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(λ,μ)-Fuzzy linear subspaces

Abstract

In this paper, we first introduce the concepts of (λ,μ)-fuzzy subfields. Then we generalize the concepts of fuzzy linear spaces, we define (λ,μ)-fuzzy linear subspaces, and we obtain some of their fundamental properties. Lastly, we list some possible directions of the extending of the present work.

1 Introduction and preliminaries

The concept of fuzzy sets was first introduced by Zadeh [1] in 1965, and then the fuzzy sets have been used in the reconsideration of classical mathematics. Recently, Yuan [2] introduced the concept of fuzzy subgroup with thresholds. A fuzzy subgroup with thresholds λ and μ is also called a (λ,μ)-fuzzy subgroup. Yao continued to research (λ,μ)-fuzzy normal subgroups, (λ,μ)-fuzzy quotient subgroups and (λ,μ)-fuzzy subrings in [35].

Nanda [6] introduced the concepts of fuzzy field and fuzzy linear space and gave some results. Biswas [7] pointed out that Proposition 4.1 of [6] was incorrect and redefined fuzzy field and fuzzy linear space. Gu and Lu [8] listed two examples to show that Proposition 4.5 in [6] is also incorrect and redefined the concept of fuzzy linear space.

In this paper, we first introduce the concepts of (λ,μ)-fuzzy subfields. Next, we generalize the concepts of fuzzy linear spaces over fuzzy fields to (λ,μ)-fuzzy linear subspaces over (λ,μ)-fuzzy subfields, and give some fundamental properties.

Let us recall some definitions and notions.

By a fuzzy subset of a nonempty set X we mean a mapping from X to the unit interval [0,1]. If A is a fuzzy subset of X, then we denote A α ={xX|A(x)α} for all α[0,1].

Throughout this paper, we always assume that 0λ<μ1.

2 (λ,μ)-fuzzy subfields

Definition 1 A fuzzy subset F of a field F is said to be a (λ,μ)-fuzzy subfield of F if k,lF, we have

  1. (1)

    F(k+l)λF(k)F(l)μ;

  2. (2)

    F(k)λF(k)μ;

  3. (3)

    F(kl)λF(k)F(l)μ;

  4. (4)

    F( k 1 )λF(k)μ, where k0.

From the previous definition, we can easily conclude that a fuzzy subfield defined by Gu [8, 9] or Biswas [7] is a (0,1)-fuzzy subfield.

Proposition 1 Let F be a (λ,μ)-fuzzy subfield of a field F, then kF, we have

  1. (1)

    F(0)λF(k)μ;

  2. (2)

    F(1)λF(k)μ, where k0;

  3. (3)

    F(0)λF(1)μ.

Proof F(0)λ=F(k+(k))λλ(F(k)F(k)μ)λ=(F(k)λ)(F(k)λ) (μλ)F(k)(F(k)μ)μ=F(k)μ. Thus we complete the proof of (1).

  1. (2)

    can be proved similarly and (3) is a corollary of (1). □

Theorem 1 Let F be a fuzzy subset of a field F. Then the following are equivalent:

  1. (1)

    F is a (λ,μ)-fuzzy subfield of F;

  2. (2)

    F α is a subfield of F, for any α(λ,μ], where F α .

Proof It is a corollary of Proposition 2.4 of [4]. □

We use F 1 , F 2 to stand for two fields in the following and define sup=0, where is the empty set.

Proposition 2 Let f: F 1 F 2 be a homomorphism, and let F 1 be a (λ,μ)-fuzzy subfield of F 1 . Then f( F 1 ) is a (λ,μ)-fuzzy subfield of F 2 , where

f( F 1 )(y)= sup x F 1 { F 1 ( x ) | f ( x ) = y } ,y F 2 .

Proof Similar to the proof of Proposition 2.7 in [4]. □

Proposition 3 Let f: F 1 F 2 be a homomorphism, and let F 2 be a (λ,μ)-fuzzy sublattice of F 2 . Then f 1 ( F 2 ) is a (λ,μ)-fuzzy sublattice of F 1 , where

f 1 ( F 2 )(x)= F 2 ( f ( x ) ) ,x F 1 .

Proof Similar to the proof of Proposition 2.8 in [4]. □

3 (λ,μ)-fuzzy linear subspaces

Definition 2 Let F be a (λ,μ)-fuzzy subfield of a field F, V be a linear space over F and V be a fuzzy subset of V. V is called a (λ,μ)-fuzzy linear subspace of V over F if for all x,yV, kF, we have

  1. (1)

    V(x+y)λV(x)V(y)μ;

  2. (2)

    V(x)λV(x)μ;

  3. (3)

    V(kx)λF(k)V(x)μ;

  4. (4)

    F(1)λV(x)μ.

Obviously, the previous definition is a generalization of fuzzy linear space defined by Gu and Tu (see Definition 3.1 in [9]).

Theorem 2 Let F be a (λ,μ)-fuzzy subfield of a field F, let V be a linear space over F, and let V be a fuzzy subset of V. V is a (λ,μ)-fuzzy linear subspace of V over F if and only if, for all x,yV, k,lF, we have

  1. (1)

    V(kx+ly)λF(k)V(x)F(l)V(y)μ;

  2. (2)

    F(1)λV(x)μ.

Proof

For all x,yV, k,lF, we have

V ( k x + l y ) λ = V ( k x + l y ) λ λ ( V ( k x ) V ( l y ) μ ) λ = ( V ( k x ) λ ) ( V ( l y ) λ ) ( μ λ ) F ( k ) V ( x ) F ( l ) V ( y ) μ .

From F(1)λV(x)μ, we know that λV(x)μ or F(1)V(x)μ. Two cases are possible:

Case 1. If λV(x)μ, then

  1. (1)

    V(x+y)λλV(x)μV(x)V(y)μ;

  2. (2)

    V(x)λλV(x)μ;

  3. (3)

    V(kx)λλV(x)μF(k)V(x)μ.

Case 2. F(1)V(x)μ, then

  1. (1)

    V(x+y)λ=V(1x+1y)λF(1)V(x)F(1)V(y)μ=V(x)V(y)μ;

  2. (2)

    V(x)λ=V(1x+0x)λλ(F(1)V(x)F(0)V(x)μ)λ(F(1)λ) (V(x)λ)(F(0)λ)(μλ)(F(1)μ)V(x)(F(1)μ)μ=V(x)μ.

  3. (3)

    V(kx)λ=V(kx+0x)λλ(F(k)V(x)F(0)V(x)μ)λ=(F(k)λ) (V(x)λ)(F(0)λ)(μλ)F(k)V(x)(F(1)μ)μ=F(k)V(x)μ.

 □

Theorem 3 Let F be a (λ,μ)-fuzzy subfield of a field F, let V be a linear space over F , and let V be a fuzzy subset of V. Then the following are equivalent:

  1. (1)

    V is a (λ,μ)-fuzzy linear subspace over F;

  2. (2)

    V α is a linear subspace over F α , for any α(λ,μ], where F α and V α .

Proof ‘(1) (2)’

Let V be a (λ,μ)-fuzzy linear subspace over F.

Take any α(λ,μ] such that F α and V α ; we need to show that kx+ly V α , x,y V α , k,l F α .

From F(k)α, F(l)α, V(x)α, V(y)α, and αμ, we conclude that V(kx+ly)λF(k)V(x)F(l)V(y)μαμ=α. Note that λ<α, we obtain that V(kx+ly)α. So kx+ly V α .

‘(1) (2)’

Conversely, let V α be a linear subspace over F α , for any α(λ,μ]. If there exist x,yV and k,lF such that V(kx+ly)λ<α=F(k)V(x)F(l)V(y)μ, then α(λ,μ], x,y V α , k,l F α . But kx+ly A α . This is a contradiction with that V α is a linear subspace over F α .

Again, if there exists xV such that F(1)λ<α=V(x)μ, then α(λ,μ], x V α and 1 F α . This is a contradiction to that V α is a linear subspace over a subfield F α of F. □

We use V 1 , V 2 to stand for two linear spaces over the same field F in the following.

Proposition 4 Let F be a (λ,μ)-fuzzy subfield of a field F, let f: V 1 V 2 be a linear transformation over F, and V 1 be a (λ,μ)-fuzzy linear subspace of V 1 over F. Then f( V 1 ) is a (λ,μ)-fuzzy linear subspace of V 2 over F, where

f( V 1 )(y)= sup x V 1 { V 1 ( x ) | f ( x ) = y } ,y V 2 .

Proof If f 1 ( y 1 )= or f 1 ( y 2 )= for any y 1 , y 2 V 2 , then (f( V 1 )(k y 1 +l y 2 ))λ0=F(k)f( V 1 )( y 1 )F(l)f( V 1 )( y 2 )μ.

Suppose that f 1 ( y 1 ), f 1 ( y 2 ) for any y 1 , y 2 V 2 , then f 1 (k y 1 +l y 2 ). So for any k,lF, we have

f ( V 1 ) ( k y 1 + l y 2 ) λ = sup t V 1 { A ( t ) | f ( t ) = k y 1 + l y 2 } λ = sup t V 1 { V 1 ( t ) λ | f ( t ) = k y 1 + l y 2 } sup x 1 , x 2 V 1 { ( V 1 ( k x 1 + l x 2 ) ) λ | f ( x 1 ) = y 1 , f ( x 2 ) = y 2 } sup x 1 , x 2 V 1 { ( F ( k ) V 1 ( x 1 ) F ( l ) V 1 ( x 2 ) ) μ | f ( x 1 ) = y 1 , f ( x 2 ) = y 2 } = ( sup x 1 V 1 { V 1 ( x 1 ) | f ( x 1 ) = y 1 } sup x 2 V 1 { V 1 ( x 2 ) | f ( x 2 ) = y 2 } ) F ( k ) F ( l ) μ = f ( V 1 ) ( y 1 ) f ( V 1 ) ( y 2 ) F ( k ) F ( l ) μ .

And for all y V 2 , we have

F ( 1 ) λ = sup x V 1 { F ( 1 ) λ | V 1 ( x ) = y } sup x V 1 { V 1 ( x ) μ | V 1 ( x ) = y } = sup x V 1 { V 1 ( x ) | V 1 ( x ) = y } μ = f ( V 1 ) ( y ) μ .

Thus f( V 1 ) is a (λ,μ)-fuzzy linear subspace of V 2 over F. □

Proposition 5 Let F be a (λ,μ)-fuzzy subfield of a field F, let f: V 1 V 2 be a linear transformation over F , and let V 2 be a (λ,μ)-fuzzy linear subspace of V 2 over F. Then f 1 ( V 2 ) is a (λ,μ)-fuzzy linear subspace of V 1 over F, where

f 1 ( V 2 )(x)= V 2 ( f ( x ) ) ,x V 1 .

Proof For any x 1 , x 2 V 1 and k,lF, we have

f 1 ( V 2 ) ( k x 1 + l x 2 ) λ = V 2 ( f ( k x 1 + l x 2 ) ) λ = V 2 ( k f ( x 1 ) + l f ( x 2 ) ) λ F ( k ) V 2 ( f ( x 1 ) ) F ( l ) V 2 ( f ( x 2 ) ) μ = F ( k ) f 1 ( V 2 ) ( x 1 ) F ( l ) f 1 ( V 2 ) ( x 2 ) μ .

And for all x V 1 , we have

f 1 ( V 2 )(x)μ= V 2 ( f ( x ) ) μF(1)λ.

So, f 1 ( V 2 ) is a (λ,μ)-fuzzy linear subspace of V 1 over F. □

4 Further research

The present work can be extended in several directions. Let us indicate some possibilities.

  1. 1.

    One can define (λ,μ)-fuzzy hypervector spaces and study their properties (definitions of fuzzy hypervector spaces can be found in [10]).

  2. 2.

    One may give the definition of ( λ 1 , μ 1 )-fuzzy linear subspaces over ( λ 2 , μ 2 )-fuzzy subfields, where 0 λ 1 < μ 1 1 and 0 λ 2 < μ 2 1. Then explore the properties of them.

  3. 3.

    One can investigate the interval-valued (type 2, lattice-valued, etc.) ( λ 1 , μ 1 )-fuzzy linear subspaces.

  4. 4.

    One can also research (λ,μ)-anti-fuzzy linear subspaces [11].

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Correspondence to Chuandong Li.

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Authors’ contributions

CL first gave the idea of generalizing the fuzzy linear space to (λ,μ)-fuzzy linear subspace, YF gave some good ideas on improving the paper, and he finished the latex version of the paper. All authors read and approved the final manuscript.

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Feng, Y., Li, C. (λ,μ)-Fuzzy linear subspaces. J Inequal Appl 2013, 369 (2013). https://doi.org/10.1186/1029-242X-2013-369

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Keywords

  • (λ,μ)-fuzzy subfield
  • (λ,μ)-fuzzy linear subspace
  • homomorphism
  • linear transformation