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

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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:


(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


, ( ), ( ) : 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 ,


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


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:


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


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


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


(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


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.


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


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


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 .


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.


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


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 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:


( , , ) ( , , ) ( , ), , , ,

, ,

, ,

( ) ( )

( )

(

)

)

(

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


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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