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

Images, , Monomorphism, Strong Monomorphism

—————————— ——————————

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

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*) )

/ *x*f *X *, µ : *X *→[ 0,1]

is a func-

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 *) ( *f*1 ,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 =η*f *. 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.

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 ) ( *p*1 , β1 ):( *f *,α )( *p*2 , 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 )(*q*1 ,δ1 ) = ( *f *,α )(*q*2 ,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 *f*1 ( *x*) =η *f *( *x*)

such that ( *p*1 , β1 )(*h *, γ )

= (*q*1 , δ1 ) and

(𝑝2 , 𝛽2) (ℎ , 𝛾 ) = ( 𝑞2, 𝛿2) .

(𝑄, 𝜉)

(*q*1 , δ1 )

(*h*, γ )

( *p*, ε )

( *p*1 , β1 )

(*Z *, θ )

(*q *2 , δ 2 )

( *p *2 , β 2 )

( *f *, α )

( *X *, µ )

fig. 3

(*u*, δ )

(*Y *, η )

fig. 4

Let 𝑖𝑓(𝑋) : 𝑓(𝑋) → 𝑌 and *i*η 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

(*i f *(*X*) , *i*η1 *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 ) )

( *f*1 , 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 *) ( *f*1 ,α1 ) , the morphism

( *f*1 ,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.

For

f ( x1 ) =

f ( x2 )

⇒ ϑ*g *( *x*1 ) = ϑ*g *( *x*2 ) [ from (1)]

⇒ *g *( *x*1 ) = *g *( *x*2 )[ since ϑ

is injective ].

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Moreover Since

g : X → J

is a homomorphism of groups,

( *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)

( *g*1 , β1 ) , ( *g *2 , β 2 ) :(*I *,ξ ) → (*C*,τ )

be fuzzy

= η*vg *( *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)

( *g*1 , 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 )

⇒ ηϑ*g *( *x*1 ) = ηϑ*g *( *x*2 )

⇒ tθg ( x1 ) = tθg ( x2 )

⇒ θ*g *( *x*1 ) = θ*g *( *x*2 )]

Moreover for all x f *X *, γη1 *f *( *x*) = γη*f *( *x*) = θ*g *( *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

( *g*1 , 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 ) = ( *f*1 ,α1 )

(9)

Hence

(*u *, *s*) (*h *, γ ) (*k *, δ ) = ( *u *, *s *) ( *f*1 ,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) ( *g*1 , β1 ) (*h *, γ ) = ( *g *2 , β 2 ) ( *h *, γ )

⇒ ( *g*1 , β1 ) ( *h *, γ ) ( *p *, ω ) = ( *g *2 , β 2 ) (*h *, γ )( *p *, ω )

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⇒ ( *g*1 , β1 )

= ( *g *2 , β 2 )

by (5)

is a fuzzy subgroup ⇒ δ is injective )

Hence ( *f*1 , 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.

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.

Now for all x f P,

𝜃𝑝1 (𝑥) = 𝜃(𝑧) where

es*.*

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

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

, *x*2

−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 )

⇒ *z*1 = *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

⇒ η*f *( *x*) = η*u*( *z*)

⇒ αµ ( *x*) = η*u*( *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 )(*q*1 ,δ1 ) = ( *f *,α )(*q*2 ,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.

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)

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 ( *i*z , *i*µ '( z ) ):( *Z *,η ' ) → (*Y *,η ) by

( *f *,α ):( *X *, µ

→ *Y *η

( *f *−1 (*Z *) , µ ' )

Hence claim 2.

−1 ( )

) ( ,

{

) can be taken as

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.

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 :

( *p*1 , β1 )(*h *, γ )

= (*q*1 , 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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