol $S^{1}$ denote a semigroup $S$ with adjoint identity if $S$ does
not have one. $\mathcal{L}$ and $\mathcal{R}$ always commute and their
composition gives a relation $\mathcal{D}$ (i. e.,
$\mathcal{L}\circ\mathcal{R}=\mathcal{R}\circ\mathcal{L}=\mathcal{D}$), while
their intersection gives a relation $\mathcal{H}$ (i. e.,
$\mathcal{L}\cap\mathcal{R}=\mathcal{H}$). If a semigroup is non-regular,
then, there is also need to understand its starred Green’s relation (which is
the generalization of the Green’s relations) in order to classify such
semigroup or study its rank properties. The starred Green’s equivalences are
also five, they are $\mathcal{L}^{*}$, $\mathcal{R}^{*}$, $\mathcal{J}^{*}$,
$\mathcal{D}^{*}$, and $\mathcal{H}^{*}$ defined as follows: Given any
semigroup $S$, $a\mathcal{L}^{*}b$ (resp., $a\mathcal{R}^{*}b$) if and only if
$a$ and $b$ are Green’s $\mathcal{L}$-related (resp., Green’s
$\mathcal{R}$-related) in some over-semigroup of $S$. $\mathcal{D}^{*}$ is a
meet of $\mathcal{L}^{*}$ and $\mathcal{R}^{*}$, while $\mathcal{H}^{*}$ is
their joint. For more properties of Green’s relation and any other unexplained
term we refer the reader to [15, 16, 17].

Below are some results concerning starred Green’s relations on the semigroups
of $\mathcal{OCP}_{n}$ and $\mathcal{ORCP}_{n}$ from [20], which shall be
needed in the subsequent sections.

######  Theorem 2.1.

[20, Theorem 1] Let $\alpha,\beta\in S$ (where
$S\in\\{\mathcal{CP}_{n},\mathcal{ORCP}_{n},\mathcal{OCP}_{n}\\})$. Then,

  * (i)

$\alpha\mathcal{L}^{*}\beta$ if and only if $\textnormal{im
}\alpha=\textnormal{im }\beta$.

  * (ii)

$\alpha\mathcal{R}^{*}\beta$ if and only if $\textnormal{ker
}\alpha=\textnormal{ker }\beta$.

  * (iii)

$\alpha\mathcal{H}^{*}\beta$ if and only if $\textnormal{ker
}\alpha=\textnormal{ker }\beta$ and $\textnormal{im }\alpha=\textnormal{im
}\beta$.

  * (iv)

$\alpha\mathcal{D}^{*}\beta$ if and only if $|\textnormal{im
}\alpha|=|\textnormal{im }\beta|$.

The following lemmas are also needed in proving Lemma 3.1 and 4.1,
respectively.

Let $D_{p}^{*}=\\{\alpha\in\mathcal{OCT}_{n}:|\textnormal{im }\alpha|=p\\}$,
and let $F_{p}^{*}=\\{\alpha\in\mathcal{ORCT}_{n}:|\textnormal{im
}\alpha|=p\\}$. Then,

######  Lemma 2.2.

[19, Lemma 3.1] If $\alpha\in D_{p}^{*}$ then $\alpha\in\langle
D_{p+1}^{*}\rangle$ for all $1\leq p\leq n-2$.

######  Lemma 2.3.

[19, Lemma 4.1] If $\alpha\in F_{p}^{*}$ then $\alpha\in\langle
F_{p+1}^{*}\rangle$ for all $1\leq p\leq n-2$.

##  3 Rank of $\mathcal{OCP}_{n}$

In this section, we compute the rank of the semigroup of order-preserving
partial contraction mappings, $\mathcal{OCP}_{n}$.

For $1\leq p\leq n$, let

$K_{p}=\\{\alpha\in\mathcal{OCP}_{n}:|\textnormal{im }\alpha|=p\\}$ (4)

and

$\mathcal{OCP}_{n,p}=\\{\alpha\in\mathcal{OCP}_{n}:|\textnormal{im
}\alpha|\leq p\\}.$ (5)

Observe that $K_{p}\subseteq\mathcal{OCP}_{n,p}$ for all $p$.

We now proof the following lemma which is indispensable in finding the rank of
$\mathcal{OCP}_{n,n-1}$.

######  Lemma 3.1.

$K_{p}\subseteq\langle K_{p+1}\rangle$, for all $1\leq p\leq n-2$.

###### Proof.

Let $\alpha\in K_{p}$ be as expressed in equation (3) where $1\leq p\leq n-2$.
Then, we consider the following cases:

Case I: Suppose $\textnormal{im }\alpha$ is convex. If $\textnormal{dom
}\alpha=[n]$, then $\alpha\in\mathcal{OCT}_{n}$ and the result follows from
Lemma 2.2. Now, suppose $\textnormal{dom }\alpha\subset[n]$ and let $c$ be in
$[n]\backslash\textnormal{dom }\alpha$. If $c<\min A_{1}$ and $b_{p}<n$, then
define $\beta$ and $\gamma$, respectively as:

$\left(\begin{array}[]{ccccc}c&A_{1}&A_{2}&\cdots&A_{p}\\\
1&2&3&\cdots&p+1\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}2&3&\cdots&p+1&p+2\\\
b_{1}&b_{2}&\cdots&b_{p}&b_{p}+1\end{array}\right);$ (6)

if $c<\min A_{1}$ and $b_{p}=n$ (notice that since $\textnormal{im }\alpha$ is
convex and $|\textnormal{im }\alpha|\leq n-2$, then $b_{p}=n$ would imply that
$b_{1}>2$), then define $\beta$ and $\gamma$, respectively as:

$\left(\begin{array}[]{ccccc}c&A_{1}&A_{2}&\cdots&A_{p}\\\
2&3&4&\cdots&p+2\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}1&3&4&\cdots&p+2\\\
b_{1}-1&b_{1}&b_{2}&\cdots&b_{p}\end{array}\right);$ (7)

if $\max A_{i}<c<\min A_{i+1}$ ($i\in\\{1,2,\ldots,p-1\\}$) and $b_{p}<n$,
then define $\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{ccccccc}A_{1}&\cdots&A_{i}&c&A_{i+1}&\cdots&A_{p}\\\
1&\cdots&i&i+1&i+2&\cdots&p+1\end{array}\right)\textrm{ and }$ (10)
$\displaystyle\left(\begin{array}[]{ccccccc}1&\cdots&i&i+2&\cdots&p+1&p+2\\\
b_{1}&\cdots&b_{i}&b_{i+1}&\cdots&b_{p}&b_{p}+1\end{array}\right);$ (13)

if $\max A_{i}<c<\min A_{i+1}$ ($i\in\\{1,2,\ldots,p-1\\}$) and $b_{p}=n$,
then define $\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{ccccccc}A_{1}&\cdots&A_{i}&c&A_{i+1}&\cdots&A_{p}\\\
2&\cdots&i+1&i+2&i+3&\cdots&p+2\end{array}\right)\textrm{ and }$ (16)
$\displaystyle\left(\begin{array}[]{ccccccc}1&2&\cdots&i+1&i+3&\cdots&p+2\\\
b_{1}-1&b_{1}&\cdots&b_{i}&b_{i+1}&\cdots&b_{p}\end{array}\right);$ (19)

if $\min A_{i}<c<\max A_{i}$ ($i\in\\{1,2,\ldots,p\\}$) and $b_{p}<n$, then
define $\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{ccccccc}A_{1}&\cdots&\\{\min
A_{i},\ldots,c-1\\}&\\{c+1,\ldots,\max A_{i}\\}&A_{i+1}&\cdots&A_{p}\\\
1&\cdots&i&i+1&i+2&\cdots&p+1\end{array}\right)\textrm{ and }$ (22)
$\displaystyle\left(\begin{array}[]{cccccc}1&\cdots&\\{i,i+1\\}&\cdots&p+1&p+2\\\
b_{1}&\cdots&b_{i}&\cdots&b_{p}&b_{p}+1\end{array}\right);$ (25)

if $\min A_{i}<c<\max A_{i}$ ($i\in\\{1,2,\ldots,p\\}$) and $b_{p}=n$, then
define $\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{cccccc}A_{1}&\cdots&\\{\min
A_{i},\ldots,c-1\\}&\\{c+1,\ldots,\max A_{i}\\}&\cdots&A_{p}\\\
2&\cdots&i+1&i+2&\cdots&p+2\end{array}\right)\textrm{ and }$ (28)
$\displaystyle\left(\begin{array}[]{ccccccc}1&2&\cdots&\\{i+1,i+2\\}&i+3&\cdots&p+2\\\
b_{1}-1&b_{1}&\cdots&b_{i}&b_{i+1}&\cdots&b_{p}\end{array}\right);$ (31)

if $\max A_{p}<c$ and $b_{p}<n$, then define $\beta$ and $\gamma$,
respectively as:

$\left(\begin{array}[]{ccccc}A_{1}&A_{2}&\cdots&A_{p}&c\\\
1&2&\cdots&p&p+1\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}1&2&\cdots&p&p+2\\\
b_{1}&b_{2}&\cdots&b_{p}&b_{p}+1\end{array}\right);$ (32)

and if $\max A_{p}<c$ and $b_{p}=n$, then define $\beta$ and $\gamma$,
respectively as:

$\left(\begin{array}[]{ccccc}A_{1}&A_{2}&\cdots&A_{p}&c\\\
2&3&\cdots&p+1&p+2\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}1&2&3&\cdots&p+1\\\
b_{1}-1&b_{1}&b_{2}&\cdots&b_{p}\end{array}\right).$ (33)

Then, it is not difficult to verify that $\beta,\gamma\in K_{p+1}$ in all the
cases and $\beta\gamma=\alpha$.

Case II: Suppose $\textnormal{im }\alpha$ is not convex, i. e., there exists
$i\in\\{1,2,\ldots,p-1\\}$ such that $b_{i+1}-b_{i}\geq 2$. Then, since
$\alpha$ is a contraction, we must have $\min A_{i+1}-\max A_{i}\geq 2$ .
Define

$\beta=\left(\begin{array}[]{cccccccc}A_{1}&A_{2}&\cdots&A_{i}&\max
A_{i}+1&A_{i+1}&\cdots&A_{p}\\\
b_{1}&b_{2}&\cdots&b_{i}&b_{i}+1&b_{i+1}&\cdots&b_{p}\end{array}\right).$

Recall that $|\textnormal{im }\alpha|\leq n-2$, therefore $|\textnormal{im
}\beta|\leq n-1$. Let $x\in[n]\backslash\textnormal{im }\beta$ and let
$A=\textnormal{im }\beta\cup\\{x\\}$. Define $\gamma$ to be the partial
identity on $A$. Then, it is easy to see that $\beta$ and $\gamma$ are in
$K_{p+1}$ and $\alpha=\beta\gamma$. ∎

######  Corollary 3.2.

$\langle K_{n-1}\rangle=\mathcal{OCP}_{n,n-1}$ for all $n\geq 3$.

###### Proof.

It follows Lemma 3.1 and the fact that $\langle
K_{n-1}\rangle\leq\mathcal{OCP}_{n,n-1}$. ∎

An element $\alpha$ in $\mathcal{OCP}_{n}$ (as well as $\mathcal{ORCP}_{n}$)
is said to have a _projection characteristic_ $(k,p)$ or belong to the set
$[k,p]$ if $|\textnormal{dom }\alpha|=k$ and $|\textnormal{im }\alpha|=p$ (see
[13]). Now, let $\alpha\in K_{n-1}$ where $n\geq 3$. Then, $|\textnormal{dom
}\alpha|$ is either $n$ or $n-1$, i. e., either $\alpha$ belongs to $[n,n-1]$
or $[n-1,n-1]$. Clearly if $\alpha\in[n,n-1]$ then $\alpha$ is a full
contraction and its kernel partition is of the form
$\\{\\{1\\},\\{2\\},\ldots,\\{i,i+1\\},\ldots,\\{n-1\\},\\{n\\}\\}$ ($1\leq
i\leq n-1$). On the other hand, if $\alpha\in[n-1,n-1]$, then $\alpha$ is one-
to-one, and so kp$(\alpha)$ is just a partition of $[n]\backslash\\{i\\}$
($i=1,2,\ldots,n$) into singleton classes.

Moreover, if $\alpha\in K_{n-1}$ and $\textnormal{dom }\alpha$ is convex, then
by contractive property of $\alpha$, $\textnormal{im }\alpha$ must also be
convex (see also Ali _et. al.,_ [20]). Therefore, $\textnormal{im }\alpha$
will either be $\\{1,2,\ldots,n-1\\}$ or $\\{2,3,\ldots,n\\}$. But if
$\textnormal{dom }\alpha$ is not convex, i. e., if kernel partition of
$\alpha$ is of the form
$\\{\\{1\\},\ldots,\\{j-1\\},\\{j+1\\},\\{j+2\\},\ldots,\\{n\\}\\}$ ($2\leq
j\leq n-1$), then $\textnormal{dom }\alpha$ has three possible choice of
images: $\\{1,2,\ldots,n-1\\}$, $\\{2,3,\ldots,n\\}$ or
$\\{1,\ldots,j-1,j+1,\ldots,n\\}$.

For $1\leq s,t\leq n$ and $1\leq i\leq n-1$, denote $P_{[n]\backslash\\{s\\}}$
to be a partition of $[n]\backslash\\{s\\}$ into singleton classes; and
$P_{\\{i,i+1\\}}$ to be a partition of $[n]$ into $n-1$ classes as
$\\{\\{1\\},\ldots,\\{i-1\\},\\{i,i+1\\},\\{i+2\\},\ldots,\\{n\\}\\}$. Also,
denote $\alpha_{s,t}$ to mean an element in $[n-1,n-1]$ whose kernel partition
is $P_{[n]\backslash\\{s\\}}$ and image, $[n]\backslash\\{t\\}$. Further,
denote $\pi_{(i,i+1),k}$ ($k=1,2$) to mean an element in $[n,n-1]$ whose
kernel partition is $P_{\\{i,i+1\\}}$ and image, $[n]\backslash\\{k\\}$. For
the purpose of illustration, consider the transformations below.
$\alpha_{n,1}=\left(\begin{array}[]{cccc}1&2&\cdots&n-1\\\
2&3&\cdots&n\end{array}\right)$,
$\alpha_{2,2}=\left(\begin{array}[]{ccccc}1&3&4&\cdots&n\\\
1&3&4&\cdots&n\end{array}\right)$ and
$\pi_{(2,3),1}=\left(\begin{array}[]{cccc}1&\\{2,3\\}&\cdots&n\\\
2&3&\cdots&n\end{array}\right)$.

Then, we have a tabular display of the elements of $K_{n-1}$ below, where the
horizontal rows represent the $\mathcal{R}^{*}-$classes and vertical columns
represent the $\mathcal{L}^{*}-$classes.

Kernel partitions / Image sets | $[n]\backslash\\{1\\}$ | $[n]\backslash\\{j\\}$ | $[n]\backslash\\{n\\}$
---|---|---|---
$P_{[n]\backslash\\{1\\}}$ | $\alpha_{1,1}$ |  | $\alpha_{1,n}$
$P_{[n]\backslash\\{j\\}}$ | $\alpha_{j,1}$ | $\alpha_{j,j}$ | $\alpha_{j,n}$
$P_{[n]\backslash\\{n\\}}$ | $\alpha_{n,1}$ |  | $\alpha_{n,n}$
$P_{\\{i,i+1\\}}$ | $\pi_{(i,i+1),1}$ |  | $\pi_{(i,i+1),n}$
(1)

where $1\leq i\leq n-1$ and $2\leq j\leq n-1$.

It is not difficult to see that the number of rows in Table (1) is $2n-1$. It
therefore follows that $K_{n-1}$ has $2n-1$ $\mathcal{R}^{*}-$classes. Thus,
we have the following lemma.

######  Lemma 3.3.

For $n\geq 3$, rank($\mathcal{OCP}_{n,n-1})\geq 2n-1$.

###### Proof.

Since any generating set of $\langle K_{n-1}\rangle$ must contain at least one
representative from each of the $\mathcal{R}^{*}-$classes, it follows that
rank($\langle K_{n-1}\rangle)\geq 2n-1$ which implies by Corollary 3.2 that
rank($\mathcal{OCP}_{n,n-1})\geq 2n-1$. ∎

######  Proposition 3.4.

Let $G=\\{\alpha_{n,1},\alpha_{1,n},\alpha_{j,j},\pi_{\\{i,i+1\\},n}:2\leq
j\leq n-1,\,1\leq i\leq n-1\\}$. Then, $G$ is a minimal generating set of
$\langle K_{n-1}\rangle$.

###### Proof.

Firstly, it is not difficult to see from the following multiplications,

$\alpha_{1,n}\cdot\alpha_{n,1}=\left(\begin{array}[]{cccc}2&3&\cdots&n\\\
1&2&\cdots&n-1\end{array}\right)\left(\begin{array}[]{cccc}1&2&\cdots&n-1\\\
2&3&\cdots&n\end{array}\right)=\left(\begin{array}[]{cccc}2&3&\cdots&n\\\
2&3&\cdots&n\end{array}\right)=\alpha_{1,1};$
$\alpha_{n,1}\cdot\alpha_{1,n}=\left(\begin{array}[]{cccc}1&2&\cdots&n-1\\\
2&3&\cdots&n\end{array}\right)\left(\begin{array}[]{cccc}2&3&\cdots&n\\\
1&2&\cdots&n-1\end{array}\right)=\left(\begin{array}[]{cccc}1&2&\cdots&n-1\\\
1&2&\cdots&n-1\end{array}\right)=\alpha_{n,n};$
$\pi_{\\{i,i+1\\},n}\cdot\alpha_{n,1}=\left(\begin{array}[]{ccccccc}1&\cdots&i-1&\\{i,i+1\\}&i+2&\cdots&n\\\
1&\cdots&i-1&i&i+1&\cdots&n-1\end{array}\right)\left(\begin{array}[]{cccc}1&2&\cdots&n-1\\\
2&3&\cdots&n\end{array}\right)$
$=\left(\begin{array}[]{ccccccc}1&\cdots&i-1&\\{i,i+1\\}&i+2&\cdots&n\\\
2&\cdots&i&i+1&i+2&\cdots&n\end{array}\right)=\pi_{\\{i,i+1\\},1};$
$\alpha_{j,j}\cdot\pi_{\\{j,j+1\\},n}=\left(\begin{array}[]{cccccc}1&\cdots&j-1&j+1&\cdots&n\\\
1&\cdots&j-1&j+1&\cdots&n\end{array}\right)\left(\begin{array}[]{ccccccc}1&\cdots&\\{j,j+1\\}&j+2&\cdots&n\\\
1&\cdots&j&j+1&\cdots&n-1\end{array}\right)$
$=\left(\begin{array}[]{cccccc}1&\cdots&j-1&j+1&\cdots&n\\\
1&\cdots&j-1&j&\cdots&n-1\end{array}\right)=\alpha_{j,n};\textnormal{ and}$
$\\\
\alpha_{j,j}\cdot\pi_{\\{j,j+1\\},1}=\left(\begin{array}[]{cccccc}1&\cdots&j-1&j+1&\cdots&n\\\
1&\cdots&j-1&j+1&\cdots&n\end{array}\right)\left(\begin{array}[]{cccccc}1&\cdots&\\{j,j+1\\}&j+2&\cdots&n\\\
2&\cdots&j+1&j+2&\cdots&n\end{array}\right)$
$=\left(\begin{array}[]{cccccc}1&\cdots&j-1&j+1&\cdots&n\\\
2&\cdots&j&j+1&\cdots&n\end{array}\right)=\alpha_{j,1};$

that $K_{n-1}\subseteq\langle G\rangle$ and as such, since $G\subseteq
K_{n-1}$, then $\langle K_{n-1}\rangle=\langle G\rangle$.

Secondly, we notice that elements of $G$ are just representatives of the
$\mathcal{R}^{*}-$classes in $K_{n-1}$. Therefore, $G$ is a minimal generating
set of $\langle K_{n-1}\rangle$. ∎

######  Corollary 3.5.

For $n\geq 3$, rank($\mathcal{OCP}_{n,n-1})=2n-1$.

###### Proof.

It is easy to see from Proposition 3.4 that the cardinality of the minimal
generating set $G$, of $K_{n-1}$ is $2n-1$. Therefore,
$\textnormal{rank}(\langle K_{n-1}\rangle)=2n-1$ which implies by Corollary
3.2 that $\textnormal{rank}(\mathcal{OCP}_{n,n-1})=2n-1$. ∎

Now we have the main result of this section.

######  Theorem 3.6.

Let $\mathcal{OCP}_{n}$ be as defined in equation (1). Then, the rank of
$\mathcal{OCP}_{n}$ is $2n$ for all $n\geq 3$.

###### Proof.

First, observe that the only element of $\mathcal{OCP}_{n}$ of height $n$ is
the identity element, $1_{[n]}$. It thus follows that
$\mathcal{OCP}_{n}=\mathcal{OCP}_{n,n-1}\cup\\{1_{[n]}\\}$. Notice also that
since $\mathcal{OCP}_{n,n-1}$ is an ideal of $\mathcal{OCP}_{n}$, then
rank$(\mathcal{OCP}_{n})=2n$, as required ∎

##  4 Rank of $\mathcal{ORCP}_{n}$

In this section, we extend the results obtained in section 3 to compute the
rank of the semigroup of order-preserving or order-reversing partial
contraction mappings, $\mathcal{ORCP}_{n}$.

Let

$W_{p}=\\{\alpha\in\mathcal{ORCP}_{n}:|\textnormal{im }\alpha|=p\\}$ (34)

and

$\mathcal{ORCP}_{n,p}=\\{\alpha\in\mathcal{ORCP}_{n}:|\textnormal{im
}\alpha|\leq p\\},$ (35)

where $1\leq p\leq n$. Observe that $K_{p}$ as defined in equation (34)
contain all the order-preserving elements in $W_{p}$. Let $K_{p}^{*}$ denote
the set of all order-reversing elements in $W_{p}$. Then, $W_{p}=K_{p}\cup
K_{p}^{*}$. Now we have the following lemma which is an extension of Lemma
3.1.

######  Lemma 4.1.

For $1\leq p\leq n-2$, $W_{p}\subseteq\langle W_{p+1}\rangle$.

###### Proof.

Let $\alpha=\left(\begin{array}[]{cccc}A_{1}&A_{2}&\ldots&A_{p}\\\
b_{1}&b_{2}&\ldots&b_{p}\end{array}\right)$ be element of $W_{p}$ where $1\leq
p\leq n-2$. If $|\textnormal{dom }\alpha|=n$, then $\alpha$ is a full
contraction and the result follows from Lemma 2.3. Now, suppose
$|\textnormal{dom }\alpha|<n$. If $\alpha\in K_{p}$, then the result follows
from Lemma 3.1, and if $\alpha\in K_{p}^{*}$, then we proceed by cases similar
to what we had in the proof of Lemma 3.1.

Case I: If $\textnormal{im }\alpha$ is convex. Then, since $\alpha$ is order-
reversing it can be expressed as
$\left(\begin{array}[]{cccc}A_{1}&A_{2}&\cdots&A_{p}\\\
k&k-1&\cdots&k-p+1\end{array}\right)$ ($p\leq k\leq n$). Let
$c\in[n]\backslash\textnormal{dom }\alpha$. If $c<\min A_{1}$ and $p<k$, then
define $\beta$ and $\gamma$, respectively as:

$\left(\begin{array}[]{ccccc}c&A_{1}&A_{2}&\cdots&A_{p}\\\
1&2&3&\cdots&p+1\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}2&3&\cdots&p+1&p+2\\\
k&k-1&\cdots&k-p+1&k-p\end{array}\right);$ (36)

if $c<\min A_{1}$ and $p=k$, then $k\leq n-2$, and so $\beta$ and $\gamma$ can
be defined respectively as:

$\left(\begin{array}[]{ccccc}c&A_{1}&A_{2}&\cdots&A_{p}\\\
2&3&4&\cdots&p+2\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}1&3&4&\cdots&p+2\\\
k+1&k&k-1&\cdots&k-p+1\end{array}\right);$ (37)

if $\max A_{i}<c<\min A_{i+1}$ ($i\in\\{1,2,\ldots,p-1\\}$) and $p<k$, then
define $\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{ccccccc}A_{1}&\cdots&A_{i}&c&A_{i+1}&\cdots&A_{p}\\\
1&\cdots&i&i+1&i+2&\cdots&p+1\end{array}\right)\textrm{ and }$ (40)
$\displaystyle\left(\begin{array}[]{ccccccc}1&\cdots&i&i+2&\cdots&p+1&p+2\\\
k&\cdots&k-i+1&k-i&\cdots&k-p+1&k-p\end{array}\right);$ (43)

if $\max A_{i}<c<\min A_{i+1}$ ($i\in\\{1,2,\ldots,p-1\\}$) and $p=k$, then
define $\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{ccccccc}A_{1}&\cdots&A_{i}&c&A_{i+1}&\cdots&A_{p}\\\
2&\cdots&i+1&i+2&i+3&\cdots&p+2\end{array}\right)\textrm{ and }$ (46)
$\displaystyle\left(\begin{array}[]{ccccccc}1&2&\cdots&i+1&i+3&\cdots&p+2\\\
k+1&k&\cdots&k-i+1&k-i&\cdots&k-p+1\end{array}\right);$ (49)

if $\min A_{i}<c<\max A_{i}$ ($i\in\\{1,2,\ldots,p\\}$) and $p<k$, then define
$\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{ccccccc}A_{1}&\cdots&\\{\min
A_{i},\ldots,c-1\\}&\\{c+1,\ldots,\max A_{i}\\}&A_{i+1}&\cdots&A_{p}\\\
1&\cdots&i&i+1&i+2&\cdots&p+1\end{array}\right)\textrm{ and }$ (52)
$\displaystyle\left(\begin{array}[]{ccccccc}1&\cdots&\\{i,i+1\\}&i+2&\cdots&p+1&p+2\\\
k&\cdots&k-i+1&k-i&\cdots&k-p+1&k-p\end{array}\right);$ (55)

if $\min A_{i}<c<\max A_{i}$ ($i\in\\{1,2,\ldots,p\\}$) and $p=k$, then define
$\beta$ and $\gamma$, respectively as:

$\displaystyle\left(\begin{array}[]{ccccccc}A_{1}&\cdots&\\{\min
A_{i},\ldots,c-1\\}&\\{c+1,\ldots,\max A_{i}\\}&A_{i+1}&\cdots&A_{p}\\\
2&\cdots&i+1&i+2&i+3&\cdots&p+2\end{array}\right)\textrm{ and }$ (58)
$\displaystyle\left(\begin{array}[]{cccccccc}1&2&\cdots&\\{i+1,i+2\\}&i+3&\cdots&p+1&p+2\\\
k+1&k&\cdots&k-i+1&k-i&\cdots&k-p+2&k-p+1\end{array}\right);$ (61)

if $\max A_{p}<c$ and $p<k$, then we define $\beta$ and $\gamma$, respectively
as:

$\displaystyle\left(\begin{array}[]{ccccc}A_{1}&A_{2}&\cdots&A_{p}&c\\\
1&2&\cdots&p&p+1\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}2&3&\cdots&p&p+2\\\
k&k-1&\cdots&k-p+1&k-p\end{array}\right)$ (66)

and if $\max A_{p}<c$ and $p=k$, then define $\beta$ and $\gamma$,
respectively as:

$\left(\begin{array}[]{ccccc}A_{1}&A_{2}&\cdots&A_{p}&c\\\
2&3&\cdots&p+1&p+2\end{array}\right)\textrm{ and
}\left(\begin{array}[]{cccccccc}1&2&3&\cdots&p+1\\\
k+1&k&k-1&\cdots&k-p+1\end{array}\right).$ (67)

Then, it is easy to see that $\beta,\gamma\in W_{p+1}$, and in all the cases
one can see that $\alpha=\beta\gamma$.

Case II: If $\textnormal{im }\alpha$ is not convex. Then $\alpha$ is of the
form $\left(\begin{array}[]{cccc}A_{1}&A_{2}&\cdots&A_{p}\\\
b_{p}&b_{p-1}&\cdots&b_{1}\end{array}\right)$ where $b_{1}<b_{2}<\ldots<b_{p}$
and $b_{i}-b_{i+1}\geq 2$ for some $i\in\\{1,2,\ldots,p-1\\}$. By contractive
property of $\alpha$ we must have $\min A_{i+1}-\max A_{i}\geq 2$ for such
$i$. Now, define

$\beta=\left(\begin{array}[]{cccccccc}A_{1}&A_{2}&\cdots&A_{i}&\max
A_{i}+1&A_{i+1}&\cdots&A_{p}\\\
b_{p}&b_{p-1}&\cdots&b_{p-i+1}&b_{p-i+1}-1&b_{p-i}&\cdots&b_{1}\end{array}\right).$

Observe that since $|\textnormal{im }\alpha|\leq n-2$, then $|\textnormal{im
}\beta|\leq n-1$. Let $c\in[n]\backslash\textnormal{im }\beta$ and let
$B=\textnormal{im }\beta\cup\\{c\\}$. Define $\gamma$ to be the partial
identity on $B$. Then one can easily verify that $\beta$ and $\gamma$ are in
$W_{p+1}$ and $\alpha=\beta\gamma$, as required.

∎

The following corollary is immediate:

######  Corollary 4.2.

For $n\geq 3$, $\langle W_{n-1}\rangle=\mathcal{ORCP}_{n,n-1}$.

###### Proof.

The proof follows from Lemma 4.1 and the fact that $\langle
W_{n-1}\rangle\leq\mathcal{ORCP}_{n-1}$. ∎

Recall the definitions of $\alpha_{s,t}$ and $\pi_{\\{i,i+1\\},k}$ (for some
$i,k,s,t$ in $\\{1,2,\ldots n\\}$) after Corollary 3.2, let $\alpha^{*}_{s,t}$
and $\pi^{*}_{\\{i,i+1\\},k}$ denote the order-reversing counterparts of
$\alpha_{s,t}$ and $\pi_{\\{i,i+1\\},k}$, respectively (which exist if
$t\in\\{1,n\\}$ $(1\leq s\leq n)$, or $t=s-n+1$ for some
$s\in\\{2,3,\ldots,n-1$). For the purpose of illustration, consider the
transformation $\alpha_{2,1}=\left(\begin{array}[]{ccccc}1&3&4&\cdots&n\\\
2&3&4&\cdots&n\end{array}\right)$. Then,
$\alpha^{*}_{2,1}=\left(\begin{array}[]{ccccc}1&3&4&\cdots&n\\\
n&n-1&n-2&\cdots&2\end{array}\right)$.

We now have elements of $W_{n-1}$ displayed in the table below, where $1\leq
i\leq n-1$ and $2\leq j\leq n-1$.

Kernel Partition / Image sets | $[n]\backslash\\{1\\}$ | $[n]\backslash\\{j\\}$ | $[n]\backslash\\{n-j+1\\}$ | $[n]\backslash\\{n\\}$
---|---|---|---|---
$P_{[n]\backslash\\{1\\}}$ | $\alpha_{1,1}$ |  |  | $\alpha_{1,n}$
$\alpha_{1,1}^{*}$ |  |  | $\alpha_{1,n}^{*}$
$P_{[n]\backslash\\{j\\}}$ | $\alpha_{j,1}$ | $\alpha_{j,j}$ |  | $\alpha_{j,n}$
$\alpha_{j,1}^{*}$ |  | $\alpha_{j,n-j+1}^{*}$ | $\alpha_{j,n}^{*}$
$P_{[n]\backslash\\{n\\}}$ | $\alpha_{n,1}$ |  |  | $\alpha_{n,n}$
$\alpha_{n,1}^{*}$ |  |  | $\alpha_{n,n}^{*}$
$P_{\\{i,i+1\\}}$ | $\pi_{(i,i+1),1}$ |  |  | $\pi_{(i,i+1),n}$
$\pi_{(i,i+1),1}^{*}$ |  |  | $\pi_{(i,i+1),n}^{*}$
(2)

Notice that by Theorem 2.1 each row in Table (2) represents a unique
$\mathcal{R}^{*}-$class in $W_{n-1}$, while columns represents
$\mathcal{L}^{*}-$classes. Therefore, the number of $\mathcal{R}^{*}-$classes
in $W_{n-1}$ is $2n-1$. The following lemmas follows immediately.

######  Lemma 4.3.

For $n\geq 3$, rank($\langle W_{n-1}\rangle)\geq 2n-1$.

###### Proof.

The proof is similar to the proof of Corollary 3.3. ∎

######  Lemma 4.4.

For $n\geq 3$,
$K_{n-1}\cup\\{\alpha_{n,n}^{*},\alpha_{1,1}^{*},\alpha_{j,n-j+1}^{*}:\,2\leq
j\leq n-1\\}$ generates $W_{n-1}$.

###### Proof.

Using routine multiplication we see that

$\displaystyle\alpha_{1,n}\cdot\alpha_{n,n}^{*}$ $\displaystyle=$
$\displaystyle\alpha_{1,n}^{*},$
$\displaystyle\alpha_{j,n}\cdot\alpha_{n,n}^{*}$ $\displaystyle=$
$\displaystyle\alpha_{j,n}^{*},$
$\displaystyle\alpha_{n,1}\cdot\alpha_{1,1}^{*}$ $\displaystyle=$
$\displaystyle\alpha_{n,1}^{*},$
$\displaystyle\alpha_{j,1}\cdot\alpha_{1,1}^{*}$ $\displaystyle=$
$\displaystyle\alpha_{j,1}^{*},$
$\displaystyle\pi_{\\{i,i+1\\},n}\cdot\alpha_{n,n}^{*}$ $\displaystyle=$
$\displaystyle\pi_{\\{i,i+1\\},n}^{*}\textrm{ and}$
$\displaystyle\pi_{\\{i,i+1\\},1}\cdot\alpha_{1,1}^{*}$ $\displaystyle=$
$\displaystyle\pi_{\\{i,i+1\\},1}^{*}.$

Therefore, $W_{n-1}\subseteq\langle
K_{n-1}\cup\\{\alpha_{n,n}^{*},\alpha_{1,1}^{*},\alpha_{j,n-j+1}^{*}\\}\rangle$.
Hence,
$K_{n-1}\cup\\{\alpha_{n,n}^{*},\alpha_{1,1}^{*},\alpha_{j,n-j+1}^{*}\\}$
generates $W_{n-1}$. ∎

The next proposition gives us a minimal generating set of $W_{n-1}$.

######  Proposition 4.5.

Let $M=\\{\alpha_{1,n},\pi_{\\{i,i+1\\},n},\alpha_{j,n-j+1}^{*}:1\leq j\leq
n,\,1\leq i\leq n-1\\}$. Then $M$ is a minimal generating set of $W_{n-1}$.

###### Proof.

By Lemma 4.3, if $B$ is any generating set of $W_{n-1}$, then $B$ must have at
least $2n-1$ elements. Furthermore, notice that $|M|=2n-1$. Thus, to show $M$
is a minimal generating set of $W_{n-1}$ we only need to show that $M$
generates $W_{n-1}$. Let
$Z=\\{\alpha_{n,n}^{*},\alpha_{1,1}^{*},\alpha_{j,n-j+1}^{*}:\,2\leq j\leq
n-1\\}$. Then, by Lemma 4.4, $K_{n-1}\cup\ Z$ generates $W_{n-1}$. Now, recall
that by Lemma 3.4, the set
$G=\\{\alpha_{n,1},\alpha_{1,n},\alpha_{j,j},\pi_{\\{i,i+1\\},n}:2\leq j\leq
n-1,\,1\leq i\leq n-1\\}$ generates $K_{n-1}$. Therefore, $G\cup
Z=\\{\alpha_{n,1},\alpha_{1,n},\alpha_{j,j},\pi_{\\{i,i+1\\},n},\alpha_{n,n}^{*},\alpha_{1,1}^{*},\alpha_{j,n-j+1}^{*}:2\leq
j\leq n-1,\,1\leq i\leq n-1\\}$ generates $W_{n-1}$.

However, not all elements in $G\cup Z$ are needed in order to generate
$W_{n-1}$, for, if $\alpha_{n,1}$ is replaced with $\alpha_{n,1}^{*}$, then we
observe that

$\displaystyle\alpha_{n,1}^{*}\cdot\alpha_{1,n}$ $\displaystyle=$
$\displaystyle\alpha_{n,n}^{*},$
$\displaystyle\alpha_{1,n}\cdot\alpha_{n,1}^{*}$ $\displaystyle=$
$\displaystyle\alpha_{1,1}^{*},$
$\displaystyle\alpha_{j,n-j+1}^{*}\cdot\alpha_{n-j+1,j}^{*}$ $\displaystyle=$
$\displaystyle\alpha_{j,j}\textnormal{ and}$
$\displaystyle\alpha_{n,1}^{*}\cdot\alpha_{1,n}\cdot\alpha_{n,1}^{*}$
$\displaystyle=$ $\displaystyle\alpha_{n,1}.$

Meaning that $\alpha_{n,n}^{*},\alpha_{1,1}^{*},\alpha_{n,1}$ and
$\alpha_{j,j}\,(2\leq j\leq n-1)$ are redundant elements.

Hence, $\\{\alpha_{1,n},\pi_{\\{i,i+1\\},n},\alpha_{j,n-j+1}^{*}:1\leq j\leq
n,\,1\leq i\leq n-1\\}$ generates $W_{n-1}$ as required. ∎

######  Corollary 4.6.

For $n\geq 3$, rank$\langle W_{n-1}\rangle=2n-1$.

###### Proof.

Follows from Proposition 4.5. ∎

We now have the rank of $\mathcal{OCP}_{n,n-1}$ in the following proposition.

######  Proposition 4.7.

Rank($\mathcal{OCP}_{n,n-1})=2n-1$ for all $n\geq 3$.

###### Proof.

Follows from Corollaries 4.2 and 4.6. ∎

Next, it is not difficult to see that the top $\mathcal{J}$-class of
$\mathcal{ORCP}_{n}$ (i. e., the $J_{n}$ class) contain only the elements,
$1_{[n]}$ and $1^{*}_{[n]}$, where
$1_{[n]}=\left(\begin{array}[]{ccccc}1&2&3&\cdots&n\\\
1&2&3&\cdots&n\end{array}\right)$. Thus, $\mathcal{ORCP}_{n}$ is a disjoint
union of $\mathcal{ORCP}_{n,n-1}$ and $\\{1_{[n]},1^{*}_{[n]}\\}$.
Consequently, we have

######  Theorem 4.8.

Let $\mathcal{ORCP}_{n}$ be as defined in equation (2). Then, the rank of
$\mathcal{ORCP}_{n}$ is $2n$ for all $n\geq 3$.

###### Proof.

First, notice that since $\mathcal{ORCP}_{n,n-1}$ is an ideal of
$\mathcal{ORCP}_{n}$, then it cannot generate any element in the upper
$\mathcal{J}$-class. Furthermore, by simple multiplication one can show that
$(1^{*}_{[n]})^{2}=1_{[n]}$, $(1_{[n]})^{2}=1_{[n]}$,
$1^{*}_{[n]}1_{[n]}=1^{*}_{[n]}$ and $1_{[n]}1^{*}_{[n]}=1^{*}_{[n]}$. Thus,
$1^{*}_{[n]}$ generates $\\{1_{[n]},1^{*}_{[n]}\\}$ and
$\\{1_{[n]},1^{*}_{[n]}\\}$ generates no other element in the lower
$\mathcal{J}$-class. Consequently, we have
rank($\mathcal{ORCP}_{n})=\textnormal{rank}(\mathcal{ORCP}_{n,n-1})+\textnormal{rank}(\\{1_{[n]},1^{*}_{[n]}\\})$,
which is $2n$. ∎

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An improvement of a saddle point theorem and some of its applications

BIAGIO RICCERI

Dedicated to the memory of Professor Wataru Takahashi

Abstract. In this paper, we establish an improved version of a saddle point
theorem ([4]) removing a weak lower semicontinuity assumption at all. We then
revisit some of the applications of that theorem in the light of such an
improvement. For instance, we obtain the following very general result of
local nature: Let $(H,\langle\cdot,\cdot\rangle)$ be a real Hilbert space and
$\Phi:B_{\rho}\to H$ a $C^{1,1}$ function, with $\Phi(0)\neq 0$. Then, for
each $r>0$ small enough, there exist only two points points $x^{*},u^{*}\in
S_{r}$, such that

$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$

for all $x\in B_{r}\setminus\\{x^{*}\\}$,

$\|\Phi(u^{*})-u^{*}\|=\hbox{\rm dist}(\Phi(u^{*}),B_{r})$

and

$\|\Phi(x)-u^{*}\|<\|\Phi(x)-x\|$

for all $x\in B_{r}\setminus\\{u^{*}\\}$, where

$B_{r}=\\{x\in H:\|x\|\leq r\\}$

and

$S_{r}=\\{x\in H:\|x\|=r\\}\ .$

Keywords. Saddle point; Hilbert space; ball; $C^{1,1}$ function; variational
inequality; best approximation point.

2010 Mathematics Subject Classification. 41A50, 41A52, 47J20, 49J35, 49J40.

In the sequel, $(H,\langle\cdot,\cdot\rangle)$ is a real Hilbert space. For
each $r>0$, set

$B_{r}=\\{x\in H:\|x\|\leq r\\}$

and

$S_{r}=\\{x\in H:\|x\|=r\\}\ .$

The aim of this paper is to establish the following result jointly with two
meaningful applications of it:

THEOREM 1. - Let $Y$ be a non-empty closed convex set in a Hausdorff real
topological vector space, let $\rho>0$ and let $J:B_{\rho}\times Y\to{\bf R}$
be a function satisfying the following conditions:

$(a_{1})$ for each $y\in Y$, the function $J(\cdot,y)$ is $C^{1}$ and
$J^{\prime}_{x}(\cdot,y)$ is Lipschitzian with constant $L$ (independent of
$y$) ;

$(a_{2})$ $J(x,\cdot)$ is upper semicontinuous and concave for all $x\in
B_{\rho}$ and $J(x_{0},\cdot)$ is sup-compact for some $x_{0}\in B_{\rho}$;

$(a_{3})$ $\delta:=\inf_{y\in Y}\|J^{\prime}_{x}(0,y)\|>0\ .$

Then, for each
$r\in\left]0,\min\left\\{\rho,{{\delta}\over{2L}}\right\\}\right]$ and for
each non-empty closed convex $T\subseteq Y$, there exist $x^{*}\in S_{r}$ and
$y^{*}\in T$ such that

$J(x^{*},y)\leq J(x^{*},y^{*})<J(x,y^{*})$

for all $x\in B_{r}\setminus\\{x^{*}\\}$, $y\in T$ .

PROOF. Fix $r\in\left]0,\min\left\\{\rho,{{\delta}\over{2L}}\right\\}\right]$
and a non-empty closed convex $T\subseteq Y$. Consider the function
$\varphi:B_{r}\times T\to{\bf R}$ defined by

$\varphi(x,y)={{L}\over{2}}\|x\|^{2}+J(x,y)$

for all $(x,y)\in B_{r}\times T$. Notice that, for each $y\in T$, the function
$\varphi(\cdot,y)$ is continuous and convex in $B_{r}$ (see the proof of
Corollary 2.3 of [3]). Consequently, if we consider $B_{r}$ endowed with the
relative weak topology, the function $\varphi$ satisfies the assumptions of a
classical minimax theorem ([1], Theorem 2) from which we infer

$\sup_{T}\inf_{B_{r}}\varphi=\inf_{B_{r}}\sup_{T}\varphi\ .$ $None$

The function $x\to\sup_{y\in T}\varphi(x,y)$ (resp. $y\to\inf_{x\in
B_{r}}\varphi(x,y)$) is weakly lower semicontinuous (resp. sup-compact).
Therefore, there exist $x^{*}\in B_{r}$ and $y^{*}\in T$ such that

$\sup_{y\in T}\varphi(x^{*},y)=\inf_{x\in B_{r}}\sup_{y\in T}\varphi(x,y)\ ,$
$\inf_{x\in B_{r}}\varphi(x,y^{*})=\sup_{y\in T}\inf_{x\in B_{r}}\varphi(x,y)\
.$

So, in view of $(1)$, we obtain

$\varphi(x^{*},y)\leq\varphi(x^{*},y^{*})\leq\varphi(x,y^{*})$ $None$

for all $x\in B_{r}$, $y\in T$. Notice that the equation

$J^{\prime}_{x}(x,y^{*})+Lx=0$

has no solution in the interior of $B_{r}$. Indeed, let $\tilde{x}\in
B_{\rho}$ be such that

$J^{\prime}_{x}(\tilde{x},y^{*})+L\tilde{x}=0\ .$

Then, in view of $(a_{1})$, we have

$\|L\tilde{x}+J^{\prime}_{x}(0,y^{*})\|\leq\|L\tilde{x}\|\ .$

In turn, using the Cauchy-Schwarz inequality, this readily implies that

$\|\tilde{x}\|\geq{{\|J^{\prime}_{x}(0,y^{*})\|}\over{2L}}\geq{{\delta}\over{2L}}\geq
r\ .$

From this remark, we infer that the set of all global minima of the function
$\varphi(\cdot,y^{*})$ is contained in $S_{r}$ and so, being convex, it
reduces to $x^{*}$ (recall that $X$ is a Hilbert space), in view of $(2)$.
Therefore, for every $x\in B_{r}\setminus\\{x^{*}\\}$, $y\in T$, from $(2)$ we
obtain

${{1}\over{2}}\|x^{*}\|^{2}+J(x^{*},y)\leq{{1}\over{2}}\|x^{*}\|^{2}+J(x^{*},y^{*})<{{1}\over{2}}\|x\|^{2}+J(x,y^{*})\leq{{1}\over{2}}\|x^{*}\|^{2}+J(x,y^{*})$

and so

$J(x^{*},y)\leq J(x^{*},y^{*})<J(x,y^{*})\ .$

The proof is complete. $\bigtriangleup$

REMARK 1. - Theorem 1 was obtained in [4] ([4], Theorem 2.1) under the
following additional assumption: for each $y\in Y$, the function $J(\cdot,y)$
is weakly lower semicontinuous. This was due to the fact that, instead of
applying the classical minimax theorem in [1] to $\varphi$ (as we did above),
we applied Theorem 1.2 of [2] to $J$.

We can now revisit two applications of Theorem 1. The first one concerns
variational inequalities.

THEOREM 2. - Let $\rho>0$ and let $\Phi:B_{\rho}\to H$ be a $C^{1}$ function
whose derivative is Lipschitzian with constant $\gamma$. Set

$\theta:=\sup_{x\in B_{\rho}}\|\Phi^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$M:=2(\theta+\rho\gamma)$

and assume also that

$\sigma:=\inf_{y\in
B_{\rho}}\sup_{\|u\|=1}|\langle\Phi(0),u\rangle-\langle\Phi^{\prime}(0)(u),y\rangle|>0\
.$

Then, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{2M}}\right\\}\right]$, there
exists a unique $x^{*}\in S_{r}$ such that

$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$

for all $x\in B_{r}\setminus\\{x^{*}\\}$.

PROOF. Consider the function $J:B_{\rho}\times B_{\rho}\to{\bf R}$ defined by

$J(x,y)=\langle\Phi(x),x-y\rangle$

for all $x,y\in B_{\rho}$. Of course, for each $y\in B_{\rho}$, the function
$J(\cdot,y)$ is $C^{1}$ and one has

$\langle
J^{\prime}_{x}(x,y),u\rangle=\langle\Phi^{\prime}(x)(u),x-y\rangle+\langle\Phi(x),u\rangle$

for all $x\in B_{\rho},u\in H$. Fix $x,v\in B_{\rho}$ and $u\in S_{1}$. We
then have

$|\langle J^{\prime}_{x}(x,y),u\rangle-\langle
J^{\prime}_{x}(v,y),u\rangle|=|\langle\Phi(x)-\Phi(v),u\rangle+\langle\Phi^{\prime}(x)(u),x-y\rangle-\langle\Phi^{\prime}(v)(u),v-y\rangle|$
$\leq\|\Phi(x)-\Phi(v)\|+|\Phi^{\prime}(x)(u)-\Phi^{\prime}(v)(u),v-y\rangle+\langle\Phi^{\prime}(x)(u),x-v\rangle|$
$\leq\theta\|x-v\|+2\rho\|\Phi^{\prime}(x)-\Phi^{\prime}(v)\|_{{\cal
L}(H)}+\theta\|x-v\|$ $\leq 2(\theta+\rho\gamma)\|x-v\|\ .$

Hence, the function $J^{\prime}_{x}(\cdot,y)$ is Lipschitzian with constant
$M$. At this point, we can apply Theorem 1 taking $Y=B_{\rho}$ with the weak
topology. Therefore, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{2M}}\right\\}\right]$, there
exist $x^{*}\in S_{r}$ and $y^{*}\in B_{r}$ such that

$\langle\Phi(x^{*}),x^{*}-y\rangle\leq\langle\Phi(x^{*}),x^{*}-y^{*}\rangle<\langle\Phi(x),x-y^{*}\rangle$
$None$

for all $x,y\in B_{r}$, with $x\neq x^{*}$. Notice that $\Phi(x^{*})\neq 0$.
Indeed, if $\Phi(x^{*})=0$, we would have

$\|\Phi(0)\|=\|\Phi(0)-\Phi(x^{*})\|\leq\theta r$

and hence, since $\sigma\leq\|\Phi(0)\|$, it would follow that

$r\leq{{\|\Phi(0)\|}\over{2M}}<{{\|\Phi(0)\|}\over{\theta}}\leq r\ .$

From the first inequality in $(3)$, taking $y=x^{*}$, we get
$0\leq\langle\Phi(x^{*}),x^{*}-y^{*}\rangle$. So, in view of the strict
inequality, we infer that $x^{*}=y^{*}$ (since, otherwise, we could take
$x=y^{*}$, obtaing a contradiction). Thus, $(3)$ actually reads

$\langle\Phi(x^{*}),x^{*}-y\rangle\leq 0<\langle\Phi(x),x-x^{*}\rangle$

for all $x,y\in B_{r}$, with $x\neq x^{*}$. In particular, we infer that
$x^{*}$ is the unique global minimum in $B_{r}$ of the linear functional
$y\to\langle\Phi(x^{*}),y\rangle$. Hence

$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$

for all $x\in B_{r}\setminus\\{x^{*}\\}$. Finally, to show the uniqueness of
$x^{*}$, argue by contradiction, supposing that there is another $\tilde{x}\in
S_{r}$, with $\tilde{x}\neq x^{*}$, such that

$\max\\{\langle\Phi(\tilde{x}),\tilde{x}-x\rangle,\langle\Phi(x)),\tilde{x}-x\rangle\\}<0$

for all $x\in B_{r}\setminus\\{\tilde{x}\\}$. So, we would have at the same
time

$\langle\Phi(\tilde{x}),\tilde{x}-x^{*}\rangle<0$

and

$\langle\Phi(\tilde{x}),x^{*}-\tilde{x}\rangle<0\ ,$

an absurd, and the proof is complete. $\bigtriangleup$

REMARK 2. - Theorem 2 was obtained in [4] ([4], Theorem 2.2) under the
following additional assumption: for each $y\in B_{\rho}$, the function
$x\to\langle\Phi(x),x-y\rangle$ is weakly lower semicontinuous.

A remarkable corollary of Theorem 2 is as follows:

THEOREM 3. - Let $\rho>0$ and let $\Phi:B_{\rho}\to H$ be a $C^{1}$ function
with Lipschitzian derivative.

Then, the following assertions are equivalent:

$(i)$ for each $r>0$ small enough, there exists a unique $x^{*}\in S_{r}$ such
that

$\max\\{\langle\Phi(x^{*}),x^{*}-x\rangle,\langle\Phi(x),x^{*}-x\rangle\\}<0$

for all $x\in B_{r}\setminus\\{x^{*}\\}$ ;

$(ii)$ $\Phi(0)\neq 0$ .

PROOF. The implication $(ii)\to(i)$ is obvious. So, assume that $(i)$ holds.
Notice that the function
$y\to\sup_{\|u\|=1}|\langle\Phi(0),u\rangle-\langle\Phi^{\prime}(0)(u),y\rangle|$
is continuous in $H$ and takes the value $\|\Phi(0)\|>0$ at $0$. Therefore,
for a suitable $r^{*}\in]0,\rho]$, we have

$\inf_{y\in
B_{r^{*}}}\sup_{\|u\|=1}\langle\Phi(0),u\rangle-\langle\Phi^{\prime}(0)(u),y\rangle|>0\
.$

Now, we can apply Theorem 2 to the restriction of the function $\Phi$ to
$B_{r^{*}}$, and $(i)$ follows. $\bigtriangleup$

Also, it is worth noticing the following further corollary of Theorem 2:

THEOREM 4. - Let $\rho>0$ and let $\Psi:B_{\rho}\to H$ be a $C^{1}$ function
whose derivative vanishes at $0$ and is Lipschitzian with constant
$\gamma_{1}$. Set

$\theta_{1}:=\sup_{x\in B_{\rho}}\|\Psi^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$M_{1}:=2(\theta_{1}+\rho\gamma_{1})$

and let $w\in H$ satisfy

$\|w-\Psi(0)\|\geq 2M_{1}\rho\ .$ $None$

Then, for each $r\in]0,\rho]$, there exists a unique $x^{*}\in S_{r}$ such
that

$\max\\{\langle\Psi(x^{*})-w,x^{*}-y\rangle,\langle\Psi(y)-w,x^{*}-y\rangle\\}<0$

for all $y\in B_{r}\setminus\\{x^{*}\\}$.

PROOF. Set $\Phi:=\Psi-w$. Apply Theorem 2 to $\Phi$. Since
$\Phi^{\prime}=\Psi^{\prime}$, we have $M=M_{1}$. Since $\Phi^{\prime}(0)=0$,
we have $\sigma=\|\Phi(0)\|$ and so, in view of $(4)$,

$\rho\leq{{\sigma}\over{2M}}$

and the conclusion follows. $\bigtriangleup$

The second application of Theorem 1 is as follows:

THEOREM 5. - Let $Y\subseteq H$ be a closed bounded convex set, let $\rho>0$
and let $f:B_{\rho}\to H$ be a $C^{1}$ function whose derivative is
Lipschitzian with constant $\gamma$. Moreover, let $\eta$ be the Lipschitz
constant of the function $x\to x-f(x)$, set

$\theta:=\sup_{x\in B_{\rho}}\|f^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$L:=2\left(\eta+\theta+\gamma\left(\rho+\sup_{y\in Y}\|y\|\right)\right)$

and assume that

$\sigma:=\inf_{y\in Y}\sup_{\|u\|=1}|\langle f^{\prime}(0)(u),y\rangle-\langle
f(0),u\rangle|>0\ .$

Then, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$ and for each
non-empty closed convex set $T\subseteq Y$, there exist $x^{*}\in S_{r}$ and
$y^{*}\in T$ such that

$\|x^{*}-f(x^{*})\|^{2}+\|f(x)-y^{*}\|^{2}-\|x-f(x)\|^{2}<\|f(x^{*})-y^{*}\|^{2}=(\hbox{\rm
dist}(f(x^{*}),T))^{2}$ $None$

for all $x\in B_{r}\setminus\\{x^{*}\\}$ .

PROOF. Consider the function $J:B_{\rho}\times Y\to{\bf R}$ defined by

$J(x,y)=\|f(x)-x\|^{2}-\|f(x)-y\|^{2}$

for all $x\in B_{\rho}$, $y\in Y$. Clearly, for each $y\in Y$, $J(\cdot,y)$ is
of class $C^{1}$. Moreover, one has

$\langle J^{\prime}_{x}(x,y),u\rangle=2\langle x-f(x),u\rangle-2\langle
f^{\prime}(x)(u),x-y\rangle$

for all $x\in B_{\rho}$, $u\in H$. Fix $x,v\in B_{\rho}$ and $u\in H$, with
$\|u\|=1$. We have

${{1}\over{2}}|\langle
J^{\prime}_{x}(x,y)-J^{\prime}_{x}(v,y),u\rangle|=|\langle
x-f(x)-v+f(v),u\rangle-\langle f^{\prime}(x)(u),x-y\rangle+\langle
f^{\prime}(v)(u),v-y\rangle|$ $\leq\eta\|x-v\|+|\langle
f^{\prime}(x)(u),x-v\rangle+\langle
f^{\prime}(x)(u)-f^{\prime}(v)(u),v-y\rangle|$
$\leq\eta\|x-v\|+\|f^{\prime}(x)(u)\|\|x-v\|+\|f^{\prime}(x)(u)-f^{\prime}(v)(u)\|\|v-y\|\leq\left(\eta+\theta+\gamma\left(\rho+\sup_{y\in
Y}\|y\|\right)\right)\|x-v\|\ .$

Therefore, the function $J_{x}^{\prime}(\cdot,y)$ is Lipschitzian with
constant $L$. Moreover, we clearly have

$\inf_{y\in Y}\|J^{\prime}_{x}(0,y)\|=2\sigma\ .$

So, if we fix
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$ and a non-
empty closed convex set $T\subseteq Y$, by Theorem 1, there exist $x^{*}\in
B_{r}$ and $y^{*}\in T$ such that

$\|f(x^{*})-x^{*}\|^{2}-\|f(x^{*})-y\|^{2}\leq\|f(x^{*})-x^{*}\|^{2}-\|f(x^{*})-y^{*}\|^{2}<\|f(x)-x\|^{2}-\|f(x)-y^{*}\|^{2}$
$None$

for all $x\in B_{r}\setminus\\{x^{*}\\}$, $y\in T$. Clearly, $(6)$ is
equivalent to $(5)$, and the proof is complete. $\bigtriangleup$ REMARK 2. -
Theorem 5 was obtained in [3] ([3], Corollary 2.5) assuming, in addition, that
$f$ is sequentially weakly-strongly continuous.

Here is a remarkable consequence of Theorem 5.

THEOREM 6. - Let $\rho>0$ and let $f:B_{\rho}\to H$ be a $C^{1}$ function
whose derivative is Lipschitzian with constant $\gamma$. Moreover, let $\eta$
be the Lipschitz constant of the function $x\to x-f(x)$, set

$\theta:=\sup_{x\in B_{\rho}}\|f^{\prime}(x)\|_{{\cal L}(H)}\ ,$
$L:=2(\eta+\theta+2\gamma\rho)$

and assume that

$\sigma:=\inf_{y\in B_{\rho}}\sup_{\|u\|=1}|\langle
f^{\prime}(0)(u),y\rangle-\langle f(0),u\rangle|>0\ .$

Then, for each
$r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$, there
exists a unique $x^{*}\in S_{r}$ such that

$\|f(x^{*})-x^{*}\|=\hbox{\rm dist}(f(x^{*}),B_{r})$ $None$

and

$\|f(x)-x^{*}\|<\|f(x)-x\|$ $None$

for all $x\in B_{r}\setminus\\{x^{*}\\}$.

PROOF. Fix $r\in\left]0,\min\left\\{\rho,{{\sigma}\over{L}}\right\\}\right]$.
Applying Theorem 5 with $Y=B_{\rho}$ and $T=B_{r}$, we obtain $x^{*}\in S_{r}$
and $y^{*}\in B_{r}$ such that

$\|x^{*}-f(x^{*})\|^{2}+\|f(x)-y^{*}\|^{2}-\|x-f(x)\|^{2}<\|f(x^{*})-y^{*}\|^{2}=(\hbox{\rm
dist}(f(x^{*}),B_{r}))^{2}$ $None$

for all $x\in B_{r}\setminus\\{x^{*}\\}$ . From this, we infer that
$y^{*}=x^{*}$. Actually, if $y^{*}\neq x^{*}$, we could take $x=y^{*}$ in
$(9)$, obtaining

$\|x^{*}-f(x^{*})\|<\hbox{\rm dist}(f(x^{*}),B_{r})$

which is absurd. Now, $(7)$ and $(8)$ follow directly from $(9)$. Finally,
concerning the uniqueness of $x^{*}$, assume that $x_{0}\in B_{r}$ is such
that

$\|f(x)-x_{0}\|<\|f(x)-x\|$

for all $x\in B_{r}\setminus\\{x_{0}\\}$. Then, if $x_{0}\neq x^{*}$, we would
have

$\|f(x^{*})-x_{0}\|<\|f(x^{*})-x^{*}\|$

which is absurd in view of $(7)$. $\bigtriangleup$

Finally, reasoning as in the proof of Theorem 3, we get the following
corollary of Theorem 6:

THEOREM 7. - Let $\rho>0$ and let $f:B_{\rho}\to H$ be a $C^{1}$ function with
Lipschitzian derivative.

Then, the following assertions are equivalent:

$(i)$ for each $r>0$ small enough, there exists a unique $x^{*}\in S_{r}$ such
that

$\|f(x^{*})-x^{*}\|=\hbox{\rm dist}(f(x^{*}),B_{r})$

and

$\|f(x)-x^{*}\|<\|f(x)-x\|$

for all $x\in B_{r}\setminus\\{x^{*}\\}$ ;

$(ii)$ $f(0)\neq 0$ .

Acknowledgement. The author has been supported by the Università di Catania,
PIACERI 2020-2022, Linea di intervento 2, Progetto “MAFANE” and by the Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References

[1] K. FAN, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A., 39 (1953), 42-47.

[2] B. RICCERI, On a minimax theorem: an improvement, a new proof and an
overview of its applications, Minimax Theory Appl., 2 (2017), 99-152.

[3] B. RICCERI, Applying twice a minimax theorem, J. Nonlinear Convex Anal.,
20 (2019), 1987-1993.

[4] B. RICCERI, A remark on variational inequalities in small balls, J.
Nonlinear Var. Anal., 4 (2020), 21-26.

Department of Mathematics and Informatics

University of Catania

Viale A. Doria 6

95125 Catania, Italy

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