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\begin{document}

\title{Convex Fuzzy Cones of Hypervector Spaces%\thanks{Grants or other notes
%about the article that should go on the front page should be
%placed here. General acknowledgments should be placed at the end of the article.}
}
%\subtitle{Do you have a subtitle?\\ If so, write %it here}

%\titlerunning{Short form of title}        % if too long for running head

\author{O. R. Dehghan \and R. Ameri}

%\authorrunning{Short form of author list} % if too long for running head

\institute{O. R. Dehghan \at Department of Mathematics, Faculty of Basic Sciences,\\
           University of Bojnord, Bojnord, Iran\\
           \email{dehghan@ub.ac.ir}
\and
           R. Ameri \at School of Mathematics, Statistics and Computer Sciences,\\
           University of Tehran, Tehran, Iran\\
           \email{rameri@ut.ac.ir}}

\date{Received: date / Accepted: date}
% The correct dates will be entered by the editor


\maketitle

\begin{abstract}
The aim of this paper is to study some main properties of convex fuzzy cones of hypervector spaces. In this regard, a characterization of fuzzy subhyperspaces is presented based on ordinary subhyperspaces. Also the convex cone spanned by non-empty subsets of real hypervector spaces is obtained. Moreover, by introducing the notion of fuzzy cone, the smallest fuzzy subhyperspace of $V$ containing $\mu$ and the largest fuzzy subhyperspace of $V$ contained in $\mu$ is achieved, for a convex fuzzy cone $\mu$ of real hypervector space $V$.
\keywords{Hypervector space \and convex cone \and fuzzy subhyperspace \and fuzzy cone \and convex fuzzy cone}
% \PACS{PACS code1 \and PACS code2 \and more}
\subclass{20N20 \and 15A03}
\end{abstract}
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\section{Introduction}
The theory of algebraic hypperstructures was founded by Marty \cite{Marty} in 1934 and has been studied in various branches of algebra (for example see \cite{Corsini 1}, \cite{Corsini 2}, \cite{Davvaz book} and \cite{Vougiouklis}). The concept of hypervector space was introduced by Tallini \cite{Tallini 1} in 1990 and has been investigated by herself, Ameri \cite{Ameri Dehghan 1} and the author \cite{Dehghan Ameri Ebrahimi} and \cite{Sedghi Dehghan Norouzi}.

The well-known concept of fuzzy subset of a non-empty set $X$ was introduced by Zadeh \cite{Zadeh} in 1965. Rosenfeld \cite{Rosenfeld} applied the theory of fuzzy subsets to the groups and then many authors studied fuzzy algebraic structures (\cite{Mordeson book 1}, \cite{Mordeson book 2}, \cite{Rosenfeld book}). Especially, fuzzy vector spaces have been studied by Katsaras and Liu \cite{Katsaras}, Nanda \cite{Nanda}, Malik and Mordeson \cite{Malik Mordeson} and Kumar \cite{Kumar}.

After the development of fuzzy theory in algebraic hyperstructures (\cite{Davvaz book 2}), Ameri \cite{Ameri FHVS over VF} introduced a view of fuzzy hypervector spaces over valued fields in 2005 and studied some of their properties (\cite{Ameri Dehghan 2}, \cite{Ameri Dehghan 3}, \cite{Ameri Dehghan 4}, \cite{Ameri Dehghan 5}).

One of the important theorems connecting algebraic structures and their fuzzy cases: ``A fuzzy subset of an algebraic structure is a fuzzy substructure if and only if all of its level subsets are substructure", has been proved in hypervector spaces (\cite{Ameri FHVS over VF}). In section \ref{Sec. Charac. F. SHS} we present a different characterization of fuzzy subhyperspaces based on ordinary subhyperspaces and show that, under special conditions, an affine fuzzy subset of a hypervector space is a finite intersection of fuzzy superhyperplanes. The concepts of convex and balanced subsets of hypervector spaces were studied by the author in \cite{Dehghan Affine Convex} and \cite{Dehghan Balanced Absorbing}. In section \ref{Sec. Conv. Cone} we study some properties of another special subset (cone) of real hypervector spaces and obtain the convex cone spanned by non-empty subsets of real hypervector spaces as well as the smallest subhyperspace of $V$ containing $C$ and the largest subhyperspace of $V$ contained in $C$, for a convex cone $C$ of real hypervector space $V$. The concepts of convex fuzzy subsets of hypervector spaces was studied by the author in \cite{Dehghan Affine Convex}. In section \ref{Sec. Conv. F. Cone} by introducing the notion of fuzzy cone of real hypervector spaces, as an special kind of convex fuzzy subsets, we investigate some properties of convex fuzzy cones of such spaces. Also we achieve the smallest fuzzy subhyperspace of $V$ containing $\mu$ and the largest fuzzy subhyperspace of $V$ contained in $\mu$, for a convex fuzzy cone $\mu$ of real hypervector space $V$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Preliminaries}

In this section we present some definitions and previous proved properties that we shall use in later.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition}\label{D HVS}\cite{Tallini 1}
Let $K$ be a field, $(V,+)$ be an Abelian group and $P_{\ast}(V)$ be the set of all non-empty subsets of $V$. We define a hypervector space over $K$ to be the quadruplet $(V,+,\circ ,K)$, where $``\circ"$ is an external hyperoperation
\[\circ:K\times V\ \ \longrightarrow \ \ P_{\ast}(V),\]
such that for all $a,b\in K$ and $x,y\in V$ the following conditions hold:
\begin{enumerate}
\item[(H$_{1}$)] $a\circ (x+y)\subseteq a\circ x+a\circ y$, right distributive law,
\item[(H$_{2}$)] $(a+b)\circ x\subseteq a\circ x+b\circ x$, left distributive law,
\item[(H$_{3}$)] $a\circ (b\circ x)=(ab)\circ x$,
\item[(H$_{4}$)] $a\circ (-x)=(-a)\circ x=-(a\circ x)$,
\item[(H$_{5}$)] $x\in 1\circ x$,
\end{enumerate}
\noindent where in (H$_{1}$), $a\circ x+a\circ y=\{p+q:p\in a\circ x, q\in a\circ y\}$. Similarly it is in (H$_{2}$). Also in (H$_{3}$), $a\circ (b\circ x)=\bigcup\limits_{t\in b\circ x} a\circ t$.

$V$ is called strongly right distributive, if we have equality in (H$_{1}$). In a similar way we define the strongly left distributive hypervector spaces. $V$ is called strongly distributive, if it is strongly right and left distributive.

A non-empty subset $W$ of $V$ is called a subhyperspace of $V$, denoted by $W\leqslant V$, if $W$ is itself a hypervector space with the external hyperoperation on $V$, i.e. for all $a\in K$ and $x,y\in W$, $x-y\in W$ and $a\circ x\subseteq W$.
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{example}\label{Ex 3}\cite{Tallini 1}
In $(\mathbb{R}^{2},+)$ define the external hyperoperation $\circ:\mathbb{R}\times \mathbb{R}^{2}\longrightarrow P_{\ast} (\mathbb{R}^{2})$ by setting:
\[
a\circ x=\left\{
\begin{array}{l}
line\ ox \ \ \ if\ x\neq (0,0), \\
\left\{(0,0)\right\} \ \ \ if \ x=(0,0).
\end{array}
\right.
\]
Then $(\mathbb{R}^{2},+,\circ,\mathbb{R})$ is a hypervector space.
\end{example}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{example}\label{EX 1}\cite{Ameri Dehghan 3}
In classical vector space $(\mathbb{R}^{3},+,.,\mathbb{R})$ we define the external hyperoperation $\circ: \mathbb{R}\times
\mathbb{R}^{3} \longrightarrow P_{\ast}(\mathbb{R}^{3})$ by $a\circ (x_{0},y_{0},z_{0})=L$, where $L$ is a line with the parametric
equations:
\[
L:\left\{
\begin{array}{l}
x=ax_{0}, \\
y=ay_{0}, \\
z=t.
\end{array}
\right.
\]
Then $V=(\mathbb{R}^{3},+,\circ ,\mathbb{R})$ is a strongly left distributive hypervector space and the $z$-axis is a subhyperspace
of $V$.
\end{example}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In the sequel of this paper, $V$ denotes a hypervector space over the field $K$, unless otherwise is specified. Also, the zero of $V$ is denoted by $\underline{0}$.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition} \label{D FHVS}\cite{Ameri FHVS over VF}
A fuzzy subset $\mu$ of $V$, $\mu\in FS(V)$, is called a fuzzy subhyperspace of $V$, if for all $a\in K$ and $x,y\in V$, the following conditions are satisfied:
\begin{enumerate}
\item[1)] $\mu (x+y)\geq \mu (x)\wedge \mu (y)$,
\item[2)] $\mu (-x)\geq \mu (x)$,
\item[3)] $\bigwedge\limits_{t\in a\circ x}\mu (t)\geq \mu(x)$.
\end{enumerate}
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{example}\label{EX 2}\cite{Dehghan Affine Convex}
Consider the hypervector space $V=(\mathbb{R}^{3},+,\circ ,\mathbb{R})$ in Example \ref{EX 1}. Define a fuzzy subset $\mu$ of
$V$ by the following:
\[
\mu (x,y,z)=\left\{
\begin{array}{cl}
t_1 & \ (x,y,z)\in \{0\}\times\{0\}\times\mathbb{R}, \\
t_2 & \ (x,y,z)\in \mathbb{R}\times \left\{ 0\right\} \times \mathbb{R} \backslash \{0\} \times \{0\} \times \mathbb{R}, \\
t_3 & \ otherwise,
\end{array}
\right.
\]
where $0\leq t_{3}<t_{2}<t_{1}\leq 1$. Then $\mu$ is a fuzzy subhyperspace of $V$.
\end{example}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 3}\cite{Ameri FHVS over VF}
$\mu\in FS(V)$ is a fuzzy subhyperspace of $V$ iff $\mu_\alpha=\{x\in V: \mu(x)\geq \alpha\}$ is a subhyperspace of $V$, for all $\alpha\in Im\ \mu$. $\mu_\alpha$ is called an $\alpha$-level subhyperspace of $V$.
\end{theorem}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition}\label{D sum} \cite{Ameri Dehghan 4}
Let $\mu ,\mu _{1},\mu _{2}$ be fuzzy subsets of $V$ and $a\in K$. We define $\mu _{1}+\mu _{2}$ and $a \circ \mu$ to be the fuzzy
subsets of $V$, whose membership functions are given by:
\[
\left( \mu _{1}+\mu _{2}\right) (x)=\bigvee_{x=x_{1}+x_{2}}\left(\mu _{1}(x_{1})\wedge\mu _{2}(x_{2})\right),
\]
and
\[
(a\circ \mu )(x)=\left\{
\begin{array}{cl}
\bigvee\limits_{x\in a\circ }\mu(t) & \exists t\in V;x\in a\circ t,\\
0 & otherwise.
\end{array}
\right.
\]
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Characterization of Fuzzy Subhyperspaces}\label{Sec. Charac. F. SHS}
In this section we show that the fuzzy subhyperspaces introduced by Ameri \cite{Ameri FHVS over VF} can be characterized by ordinary subhyperspaces. Also the relation between affine fuzzy subsets of hypervector spaces and fuzzy superhyperplanes is investigated.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------

A subset $S$ of $V$ is called linearly independent (\cite{Ameri Dehghan 1}) if for every vectors $v_{1},\ldots ,v_{n}$ in $S$, and $c_{1},\ldots ,c_{n}\in
K$, $\underline{0}\in c_{1}\circ v_{1}+\cdots +c_{n}\circ v_{n}$, implies that $c_{1}=\cdots =c_{n}=0$. $S$ is called linearly
dependent if it is not linearly independent. A basis for $V$ is a linearly independent subset of $V$ such that spans $V$, i.e.
$V=\left\langle S\right\rangle$, where
\begin{eqnarray*}
\left\langle S\right\rangle &=&\left\{t\in V:t\in \sum\limits_{i=1}^{n}a_i\circ s_i,a_i\in K,s_i\in S,n\in\mathbb{N}\right\}\\
&=&\left\{t_{1}+\cdots +t_{n}:t_{i}\in a{_{i}}\circ {s_{i}},a_{i}\in K,{s_{i}}\in S,n\in \mathbb{N}\right\}.
\end{eqnarray*}
We say that $V$ is finite-dimensional if it has a finite basis. If $V$ is strongly left distributive, invertible ($V$ is said to be
invertible if and only if $u\in a\circ v$ implies that $v\in a^{-1}\circ u$) and finite-dimensional, then every two basis of $V$
have the same cardinality. In this case the cardinal of any basis of $V$ is called the dimension of $V$ and denoted by $\dim V$. Note that some hypervector spaces $V$ (some set $W$ of vectors) may not have any collection of linearly independent vectors. Such hypervector space (set) is called independentless. Clearly if $V$ is independentless, then $V$ has not any basis and for such hypervector spaces dimension is not defined.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{lemma}\label{L 1}
Let $V$ be finite-dimensional. If $U$ and $W$ are subhyperspaces of $V$ such that $U\varsubsetneq W$ and $\dim W-\dim U\geqslant 2$, then there exist subhyperspaces $E_1,\ldots,E_k$ of $V$, $k=\dim W-1$, such that $U\varsubsetneq E_1\varsubsetneq\cdots \varsubsetneq E_k\varsubsetneq W$ and $\dim E_d=\dim U+d$, $1\leq d\leq k$.
\end{lemma}
%---------------------------------------------------------------------
\begin{proof}
Let $\beta$ be a basis for $U$ and $x_1\in W\setminus U$. Put $E_1=\langle\beta\cup\{x_1\}\rangle$. Then $E_1$ is a subhyperspace of $V$ such that $U \varsubsetneq E_1\varsubsetneq W$ and $\dim E_1=\dim U+1$. If $\dim W=\dim E_1+1$, then there is nothing to prove. If $\dim W-\dim E_1\geq 2$, then similarly by considering $E_2=\langle\beta\cup\{x_1,x_2\}\rangle$, where $x_2\in W\setminus E_1$, $E_2$ is a subhyperspace of $V$ such that $U\varsubsetneq E_1\varsubsetneq E_2\varsubsetneq W$ and $\dim E_2=\dim E_1+1=\dim U+2$. By repeating the above, the proof is completed.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The properties of affine subsets and affine fuzzy subsets of hypervector spaces have been studied (\cite{Dehghan Affine Convex}). A non-empty subset $E$ of $V$ is called affine if $a\circ x+(1-a)\circ y\subseteq E$, for all $x,y\in E$, $a\in K$. It is clear that every subhyperspace of $V$ is an affine subset of $V$. A fuzzy subset $\mu$ of $V$ is said to be affine if $a\circ\mu +(1-a)\circ \mu \subseteq \mu$, for all $a\in K$.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 1}\cite{Dehghan Affine Convex}
Let $V$ be a hypervector space over the field $K$. Then
\begin{enumerate}
\item[1)] every fuzzy subhyperspace of $V$ is an affine fuzzy subset of $V$;
\item[2)] $\mu \in FS(V)$ is affine if and only if $\mu_{\alpha}=\{x\in V:\mu(x)\geq\alpha\}$ is affine, for all $\alpha\in [0,1]$;
\item[3)] if $V$ is strongly right distributive and $\mu$ is an affine fuzzy subset of $V$, such that $\mu (\underline{0})=\bigvee\limits_{t\in V}\mu (t)$, then $\mu$ is a fuzzy subhyperspace of $V$.
\item[4)] if $V$ is strongly right distributive and $\mu\in FS(V)$, then $\mu$ is affine if and only if it is the translation of a fuzzy subhyperspace $\nu$ of $V$, i.e. $\mu=\nu+x_0$ for some $x_0\in V$, where $(\nu+x_0)(x)=\nu(x-x_0)$.
\end{enumerate}
\end{theorem}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 2} \cite{Ameri Dehghan 1}
Let $V$ be invertible. Then for every $v_1,\ldots,v_n\in V$, either $v_1,\ldots,v_n$ are linearly independent or for some $1\leq j \leq n$, $v_j$ is in a linear combination of the others.
\end{theorem}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}
Let $V$ be invertible $n$-dimensional strongly right distributive and $\mu\in FS(V)$. Then $\mu$ is a fuzzy subhyperspace of $V$ if and only if there exist nested subhyperspaces $\{\underline{0}\}=V_0\varsubsetneq V_1\varsubsetneq\cdots\varsubsetneq V_n=V$ of $V$ and $\alpha_0,\alpha_1,\ldots,\alpha_n\in[0,1]$ such that $\dim V_i=d_i$ $(0\leq i\leq n)$, $0\leq d_0\leq d_1\leq\cdots\leq d_n=n$, $1\geq \alpha_0\geq \alpha_1\geq\cdots\geq\alpha_n\geq 0$ and $\mu=\alpha_0 1_{V_0}\vee \cdots\vee\alpha_n 1_{V_n}$.
\end{theorem}
%---------------------------------------------------------------------
\begin{proof}
Let $\mu=\alpha_0 1_{V_0}\vee \cdots\vee\alpha_n 1_{V_n}$ as stated above. Then every level subset of $\mu$ is a subhyperspace and so is an affine subset of $V$. Thus by Theorem \ref{Thm 1}(2), $\mu$ is an affine fuzzy subset of $V$. Also
\begin{eqnarray*}
\mu(\underline{0}) &=&(\alpha_0 1_{V_0}\vee \ldots \vee \alpha_n 1_{V_n})(\underline{0})\\
&=&(\alpha_0 1_{V_0})(\underline{0})\vee \ldots \vee (\alpha_n 1_{V_n})(\underline{0})\\
&=&\alpha_0\vee \ldots \vee \alpha_n\\
&=&\alpha_0\\
&=&\bigvee\limits_{x\in V}\mu (x).
\end{eqnarray*}
Hence by Theorem \ref{Thm 1}(3), $\mu$ is a fuzzy subhyperspace of $V$.\\
Conversely, let $\mu$ be a fuzzy subhyperspace of $V$. At first, we prove that $\mu$ can achieve at most $n$ different values on $V\setminus \{\underline{0}\}$. Suppose $\{x_0,\ldots,x_n\}\subseteq V\setminus \{\underline{0}\}$ is not independentless and $\mu(x_0)<\mu(x_1)<\cdots<\mu(x_n)$. If $x_0\in \langle x_1,\ldots,x_n\rangle$, then $x_0\in a^\prime_1\circ x_1+\cdots+a^\prime _n\circ x_n$, for some $a^\prime_1,\ldots,a^\prime_n\in K$. By Definition \ref{D FHVS}, it follows that:
\[\mu(x_0)\geq \bigwedge\limits_{t\in a^\prime_1\circ x_1+\cdots+a^\prime_n\circ x_n}\mu (t)\geq\mu(x_1)\wedge \cdots\wedge \mu(x_n),\]
which is impossible. Thus $x_0\notin \langle x_1,\ldots,x_n\rangle$. Similarly one can show that
\[x_1\notin \langle x_2,\ldots,x_n\rangle, \ldots, x_{n-1}\notin \langle x_n\rangle.\]
Since all $x_i\neq\underline{0}$, by Theorem \ref{Thm 2}, it follows that:
\begin{eqnarray*}
\dim\langle x_0,\ldots,x_n\rangle&=&1+\dim \langle x_1,\ldots,x_n\rangle\\
&=&\cdots=n+\dim\langle x_n\rangle\\
&=&n+1,
\end{eqnarray*}
which is impossible, since $\dim V=n$. Consequently, $|Im\ \mu|\leq n+1$, $\mu$ achieves $n$ values at vectors of $V\setminus \{\underline{0}\}$ and the maximum is attained at $\underline{0}$.\\
Set $\mu(V)=\{\beta_0,\ldots,\beta_k\}$, where $k\leq n$, $\beta_i>\beta_{i+1}$, $0\leq i\leq k-1$. Put $W_i=\mu_{\beta_i}$, for any $0\leq i\leq k$. It is clear that $W_k=V$. Then by Theorem \ref{Thm 3}, for all $0\leq i\leq k$, $\mu_{\beta_i}$ is a subhyperspace of $V$, such that $W_i\varsubsetneq W_{i+1}$ and so $\dim W_i<\dim W_{i+1}$, $0\leq i\leq k-1$. Now put $\dim W_i=d_i$, $0\leq i\leq k$. Then $0\leq d_0\leq d_1\leq\cdots\leq d_k=n$. For each $0\leq i\leq k$, suppose $V_{d_i}=W_i$. Then by Lemma \ref{L 1}, for any $d\in\{0,\ldots,n\}\setminus\{d_0,\ldots,d_k\}$, there exists a subhyperspace $V_d$ of dimension $d$ such that the family $\{V_d\}_{d\in\{0,\ldots,n\}}$ is nested, i.e. for all $0\leq d\leq n-1$, $V_d\varsubsetneq V_{d+1}$. Now put
\[
\alpha _{d}=\left\{
\begin{array}{ll}
\mu (\underline{0})=\beta _{0} & d=d_{0} \\
\beta _{1} & d_{0}<d\leq d_{1} \\
\vdots  & \\
\beta _{k} & d_{k-1}<d\leq d_{n}=n
\end{array}
\right.
\]
Let $x\in V$. If $d$ is the smallest integer such that $x\in V_d$, then $x\in W_{l+1}\setminus W_l$, for some $0\leq l\leq k-1$, such that $d_l<d\leq d_{l+1}$. Thus $\beta_{l+1}\leq \mu(x)<\beta_l$ and so $\mu(x)=\beta_{l+1}$. On the other hand,
\begin{eqnarray*}
\alpha_0 1_{V_0}(x)\vee \cdots\vee\alpha_n 1_{V_n}(x)&=&\alpha_d 1_{V_d}(x)\vee \cdots\vee\alpha_n 1_{V_n}(x)\\
&=&\alpha_d \vee \cdots\vee\alpha_n\\
&=&\alpha_d\\
&=&\beta_{l+1}.
\end{eqnarray*}
Thus the theorem is proved.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In the above decomposition of $\mu$, $n+1$ subhyperspace $V_i$ are nested and some of these levels coincide, so we can leave out those with the smaller dimension and keep the largest subhyperspace with the same level. This gives the following formulation of representation theorem.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{corollary}\label{C 1}
Let $V$ be invertible $n$-dimensional strongly right distributive and $\mu\in FS(V)$. Then $\mu$ is a fuzzy subhyperspace of $V$ if and only if there exist nested subhyperspaces $V_0\varsubsetneq V_1 \varsubsetneq \cdots \varsubsetneq V_k=V$ of $V$, $k\leq n$ and $\alpha_0,\alpha_1,\ldots,\alpha_k\in[0,1]$ such that $\dim V_d<\dim V_{d+1}$, $0\leq d\leq k-1$, $\alpha_0>\alpha_1>\cdots> \alpha _k$ and $\mu=\alpha_0 1_{V_0}\vee \cdots\vee\alpha_k 1_{V_k}$. In this decomposition $\mu$ can not obtain with less components.
\end{corollary}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition}
Let $\mu=\bigvee\limits_{i=0}^n \alpha_i 1_{V_i}$ be a fuzzy subhyperspace of $V$. Then $\mu$ is called a fuzzy superhyperspace if and only if $\dim V_{n-1}=(\dim V)-1$, i.e. there is a non-trivial $(n-1)$-dimensional component in the decomposition of $\mu$. $\mu\in FS(V)$ is called a fuzzy superhyperplane if and only if it is the translation of some fuzzy superhyperspace.
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 4}\cite{Dehghan Linear Functionals}
Let $V$ be invertible $n$-dimensional strongly left distributive and $W\leqslant V$ such that $\dim W=d$. Then $W$ is the intersection of $(n-d)$ superhyperspaces in $V$ (any subhyperspace of dimension $n-1$ is called a superhyperspace).
\end{theorem}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}
Let $V$ be invertible $n$-dimensional strongly right distributive. Then every affine fuzzy subset $\mu$ of $V$ is a finite intersection of fuzzy superhyperplanes.
\end{theorem}
%---------------------------------------------------------------------
\begin{proof}
By Theorem \ref{Thm 1}(4), $\mu=\mu^\prime+x_0$, where $\mu^\prime$ is a fuzzy subhyperspace of $V$ and $x_0\in V$. Thus by Corollary \ref{C 1} there exist subhyperspaces $V_0\varsubsetneq V_1 \varsubsetneq \cdots \varsubsetneq V_k=V$ of $V$, $k\leq n$ and $1\geq\alpha_0>\alpha_1>\cdots>\alpha_k\geq 0$ such that $\mu^\prime=\bigvee\limits_{i=0}^k \alpha_i 1_{V_i}$. If $\dim V_{k-1}=n-1$, then $\mu^\prime$ is a fuzzy superhyperspace and the proof is completed. If $0\leq\dim V_{k-1}<n-1$, then by Theorem \ref{Thm 4}, $V_{k-1}=W_1\cap\cdots\cap W_p$, where $W_1,\ldots,W_p$ are superhyperspaces of $V$. For all $1\leq i\leq p$ put
\[\mu^\prime_i=\alpha_0 1_{V_0}\vee \cdots\vee\alpha_{k-1} 1_{W_i}\vee\alpha_k 1_{V_k}.\]
Then for all $1\leq i\leq p$, $\mu^\prime_i$ is a fuzzy superhyperspace.\\
Now let $x\in E$ and $l$ be the smallest number that $x\in V_l$. If $l\leq k-1$, then for all $1\leq i\leq p$, $\mu^\prime_i(x)=\alpha_l$ and so $\bigwedge\limits_{i=1}^p \mu^\prime _i(x)=\alpha_l=\mu^\prime(x)$. Thus in this case $\mu^\prime=\bigcap\limits_{i=1}^p\mu^\prime_i$. If $l=k$, then $x\notin V_{k-1}$ and $x\notin W_{i_0}$, for some $1\leq i_0\leq p$ and so $\bigwedge\limits_{i=1}^p \mu^\prime _i(x)\leq\mu^\prime_{i_0}(x)=\alpha_k$. Thus $\bigwedge\limits_{i=1}^p \mu^\prime _i(x)=\alpha_k=\mu^\prime_i(x)$, since $\alpha_k$ is the smallest value in $Im\ (\bigwedge\limits_{i=1}^p \mu^\prime _i)$. Hence $\mu^\prime=\bigwedge\limits_{i=1}^p \mu^\prime _i$ and so
\[\mu=(\bigwedge\limits_{i=1}^p \mu^\prime _i)+x_0=\bigwedge\limits_{i=1}^p (\mu^\prime _i+x_0),\]
and the proof is completed.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convex Cones}\label{Sec. Conv. Cone}
The concepts of convex and balanced subsets of hypervector spaces were studied by the author in \cite{Dehghan Affine Convex} and \cite{Dehghan Balanced Absorbing}. In this section some properties of another special subset, cone, of real hypervector spaces are studied. We see that in a real hypervector space, every subhyperspace is a cone as well as any cone is balanced. Also the convex cone spanned by non-empty subsets of real hypervector spaces is obtained. Moreover, the smallest subhyperspace of $V$ containing $C$ and the largest subhyperspace of $V$ contained in $C$ is achieved, for a convex cone $C$ of real hypervector space $V$.\\
Let $K$ be a field. Then the mapping $\left\vert \ \right\vert :K\rightarrow \mathbb{R}$ is called a valuation on $K$ if and only
if for all $a,b \in K$, the followings hold:
\begin{enumerate}
  \item [1)] $\left\vert a \right\vert \geq 0$ and we have equality iff $a=0$;
  \item [2)] $\left\vert ab \right\vert =\left\vert a \right\vert \left\vert b \right\vert$;
  \item [3)] $\left\vert a+b \right\vert \leq \left\vert a \right\vert + \left\vert b \right\vert$.
\end{enumerate}
A field $K$ together with a valuation is called a valued field.\\
Let $V$ be a hypervector space over valued field $K$. Then a non-empty subset $E$ of $V$ is called
\begin{enumerate}
  \item convex if $a\circ x+(1-a)\circ y\subseteq E$, for all $x,y\in E$, $a\in K$, $|a|\leq 1$,
  \item balanced if $a\circ E\subseteq E$, for all $a\in K$, $|a|\leq 1$.
\end{enumerate}
A particular type of convex sets in real hypervector spaces is the convex cones.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition}
Let $V$ be a real hypervector space. Then $C\subseteq V$ is called a cone if
\[
\forall a\in\mathbb{R}^+,\ \forall x\in C,\ a\circ x\subseteq C.
\]
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
It is clear that in a real hypervector space $V$ every subhyperspace is a cone as well as any cone is balanced, with the usual absolute value on $\mathbb{R}$. The next examples show, in general, there is not any other relation between the concepts cone, subhyperspace and balanced subset of real hypervector spaces.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{example}\label{Ex 4}
Consider the hypervector space $(\mathbb{R}^2,+,\circ,\mathbb{R})$ in Example \ref{Ex 3}, with any valuation on $\mathbb{R}$. Then for all $r_1,r_2\in \mathbb{R}^+$, such that $r_1<r_2$, the subset $B=\{(x,y)\in\mathbb{R}^2; r_{1}x\leq y\leq r_{2}x\}\subseteq \mathbb{R}^2$ is balanced and a cone, while it is not a subhyperspace.
\end{example}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{example}\label{Ex 5}
Consider the real hypervector space $V=(\mathbb{R}^2,+,\circ,\mathbb{R})$, where $\circ:\mathbb{R}\times \mathbb{R}^{2} \longrightarrow P_{\ast}(\mathbb{R}^{2})$ defined by $a\circ (x,y)=ax\times \mathbb{R}$ with the usual absolute value $|\cdot|$ on $\mathbb{R}$. Then the subset $B=\{(x,y)\in\mathbb{R}^2; 0\leq x\leq 1\}\subseteq \mathbb{R}^2$ is balanced which is not a subhyperspace of $\mathbb{R}^2$ or a cone.
\end{example}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{proposition}
Let $V$ be a real hypervector space. Then
\begin{enumerate}
  \item For any $x\in V$, the set $C=\mathbb{R}^+\circ x=\bigcup\limits_{a\in \mathbb{R}^+}a\circ x$ is a cone of $V$.
  \item If $C$ is a cone, then $C=\bigcup\limits_{a\in\mathbb{R}^+, x\in C}a\circ x$.
\end{enumerate}
\end{proposition}
%---------------------------------------------------------------------
\begin{proof}
1) If $b\in \mathbb{R}^+$ and $t\in \mathbb{R}^+\circ x$, then $t\in a_0\circ x$, for some $a_0\in \mathbb{R}^+$ and so $b\circ t\subseteq b\circ(a_0\circ x)=(ba_0)\circ x\subseteq \mathbb{R}^+\circ x=C$. Thus $\mathbb{R}^+\circ x$ is a cone.\\
2) If $x_0\in C$, then $x_0\in 1\circ x_0\subseteq \bigcup\limits_{a\in\mathbb{R}^+, x\in C}a\circ x$. Conversely, $a\circ x \subseteq C$, for all $a\in\mathbb{R}^+$ and $x\in C$. Thus $\bigcup\limits_{a\in\mathbb{R}^+, x\in C}a\circ x\subseteq C$.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition}
A convex cone in a real hypervector space $V$ is a cone which is a convex set.
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{example}
The set $C=\{(x,y)\in \mathbb{R}^2:\ x\geq 0\}$ is a convex cone of the hypervector space in Example \ref{Ex 5}, which is not a subhyperspace of $\mathbb{R}^2$.
\end{example}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 5}
A subset $C$ of real hypervector space $V=(V,+,\circ,\mathbb{R})$ is a convex cone if and only if it is closed under addition and positive external hyperoperation, i.e.
\[\forall x,y\in C, a\in\mathbb{R}^+;\ x+y\in C\ and\ a\circ x\subseteq C.\]
\end{theorem}
%---------------------------------------------------------------------
\begin{proof}
Let $C$ be a convex cone and $x,y\in C$. Then $\frac{1}{2}\circ(x+y)\subseteq \frac{1}{2}\circ x+\frac{1}{2}\circ y\subseteq C$. Thus $x+y\in 1\circ(x+y)=2\circ(\frac{1}{2}\circ(x+y))\subseteq C$. Conversely, if $x,y\in C$ and $a\in\mathbb{R}^+$ such that $|a|\leq 1$, then $(1-a)\circ x\subseteq C$ and $a\circ y\subseteq C$. Hence $(1-a)\circ x+a\circ y\subseteq C$, which means that $C$ is convex.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{corollary}
A subset $C$ of real hypervector space $V=(V,+,\circ,\mathbb{R})$ is a convex cone if and only if $a_1\circ x_1+\cdots+a_n\circ x_n\subseteq C$, for all $a_1,\dots,a_n\in \mathbb{R}^+$ and $x_1,\ldots,x_n\in C$.
\end{corollary}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{corollary}
Let $S$ be a non-empty subset of real hypervector space $V=(V,+,\circ,\mathbb{R})$. Then
\[C=\left\{t\in \sum_{i=1}^{n}a_i\circ x_i: a_i\in\mathbb{R}^+, x_i\in S, n\in \mathbb{N}\right\}\]
is the smallest convex cone containing $S$.
\end{corollary}
%---------------------------------------------------------------------
\begin{proof}
Let $a\in\mathbb{R}^+$ and $t\in C$. Then
\[a\circ t\subseteq a\circ\left(\sum_{i=1}^{n}a_i\circ x_i\right)\subseteq\sum_{i=1}^{n}(aa_i)\circ x_i\subseteq C,\]
and so $C$ is a cone. Also if $t,t^\prime\in C$ and $a\in\mathbb{R}$ such that $|a|\leq 1$, then $t\in\sum_{i=1}^{n}a_i\circ x_i$ and $t^\prime\in\sum_{j=1}^{m}a^\prime_j\circ x^\prime_j$, where $a_i,a^\prime_j\in\mathbb{R}^+$ and $x_i,x^\prime_j\in S$. Thus
\begin{eqnarray*}
a\circ t+(1-a)\circ t^\prime &\subseteq &a\circ\sum\limits_{i=1}^{n}a_{i}\circ x_{i}+(1-a) \circ \sum\limits_{j=1}^{m} a^\prime_{j} \circ x^\prime_{j} \\
&\subseteq &\sum\limits_{i=1}^{n}(aa_{i})\circ x_{i}+\sum\limits_{j=1}^{m}((1-a)a^\prime_{j})\circ x^\prime_{j} \\
&\subseteq &C.
\end{eqnarray*}
Hence $C$ is convex. Also for all $x\in S$, $x\in 1\circ x$ and so $x\in C$, which implies that $S\subseteq C$. Now suppose $C^\prime$ is a convex cone containing $S$ and $t\in C$. Then $t\in \sum_{i=1}^{n}a_i\circ x_i$, for some $a_i\in\mathbb{R}^+$, $x_i\in S$ and $n\in \mathbb{N}$. Thus $t\in\sum_{i=1}^{n}a_i\circ x_i\subseteq C^\prime$, by Theorem \ref{Thm 5}, and so $C\subseteq C^\prime$.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
If $S$ is a non-empty subset of real hypervector space $V=(V,+,\circ,\mathbb{R})$, then the convex cone
\[Cone\ S=\left\{t\in \sum_{i=1}^{n}a_i\circ x_i: a_i\in\mathbb{R}^+, x_i\in S, n\in \mathbb{N}\right\}\]
is called the convex cone spanned by $S$.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 7}
Let $C$ be a convex cone of real hypervector space $V=(V,+,\circ,\mathbb{R})$ containing $\underline{0}$. Then $C-C=\{x-y: x,y\in C\}$ is the smallest subhyperspace of $V$ containing $C$ and $(-C)\cap C$ is the largest subhyperspace of $V$ contained in $C$.
\end{theorem}
%---------------------------------------------------------------------
\begin{proof}
By Theorem \ref{Thm 5}, $C$ is closed under addition and positive external hyperoperation. If $x-y,x^\prime-y^\prime\in C-C$, then $(x-y)-(x^\prime-y^\prime)=(x+y^\prime)-(y+x^\prime)\in C-C$. Now let $x-y\in C-C$ and $a\in \mathbb{R}$. If $a\geq 0$, then $a\circ (x-y)\subseteq a\circ x-a\circ y\subseteq C-C$. If $a<0$, then
\begin{eqnarray*}
a\circ (x-y) &=& (-a)\circ ((-1)\circ(x-y))\\
&\subseteq & (-a)\circ ((-1)\circ x+(-1)\circ(-y))\\
&=& (-a)\circ (1\circ y-1\circ x)\\
&\subseteq & (-a)\circ (1\circ y)-(-a)\circ (1\circ x)\\
&=& (-a)\circ y-(-a)\circ x\\
&\subseteq & C-C.
\end{eqnarray*}
Thus $C-C\leqslant V$. Clearly, $C\subseteq C-C$. Now if $C^\prime$ is a subhyperspace of $V$ containing $C$, then for all $x-y\in C-C$, $x-y\in C^\prime-C^\prime\subseteq C^\prime$, and so $C-C\subseteq C^\prime$, i.e. $C-C$ is the smallest subhyperspace of $V$ containing $C$. Similarly, the second part of the theorem is proved.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convex Fuzzy Cones}\label{Sec. Conv. F. Cone}
The concepts of convex fuzzy subsets of hypervector spaces was studied by the author in \cite{Dehghan Affine Convex}. In this section by introducing the notion of fuzzy cones of real hypervector spaces, as an special kind of convex fuzzy subsets, some properties of convex fuzzy cones of such spaces are studied. Also the smallest fuzzy subhyperspace of $V$ containing $\mu$ and the largest fuzzy subhyperspace of $V$ contained in $\mu$ is achieved, for a convex fuzzy cone $\mu$ of real hypervector space $V$.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Let $V$ be a hypervector space over valued field $K$. Then a fuzzy subset $\mu$ of $V$ is called convex if $a\circ\mu+(1-a)\circ\mu \subseteq\mu$, for all $a\in K$ with $\left\vert a\right\vert \leq 1$.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 6}\cite{Dehghan Affine Convex}
Let $V$ be a hypervector space over valued field $K$ and $\mu\in FS(V)$. Then the following conditions are equivalent:
\begin{enumerate}
  \item $\mu$ is convex;
  \item $\forall x,y\in V$, $\forall a\in K$, $|a|\leq 1$, $\bigwedge\limits_{t\in a\circ x+(1-a)\circ y}\mu(t) \geq\mu(x) \wedge \mu(y)$;
  \item for all $\alpha\in[0,1]$, $\mu_{\alpha}$ is a convex subset of $V$.
\end{enumerate}
\end{theorem}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{proposition}\label{P 1}\cite{Dehghan Affine Convex}
Let $V$ be a hypervector space over valued field $K$ and $\left\{\mu_{i}\right\}_{i\in I}$ be a family of convex fuzzy subsets of $V$. Then $\bigcap\limits_{i\in I}\mu_{i}$ is a convex fuzzy subset of $V$.
\end{proposition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition}
A fuzzy subset $\mu$ of real hypervector space $V$ is called a fuzzy cone if
\[\forall x\in V,\ a\in \mathbb{R}^+,\ \bigwedge_{t\in a\circ x}\mu(t)\geq \mu(x).\]
$\mu\in FS(V)$ is called convex fuzzy cone if it is convex and fuzzy cone.
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{proposition}\label{D fuzzy cone}
If $\mu$ is a fuzzy cone of real hypervector space $V$, then $\bigwedge\limits_{t\in a\circ x}\mu(t)=\mu(x)$, for all $x\in V$, $a\in \mathbb{R}^+$.
\end{proposition}
%---------------------------------------------------------------------
\begin{proof}
If $x\in V$ and $a\in \mathbb{R}^+$, then
\begin{eqnarray*}
\bigwedge\limits_{t\in a\circ x}\mu(t) &\geq& \mu(x)\\
&\geq& \bigwedge\limits_{y\in 1\circ x}\mu(y)\\
&=& \bigwedge\limits_{y\in a^{-1}\circ(a\circ x)}\mu(y)\\
&=& \bigwedge\limits_{y\in a^{-1}\circ z, z\in a\circ x}\mu(y)\\
&\geq& \bigwedge\limits_{z\in a\circ x}\mu(z).
\end{eqnarray*}
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{proposition}\label{P 2}
If $\mu$ is a fuzzy subset of real hypervector space $V$, then the following conditions are equivalent:
\begin{enumerate}
  \item $\mu$ is a convex fuzzy cone;
  \item $\forall x,y\in V$, $\forall a,a^\prime\in\mathbb{R}^+$, $|a|\leq 1$, $\bigwedge\limits_{t\in a\circ x+(1-a)\circ y}\mu(t)\geq\mu(x)\wedge\mu(y)$ and $\bigwedge\limits_{t\in a^\prime\circ x}\mu(t)\geq\mu(x)$;
  \item for all $\alpha\in[0,1]$, $\mu_{\alpha}$ is a convex cone of $V$.
\end{enumerate}
\end{proposition}
%---------------------------------------------------------------------
\begin{proof}
$1\Leftrightarrow 2)$ It follows from Theorem \ref{Thm 6} and Definition \ref{D fuzzy cone}.\\
$2\Leftrightarrow 3)$ By Theorem \ref{Thm 6}, $\mu$ is convex if and only if $\mu_{\alpha}$ is convex. Now if $\mu$ is a fuzzy cone, $\alpha\in[0,1]$, $y\in\mu_{\alpha}$ and $z\in a\circ y$, then $\mu(z)\geq \bigwedge\limits_{t\in a\circ y}\mu(t)\geq \mu(y) \geq \alpha$ and so $z\in\mu_\alpha$, i.e. $a\circ \mu_\alpha \subseteq \mu_\alpha$, which means that $\mu_\alpha$ is a cone. Conversely, let $a\in\mathbb{R}^+$ such that $0<a\leq 1$, $x\in V$ and $t\in a\circ x$. Put $\mu(x)=\alpha$. Then $x\in\mu_\alpha$ and so $a\circ x\subseteq \mu_\alpha$. Thus $t\in\mu_\alpha$ and $\mu(t)\geq\alpha$. Hence $\bigwedge\limits_{t\in a\circ x} \mu(t) \geq\alpha$.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{proposition}
The intersection of an arbitrary family $\{\mu_i\}_{i\in I}$ of (convex) fuzzy cones is a (convex) fuzzy cone.
\end{proposition}
%---------------------------------------------------------------------
\begin{proof}
By Proposition \ref{P 1}, $\bigcap\limits_{i\in I}\mu_{i}$ is convex. Also
\begin{eqnarray*}
\bigwedge\limits_{t\in a\circ x}\left(\bigcap\limits_{i\in I}\mu_i\right)(t) &=& \bigwedge\limits_{t\in a\circ x}\left(\bigwedge \limits_{i\in I}\mu_i(t)\right)\\
&=& \bigwedge\limits_{i\in I}\left(\bigwedge \limits_{t\in a\circ x}\mu_i(t)\right)\\
&\geq& \bigwedge\limits_{i\in I}\mu_i(x)\\
&=& \left(\bigcap\limits_{i\in I}\mu_i\right)(x),
\end{eqnarray*}
for all $a\in \mathbb{R}^+$ and $x\in V$.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Similar to other structures, the previous proposition allows us to define the cone of a fuzzy subset.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{definition}
Let $\mu$ be a fuzzy subset of real hypervector space $V$. Then the convex fuzzy cone of $\mu$, denoted by $Cone\ \mu$, is the infimum of all convex fuzzy cones of $V$ containing $\mu$, i.e. the smallest convex fuzzy cone of $V$ containing $\mu$.
\end{definition}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}\label{Thm 8}
Let $V$ be an invertible $n$-dimensional strongly right distributive real hypervector space and $\mu$ be a convex fuzzy cone of $V$ such that $\mu(\underline{0})=\bigvee \limits _{x\in V}\mu(x)$. Then $\mu-\mu\in FS(V)$ defined by
\[(\mu-\mu)(x)=\bigvee\limits_{x=x_1-x_2}(\mu(x_1)\wedge\mu(x_2))\]
is the smallest fuzzy subhyperspace of $V$ containing $\mu$.
\end{theorem}
%---------------------------------------------------------------------
\begin{proof}
At first we shall construct the smallest fuzzy subhyperspace of $V$ from $\mu$ (the intersection of all fuzzy subhyperspaces of $V$ containing $\mu$) and then show that it is equal to $\mu-\mu$.\\
By Proposition \ref{P 2}, for all $\alpha \in [0,1]$, $\mu_\alpha$ is a convex cone containing $\underline{0}$, since $\mu (\underline{0})=\bigvee\limits_{x\in V}\mu(x)$. Suppose $V_\alpha=\langle\mu_\alpha\rangle$. Then $\{V_\alpha\}_{\alpha\in I}$ is a chain of at most $n+1$ different subhyperspaces of $V$ with the largest element $V=V_0=\langle\mu_0\rangle$. Denote this chain by $W_0\subseteq W_1\subseteq\cdots\subseteq W_k$, where $k\leq n$. Now we partition $I$ in $k+1$ subsets
\[I_i=\{\alpha\in I:\ V_\alpha=W_i\},\ \ 0\leq i\leq k.\]
Put $\alpha_i=\bigvee I_i$, for all $0\leq i\leq k$. Then by Representation Theorem (Corollary \ref{C 1}), $\nu=\bigvee\limits _{i=0}^k \alpha_i 1_{W_i}$ is a fuzzy subhyperspace of $V$. Also if $x\in V$ and $\mu(x)=\alpha$, then $\alpha\in I_i$, for some $0\leq i\leq k$ and so
\[\nu(x)\geq \alpha_i=\bigvee I_i\geq\alpha=\mu(x).\]
Now by construction of $\alpha_i$'s and $W_i$'s it follows that $\nu$ is the smallest fuzzy subhyperspace of $V$ containing $\mu$.\\
Finally, we show that $\nu=\mu-\mu$. Let $x\in W_i\setminus W_{i-1}$, for some $0\leq i\leq k$. Then $\nu(x)=\alpha_i$ and by Theorem \ref{Thm 7}, $x=y_0-z_0$ for some $y_0,z_0\in\mu_\alpha$. Thus
\begin{eqnarray*}
(\mu-\mu)(x) &=& \bigvee\limits_{x=y-z}(\mu(y)-\mu(z))\\
&\geq& \mu(y_0)-\mu(z_0)\\
&\geq& \alpha\wedge\alpha\\
&=& \alpha.
\end{eqnarray*}
Hence $(\mu-\mu)(x)\geq\bigvee I_i=\alpha_i=\nu(x)$. Note that for all $\alpha\in I_{i-1}$, there exist no $y,z\in\mu_\alpha$ such that $x=y-z$, since $x\notin W_{i-1}$. By repeating the above it follows that for all $\alpha>\alpha_i$, there are no $y,z\in\mu_\alpha$ such that $x=y-z$. Then $(\mu-\mu)(x)=\alpha_i=\nu(x)$.
\end{proof}
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{theorem}
Let $V$ be an invertible $n$-dimensional strongly right distributive real hypervector space and $\mu$ be a convex fuzzy cone of $V$ such that $\mu(\underline{0})=\bigvee \limits _{x\in V}\mu(x)$. Then $\mu\wedge(-\mu)\in FS(V)$ defined by $(\mu\wedge(-\mu))(x)= \mu(x)\wedge\mu(-x)$ is the largest fuzzy subhyperspace of $V$ contained in $\mu$.
\end{theorem}
%---------------------------------------------------------------------
\begin{proof}
The proof is similar to the proof of Theorem \ref{Thm 8}.\\
By Proposition \ref{P 2}, for all $\alpha \in [0,1]$, $\mu_\alpha$ is a convex cone containing $\underline{0}$, since $\mu (\underline{0})=\bigvee\limits_{x\in V}\mu(x)$. From Theorem \ref{Thm 7}, there exists a largest subhyperspace $V_\alpha\subseteq \mu_\alpha$. Let $\{V_\alpha\}_{\alpha\in I}$ be the chain of subhyperspaces of $V$ and $W_0\subseteq W_1\subseteq\cdots\subseteq W_k$, $k\leq n$, be the at most $n+1$ different ones. Here we have again a partition of $I$ in $I_i=\{\alpha\in I:\ V_\alpha= W_i\}$, $0\leq i\leq k$. Suppose $\alpha_i=\bigvee I_i$, $0\leq i\leq k$ and $\nu=\bigvee\limits _{i=0}^k \alpha_i 1_{W_i}$. Then by Representation Theorem (Corollary \ref{C 1}) and by construction of $\alpha_i$'s and $W_i$'s, $\nu$ is the largest fuzzy subhyperspace of $V$ contained in $\mu$. Now we show that $\nu=\mu\cap(-\mu)$. Let $x\in W_i\setminus W_{i-1}$, for some $0\leq i\leq k$. Then $\nu(x)=\alpha_i\in I_i$ and by Theorem \ref{Thm 7}, $W_i=\mu_{\alpha_i}\cap(-\mu_{\alpha_i})$. Thus
\[(\mu\cap(-\mu))(x)=\mu(x)\wedge\mu(-x))\geq \alpha_i\wedge\alpha_i=\alpha_i=\nu(x).\]
It is clear that for all $\alpha\in I_{i-1}$, $\mu(x)<\alpha$ or $\mu(-x)<\alpha$. Thus in this case $\bigwedge I_{i-1}=\alpha_i$ and so $(\mu\wedge(-\mu))(x)=\alpha_i=\nu(x)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%\begin{acknowledgements}
%If you'd like to thank anyone, place your comments here
%and remove the percent signs.
%\end{acknowledgements}


% Authors must disclose all relationships or interests that
% could have direct or potential influence or impart bias on
% the work:
\Section*{Conflict of interest}

The authors declare that they have no conflict of interest.


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