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 , 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

International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1057

ISSN 2229-5518

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 ].

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

ISSN 2229-5518

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  ]

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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ISSN 2229-5518

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

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

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

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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) ]