 Open Access
 Total Downloads : 1136
 Authors : P.M.Sithar Selvam, T.Priya, T.Ramachandran
 Paper ID : IJERTV1IS5026
 Volume & Issue : Volume 01, Issue 05 (July 2012)
 Published (First Online): 02082012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Anti Fuzzy Subalgebras and Homomorphism of CI Algebras
P.M.Sithar Selvam T.Priya T.Ramachandran
Department of Mathematics 
Department of Mathematics 
Department of Mathematics 
PSNACET,Dindigul624 622 
PSNACET,Dindigul624 622 
Govt. Arts College,Karur, 
Abstract
In this paper, we introduce the concept of Anti fuzzy subalgebras of CIalgebras.Also we discussed about ideals in CIalgebra under homomorphism and anti homomorphism and some of its properties.we proved that if Âµ and are anti fuzzy ideals in a CI algebra X, then Âµ x is an anti fuzzy ideal in X x X and few more results in Cartesian product.
Keywords
CIalgebra, fuzzy ideal, Anti fuzzy ideal,fuzzy sub algebra , Anti fuzzy sub algebra, Homomorphism, Anti Homomorphism, Cartesian Products.
AMS Subject Classification (2000): 20N25, 03E72, 03F055,06F35, 03G25.

Introduction
Y.Imai and K.Iseki introduced two classes of abstract algebras : BCKalgebras and BCI algebras [6,7]. It is known that the class of BCKalgebras is a proper subclass of the class of BCIalgebras. In [4,5]Q.P.Hu and X .Li introduced a wide class of abstract BCHalgebras. They have shown that the class of BCI algebras. J.Neggers, S.S.Ahn and H.S.Kim introduced Qalgebras which is generalization of BCK / BCI algebras and obtained several results. In K.Megalai and A.Tamilarasi introduced a class of abstract algebras : TMalgebras , which is a generalisation of Q / BCK / BCI / BCH algebras. In [9] B.L.Meng introduced the notion of a CIalgebra as a generation of a BEalgebra. The concept of fuzzification of ideals in CIalgebra have introduced by Samy.M.Mostafa[14] and the concept of anti fuzzy ideals of CIalgebra have introduced by Priya
T. and Sithar Selvam P.M. [12].R.Biswas introduced the concept of Anti fuzzy subgroups of groups[2]. Modifying his idea, in this paper we apply the idea in CI algebras . We introduce the notion of anti fuzzy ideals,anti fuzzy subalgebras of CIalgebras and studied some of its properties under Homomorphism and Cartesian products.

Preliminaries
In this section we site the fundamental definitions that will be used in the sequel.
Definition 2.1 [9] An algebraic system ( X,*,1) of type (2,0) is called a CI algebra if it satisfies the following axioms.
1. x * x = 1 (2.1)
2. 1 * x = x, (2.2)
3. x * (y * z ) = y * (x * z) , for all x, y , z X (2.3)
In X we can define a binary operation by x y if and only if x * y = 1 for all x, y X . (2.4)
Example 2.1
Let X = {1, 2, 3, 4 } be a set with a binary operation * defined by the following table
*
1
2
3
4
1
1
2
3
4
2
1
1
2
4
3
1
1
1
4
4
1
2
3
1
Then ( X , * , 1 ) is a CIalgebra.
Remark : In an CI algebra, the following identities are true:
4. y * ( ( y * x ) * x ) = 1. (2.5)
5. ( x * 1) * (y * 1) = ( x * y) * 1. (2.6)
Definition 2.2 [14] Let ( X , * , 1) be a CI algebra. A non empty subset I of X is called an ideal of X if it satisfies the following conditions

If x X and a I , then x * a I, (i.e) X * I I
(2.7)

If x X and a,b I , then ( a * ( b * x )) * x I .
(2.8)
Let X be a CI algebra. Then (i) Every ideal of X contains 1.(ii) If I is an ideal of X, then (a * x) * x I for all a I and x X .
Definition 2.3 Let X be a nonempty set. A fuzzy subset
of the set X is a mapping : X [0, 1].
Definition 2.4 [14]Let X be a CI algebra. A fuzzy set
in X is called a fuzzy ideal of X if
(i) ( x * y) (y) , for all x, y X . (2.9)
(ii) ( (x * ( y * z ) ) *z ) min{(x) , (y)}, for all x, y, z X. (2.10)
Definition 2.5 A fuzzy set of a CIalgebra X is called an anti fuzzy ideal of X, if
(i) ( x * y) (y) , for all x, y X. (2.11)
(ii) ((x * ( y * z ) *z ) max{ (x),(y)}, for all x, y, z X. (2.12)
Definition 2.6 A non empty subset S of a CI – algebra X is said to be a sub algebra of X if x * y S whenever x, y S.


Fuzzy Sub algebra and Anti Fuzzy Sub algebra of CI algebra
Definition 3.1[11] A fuzzy set in a CIalgebra X is called a fuzzy Sub algebra of X if
(x * y) min{(x ), ( y)},for all x,y X.
Definition 3.2[11] A fuzzy set in a CI algebra X is called an Anti fuzzy sub algebra of X if
(x * y) max{(x), (y)}, for all x,y X.
Remark:
Every anti fuzzy ideal of a CIalgebra X is an anti fuzzy sub algebra if x = y for any x,y X.
Theorem 3.1
If is an anti fuzzy sub algebra of a CI algebra X, then (1) (x) , for any x X.
Proof
Since x * x = 1 for any x X , then
(1) = (x * x)
max { (x) , (x)}
= (x)
(1) (x).
Definition 3.3 [11] Let be a fuzzy set of X. For a fixed t[0,1], the set t ={x X (x) t} is called the lower level subset of .
Clearly t t = X for t[0,1] if t1 < t2 ,then t1 t2.
Theorem 3.2
A fuzzy set Âµ of a CI algebra X is an anti fuzzy subalgebra if and only if for every t [0,1] , Âµt is either empty or a sub algebra of X.
Proof:
Assume that Âµ is an anti fuzzy sub algebra of X and
Âµt . Then for any x,y Âµt , we have
Âµ(x * y) max { (x), (y)} t. Therefore x*y Âµt. Hence Âµt is a sub algebra of X. Now Let x,y X.
Take t = max { (x), (y)}. Then by assumption Âµt is a sub algebra of X implies x* y Âµt.
Therefore Âµ(x * y) t = max { (x), (y)}. Hence Âµ is an Anti fuzzy sub algebra of X.
Theorem 3.3
Any sub algebra of a CI algebra X can be realized as a level sub algebra of some Anti fuzzy sub algebra of X.
Proof:
Let A be a sub algebra of a given CI algebra X and let Âµ be a fuzzy set in X defined by
Âµ (x) =
Where t [0,1] is fixed. It is clear that Âµt =A.
Now we prove such defined Âµ is an anti fuzzy sub algebra of X.
Let x,y X. If x,y A, then x * y A. Hence (x) = (y)= (x*y) = t and
Âµ(x*y) max{ (x), (y)} If x,y A,then (x) = (y) = 0 and
Âµ(x * y)max { (x),(y)}= 0.
If at most one of x,y A, then at least one of (x) and
(y) is equal to 0.
Therefore, max{ (x), (y)}= 0 so that Âµ(x * y) 0, which completes the proof.
Theorem 3.4
Two level sub algebras Âµs, Âµt (s < t) of an anti fuzzy sub algebra are equal iff there is no xX such that s Âµ(x) < t.
Proof
Let Âµs = Âµt for some s < t.If there exist xX such that s Âµ(x) < t , then Âµt is a proper subset of Âµs, which is
Âµf (( x *( y * z )) * z ) = Âµ ( f( ( x *( y * z )) * z ))
= Âµ ( f( x *( y * z ) ) * f (z))
= Âµ ( ( f( x ) * f( y * z ) ) * f(z))
= Âµ ((f(x)*(f(y) * f(z)))* f (z) )
max{ Âµ ( f( x ) ) , Âµ ( f( y ) ) }
= max { Âµ ( x ), Âµ ( y ) }
a contradiction. f f
Conversely, assume that there is no xX such that s
Âµ(x) < t.
If x Âµs, then Âµ(x) s and Âµ(x) t,
Since Âµ(x) does not lie between s and t. Ths x Âµt , which gives
Âµs Âµt , Also Âµt Âµs. Therefore Âµs = Âµt.

Homomorphism and Anti Homomorphism of CI algebra
In this section, we discussed about ideals in CI algebra under homomorphism and anti homomorphism and some of its properties.
Definition 4.1 Let (X,*,1) and ( Y , ,1` ) be CI algebras. A mapping f: X Y is said to be a homomorphism if f( x * y) = f(x) f(y) for all x,y X.
Definition 4.2 Let (X,*,1) and ( Y , ,1`) be CI algebras. A mapping f: X Y is said to be a anti homomorphism if f( x * y) = f(y) f(x) for all x,y X.
Definition 4.3 Let f: X X be an endomorphism and Âµ be a fuzzy set in X. We define a new fuzzy set in X by Âµf in X as Âµf (x) = Âµ (f(x)) for all x in X.
Definition 4.4 For any homomorphism f: X Y, the set
{xX / f(x) = 1} 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).
Theorem 4.1
Let f be an endomorphism of a CI algebra X. If Âµ is an anti fuzzy ideal of X, then so is Âµf .
Proof:
Âµf ( x * y ) = Âµ ( f ( x * y ))
= Âµ (f(x) * f (y))
Âµ (f(y)) = Âµf (y) , for all x,y X.
Let x,y,z X.
Then
Âµf (( x *( y * z )) * z ) max { Âµf ( x ), Âµf ( y ) }
Hence Âµf is an anti fuzzy ideal of X.
Theorem 4.2
Let f: X Y be an epimorphism of CI algebra. If Âµf is an anti fuzzy ideal of X, then Âµ is an anti fuzzy ideal of Y.
Proof:
Let y Y.Then there exists x X such that f( x ) = y. Let y1,y2 ,y3 Y.
Âµ ( y1 y2 ) = Âµ ( f ( x1 ) f( x2 ) )
= Âµ ( f ( x1 * x2 ) )
= Âµf (x1 * x2 )
Âµf ( x2 ) = Âµ ( f ( x2 ) ) = Âµ ( y2 )
Âµ (y1 y2 ) Âµ (y2 ) Then
Âµ((y1 (y2 y3)) y3) = Âµ([f(x1) ( f(x2)f(x3))] f(x3))
= Âµ( [ f(x1) f(x2 * x3) ] f(x3) )
= Âµ( f [ x1 * ( x2 * x3 ) ] f(x3) )
= Âµ( f ( [ x1 * ( x2 * x3 ) ] * x3 ) )
= Âµf ( [ x1 * ( x2 * x3 ) ] * x3 )
max { Âµf ( x1 ), Âµf ( x2 ) }
= max { Âµ ( f( x1 ) ) , Âµ ( f( x2) ) }
= max { Âµ ( y1 ), Âµ ( y2 ) }
Âµ ( ( y1 (y2 y3) ) y3) max { Âµ ( y1 ), Âµ ( y2 ) } Hence Âµ is an anti fuzzy ideal of Y.
Theorem 4.3
Let f: X Y be a homomorphism of CI algebra. If Âµ is an antifuzzy ideal of Y then Âµf is an anti fuzzy ideal of X.
Proof:
Let x,y,z X.
Âµf ( x * y ) = Âµ ( f ( x * y ) )
= Âµ ( f( x ) f ( y ) )
Âµ ( f( y ) ) = Âµf (y) .
Âµf ( x * y ) Âµf ( y ). Then
Âµf (( x *( y * z )) * z ) = Âµ ( f ( ( x *( y * z )) * z ) )
= Âµ ( f( x *( y * z ) ) f (z))
= Âµ ( ( f( x ) f( y * z ) ) f (z))
= Âµ ( ( f( x ) ( f( y ) f( z ) ) ) f (z) )
max { Âµ ( f( x ) ) , Âµ ( f( y ) ) }
= max { Âµf ( x ), Âµf ( y ) }
Âµf (( x *( y * z )) * z ) max { Âµf ( x ), Âµf ( y ) } Hence Âµf is an anti fuzzy ideal of X.
Theorem 4.4 Let (X ,*, 1) and ( Y , ,1`) be CI algebras. A mapping f: X Y is a homomorphism of CIalgebra, Then Ker(f) is an ideal.
Proof:
It is clear that 1 ker f.
By [14] it is enough to prove that (x * y) * z ker (f )
x * z ker f , for all x,z X and y ker f. Let (x * y) * z ker (f ) & y ker (f ) .
Then f ( (x * y ) * z) = 1 & f ( y ) = 1 Since 1 = f ((x * y) * z)
= f ( x ) f ( y * z )
= f( x ) ( f( y ) f( z ) )
= f( x ) (1 f( z ))
= f( x ) f( z )
= f (x * z)
x * z ker ( f ) Hence ker f is an ideal.

Cartesian Product of Anti Fuzzy ideals of CI algebras
In this section, we introduce the concept of Cartesian product of anti fuzzy ideals of CIalgebra.
Definition 5.1 Let Âµ and be the fuzzy sets in X. The Cartesian product Âµ x : X x X [0,1] is defined by
( Âµ x ) ( x, y) = min {(x),(y)}, for all x, y X.
Definition 5.2 Let Âµ and be the anti fuzzy sets in X. The Cartesian product Âµ x : X x X [0,1] is defined by ( Âµ x ) ( x, y) = max {(x), (y)} , for all x, y X.
= max { Âµ ( x1 * y1 ), ( x2 * y2 ) }
max { Âµ ( y1 ) , ( y2 ) }
= ( Âµ x ) ( y1 , y2 )
( Âµ x ) ( ( x1 , x2 ) * ( y1 , y2 ) ) ( Âµ x ) ( y1 , y2 )
(Âµ x ) { ( ( x1 , x2 ) * ( ( y1 * y2 ) * ( z1, z2 ) ) ) * ( z1, z2 )}
= (Âµ x ) { [ ( x1, x2 ) * ( y1 * z1 , y2 * z2 ) ] * ( z1, z2 ) }
= (Âµ x ) {[ ( x1 * ( y1 * z1 )) , x2 * (y2 * z2 ) ] * ( z1, z2 ) }
= (Âµ x ) { ( x1 * ( y1 * z1 )) * z1 , ( x2 * ( y2 * z2 )) * z2 )}
= max{Âµ(( x1 *(y1* z1 )) * z1 ), ( ( x2 * ( y2 * z2 )) * z2 )}
max { max { Âµ( x1 ), Âµ ( y1 )}, max{ ( x2 ), ( y2 )}}
= max { max { Âµ( x1 ), ( x2 ) } , max { Âµ ( y1 ), ( y2 ) } }
= max { (Âµ x ) ( x1 , x2 ) , (Âµ x ) ( y1 , y2 ) }
( Âµ x ){(( x1 , x2 ) * ( ( y1 * y2 ) * ( z1, z2 ) ) )* (z1, z2 )}
max { (Âµ x ) ( x1 , x2 ) , (Âµ x ) ( y1 , y2 ) } Hence, Âµ x is an antifuzzy ideal in X x X.
Theorem 5.2:
Let & be fuzzy sets in a CI algebra X such that x
is an Antifuzzy ideal of X x X . Then

Either (1) (x) (or) (1) (x) for all x X

If (1) (x) for all x X , then either (1) (x) (or) (1) (x)

If (1) (x) for all x X, then either (1) (x) (or) (1) (x).

Either or is an anti fuzzy ideal of X.
Proof:
Let x be an anti fuzzy ideal of X x X .
Therefore (Âµ x ) ( ( x1 , x2 ) * ( y1 , y2 ) ) ( Âµ x ) ( y1 , y2 ) and
( Âµ x ) {(( x1 , x2 ) * ( ( y1 * y2 ) * ( z1, z2 )) ) * ( z1, z2 )}
max { (Âµ x ) ( x1 , x2 ) , (Âµ x ) ( y1 , y2 ) }
for all ( x1, x2 ),( y1, y2 ) & ( z1, z2 ) X x X.

Suppose that (1) > (x) and (1) > (x) for some x, y X
Then ( x ) (x ,y) = max{ (x) , (y) }
< max { (1) , (1) }
Theorem 5.1
contradiction.[12]
= ( x ) (1,1) , Which is a
If Âµ and are anti fuzzy ideals in a CI algebra X, then Âµ x is an anti fuzzy ideal in X x X .
Proof:
Let ( x1, x2) , ( y1, y2) , ( z1, z2) X x X.
(Âµ x ) (( x1 , x2 ) * ( y1 , y2 )) =( Âµ x ) ( x1 * y1 , x2 * y2 )
Therefore (1) (x) and (1) (x) for all x X.

Assume that there exists x ,y X such that
(1) > (x) and (1) > (x).
Then ( x ) (1,1) = max { (1), (1) } = (1) and hence
( x ) (x , y) = max { (x),(y) } < (1) = ( x ) (1,1 ) Which is a contradiction.[12]
Hence if (1) (x) for all x X, then either
(1) (x) (or) (1) (x).
Similarly, we can prove that if (1) (x) for all x X ,then either (1) (x) or (1) (y), which yields (iii) .
(iv) First we prove that is an anti fuzzy ideal of X
Since by (i)either (1) (x) (or) (1) (x) for all x X.
Assume that (1) (x), for all x X.
It follows from (iii) that either (1) (x) or (1) (x). If (1) (x) for any x X, then
(x) = max { (1) , (x)} = ( x ) (1, x)
( x * y) = max { (1), (x *y)}
= ( x ) ( 1, x * y )
= ( x ) ( 1 * 1, x * y )
= ( x ) ( (1, x ) * (1,y ) )
( x ) ( (1, y)
= (y )
(x * y) (y)
((x * (y * z))* z) = max { Âµ(1), ((x * (y * z))* z) }
= ( Âµ x ) (1,( x * (y * z))* z )
= ( Âµ x ) { 1 * 1, (x * (y * z))* z) }
= ( Âµ x ) { ( 1, x * (y * z) ) * (1, z) }
= ( Âµ x ) { ( 1 * 1 , x * (y * z) ) * (1, z) }
= ( Âµ x ){ [(1, x) * (1,y * z) )] * (1, z) }
= (Âµ x ) {[(1, x) * (1*1,y*z)] * (1, z) }
= (Âµ x ) {[(1, x) * ((1,y) * (1,z))] * (1, z)}
max { (Âµ x ) ( 1 , x ) , (Âµ x ) ( 1 , y ) }
= max { (x) , (y) }
((x * (y * z))* z) max { (x) , (y) } Hence is an Anti fuzzy ideal of X.
Next we will prove that is an anti fuzzy ideal of X. Let (1) (x)
Since by (ii), either (1) (x) or (1) (x). Assume that (1) (x),then
(x) = max { (x) , (1)} = ( x ) ( x,1 )
( x * y) = max {( x * y ), (1) }
= ( x ) ( x * y,1 )
= ( x ) ( x * y ,1 * 1 )
= ( x ) ( ( x,1 ) * ( y,1) )
( x )( y,1 )}
= (y)
Âµ ((x * (y * z))* z) = max { Âµ ( (x * (y * z)) * z) , (1) }
= ( Âµ x ) { ( x * (y * z) )* z , 1}
= ( Âµ x ) { (x * (y * z))* z), 1 * 1 }
= ( Âµ x ) { ( x * (y * z),1) * (z,1) }
= ( Âµ x ) { ( x * (y * z),1 *1) * (z,1)}
= ( Âµ x ){ [(x,1) * ( y * z,1)] * (z,1)}
= (Âµ x ) {[(x,1) * (y * z,1 * 1)] * (z,1)}
= (Âµ x ) {[(x,1) * ((y,1) * (z,1))] * (z,1)}
max { (Âµ x ) ( x , 1) , (Âµ x ) ( y , 1 ) }
= max { Âµ(x) , Âµ(y) }
Âµ ((x * (y * z))* z) max { Âµ(x) , Âµ(y) } Hence Âµ is an Anti fuzzy ideal of X.
Conclusion
In this article we have discussed anti fuzzy ideal, Anti fuzzy sub algebra of CIalgebras under homomorphism and Anti homomorphism, Cartesian Products. It has been observed that the CIalgebra as a generation of BE algebra. These concepts can further be generalized.
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