%&latex
%  File: OSF3.tex
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%  Ordering of Fuzzy sets -- A brief survey and new results
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%
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%  Kurano note 1998-12-07
%  Input at 98-12-13 by Yasuda
%  accepted at 99-03-18 JORSJ
%  1st revised at 99-05-15 by Nakagami
%  2nd revised at 99-11-15 by Nakagami


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%----------------------Definition------------------------
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% \def\T#1{{\tilde{#1}}}
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%\newcommand{\succcurlyeq}{{\preccurlyeq}}
\newcommand{\af}{{\alpha}}
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%----------------------Definition------------------------
 
 
\begin{document}%----------------------------------------
 
\begin{center}
\begin{Large}
{\bf Ordering of convex fuzzy sets \\ --- A brief survey and new results}
\end{Large}
\end{center}
 
\baselineskip 5.6mm
 
\vspace{2ex}

\begin{center}

M. Kurano(Chiba Univ.) \ 
M. Yasuda(Chiba Univ.) 

J. Nakagami(Chiba Univ.) \ and \ 
Y. Yoshida(Kitakyushu Univ.)


\end{center}

 
\medskip
 
%\begin{small}
%{\it
%${}^{\dag}${Faculty of Education,}
%${}^{\ddag}${Faculty of Science, Chiba University,}
%Yayoi-cho,Inage-ku,Chiba 263-8522, Japan.
%${}^{\S}${Faculty of Economics and Business Administration,
%Kitakyushu University,}
%Kokuraminami, Kitakyushu 802-8577, Japan
%}
%\end{small}
 
% \vspace{2ex}
% \begin{small}
% \noindent
% \hspace*{3cm} 千葉大学教育学部　　蔵野正美　(Masami KURANO)\\
% \hspace*{3cm} 千葉大学理学部　　　安田正實　(Masami YASUDA)\\
% \hspace*{3cm} 千葉大学理学部　　　中神潤一　(Jun-ichi NAKAGAMI)\\
% \hspace*{3cm} 北九州大学経済学部　吉田祐治　(Yuji YOSHIDA)
% \end{small}
 
 
\begin{abstract}

Concerning with the topics of a fuzzy max order, 
a brief survey on ordering of fuzzy numbers is presented in this article,
and we will consider an extension to that of fuzzy sets.
An extension of the fuzzy max order as
a pseudo order is investigated and defined on a class
of fuzzy sets on  $\BR^n (n \ge 1)$.
This order is developed by using a non-empty closed convex cone and
characterized by the projection into its dual cone.
Especially a structure of the lattice can be illustrated with the class
of rectangle-type fuzzy sets.
 
\end{abstract}
 
\noindent
{\it Keywords} : 
Fuzzy set, fuzzy max order, pseudo order, $n$-dimensional fuzzy set,
rectangle-type fuzzy set.
 
%--------------------- Section 1  -----------------------------
\section*{1. Introduction }
Fuzzy set theory has made applications in many fields of
management science, operations research and statistics
(cf.\cite{KG88, LY95, S98}), 
in which ordering or ranking fuzzy sets
is a fundamental problem in fuzzy optimization or
fuzzy decision making. 
Many methods of ordering fuzzy numbers have been proposed 
in the literature 
(see, for example, the survey paper, Bortolan and Degani\cite{BD85}). 
Also, in multiple criteria decision making several procedures 
for ranking fuzzy multicriteria alternatives are investigated 
(for example, see \cite{CL93, S93}).
Each method has its own advantages and disadvantages, so that
the ordering method should be chosen to be suitable for the particular
problem.
Among various ordering methods, a partial order on the set of fuzzy
numbers, called the fuzzy max order, introduced
by Ram{\'\i}k and {\u R}im{\'a}nek\cite{RR85} 
is very interesting in the concerns of pure mathematics
because it is a natural extension of the order over real numbers and includes
many theoretical and applicable potentials.
 
In this paper, concentrating on the fuzzy max order, we present a brief
survey
on ordering fuzzy sets in a real line $\BR$, 
which motivates our new attempt
of ordering high-dimensional fuzzy sets. The fuzzy max order of fuzzy
numbers
is extended to a pseudo order on a class of fuzzy sets defined on an
$n$-dimensional Euclidean space $\BR^n$.
 
The pseudo order for fuzzy sets is defined 
by a non-empty closed convex cone
$K$ in $\BR^n$ and characterized by the projection into its dual cone
$K^+$. Also, the structure of a lattice is discussed for the class of
rectangle-type fuzzy sets.
So, we can imagine the much wider application to the fuzzy
optimization problem.
Our idea of the motivation originates from a set-relation in $\BR^n$
given by Kuroiwa\cite{K98}, Kuroiwa, Tanaka and Ha\cite{KTH97}, 
in which various types
of set-relations in $\BR^n$ are used in set-valued optimizations.
 
The outline for this paper is as follows: The next section contains some
notations and basic concepts of fuzzy set theory
referring to the text books (cf. \cite{DP80, N89});
several methods of ordering fuzzy sets of a real line are surveyed
concentrating on the fuzzy max order and its related topics in the
third section; a pseudo order on the class of fuzzy sets on $\BR^n$ is
originally introduced as an extension of the fuzzy max order.
Its characterization and the structure of a lattice is considered
in the fourth section.



%--------------------- Section 2  -----------------------------
\section*{2. Notations and basic concepts}
In this section we describe notations and basic concepts
of fuzzy set theory (cf. \cite{DP80, KG88, N89, SW70, Z65}). 

Let $\BR$ be the set of all real numbers and
$\BR^n$ an $n$-dimensional Euclidean space.
We write a fuzzy set on $\BR^n$ by its membership function
$\T{s} : {\BR^n} \to [0,1]$
(cf. \cite{N89, Z65}).
The $\af$-cut ($\af \in [0,1]$) of the fuzzy set $\T{s}$ on $\BR^n$
is defined as
$$
\T{s}_{\af} := \{ x \in \BR^n \mid \T{s}(x) \geq \af \}
            \ (\af > 0) \quad \mbox{and} \quad
\T{s}_{0} := \mbox{\rm cl} \{ x \in {\BR^n} \mid \T{s}(x) > 0 \},
$$
where {\rm cl} denotes the closure of the set.
A fuzzy set $\T{s}$ is called convex if
$$
\T{s}(\ld x + (1- \ld )y) \ge \T{s}(x) \wedge \T{s}(y)
\quad x,y \in {\BR^n}, \ \ld \in [0,1],
$$
where $a \wedge b := \min\{a,b\}$.
Note that $\T{s}$ is convex if and only if the $\af$-cut $\T{s}_{\af}$ is
a convex set for all $\af \in [0,1]$.
Let $\C{F}({\BR^n})$ be the set of all convex fuzzy sets
whose membership functions $\T{s} : {\BR^n} \to [0,1]$
are upper-semicontinuous and
normal ($\sup_{x \in \BR^n} \T{s}(x) = 1$)
and have a compact support.
When the one-dimensional case $n=1$,
the fuzzy sets are called fuzzy numbers and
$\C{F}(\BR)$ denotes the set of all fuzzy numbers.
 
Let $\C{C}(\BR^n)$ be the set of all compact convex subsets of $\BR^n$,
and $\C{C}_{r}(\BR^n)$ be the set of all rectangles in $\BR^n$.
For $\T{s} \in \C{F}(\BR^n)$,
we have $\T{s}_{\af} \in \C{C}(\BR^n)$ ($\af \in [0,1]$).
We write a rectangle in $\C{C}_{r}(\BR^n)$ by
$$
[x,y] =
[x_1,y_1] \times [x_2,y_2] \times \cdots \times [x_n,y_n]
$$
for $x = (x_1,x_2,\cdots,x_n), y = (y_1,y_2,\cdots,y_n)
\in \BR^n$ with $x_{i} \le y_{i} \ (i=1,2,\cdots,n)$.
For the case of $n=1$, $\C{C}(\BR) = \C{C}_{r}(\BR)$
and it denotes the set of all bounded closed intervals.
When $\T{s} \in \C{F}(\BR^n)$ satisfies
$\T{s}_{\af} \in \C{C}_{r}(\BR^n)$ for all $\af \in [0,1]$,
$\T{s}$ is called a rectangle-type.
We denote by $\C{F}_{r}(\BR^n)$
the set of all rectangle-type fuzzy sets on $\BR^n$.
Obviously $\C{F}_{r}(\BR) = \C{F}(\BR)$.
 
 
Here we give the extension principle introduced by
Zadeh which provides a general method
for fuzzification of non-fuzzy mathematical concepts.
 
\medskip
\noindent
{\bf The extension principle} (cf. \cite{DP80}): 

\smallskip
\noindent
{\sl
Let $f$ be a map from $\BR^n$ to $\BR$ such that
$y = f(x_1,\cdots,x_n)$.
It allows us to induce a map from 
$n$ fuzzy sets $\T{s_i}$ on $\BR$ 
to a fuzzy set $\T{s}$ on $\BR$ through $f$ such that
$$
 \T{s}(y) := \sup_{\scriptstyle
                   {x_1,\cdots,x_n \in \BR} \atop  {y = f(x_1,\cdots,x_n)} 
                   }
   \{   \min(\T{s_1}(x_1), \cdots, \T{s_n}(x_n))   \}
$$
where \quad $\T{s}(y) :=0$ \ if \ $f^{-1}(y) = \phi$. 
}

\medskip

Applying the extension principle, the addition and the scalar
multiplication on $\BR$ are extended to those on $\C{F}(\BR)$
as follows:
 
For
$\T{s}, \T{r} \in \C{F}(\BR)$ and $\ld \geq 0$,
$$
(\T{s} + \T{r})(x) :=
  \sup_{\scriptstyle{x_1, x_2 \in \BR}\atop {\ x_1 + x_2 = x}}
     \{ \T{s}(x_1) \wedge \T{r}(x_2) \} ,
\leqno{(2.1)}
$$
\vspace*{-5mm}
$$
  ( \ld \T{s})(x) :=
     \left\{
       \begin{array}{ll}
        \T{s}(x/\ld) & \mbox{if} \ \ld>0 \\
        I_{\{0\}}(x) & \mbox{if} \ \ld=0
       \end{array}
       \quad (x \in \BR),
     \right.
\leqno{(2.2)}
$$
where $I_{\{\cdot\}}(\cdot)$ is an indicator.
By using set operations $A + B := \{ x+y \mid x \in A, y \in B\}$
and $\ld A := \{ \ld x \mid x \in A \}$
for any non-empty sets $A, B \subset \BR$,
the following holds immediately:
$$
(\T{s} + \T{r})_{\af} = \T{s}_{\af} + \T{r}_{\af}
\quad \mbox{and} \quad
(\ld \T{s})_{\af} = \ld \T{s}_{\af}
\quad (\af \in [0,1]).
\leqno{(2.3)}
$$

Also, for $\T{s}, \T{r} \in \C{F}(\BR)$, 
$\widetilde{\max}(\T{s},\T{r})$ and $\widetilde{\min}(\T{s},\T{r})$ 
are defined by
$$
\widetilde{\max}(\T{s},\T{r})(y) :=
\sup_{\scriptstyle{ x_1, x_2 \in \BR} \atop {y=\max(x_1,x_2)}}
 \{ \T{s}(x_1) \wedge \T{r}(x_2) \}
\leqno{(2.4)}
$$
\vspace*{-2mm}
and
$$
\widetilde{\min}(\T{s},\T{r})(y) :=
\sup_{\scriptstyle{ x_1, x_2 \in \BR} \atop {y=\min(x_1,x_2)}}
 \{ \T{s}(x_1) \wedge \T{r}(x_2) \}
\leqno{(2.5)}
$$
%\vspace*{-3mm}
respectively. The images of 
$\widetilde{\max}(\T{s},\T{r})$ and $\widetilde{\min}(\T{s},\T{r})$ 
are illustrated as follows:

\bigskip

%%%%%  Figure 1 %%%%%
 \begin{center}
\includegraphics[width=12cm,height=5cm,keepaspectratio]{figure1.eps}
  \vskip-4.9cm
  \hskip-8.5cm $1$
  \vskip0.5cm 
  \hskip6mm $\widetilde{s}$
  \hskip3.8cm $\widetilde{r}$
  
  \hskip75mm 
  
  \hskip75mm 
  \vskip15mm
  
  \hskip-8.5cm 0 
  
 Fig.1: $\widetilde{s}$ and $\widetilde{r}$

\includegraphics[width=12cm,height=5cm,keepaspectratio]{figure2.eps}
  \vskip-4.9cm
  \hskip-8.5cm $1$
  \vskip0.5cm 
  \hskip6mm 
  \hskip3.8cm 
  
  \hskip75mm \rule{10pt}{1pt} $\widetilde{\max}(\widetilde{s},\widetilde{r})$
  
  \hskip75mm \rule{10pt}{0.2pt} $\widetilde{\min}(\widetilde{s},\widetilde{r})$
  \vskip15mm
  
  \hskip-8.5cm 0 
  
 Fig.2: $\widetilde{\max}(\widetilde{s},\widetilde{r})$ 
          and $\widetilde{\min}(\widetilde{s},\widetilde{r})$ 
\end{center}

\medskip

We need a representative theorem
(cf. \cite{DP80, N89}) 
which is a basic tool for the fuzzy interval analysis.

\medskip

\noindent
{\bf The representative theorem}:
\vspace*{-2mm}
{\sl
\begin{itemize}
 \item[(i)]
    For any $\T{s} \in \C{F}(\BR^n)$,
    $\displaystyle  \T{s}(x) = \sup_{\alpha \in [0,1]}
                 \{ \alpha \wedge {\bf 1}_{\T{s}_\alpha}(x)\} ,
                 \quad x \in \BR.$
\vspace*{-3mm}
 \item[(ii)]
    Conversely, \ for a family of subsets
    $\{ D_\alpha \in \C{C}(\BR^n) \mid 0 \le \alpha \le 1\}$
    with $D_\alpha \subset D_{\alpha'}$ \\ 
    for $\alpha' \le \alpha$ and 
    $\displaystyle \cap_{\alpha' < \alpha} D_{\alpha'} = D_\alpha$,
    \ \ we set 
    $\displaystyle \T{s}(x) := \sup_{\alpha \in [0,1]}
                 \{ \alpha \wedge {\bf 1}_{D_\alpha}(x) \},
                 \ x \in \BR$, \\  
    then $\T{s}$ 
    belongs to $\C{F}(\BR^n)$ and satisfies $\T{s}_\alpha = D_\alpha,
    \ \alpha \in [0,1]$.
 
\end{itemize}
 }



%--------------------- Section 3  -----------------------------
\section*{3. A brief survey on ordering of fuzzy numbers}
In this section, we give a brief survey on method for ordering
fuzzy numbers which is mainly concerning to the fuzzy max order.


%--------------------- Section 3.1  -----------------------------
\subsection*{3.1. Fuzzy max order}
The following binary relation $\inc$ has been formulated
first by  Ram{\'\i}k and {\u R}im{\'a}nek\cite{RR85}.
  Let $\T{s}$ and $\T{r}$ be two fuzzy numbers. Then
$\T{s} \ \inc \ \T{r}$ if and only if $\sup \T{s}_\alpha \le \sup \T{r}_\alpha $
and $\inf \T{s}_\alpha \le \inf \T{r}_\alpha $ for each $\alpha
\in [0,1]$, 
where $\T{s}_\alpha$ and $\T{r}_\alpha$ are $\alpha$-cuts of
$\T{s}$ and $\T{r}$ respectively. 
Without any confusion we put 
$\T{s}_\alpha := [ \inf\T{s}_\alpha, \sup\T{s}_\alpha ]$ and 
$\T{r}_\alpha := [ \inf\T{r}_\alpha, \sup\T{r}_\alpha ]$.
 Obviously the binary relation $\inc $ satisfies the axioms of
a partial order relation on $\C{F}(\BR)$ and is called
the fuzzy max order.
 
  Some properties of the relation $\inc$ are investigated in
 Ram{\'\i}k and {\u R}im{\'a}nek\cite{RR85}.
 
 \medskip
 \noindent
{\bf Proposition 1}. \ 
{\sl 
Let $\T{s}, \T{r}$ be fuzzy numbers.
\begin{itemize}
 \item[(i)] The inequality $\T{s} \ \inc \ \T{r}$ if and only if there are
   $m, n , t^* \in \BR$ with $m \le t^* \le n$,
   $\T{s}(m) = \T{r}(n) = 1$ and
   $ \T{s}(t) \ge \T{r}(t) $ for any $ t \le t^*$ and
   $ \T{s}(t) \le \T{r}(t) $ for any $ t > t^*$.
 \item[(ii)]
   The following three conditions (a) to (c) are equivalent: \\
    (a) $\T{s} \ \inc \ \T{r}$, \quad
    (b) $\T{\max}\{\T{s},\T{r}\} = \T{r}$, \quad
    (c) $\T{\min}\{\T{s},\T{r}\} = \T{s}$, \\
    where $\widetilde{\min}$ and $\widetilde{\max}$ are defined in
 (2.4) and (2.5).
\end{itemize}
 }

For the fuzzy max order on fuzzy numbers, Congxin and Cong\cite{CC97}
proved that the bounded set of fuzzy numbers must have supremum
and infimum. The basic proof is as follows.
For any sequence of fuzzy numbers 
$\displaystyle{ \{ \T{s}_n \}_{n=1}^\infty }$, let
$\displaystyle{ 
   \underline{D}_\alpha := \lim_{\alpha' \uparrow \alpha} 
   \sup_{n \ge 1} \inf \T{s}_{n\alpha'} }$
and
$\displaystyle{ 
   \overline{D}_\alpha :=  \lim_{\alpha' \uparrow \alpha} 
   \sup_{n \ge 1} \sup \T{s}_{n\alpha'} }$, 
where $\T{s}_{n\alpha}$ is the $\af$-cut of $\T{s}_{n}$. 
Then, the family of closed subsets
  $\{D_\alpha := [ \underline{D}_\alpha , \overline{D}_\alpha ] \mid
    \alpha \in [0,1]\} $
satisfies condition (ii) of the representation theorem, 
so that $\T{s}$ defined by
$$
  \T{s}(x) := \sup_{\alpha \in [0,1]} \{ \alpha \wedge {\bf 1}_{D_\alpha}(x) \}
  \quad (x \in \BR)
$$
belongs to $\C{F}(\BR)$ and 
$\displaystyle{ \T{s} = \sup_{n \ge 1} \T{s}_n }$.
 
Similarly, the infimum of 
$\displaystyle{ \{ \T{s}_n \}_{n=1}^{\infty} }$ is constructed.
These results derived the interesting mathematical fact that
the continuous fuzzy-valued function on a closed interval has
a maximum and minimum.
Also, the structure of lattice for fuzzy numbers is discussed in Zhang
and Hirota\cite{ZH97}.

To be suitable for computations and treatments, a class of fuzzy numbers,
called an 
$L$-$R$-fuzzy number, is introduced in many text books.
 
  Let $L, R : [0, \infty) \rightarrow [0,1]$ be two non-increasing
and not constant functions with $L(0) = R(0) = 1 $ and
$ L(x_0) = R(x_0) =0 $ for some $ x_0 > 0$.
A fuzzy number $\T{s}$ is called an $L$-$R$-fuzzy number if there exist
real numbers $m, n (m\le n), \alpha, \beta (\alpha, \beta >0)$
such that
$$
 \T{s}(x) = \left\{
             \begin{array}[c]{ll}
             L(\frac{m-x}{\alpha}) & {\textrm {for}}\quad  x \le m, \\
             1                     & {\textrm {for}}\quad  m \le x \le n, \\
             R(\frac{x-n}{\beta})  & {\textrm {for}}\quad  n \le x. \\
             \end{array}
            \right.
$$
Given functions $L$, $R$ with the properties in the above definition,
the $L$-$R$-fuzzy number specified by $m,n,\alpha, \beta$ is
denoted by an ordered tetradic $(m, n, \alpha, \beta)_{L{\textrm - }R}$,
which includes the triangular and trapezoidal fuzzy numbers.
Then, the fuzzy max order on the set of $L$-$R$-fuzzy numbers
is characterized by inequalities of the elements.
(cf. \cite{RR85}).
 
In particular, the symmetric fuzzy number
$\T{s}=(m,m,\alpha,\alpha)_{L{\textrm-}L}$ is called an $L$-fuzzy number
and denoted by $(m,\alpha)_L$.
Furukawa\cite{F93} extended the $L$-fuzzy number $(m,\alpha)_L$ 
with $\alpha >0$ 
to the case of $\alpha \in \BR$
and proved that for $\af\beta \ge 0$ 
the fuzzy max order $ (m,\alpha)_L \ \inc \ (n,\beta)_L $ if and only if 
$ x_0 \vert \alpha - \beta \vert \le n-m $ where
$x_0$ is the zero point of $L$.
Moreover Furukawa\cite{F93} introduced
the linear operations on the set of extended $L$-fuzzy numbers by
$$
 (m,\alpha)_L \oplus (n,\beta)_L = (m+n,\alpha+\beta)_L ,
$$
$$\lambda(m,\alpha)_L = (\lambda m,\lambda \alpha)_L \quad 
\mbox{for any scalar \ }  \lambda \in \BR .
$$

The fuzzy max order is proved to be adapted to the above
operations.
Also Furukawa \cite{F94} introduced a parametric order relation on
$L$-fuzzy numbers which is an extension of the fuzzy max order
and its total order relation.
The fuzzy optimization problems related to the fuzzy max order are
dealed with many authors; e.g., 
Furukawa\cite{F94}, 
Kurano et al.\cite{KYNY96}, Yoshida\cite{Y98} and others.


%--------------------- Section 3.2  -----------------------------
\subsection*{3.2. Other ordering methods}

Besides the fuzzy max order, a large collection of methods for
ordering fuzzy sets of linear line has been developed in the literature.
A simple method for ordering fuzzy numbers consists of the definition of
ordering or ranking function.
Let $ f : \C{F}(\BR) \rightarrow \BR $, 
where a natural ranking relation $\le$ on $\BR$ is defined. 
Then, using this ranking function,
we can define the order relation $\inc$ on $\C{F}(\BR)$ as follows: 
$$
   \T{s} \ \inc \ \T{r} \quad \quad \textrm{implies} \quad \quad
   f(\T{s}) \le f(\T{r}) \quad\quad \textrm{for} \quad\quad
   \T{s}, \T{r} \in \C{F}(\BR).
$$
If, for all $\T{s}, \T{r} \in \C{F}(\BR)$,
$f(\T{s} + \T{r}) = f(\T{s}) + f(\T{r}) $
and
$f(\lambda \T{s} ) = \lambda f(\T{s})$ for
$\lambda \ge 0$ are satisfied, it is called a linear ranking function.
As a simple ranking function, we can construct a family of ranking
procedures as follows:
$$
  f_p(\T{s}) :=
  \int_0^1 (p_\alpha \inf \T{s}_\alpha +q_\alpha \sup \T{s}_\alpha)
  \, d\alpha,
\mbox{\ \ where\ } \int_0^1 (p_\alpha  +q_\alpha)  \, d\alpha =1.
$$
Several variants of this family are discussed by the authors; 
see, e.g., 
Adamo\cite{A80}, Campos, Bonzalez and Vila\cite{CGV92}, 
Fortemps and Roubens\cite{FR96} and  Yager\cite{Y81}.
Gonzalez and Vila\cite{GV92} defined the dominance relations
by means of a ranking function evaluated into $\BR^n$.
For a $\BR^n$-valued ranking function
$f: \C{F}(\BR) \rightarrow \BR^n$, the order relation $\inc$ on $\C{F}(\BR)$
is defined using an order relation $\inc_n$ on 
$\BR^n$(defined in the next section)
 as follows:
$$
     \T{s} \ \inc \ \T{r} \quad \quad \textrm{implies} \quad \quad
   f(\T{s}) \ \inc_n \ f(\T{r}).
$$
From flexibility of order relations on $\BR^n$, this seems to be
useful in the suitable optimization problem of fuzzy decision
making.
 
Another approach to ordering fuzzy numbers is discussed by using 
the fuzzy ordering
(cf. \cite{Z71}).
From the point of view of possibility theory, Dubois and Prade\cite{DP83} 
defined four fuzzy relations on $\C{F}(\BR)$:
$\textrm{Pos}(\T{s} \ \inc \ \T{r}),
 \textrm{Pos}(\T{s} \prec \T{r}),
 \textrm{Nes}(\T{s} \ \inc \ \T{r}) \ \textrm{and} \ 
 \textrm{Nes}(\T{s} \prec \T{r})$.
%
% Also, Horiuchi and Tamura\cite{HT98} 
% defined the fuzzy ordering on $\C{F}(\BR)$
% by a grading function similar to the fuzzy ranking function.


%--------------------- Section 4  -----------------------------
\section*{4. On an extension to a pseudo order on $\C{F}(\BR^n)$}
 
In this section, we extend the fuzzy max order on $\C{F}(\BR)$
to a pseudo order on $\C{F}(\BR^n)$.
 
%--------------------- Section 4.1  -----------------------------
\subsection*{4.1.  A pseudo order on $\C{F}(\BR^n)$}
 
We will review a vector ordering on $\BR^n$ 
by a non-empty convex cone $K \subset \BR^n$.
Using this $K$,
we can define a pseudo order relation $\inc_K$ on $\BR$
by $x \ \inc_K \ y$ if and only if $y-x \in K$.
Let $\BR^n_+$ be the subset of
entrywise non-negative elements in $\BR^n$.
When $K = \BR^n_+$,
the order $\inc_K$ will be denoted by $\inc_n$ and
$x \ \inc_n \ y$ means that $x_i \leq y_i$ for all $i=1,2,\cdots,n$,
where
$x=(x_1,x_2,\cdots,x_n)$ and
$y=(y_1,y_2,\cdots,y_n) \in \BR^n$.
 
%%%%  The begin of section 2 in the pseudo2.tex %%%%
 
First we introduce a binary relation on
$\C{C}(\BR^n)$,
by which a pseudo order on $\C{F}(\BR^n)$ is given.
Henceforth we assume that the convex cone $K \subset \BR^n$
is given.
We define a binary relation $\preccurlyeq_K$ on $\C{C}(\BR^n)$ 
by abuse of notation.
For $A,B \in \C{C}(\BR^n)$,
$A \preccurlyeq_K B$ means the following (C.a) and (C.b)
(cf.~\cite{K98, KTH97}):
\begin{itemize}
\item [(C.a)]
For any $x \in A$,
there exists $y \in B$ such that $x \preccurlyeq_K y$.
\item [(C.b)]
For any $y \in B$,
there exists $x \in A$ such that $x \preccurlyeq_K y$.
\end{itemize}
 
\vspace{2ex}
\noindent
{\bf Lemma 4.1.} \
{\sl
The binary relation $\preccurlyeq_K$ is a pseudo order on $\C{C}(\BR^n)$.
}
 
\vspace{2ex}
\noindent
\Pf
It is trivial that $A \preccurlyeq_K A$ for $A \in \C{C}(\BR^n)$.
Let $A,B,C \in \C{C}(\BR^n)$ such that $A \preccurlyeq_K B$ and $B
\preccurlyeq_K C$.
We will check $A \preccurlyeq_K C$ by two cases (C.a) and (C.b).
Case(C.a):
Since $A \preccurlyeq_K B$ and $B \preccurlyeq_K C$,
for any $x \in A$ there exists $y \in B$
such that $x \preccurlyeq_K y$ and
there exists $z \in C$ such that $y \preccurlyeq_K z$.
Since $\preccurlyeq_K$ is a pseudo order on $\BR^n$,
we have $x \preccurlyeq_K z$.
Therefore it holds that for any $x \in A$
there exists $z \in C$ such that $x \preccurlyeq_K z$.
Case(C.b): Since $A \preccurlyeq_K B$ and $B \preccurlyeq_K C$,
for any $z \in C$ there exists $y \in B$
such that $y \preccurlyeq_K z$
and there exists $x \in A$ such that $x \preccurlyeq_K y$.
Since $\preccurlyeq_K$ is a pseudo order on $\BR^n$,
we have $x \preccurlyeq_K z$.
Therefore it holds that for any $z \in C$
there exists $x \in A$ such that $x \preccurlyeq_K z$.
From the above (C.a) and (C.b), we obtain $A \preccurlyeq_K C$.
Thus the lemma holds.
\qed
 
\vspace{2ex}
When $K = \BR^n_+$,
the binary relation $\preccurlyeq_K$ on $\C{C}(\BR^n)$
will be written simply by $\preccurlyeq_n$ and
for $[x,y], [x',y'] \in \C{C}_r(\BR^n)$,
$[x,y] \preccurlyeq_n [x',y']$ means
$x \preccurlyeq_n x'$ and $y \preccurlyeq_n y'$.
 
Next, we introduce a binary relation $\preccurlyeq_K$ on $\C{F}(\BR^n)$:
Let $\T{s},\T{r} \in \C{F}(\BR^n)$. The relation
$\T{s} \preccurlyeq_K \T{r}$ means the following (F.a) and (F.b):
\begin{itemize}
\item [(F.a)]
For any $x \in \BR^n$, there exists $y \in \BR^n$
such that $x \preccurlyeq_K y$ and $\T{s}(x) \leq \T{r}(y)$.
\item [(F.b)]
For any $y \in \BR^n$, there exists $x \in \BR^n$
such that $x \preccurlyeq_K y$ and $\T{s}(x) \geq \T{r}(y)$.
\end{itemize}

Note that the notation $\inc_K$ denotes the binary relation on 
$\BR^n, \C{C}(\BR^n), \C{F}(\BR^n)$ 
with some abuse of notation.

\vspace{2ex}
\noindent
{\bf Lemma 4.2.} \
{\sl
The binary relation $\preccurlyeq_K$ is a pseudo order on $\C{F}(\BR^n)$.
}
 
\vspace{2ex}
\noindent
\Pf
It is trivial that $\T{s} \preccurlyeq_K \T{s}$
for $\T{s} \in \C{F}(\BR^n)$.
Let $\T{s},\T{r},\T{p} \in \C{F}(\BR^n)$
such that $\T{s} \preccurlyeq_K \T{r}$ and $\T{r} \preccurlyeq_K \T{p}$.
We will check $\T{s} \preccurlyeq_K \T{p}$ 
by following two cases (F.a) and (F.b).
Case(F.a): Since $\T{s} \preccurlyeq_K \T{r}$ and $\T{r} \preccurlyeq_K
\T{p}$,
for any $x \in \BR^n$ there exists $y \in \BR^n$
such that $x \preccurlyeq_K y$ and $\T{s}(x) \leq \T{r}(y)$,
and there exists $z \in \BR^n$ such that $y \preccurlyeq_K z$
and $\T{r}(y) \leq \T{p}(z)$.
Since $\preccurlyeq_K$ is a pseudo-order on $\BR^n$,
we have $x \preccurlyeq_K z$ and $\T{s}(x) \leq \T{p}(z)$.
Therefore it holds that for any $x \in \BR^n$
there exists $z \in \BR^n$ such that $x \preccurlyeq_K z$
and $\T{s}(x) \leq \T{p}(z)$.
Case(F.b): Since $\T{s} \preccurlyeq_K \T{r}$ and $\T{r} \preccurlyeq_K \T{p}$,
for any $z \in \BR^n$ there exists $y \in \BR^n$
such that $y \preccurlyeq_K z$ and $\T{r}(y) \leq \T{p}(z)$,
and there exists $x \in \BR^n$ such that $x \preccurlyeq_K y$
and $\T{s}(x) \leq \T{r}(y)$.
Since $\preccurlyeq_K$ is a pseudo-order on $\BR^n$,
we have $x \preccurlyeq_K z$.
Therefore it holds that for any $z \in \BR^n$
there exists $x \in \BR^n$ such that $x \preccurlyeq_K z$
and $\T{s}(x) \leq \T{p}(z)$.
From the above (F.a) and (F.b), we obtain $\T{s} \preccurlyeq_K \T{p}$.
Thus the lemma holds.
\qed
 
\vspace{2ex}
The following lemma implies the correspondence
between the pseudo order on $\C{F}(\BR^n)$ for fuzzy sets and
the pseudo order on $\C{C}(\BR^n)$ for the $\af$-cuts.
 
\vspace{2ex}
\noindent
{\bf Lemma 4.3.} \
{\sl
Let $\T{s},\T{r} \in \C{F}(\BR^n)$.
$\T{s} \preccurlyeq_K \T{r}$ on $\C{F}(\BR^n)$ if and only if
$\T{s}_{\af} \preccurlyeq_K \T{r}_{\af}$ on $\C{C}(\BR^n)$
for all $\af \in (0,1]$.
}
 
\vspace{2ex}
\noindent
\Pf
Let $\T{s},\T{r} \in \C{F}(\BR^n)$ and $\af \in (0,1]$.
Suppose $\T{s} \preccurlyeq_K \T{r}$ on $\C{F}(\BR^n)$.
Then, two cases (a) and (b) are considered.
Case(a): Let $x \in \T{s}_{\af}$. Since $\T{s} \preccurlyeq_K \T{r}$,
there exists $y \in \BR^n$
such that $x \preccurlyeq_K y$ and $\af \le \T{s}(x) \le \T{r}(y)$.
Namely $y \in \T{r}_{\af}$.
Case(b): Let $y \in \T{r}_{\af}$. Since $\T{s} \preccurlyeq_K \T{r}$,
there exists $x \in \BR^n$
such that $x \preccurlyeq_K y$ and $\T{s}(x) \ge \T{r}(y) \ge \af$.
Namely $x \in \T{s}_{\af}$.
Therefore we get $\T{s}_{\af} \preccurlyeq_K \T{r}_{\af}$
on $\C{C}(\BR^n)$ for all $\af \in (0,1]$
from the above (a) and (b).
 
On the other hand,
suppose $\T{s}_{\af} \preccurlyeq_K \T{r}_{\af}$ on $\C{C}(\BR^n)$
for all $\af \in (0,1]$.
Then, two cases (a') and (b') are considered.
Case(a'): Let $x \in \BR^n$. Put $\af = \T{s}(x)$.
If $\af=0$, then $x \preccurlyeq_K x$ and $\T{s}(x) = 0 \le \T{r}(x)$.
While, if $\af>0$, then $x \in \T{s}_{\af}$.
Since $\T{s}_{\af} \preccurlyeq_K \T{r}_{\af}$,
there exists $y \in \T{r}_{\af}$ such that $x \preccurlyeq_K y$.
And we have $\T{s}(x) = \af \le \T{r}(y)$.
Case(b'): Let $y \in \BR^n$. Put $\af = \T{r}(y)$.
If $\af=0$, then $x \preccurlyeq_K x$ and $\T{s}(x) \ge 0 = \T{r}(y)$.
While, if ${\af}>0$, then $y \in \T{r}_{\af}$.
Since $\T{s}_{\af} \preccurlyeq_K \T{r}_{\af}$,
there exists $x \in \T{s}_{\af}$ such that $x \preccurlyeq_K y$.
And we have $\T{s}(x) \ge \af = \T{r}(y)$.
 
Therefore we get $\T{s} \preccurlyeq_K \T{r}$ on $\C{F}(\BR^n)$
from the above Case (a') and (b').
Thus we obtain this lemma.
\qed
 
\vspace{2ex}
For the case of $K = \BR_+$,
Lemma 4.3 says that the order relation $\inc_1$ on $\C{F}(\BR)$ 
(That is, $n=1$) 
is the fuzzy max order 
mentioned in Section 3.

Define the dual cone of a cone $K$ by
$$
K^{+} := \{ a \in \BR^n \mid a \cdot x \ge 0 \
\mbox{for all} \ x \in K \},
$$
where $x \cdot y$ denotes the inner product on $\BR^n$
for $x,y \in \BR^n$.
For a subset $A \subset \BR^n$ and $a \in \BR^n$,
we define
$$
a \cdot A := \{ a \cdot x \mid x \in A \} \ (\subset \BR).
\leqno{(4.1)}
$$
The equation (4.1) means that $a\cdot A$ is the projection of $A$
on the parallel line with the vector $a$ if $a \cdot a = 1$.
It is trivial that $a \cdot A \in \C{C}(\BR)$
if $A \in \C{C}(\BR^n)$ and $a \in \BR^n$.
 
\vspace{2ex}
\noindent
{\bf Lemma 4.4.} \
{\sl
Let $A,B \in \C{C}(\BR^n)$.
$A \preccurlyeq_K B$ on $\C{C}(\BR^n)$ if and only if
$a \cdot A \preccurlyeq_{1} a \cdot B$ on $\C{C}(\BR)$
for all $a \in K^{+}$.
}

\vspace{2ex}
\noindent
\Pf
Suppose $A \preccurlyeq_K B$ on $\C{C}(\BR^n)$.
Consider the two cases (a) and (b).
Case(a): For any $x \in A$, there exists $y \in B$
such that $x \preccurlyeq_K y$.
Then $y-x \in K$. If $a \in K^{+}$, then $a \cdot (y-x) \ge 0$
and i.e. $a \cdot x \le a \cdot y$.
Case(b): For any $y \in B$, there exists $x \in A$
such that $x \preccurlyeq_K y$.
Then $y-x \in K$. If $a \in K^{+}$, then $a \cdot (y-x) \ge 0$
and i.e. $a \cdot x \le a \cdot y$.
From the above cases (a) and (b),
we have that
$a \cdot A \preccurlyeq_{1} a \cdot B$.
 
On the other hand, to prove the inverse statement,
we assume that $A \preccurlyeq_K B$ on $\C{C}(\BR^n)$ does not hold.
Then we have the following two cases (i) and (ii).
Case(i): There exists $x \in A$ such that $y-x \not\in K$
for all $y \in B$.
Then $B \cap (x+K) = \emptyset$.
Since $B$ and $x+K$ are closed convex, by the separation theorem
there exists  $a \in \BR^n$ ($a \neq 0$)
such that $a \cdot y < a \cdot x + a \cdot z$ for all $y \in B$
and all $z \in K$.
Now, we suppose that there exists $z \in K$
such that $a \cdot z < 0$.
Then $\ld z \in K$ for all $\ld \ge 0$ since $K$ is a cone,
and so we have
$a \cdot x + a \cdot \ld z = a \cdot x + \ld
a \cdot z \to - \infty$ as $\ld \to \infty$.
This contradicts $a \cdot y < a \cdot x + a \cdot z$.
Therefore we obtain $a \cdot z \ge 0$ for all $z \in K$, 
which implies $a\in K^+$.
Especially taking $z = 0 \in K$,
we get $a \cdot y < a \cdot x$ for all $y \in B$.
This contradicts $a \cdot A \preccurlyeq_{1} a \cdot B$.
Case(ii): There exists $y \in B$ such that $y-x \not\in K$
for all $x \in A$.
Then we derive a contradiction in the similar way to the case (i).
Therefore the inverse statement holds from the results of
the above (i) and (ii).
The proof of this lemma is completed.
\qed
 
\vspace{2ex}
For $a \in \BR^n$ and $\T{s} \in \C{F}(\BR^n)$,
applying the representation theorem 
we define a fuzzy number $a \cdot \T{s} \in \C{F}(\BR)$ by
$$
  a \cdot \T{s}(x) := \sup_{\af \in [0,1]}
  \{ \af \wedge 1_{a \cdot \T{s}_{\af}}(x) \},
  \quad x \in \BR.
\leqno{(4.2)}
$$
 
The following theorem gives the correspondence
between the pseudo order $\preccurlyeq_K$ on $\C{F}(\BR^n)$ and
the fuzzy max order $\preccurlyeq_1$ on $\C{F}(\BR)$.
 
 
\vspace{2ex}
\noindent
{\bf Theorem 4.1.} \
{\sl
For $\T{s},\T{r} \in \C{F}(\BR^n)$,
$\T{s} \preccurlyeq_K \T{r}$ \ if and only if \
$a \cdot \T{s} \preccurlyeq_{1} a \cdot \T{r}$ for all $a \in K^{+}$.
}
 
\vspace{2ex}
\noindent
\Pf
By (4.2) and the representative theorem, 
we have 
$(a \cdot \T{s})_{\af} = a \cdot \T{s}_{\af}$
for all $\af \in [0,1]$. 
On the other hand, 
from Lemmas 4.3 and 4.4, $\T{s} \preccurlyeq_K \T{r}$ if and only if
$a \cdot \T{s}_{\af} \preccurlyeq_{1} a \cdot \T{r}_{\af}$
for all $a \in K^{+}$.
Thus, noting the definition of the max order $\inc_1$ on $\C{F}(\BR)$, 
Theorem 4.1 follows. 
\qed
 
\vspace{2ex}
For $\{\T{s}_{k}\}_{k=1}^{\infty} \subset \C{F}(\BR^n)$
and $\T{s} \in \C{F}(\BR^n)$,
$\displaystyle{
\lim_{k \to \infty} \T{s}_{k} = \T{s}
}$
means that
$\displaystyle{
\sup_{\af \in [0,1]} \rho_n ( \T{s}_{k\af},\T{s}_{\af})
\to  0
}$
$(k \to \infty)$,
where $\T{s}_{k\af}$ is the $\af$-cut of $\T{s}_{k}$ and
$\rho_n$ is the Hausdorff metric on $\C{C}(\BR^n)$.
 
\vspace{2ex}
\noindent
{\bf Lemma 4.5.} \
{\sl
Let $\{\T{s}_{k}\}_{k=1}^{\infty} \subset \C{F}(\BR)$
and $\T{s} \in \C{F}(\BR)$
such that $\T{s}_{k} \preccurlyeq_{1} \T{s}_{k+1}$ ($k \ge 1$)
and $\lim_{k \to \infty} \T{s}_{k} = \T{s}$. \
Then we have 
$\T{s}_{1} \preccurlyeq_{1} \T{s}$.
}
 
\vspace{2ex}
\noindent
\Pf Trivial.
\qed
 
\vspace{2ex}
\noindent
{\bf Theorem 4.2.} \
{\sl
Let $\{\T{s}_{k}\}_{k=1}^{\infty} \subset \C{F}(\BR^n)$
and $\T{s} \in \C{F}(\BR^n)$
such that $\T{s}_{k} \preccurlyeq_K \T{s}_{k+1}$ ($k \ge 1$)
and $\lim_{k \to \infty} \T{s}_{k} = \T{s}$. \
Then we have $\T{s}_{1} \preccurlyeq_K \T{s}$.
}
 
\vspace{2ex}
\noindent
\Pf
From Theorem 4.1, for all $a \in K^{+}$ it holds that 
$a \cdot \T{s}_{k} \preccurlyeq_{1} a \cdot \T{s}_{k+1}$ ($k \ge 1$). 
Also, since  $(a \cdot \T{s}_k)_\af =  a \cdot \T{s}_{k\af}$ 
from (4.2) and 
$\rho_1 ( a \cdot \T{s}_{k\af}, a \cdot \T{s}_{\af})
\le 
 \|a\| \rho_n (\T{s}_{k\af}, \T{s}_{\af})$
for all $k \ge 1$, 
we get 
$\lim_{k \to \infty} a \cdot \T{s}_{k} =  a \cdot \T{s}$ 
where $\|a\|$ is a norm of $a$.
By Lemma 4.5, it holds that 
$a \cdot \T{s}_{1} \preccurlyeq_{1} a \cdot \T{s}$ for all $a \in K^{+}$.
From Theorem 4.1, we have $\T{s}_{1} \preccurlyeq_K \T{s}$.
\qed
 
\vspace{2ex}
\noindent
{\bf Remark.} \
Let consider a continuous map $g : [0,1] \to \C{F}(\BR^n)$.
A point $x_0$ is said to be efficient if $x_0 \in [0,1]$
and $g(x_0) \preccurlyeq_K g(x)$ for some $x \in [0,1]$ implies
$g(x) = g(x_0)$.
Then, by applying the same idea as in Lemma 3.2 of
Furukawa\cite{F80},
we observe that there exists at least one efficient point in $[0,1]$.
In fact, considering, if necessary,
a partial order $\preccurlyeq_K$ on the class of the quotient sets
with respect to the equivalence relation $\sim_K$
defined by
$\T{s} \sim_K \T{r}$ if and only if
$\T{s} \preccurlyeq_K \T{r}$ and $\T{r} \preccurlyeq_K \T{s}$,
we can assume that $\preccurlyeq_K$ is a partial order on $\C{F}(\BR^n)$.
By Theorem 4.2 and the continuity of $g$,
$\{ g(x) \mid x \in [0,1] \}$
can be proved to be an inductively ordered set.
So, by Zorn's lemma
$\{ g(x) \mid x \in [0,1] \}$
has an efficient element.


% --------------------- Section 4.2  -----------------------------
\subsection*{4.2. Further results}
 
 
In this section, we shall investigate the above pseudo order $\preccurlyeq_K$
on $\C{F}_r(\BR^n)$ for a polyhedral cone $K$ with
$K^+ \subset \BR^n$.
To this end, we need the following lemma.
 
\vspace{2ex}
\noindent
{\bf Lemma 4.6.} \
Let $a, b \in \BR_+^n$ and $A \in \C{C}_r(\BR^n)$. \
Then for any scalars $\ld_1, \ld_2 \geq 0$, it holds
$$
(\ld_1 a + \ld_2 b)\cdot A = \ld_1(a\cdot A) + \ld_2(b\cdot A),
\leqno{(4.3)}
$$
where the arithmetic in (4.3) is defined in (4.1).
 
\vspace{2ex}
\noindent
\Pf
Let $\ld_1 a\cdot x + \ld_2 b \cdot y
 \in
 \ld_1(a\cdot A) + \ld_2(b\cdot B)$
with $x, y \in A$.
It suffices to show that
$\ld_1 a\cdot x + \ld_2 b \cdot y \in
(\ld_1 a + \ld_2 b)\cdot A$.
Define
$z = (z_1,z_2,\cdots,z_n)$ by
$$
  z_i :=
     \left\{
       \begin{array}{ll}
  (\ld_1 a_i x_i + \ld_2 b_i y_i)/(\ld_1 a_i + \ld_2 b_i)
        &  \mbox{if} \quad (\ld_1 a_i + \ld_2 b_i)>0    \\
  x_i
        & \mbox{if} \quad (\ld_1 a_i + \ld_2 b_i)=0 
       \end{array}
         \quad (i=1,2,\cdots,n).
     \right.
$$
Then, clearly
$(\ld_1 a + \ld_2 b)\cdot z = \ld_1 a\cdot x + \ld_2 b\cdot y$.
Since $A \in \C{C}_r(\BR^n) , z \in A$, so that
$\ld_1 a\cdot x + \ld_2 b\cdot y \in 
(\ld_1 a + \ld_2 b)\cdot A$.
\qed
 
\vspace{2ex}
Henceforth, we assume that $K$ is a polyhedral convex cone
with $K^+ \subset \BR^n_+$, i.e.,
there exist vectors $b^i \in \BR_+^n (i=1,2,\cdots,m)$ such that
$$
K = \{ x \in \BR^n \mid b^i\cdot x \geq 0
                   \mbox{\ for all \ } i=1,2,\cdots,m \}.
$$
Then, it is well-known (cf.~\cite{SW70}) that $K^+$ is expressed as
$$
K^+ = \{ x \in \BR^n \mid x = \sum_{i=1}^{m} \ld_i b^i, \  \ld_i \geq 0,
                   \ i=1,2,\cdots,m \}.
$$
The above dual cone $K^+$ is denoted simply by
$$
K^+ = \mbox{cone convex}\{ b^1,b^2,\cdots,b^m \}.
$$
The pseudo order $\preccurlyeq_K$ on $\C{C}_r(\BR^n)$
is characterized by the pseudo order $\inc_1$ on $\C{C}_r(\BR)$.
 
\vspace{2ex}
\noindent
{\bf Corollary 4.1.} \
Let $ K^+ = \mbox{cone convex}\{ b^1,b^2,\cdots,b^m \}$
with $b^i \in \BR^n_+$.
Then, for $A, B \in \C{C}_r(\BR^n)$,
$A \preccurlyeq_K B$ if and only if
$b^i\cdot A \preccurlyeq_1 b^i\cdot B$
for all $i=1,2,\cdots,m$. 

\vspace{2ex}
\noindent
\Pf
We assume that
$b^i\cdot A \preccurlyeq_1 b^i\cdot B$
for all $i=1,2,\cdots,m$.
For any $a \in K^+$, there exist $\ld_i \geq 0$ with
$ a = \sum_{i=1}^{m}\ld_i b^i $.
From Lemma 4.1, we have:
$$
a\cdot A = \sum_{i=1}^{m}\ld_i(b^i \cdot A) \ \preccurlyeq_1 \
         \sum_{i=1}^{m}\ld_i(b^i \cdot B)
         = a\cdot B.
$$
Thus, by Lemma 4.4, $A \ \preccurlyeq_K \ B$ follows.
By applying Lemma 4.4 again, the `only if' part of Corollary holds.
\qed
 
\vspace{2ex}
\noindent
{\bf Lemma 4.7.} \
Let $a, b \in \BR_+^n$ and $\T{s} \in \C{F}_r(\BR^n)$.
Then, for any $\ld_1, \ld_2 \geq 0$,
$$
(\ld_1 a + \ld_2 b)\cdot \T{s}
= \ld_1(a\cdot \T{s}) + \ld_2(b\cdot \T{s}),
\leqno{(4.4)}
$$
where the arithmetic in (4.4) is given in (2.1), (2.2) and (4.2).
 
\vspace{2ex}
\noindent
\Pf
For any $\af \in [0,1]$,
it follows from the definition and Lemma 4.6 that
\begin{align*}
((\ld_1 a + \ld_2 b)\cdot \T{s})_\af
      &= (\ld_1 a + \ld_2 b)\cdot \T{s}_\af
       = \ld_1(a\cdot \T{s}_\af) + \ld_2(b\cdot \T{s}_\af) \\
      &= \ld_1(a\cdot \T{s})_\af + \ld_2(b\cdot \T{s})_\af
       = (\ld_1(a\cdot \T{s}) + \ld_2(b\cdot \T{s}))_\af.
\end{align*}
The last equality follows from (2.3).
The above shows that (4.4) holds.
\qed
 
\vspace{2ex}
The main result in this section is given in the following.
 
\vspace{2ex}
\noindent
{\bf Theorem 4.3.} \
Let $K^+ = \mbox{cone convex} \{ b^1,b^2,\cdots,b^m \}$
with $b^i \in \BR^n$.
Then, for $\T{s}, \T{r} \in \C{F}_r(\BR^n)$,
$$
\T{s} \ \preccurlyeq_K \ \T{r}
\mbox{\quad if and only if \quad}
b^i \cdot \T{s} \ \preccurlyeq_1 \ b^i \cdot \T{r}
\mbox{\ for \ }
i=1,2,\cdots,m.
$$
 
\noindent
\Pf
It suffices to prove the `if' part of Theorem 4.3.
For any $a \in K^+$,
there exist $\ld_i \geq 0$ with
$a = \sum_{i=1}^{m} \ld_i b^i$.
Applying Lemma 4.7, we have
$$
a \cdot \T{s} =
\sum_{i=1}^{m} \ld_i (b^i \cdot \T{s}) \ \preccurlyeq_1 \
\sum_{i=1}^{m} \ld_i (b^i \cdot \T{r}) =
a \cdot \T{r},
$$
From Theorem 4.1, $\T{s} \ \preccurlyeq_K \ \T{r}$ follows.
\qed
 
\vspace{2ex}
%--------------------- Figure 3 ----------------------------
% \begin{figure}[htbp]
        \begin{center}
%\includegraphics{p1.eps}
 \includegraphics[trim=0mm 1mm 1mm 1mm, width=.8\linewidth]{p1.eps}
% \includegraphics[trim=1mm 5mm 1mm 5mm, width=.8\linewidth]{p1.eps}

\vskip -8.5cm
\hskip -2cm
$\T{max}(\T{s},\T{r})$

\vskip 0.2cm %3.1cm
\hskip -9.6cm %-8.1cm
$\T{r}$

\vskip 2.8cm %1.0cm
\hskip 2.1cm %5.9cm
$\T{s}$

\vskip 1.0cm %1.5cm
\vskip 2.7cm %2.7cm
Fig 3:  $\T{\max}(\T{s},\T{r})$

\end{center}
%        \caption{$\T{v} = \sup(\T{s},\T{r})$}
%        \label{fig:p1.eps}
% \end{figure}

\bigskip

Zhang and Hirota\cite{ZH97} described the structure of
the fuzzy number lattice $(\C{F}_r(\BR), \preccurlyeq_1)$.
When $K = \BR^n$, $K^+ = \BR^n$ and
$ K^+ = \mbox{cone convex}\{ e^1, e^2, \cdots, e^n \}$.
So that, by Theorem 4.3, we see that for
$\T{s}, \T{r} \in \C{F}_r(\BR^n)$,
$\T{s} \preccurlyeq_n \T{r}$ means
$e^i\cdot \T{s} \preccurlyeq_1 e^i\cdot \T{r}$ for all $i=1,2,\cdots,n$.
Therefore, by applying the same method as Zhang and Hirota\cite{ZH97},
we can describe the structure of
the fuzzy set lattice $(\C{F}_r(\BR^n), \preccurlyeq_n)$.
Figure 3 illustrates $\T{\max}(\T{s},\T{r})$ for
$\T{s}, \T{r} \in \C{F}_r(\BR^2)$.
 

\subsection*{Acknowledgement}
Authors would express their thanks to the referees 
for several comments and corrections which led to 
a considerable improvement of the original version. 
Also, they are grateful to Associate Professor H. Kuwano 
of Kanazawa Gakuin University 
for useful discussions related to this paper. 

 
%%%%  The end of text pseudo2,tex %%%%
 
 
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\begin{flushright}

Masami Kurano \\
Department of Mathematics \\
Faculty of Education   \\
Chiba University  \\
Yayoi-cho, Inage-ku \\
Chiba \ 263-8522, Japan

\end{flushright}



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