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) )
/ xf 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 ) ( 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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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.
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 )
( 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
(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.
For
f ( x1 ) =
f ( x2 )
⇒ ϑg ( x1 ) = ϑg ( x2 ) [ from (1)]
⇒ g ( x1 ) = g ( x2 )[ since ϑ
is injective ].
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ISSN 2229-5518
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)
( 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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International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1059
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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.
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
, 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.
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 ( 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
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 :
( 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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