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Advances in Fuzzy Systems

Volume 2008 (2008), Article ID 295649, 9 pages

http://dx.doi.org/10.1155/2008/295649

## Fuzzy Hypervector Spaces

Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, 47415-453 Babolsar, Iran

Received 19 September 2007; Revised 27 February 2008; Accepted 11 May 2008

Academic Editor: Adel Alimi

Copyright © 2008 R. Ameri and O. R. Dehghan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this paper is the generalization of the notion of fuzzy vector spaces to fuzzy hypervector spaces. In this regard, by considering the notion of fuzzy hypervector spaces, we characterized a fuzzy hypervector space based on its level sub-hyperspace. The algebraic nature of fuzzy hypervector space under transformations is studied. Certain conditions are obtained under which a given fuzzy hypervector space can or cannot be realized as a union of two fuzzy hypervector spaces such that none is contained in the other. The construction of a fuzzy hypervector space generated by a given fuzzy subset of a hypervector space is given. The set of all fuzzy cosets of a fuzzy hypervector space is shown to be a hypervector space. Finally, a fuzzy quotient hypervector space is defined and an analogue of a consequence of the “fundamental theorem of homomorphisms” is obtained.

#### 1. Introduction

The notion of a hypergroup was introduced by Marty in 1934 [1]. Since then many researchers have worked on hyperalgebraic structures and developed this theory (for more see [2–4]). In 1990, Tallini introduced the notion of hypervector spaces (see [5, 6]) and studied basic properties of them.

The concept of a fuzzy subset of a nonempty set was introduced by Zadeh in 1965 [7] as a function from a nonempty set into the unit real interval . Rosenfeld [8] applied this to the theory of groupoids and groups and then many researchers developed it in all the fields of algebra. The concepts of a fuzzy field and a fuzzy linear space over a fuzzy field were introduced and discussed by Nanda [9]. In 1977, Katsaras and Liu [10] formulated and studied the notion of fuzzy vector subspaces over the field of real or complex numbers.

Recently fuzzy set theory has been well developed in the context of hyperalgebraic structure theory (see, e.g., [11–18]). In [11], the first author introduced and studied the notion of fuzzy hypervector space over valued fields. In this paper, we follow [11, 12] and study more properties of fuzzy hypervector spaces. The paper provides the suitable tools to define and study the properties of fuzzy hypervector spaces, as a generalization of fuzzy vector spaces, and hence can be considered as an application of fuzzy sets to hyperstructure theory. In this regard, we study the algebraic nature of fuzzy hypervector spaces under transformation. Also we introduce and study fuzzy quotient of fuzzy hypervector spaces. In Section 2, some basic definitions and results of hypervector spaces and fuzzy sets which will be used in next sections are given. In Section 3, the union of fuzzy sub-yperspaces are investigated. It is shown that every fuzzy sub-hyperspace can be written as union of two distinct fuzzy sub-hyperspaces.

In Section 4, the notion of a fuzzy sub-hyperspaces generated by a fuzzy subset is studied. Finally, in Section 5, a fuzzy coset of a fuzzy hypervector space is defined. For a fuzzy sub-hyperspace of a hypervector space, the following results are established:

(i)the set of all fuzzy cosets of in is a hypervector space;(ii), where

Finally, a fuzzy quotient hyperspace is defined and it is shown that each fuzzy sub-hyperspace of has a correspondence in a natural way to a fuzzy sub-hyperspaces of

#### 2. Preliminaries

In this section, we present some definitions and simple properties of hypervector spaces and fuzzy subsets, that we will use later.

A map is called a *hyperoperation* or join
operation, where is the set of all nonempty subsets of .
The join operation is extended to subsets of in natural way, so that is given byThe notations and are used for and ,
respectively. Generally, the singleton is identified by its element .

*Definition 1 (See [5]). * Let be a field and an Abelian group. A hypervector space over is a quadruple ,
where “” is a mapping:such that for all and the following conditions hold:

(H_{1})(H_{2})(H_{3})(H_{4})(H_{5})

*Remark 1. *(i) In
the right-hand side of the right distributive
law (), the sum is meant in the sense of Frobenius, that is we consider the
set of all sums of an element of with an element of .
Similarly, it is in the left distributive law ().

(ii) We say that is antileft distributive ifand strongly left distributive
ifIn a similar way, we define the antiright
distributive and strongly right distributive hypervector spaces,
respectively. The hypervector space is called strongly distributive if it is both
strongly left and strongly right distributive.

(iii) The left-hand side of the associative law () means the set-theoretical union of all the sets ,
where runs over the set ,
that is,
(iv) Let ,
where is the zero of .
In [5], it is shown
that if is either strongly right or strongly left
distributive, then is a subgroup of .

*Example 1. *In ,
we define the scalar product in by settingwhere is the line through the point and .
Then is a strongly left distributive hypervector
space.

In the sequel of this article, unless otherwise specified, we assume that is a hypervector space over the field

*Definition 2. *A nonempty subset of is a sub-hyperspace if is itself a hypervector space with the
hyperoperation on ,
that is,In this case,
one writes
.

Proposition 1. *The intersection of a family
of sub-hyperspaces is a sub-hyperspace.*

*Proof. *Straightforward.

*Definition 3. *If is a nonempty subset of ,
then the linear span of is the smallest sub-hyperspace of containing ,
that is,

Lemma 1 (See [12]). * If is a nonempty subset of ,
then *

*Definition 4. *A subset of is called linearly independent if for
every vector in ,
and , ,
implies that .
A subset of is called linearly dependent if it is
not linearly independent.

*Definition 5. *A basis for is a linearly independent subset of which linearly spans , that is, .
One says that is finite dimensional if it has a
finite basis.

*Remark 2. *Note that some hypervector spaces (some set of vectors) may not have any collection of
linearly independent vectors. Such hypervector space (set) is called independentless. Clearly if is independentless, then has not any basis; and for such hypervector
spaces, dimension is not defined. In this case we say that is dimensionless.
The hypervector space in Example 1 is a nontrivial example of an
independentless hypervector space, since belongs to every line through the .

*Definition 6. *A hypervector space over is said to be -invertible or shortly invertible if
and only if implies that ,
for

*Remark 3. *Let be a hypervector space and
a sub-hyperspace of .
Consider the quotient Abelian group .
Define the ruleThen it is easy to verify that is a hypervector space over and it is called the quotient hypervector
space of over .

Theorem 1 (See [19]). * Let be strongly left distributive and invertible.
If is finite dimensional and is sub-hyperspace of ,
then the following hold: *

(i)* is finite dimensional and *(ii)

*Definition 7. *Let and be hypervector spaces over .
A mapping is called

(i) weak linear transformation if and only if
(ii) (inclusion) linear transformation if and only if
(iii) good transformation if and only if

*Definition 8. *Let be a linear transformation. Then the kernel of is denoted by and defined by

*Definition 9. *(i) For a fuzzy subset of ,
the level subset is defined by
(ii) The image of is denoted by and is defined by
(iii) If and ,
then by ,
one means

*Definition 10. *(i) (Extension principle) let be a mapping, , and .
Then,
and ,
respectively, are defined:
(ii)

*Definition 11. *Let be a mapping and .
Then is called -invariant if

Clearly if is -invariant, then

*Definition 12. *Let be a field and .
Suppose the following conditions hold:

(i), for all ,(ii), for all ,(iii), for all ,(iv),

Then, call a fuzzy field in and denote it by .

Obviously, Definition 12 is a generalization of the classical field notion.

*Definition 13 (See [11]). * Let be a hypervector space over a field and
a fuzzy field of .
A fuzzy set of is said to be a fuzzy hypervector space of over fuzzy field ,
if for all and all ,
the following conditions are satisfied:

(i)(ii)(iii)(iv).

Obviously, Definition 13 is a generalization of the concept of a fuzzy vector space and also of the classical notion of a hypervector space (in sense of Tallini [5]). If we consider the characteristic function of then is called a fuzzy sub-hyperspace of .

*Definition 14. *Let be a nonempty collection of fuzzy
sub-hyperspaces of .
Then the fuzzy subset of is defined by the following:

Proposition 2. *The intersection of a family
of fuzzy sub-hyperspaces is a fuzzy sub-hyperspace.*

Proposition 3 (See [11]). * Let be a strongly left distributive hypervector
space over the field and
a fuzzy field. Let .
Then is a fuzzy hypervector space over if and only if **and for all and *

Proposition 4 (See [11]). * Let and .
Then is a fuzzy hypervector space over if and only if is a hypervector space over the field for all and . is called a level sub-hyperspace of .*

Proposition 5 (See [11]). * Let and be strongly left distributive hypervector
spaces, and a good transformation. Let and be fuzzy sub-hyperspaces over .
Then and are fuzzy sub-hyperspaces over .*

#### 3. Union of Fuzzy Sub-Hyperspaces

We start this section with some basic properties of fuzzy hypervector spaces.

Proposition 6. *If is a fuzzy hypervector space over the fuzzy
field ,
then *

*Proof. *By () so .
On the other hand, using Definition 13, we haveThus .

Theorem 2. *Let be a fuzzy sub-hyperspace of and let
(with ) any two level sub-hyperspaces of .
Then if and only if there is no in such that .*

*Proof. *Straightforward..

From Theorem 2 it follows that the level sub-hyperspaces of a fuzzy sub-hyperspace of need not be distinct. If such that , then the family of level sub-hyperspaces of consists of and we have the chain

Proposition 7. *Let be a proper sub-hyperspace of .
Then the fuzzy subset of defined by **where is a fuzzy sub-hyperspace of *

*Proof. *Obvious.

Theorem 3. *Let and be hypervector spaces over the field ,
and let
be a mapping. Let and be fuzzy sub-hyperspaces over such that **Then *

(i)* and the chain of level sub-hyperspaces of is*(ii)* and the chain of level sub-hyperspaces of is*

*Proof. *(i) Clearly ,
sinceAlso for all because Hence the chain of level
sub-hyperspaces of is

(ii) Clearly ,
sinceAlso for all becauseHence the chain of level
sub-hyperspaces of is

Theorem 4. *Let and be strongly left-distributive hypervector
spaces over the field
and
a good transformation. Then the mapping defines a one-to-one correspondence between
the set of all -invariant fuzzy sub-hyperspaces of and the set of all fuzzy sub-hyperspaces of .*

*Proof. *
It immediately follows from
Proposition 5 and the following results:

(i) where is a -invariant fuzzy sub-hyperspace of

(ii) where is any fuzzy sub-hyperspace of

For sub-hyperspaces and of , it is easy to verify that is a sub-hyperspace of if and only if or In the sequel, an attempt is made to study the following problem. Is it possible to realize a fuzzy sub-hyperspace as a union of two fuzzy sub-hyperspaces such that none of them is contained in the other?

It is shown that there exist fuzzy sub-hyperspaces such that their union is a fuzzy sub-hyperspace, but none of them is contained in the other. Furthermore, it turns out that an answer to the above problem depends on the image set of the underlying fuzzy sub-hyperspace. The complete story is explained in Theorems 5, 6, 7, and 8.

We begin by offering an example showing that the union of two fuzzy sub-hyperspaces need not be so.

*Example 2. *In
Example 1, setObviously, and are sub-hyperspaces of such that for all Choose numbers ,
such that Define fuzzy subsets and byThen and Thus from Proposition 4, it follows that and are fuzzy sub-hyperspaces of .
Clearly ,
given byis not a fuzzy sub-hyperspace of since is not a sub-hyperspace of

Theorem 5. *Let be a fuzzy sub-hyperspace of such that ,
where If ,
where and are fuzzy sub-hyperspaces of ,
then either or *

*Proof. *Since is a fuzzy sub-hyperspace of ,
the level fuzzy sub-hyperspace is a sub-hyperspace of such thatSo either or Thus either or

Next we give an example showing that the union of two fuzzy sub-hyperspaces, such that none is contained in the other, may not be a fuzzy sub-hyperspace.

*Example 3. *Let and be the same as in Example 2. Define fuzzy
subsets and of byThensuch that and Thus from Proposition 4, it follows that ,
and are fuzzy sub-hyperspaces of .
However, and

Lemma 2. *Let be a fuzzy sub-hyperspace of .
If for some then *

Theorem 6. *Let be a fuzzy sub-hyperspace of such that ,
where If ,
where and are fuzzy sub-hyperspaces of ,
then either or *

*Proof. *To
obtain a proof by contradiction, assume that and for some ThenThereforesince So thus by Lemma 2 we obtain
thatand similarlyHenceAgain, the desired contradiction.

Theorem 7. *Let be a fuzzy sub-hyperspace of such that Then there exist fuzzy sub-hyperspaces and of such that , and *

*Proof. *Let ,
where and Choose such thatDefine fuzzy subsets and of byClearly ,
and are fuzzy sub-hyperspaces of such that .
However, and

Theorem 8. *If is a fuzzy sub-hyperspace of such that where and then there exist fuzzy sub-hyperspaces and of such that and *

#### 4. Fuzzy Sub-Hyperspace Generated by a Fuzzy Subset

In this section, we give the construction of the fuzzy sub-hyperspace generated by a fuzzy subset of . It is possible that may be strictly greater than . For any we will write for the sub-hyperspace generated by in .

Theorem 9. *Let be a fuzzy subset of such that Define sub-hyperspaces by **where and Then the fuzzy subset of defined by **is the fuzzy sub-hyperspace
generated by in *

*Proof. *By
Proposition 4, we obtain that is a fuzzy sub-hyperspace of Now where and for whereLet be any fuzzy sub-hyperspace of containing It is sufficient to prove thatLet Then there exist and such that NowHence

Let ,
Then there exist and such that NowHence for all as desired.

#### 5. Fuzzy Cosets

The following theorem will serve as a guiding factor in defining a fuzzy coset of a fuzzy sub-hyperspace.

Theorem 10. *(i) Let be a fuzzy sub-hyperspace of and Then the fuzzy subset **is a fuzzy sub-hyperspace of **(ii) If is a sub-hyperspace of and is a fuzzy sub-hyperspace of such that only when then there exists a fuzzy sub-hyperspace of such that and *

*Proof. *(i) It
is easy to verify that is well defined. We now show that is a fuzzy sub-hyperspace of Let and Thenand if ,
thenthusTherefore
(ii) Define a fuzzy subset of byClearly is a fuzzy sub-hyperspace of Now becauseMoreover since

*Definition 15. *Let be a fuzzy sub-hyperspace of For the fuzzy subset of defined byis called the fuzzy coset
determined by and The set of all fuzzy cosets of is denoted by , that is,

Proposition 8. *Let be a fuzzy sub-hyperspace of Then the set of all fuzzy cosets of with operation **and external
operation **is a hypervector space over the
field *

*Proof. *It is
easy to verify that “” and “” are well defined. Let and Then

(H_{1})
(H_{2})
(H_{3})
(H_{4})
(H_{5}) for all ,
since

The following lemma will serve as a powerful tool in proving Theorem 11 regarding the of the hypervector space

Lemma 3. *Let be a fuzzy sub-hyperspace of Then *

*Proof. *Let Then for all .
If then by Lemma 2. If then where Hence Thus in either case, we have shown that for all .
That is, for all .
Consequently The converse is straightforward.

Theorem 11. *Let be a fuzzy sub-hyperspace of and Then *

*Proof. *If for all ,
then, for all .
So that and HenceSo we assume that is not constant. Let and Letbe a basis of such that is a basis of Then is a basis of We will show that the set is a basis of over the field

Clearly whenever , becauseNext we prove that the set generates over For this, let .
Then by Lemma 3. So that is a nonzero element of Hence there exist such thatthusAlso is linearly independent over because if where then such thatso there exist such thatthus there exist such thatHence

Theorem 12. *Let be a fuzzy sub-hyperspace of and Then *

(a)* the fuzzy subset of defined byis a fuzzy subhyperspace of ;*(b)*the mapis an onto good transformation with kernel ,
where *

*Proof. *(a)
(i)
(ii)

(iii) for all thus

(b) For every and thus is an onto good
transformation. Moreover, ,
wherethereforebut soThus ,
where

*Definition 16. *Let be a fuzzy sub-hyperspace of .
The fuzzy subset of is called the fuzzy quotient hypervector space
determined by

Finally, we establish for fuzzy sub-hyperspace an analogu of a consequence of the “fundamental theorem of homomorphisms.”

Theorem 13. *Let be a fuzzy sub-hyperspace of Then each fuzzy sub-hyperspace of corresponds in a natural way to a fuzzy
sub-hyperspace of *

*Proof. *Let be any fuzzy sub-hyperspace of Then it is easy to see that the fuzzy subset of defined byis a fuzzy sub-hyperspace of

#### 6. Conclusion

As discussed above, the notion of a fuzzy hypervector space is a generalization of fuzzy vector spaces and the paper provides the basic notions and results to study the fuzzy hypervector spaces. We hope that this paper encourages the researchers to study the more properties of fuzzy hypervector spaces and its application.

#### Acknowledgments

This research is partially supported by the Fuzzy Systems and Its Applications Center of Excellence, Shahid Bahonar University of Kerman; and by Research Center in Algebraic Hyperstructures and Fuzzy Mathematics, University of Mazandaran, Babolsar.

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