Intuitionisitic Fuzzy B-algebras

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1 Research Joural of pplied Scieces, Egieerig ad Techology 4(21: , 2012 ISSN: Maxwell Scietific Orgaizatio, 2012 Submitted: December 18, 2011 ccepted: pril 23, 2012 Published: November 01, 2012 Ituitioisitic Fuzzy -algebras Jiayi Peg School of Mathematics ad Iformatio Sciece, Neijiag Normal Uiversity, Neijiag, O hia bstract: The otio of ituitiositic fuzzy -algebras is itroduced ad their some properties are ivestigated. How to deal with the homomorphic image ad iverse image of ituitiositic fuzzy ideals are -algebras. The relatios betwee a ituitiositic fuzzy -algebra ad a ituitiositic fuzzy -algebra i the product -algebras are give. Keywords: -algebra, homomorphism, ituitiositic fuzzy -algebra, product -algebra INTRODUTION I 1966, (Imai ad Iseki, 1966 defied a class of algebras of type (2, 0 amed K-algebras. t the same time, (Iseki, 1966 geeralized aother class of algebras of type (2, 0 called I-algebras. Hu ad Li (1983 itroduced the cocept of H-algebras ad gave examples of proper H-algebras (Hu ad Li, Neggers ad Kim (2001 itroduced ad ivestigated a class of algebras, viz., the class of -algebras, which is related to several classes of algebras of iterest such as H/I/K- algebras ad which seems to have rather ice properties without beig excessively complicated otherwise. Xi (1991 applied the otio of fuzzy sets (Zadeh, 1966 to K/I-algebras. Ju (1993, Meg, (1994 ad Ju ad Roh (1994 ivestigated fuzzy ideals of K/I-algebras. Youg et al. (2002 itroduced otios of fuzzy -algebras ad ivestigated its properties. I 1986, (taassov, 1986 itroduced the cocept of ituitioistic fuzzy sets, which is a sigificat extesio of fuzzy set theory by Zadeh (1966.Li ad Wag (2000 studied ituitioistic fuzzy group ad its homomorphisic image ad subsequetly discussed the product ad extesio priciple of the ituitioistic fuzzy sets (Li ad Wag, I this study, we itroduce otio of ituitioistic fuzzy -algebras ad discuss their properties. Some properties of the homomorphic (isomorphic image ad iverse image of ituitioistic fuzzy -algebras will be studied, the otios of ituitioistic fuzzy relatio, strogest ituitioistic fuzzy relatio ad artesia product of ituitioisitic fuzzy sets will be give, some relatios betwee ituitioistic fuzzy -algebras ad its product -algebras are exposed. PRELIMINRIES Defiitio 1: taassov (1986 Let X be a oempty set. set = {< x, : (x < (x> x, X } is said to a ituitiositic fuzzy set of X if mappig : : [0, 1] ad < : X [0, 1] satisfy 0 # : (x + < (x # 1 for all x, X. collectio of ituitiositic fuzzy sets of X is deoted by IFS(X. Let,, IFS(X. = {< x, : (x < (x> x, X } ad = {< x, : (x < (x> x, X }.osider the relatio betwee ad as follow: f if ad oly if : (x < : (x ad < (x $ < (x for all x, X. = if ad oly if : (x = : (x ad < (x = < (x for all x, X. = { < x, ( x ( x v ( x > x X} = { < x, ( x ( x, v ( x v ( x > x X} Otherwise, I i = x, ( x, v ( x x X j j I i = x, ( x, v ( x x X j j ~ = {< x, : (x 1! : (x > x, X } = {< x, 1! < (x, < (x> x, X } Defiitio 2: (Li ad Wag, 2000 Let f be a mappig from the set X to the set Y ad = {< x, : (x, >*x, X }. a ituitiositic fuzzy set of Y. The iverse image of, deoted by f!1 (, is the ituitiositic fuzzy set of X defied by: f!1 ( {+x, : (f(x, v (f(x, x0x} oversely, let = {< x, : (x, < (x, > x, X }. be a ituitiositic fuzzy set of X. The the image of, deoted by f (, is the ituitiositic fuzzy set of Y give by: 4200

2 Res. J. ppl. Sci. Eg. Techol., 4(21: , 2012 where, f( {+y, f(: (y, f f ( ( y ( ν ( y $f (v (y, y0y} ( x, f ( y f x ( y = 0, f ( y ν ( x, f ( y f x ( y = 1, f ( y Defiitio 3: Neggers ad Kim (2001 algebra of type (X;*,0 is called a -algebra if it satisfies the followig axioms: (1 x * x = 0 (2 x * 0 = x (H3 (x* y* z = x* (z* (0 * y for all x,y, z, X. I Youg et al. (2002, the cocept of fuzzy - algebras is itroduced. fuzzy set : i -algebra X is said to be a ati-fuzzy -algebra of X if for all x, y, X, :(x*y # :(x :(y. INTUITIONSITI FUZZY -LGERS Defiitio 1: Let (X; *, 0 be a -algebra. ituitiositic fuzzy set = {< x, : (x, < (x, > x, X } of X is said to be a ituitiositic fuzzy -algebra if it satisfies : (x*y v : (x : (y ad < (x*y # < (xw< (y for all x, y, X: Propositio 1: Let = {< x, : (x, < (x, > x, X } be a ituitiositic fuzzy -algebra. The, : (0 $, : (x ad < (0 # < (x, for all x, X. Proof: Sice x*x = 0 for all x, X, we havet hat : (0 = : (x*x $ : (xv: (x = : (x ad < (0 # < (xw< (x = < (x < (0 #.This completes the proof. For ay elemets x ad y of X, let us write x* y for x*( *(x*(x*y where, x occurs times. Propositio 2: Let = {< x, : (x, < (x, > x, X } be a ituitiositic fuzzy -algebra ad let, N. The for all: ( x* x ( x ad ν ( x* x ν ( x Wheever is odd ( x* x = ( x ad ν( x* x = ν( x Wheever is eve. Proof: Let x, X ad assume that is odd. The =!1 for some positive iterger k. Observe that : (x*x = : (0 $ : (x ad < (x*x = < (0 # < (x. Suppose that ( x* x ( x ad ν( x* x ν( x for a positive iteger k. The: ( x* x = ( x* x ( x*( x*( x* x 2( k = ( x* x ( x ν ( x* x = ν ( x* x = ν ( x*( x*( x* x 2( k = ν ( x* x ν ( x which proves (1. Similarly we obtai the secod part. Propositio 3: Let = {< x, : (x, < (x, > x, X } be a ituitiositic fuzzy -algebra, the for all x, y, X: (IF1 : (0*x $ : (x ad < (0*x # < (x, (IF2 : (x*(0*y $ : (x : (y ad < (x*(0*y # < (x < (y Proof: For ay x, y, X, we have : (0*x $ : (0 : (x, = : (x ad : (x*(0*y $ : (x : (x (0*y $ : (x : (y. Similarly we have that < (0*x # < (x ad < (x* (0*y# < (x < (y. orollary 4: Let = {< x, : (x, < (x, > x, X }be a ituitiositic fuzzy -algebra, the for all x, X, : (0*x = : (x ad < (0*x = < (x. Proof: Sice x = 0* (0*x, if is a ituitiositic fuzzy - algebra, the < (x = < (0*(0*x # < (0 < (0*x = <(0*x, i.e., < (0*x = < (x. Similarly we ca prove that : (0*x = : (x. for all x, X. Theorem 5: If a ituitiositic fuzzy set of X satisfies (IF1 ad (IF2, the is a ituitiositic fuzzy - algebra. Proof: Suppose that satisfies (IF1 ad (IF2 ad let x, y, X. The, < (x * y = < (x * (0 * (0 * y 4201

3 Res. J. ppl. Sci. Eg. Techol., 4(21: , 2012 # < (x < (0*y = < (xw< (y Similarly we have that : (x * y $ : (x v : (y Hece is a ituitiositic fuzzy -algebra. Theorem 6: If ad are ituitiositic fuzzy -algebra of X, the so is 1. Proof: For all x, y, X, we get: ( x* y = ( x* y ( x* y [ ( x ( y] [ ( x ( y] = [ ( x ( x] [ ( y ( y] = ( x ( y similarly, we have that: ν ( x* y v ( x v ( y ad hece 1 is a ituitiositic fuzzy -algebra. orollary 7: Let i be ituitiositic fuzzy -algebra of X for all i, J. The I i is also a ituitiositic fuzzy - algebra of X. Similarly, we have: i j Theorem 8: Let ad are ituitiositic fuzzy - algebras of X. The ~ ad are ituitiositic fuzzy -algebras of X. Defiitio 2: ituitiositic fuzzy set of X has sup-if property if, for ay TfX, there exist x 0, y 0,, T such that : (x 0 = SUP z,t : (z ad < (y 0 = if z,t < (z. Theorem 9: Let f be a homomorphism from a -algebra X ito a -algebra Y ad a ituitiositic fuzzy -algebra of X with sup-if property. The the image f( of is a ituitiositic fuzzy -algebra of Y. Proof: Let = {< x, : (x, < (x, > x, X } ad let y 1, y 2, Y. we cosider the followig two cases: ase (1: If f!1 (y 1 = Ø or f!1 (y 2 = Ø, the f!1 (y 1 * y 2 = Ø. d so f(: (y 1 * y 2 = 0 ad $f (< (y 1 * y 2 = 1. Thus f(: (y 1 * y 2 = 0 = f(: (y 2 ad: f $ ( ν ( y * y = 1 = f $ ( ν ( y f $ ( ν ( y ase (2: If f!1 (y 1 = Ø ad f!1 (y 2 = Ø, the let x 10, x 20, X such that : (x 10 = if ( z, : (x 20 = if ( z ad: zef ( y1 zef ( y2 : (x 10 * x 20 if ( z. The f(: (y 1 *y 2 zef ( y1* y2 = sup ( z = ( x * x ( x ( x z f ( y1* y = ( if ( z ( if ( z = f ( ( y f ( ( y z f ( y1 z f ( y2 Similarly, we have that: f$ ( ( y * y f$ 1 2 ν 1 2 ( v ( y f $ ( v ( y 1 2 Therefore, f( is a ituitiositic fuzzy -algebra of Y. Theorem 10: Let f be a homomorphism from a -algebra X oto a -algebra Y ad a ituitiositic fuzzy - algebra of Y. The the preimage f!1 ( of is a ituitiositic fuzzy -algebra of X. Proof: Let x,y, X. Sice is a ituitiositic fuzzy - algebra of Y, we have that: 1 ( ( x* y = ( f ( x* y f = ( f ( x* f ( y ( f ( x ( f ( y = ( x ( y Similarly, f ( f ( ν ( x* y ν ( x ν ( y f ( f ( f ( Hece f!1 ( is a ituitiositic fuzzy -algebra of X. orollary 11: Let f be a homomorphism from a -algebra X oto a -algebra Y. The the followig coclusios hold: If for all j, J, j, are ituitiositic fuzzy -algebras of X, the f ( I j is ituitiositic fuzzy -algebra of Y. If for all t, T, t are ituitiositic fuzzy -algebras of Y, the f 1( I t is ituitiositic fuzzy - t T algebras of X

4 Res. J. ppl. Sci. Eg. Techol., 4(21: , 2012 If is a ituitiositic fuzzy -algebra of X, the f( ad f( are ituitiositic fuzzy -algebra of Y. If is a ituitiositic fuzzy -algebra of Y, the f!1 ( ad f!1 ( are ituitiositic fuzzy -algebras of X. Theorem 12: Let f be a isomorphism from a -algebra X oto a -algebra Y. If is a ituitiositic fuzzy - algebra of X, the f!1 ( f( =. Proof: For ay x, X, let f(x = y, sice f is a isomorphism, f!1 (y = {x}. Thus = f(: (f(x = f(: (x = f(: (y= ( x = ( x ad: x f ( y where, : = : (xv: (y, < = < (xw< (y. The is a biary ituitiositic fuzzy relatio o X. Theorem 16: Let = {< x, : (x, < (x,> x, X } ad = {< x, : (x, < (x,> x, X } be ituitiositic fuzzy - algebra of X. The is a ituitiositic fuzzy - algebra of X X. Proof: Sice, are ituitiositic fuzzy -algebras of X, we have: (( xy, *( x', y' = ( x* x', y* y' f ( f $ ( ν ( x f $ ( ν ( f ( x = f $ ( ν ( y = ( x* x' ( y* y' = ν ( = x ν ( x 1 x f therefore f!1 (f( = (. orollary 13: Let f be a isomorphism from a -algebra X oto a -algebray. If is a ituitiositic fuzzy - algebra of Y, the f!1 (f( = (. orollary 14: Let f : X X be a automorphism. If is a ituitiositic fuzzy -algebra of X, the: f ( = f ( = ituitiositic fuzzy set: R = {+(x, y, : R (x, y, v R (x, y,} x0x, y0y}0 IFS(X Y is called a biary ituitiositic fuzzy relatio (Lei et al., 2005 from X ito Y. biary ituitiositic fuzzy relatio from X ito Y is said to be a biary ituitiositic fuzzy relatio o X if X = Y. Defiitio 3: Let = {< x, : (x, < (x,> x, X } IFS[X] biary ituitiositic fuzzy relatio: { (,, R(,, νr(,, } R = x y x y x y x y X o X is called a ituitiositic fuzzy relatio o if : R (x, y # : (x v: (y ad < R (x,y $ < (xw< (y for all x, y, X. Lemma 15: Let = {< x, : (x, < (x,> x, X } ad = {< x, : (x, < (x,> x, X } be ituitiositic fuzzy sets of X. artesia product of ad defied by: { (,, (,, ν (,, } xy xy xy xy X for all: [ ( x ( x'] [ ( y ( y' ] = [ ( x ( y] [ ( x' ( y'] = ( xy, ( x', y' ( xy,,( x', y' X X Similarly, ν (( xy, *( x', y' ν ( xy, ν ( x', y' for all (x, y, ( x', y' X X. Hece is a ituitiositic fuzzy -algebra of X X. Theorem 17: Let = {< x, : (x, < (x,> x, X }ad = {< x, : (x, < (x,> x, X } be ituitiositic fuzzy sets of a -algebra X such that is a ituitiositic fuzzy - algebra of X X. The, Either : (x # : (0 or : (x # : (0 for all x, X. Either < (x # < (0 or < (x $< (0 for all x, X. If : (x # : (0 for all x, X, the : (x # : (0 or : (x : (0. If < (x # < (0 for all x, X, the < (x $ < (0 or < (x $ < (0. If : (x # : (0 for all x, X, the : (x # : (0 or : (x # : (0. If < (x # < (0 for all x, X, the < (x $ < (0 or < (x $ < (0. Either : or : is a fuzzy -algebra (Youg et al., 2002 of X. Either < or < is a ati-fuzzy -algebra of X. 4203

5 Res. J. ppl. Sci. Eg. Techol., 4(21: , 2012 Proof: (i Suppose that : (x # : (0 ad : (y > : (0 for some x, y, X. The: ( xy, = ( x ( y > ( 0 ( 0 = ( 00,. This is a cotradictio. Hece (i holds. (ii is by similar method to part (i. (iii ssume that there exist x, y, X. such that : (x > : (0 ad : ( y >: (0. The ( xy, = ( x ( y > ( 0 = ( 0 = ( 00,, which is a cotradictio. Hece (iii holds. (vi, (v ad (vi are by similar method to part (iii. (vii Sice by (i either : (x # : (0 or : (x > : (0 for all x, X, without loss of geerality we may assume that : (x # : (0 for all x, X. From (v, it follows that : (x # : (0 or : (x > : (0. If : (x > : (0, the : (0, x = : (0v: (x = : (x. Let (x 1, x 2, (y 1, y 2,, X X. Sice is a ituitiositic fuzzy -algebra of, X X we have: (IF3 (( x1, x2 *( y1, y2 (( x, x ( y, y = [ ( x ( x ] [ ( y ( y ] If we take x 1 = y 2 = 0, the : (x 2 * y 2 = : (0, x 2 * y 2 $[ : (0v: (x 2 ]v[ : (0v: (y 2 ] = : (x 2 v: (y 2 This proves that : is a fuzzy -algebra of X. Now we cosider the case : (y # : (0 for all x, X. Suppose that : (y > : (0 for some y, X. The : (0 < : (y # : (0. Sice : (x # : (0 for all x, X, we have : (0 > : (x. for all x, X. Hece : (x,0 = : (xv: (0 = : (x. Takig x 2, y 2 = 0 i (IF3, the: : (x 1 *y 1 = : (x 1 *y 1, 0 : ((x 1,0*( y 1,0 [ ( x ( 0] [ ( y ( 0] 1 1 = ( x ( y 1 1 Therefore : is a fuzzy -algebra of X. (viii is by similar method to part (vii ad the proof is completed. From the proof of Theorem 17 (vii ad Theorem 17 (viii, the followig results hold up. Theorem 18: Let = {< x, : (x, < (x,> x, X } ad = {< x, : (x, < (x,> x, X }be ituitiositic fuzzy sets of a -algebra X such that is a ituitiositic fuzzy - algebra of X X. The: If : (x # : (0 v: (0 ad < (x # < (0w< (0 for all x, X, the is a ituitiositic fuzzy -algebra of X. If : (x # : (0 v: (0 ad wfor all x, X, the is a ituitiositic fuzzy -algebra of X. Defiitio 4: Let = {< x, : (x, < (x,> x, X }, IFS(X. ituitiositic fuzzy relatio: { (,, R(,, νr(,, } R = x y x y x y x y X o X is called a strogest ituitiositic fuzzy relatio o if: : (x,y = : (xv: (y ad < R (x, y = < (xw< (y for all x, y, X. Propositio 19: For a give ituitiositic fuzzy set = {< x, : (x, < (x,> x, X } of a -algebra X, let R be a strogest ituitiositic fuzzy relatio o. If R is a ituitiositic fuzzy -algebra of X X, the : (x $ : (0 ad < (x $ < (0 for all x, X. Proof: Sice R is a ituitiositic fuzzy -algebra, we have : R (x, x # : R (0,0 ad < R (x,x $ < R (0,0 for all x, X, that is, v ad < (x v< (x $ < (0 w< (0. So : (x # : (0 ad < (x $ < (0 for all x, X. Theorem 20: Let = {< x, : (x, < (x,> x, X } Let be a ituitiositic fuzzy set of -algebra X ad: { (,, R(,, νr(,, } R = x y x y x y x y X strogest ituitiositic fuzzy relatio o. The is a ituitiositic fuzzy -algebra of X if ad oly if R is a ituitiositic fuzzy -algebra of X X. Proof: ssume that is a ituitiositic fuzzy -algebra of X. The: (( x, x *( y, y = ( x * y, x * y R R

6 Res. J. ppl. Sci. Eg. Techol., 4(21: , 2012 = ( x * y ( x * y [ ( x ( y ] [ ( x ( y ] = [ ( x ( x ] [ ( y ( y ] = ( x, x ( y, y R 1 2 R 1 2 for all (x 1, x 2, (y 1, y 2, X X. Similarly, we have that < R (x 1, x 2,*(y 1, y 2 # < R (x 1, x 2 w< R (y 1, y 2 for all (x 1, x 2, (y 1, y 2, X X. Hece R is a ituitiositic fuzzy -algebra of X X. oversely, suppose that R is a ituitiositic fuzzy - algebra of X X ad let x, y, X, The, : (x*y = : R (x*y, x*y = : R ((x, x*, (y, y $ : R (x, x* v = : R ( y, y =: R (x v: R (y Similarly, we have < (x*y # < (x w< (y. d completes the proof. REFERENES taassov, K., Ituitioistic fuzzy sets. Fuzzy Sets Syst., 20(1: Hu, Q.O. ad X. Li, O H-algebras. Math. Sem. Notes, 11: Hu, Q.O. ad X. Li, O proper H-algebras. Math. Japa., 30: Imai, Y. ad K. Iseki, O axiom systems of propositioal calculi XIV. Proc. Japa cad., 41: Iseki, K., algebra related with a propositioal calculus XIV. Proc. Japa cad., 42: Ju, Y.., haracterizatio of fuzzy ideals by their level ideals i K (I-algebras. Math. Japa., 38: Ju, Y.. ad E.H. Roh, Fuzzy commutative ideals of K-algebras. Fuzzy Sets Syst., 64: Li, X.P. ad G.J. Wag, Ituitioistic fuzzy group ad its homomorphisic image. Fuzzy Syst. Math., 14(1: Li, X.P. ad G.J. Wag, The extesio operatios of the ituitioistic fuzzy sets. Fuzzy Syst. Math., 16(1: Lei, Y.J.,.S. Wag ad Q.G. Miao, O the ituitiositic fuzzy relatios with compositioal operatios. Syst. Eg. Theor. Pract., 2: Neggers, J. ad H.S. Kim, O -algebras. It. J. Math. Sci., 27: Meg, J., Fuzzy ideals of I-algebras. SE ull. Math., 18: Xi, O.G., Fuzzy K-algebras. Math. Japa., 36: Youg,.J., H.R. Eu ad J. hi, O fuzzy - algebras. zechoslovak Math. J., 52(127: Zadeh, L.., Fuzzy sets. Iform. otr., 8: