R. Muthuraj1 and G. Santha Meena2...Some Characterization of Bipolar L-Fuzzy ℓ - HX group R....

Some Characterization of Bipolar L-Fuzzy ℓ - HX group

R. Muthuraj1 and G. Santha Meena

2

1 PG & Research Department of Mathematics

H.H. The Rajah’s College

Pudukkottai – 622 001, Tamilnadu, India

E-mail: [email protected]

2 Department of Mathematics

PSNA College of Engineering and Technology

Dindigul – 624 622, Tamilnadu, India

E-mail:[email protected]

Abstract

In this paper, we define a new algebraic structure of Bipolar L-fuzzy ℓ - HX group and some

related properties are investigated. We also discuss the properties of Bipolar L-fuzzy sub ℓ - HX

group under ℓ - HX group homomorphism and ℓ - HX group anti homomorphism.

Keywords

ℓ - HX group, Bipolar L-fuzzy ℓ- HX group, ℓ - HX group homomorphism, ℓ - HX group anti

homomorphism, image and pre-image of bipolar L-fuzzy sets are discussed.

1. Introduction

The concept of fuzzy sets was initiated by Zadeh[1] . Then it has become a vigorous area of

research in engineering, medical science, social science, graph theory etc. Rosenfeld[2] gave the

idea of fuzzy subgroups. In fuzzy sets the membership degree of elements range over the interval

[0,1]. The membership degree expresses the degree of belongingness of elements to a fuzzy set.

The membership degree 1 indicates that an element completely belongs to its corresponding

fuzzy set and membership degree 0 indicates that an element does not belong to fuzzy set. The

membership degrees on the interval (0, 1) indicate the partial membership to the fuzzy set.

Sometimes, the membership degree means the satisfaction degree of elements to some property

or constraint corresponding to a fuzzy set. Li Hongxing[3] introduced the concept of HX group

and the authors LuoChengzhong , MiHonghai, Li Hongxing[4] introduced the concept of fuzzy

HX group. The author W.R. Zhang[5] commenced the concept of bipolar fuzzy sets as a

generalization of fuzzy sets in 1994. K.M. Lee[6] introduced Bipolar-valued fuzzy sets and their

International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

137

mailto:[email protected]

operations. In case of Bipolar-valued fuzzy sets membership degree range is enlarged from the

interval [0, 1] to [−1, 1]. In a bipolar-valued fuzzy set, the membership degree 0 means that the

elements are irrelevant to the corresponding property, the membership degree (0,1] indicates that

elements somewhat satisfy the property and the membership degree [−1,0) indicates that

elements somewhat satisfy the implicit counter-property. G.S.V. SatyaSaibaba[7] initiated the

study of L-fuzzy lattice ordered groups and introduced the notions of L-fuzzy sub ℓ - HX group.

J.A Goguen[8] replaced the valuation set [0,1] by means of a complete lattice in an attempt to

make a generalized study of fuzzy set theory by studying L-Fuzzy sets. R. Muthuraj,

M.Sridharan[9]introduced Homomorphism and anti-homomorphism on bipolar fuzzy sub HX

groups.R. Muthuraj, T. Rakesh Kumar[10]defined some characterization of L – fuzzy ℓ - HX

group. In this paper we define a new concept of Bipolar of L – fuzzy sub ℓ - HX group and

study some of their related properties.

2. Preliminaries

In this section, we site the fundamental definitions that will be used in the sequel. Throughout

this paper, G = (G , .) is a group, e is the identity element of G, and xy, we mean x .y

Definition 2.1

In 2G – {}, a non-empty set (ϑ⊂2

G – {}, . , ≤ )is called as a ℓ - HX group or lattice ordered HX

group on G, if the following conditions are satisfied.

i) (ϑ, . ) is a HX group

ii) (ϑ,≤ ) is a lattice.

Definition 2.2

A Bipolar L-fuzzy setµ in G is a bipolar L-fuzzy subgroup of G if for all x,yG.

i) µ+ (xy ) ≥µ

+ (x ) ˄ µ

+ (y )

ii) µ- (xy ) ≤ µ

- (x ) ˅ µ

- (y )

iii) µ+ (x

-1) = µ

+ (x ) , µ

- (x

-1) = µ

- (x ).


138

Definition 2.3

Let µbe a bipolar L-fuzzy subset defined on G. Let ϑ ⊂ 2G – {} be a ℓ - HX group on G. A

bipolar L-fuzzy set µ

defined on ϑ is said to be a bipolar L-fuzzy sub ℓ - HX group on ϑ if for

all A,Bϑ.

i) (µ)

+ (AB) ≥ (

µ)

+ (A) ˄ (

µ)

+ (B)

ii) (µ)

- (AB) ≤ (

µ)

- (A) ˅ (

µ)

- (B)

iii) (µ)

+ (A) = (

µ)

+ (A

-1)

iv) (µ)

- (A) = (

µ)

- (A

-1)

v) (µ)

+ (A˅B) ≥ (

µ)+ (A)˄ (

µ)

+ (B)

vi) (µ)

- (A˅B) ≤ (

µ)

- (A) ˅ (

µ)

- (B)

vii) (µ)

+ (A˄B) ≥ (

µ)+ (A) ˄ (

µ)

+ (B)

viii) (µ)

- (A˄B) ≤ (

µ)

- (A) ˅ (

µ)

- (B)

Where(µ)

+ (A) = max{ µ

+ (x ) / for all xA⊆G }

and

(µ)

- (A) = min{ µ

- (x ) /for all xA⊆ G }

Example 2.1

Let G={Z10-{0}, .10} be a group and define a bipolar L- fuzzy set µ on G as µ+(1)=0.8,

µ+(3)=0.5, µ

+(7)=0.5, µ

+(9)=0.4 and µ

-(1)= - 0.9, µ

-(3)= - 0.4, µ

-(7)= - 0.4, µ

-(9)= - 0.3

By routine computations, it is easy to see that µ is a bipolar L-fuzzy sub group of G.

Let ϑ = {{3, 7}, {1, 9}} be a ℓ - HX group of G.

Let us consider A = {3, 7}, B = {1, 9}.

.10 A B ˄ A B ˅ A B

A B A A A B A A A

B A B B B B B A B

Define(µ)

+ (A) = max{µ

+ (x ) / for all xA ⊆G }


139

and

(µ)

- (A) = min{µ

- (x ) / for all xA⊆ G }

Now (µ)

+(A)= (

µ)

+({3,7}) = max{µ

+(3), µ

+ (7)} = max{0.5,0.5} = 0.5

(µ)

+(B)= (

µ)

+({1,9}) = max{µ

+(1), µ

+ (9)} = max{0.8,0.4} = 0.8

(µ)

+(AB) = (

µ)+(A) = 0.5

(µ)

+(A˄B) = (

µ)

+(B) = 0.8

(µ)

+(A˅B) = (

µ)

+(A) = 0.5

(µ)

-(A)= (

µ)

-({3,7}) = min{µ

-(3), µ

- (7)} = min{- 0.4, - 0.4} = - 0.4

(µ)

-(B)= (

µ)

-({1,9}) = min{µ

-(1), µ

- (9)} = min{ -0.9, - 0.3} = - 0.9

(µ)

-(AB)= (

µ)

-(A) = - 0.4

(µ)

-(A˄B)= (

µ)

-(B) = - 0.9

(µ)

-(A˅B)= (

µ)

-(A) = - 0.4

By routine computations, it is easy to see that µ is a bipolar L-fuzzy sub ℓ - HX group of ϑ .

Definition 2.4

Let G1 and G2 be any two groups. Letϑ1⊂ 2G

1 – {} and ϑ2⊂ 2G

2 – {} be any two ℓ - HX groups

defined on G1 and G2 respectively. Let µ= (µ+, µ

-) and α= (α

+, α

-) are bipolar L-fuzzy subsets in

G1 and G2 respectively. Let µ=((

µ)

+,(

µ)

-)and

=((

)

+, (

)

-) are bipolar L-fuzzy subsets in

ϑ1andϑ2 respectively. Let f: ϑ1 → ϑ2be a mapping then the image f(µ) of

µ is the bipolar L-

fuzzy subset

f(µ)=( (f(

µ))

+, (f(

µ))

-) of ϑ2 defined as for each U ϑ2

(f ())

+ (U) = sup {(

)

+(X): X f

-1(U)} , if f

-1(U) ≠

0 , otherwise

and

(f ())

- (U) = sup {(

)

-(X): X f

-1(U)} , if f

-1(U) ≠

0 , otherwise


140

And the pre-image f-1

()of

under f is the bipolar L-fuzzy subset of ϑ1 defined as for each

X ϑ1 ,( (f-1

())

+(X) = (

)

+(f(X)), (f

-1(

))

- (X) = (

)

- (f(X)).

Definition 2.5

Let G1 and G2 be any two groups. Let ϑ1⊂ 2G

1 – {} and ϑ2⊂ 2G

2 – {} be any two ℓ - HX

groups defined on G1 and G2 respectively. Let µ

be a bipolar L-fuzzy sub ℓ - HX group of a

ℓ - HX groupinϑ1. Let f: ϑ1 → ϑ2 be a function. Then µ is called f-invariant if the following

conditions are satisfied.

i) ((f(A))+ = ((f(B))

+ implies that (

µ)+(A) = (

µ)

+(B)

ii) ((f(A))- = ((f(B))

- implies that (

µ)

-(A) = (

µ)

-(B)

3. Properties of Bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group

In this section, We discuss some of the properties of Bipolar L-fuzzy sub ℓ - HX group of a

ℓ - HX group.

Theorem 3.1

Let G be a group. Let µbe a bipolar L-fuzzy subset defined on G. Let a non-empty set

ϑ ⊂2G – {} be an ℓ - HX group on G. If

µ =((

µ)

+,(

µ)

-) is a bipolar L-fuzzy sub ℓ - HX group

of a ℓ - HX group ϑ, then for all A,B in ϑ,

i) (µ)

+(AB

-1) ≥ (

µ)+ (A) ˄ (

µ)

+(B)

ii) (µ)

- (AB

-1) ≤ (

µ)

- (A) ˅ (

µ)

- (B)

Proof:

Let µ =((

µ)

+,(

µ)

-) be a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ, then for all A,B

in ϑ,

i) (µ)

+(AB

-1) ≥ (

µ)+ (A) ˄ (

µ)

+ (B

-1)

= (µ)+ (A)˄ (

µ)

+ (B)

(µ)

+(AB

-1) ≥(

µ)

+ (A)˄ (

µ)

+ (B)

ii) (µ)

-(AB

-1)≤(

µ)

- (A) ˅ (

µ)

- (B

-1)

=(µ)

- (A) ˅ (

µ)

- (B)


141

(µ)

-(AB

-1) ≤ (

µ)

- (A) ˅ (

µ)

- (B)

Theorem 3.2

If µ=((

µ)

+,(

µ)

-) is a bipolar L-fuzzy subset of a ℓ - HX group ϑ and for all A,B ϑ

i) (µ)

+(AB

-1) ≥ (

µ)+(A) ˄ (

µ)

+ (B)

ii) (µ)

-(AB

-1) ≤ (

µ)

- (A) ˅ (

µ)

- (B)

iii) (µ)

+(A ˅ B) ≥ (

µ)

+(A) ˄ (

µ)

+(B)

iv) (µ)

-(A ˅ B) ≤ (

µ)

- (A) ˅ (

µ)

-(B)

v) (µ)

+(A ˄ B) ≥ (

µ)

+(A) ˄ (

µ)

+ (B)

vi) (µ)

-(A ˄ B) ≤ (

µ)

- (A) ˅ (

µ)

-(B)

Then µ =((

µ)

+,(

µ)

-)is a bipolar L-fuzzy sub ℓ - HX group of aℓ - HX group ϑ.

Proof:

Let µ =((

µ)

+,(

µ)

-) is a bipolar L-fuzzy subset of a ℓ - HX group ϑ and for all A,Bϑ,

Now, (µ)+(A) = (

µ)

+(E(A

-1)

-1)

≥ (µ)

+(E) ˄(

µ)

+( A

-1)

= (µ)+( A

-1)

(µ)

+(A) ≥ (

µ)

+( A

-1)

Also (µ)

+( A

-1) = (

µ)

+(E(A

-1)

≥ (µ)

+(E) ˄(

µ)

+( A)

= (µ)+( A)

(µ)

+( A

-1) ≥ (

µ)

+( A)

Hence (µ)

+(A) = (

µ)

+( A

-1)

And, (µ)

-(A) = (

µ)

- (E(A

-1)

-1)

≤ (µ)

- (E)˅(

µ)

-( A

-1)

= (µ)

-( A

-1)

(µ)

-(A) ≤ (

µ)

-( A

-1)

Also (µ)

-( A

-1) = (

µ)

-(E(A

-1)

≤ (µ)

-(E) ˅(

µ)

-( A)

= (µ)

-( A)


142

(µ)

-( A

-1) ≤ (

µ)

-( A)

Hence (µ)

-(A) = (

µ)

-( A

-1)

Now, (µ)

+(AB) = (

µ)

+(A(B

-1)

-1)

≥ (µ)

+(A) ˄ (

µ)+( B

-1)

≥ (µ)

+(A) ˄ (

µ)+( B)

Therefore, (µ)

+(AB) ≥ (

µ)

+(A) ˄ (

µ)

+( B)

And, (µ)

-(AB) = (

µ)

-(A(B

-1)

-1)

≤ (µ)

-(A) ˅(

µ)

-( B

-1)

≤ (µ)

-(A) ˅(

µ)

-( B)

Therefore, (µ)

-(AB) ≤ (

µ)

-(A) ˅(

µ)

-( B)

By using the condition, iii),iv),v),vi),vii),viii),ix) and x)µ =((

µ)+,(

µ)

-) is a bipolar L-fuzzy sub

ℓ - HX group of a ℓ - HX group ϑ.

Theorem 3.3

Let µ =((

µ)

+,(

µ)

-) be a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ then

i) (µ)

+(A) ≤ (

µ)

+(E) for all Aϑ and E is the identity element of ϑ.

ii) (µ)

-(A) ≥ (

µ)

-(E) for all A ϑ and E is the identity element of ϑ.

iii) The subset H={Aϑ / (µ)

+(A) =(

µ)

+(E)} is a sub ℓ - HX group of ϑ.

iv) The subset H={Aϑ / (µ)

-(A) =(

µ)

+(E)} is a sub ℓ - HX group of ϑ.

Proof i) Let Aϑ

(µ)

+ (A) = (

µ)

+ (A) ˄ (

µ)

+ (A)

= (µ)+ (A) ˄ (

µ)

+ (A

-1)

≤ (µ)

+ (AA

-1)

= (µ)+(E)

(µ)

+ (A) ≤ (

µ)

+(E) for all Aϑ

ii) Let Aϑ

(µ)

- (A) = (

µ)

- (A) ˅ (

µ)

- (A)

= (µ)

- (A) ˅ (

µ)

- (A

-1)

≥ (µ)

- (AA

-1)


143

= (µ)

-(E)

(µ)

- (A) ≥ (

µ)

-(E) for all Aϑ

iii)Let H={Aϑ/(µ)

+(A)=(

µ)

+(E)}clearly, H is non-empty as EH Let A,B H, then

(µ)

+(A)= (

µ)

+(B)=(

µ)

+(E)

Now, (µ)+(AB

-1) ≥ (

µ)

+(A) ˄ (

µ)+(B

-1)

= (µ)+(A) ˄ (

µ)

+(B)

= (µ)+(E) ˄ (

µ)

+(E)

= (µ)+(E)

(µ)

+(AB

-1) ≥ (

µ)

+(E)

That is, (µ)

+(AB

-1) ≥ (

µ)

+(E) and by part (i)

(µ)

+(AB

-1) ≤ (

µ)

+(E)

Hence, (µ)+(AB

-1) = (

µ)+(E)

Hence, (µ)+(AB

-1) = (

µ)+(E) then AB

-1H

Now, (µ)+(A ˅ B) ≥ (

µ)

+(A) ˄ (

µ)

+(B)

= (µ)+(A) ˄ (

µ)

+(B)

= (µ)+(E) ˄ (

µ)

+(E)

= (µ)+(E)

(µ)

+(A ˅ B) ≥ (

µ)

+(E)

That is, (µ)

+(A ˅ B) ≥ (

µ)


(µ)

+(A ˅ B) ≤ (

µ)

+(E)

Hence, (µ)

+(A ˅ B) = (

µ)+(E)

Hence, (µ)

+(A ˅ B) = (

µ)+(E) then A ˅ B H

Now, (µ)+(A ˄ B) ≥ (

µ)

+(A) ˄ (

µ)

+(B)

= (µ)+(A) ˄ (

µ)

+(B)

= (µ)+(E) ˄ (

µ)

+(E)

= (µ)+(E)

(µ)

+(A ˄ B) ≥ (

µ)

+(E)

That is, (µ)

+(A ˄ B) ≥ (

µ)


(µ)

+(A ˄ B) ≤ (

µ)

+(E)


144

Hence, (µ)+(A ˄ B) = (

µ)+(E)

Hence, (µ)+(A ˄ B) = (

µ)+(E) then A ˄ B H

Clearly, H is a sub ℓ - HX group of ϑ.

iv) Let H={Aϑ/(µ)

-(A)=(

µ)

-(E)} clearly,H is non-empty as EH Let A,BH, then

(µ)

-(A) = (

µ)

-(B) = (

µ)

-(E)

Now, (µ)

-(AB

-1) ≤ (

µ)

-(A) ˅ (

µ)

-(B

-1)

= (µ)

-(A)˅ (

µ)

-(B)

= (µ)

-(E)˅(

µ)

-(E)

= (µ)

-(E)

(µ)

-(AB

-1) ≤ (

µ)

-(E)

That is, (µ)

-(AB

-1) ≤ (

µ)

-(E) and by part (ii)

(µ)

-(AB

-1) ≥ (

µ)

-(E)

Hence, (µ)

-(AB

-1) = (

µ)

-(E)

Hence, (µ)

-(AB

-1) = (

µ)

-(E) then AB

-1H

Now, (µ)

-(A ˅ B) ≤ (

µ)

-(A) ˅ (

µ)

-(B)

= (µ)

-(A) ˅ (

µ)

-(B)

= (µ)

-(E) ˅ (

µ)

-(E)

= (µ)

-(E)

(µ)

-(A ˅ B) ≤ (

µ)

-(E)

That is, (µ)

-(A ˅ B) ≤ (

µ)


(µ)

-(A ˅ B) ≥ (

µ)

-(E)

Hence, (µ)

-(A ˅ B) = (

µ)

-(E)

Hence, (µ)

-(A ˅ B) = (

µ)

-(E) then A ˅ B H

Now, (µ)

-(A ˄ B) ≤ (

µ)

-(A) ˅ (

µ)

-(B)

= (µ)

-(A) ˅ (

µ)

-(B)

= (µ)

-(E) ˅ (

µ)

-(E)

= (µ)

-(E)

(µ)

-(A ˄ B) ≤ (

µ)

-(E)

That is, (µ)

-(A ˄ B) ≤ (

µ)



145

(µ)

-(A ˄ B) ≥ (

µ)

-(E)

Hence, (µ)

-(A ˄ B) = (

µ)

-(E)

Hence, (µ)

-(A ˄ B) = (

µ)

-(E) then A ˄ B H

Clearly, H is a sub ℓ - HX group of ϑ.

Theorem 3.4

Letµ=((

µ)

+,(

µ)

-) be a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ

with identity E. Then

i) (µ)

+(AB

-1) = (

µ)

+(E) ⇒ (

µ)

+(A) = (

µ)+(B), for all A,B in ϑ

ii) (µ)

-(AB

-1) = (

µ)

-(E) ⇒ (

µ)

-(A) = (

µ)

-(B), for all A,B in ϑ

Proof:

i) Let µ =((

µ)+,(

µ)

-)be a bipolar L – fuzzy sub ℓ - HXgroup of a ℓ - HX group ϑ with

identity E and (µ)

+(AB

-1)= (

µ)

+(E) Then, for all A,B in ϑ.

(µ)

+(A) = (

µ)

+(A(B

-1B))

=(µ)

+((A(B

-1)B)

≥ (µ)

+(AB

-1) ˄ (

µ)

+(B)

=(µ)

+(E) ˄ (

µ)

+(B)

=(µ)

+(B)

That is, (µ)

+(A) ≥ (

µ)

+(B)

That is, (µ)

+(AB

-1)=(

µ)

+(E) ⇒(

µ)

+(A) ≥ (

µ)

+(B)

Since, (µ)

+(BA

-1) = (

µ)

+((BA

-1)

-1)

= (µ)+(AB

-1)

= (µ)+(E)

(µ)

+(BA

-1) = (

µ)

+(E)

That is, (µ)

+(BA

-1) = (

µ)

+(E) ⇒(

µ)

+(B) ≥ (

µ)+(A)

Hence, (µ)+(A) = (

µ)

+(B)

ii)Let µ =((

µ)

+,(

µ)

-)be a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ with

identity E and (µ)

-(AB

-1)= (

µ)

-(E) Then, for all A,B in ϑ

(µ)

-(A) = (

µ)

-(A(B

-1B))


146

= (µ)

-((A(B

-1)B)

≤ (µ)

-(AB

-1) ˅(

µ)

-(B)

= (µ)

-(E) ˅(

µ)

-(B)

= (µ)

-(B)

That is, (µ)

-(A) ≤ (

µ)

-(B)

That is, (µ)

-(AB

-1) = (

µ)

-(E) ⇒(

µ)

-(A) ≤ (

µ)

-(B)

Since, (µ)

-(BA

-1) = (

µ)

-((BA

-1)

-1)

= (µ)

-(AB

-1)

= (µ)

-(E)

(µ)

-(BA

-1) = (

µ)

-(E)

That is, (µ)

-(BA

-1) = (

µ)

-(E) ⇒(

µ)

-(B) ≤(

µ)

-(A)

Hence, (µ)

-(A) = (

µ)

-(B)

4.Properties of Bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group under ℓ - HX group

homomorphism and ℓ - HX group anti homomorphism.

In this section we discuss some of the properties of bipolar L-fuzzy sub ℓ - HX group of a

ℓ - HX group under ℓ - HX group homomorphism and ℓ - HX group anti homomorphism.

Theorem 4.1

Let ϑ and ϑ' be any two ℓ - HX group on G and G

'. Let

µ =((

µ)

+,(

µ)

-) be a bipolar L-fuzzy sub

ℓ - HX group of a ℓ - HX group ϑ. Let f:ϑ → ϑ' be an onto ℓ - HX group homomorphism. Then

f(µ)=((f(

µ))

+ ,( f(

µ))

-) is a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ

' if

µ =((

µ)

+,(

µ)

-) has supremum property and

µ =((

µ)

+,(

µ)

-) is f-invariant.

Proof:

Let µ=((

µ)+,(

µ)

-) be a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ. For all A,B in ϑ.

i) (f(µ))

+((f(A)f(B))= (f(

µ))

+(f(AB))

=(µ)

+(AB)

≥(µ)

+(A) ˄(

µ)

+(B)

≥ (f(µ))

+(f(A)) ˄ (f(

µ))

+(f(B))


147

(f(µ))

+((f(A)f(B)) ≥(f(

µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

ii) (f(µ))

-((f(A)f(B)) = (f(

µ))

-(f(AB))

= (µ)

-(AB)

≤(µ)

-(A) ˅(

µ)

-(B)

≤(f(µ))

-(f(A)) ˅(f(

µ))

-(f(B))

(f(µ))

-((f(A)f(B)) ≤(f(

µ))

-(f(A)) ˅(f(

µ))

-(f(B))

iii) (f(µ))

+(f(A)

-1) =(f(

µ))

+(f(A

-1))

= (µ)

+(A

-1)

= (µ)

+(A)

= (f(µ))

+(f(A)

(f(µ))

+(f(A)

-1) =(f(

µ))

+(f(A))

iv) (f(µ))

-(f(A)

-1) =(f(

µ))

-(f(A

-1))

= (µ)

-(A

-1)

=(µ)

-(A)

= (f(µ))

-(f(A)

(f(µ))

-(f(A)

-1)=(f(

µ))

-(f(A))

v) (f(µ))

+((f(A)˅f(B)) =(f(

µ))

+(f(A˅B))

=(µ)+( A˅B)

≥ (µ)

+(A) ˄ (

µ)

+(B)

≥(f(µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

(f(µ))

+((f(A)˅f(B)) ≥(f(

µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

vi) (f(µ))

-((f(A)˅f(B)) =(f(

µ))

-(f(A˅B))

=(µ)

-( A˅B)

≤(µ)

-(A) ˅ (

µ)

-(B)

≤(f(µ))

-(f(A)) ˅(f(

µ))

-(f(B))

(f(µ))

-((f(A)˅f(B)) ≤(f(

µ))

-(f(A)) ˅(f(

µ))

-(f(B))

vii)(f(µ))

+((f(A)˄f(B)) = (f(

µ))

+(f(A˄B))

=(µ)

+( A˄B)

≥ (µ)

+(A) ˄ (

µ)

+(B)


148

≥(f(µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

(f(µ))

+((f(A)˄f(B)) ≥ (f(

µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

viii)(f(µ))

-((f(A)˄f(B)) = (f(

µ))

-(f(A˄B))

=(µ)

-( A˄B)

≤ (µ)

-(A) ˅ (

µ)

-(B)

≤(f(µ))

-(f(A)) ˅ (f(

µ))

-(f(B))

(f(µ))

-((f(A)˄f(B))≤(f(

µ))

-(f(A)) ˅ (f(

µ))

-(f(B))

Hence f(µ)=((f(

µ))

+ ,( f(

µ))

-) is a bipolar L-fuzzy sub ℓ - HX group of a ℓ

- HX group ϑ'

Theorem 4.2


'. Let

=((

)

+, (

)

-) be a bipolar

L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ'. Let f: ϑ → ϑ

' be a ℓ - HX group

homomorphism. Then f-1

() = ((f

-1(

))

+ , (f

-1(

)

-) is a bipolar L-fuzzy sub ℓ - HX group of

aℓ - HX group ϑ.

Proof:

Let =((

)

+, (

)

-) be a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ

'.

i) (f-1

())

+(AB) =(

)

+(f(AB))

=()

+(f(A) f(B))

≥()

+(f(A))˄ (

)

+(f(B))

≥(f-1

())

+(A) ˄ (f

-1(

))

+(B)

(f-1

())

+(AB)≥(f

-1(

))

+(A) ˄ (f

-1(

))

+(B)

ii) (f-1

())

-(AB)=(

)

-(f(AB))

=()

-(f(A) f(B))

≤()

-(f(A)) ˅(

)

+(f(B))

≤(f-1

())

-(A) ˅ (f

-1(

))

-(B)

(f-1

())

-(AB) ≤(f

-1(

))

-(A) ˅ (f

-1(

))

-(B)

iii) (f-1

())

+(A

-1) = (

)

+(f(A

-1))

= ()

+(f(A)

-1)

=()

+(f(A))


149

=(f-1

())

+(A)

(f-1

())

+(A

-1) =(f

-1(

))

+(A)

iv) (f-1

())

-(A

-1) = (

)

-(f(A

-1))

=()

-(f(A)

-1)

= ()

-(f(A))

= (f-1

())

-(A)

(f-1

())

-(A

-1) = (f

-1(

))

-(A)

v) (f-1

())

+(A˅B)=(

)

+(f(A˅B))

= ()

+(f(A) ˅ f(B))

≥()

+(f(A))˄ (

)

+(f(B))

≥(f-1

())

+(A) ˄ (f

-1(

))

+(B)

(f-1

())

+(A˅B) ≥(f

-1(

))

+(A) ˄ (f

-1(

))

+(B)

vi) (f-1

())

-(A˅B) =(

)

-(f(A˅B))

= ()

-(f(A) ˅ f(B))

≤ ()

-(f(A))˅ (

)

-(f(B))

≤ (f-1

())

-(A) ˅(f

-1(

))

-(B)

(f-1

())

-(A˅B) ≤ (f

-1(

))

-(A)˅ (f

-1(

))

-(B)

vii) (f-1

())

+(A˄B) = (

)

+(f(A˄B))

= ()

+(f(A) ˄ f(B))

≥()

+(f(A))˄ (

)

+(f(B))

≥(f-1

())

+(A) ˄ (f

-1(

))

+(B)

( f-1

())+(A˄B) ≥(f

-1())

+(A) ˄ (f

-1())

+(B)

viii) (f-1

())

-(A˄B) = (

)

-(f(A˄B))

= ()

-(f(A) ˄ f(B))

≤ ()

-(f(A)) ˅(

)

-(f(B))

≤(f-1

())

-(A) ˅ (f

-1(

))

-(B)

(f-1

())

-(A˄B) ≤(f

-1(

))

-(A) ˅ (f

-1(

))

-(B)

Hence f-1

() = ((f-1

())+ , (f

-1()

-) is a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ.


150

Theorem 4.3


'. Let

µ =((

µ)

+,(

µ)


ℓ - HX group of aℓ - HX group ϑ. Let f: ϑ → ϑ' be an onto ℓ - HX group anti homomorphism.

Then f(µ)=((f(

µ))

+ ,( f(

µ))

-) is a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ

' if

µ

=((µ)

+,(

µ)

-) has supremum property and

µ =((

µ)+,(

µ)

-) is f-invariant.

Proof:

Letµ =((

µ)+,(

µ)

-) be a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ for all A,B in ϑ.

i) f(µ)

+((f(A) f(B)) = (f(

µ))

+(f(BA))

= (µ)

+(BA)

≥(µ)

+(B)˄ (

µ)

+(A)

≥(µ)

+(A)˄ (

µ)+(B)

≥(f(µ))

+( f(A))˄ (f(

µ))

+ (f(B))

ii) f(µ)

-((f(A) f(B))= (f(

µ))

-(f(BA))

= (µ)

-(BA)

≤(µ)

-(B)˅ (

µ)

-(A)

≤(µ)

-(A) ˅(

µ)

-(B)

≤ (f(µ))

-( f(A)) ˅ (f(

µ))

- (f(B))

iii) (f(µ))

+(f(A)

-1)=(f(

µ))

+(f(A

-1))

= (µ)+(A

-1)

=(µ)

+(A)

= (f(µ))

+(f(A)

(f(µ))

+(f(A)

-1)=(f(

µ))

+(f(A))

iv) (f(µ))

-(f(A)

-1) =(f(

µ))

-(f(A

-1))

= (µ)

-(A

-1)

= (µ)

-(A)

=(f(µ))

-(f(A)

(f(µ))

-(f(A)

-1) =(f(

µ))

-(f(A))

v) (f(µ))

+((f(A)˅f(B) = (f(

µ))

+(f(A˅B))

=(µ)

+( A˅B)


151

≥ (µ)

+ (A) ˄ (

µ)

+(B)

≥ (f(µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

(f(µ))

+((f(A)˅f(B)) ≥ (f(

µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

vi) (f(µ))

-((f(A)˅f(B)) = (f(

µ))

-(f(A˅B))

=(µ)

-( A˅B)

≤(µ)

- (A) ˅ (

µ)

-(B)

≤ (f(µ))

-(f(A)) ˅ (f(

µ))

-(f(B))

(f(µ))

-((f(A)˅f(B)) ≤ (f(

µ))

-(f(A)) ˅ (f(

µ))

-(f(B))

vii) (f(µ))

+((f(A)˄f(B)) = (f(

µ))

+(f(A˄B))

=(µ)

+( A˄B)

≥ (µ)

+ (A) ˄ (

µ)+(B)

≥ (f(µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

(f(µ))

+((f(A)˄f(B)) ≥ (f(

µ))

+(f(A)) ˄ (f(

µ))

+(f(B))

viii) (f(µ))

-((f(A)˄f(B)) = (f(

µ))

-(f(A˄B))

=(µ)

-( A˄B)

≤ (µ)

- (A) ˅ (

µ)

-(B)

≤(f(µ))

-(f(A)) ˅ (f(

µ))

-(f(B))

(f(µ))

-((f(A)˄f(B)) ≤(f(

µ))

-(f(A)) ˅ (f(

µ))

-(f(B))

Hence f(µ)=((f(

µ))

+ ,( f(

µ))

-) is a bipolar L-fuzzy sub ℓ - HX group of an ℓ - HX group ϑ

'.

Theorem 4.4


'. Let

=((

)

+, (

)


ℓ - HX group of aℓ - HX group ϑ'. Let f: ϑ → ϑ

' be a ℓ - HX group anti homomorphism. Thenf

-

1(

) = ((f

-1(

))

+ , (f

-1(

)

-) is a bipolar L-fuzzy sub ℓ - HX group of aℓ - HX group ϑ.

Proof:

Let =((

)

+, (

)

-)be a bipolar L-fuzzy sub ℓ - HX group of aℓ - HX group ϑ

'.

i) (f-1

())

+(AB)= (

)

+(f(AB))

= ()

+(f(B) f(A))

≥ ()

+(f(B))˄(

)

+(f(A))

≥ ()

+(f(A))˄(

)

+(f(B))


152

≥( f-1

())

+(A) ˄ ( f

-1(

))

+(B)

(f-1

())

+(AB) ≥( f

-1(

))

+(A) ˄ ( f

-1(

))

+(B)

ii) (f-1

())

-(AB) = (

)

-(f(AB))

= ()

-(f(B) f(A))

≤()

-(f(B)) ˅(

)

-(f(A))

≤()

-(f(A)) ˅ (

)

-(f(B))

≤( f-1

())

-(A) ˅ ( f

-1(

))

-(B)

(f-1

())

-(AB) ≤( f

-1(

))

-(A) ˅ ( f

-1(

))

-(B)

iii) (f-1

())

+(A

-1) = (

)

+(f(A

-1))

= ()

+(f(A)

-1)

= ()

+(f(A))

= (f-1

())

+(A)

(f-1

())

+(A

-1) = (f

-1(

))

+(A)

iv) (f-1

())

-(A

-1) = (

)

-(f(A

-1))

= ()

-(f(A)

-1)

= ()

-(f(A))

= (f-1

())

-(A)

(f-1

())

-(A

-1) = (f

-1(

))

-(A)

v) (f-1

())

+(A˅B)= (

)

+(f(A˅B))

= ()

+(f(A) ˅ f(B))

≥()

+(f(A))˄ (

)

+(f(B))

≥(f-1

())

+(A) ˄ (f

-1(

))

+(B)

(f-1

())+(A˅B) ≥(f

-1())

+(A) ˄ (f

-1())

+(B)

vi) (f-1

())

-(A˅B) = (

)

-(f(A˅B))

= ()

-(f(A) ˅ f(B))

≤()

-(f(A))˅ (

)

-(f(B))

≤(f-1

())

-(A) ˅(f

-1(

))

-(B)

(f-1

())

-(A˅B) ≤ (f

-1(

))

-(A)˅ (f

-1(

))

-(B)

vii) (f-1

())

+(A˄B) = (

)

+(f(A˄B))


153

= ()

+(f(A) ˄ f(B))

≥()

+(f(A))˄ (

)

+(f(B))

≥(f-1

())

+(A) ˄ (f

-1(

))

+(B)

(f-1

())+(A˄B) ≥(f

-1())

+(A) ˄ (f

-1())

+(B)

viii) (f-1

())

-(A˄B) = (

)

-(f(A˄B))

= ()

-(f(A) ˄ f(B))

≤ ()

-(f(A)) ˅(

)

-(f(B))

≤(f-1

())

-(A) ˅ (f

-1(

))

-(B)

(f-1

())-(A˄B) ≤(f

-1())

-(A) ˅ (f

-1())

-(B)

Hence f-1

() = ((f

-1(

))

+ , (f

-1(

)

-) is a bipolar L-fuzzy sub ℓ - HX group of a ℓ - HX group ϑ.

V.conclusion

In this paper we define a new algebraic structures of bipolar L-fuzzy ℓ - HX group and

discussed some of its related properties. Characterizations of bipolar L-fuzzy ℓ - HX group

of aℓ - HX group under ℓ - HX group homomorphism and ℓ - HX group anti homomorphism

are given.

REFERENCES

[1] Zadeh, L.A., 1965, “Fuzzy Sets”, Inform and Control, 8, pp. 338-365.

[2] Rosenfeld, A., 1971, “Fuzzy Groups”, J. Math. Anal. Appl. 35, pp. 512-517.

[3] Li Hongxing, 1987, “ HX group”, BUSEFAL, 33, pp. 31 – 37.

[4] LuoChengzhong , MiHonghai, Li Hongxing, 1989, “Fuzzy HX group”, BUSEFAL 41 –

14, pp. 97 – 106.

[5] Zhang, W.R., 1998, “Bipolar fuzzy sets”, Proc. of FUZZ-IEEE, pp: 835-840.

[6] Lee, K.M., 2000, “Bipolar-valued fuzzy sets and their operations”, Proc. Int. Conf. on

Intelligent Technologies, Bangkok, Thailand , pp. 307– 312.

[7] Satyasaibaba, G.S.V., 2008, “Fuzzy lattice ordered groups”, South east Asian Bulletin of

Mathematics, 32, pp. 749-766.

[8] Goguen, J.A., 1967, “ L-Fuzzy sets”, J. Math Anal. Appl., 18, pp. 145-174.

[9] Muthuraj, R., Sridharan, M., 2013, “Homomorphism and anti homomorphism of bipolar


154

fuzzy sub HX groups”, General Mathematical Notes, Volume 17, No. 2, pp. 53-65.

[10] Muthuraj, R., Rakesh kumar, T., 2016, “Some Characterization of L-Fuzzy ℓ- HX

group”, International Journal of Engineering Associates , No. 38 , Volume 5,Issue 7, pp.

38-41.


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