Gen. Math. Notes, Vol. 17, No. 2, August, 2013, pp. 53-65
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Homomorphism and Anti Homomorphism on
Bipolar Fuzzy Sub HX Groups
R. Muthuraj1 and M. Sridharan2
1
PG & Research Department of Mathematics
H.H. The Rajah’s College
Pudukkottai – 622 001, Tamilnadu, India
E-mail: rmr1973@yahoo.co.in
2
Department of Mathematics
PSNA College of Engineering and Technology
Dindigul – 624 622, Tamilnadu, India
E-mail: msridharan1972@gmail.com
(Received: 14-4-13 / Accepted: 29-5-13)
Abstract
In this paper, we introduce the concept of an image, pre image of a bipolar
fuzzy subsets and discuss in detail a series of homomorphic and anti
homomorphic properties of bipolar fuzzy sub groups.
Keywords: Fuzzy set, HX group, Fuzzy HX group, Bipolar-valued fuzzy set,
Bipolar fuzzy HX group, Homomorphism and Anti homomorphism of fuzzy HX
group, Image and pre-image of bipolar fuzzy sets are discussed.
1
Introduction
The concept of fuzzy sets was initiated by Zadeh [16]. Then it has become a
vigorous area of research in engineering, medical science, social science, graph
54
R. Muthuraj et al.
theory etc. Rosenfeld [12] gave the idea of fuzzy subgroup. 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 Luo Chengzhong, Mi Honghai, Li Hongxing [4] introduced the concept of
fuzzy HX group. The author W.R. Zhang [14], [15] commenced the concept of
bipolar fuzzy sets as a generalization of fuzzy sets in 1994. In case of Bipolarvalued 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 counterproperty. M. Marudai, V. Rajendran [5] introduced the pre image of bipolar Q
fuzzy subgroup. In this paper we define the image and pre image of a bipolar
fuzzy subgroup and bipolar fuzzy sub HX group and discuss some of its
properties.
2
Preliminaries
In this section, we site the fundamental definitions that will be used in the sequel.
Throughout this paper, G = (G, *) be a finite group, e is the identity element of G,
and xy we mean x * y.
Definition 2.1[9]: Let X be any non-empty set. A fuzzy subset µ of X is a function
µ : X → [0,1].
Definition 2.2[3]: In 2G − {φ}, a non-empty set ϑ ⊂ 2G −{φ} is called a HX group
on G, if ϑ is a group with respect to the algebraic operation defined by
AB = {ab / a ∈ A and b ∈ B}, which its unit element is denoted by E.
Definition 2.3[10]: Let µ be a fuzzy subset defined on G. Let ϑ ⊂ 2G −{φ} be a
HX group on G. A fuzzy set λµ defined on ϑ is said to be a fuzzy subgroup induced
by µ on ϑ or a fuzzy HX subgroup on ϑ if for any A, B ∈ϑ,
i.
ii.
λµ (AB) ≥ min { λµ (A), λµ (B)}
λµ (A−1) = λµ (A) .
where, λµ (A) = max { µ (x) / for all x ∈ A⊂ G }.
Homomorphism and Anti Homomorphism on…
55
Definition 2.4[10]: Let G be a non-empty set. A bipolar-valued fuzzy set or
bipolar fuzzy set µ in G is an object having the form µ = {⟨x, µ+ (x), µ− (x) ⟩ / for
all x∈G},where µ+ : G → [0,1] and µ− : G → [-1,0] are mappings. The positive
membership degree µ+ (x) denotes the satisfaction degree of an element x to the
property corresponding to a bipolar-valued fuzzy set µ ={⟨x, µ+ (x), µ− (x)⟩ / for
all x∈G} and the negative membership degree µ− (x) denotes the satisfaction
degree of an element x to some implicit counter property corresponding to a
bipolar-valued fuzzy set µ = {⟨x, µ+ (x), µ− (x) ⟩ / for all x∈G}. If µ+ (x) ≠ 0 and
µ− (x) = 0, it is the situation that x is regarded as having only positive satisfaction
for µ = {⟨x, µ+ (x), µ− (x)⟩ / for all x∈G}. If µ+ (x) = 0 and µ− (x) ≠ 0, it is the
situation that x does not satisfy the property of µ ={⟨x, µ+ (x), µ− (x)⟩ / for all
x∈G}, but somewhat satisfies the counter property of µ ={⟨x, µ+ (x), µ− (x)⟩ / for
all x∈G}. It is possible for an element x to be such that µ+ (x) ≠ 0 and µ− (x) ≠ 0
when the membership function of property overlaps that its counter property over
some portion of G. For the sake of simplicity, we shall use the symbol µ = (µ+,µ−
) for the bipolar-valued fuzzy set µ = {⟨x, µ+ (x), µ− (x)⟩ / for all x∈G}.
Definition 2.5[10]: A bipolar-valued fuzzy set or bipolar fuzzy set µ in G is a
bipolar fuzzy subgroup of G. if for all x, y ∈G,
i. µ+ (xy) ≥ min {µ+ (x), µ+ (y) ,
ii. µ− (xy) ≤ max{µ− (x), µ− (y)},
iii. µ+ (x−1) = µ+ (x) , µ− (x−1) = µ− (x) .
Definition 2.6[10]: Let ϑ be a non-empty set. A bipolar-valued fuzzy set or
bipolar fuzzy set [BFS] λµ in ϑ is an object having the form λµ = {⟨A, λµ+ (A), λµ−
(A)⟩ / for all x∈A⊂G}, where λµ+ : ϑ → [0,1] and λµ− : ϑ → [-1,0] are mappings.
The positive membership degree λµ+ (A) denotes the satisfaction degree of an
element A to the property corresponding to a bipolar-valued fuzzy set λµ = {⟨A,
λµ+ (A), λµ− (A)⟩ / for all x ∈ A ⊂ G } and the negative membership degree λµ−(A)
denotes the satisfaction degree of an element A to some implicit counter property
corresponding to a bipolar-valued fuzzy set λµ ={⟨A, λµ+ (A), λµ− (A)⟩ / for all x ∈
A ⊂ G}. If λµ+ (A) ≠ 0 and λµ− (A) = 0, it is the situation that A is regarded as
having only positive satisfaction for λµ = {⟨A, λµ+ (A), λµ− (A)⟩ / for all x ∈ A ⊂ G}.
If λµ+ (A) = 0 and λµ− (A) ≠ 0, it is the situation that A does not satisfy the property
of λµ = {⟨A, λµ+ (A), λµ− (A)⟩ / for all x ∈ A ⊂ G}, but somewhat satisfies the
counter property of λµ = {⟨A, λµ+ (A), λµ− (A)⟩ / for all x ∈ A ⊂ G}. It is possible
for an element A to be such that λµ+ (A) ≠ 0 and λµ− (A) ≠ 0 when the membership
function of property overlaps that its counter property over some portion of ϑ.
For the sake of simplicity, we shall use the symbol λµ = (λµ+, λµ− ). For the
bipolar-valued fuzzy set λµ = {⟨A, λµ+ (A), λµ− (A)⟩ / for all x∈A ⊂ G}.
Definition 2.7[10]: Let µ be a bipolar fuzzy subset defined on G. Let ϑ ⊂ 2G −{φ}
be a HX group on G. A bipolar fuzzy set λµ defined on ϑ is said to be a bipolar
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R. Muthuraj et al.
fuzzy subgroup induced by µ on ϑ or a bipolar fuzzy HX subgroup on ϑ , if for
any A , B ∈ ϑ,
i. λµ+ (AB) ≥ min{ λµ+ (A), λµ+ (B)},
ii. λµ− (AB) ≤ max{ λµ− (A), λµ− (B)},
iii. λµ+ (A−1) = λµ+ (A) , λµ− (A−1) = λµ− (A).
Where,
λµ+ (A) = max {µ+ (x) / for all x ∈ A ⊂ G}
λµ−(A) = min {µ− (x) / for all x ∈ A ⊂ G}.
and
Definition 2.8[13]: A mapping f from a group G1 to a group G2 is said to be a
homomorphism if f (xy) = f(x) f(y) for all x , y ∈ G1.
Definition 2.9[13]: A mapping f from a group G1 to a group G2 (G1 and G2 are
not necessarily commutative) is said to be an anti homomorphism if f (xy) = f(y)
f(x) for all x,y∈ G1.
Definition 2.10[9]: A mapping f from a HX group ϑ1 to a HX group ϑ2 is said to
be a homomorphism if f (AB) = f(A) f(B) for all A,B ∈ ϑ1.
Definition 2.11[9]: A mapping f from a HX group ϑ1 to a HX group ϑ2 (ϑ1 and ϑ2
are not necessarily commutative) is said to be an anti homomorphism if f (AB) =
f(B) f(A) for all A,B ∈ ϑ1.
3
Image and Pre-Image of a Bipolar Fuzzy Sub Group
of a Group under Homomorphism and Anti
Homomorphism
In this section, we introduce the notion of image and pre-image of the bipolar
fuzzy subgroup of a group, and discuss some of its properties. Throughout this
section, We mean that G1 and G2 are finite groups and e1 and e2 are the identity
elements of G1 and G2 respectively, and xy we mean x * y.
Definition 3.1: Let f be a mapping from a group G1 to a group G2 and let µ, φ be
fuzzy subsets in G1 and G2 respectively. Then the image f(µ) of µ is the fuzzy
subset of G2 defined by for u∈ G2
(f(µ))(u) = max { µ(x) : x∈ f –1(u)}, if f –1(u) ≠φ
0
, Otherwise
and the pre-image f –1(φ) of φ under f is the fuzzy subset of G1 defined by for
x∈ G1 , ( f –1(φ)) (x) = φ (f(x)) .
Homomorphism and Anti Homomorphism on…
57
Definition 3.2: Let f be a mapping a group G1 to a group G2 and let µ = ( µ+,µ– )
and φ = ( φ +, φ – ) are bipolar fuzzy subsets in G1 and G2 respectively. Then the
image f (µ) of µ is the bipolar fuzzy subset f(µ) = ( (f(µ))+,(f (µ))– ) of G2 defined
by for u∈ G2,
(f(µ))+ (u) = max { µ+ (x): x∈ f –1(u)}, if f –1(u) ≠φ
0
, Otherwise
and
(f(µ))– (u) =
max { µ– (x): x∈ f –1(u)}, if f –1(u) ≠φ
0
, Otherwise
and the pre-image f –1(φ) of φ under f is the bipolar fuzzy subset of G1 defined
by for x∈ G1 , ( f –1(φ)) + (x) = φ + (f(x)) , (f –1(φ)) – (x) = φ – (f(x)).
Theorem 3.1: Let f be a homomorphism from a group G1 into a group G2 . If µ =
( µ+,µ– ) be a bipolar fuzzy subgroup of G1 then f (µ) , the image of µ under f , is
a bipolar fuzzy subgroup of G2.
Proof: Let µ = ( µ+,µ– ) be a bipolar fuzzy subgroup of G1
µ− : G1 → [−1,0] are mappings.
,
µ+ : G1 → [0,1] and
Let u,v ∈ G2, since f is homomorphism and so there exist x,y ∈ G1 such that f(x)
= u and f(y) = v it follows that xy ∈ f –1(uv).
Now,
(f(µ))+ (uv) = max {µ+ (z): z = xy∈ f –1 (uv) }
≥ max {µ+ (xy): x∈ f –1 (u), y∈ f –1 (v) }
≥ max {min {µ+ (x), µ+ (y)}: x∈ f –1 (u), y∈ f –1 (v) }
= min {max{µ+ (x) : x∈ f –1 (u)}, max{µ+ (y) : y∈ f –1 (v)}
= min { (f(µ))+ (u) , (f(µ))+ (v) }
Therefore, (f(µ))+ (uv) ≥ min { (f(µ))+ (u) , (f(µ))+ (v) }
And
(f(µ))– (uv) = max {µ– (z): z = xy∈ f –1 (uv) }
≤ max {µ– (xy): x∈ f –1 (u), y∈ f –1 (v) }
≤ max {max {µ– (x), µ– (y)}: x∈ f –1 (u), y∈ f –1 (v) }
= max {max {µ– (x) : x∈ f –1 (u)}, max {µ– (y) : y∈ f –1 (v)}
= max { (f(µ))– (u) , (f(µ))– (v) }
Therefore, (f(µ))– (uv) ≤ max { (f(µ))– (u) , (f(µ))– (v) }
Now,
(f(µ))+ (u–1) = max {µ+ (x): x∈ f –1 (u–1) }
= max {µ+ (x–1): x–1∈ f –1 (u)}
= (f(µ))+ (u)
And
(f(µ))– (u–1) = max {µ– (x): x∈ f –1 (u–1) }
58
R. Muthuraj et al.
= max {µ– ( x–1): x–1∈ f –1 (u)}
= (f(µ))– (u)
Therefore f(µ) is a bipolar fuzzy subgroup of G2 .
Hence, if µ be a bipolar fuzzy subgroup of G1 then f(µ) is a bipolar fuzzy
subgroup of G2.
Theorem 3.2: The homomorphic pre-image of a bipolar fuzzy subgroup φ = ( φ+,
φ– ) of a group of G2 is a bipolar fuzzy subgroup of a group G1.
Proof: Let φ = ( φ+, φ– ) be a bipolar fuzzy subgroup of G2, φ + : G2 → [0,1]
and φ − : G2 → [-1,0] are mappings.
Now,
(f –1(φ))+ (xy)
= φ+ (f(xy))
= φ+ (f(x) f(y))
+
≥ min { φ (f(x)) , φ+ (f(y))}
= min {(f –1(φ))+ (x) , (f –1(φ))+ (y))}
Therefore, (f –1(φ))+ (xy) ≥ min {( f –1(φ))+ (x) , (f –1(φ))+ (y))}
And
(f –1(φ))– (xy)
= φ– (f(xy))
= φ– (f(x)f(y))
≤ max { φ– (f(x)), φ– (f(y))}
= max {( f –1(φ))– (x), (f –1(φ))– (y))}
Therefore, (f –1(φ))– (xy) ≤ max {( f –1(φ))– (x), (f –1(φ))– (y))}
Now,
(f –1(φ))+ (x–1) =
=
=
=
φ+ (f(x–1))
φ+ (f(x)–1)
φ+ (f(x))
(f –1(φ))+ (x)
And
(f –1(φ))– (x–1) = φ– (f(x–1))
= φ– (f(x)–1)
= φ– (f(x))
–1
–
= (f (φ)) (x)
Therefore, f –1(φ) is a bipolar fuzzy subgroup of G1.
Hence, if φ be a bipolar fuzzy subgroup of G2 then f
subgroup of G1.
–1
(φ) is a bipolar fuzzy
Theorem 3.3: Let f be an anti homomorphism from a group G1 into a group G2. If
µ = ( µ+,µ– ) is a bipolar fuzzy subgroup of G1 then f(µ) ,the image of µ under f
, is a bipolar fuzzy subgroup of a group G2.
Homomorphism and Anti Homomorphism on…
59
Proof: Let µ = ( µ+,µ– ) be a bipolar fuzzy subgroup of G1 , µ+ : G1 → [0,1] and
µ− : G1→ [−1,0] are mappings.
Let u, v ∈ G2, since f is an anti homomorphism and so there exist x,y ∈ G1 such
that f(x) = u and f(y) = v it follows that xy ∈ f –1(vu).
–1
(f(µ))+ (uv) = max {µ+ (z): z = xy∈ f (vu) }
–1
–1
≥ max {µ+ (xy): x∈ f (u) , y∈ f (v) }
–1
–1
≥ max {min {µ+ (x), µ+ (y)}: x∈ f (u) , y∈ f (v) }
–1
–1
= min {max {µ+ (x) : x∈ f (u)},max {µ+ (y) : y∈ f (v)}}
= min { (f(µ))+ (u) , (f(µ))+ (v) }
+
Therefore, (f(µ)) (uv) ≥ min { (f(µ))+ (u) , (f(µ))+ (v) }
Now,
And
–
–
–1
= max {µ (z): z = xy∈ f (vu) }
–
–1
–1
≤ max {µ (xy): x∈ f (u) , y∈ f (v) }
–
–
–1
–1
≤ max {max {µ (x), µ (y)}: x∈ f (u) , y∈ f (v) }
–
–1
–
–1
= max {max {µ (x) : x∈ f (u)}, max {µ (y) : y∈ f (v)}}
–
–
= max { (f(µ)) (u) , (f(µ)) (v) }
–
–
–
Therefore, (f(µ)) (uv) ≤ max { (f(µ)) (u) , (f(µ)) (v) }
Now,
And
(f(µ)) (uv)
–1
(f(µ))+ (u )=
=
=
– –1
(f(µ)) (u ) =
=
=
–1
–1
max {µ+ (x): x∈ f (u ) }
–1
–1
–1
max {µ+ (x ): x ∈ f (u)}
(f(µ))+ (u)
–
–1 –1
max {µ (x): x∈ f (u ) }
–
–1
–1
–1
max {µ ( x ): x ∈ f (u)}
–
(f(µ)) (u)
Therefore, f(µ) is a bipolar fuzzy subgroup of G2.
Hence, if µ be a bipolar fuzzy subgroup of G1
subgroup of G2.
then f(µ) is a bipolar fuzzy
Theorem 3.4: The anti homomorphic pre-image of a bipolar fuzzy subgroup φ =
–
(φ+, φ ) of a group G2 is a bipolar fuzzy subgroup of a group G1.
–
Proof: Let φ = ( φ+, φ ) be a bipolar fuzzy subgroup of G2
and φ − : G2 → [−1,0] are mappings.
Now,
(f
–1
(φ))+ (xy) = φ+ (f(xy))
= φ+ (f(y) f(x))
≥ min { φ+ (f(y)) , φ+ (f(x))}
–1
–1
= min {(f (φ))+ (y) , (f (φ))+ (x))}
–1
–1
= min {(f (φ))+ (x) , (f (φ))+ (y))}
,
φ + : G2 → [0,1]
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R. Muthuraj et al.
Therefore, (f
–1
–1
(φ))+ (xy) ≥ min{(f
–
–1
–1
(φ))+ (x) , (f (φ))+ (y))}
–
(φ)) (xy) = φ (f(xy))
–
= φ (f(y)f(x))
–
–
≤ max { φ (f(y)), φ (f(x))}
–1
–
–1
–
= max {(f (φ)) (y), (f (φ)) (x))}
–1
–
–1
–
= max {(f (φ)) (x), (f (φ)) (y))}
–1
–
–1
–
–1
–
Therefore, (f (φ)) (xy) ≤ max {(f (φ)) (x), (f (φ)) (y))}
And
Now,
And
(f
–1
–1
–1
(φ))+ (x ) = φ+ (f(x ))
–1
= φ+ (f(x) )
= φ+ (f(x))
–1
= (f (φ))+ (x)
–1
– –1
–
–1
(f (φ)) (x ) = φ (f(x ))
–
–1
= φ (f(x) )
–
= φ (f(x))
–1
–
= (f (φ)) (x)
(f
–1
Therefore, f (φ) is a bipolar fuzzy subgroup of G1.
Hence, if φ be a bipolar fuzzy subgroup of G2 then f
subgroup of G1.
–1
(φ) is a bipolar fuzzy
4
Image and Pre-Image of a Bipolar Fuzzy HX Group
of a HX Group under Homomorphism and Anti
Homomorphism
In this section, we introduce the notion of image and pre-image of the bipolar
fuzzy sub HX group of a HX group, and discuss some of its properties.
Throughout this section, We mean that ϑ1 and ϑ2 are HX groups and E1 and E2
are the identity elements of ϑ1 and ϑ2 respectively, and XY we mean X *Y.
Definition 4.1: Let G1 and G2 be any two groups. Let ϑ1 ⊂ 2G1 −{φ} and ϑ2 ⊂
–
2G2 −{φ} are HX groups defined on G1 and G2 respectively. Let µ = ( µ+,µ )
–
and φ = (φ +, φ ) are bipolar fuzzy subsets in G1 and G2 respectively, let λµ = (
–
–
λµ+ , λµ ) , and λφ = (λφ+ , λφ ) are bipolar fuzzy subsets defined on ϑ1 and ϑ2
respectively induced by µ and φ. Let f : ϑ1→ ϑ2 be a mapping then the image f
–
(λµ) of λµ is the bipolar fuzzy subset (f (λµ) = ( (f (λµ))+ , (f (λµ)) ) of ϑ2 defined
by for U∈ ϑ2 ,
(f(λµ))+ (U) =
And
max { (λµ)+ (X) : X∈ f
0
–1
–1
(U) } , if f (U) ≠φ
, otherwise
Homomorphism and Anti Homomorphism on…
–
(f(λµ)) (U)) =
–
61
max { (λµ) (X) : X∈ f
0
–1
–1
(U) }, if f (U) ≠φ
, otherwise
–
also the pre-image f 1(λφ ) of λφ under f is bipolar fuzzy subset of ϑ1 defined by
–
–
–
–
(f 1(λφ))+ (X) = λφ+ (f(X)) , (f 1(λφ)) (X) = λφ (f(X)).
Theorem 4.1: Let f be a homomorphism from a HX group ϑ1 into a HX group ϑ2.
–
If λµ = ( λµ+ , λµ ) is a bipolar fuzzy sub HX group on ϑ1 then f (λµ), the image of
λµ under f, is a bipolar fuzzy sub HX group of ϑ2.
–
Proof: Let µ = ( µ+,µ ) be a bipolar fuzzy subset of G1 , µ+ : G1→ [0,1] and
µ− : G1 → [−1,0] are mapping, and λµ is a bipolar fuzzy sub HX group on ϑ1 .
Let U, V ∈ ϑ2, since f is homomorphism and so there exist X,Y ∈ ϑ1 such that
–1
f(X) = U and f(Y) = V it follows that XY ∈ f (UV).
Now,
–1
(f (λµ))+ (UV) = max { λµ+ (Z) : Z = XY∈ f (UV) }
–1
–1
≥ max { λµ+ (XY) : X∈ f (U), Y∈ f (V) }
–1
–1
≥ max {min { λµ+ (X) , λµ+ (Y)}: X∈ f (U), Y∈ f (V) }
–1
–1
= min{max {λµ+ (X) : X∈ f (U)},max{λµ+ (Y) :Y∈ f (V)}}
= min { (f (λµ))+ (U) , (f (λµ))+ (V) }
Therefore, (f (λµ))+ (UV) ≥ min{ (f (λµ))+ (U) , (f (λµ))+ (V) }
And,
–
–
–1
(f (λµ)) (UV) = max { λµ (Z) : Z = XY∈ f (UV) }
–
–1
–1
≤ max { λµ (XY) : X∈ f (U), Y∈ f (V) }
–
–
–1
–1
≤ max {max{ λµ (X), λµ (Y)}: X∈ f (U), Y∈ f (V) }
–
–1
–
–1
= max {max{ λµ (X): X∈ f (U)},max{ λµ (Y): Y∈ f (V)}}
–
–
= max { (f (λµ)) (U) , (f (λµ)) (V) }
–
–
–
Therefore, (f (λµ)) (UV) ≤ max{ (f (λµ)) (U) , (f (λµ)) (V) }
–1
Now, (f (λµ))+ (U )
–
–1
–1
–1
= max{ λµ+ (X): X∈ f (U ) }
–1
–1
–1
= max{ λµ+ (X ): X ∈ f (U)}
= (f (λµ))+ (U)
–
–1
–1
And (f (λµ)) (U ) = max { λµ (X): X∈ f (U ) }
–
–1
–1
–1
= max { λµ ( X ): X ∈ f (U)}
–
= (f (λµ)) (U)
Therefore, f (λµ) is a bipolar fuzzy sub HX group of ϑ2.
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R. Muthuraj et al.
Hence, if λµ be a bipolar fuzzy sub HX group on ϑ1 then f (λµ) is a bipolar fuzzy
sub HX group of ϑ2.
Theorem 4.2: The homomorphic pre-image of a bipolar fuzzy sub HX group λφ =
–
(λφ+ , λφ ) of a HX group ϑ2 is a bipolar fuzzy sub HX group of a HX group
ϑ1.
–
Proof: Let φ = (φ+, φ ) be a bipolar fuzzy subset of G2 , φ + : G2 → [0,1] and
φ − : G2 → [−1,0] are mappings, and λφ be a bipolar fuzzy sub HX group on ϑ2 .
–1
(λφ))+ (XY) = λφ+ (f(XY))
= λφ+ (f(X) f(Y))
≥ min { λφ+ (f(X)) , λφ+ (f(Y))}
–1
–1
= min {(f (λφ))+ (X) , (f (λφ))+ (Y))}
–1
–1
–1
Therefore, (f (λφ))+ (XY) ≥ min{( f (λφ))+ (X) , (f (λφ))+ (Y))}
Now, (f
–1
–
–
(λφ)) (XY)= λφ (f(XY))
–
= λφ (f(X)f(Y))
–
–
≤ max { λφ (f(X)), λφ (f(Y))}
–1
–
–1
–
= max {( f (λφ)) (X), (f (λφ)) (Y))}
–1
–
–1
–
–1
–
Therefore, (f (λφ)) (XY) ≤ max{(f (λφ)) (X), (f (λφ)) (Y))}
And
(f
–1
–1
–1
(λφ))+ (X ) = λφ+ (f(X ))
–1
= λφ+ (f(X) )
+
= λφ (f(X))
–1
= (f (λφ))+ (X)
–1
–
–1
–
–1
And (f (λφ)) (X ) = λφ (f(X ))
–
–1
= λφ (f(X) )
–
= λφ (f(X))
–1
–
= (f (λφ)) (X)
Now, (f
–1
Therefore, f (λφ) is a bipolar fuzzy sub HX group of ϑ1.
Hence, if λφ be a bipolar fuzzy sub HX group on ϑ2 then f
fuzzy sub HX group of ϑ1.
–1
(λφ) is a bipolar
Theorem 4.3: Let f be an anti homomorphism from a HX group ϑ1 into a HX
–
group ϑ2 . If λµ = ( λµ+ , λµ ) is a bipolar fuzzy sub HX group of ϑ1 then f (λµ) ,
the image of λµ under f, is a bipolar fuzzy sub HX group of ϑ2.
–
Proof: Let µ = ( µ+,µ ) be a bipolar fuzzy subgroup of G1, µ+ : G1→ [0,1] and
µ− : G1 → [−1,0] are mappings, and λµ is a bipolar fuzzy sub HX group of ϑ1.
Homomorphism and Anti Homomorphism on…
63
Let U, V ∈ ϑ2, since f is an anti homomorphism and so there exist X,Y ∈ ϑ1 such
–1
that f(X) = U and f(Y) = V it follows that XY ∈ f (VU).
Now,
–1
(f (λµ))+ (UV) = max { λµ+ (Z): Z = XY∈ f (VU) }
–1
–1
≥ max { λµ+ (XY): X∈ f (U) , Y∈ f (V)}
–1
–1
≥ max {min { λµ+ (X), λµ+ (Y)}: X∈ f (U), Y∈ f (V)}
–1
–1
= min { max{ λµ+ (X): X∈ f (U)}, max{ λµ+ (Y):Y∈ f (V)}}
= min { (f (λµ))+ (U) , (f (λµ))+ (V) }
Therefore, (f (λµ))+ (UV) ≥ min{ (f (λµ))+ (U) , (f (λµ))+ (V) }
And
–
–
–1
(f (λµ)) (UV) = max { λµ (Z): Z = XY∈ f (VU) }
–
–1
–1
≤ max { λµ (XY): X ∈ f (U) , Y∈ f (V) }
–
–
–1
–1
≤ max {max { λµ (X), λµ (Y)}: X∈ f (U), Y∈ f (V) }
–
–1
–
–1
= max {max{λµ (X): X∈ f (U)}, max{λµ (Y):Y∈ f (V)}}
–
–
= max { (f (λµ)) (U) , (f (λµ)) (V) }
–
–
–
Therefore, (f (λµ)) (UV) ≤ max { (f (λµ)) (U) , (f (λµ)) (V) }
–1
–1
–1
–1
–1
Now, (f (λµ))+ (U ) = max { λµ+ (X): X∈ f (U ) }
–1
–1
–1
= max { λµ+ (X ): X ∈ f (U)}
= (f (λµ))+ (U)
–
–1
–
And (f (λµ)) (U ) = max { λµ (X): X∈ f (U ) }
–
–1
–1
–1
= max { λµ ( X ): X ∈ f (U)}
–
= (f (λµ)) (U)
Therefore, f (λµ) is a bipolar fuzzy sub HX group of ϑ2.
Hence, if λµ be a bipolar fuzzy sub HX group on ϑ1 then f (λµ) is a bipolar fuzzy
sub HX group of ϑ2.
Theorem 4.4: The anti homomorphic pre-image of a bipolar fuzzy sub HX group
–
λφ = ( λφ+ , λφ ) of a HX group ϑ2 is a bipolar fuzzy sub HX group of a HX group
ϑ1.
–
Proof: Let φ = ( φ+, φ ) be a bipolar fuzzy subgroup of G2 , φ + : G2 → [0,1]
and φ − : G2 → [−1,0] are mappings and λφ be a bipolar fuzzy sub HX group on
ϑ2 .
Now, (f
–1
(λφ))+ (XY) = λφ+ (f(XY))
= λφ+ (f(Y) f(X))
≥ min { λφ+ (f(Y)) , λφ+ (f(X))}
–1
–1
= min {(f (λφ))+ (Y) , (f (λφ))+ (X))}
64
R. Muthuraj et al.
–1
–1
= min {(f (λφ))+ (X) , (f (λφ))+ (Y))}
–1
–1
–1
Therefore, (f (λφ))+ (XY) ≥ min {(f (λφ))+ (X) , (f (λφ))+ (Y))}
(f
–1
Now, (f
–1
And (f
–1
–
–
(λφ)) (XY) = λφ (f(XY))
–
= λφ (f(Y)f(X))
–
–
≤ max { λφ (f(Y)), λφ (f(X))}
–1
–
–1
–
= max {(f (λφ)) (Y), (f (λφ)) (X))}
–1
–
–1
–
= max {(f (λφ)) (X), (f (λφ)) (Y))}
–1
–
–1
–
–1
–
Therefore, (f (λφ)) (XY) ≤ max {(f (λφ)) (X), (f (λφ)) (Y))}
And
–1
–1
(λφ))+ (X ) = λφ+ (f(X ))
–1
= λφ+ (f(X) )
= λφ+ (f(X))
–1
= (f (λφ))+ (X)
–
–1
–
–1
(λφ)) (X ) = λφ (f(X ))
–
–1
= λφ (f(X) )
–
= λφ (f(X))
–1
–
= (f (λφ)) (X)
–1
Therefore, f (λφ) is a bipolar fuzzy sub HX group of ϑ1.
Hence, if λφ be a bipolar fuzzy sub HX group on ϑ2 then f
fuzzy sub HX group of ϑ1.
–1
(λφ) is a bipolar
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