Complex Intuitionistic Fuzzy Group

Global Jounal of Pue and pplied Mathematics. ISSN 0973-1768 Volume 1, Numbe 6 (016, pp. 499-4949 Reseach India Publications http://www.ipublication.com/gjpam.htm Comple Intuitionistic Fuzzy Goup Rima l-husban a, bdul Razak Salleh b and bd Ghafu Bin hmad b a, b,c School of Mathematical Sciences, Faculty of Science and Technology, Univesiti Kebangsaan Malaysia, 43600 UKM Bangi, Selango DE, Malaysia. bstact The aim of this study is to intoduce the notion of comple intuitionistic fuzzy goups based on the notion of comple intuitionistic fuzzy space this concept genealeis the concept of intuitionistic fuzzy space fom the eal ange of membeship function and non-membeship[0,1], to comple ange of membeship function and, non-membeship unit disc in the comple plane. This genealization leads us to intoduce and study the appoach of intuitionistic fuzzy goup theoy that studied by Fathi (009 in the ealm of comple numbes. In this pape, a new theoy of comple intuitionistic fuzzy goup is developed. lso, some notions, definitions and eamples elated to intuitionistic fuzzy goup ae genealized to the ealm of comple numbes. lso a elation between comple intuitionistic fuzzy goups and classical intuitionistic fuzzy subgoups is obtained and studied. Keywods Comple intuitionistic fuzzy space, comple intuitionistic fuzzy binay opeation, comple intuitionistic fuzzy goup, comple intuitionistic fuzzy subgoup. INTRODUCTION The theoy of intuitionistic fuzzy set is epected to play an impotant ole in moden mathematics in geneal as it epesents a genealization of fuzzy set. The notion of intuitionistic fuzzy set was fist defined by tanassov (1986 as a genealization of Zadeh s (1965 fuzzy set. fte the concept of intuitionistic fuzzy set was intoduced, seveal papes have been published by mathematicians to etend the classical

4930 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad mathematical concepts and fuzzy mathematical concepts to the case of intuitionistic fuzzy mathematics. The study of fuzzy goups was fist stated with the intoduction of the concept of fuzzy subgoups by Rosenfeld (1971. nthony and Shewood (1979 edefined fuzzy subgoups using the concept of tiangula nom. In his emakable pape Dib (1994 intoduced a new appoach to define fuzzy goups using his definition of fuzzy space which seves as the univesal set in classical goup theoy. Dib (1994 emaked the absence of the fuzzy univesal set and discussed some poblems in Rosenfeld s appoach. In the case of intuitionistic fuzzy mathematics, thee wee some attempts to establish a significant and ational definition of intuitionistic fuzzy goup. Zhan and Tan (004 defined intuitionistic fuzzy subgoup as a genealization of Rosenfeld s fuzzy subgoup. By stating with a given classical goup they define intuitionistic fuzzy subgoup using the classical binay opeation defined ove the given classical goup. We intoduce the notion of intuitionistic fuzzy goup based on the notion of intuitionistic fuzzy space and intuitionistic fuzzy function defined by Fathi and Salleh (008 a, b, which will seve as a univesal set in the classical case. Fathi and Salleh (009 intoduced the notion of intuitionistic fuzzy space which eplace the concept of intuitionistic fuzzy univesal set in the odinay case. The notion of intuitionistic fuzzy goup as a genealization of fuzzy goup based on fuzzy space by Dib in 1994 was intoduced by Fathi and Salleh (009. Husban and Salleh (016 genealized the concept of fuzzy space to the concept of comple fuzzy space and intoduced the concept of comple fuzzy goup and comple fuzzy subgoup. In 1989, Buckley incopoated the concepts of fuzzy numbes and comple numbes unde the name fuzzy comple numbes (Buckley 1989. This concept has become a famous eseach topic and goal fo many eseaches (Buckley 1989, Ramot et al. 00, lkoui and Salleh 01, 013. Howeve, the concept given by Buckley has diffeent ange compaed to the ange of Ramot et al. s definition fo comple fuzzy set Buckley s ange goes to the inteval[0, 1], while Ramot et al. s ange etends to the unit cicle in the comple plane. Tami and Kandel (011 developed an aiomatic appoach fo popositional comple fuzzy logic. Besides, it eplains some constaints in the concept of comple fuzzy set that was given by Ramot et al. (003. Moeove, many eseaches have studied developed, impoved, employed and modified Ramot et al. s appoach in seveal fields (Zhang et al. 009, Tami et al. 011, Zhifei et al 01. lkoui and Salleh (01 intoduced a new concept of comple intuitionistic fuzzy set, which is genealized fom the innovative concept of a comple fuzzy set by adding the non-membeship tem to the definition of comple fuzzy set. The novelty of comple intuitionistic fuzzy set lies in its ability fo membeship and non-membeship functions to achieve moe ange of values (lkoui and Salleh 013.

Comple Intuitionistic Fuzzy Goup 4931 In this pape we incopoate two mathematical fields on intuitionistic fuzzy set, which ae intuitionistic fuzzy algeba and comple intuitionistic fuzzy set theoy to achieve a new algebaic system, called, comple intuitionistic fuzzy goup based on comple intuitionistic fuzzy space. We will use Fathi and Salleh (009 appoach and genealize it to comple ealm by following the appoach of Ramot et al. (00. Theefoe, the concept of comple intuitionistic fuzzy space is intoduced. This concept genealizes the concept of intuitionistic fuzzy space. The comple fuzzy intuitionistic Catesian poduct, comple intuitionistic fuzzy functions, and comple intuitionistic fuzzy binay opeations in intuitionistic fuzzy spaces and intuitionistic fuzzy subspaces, as well, ae defined. Having defined comple intuitionistic fuzzy spaces and comple intuitionistic fuzzy binay opeations, the concepts of comple intuitionistic fuzzy goup, comple intuitionistic fuzzy subgoup ae intoduced and the theoy of intuitionistic comple fuzzy goups is developed. PRELIMINRIES This section is set to eview the intoductoy definitions, eplanations and outcomes that ae consideed mandatoy to undestand the pupose of this pape. Definition.1 (tanassov 1986 an intuitionistic fuzzy set in a non-empty set U (a univese of discouse is an object having the fom:,, : U whee, the functions ( : U [0,1] and ( : U [0,1], denote the degee of membeship and degee of non-membeship of each element espectively, and 0 ( ( 1 fo all U. U to the set Definition. (Fathi 009 Let X be a nonempty set. n intuitionistic fuzzy space denoted by ( X, I [0,1], I [0,1] is the set of all odeed tiples (, I, I, whee (, I, I {(,, s:, s I} with s 1 and X. The odeed tiplet (, I, I is called an intuitionistic fuzzy element of the intuitionistic fuzzy space ( X, I, I and the condition, s I with s 1 will be efeed to as the intuitionistic condition. Definition.3 (Fathi 009 Let G be a nonempty set. an intuitionistic fuzzy goup is an intuitionistic fuzzy monoid in which each intuitionistic fuzzy element has an invese. That is, an intuitionistic fuzzy goupoid ( G, I, I, F ( F, f, f is an intuitionistic fuzzy goup iff the following conditions ae satisfied: y y

493 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad (,,,(,,,(,, (,,, (1 ssociativity: fo any choice of I I y I I z I I G I I F ((, I, I F( y, I, I F( z, I, I (, I, I F( ( y, I, I F ( z, I, I. ( Eistence of an identity: thee eists an intuitionistic fuzzy element eists an intuitionistic fuzzy element ( e, I, I (G, I, I such that fo all ( X, I, I in (( G, I, I, F : ( e, I, I F(, I, I (, I, I F( e, I, I (, I, I. (3 Eistence of an invese: fo evey intuitionistic fuzzy element (, I, I in (( G, I, I, F thee eists an intuitionistic fuzzy element ( 1, I, I in (( G, I, I, F such that: 1 1 1 (, I, I F(, I, I (, I, I F(, I, I (, I, I ( e, I, I Definition.4 (Ramot et al. 00 comple fuzzy set, defined on a univese of discouse U, is chaacteized by a membeship function (, that assigns to any element U a comple-valued gade of membeship in. By definition, the values of ( ae thus of the fom, may eceive all lying within the unit cicle in the comple plane, and i e, whee i 1, each of ( and ( ae both eal-valued, and [0, 1]. The comple fuzzy set may be epesented i ( as the set of odeed pais U e U (, : (, :. Definition.5 (lkoui & Salleh 01 comple tanassov s intuitionistic fuzzy set, defined on a univese of discouse U, is chaacteized by membeship and nonmembeship functions ( and (, espectively, that assign to any element U a comple-valued gade of both membeship and non-membeship in. By, (, and thei sum may eceive all that lies within definition, the values of the unit cicle in the comple plane, and ae of the fom fo membeship function in and function in, whee 1 i ( ( e i k e fo non-membeship i, each of ( and k ae eal-valued and both belong to the inteval [0, 1] such that 0 ( k ( 1, also and ae eal valued. We epesent the comple tanassov s intuitionistic fuzzy set as

Comple Intuitionistic Fuzzy Goup 4933,, : U, whee ( : U a a C, a 1, :, 1 U a a C a, 1. Definition.6 (lkoui & Salleh 01. The complement, union and intesection two CIFSs, (, ( : U and B, (, ( : U of discouse U, ae defined as follows: in a univese B B (i Union of comple intuitionistic fuzzy sets: B, B(, B( : U i whee ma[, ] e B B (ma(, B i(min(, B and min[ k, k ] e. B B (ii Intesection of two comple intuitionistic fuzzy sets: B, B(, B( : U i(min(, B whee B min[, B ] e i(ma( (, ( B and B ma[ k, kb] e. Definition.7 (lkoui & Salleh 013 Let,..., 1 be comple tanassov s n intuitionistic fuzzy sets in X,..., 1 X n, espectively. The Catesian poduct 1... n is a comple intuitionistic fuzzy set defined by... (,...,, (,...,, (,..., : (,..., X... X 1 n 1 n 1... n 1 n 1... n 1 n 1 n 1 n whee (,..., [min( (,..., ( e imin( (,..., ( 1 n 1 n 1... 1 1 1 ] n n n n ima( (,..., (... ] 1 1 n n nd ( 1,..., n [ma( ( 1,..., ( n e. 1 n 1 n In the case of two comple tanassov s intuitionistic fuzzy sets and B in X and Y espectively, we have B (, y, (, y, (, y : (, y X Y B B imin(, ( y whee (, y min( ( B B, B( y e,

4934 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad ima(, ( y and (, y ma( ( B B, B( y e. Definition.8 (lkoui & Salleh 013 If and B ae comple tanassov s intuitionistic fuzzy sets in a univese of discouse U, whee and i i {(,, k e k e } i B i B {(,, k e k e }, then B B 1. B if and only if and k k, fo amplitude B B tems and the phase tems (aguments and, fo all U.. B if and only if and k k, fo amplitude B B tems and the phase tems (aguments and, fo all U. B B k k k k B B Definition.9 ( Husban & Salleh 016 comple fuzzy space, denoted by ( X, E, whee E is the unit disc, is set of all odeed pais X, i.e., ( X, E {(, E : X }. We can wite i 1, [0, 1], and [0, ]. The odeed pai (, E fuzzy element in the comple fuzzy space ( X E { }, (, E (, e : e E whee,. is called a comple Definition.10 ( Husban & Salleh 015 The comple fuzzy subspace U of the i comple fuzzy space ( X, E is the collection of all odeed pais (, e whee U 0 fo some U0 X and i e is a subset of E, element beside the zeo element. If it happens that U, 0 U 0 0. The comple fuzzy subspace U is denoted by { } 0 which contains at least one then 0 and U (, e : U (, e : X, 0 1, 0 { is called the suppot of U, that is U 0 U : 0 and 0. and denoted by SU U. 0 }

Comple Intuitionistic Fuzzy Goup 4935 Definition.11 ( Husban & Salleh 016 comple fuzzy binay opeation F ( F, on the comple fuzzy space ( X, E is a comple fuzzy function ƒ y fom ( X, E ( X, E ( X, E with comembeship functions ƒ y satisfying: i (i ƒ i y e, s e s 0if 0, s 0, s 0 and 0, (ii ƒy ae onto, i.e., ƒ y ( E E E,, y X. Definition.1 ( Husban & Salleh 016 comple fuzzy algebaic system (( X, E, F is called a comple fuzzy goup if and only if fo evey (, E, ( y, E, ( z, E ( X, E (i ssociativity. the following conditions ae satisfied: (( E ( y E ( z E ( E ( y F, F, F,, F (, E ( z, E, (ii Eistence of an identity element ( e, E, fo which (, E F ( e, E ( e, E F (, E (, E, (iii Evey comple fuzzy element (, E has an invese 1 (, E 1 1 (, E F (, E (, E F (, E ( e, E. such that COMPLEX INTUITIONISTIC FUZZY SPCE In this section we genealize Fathi notion of intuitionistic fuzzy spaces (009 to the case of comple intuitionistic fuzzy spaces and its popeties. Husban (016 defined a patial ode ove E E, whee Let L E E, whee E unit disc E unit disc.. Define a patial ode on L, in tems of the 1 1 patial ode on E 1 1, as follows: Fo evey ( e, e, ( s e s, s e s E : 1 1 (i ( 1, ( s e e s1e, s s e, if 1 s1,, 1 s 1 s,, s wheneve s s 1 and s s 1 i i i s i 1 s (ii (0 e, 0 e ( s1e, s e wheneve s 0, 0 1 s o s 1 0, s 0. Thus the Catesian poduct L E E is a distibutive, not complemented lattice. The opeation of infimum and supemum in L ae given espectively by:

4936 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad 1 s 1 s i( i 1 s1 s 1e, e 1e, e 1 1e, e ( ( s s ( s s 1 s 1 s i i 1 s1 s 1e, e 1e, e 1 1e, e ( ( s s ( s s. The scaling facto, [0, ] is used to confine the pefomance of the phases with in the inteval [0, ], and the unit disc. Hence, the phase tems may epesent the fuzzy set infomation, and phase tems values in this epesentation case belong to the inteval [0, 1] and satisfied fuzzy set estiction. So, will always be consideed equal to { } Remak.1 The comple intuitionistic fuzzy set (, e, e : X in X i will be denoted by o simply,, whee, ( e and ( e. The suppot of the comple intuitionistic fuzzy set in X is the subset of X defined by: (, e, e : X} { } 0 X : 0, 0 and 1,. Definition 3.1 Let X be a nonempty set. comple intuitionistic fuzzy space denoted by odeed tiples with s 1 ( X, E, E, whee (, E, E whee, [0, ] s E is unit disc in comple plane is the set of all s s (, E, E { (, e, s e : e, se E } and X. The odeed tiple (, E, E is called a comple intuitionistic fuzzy element of the comple intuitionistic fuzzy spaces ( X, E, E and the condition, s, as the comple intuitionistic condition. i e se E i s with e se 1 will be efeed to Theefoe a comple intuitionistic fuzzy space is an (odinay set with odeed tiples. In each tiplet the fist component indicates the (odinay element while the second and the thid components indicate its set of possible a comple valued gade of both membeship ( i e whee epesents an amplitude tem and epesents a i phase tem. and non-comembeship values espectively. ( e whee epesents an amplitude tem and epesents a phase tem. Definition 3. Let U 0 be the suppot of a given subset U of X. comple intuitionistic fuzzy subspace U of the comple intuitionistic fuzzy space (,, X E E is the collection of all odeed tiples (, i e, e, whee, U and e, e ae subsets of E such that i e contains at least one element beside the

Comple Intuitionistic Fuzzy Goup 4937 zeo element and U, then i e contains at least one element beside the unit. If 0 0 and 0 and 1,. The odeed tiple (, e, e will be called a comple intuitionistic fuzzy element of the comple intuitionistic fuzzy subspace U. The empty comple fuzzy subspace denoted by is defined to be: {(, E, E : }. Remak Fo the sake of simplicity, thoughout this pape by saying a comple intuitionistic fuzzy space X we mean the comple intuitionistic fuzzy space ( X, E, E. Let X be a comple intuitionistic fuzzy space and be a comple intuitionistic fuzzy subset of X. The comple intuitionistic fuzzy subset induces the following comple intuitionistic fuzzy subspaces: i The lowe- uppe comple intuitionistic fuzzy subspace induced by e : i {(, i i H e e, e : } clu The uppe-lowe comple intuitionistic fuzzy subspace induced by { },{(0,0} i H e e, {(1,1 } e : clu 0 0 i e : i The finite comple intuitionistic fuzzy subspace induced by e is denoted by: { }. H e e i i 0 (,{(0,0, i e },{,(1,1} : 0 The union and intesection of two comple intuitionistic fuzzy subspaces U {(, i e, e : U0} and {(, s s V se, se : U0} of the comple intuitionistic fuzzy space X ae given in the following definition: Definition 3.3 The union U V and intesection U V is given espectively by i( s i( s {(,, : } U V s e s e U V { } U V s e s e U V i( s i( s (,, :. Definition 3.4 The comple intuitionistic fuzzy Catesian poduct of the comple intuitionistic fuzzy space, denoted by ( X, E, E ( Y, E, E is defined by ( X, E, E ( Y, E, E (( X Y, E E, E E { } ( y, E E, E E : X y Y. Definition 3.5 The comple intuitionistic fuzzy Catesian poduct of the comple { } intuitionistic fuzzy subspaces U (, e, e : U and V ( y, se, se : y V } { s s

4938 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad of the comple intuitionistic fuzzy spaces (,, and (,, X E E Y E E espectively is the comple intuitionistic fuzzy subspaces of the comple intuitionistic fuzzy Catesian poduct given by U (( X Y, E E, E E which is denoted by U V and { } (, y ( y ((,, y, y : (,. V y s e s e y U V Definition 3.6 comple intuitionistic fuzzy elation ρ fom a comple intuitionistic fuzzy space X to a comple intuitionistic fuzzy space Y is a subset of the comple intuitionistic fuzzy Catesian poduct (,, (,,. X E E Y E E comple intuitionistic fuzzy elation fom a comple intuitionistic fuzzy space X into itself is called a comple intuitionistic fuzzy elation in the comple intuitionistic fuzzy space X. Fom the above definition we note that a comple intuitionistic fuzzy elation fom a comple intuitionistic fuzzy space X to a intuitionistic fuzzy space Y is simply a collection of comple intuitionistic fuzzy subsets of the comple intuitionistic fuzzy Catesian poduct an comple intuitionistic fuzzy elation. (,, (,, X E E Y E E, also ( X, E, E ( Y, E, E is itself Definition 3.7 comple intuitionistic fuzzy function between two comple intuitionistic fuzzy spaces X and Y is a comple intuitionistic fuzzy elation F fom the X to Y satisfying the following conditions: (i Fo evey X with with w e, z e E some F s e, se E, thee eists a unique element y Y w i z w s z such that (,,(,,(, i i i i 1 1 1 1 1 1 1 1 (ii If (, y,( e, w e w, ( s e s, z e z F, and i i w i s i z y e w e se z e fo ((, y,( e, w e, ( s e, z e B F then y y 1 1 1 1 (iii If ((,,( i, i w s z, ( i y e w e s e, z e i F, and 1 1 1 1 ((,, ( i,, (, i w i y e w e s e s z e i z B F then ( 1,( 1 implies ( w1 w,( w 1 w ( z z,( 1 and s s ( s 1 s z z 1 1, implies

Comple Intuitionistic Fuzzy Goup 4939, then 0, 0 1 1 1 1 (iv If (,,( i, i w, ( i y e w e s e s, z e i z F 1 1 1 1 implies w 0, w 0, s 1, s 1 implies z 0, z 0 and 1, implies w 1, w 1, s 0, s 0 implies z 1, z Conditions (i and (ii imply that thee eists a unique (odinay function fom X to Y, namely F : X Y and that fo evey X thee eists unique (odinay functions fom E to E, namely f f E E On the othe hand conditions, :. (iii and (iv ae espectively equivalent to the following conditions: (i f, f ae non-deceasing on (ii 0i 3 4 i f ( 0e 0 f (1 e and E f (1 e 1 f (1 e. 0i 0 i That is a comple intuitionistic fuzzy function between two comple intuitionistic fuzzy space X and Y is a function F fom X to Y chaacteize by the odeed tiple: ( F,{ },{ f } f X X whee, F is a function fom X to Y and { f },{ f } X X ae family of functions fom E to E satisfying the conditions (i and (ii such that the image of any comple intuitionistic fuzzy subset of the comple intuitionistic fuzzy space X unde F is the intuitionistic fuzzy subset F( of the intuitionistic fuzzy space Y defined by: F y ( f ( e, f ( e, i 1 1 F ( y F ( y (0,1 1 if F ( y 1, if F ( y We will call the functions f, f the co-membeship functions and the cononmembeship functions, espectively. The comple intuitionistic fuzzy function F will be denoted by: F F, f, f.

4940 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad COMPLEX INTUITIONISTIC FUZZY GROUPS Hee, we intoduce the concept of comple intuitionistic fuzzy goup and study some of its popeties. Fist we stat by defining comple intuitionistic fuzzy binay opeation on a given comple intuitionistic fuzzy space. Definition 4.1 comple intuitionistic fuzzy binay opeation F on a comple intuitionistic fuzzy space (, X, E E is a comple intuitionistic fuzzy function F : ( X, E, E ( X, E, E ( X, E, E with co-membeship functions f y and co-nonmembeship functions f y satisfying: i i s (i f ( e, se 0 y iff 0, 0 and s 0, s 0 and f w z ( we, ze 1 iff w 1, and z 1,. (ii f, f ae onto. That is, y y w z f y E E E and Thus fo any two comple intuitionistic fuzzy elements y f E E E (. y (,,, (,, E E y E E of the comple intuitionistic fuzzy space X and any comple intuitionistic fuzzy binay opeation F ( F, f, f defined on a comple intuitionistic fuzzy space X, the y y action of the comple intuitionistic fuzzy binay opeation F ( F, f, f ove the comple intuitionistic fuzzy spaces X is given by: (, E, E F ( y, E, E F (, y, f y ( E E, f y ( E E ( F (, y, E, E. comple intuitionistic fuzzy binay opeation is said to be unifom if the associated co-membeship and co-nonmembeship functions ae identical. That is, f y fy f fo all, y X. y y Definition 4. comple intuitionistic fuzzy goupoid, denoted by ( X, E, E ; F is a comple intuitionistic fuzzy space X, E, E togethe with a comple intuitionistic fuzzy binay opeation F defined ove it. unifom (left unifom, ight unifom comple intuitionistic fuzzy goupoid is a comple intuitionistic fuzzy goupoid with unifom (left unifom, ight unifom comple intuitionistic fuzzy binay opeation. The net theoem gives a coespondence elation between comple intuitionistic fuzzy goupoid, intuitionistic fuzzy goupoid and odinay (fuzzy goupiods.

Comple Intuitionistic Fuzzy Goup 4941 Theoem 4.1 To each comple intuitionistic fuzzy goupoid is an associated comple fuzzy goupoid ((,, ((,, ; X E E F thee X E F whee F ( F, f which is isomophic to the comple intuitionistic fuzzy goupoid by the coespondence (, E, E (, E Poof. Let ((,, ; X E E F be given comple intuitionistic fuzzy goupoid. Now edefine F ( F, f, f to be F ( F, f since y y y f satisfies the aioms of y comple fuzzy comembeship function F ( F, f will be comple fuzzy binay opeation in the sense of Husban (016. s a consequence of the above theoem and by using Theoem 4. of Fathi et al. (009 and by using Theoem 6.1 Husban et al. (016 we obtain the following coollay s. y y Coollay 4.1 To each comple intuitionistic fuzzy goupoid ((,, ; X E E F thee is an associated intuitionistic fuzzy goupoid ((X, I, I, F whee ((X, I, I, F which is isomophic to the comple intuitionistic fuzzy goupoid by the coespondence (, E, E (, I, I. Coollay 4. To each comple intuitionistic fuzzy goupoid is an associated fuzzy goupoid ((,, ((,, ; X E E F thee X I F whee F ( F, f which is isomophic to the comple intuitionistic fuzzy goupoid by the coespondence (, E, E (, I. Coollay 4.3 To each comple intuitionistic fuzzy goupoid y ((,, ; X E E F thee is an associated (odinay goupoid ( X, F which is isomophic to the comple intuitionistic fuzzy goupoid by the coespondence (, E, E. Definition 4.3 The odeed pai ( U; F is said to be a comple intuitionistic fuzzy subgoupoid of the comple intuitionistic fuzzy goupoid ((,, ; X E E F iff U is a comple intuitionistic fuzzy subspace of the comple intuitionistic fuzzy space X and U is closed unde the comple intuitionistic binay opeation F.

494 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad Definition 4.4 comple intuitionistic fuzzy semigoup is a comple intuitionistic fuzzy goupoid that is associative. comple intuitionistic fuzzy monoid is comple intuitionistic fuzzy semi-goup that admits an identity. fte defining the concepts of comple intuitionistic fuzzy goupoid, comple intuitionistic fuzzy semi-goup and comple intuitionistic fuzzy monoid we intoduce now the notion of comple intuitionistic fuzzy goup. Definition 4.5 comple intuitionistic fuzzy goup is a comple intuitionistic fuzzy monoid in which each comple intuitionistic fuzzy element has an invese. That is, a comple intuitionistic fuzzy goupoid (( G, E, E ; fuzzy goup iff the following conditions ae satisfied: (i (ii (iii ssociativity: (,,,(,,,(,, (,, ; E E y E E z E E G E E F F is a comple intuitionistic ((, E, E F( y, E, E F( z, E, E (, E, E F( ( y, E, E F( z, E, E Eistence of an identity: thee eists a comple intuitionistic fuzzy element ( e, E, E ( G, E, E such that fo all (,, X E E in ( G, E, E ; F : ( e, E, E F(, E, E (, E, E F( e, E, E (, E, E. Eistence of an invese: fo evey comple intuitionistic fuzzy element (, E, E in element 1 (, E, E (,, ; : G E E F thee eists a comple intuitionistic fuzzy in ((,, ; G E E F such that: 1 1 (, E, E F(, E, E (, E, E F(, E, E ( e, E, E. comple intuitionistic fuzzy goup ((,,, G E E F is called an abelian (commutative comple intuitionistic fuzzy goup if and only if fo all (,,, (,, (,, ; E E y E E G E E F (, E, E F( y, E, E ( y, E, E F(, E, E. By the ode of a comple intuitionistic fuzzy goup we mean the numbe of comple intuitionistic fuzzy elements in the comple intuitionistic fuzzy goup. comple intuitionistic fuzzy goup of infinite ode is an infinite comple intuitionistic fuzzy goup.

Comple Intuitionistic Fuzzy Goup 4943 Theoem 4. To each comple intuitionistic fuzzy goup (( X, E, E ; F thee is an associated comple fuzzy goup ((,, X E F whee F ( F, f which is isomophic to the comple intuitionistic fuzzy goup by the coespondence (, E, E (, E. y Coollay 4.4 To each comple intuitionistic fuzzy goup ((,, ; X E E F thee is an associated intuitionistic fuzzy goup ((X, I, I, F whee ((X, I, I, F which is isomophic to the comple intuitionistic fuzzy goup by the coespondence (, E, E (, I, I. Coollay 4.5 To each comple intuitionistic fuzzy goup an associated fuzzy goup ((,, comple intuitionistic fuzzy goup by the coespondence ((,, ; X E E F thee is X I F whee F ( F, f which is isomophic to the y (, E, E (, I. Coollay 4.6 To each comple intuitionistic fuzzy goup ((,, ; X E E F thee is an associated (odinay goup ( X, F which is isomophic to the comple intuitionistic fuzzy goup by the coespondence (, E, E. s a esult of pevious theoem (which we will efe to as the associativity theoem a sufficient and necessay condition fo comple intuitionistic fuzzy goup is given in the following coollay Eample 4.1 Conside the set G a. Define the comple intuitionistic fuzzy binay opeation F ( F, f, f ove the comple intuitionistic fuzzy space that: F( a, a y a and y f a ( s s i (,, ( s, a e s e e Thus, the comple intuitionistic fuzzy space comple intuitionistic fuzzy goup ((,, ; G E E F. ( G, E, E such f ( e, se ( s e aa s (, s ( G, E, E togethe with F define a Eample 4. Conside the set 3 0,1,. Define the comple intuitionistic fuzzy binay opeation F ( F, f, f ove the comple intuitionistic fuzzy space ( E, E as follows: 3, F(, y y 3, whee, 3 y y efes to addition modulo 3 and s i ( s s i ( s y f ( e, se se, f ( e, se s e. y

4944 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad Then (, E, E ; F is a comple intuitionistic fuzzy goup. 3 The net theoem follows diectly fom the definition of comple intuitionistic fuzzy goup and the associativity theoem. Theoem 4.3 Fo any comple intuitionistic fuzzy goup (( G, E, E ; F, the following ae tue: (1 The comple intuitionistic fuzzy identity element is unique ( The invese of each comple intuitionistic fuzzy element (, E, E ( ( G, E, E ; F is unique. (3 (4 1 1 (, E, E (, E, E. Fo al l (, E, E,( y, E, E ( G, E, E ; F, 1 1 1 (, E, E F( y, E, E ( y, E, E F(, E, E (5 Fo all (,, (,,, (,, (,, ; E E F y E E z E E G E E F If E E F y E E z E E F y E E then E E z E E (,, (,, (,, (,,, (,, (,, If y E E F E E y E E F z E E then E E z E E (,, (,, (,, (,,, (,, (,,. Poof. The poof is staightfowad and is simila to the odinay and comple fuzzy case. Definition 4.4 Let S be a comple intuitionistic fuzzy subspace of the comple intuitionistic fuzzy space ( G, E, E. The odeed pai ( S; F is a comple intuitionistic fuzzy subgoup of the comple intuitionistic fuzzy goup ((,,,, G E E F denoted by( S; F (( G, E, E ; F, if ( S; F defines a comple intuitionistic fuzzy goup unde the comple intuitionistic fuzzy binay opeation F. Obviously, if ( S; F is a comple intuitionistic fuzzy subgoup of and ( T; F is a comple intuitionistic fuzzy subgoup of ( ;, (( G, E, E ; F S F then T; F is a comple intuitionistic fuzzy subgoup of (( G, E, E ; F. lso if (( G, E, E ; F is a

Comple Intuitionistic Fuzzy Goup 4945 comple intuitionistic fuzzy goups with a comple intuitionistic fuzzy identity then both {( e, E, E ; F } and ( G, E, E ; F ae comple intuitionistic fuzzy subgoups of comple intuitionistic fuzzy goup (( G, E, E ; F. { } s s Theoem 4.4 Let S (, se, se : S0 be a comple intuitionistic fuzzy subspace of the comple intuitionistic fuzzy space ( G, E, E. Then ( S; F is a comple intuitionistic fuzzy subgoup of the comple intuitionistic fuzzy goup ( G, E, E ; F iff: 1. ( S; F is an fuzzy subgoup of the fuzzy goup ( G; F, fo all y S. i(. fy y Fy fy y Fy s fys ye s Fy e and s fysye s Fye Poof. If (1 and ( ae satisfied, then: (i The comple intuitionistic fuzzy subspace S is closed unde F :, o i i i y i y Let (, s e, s e, ( y, s e, s e y y be in S then y y y y (, s e, se F(, s y e, sye sy ( F(, y, f y ( se, sye, fy ( se, sye (( Fy, s e, s e S. (ii S; F satisfies the conditions of comple intuitionistic fuzzy goup: y y (a Let (, s e, s e, ( y, s e, s e and ( z, s e, s e be in S then: ( y y z (, s e, s e F( y, s e, s e F( z, s e, s e y ( y y y y z z F(, y, f ( s e, s e, f ( s e, s e F( z, s e, s e Fy i i i i (( Fy, s, s F( z, s, s (b Since o ; Fy Fy z z Fye Fye e e z z FyFz ( Fy ( Fz, s, Fy Fz Fy Fz y z y y y y z z e s e FyFz F yfz F yfz (( F( yfz, s e, s e F yfz F yfz y y z z (, s e, s e F( ( y, s e, s e F( z, s e, s e y y S F is a fuzzy subgoup of the fuzzy goup ( G, F then o contains the identity e. That is, ( e, s e, s e S Thus: e e z z Fy z z S

4946 Rima l-husban, bdul Razak Salleh and bd Ghafu Bin hmad Fe Fe ( Fe, s Fee, sfee ef ef ( ef, s efe, sef e (,, ef ef ef s e s e e e e e (, s e, s e F ( e, s e, s e F(, e, f ( s e, s e, f ( s e, s e e e e e e e ef ef e e ( e, s e, s e F(, s e, s e e e (,, s e s e Similaly we can show that (, i e, i e (, i i i i e s e s e F s e, s e (, s e, s e e e (c gain, since ( S F is an (fuzzy subgoup of the fuzzy goup ((,, ; o, 1 G E E F then S o contains the invese element 1 1 1 each (, s 1e, s 1e S, then: fo each S, that is, fo 1 1 i 1 1 i 1 i 1 1 e e 1e 1e 1 1e 1e 1 e 1e (, s, s F(, s, s F(,, f ( s, s, f ( s, s 1 F 1, 1e, 1 F F ( F s s e F 1 1 1 F 1 F ( F, s 1 e, s 1 e F F 1 1 1, 1e, 1e, ( s s F ( s e, s e e e ( e, s e, s e. e e 1 1 1 Similaly we can show that (, i, i (, i e, i (, i i s e s e F s e s e e s e, s e e. 1 1 e e By (i and (ii we conclude that ( S; F is a comple intuitionistic fuzzy subgoup of G E E F Convesely if ( S; F is a comple intuitionistic fuzzy subgoup of (,, ;. ((,, ; G E E F then (1 holds by the associativity theoem. { } s s Coollay 4.7 Let S (, se, se : S0 be a comple intuitionistic fuzzy subspace of the comple intuitionistic fuzzy space (,,. G E E Then ( S; F is a comple intuitionistic fuzzy subgoup of the comple intuitionistic fuzzy goup ( G, E, E ; F iff: 1 ( S; F is an (odinay subgoup of the goup ( G; F i( fy y Fy fy y Fy s fys ye s Fy e and s fysye s Fye, fo all y S., o

Comple Intuitionistic Fuzzy Goup 4947 Eample 4.3 Let (( G, E, E ; F be defined as in Eample 4.1. Conside the comple intuitionistic fuzzy subspace 1.4 {(, i i, } S a e e such that 0, 1. Then S, F defines a comple intuitionistic fuzzy subgoup of ( G, E, E ; F. If we { } conside 1.3 1.7 (, i i S e, e such that 0, 1 and, then ( S, F defines a comple intuitionistic fuzzy subgoup of G E E F, whee, S S. That is, a comple intuitionistic fuzzy goup can ((,, ; have moe than one comple intuitionistic fuzzy subgoup, which is diffeent fom the case of odinay goups, since an odinay tivial goup can only have one subgoup, namely the goup itself. Eample 4.4 Let ((, E, E ; F be defined as in Eample 4.. Conside the comple fuzzy subspace 3 1.4 1.3 1.7 {( 0, e, e i,(1, e i, e i } Z such that 0 1 and 0 1. Then ( Z, F is not a comple intuitionistic fuzzy subgoup of (( G, E, E ; F, since Z is not closed unde F That is: i i i i i i ( 1.4 1.3 1.7 1.7 0, e, e F 1, e, e 1, e,, e Z. If, 1 and, 0, then Z (0, E, E,(1, E, E comple intuitionistic fuzzy subgoup of (( G, E, E ; F. togethe with F defines a CONCLUSION In this study, we have genealized the study initiated in (Fathi 009 about fuzzy intuitionistic goups to the comple intuitionistic fuzzy goup. In this pape we define the notion of a comple intuitionistic fuzzy goup using the notion of a comple intuitionistic fuzzy space and comple intuitionistic fuzzy binay opeation REFERENCE [1] l-husban,, Salleh,. R. & Hassan. N. 015. Comple fuzzy nomal subgoup. The 015 UKM FST Postgaduate Colloquium. IP Confeence Poceedings, 1678, 060008. [] l-husban,, & Salleh,. R. 016. Comple fuzzy goup based on comple fuzzy space. Global Jounal of Pue and pplied Mathematics (1: 1433-1450.

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