Advances in Linear Algebra & Matrix Theory, 2016, 6, 158-168 http://www.scirp.org/journal/alamt ISSN Online: 2165-3348 ISSN Print: 2165-333X
L-Fuzzy Vector Subspaces and Its Fuzzy Dimension Chun’e Huang1*, Yan Song2, Xiruo Wang2 College of Biochemical Engineering, Beijing Union University, Beijing, China Mudanjiang Normal University, Mudanjiang, China
1 2
How to cite this paper: Huang, C.E, Song, Y. and Wang, X.R. (2016) L-Fuzzy Vector Subspaces and Its Fuzzy Dimension. Advances in Linear Algebra & Matrix Theory, 6, 158-168. http://dx.doi.org/10.4236/alamt.2016.64015 Received: November 8, 2016 Accepted: December 25, 2016 Published: December 28, 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access
Abstract In this paper, we introduce the definition of L-fuzzy vector subspace, define its dimension by an L-fuzzy natural number. For a finite-dimensional L-fuzzy vector subspace, we prove that the equality dim E + E + dim E ∩ E = dim E + dim E
(
1
2
)
(
1
2
)
1
2
holds without any restricted conditions. At the same time, we deduce that the forf = f + dim kef mula dim im dim E holds.
(
)
(
)
Keywords L-Fuzzy Sets, L-Fuzzy Vector Subspace, L-Fuzzy Dimension
1. Introduction Firstly, fuzzy vector subspace was introduced by Katsaras and Liu [1]. Then its properties and characters were investigated (see [2] [3] [4] [5], etc). The dimension of a fuzzy vector space was defined as a n-tuple by Lowen [6]. Subsequently, it was defined as a non-negative real number or infinity by Lubczonok [5], and proved that the formula
(
)
(
)
dim E1 + E 2 + dim E1 ∩ E 2 = dim E1 + dim E 2
(1)
is valid under certain conditions, where E1 and E 2 are fuzzy vector spaces. Recently, basis and dimension of a fuzzy vector space were redefined as a fuzzy set and a fuzzy natural number by Shi and Huang [7], respectively. Under the definitions, more properties of (crisp) vector spaces were correct in fuzzy vector spaces. In this paper, we generalize the results in [7] to L lattice, and prove that some formulas still hold in the lattice L. In particular, we present the definition of L-fuzzy vector subspace and its -fuzzy dimension. The L-fuzzy dimension of a finite dimensional fuzzy vector subspace is a fuzzy natural number. We prove that (1) holds without any reDOI: 10.4236/alamt.2016.64015 December 28, 2016
C. E Huang et al.
(
)
(
)
f + dim im f = stricted conditions and dim kef dim E holds.
2. Preliminaries Given a set X and a completely distributive lattice L, we denote the power set of X and the set of all L-fuzzy sets on X (or L-sets for short) by 2 X and LX , respectively . For any A ⊆ X , we denote the cardinality of A by A . An element a in L is called a prime element if a ≥ b∧c implies a ≥ b or a ≥ c . a in L is called co-prime if a ≤ b∨c implies a ≤ b or a ≤ c [8]. The set of non-
unit prime elements in L is denoted by P ( L ) . The set of non-zero co-prime elements in L is denoted by J ( L ) . The binary relation < in L is defined as follows: for a, b ∈ L , a < b if and only if for every subset D ⊆ L , the relation b ≤ sup D always implies the existence of
d ∈ D with a ≤ d [9]. {a ∈ L : a < b} is called the greatest minimal family of b in the sense of [10], denoted by β ( b ) , and β *= ( b ) β ( b ) ∩ J ( L ) . Moreover, for b ∈ L , * op we define α ( b ) = and α = b a ∈ L : a < b ( ) α ( b ) ∩ P ( L ) . In a completely distri{ } butive lattice L , there exist α ( b ) and β ( b ) for each b ∈ L , and b = ∨ β (b ) = ∧α ( b ) (see [10]). In [10], Wang thought that β ( 0 ) = {0} and α (1) = {1} . In fact, it should be that β ( 0 ) = ∅ and α (1) = ∅ . Throughout this paper, L denotes a completely distributive lattice, and E is a crisp vector space. We often do not distinguish a crisp subset A of E and its characteristic function χ A . If A ∈ LX and a ∈ L , we can define
{
A[ a ] = { x ∈ X : A ( x ) ≥ a} ,
{
}
A( a ) =x ∈ X : a ∈ β ( A ( x ) ) .
}
a (a) A[ ] =x ∈ X : a ∈ { x ∈ X : A ( x ) a}. / α ( A ( x )) , A =
Some properties of these cut sets can be found in [11]-[16].
In [17] Shi introduced the concept of L-fuzzy natural numbers(denoted by ( L ) ), defined their operations and discussed the relation of α -cut sets. We simply recall as
follows: for any λ , µ ∈ ( L ) , a ∈ L , (1) ( λ + µ )( a ) ⊆ λ( a ) + µ( a ) ⊆ λ[ a ] + µ[ a ] ⊆ ( λ + µ )[ a ] ; (2)
( λ + µ )(
a)
[a] a a a a ⊆ λ ( ) + µ ( ) ⊆ λ[ ] + µ[ ] ⊆ (λ + µ ) ;
(3) For any λ , µ ∈ ( L ) and a ∈ P ( L ) , it follows that
( λ + µ )(
a)
=λ ( ) + µ ( ) . a
a
3. L-Fuzzy Vector Subspaces Definition 3.1. L-fuzzy vector subspace is a pair E = ( E , µ ) where E is a vector space on field F , µ : E → L is a map with the property that for any x, y ∈ E , k , l ∈ F , we have µ ( kx + ly ) ≥ µ ( x )∧ µ ( y ) .
In this definition, when L = [ 0,1] , L-fuzzy vector subspace is exactly the fuzzy vector subspace defined in [1]. We denote the family of L-fuzzy vector subspaces by LFVS . Let E = ( E , µ ) be a member of LFVS , we denote
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{
E[ a ] = µ[ a] = { x ∈ E : µ ( x ) ≥ a} ,
{
}
E ( a ) = µ( a ) =x ∈ E : a ∈ β ( µ ( x ) ) .
}
a E [ ] = µ [ a] =x ∈ E : a ∈/ α ( µ ( x ) ) , E ( a ) = µ (a) = { x ∈ E : µ ( x ) a} .
We can obtain some properties of LFVS analogous to fuzzy vector subspaces as follows.
Theorem 3.2. Let E = ( E , µ ) be a member of LFVS , then (1) µ ( 0 ) = sup µ ( x ) . x∈E
(2) For any k ∈ F \ {0} and x ∈ E , µ ( kx ) = µ ( x ). The prove is trivial and omitted. Remark: Since L is a completely distributive lattice, the property that if µ ( x ) ≠ µ ( y ) , then µ ( x + y ) = µ ( x )∧ µ ( y ) not holds for LFVS . This can be seen from the following example. Example 3.3. Let L be a completely distributive lattice with four elements as follows.
(
Let E = 2 , µ
)
be an L-fuzzy vector subspace on 2 where µ is defined by
1, a, µ ( x) = b, 0,
x = ( 0, 0 ) . x ∈ {( y, 0 ) : y ∈ \ {0}}.
x ∈ {( 0, y ) : y ∈ \ {0}}.
otherwise.
We can easily check E is an L-fuzzy vector subspace on 2 . Suppose that x = ( 3, 2 ) and = y ( 0, −2 ) , then µ ( x + y ) = µ ( 3, 0 ) = a > µ ( x )∧ µ ( y ) = 0 ∧ b = 0.
This example illustrates for L-fuzzy vector subspace µ ( x ) ≠ µ ( y ) ,
µ ( x + y ) > µ ( x )∧ µ ( y ) . Theorem 3.4. Let E be a vector space, µ ∈ LE and E = ( E , µ ) . Then the following statements are equivalent: (1) E is an L-fuzzy vector subspace. (2) (a) For all x ∈ E and k ∈ F , µ ( kx ) ≥ µ ( x ) . (b) For any x, y ∈ E , µ ( x + y ) ≥ µ ( x )∧ µ ( y ) . (3) For any x1 , , xr ∈ E and k1 , , kr ∈ F , where r is a finite natural number, we have
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r
r
i =1
µ ∑ki xi ≥ ∧ µ ( xi ) . i =1
The prove is trivial and omitted. In the following paper, the vector spaces we discuss are finite-dimensional. For their
L-fuzzy vector subspaces, the following observation will be useful. Remark: Let E = ( E , µ ) be a member of LFVS . Suppose that = µ ( E ) {µ ( x ) : x ∈ E} . Since E is finite-dimensional vector space, denotes dim E = n , then µ ( E ) is a finite subset of L. In the fact, let B be a basis of E , then B = n . Suppose that µ ( E ) is infinite, then for all a ∈ L , the total number of E[ a ] is infinite. Since B ∩ E[ a ] is a basis of E[a] , we have E[= B ∩ E[ a ] . Again since B is finite, the total number of E[ a ] is a] also finite. It contradicts with the hypothesis. Therefore µ ( E ) is a finite subset of L with at most 2n + 1 values; 2n values which can be attained at the vectors of E \ {0} and the maximum which is attained at 0.
Theorem 3.5. Let E be a vector space, µ ∈ LE and E = ( E , µ ) . Then the following statements equivalent: (1) E is an L-fuzzy vector subspace. (2) For all a ∈ L , E[ a ] is a vector space. (3) For all a ∈ J ( L ) , E[ a ] is a vector space. (4) For all a ∈ L , E [ a ] is a vector space.
(5) For all a ∈ P ( L ) , E [ a ] is a vector space. (6) For all a ∈ P ( L ) , E ( a ) is a vector space. Proof. We prove (1) ⇔ ( 4 ) and (1) ⇔ ( 6 ) , the others can be proved analogously. (1) ⇒ ( 4 ) We show that E [a] is a vector space as follows. Suppose that x, y ∈ E [a] ,
then a ∈ α ( µ ( x )∧ µ ( y ) ) . / α ( µ ( x )) ∪ α ( µ ( y )) = / α ( µ ( y ) ) , i.e. a ∈ / α ( µ ( x ) ) and a ∈ Since E = ( E , µ ) be an L-fuzzy vector subspace, then α ( µ ( x )∧ µ ( y ) ) ⊇ α ( µ (( kx + ly ) ) , [a] [a] we have a ∈ is a vector space. / α ( µ ( kx + ly ) ) , this means kx + ly ∈ E . Therefore E
Suppose that for all a ∈ L , E [ ] is a vector space. Let x, y ∈ E and k , l ∈ F . Since E [ a ] is a vector space, then kx + ly ∈ E [ a ] if and only if
( 4 ) ⇒ (1)
a
a a x ∈ E [ ] and y ∈ E [ ] . We have
(
)
a ∧ a∧ E [ ] ( kx + ly ) µ ( kx + ly ) = a∈L
( (
a a = ∧ a∨ E [ ] ( x )∧ E [ ] ( y ) a∈L
( (
)) ( (
))
a a = ∧ a∨ E [ ] ( x ) ∧ ∧ a∨ E [ ] ( y ) a∈L
a∈L
))
= µ ( x )∧ µ ( y ) . Therefore E is an L-fuzzy vector subspace.
(1) ⇒ ( 6 ) Suppose that x, y ∈ E ( a ) , then µ ( x ) a and µ ( y ) a . Since a ∈ P ( L ) , then µ ( x )∧ µ ( y ) a . Because E = ( E , µ ) is an L-fuzzy vector subspace, we can a have µ ( kx + ly ) a , this implies kx + ly ∈ E ( ) . Thus E ( a ) is a vector space. ( 6 ) ⇒ (1) Let x, y ∈ E and k , l ∈ F . Since E ( a ) is a vector space, then 161
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kx + ly ∈ E (
a)
if and only if x ∈ E ( ) and y ∈ E ( ) . We have the following implications. a
a
( a∨E ( ) ) ( kx + ly ) = ∧ ( a∨( E ( ) ( x )∧ E ( ) ( y ) ) ) ( ) ∧ a∨ E ( ) x ∧ ∧ a∨ E ( ) y = ( )) ( ) ( ( )) ( )(
µ ( kx + ly ) = ∧
a
a∈P ( L )
a
a
a∈P L
a
a
a∈P L
a∈P L
= µ ( x )∧ µ ( y ) .
Therefore E is an L-fuzzy vector subspace. Theorem 3.6. Let E be a vector space, µ : E → L be a map, E = ( E , µ ) , and for
all a, b ∈ L, β ( a∧b ) =∩ β ( a ) β ( b ) . Then the following statements equivalent: (1) E is an L-fuzzy vector subspace. (2) For all a ∈ L , E ( a ) is a vector space. Proof. (1) ⇒ ( 2 ) Suppose that x, y ∈ E ( a ) , then a ∈ β ( µ ( x ) ) and a ∈ β ( µ ( y ) ) , i.e. a ∈ β ( µ (( x ) ) ∩ β ( µ ( y ) ) . Since for all a, b ∈ L, β ( a∧b ) =∩ β ( a ) β ( b ) and E
is an L-fuzzy vector subspace, we can know a ∈ β ( µ ( x )∧ µ ( y ) ) ⊆ β ( µ ( ax + by ) ) , this implies ax + by ∈ E ( a ) . Therefore E ( a ) is a vector space. ( 2 ) ⇒ (1) Suppose that for all a ∈ L , E( a ) is a vector space. Let x, y ∈ E and
k , l ∈ F . Since E ( a ) is a vector space, then kx + ly ∈ E ( a ) if and only if x ∈ E ( a ) and y ∈ E ( a ) . We have
)
(
∨ a∧ E ( a ) ( kx + ly ) µ ( kx + ly ) = a∈L
( (
= ∨ a∧ E ( a ) ( x ) ∧ E ( a ) ( y ) a∈L
=
( ∨ ( a∧E a∈L
(a)
))
( x ) ) )∧ ∨ ( a∧ E( a ) ( y ) )
= µ ( x )∧ µ ( y ) .
a∈L
Therefore E is an L-fuzzy vector subspace. We can define the operations between two L-fuzzy vector subspaces analogous to fuzzy vector subspaces.
Definition 3.7. Let E1 =
E , µ1 ) and E 2 ( E , µ2 ) (=
be two L-fuzzy vector subspaces on E . Define the intersection of E1 and E 2 to be E1 ∩ E 2 = ( E , µ1 ∧µ2 ) . Define the sum of E1 and E 2 to be E1 + E= ( E , µ1 + µ2 ) where µ1 + µ2 is defined by for 2 all x ∈ E
∨ ( µ1 ( x1 ) ∧ µ2 ( x2 ) ) ( µ1 + µ2 )( x ) = x= x + x 1
2
= ∨ ( µ1 ( x1 ) ∧ µ2 ( x − x1 ) ) . x1∈E
Definition 3.8. Let E1 =
E1 , µ1 ) and E 2 ( E2 , µ2 ) (=
be two members of LFVS and E = E1 ⊕ E2 . We define the direct sum of E1 and E2 to be E1 ⊕ E= ( E , µ1 ⊕ µ2 ) 2 where µ1 ⊕ µ2 is defined by for all x ∈ E , x =x1 ⊕ x2 , xi ∈ Ei , i = 1, 2
( µ1 ⊕ µ2 )( x ) = ( µ1 ⊕ µ2 )( x1 ⊕ x2 ) = µ1 ( x1 )∧µ2 ( x2 ) . 162
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E , µ1 ) and E 2 ( E , µ2 ) (=
Theorem 3.9. Let E1 =
be two members of LFVS on
E . We have
(1) E1 ∩ E 2 is a member of LFVS on E . (2) E1 + E 2 is a member of LFVS on E . The proof of the theorem is trivial and it is omitted. Theorem 3.10. Let E1 = ( E , µ1 ) and E 2 = ( E , µ2 ) be the members of LFVS . We
have
( E ∩ E )[ ] = ( E )[ ] ∩ ( E )[ ] .
(1) For all a ∈ L ,
1
2
1
a
[a]
2
a
[a]
a
[a]
( E ∩ E ) = ( E ) ∩ ( E ) . ( ) ( ) ( ) a ∈ P ( L ) , ( E ∩ E ) = ( E ) ∩ ( E ) . ( ) ( ) ( ) a ∈ P ( L ) , ( E + E ) = ( E ) + ( E ) .
(2) For all a ∈ L ,
1
2
1
2
a
(3) For any
1
2
1
2
a
2
a
(4) For any
a
1
a
a
1
2
Proof. The proofs of (1) and (2) are easy by the definition of E1 ∩ E 2 and the properties of L-fuzzy sets. (3) For any a ∈ P ( L ) , we have
(
x ∈ E1 ∩ E 2
)
(a)
⇔ µ1 ( x )∧ µ2 ( x ) a
⇔ µ1 ( x ) a and µ2 ( x ) a
( )
⇔ x ∈ E1
(a)
( )
∩ E 2
(a)
(4) By the definition of the sum of L-fuzzy vector subspaces, for any a ∈ P ( L ) we have
(
x ∈ E1 + E 2
)
(a)
⇔
∨
x= x1 + x2
( µ1 ( x1 )∧µ2 ( x2 ) ) a
⇔ ∃x1 , x2 ∈ E and x = x1 + x2 , such that µ1 ( x1 )∧ µ2 ( x2 ) a. ⇔ ∃x1 , x2 ∈ E , µ1 ( x1 ) a and µ2 ( x2 ) a. ⇔ x = x1 + x2 ∈ E1( ) + E 2( ) . a
a
Theorem 3.11. Let E1 = ( E , µ1 ) and E 2 = ( E , µ2 ) be two members of LFVS .
Suppose that for any a, b ∈ L , we have β ( a∧ b ) β ( a ) ∩ β ( b ) . Then = (1) ( E ∩ E ) = ( E ) ∩ ( E ) , 1
(2)
(
2
E1 + E 2
(a)
1
(a)
2
(a)
)( ) ( )( ) ( )( ) . a
= E1
a
+ E 2
a
The prove is trivial and omitted.
4. Fuzzy Dimension of L-Fuzzy Vector Subspaces Definition 4.1. Let ( L ) be the family of L-fuzzy natural number. The map dim : LFVS → ( L ) is defined by
(
dim E ( n ) = ∨ a∧dim E[ a ] a∈L
) (n)
is called the L-fuzzy dimensional function of the L-fuzzy vector subspace E , and dim E is called the L-fuzzy dimension of E , it is an L-fuzzy natural number. We 163
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usually use another form of dim E as follows.
{
}
dim E ( n ) = ∨ a ∈ L : dim E[ a ] ≥ n . Theorem 4.2. For each E ∈ LFVS and n ∈ , we have
(
dim E ( n ) = ∨ a∧dim E ( a ) a∈L
) ( n ) =∨{a ∈ L : dim E( ) ≥ n} a
(
)
Proof. For any n ∈ , let λ = ∨ a∧dim E ( a ) ( n ) . Obviously λ ≤ dim E ( n ) . Next a∈L
(
)
we show that λ ≥ dim E ( n ) . Suppose that b ∈ L and b ∈ β dim E ( n ) , then there exists a ∈ L and dim E[ a ] ≥ n such that b ∈ β ( a ) . In this case,
{
}
n ≤ dim E[a] ≤ dim E (b ) ≤ dim E[b] which implies λ = ∨ a ∈ L : dim E ( a ) ≥ n ≥ b . Thus we have
{
}
λ ≥ ∨ b b ∈ β ( dim E ( n ) ) =dim E ( n ) . This completes the proof.
Theorem 4.3. Let the pair E = ( E , µ ) be a member of LFVS . Then for any a ∈ L,
( dim E )( ) ≤ dim E[ ] ≤ ( dim E )[ ] . a
a
a
If β ( a∧ = b ) β ( a ) ∩ β ( b ) for all a, b ∈ L , then
( dim E )( ) ≤ dim E( ) ≤ dim E[ ] ≤ ( dim E )[ ] . a
a
a
a
( dim E )[ ] = dim E[ ] for any a ∈ J ( L ) . Proof. In order to prove ( dim E ) ≤ dim E ( ) . Suppose that n ≤ ( dim E ) , then ( ) ( ) a ∈ β ( dim E ( n ) ) . Since β is a preserve-union map, there is b ∈ L and In particular,
a
a
a
a
a
n ≤ dim E[b] , such that a ∈ β ( b ) . Because E[b] ⊆ E ( a ) ⊆ E[a] , thus n ≤ dim E ( a ) . Therefore
( dim E )( ) ≤ dim E( ) . a
a
dim E( a ) ≤ dim E[a] is obvious. Moreover, we can obtain that dim E[a] ≤ ( dim E )
[a]
( )
from the definition of dim E .
(
In order to prove for any a ∈ J ( L ) , dim E
( dim E )[ ] ⊆ dim E[ ] . Since the set a
(
a
a
µ ( E ) is finite, for any a ∈ J ( L ) we have
a
n ≤ dim E
dim E[ ] , we only need to show )[ ] =
)[ ] ⇒ dim E ( n ) ≥ a a
{
}
⇒ ∨ b ∈ L : dim E[b] ≥ n ≥ a ⇒ ∃a ≤ b, such that n ≤ dim E[b] ⇒ n ≤ dim E[ a ]
Therefore 164
( dim E )[ ] = dim E[ ] . a
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Theorem 4.4. Let E = ( E , µ ) be a member of LFVS . Then
( dim E )
(
)
( dim E ) = dim E ( ) for any a ∈ P ( L ) . ( ) ( dim E ) ≤ dim E ( ) can be proved from the following implications. (a)
In particular,
Proof.
[a] a a ≤ dim E ( ) ≤ dim E [ ] ≤ dim E .
(a)
a
a
a
(
n ≤ dim E
)
(a)
⇔ dim E ( n ) a
{
}
⇔ ∨ b ∈ L : dim E[b] ≥ n a ⇔ ∃b a, such that n ≤ dim E[b] (a) ⇒ dim E= dim E[b] ≥ n. ba
(
a Let a ∈ P ( L ) . In order to show dim E ( ) ≤ dim E
)
(a)
, we need to show that
dim E[b] ≤ ∨ dim E[b] . Suppose that n ≤ dim E[b] . Since the number of E[a] ba ba ba is finite, then when b a, the number of E is finite, denotes E , E , , E , [b ]
where ai a for any i ∈ {1, 2, , r} . Thus
[ a1 ]
r
E[b] = E[a ] .
ba
i =1
we have c = a1 ∧ a 2 ∧ ∧ a r a. Further we have
r
E[a ] ⊆ E [c] . Thus for any i =1
i
(
( dim E )
(a)
[ ar ]
Since a ∈ P ( L ) , then
i
r = n ≤ dim dim E[b] ≤ ∨= dim E E[b] dim E[ ai ] ≤ dim E[c] ≤ b∨ [b ] a ba i =1 ba Therefore for any a ∈ P ( L ) ,
[ a2 ]
)
( dim E )
(a)
.
a = dim E ( ) .
[a] a a a in the followdim E ( ) ≤ dim E [ ] is obvious. We show that dim E [ ] ≤ dim E )
(
)
ing implications. a dim E [ ] dim =
b b E ( ) ≤ ∧ dim E ( ) a∈α ( b ) b∈P ( L )
a∈α ( b ) b∈P ( L )
(
= ∧ dim E a∈α ( b ) b∈P ( L )
)
(b )
a (dim E )[ ] . =
Theorem 4.5. Let E1 = ( E , µ1 ) and E 2 = ( E , µ2 ) be two L-fuzzy vector subspaces.
Then the following equality holds
(
)
(
)
dim E1 + E 2 + dim E1 ∩ E 2 = dim E1 + dim E 2 .
Proof. We denote the sum of E1 and E 2 by E1 + E 2 = ( E , µ ) . From Theorem 11, we know that E1 + E 2 is a L-fuzzy vector subspace. By the properties of L-fuzzy natural numbers, Theorem 12 and the dimensional formulation of vector spaces, we know for any a ∈ P ( L ) ,
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( dim ( E + E ) + dim ( E ∩ E )) ( ) ( ) = ( dim ( E + E ) ) + ( dim ( E ∩ E ) ) (a)
1
2
1
2
a
1
a
2
(
= dim E1 + E 2
(
1
)
(a)
2
(
+ dim E1 ∩ E 2
)
(
)
(a)
)
a a a a = dim E1( ) + E 2( ) + dim E1( ) ∩ E 2( )
(
) (
a a a a a a = dim E1( ) + dim E 2( ) − dim E1( ) ∩ E 2( ) + dim( E1( ) ∩ E 2( )
(a)
)
(a)
= dim E1 + dim E2
(
)
(
)
Therefore dim E1 + E 2 + dim E1 ∩ E 2 = dim E1 + dim E 2 . Definition 4.6. Suppose that E = ( E , µ ) is an L-fuzzy vector subspace. A map
f : E → E is called an L-fuzzy linear transformation, if it satisfies the following conditions: (1) f is a linear map on E . (2) For all x ∈ E , µ ( f ( x ) ) ≥ µ ( x ) . Theorem 4.7. Suppose that E = ( E , µ ) is an L-fuzzy vector subspace, f is an f = kerf , µ L-fuzzy linear transformation on E , then ker and kerf are L-fuzzy vector subspaces. im f = imf , µ
(
imf
(
)
)
The prove is trivial and omitted. Theorem 4.8. Suppose that E = ( E , µ ) is an L-fuzzy vector subspace, f : E → E
is an L-fuzzy linear transformation, then
(
)
(
)
f + dim im f = dim ker dim E
Proof. Suppose that ϕ is a linear transformation on (crisp) vector spaces V , then
the equality dim ( imϕ ) + dim ( kefϕ ) = dimV holds. Hence, for all a ∈ P ( L ) , we have
f ) f ) ( dim (im f ) + dim ( ker ) = ( dim(im f ) )( ) + ( dim ( ker ) (a)
a
f ) ( ) + dim ( ker dim ( E ( ) ∩ imf ) + dim ( E ( ) ∩ keff ) (a)
f = dim im = Since f
a E ( )
(a)
(a)
a
a
is a linear transformation on E ( a ) , we have
f ) ( dim (im f ) + dim ( ker )
(a)
=dim ( imf
= dim = E ( ) a
(
)
(
) + dim ( kef f ( ) ( dim E ) .
a E ( )
a E ( )
)
a
)
f + dim im f = Therefore dim ker dim E . .
5. Conclusion In this paper, L-fuzzy vector subspace is defined and showed that its dimension is an
L-fuzzy natural number. Based on the definitions, some good properties of crisp vector spaces are hold in a finite-dimensional L-fuzzy vector subspace. In particular, the equality dim E1 + E 2 + dim E1 ∩ E 2 = dim E1 + dim E 2 holds without any restricted
(
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)
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(
)
(
)
f + dim kef f = conditions. At the same time, dim im dim E holds.
Acknowledgements The authors would like to thank the reviewers for their valuable comments and suggestions.
Fund The project is by the Science & Technology Program of Beijing Municipal Commission of Education (KM201611417007), the NNSF of China (11371002), the academic youth backbone project of Heilongjiang Education Department (1251G3036), the foundation of Heilongjiang Province (A201209).
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