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Some Results on Intuitionistic Fuzzy Ideals in BCK-Algebras
B. Satyanarayana1 and R. Durga Prasad2
1,2Department of Applied Mathematics, Acharya Nagarjuna University Campus, Nuzvid-521201, Krishna (District), Andhra Pradesh, India
1E-mail: [email protected]
2E-mail: [email protected]
(Received: 20-11-10 /Accepted: 7-4-11)
Abstract
In this paper, we give some results on the intuitionistic fuzzy implicative ideals, intuitionistic fuzzy positive implicative ideals, intuitionistic fuzzy commutative ideals.
Keywords: BCK-algebra, Fuzzy (implicative, positive implicative and commutative) ideal.
1 Introduction
After the introduction of the concept of fuzzy sets by Zadeh [12] several researches were conducted on the generalizations of the notion of fuzzy sets. The idea of
“intuitionistic fuzzy set” was first published by Atanassov [1, 2] as a generalization of the notion of fuzzy set. The first author (together with Hong, Kim, Meng, Roh and Song) [3, 5, 6, 7] considered the fuzzification of ideals and sub- algebras in BCK- algebras (cf. [3, 4, 5, 6). In this paper we give some results on the intuitionistic fuzzy implicative ideals, intuitionistic fuzzy positive implicative ideals, intuitionistic fuzzy commutative ideals.
2 Preliminaries
First we present the fundamental definitions. By a BCK-algebra (see [7, 8, 9]) we mean a nonempty set X with a binary operation * and a constant 0 satisfying the axioms:
(BCK-1) ((x∗y)∗(x∗z))≤(z∗y), (BCK-2) (x∗(x∗y))≤y,
(BCK-3) x≤x,
(BCK-4) x≤y and y≤ximply that x=y, (BCK-5) 0≤x
for all x,y,z∈X.
A partial ordering “≤” on X can be defined by x≤yif and only ifx∗y=0. In any BCK-algebra X the following holds:
(P1) x∗0=x (P2) x∗y≤x
(P3) (x∗y)∗z=(x∗z)∗y (P4) (x∗z)∗(y∗z)≤x∗y (P5) x∗(x∗(x∗y))=x∗y
(P6) x≤y⇒x∗z≤y∗z andz∗y≤z∗x, for all x,y,z∈X.
A BCK-algebra X is said to be implicative if x =x∗(y∗x), for all x,y X∈ .
A BCK-algebra X is said to be positive implicative if (x∗y)∗z=(x∗z)∗(y∗z) for all x,y,z∈X.
A BCK-algebra X is said to be commutative if x∗(x∗y)=y∗(y∗x) for all X.
z y,
x, ∈
A non-empty subset I of a BCK-algebra X is called an ideal of X, (I1)0∈I
(I2) x∗y and y∈I imply that x∈I for allx,y∈X.
A non-empty subset I of a BCK-algebra X is said to be sub-algebra of X if X
y
x∗ ∈ whenever x,y∈X
A non-empty subset I of a BCK-algebra X is called an implicative ideal of X if it satisfies (I1) and (I ) 3 (x∗(y∗x))∗z∈Iand z∈I imply x∈I for allx,y,z∈X. A non-empty subset I of a BCK-algebra X is called a commutative ideal of X if it satisfies (I1) and (I4) (x∗y)∗z∈Iandz∈Iimply x∗(y∗(y∗x))∈I for x,y,z∈X. A non-empty subset I of a BCK-algebra X is said to be positive implicative ideal of X if it satisfies (I1) and (I5) (x∗y)∗z∈Iandy∗z∈I imply x∗z∈I for allx,y,z∈X. Let µ and λ be the fuzzy sets in a set X. For s, t ε [0, 1], the set U (µ, s) = {x∈X/ µ(x) ≥ s} is called a upper level of µ and the set L (λ, t) = {
x ∈ X
/ λ(x) ≤ t} is called a lower level of λ.An intuitionistic fuzzy set A in a non-empty set X is an object having the form X}
(x)/x λ (x), µ {x,
A= A A ∈ , where the function µA :X→[0,1] and λA:X→[0,1]
denoted the degree of membership (namely µ(x) ) and the degree of non membership (namely λ(x) ) of each element
x ∈ X
to the set A respectively, and1 (x) λ (x) µ
0≤ A + A ≤ for all
x ∈ X
. For the sake of simplicity, we shall use the symbol A =(X,µA,λA) or A=(µA,λA).Definition 2.1. Let A=(µA,λA) and B=(µB,λB)be intuitionistic fuzzy sets in X.
Then
(i) A {(x,µ (x),µA(x))/x X}
_
A ∈
=
(ii) ◊A {(x,λA(x),λA(x))/x X}.
_ ∈
=
In what follows, let X denote a BCK-algebra unless otherwise specified.
Definition 2.2. An IFS A=(X,µA,λA) in X is an intuitionistic fuzzy sub-algebra of X if it satisfies
(IFS 1) µA(x∗y)≥min{µA(x),µA(y)}
(IFS 2)λA(x∗y)≤max{λA(x),λA(y)}for all x,y∈X.
Example 2.3. Consider a BCK-algebra X = {0, a, b, c} with the following Cayley table:
0 0 0 0
0 0 0 0 0
0
|
*
c c c
b a
b
a a
c b a
c b a
Let A=(X,µA,λA) be an IFS in X defined by µA(0)=µA(a)=µA(c)=0.7>0.3=µA(b) and
λA(0)=λA(a)=λA(c)=0.2<0.5=λA(b).
Then A=(X,µA,λA)is an IF subalgebra of X.
Proposition 2.4. LetA=(X,µA,λA) be an intuitionistic fuzzy sub-algebra of X, then µ (x)
µA(0)≥ A and
λ
A(0) ≤ λ
A(x)
for allx ∈ X.
Definition 2.5. An IF A=(X,µA,λA) in X is an intuitionistic fuzzy ideal (IF-ideal) of X if it satisfies
(IF1) µA(0)≥µA(x)and
λ
A(0) ≤ λ
A(x)
(IF2)µA(x)≥min{µA(x∗y),µA(x)}
(IF3)λA(x)≤min{λA(x∗y),λA(y)}, for all x,y∈X.
Theorem 2.6. [4]Let A=(X,µA,λA) be an intuitionistic fuzzy ideal of X. If x≤ y in X, then
(y), µ (x)
µA ≥ A λA(x)≤λA(y),
that is µAis order-reversing and λAis order-preserving.
Theorem 2.7. [4]Every intuitionistic fuzzy ideal of X is an intuitionistic fuzzy sub- algebra of X.
Theorem 2.8. [4]A=(X,µA,λA) is an intuitionistic fuzzy ideal of X if and only if for µ (z)}
min{µ (y), µ (x)
z y x X, z y,
x, ∈ ∗ ≤ ⇒ A ≥ A A and λA(x)≤max{λA(y),λA(z)}. Proposition 2.9. [4]A=(X,µA,λA) is an intuitionistic fuzzy ideal of X if and only if the non-empty upper s-level cut U(µA;s) and the non-empty lower t-level cut
t)
L(λA; are ideals of X, for anys,t∈[0,1].
Corollary 2.10. A =(X,µA,λA) is an intuitionistic fuzzy subalgebra of X if and only if the non-empty upper s-level cut U(µA;s) and the non-empty lower t-level cut
t)
L(λA; are sub-algebras of X, for anys,t∈[0,1].
Proposition 2.11. [11]In a BCK-algebra X, the following holds, for allx,y,z∈X, (i) ((x∗z)∗z)∗(y∗z)≤(x∗y)∗z.
(ii) (x∗z)∗(x∗(x∗z))=(x∗z)∗z
(iii) (x∗(y∗(y∗x)))∗(y∗(x∗(y∗(y∗x))))≤x∗y.
3 Main Results
In this section we present the results on the intuitionistic fuzzy implicative ideals, intuitionistic fuzzy positive implicative ideals and intuitionistic fuzzy commutative ideals.
Definition 3.1. [11]An IFS A=(X,µA,λA) in a BCK-algebra X is an intuitionistic fuzzy implicative ideal (IFI-ideal) of X if it satisfies
(IFI 1) µA(0)≥µA(x)and λA(0)≤λA(x) (IFI 2)µA(x)≥min{µA((x∗(y∗x))∗z),µA(z)}
(IFI 3)λA(x)≤max{λA((x∗(y∗x))∗z),λA(z)}, for all x,y,z∈X.
Definition 3.2. [11]An IFS A=(X,µA,λA) in X is an intuitionistic fuzzy commutative ideal (IFCI-ideal) of X if it satisfies
(IFCI 1) µA(0)≥µA(x)and λA(0)≤λA(x)
(IFCI 2)µA(x∗(y∗(y∗x))≥min{µA((x∗y)∗z),µA(z)}
(IFCI 3)λA(x∗(y∗(y∗x))≤max{λA((x∗y)∗z),λA(z)}for all x,y,z∈X.
Definition 3.3. [11]An IFS A=(X,µA,λA) in a BCK-algebra X is an intuitionistic fuzzy positive implicative ideal (IFPI-ideal) of X if it satisfies
(IFPI 1) µA(0)≥µA(x)and λA(0)≤λA(x)
(IFPI 2)µA(x∗z)≥min{µA((x∗y)∗z),µA(y∗z)}
(IFPI 3)λA(x∗z)≤max{λA((x∗y)∗z),λA(y∗z)}for all x,y,z∈X.
Theorem 3.4. An intuitionistic fuzzy ideal A =(X,µA,λA) of X is an intuitionistic fuzzy implicative if and only if A is both intuitionistic commutative and intuitionistic fuzzy positive implicative.
Proof: Assume that A =(X,µA,λA) is an intuitionistic fuzzy implicative ideal of X.
By (2.11(i) and 2.8), we have
min{µA((x∗y)∗z),µA(y∗z)}≤µA((x∗z)∗z)
=µA((x∗z)∗(x∗(x∗z))) ( by 2.11(ii)) =µA(x∗z) ( by [11, 3.7(iii)])
and max{λA((x∗y)∗z),λA(y∗z)}≥ λA((x∗z)∗z) =λA((x∗z)∗(x∗(x∗z))) =λA(x∗z),for allx,y,z∈X.
Then A=(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of X. And by theorem 2.6, 2.11(iii) and 3.7(iii),
x))) (y (y µ (x
x))))) (y
(y (x (y x))) (y (y µ ((x
y)
µA(x∗ ≤ A ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = A ∗ ∗ ∗ and
x))).
(y (y λ (x
x))))) (y
(y (x (y x))) (y (y λ ((x
y)
λA(x∗ ≥ A ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = A ∗ ∗ ∗
It follows from [11, 4.6] that A=(X,µA,λA) is an intuitionistic fuzzy commutative.
Conversely, suppose that A=(X,µA,λA) is both intuitionistic fuzzy positive implicative and intuitionistic fuzzy commutative.
Since,(y∗(y∗x))∗(y∗x)≤x∗(y∗x), it follows from theorem 2.6.
x)) (y µ (x
x)) (y x)) (y
µA(y∗ ∗ ∗ ∗ ≥ A ∗ ∗ and λA(y∗(y∗x))∗(y∗x))≤λA(x∗(y∗x)).
Using [11, 5.8], we have
x)) (y µ (y
x)) (y x)) (y
µA(y∗ ∗ ∗ ∗ = A ∗ ∗ and
x)).
(y λ (y x)) (y x)) (y
λA(y∗ ∗ ∗ ∗ = A ∗ ∗ Therefore
x)) (y µ (y
x)) (y
µA(x∗ ∗ ≤ A ∗ ∗ and λA(x∗(y∗x))≥λA(y∗(y∗x))… (1) On the other hand since x∗y≤x∗(y∗x), we have, by theorem 2.6
) x) (y µ (x
y)
µA(x∗ ≥ A ∗ ∗ and λA(x∗y)≤λA(x∗(y∗x)).
Since A=(X,µA,λA) is an intuitionistic fuzzy commutative ideal of X, by [11, 4.7]
we have
) x)) (y (y µ (x
y)
µA(x∗ = A ∗ ∗ ∗ and λA(x∗y)=λA(x∗(y∗(y∗x))).
Hence
) x)) (y (y µ (x
x)) (y
µA(x∗ ∗ ≤ A ∗ ∗ ∗ andλA(x∗(y∗x))≥λA(x∗(y∗(y∗x)))…(2) Combining (1 ) and (2), we obtain
µ (x) x))}
(y µ (y
x))), (y (y min{µ (x
x) (y
µA(x∗ ∗ ≤ A ∗ ∗ ∗ A ∗ ∗ ≤ A and
λA(x∗(y∗x)≥max{λA(x∗(y∗(y∗x))),λA(y∗(y∗x))}≥λA(x).
So A=(X,µA,λA) is an intuitionistic fuzzy implicative ideal of X . The proof is complete.
Theorem 3.5. If A =(X,µA,λA) is an intuitionistic fuzzy ideal of X with the following conditions holds
(i) µA(x∗y)≥min{µA(((x∗y)∗y)∗z),µA(z)}
(ii)λA(x∗y)≤max{λA(((x∗y)∗y)∗z),λA(z)}, for allx,y,z∈X. Then A is intuitionistic fuzzy positive implicative ideal of X.
Proof: Suppose A=(X,µA,λA) is intuitionistic fuzzy ideal of X.
with condition (i) and (ii). Using (P3) and (P4), we have
z, y) (x y z) (x z)) (y z) z)
((x∗ ∗ ∗ ∗ ≤ ∗ ∗ = ∗ ∗ for all
x,y,z ∈ X
,therefore by theorem 2.6
µA(((x∗z)∗z)∗(y∗z)))≥µA((x∗y)∗z) And
λA(((x∗z)∗z)∗(y∗z)))≤λA((x∗y)∗z).
Now
} z) µ (y z)), (y z) z) min{µ (((x
z)
µA(x∗ ≥ A ∗ ∗ ∗ ∗ A ∗ ≥min{µA((x∗y)∗z),µA(y∗z)}, for all x,y,z∈X
and
} z) λ (y z)), (y z) z) max{λ (((x
z)
λA(x∗ ≤ A ∗ ∗ ∗ ∗ A ∗
≤max{λA((x∗y)∗z),λA(y∗z)},for all x,y,z∈X.
Hence A=(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of X.
Lemma 3.6. Let A=(X,µA,λA) be a fuzzy ideal of X, then A is an intuitionistic fuzzy positive implicative ideal of X if and only if
) z) y) µ ((x
z)) (y z)
µA((x∗ ∗ ∗ ≥ A ∗ ∗ and λA((x∗z)∗(y∗z))≤λA((x∗y)∗z)), for all x,y,z∈X.
Proof: Suppose that A=(X,µA,λA) is a fuzzy ideal of X and )
z) y) µ ((x
z)) (y z)
µA((x∗ ∗ ∗ ≥ A ∗ ∗ and λA((x∗z)∗(y∗z))≤λA((x∗y)∗z)), for all x,y,z∈X Therefore
} z) µ (y z)), (y z) min{µ ((x
z)
µA(x∗ ≥ A ∗ ∗ ∗ A ∗ ≥min{µA((x∗y)∗z),µA(y∗z)}
} z) λ (y z)), (y z) max{λ ((x
z)
λA(x∗ ≤ A ∗ ∗ ∗ A ∗ ≤max{λA((x∗y)∗z),λA(y∗z)},
for allx,y,z∈X. Thus A is an intuitionistic fuzzy positive implicative ideal of X.
Conversely, assume that A=(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of X implies that A=(X,µA,λA) is an IF-ideal of X.
Leta =x∗(y∗z) and b=x∗y,
Since((x∗(y∗z))∗(x∗y))≤y∗(y∗z),
we have that
µA((a∗b)∗z)=µA(((x∗(y∗z))∗(x∗y)∗z)≥µA((y∗(y∗z))∗z)=µA(0) and so,
) z z) (y µ ((x
z)) (y z)
µA((x∗ ∗ ∗ = A ∗ ∗ ∗ =µA(a∗z) z)}
µ (b z), b)
min{µA((a∗ ∗ A ∗
≥ ≥min{µA(0),µA(b∗z)}
).
z y) µ ((x
z)
µA(b∗ = A ∗ ∗
= Therefore
z), y) µ ((x
z)) (y z)
µA((x∗ ∗ ∗ ≥ A ∗ ∗ for allx,y,z∈X. And
) 0 λ ( z) z)) (y λ ((y z) y) (x z)) (y λ (((x z) b)
λA((a∗ ∗ = A ∗ ∗ ∗ ∗ ∗ ≤ A ∗ ∗ ∗ = A And so,
) z z) (y λ ((x
z)) (y z)
λA((x∗ ∗ ∗ = A ∗ ∗ ∗ =λA(a∗z) z)}
λ (b z), b)
max{λA((a∗ ∗ A ∗
≤ ≤max{λA(0),λA(b∗z)}
).
z y) λ ((x
z)
λA(b∗ = A ∗ ∗
= Therefore
z), y) λ ((x
z)) (y z)
λA((x∗ ∗ ∗ ≤ A ∗ ∗ for all x,y,z∈X.
Thus
), z) y) µ ((x
z)) (y z)
µA((x∗ ∗ ∗ ≥ A ∗ ∗ λA((x∗z)∗(y∗z))≤λA((x∗y)∗z)), for all x,y,z∈X.
Theorem 3.7. If A=(X,µA,λA) is intuitionistic fuzzy positive implicative ideal of X then (PI 1) for any
x,y,a,b X, ((x y) y) a b µ (x y) min{µ (a),µ (b)}
A A A
∈ ∗ ∗ ∗ ≤ ⇒ ∗ ≥
and
λ (b)}
max{λ (a), y)
λA(x∗ ≤ A A . (PI 2) For any
x,y,z,a,b X, ((x y) z) a b µ ((x ) (y z)) min{µ (a),µ (b)}
A z A A
∈ ∗ ∗ ∗ ≤ ⇒ ∗ ∗ ∗ ≥
and
λA((x∗ ∗ ∗z) (y z)) max{λ≤ A(a),λA(b)}.
Proof: Suppose, A=(X,µA,λA) is intuitionistic fuzzy positive implicative ideal of X.
(PI1). Let x,y,z∈X be such that((x∗y)∗y)∗a≤b. Using 2.6, we have
µ (b)}
min{µ (a), y)
y)
µA((x∗ ∗ ≥ A A and λA((x∗y)∗y)≤max{λA(a),λA(b)}.
It follows that
} y) µ (y
y), y) min{µ ((x
y)
µA(x∗ ≥ A ∗ ∗ A ∗ =min{µA((x∗y)∗y),µA(0)}
y) y) ((x µA ∗ ∗
= ≥min{µA(a),µA(b)}.
And
} y) λ (y
y), y) max{λ ((x
y)
λA(x∗ ≤ A ∗ ∗ A ∗ λ (0)}
y), y)
max{λA((x∗ ∗ A
= =λA((x∗y)∗y) ≤max{λA(a),λA(b)}.
(ii) Now let x,y,z∈X be such that((x∗y)∗z)∗a ≤b.
Since A =(X,µA,λA) intuitionistic fuzzy positive implicative ideal of X, it follows from known lemma 3.6,
µA((x∗ ∗ ∗z) (y z)) µ≥ A((x y) z) min{µ∗ ∗ ≥ A(a),µA(b)}
and
λA((x∗ ∗ ∗z) (y z)) λ≤ A((x y) z) max{λ∗ ∗ ≤ A(a),λA(b)}
This completes the proof.
Theorem.3.8. Let A=(X,µA,λA) be IFS in X satisfying the condition }
µ (b) min{µ (a),
y) µ (x
b a y) y)
((x∗ ∗ ∗ ≤ ⇒ A ∗ ≥ A A
and
}, λ (b) max{λ (a),
y)
λA(x∗ ≤ A A
for any x,y,a,b∈X,Then A=(X,µA,λA) intuitionistic fuzzy positive implicative ideal of X.
Proof: First we prove that A=(X,µA,λA) is an IF-ideal of X.
Let x,y,z∈X be such thatx∗y≤z.
Then(((x∗0)∗0)∗y)∗z=(x∗y)∗z)=0, that is(((x∗0)∗0)∗y)≤z
Since, for x,y,a,b X∈ ,
((x y) y) a b ∗ ∗ ∗ ≤ ⇒ µA(x y) min{µ∗ ≥ A(a),µA(b)}
and
λ (b)}
max{λ (a), y)
λA(x∗ ≤ A A
Puty=0,a=y,b=z,
we get
µ (z)}
min{µ (y), µ(x 0)
µA(x)= ∗ ≥ A A and
(z)}.
λ max{λ (y), 0)
λ (x
λA(x)= A ∗ ≤ A A
It follows that A =(X,µA,λA ) is IF- ideal of X.
Note that
0 0 y)) y) ((x y) y)
(((x∗ ∗ ∗ ∗ ∗ ∗ =
implies
X.
y x, 0, y)) y) ((x y) y)
(((x∗ ∗ ∗ ∗ ∗ ≤ ∀ ∈
From hypothesis we have
) y y) µ ((x
µ (0)}
y), y) min{µ ((x
y)
µA(x∗ ≥ A ∗ ∗ A = A ∗ ∗ and
) y y) λ ((x
λ (0)}
y), y) max{λ ((x
y)
λA(x∗ ≤ A ∗ ∗ A = A ∗ ∗ .
And so A=(X,µA,λA) is intuitionistic fuzzy positive implicative ideal of X.
Theorem 3.9. Let A=(X,µA,λA) be an IFS in X satisfying ((x∗y)∗z)∗a≤bimply }
(b) µ (a), min{µ z))
(y y) ((x
µA ∗ ∗ ∗ ≥ A A and λA((x∗y)∗(y∗z))≤max{λA(a),λA(b)}
for any x,y,z,a,b∈X.
ThenA=(X,µA,λA)is an intuitionistic fuzzy positive implicative ideal of X.
Proof: Let x,y,a,b∈X be such that((x∗y)∗y)∗a≤b, that is
0 b a) y) y)
(((x∗ ∗ ∗ ∗ = therefore
)) y (y y) µ ((x
0) y) µ ((x
y)
µA(x∗ = A ∗ ∗ = A ∗ ∗ ∗ ≥min{µA(a),µA(b)}
And
λA(x∗y)=λA((x∗y)∗0)=λA((x∗y)∗(y∗y)) ≥min{λA(a),λA(b)}.
It follows from 3.8, A =(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of X.
Theorem 3.10. Let A=(X,µA,λA) be an intuitionistic fuzzy positive implicative ideal of BCK-algebra X, then so is A (X,µ= A,µA).
Proof: We have µA(0)≥µA(x)⇒1−µA(0)≥1−µA(x)⇒µA(0)≤µA(x),∀x∈X. Consider for anyx,y,z∈X,
} z) µ (y
z), y) min{µ ((x
z)
µA(x∗ ≥ A ∗ ∗ A ∗
⇒1−µA(x∗z)≥min{1−µA((x∗y)∗z),1−µA(y∗z)}
⇒µA(x∗z)≤1−min{1−µA((x∗y)∗z),1−µ(y∗z)}
⇒µA(x∗z)≤max{µA((x∗y)∗z),µA(y∗z)}
Hence A (X,µ= A,µA)is an intuitionistic fuzzy positive implicative ideal of BCK- algebra X.
Theorem 3.11. Let A=(X,µA,λA) be an intuitionistic fuzzy positive implicative ideal of BCK-algebra X then so is ◊A=(X,λA,λA) .
Proof: We have λA(0)≤λA(x)⇒1−λA(0)≤1−λA(x)⇒λA(0)≥λA(x),∀x∈X.
Consider for anyx,y,z∈X
} z) λ (y z), y) max{λ ((x
z)
λA(x∗ ≤ A ∗ ∗ A ∗
⇒1−λA(x∗z)≤max{1−λA((x∗y)∗z),1−λA(y∗z)}
⇒λA(x∗z)≤1−max{1−λA((x∗y)∗z),1−λA(y∗z)}
⇒λA(x∗z)≥min{λA((x∗y)∗z),λA(y∗z)}.
Hence ◊A =(X,λA,λA)is an intuitionistic fuzzy positive implicative ideal of BCK- algebra X.
Theorem 3.12. A=(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of BCK-algebra X if and only if A (X,µ= A,µA)and ◊A=(X,λA,λA)are intuitionistic fuzzy positive implicative ideal of BCK-algebra.
Theorem 3.13. A=(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of BCK-algebra X if and only if the non-empty upper s-level cut U(µA;s) and the non- empty lower t-level cut L(λA;t)are PI-ideals of X, for anys,t∈[0,1].
Proof: Suppose A =(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of X and U(µA;s)≠φ for any s [0,1]∈ . It is clear that for any x∈X,
µ (x)
µA(0)≥ A ⇒µA(0)≥µA(x)≥s ⇒µA(0)≥simplies 0 U(µ ;s)
A
.
∈
Furthermore if (x∗y)∗z∈U(µA;s),y∗z∈U(µA;s) implies
s z)) y)
µA((x∗ ∗ ≥ and µA(y∗z)≥s.
Therefore
s s}
min{s, z)}
µ (y z), y) min{µ ((x
z)
µA(x∗ ≥ A ∗ ∗ A ∗ ≥ =
implies x z U(µ∗ ∈ A;s).
This shows that U(µA;s)is positive implicative ideal of X.
Similarly, we can prove L(λA,t) is positive implicative ideal of X,∀s,t∈[0,1]
Conversely, assume that for any s,t [0,1]∈ , U(µA;s) and L(λA,t) are either empty or positive implicative ideals of X.
Put µA(x)=s, λA(x)=t for anyx∈X.
Since 0∈U(µA;s)⇒µA(0)≥s=µA(x) and 0 L(λ∈ A,t)⇒λA(0) t≤ =λA(x)
thus
µ (x)
µA(0)≥ A and λA(0)≤λA(x)for all
x ∈ X
.Now we only need to show that (IFPI 3),
then take s1 =min{µA((x∗y)∗z),µA(y∗z)}⇒(x∗y)∗z,y∗z∈U(µA;s1). Since U(µA;s1) is implicative ideal of X
we have
1 A
1
A;s ) µ (x z) s U(µ
z
y∗ ∈ ⇒ ∗ ≥ =min{µA((x∗y)∗z),µA(y∗z)}. Therefore
} z) µ (y
z), y) min{µ ((x
z)
µA(x∗ ≥ A ∗ ∗ A ∗ for allx,y,z∈X
Similarly we can prove λA(x∗z)≤min{λA((x∗y)∗z),λA(y∗z)}for all x,y,z∈X.
Hence A=(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of BCK- algebra X.
Theorem 3.14. A=(X,µA,λA) is an intuitionistic fuzzy implicative or commutative ideals of BCK-algebra X if and only if the non-empty upper s-level cut U(µA;s) and the non-empty lower t-level cut L(λA;t)are implicative or commutative ideals of X, for anys,t∈[0,1].
Corollary 3.15. A=(X,µA,λA) is an intuitionistic fizzy implicative ideal of BCK- algebra X if and only if the non-empty upper s-level cut U(µA;s) and the non-empty
lower t-level cut L(λA;t)are both commutative and positive ideals of X, for anys,t∈[0,1].
Corollary 3.16. A =(X,µA,λA) is an intuitionistic fuzzy commutative and intuitionistic fuzzy positive implicative ideals of BCK-algebra X if and only if the non- empty upper s-level cut U(µA;s) and the non-empty lower t-level cut L(λA;t)are implicative ideals of X, for anys,t∈[0,1].
Theorem 3.17. Let A=(X,µA,λA) be an IFS of a BCK-algebra. If A is an intuitionistic fuzzy positive implicative ideal of X then the set
µ (0)}
X/µ (x) {x
J= ∈ A = A and K={x∈X/λA(x)=λA(0)}are an PI-ideal of X.
Proof: Assume thatA=(X,µA,λA) intuitionistic fuzzy positive implicative ideal of X. Since, µA(0)=µA(0)⇒0∈J.
If (x∗y)∗z,y∗z∈J⇒µA((x∗y)∗z)=µA(0)and µA(y∗z)=µA(0).
Since
} z) µ (y
z), y) min{µ ((x
z)
µA(x∗ ≥ A ∗ ∗ A ∗ =min{µA(0),µA(0)}=µA(0), but,
µ (0).
z)
µA(x∗ ≤ A Therefore, µA(x∗z)=µA(0)⇒x∗z∈J.
Thus, J is an implicative ideal of X andλA(0)=λA(0)⇒0∈K
If (x∗y)∗z,y∗z∈K Then
λ (0) z) y)
λA((x∗ ∗ = A And
λ (0).
z) λA(y∗ = A Since,
} z) λ (y z), y) max{λ ((x
z)
λA(x∗ ≤ A ∗ ∗ A ∗ =max{λA(0),λA(0)}=λA(0) but,
λ (0) z)
λA(x∗ ≥ A .
Therefore, λA(x∗z)=λA(0)⇒x∗z∈K. Thus, K is an implicative ideal of X
Theorem 3.18. (Extension property for intuitionistic fuzzy positive implicative ideals) Let A=(X,µA,λA) and B=(X,µB,λB) are two fuzzy ideals of X such that
A(0) = B(0) and A B⊆ (that is µA(0)=µB(0),λA(0)=λB(0) and µA(x)≤µB(x),
X x λ (x),
λA(x)≥ B ∀ ∈ ). If A=(X,µA,λA) is an intuitionistic fuzzy positive implicative ideal of X,, then so is B.
Proof: Suppose that A=(X,µA,λA) is intuitionistic fuzzy positive implicative ideal of X
)) z (y z)) y) ((x z) µ (((x z)) y) ((x z)) (y z)
µB(((x∗ ∗ ∗ ∗ ∗ ∗ = B ∗ ∗ ∗ ∗ ∗ ∗ (by P2)
=µB(((x∗((x∗y)∗z))∗z)∗(y∗z)) (by P2)
≥µA(((x∗((x∗y)∗z))∗z)∗(y∗z)) (SinceµA⊆µB) ≥µA(((x∗((x∗y)∗z))∗y)∗z) (by lemma 3.6) =µA(((x∗y)∗((x∗y)∗z)∗z) (by P2)
=µA(((x∗y)∗z)∗((x∗y)∗z)) (by P2) =µA(0)=µB(0) (by BCK-3 ).
It follows from (F1) and (F2) that
)}
z y) µ ((x
z)), y) ((x z) (y z) min{µ (((x
z)) (y z)
µB((x∗ ∗ ∗ ≥ B ∗ ∗ ∗ ∗ ∗ ∗ B ∗ ∗
≥min{µB(0),µB((x∗y)∗z))}=µB((x∗y)∗z)for all x,y,z∈X.
Therefore, for any x,y,z∈X, µB((x∗z)∗(y∗z))≥µB((x∗y)∗z) and )) z (y z)) y) ((x z) λ (((x z)) y) ((x z)) (y z)
λB(((x∗ ∗ ∗ ∗ ∗ ∗ = B ∗ ∗ ∗ ∗ ∗ ∗ (by P2)
=λB(((x∗((x∗y)∗z))∗z)∗(y∗z)) (by P2) ≤λA(((x∗((x∗y)∗z))∗z)∗(y∗z)) (SinceλB ⊆λA) ≤λA(((x∗((x∗y)∗z))∗y)∗z) (by 3.6) =λA(((x∗y)∗((x∗y)∗z)∗z)
=λA(((x∗y)∗z)∗((x∗y)∗z))
=λA(0)=λB(0) (by BCK-3)
It follows from (F1) and (F2) that
λB((x∗z)∗(y∗z))≤max{λB(((x∗z)∗(y∗z)∗((x∗y)∗z)),λB((x∗y)∗z)}
≤max{λB(0),λB((x∗y)∗z))}=λB((x∗y)∗z) for all x,y,z∈X.
Therefore λB((x∗z)∗(y∗z))≤λB((x∗y)∗z),for all x,y,z∈X.
Hence B=(X,µB,λB) is an intuitionistic fuzzy positive implicative ideal of X.
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