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1. P.M.Sitharselvam , T.Priya, T.Ramachandran/ International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1286-1289
Anti Q-Fuzzy KU - Ideals In KU- Algebras And Its Lower Level
Cuts
1
P.M.Sitharselvam , 2 T.Priya, 3 T.Ramachandran
1,2
Department of Mathematics, PSNA College of Engineering and Technology,
Dindigul-624 622, Tamilnadu , India.
3
Department of Mathematics, Government Arts College(Autonomous),
Karur– 639 005,Tamilnadu , India
Abstract
In this paper, we introduce the concept of 2. 0 * x = x
Anti Q-fuzzy KU-ideals of KU-algebras, lower level 3. x * 0 = 0
cuts of a fuzzy set and prove some results . We show 4. x * y = 0 = y * x implies x = y , for all x, y , z  X .
that a Q-fuzzy set of a KU-algebra is a KU-ideal if In X we can define a binary operation  by
and only if the complement of this Q-fuzzy set is an x  y if and only if y * x = 0. Then (X,*,0) is a
anti Q-fuzzy KU-ideal. Also we discussed few results KU-algebra if and only if it satisfies that
of Anti Q-fuzzy subalgebras of KU-algebra. i. (y * z) * (x * z) ≤ (x * y)
ii. 0 ≤ x
Keywords:- KU-algebra,fuzzy KU- ideal,Anti Q- iii. x ≤ y , y ≤ x implies x = y.
fuzzy KU-ideal, Q-fuzzy subalgebra, Anti Q- fuzzy iv. x  y if and only if y * x = 0, for all x, y , z  X.
subalgebra, lower level cuts. In a KU-algebra,the following identities are true[14]:
AMS Subject Classification (2000): 20N25, 03E72, 1. z * z = 0.
03F055 , 06F35, 03G25. 2. z * (x * z) = 0
3. x  y  y * z ≤ x * z
1.Introduction 4. z * ( y * x ) = y * (z * x)
Y.Imai and K.Iseki introduced two classes of 5. y * [ (y * x) * x ] = 0 , for all x, y , z  X.
abstract algebras : BCK-algebras and BCI – Example 2.1
algebras[6,7]. It is known that the class of BCK- Let X = { 0 , a , b, c, d } be a set with a binary
algebras is a proper subclass of the class of BCI- operation * defined by the following table
algebras. Q.P.Hu and X .Li introduced a wide class of
abstract BCH-algebras[4,5]. They have shown that it * 0 a b c d
is the class of BCI-algebras. J.Neggers, S.S.Ahn and 0 0 a b c d
H.S.Kim introduced Q-algebras which is a 0 0 b c d
generalization of BCK / BCI algebras and obtained b 0 a 0 c c
several results[10].W.A.Dudek and Y.B.Jun studied c 0 0 b 0 b
fuzzy ideals and several fuzzy structures in BCC- d 0 0 0 0 0
algebras are considered[2,3]. C.Prabpayak and
U.Leerawat introduced a new algebraic structure
Then Clearly ( X , * , 0 ) is a KU-algebra.
which is called KU-algebras and investigated some
properties[11].Samy M.Mostafa and Mokthar A.
Definition 2.2 [11]
Abdel Naby introduced fuzzy KU-ideals in KU-
Let ( X , * , 0) be a KU-algebra. A non empty subset
algebras[14].R.Biswas introduced the concept of Anti
I of X is called KU- ideal of X if it satisfies the
fuzzy subgroups of groups[1]. Modifying his idea, in
following conditions
this paper we apply the idea of KU-algebras . We
(1) 0I
introduce the notion of Anti Q-fuzzy KU-ideals of
KU-algebras and Anti Q-fuzzy subalgebras of KU- (2) x * (y * z)  I and y  I  x * z  I
algebras. for all x , y, z  X .
Remark: From Example 2.1, It is clear that I1 = {0,a}
2. Preliminaries
and I2 = {0,a,b} are KU-ideals of X.
In this section we site the fundamental
Definition 2.3 [16]
definitions that will be used in the sequel.
Definition 2.1[11] Let X be a non-empty set. A fuzzy subset 
A non empty set X with a constant 0 and a of the set X is a mapping  : X [0, 1].
binary operation * is called a KU-algebra if it Definition 2.4 [8,9]
satisfies the following axioms. Let Q and G be any two sets.A mapping
1. (x * y) *[(y * z) * (x * z) ] = 0 : G x Q [0, 1] is called a Q –fuzzy set in G.
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2. P.M.Sitharselvam , T.Priya, T.Ramachandran/ International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1286-1289
(ii) µc ( x * z, q) = 1 - µ(x * z, q)
Definition 2.5 ≤ 1 – min { µ ( x * ( y * z) , q) , µ ( y ,q) }
A Q- fuzzy set  in X is called a Q-fuzzy = 1 – min{1- µc ( x * (y * z) , q) ,1 - µc(y,q) }
KU- ideal of X if = max { µc ( x * ( y * z) , q) , µc ( y , q) }
(i) (0,q) ≥ (x,q) That is,
(ii) (x* z ,q) ≥ min{(x *( y * z),q), (y,q)}, µc ( x * z , q) ≤ max { µc ( x* ( y * z) , q) , µc ( y , q) }.
for all x,y, z X and q ∈ Q. Thus µc is an anti Q-fuzzy KU-ideal of X. The
converse also can be proved similarly.
Definition 2.6 [9]
If µ be a Q-fuzzy set in set X then the Theorem 3.3
complement denoted by µc is the Q-fuzzy subset of X Let µ be an anti Q-fuzzy KU-ideal of
given by µc (x,q) = 1-µ(x,q), for all x,y X and q ∈ Q. KU – algebra X. If the inequality x * y ≤ z , then
µ(y,q) ≤max{µ(x,q),µ(y,q)} for all x,y,zX and q∈Q.
3.ANTI Q-FUZZY KU-IDEALS
Definition 3.1 Proof:
Assume that the inequality x * y ≤ z for all
A Q-fuzzy set  of a KU-algebra X is called
an anti Q-fuzzy KU-ideal of X, if x,y,z  X and q ∈ Q. Then z * ( x * y) = 0 .
Now, µ ( y , q ) = µ ( 0 * y , q )
(i) (0,q) ≤ (x,q)
≤ max { µ ( 0 * (x * y) , q ), µ (x,q) }
(ii) (x *z,q) ≤ max {  ((x * (y * z)) , q ), ( y , q )},
= max { µ (x * y,q ), µ (x,q)}
for all x,y,z X and q ∈ Q.
≤ max{max{ (x * (z * y) , q ), ( z , q )} , µ(x,q)}
Example 3.1
Let X = { 0 , 1 , 2, 3, 4 } be a set with a binary = max{max{ (z * (x * y) , q ), ( z , q )} , µ(x,q)}
operation * defined by the following table = max { max {  ( 0, q ), ( z , q )} , µ(x,q)}
* 0 1 2 3 4 = max { ( z , q )} , µ(x,q)}
0 0 1 2 3 4  µ (y,q) ≤ max {µ(x,q),µ(y,q)}.
1 0 0 2 2 4
2 0 0 0 1 4 Theorem 3.4
If µ is an anti Q-fuzzy KU-ideal of KU–
3 0 0 0 0 4
4 0 0 0 1 0 algebra X, then for all x, y  X and q ∈ Q,
µ ( x * (x * y), q ) ≤ µ ( y , q )
Then Clearly ( X , * , 0 ) is a KU-algebra.
Let t0, t1, t2 ∈ [ 0, 1] be such that t0 < t1 < t2 .Define a Q- Proof:
fuzzy set µ : X x Q [0, 1] by µ(0,q) = t0, µ(1,q) = t1 = Let x,y ∈ X and q ∈ Q.
µ(2,q), µ(3,q) = t2 = µ(4,q) ,by routine calculations µ is µ (x*(x*y),q) ≤ max{ µ(x* (y*( x * y)), q ), µ( y, q) }
an anti Q-fuzzy KU- ideal of X where q∈Q. = max { µ ( x * (x * ( y * y)), q ), µ( y, q) }
= max { µ ( x * (x * 0), q ), µ( y, q) }
Theorem 3.1 = max { µ ( x * 0, q ), µ( y, q) }
Every Anti Q-fuzzy KU- ideal  of a KU- = max { µ ( 0, q ), µ( y, q) }
algebra X is order preserving. = µ (y,q)
 µ ( x * (x * y), q ) ≤ µ ( y , q )
Proof
Let  be an anti Q-fuzzy KU- ideal of a KU- 4.LOWER LEVEL CUTS IN ANTI Q-
algebra X and let x , y  X and q ∈ Q. FUZZY KU-IDEALS OF KU-ALGEBRA
Then (x,q) =  (0 * x , q)
≤ max { (0 * (y * x) , q) ,  (y,q)} Definition 4.1 [12]
= max {  (0 * 0 , q) ,  (y,q) } Let  be a Q-fuzzy set of X. For a fixed t 
=  (y,q) [0, 1], the set t ={x  X (x,q) ≤ t for all q ∈ Q} is
(x,q) ≤  (y,q). called the lower level subset of .
Clearly t  t =X for t[0,1] if t1 < t2 , then t1  t2.
Theorem 3.2
 is a Q-fuzzy KU-ideal of a KU-algebra X if Theorem 4.1
and only if µc is an anti Q-fuzzy KU-ideal of X. If µ is an anti Q-fuzzy KU-ideal of KU-
Proof : algebra X,then µt is an KU-ideal of X for every
Let µ be a Q-fuzzy KU- ideal of X and let x , y , z  t [0,1] .
X and q ∈ Q. Proof
(i) c
µ (0,q) = 1 - µ(0,q) Let µ be an anti Q-fuzzy KU-ideal of KU-
≤ 1 - µ ( x,q) algebra X.
= µc ( x,q) (i) Let x ∈ µt  µ( x, q ) ≤ t.
That is µc(0,q) ≤ µc ( x,q) . µ(0,q) = µ ( y * 0, q )
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3. P.M.Sitharselvam , T.Priya, T.Ramachandran/ International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1286-1289
≤ max { µ ( y * (x * 0),q ) , µ(x,q) }  µ (x,q) ≤ t .
= max { µ ( y * 0),q ) , µ(x,q) }  x ∈ µt.
= max { µ (0),q ) , µ(x,q) } Therefore A(a,b) ⊆ µt .
= µ(x,q) ≤ t. Conversely suppose that A(a,b) ⊆ µt .
 0 ∈ µt . Obviously 0 ∈ A(a,b) ⊆ µt for all a,b ∈ X.
( ii) Let x * (y * z) ∈ µt and y ∈ µt , for all x,y,z  X Let x,y,z ∈ X be such that x * (y * z) ∈ µt and y ∈ µt .
and q ∈ Q. Since (x * ( y * z )) * (y * ( x * z )) = (x * ( y * z )) *
 µ (x * (y * z) ,q ) ≤ t and µ ( y,q ) ≤ t. ( x * ( y * z )) = 0.
µ ( ( x * z),q ) ≤ max {  ((x * (y * z)) , q ), ( y , q )} We have x * z ∈ A ( x * ( y * z ) , y ) ⊆ µt .
≤ max {t,t} = t.  µt is an KU- ideal of X.
 x * z ∈ µt. Hence,by theorem 4.2, µ is an anti Q-fuzzy KU-ideal
Hence µt is an KU- ideal of X for every t [0,1]. of X.
Theorem 4.2 Theorem 4.4
Let µ be a Q-fuzzy set of KU- algebra X.If Let µ be a Q-fuzzy set in KU-algebra X.If µ
for each t ∈ [0,1], the lower level cut µt is an KU- is an anti Q-fuzzy KU-ideal of X then
ideal of X, then µ is an anti Q- fuzzy KU-ideal of X. ∀ 𝑡 ∈ 0,1 𝜇𝑡 ≠ ∅  𝜇𝑡 = 𝐴 𝑎, 𝑏 .
𝑎,𝑏 ∈ 𝜇 𝑡
Proof
Let µt be an KU-ideal of X. Proof :
If µ(0,q) > µ(x,q) for some x ∈ X and q ∈ Q.Then Let t ∈ [0,1] be such that 𝜇 𝑡 ≠ ∅. Since 0 ∈ µt , we
1
µ(0,q) > t0 > µ(x,q) by taking t0 = { µ(0,q) + µ(x,q) }.
2 have 𝜇 𝑡 ⊆ 𝑎 ∈ 𝜇 𝑡 𝐴 𝑎, 0 ⊆ 𝑎,𝑏 ∈ 𝜇 𝑡 𝐴 𝑎, 𝑏 .
Hence 0 ∉ µt0 and x ∈ µt0 , which is a contradiction. Now, let x ∈ 𝑎,𝑏 ∈ 𝜇 𝑡 𝐴 𝑎, 𝑏 .
Therefore, µ(0,q) ≤ µ(x,q). Then there exists (u,v) ∈ µt such that x ∈ A (u,v) ⊆ µt
Let x,y,z ∈ X and q ∈ Q be by theorem 4.3. Thus 𝑎,𝑏 ∈ 𝜇 𝑡 𝐴 𝑎, 𝑏 ⊆ 𝜇 𝑡 .
µ( ( x * z ) , q ) > max{ µ (x * ( y * z) , q), µ(y,q)}  𝜇 𝑡 = 𝑎,𝑏 ∈ 𝜇 𝑡 𝐴 𝑎, 𝑏 .
1
Taking t1 = { µ((x * z ) , q) + max {µ (x * ( y * z)
2
, q), µ (y,q)}} and µ((x * z ) , q) > t1 > max{µ (x * ( y 5.Anti Q- Fuzzy Sub algebra of KU algebra
* z) , q), µ(y,q)} . Definition 5.1 [11]
It follows that ( x * ( y * z)),y ∈ µt1 and x * z ∉ µt1. A nonempty subset S of a KU-algebra X is
This is a contradiction. said to be KU-subalgebra of X , if x , y  S implies
Hence µ((x * z), q) ≤ max{µ (x * ( y * z) , q), µ(y,q)} x * y  S.
Therefore µ is an anti Q-fuzzy KU-ideal of X.
Definition 5.2
Definition 4.2 [13] A Q-fuzzy set  in a KU-algebra X is called
Let X be an KU- algebra and a,b ∈ X.We can a Q-fuzzy Subalgebra of X if (x * y , q ) ≥ min{(x
define a set A(a,b) by A(a,b) ={ x ∈ X / a* ( b* x) = 0 }. , q), ( y , q)},for all x,y X and q ∈ Q.
It is easy to see that 0,a,b ∈ A(a,b) for all a,b ∈ X.
Definition 5.3 [15]
Theorem 4.3 A Q-fuzzy set  in a KU-algebra X is called
Let µ be a Q-fuzzy set in KU-algebra X.Then
an Anti Q-fuzzy sub algebra of X if (x * y , q) ≤
µ is an anti Q- fuzzy KU- ideal of X iff µ satisfies the
max{(x , q), (y , q)}, for all x,y X and q ∈ Q.
following condition.
Example 5.1
∀ 𝑎, 𝑏 ∈ 𝑋 , ∀ 𝑡 ∈ 0,1
From example 3.1 , Let t0, t1, t2 ∈ [ 0, 1] be
𝑎, 𝑏 ∈ 𝜇 𝑡  A a, b ⊆ 𝜇 𝑡
such that t0 < t1 < t2 .Define a Q-fuzzy set µ : X x Q
Proof:
Assume that µ is an anti Q-fuzzy KU- ideal [0, 1] by µ(0,q) = t0, µ(1,q) = t1 = µ(2,q),
of X. µ(3,q) = t2 = µ(4,q) ,By routine calculations µ is an
Let a,b ∈ µt. Then µ (a,q ) ≤ t and µ (b,q ) ≤ t. anti Q-fuzzy subalgebra of X where q∈Q.
Let x ∈ A (a,b). Then a * ( b * x ) = 0. Theorem 5.1
Now, If  is an anti Q-fuzzy sub algebra of a KU-
µ (x,q) = µ (0 * x , q) algebra X, then (0,q) ≤  (z,q) , for any z  X and q
≤ max { µ (0 * (b * x) , q) , µ ( b,q )} ∈ Q.
= max { µ ((b * x) , q) , µ ( b,q )} Proof
≤ max { max {µ (b * (a * x) , q) , µ (a,q)} , µ(b,q) } Since z * z = 0 for any x  X and q ∈ Q, then
= max { max {µ (a * (b * x) , q) , µ (a,q)} , µ(b,q) } ( 0, q ) =  ( z * z, q )
= max { max {µ (0,q) , µ (a,q)} , µ(b,q) } ≤ max { ( z, q ) ,  ( z, q )} =  (z,q)
= max { µ (a,q)} , µ(b,q) }≤ max { t , t} = t  ( 0, q ) ≤  ( z, q ).
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4. P.M.Sitharselvam , T.Priya, T.Ramachandran/ International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1286-1289
Theorem 5.2 Conclusion
A Q-fuzzy set µ of a KU– algebra X is an In this article we have discussed anti Q-fuzzy
anti Q-fuzzy subalgebra if and only if for every KU- ideal of KU-algebras and its lower level cuts in
t [0,1] , µt is either empty or a KU-sub algebra of X. detail.These concepts can further be generalized.
Proof: REFERENCES :
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Therefore µs = µt.
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