International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1056

ISSN 2229-5518

Images and Inverse images in the Category of

Fuzzy groups

P.Vijayalakshmi, Dr. P. Alphonse Rajendran

Abstract— Ever since fuzzy sets were introduced by Lotfi Zadeh in the year 1965 [ 1 ], many algebraic structures were introduced by many authors . One such structure is fuzzy groups introduced in [ 2 ] and [ 3 ]. In [ 4 ] the authors introduced a novel definition of fuzzy group homomorphism between any two fuzzy groups and gave element wise characterization of some special morphisms in the category of fuzzy groups

Index Terms— Epimorphism, Epimorphic images, Fuzzy sub group, Fuzzy group homomorphism, Images, Inverse

Images, , Monomorphism, Strong Monomorphism

1 INTRODUCTION

—————————— ——————————
In this article we prove that the category of fuzzy groups has epimorphic images and inverse images. We begin with the following definitions.
In [3], Azriel Rosenfeld has defined a fuzzy subgroup 𝜇 on a

2 DEFINITIONS

Definition 2.1

A fuzzy morphism
fig. 1

( f ,α ) :( X , µ ) → (Y ,η )

is called a
group S where µ : S→[ 0 ,1] is a function as one which satis-

−1

monomorphism in F if for all pairs of fuzzymorphisms
fies

µ ( xy

) ≥ min{ µ ( x ) , µ ( y)} for all

x , y fS.

( g , β ) and

Equivalently by proposition 5.6 in [ 3 ]

(h, δ ) :(Z ,θ ) → ( X , µ ), ( f ,α ) ( g , β ) = ( f ,α ) (h, δ ) im-

(i)

µ ( xy )

≥ min{ µ ( x ) , µ ( y)}

plies that

( g , β ) = (h, δ ) {( i.e). ( f ,α )

is left cancellable in

(ii) 𝜇(𝑥 −1) = 𝜇(𝑥)

We take this as the definition of a fuzzy group. However in
our notation and terminology for fuzzy sets a fuzzy group in this article will be a pair

F}.

A monomorphism is called a strong monomorphism if

(𝑓, 𝛼 ) is injective.

( X , µ )

= {( x , µ ( x) )

/ xf X , µ : X →[ 0,1]

is a func-

Definition 2.2

tion } , where
(i) X is a group and

Let ( f ,α ) :( X , µ ) → (Y ,η ) be a given fuzzy group homo-

morphism and (𝑢 , 𝛿 ): (𝐼, 𝜉 ) → (𝑌, 𝜂 ) be a fuzzy sub-
(ii)

µ ( xy −1 )

≥ min{ µ ( x ) , µ ( y)}

for all x, y X
group of (𝑌, 𝜂). Then (𝐼, 𝜉) is called an image of ( f ,α ) if

Let ( X , µ ) and

(Y ,η )

be fuzzy groups. Then a fuzzy group

(i) ( f ,α ) = (u , s ) ( f1 ,α1 ) for some fuzzy group homomor-

homomorphism from ( X , µ )

into (Y ,η )

is a pair

( f ,α )

phism (𝑓1 , 𝛼1 ): (𝑋, 𝜇) → (𝐼, 𝜉)

where

f : X Y

is a group homomorphism (in the crisp
sense) and

α : µ ( X ) → η (Y )

is a function such that

(ii) If (v, t ) : ( J , θ ) → (Y , η )

is any fuzzy subgroup of (Y ,η )

αµ =ηf . Equivalently ( f ,α ) : ( X , µ ) (Y ,η ) is a fuzzy

such that

( f , α ) = (v, t ) ( g, β )

for some fuzzy group homo-
group homomorphism (or fuzzy morphism) if

f : X Y is

morphism ( g , β ) :( X , µ ) → ( J ,θ ) then there exists a

a homomorphism (crisp ) of groups and the following dia- gram commutes.
fuzzy group homomorphism (ℎ , 𝛾 ): (𝐼, 𝜉) → (𝐽 , 𝜃 )

such that (u, s) = (v, t ) (h, γ ).

µ ( X ) α

η (Y )

µ η

f

X Y

————————————————

P.Vijayalakshmi, Research Scholar, Periyar Maniammai University, Peri- yar Nagar, Vallam, Thanjavur, Tamil Nadu, India, Mobile no:9790035790.

Dr. P. Alphonse Rajendran, Former Prof.(Dean SHSM),Department of

Mathematics, Periyar Maniammai University,Periyar Nagar, Vallam,

Thanjavur, Tamil Nadu, India. Mobile no:9790035789

fig. 2

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Definition 2.3 : Let (𝑓, 𝛼) : (𝑋 , 𝜇) → (𝑌 , 𝜂) be a given fuzzy group homomorphism

Proof. We prove the theorem via two lemmas.

Lemma 3.2: The category of fuzzy groups has images.

and (𝑢, 𝛿) : (𝑍 , 𝜃) → (𝑌 , 𝜂) be a fuzzy sub group. An
object (𝑃, 𝜖) in F is called the inverse image of (𝑍, 𝜃)

Proof. Let ( f ,α ) : ( X , µ ) → (Y ,η )

group homomorphism.

𝑓(𝑥)

be any given fuzzy

by ( f ,α ) if there exists morphisms (𝑝1, 𝛽1) : �𝑃, 𝜖 � →

Let 𝑓(𝑋) = {

�𝑥 ∈ 𝑋} .

(𝑍, 𝜃) and (𝑝2 , 𝛽2 ) : �𝑃, 𝜖 � → (𝑋 , 𝜇) such that

Define

f1 : X f ( X ) , as

f1 ( x) = f ( x)

for all x f X .

(i).

(u , δ ) ( p1 , β1 ):( f ,α )( p2 , β 2 ) and

𝜂1 : 𝑓(𝑋) → [ 0 , 1 ] as 𝜂1 𝑓(𝑥) = 𝜂𝑓(𝑥) and

(ii). if there exists morphisms (𝑞1, 𝛿1) : �𝑄, 𝜉 � → (𝑍 , 𝜃)

and (𝑞2, 𝛿2) : �𝑄, 𝜉 � → (𝑋 , 𝜇) such that

(u ,δ )(q1 ,δ1 ) = ( f ,α )(q2 ,δ 2 ) then there exists a unique

fuzzy group homomorphism(ℎ , 𝛾): ( 𝑄 , 𝜉 ) → (𝑝 , 𝜀 )

𝛼1

: 𝜇(𝑋) → 𝜂1

𝑓(𝑋) as α1 µ ( x) =η1 f1 ( x) =η f ( x)

such that ( p1 , β1 )(h , γ )

= (q1 , δ1 ) and

(𝑝2 , 𝛽2) (ℎ , 𝛾 ) = ( 𝑞2, 𝛿2) .

(𝑄, 𝜉)

(q1 , δ1 )

(h, γ )

( p, ε )

( p1 , β1 )

(Z , θ )

(q 2 , δ 2 )

Remark 2.4:

( p 2 , β 2 )

( f , α )

( X , µ )

fig. 3

(u, δ )

(Y , η )

fig. 4

Let 𝑖𝑓(𝑋) : 𝑓(𝑋) → 𝑌 and iη f ( X ) :η1 f ( X ) → η (Y )

respective inclusion maps. Then

( f ( X ) ,η1 ) is a fuzzy subgroup of (Y ,η ) . In fact

(i f ( X ) , iη f ( X ) ) ) is a strong monomorphism.

be the
(a) Since (𝐼, 𝜉 ) and

( J ,θ )

are fuzzy subgroups of

Claim.

(i f (X) , iη1 f (X) ) : ( f (X) , η1) → (Y , η )

is an image

(Y ,η ) , (u , s ) and (v , t ) are strong monomorphisms.

(b) From (ii) we have 𝑣ℎ = 𝑢 is injective and

tγ = s

of ( f ,α ) .

Now from the definitions, it follows that for all x f X ,

is also injective , both

h and

γ are injective. Thus

(h , γ )

i f ( X )

f1 ( x) = i f ( X )

f ( x) = f ( x)

is a strong monomorphism.[definition of strong monomor-
phism]

so that 𝑖𝑓(𝑋) ° 𝑓1 = 𝑓. Similarly

iη f ( X )

α1 = α

so that

(i f ( X ) , iη f ( X ) )

( f1 , α1 ) = ( f , α ).

(c) Since (v , t )

is a (strong ) monomorphism. (h , γ )

with
Thus condition (i) of definition 1.2 is satisfied.
the above property in (ii) is unique.
A category A is said to have images if every morphism

Suppose there exists a morphism ( g , β ) :( X , µ ) → ( J ,θ )

in that category has an image . Moreover, if in the factoriza-
and a strong monomorphism

(ϑ , t ):( J ,θ ) → (Y ,η )

such
tion

( f ,α ) = (u , s ) ( f1 ,α1 ) , the morphism

( f1 ,α1 ) is

that (ϑ , t ) ( g , β ) = ( f ,α )

(1)
always an epimorphism, then the category A is said to have
epimorphic images. We now prove
Define ℎ: 𝑓(𝑋) → 𝐽 by

h f ( x) = g ( x)

(2)
Then h
is well defined.

3 THEOREMS

For

f ( x1 ) =

f ( x2 )

Theorem 3.1: The category of fuzzy groups has epimorphic images.

ϑg ( x1 ) = ϑg ( x2 ) [ from (1)]

g ( x1 ) = g ( x2 )[ since ϑ

is injective ].

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International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1058

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Moreover Since

g : X J

is a homomorphism of groups,

Lemma 3.3: Let

( f ,α ) ( X , µ ) → (Y ,η )

be a fuzzy group

h is also a homomorphism of groups.

homomorphism and let
Also for all

x f X ,

ϑhf ( x) =ϑg ( x)

(by (2))

( X , µ ) (f1 , α1 ) → (I , ξ ) (u ,s ) → (Y , η )

= f ( x)

( by (1) )

be a factorization of ( f ,α ) through its image

⟹ 𝑣ℎ = 𝑖𝑓(𝑋) (3)

Again we define 𝛾: 𝜂1 �𝑓(𝑋)� → 𝜃(𝐽) as follows

(u, s) : (I , ξ ) → (Y , η ). Then ( f1

,α1

) is an epimorphism.

Given x f X, η1 f ( x) = ηf ( x)

proof. Let

( g1 , β1 ) , ( g 2 , β 2 ) :(I ,ξ ) → (C,τ )

be fuzzy

= ηvg ( x) (by (1) )

group homomorphism such that

= tθg ( x)

[ since ηϑ = t θ ]

�𝑡1 , 𝛽1 � �𝑓1 , 𝛼1� = �𝑡2 , 𝛽2 � (𝑓1 , 𝛼1) (7)

Hence for each x f X , there is a unique θg(x) f θ ( J )[

Let (h , γ ) : (E,θ ) → (I , ξ ) be the equalizer for

since t

is injective] such thatη1 f ( x) = tθg ( x)

(4)

( g1 , β1 ) and ( g 2 , β 2 ) [ This exists by [4] ]

So define

𝛾 ∶ 𝜂1 𝑓(𝑋) → 𝜃 (𝐽) by 𝛾𝜂1𝑓(𝑥) = 𝜃𝑡(𝑥) (5)

[γ is well defined since

ηf ( x1 ) =η f ( x2 )

ηϑg ( x1 ) = ηϑg ( x2 )

tθg ( x1 ) = tθg ( x2 )

θg ( x1 ) = θg ( x2 )]

Moreover for all x f X , γη1 f ( x) = γηf ( x) = θg ( x) [ defini-

tion of
and so

γ ]

γη1 =θh .

= θhf ( x) [since

g = hf ]

Thus

(h , γ ) :( f ( x) ,η1 ) → ( J ,θ )

is a fuzzy group homo-
morphism.

Then
fig. 6

( g1 , β1 ) (h , γ ) = ( g 2 , β 2 ) (h , γ )

(8)
Now from (1) and the definition of an equalizer there exists a
unique fuzzy group homomorphism

(𝑘 , 𝛿 ): (𝑋 , 𝜇 ) → (𝐸, 𝜃) such that

(h , γ ) (k , δ ) = ( f1 ,α1 )

(9)
Hence

(u , s) (h , γ ) (k , δ ) = ( u , s ) ( f1 ,α1 ) = ( f ,α ) (10)

Thus

( f ,α ) factors through

(E ,θ ) .

fig. 5

Finally for all 𝑥 ∈ 𝑋 , 𝑡𝛾�𝜂1 𝑓(𝑥)� = 𝑡𝜃𝑡(𝑥)

[ by (5)]

Therefore by the definition of an image , there exists a unique fuzzy group homomorphism ( p , ω ): (I ,ξ ) → (E ,θ ) such

that �𝑢 , 𝑠� �ℎ , 𝛾�(𝑝, 𝜔) = (𝑘 , 𝑠 )

= ηϑg ( x) [ sin ce

tθ = ηv]

= η f ( x) [ sin ce vg = f ]

= η1 f ( x)

which implies that
𝑡𝛾 = (𝑖𝑓(𝑥) , 𝑖𝑚𝑓(𝑥) ) (6)
from (3) and (6) we have
(𝑣, 𝑡)(ℎ, 𝛾) = (𝑖𝑓(𝑥) , 𝑖𝑚𝑓(𝑥) )
Thus condition (ii) of definition 1.2 is also satisfied.
Thus the category of fuzzy groups say F has images.
This implies that
fig. 7

(h , γ ) ( p , ω ) = identity on

(I , ξ )

(11)

Now from (2) ( g1 , β1 ) (h , γ ) = ( g 2 , β 2 ) ( h , γ )

⇒ ( g1 , β1 ) ( h , γ ) ( p , ω ) = ( g 2 , β 2 ) (h , γ )( p , ω )

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⇒ ( g1 , β1 )

= ( g 2 , β 2 )

by (5)
is a fuzzy subgroup δ is injective )

Hence ( f1 , α1 ) : ( X , µ ) → (I , ξ )

is an epimorphism.
Thus given
x f P, there is a unique 𝜃(𝑧) ∈ 𝜃(𝑍) ( z need
Proof of the Theorem. From 3.2 and Lemma 3.3, it follows that F has epimorphic images.

Remark 3.4: We can prove that any two images are isomor- phic fuzzy groups. Hence for practical purposes the image of

not be unique ) such that

𝛼𝜖(𝑥) = 𝛿𝜃(𝑧).

Define 𝛽1 : 𝜖(𝑃) → 𝜃(𝑍) as 𝛽1 𝜖(𝑥) = 𝜃(𝑧) if

𝛼𝜖(𝑥) = 𝛿𝜃(𝑧) (13)

(𝑓, 𝛼) : (𝑋 , 𝜇) → (𝑌 , 𝜂) will be taken as

where η1 is the restriction of η .

( f ( X ) ,η1 )

Claim 1. (𝑝1 , 𝛽1 ) : �𝑃, 𝜖 � → (𝑍 , 𝜃) is a fuzzy group

homomorphism.

Theorem 3.5: The category of fuzzy groups has inverse imag-

Now for all x f P,

𝜃𝑝1 (𝑥) = 𝜃(𝑧) where

es.

f ( x) = u( z) [ by (12)]

Proof.

Let

P = {

x f X

f (x) f u(Z)}

Then P is a subgroup of X. For if

x1 , x2 f P

and

x1 = u( z1 ) , x2 = u( z 2 )

−1

where z1 , z 2 fZ

−1

, then

−1

𝑥1𝑥2

= 𝑢(𝑧1)( 𝑢(𝑧2))

−1

= 𝑢(𝑧1 )𝑢(𝑧2 )

= 𝑢(𝑧1 𝑧2

) ∈ 𝑢(𝑍)

so that x1

, x2

−1 f P

and hence P is a subgroup of X.
fig. 8
Define 𝜖 : 𝑃 → [0 , 1] as 𝜖(𝑥) = 𝜇(𝑥) , for all

x f P ,

Also 𝛼𝜖(𝑥) = 𝛿𝜃(𝑧) so that 𝛽1𝜖(𝑥) = 𝜃(𝑧) by (2)

that is 𝜖 = 𝜇 𝑃 . Then (𝑃, 𝜖 ) is a fuzzy subgroup of
Hence 𝜃𝑝1 (𝑥) = 𝜃(𝑧) where f(x) = u(z) from (3).
Thus for all 𝑥 ∈ 𝑃, 𝛽 𝜖(𝑥) = 𝜃𝑃 (𝑥)
that

( X , µ ) .

Consider�𝑖𝑃 , 𝑖𝜖(𝑃) � : �𝑃 , 𝜖 � → (𝑋 , 𝜇 ). From the

1

𝛽1 𝜖 = 𝜃𝑃1.

Hence the claim 1.

1 so

definition of 𝜖 , we see that for all

x f P,

𝜇 𝑖𝑃 (𝑥) =

Claim 2. �𝑢 , 𝛿 � �𝑝1 , 𝛽1� = �𝑓, 𝛼 � (𝑖𝑃 , 𝑖𝜖(𝑃) )

Now for all 𝑥 ∈ 𝑃, 𝑢𝑝1 (𝑥) = 𝑢(𝑧), if f(x) = u(z) by (1)

𝜇(𝑥) = 𝜖(𝑥) = 𝑖𝜖(𝑃) 𝜖(𝑥) so that 𝜇 𝑖𝑃 = 𝑖𝜖(𝑃) 𝜖.

Hence �𝑖𝑃 , 𝑖𝜖(𝑃) � : �𝑃 , 𝜖 � → (𝑋 , 𝜇 ) is a fuzzy

group homomorphism.

Moreover since 𝑖𝜖(𝑃) ∶ 𝜖(𝑃) → 𝜇 (𝑋) is injective.

�𝑖𝑃 , 𝑖𝜖(𝑃) � : �𝑃 , 𝜖 � → (𝑋 , 𝜇 ) is a fuzzy subgroup.

Next we define a fuzzy group homomorphism

(𝑝1 , 𝛽1) : �𝑃, 𝜖 � → (𝑍 , 𝜃) as follows.

= f(x)

= 𝑓 𝑖𝑃 (𝑥)

⇒ 𝑢 𝑝1 = 𝑓 𝑖𝑃 (A)

Also for all 𝑥 ∈ 𝑃, 𝛿𝛽1 𝜖(𝑥) = 𝛿𝜃(𝑧)

where 𝛽1𝜖(𝑥) = 𝜃(𝑧)

if 𝛼𝜖(𝑥) = 𝛿𝜃(𝑧) by (2)

Hence 𝛿𝛽1 𝜖(𝑥) = 𝛿𝜃(𝑧)

= 𝛼𝜖(𝑥)

= 𝛼 𝑖𝜖(𝑃) 𝜖(𝑥) for all 𝑥 ∈ 𝑃.

Now x f P
tion of P ).

𝑓(𝑥) = 𝑢(𝑧) for some z f Z ( by defini-

Therefore 𝛿𝛽1 = 𝛼 𝑖𝜖(𝑃) (B)

From (A) and (B) we get claim (2)
Moreover

u( z1 ) = u ( z 2 )

z1 = z 2

since

u is injective.

Thus x f P

f ( x) = u( z) .

there is a unique z f Z
such that

Define 𝑝1 (𝑥) = 𝑧 if

f ( x) = u( z)

(12)

Again x f P

f ( x) = u( z), z f Z

ηf ( x) = ηu( z)

αµ ( x) = ηu( z)

[ since αµ = ηf ]

⇒ 𝛼𝜖(𝑥) = 𝜂𝑢(𝑧) [ since 𝜖 = 𝜇 𝑃]

⇒ 𝛼𝜖(𝑥) = 𝛿𝜃(𝑧) [ since 𝜂𝑢 = 𝛿𝜃]

Also 𝛿𝜃(𝑧1) = 𝛿𝜃(𝑧2 ) ⇒ 𝜃(𝑧1) = 𝜃(𝑧2) (since (𝑢, 𝛿)

fig. 9

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Suppose there exists (𝑞1, 𝛿1) : �𝑄, 𝜉 � → (𝑍 , 𝜃) and

(𝑞2, 𝛿2) : �𝑄, 𝜉 � → (𝑋 , 𝜇) such that

(u ,δ )(q1 ,δ1 ) = ( f ,α )(q2 ,δ 2 )

We define (ℎ, 𝛾) : (𝑄 , 𝜉) → (𝑃 , 𝜖) as follows

Let 𝑡 ∈ 𝑄. Then𝑞2(𝑡) ∈ 𝑋 and 𝑞1(𝑡) ∈ 𝑍 such

that 𝑓𝑞2(𝑡) = 𝑢𝑞1(𝑡) ∈ 𝑢(𝑧).

(14)

= 𝛼𝛿2(𝜖(𝑡)) = 𝛿𝛿1(𝜖(𝑡))

[ since 𝛿 is injective ]

We conclude that 𝛽1𝛿2�𝜖(𝑡)� = 𝛿1(𝜖(𝑡)) (19)

From (7) and (8) we conclude that 𝛽1𝛾(𝜖(𝑡)) = 𝛿1(𝜖(𝑡))

(Since δ is injective )

In other words 𝛽1𝛾 = 𝛿1 (20)

Thus the category of fuzzy groups has inverse images.

Note 3.6:

Hence

q2 (t ) f P (by definition of P ) and

As in any category we can prove that any two images / in-

𝑝1 𝑞2 (𝑡) = 𝑞1(𝑡) by definition of 𝑝1 .

verse images are isomorphic.
Define ℎ : 𝑄 → 𝑃 as

h(t ) = q2

(t )

(15)

Remark 3.7: Since any two inverse images can be proved to

be isomorphic fuzzy groups, from the construction in proposi-
Also define 𝛾 ∶ 𝜉(𝑄) → 𝜖(𝑃) by

𝛾𝜉(𝑡) = 𝛿2𝜉(𝑡) 𝑡 ∈ 𝑄

tion 3.1.23. it follows that the inverse image of a fuzzy sub- group ( iz , iµ '( z ) ):( Z ,η ' ) → (Y ,η ) by

( f ,α ):( X , µ

Y η

( f −1 (Z ) , µ ' )

Hence claim 2.

−1 ( )

) ( ,

{

) can be taken as

Claim 3 :

where 𝑓
𝑧 =
𝑥 ∈ 𝑋 𝑡𝑖𝑣𝑔𝑎 𝑓(𝑥) ∈ 𝑍} ( set theoretic inverse
γ is well defined. (that is we have to prove that

𝛿2𝜉(𝑡) belongs to P)

image ) and

µ ' is the inclusion.

REFERENCES


fig. 10

Now for all 𝑡 ∈ 𝑄, 𝛿2𝜉(𝑡) = 𝜇𝑞2(𝑡)

[ since 𝛿2𝜉 = 𝜇𝑞2 ] = 𝜖𝑞2 (𝑡)

[ since 𝑞2(𝑡) ∈ 𝑃 and 𝜖 = 𝑃 𝑎𝑎𝑎 𝑠𝑠 𝛿2𝜉(𝑡) belongs

[1] L.A. Zadeh, Fuzzy sets, Information and control, 1965, 8: 338 - 353.

[2] J.M. Anthony and Sherwood, Fuzzy Groups Redefined, Journal of

Mathematical Analysis and Applications 69, 124-130 (1979).

[3] Azriel Rosen Feld, Fuzzy Groups, Journal of Mathematical Analysis and Applications 35, 512-517 (1971).

[4] P.Vijayalakshmi, P. Geetha, A. Kalaivani, Category of Fuzzy Groups, Two day International conference on Algebra and its Applications (December 14 and 15 2011, Pp 337-343).

[5] George Boj Adziev And Maria Boj Adziev Fuzzy Sets, Fuzzy Logic, Applications, Advances in Fuzzy Systems-Applications

and Theory,Vol5, World Scientific Publishing Company, 1995.

[6] Horst Schubert, Categories, Springer-Verlag, Berlin Heidelberg

Newyork 1972.

to 𝜖(𝑃) [since 𝑞2(𝑡) ∈ 𝑃]

Thus γ is well defined.
Moreover for all 𝑡 ∈ 𝑄

𝛾𝜉(𝑡) = 𝛿2𝜉(𝑡)

= 𝜇𝑞2(𝑡) [ since (𝑞2 , 𝛿2) : �𝑄, 𝜉 � → (𝑋 , 𝜇) is a

fuzzy group homomorphism]

= 𝜖𝑞2 (𝑡) [ since

q (t ) f P and 𝜖 = 𝜇 𝑃 ]

= 𝜖ℎ(𝑡) [ by definition of h )
Hence 𝛾𝜉 = 𝜖ℎ so that (ℎ, 𝛾) : (𝑄 , 𝜉) → (𝑃 , 𝜖) is a
fuzzy group homomorphism. Finally by definition of h and

γ we have 𝑖𝑃 ℎ = 𝑞2 and 𝑖𝜖(𝑃) 𝛾 = 𝛿2 so that

( 𝑖𝑃 , 𝑖𝜖(𝑃) ) (ℎ , 𝛾 ) = ( 𝑞2 , 𝛿2) (16)

Claim 4 :

( p1 , β1 )(h , γ )

= (q1 , δ1 )

Now for all 𝑡 ∈ 𝑄, 𝑝1 ℎ(𝑡) = 𝑝1 𝑞2( ( by definition of h )

= 𝑞1(𝑡)

and hence 𝑝1 ℎ = 𝑞1 (17)

Also for all 𝑡 ∈ 𝑄, 𝛽1𝛾(𝜖(𝑡)) = 𝛽1 𝛿2(𝜖(𝑡)) (18)

and 𝛿𝛽1𝛿2(𝜖(𝑡)) = 𝛼 𝑖ϵ(P) 𝛿2(𝜖(𝑡))

[ since 𝛿𝛽1 = 𝛼 𝑖 𝜖(𝑃) ]

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