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In this paper, we define the concepts of a QSmarandache nfold strong ideal anda QSmarandache fuzzy (strong, nfold strong) ideal of a BH algebra .Also, we study some properties of these fuzzy ideals.TRANSCRIPT

International Journal of Science and Research (IJSR) ISSN (Online): 23197064
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 nFold Strong Ideal of a BHAlgebra
Shahrezad Jasim Mohammed
Abstract: In this paper, we define the concepts of a QSmarandache nfold strong ideal anda QSmarandache fuzzy (strong, nfold strong) ideal of a BH algebra .Also, we study some properties of these fuzzy ideals
Keywords: BCKalgebra , BCI/BCHalgebras, BHalgebra , Smarandache BHalgebra, QSmarandache 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 BCKalgebras [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 BCHalgebras [13]. In 2001, Q. Zhang, E. H.
Roh and Y. B. Jun studied the fuzzy theory in a BHalgebra
[10] . In 2006,C.H. Park introduced the notion of an interval
valued fuzzy BHalgebra in a BHalgebra [2].In 2009, A. B.
Saeid and A. Namdar, introduced the notion of a
Smarandache BCHalgebra and QSmarandache ideal of a
Smarandache BCHalgebra [1].In 2012, H. H .Abbass
introduced the notion of a QSmarandache 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 (nfold 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 BHalgebra[4].In the
same year, H. H. Abbass and S. J. Mohammed introduced
the QSmarandache fuzzy completely closed ideal with
respect to an element of a BHalgebra[6] . In this paper, we
define the concepts of QSmarandachenfold strong ideal
anda QSmarandache fuzzy (strong, nfold 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 BCIalgebra ,a BHalgebra , a BH*algebra,a
normal BHalgebra, fuzzy strong ideal, fuzzynfold strong
ideala Smarandache BHalgebra,( QSmarandache ideal, Q
Smarandache fuzzyclosedideal, QSmarandache fuzzy
completely closed ideal and QSmarandache fuzzy ideal of
BHalgebra
Definition 1 (see[11]).A BCIalgebra 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 BCKalgebra is a BCIalgebra
satisfying the axiom
v. 0 * x = 0 for all x X.
Definition 3(see[9]). A BHalgebra 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 BHalgebraX is called a BH*
algebraif (x*y)*x=0 for all x,yX
Definition5.(see[9]). A BHalgebra X is said to be a
normalBHalgebra 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.
Definition 6.(see[5]) . Let X be a BHalgebra.Then the set
X+={xX:0*x=0} is called the BCApart of X .
Remark 1(see[6]).Let X and Y be BHalgebras. 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 BHalgebraand n be a
positive integer. A nonempty subset I of X is called a nfold
strong ideal of X if it satisfies: i. 0 I , ii. y I and (x* y)*z
nI x*znI, x ,z X.
Definition 8.(see[6]).A Smarandache BHalgebra is defined
to be a BHalgebra X in which there exists a proper subset Q
of X such that
i. 0 Q and Q 2 . ii. Q is a BCKalgebra under the operation of X.
Definition9 (see[6]). Let X be a Smarandache BHalgebra .
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

International Journal of Science and Research (IJSR) ISSN (Online): 23197064
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
Definition 10(see[7]). Let be a fuzzy set in X, for all t[0,1]. The set t{xX, (x) t} is called a level subset of .
Definition11(see [4)].Let A and B be any two sets, be any fuzzy set in A and f : A B be any function. Set1(y) = {x A f (x) = y} for y B. The fuzzy set in B defined by
(y) ={0 otherwisesup {(x) x 1(y)} if 1 (y)
for all y B, 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 all x X is called the preimage of under f and is denoted by 1().
Definition13.(see[4].A fuzzy set in a BHalgebra X is 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].A fuzzy set in a BHalgebra X is called a fuzzy nfold strong idealof X if i. For all xX, (0) (x). ii. (x zn) min{ ((x y) zn); (y)}, x, y X.
Definition15(see [6]).A fuzzy subset of a Smarandache BHalgebra X is said to be a QSmarandachefuzzy idealif
and only if :
i. For all xX, (0) (x). ii.For all x Q, yX, (x) min{ (x*y), (y)}. Is said to be closed if (0*x) (x), for all xX.
Definition16(see[6]).Let X be a Smarandache BHalgebra
and be a QSmarandache fuzzy ideal of X. Then is called a QSmarandachefuzzy completely closed ideal if
(x*y) min{ (x), (y)},x,yX.
Proposition1(see[6]).Let X be a BH*algebra ,and be a QSmarandache fuzzy ideal. Then
i. is a QSmarandache fuzzy closed ideal of X . ii. is a QSmarandache fuzzy completely closed ideal of X, if X*X/{0} Q
3. The Main Results
In this paper, we give the concepts a QSmarandache nfold
strong ideal and a QSmarandache fuzzy (strong, nfold
strong) ideal of a BH algebra .Also, we give some
properties of these fuzzy ideals
Definition 1 .A fuzzy subset of a BHalgebra X is called a
QSmarandache fuzzy strong ideal ,iff i. (0) (x) )
xX
ii. (x*z) min{ ((x*y)*z), (y)},x,zQ.
Example 1: The set X={0,1,2,3} with the following
operation table
* 0 1 2 3
0 0 0 2 2
1 1 0 1 2
2 2 2 0 0
3 3 2 1 0
is a BHalgebra Q={0,1} is a BCKalgebra .Then (X,*,0) is a Smarandache BHalgebra.The fuzzy set which is defined by:
(x) = 0.5 x = 0,3 0.4 x = 1,2
is a QSmarandache fuzzy strong ideal , since: i.(0) = 0.5 (x) xX, ii. (x*z) min{ ((x*y)*z), (y)},x, zQ.
But the fuzzy set(x) = 0.5 x = 0,2,3
0.4 x = 1
is not a QSmarandache fuzzy strong ideal
since(1*0)=(1)=0.4 < min{ (1*3)*0), (3)}=0.5
Proposition1.Every QSmarandache fuzzy strong ideal of a
Smarandache BHalgebra X is a QSmarandache 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)]
When z=0(x*0) min{((x*y)*0) , (y)}(x) min{(x*y) , (y)} is a QSmarandache fuzzy ideal of X.
Proposition2. Let Q1 and Q2 be a BCKalgebras contained in a Smarandache BH algebra X and Q1Q2. Let be Q2Smarandache fuzzy strong ideal of X then is a Q1Smarandache fuzzy strong ideal of X .
Proof :Let be a Q2Smarandache fuzzy strong ideal of X . i. Let x X (0) (x) . [Since is a Q2 Smarandache fuzzy strong ideal. By definition 1(i)]
ii. Let x,z Q1,yX x,z Q2 (x*z) min{ ((x*y)*z) ,(y)} [Since is a Q2 Smarandache fuzzy strong ideal. By definition 1(ii)]
is a Q1Smarandache fuzzy strong ideal of X.
Proposition3. Every fuzzy strong ideal of a Smarandache
BHalgebra X is a QSmarandache 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 QSmarandache fuzzy strong ideal of X.
Theorem 1. Let X be SmarandacheBHalgebra and let be a fuzzy set.Then is a QSmarandache fuzzy strong ideal if and only if (x) = (x) / (0) is a QSmarandache fuzzy strong ideal.
Proof: Let be a QSmarandache fuzzy strong ideal, 1) (0)= (0) / (0), (0)=1 (0) (x) xX
Paper ID: SUB151537 1517

International Journal of Science and Research (IJSR) ISSN (Online): 23197064
Index Copernicus Value (2013): 6.14  Impact Factor (2013): 4.438
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2) (x*z)= (x*z)/ (0) min{ ((x*y)*z), (y)}/ (0)} [Since is a QSmarandache fuzzy strong ideal. .By definition1(ii)]
min{ ((x*y)*z) / (0) , (y) / (0)} min{ ((x*y)*z), (y)} (x*z) min{ ((x*y)*z), (y)} is a QSmarandache fuzzy strong ideal.Conversely.Let be a QSmarandache fuzzy strong ideal. i. (0)= (0). (0), (0) (x). (0) (0) (x) xX ii. (x*z)= (x*z) . (0) min{ (x*(y*z)), (y)}. (0) [Since ' is a QSmarandache 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 QSmarandache fuzzy strong ideal. Proposition 4.Let X be a BH*algebra ,and be a QSmarandache fuzzy strong ideal. Then
i. is a QSmarandache fuzzy closed idealof X. ii. is a QSmarandache fuzzy completely closed ideal of X,if X*X/{0} Q.
Proof :Let be a fuzzy strong ideal of X. [ByProposition1]
is a QSmarandache fuzzy ideal of X. i. is a QSmarandache fuzzy closed ideal of X. [By proposition 1]
ii. Let x,y X is a QSmarandache fuzzy completely closed ideal of X.
[By proposition 1].
Theorem 2.Let A be a nonempty subset of a Q
Smarandache BHalgebra X and let be a fuzzy set in X defined by:
(x)={2 otherwise1 xQ
where 1 > 2 in [0, 1].Then is a QSmarandache fuzzy 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 QSmarandache fuzzy strong ideal of X.
Theorem 3. Let X be BHalgebra and let be a fuzzy set.Then is a QSmarandache fuzzy strong ideal if and only if '(x)= (x)+1 (0) is a QSmarandache fuzzy strong ideal.
Proof: Let be a QSmarandache fuzzy strong ideal, i. '(0)= (0)+1 (0), '(0)=1 '(0) '(x) xX ii. '(x*z)= (x*z)+1 (0) min{ ((x*y)*z)), (y)}+1 (0)[Since is a QSmarandache 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 QSmarandache fuzzy strong ideal.
ConverselyLet ' be a QSmarandache fuzzy strong ideal. 1) (0)= '(0)1+ (0), (0) '(x)1+ (0) (0) (x) xX 2) (x*z)= '(x*z)1+ (0) min{ '((x*y)*z)), '(y)}1+ (0) [Since ' is a QSmarandache 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 QSmarandache fuzzy strong ideal.
Theorem4. Let X be a Smarandache BHalgebra such that X
= X+ and be a QSmarandachestrong ideal of X. Then is a QSmarandache closed ideal of X .
Proof :Let be a QSmarandache strong ideal I is a QSmarandache ideal of X. [Byproposition1]
Now, let x I (0*x) = (0) (x) [By definition 6] is a QSmarandache closed ideal of X .
Proposition5. Let X be a Smarandache normal BHalgebra
such that X=X+ and let be a QSmarandache fuzzy strong ideal such that x*y Q,x,yX and y0.Then is a QSmarandache fuzzy completely closed ideal of X.
Proof: Let I be a QSmarandache fuzzy strong ideal of X. is a QSmarandache 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)}[By definition 3(ii)] = min{ (0*y), (x)} [By definition 5(ii)] = min{ (0), (x)} = (x) (x*y) min{ (y), (x)} is a QSmarandache fuzzy completely closed ideal.
Proposition 6.Let X be a normal BHalgebra such that
X*X/{0} Q .Then every QSmarandache fuzzy strong ideal and closed of X is a QSmarandache fuzzy completely
closed ideal of X.
Proof :Let be a QSmarandache fuzzy strong ideal of X.
Paper ID: SUB151537 1518

International Journal of Science and Research (IJSR) ISSN (Online): 23197064
Index Copernicus Value (2013): 6.14  Impact Factor (2013): 4.438
Volume 4 Issue 2, February 2015
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is a QSmarandache 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 QSmarandache fuzzy completely closed ideal of X.
Remark 1. Let be a fuzzy set of a Smarandache BHalgebra X and w X .The set {xX: (w) (x)} is denoted by (w) . Theorem 5.Let X be a Smarandache BHalgebra, wX and is a QSmarandache fuzzy strong ideal of X. Then(w) is a QSmarandachestrong ideal of X.
Proof :Let be a QSmarandache fuzzy strong ideal of X. To prove that (w) is a QSmarandache 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 QSmarandache strong ideal of X.
Corollary1.Let X be a Smarandache BHalgebra . Then is a QSmarandache fuzzy strong ideal of X if and only if t is a QSmarandache strong ideal of X, for all t [0,sup
Xx
(x)]
Proof :Let t [0,supXx
(x)]. To prove that t is a Q
Smarandache strong ideal of X.Since is a QSmarandache 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 QSmarandache strong ideal of X. Conversely,
To prove that is a QSmarandache fuzzy strong ideal of X. Since t is a QSmarandache strong ideal of X. Let t =sup
Xx
(x) ,x,zQ and(x*y)*z, y t
x*z t[By definition9]
(x*z) t (x*z) = t [Since t =Xx
sup(x)]
Similarly,((x*y)*z)=t and (y)=t t = min {((x*y)*z),(y)} (x*z) min {((x*y)*z),(y)} is a QSmarandache fuzzy strong ideal of X.
Proposition7 .Let f : (X, *, 0) (Y, , 0 ) be a Smarandache BHepimorphism. If is a QSmarandache fuzzy strong ideal of X, then f () is a f (Q)Smarandache fuzzy strong ideal of Y .
Proof :Let be a QSmarandache 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. Let y1,y3 f (Q), y2 Y, there exists x1, x3 Q and
x2 X such that y1= 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.
Theorem 6.Let f :(X,*,0)(Y, , 0 )be a Smarandache BHepimorphism.Ifis a Q Smarandache fuzzy strong ideal of Y, then 1() is a1(Q)Smarandache fuzzy strong ideal of X
Proof :i. Let x X. Since f (x) Y and is a QSmarandachestrong 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 QSmarandache fuzzy strong ideal of X .
Definition 2. Let X be a Smarandache BHalgebra and n be
apositive integer. A nonempty subset I of X is called a
Smarandachenfold strong ideal of X related toQ ( or
briefly, Q Smarandachenfold strong ideal of X) if it
satisfies: i. 0 I , ii. y I and (x* y)*znI x*z I, x ,z Q.
Example 2.Consider the set I={0,3} in example 1 is a Q
Smarandachenfold strongideal of X.But the set I={0,2,3} is
not a QSmarandachenfold strongideal since(1*3)*0n=
2I,but 1*0n=1 I.
Remark 2. Every nfold strong ideal of a Smarandache BH
algebra X is a QSmarandachenfold strong ideal of X.
Remark3.Every QSmarandachenfold strong ideal of a
Smarandache BHalgebra X is a QSmarandachestrong ideal
of
X.Remark4.Every QSmarandachenfold strong ideal of a
Smarandache BHalgebra X is a QSmarandache ideal of X.
Proposition8 . Let X be a Smarandache BHalgebra . Then
every QSmarandachenfoldstrong ideal which is contained
in Q is a QSmarandache completely closed ideal of X.
Proof :Let I be a QSmarandachenfoldstrong ideal of X I is a QSmarandache ideal of X. [By remark4]
Now, let x,yI x,yQ x*yQ [Since I Q ]
Paper ID: SUB151537 1519

International Journal of Science and Research (IJSR) ISSN (Online): 23197064
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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)*0nI and xI. (x*y)*0n I x*y I[By definition 3(ii)]
I is a QSmarandache completely closed ideal of X.
Definition 3.A fuzzy subset of a Smarandache BHalgebra
Xand n be a
positive integer is called a QSmarandache fuzzy nfold
strong ideal ,iff i. (0) (x) ) xX. ii. (x*zn) min{
((x*y)*zn), (y)},x,zQ.
Example3.Consider thefuzzy set which is defined by:
(x) = 0.5 x = 0,2 0.4 x = 1,3
is a QSmarandache fuzzy nfoldstrong ideal , since: i.(0) = 0.5 (x) xX, ii. (x*z1) min{ ((x*y)*z1), (y)},x, zQ.
But the fuzzy set(x) = 0.5 x = 0,2,3
0.4 x = 1
is not a QSmarandache fuzzy nfoldstrong ideal
since(1*03)=(1)=0.4 < min{ (1*3)*03), (3)}=0.5
Remark5. Every fuzzy nfold strong ideal of a Smarandache
BHalgebra X is a QSmarandache fuzzy nfold strong ideal
of X.
Proposition 9 .Every QSmarandache fuzzy nfold strong
ideal of a Smarandache BHalgebra X is a QSmarandache
fuzzy strong ideal of X.
Proof : Let be a QSmarandache fuzzy nfold strong ideal of X .i. Let x X (0) (x) .[By definition 3(i)] ii. let x,zQ and y X (x*zn) min{((x*y)*zn) , (y)}[By definition 3(ii)] When n=1(x*z) min{((x*y)*z) , (y)} is a QSmarandache fuzzy strong ideal of X.
Theorem 7. Let f :(X,*,0)(Y, , 0 )be a Smarandache BHepimorphism. If is a QSmarandache fuzzy nfold strongideal of Y, then 1() is a1(Q)Smarandache fuzzy nfold strongideal of X.
Proof :i.Let x X. Since f (x) Y and is a QSmarandache fuzzy nfold strongideal of Y.
(1())(0)= ( f (0))= (0 ) ( f (x)) =(1())(x) ii. Let x,z 1(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 QSmarandache fuzzynfoldstrong ideal of X .
Theorem 8.Let X be Smarandache BHalgebra. If is a fuzzy set such that Q=
X={xX: (x)=(0)}and (0) (x) xX, then is a QSmarandache fuzzy nfold strong ideal of X.
Proof :Let be a fuzzy set of X, such that Q =X and (0) (x) xX. i. (0)(x) xX. ii. Let x,zQ and yX . (0)(y) and (0) x y zn[Since(0)(x) xX]
(0) min{((x*y)*zn),(y)} But (x*zn)=(0)[Since Q= X] (x*zn) min{((x*y)*zn),(y)} is a QSmarandache fuzzy n fold strong ideal of X.
Proposition10.Let {:} be a family of QSmarandache fuzzy n fold strongideals of a Smarandache BHalgebra X.
Then
is a fuzzy nfold strongideal of X.
Proof :Let {:}be a family of QSmarandache fuzzy n foldstrong ideals of X.
i. Let xX.
(0) = inf{ (0), }inf {(x), }
[Since is a QSmarandache fuzzy nfoldideal, . By definition 3(i)]
=
(x)
(0)
(x)
ii. Let x,z Q and yX
(x*zn) = inf{ (x*z
n), } inf{
min{((x*y)*zn), (y)}, }[Since is a Q
Smarandache fuzzy nfoldstrong ideal, . By definition3(ii)]= min{ inf{ ((x*y)*z
n), }, inf{ (y),
} }
= min{
((x*y)*zn),
(y)
(x*zn)min{
((x*y)*zn),
(y)}
is a QSmarandache fuzzy nfoldstrong ideal of X.
Proposition 11.Let {: } be a chain of QSmarandache fuzzy nfold strongideals of a Smarandache BHalgebra X.
Then is a QSmarandache fuzzy nfoldstrong ideal
of X.
Proof :Let {: } be a chain of QSmarandache fuzzy nfold strongideals of X.
i. Let xX. (0)sup{ (0),} sup{
(x),} [Since is a QSmarandache fuzzy nfoldstrong ideal,.Bydefinition 3(i)]
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=
(x)
(0)
(x) xX .
ii. Let x,zQ,yX.
(x*zn) = sup{ (x),}
sup{min{((x*y)*zn),(y)}, }
[Since is a QSmarandache fuzzy nfoldstrong ideal, . By definition 3(ii)] But { ,} is a chain there exist, j such that sup{ min{((x*y)*z
n), (y)}, } = min{j((x*y)*z
n),
j(y)} =min{sup{((x*y)*z
n),}, sup{(y), }}
(x*zn) min{j((x*y)*z
n) , j(y)}
min{sup{((x*y)*zn),}, sup{(y), }= min{
((x*y)*z
n),
(y) }
(x*z
n) min{
((x*y)*z
n),
(y)}
is a QSmarandache fuzzy nfold strong ideal of X.
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Paper ID: SUB151537 1521