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
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