Cubic fuzzy H-ideals in BF-Algebras

Transcription

1 OSR Joural of Mathematcs (OSR-JM) e-ssn: p-ssn: X Volume ssue 5 Ver (Sep - Oct06) PP 9-96 wwwosrjouralsorg Cubc fuzzy H-deals F-lgebras Satyaarayaa Esraa Mohammed Waas ad U du Madhav 3 Departmet of Mathematcs charya Nagarjua Uversty Nagarjua Nagar-5 50 dhra Pradesh da 3 Departmet of Mathematcs Krsha Uversty PG Cetre Nuzvd-5 0 Krsha (Dt) dhra Pradesh da bstract: ths paper we troduce the oto of cubc fuzzy H-deals F-algebras ad prove some terestg propertes troducto Zadeh [8] has troduced the cocept of fuzzy subsets 965 Ths cocept has bee wdely adopted ad appled to may dscples Zha ad Ta [] troduced the oto of fuzzy H-deals CK-algebras ad Satyaarayaa etal ([ 5] [6 ]) studed tutostc fuzzy H-deals CK-algebras Ju etal [] troduced the oto of cubc sets ths paper we troduce the oto of cubc fuzzy H-deals F-algebras ad vestgate some of ts propertes Defto ([4] [7]) F-algebra s a o-empty set X wth a costat 0 ad a bary operato satsfyg the followg axoms: () x x 0 () x 0 x () 0 (x y) (y x) for all x y X Defto subset of a F-algebra (X 0) s called a deal of X f for ay xy X () 0 () x y ad y x y Defto 3 [] o-empty subset of X s called a H-deal of X f () 0 () x yz) ad y x z Sce x 0 x t s clear that every H-deal s a deal Defto 4[] fuzzy subset a F-algebra X s called a fuzzy H- deal of X f () (0) (x) () (x z) m{(x (y z))(y)} for all xyz X Sce x 0 x t s clear that every fuzzy H-deal s a fuzzy deal The determato of maxmum ad mmum betwee two real umbers s very smple but t s ot smple for two tervals swas [] descrbed a method to fd max/sup ad m/f betwee two tervals ad set of tervals y a terval umber a o [0 ] we mea a terval a a where 0 a a The set of all closed subtervals of [0 ] s deoted by D[0 ] The terval a a s detfed wth the umber a [0 ] For a terval umberg d put a a b D[0] We defe f a m a m b sup a max a max b = m a b a b () a a m( a a ) DO: 09790/ wwwosrjouralsorg 9 Page

2 Cubc fuzzy H-deals F-lgebras m a a m b b () a a max( a a ) =max a b a b max a a maxb b () a b a a a b b a a b (v) a a a a ad b b (v) a a a a ad b b (v) a a a a ad b b (v) md m[a b ] [ma mb ] where 0 m Obvously D[0] forms a complete lattce wth[0 0] as ts least elemet ad [ ] as ts greatest elemet [0] Zarad ad orumad defed aother type of fuzzy set called terval-valued fuzzy set (-v FS) The membershp value of a elemet of ths set s ot a sgle umber t s a terval ad ths terval s a sub-terval of the terval [0] Let D [0 ] be the set of a subtervals of the terval [0 ] The oto of terval-valued fuzzy set was frst troduced by Zadeh as a exteso of fuzzy set terval-valued fuzzy set s a fuzzy set whose membershp fucto s may-valued ad form a terval the membershp scale Ths dea gves the smplest method to capture the mpresso of the membershp grade for a fuzzy set Let X be a gve oempty set terval-valued fuzzy set (brefly -v fuzzy set) o X s defed by x [ (x) (x)] : x X Where (x) ad (x) are fuzzy sets of X such that (x) (x) for all x X Let (x) = [ (x) (x)] the (x (x)) : x X Where : X D[0] Defto 5 Cosder two elemets D D D[0] f D [a a ] r r m(d D ) [m(a a )m(a a )] whch s deoted by D D D [a a ] D[0] for =34 the we defe ad D [a a ] the Thus f r rsup (D ) [sup(a )sup(a )]e D [ a a ] Now we call D D f ad oly f a a ad a a Smlarly the relatos D D ad D = D are defed ased o the (terval-valued fuzzy sets Ju et al [] troduced the oto of cubc sets ad vestgated several propertes Defto 6 Let X be a o-empty set cubc set X s a Structure whch s brefly deoted by _ + = λ where = s a terval valued fuzzy set X ad λ s fuzzy set X = λ be cubc set X where X s F subalgebra the the set s cubc F Defto 7 Let subalgebra over the bary operato f t satsfes the followg codtos (xy) rm{ (x) (y)} () λ (xy) max{λ (x) λ (y)} () = λ s a cubc F subalgebra X the for all x X Proposto 8 f (0) (x) Thus respectvely (0) (x) ad (0)ad λ (0) are the upper bouds ad lower bouds of (x) ad λ (x) DO: 09790/ wwwosrjouralsorg 93 Page

3 Cubc fuzzy H-deals F-lgebras Cubc Fuzzy H-deals F-algebras ths secto we apply the cocept of cubc fuzzy set to H-deal of F-algebras ad troduced the otos of cubc fuzzy H-deals of F-algebras ad vestgate some of ts related propertes Defto Let (X ) be cubc fuzzy set X where X s a F algebra the the set s cubc fuzzy deal over the bary operato t satsfes the followg codtos: (CF) (0) (x) ad (0) (x) (CF) (x) r m{ (x y) (y)} (CF3) (x) max{ (xy) (y)} for all x yx Defto o empty sub set of F-algebra X s called a H-deal of X f () 0 () x (yz) ad y x z Defto 3 cubc fuzzy set (X ) a F-algebra X s called a cubc fuzzy H-deal of X f (CFH ) (0) (x) ad λ (0) λ (x) (x z) rm{ (x (yz)) (y)} ad (CFH ) λ (xz) max{λ (x (y z)) λ (y)} x y zx (CFH 3) Proposto 4 Every cubc fuzzy H-deal (X ) s a cubc fuzzy deal Proof: y settg z 0 (CFH ) ad (CFH 3) we get (x) r m{ (xy) (y)} ad λ (x) max{λ (x y)λ (y)} for all xy X (X ) s a cubc fuzzy deal of X Therefore Theorem 5 Let (X ) be a cubc fuzzy H-deal of X f there s a sequece {x } X such that () lm (x ) [] the (0) = [] ad () lm λ (x ) 0 the λ (0) 0 Proof: Sce (0) (x) for all x X (0) (x ) for every postve teger [] (0) lm (x ) [] Therefore Cosder Hece (0) = [] Sce λ (0) λ (x) for all x X Thus λ (0) λ (x ) for every postve teger Now λ (0) 0 0 λ (0) lm λ (x ) =0 Hece Theorem 6 cubc fuzzy set (X ) X s a cubc fuzzy H-deal of X f ad oly f λ are fuzzy H-deals of X Proof: Let ad λ are fuzzy H-deals of X ad xyz X The by defto (0) (x) (0) (x) (x z) m{ (x (y z)) (y)} (x z) m{ (x (y z)) (y)} λ (x z) max{λ (x (y z))λ (y)} (x z) = [ (x z) (x z)] Now DO: 09790/ wwwosrjouralsorg 94 Page [m{ (x (yz)) (y)} m{ (x (y z)) (y)}] r m{[ (x (yz)) (x (y z))][ (y) (y)]} r m{ (x (yz)) (y)} Therefore s cubc fuzzy H-deal of X ad

4 Coversely assume that (X ) s cubc fuzzy H-deal of X For ay xyz X [ (x z) (x z)] = (x z) r m{ (x (yz)) (y)} Thus (x z) m{ (x (y z)) (y)} (x z) m{ (x (y z)) (y)} λ (x z) max{λ (x (y z))λ (y)} Hece ad λ are fuzzy H-deals of X Theorem 7 Let atural umbers) The () Cubc fuzzy H-deals F-lgebras r m{[ (x (yz)) (x (y z))][ (y) (y)]} [m{ (x (yz)) (y)} m{ (x (y z)) (y)}] (X ) be a cubc fuzzy H-deal of F-algebra X ad let (the set of ( x x) (x) for ay odd umber () () λ ( xx) λ (x) for ay odd umber ( x x) (x) for ay eve umber (v) for ay eve umber λ ( x x) λ (x) (X ) Theorem 8 f be a cubc fuzzy H-deal of F-algebra X the the o empty upper s-level cut U( ;s) ad o-empty lower t-level cut L(λ ;t) are H-deals of X for ay sd[0] ad t [0] Proof: proof s straght forward (X ) J={x X/ (x) (0)} ad K {x X/ λ (x) λ (0)} are H-deals of X mples 0J ad 0 K So x (y z) J y J (x (y z)) (0) ad (y) (0) Corollary 9 Let sets Proof: Sce 0 X (0) (0) ad λ (0) λ (0) J ad K Let Sce be cubc fuzzy set f s a cubc fuzzy H-deal of F-algebra X the the (x z) rm{ (x (y z)) ( y)} r m{ (0) (0)} (0) (x z) (0) but (0) (xz) t follows that xz J for all xyz X Thus J s H-deal of X Let x (yz) K yk λ (x (yz)) λ (0) ad λ (y) λ (0) Sce λ (xz) max{λ (x (y z)) λ (y)} max{tt}=t but λ (0) λ (x z) Therefore xz K for all xyz X Thus K s H-deal of X Theorem 0 Let (X ) be a cubc fuzzy deal of F-algebra X f (x y) (x)ad λ (x y) λ (x) for ay xy X (X ) deal of F-algebra X (X ) the be a cubc fuzzy H- Proof: Let be a cubc fuzzy deal of F-algebra X f for ay xy X We have by hypothess (x y) (x)ad λ (x y) λ (x) rm{ (x (y z)) (y)} rm{ ((x z) (y z)) (y z)} DO: 09790/ wwwosrjouralsorg 95 Page

5 (yz) (y z) rm{ (x (y z)) (y)} ad max{λ (x (y z))λ (y)} max{λ ((x z) (y z))λ (y z)} λ (y z) λ (y z) max{λ (x (y z)) λ (y)} Hece (X ) s a cubc fuzzy H-deal of F-algebra X Cubc fuzzy H-deals F-lgebras ( λ ) Defto Let f be a mappg from a set X to a set Y Let be cubc fuzzy set Y The the verse mage of s defed as f () {(xf ( )f (λ )) / x X} wth the membershp fucto ad o-membershp fucto respectvely gve by f ( )(x) = (f(x)) ad f (λ )(x) = λ (f(x)) t - ca be show that f () s cubc fuzzy set Theorem Let f : X Y be a homomorphsm of F-algebras f ( λ ) s a cubc fuzzy H-deal of Y the the pre mage f () {(xf ( )f (λ )) / x X} of uder f s a cubc fuzzy H-deal of X Proof: ssume that Cosder f ( )(0) (f(0)) (f(x)) = f ( )(x) ad ( λ ) s a cubc fuzzy H-deal of Y Let xyz X f(x) f(y) f(z) Y f (λ )(0) λ (f(0)) λ (f(x)) = f (λ )(x) Thus f ( )(x z)= (f(x z) rm{ (f(x (yz)) (f(y))} = rm{f ( )(x (yz))f ( )(y)} d f (λ )(x z)=λ (f(x z) max{λ (f(x (y z))λ (f(y))} = max{f (λ )(x (y z))f (λ )(y)} Therefore f () {(xf ( )f (λ )) / x X} s a cubc fuzzy H-deal of Y Refereces [] swas R Rosefeld s fuzzy subgroups wth terval valued membershp fucto Fuzzy Sets ad Systems (63) (994) [] Ju Y Km CS ad Yag KO Cubc sets Fuzzy Math form 4() (0) [3] Ju Y Km CS ad Kag MS Cubc sub algebras ad deals of CK/C algebras Far East Joural of Mathematcal Sceces 44()(00)39-50 [4] Neggers J h SS ad Km HS O F-algebra teratoal Joural of Math Sc 7 (00) [5] Satyaarayaa du Madhav U ad Durga Prasad R O tutostc fuzzy H-deals CK-algebras teratoal Joural of lgebra 4 (5)(00) [6] Satyaarayaa du Madhav U ad Durga Prasad R O foldess of tutostc fuzzy H-deals CK-algebras teratoal Mathematcal Form 5(45)( 00) 05- [7] Satyaarayaa Ramesh D ad Pragath Kumar Y terval valued tutostc Fuzzy Homomorphsm of F-algebras Mathematcal Theory ad Modelg 3 (03) 6-65 [8] Zadeh L Fuzzy sets form Cotrol (965) [9] Zadeh L The cocept of a lgustc varable ad ts applcato to approxmate reasog formato Sc d Cotrol 8 (975) Z99-49 [0] Zarad ad orumad Saed terval-valued fuzzy F-algebras t J Cotemp Math Sceces 4 (7) (009) [] Zha J ad Ta Z Fuzzy H-deals CK-algebras Southeast sa ullet of Mathematcs 9 (005) DO: 09790/ wwwosrjouralsorg 96 Page