# Fuzzy Hypervector Spaces

Authors: R. Ameri; O. R. Dehghan
Journal: Advances in Fuzzy Systems
 (2008)

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## 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.


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## Body

## 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 X into the unit real interval I=[0,1].
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 Section4, 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 setVμ of all fuzzy cosets of μ in V is a hypervector space;(ii)
dimVμ=dimV−dimμt,
where(1)μ(t)={x∈V:μ(x)=μ(0¯)}.Finally, a fuzzy quotient hyperspace is defined and it
is shown that each fuzzy sub-hyperspace ofVμ has a correspondence in a natural way to a
fuzzy sub-hyperspaces of V.


## 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◦:H×H→P∗(H) is called a hyperoperation or join
operation, where P∗(H) is the set of all nonempty subsets of H.
The join operation is extended to subsets of H in natural way, so that A◦B is given by(2)A∘B=⋃{a∘b:a∈A,b∈B}.The notations a◦A and A◦a are used for {a}◦A and A◦{a},
respectively. Generally, the singleton {a} is identified by its element a.Definition 1 (See [5]).
LetK be a field and (V,+)  an Abelian group. A hypervector space over K is a quadruple (V,+,◦,K),
where “◦” is a mapping:(3)∘:K×V⟶P∗(V),such that for all a,b∈K and x,y∈V the following conditions hold:
(H1)
a◦(x+y)⊆a◦x+a◦y,(H2)
(a+b)◦x⊆a◦x+b◦x,(H3)
a◦(b◦x)=(ab)◦x,(H4)
a◦(−x)=(−a)◦x=−(a◦x),(H5)
x∈1◦x.Remark 1.
(i) In
the right-hand side of the right distributive
law (H1), the sum is meant in the sense of Frobenius, that is we consider the
set of all sums of an element of a◦x with an element of a◦y.
Similarly, it is in the left distributive law (H2).
(ii) We say that(V,+,◦,K) is antileft distributive if(4)∀a,b∈K,∀x∈V,(a+b)∘x⊇a∘x+b∘x,and strongly left distributive
if(5)∀a,b∈K,∀x∈V,(a+b)∘x=a∘x+b∘x,In a similar way, we define the antiright
distributive and strongly right distributive hypervector spaces,
respectively. The hypervector space (V,+,◦,K) is called strongly distributive if it is both
strongly left and strongly right distributive.
(iii) The left-hand side of the associative law (H3) means the set-theoretical union of all the sets a◦y,
where y runs over the set b◦x,
that is,(6)a∘(b∘x)=⋃y∈b∘xa∘y.
(iv) Let ΩV=0◦0¯,
where 0¯ is the zero of (V,+).
In [5], it is shown
that if V is either strongly right or strongly left
distributive, then Ω is a subgroup of (V,+).Example 1.
In(ℝ2,+),
we define the scalar product  in ℝ by setting(7)∀a∈ℝ,∀x∈ℝ2:a∘x={ox¯ifx⁢≠0¯,{0¯}ifx=0¯,where ox¯ is the line through the point x and 0¯=(0,0).
Then (ℝ2,+,◦,ℝ) is a strongly left distributive hypervector
space.In the sequel of this article, unless otherwise
specified, we assume thatV is a hypervector space over the field K.Definition 2.
A nonempty subsetW of V is a sub-hyperspace if W is itself a hypervector space with the
hyperoperation on V,
that is,(8)W≠∅,∀x,y∈W⟹x−y∈W,∀a∈K,∀x∈W⟹a∘x⊆W.In this case,
one writes
W≤V.Proposition 1.
The intersection of a family
of sub-hyperspaces is a sub-hyperspace.Proof.
Straightforward.Definition 3.
IfS is a nonempty subset of V,
then the linear span of S is the smallest sub-hyperspace of V containing S,
that is,(9)L(S)=〈S〉=⋂S⊆W≤VW.Lemma 1 (See [12]).
IfS is a nonempty subset of V,
then(10)L(S)={t∈∑i=1nai∘si,⁢ai∈K,⁢si∈S,⁢n∈ℕ}.Definition 4.
A subsetS of V is called linearly independent if for
every vector v1,v2,…,vn in S,
and c1,…,cn∈K, 0¯∈c1◦v1+⋯+cn◦vn,
implies that c1=c2=⋯=cn=0.
A subset S of V is called linearly dependent if it is
not linearly independent.Definition 5.
A basis forV is a linearly independent subset β of V which linearly  spans V, that is, L(β)=V.
One says that V is finite dimensional if it has a
finite basis.Remark 2.
Note that some hypervector spacesV (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. In this case we say that V is dimensionless.
The hypervector space (ℝ2,+,◦,ℝ) in Example 1 is a nontrivial example of an
independentless hypervector space, since 0¯ belongs to every line through the 0¯.Definition 6.
A hypervector spaceV over K is said to be K-invertible or shortly invertible if
and only if u∈a◦v implies that v∈a−1◦u,
for u,v∈V,a∈K∖{0}.Remark 3.
Let(V,+,◦,K) be a hypervector space and
W a sub-hyperspace of V.
Consider the quotient Abelian group (V/W,+).
Define the rule(11)∗:K×V/W⟶P∗(V/W)(a,v+W)⟼a∘v+W.Then it is easy to verify that (V/W,+,∗,K) is a hypervector space over K and it is called the quotient hypervector
space of V over W.Theorem 1 (See [19]).
LetV be strongly left distributive and invertible.
If V is finite dimensional and W is sub-hyperspace of V,
then the following hold:
(i)
W is finite dimensional and dimW≤dimV,(ii)
dimV/W=dimV−dimW.Definition 7.
LetV and W be hypervector spaces over K.
A mapping T:V→W is called
(i) weak linear transformation if and only if(12)T(x+y)=T(x)+T(y),T(a∘x)∩a∘T(x)≠∅,
(ii) (inclusion) linear transformation if and only if(13)T(x+y)=T(x)+T(y),T(a∘x)⊆a∘T(x),
(iii) good transformation if and only if(14)T(x+y)=T(x)+T(y),T(a∘x)=a∘T(x).Definition 8.
LetT:V→W be a linear transformation. Then the kernel of T is denoted by kerT and defined by(15)ker⁡T={x∈V:T(x)∈ΩW}.Definition 9.
(i) For a fuzzy subsetμ of X,
the level subset μt is defined by(16)μt={x∈X:μ(x)≥t},t∈[0,1].
(ii) The image of μ is denoted by Im(μ) and is defined by(17)Im(μ)={μ(x):x∈X}.
(iii) If μ∈FS(X) and A⊆X,
then by μ(A),
one means
(18)μ(A)=⋀a∈Aμ(a).Definition 10.
(i) (Extension principle) letf:X→Y be a mapping, μ∈FS(X), and ν∈FS(Y).
Then,
f(μ)∈FS(Y) and f−1(ν)∈FS(X),
respectively, are defined:(19)f(μ)(y)={⋁x∈f−1(y)μ(x)iff−1(y)≠∅,0otherwise,
(ii)
(20)f−1(ν)(x)=ν(f(x)),∀x∈X.Definition 11.
Letf:X→Y be a mapping and μ∈FS(X).
Then μ is called f-invariant if(21)∀x,y∈X,f(x)=f(y)⟹μ(x)=μ(y).Clearly ifμ is f-invariant, then f−1(f(μ))=μ.Definition 12.
LetK be a field and ν∈FS(K).
Suppose  the following  conditions hold:
(i)
ν(a+b)≥ν(a)⋀ν(b), for all a,b∈K,(ii)
ν(−a)≥ν(a), for all a∈K,(iii)
ν(ab)≥ν(a)⋀ν(b),  for all a,b∈K,(iv)
ν(a−1)≥ν(a),∀a∈K∖{0},
Then, callν a fuzzy field in K and denote it by νK.Obviously, Definition12 is a generalization of the
classical field notion.Definition 13 (See [11]).
LetV be a hypervector space over a field K and
ν  a fuzzy field of K.
A fuzzy set μ of V is said to be a fuzzy hypervector space of V over fuzzy field νK,
if for all x,y∈V and all a∈K,
the following conditions are satisfied:
(i)
μ(x+y)≥μ(x)⋀μ(y),(ii)
μ(−x)≥μ(x),(iii)
⋀y∈a◦xμ(y)≥ν(a)⋀μ(x),(iv)
ν(1)≥μ(0¯).Obviously, Definition13 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 ν=χK, the characteristic function of K, then μ is called a fuzzy sub-hyperspace of V.Definition 14.
Let{μi}i∈I be a nonempty collection of fuzzy
sub-hyperspaces of V.
Then the fuzzy subset ∩i∈Iμi of V is defined by the following:(22)(⋂i∈Iμi)(x)=⋀i∈Iμi(x).Proposition 2.
The intersection of a family
of fuzzy sub-hyperspaces is a fuzzy sub-hyperspace.Proposition 3 (See [11]).
LetV be a strongly left distributive hypervector
space over the field K and
νK  a fuzzy field. Let μ∈FS(V).
Then μ is a fuzzy hypervector space over νK if and only if(23)⋀z∈a∘x+b∘yμ(z)≥(ν(a)∧μ(x))∧(ν(b)∧μ(y)),and ν(1)≥μ(x), for all a,b∈K and x,y∈V.Proposition 4 (See [11]).
Letμ∈FS(V) and ν∈FS(K).
Then μ is a fuzzy hypervector space over νK if and only if μα is a hypervector space over the field να, for all α∈Im(ν) and ν(1)≥μ(0¯). μα is called a level sub-hyperspace of V.Proposition 5 (See [11]).
LetV and W be strongly left distributive hypervector
spaces, and T:V→W a good transformation. Let μV and ηW be fuzzy sub-hyperspaces over νK.
Then T(μ)W and T−1(η)V are fuzzy sub-hyperspaces over νK.


## 3. Union of Fuzzy Sub-Hyperspaces

We start this section with some
basic properties of fuzzy hypervector
spaces.Proposition 6.
Ifμ is a fuzzy hypervector space over the fuzzy
field νK,
then(24)⋀y∈1∘xμ(y)=μ(x).Proof.
By (H5) x∈1◦x, so ⋀y∈1◦xμ(y)≤μ(x).
On the other hand, using Definition 13, we have(25)⋀y∈1∘xμ(y)≥νK(1)∧μ(x)=1∧μ(x)=μ(x).Thus ⋀y∈1◦xμ(y)=μ(x).Theorem 2.
Letμ be a fuzzy sub-hyperspace of V and let
μt1,μt2 (with t1<t2) any two level sub-hyperspaces of μ.
Then μt1=μt2 if and only if there is no x in V such that t1≤μ(x)<t2.Proof.
Straightforward..From Theorem2 it follows that the level
sub-hyperspaces of a fuzzy sub-hyperspace μ of V need not be distinct. If Im(μ)={t0,…,tn} such that t0>t1>⋯>tn,
then the family of level sub-hyperspaces of μ consists of {μt1,…,μtn} and we have the chain μt1⊆⋯⊆μtn=V.Proposition 7.
LetW be a proper sub-hyperspace of V.
Then the fuzzy subset μ of V defined by(26)μ(x)={1ifx∈W,totherwise,where t∈[0,1) is a fuzzy sub-hyperspace of V.Proof.
Obvious.Theorem 3.
LetV and W be hypervector spaces over the field K,
and let
f:V→W be a mapping. Let μV and ηW be fuzzy sub-hyperspaces over νK such that(27)Im(μ)={t0,…,tn},witht0>t1>⋯>tn,Im(η)={s0,…,sm},withs0>s1>⋯>sm.Then
(i)
Imf(μ)⊆Im(μ) and the chain of level sub-hyperspaces of f(μ) is(28)f(μt1)⊆f(μt2)⊆⋯⊆f(μtn)=W.(ii)
Imf−1(η)=Im(η) and the chain of level sub-hyperspaces of f−1(η) is(29)f−1(ηs1)⊂f−1(ηs2)⊂⋯⊂f−1(ηsm)=V.Proof.
(i) ClearlyImf(μ)⊆Im(μ),
since(30)f(μ)(y)=⋁x∈f−1(y)μ(x),∀y∈W.Also f(μti)=(f(μ))ti, for all i=0,…,n, because (31)y∈f(μti)⟺∃x∈f−1(y),x∈μti⟺∃x∈f−1(y),μ(x)≥ti⟺⋁z∈f−1(y)μ(z)≥ti⟺f(μ)(y)≥ti⟺y∈(f(μ))ti. Hence the chain of level
sub-hyperspaces of f(μ) is f(μt1)⊆f(μt2)⊆⋯⊆f(μtn)=W.
(ii) ClearlyImf−1(η)=Im(η),
since(32)(f−1(η))(x)=η(f(x)),∀x∈V.Also f−1(ηsi)=(f−1(η))si, for all i=0,…,m, because(33)y∈f−1(ηsi)⟺f(x)∈ηsi⟺η(f(x))≥si⟺(f−1(η))(x)≥si⟺x∈(f−1(η))siHence the chain of level
sub-hyperspaces of f−1(η) is(34)f−1(ηs1)⊂f−1(ηs2)⊂⋯⊂f−1(ηsm)=V.Theorem 4.
LetV and W be strongly left-distributive hypervector
spaces over the field K
and
T:V→W a good transformation. Then the mapping μ↦f(μ) defines a one-to-one correspondence between
the set of all f-invariant fuzzy sub-hyperspaces of V and the set of all fuzzy sub-hyperspaces of W.Proof.
It immediately follows from
Proposition5 and the following results:
(i)f−1(f(μ))=μ, where μ is a f-invariant fuzzy sub-hyperspace of V,
(ii)f(f−1(η))=η, where η is any fuzzy sub-hyperspace of W.For sub-hyperspacesW1 and W2 of V,
it is easy to verify that W1∪W2 is a sub-hyperspace of V if and only if W1⊆W2 or W2⊆W1. 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 Theorems5,
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
Example1, set(35)W1={(b,0):b∈ℝ},W2={(0,c):c∈ℝ}.Obviously, (W1,+,◦,ℝ) and (W2,+,◦,ℝ) are sub-hyperspaces of V=(ℝ2,+,◦,ℝ), such that for all a,b,c∈ℝ,(36)a∘(b,0)={W1ifb⁢≠0,{0¯}ifb=0,a∘(0,c)={W2ifc⁢≠0,{0¯}ifc=0.Choose numbers ti∈[0,1],0≤i≤3,
such that t0>t1>t2>t3. Define fuzzy subsets μ1 and μ2 by(37)μ1(x)={t0ifx∈W1,t3otherwise,μ2(x)={t1ifx∈W2,t2otherwise.Then μ1t0=W1,μ1t3=ℝ2,μ2t1=W2  and μ2t2=ℝ2. Thus from Proposition 4, it follows that μ1 and μ2 are fuzzy sub-hyperspaces of V.
Clearly μ=μ1∪μ2,
given by(38)(μ1∪μ2)(x)={t0ifx∈W1,t1ifx∈W2∖{0¯},t2ifx∈V∖W1∪W2,is not a fuzzy sub-hyperspace of V, since μt1=W1∪W2 is not a sub-hyperspace of V.Theorem 5.
Letμ be a fuzzy sub-hyperspace of V such that Im(μ)={α},
where α∈[0,1]. If μ=μ1∪μ2,
where μ1 and μ2 are fuzzy sub-hyperspaces of V,
then either μ1⊆μ2 or μ2⊆μ1.Proof.
Sinceμ=μ1∪μ2 is a fuzzy sub-hyperspace of V,
the level fuzzy sub-hyperspace (μ1∪μ2)α is a sub-hyperspace of V such that(39)(μ1∪μ2)α={x∈V:(μ1∪μ2)(x)≥α}={x∈V:μ1(x)∨μ2(x)≥α}={x∈V:μ1(x)≥α}∪{x∈V:μ2(x)≥α}=μ1α∪μ2α.So either μ1α⊆μ2α or μ2α⊆μ1α. Thus either μ1⊆μ2 or μ2⊆μ1.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.
LetV and  ti be the same as in Example 2. Define fuzzy
subsets μ1 and μ2 of V by(40)μ1(x)={t0ifx=0¯,t3otherwise,μ2(x)={t1ifx=0¯,t2otherwise.Then(41)(μ1∪μ2)(x)={t0ifx=0¯,t2otherwise,such that μ1t0={0¯},μ1t3=ℝ2,μ2t1={0¯},(μ1∪μ2)t0={0¯} and (μ1∪μ2)t2=ℝ2. Thus from Proposition 4, it follows that μ1,μ2,
and μ1∪μ2 are fuzzy sub-hyperspaces of V.
However, μ1⊈μ2 and μ2⊈μ1.Lemma 2.
Letμ be a fuzzy sub-hyperspace of V.
If μ(x)<μ(y), for some x,y∈V, then(42)μ(x−y)=μ(x)=μ(y−x).Theorem 6.
Letμ be a fuzzy sub-hyperspace of V such that Im(μ)={0,t},
where t∈(0,1]. If μ=μ1∪μ2,
where μ1 and μ2 are fuzzy sub-hyperspaces of V,
then either μ1⊆μ2 or μ2⊆μ1.Proof.
To
obtain a proof by contradiction, assume thatμ1(x)>μ2(x) and μ2(y)>μ1(y), for some x,y∈V. Then(43)μ(x)=(μ1∪μ2)(x)=μ1(x)>μ2(x)≥0,μ(y)=(μ1∪μ2)(y)=μ2(y)>μ1(y)≥0.Therefore(44)μ(x)=t=μ1(x)=μ(y)=μ2(y),since Im(μ)={0,t}. So μ1(y)<μ1(x), thus by Lemma 2 we obtain
that(45)μ1(x−y)=μ1(y)<μ2(y)=t,and similarly(46)μ2(x−y)=μ2(x)<μ1(x)=t.Hence(47)μ(x−y)=μ1(x−y)∨μ2(x−y)<t.Again, μ(x−y)=μ(x)⋀μ(y)=t, the desired contradiction.Theorem 7.
Letμ be a fuzzy sub-hyperspace of V such that 3≤|Im(μ)|<∞. Then there exist fuzzy sub-hyperspaces μ1 and μ2 of V such that μ=μ1∪μ2, μ1⊈μ2 and μ2⊈μ1.Proof.
LetIm(μ)={t0,t1,…,tn},
where 2≤n<∞, and t0>t1>⋯>tn. Choose s1,s2∈[0,1] such that(48)1≥t0>s1>t1>s2>t2>t3>⋯>tn.Define fuzzy subsets μ1 and μ2 of V by(49)μ1(x)={t0ifx∈μt0,s2ifx∈μt1∖μt0,μ(x)otherwise,μ2(x)={s1ifx∈μt0,t1ifx∈μt1∖μt0,μ(x)otherwise.Clearly μ1,μ2,
and μ1∪μ2 are fuzzy sub-hyperspaces of V such that μ=μ1∪μ2.
However, μ1⊈μ2 and μ2⊈μ1.Theorem 8.
Ifμ is a fuzzy sub-hyperspace of V such that Im(μ)={t0,t1}, where t0,t1∈(0,1] and t0>t1, then there exist fuzzy sub-hyperspaces μ1 and μ2 of V such that μ=μ1∪μ2,μ1⊈μ2 and μ2⊈μ1.


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

In this section, we give the construction of the fuzzy
sub-hyperspaceμ¯ generated by a fuzzy subset μ of V.
It is possible that |Im(μ)| may be strictly greater than |Im(μ¯)|.
For any W⊆V, we will write 〈W〉 for the sub-hyperspace generated by W in V.Theorem 9.
Letμ be a fuzzy subset of V such that |Im(μ)|<∞. Define sub-hyperspaces Vi,0≤i≤k, by(50)V0=〈{x∈V:μ(x)=⋁z∈Vμ(z)}〉,Vi=〈Vi−1,{x∈V:μ(x)=⋁z∈V∖Vi−1μ(z)}〉,where k≤|Im(μ)| and Vk=V. Then the fuzzy subset μ¯ of V defined by(51)μ¯(x)={⋁z∈Vμ(z)ifx∈V^0=V0,⋁z∈V∖Vi−1μ(z)ifx∈V^i=Vi∖Vi−1is the fuzzy sub-hyperspace
generated by μ in V.Proof.
By
Proposition4, we obtain that μ¯ is a fuzzy sub-hyperspace of V. Now V0=〈S0〉, where S0={x∈V:μ(x)=∨z∈Vμ(z)} and for i=1,…,k,Vi=〈Vi−1,Si〉, where(52)Si={x∈V:μ(x)=⋁z∈V∖Vi−1μ(z)}.Let η be any fuzzy sub-hyperspace of V containing μ. It is sufficient to prove that(53)η(x)≥μ¯(x)≥μ(x)for allx∈V^i,⁢i=1,…,k.Let x∈V^0. Then there exist a01,a02,…,a0m∈K and s01,s02,…,s0m∈S0, such that x∈a01◦s01+⋯+a0m◦s0m. Now(54)η(x)≥⋀t∈a01∘s01+⋯+a0m∘s0mμ(t)≥μ(s01)∧μ(s02)∧⋯∧μ(s0m)(sinceη⁢isafuzzysub-hyperspace)=⋁z∈Vμ(z)≥μ(x).Hence η(x)≥μ¯(x)≥μ(x),∀x∈V0.
Letx∈V^i,i=1,…,k,
Then there exist ai1,ai2,…,aim∈K and si1,si2,…,sim∈Si such that x∈ai1◦si1+⋯+aim◦sim. Now(55)η(x)≥⋀t∈ai1∘si1+⋯+aim∘simμ(t)≥μ(si1)∧μ(si2)∧⋯∧μ(sim)(sinceη⁢isafuzzysub-hyperspace)=⋁z∈V∖Vi−1μ(z)≥⋁z∈V^iμ(z)(sinceV^i⊆V∖Vi−1)≥μ(x).Hence η(x)≥μ¯(x)≥μ(x), for all x∈V^i, 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) Letμ be a fuzzy sub-hyperspace of V and t=μ(0¯). Then the fuzzy subset(56)μ∗:V/μt⟶[0,1],μ∗(x+μt)=μ(x)is a fuzzy sub-hyperspace of V/μt.
(ii) IfW is a sub-hyperspace of V and η is a fuzzy sub-hyperspace of V/W such that μ∗(x+W)=η(W) only when x∈W, then there exists a fuzzy sub-hyperspace μ of V such that μt=W(t=μ(0¯)) and η=μ∗.Proof.
(i) It
is easy to verify thatμ∗ is well defined. We now show that μ∗ is a fuzzy sub-hyperspace of V/μt. Let x,x1,x2∈V and a∈K. Then(57)μ∗((x1+μt)−(x2+μt))=μ∗(x1−x2+μt)=μ(x1−x2)≥μ(x1)∧μ(x2)≥μ∗(x1+μt)∧μ∗(x2+μt),and if s∈a◦x,
then(58)μ∗(s+μt)=μ(s)≥⋀t∈a∘xμ(t)≥μ(x)=μ∗(x+μt),thus(59)⋀s∈a∘xμ∗(s+μt)≥μ∗(x+μt).Therefore(60)⋀t∈a∗(x+μt)=a∘x+μtμ(t)≥μ∗(x+μt).
(ii) Define a fuzzy subset μ of V by(61)μ(x)=η(x+W),∀x∈V.Clearly μ is a fuzzy sub-hyperspace of V. Now μt=W, because(62)x∈μt⟺μ(x)=t=μ(0¯)⟺η(x+W)=η(W)⟺x∈W.Moreover η=μ∗, since(63)μ∗(x+W)=μ∗(x+μt)=μ(x)=η(x+W).Definition 15.
Letμ be a fuzzy sub-hyperspace of V. For x∈V, the fuzzy subset μx∗ of V defined by(64)μx∗(z)=μ(x−z),∀z∈V,is called the fuzzy coset
determined by x and μ. The set of all fuzzy cosets of μ is denoted by Vμ, that is, Vμ={μx∗:x∈V}.Proposition 8.
Letμ be a fuzzy sub-hyperspace of (V,+,◦,K). Then the set of all fuzzy cosets of μ,Vμ={μx∗:x∈V}, with operation(65)⊕:Vμ×Vμ⟶Vμ,μx∗⊕μy∗=μx+y∗and external
operation(66)⊚:K×Vμ⟶P∗(Vμ),a⊚μx∗=μa∘x∗={μt∗:t∈a∘x}is a hypervector space over the
field K.Proof.
It is
easy to verify that “+” and “◦” are well defined. Let a,b∈K and x,y∈V. Then
(H1)
(67)a⊚(μx∗⊕μy∗)=a⊚(μx+y∗)={μt∗:t∈a∘(x+y)}⊆{μt∗:t∈a∘x+a∘y}={μt∗:t=t1+t2,t1∈a∘x,t2∈a∘y}={μt1+t2∗:t1∈a∘x,t2∈a∘y}={μt1∗⊕μt2∗:t1∈a∘x,t2∈a∘y}={μt1∗:t1∈a∘x}⊕{μt2∗:t2∈a∘y}=(a⊚μx∗)⊕(a⊚μy∗);
(H2)
(68)(a+b)⊚μx∗={μt∗:t∈(a+b)∘x}⊆{μt∗:t∈a∘x+b∘x}={μt∗:t=t1+t2,t1∈a∘x,t2∈b∘x}={μt1+t2∗:t1∈a∘x,t2∈b∘x}={μt1∗⊕μt2∗:t1∈a∘x,t2∈b∘x}={μt1∗:t1∈a∘x}⊕{μt2∗:t2∈b∘x}=(a⊚μx∗)⊕(b⊚μx∗);
(H3)
(69)a⊚(b⊚μx∗)=⋃μt∗∈b⊚μx∗a⊚μt∗=⋃t∈b∘xa⊚μt∗=⋃t∈b∘x{μs∗:s∈a∘t}={μs∗:s∈a∘(b∘x)}={μs∗:s∈(ab)∘x}=(ab)⊚μx∗;
(H4)
(70)a⊚(−μx∗)=a⊚(μ−x∗)={μt∗:t∈a∘(−x)}={μt∗:t∈(−a)∘x}=(−a)⊚μx∗=−(a⊚μx∗);
(H5) for all 	 μx∗∈Vμ,
μx∗∈1⊚μx∗={μt∗:t∈1◦x}, since x∈1◦x.The following lemma will serve as a powerful tool in
proving Theorem11 regarding the dim of the hypervector space Vμ.Lemma 3.
Letμ be a fuzzy sub-hyperspace of V. Then(71)∀x∈V,μ(x)=μ(0¯)⟺μx∗=μ0¯∗.Proof.
Letμ(x)=μ(0¯). Then for all z∈V,μ(z)≤μ(x)=μ(0¯).
If μ(z)<μ(x)=μ(0¯), then μ(z−x)=μ(z), by Lemma 2. If μ(z)=μ(x)=μ(0¯), then z,x∈μt, where t=μ(0¯). Hence μ(z−x)=t=μ(0¯)=μ(z). Thus in either case, we have shown that for all  z∈V,μ(z−x)=μ(z).
That is, for all z∈V,μ(x−z)=μ(z).
Consequently μx∗=μ0¯∗. The converse is straightforward.Theorem 11.
Letμ be a fuzzy sub-hyperspace of V and t=μ(0¯). Then(72)dim⁡Vμ=dim⁡V/μt.Proof.
If for allz∈V,μ(z)=α,
then, for all z∈V,μ(z)=μ(0¯).
So that Vμ=μ0¯∗ and μt=V. Hence(73)dim⁡Vμ=0=dim⁡V/μt.So we assume that μ is not constant. Let dimV=n and dimμt=m. Let(74){v1,v2,…,vm,x1,x2,…,xr},r=n−m,be a basis of V such that {v1,v2,…,vm} is a basis of μt. Then {x1+μt,x2+μt,…,xm+μt} is a basis of V/μt. We will show that the set β∗={μx1∗,μx2∗,…,μxr∗} is a basis of Vμ over the field K.
Clearlyμxi∗≠μxj∗ whenever i≠j,  because(75)μxi∗=μxj∗⟹μxi−xj∗=μ0¯∗,⟹μ(xi−xj)=μ(0¯),byLemma3,⟹xi−xj∈μt,⟹xi+μt=xj+μt,⁢acontradiction.Next we prove that the set β∗ generates Vμ over K. For this, let μx∗∈Vμ,μx∗≠μ0¯∗.
Then μ(x)≠μ(0¯), by Lemma 3. So that x+μt is a nonzero element of V/μt. Hence there exist a1,…,ar∈K such that(76)x+μt∈a1∗(x1+μt)+⋯+ar∗(xr+μt)=a1∘x1+μt+⋯+ar∘xr+μt=a1∘x1+⋯+ar∘xr+μt,⟹∃t1,…,tr,⁢ti∈ai∘xi:x+μt=t1+⋯+tr+μt,⟹x−(t1+⋯+tr)∈μt,⟹μ(x−t1−⋯−tr)=μ(0¯),⟹μ(x−t1−⋯−tr)∗=μ0¯∗,byLemma3,thus(77)μx∗=μt1+⋯+tr∗=μt1∗⊕⋯⊕μtr∗∈a1⊚μx1∗⊕⋯⊕ar⊚μxr∗.Also β∗ is linearly independent over K, because if μ0¯∗∈a1⊚μx1∗⊕⋯⊕ar⊚μxr∗, where a1,…,ar∈K, then ∃μti∗∈a1⊚μxi∗,1≤i≤r, such that(78)μ0¯∗=μt1∗⊕⋯⊕μtr∗=μt1+⋯+tr∗,⟹μ(t1+⋯+tr)=μ(0¯),⟹t1+⋯+tr∈μt,so there exist b1,…,bm∈K, such that(79)t1+⋯+tr∈b1∘v1+⋯+bm∘vm,thus there exist sj∈bj◦vj,1≤j≤m, such that(80)t1+⋯+tr=s1+⋯+sm,⟹0¯=t1+⋯+tr−s1−⋯−sm,⟹0¯∈a1∘x1+⋯+ar∘xr−b1∘v1−⋯−bm∘vm,⟹a1=⋯=ar=b1=⋯=bm=0.Hence(81)dim⁡Vμ=r=n−m=dim⁡V−dim⁡μt=dim⁡V/μt.Theorem 12.
Letμ be a fuzzy sub-hyperspace of V and t=μ(0¯). Then
(a)
the fuzzy subsetμ′ of Vμ defined by(82)∀μx∗∈Vμ,μ′(μx∗)=μ(x)is a fuzzy subhyperspace of Vμ;(b)
the map(83)f:V⟶Vμ,x⟼μx∗is an onto good transformation with kernel μt,
where t=μ(0¯).Proof.
(a)
(i)(84)μ′(μx∗⊕μy∗)=μ′(μx+y∗)=μ(x+y)≥μ(x)∧μ(y)=μ′(μx∗)∧μ′(μy∗),
(ii) μ′(−μx∗)=μ′(μ−x∗)=μ(x)≥μ(−x)=μ′(μx∗),
(iii) for allμt∗∈a⊚μx∗,(85)μ′(μt∗)=μ(t)≥⋀s∈a∘xμ(s)≥μ(x)=μ′(μx∗),thus ⋀μt∗∈a⊚μx∗μ′(μt∗)≥μ′(μx∗).
(b) For everya∈K and x,y∈V,(86)f(x+y)=μx+y∗=μx∗⊕μy∗=f(x)⊕f(y),f(a∘x)=μa∘x∗=a⊚f(x),thus f is an onto good
transformation. Moreover, kerf={x∈V:f(x)∈ΩVμ},
where(87)ΩVμ=0⊚μ0¯∗={μt∗:t∈0∘0¯=ΩV},therefore(88)ker⁡f={x∈V:μx∗=μt∗,forsomet∈0∘0¯}={x∈V:μ(x)=μ(t),forsomet∈0∘0¯},but μ(t)≥⋀s∈0◦0¯μ(s)≥μ(0¯), so(89)ker⁡f={x∈V:μ(x)≥μ(0¯)}.Thus kerf=μt,
where t=μ(0¯).Definition 16.
Letμ be a fuzzy sub-hyperspace of V.
The fuzzy subset μ′ of Vμ is called the fuzzy quotient hypervector space
determined by μ.Finally, we establish for fuzzy sub-hyperspace an
analogu of a consequence of the “fundamental theorem of
homomorphisms.”Theorem 13.
Letμ be a fuzzy sub-hyperspace of V. Then each fuzzy sub-hyperspace of Vμ corresponds in a natural way to a fuzzy
sub-hyperspace of V.Proof.
Letμ′ be any fuzzy sub-hyperspace of Vμ. Then it is easy to see that the fuzzy subset μ of V defined by(90)∀x∈V,μ(x)=μ′(μx∗),is a fuzzy sub-hyperspace of V.


## 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.



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*Source: 295649-2008-08-06.xml*