r. muthuraj1 and g. santha meena2...some characterization of bipolar l-fuzzy ℓ - hx group r....
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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
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 ).
International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com
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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 }
International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com
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
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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)
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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)
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(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)
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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)
+(E) and by part (i)
(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)
+(E) and by part (i)
(µ)
+(A ˄ B) ≤ (
µ)
+(E)
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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)
-(E) and by part (ii)
(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)
-(E) and by part (ii)
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(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))
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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))
International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com
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(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)
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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 ϑ and ϑ' be any two ℓ - HX group on G and G
'. 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))
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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.
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Theorem 4.3
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 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)
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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 ϑ and ϑ' be any two ℓ - HX group on G and G
'. Let
=((
)
+, (
)
-) be a bipolar L-fuzzy sub
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))
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≥( 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))
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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.
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[3] Li Hongxing, 1987, “ HX group”, BUSEFAL, 33, pp. 31 – 37.
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[5] Zhang, W.R., 1998, “Bipolar fuzzy sets”, Proc. of FUZZ-IEEE, pp: 835-840.
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fuzzy sub HX groups”, General Mathematical Notes, Volume 17, No. 2, pp. 53-65.
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