Complex Fuzzy Normal Subgroup
Descripción
Complex fuzzy normal subgroup Abdallah Al-Husban, Abdul Razak Salleh, and Nasruddin Hassan Citation: AIP Conference Proceedings 1678, 060008 (2015); doi: 10.1063/1.4931335 View online: http://dx.doi.org/10.1063/1.4931335 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1678?ver=pdfcov Published by the AIP Publishing Articles you may be interested in C-normal and C-permutable subgroups in finite groups AIP Conf. Proc. 1635, 383 (2014); 10.1063/1.4903611 Complex fuzzy soft multisets AIP Conf. Proc. 1614, 955 (2014); 10.1063/1.4895330 Complex intuitionistic fuzzy sets AIP Conf. Proc. 1482, 464 (2012); 10.1063/1.4757515 Upper Bounds for the Level of Normal Subgroups of Hecke Groups AIP Conf. Proc. 1389, 337 (2011); 10.1063/1.3636733 Determination of Genus of Normal Subgroups of Discrete Groups AIP Conf. Proc. 1281, 1148 (2010); 10.1063/1.3497859
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Complex Fuzzy Normal Subgroup Abdallah Al-Husbana, Abdul Razak Sallehb and Nasruddin Hassanc a,b,c
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 43600 UKM Bangi Selangor DE, MALAYSIA
Abstract. In this paper, we continue the study of complex fuzzy groups by introducing the notion of complex fuzzy normal subgroup based on complex fuzzy space as a generalisation of fuzzy normal subgroup in the sense of Dib. Keywords: Complex fuzzy space, complex fuzzy subspace, complex fuzzy group, complex fuzzy normal subgroup. PACS: 02.10. De.
INTRODUCTION The study of fuzzy groups was first started with the introduction of the concept of fuzzy subgroups by Rosenfeld [2]. In 1979, Anthony and Sherwood [8] redefined fuzzy subgroups using the concept of triangular norm. In his remarkable paper Dib [10] introduced a new approach to define fuzzy groups using his definition of fuzzy space, which serves as the universal set in classical group theory. Dib remarked the absence of the fuzzy universal set and discussed some problems in Rosenfeld's approach.The notion of fuzzy normal subgroup was first initiated by Abdul Razak [1] to continue the theory of fuzzy groups obtained by Dib. In 1998, Dib and Hassan [11] introduced and discussed the fuzzy normal group in a similar manner to Abdul Razak [1]. In1989, Buckley [9] incorporated the concepts of fuzzy numbers and complex numbers under the name fuzzy complex numbers. This concept has become a famous research topic and a goal for many researchers ([4], [7], [9]). However, the concept given by Buckley has different range compared to the range of Ramot et al.’s [4] definition for complex fuzzy set (CFS). Buckley’s range goes to the interval [0,1], while Ramot et al.’s range extends to the unit circle in the complex plane. In [5, 6] they developed an axiomatic approach for propositional complex fuzzy logic. Besides, it explains some constraints in the concept of CFS that was given by Ramot et al. [3]. In this paper we continue the study of complex fuzzy groups by generalising the notion of the fuzzy normal subgroup to the complex fuzzy normal subgroup case.
PRELIMINARIES In this section, we recall some definitions which will be used in the paper. Definition 2.1 [12] A fuzzy set A in a universe of discourse U is characterised by a membership function PA (x ) that takes values in the interval [0,1]. Definition 2.2 [4] A complex fuzzy set (CFS) A, defined on a universe of discourse U is characterised by a membership function PA (x ) , that assigns to any element x U a complex-valued grade of membership in A. By definition, the values of PA (x ) , may receive all lying within the unit circle in the complex plane, and are thus of the iZ ( x ) A
each of PA (x ) and ZA (x ) are both real-valued, and rA (x ) [0,1] . The CFS A may be represented as the set of ordered pairs form P A ( x)
rA ( x) e
, where
A
i
1 ,
{(x, r ( x)) : x U} {(x, r ( x) e Z i
A
A
A ( x)
) : x U}.
The 2015 UKM FST Postgraduate Colloquium AIP Conf. Proc. 1678, 060008-1–060008-7; doi: 10.1063/1.4931335 © 2015 AIP Publishing LLC 978-0-7354-1325-2/$30.00
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Let X be a given nonempty set and let A be a complex fuzzy set of X The complex fuzzy set A can be identified with its membership function PA : X o {a : | a | d 1 1} defined by P A ( x )
rA ( x ) e
iDZ ( x ) A
1 , each of rA (x ) and ZA (x ) are both real-valued, and Z A ( x), rA ( x) [0,1], D [0, 2S ], and S
i
1
where is unit
circle with the usual order of complex numbers. The scaling factor D , D [0, 2S ], is used to confine the performance of the phases within the interval [0, 2S ] and the unit circle. Hence, the phase term T may represent the FS information, and phase term values in this representation case belong to the interval [0,1] and satisfies FS restriction. So, D in this paper will always be considered equal to 2S . Let S 1 be the unit circle. Then S 1 S 1 is the Cartesian product S 1 u S 1 with partial order defined by: (i) ( r1e
i Tr
1
, r2e
i Tr
2
) d (s1e
i Ts
1
, s 2e
i Ts
2
) iff r1 d s1 and r2 d s 2 , T r d T s and T r d Ts whenever s1 z 0 and 1 1 2 2
s 2 z 0 for all r1 , s1 ,Tr1 Ts1 S and r2 , s 2 , Tr2 Ts 2 S 1 , 1
(ii)
(0 e
i T1
,0e
i T2
)
(s1e
i Ts 1
, s 2e
i Ts
2
)
whenever s1 z 0 or s 2 z 0 and T1 z 0 or T2 z 0 for every s1 T1 S 1 and
s 2 , T2 S 1. Definition 2.3 A complex fuzzy space, denoted by (X , S 1 ) where S 1 is the unit circle, is set of all ordered pairs (X , S 1 ), x X i
1 ,
i.e., ( X , S1 )
{( x, S1 ) : x X }.
{(x ,
We can write ( x , S 1 )
re
i DT
) : re
i DT
S
}, where
1
r [0,1] T [0,1] and D [0,1]. The ordered pair (x , S 1 ) is called a complex fuzzy element in the
complex fuzzy space (X , S 1 ), Let U 0 denote the support of U , that is U 0
{x U : r ! 0 and
T ! 0}.
Therefore, the complex fuzzy space is an (ordinary) set of ordered pairs. In each pair the first component indicates the (ordinary) element and the second component indicates a set of possible complex membership values ( reiT where r represents an amplitude term and T represents a phase term). Definition 2.4 The complex fuzzy subspace U of the complex fuzzy space (X , S 1 ) is the collection of all ordered pairs (x , rx e i T ) where x U 0 for some U 0 X and r x e x
i Tx
element beside the zero element. If it happens that x U 0 , then rx U is denoted by U
{( x, rx e
iT x
is a subset of S 1 , which contains at least one 0 and T x
0 . The complex fuzzy subspace
) : x U 0}, U 0 is called the support of U and denoted by SU
U 0 . Any
empty complex fuzzy subspace is defined as
{( x, x
i 0x
0x e
) : x }, i.e S
.
Definition 2.5 Let (X , S 1 ) and (Y , S 1 ) be complex fuzzy spaces. The complex fuzzy function F from the complex fuzzy space (X , S 1 ) into the complex fuzzy space (Y , S 1 ) is defined as an ordered pair F where F : X o Y is a function and {ƒ x }
x X
(i)
f x is nondecreasing on S 1
(ii)
f x (0e i T )
0 if T
(F , ^ƒ x`xX ),
is a family of functions f x : S o S satisfying the conditions
1 and f x (1e i T ) 1 if T
1
1
0,
such that the image of any S 1 complex fuzzy subset A of X under F is the complex fuzzy subset F (A ) of Y defined by
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iT ° i T 1 f x (re ) F ( A ) y ®r e F ( y ) °¯ 0
We write F
F , ƒx
1
if F ( y )
1
1
for any F ( y ) S , y Y
Ø
F , ƒ x : (X , S 1 ) o (Y , S 1 ) to denote the complex fuzzy function from (X , S 1 ) to (Y , S 1 ) , and we
call the functions ƒ x , x X
F
1
if F ( y ) z Ø
the co-membership functions associated to F . A complex fuzzy function
is said to be uniform if the co-membership functions f x
are identical. That is, f x
for all
f
x X . F , ƒ x : (X , S 1 ) o (Y , S 1 ) acts on the complex fuzzy element ( x , S 1 ) of
Every complex fuzzy function F
(X , S 1 ) as follows: F (x , S 1 )
(F (x ),
f x (S 1 ))
(F (x ),
S 1 ).
Definition 2.6 Let (X , S 1 ) , (Y , S 1 ) and ( Z , S 1 ) be complex fuzzy spaces. The complex fuzzy function F from 1
1
(X , S )
(Y , S )
( X uY , S
1
1 S ) into ( Z , S ) is defined by the ordered pair
1
(F , {ƒ
xy
}( x , y )X uY
)
where
F : X uY o Z is a function and {ƒ xy }( x , y )X uY is a family of functions ƒ xy : S 1 u S 1 o S 1 satisfying the conditions :
(i)
f xy is non-decreasing on S
(ii)
f xy (0 e i T , 0 e i T )
1
S
0 if T
1
0, and f xy (1e i T , 1e i T ) 1 if T
C of X uY under
such that the image of any S 1 u S 1 -complex fuzzy subset F (C ) of Z defined by
F C z
°( x , y )F 1 z f xy C ( x , y ) , ® ° 0, ¯
Definition 2.7 A complex fuzzy binary operation F
0
if F
1
if F
1
(z ) z Ø (z )
F is the complex fuzzy subset
for every z Z .
Ø
(F , ƒ xy ) on the complex fuzzy space (X , S 1 ) is a complex
fuzzy function from (X , S 1 ) u (X , S 1 ) o (X , S 1 ) with co-membership functions ƒ xy satisfying: i Tr
, se
i Ts
) z 0 if r z 0, s z 0 , Ts z 0 and T r z 0,
(i)
ƒ xy (re
(ii)
ƒ xy is onto, i.e., ƒx y (S 1 S 1 )
The complex fuzzy binary operation F
S 1, x , y X .
(F , ƒ xy ) on a set X is a complex fuzzy function from X u X to X
and is said to be uniform if F is a uniform complex fuzzy function. We call the pair ((X , S 1 ), F ) of complex fuzzy algebraic system.
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Definition 2.8 A complex fuzzy algebraic system ((X , S 1 ), F ) is called a complex fuzzy group if and only if for every (x , S 1 ),( y , S 1 ),(z , S 1 ) (X , S 1 ) the following conditions are satisfied: (i) Associative:
((x , S
1
) F ( y , S 1 )) F (z , S 1 )
i.e., (x F y ) F z , S 1
(x , S 1 ) F
(( y , S
x F ( y F z ),
1
) F (z , S 1 ) ),
S 1 .
(ii) It has an identity (e , S 1 ) , for which
x , S F e , S e , S F x , S x , S , 1
1
1
i.e, (x Fe , S 1 )
1
(e F x , S 1 )
1
(x , S 1 )
1
(iii) Every complex fuzzy element ( x , S ) has an inverse ( x , S 1 ) 1 such that
x
, S 1 F x , S 1
Denote (x , S 1 ) 1
x F
x
1
, S 1
1
F
x , S e , S . 1
1
1
y , S 1
( y , S ) , then we have
y F x ,
S 1
e , S . 1
From (i), (ii) and (iii), it follows that ( X , F ) is a fuzzy group. Therefore, we can write x 1 y and then ( x, S1 ) 1 ( y, S1 ).
{(x , rx e i Tx ) : x U 0} be a complex fuzzy (X , S 1 ). (U , F ) is called a complex fuzzy subgroup of the complex fuzzy group ((X , S 1 ); F ) if:
Definition 2.9 Let ((X , S 1 ), F ) be a complex fuzzy group and let U subspace of (i)
F is closed on the complex fuzzy subspace U, i.e., iT
( x, rx eiTx ) F ( y, ry e y ) ( xF y, rxF y e (ii)
iT x F y
)
(xF y, (rx f x y ry ) e
i (T x f x yT y )
)
(U , F ) satisfies the conditions of a complex fuzzy group.
COMPLEX FUZZY NORMAL SUBGROUP In this section, we introduce the notion of the associative complex fuzzy subgroup and a complex fuzzy normal subgroup. Let (( X , S 1 ); F ) be a complex fuzzy group and U
(( X , S1 ), F ).
{(x , rx e i Tx ) : x U 0}
Contrary to the ordinary case, the complex fuzzy elements
be a complex fuzzy subgroup of iT x
( x, rx e )
of the complex fuzzy
1
subgroup (U , F ) are not necessary associative with the complex element (X , S ) of the complex fuzzy group
(( X , S1 ), F ). That is; aF (b Fc ) z (aFb ) Fc ,
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where a , b and c are some complex fuzzy elements U or (X , S 1 ) such that one or two of these complex fuzzy elements belong to U . Example 3.1 Let X
F
{1, 1, i, i} and S 1 be a unit circle Let the complex fuzzy binary operation
(F , f xy ) on ( X , S1 ) where F : X u X o X be the ordinary multiplication of complex numbers .The
comemership functions f xy may have the following forms:
r .s e i (Tr Ts ) , ° iT ° iT D f 11 ( re r , s e s ) ® i (T T ) ° r .s e r s 1 1 , ° 1D ¯ f 11 ( re
i Tr
, se
i Ts
)
f 11 ( re
i Tr
, se
i Ts
and the other co-membership function are defined by the r .s e
U
0 E D 1. It is easy to check that
{(1, [0, D e 1.3S i ] ), (1, [0, E e
i
S
i (Tr Ts )
rs ! D
2
rs d E
2
rs ! E
2
where D and E are given fixed real number
(( X , S1); F )
]} is a complex fuzzy subgroup of
2
2
)
rs e i (Tr Ts ) , ° ° D ® °1 1 E 1 ( rs e i (Tr Ts ) 1), ° ¯ 1 DE
such that
rs d D
is a complex fuzzy group and
((X , S 1 ), F )
Moreover, ( X , F ) is a
complex fuzzy subspaces group of ((X , S ), F ). Now, we notice that 1
Si
((1, [0, 1 e
1.3S i
]) F ((1, [0, D e
S
]) F ( 1, [0, E e
2
i
S § ª 1 E § 2D E · 2 i º · ¨ 1, «0, 1 ¨1 ¸ e »¸, 1D E © 1D ¹ ¬ ¼¹ ©
])
and Si
(1, [0, 1eS i ]) F ((1, [0, D e1.3S i ]) F ( 1, [0, E e 2 ]))
S 2 § ª (1 E ) º 2 i · e . 1, «0, 1 ¨ ¸ » ¨ ¸ 1D E ¼ ¬ © ¹
That is, the complex fuzzy element of U are not associative with complex fuzzy elements of ( X , S 1 ). Definition 3.1 An associative complex fuzzy subgroup (U , F ) of a complex fuzzy group
((X , S 1 ), F )
is a
complex fuzzy group (U , F ) of ((X , S ), F ) in which complex fuzzy elements of U are associative with the 1
complex fuzzy elements of ((X , S 1 ), F ) for any arbitrary choice of complex fuzzy elements of U and ( X , S ). 1
Example 3.2 Let X
{1, 1, i , i }. Define the complex fuzzy binary operation F
1 (F , f xy ) on (X , S ) such
that F : X u X o X is the ordinary multiplication of complex numbers and the co-membership function are given for all x , y X by
f xy (reiTr , seiTs )
(r s)ei (Tr Ts ) .
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({(1, [0,
Obviously the complex fuzzy spaces U
0.5 e2S i ]), (1, [0, 0.5 eS i ] )}; F
)
defines an
associative complex fuzzy subgroup of ((X , S 1 ), F ) under F .
((X , S 1 ), F )
Theorem 3.1 Let
iT
operation for which f (1e , re
be a complex fuzzy group and F
i Tr
)
f (re
i Tr
,1e
iT
)
re
i Tr
(F , f ) be a uniform complex fuzzy binary
. Then every complex fuzzy subgroup of the complex
fuzzy group ((X , S ), F ) is an associative fuzzy subgroup. 1
Proof Let (U , F ) be a complex fuzzy subgroup of the complex fuzzy group ((X , S 1 ); F ) Consider the complex 1
1
fuzzy elements (x , S 1 ) , ( y , S 1 ) and ( z , S ) of U or ( X , S ). Now using the properties of f
and the
associativity of F we have: 1
1
1
1
(x , S ) F (( y , S ) F (z , S ))
1
1
(x , S ) F (( yFz , f (S u S ) ) 1
1
( x, S ) F (( yFz, S )
1
1
( xF ( yFz ), f ( S u S ))
(( xFy) Fz, S1 ) (( x, S1 ) F ( y, S1 ) ) F ( z, S1 ). which proves the associativity of complex fuzzy elements of U with complex fuzzy elements of ( X , S 1 ) under F . Definition 3.2 The complex fuzzy subgroup U of the complex fuzzy group ((X , S 1 ); F ), is called a complex fuzzy normal subgroup if (i)
U is associative in ((X , S 1 ); F ).
(ii)
(x , S 1 ) U
U (x , S 1 ), such that x X .
Example 3.3 Consider the complex fuzzy space (X , S 1 ) where X Define the complex fuzzy binary operations F f xy (re i Tr , se i Ts )
(F , f xy ), where
(r s ) e i (Tr Ts ) , for all x , y X
S 3 , the set of all permutations on {1, 2, 3}.
F is the composition of permutations and
S 3 and for all 0 d r , s d 1, Tr ,Ts [0, 2S ]. It is clear
that ((X, S 1 ); F ) is a complex fuzzy group. Let U 0 {H , D (12)}, where H is the identity permutation, then it is easy to see that U {(H , [0, 0.5e 2S i ], (D ,[0, 0.5e S i ])} is an associative complex fuzzy subgroup in
((X , S 1 ); F )
but it is not a complex fuzzy normal subgroup in ((X , S ); F ). 1
1
iT
Theorem 3.2 If U {(z , rz e z ) : z U 0} is a complex fuzzy subgroup of the complex fuzzy group ((X , S ); F ) then U is a complex fuzzy normal subgroup if and only if (i)
(U 0 , F ) is an (ordinary )normal subgroup of the ordinary group ( X , F ).
(ii)
f xz (S 1 , rz e
Proof Assume U
i Tz
)
f z cx (rz ce
{(z , r e T i
z
z
i Tz c
, S 1) : x F z
z c F x and x , z , z c U 0 where x X .
) : z U 0} is a complex fuzzy normal subgroup ((X , S 1 ); F ), from the
correspondence theorem we have (U 0 , F ) is an ordinary normal subgroup of the ordinary group (X , F ). Using the normality of U we have (x , S 1 ) U
U (x , S 1 ) : x X , that is,
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{(xFz , f xz (S 1 , rz e i T )) : z
U 0}
{(zFx , f xz (rz e i T , S 1 ) : z
Therefore for every z U 0 there exists z c U 0 such that xFz
U 0}.
z cFx . In other words xFU 0
U 0 Fx . Hence
U 0 is an ordinary normal subgroup of ordinary group ( X , F ) which proves that (i) and (ii) follows directly from the definition. The other part of the proof is direct.
CONCLUSION In this paper, we studied complex fuzzy groups. Also we define the notion of complex fuzzy normal subgroup as a generalisation of fuzzy normal subgroup defined in [11, 12] using the notion complex fuzzy group based on complex fuzzy space. We have also defined an associative complex fuzzy subgroup.
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