smarandache fuzzy strong ideal and smarandache fuzzy n-fold strong ideal of a bh-algebra

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International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 2, February 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Smarandache Fuzzy Strong Ideal and Smarandache Fuzzy n-Fold Strong Ideal of a BH-Algebra Shahrezad Jasim Mohammed Abstract: In this paper, we define the concepts of a Q-Smarandache n-fold strong ideal anda Q-Smarandache fuzzy (strong, n-fold strong) ideal of a BH- algebra .Also, we study some properties of these fuzzy ideals Keywords: BCK-algebra , BCI/BCH-algebras, BH-algebra , Smarandache BH-algebra, Q-Smarandache fuzzystrong ideal. 1. Introduction In 1965, L. A. Zadeh introduced the notion of a fuzzy subset of a set as a method for representing uncertainty in real physical world [7] . In 1991, O. G. Xi applied the concept of fuzzy sets to the BCK-algebras [8] . In 1993, Y. B. Jun introduced the notion of closed fuzzy ideals in BCI - algebras[11].In 1999,Y.B.Jun introduced the notion of fuzzy closed ideal in BCH-algebras [13]. In 2001, Q. Zhang, E. H. Roh and Y. B. Jun studied the fuzzy theory in a BH-algebra [10] . In 2006,C.H. Park introduced the notion of an interval valued fuzzy BH-algebra in a BH-algebra [2].In 2009, A. B. Saeid and A. Namdar, introduced the notion of a Smarandache BCH-algebra and Q-Smarandache ideal of a Smarandache BCH-algebra [1].In 2012, H. H .Abbass introduced the notion of a Q-Smarandache fuzzy closed ideal with respect to an element of a Smarandache BCH- algebra [5].In the seam year, E.M. kim and S. S. Ahn defined the notion of a fuzzy (n-fold strong) ideal of a BH- algebra[3]. In 2013, E.M. kim and S. S. Ahn defined the notion of a fuzzy (strong) ideal of a BH-algebra[4].In the same year, H. H. Abbass and S. J. Mohammed introduced the Q-Smarandache fuzzy completely closed ideal with respect to an element of a BH-algebra[6] . In this paper, we define the concepts of Q-Smarandachen-fold strong ideal anda Q-Smarandache fuzzy (strong, n-fold strong) ideal of a Smarandache BH- algebra .Also, we study some properties of these fuzzy ideals 2. Preliminaries In this section, we give some basic concept about a BCK- algebra ,a BCI-algebra ,a BH-algebra , a BH*-algebra,a normal BH-algebra, fuzzy strong ideal, fuzzyn-fold strong ideala Smarandache BH-algebra,( Q-Smarandache ideal, Q- Smarandache fuzzyclosedideal, Q-Smarandache fuzzy completely closed ideal and Q-Smarandache fuzzy ideal of BH-algebra Definition 1 (see[11]).A BCI-algebra is an algebra (X,*,0), where X is a nonempty set, "*" is a binary operation and 0 is a constant, satisfying the following axioms : for all x, y, z X: i. (( x * y) * ( x * z) ) *( z * y)= 0, ii. (x*(x*y))*y = 0, iii. x * x = 0, iv. x * y =0 and y * x = 0 x = y. Definition 2 (see [8]). A BCK-algebra is a BCI-algebra satisfying the axiom v. 0 * x = 0 for all x X. Definition 3(see[9]). A BH-algebra is a nonempty set X with a constant 0 and a binary operation "*" satisfying the following conditions:i.x*x=0, xX ii. x*0 =x, xX. iii. x*y=0 and y*x =0 x = y,:for all x,y X. Definition4.(see[4]).A BH-algebraX is called a BH*- algebraif (x*y)*x=0 for all x,yX Definition5.(see[9]). A BH-algebra X is said to be a normalBH-algebra if it satisfying the following condition: i. 0*(x*y)= (0*x)*(0*y) ,x,yX . ii. (x*y)*x = 0* y ,x,yX. iii. (x*(x*y))*y = 0 , x,yX. Definition 6.(see[5]) . Let X be a BH-algebra.Then the set X + ={xX:0*x=0} is called the BCA-part of X . Remark 1(see[6]).Let X and Y be BH-algebras. A mapping f : XY is called a homomorphism if f(x*y)=f(x)* ` f(y) for all x, yX. A homomorphism f is called a monomorphism (resp., epimorphism) if it is injective (resp., surjective). For any homomorphism f:X Y, the set { xX : f(x)=0'} is called the kernel of f, denoted by Ker(f), and the set { f(x) : xX} is called the image of f, denoted by Im(f). Notice that f(0)=0' for all homomorphism f. Definition7 (see[3]). Let X be a BH-algebraand n be a positive integer. A nonempty subset I of X is called a n-fold strong ideal of X if it satisfies: i. 0 I , ii. y I and (x* y)*z n I x*z n I, x ,z X. Definition 8.(see[6]).A Smarandache BH-algebra is defined to be a BH-algebra X in which there exists a proper subset Q of X such that i. 0 Q and |Q| 2 . ii. Q is a BCK-algebra under the operation of X. Definition9 (see[6]). Let X be a Smarandache BH-algebra . A nonempty subset I of X is called a Smarandache strong ideal of X related toQ ( or briefly, Q- Smarandache strong ideal of X) if it satisfies: i. 0 I , ii. y I and (x* y)*z I x*z I, x ,z Q. Paper ID: SUB151537 1516

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In this paper, we define the concepts of a Q-Smarandache n-fold strong ideal anda Q-Smarandache fuzzy (strong, n-fold strong) ideal of a BH- algebra .Also, we study some properties of these fuzzy ideals.

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International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 2, February 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Smarandache Fuzzy Strong Ideal and Smarandache Fuzzy n-Fold Strong Ideal of a BH-Algebra Shahrezad Jasim Mohammed Abstract:I n this paper, we define the concepts of a Q-Smarandache n-fold strong ideal anda Q-Smarandache fuzzy (strong, n-fold strong) ideal of a BH- algebra .Also, we study some properties of these fuzzy ideals Keywords: BCK-algebra , BCI/BCH-algebras, BH-algebra , Smarandache BH-algebra, Q-Smarandache fuzzystrong ideal. 1.Introduction In 1965, L. A. Zadeh introduced the notion of a fuzzy subset ofasetasamethodforrepresentinguncertaintyinreal physical world [7] . In 1991, O. G. Xi applied the concept of fuzzysetstotheBCK-algebras[8].In1993,Y.B.Jun introducedthenotionofclosedfuzzyidealsinBCI-algebras[11].In 1999,Y.B.Jun introduced the notion of fuzzy closed ideal in BCH-algebras [13]. In 2001, Q. Zhang, E. H. Roh and Y. B. Jun studied the fuzzy theory in a BH-algebra [10] . In 2006,C.H. Park introduced the notion of an interval valued fuzzy BH-algebra in a BH-algebra [2].In 2009, A. B. SaeidandA.Namdar,introducedthenotionofa SmarandacheBCH-algebraandQ-Smarandacheidealofa SmarandacheBCH-algebra[1].In2012,H.H.Abbass introducedthenotionofaQ-Smarandachefuzzyclosed idealwithrespecttoanelementofaSmarandacheBCH-algebra[5].Intheseamyear,E.M.kimandS.S.Ahn defined the notionofafuzzy(n-foldstrong)idealofaBH-algebra[3].In2013,E.M.kimandS.S.Ahndefinedthe notionofafuzzy(strong)idealofaBH-algebra[4].Inthe sameyear,H.H.AbbassandS.J.Mohammedintroduced theQ-Smarandachefuzzycompletelyclosedidealwith respect to an element of a BH-algebra[6] . In this paper,we definetheconceptsofQ-Smarandachen-foldstrongideal anda Q-Smarandache fuzzy (strong, n-fold strong) ideal of a SmarandacheBH-algebra.Also,westudysomeproperties of these fuzzy ideals 2.Preliminaries Inthissection,wegivesomebasicconceptaboutaBCK-algebra,aBCI-algebra,aBH-algebra,aBH*-algebra,a normalBH-algebra,fuzzystrongideal,fuzzyn-foldstrong idealaSmarandacheBH-algebra,(Q-Smarandacheideal,Q-Smarandachefuzzyclosedideal,Q-Smarandachefuzzy completelyclosedidealandQ-Smarandachefuzzyidealof BH-algebra Definition 1 (see[11]).A BCI-algebra is an algebra (X,*,0), where X is a nonempty set, "*" is a binary operation and 0 is a constant, satisfying thefollowing axioms: for allx,y, ze X: i. (( x * y) * ( x * z) ) *( z * y)= 0, ii. (x*(x*y))*y = 0,iii. x * x = 0, iv. x * y =0 and y * x = 0 x = y. Definition2(see[8]).ABCK-algebraisaBCI-algebra satisfying the axiomv. 0 * x = 0 for all x eX. Definition3(see[9]).ABH-algebraisanonemptysetX withaconstant0andabinaryoperation"*"satisfyingthe following conditions:i.x*x=0, xeX ii. x*0 =x, xeX.iii. x*y=0 and y*x =0 x = y,:for all x,ye X. Definition4.(see[4]).ABH-algebraXiscalledaBH*-algebraif (x*y)*x=0 for all x,yX Definition5.(see[9]).ABH-algebraXissaidtobea normalBH-algebra if it satisfying the following condition:i.0*(x*y)= (0*x)*(0*y) ,x,yX . ii. (x*y)*x = 0* y ,x,yX. iii. (x*(x*y))*y = 0 , x,y X. Definition6.(see[5]).LetXbeaBH-algebra.Thentheset X+={xeX:0*x=0} is called the BCA-part of X . Remark 1(see[6]).Let X and Y be BH-algebras. A mapping f : XY is called a homomorphism if f(x*y)=f(x)*`f(y) for all x,yeX.Ahomomorphism fis called amonomorphism (resp., epimorphism) if it is injective (resp., surjective). For anyhomomorphismf:XY,theset{xeX:f(x)=0'}is called the kernel of f, denoted by Ker(f), and the set { f(x) : xeX} is called the image of f, denoted by Im(f). Notice that f(0)=0' for all homomorphism f. Definition7(see[3]).LetXbeaBH-algebraandnbea positive integer. A nonempty subset I of X is called a n-fold strong ideal of X ifitsatisfies:i.0I,ii.yIand(x* y)*znI x*znI, x ,z X. Definition 8.(see[6]).A Smarandache BH-algebra is defined to be a BH-algebra X in which there exists a proper subset Q of X such that i. 0 Q and |Q| 2 . ii. Q is a BCK-algebra under the operation of X. Definition9(see[6]).LetXbeaSmarandacheBH-algebra. AnonemptysubsetIofXiscalledaSmarandache strong ideal of X related toQ ( or briefly, Q- Smarandache strong ideal of X) if it satisfies: i. 0 I , ii. y I and (x*y)*z I x*z I, x ,z Q. Paper ID: SUB151537 1516International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 2, February 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Definition10(see[7]).LetbeafuzzysetinX,forall te[0,1].Thesett={xeX,(x)>t}iscalledalevel subset of . Definition11(see [4)].Let A and B be any two sets, be any fuzzysetinAandf:ABbeanyfunction.Set1(y)= {x A| f (x) = y} for y B. The fuzzy set in B defined by (y) ={0 otherwisesup {(x)| x 1(y)} if1 (y) for all yB, is called the image of under f and is denoted by f (). Definition 12(see[ 4].Let A and B be any two sets, f : A B be any function and be any fuzzy set in f (A). The fuzzy set in A defined by: (x) = ( f (x)) for allx X is called the preimageof under f and is denoted by 1(). Definition13.(see[4].AfuzzysetinaBH-algebraXis called a fuzzy strong ideal of X if i.For all xX, (0) (x). ii. (x z) min{ ((x y) z) , (y)}, x; y X. Definition14.(see[5].AfuzzysetinaBH-algebraXis called a fuzzy n-fold strong idealof X if i. For all xX, (0) (x).ii. (x zn) min{ ((x y) zn); (y)}, x, y X. Definition15(see[6]).AfuzzysubsetofaSmarandache BH-algebraXissaidtobeaQ-Smarandachefuzzy idealif and only if : i. For all xX, (0) (x).ii.For all x Q, yeX, (x) min{ (x*y), (y)}. Is said to be closed if (0*x) (x), for all xX. Definition16(see[6]).LetXbeaSmarandacheBH-algebra andbeaQ-SmarandachefuzzyidealofX.Thenis calledaQ-Smarandachefuzzycompletelyclosedidealif (x*y) > min{ (x), (y)},x,yeX. Proposition1(see[6]).Let X be a BH*-algebra ,and be a Q-Smarandache fuzzy ideal. Then i. is a Q-Smarandache fuzzy closed ideal of X . ii. is a Q-Smarandache fuzzy completely closed ideal of X, if X*X/{0} _Q 3.The Main Results In this paper,we give the concepts a Q-Smarandache n-fold strongidealandaQ-Smarandachefuzzy(strong,n-fold strong)idealofaBH-algebra.Also,wegivesome properties of these fuzzy ideals Definition1.AfuzzysubsetofaBH-algebraXiscalleda Q-Smarandache fuzzystrongideal,iffi.(0)>(x)) xeX ii. (x*z) > min{ ((x*y)*z), (y)},x,zeQ. Example1:ThesetX={0,1,2,3}withthefollowing operation table *0123 00022 11012 22200 33210 isaBH-algebraQ={0,1}isaBCK-algebra.Then (X,*,0) is a Smarandache BH-algebra.The fuzzy set which is defined by: (x) = 0.5 x = 0,3 0.4 x = 1,2 isaQ-Smarandachefuzzystrongideal,since:i.(0)=0.5 >(x) xeX, ii. (x*z) > min{ ((x*y)*z), (y)},x, zeQ. But the fuzzy set(x) = 0.5 x = 0,2,3 0.4 x = 1 isnotaQ-Smarandachefuzzystrongideal since(1*0)=(1)=0.4 < min{ (1*3)*0), (3)}=0.5 Proposition1.EveryQ-Smarandachefuzzystrongidealofa Smarandache BH-algebra X is a Q-Smarandache fuzzy ideal of X. Proof :Let be a fuzzy strong ideal of X . i.Let x X (0) (x) .[By definition 1(i)] ii. let x, z X and y X x,z Q (x*z) min{((x*y)*z) , (y)}[By definition 1(ii)] Whenz=0(x*0)min{((x*y)*0),(y)}(x) min{(x*y) , (y)} is a Q-Smarandache fuzzy ideal of X. Proposition2.LetQ1andQ2beaBCK-algebrascontained inaSmarandacheBH-algebraXand Q1_Q2.Letbe Q2-SmarandachefuzzystrongidealofXthenisaQ1-Smarandache fuzzy strong ideal of X . Proof :Let be a Q2-Smarandache fuzzy strong ideal of X . i. Let x X (0) (x) . [SinceisaQ2-Smarandachefuzzystrongideal.By definition 1(i)] ii.Let x,z Q1,yX x,z Q2 (x*z) min{ ((x*y)*z) ,(y)}[SinceisaQ2-Smarandachefuzzystrongideal.By definition 1(ii)] is a Q1-Smarandache fuzzy strong ideal of X. Proposition3.EveryfuzzystrongidealofaSmarandache BH-algebra X is a Q-Smarandache fuzzy strong ideal of X. Proof :Let be a fuzzy strong ideal of X . i. Let x X (0) (x) .[By definition 13(i)] ii. let x, z X and y X x,z Q (x*z) min{((x*y)*z) , (y)}[By definition13(ii)] is a Q-Smarandache fuzzy strong ideal of X. Theorem 1.Let X be SmarandacheBH-algebra and let be a fuzzy set.Then is a Q-Smarandache fuzzy strong ideal if andonlyif(x)=(x)/(0)isaQ-Smarandachefuzzy strong ideal. Proof: Let be a Q-Smarandache fuzzy strong ideal, 1) (0)= (0) / (0), (0)=1 (0) (x) xeX Paper ID: SUB151537 1517International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 2, February 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY 2) (x*z)= (x*z)/ (0) min{ ((x*y)*z), (y)}/ (0)} [SinceisaQ-Smarandachefuzzystrongideal..By definition1(ii)] min{ ((x*y)*z) / (0) , (y) / (0)} min{ ((x*y)*z), (y)} (x*z) min{ ((x*y)*z), (y)} isaQ-Smarandachefuzzystrongideal.Conversely.Let be a Q-Smarandache fuzzy strong ideal. i. (0)= (0). (0), (0) (x). (0) (0) (x) xeX ii.(x*z)=(x*z).(0)min{(x*(y*z)),(y)}.(0) [Since ' is a Q-Smarandache fuzzy ideal.By definition1(i)] min{ ((x*y)*z) . (0) , (y) . (0)} min{ ((x*y)*z), (y)} (x) min{ ((x*y)*z), (y)} is a Q-Smarandache fuzzy strong ideal. Proposition4.LetXbeaBH*-algebra,andbeaQ-Smarandache fuzzy strong ideal. Then i. is a Q-Smarandache fuzzy closed idealof X. ii.isaQ-Smarandachefuzzycompletelyclosedidealof X,if X*X/{0} _Q. Proof :Let be a fuzzy strong ideal of X. [ByProposition1] is a Q-Smarandache fuzzy ideal of X. i.isaQ-SmarandachefuzzyclosedidealofX.[By proposition 1] ii.Letx,yXisaQ-Smarandachefuzzycompletely closed ideal of X. [By proposition 1]. Theorem2.LetAbeanon-emptysubsetofaQ-SmarandacheBH-algebraXandletbeafuzzysetinX defined by: (x)={2 otherwise1 xQ where1>2 in[0,1].ThenisaQ-Smarandachefuzzy strong ideal X Proof : Let be a fuzzy set of X. i. 0 Q (0) = 1. (0) (x) [Since1>2 ] ii. let x,zQ and y X x*zQ (x*z) = 1 Then we have for cases. Case1 If ((x*y)*z)= 1and (y) =1 min{((x*y)*z) ,(y)} =1 (x*z) min{((x*y)*z) ,(y)} Case2 If ((x*y)*z)= 2and (y) =1 min{((x*y)*z) ,(y)} =2 (x*y) = min{(x*y) ,(y)} Case3 If ((x*y)*z)= 1and (y) =2 min{((x*y)*z) ,(y)} =2 (x*y) = min{(x*y) ,(y)} Case3 If ((x*y)*z)= 2and (y) =2 min{((x*y)*z) ,(y)} =2 (x*y) = min{(x*y) ,(y)} is a Q-Smarandache fuzzy strong ideal of X. Theorem3.LetXbeBH-algebraandletbeafuzzy set.Then is a Q-Smarandache fuzzy strong ideal if and only if'(x)=(x)+1-(0)isaQ-Smarandachefuzzystrong ideal. Proof: Let be a Q-Smarandache fuzzy strong ideal, i.'(0)= (0)+1- (0), '(0)=1 '(0) '(x) xeX ii.'(x*z)= (x*z)+1- (0) min{((x*y)*z)),(y)}+1-(0)[SinceisaQ-Smarandache fuzzy strong ideal. .By definition1] min{((x*y)*z))+1-(0),(y)+1-(0)}min{

'((x*y)*z)), '(y)}'(x*z) min{ '((x*y)*z)), '(y)} ' is a Q-Smarandache fuzzy strong ideal. ConverselyLet ' be a Q-Smarandache fuzzy strong ideal. 1)(0)='(0)-1+(0),(0)'(x)-1+(0)(0) (x) xeX 2) (x*z)= '(x*z)-1+ (0) min{'((x*y)*z)),'(y)}-1+(0)[Since'isaQ-Smarandache fuzzy strong ideal. By definition1] min{'((x*y)*z)) -1+ (0), '(y) -1+ (0)} min{ ((x*y)*z)), (y)} (x) min{ ((x*y)*z)), (y)} is a Q-Smarandache fuzzy strong ideal. Theorem4. Let X be a Smarandache BH-algebra such that X = X+ and be a Q-Smarandachestrong ideal of X. Then is a Q-Smarandache closed ideal of X . Proof :Let be a Q-Smarandache strong ideal I is a Q-Smarandache ideal of X. [Byproposition1] Now, let x I (0*x) = (0) (x) [By definition 6] is a Q-Smarandache closed ideal of X . Proposition5.LetXbeaSmarandachenormalBH-algebra suchthatX=X+andletbe aQ-Smarandachefuzzy strong idealsuchthatx*yQ,x,yXandy0.ThenisaQ-Smarandache fuzzy completely closed ideal of X. Proof: Let I be a Q-Smarandache fuzzy strong ideal of X. is a Q-Smarandache fuzzy ideal of X. [By remark3 ] Now, let x,y X x*y Q [Since x*y Q,x,y X ] We have (x*y)=((x*y)*0)>min{(((x*y)*x)*0)),(x)}[By definition 3(ii)] =min{((x*y)*x),(x)}[Bydefinition3(ii)]=min{ (0*y), (x)} [By definition 5(ii)] = min{ (0), (x)} = (x) (x*y) > min{ (y), (x)} is a Q-Smarandache fuzzy completely closed ideal. Proposition6.LetXbeanormalBH-algebrasuchthat X*X/{0}_Q.TheneveryQ-Smarandachefuzzystrong ideal and closed of X is a Q-Smarandache fuzzy completely closed ideal of X. Proof :Let be a Q-Smarandache fuzzy strong ideal of X. Paper ID: SUB151537 1518International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 2, February 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY is a Q-Smarandache fuzzy ideal of X. [By proposition1] Now, let x, y X. x*y Q [Since X*X /{0} _Q] (x*y) min{((x*y)*x)*0) , (x)} [By definition1 ] = min{(0*y) , (x)}[By definition 5(ii)] min{(y) , (x)}[By definition14] is a Q-Smarandache fuzzy completely closed ideal of X. Remark1.LetbeafuzzysetofaSmarandacheBH-algebra X and w X .The set {xX: (w) (x)} is denoted by (w) . Theorem 5.Let XbeaSmarandacheBH-algebra,wXand is a Q-Smarandache fuzzy strong ideal of X. Then(w) is a Q-Smarandachestrong ideal of X. Proof :Let be a Q-Smarandache fuzzy strong ideal of X. To prove that (w) is a Q-Smarandache strong ideal of X. 1) Let x (w) (0) (x) [By definition 1(i)] (0) (w) 0 (w)2) Let x,z Q, y (w) and (x*y)*z(w). (w) (y) and (w) ((x*y)*z) (w) min{ (y) , ((x*y)*z)} But (x*z) min{((x *y)*z), (y)} [By definition 1(ii)] (w) (x*z) x*z(w) (w) is a Q-Smarandache strong ideal of X. Corollary1.Let X be a Smarandache BH-algebra . Then is a Q-Smarandache fuzzy strong ideal of X if and only if t is a Q-Smarandache strong ideal of X, for all t [0,supX xe(x)] Proof:Lett[0, supX xe(x)].ToprovethattisaQ-Smarandache strong ideal ofX.Since is a Q-Smarandache fuzzy strong ideal of X. Now, let y t and x*(y*z) t(y) t and ((x*y)*z)) t . To prove that x*z t We have (x*z) min{((x*y)*z)) ,(y)} [By definition 1] Since ((x*y)*z)) t and (y) t min{((x*y)*z)) ,(y)} t(x*z) t x*z t t is a Q-Smarandache strong ideal of X. Conversely, To prove that is a Q-Smarandache fuzzy strong ideal of X. Since t is a Q-Smarandache strong ideal of X. Let t =supX xe(x) ,x,zQ and(x*y)*z, y t x*z t[By definition9] (x*z) t (x*z) = t [Since t =X xesup(x)] Similarly,((x*y)*z)=t and (y)=t t = min {((x*y)*z),(y)}(x*z) min {((x*y)*z),(y)} is a Q-Smarandache fuzzy strong ideal of X. Proposition7.Letf:(X,*,0)(Y, -', 0' )bea SmarandacheBH-epimorphism.IfisaQ-Smarandache fuzzystrongidealofX,thenf()isaf(Q)-Smarandache fuzzy strong ideal of Y . Proof :Let be a Q-Smarandache fuzzy strong ideal of X. i. Let y f () such thaty = f (x) .( f ())( 0' )=sup {(x) x 1( 0' )} =(0) (x) [By definition 1(i)] =( f ())( f (x)) = ( f ())(y)( f ())( 0' )( f ())(y)ii.Lety1,y3f(Q),y2Y,thereexistsx1, x3Qand x2Xsuchthaty1=f(x1), y3=f(x3)andy2=f(x2)(f ())(y1 y3 ) = sup{(x1*x3) x 1(y1 y3)} (f())(y1*y1)(x1 x3)min{ ((x1 x2)*x3),(x2)} [By definition 1(ii)] =min{(f())(f((x1*x2)*x3)),(f())(x2)}=min{(f())((f (x1) -' f (x2)) -' f(x3)),(f ())(f (x2)}= min{(f ())(y1-'y2)-'y3),( f ())(y2)} ( f ())(y1 ) min { ( f ())((y1-'y2) -'y2),( f ())(y2 )} f () is a f (Q)-Smarandache fuzzy strong ideal of Y. Theorem6.Letf:(X,*,0)(Y, -', 0' )beaSmarandache BH-epimorphism.Ifis a Q -Smarandache fuzzy strong ideal of Y, then 1() is a1(Q)-Smarandache fuzzy strong ideal of X Proof:i.LetxX.Sincef(x)YandisaQ-Smarandachestrong fuzzy ideal of Y . (1())(0)= ( f (0))= ( 0' ) ( f (x)) =(1())(x)ii. Let x 1(Q), y X .

1()(x*z)=( f (x*z)) [By definition 12] min{( f (x)-'f (y))-'f (z)), ( f (y))} [By remark1 ]=min{(f((x*y)*z),(f(y))} 1()(x)min{1()((x*y)*z),1()(y)}[By definition1] 1() is a Q-Smarandache fuzzy strong ideal of X . Definition 2. Let X be a Smarandache BH-algebra and n be apositiveinteger.AnonemptysubsetIofXiscalleda Smarandachen-foldstrongidealofXrelatedtoQ(or briefly,Q-Smarandachen-foldstrongidealofX)ifit satisfies: i. 0 I , ii. y I and (x* y)*znI x*z I, x ,z Q. Example2.ConsiderthesetI={0,3}inexample1isaQ-Smarandachen-fold strongideal of X.But the set I={0,2,3} is notaQ-Smarandachen-foldstrongidealsince(1*3)*0n= 2eI,but 1*0n=1 I. Remark 2. Every n-fold strong ideal of a Smarandache BH-algebra X is a Q-Smarandachen-fold strong ideal of X. Remark3.EveryQ-Smarandachen-foldstrongidealofa Smarandache BH-algebra X is a Q-Smarandachestrong ideal ofX.Remark4.EveryQ-Smarandachen-foldstrongidealofa Smarandache BH-algebra X is a Q-Smarandache ideal of X. Proposition8.LetXbeaSmarandacheBH-algebra . Then everyQ-Smarandachen-foldstrongidealwhichiscontained in Q is a Q-Smarandache completely closed ideal of X. Proof :Let I be a Q-Smarandachen-foldstrong ideal of X I is a Q-Smarandache ideal of X. [By remark4] Now, let x,yI x,yQ x*yQ [Since I_ Q ] Paper ID: SUB151537 1519International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 2, February 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Where((x*y)*x)*0=((x*x)*y)*0[Since(x*y)*z=(x*z)*y. By definition 1(iii)] =(0*y)*0 [By definition 3(i)] =0*y [By definition 3(ii)] =0I [Since 0*x=0.By definition 2(v)] ((x*y)*x)*0nIandxI.(x*y)*0nIx*yI[By definition 3(ii)] I is a Q-Smarandache completely closed ideal of X. Definition3.AfuzzysubsetofaSmarandacheBH-algebra Xand n be a positiveintegeriscalledaQ-Smarandache fuzzyn-fold strong ideal ,iff i. (0) > (x) ) xeX. ii. (x*zn) > min{ ((x*y)*zn), (y)},x,zeQ. Example3.Consider thefuzzy set which is defined by: (x) = 0.5 x = 0,2 0.4 x = 1,3 is a Q-Smarandache fuzzy n-foldstrong ideal , since: i.(0) = 0.5>(x)xeX,ii.(x*z1)>min{((x*y)*z1),(y)},x, zeQ. But the fuzzy set(x) = 0.5 x = 0,2,3 0.4 x = 1 isnotaQ-Smarandachefuzzyn-foldstrongideal since(1*03)=(1)=0.4 < min{ (1*3)*03), (3)}=0.5 Remark5. Every fuzzy n-fold strong ideal of a Smarandache BH-algebra X is a Q-Smarandache fuzzy n-fold strong ideal of X. Proposition9.EveryQ-Smarandachefuzzyn-foldstrong idealofaSmarandacheBH-algebraXisaQ-Smarandache fuzzy strong ideal of X. Proof : Let be a Q-Smarandache fuzzy n-fold strong ideal of X .i. Let x X (0) (x) .[By definition 3(i)] ii.letx,zQandyX(x*zn)min{((x*y)*zn), (y)}[By definition 3(ii)] When n=1(x*z) min{((x*y)*z) , (y)} is a Q-Smarandache fuzzy strong ideal of X. Theorem7.Letf:(X,*,0)(Y, -', 0' )beaSmarandache BH-epimorphism.IfisaQ-Smarandachefuzzyn-fold strongideal of Y, then 1() is a1(Q)-Smarandache fuzzy n-fold strongideal of X. Proof :i.Let x X. Since f (x) Y and is a Q-Smarandache fuzzy n-fold strongideal of Y. (1())(0)= ( f (0))= ( 0' ) ( f (x)) =(1())(x)ii. Let x,z1(Q), y X .

1()(x*zn)=( f (x*zn)) [By definition11] min{( f (x)-'f (y))-'f (zn)), ( f (y))} [By remark1]=min{( f ((x * y)*zn), ( f (y))} 1()(x*zn)min{1()((x*y)*zn ), 1()(y)} 1() is a Q-Smarandache fuzzyn-foldstrong ideal of X . Theorem8.LetXbeSmarandacheBH-algebra.Ifisa fuzzy set such that Q=X={xeX:(x)=(0)}and(0)(x) xX,then is aQ-Smarandache fuzzy n-fold strong ideal of X. Proof:LetbeafuzzysetofX,suchthatQ=Xand (0)(x) xX. i. (0)(x) xX. ii. Let x,zQ and yeX . (0)(y)and(0) x yzn[Since(0)(x) xX] (0) min{((x*y)*zn),(y)} But (x*zn)=(0)[Since Q= X] (x*zn)min{((x*y)*zn),(y)}isaQ-Smarandache fuzzy n- fold strong ideal of X. Proposition10.Let {:e} be a family of Q-Smarandache fuzzyn-foldstrongidealsofaSmarandacheBH-algebraX. Then oe o is a fuzzy n-fold strongideal of X. Proof :Let {:e}be a family of Q-Smarandache fuzzy n- foldstrong ideals of X. i. Let xeX. oe o(0) = inf{ (0), e}>inf {(x), e} [SinceisaQ-Smarandachefuzzyn-foldideal,e.By definition 3(i)]= oe o(x) oe o(0) > oe o(x) ii. Let x,z Q and yeX oe o(x*zn)=inf{(x*zn),e}>inf{ min{((x*y)*zn),(y)},e}[SinceisaQ-Smarandachefuzzyn-foldstrongideal,e.By definition3(ii)]=min{inf{((x*y)*zn),e},inf{(y), e } } = min{ oe o ((x*y)*zn), oe o(y) oe o(x*zn)>min{ oe o((x*y)*zn), oe o(y)} oe o is a Q-Smarandache fuzzy n-foldstrong ideal of X. Proposition 11.Let {: e} be a chain of Q-Smarandache fuzzyn-foldstrongidealsofaSmarandacheBH-algebraX. Then oeoisaQ-Smarandachefuzzyn-foldstrongideal of X. Proof :Let {: e} be a chain of Q-Smarandache fuzzy n-fold strongideals of X.i.LetxeX. oeo(0)=sup{(0),e}>sup{ (x),e} [SinceisaQ-Smarandachefuzzyn-foldstrong ideal,e.Bydefinition 3(i)]Paper ID: SUB151537 1520International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 2, February 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY = oeo(x) oeo(0) > oeo(x) xeX .ii.Letx,zQ,yX. oeo(x*zn)=sup{(x),e}> sup{min{((x*y)*zn),(y)}, e} [SinceisaQ-Smarandachefuzzyn-foldstrongideal, e. By definition 3(ii)] But { ,e} is a chain there exist, je such thatsup{min{((x*y)*zn),(y)},e}=min{j((x*y)*zn), j(y)} =min{sup{((x*y)*zn),e}, sup{(y), e}} oeo(x*zn) min{j((x*y)*zn) , j(y)} >min{sup{((x*y)*zn),e},sup{(y),e}=min{ oeo((x*y)*zn), oeo(y) } oeo(x*zn)>min{ oeo((x*y)*zn), oeo(y)} oeo is a Q-Smarandache fuzzy n-fold strong ideal of X. References [1]A.B.SaeidandA.Namdar,"SmarandacheBCH-algebras",WorldAppliedSciencesJournal7(11), 1818-4952, pp.77-83, 2009. [2]C.H.Park,"Interval-valuedFuzzyIdealinBH-algebras",Advanceinfuzzysetandsystems1(3), pp.231240, 2006. [3]E. M.kim and S.S. Ahn, " On fuzzy n-fold strong ideals ofBH-algebras",J.Appl.Math.&InformaticsVol. 30,No.3 4, pp. 665 - 676, 2012 . [4]E.M.kimandS.S.Ahn,"FuzzystrongidealsofBH-algebraswithdegreesintheinterval(0;1]",J.Appl. Math.&InformaticsVol.31,No.1-2,pp.211220, 2013. [5]H.H.Abbass,"TheQ-Smarandacheclosedidealand Q-SmarandacheFuzzyClosedIdealwithRespecttoan ElementofaQ-SmarandacheBCH-Algebra",Journal kerbala university, vol. 10, No.3 scientific, pp. 302-312, 2012 . [6]H.H.AbbassandS.J.Mohammed"OnaQ-SmarandacheFuzzyCompletelyClosedIdealwith RespecttoanElementofaBH-algebra",Kufa University, M.Sc. thesis, 2013. [7] L.A.Zadeh,"Fuzzysets",Informationandcontrol ,Vol. 8, PP. 338-353, 1965.[8]O. G. Xi, "Fuzzy BCK-algebra", Math. Japonica 36, No. 5, pp.935942, 1991. [9]Q.Zhang,Y.B.JunandE.H.Roh,"Onthebranchof BH-algebras",ScientiaeMathematicaeJaponicae Online, Vol.4 , pp.917- 921, 2001. [10] Q.Zhang,E.H.RohandY.B.Jun,"OnFuzzyBH-algebras", J.Huanggang, Normal Univ. 21(3), pp.1419, 2001. [11] Y. B. Jun, "Closed Fuzzy ideals in BCI-algebras", Math. Japon.ica 38, No. 1, pp.199202, 1993.[12] Y. B. Jun, E. H. Roh and H. S. Kim,"On BH-algebras", Scientiae Mathematicae , 1(1) , pp.347354, 1998. [13] Y.B.Jun,"FuzzyClosedIdealsandFuzzyfiltersin BCH-algebras",J.FuzzyMath.7,No.2,pp.435444, 1999.Paper ID: SUB151537 1521