Download - Intuitionistic Fuzzy Groups
Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 12, Number 6 (2016), pp. 4929-4949
© Research India Publications
http://www.ripublication.com/gjpam.htm
Complex Intuitionistic Fuzzy Group
Rima Al-Husbana, Abdul Razak Sallehb and Abd Ghafur Bin Ahmadb
a, b,c School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia.
Abstract
The aim of this study is to introduce the notion of complex intuitionistic fuzzy
groups based on the notion of complex intuitionistic fuzzy space this concept
generaleis the concept of intuitionistic fuzzy space from the real range of
membership function and non-membership[0,1] , to complex range of
membership function and, non-membership unit disc in the complex plane.
This generalization leads us to introduce and study the approach of
intuitionistic fuzzy group theory that studied by Fathi (2009) in the realm of
complex numbers. In this paper, a new theory of complex intuitionistic fuzzy
group is developed. Also, some notions, definitions and examples related to
intuitionistic fuzzy group are generalized to the realm of complex numbers.
Also a relation between complex intuitionistic fuzzy groups and classical
intuitionistic fuzzy subgroups is obtained and studied.
Keywords Complex intuitionistic fuzzy space, complex intuitionistic fuzzy
binary operation, complex intuitionistic fuzzy group, complex intuitionistic
fuzzy subgroup.
INTRODUCTION
The theory of intuitionistic fuzzy set is expected to play an important role in modern
mathematics in general as it represents a generalization of fuzzy set. The notion of
intuitionistic fuzzy set was first defined by Atanassov (1986) as a generalization of
Zadeh’s (1965) fuzzy set. After the concept of intuitionistic fuzzy set was introduced,
several papers have been published by mathematicians to extend the classical
4930 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
mathematical concepts and fuzzy mathematical concepts to the case of intuitionistic
fuzzy mathematics.
The study of fuzzy groups was first started with the introduction of the concept of
fuzzy subgroups by Rosenfeld (1971). Anthony and Sherwood (1979) redefined fuzzy
subgroups using the concept of triangular norm. In his remarkable paper Dib (1994)
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 (1994) remarked the
absence of the fuzzy universal set and discussed some problems in Rosenfeld’s
approach. In the case of intuitionistic fuzzy mathematics, there were some attempts to
establish a significant and rational definition of intuitionistic fuzzy group. Zhan and
Tan (2004) defined intuitionistic fuzzy subgroup as a generalization of Rosenfeld’s
fuzzy subgroup. By starting with a given classical group they define intuitionistic
fuzzy subgroup using the classical binary operation defined over the given classical
group. We introduce the notion of intuitionistic fuzzy group based on the notion of
intuitionistic fuzzy space and intuitionistic fuzzy function defined by Fathi and Salleh
(2008 a, b), which will serve as a universal set in the classical case. Fathi and Salleh
(2009) introduced the notion of intuitionistic fuzzy space which replace the concept
of intuitionistic fuzzy universal set in the ordinary case. The notion of intuitionistic
fuzzy group as a generalization of fuzzy group based on fuzzy space by Dib in 1994
was introduced by Fathi and Salleh (2009). Husban and Salleh (2016) generalized the
concept of fuzzy space to the concept of complex fuzzy space and introduced the
concept of complex fuzzy group and complex fuzzy subgroup.
In 1989, Buckley incorporated the concepts of fuzzy numbers and complex numbers
under the name fuzzy complex numbers (Buckley 1989). This concept has become a
famous research topic and goal for many researchers (Buckley 1989, Ramot et al.
2002, Alkouri and Salleh 2012, 2013). However, the concept given by Buckley has
different range compared to the range of Ramot et al.’s definition for complex fuzzy
set Buckley’s range goes to the interval[0, 1] , while Ramot et al.’s range extends to
the unit circle in the complex plane. Tamir and Kandel (2011) developed an
axiomatic approach for propositional complex fuzzy logic. Besides, it explains some
constraints in the concept of complex fuzzy set that was given by Ramot et al. (2003).
Moreover, many researchers have studied developed, improved, employed and
modified Ramot et al.’s approach in several fields (Zhang et al. 2009, Tamir et al.
2011, Zhifei et al 2012). Alkouri and Salleh (2012) introduced a new concept of
complex intuitionistic fuzzy set, which is generalized from the innovative concept of
a complex fuzzy set by adding the non-membership term to the definition of complex
fuzzy set. The novelty of complex intuitionistic fuzzy set lies in its ability for
membership and non-membership functions to achieve more range of values (Alkouri
and Salleh 2013).
Complex Intuitionistic Fuzzy Group 4931
In this paper we incorporate two mathematical fields on intuitionistic fuzzy set, which
are intuitionistic fuzzy algebra and complex intuitionistic fuzzy set theory to achieve
a new algebraic system, called, complex intuitionistic fuzzy group based on complex
intuitionistic fuzzy space. We will use Fathi and Salleh (2009) approach and
generalize it to complex realm by following the approach of Ramot et al. (2002).
Therefore, the concept of complex intuitionistic fuzzy space is introduced. This
concept generalizes the concept of intuitionistic fuzzy space. The complex fuzzy
intuitionistic Cartesian product, complex intuitionistic fuzzy functions, and complex
intuitionistic fuzzy binary operations in intuitionistic fuzzy spaces and intuitionistic
fuzzy subspaces, as well, are defined. Having defined complex intuitionistic fuzzy
spaces and complex intuitionistic fuzzy binary operations, the concepts of complex
intuitionistic fuzzy group, complex intuitionistic fuzzy subgroup are introduced and
the theory of intuitionistic complex fuzzy groups is developed.
PRELIMINARIES
This section is set to review the introductory definitions, explanations and outcomes
that are considered mandatory to understand the purpose of this paper.
Definition 2.1 (Atanassov 1986) an intuitionistic fuzzy set A in a non-empty set U (a
universe of discourse) is an object having the form: , ( ), ( ) :A AA x x x x U ,
where
the functions ( ) : [0,1]A x U and ( ) : [0,1]A x U , denote the degree of
membership and degree of non-membership of each element x U to the set A
respectively, and 0 ( ) ( ) 1A Ax x for all .x U
Definition 2.2 (Fathi 2009) Let X be a nonempty set. An intuitionistic fuzzy space
denoted by , [0,1] , ( )[0,1]X I I is the set of all ordered triples ( , , ),x I I where
( , , ) , , ) : , }{(x I I x r s r s I with 1r s and .x X The ordered triplet
), ( ,x I I is called an intuitionistic fuzzy element of the intuitionistic fuzzy space
), ( ,X I I and the condition ,r s I with 1r s will be referred to as the
“intuitionistic condition”.
Definition 2.3 (Fathi 2009) Let G be a nonempty set. an intuitionistic fuzzy group is
an intuitionistic fuzzy monoid in which each intuitionistic fuzzy element has an
inverse. That is, an intuitionistic fuzzy groupoid ( , , ), ( , , )( )xy xyG I I F f fF is an
intuitionistic fuzzy group iff the following conditions are satisfied:
4932 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
(1) Associativity: for any choice of ( , , ), ( , , ), ( , , ) ( , , ), )(x I I y I I z I I G I I F
( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , )( ) ( ).x I I y I I z I I x I I y I I z I IF F F F
(2) Existence of an identity: there exists an intuitionistic fuzzy element exists an
intuitionistic fuzzy element , , ( ) (G, ), e I I I I such that for all ), ( ,X I I in
(( , , ), ) : G I I F
( , , ) ( , , ) ( , , ) ( , , )
( , , ).
e I I x I I x I I e I Ix I I
F F
(3) Existence of an inverse: for every intuitionistic fuzzy element ), ( ,x I I in
, , ( ), ( )G I I F there exists an intuitionistic fuzzy element 1, ),( x I I
in
, , ( ), ( )G I I F such that:
1 1 1( , , ) ( , , ) ( , , ) (x, , )( , , )
( , , )
x I I x I I x I I I I x I Ie I I
F F
Definition 2.4 (Ramot et al. 2002) A complex fuzzy set A, defined on a universe of
discourse U, is characterized by a membership function ( )A x , that assigns to any
element x U a complex-valued grade of membership in A. By definition, the
values of ( )A x , may receive all lying within the unit circle in the complex plane, and
are thus of the form ( )
( ) ( )A AAi xx r x e
, where 1i , each of ( )Ar x and ( )A x
are both real-valued, and ( ) [0, 1]Ar x . The complex fuzzy set A may be represented
as the set of ordered pairs ( )( , ( )) : ( , ( ) ) : .Ai xAA x x x U x r x e x U
Definition 2.5 (Alkouri & Salleh 2012) A complex Atanassov’s intuitionistic fuzzy set A, defined on a universe of discourse U, is characterized by membership and non-
membership functions ( )A x and ( )A x , respectively, that assign to any element
x U a complex-valued grade of both membership and non-membership in A. By
definition, the values of ( )A x , ( )A x , and their sum may receive all that lies within
the unit circle in the complex plane, and are of the form ( )
( ) ( ) AA A
i xx r x e
for membership function in A and ( )
( ) ( ) AA A
i xx k x e for non-membership
function in A, where 1i , each of ( )Ar x and ( )Ak x are real-valued and both
belong to the interval [0, 1] such that 0 ( ) ( ) 1A Ar x k x , also ( )A
x and ( )A
x
are real valued. We represent the complex Atanassov’s intuitionistic fuzzy set A as
Complex Intuitionistic Fuzzy Group 4933
, ( ), ( ) : A AA x x x x U ,
where ( ) : , 1 ,A x U a a C a ( ) : , 1A x U a a C a , ( ) ( ) 1.A Ax x
Definition 2.6 (Alkouri & Salleh 2012). The complement, union and intersection two
CIFSs , ( ), ( ) :A A
A x x x x U and , ( ), ( ) :B B
B x x x x U in a universe
of discourse ,U are defined as follows:
(i) Union of complex intuitionistic fuzzy sets:
, ( ), ( ) :A B A BA B x x x x U
where (max( ( ), ( )))( ) max[ ( ), ( )] A Bx x
BA B Aix r x r x e
and (min( ( ), ( )))( ) min[ ( ), ( )] A Bx x
BA B Aix k x k x e
.
(ii) Intersection of two complex intuitionistic fuzzy sets:
, ( ), ( ) :A B A BA B x x x x U
where (min( ( ), ( )))
( ) min[ ( ), ( )] A Bx xBA B A
ix r x r x e
and(max( (x), (x)))
( ) max[ ( ), ( )] A BBA B A
ix k x k x e
.
Definition 2.7 (Alkouri & Salleh 2013) Let 1,..., nA A be complex Atanassov’s
intuitionistic fuzzy sets in 1,..., nX X , respectively. The Cartesian product 1 ... nA A
is a complex intuitionistic fuzzy set defined by
1 11 1 1 1 1 1... ...... ( ,..., ), ( ,..., ), ( ,..., ) : ( ,..., ) ...
n nn n n n n nA A A AA A x x x x x x x x X X
11
1 11 1
min( ( ),..., ( ))
... ]where ( ,..., ) [min( ( ),..., ( ))n
n
n nn n
i x xA AA A A A
x x x x e
And 1
1
1 11 1
max( ( ),..., ( ))
... ]( ,..., ) [max( ( ),..., ( ))n
n
n nn n
i x xA AA A A A
x x x x e
.
In the case of two complex Atanassov’s intuitionistic fuzzy sets A and B in X and Y
respectively, we have
( , ), ( , ), ( , ) : ( , ) A B A BA B x y x y x y x y X Y
min( ( ), ( ))
where ( , ) min( ( ), ( )) A Bi x y
A BA B x y x y e ,
4934 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
and max( ( ), ( ))
( , ) max( ( ), ( )) A Bi x y
A BA B x y x y e .
Definition 2.8 (Alkouri & Salleh 2013) If A and B are complex Atanassov’s
intuitionistic fuzzy sets in a universe of discourse U,
where ( )( )
{( , ( ) , ( ) }A Ar AkA i xi xA x r x e k x e
and ( )( )
{( , ( ) , ( ) }B Br BkB i xi xA x r x e k x e
, then
1. A B if and only if ( ) ( ) and ( ) ( ),A B A Br x r x k x k x for amplitude
terms and the phase terms (arguments) ( ) ( )A Br rx x and ( ) ( )
A Bk kx x , for all
.x U
2. A B if and only if ( ) ( ) and ( ) ( ),A B A Br x r x k x k x for amplitude
terms and the phase terms (arguments) ( ) ( )A Br rx x and ( ) ( )
A Bk kx x , for all
.x U
Definition 2.9 ( Husban & Salleh 2016) A complex fuzzy space, denoted by
2, ,( )X E where 2E is the unit disc, is set of all ordered pairs ,x X i.e.,
2 2(, , ) :)( }.{X E x E x X We can write 2 2( , ) ( , ) :{ },r ri ix E x re re E where
1i , [0, 1],r and [0,2 ]. The ordered pair 2( , )x E is called a complex
fuzzy element in the complex fuzzy space 2, .( )X E
Definition 2.10 ( Husban & Salleh 2015) The complex fuzzy subspace U of the
complex fuzzy space 2( , )X E is the collection of all ordered pairs ( , )xixx r e
where
0x U for some
0U X and xi
xr e
is a subset of 2 ,E
which contains at least one
element beside the zero element. If it happens that 0 ,x U
then 0 andxr
0x . The complex fuzzy subspace U is denoted by
0( ) : ( , ) : , 0 1,, 0 2 { }x xi ix xU x r e x U rx x X re
0U is called the support of ,U that is 0 :{U x U 0 and 0 .}r and denoted
by 0.SU U
Complex Intuitionistic Fuzzy Group 4935
Definition 2.11 ( Husban & Salleh 2016) A complex fuzzy binary operation ,( )ƒxyF F
on the complex fuzzy space 2( , )X E
is a complex fuzzy function
from 2 2 2( , ) ( , ) ( , )X E X E X E with comembership functions ƒxy satisfying:
(i) ƒ , 0( )sr iixy re s e
if 0, 0 , 0sr s and 0,r
(ii) ƒ arexy onto, i.e.,
2 2 2( )ƒ ,xy E E E , .x y X
Definition 2.12 ( Husban & Salleh 2016) A complex fuzzy algebraic
system 2( , ),( )X E F is called a complex fuzzy group if and only if for every
2 2 2 2( , ), ( , ), ( , ) ( , )x y z E XE E E the following conditions are satisfied:
(i) Associativity.
2 2 2 2 2 2( ( ( (, ) , ) , ) , ) , ) , ) ,( (( ) ( )x E y E z E x E y FE z EF F F
(ii) Existence of an identity element 2( , )e E , for which
2 2 2 2 2, ) , ) , ) , ) , )( ( ( ( ,(x E e E e EF x E x EF
(iii) Every complex fuzzy element 2 ( , ) x E has an inverse 1 2( , )x E
such that
2 1 2 1 2 2 2 , ) , ) ,( ( ( () , ) ,( )x E x E x E x EF EF e .
COMPLEX INTUITIONISTIC FUZZY SPACE
In this section we generalize Fathi notion of intuitionistic fuzzy spaces (2009) to the
case of complex intuitionistic fuzzy spaces and its properties. Husban (2016) defined
a partial order over 2 2,E E where
2 unit discE .
Let 2 2,E EL where
2 unit discE . Define a partial order on L , in terms of the
partial order on 2E , as follows: For every 21 1 2 2
1 2 1 2( , ), ( , )rr s sii i ir r s s Ee e e e
:
(i) 21 1 2
1 2 1 2, ( ) ( , )rr s sii i ie e e er r s s
, if 1 1 2 21 1 2 2, , , ,r s r sr s r s whenever
1 21 2 and s ss s
(ii) 1 2
1 2(0 , 0 ) ( , )s sii iie e s se e whenever
11 0 , 0ss or 22 0.0, ss
Thus the Cartesian product 2 2 L E E is a distributive, not complemented lattice.
The operation of infimum and supremum in L are given respectively by:
4936 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
2 2 21 1 2 1 1
2 21 1 2 1 1 2
1 2 1 2 1 1 2 2
1 2 1
( )( )
( )( )
2 1 1 2 2
, , ,
, , ,
( ) ( ) ( )
( ) ( ) ( ).
r r s
s
r s s r s
r rr s s r s
i ii i i i
i ii i i i
e e e e e e
e e e e e e
r r s s r s r s
r r s s r s r s
The scaling factor , [0,2 ] is used to confine the performance of the phases
with in the interval [0,2 ] , and the unit disc. Hence, the phase terms may represent
the fuzzy set information, and phase terms values in this representation case belong to
the interval [0, 1] and satisfied fuzzy set restriction. So, will always be considered
equal to 2
Remark.1 The complex intuitionistic fuzzy set ( , , ) :{ }r ri iA x re r e x X
in X
will be denoted by or simply , ( ), ( )( )A x A x xA where, ( )ri
A x re
and
( ) .ri
A x r e
The support of the complex intuitionistic fuzzy set
( , , ) : }r ri i
A x re re x X
in X is the subset A of X defined by:
0 : 0, 0 and 1, 2 .{ }r rA x X r r
Definition 3.1 Let X be a nonempty set. A complex intuitionistic fuzzy space
denoted by 2 2( , , )X E E , where 2E is unit disc in complex plane is the set of all
ordered triples 2 2( , , )x E E where
2 2 2( , , ) , , ) : ,( }{ s sr ri ii ix E E x r e s e re se E
with 1r s , [0,2 ]r s and .x X The ordered triple 2 2( , , )x E E is called a
complex intuitionistic fuzzy element of the complex intuitionistic fuzzy spaces
2 2( , , )X E E and the condition, 2, sr iire se E
with 1sr iire se will be referred to
as the complex intuitionistic condition.
Therefore a complex intuitionistic fuzzy space is an (ordinary) set with ordered
triples. In each triplet the first component indicates the (ordinary) element while the
second and the third components indicate its set of possible a complex–valued grade
of both membership (ire
where r represents an amplitude term and represents a
phase term). and non-comembership values respectively. ( ire where r represents
an amplitude term and represents a phase term).
Definition 3.2 Let 0 U be the support of a given subset U of X . A complex intuitionistic fuzzy subspace U of the complex intuitionistic fuzzy space
2 2, ,( ) X E E is the collection of all ordered triples , , ),( r ri ix re re where, x U and
,r ri ire re are subsets of 2E such that rire contains at least one element beside the
Complex Intuitionistic Fuzzy Group 4937
zero element and rire contains at least one element beside the unit. If 0x U , then
0 and 0rr and 1 , 2rr . The ordered triple ( ), ,r ri ix re re will be
called a complex intuitionistic fuzzy element of the complex intuitionistic fuzzy
subspace U . The empty complex fuzzy subspace denoted by is defined to be:
2 2( , , ) :{ }.x E E x
Remark For the sake of simplicity, throughout this paper by saying a complex
intuitionistic fuzzy space X we mean the complex intuitionistic fuzzy space
2 2( , , ).X E E
Let X be a complex intuitionistic fuzzy space and be a complex intuitionistic fuzzy
subset of .X The complex intuitionistic fuzzy subset A induces the following
complex intuitionistic fuzzy subspaces:
The lower- upper complex intuitionistic fuzzy subspace induced by :rire
0( , , ) : ( ) { }r r ri iclu
iH x re re x Are
The upper-lower complex intuitionistic fuzzy subspace induced by rire :
0,{(0,0)} , {(1,1)( ) } : {( ) }r r ri ic
iluH x re re x Are
The finite complex intuitionistic fuzzy subspace induced by rire is denoted by:
0 0( ,{(0,0), },{ ,(1,1)}) : ( ) { }.r rr ii iH x re re x Are
The union and intersection of two complex intuitionistic fuzzy subspaces
U ,( ),{ r ri ix re re 0: }x U and 0( ) :, ,{ }s si ix se s UeV x
of the complex
intuitionistic fuzzy space X are given in the following definition:
Definition 3.3 The union U V and intersection U V is given respectively by
( ) ( )
( ) ( )
( , , ) :
( , , ) : .
{ }
{ }
r s r s
r s r s
i ix x x x
i ix x x x
U V x r s e r s e x U V
U V x r s e r s e x U V
Definition 3.4 The complex intuitionistic fuzzy Cartesian product of the complex
intuitionistic fuzzy space, denoted by 2 2 2 2( , , ) ( , , )X E E Y E E is defined by
2 2 2 2 2 2 2 2
2 2 2 2
( , , ) ( , , ) ( ), ,
( ), , : .( )
( )
{ }
X E E Y E E X Y E E E E
x y E E E E x X y Y
Definition 3.5 The complex intuitionistic fuzzy Cartesian product of the complex
intuitionistic fuzzy subspaces ( , , ) :{ }r ri iU x r e r e x U
and ( , , ) : }{ s si iV y se se y V
4938 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
of the complex intuitionistic fuzzy spaces2 2 2 2( , , ) and ( , , )X E E Y E E respectively is
the complex intuitionistic fuzzy subspaces of the complex intuitionistic fuzzy
Cartesian product 2 2 2 2( ), ,( )X Y E E E E which is denoted by U V and
given by
( , ) ( )(( , ), , ) : ( , ) .{ }x y x yi i
x y x yV x y r s e r s e x y U VU
Definition 3.6 A complex intuitionistic fuzzy relation ρ from a complex intuitionistic
fuzzy space X to a complex intuitionistic fuzzy space Y is a subset of the complex
intuitionistic fuzzy Cartesian product 2 2 2 2, , , , ( ) ( ).X E E Y E E A complex
intuitionistic fuzzy relation from a complex intuitionistic fuzzy space X into itself is
called a complex intuitionistic fuzzy relation in the complex intuitionistic fuzzy space
.X
From the above definition we note that a complex intuitionistic fuzzy relation from a
complex intuitionistic fuzzy space X to a intuitionistic fuzzy space Y is simply a
collection of complex intuitionistic fuzzy subsets of 2 2 2 2, , , , ( ) ( )X E E Y E E , also
the complex intuitionistic fuzzy Cartesian product 2 2 2 2( , , ) ( , , )X E E Y E E is itself
an complex intuitionistic fuzzy relation.
Definition 3.7 A complex intuitionistic fuzzy function between two complex
intuitionistic fuzzy spaces X and Y is a complex intuitionistic fuzzy relation F
from the X to Y satisfying the following conditions:
(i) For every x X with 2, sr iire se E
, there exists a unique element y Y
with 2, w zi iw e z e E
such that ( , ), ( , ), ( , )( )w sr zi ii ix y re w e se z e A for
some A F
(ii) If 1 1 1 1
1 1 1 1( , ), ( , ), ( , ) ( )r w s zi i i ix y re w e s e z e A F , and
2 2 2 2
2 2 2 2( , ), ( , ), ( , )( )r w s zi i i ix y r e w e s e z e B F then y y
(iii) If 1 1 1 1
1 1 1 1, ) , ), ( ,( , ( ))( r w s zi i i ix y re w e s e z e A F , and
2 2 2 2
2 2 2 2( , ), ( , ), ( , )( )r w s zi i i ix y r e w e s e z e B F then
1 21 2( ) ,( )r rr r
implies 1 21 2( , () )w ww w and
1 21 2 , )( s ss s implies
1 21 2( , () )z zz z
Complex Intuitionistic Fuzzy Group 4939
(iv) If 1 1 1 1
1 1 1 1( , ), ( , ), ( , )( )r w s zi i i ix y re w e s e z e A F , then 0, 0rr
implies 0, 0ww , 1, 1ss implies 0 , 0zz and 1, 2rr
implies 1, 1ww , 0 , 0ss implies 1 , 2zz
Conditions (i) and (ii) imply that there exists a unique (ordinary) function from X to
,Y namely : F X Y and that for every x X there exists unique (ordinary)
functions from 2E to 2E , namely 2 2 , : .x xf f E E On the other hand conditions
(iii) and (iv) are respectively equivalent to the following conditions:
(i) ,x xf f are non-decreasing on 2E
(ii) 3
40( )0 0 (1 )i
x xif e f e
and 0 0(1 ) 1 (1 ).i
x xif e f e
That is a complex intuitionistic fuzzy function between two complex intuitionistic
fuzzy space X and Y is a function F from X to Y characterize by the ordered
triple:
( ),{ } ,{ }( )x X xx x XF x ff
where, ( )F x is a function from X to Y and { } ,{ }x X x Xx xf f are family of
functions from 2E to 2E satisfying the conditions (i) and (ii) such that the image of
any complex intuitionistic fuzzy subset A of the complex intuitionistic fuzzy space
X under F is the intuitionistic fuzzy subset ( ) F A of the intuitionistic fuzzy space
Y defined by:
1 1
1
( ) ( )
1
( ), (
(0,1)
) if ( )
if (, )
( ),r ri ixx
x F y x F yf re f re
F A yF y
F y
We will call the functions ,x xf f the co-membership functions and the co-
nonmembership functions, respectively. The complex intuitionistic fuzzy function F
will be denoted by: , , .( )x xF F f f
4940 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
COMPLEX INTUITIONISTIC FUZZY GROUPS
Here, we introduce the concept of complex intuitionistic fuzzy group and study some
of its properties. First we start by defining complex intuitionistic fuzzy binary
operation on a given complex intuitionistic fuzzy space.
Definition 4.1 A complex intuitionistic fuzzy binary operation F on a complex
intuitionistic fuzzy space 2 2),( ,X E E is a complex intuitionistic fuzzy function
2 2 2 2 2 2: ( , , ) ( , , ) ( , , )F X E E X E E X E E with co-membership functions xyf and
co-nonmembership functions xyf satisfying:
(i) ,( ) 0sry
iixf re se
iff 0, 0rr and 0, 0ss and ( , ) 1w zix
iy we zef
iff 1, 2ww and 1, 2 .zz
(ii) ,xy xyf f are onto. That is, 2 2 2( )xyf E E E and 2 2 2( ) .xyf E E E
Thus for any two complex intuitionistic fuzzy elements 2 2 2 2, , , , ( ) ),(x E E y E E of
the complex intuitionistic fuzzy space X and any complex intuitionistic fuzzy binary
operation ( , , )xy xyfF F f defined on a complex intuitionistic fuzzy space ,X the
action of the complex intuitionistic fuzzy binary operation ( , , )xy xyfF F f over the
complex intuitionistic fuzzy spaces X is given by:
2 2 2 2 2 2 2 2
2 2
, ,( ) ( ) ( , ), ( ), ( ), ,
( , ), ,
( )
( ).
xy xyF x y f fx E E F y E E E E E E
F x y E E
A complex intuitionistic fuzzy binary operation is said to be uniform if the associated
co-membership and co-nonmembership functions are identical. That is, xy xyff f
for all , .x y X
Definition 4.2 A complex intuitionistic fuzzy groupoid, denoted by 2 2, , ( ); ( )X E E F
is a complex intuitionistic fuzzy space 2 2, ,X E E together with a complex
intuitionistic fuzzy binary operation F defined over it. A uniform (left uniform, right
uniform) complex intuitionistic fuzzy groupoid is a complex intuitionistic fuzzy
groupoid with uniform (left uniform, right uniform) complex intuitionistic fuzzy
binary operation.
The next theorem gives a correspondence relation between complex intuitionistic fuzzy groupoid, intuitionistic fuzzy groupoid and ordinary (fuzzy) groupiods.
Complex Intuitionistic Fuzzy Group 4941
Theorem 4.1 To each complex intuitionistic fuzzy groupoid 2 2, , ( ); ( )X E E F there
is an associated complex fuzzy groupoid 2( , ),( )X E F where ( , )xyFF f which is
isomorphic to the complex intuitionistic fuzzy groupoid by the correspondence 2 2 2( , , ) ( , )x E E x E
Proof. Let 2 2, , ( ); ( )X E E F be given complex intuitionistic fuzzy groupoid. Now
redefine , )( ,x xyyF F f f to be ( , )xyF F f since xy
f satisfies the axioms of
complex fuzzy comembership function ( , )xyF F f will be complex fuzzy binary
operation in the sense of Husban (2016).
As a consequence of the above theorem and by using Theorem 4.2 of Fathi et al.
(2009) and by using Theorem 6.1 Husban et al. (2016) we obtain the following
corollary’s.
Corollary 4.1 To each complex intuitionistic fuzzy groupoid 2 2, , ( ); ( )X E E F there
is an associated intuitionistic fuzzy groupoid , , ), ( ) (X I I F where , , ), ( ) (X I I F which is isomorphic to the complex intuitionistic fuzzy groupoid by the
correspondence 2 2( , , ) ( , , ).x E E x I I
Corollary 4.2 To each complex intuitionistic fuzzy groupoid 2 2, , ( ); ( )X E E F there
is an associated fuzzy groupoid ( , ),( )X I F where ( , )xyFF f which is isomorphic
to the complex intuitionistic fuzzy groupoid by the correspondence 2 2( , , ) ( , ).x E E x I
Corollary 4.3 To each complex intuitionistic fuzzy groupoid 2 2, , ( ); ( )X E E F there
is an associated (ordinary) groupoid ( , )X F which is isomorphic to the complex
intuitionistic fuzzy groupoid by the correspondence 2 2( , , ) .x E E x
Definition 4.3 The ordered pair ( ); U F is said to be a complex intuitionistic fuzzy
subgroupoid of the complex intuitionistic fuzzy groupoid 2 2, , ( ); ( )X E E F iff U is
a complex intuitionistic fuzzy subspace of the complex intuitionistic fuzzy space X
and U is closed under the complex intuitionistic binary operation F .
4942 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
Definition 4.4 A complex intuitionistic fuzzy semigroup is a complex intuitionistic
fuzzy groupoid that is associative. A complex intuitionistic fuzzy monoid is complex
intuitionistic fuzzy semi-group that admits an identity.
After defining the concepts of complex intuitionistic fuzzy groupoid, complex
intuitionistic fuzzy semi-group and complex intuitionistic fuzzy monoid we introduce
now the notion of complex intuitionistic fuzzy group.
Definition 4.5 A complex intuitionistic fuzzy group is a complex intuitionistic fuzzy
monoid in which each complex intuitionistic fuzzy element has an inverse. That is, a
complex intuitionistic fuzzy groupoid 2 2, , ( ) ;( )EG E F is a complex intuitionistic
fuzzy group iff the following conditions are satisfied:
(i) Associativity:
2 2 2 2 2 2 2 2( , , ), ( , , ), ( , , ) ( , , );( )x E E y E E z E E G E E F
2 2 2 2 2 2 2 2 2 2 2 2( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , )( ) ( )x E E F y E E F z E E x E E F y E E F z E E
(ii) Existence of an identity: there exists a complex intuitionistic fuzzy element
2 2 2 2( , , ) ( , , )e E E G E E such that for all 2 2( , ),X E E in
2 2( , , ); :( )G E E F
2 2 2 2 2 2 2 2 2 2( , , ) ( , , ) ( , , ) ( , , ) ( , , ).e E E F x E E x E E F e E E x E E
(iii) Existence of an inverse: for every complex intuitionistic fuzzy element
2 2( ), ,Ex E in 2 2( , , ); :( )G E E F there exists a complex intuitionistic fuzzy
element 1 2 2, ), ( E Ex
in 2 2, ,( );( )G E E F such that:
2 2 1 2 2 1 2 2 2 2 2 2( , , ) ( , , ) ( , , ) ( , , ) ,( , ).x E E F x E E x E E F x E E e E E
A complex intuitionistic fuzzy group 2 2( , , ),( )G E E F is called an abelian
(commutative) complex intuitionistic fuzzy group if and only if for all 2 2 2 2 2 2, , , ( , ( ) ( );, ) , , ( )x E E y E E G E E F
2 2 2 2 2 2 2 2( , , ( , , ) ( , , ) ( )) ., ,x E E F y E E y E E F x E E
By the order of a complex intuitionistic fuzzy group we mean the number of complex
intuitionistic fuzzy elements in the complex intuitionistic fuzzy group. A complex
intuitionistic fuzzy group of infinite order is an infinite complex intuitionistic fuzzy
group.
Complex Intuitionistic Fuzzy Group 4943
Theorem 4.2 To each complex intuitionistic fuzzy group 2 2, , ( ); ( )X E E F there is an
associated complex fuzzy group 2( , ),( )X E F where ( , )xyFF f which is isomorphic
to the complex intuitionistic fuzzy group by the correspondence 2 2 2( , , ) ( , ).x E E x E
Corollary 4.4 To each complex intuitionistic fuzzy group 2 2, , ( ); ( )X E E F there is an
associated intuitionistic fuzzy group , , ), ( ) (X I I F where , , ), ( ) (X I I F which is isomorphic to the complex intuitionistic fuzzy group by the correspondence
2 2( , , ) ( , , ).x E E x I I
Corollary 4.5 To each complex intuitionistic fuzzy group 2 2, , ( ); ( )X E E F there is
an associated fuzzy group ( , ),( )X I F where ( , )xyFF f which is isomorphic to the
complex intuitionistic fuzzy group by the correspondence 2 2( , , ) ( , ).x E E x I
Corollary 4.6 To each complex intuitionistic fuzzy group 2 2, , ( ); ( )X E E F there is an
associated (ordinary) group ( , )X F which is isomorphic to the complex
intuitionistic fuzzy group by the correspondence 2 2( , , ) .x E E x
As a result of previous theorem (which we will refer to as the associativity theorem) a
sufficient and necessary condition for complex intuitionistic fuzzy group is given in
the following corollary
Example 4.1 Consider the set .G a Define the complex intuitionistic fuzzy binary
operation ( , , )xy xyfF F f over the complex intuitionistic fuzzy space 2 2)( , ,G E E such
that: ( , )F a a a and ( , )
, ) ( ) ,( s r sra
i iia r e s e r ef s
( , )
, ) (( )s r sr iia
iaf re se r s e
Thus, the complex intuitionistic fuzzy space 2 2)( , ,G E E together with F define a
complex intuitionistic fuzzy group 2 2( , , ); ( )G E E F .
Example 4.2 Consider the set 3 0,1,2 . Define the complex intuitionistic fuzzy
binary operation ( , , )xyxyF F f f over the complex intuitionistic fuzzy space
2 2
3 ), ( ,E E as follows:
3( ), F x y x y , where, 3 refers to addition modulo 3 and
( ) ( )( , ) , ( , ) .
i i i ii is r sx
r sry
srxyf re se r se f re se r se
4944 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
Then 2 2
3( , , ); ( )E E F is a complex intuitionistic fuzzy group.
The next theorem follows directly from the definition of complex intuitionistic fuzzy
group and the associativity theorem.
Theorem 4.3 For any complex intuitionistic fuzzy group 2 2, ,( );( ),G E E F the
following are true: (1) The complex intuitionistic fuzzy identity element is unique (2) The inverse of each complex intuitionistic fuzzy element 2 2, ,( )x E E 2 2, ,( ); ( )G E E F is unique.
(3) 1 1 2 2 2 2( ) , , ( , , ).( )x E E x E E
(4) 2 2 2 2 2 2, , , ( , ,For al ) , , ,l ( ) ( );( )x E E y E E G E E F
2 2 1 2 2 1 2 2 1 2 2 ( , , ) ( , , ) ( , , ) ( , , )( )x E E F y E E y E E F x E E
(5) For all 2 2 2 2 2 2 2 2( , , ) ( , , ), ( , , ) ( , , ); ( )x E E F y E E z E E G E E F
2 2 2 2 2 2 2 2 2 2 2 2 ( , , ) ( , , ) ( , , ) ( , , ), ( , , ) ( , , )If x E E F y E E z E E F y E E then x E E z E E
2 2 2 2 2 2 2 2 2 2 2 2( , , ) ( , , ) ( , , ) ( , , ), ( , , ) ( , , ).If y E E F x E E y E E F z E E then x E E z E E
Proof. The proof is straightforward and is similar to the ordinary and complex fuzzy
case.
Definition 4.4 Let S be a complex intuitionistic fuzzy subspace of the complex
intuitionistic
fuzzy space 2 2( , , ).G E E The ordered pair ( ); S F is a complex intuitionistic fuzzy
subgroup of the complex intuitionistic fuzzy group 2 2, ,( ),( ),G E E F denoted
by 2 2; , , ; ( )( ) ( )S F G E E F , if ( ); S F defines a complex intuitionistic fuzzy group
under the complex intuitionistic fuzzy binary operation .F
Obviously, if ( ); S F is a complex intuitionistic fuzzy subgroup of 2 2, , ; (( ) )G E E F
and ( ); T F is a complex intuitionistic fuzzy subgroup of ( ); ,S F then ; T F is a
complex intuitionistic fuzzy subgroup of 2 2, , ; (( ) )G E E F . Also if 2 2, , ; (( ) )G E E F is a
Complex Intuitionistic Fuzzy Group 4945
complex intuitionistic fuzzy groups with a complex intuitionistic fuzzy identity then
both 2 2, ,( ); { }e E E F and 2 2, , ; ( )( )G E E F are complex intuitionistic fuzzy subgroups
of complex intuitionistic fuzzy group 2 2, , ; (( ) )G E E F .
Theorem 4.4 Let 0( , :, ){ }s sx xi i
x xe sS x s xe S be a complex intuitionistic fuzzy
subspace of the complex intuitionistic fuzzy space 2 2, ,( ).G E E Then ( ); S F is a complex
intuitionistic fuzzy subgroup of the complex intuitionistic fuzzy group 2 2, , ;( )( )G E E F
iff:
1. ( ); S F is an fuzzy subgroup of the fuzzy group ( ); G F
2. ( ) and xy xFy xy xFx y y yxif f
x xy y x xy yxFii
Fi
y x ys f s s e se f e es s , for all ., ox y S
Proof. If (1) and (2) are satisfied, then:
(i) The complex intuitionistic fuzzy subspace S is closed under :F
Let , , , , ,( )( ) yx yx ii ix y
ix yx s s y s s ee e e be in S then
( , , , , ( , ), , , ,
( ),
) (
,
( ) ) (( )
) .(
yy y yx x x xix x y y xy x y xy x y
xFy xFy
i i ii i i iy
i i
x s s F x s s e s F x y f se e e e s e e e
e e
f s s
xFy s s S
(ii) ; S F satisfies the conditions of complex intuitionistic fuzzy group:
(a) Let ( , , ), ( , , )y yx xx y
ix y
i iix s s y se e e es and ), ,( i i
z zez s s e be in S then:
( ) ( )
( , , ) ( , , ) ( , , )
( , ), , , , ( , , )
( ), , ( , , )
( ) , ,(
( )
(
)
)
(
y yx x x z
y yx
xFy xFy
xFy Fz
x z z
z z
x x yi ii i i i
i ii i i i
i i i i
y x z
xy x y xy x y z z
z zxFy xFy
xFy Fz y z
i
xF F
x s s F y s s F z s s
F x y f s s f s s F z s s
xFy s s F z s s
xFy Fz s
e e e e e e
e e e e e e
e e e e
e s e
( )( ), ,
( , , ) ( , , ) ( , , )
)
( )
( )y
xFy
yx x z
Fz
xF yFz
z
xF yFz
i
i i
i ii
xF yFz xF yFz
x x y yi
zi
zi
e e
e
xF yFz s s
x s s F y s s F ze e ese e s
(b) Since ; oS F is a fuzzy subgroup of the fuzzy group ( ), G F then oS
contains the identity e. That is, ( , , ) i ie ee s se e S Thus:
4946 Rima Al-Husban, Abdul Razak Salleh and Abd Ghafur Bin Ahmad
( , , ) ( , , ) ( , ), , , ,
, ,
, ,
( ) ( )
( )
(
)
)
(
xFe xFe
eFx
x x e e x
e
e x e
Fx
x x e e xe x e xe x e
xFe xFe
eF
i i i i i i
x eF
i i
x
i i
i i
e e ex s s F e s s F x e f s s f s s
xFe s s
eFx s s
e e e e e
e e
e e
, ,
, , , ,
( ) ( )
( )eFx e
e e x x
Fxi i
i i
eFx eFx
e e xi
xi
eFx s s
e s s F x s
e e
e e e es
, ,( )x xx x
i ies s ex
Similarly we can show that ( , , ) ( , , ) ( , , )e e x x x xi i ie e x x x x
i i iee s s e e eF x s se es x s
(c) Again, since ( ), oS F is an (fuzzy) subgroup of the fuzzy group
2 2( , , ); ( )G E E F then oS contains the inverse element 1 x for each x S , that is, for
each 1
1
1
11( , , ) ,x x
x x
i ie ex s s S
then:
1 1 1 1
1 1 1 1 1 11
1
1 1
11 1
1
( , , ( , , ( , ), , , ,
, ,
) ) ( ) ( )
(
( )x x x xx x x x
xFx
ii i iii i ix x xx x x x xx xxx
xFx xFx
i
e e e e e e ex s s F x s s F x x f s s f s s
ex
e
eFx s s
1
1 1
1 1
1 1
1 1
1
1
, ,
, , ,
)
( )
( ) (
xFx
x Fx x Fx
x x
x Fx x Fx
x
i
x
i i
i i
x Fx s se
x s s Fe e x
e
,
( , , ).
)x x
e e
i i
i i
x x
e e
s s
e
e e
e es s
Similarly we can show that 1 1
1 1
1( , , ) ( , , ) ( , , )x x ex x ei i i i i
x x e ex xix s s F x se e s e se e ese
.
By (i) and (ii) we conclude that ( ); S F is a complex intuitionistic fuzzy subgroup of
2 2( , , ); .( )G E E F Conversely if ( ; )S F is a complex intuitionistic fuzzy subgroup of
2 2( , , ); ( )G E E F then (1) holds by the associativity theorem.
Corollary 4.7 Let 0( , :, ){ }s sx xi ix xe sS x s xe S
be a complex intuitionistic fuzzy
subspace of the complex intuitionistic fuzzy space 2 2, ,( ).G E E Then ( ); S F is a complex
intuitionistic fuzzy subgroup of the complex intuitionistic fuzzy group 2 2, , ;( )( )G E E F iff: 1 ( ); S F is an (ordinary) subgroup of the group ( ); G F
2 ( ) and xy xFy xy xFx y y yxif f
x xy y x xy yxFii
Fi
y x ys f s s e se f e es s , for all ., ox y S
Complex Intuitionistic Fuzzy Group 4947
Example 4.3 Let 2 2( , , ); ( )G E E F be defined as in Example 4.1. Consider the complex
intuitionistic fuzzy subspace 1.4, ,( ){ }i iS a e e such that 0 , 1. Then
, S F defines a complex intuitionistic fuzzy subgroup of 2 2( , , ); .( )G E E F If we
consider 1.3 1.7, ( ),{ }i ie eS such that 0 , 1 and
, then ( ), S F defines a complex intuitionistic fuzzy subgroup of
2 2( , , );( )G E E F , where, S S . That is, a complex intuitionistic fuzzy group can
have more than one complex intuitionistic fuzzy subgroup, which is different from
the case of ordinary groups, since an ordinary trivial group can only have one
subgroup, namely the group itself.
Example 4.4 Let 2 2
3( , , ); ( )E E F be defined as in Example 4.2. Consider the
complex fuzzy subspace 1.4 1.3 1.7
0, , ), (1, ),({ }i i i ie e eZ e such that
0 1 and 0 1 . Then ( ), Z F is not a complex intuitionistic fuzzy
subgroup of 2 2( , , ); ( ),G E E F since Z is not closed under F That is:
1.4 1.3 1.7 1.70, , 1, , 1, · , · , .)( ) ()( i i i i i ie e e e e eF Z
If , 1 and , 0 , then 2 2 2 2(0, , ), (1, , )Z E E E E together with F defines a
complex intuitionistic fuzzy subgroup of 2 2( , , ); .( )G E E F
CONCLUSION
In this study, we have generalized the study initiated in (Fathi 2009) about fuzzy
intuitionistic groups to the complex intuitionistic fuzzy group. In this paper we define
the notion of a complex intuitionistic fuzzy group using the notion of a complex
intuitionistic fuzzy space and complex intuitionistic fuzzy binary operation
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