Glbal Jurnal f Pure and Applied Mathematics. ISSN 973-768 Vlume, Number (6, pp. 433-45 Research India Publicatins http://www.ripublicatin.cm Cmple Fuzz Grup Based n Cmple Fuzz Space Abdallah Al-Husban and Abdul Razak Salleh Schl f Mathematical Sciences, Facult f Science and Technlg, Universiti Kebangsaan Malasia, 436 UKM Bangi, Selangr DE, Malasia. Abstract In this paper the cncept f cmple fuzz space and cmple fuzz binar peratin are presented and develped. This cncept generalises the cncept f fuzz space frm the real range f membership functin, [,], t cmple range f membership functin, unit disc in the cmple plane. This generalisatin leads us t intrduce and stud the apprach f fuzz grup ther in the realm f cmple numbers. A new ther f cmple fuzz grup is intrduced and develped. Kewrds: Cmple fuzz space, cmple fuzz subspace, cmple fuzz binar peratin, cmple fuzz grup, cmple fuzz subgrup.. Intrductin In the last fur decades, the fuzz grup ther was develped as fllws. Rsenfeld (97 intrduced the cncept f fuzz subgrup f a grup. In 975 Negita and Ralescu cnsidered a generalisatin f Rsenfeld s definitin in which the unit interval [, l] was replaced b an apprpriate lattice structure. In 979 Anthn and Sherwd redefined fuzz subgrup f a grup using the cncept f triangular nrm previusl defined b Schweizer and Sklar (96. Several mathematicians fllwed the Rsenfeld-Anthn-Sherwd apprach in investigating fuzz grup ther (Akgiil 988; Anthn & Sherwd 98; Das 98; Liu 98; Sessa 984; Sherwd 983. Yussef and Dib (99 intrduced a new apprach t define fuzz grupid and fuzz subgrupid. In the absence f the cncept f fuzz universal set, frmulatin f the intrinsic definitin fr fuzz grup and fuzz subgrup is nt evident. This is cnsidered as a purpse t intrduce fuzz universal set and fuzz binar peratin b Yussef and Dib (99 and Dib (994. The main difference between the Rsenfeld s apprach and that f Yussef and Dib (99 is the
434 Abdallah Al-Husban and Abdul Razak Salleh replacement f the t-nrm f with a famil f c-membership functins { f :, X }. Dib (994 intrduced the cncept f fuzz space. Fathi and Salleh (9 generalised the cncept f fuzz space t the cncept f intuitinitic fuzz space.this cncept replaces the cncept f universal set in the rdinar case. The fuzz Cartesian prduct, fuzz functins, and fuzz binar peratins in fuzz spaces and fuzz subspaces, as well, were defined b Dib (994. After having defined fuzz space and fuzz binar peratin, the cncepts f fuzz grup, fuzz subgrup, and fuzz hmmrphism were intrduced and the ther f fuzz grups was cnstructed in the same paper. In 998, Dib and Hassan intrduced the cncept f fuzz nrmal subgrup. In 989 Buckle incrprated the cncepts f fuzz numbers and cmple numbers under the name fuzz cmple numbers. This cncept has becme a famus research tpicand a gal fr man researchers (Buckle 989; Ramtet al. ; Qiu & Shu 8. Hwever, the cncept given b Buckle has different range cmpared t the range f Ramt et al. s definitin fr cmple fuzz set (CFS.Buckle s range ges t the interval [,], while Ramt et al. s range etends t the unit circle in the cmple plane. Tamir et al. (, develped an aimatic apprach fr prpsitinal cmple fuzz lgic. Besides, it eplains sme cnstraints in the cncept f CFS that was given b Ramt et al. (3. Mrever, man researchers have studied, develped, imprved, empled and mdified Ramt et al. s apprach in several fields (Tamir et al. ; Zhang et al. 9; Jun et al.. In particular, the cncept f fuzz set spreads in man mathematical fields, such as tplg, algebra and cmple numbers. In this paper we incrprate tw mathematical fields n fuzz set, which are fuzz algebra and cmple fuzz set ther t achieve a new algebraic sstem, called cmple fuzz grup based n cmple fuzz space. We will use the apprach f Yussef and Dib (99 and generalise it t cmple realm b fllwing the apprach f Ramt et al. ( and Tamir et al. ( t intrduce the cncept f cmple fuzz space. This cncept generalises the cncept f fuzz space f Yussef and Dib. Having defined cmple fuzz spaces and cmple fuzz binar peratins, the cncepts f cmple fuzz grup and cmple fuzz subgrup are intrduced and the ther f cmple fuzz grups is develped. In particular, the cncept f fuzz set spreads in man pure mathematical fields, such as tplg (Lwen 976, algebra (Hungerfrd 974 and cmple numbers (Buckle 989. Our aim in this stud is t incrprate tw mathematical fields n fuzz set, which are fuzz algebra and cmple fuzz set ther t achieve a new algebraic sstem, called cmple fuzz grup based n cmple fuzz space t generalise the cncept f fuzz space. We will use the apprach f Yssef and Dib (99 and generalise it t cmple realm b fllwing the apprach f Ramt et al. ( and Tamir et al. (. Having defined cmple fuzz space and cmple fuzz binar peratin, the cncepts f cmple fuzz grup and cmple fuzz subgrup are intrduced and the ther f cmple fuzz grup is develped.
Cmple Fuzz Grup Based On Cmple Fuzz Space 435. Preliminaries In this sectin we recall the definitins and related results which are needed in this wrk. Definitin. (Zadeh 965 A fuzz se t A in a universe f discurse U is characterised b a membership functin μ ( that takes values in the interval [, ]. A Definitin. (Dib 994 Afuzz space ( X, I = [,] is the set f all rdered pairs (, I, X ( X, I = {(, I: X }, where (, I = {(, r: r I}. The rdered pair (, I is called a fuzz element in the fuzz space (X, I. Definitin.3 (Dib 994 A fuzz s ubspace U f the fuzz space (X, I is the cllectin f all rdered pairs (, u, where U fr sme U X and u is a subset f I, which cntains at least ne element beside the zer element. If it happens that U, then u =. An empt fuzz subspace is defined as (, : U. Definitin.4 (Dib 994 A fuzz binar peratin F = ( F, f n the fuzz space ( X, I is a fuzz functin frm ( X, I ( X, I ( X, I, where F :X X X and f : I I I are functins (dented b c-membership functins that satisf (i f (, rs if r and s and (ii f are nt, i.e; f ( I I = I fr all (, X X. Definitin.5 (Dib994 A fuzz space ( X, I with fuzz binar peratin F = ( F, is called a fuzz grupid and is dented b (,,. ƒ ( X I F Definitin.6 (Dib 994 The rdered pair ( U ; F is called a fuzz subgrupid f the fuzz grupid (( X, I, F if the fuzz subspace U = {(, u : U } is clsed under the fuzz binar peratin F. Thus, fuzz grupid (( X, I, F is a fuzz grup iff fr ever (, I, (, I, ( z, I ( X, I the fllwing cnditins are satisfied: (i The binar peratin is assciative: (, I F(, I F ( z, I = (, I F (, I F ( z, I, (( F Fz, I = ( F( Fz, I. (ii There is an identit element (e, I in (( X, I ; F such that, I F e, I e, I F, I, I, Fe, I = ef, I =, I. = =
436 Abdallah Al-Husban and Abdul Razak Salleh (iii Ever fuzz element (, I has an inverse (, I, I F, I = (, I F (, I = ( e, I. Dente ( I, ( I, F, I = F, I = e, I. Frm (i, (ii and (iii, it fllws that (X, F is an rdinar grup. Therefre, we can write = and then (, I = (, I. A fuzz grup (( X, I, F is called a unifrm fuzz grup if F = ( F, ƒ is a unifrm fuzz binar peratin, i.e., ƒ ( rs, = ƒ( rs,, fr all, X. =, then we have Definitin.7 (Dib 994 A fuzz grup (( X, I, F is called a cmmutative r abelian f uzz grup if (, I F (, I = (, I F (, I, fr all fuzz elements (, I and (, I f the fuzz space ( X, I. It is clear that (( X, I, F is a cmmutative fuzz grup iff (X, F is an rdinar cmmutative grup. Definitin.8 (Ramt et al. A cmple fuzz set (CFS A, defined n a universe f discurse U, is characterised b a membership functin μ A(, that assigns t an element U a cmple-valued grade f membership in A. B definitin, the values f μ A(, ma receive all ling within the unit circle in the cmple plane, and i A are thus f the frm ( μ A = ra e ω, where i =, each f ra ( and ω A( are bth real-valued, and ra [, ]. The CFS A ma be represented as the set f rdered pairs A = (, μ ( : U A i ω A ( { ra e U} = (, :. Definitin.9 (Ramt et al. Let A and B be tw cmple fuzz sets n U and V respectivel, where {, i arg r A ( : iarg r = μa = A } and { B B = μb = rb e U} A r e U, :. The cmple fuzz unin f A and B, dented b A B, is specified b A B=, : U, { μ A B } i arg r ( i ma arg r (, arg r ( A B A B where μa B = ra B e = ma ( ra, rb e. The cmple fuzz intersectin f A and B, dented b A B, is specified b A B =, : U, { μ A B }
Cmple Fuzz Grup Based On Cmple Fuzz Space 437 where μ = =. i arg r ( i min(arg r (,arg r ( A B A B A B ra B e min ra, rb e Definitin. (Zhang et al. 9 Let A and B be tw cmple fuzz sets n X, and i arg A ( arg B let μ A( = ra(. e and. i μ B = rb e their membership functins, respectivel. We sa that A is greater than B, dented b A B r B A, if fr an X, r ( r (, and arg arg. A B A B 3. Main Results In this sectin we generalisethe ntin f a fuzz space f Dib (994 t the case f a cmple fuzz space and discuss its prperties. We als discuss sme related results. Let X be a given nnempt set and let A be a cmple fuzz set f X. The cmple fuzz set A can be identified with its membership functin i A μ A : X { a : a } defined b μ A = ra e αω, where i =, each f ra ( and ω A( are bth real-valued, and ωa, ra [,], α [, π ] if i A and A = e if A. In ur ntatin, we can write A = {(, r e } where the membership values r take their values frm the set [,], θ [, π ] and i the set A = {(, r e θ } is a cmple subset f X E, where E is the unit disc with the usual rder f cmple numbers. The scaling factr α (, π ], is used t cnfine the perfrmance f the phases within the interval (, π ], and the unit circle. Hence, the phase term θ ma represent the fuzz set infrmatin, and phase term values in this representatin case belng t the interval [, ] and satisfiesfuzz set restrictin. S, α in this paper will alwas be cnsidered equal t π. Let E be the unit disc.then E E is the Cartesain prduct E E with partial rder defined b: (i ( r, r s s ( re r e se, s e iff r s and r s, θr θ s and θ r θ s whenever s and s fr all r, s, θr, θ S E and r, s, θr, θ s E. s s e,e = ( s e, s e whenever s = rs = and θ = r (ii θ = fr ever s, E θ and s, θ E. B mentining the E -cmple fuzz subset, the assciated membership functins are meant t take their values frm the circle E. The cmple fuzz subset A will be dented b {(, A ( ; X} r simpl {(, A( }. A cmple fuzzsingletn f X with supprt X and value re E, r, θ ma be dented b i [, re αθ ]. Thrughut this paper the ntatin (, re A, where A E, means that A = re.
438 Abdallah Al-Husban and Abdul Razak Salleh Definitin 3. A cmple f uzz f unctin frm X t Y is defined as a functin F frm E t E characterised b the rdered pair is a functin frm X t Y and { ƒ } X F, ƒ, where F : X Y X is a famil f functins ƒ : E E satisfing the cnditins (i f is nndecreasing n E i (ii ƒ ( e θ i = if θ = and ƒ ( e θ = if θ = such that the image f an cmple fuzz subset A f X under F is the cmple fuzz subset F ( A f Y defined b r f if ( re F ( Ø r F ( Fre = fr an Y. if F ( = Ø We write F = ( F, ƒ : X Y t dente a cmple f uzz f unctin frm X t Y and we call the functins ƒ, X the cmembership functins assciated t F. A cmple fuzz functin F = ( F, is said t be unifrm if the cmembership functins ƒ are identical fr all ƒ X i.e. f = f fr X. Therem 3. Tw cmple f uzz f unctins F ( F, ƒ Y are equalif F = G and ƒ = g fr all X. = and G ( G g, = frm X t Prf. It is clear that if F = G and ƒ = g, fr ever X, then F = G. Cnversel, let F = G. If F G, then there eists an element X. such that F ( G(. Cnsider the cmple fuzz subset A f X defined b: π i e if = A( = i e if We have π i πi e if = F( e if = G( F( A = and GA = i i e if F( e if G(. Nw, if F ( G(. then F ( A G( B, which cntradicts the fact that F = G. i On the ther hand, if ƒ g, then there eist X. and re θ E. such that f ( re g ( re. If we cnsider the cmple fuzz subset f ( X, E
Cmple Fuzz Grup Based On Cmple Fuzz Space 439 re if = B = i e if then F = G and f ( re g ( re. This implies that F ( B G( B. The therem is thus prved. Definitin 3. A cmple fuzz functin frm X Y t Z is a functin F frm the X Y cmple fuzz Cartesian prduct X Y = ( E E f X and Y t the set f cmple fuzz subsets f Z, characterised b the rdered pair ( F, { ƒ }(, X Y, where F : X Y Z is a functin and {ƒ } (, X Y is a famil f functins ƒ : E E E satisfing the cnditins (i f is nndecreasing n E E and (ii ƒ ( e, e = ifθ = and f ( e,e = if θ =, such that the image f an E E -cmple fuzz subset C f X Y under F the cmple fuzz is subset F( C f Z defined b f ( C, if F z Ø (, F ( z F ( C z = fr ever if F ( z = Ø z Z. The cmple fuzz binar peratin F = ( F, ƒ n a set X is a cmple fuzz functin frm X X t X and is said t be unifrm if F is a unifrm cmple fuzz functin. Definitin 3.3 A cmple f uzz space, dented b ( X, E, where E is the unit disc, is a set f all rdered pairs (, E, X, i.e., ( X, E = {(, E : X }. We can write (, E = {(, re : re E }, where i =, r [, ], and θ [, π ]. The rdered pair (, E is called a cmple fuzz element in the cmple fuzz space ( X, E. Therefre, the cmple fuzz space is an (rdinar set f rdered pairs. In each pair the first cmpnent indicates the (rdinar element and the secnd cmpnent indicates a set f pssible cmple membershisp values re where r represents an amplitude term and θ represents aphaseterm. Definitin 3.4. The cmple fuzz subspace U f the cmple fuzz space ( X, E i is the cllectin f all rdered pairs (, r e θ where U fr sme U X E
44 Abdallah Al-Husban and Abdul Razak Salleh and i re θ is a subset f E, which cntains at least ne element beside the zer element. If it happens that U, then r = and θ =. The cmple fuzz subspace U is dented b U = (, r e : U. Let U dente the supprt f U, that is U = { U : r > and θ > } and dented b SU = U. An empt i cmple fuzz subspace is defined as (, = e :. i.e., S =. 4. Algebra f Cmle Fuzz Subspaces B using the definitins f unin and intersectin f Ramt (, we intrduce the fllwing definitins: Definitin 4. Let (, r U = re : U and V = (, re r : V be cmple fuzz subspaces f the cmple fuzz space ( X, E. The unin U V and the intersectin U V f cmple fuzz subspaces are defined as fllws: { i ( θr θ r r e : V U V = (, r U i ( θr θ r U V = {(, r r e : U V} The supprt f these cmple fuzz subspaces satisfies the fllwing: SU ( V = SU SV = U V SU ( V = SU SV = U V. Definitin 4. The difference U V between the cmple fuzz subspaces U {(, re : } r = U and V = { (, re : } r V is defined b i ( r r U V = ( r r e θ θ {}. Ntice that S( U V U V and equalit hlds if r r θ θ fr all U V. }, r r Definitin 4.3 The cmple fuzz subspace V = (, re r : V is cntained in the cmple fuzz subspace U = (, r r e : U and dented b V U, if V U, r < r and θr < θ r fr all V. Nte. It is clear that the empt cmple fuzz subspace is cntained in an cmple fuzz subspace U. Nw let ( X, E be a cmple fuzz space and let A be a cmple fuzz subset f X. Dente b A the subset f X cntaining all the elements with nn-zer cmple membership values in A, i.e., the cmple fuzz subset A induces the fllwing cmple fuzz subspaces:
Cmple Fuzz Grup Based On Cmple Fuzz Space 44. The lwer cmple fuzz subspace, induced b A, is given as: r H ( A = {(,[, re ]: A}. The upper cmple fuzz subspace, induced b A, is given as:,{} [ r,] : H A = re A 3. The finite cmple fuzz subspace, induced b A, is given as: r H ( A = (, {, re } : A Let ( X, E and. ( Y, E be cmple fuzz spaces. Definitin 4.4 The Cartesian pr duct f the cmple fuzz spaces ( X, E and (, X, E Y, E, defined b Y E is a cmple fuzz space, dented b ( X, E ( Y, E = ( X Y, E E = {( E E : X Y} (,,. Ever E E -cmple fuzz subset f X Y, A : X Y E E, belngs t the cmple fuzz space ( X Y, E E and ((,,E E is the cmple fuzz element f this space., E is The Cartesian prduct f tw cmple fuzz elements (, E and,,,. B using this relatin, we can write defined b ( E ( E = ((, E E ( X E ( Y E = {( E ( E : X, Y },,,,. Definitin 4.5 The Cartesian pr duct f the cmple fuzz subspaces r U = {(, re : U } and V = {(, se : } s V f the cmple fuzz spaces, Y, E, respectivel, is a cmple fuzz subspace f the cmple ( X E and fuzz Cartesian prduct ( X Y E E,, which is dented b U V : i {((,, θ r θs (, } U V = r s e : U V. 5. Cmple Fuzz Functins n Cmple Fuzz Spaces In this sectin we intrduce the definitin f a cmple fuzz functin n cmple fuzz spaces. Definitin 5. Let ( X, E and ( Y, E be cmple fuzz spaces. The cmple fuzz f unctin F frm the cmple fuzz space ( X, E int the cmple fuzz
44 Abdallah Al-Husban and Abdul Razak Salleh space Y E is defined as an rdered pair F ( F (, is a functin and {ƒ } X cnditins =, ƒ X, where F : X Y is a famil f functins ƒ : E E satisfing the E (i ƒ is nndecreasing n (ii ƒ ( i e θ = if θ = and ƒ ( i e θ = if θ =, such that the image f an E -cmple fuzz subset A f X under F is the cmple fuzz subset F ( A f Y defined b f if i re F Ø θ FA re F ( = fr an if F ( = Ø F ( E, Y. We write F = ( F, ƒ :( X, E ( Y, E t dente the cmple f uzz f unctin frm ( X, E t ( Y, E, and we call the functins ƒ, X the cmembership functins assciated t F. A cmple fuzz functin = is said t be F F,ƒ unifrm if the cmembership functins ƒ are identical fr all X. F = F, ƒ :( X, E ( Y, E acts n the Ever cmple fuzz functin cmple fuzz element (, E f ( X, E as fllws: F(, E = F, f ( E = F, E. Definitin 5. Let ( X, E,( Y, E and ( Z, E be cmple fuzz spaces. The cmple f uzz functin F frm ( X, E ( Y, E = ( X Y, E E int ( Z, E is defined b the rdered pair ( F, { ƒ }(, X Y where F : X Y Z is a functin and {ƒ } (, X Y is a famil f functins ƒ : E E E satisfing the cnditins: (i f is nn decreasing n E E, (ii ƒ ( e, e = if θ = and f ( e,e = if θ = such that the image f an E E -cmple fuzz subset C f X Y under F is the cmple fuzz subset F( C f Z defined b f (, C (,, if F ( z Ø F ( z F C z = fr ever z Z., if F ( z = Ø
Cmple Fuzz Grup Based On Cmple Fuzz Space 443 We write F = ( F, ƒ :( X Y, E E E and ƒ are called the ( Z, cmembership functins assciated t F. Ever cmple fuzz functin the cmple fuzz element F= ( F, ƒ :( X Y, E E E acts n ((,, E E f ( Z, X Y E E as fllws: (, ((,, E E = ( (,, ( E E F F f = ( F (,, E. Definitin 5.3 A cmple fuzz binar peratin F = ( F, ƒ n the cmple fuzz space ( X, E is a cmple fuzz functin frm F:( X, E ( X, E ( X, E with cmembership functins ƒ satisfing: i r i s (i ƒ ( re θ, se θ iff r, s, θ s and θr, (ii ƒ are nt, i.e., ƒ ( E E = E,, X. The cmple fuzz binar peratin F = ( F, n a set X is a cmple fuzz functin frm X X t X and is said t be unifrm if F is a unifrm cmple fuzz functin. ƒ 6. Cmple Fuzz Grupids Frm nw n, we cnsider the cmple fuzz space ( X, E. Definitin 6. Acmple fuzz space ( X, E with cmple fuzz binar peratin F = ( F, is called a cmple fuzz grupid and is dented b ( X, E, F ƒ Fr ever cmple fuzz elements (, E and (, E, we can write (, E (, E = (,, E E, which is a cmple fuzz element f ( X X E E.,. The actin f F = ( F, ƒ n this cmple fuzz element is given b, E F, E = F (, E (, E = F (((,, ( E E = ( F (,, ƒ ( E E = ( F (,, E.
444 Abdallah Al-Husban and Abdul Razak Salleh If we write F(, = F, then we have ( E F ( E = ( F E,,,. (, Therem 6. Assciated t each cm ple f uzz gr upid ( X, E ; F where F = ( F, f, is a fuzz grupid (( X, I ; F which is ismrphic t the cmple fuzz grupid ( X, E ; F b the crrespndence (, E (, I. Prf. Let (( X, E ; F be a given cmple fuzz grupid. Nw redefine F = ( F, f t be F = ( F, f. Since f satisfies the aims f fuzz cmembershisp functin, F = ( F, f will be a fuzz binar peratin in the sense f Dib. That is ( X, I ; F will define a fuzz grupid in the sense f Dib. Therem 6. T each cm ple f uzz grupid ( X, E ; F there is an assci ated (rdinar gr upid ( X, F which i s i smrphic t t he cm ple fuzz gr upid ( X, E ; F b the crrespndence (, E. (. Prf. Cnsider the cmple fuzz grupid ( X, E ; F Nw using the ismrphism (, E. we can redefine the cmple fuzz functin F = ( F, f t be F = F: X X X. That is F defines an rdinar binar peratin ver X. Thus, ( X, F is the assciated rdinar grupid. A cmple fuzz binar peratin F = ( F, ƒ n ( X, E is said t be unifrm if the assciated cmembership functins ƒ are identical, i.e., ƒ = ƒ fr a ll, X. A cmple fuzz grupid ((, E ; F is called a unifrm cmple fuzz grupid if F is unifrm. Definitin 6. The rdered pair ( U ; F is called a cmple f uzz subgrupid f the cmple fuzzgrupid (( X, E ; F, if the cmple fuzz subspace U = (, r e : U is clsed under the cmple fuzz binar peratin F. Therem 6.3 T he cmpl e f uzz subspace U = (, re : U with the cmple f uzz bi nar per atin F = ( F, ƒ is a cm ple f uzz s ubgrupid f (( X, E, F ifffr all, U : (i F U i i (ii ƒ ( r s e θ θ = r e θ. F F
Cmple Fuzz Grup Based On Cmple Fuzz Space 445 7. Cmple Fuzz Grups Having defined the cncept f cmple fuzz grupid and cmple fuzz subgrupid, we are nw in a psitin t define the cncept f cmple fuzz grup. Definitin 7. A cmple f uzz semigrup is a cmple fuzz grupid that is assciative. A cmple f uzzmnid is a cmple fuzz semigrup that admits an identit. After defining the cncept f cmple fuzzgrupid, cmple fuzz semigrup and cmple fuzz mnid, we nw intrduce the cncept f cmple fuzz grup. We call the pair ( X, E, F a cmple fuzz algebraic sstem. Definitin 7. A cmple fuzz algebraic sstem (( X, E, F is called a cmple fuzz gr up if and nl if fr ever (, E, (, E, ( z, E ( X, E the fllwing cnditins are satisfied: (i Assciativit: (, E F(, E F ( z, E = (, E F (, E F ( z, E, (ii (iii i.e.,( F Fz, E = ( F Fz E (,. There eists an identit element (, e E, fr which, E F e, E = e, E F, E =, E, i.e.,( Fe E = ( ef E = ( E,,,. Ever cmple fuzz element (, E has an inverse ( ( ( E F E = E F ( E = ( e E,,,,,. E such that (, Dente (, E (, E F, E = F, E = e, E. Frm (i, (ii and (iii, it fllws that ( X, F is a fuzz grup. Therefre, we can write = and then (, E = (, E. Frm the preceding discussin, we have the fllwing therem. =, then we have Therem 7. Assciated t each cm ple fuzz gr up (( X, E ; F, where F = ( F, f, is a f uzz grup (( X, I; F, where F = ( F, f, which is ismrphic t the cmple fuzz grup( ( X, E ; F b the crrespndence (, E (, I. Prf. The prf is similar t that f Therem 6..
446 Abdallah Al-Husban and Abdul Razak Salleh Therem 7. T each cm ple fuzz gr up (( X, E ; F there i s an assci ated (rdinar gr up ( X, F, which i s i smrphic t t he cm ple f uzz gr up (( X, E ; F b the crrespndence (, E. Prf. Similar t the prf f Therem 6.. A cmple fuzz grup ( X, E, F is called a unifrm cmple fuzz grup if ƒ F = ( F, is a unifrm cmple fuzz binar peratin, i.e., r s r s ƒ ( re, se = ƒ( re, se, frall, X. Definitin 7.3 A cmple fuzz grup (( X, E, F is called a cmmutative r abelian cm ple f uzz gr up if cmple fuzz elements It is clear that (, E ; cmmutative grup. (, E and (, E F (, E = (, E F (, E, fr all (, E f the cmple fuzz space ( X, E. ( X F is a cmmutative cmple fuzz grup iff ( X, F is a Eample 7. Cnsider the set G = { a}. Define the cmple fuzz binar peratin F = ( F, f ver the cmple fuzz space ( G, E such that: F( a, a = a and r (, s f re se ( r s e r θ s G, E tgether aa =. Thus, the cmple fuzz space with F define a (trivial cmple fuzz grup (( G, E, F. Eample 7. Cnsider the set 3 = {,, }. Define the cmple fuzz binar peratin F = ( F, f ver the cmple fuzz space ( 3, E as fllws: F (, = + 3, where + 3 refers t additin mdul 3 and r s i ( θr. θs f ( re, se ( r. s e. (, E, F is a cmple fuzz grup. = Then ( 3 ((, E, Definitin 7.4 Let X F be a cmple fuzz grup and let (, U = r e : U be a cmple fuzz subspace f ( X, E. ( UF, is called a cmple fuzz subgrup f the cmple fuzz grup (i F is clsed n the cmple fuzz subspace U, i.e., F F = F i ( θ f θ = ( F, ( rf r e (, r e (, r e ( F, r e (ii ( U, F satisfies the cnditins f an rdinar grup. ((, E, X F if:
Cmple Fuzz Grup Based On Cmple Fuzz Space 447 Therem 7. The s ubspace ( U ; F is a cm ple fuzz s ubgrup f t he cm ple fuzz grup ( X, E, F if and nl if: (i ( U, F is an rdinar subgrup f ( X, F ; (ii i( θ ƒ θ F re re = rfe = r r e ƒ (, ( ƒ. Prf. Suuppse cnditins (i and (ii are satisfied. Then (a F is clsed n the cmple fuzz subspace U. Let (, re, (, re U. Then ( (, r e F (, r e = F, ƒ ( r e, r e F = F, r e U. i (b ( U, F satisfies the cnditins f an rdinar grup. Let (, re θ, z (, re, ( z, rze U. Then z F z ( b (, re F(, re F( z, re z = ( F, rfe F( z, re z ( F Fz = ( F Fz, r( F Fze F( Fz = ( F( Fz, rf( Fz e Fz = (, re F( Fz, rfze z = (, re F(, re F( z, rze. e Fe ( b (, r e F ( e, re = ( Fe, r e F e Fe = (, r e ef = ( ef, r e ef e e = ( e, r e F (, r e. e e b3 Each (, r e has an inverse (, r e, since ( r e F r e F r e F, (, = (, F = (, F r e e = ( e, re = e F F (, r e F(, r e. Frm (a and (b we cnclude that ( U; F is a cmple fuzz subgrup f (( X, E, F.
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