(λ,μ) Fuzzy linear subspaces

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R E S E A R C H

Open Access

(λ,

μ)

-Fuzzy linear subspaces

Yuming Feng

1,2

_{and Chuandong Li}

1*

*_{Correspondence: [email protected];}

[email protected]

1_{College of Electronic Information}

Engineering, Southwest University, Chongqing, 400715, P.R. China Full list of author information is available at the end of the article

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.

Keywords: (

λ

,

μ

)-fuzzy subﬁeld; (

λ

,

μ

)-fuzzy linear subspace; homomorphism; linear transformation

1 Introduction and preliminaries

The concept of fuzzy sets was ﬁrst introduced by Zadeh [] in , and then the fuzzy sets have been used in the reconsideration of classical mathematics. Recently, Yuan [] intro-duced the concept of fuzzy subgroup with thresholds. A fuzzy subgroup with thresholdsλ andμis 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 [–].

Nanda [] introduced the concepts of fuzzy field and fuzzy linear space and gave some results. Biswas [] pointed out that Proposition . of [] was incorrect and redefined fuzzy field and fuzzy linear space. Gu and Lu [] listed two examples to show that Proposition . in [] 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 gener-alize 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 setXwe mean a mapping fromXto the unit interval [, ]. IfAis a fuzzy subset ofX, then we denoteAα={x∈X|A(x)≥α}for allα∈[, ].

Throughout this paper, we always assume that ≤λ<μ≤.

2 (

λ

,

μ

)-fuzzy subﬁelds

Definition  A fuzzy subsetFof a fieldFis said to be a (λ,μ)-fuzzy subfield ofFif∀k,l∈F, we have

() F(k+l)∨λ≥F(k)∧F(l)∧μ; () F(–k)∨λ≥F(k)∧μ; () F(kl)∨λ≥F(k)∧F(l)∧μ; () F(k–₎_∨_λ_≥_F(k)_∧_{μ, where}_k_{= .}

From the previous definition, we can easily conclude that a fuzzy subfield defined by Gu [, ] or Biswas [] is a (, )-fuzzy subfield.

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Proposition  Let F be a(λ,μ)-fuzzy subﬁeld of a ﬁeldF,then∀k∈F,we have () F()∨λ≥F(k)∧μ;

() F()∨λ≥F(k)∧μ,wherek= ; () F()∨λ≥F()∧μ.

Proof F()∨λ=F(k+ (–k))∨λ∨λ≥(F(k)∧F(–k)∧μ)∨λ= (F(k)∨λ)∧(F(–k)∨λ)∧ (μ∨λ)≥F(k)∧(F(k)∧μ)∧μ=F(k)∧μ. Thus we complete the proof of ().

() can be proved similarly and () is a corollary of ().

Theorem  Let F be a fuzzy subset of a ﬁeldF.Then the following are equivalent: () Fis a(λ,μ)-fuzzy subﬁeld ofF;

() Fαis a subﬁeld ofF,for anyα∈(λ,μ],whereFα=∅.

Proof It is a corollary of Proposition . of [].

We useF,Fto stand for two ﬁelds in the following and deﬁnesup∅= , where∅is the empty set.

Proposition  Let f :F→Fbe a homomorphism,and let Fbe a(λ,μ)-fuzzy subﬁeld

ofF.Then f(F)is a(λ,μ)-fuzzy subﬁeld ofF,where f(F)(y) =sup

x∈F

F(x)|f(x) =y

, ∀y∈F.

Proof Similar to the proof of Proposition . in [].

Proposition  Let f :F→Fbe a homomorphism,and let Fbe a(λ,μ)-fuzzy sublattice

ofF.Then f–(F)is a(λ,μ)-fuzzy sublattice ofF,where f–(F)(x) =F

f(x), ∀x∈F.

Proof Similar to the proof of Proposition . in [].

3 (

λ

,

μ

)-fuzzy linear subspaces

Definition  LetFbe a (λ,μ)-fuzzy subfield of a fieldF,Vbe a linear space overFand V be a fuzzy subset ofV.V is called a (λ,μ)-fuzzy linear subspace ofVoverFif for all x,y∈V,k∈F, we have

() V(x+y)∨λ≥V(x)∧V(y)∧μ; () V(–x)∨λ≥V(x)∧μ;

() V(kx)∨λ≥F(k)∧V(x)∧μ; () F()∨λ≥V(x)∧μ.

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

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() V(kx+ly)∨λ≥F(k)∧V(x)∧F(l)∧V(y)∧μ; () F()∨λ≥V(x)∧μ.

Proof ‘⇒’

For allx,y∈V,k,l∈F, we have

V(kx+ly)∨λ=V(kx+ly)∨λ∨λ≥V(kx)∧V(ly)∧μ∨λ

=V(kx)∨λ∧V(ly)∨λ∧(μ∨λ)

≥F(k)∧V(x)∧F(l)∧V(y)∧μ.

‘⇐’

FromF()∨λ≥V(x)∧μ, we know thatλ≥V(x)∧μorF()≥V(x)∧μ. Two cases are possible:

Case . Ifλ≥V(x)∧μ, then

() V(x+y)∨λ≥λ≥V(x)∧μ≥V(x)∧V(y)∧μ; () V(–x)∨λ≥λ≥V(x)∧μ;

() V(kx)∨λ≥λ≥V(x)∧μ≥F(k)∧V(x)∧μ. Case .F()≥V(x)∧μ, then

() V(x+y)∨λ=V(x+ y)∨λ≥F()∧V(x)∧F()∧V(y)∧μ=V(x)∧V(y)∧μ; () V(–x)∨λ=V(–x+ x)∨λ∨λ≥(F(–)∧V(x)∧F()∧V(x)∧μ)∨λ≥(F(–)∨

λ)∧(V(x)∨λ)∧(F()∨λ)∧(μ∨λ)≥(F()∧μ)∧V(x)∧(F()∧μ)∧μ=V(x)∧μ. () V(kx)∨λ=V(kx+ x)∨λ∨λ≥(F(k)∧V(x)∧F()∧V(x)∧μ)∨λ= (F(k)∨λ)∧ (V(x)∨λ)∧(F()∨λ)∧(μ∨λ)≥F(k)∧V(x)∧(F()∧μ)∧μ=F(k)∧V(x)∧μ.

Theorem  Let F be a(λ,μ)-fuzzy subﬁeld of a ﬁeldF,letVbe a linear space overF,and let V be a fuzzy subset ofV.Then the following are equivalent:

() Vis a(λ,μ)-fuzzy linear subspace overF;

() Vαis a linear subspace overFα,for anyα∈(λ,μ],whereFα=∅andVα=∅.

Proof ‘()⇒()’

LetVbe a (λ,μ)-fuzzy linear subspace overF.

Take any α∈(λ,μ] such thatFα=∅andVα=∅; we need to show thatkx+ly∈Vα, ∀x,y∈Vα,∀k,l∈Fα.

FromF(k)≥α,F(l)≥α,V(x)≥α,V(y)≥α, andα≤μ, we conclude thatV(kx+ly)∨ λ≥F(k)∧V(x)∧F(l)∧V(y)∧μ≥α∧μ=α. Note thatλ<α, we obtain thatV(kx+ly)≥α. Sokx+ly∈Vα.

‘()⇐()’

Conversely, letVαbe a linear subspace overFα, for anyα∈(λ,μ]. If there existx,y∈V

andk,l∈Fsuch thatV(kx+ly)∨λ<α=F(k)∧V(x)∧F(l)∧V(y)∧μ, thenα∈(λ,μ], x,y∈Vα,k,l∈Fα. Butkx+ly∈/Aα. This is a contradiction with thatVαis a linear subspace

overFα.

Again, if there existsx∈Vsuch thatF()∨λ<α=V(x)∧μ, thenα∈(λ,μ],x∈Vαand

 /∈Fα. This is a contradiction to thatVαis a linear subspace over a subﬁeldFαofF.

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Proposition  Let F be a(λ,μ)-fuzzy subﬁeld of a ﬁeldF, let f :V→V be a linear transformation overF,and Vbe a(λ,μ)-fuzzy linear subspace ofVover F.Then f(V)is

a(λ,μ)-fuzzy linear subspace ofVover F,where f(V)(y) =sup

x∈V_

V(x)|f(x) =y

, ∀y∈V.

Proof Iff–(y) =∅orf–(y) =∅for anyy,y∈V, then (f(V)(ky+ly))∨λ≥ =F(k)∧

f(V)(y)∧F(l)∧f(V)(y)∧μ.

Suppose thatf–_(y

)=∅,f–(y)=∅for anyy,y∈V, thenf–(ky+ly)=∅. So for any

k,l∈F, we have

f(V)(ky+ly)∨λ

=sup

t∈V_

A(t)|f(t) =ky+ly

∨λ

=sup

t∈V

V(t)∨λ|f(t) =ky+ly

≥ sup

x,x∈V

V(kx+lx)

∨λ|f(x) =y,f(x) =y

≥ sup

x,x∈V

F(k)∧V(x)∧F(l)∧V(x)

∧μ|f(x) =y,f(x) =y

=sup

x∈V

V(x)|f(x) =y

∧ sup

x∈V

V(x)|f(x) =y

∧F(k)∧F(l)∧μ

=f(V)(y)∧f(V)(y)∧F(k)∧F(l)∧μ.

And for ally∈V, we have F()∨λ= sup

x∈V

F()∨λ|V(x) =y

≥ sup

x∈V

V(x)∧μ|V(x) =y

= sup

x∈V

V(x)|V(x) =y

∧μ

=f(V)(y)∧μ.

Thusf(V) is a (λ,μ)-fuzzy linear subspace ofVoverF.

Proposition  Let F be a (λ,μ)-fuzzy subﬁeld of a ﬁeldF, let f :V→V be a linear transformation over F, and let V be a(λ,μ)-fuzzy linear subspace ofVover F.Then f–_(V

)is a(λ,μ)-fuzzy linear subspace ofVover F,where f–(V)(x) =V

f(x), ∀x∈V.

Proof For anyx,x∈Vandk,l∈F, we have

f–(V)(kx+lx)∨λ=V

f(kx+lx)

∨λ

=V

kf(x) +lf(x)

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≥F(k)∧V

f(x)

∧F(l)∧V

f(x)

∧μ

=F(k)∧f–(V)(x)∧F(l)∧f–(V)(x)∧μ.

And for allx∈V, we have

f–(V)(x)∧μ=V

f(x)∧μ≤F()∨λ.

So,f–_(V

) is a (λ,μ)-fuzzy linear subspace ofVoverF.

4 Further research

The present work can be extended in several directions. Let us indicate some possibilities. . One can deﬁne(λ,μ)-fuzzy hypervector spaces and study their properties

(deﬁnitions of fuzzy hypervector spaces can be found in []).

. One may give the deﬁnition of(λ,μ)-fuzzy linear subspaces over(λ,μ)-fuzzy

subﬁelds, where≤λ<μ≤and≤λ<μ≤. Then explore the properties

of them.

. One can investigate the interval-valued (type , lattice-valued,etc.)(λ,μ)-fuzzy

linear subspaces.

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

Competing interests

The authors declare that they have no competing interests.

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.

Author details

1_{College of Electronic Information Engineering, Southwest University, Chongqing, 400715, P.R. China.}2_{School of}

Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404100, P.R. China.

Received: 2 May 2013 Accepted: 23 July 2013 Published: 7 August 2013 References

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doi:10.1186/1029-242X-2013-369