10
IV
https://doi.org/10.22214/ijraset.2022.41471
April 2022
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com
Homomorphism of Characteristic Fuzzy Subgroup and Abelian Fuzzy Subgroup Amit Kumar Arya1, Dr. M. Z. Alam2 1
2
Research Scholar, P.G. Department of Mathematics, M . U. Bodh Gaya , Bihar Associted Proofeser , P.G. Department of Mathematics , College of Commerence , Arts & Science , Patna , Patliputra University , Patna – 20 , India
Abstract: In this paper, we have established some independent proof of homomorphism on algebra of abelian and characteristic fuzzy subgroup. The characteristic of fuzzy subgroup [13] was first introduced by P. Bhattacharya and N. P. Mukharjee in 1986. Keywords: Fuzzy subgroup, characteristic fuzzy subgroup, abelian fuzzy subgroup and normal fuzzy subgroup. I. INTRODUCTION The concept of fuzzy sets was introduced by L.A.Zadeh [15] in 1965.Study of algebraic structure was first introduced by A.Rosenfeld [1]. After that a series of researches have done in this direction P.Bhattacharya and N.P.Mukharjee[13] have defined fuzzy normal subgroup and characteristic fuzzy subgroup in 1986. In this paper we have tried to established some independent proof about the properties of fuzzy group homomorphism on algebra of characteristic fuzzy subgroup. II. PRELIMINARIES In this section, we recall and study some concepts associated with fuzzy sets and fuzzy group, which we need in the subsequent sections. A. Fuzzy Set Over the past three decades, a number of definitions of a fuzzy set and fuzzy group have appeared in the literature (cf., e.g., [15, 1, 3, 7, 10]). In [15], it has been shown that some of these are equivalent. We begin with the following basic concepts of fuzzy set, fuzzy point and fuzzy group. be a function ʄ :
Definition 2.1 [15] A fuzzy subset of of and designate by ( ).
the set of all fuzzy subset of
is sad to be fuzzy power set
Definition 2.2 [15] Support of fuzzy set. Suppose ( ) then the set { ( ) : } is said to be the image of designate by ( ). The set { : ∈ , ( ) > 0} is said to be the support of is designate by *. Definition 2.3 [15] Let designate by
,
(
) such that
( )<
( ), ∀
∈
then
is said to be contained in
is
and it is
Definition 2.4 [15] Let
[0 ,1] we defined
and
(
) as
, ∊ 0, ∊ } is called a fuzzy point. (a) =
If
is a singleton { } then
{
For any collection { , } of fuzzy subset of , where greatest lower bound (G.L.B) ⋂ ∊ of are given by (⋃
∊
) ( )=∨
∊
( ), ∀
is an index set the least upper bound (L.U.B.) ⋃
∊
and
.
(⋂ ∊ )( ) =∧ ∈ ( ), ∀ Fuzzy subgroup In this section, we discuss the concept of a fuzzy subgroup in details (c.f.,[1]).
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1641
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com Definition 2.5 Fuzzy subgroup (or ( )) Let be any group, we define the binary operation o’ and unary operation −1 on ( ) as follows, ∀A , C ∈ ( ) and ∀ ∈ (A o C ) ( ) = ∨{A (y ) ∧ C (z ) : y z = ,∀ y , z ∈ } A −1( ) = A ( −1) Proposition 2.1 [3] If A ∈ ( ), then for all ∈ (i) A ( ) ≥ A ( ) (ii) A ( ) = A ( −1)
A , then
Let
Proof (i)
(ii)
Finally,
=
A ( ) = A > A ≥A ∴ A ( ) > A A ( )= A > A > A A ( )= A
( ( ( ( ( ( ( (
) )∧A ( ) )∧A ( )=A ( ), ∀ ∈ G ) ) ) )
)
Anti fuzzy subgroup In this section we discuss the basic concepts of anti fuzzy subgroup of G ,[5] Definition 2.6 A fuzzy subset A of G is said to be anti fuzzy group of G , and is denoted as anti F (G ) if for all , c ∈ G (i) A ( . c ) ≤ max{A ( ), A(c )} (ii) A ( −1) = A ( ) Definition 2.7 Let G be any group we define the binary operation ’o’ and unary operation’−1’ on anti-fuzzy group of G as follows ,∀ A , B ∈ anti F (G ) and ∀ ∈ G (A B ) ( ) = ∧ { A (c ) ∨ B (p ) : p = , ∀ , p ∈ G } i. ii. A ( )=A ( ) ∀ ∈G Proposition 2.2 [5] Suppose A , B ∈ anti F ∀ P (G ) also A anti F P (G ) for each i ∈ I, the following holds (i) (A o B ) ( ) = ∧ ∈ {A ( ) ∨ B ( )} =∧
∈
( oA )( )=A ( (A o )( )=A (
(ii)
) ∨ B ( )}
{A ( ∀
) )
, ,
PROOF:- (i) We have , ∈ G ∈ G ( ) = ( )= e= Also ( )=( ) =e = Thus, {A ( ) ∨ B ( ) = ∧ ∈ {( A ( ) ∨ A (
(ii)
(
) ∨ B ( )}
=∧
∈
{ (A ( ) ∨ (∧ A (
=∧
∈
{ (A ( ) ∨ (A o B ) (
=∧
∈
{ A o (A o B ) (
= (A o B ) Similarly, we get ∧ ∈ {A ( )∨B (
∈G ∈G
,∀
) ∨ B ( )} )}
)
∈G
)} = (A o B ) ( ) ∀
∈G
oA )( )= ∧
∈
{A (
) ∨ A ( )}
= ∧
∈
{A (
) ∨ A ( ) ∨ A ( )}
= ∧
∈
{A (
) ∨ A ( )}
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1642
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com = A (
) ∀
,
∈G
Also, (A o
)(
)=∧
∈
{A ( ) ∨ A (
=∧
∈
{A (
=∧
∈
{A ( )∨A (
)}
)∨A ( )∨A (
)}
)}
=A ( ) , ∈G Fuzzy homomorphism In this section author have extend the properties of fuzzy homomorphism in abelian fuzzy subgroup and anti-abelian fuzzy subgroup III.
ABELIAN FUZZY SUBGROUP [6]
Definition 2.8 If A ∈ F (G ) and if A ( )=A ( ) for all , ∈ G then A is called an abelian fuzzy subgroup of G Proposition 3.1:- If : G → G be a homomorphism of group G into G . Let A ∈ F (G ) is abelian fuzzy sub group then expression that (A ) ∈ F (G ) is also an abelian fuzzy subgroup. PROOF:- Let m , n ∈ G then ( (A )) (m n ) = ∨{A (p ) : p ∈ G , (p ) = m n } > ∨{A ( ): , ∈G , ( )=m , ( )=n } = ∨{A ( ): , ∈G , ( )=m , ( ) =n } = ∨{A ( ) ∧ A ( ) : , ∈ G , ( ) = m , ( ) = n } = ∨{ A ( ) : ∈ G , ( ) = m } ∧ {∨{ A ( ) : ∈ G , ( ) = n } = (A ) (m ) ∧ (A ) (n ) = ( (A )) (m n ) ∀ m , n ∈ G Hence, (A ) ∈ F (G ) is an abelian fuzzy subgroup (ABFSG) of G . Proposition 3.2:- Let : G → G is a homomorphism of group G into a group G . If A ∈ F (G ) is an abelian fuzzy subgroup of G Then show that ( A ) ∈ F (G ) is also an abelian fuzzy subgroup of G . PROOF:- Let : G → G be homomorphism of group G into group G . Let A ∈ F (G ) be an abelian fuzzy subgroup of G . Then show ( A ) ∈ F (G ) is also an abelian fuzzy subgroup of G . Suppose , ∈ G we have ( ( A )) ( )=A ( ( )) = A ( ( ) ( )), since is a homomorphism = A ( ( ) ( )), since G is an abelian subgroup =A ( ( )) =( ( A )) ( )∀ , ∈ G . Hence, ( A ) ∈ F (G ) is an abelian fuzzy subgroup of G . Proposition 3.3:- If : G → G ʹ is a homomorphism of group G into G ʹ and g : G ʹ → G ʺ be a homomorphism of group G ʹ into group G ʺ. Let A ∈ F (G ) then show that the composition (g o ) (A ) ∈ F (G ʺ). PROOF:- Let , ∈ G ʺ. If possible, let (g o ) (G ) or (g o ) A ( ) ∧ (g o ) A ( ) = 0 < (g o ) A ( ).
(g o ) (G ) then
If we suppose (g o ) (G ) then (g o ) (G ) Implies that (g o ) (A ) = 0 = ( g o ) (A ) Again if we assume = (g o ) ( ) and = (g o ) ( ) for some , ∈ G . Also (g o ) (A ) ( ) = ∨{ A (p ) : p ∈ G , (g o ) p = } (g o ) (A ) ( ) ≥∨{A ( ) : , ∈ G , (g o ) = , (g o ) = } ≥ ∨ { A ( ) ∧ A ( ) : , ∈ G , (g o ) = , (g o ) =
}
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1643
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com = ∨{A ( ): = (g o ) A (
∈ G , (g o ) = ) ∧ (g o ) A ( )
} ∧ {∨ ((A ( )) :
∈ G , (g o
)
∈
}
Also, (g o ) A = ∨ { A ( ) : ∈ G, (g o ) = =∨{A ( ) : ∈ G, (g o ) = (g o ) A ( )
} =
}
Hence, (g o ) (A ) ∈ F (G ʺ) Proposition 3.4:- Suppose : G → G ʹ and g : G ʹ → G ʺ where and g are homomorphism of a group G into group G ʹ and from a group G ʹ into a group G ʺ respectively then the composition homomorphism (g o ) from G into G ʺ. Let A ∈ F (G ) is an abelian group then prove that (g o ) (A ) ∈ F (G ʺ) is also an abelian group. PROOF :-Let , ∈ G ʺ then we have by extension principle (g o ) (A ) ( , ) = ∨{ A ( ) : G , (g o ) = )} ≥ ∨{ A ( ) : , ∈ G , (g o ) = , (g o ) = } = ∨{ A ( ) : , ∈ G , (g o ) = , (g o ) = } Since A ∈ F (G ) is an abelian group (g o ) (A ) ( , ) = ∨{ A ( ) ∧ A ( ) : , ∈ G , (g o ) = , (g o ) = } = ∨ [{A ( ) ∈ G , (g o ) = }] ∧ [ ∨ A ∈ ( ) : ∈ G , (g o ) = ] = (g o ) (A ) ( ) ∧ (g o ) (A ) ( ) = (g o ) (A ) ( ) Hence, (g o ) A ∈ F (G ʺ) is an abelian fuzzy subgroup of G ʺ. Proposition on abelian anti fuzzy subgroup Proposition 3.5 If : G → G be a homomorphism of group G into group G . Let A ∈ anti F (G ) is abelian anti fuzzy subgroup of G , then show that A ∈ F (G ) is also abelian anti fuzzy subgroup of G . ∈G PROOF: Let 1, 1 (
A )) ( = = = <
) ∧ {A (p ) : p ∈ G , (p ) = } ∧ {A ( ): , ∈G , ( )= , ( )= } ∧ {A ( ): , ∈ G , ( )= , ( )= } ∈ ∧ {A1 (c1) ∨ A1 (d1) : d1, c1 G1, f1 (d1) = 1, , f1 (c1) =
1}
= ∧ {A ( ) : ∈ G , ( ) = ∨ (∧ ( ) : ∈ G , ( ) = }) = { (A ) ∨ (A )} ( ) = ( (A )) ( ) ∀ , ∈G Hence (A ) ∈ anti F (G ) is abelian anti-fuzzy subgroup of G Proposition 3.6:- Let : G G is a homomorphism of a group G into a group G . If A anti F (G ) is an abelian anti-fuzzy subgroup of G then show that (A ) anti F (G ) is also an abelian anti-fuzzy subgroup of G . PROOF :- Suppose : G G is a homomorphism of a group G into a group G . Let A anti F (G ) be abelian anti-fuzzy subgroup of G . Then show that (A ) anti F (G ) is also an abelian anti-fuzzy subgroup G . Let G We have ( (A )) ( )= A ( ( )) = A ( ( ) ( )) since is a homomorphism = A ( ( ) ( )) since G is an abelian subgroup = A ( ( ))
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1644
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com
Finally,
(
)
∈
= ( )( , ) anti F1 (G1) is an abelian anti-fuzzy subgroup.
Proposition 3.7: Suppose : G → G ʹ and g : G ʹ → G ʺ where and g are homomorphism of a group G into group G ʹ and from a group G ʹ into a group G ʺ respectively. Let A ∈ anti F (G ) is an abelian anti fuzzy subgroup of G then prove that the image of composition homo – morphism of fuzzy anti subgroup A of G ʺ is also an abelian anti fuzzy subgroup of G ʺ PROOF: - Let , ∈ G ʺ then we have by extension principle (g o ) (A ) ( , ) = ∧ { A ( ) : ∈ G , (g o ) = )} ≤∧{A ( ) : , ∈ G , (g o ) = , (g o ) = } = ∧{A ( ) : , ∈ G , (g o ) = (g o ) } ≤ ∧ { A ( ) ∨ A ( ) : , ∈ G , (g o ) = , (g o ) = } = ∧ [{A ( ) ∈ G , (g o ) = }] ∨ [ ∧ A ( ) : ∈ G , (g o ) ] = (g o ) (A ) ( ) ∨ (g o ) (A ) ( ) = (g o ) (A ) ( ) Finally, (g o ) A F (G ʺ) is an abelian anti fuzzy subgroup of G ʺ. IV. CHARACTERISTIC FUZZY SUBGROUP [13] DEFINITION: 4.1:- Let A be a fuzzy subgroup of G and ϕ be a function from G into itself. Now define the fuzzy subset A of G by A (d ) = A d , where d = ϕ(d ) A subgroup K of group G is called a characteristic subgroup if K = K for every automorphism ϕ of G , where K denote ϕ(k). Definition 4.2 Characteristic fuzzy subgroup: A fuzzy subgroup A on a group K is called a fuzzy characteristic subgroup of G if A ( ) = A ( ) for every automorphism ϕ of G and for all ∈G Proposition 4.1 :- Let A is a fuzzy subgroup of a group G if a. If ϕ is a homomorphism of G into itself, then A is a fuzzy subgroup of G b. If A is a fuzzy characteristic subgroup of G then A is a normal. PROOF : (i) , ∈ G then )= A ( A ( ) = A ( ) Subsequently ϕ is a homomorphism and A is a fuzzy subgroup of G . ) A ≥A ∧A ( A Also,
A
(
)= A
(
)∧A
( )
= A = A = A
=A ( ) Hence, A is a fuzzy subgroup of G . (ii) Let , ∈ G to prove that A is normal we have to show A ( )=A ( ) Let ϕ be function from G into itself definition by ϕ(z) = z , ∀ z∈G Since A is a fuzzy characteristic subgroup of G , ∴A =A ) Thus A ( )=A ( =A ( ) = A (ϕ ( ))
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1645
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com (
=A ( =A ( Hence A is normal subgroup of G .
)
)
)
V. MAIN RESULT Proposition 5.1 : Let A , C be the fuzzy subgroup of G if ( i) If ϕ is a homomorphism of G into itself, then A is a fuzzy subgroup of G (ii) If A is a fuzzy characteristic subgroup of G then A is a normal. PROOF : (i) , ∈ G then )= A ( A ( ) =A ( ) Subsequently ϕ is a homomorphism and A is a fuzzy subgroup of G . ) A ≥A ∧A ( (
A
)= A
A
Also,
(
)∧A
( )
= A = A = A
=A ( ) Hence, A is a fuzzy subgroup of G . Proposition 5.2 : Let A , C be the fuzzy subgroups of a group G . Then the following statement hold (i) If ϕ is a homomorphism of G into itself. Then A & C are G . Then show that (a) (A ∪ C ) and (b) (A ∩ C ) are fuzzy subgroup of G . (ii) If A , C are fuzzy characteristic subgroup of G , so A and C we have to show that A ∪ C and A ∩ C are also normal. Proof:(i) Let A , C ∈ F P (G ) and ϕ is a homomorphism of G into itself. Let ∈ G , we have ) = (A ∪ C ) (( ) ) (A ∪ C ) (
fuzzy are
subgroup normal
of then
= (A ∪ C ) = A
(A ∪ C ) ( (A ∪ C )
∨C
≥ A
∧A (
= A
∨C
) ∨ C ∧ A (
∧C (
)
)∨C (
)
= (A ∪ C ) ∧ (A ∪ C ) ) ≥ (A ∪ C ) ( ) ∧ (A ∪ C ) (
)
= (A ∪ C ) = (A ∪ C ) = A = A
∧C
= (A ∪ C ) = (A ∪ C ) ( ) (A Hence, ∪ C ) ∈ F (G ) Similarly, i (b) we have (A ∩ C ) ( ) = (A ∩ C ) (( = (A ∩ C ) = A
since A , C ∈ F (G )
∧C
) )
∧C
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1646
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com
i.e., (A ∩ C ) ( Also , (A ∩ C )
≥ A
∧A (
= A
∧C
) ∧ C ∧ A (
∧C (
)
)∧C (
)
= (A ∩ C ) ∧ (A ∩ C ) = (A ∩ C ) ( ) ∧ (A ∩ C ) ) ≥ (A ∩ C ) ( ) ∧ (A ∩ C ) ( ) = (A ∩ C ) = (A ∩ C ) = A = A
∧C
since A , C ∈ F (G )
∧C
= (A ∩ C ) = (A ∩ C ) ( ) (A ∩ C ) ∈ F (G ) Hence, (ii) Let , ∈ G to prove that A is normal we have to show A ( )=A ( ) Let ϕ be function from G into itself definition by ϕ(z) = z , ∀ z∈G Since A is a fuzzy characteristic subgroup of G , ∴A
=A ) Thus A ( )=A ( =A ( ) = A (ϕ ( )) ( ) ) =A ( =A ( ) Hence A is normal subgroup of G . Again, Suppose , ∈ F (G ) to prove that (A ∩ C ) is a normal fuzzy subgroup of G it is necessary to show (A ∩ C )( ) = (A ∩ C )( ) Let ϕ be the function of group G into itself defined by ϕ (z) = z ∀ ∈G Since A and C are fuzzy characteristic subgroup of G , hence be normal as we prove (A ∩ C ) = (A ∩ C ) (A ∩ C )( ) = (A ∩ C ) ( ) ) = (A ∩ C ) ( (A ) ( ) = ∩C = (A ∩ C ) (
)(
)
) = (A ∩ C ) ( Hence (A ∩ C ) ∈ F (G ) is normal. Similarly, (A ∪ C ) = (A ∪ C ) (A ∪ C ) ( ) = (A ∪ C ) ( ) = (A ∪ C )( ( ) = (A ∪ C ) = (A ∪ C ) = (A ∪ C )(
)(
)
)
)
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1647
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com Hence (A ∪ C ) ∈ F (G ) is also normal. PROPOSITION 5.3: Let A is a normal fuzzy subgroup of G and let ϕ be a homomorphism of G into itself. Then ϕ induces a homomorphism ϕ of
into itself defined by
ϕ( A ) = ϕ ( )A Proof : Let , ∈ G we have
For all
∈ (G )
A = A Then we have to show that ϕ ( )A = ϕ ( )A Since A = A we have A ( )= A ( ) ⇒ A (e) = A ( ) A ( )= A ( ) ⇒ A ( ) = A (e) A ( )=A ( ) = A (e) Implies that ( ),( )∈A ∗ Since we have ϕ (A ∗) = A ∗ Therefore ϕ( ) and ϕ( ) also belong to A ∗ Which implies that A (ϕ( )) = A (ϕ( )) = A (e) Let g ∈ G, Then ϕ ( )A (g ) = A (ϕ ( )g ) = A (ϕ( )ϕ( )ϕ( )g ) ≥ A (ϕ( ) ϕ ( ) ∧ A (ϕ( )g ) = A (ϕ( )) ∧ ) ϕ ( ) A (g ) = A (e) ∧ ϕ ( ) ∧ A (g ) = ϕ ( ) A (g ) Finally, ϕ ( ) A (g ) ≥ ϕ ( ) A (g ) …………. (i) Similarly, we can prove that ϕ ( ) A (g ) ≤ ϕ ( ) A (g ) ……………(ii) Since g ∈ G is arbitrary Hence, ϕ( )A =ϕ( )A Therefore, we find that ϕ is well defined Now we have only to show that ϕ is a homomorphism Let , ∈G . Since ϕ is homomorphism ϕ( )=ϕ( )ϕ( ) ϕ( )A =ϕ( )ϕ( )A . ϕ( ) A = ϕ ( )A . ϕ ( )A . = ϕ( A . A ) = ϕ( A ) . ϕ ( A ).
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1648
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IV Apr 2022- Available at www.ijraset.com Hence ϕ is a homomorphism. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
A.Rosenfeld.: Fuzzy group. J.Math.Anal.Appl.35,512-517 (1971). S.Sebastian and S.Babunder. Fuzzy groups and group homomorphism.Fuzzy sets and systems.8,397-401 (1996). S. Abou-zaid, On fuzzy subgroup, Fuzzy sets and systems,55. 1993, pp. 237–240. N.Ajmal and A.S.Prajapati,Fuzzy cosets and fuzzy normal subgroup, Inform.Sci, 64 (1992) 17-25. R.Biswas,Fuzzy subgroup and anti fuzzy subgroup.Fuzzy sets and systems.35,121-124 (1990). D.S.Malik,and J.N.Mordeson,Fuzzy subgroup and abelian group.Chinese.J.Math(Taipei).19,129-145 (1991). J.M.Anthony,and H.Sherwood,Fuzzy subgroup redefined,J.Math.Anal.Appl.69,124-130 (1979). J.M.Anthony,and H.Sherwood,A characterization of fuzzy subgroup, J.Math.Anal.Appl,69 (1979) 297-305. P.S.Das,Fuzzy groups and level subgroups J.Math.Anal.Appl.,84 (1981) 264-269. V.N.Dixit,R.kumar,and N.Ajmal,Level subgroups and union of fuzzy subgroups. Fuzzy Sets and Systems, 37 (1990) 359-371. and Technology, 1994, Vol. 2, pp. 87-98. M.S.Eroglu, The homomorphic image of a fuzzy subgroup is always a fuzzy subgroup, Fuzzy Sets and Systems, 33 (1989) 255-256. I.J.Kumar,P.K.Saxena,and P.Yadav,Fuzzy normal subgroups and fuzzy quotients, Fuzzy sets and systems, 46 (1992) 121-132. N.P.Mukherjee,and P.Bhattacharya,Fuzzy normal subgroups and fuzzy cosets: Information Science, 34 (1984) 225-239. M.T.A.Osman, On the direct product of fuzzy subgroups, Fuzzy sets and sysems, 12 (1984) 87-91. L.A.Zadeh. Fuzzy sets, Inform.Control, 8 (1965) 338-353. S.M.A.Zaidi, and Q.A.Ansari,: Some results of categories of L-fuzzy subgroups.Fuzzy sets and systems, 64 (1994) 249-256. Information Technology, 2012, Vol. 2, pp. 527-531. automata. Soft Computing, 2013 (Communicated). Y.Yu, A theory of isomorphism of fuzzy groups, Fuzzy system and Math.2,(1988),57-68.
©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 |
1649