Cmjv03i01p0083 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

On Intervel-Valued Intuitionistic Fuzzy Ideals of BF-Algebras B. SATYANARAYANA, D. RAMESH and R. DURGA PRASAD Department of Applied Mathematics Acharya Nagarjuna University Campus Nuzvid-521 201, Krishna (District) Andhara Pradesh, India. (Received on: 18th January, 2012) ABSTRACT The notion of interval-valued intuitionistic fuzzy sets was first introduced by Atanassov and Gargov in 1989 as a generalization of both interval-valued fuzzy sets and intuitionistic fuzzy sets. In 2010, Satyanarayana with others applied the concept of intervalvalued intuitionistic fuzzy sets to BF-subalgebras. In this paper we introduce the notion of interval-valued intuitionistic fuzzy ideals of BF-algebras and investigate some interesting properties. Mathematics Subject Classification: 03B52, 03B25, 06F35, 94D05 Keywords: BF-algebras, Interval-valued intuitionistic fuzzy sets, intuitionistic fuzzy ideals.

1. INTRODUCTION AND PRELIMINARIES The notion of interval-valued fuzzy sets was first introduced by Zadeh11 as an extension of fuzzy sets. An interval-valued fuzzy sets is a fuzzy set whose membership function is many-valued and form an interval in the membership scale. This idea gives the simplest method to capture the imprecision of the membership grade for a fuzzy set. On the other hand, Atanassov2 introduced the nation of intuitionistic fuzzy sets as an extension of fuzzy set which not only a membership degree is given, but also

a non-membership degree is involved. Atanassove and Gargov3 introduced the notion of interval-valued intuitionistic fuzzy sets which is a generalization of both intuitionistic fuzzy sets and interval-valued fuzzy sets. In8 Sathyanarayana with others applied the concept of interval-valued intuitionistic fuzzy sets to BF-subalgebras. In this paper we introduce the notion of interval-valued intuitionistic fuzzy ideals of BF-algebras and investigate some interesting properties. By a BF-algebra we mean an algebra satisfying the axioms:

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

(1). x ∗ x = 0 , (2). x ∗ 0 = x , (3). 0 ∗ (x ∗ y) = y ∗ x , for all x, y ∈ X Throughout this paper, X is a BF-algebra. Example 1.1 Let R be the set of real number and let A = (R, ∗, 0) be the algebra with the operation ∗ defined by

 x, if y = 0  x ∗ y =  y, if x = 0 0, otherwise  Definition 1.2 The subset I of X is said to be an ideal of X , if (i) 0 ∈ I and (ii) x ∗ y ∈ I and y ∈ I ⇒ x ∈ I . We now review some fuzzy logic concepts. A fuzzy set in X is a function µ : X → [0, 1] . For fuzzy sets µ and λ of X and s, t ∈ [0, 1] .

The sets U(µ; t ) = {x ∈ X : µ(x) ≥ t}is called upper t-level cut of µ and

L(λ ; s ) = {x ∈ X : λ(x) ≤ s} is called lower s-level cut of λ . The fuzzy set µ in X is called a fuzzy sub algebra of X , if µ(x ∗ y) ≥ min{µ(x), µ(y)} , for all x, y ∈ X .

An intuitionistic fuzzy set (shortly, IFS) in a non-empty set X is an object having the form A = {(x, µ A (x), λ A (x) ) : x ∈ X} , where the function µ A : X → [0, 1] and

λ A : X → [0, 1] denote the degree of membership (namely µ A (x) ) and the degree

of non membership (namely λ A (x) ) of each element x ∈ X to the set A respectively such that 0 ≤ µ A (x) + λ A (x) ≤ 1 for all x ∈ X . For the sake of simplicity we use the symbol form A = (X, µ A , λ A ) or

A = ( µA, λA ) .

By an interval number D on [0, 1] , we

mean

[

−

]

interval a − , a + ,

an +

where

0 ≤ a ≤ a ≤ 1 . The set of all closed subintervals of [0, 1] is denoted by D[0, 1] .

[

]

For interval numbers D1 = a 1− , b1+ , − 2

+ 2

]

D 2 = a , b . We define

• D1 ∩ D 2 = min(D1 , D 2 ) =min

([a , b ], [a , b ]) = [min{a , a }min{b − 1

+ 1

− 1

− 2 + 2

+ 2

− 1

, b +2

}]

• D1 ∪ D 2 = max(D1 , D 2 ) =max

([a , b ], [a , b ]) = [max{a , a }max{b − 1

+ 1

− 1

− 2 − 2

+ 2

+ 1

, b +2

}]

[

D1 + D 2 = a1− + a −2 − a1− .a−2 , b1+ + b+2 − b1+ .b+2 and put − − + + • D1 ≤ D 2 ⇔ a 1 ≤ a 2 and b1 ≤ b 2

• D1 = D 2 ⇔ a 1− = a −2 and b1+ = b +2 , • D1 < D 2 ⇔ D1 ≤ D 2 and D1 ≠ D 2

mD = m[a1− , b1+ ] = [ma1− , mb1+ ] , where • 0 ≤ m ≤1.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

]

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

Obviously (D[0,1], ≤, ∨, ∧ ) form a complete lattice with [0, 0] as its least element and [1, 1] as its greatest element. Let L be a given nonempty set. An interval-valued fuzzy set (briefly, i-v fuzzy set) on L is defined by B − + B = x, [µ B (x), µ B (x)] : x ∈ L , Where

{(

)

}

µ −B (x) and µ +B (x) are fuzzy sets of L such that µ −B (x)

≤ µ +B (x) for all x ∈ L .

[

]

~ (x) = µ − (x), µ + (x) , Let µ B B B

~ (x)) : x ∈ L} then B = {(x, µ B ~ where µ B : L → D[0, 1] . ~ ,~ A mapping A = (µ A λA ) : L → D[0, 1] × D[0, 1] is called

an interval-valued intuitionistic fuzzy set (i-v IF + set, in short) in L if 0 ≤ µ A (x) + λ +A (x) ≤ 1

intuitionistic fuzzy ideal (shortly i-v IF ideal) of BF-algebra X if satisfies the following inequalities ~ (0) ≥ µ ~ (x) (i-v IF1) µ

A A ~ ~ and λ (0) ≤ λ (x) A A ~ (i-v IF2) µ

~ (x ∗ y ), µ ~ (y)} (x) ≥ min{ µ A A

}

(i-v IF2) λ (x) ≤ max λA (x∗y), λA (y) , A for all x, y, z ∈ X .

Example

2.2

Consider a BF-algebra X = {0, a, b, c} with following table

∗ 0 a b c

0 0 a b c

a a 0 c b

b b c 0 a

c c b a 0

− and 0 ≤ µ A (x) + λ −A (x) ≤ 1 for all x ∈ L + (that is, A + = (X, µ A , λ +A ) and

A − = (X, µ −A , λ −A ) are intuitionistic fuzzy sets), where the mappings

~ (x) = [µ − (x), µ + (x)] : L → D[0, 1] µ A A A ~ − and λ A (x) = [λ A (x), λ +A (x)] : L → D[0, 1] denote the degree of membership (namely ~ (x)) and degree of non-membership µ A

(namely λ A (x)) of each element x ∈ L to A respectively. 2. INTERVAL-VALUED INTUITIONISTIC FUZZY IDEALS OF BF-ALGEBRAS Definition 2.1: An interval-valued IFS

~ ,~ A = (X, µ A λ A ) is called interval-valued

Let A be an interval valued fuzzy set in X ~ (0) = µ ~ (a) = [0.6, 0.7] and defined by µ A A

~ (b) = µ ~ (c) = [0.2, 0.3] , µ A A ~ ~ λ A (0) = λ A (a) = [0.1, 0.2] , ~ ~ λ A (b) = λ A (c) = [0.5, 0.7] . It is easy to verify that A is an interval valued intuitionistic fuzzy ideal of X . ~

~ , λ ) be an Theorem 2.3: Let A = (X, µ A A i-v IFS then A is an i-v IF ideal of X

⇔ A − = (X, µ −A , λ −A ) and A + = (X, µ +A , λ +A ) are IF ideal of X . ~

~ , λ ) is an i-v Proof: Assume A = (X, µ A A IF ideal of X .

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

~ Since µ

~ (x) and (0) ≥ µ A A ~ ~ λ (0) ≤ λ (x) , for all x ∈ X ⇔ A A − + + [µ (0), µ (0)] ≥ [µ − A (x), µ A (x)] and A A + [λ − (0), λ + (0)] ≤ [λ − A (x), λ A (x)], for all A A x ∈ X ⇔ µ − (0) ≥ µ − (x), A A − − λ (0) ≤ λ (x) and µ + (0) ≥ µ + (x), A A A A + + λ A (0) ≤ λ A (x)] , for all x ∈ X ~ (x) ≥ min{ µ ~ (x ∗ y ), µ ~ (y)} Since µ A A A

(x ∗ y ), µ +A (x ∗ y) , µ −A (y),µ +A (y)  = min µ −      A = [min{ µ A− (x ∗ y ), µ A− (y)}, min{ µ A+ (x ∗ y ), µ A+ (y)}] + ⇔ [µ − A (x), µ A (x)]   + −  +  ≥ minµ − A (x∗y ), µ A (y) , minµ A (x∗y ), µ A (y)     

{

}

− ⇔ µ − (x) ≥ min µ − A (x ∗ y ), µ A (y) and A + (x ∗ y ), µ + (y) , for all µ + (x) ≥ min µ A A A x, y ∈ X.

{

}

{[

][

= max λ A− (x ∗ y), λ A+ (x ∗ y) , λ A− (y),λ A+

{λ

=  max 

− A

(x ∗ y ), λ −A (y) }, max

{λ

+ A

(x ∗ y ), λ +A (y) }

Therefore,  λ − (x), λ + (x)  ≤ A  A 

{

}

{

}

 max λ − (x ∗ y ), λ − (y) , max λ + (x ∗ y ), λ + (y)  A A A A  

⇒ λ −A (x) ≤ max{λ −A (x ∗ y ), λ −A (y)} and

{

}

+ ( ) + λ+ A (x) ≤ max λ A x ∗ y , λ A (y) , for all ~ , ~λ ) is an i-v x, y ∈ X . Hence A = (X, µ A A

− IF ideal ⇔ A − = (X, µ A , λ −A ) and

A + = (X, µ +A , λ +A ) are IF ideals. ~

~ , λ ) be an Theorem 2.4: Let A = (X, µ A A ~

~ , λ ) is an i-v IF set of X . Then A = (X, µ A A i-v IF ideal of X if and only if ◊ ~ ~ A = (X, (λ A ) c , λ A ) is an i-v IF ideal of X .

⇔ ◊ A − = (X, (λ−A ) c , λ −A ) +

A =

(X, (λ A+ ) c , λ +A ) are

and ◊

IF ideals of X .

~ ) c (0) ≥ (λ ) c (x) A A ~ ~ ⇔ [1, 1] - (λ A )(0) ≥ [1, 1] - (λ A )(x) − (x), λ + (x) ⇔ [1, 1] − λ −A (0), λ +A (0) ≥ [1, 1] - λ A A     ⇔ 1− λ −A (0), 1− λ +A (0) ≥ 1 − λ A− (x),1 − λ +A (x)  

Proof: We have (λ

[

} (y)]}

Similarly λ A (x) ≤ max λ A (x ∗ y ), λ A (y)

]

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

⇔ (λ − ) c (0), (λ + ) c (0) ≥ (λ − ) c (x), (λ + ) c (x)  A   A  A A ⇔ (λ −A ) c (0) ≥ (λ −A ) c (x) and (λ +A ) c (0) ≥ (λ +A ) c (x)

{

}

~ c c ( ) ~ c A ) (x) ≥ min (λ A ) x ∗ y , (λ A ) (y) ~ ~ ~ ⇔ [1, 1] − λ (x) ≥ min [1, 1]− λ (x ∗ y ), [1, 1] − λ A (y) A ~

And (λ

{

}

− (x), λ + (x) ⇔ [1, 1] −  λ A A   (x ∗ y ), λ +A (x ∗ y ), [1, 1]− λ −A (y), λ +A (y)  ≥ min [1, 1]− λ −  A    

[

{[

]

][

]}

⇔ 1 − λ −A (x).1 − λ −A (x) ≥ min 1 − λ A− (x ∗ y ), 1 − λ A+ (x ∗ y ) , 1 − λ A− (y), 1 − λ +A (y) ⇔ [(λ −A ) c (x), (λ A+ ) c (x)] ≥ min [(λ − ) c (x ∗ y ), (λ − ) c (y)], [(λ + ) c (x ∗ y ), (λ + ) c (y)] A A A A − c − c − c Therefore, (λ A ) (x) ≥ min [(λ ) (x ∗ y ), (λ ) (y)] and A A c( ) + c (λ +A ) c (x) ≥ min [(λ + A ) x ∗ y , (λ A ) (y)] , for all x, y ∈ X.

{ {

{

} }

~ , λ ) be an Theorem 2.5: Let A = (X, µ A A ~

~ , λ ) is i-v IF set of X . Then A = (X, µ A A an i-v IF ideal of X. ⇔ ◊A = (X, µ~ A , (µ~ A ) c ) is an i-v IF ideal of X ⇔ ◊A − = (X, µ −A , (µ −A ) c ) and

◊A + = (X, µ +A , (µ +A ) c ) are IF ideals of X. Theorem 2.6: Let A1 and A 2 are i-v IF ideals of X . Then A1 ∩ A 2 is also an i-v IF ideal of X . Corollary 2.7: Let {A \ i ∈ A} be a family

of i-v IF ideal of X . Then I A is also an i

i∈I

i-v IF ideal of X .

}

~ , λ ) is an i-v Theorem 2.8: If A = (X, µ A A IF ideal of X , then the sets ~ (x )= µ ~ (0) and X ~ = x∈X/µ A A

µA

{

}

{

}

~ ~ X ~ = x∈X/ λ A (x )= λ A (0 ) are ideals λA of X . Proof:

Since

~ ~ (0) = µ ~ (0) and ~ µ λ (0) = λ (0) ⇒ A A A A 0 ∈ X ~ and 0 ∈ X ~ µA µA If (x ∗ y), y ∈ X ~ ⇒ µA ~ (x ∗ y) = µ ~ (0 ), µ ~ (y) = µ ~ (0) µ A A A A

and so

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

{

~ ~ ~ µ A (x) ≥ min µ A (x ∗ y), µ A (y) ~ ~ ~ = min µ A (0), µ A (0) = µ A (0)

{

}

− (ii) A − = (X, µ A , λ −A ) and

~ implies µ

~ (0) but (x) ≥ µ A A ~ (x) ≤ µ ~ (0) implies µ ~ (x) = µ ~ (0) µ A A A A implies x ∈ X ~ that is µA (x ∗ y), y ∈ X ~ ⇒ x ∈ X ~ . µA µA Therefore, X µ ~ is an ideal of X . Similarly A X ~ is an ideal of X .

λA

~ , λ ) be an Definition 2.9: Let A = (X, µ A A i-v IFS in X . For [s1 , s 2 ], [t 1 , t 2 ] ∈ D[0, 1] then the set

~ ; [s , s ]) = {x ∈ X/µ ~ (x) ≥ [s , s ]} U (µ A 1 2 A 1 2 ~ and the set is called i-v upper level cut of µ A ~ ~ L(λ A ; [t 1 , t 2 ]) = {x ∈ X/ λ A (x) ≤ [t 1 , t 2 ]} ~ is called i-v lower level cut of λ A . Note: ~ ; [s , s ]) = {x ∈ X/µ ~ (x) ≥ [s , s ]} (i) U (µ A 1 2 A 1 2 − = {x ∈ X/[µ A (x), µ +A (x)] ≥ [s1 , s 2 ]} − = {x ∈ X/µ A (x) ≥ s1 and − + = U (µ A ; s1 ) I U(µ A ; s2 )

µ A+ (x)

≥ s2}

+ (ii) L(λA ; [t1 , t 2 ]) = L(λ −A ; t1 ) I L(λ A ; t 2 ).

~ , λ ) be an Theorem 2.10: Let A = (X, µ A A i-v IF set of X , Then the following conditions are equivalent. (i) A is an i-v IF ideal of X .

A + = (X, µ A+ , λ +A ) are IF ideals of X . (iii) The non-empty sets

U (µ −A ; s1 ), L(λ −A ; t 1 ) and U (µ +A ; s 2 ), L(λ +A ; t 2 )

~ ; [s , s ]) (iv) The non-empty sets U (µ A 1 2 ~

and L( λ A ; [t 1 , t 2 ]) are i-v ideals of X .

~ Proposition 2.11: If A = (µ

~ , λ ) is i-v A A

IF ideal of X , then the level subsets

~ ; ~s ) and L(~ U (µ λ ; ~s ) are i-v ideals of A A ~ ~ ) ∩ Im(~ for every s ∈ Im(µ λ ) X A A ~ ~ ⊆ D[0, 1] , where Im(µ ) and Im(λ ) A A ~ ~ and λ , are sets of values of µ A A

respectively. Proposition 2.12: If for all non-empty i-v

~ ~ ; s ) of A A ~ ,~ λ ) are ideals of X , an i-v IF set A = (µ A A then A is i-v intuitionistic fuzzy ideal of X . ~ ; ~s ) and L( λ level subsets U (µ

~ , λ ) be an iDefinition 2.13: Let A = (µ A A

~ v IF set on X and ~ s , t ∈ D[0,1] such that

~s + ~t ≤ [1, 1] . Then the set ~ (~s , t ) ~ (x), X = {x∈X / ~s ≤ µ A A ~ ~ ~ ~ λ A (x) ≤ t is called an ( s , t ) level subset of A . The set of all

}

~ ~ ) × Im(~λ ) such that (~s , t ) ∈ Im(µ A A

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B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

~ ~ (~s1 , t1 ) (~s , t ) ⊆X Then X , i. e ., A A ~ (~s , t ) . This implies that x, y ∈ X A ~ ~ (~s , t ) (~s , t ) (x ∗ y), y ∈ X since X A A

~s + ~t ≤ [1, 1] is called the image of ~ ,~ A = (µ A λ A ) . Obviously, ~ ~ (~s , t ) ~ , ~s ) ∩ L(~ X = U(µ λ , t ). A A A Theorem

2.14:

i-v

set

~ ,~ A = (µ λ ) of X is an i-v IF ideal of A A ~ (~s , t ) is an ideal of X , X if and only if X A ~ ~ ) × Im(~λ ) with for every ( ~ s , t ) ∈ Im(µ A A ~s + ~t ≤ [1, 1] .

is an ideal of X . Hence we deduce that

~ (x) ≥ ~s = min{µ ~ (x ∗ y), µ ~ (y)} , µ A A A

~ ~ ~ ~ λ (x) ≤ t = max{λ (x ∗ y), λ (y)} . A A A ~ ~ This shows that A = (µ , λ ) is an i-v IF A A ideal of X .

Proof: We only need to prove the necessity, because the sufficient is trivial.

~ (~s , t ) is an ideal of X and A ~ ,~ A = (µ λ ) an i-v IF set on X . It is easy A A ~ (0) ≥ µ ~ (x) and to see that µ A A ~ ~ λ (0) ≤ λ (x) . A A Assume that

Theorem 2.15: Let A = (~ α ,λ

where the functions

~ ~ α ∧ β : X → D[0, 1] and A B ~ ~ λ ∨ υ : X → D[0, 1] defined by A B

~ ~ A(x ∗ y) = (~s , t ) and A(y) = (~s1 , t1 ) i. e ~ ~ (x ∗ y) = ~s , ~ µ λ A (x ∗ y) = t , A ~ ~ (y) = ~s and ~ µ λ A (y) = t1 . Without loss A 1

∀ x ∈ X, ∀ x ∈ X,

generality,

We may assume that ~ ~ ~ ~ ~ ~ ( s , t ) ≤ ( s1 , t1 ) , i.e ~s ≤ ~s1 and t1 ≤ t .

Proof: For every x ∈ X , we have

(~ α

{ {

) and

~ B = (β , ~ υ ) be i-v intuitionistic fuzzy B B ideals of X . Then the generalized Cartesian ~ ~ ∧~ product A × B = (α β , λ ∨ ~υ ) , A B A B

Consider x ∗ y, y ∈ X such that

(~α A ∧ β~B )(x) = min{ ~α A (x), β~ B (x)} ,

(~λ A ∨ ~υ B )(x) = max{ ~λ A (x), ~υ B (x)} ,

is an i-v IF ideal of X .

} }

~ ~ ∧ β )(0) = min ~ α A (0), β B (0) A B ~ ≥ min ~ α A (x), βB (x) ~ = (~ α A ∧ βB )(x) ,

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

~ (λ

{ {

} }

~ ∨~ υ )(0) = max λ A (0), ~υB (0) A B ~ ≤ max λ A (x), ~υ B (x) ~ = (λ ∨ ~υ )(x) A

Hence, for x, y ∈ X , we deduce that

{ { { { {

}

~ ~ (~ α A ∧ β B )(x) = min ~ α A (x), β B (x) ~ ~ ≥ min min{~ α A (x ∗ y), ~ α A (y)}, min β B (x ∗ y), β B (y) ~ ~ = min min ~ α A (x ∗ y), βB (x ∗ y) , min ~ α A (y), β B (y) ~ ~ = min (~ α A ∧ βB )(x ∗ y), (~ α A ∧ βB (y) ~ ~ (λ ∨ ~ υ )(x) = max{λ (x), ~ υ (x)} B A B A ~ ~ ≤ max max λ A (x ∗ y), λ A (y) , max{~υ B (x ∗ y), ~υ B (y)} ~ ~ = max max λ A (x ∗ y), ~υ B (x ∗ y) , max λ A (y), ~υ B (y) ~ ~ = max (λ A ∧ ~υ B )(x ∗ y), (λ A ∧ ~υ B (y) . ~ ~ ∧~ This shows that A × B = (α β , λ ∨ ~υ ) is an i-v intuitionistic fuzzy ideal of X . A B A B

{ { { { {

3. ARTINIAN AND NOETHERIAN BFALGEBRAS Definition 3.1: A BF-algebra X is said to be Noetherian if every ideal of X is finitely generated. We say that X satisfies the ascending chain condition oni-v IF ideal if for every ascending sequence I1 ⊆ I 2 ⊆ I 3 ⊆ ............ of ideals of X there exist natural numbers ‘n’ such that I n = I K for all n ≥ K . We call X satisfies the interval-valued intuitionistic fuzzy ascending chain condition if for every ascending sequence A1 ⊆ A 2 ⊆ .... of interval-valued intuitionistic fuzzy ideal in X , there exist natural number ‘n’ such that ~ =µ ~ , for all n ≥ k. µ n k

{

}

}} }}

{ }

} }}

The following lemma is obvious.

~ , λ ) be an i-v Lemma 3.2: Let A = (µ A A intuitionistic fuzzy ideal of X and let

~s , ~t ∈ Im(µ ~ ) and ~s , ~t ∈ Im(~λ ) then A A ~ ; ~t ) ⇔ ~s = ~t and ~ ; ~s ) = U (µ U (µ A A ~ ~ ~ ~ ~ L(λ A ; s ) = L(λ A ; t ) ⇔ ~s = t .

Theorem 3.3: Let X be a BF-algebra .Then every i-v intuitionistic fuzzy ideal of X has finite number of values if and only if X is Artinian. Proof: Suppose that every i-v IF ideal of X has finite number of values but X is not Artinian .Then there exist a strictly descending chain

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

X = A 0 ⊃ A1 ⊃ A 2 ⊃ .......

prove (i-vIF-2) and (i-vIF-3), we consider an interval-valued intuitionistic fuzzy set

of ideals of X . Conditions (i-vIF-1) of definitions 2.1 hold obviously. In order to

~ ,~ A = (µ A λ A ) Given by

 1 n  , if x ∈ A n \ A n +1 , n = 0,1......     n + 1 n + 1 ~ µ A (x) :=  ∞  [1, 1] if x ∈ I A n ,  n =0 ~ ~ (x), ∀x ∈ X λ A (x) := [1, 1] − µ A Let x, y ∈ X, then (x ∗ y) , y ∈ A

n +1

n = 0,1,2.......... and either (x ∗ y) ∉ A or y ∉ A n +1 .We now let n +1 x * y, y ∈ A \ A , for k ≤ n . n n +1

for

some

Then we can observe that every subset of D[0, 1] contains either a strictly increasing sequence or strictly decreasing sequence. Now, let [s1 , t1 ] < [s 2 , t 2 ] < [s3 , t 3 ] < .........

~ ). be a strictly increasing sequence in Im(µ A ~

Then it follows that

~ (x) =  1 , n  ≥  1 , k  µ A  n + 1 n + 1  k + 1 k + 1 ~ (x ∗ y), µ ~ (y)). ≥ min(µ A A

~ is an i-v fuzzy ideal of X Thus µ A

~ has infinite number of different and µ A ~

values. In a similar way, λ A is i-v fuzzy

ideal of X and λ A has infinite number of

~ , λ ) is an different values. Hence A = (µ A A i-v IF ideal of X and has infinite number of different values. This contradiction proves that the BF-algebra X is Artinian. Conversely, if BF-algebra X is Artinian ~

~ , λ ) be an i-v IF then we can let A = (µ A A

~ ) is finite. ideal of X . Suppose that Im(µ A

Then U(µ A ; [s1, t1]) ⊃ U(µ A ; [s2 , t 2 ] ⊃ U(µA ; [s3 , t 3 ] ⊃ ...... is strictly decreasing chain of ideals of X . Since X in Artesian, there exist a natural number i such that

~ ; [s , t ]) = U(µ ~ ; [s , t ] for all U (µ A i i A i +n i +n ~ n ≥ 1 . Since [si , t i ] ∈ Im(µ A ) for all i, by applying

Lemma

3.2,

have

s i = s i+n , t i = t i+n for all n ≥ 1 , which is contradiction, since si , t i are distinct. This ~ ) is finite .On the other shows that Im(µ A hand, if [s1 , t 1 ] > [s 2 , t 2 ] > [s 3 , t 3 ] > ...... is ~ ), strictly decreasing sequence in Im(µ A ~ ~ then L( λ A ; [s1 , t 1 ]) ⊇ L( λ A ; [s 2 , t 2 ] ~ ⊇ L(λ A ; [s 3 , t 3 ] ⊇ ....... is and descending chain of ideals of X . Since X is Artinian, there exist a natural number j such that

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~ ~ L(λ A ; [s j , t j ]) = L(λ A ; [s j+ n , t j+ n ] for all ~ n ≥ 1 . Since [s j , t j ] ∈ Im(λ A ) for all j , by Lemma 3.2,

s j = s j+ n , t j = t j+ n for all

n ≥ 1 , which is again contradiction, since ~ s j , t j are distinct. This shows that Im(λ A ) is finite. Theorem 3.4 Let a BF-algebra X be Artinian. If

~ ,~ A = (µ λ ) is an i-v IF A A

ideal of X . Then U ~

L~ λ

~ ) and = Im(µ A

~ = Im(λ A ) , Where

U~ µ

and

are families of all level ideals of X

~ with respect to µ

and λ

respectively.

Theorem 3.5 Let BF-algebra X be Artinian.

~ If A = (µ

~ , λ ) and B = ( ~υ , ~ η ) are A B B

i-v IF ideal of X , Then the following statements hold:

~ ~ µ = U~ υ and Im(µ A ) = Im(υB ) A B ~ = ~υ , if and only if µ A B

(i) U~

(ii)

L~ λ

= L~ η

B A ~ Im(λ ) = Im(~ η ) A B ~ if and only if λ = ~ η . A B

and

= ~υ , then U ~ = U ~ µ υ B A B ~ ~ and Im(µ ) = Im(υ ) . Now suppose A B ~ ) = Im(~υ ) . that U ~ = U ~ and Im(µ µ υ A B A B ~ Proof: (i) If µ

theorem

3.3

and

~ ) = Im(~υ ) are finite and Im(µ A B

Uµ ~

~ ) , U~ = Im(µ A υ

3.4,

= Im(~υ B ) .

~ ) = {~t , ~t ........~t } Im( µ A 1 2, n ~ ~ ~ ~ and Im(υ ) = { s , s ,...... s }, Where B 1 2 n ~ ~ ~ ~ t < t < t .......... < t and 1 2 3 n ~s < ~s < ........~s . This shows that 1 2 n ~ ~ t = s for all i. We now prove that i i ~ ~ ~ U(µ ; t ) = U(~υ ; t ) for all i. Note that A i B i ~ ; ~t ) = X = U(~υ ; ~t ) . Consider U(µ A 1 B 1 ~ ; ~t ) and U(~υ ; ~t ). Suppose that U( µ A 2 B 2 ~ ~ ~ ~ U(µ ; t ) ≠ U( υ , t ). A 2 B 2 ~ ~ ~ Then U(µ ; t ) = U(~ υ , t ) for some A 2 B k ~ ~ ; ~t ) for k > 2 and U(~ υ , t ) = U(µ B 2 A j some j > 2 . If there exist x ∈ X such that ~ (x) = ~t , then µ A 2 Let

~ (x) < ~t , ∀ j > 2 µ A j ~ ; ~t ) = U(~υ ; ~t ) , Since U(µ A 2 B k ~ ~ . This implies that x ∈ U( υ ; t ) B K

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(1)

B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

~υ (x) ≥ ~t > ~t , k > 2 . Thus B K 2 ~ ~ x ∈ U( υ ; t ) . Since B 2 ~ ~ ~ ; ~t ) , x ∈ U(µ ~ ; ~t ) . U( υ ; t ) = U(µ B 2 A j A j ~ (x) < ~t for some j > 2 → (2) Hence µ A j Clearly (1) and (2) contradict each other. ~ ; ~t ) = U(~υ ; ~t ) . Hence U(µ

B 2

Continuing in this way, we eventually obtain ~ ; ~t ) = U(~υ ; ~t ) for all i. U(µ

A i B i ~ (x) = ~t Now let x ∈ X . Suppose that µ A i ~ ~ for some i. Then x ∉ U(µ ; t ) for all A j i + n ≤ j ≤ n . This implies that ~ x ∉ U(~υ ; t ) for all i + 1 ≤ j ≤ n . But B j ~ then we have ~ υ (x) < t for some B j i + 1 ≤ j ≤ n . Suppose we have that ~υ (x) = ~t for some i ≤ m ≤ n . B m ~ If i ≠ m , then x ∈ U(~ υ ; t ) . On the B i ~ (x) = ~t , other hand, since µ A i ~ ~ ~ ~ x ∈ U(µ ; t ) = U( υ ; t ) . Thus we A i B i arrive a contradiction. Hence i = m and ~ (x) = ~t = ~υ (x) . Consequently µ A i B ~ ~ µ =υ . A B (ii) The proof is similar and omitted. We now characterize the Noetherian BFalgebra in the following theorem. Theorem 3.6: A BF-algebra X is Noetherian if and only if the set of values of i-v

Intuitionistic fuzzy ideal of X are well ordered subsets of D[0, 1] .

~ Proof: Suppose that A = (µ

~ , λ ) is an A

i-v intuitionistic fuzzy ideal of X whose set of values is not a well ordered subset of D[0, 1] . Then there exist a strictly decreasing sequence [s , t ] such that

n n ~ µ (x ) = [s , t ] . Denote sequence by A n n n ~ (x ) ≥ [s , t ]} . U the set {x ∈ G / µ n A n n n Then U ⊂ U ⊂ U ...... is a strictly 1 2 3 ascending chain of ideals of X . This clearly contradicts that X is Noetherian. Hence, ~ ) must be a well-ordered subset of Im(µ

A D[0, 1] .

Similarly, for Im(λ

Conversely, assume that the set of values of i-v intuitionistic fuzzy ideal of X is a well ordered subset of D[0, 1] and X is not Noetherian BF-algebra. Then there exist strictly ascending chain

U ⊂ U ⊂ U ...... .. 1 2 3 of ideals of X .

→

(∗)

Define an i-v IF set

~ ,~ A = (µ λ ) on X by putting A A

 1 1   k + 1 , k  for x ∈ A K \ A K − 1 ,   ~ µ (x) :=  ∞ A  [0, 0 ] for x ∉ U A , K  K =1

~ ~ (x) , λ (x) := [1, 1] − µ A A

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B. Satyanarayana, et al., J. Comp. & Math. Sci. Vol.3 (1), 83-96 (2012)

Then, we can easily prove that

~ ,~ A = (µ A λ A ) is an i-v IF ideal of X .

Since the ascending chain (∗) is not terminating, A has strictly descending sequence of values, this contradicts that the value set of an i-v IF ideal is well ordered. Consequently, X is Noetherian. Finally, we state a theorem of Noetherian BF-algebra. Since the proof is straightforward, we omit the proof. Theorem 3.7: If X is a Noetherian BFalgebra, then every i-v IF ideal of X is finite valued. 4. FULLY INVARIANT AND CHARACTERISTIC INTERVALVALUED INTUITIONISTIC FUZZY IDEALS OF BF-ALGEBRAS Definition 4.1 An ideal F of BF-algebra X is said to be a fully invariant ideal if f(F) ⊆ F for all f ∈ End(X) , where End(X) is set of all endomorphism of X , An i-v intuitionistic

Proof:

can

easily

seen

that

~ ,∨~ I A = (∧µ λ ) is i-v IF ideal of X . i Ai Ai Let x ∈ X and f ∈ End(X). Then f ∧ µ ~  ~  i∈I A  (x) = ( ∧ i∈I µ A )(f(x)) i  i

~ = inf{µ ~ ≤ inf{µ

(f(x)) / i ∈ I}

(x) / i ∈ I} Ai ~ )(x), = (∧ µ i∈I A i

~ f ~ ∨  i∈I λ A  (x) = (∨ i∈I λ A )(f(x)) i  i ~ = sup{ λ (f(x)) / i ∈ I} Ai ~ ≤ sup{ λ (x) / i ∈ I} Ai ~ = (∨ λ )(x). i∈I A i Hence

~ ~ ,∨ A = (∧ µ λ ) i i∈I A i i∈I A i is an

~ , λ ) of X is called a fuzzy ideal A = (µ A A

~ f (x) = µ ~ (f(x)) ≤ µ ~ (x) fully invariant if µ A A A

i-v IF fully invariant ideal of X .

and λ Af (x) = λ A (f(x)) ≤ λ A (x) x ∈ X and f ∈ End(X) .

Theorem 4.3: Let H be nonempty subsets ~ ,~ of BF-algebra X and A = (µ A λ A ) be an iv IF ideals defined by

for

all

Theorem 4.2 If {Ai / i ∈ I} is a family of iv IF fully invariant ideals of X , then

~ ~ ,∨ Ii∈I A i = (∧ i∈I µ A i i∈I λ A i ) is an i-v

IF fully invariant ideal of X , where

~ (x) = inf{µ ~ (x) / i ∈ I, x ∈ X} ∧ µ A Ai i∈I i ~ ~ ∨ λ (x) = sup{ λ (x) / i ∈ I, x ∈ X}. A Ai i∈I i

i∈I

~ (x) = [s 2 , t 2 ] if x ∈ H µ  A  [s1 , t 1 ] otherwise, [α , β ] if x ∈ H ~ λ A (x) =  2 2  [α1 , β1 ] otherwise, Where [0, 0] ≤ [s 1 , t 1 ] < [s 2 , t 2 ] ≤ [1, 1], [0, 0] ≤ [α 2 , β 2 ] < [α 1 , β 1 ] ≤ [1, 1],

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

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[0, 0] ≤ [si , t i ] + [α i , β i ] ≤ [1,1] for i = 1, 2 . If H is an i-v IF fully invariant ideal of X , ~ ,~ then A = (µ A λ A ) is an i-v IF fully invariant ideal of X . Proof:

can

easily

see

that

~ ,~ A = (µ A λ A ) is an i-v IF ideal of X . Let x ∈ X and f ∈ End(X) . If x ∈ H, then f(x) ∈ f(H) ⊆ H . Thus we have

~ f (x) = µ ~ (f(x)) ≤ µ ~ (x) = [s , t ] , µ A A A 2 2 ~f ~ ~ λ A (x) = λ A (f(x)) ≤ λ A (x) = [α 2 , β 2 ] . For if otherwise, then we have

i-v

ideal

~ ,~ A = (µ A λ A ) of X has the same type as an ~ ,~ i-v IF ideal B = (µ λ ) of X if there B

~ ~ (x)) = µ ~ (φ (x)), f(~ f(µ λ A (x)) = λ B (φ (x)) A B

for all x ∈ X . Then, it is clearly that f is a surjective homomorphism. Also, f is ~ (x)) = f(µ ~ (y)) for injective because f(µ A A

~ (φ (x)) = µ ~ (φ (y)) , all x, y ∈ X implies µ B B

where

~ (x) = µ ~ (y) . Likewise, from µ A B ~

~ , λ ) is Hence A = (µ A A ~

~ , λ ) . This completes isomorphic to B = (µ B B the proof.

~ ,λ ) Thus, we have verified that A = (µ A A is an i-v IF fully invariant ideal of X . An

all x ∈ X Let f : A(X) → B(X) be a mapping defined by f(A(x)) = B(φ (x)) for all x ∈ X ,i.e,

for all x ∈ X .

~f ~ ~ λ A (x) = λ A (f(x)) ≤ λ A (x) = [α1 , β1 ] .

4.4

~ ~ (x) ≥ µ ~ (φ (x)), ~ µ λ A (x) ≥ λ B (φ (x)) for A B

~ ~ ~ ~ f( λ A (x)) = f( λ A (y)) we conclude λA (x) = λB (y)

~ f (x) = µ ~ (f(x)) ≤ µ ~ (x) = [s , t ] µ A A A 1 1

Definition

~ , λ ) , then there exist same type as B = (µ B B φ ∈ End(X) such that

exist f ∈ End(X) such that A = B o f , i. e

~ ~ (x) ≥ µ ~ (f(x)), ~ µ λ A (x) ≥ λ B (f(x)) for all A B x∈X.

Theorem 4.5: Interval-valued intuitionistic fuzzy ideals of X have same type if and only if they are isomorphic.

Definition 4.6: An ideal C of X is said to be characteristic if f(C) = C for all f ∈ Aut(X) , where Aut(X) is the set of all automorphisms of X . An i-v IF ideal ~ ,~ A = (µ A λ A ) of X is called characteristic

~ (f(x) = µ ~ (x) and λ (f(x)) = λ (x) if µ A A A A for all x ∈ X and f ∈ Aut(X) . ~

~ , λ ) be an i-v Lemma 4.7: Let A = (µ A A intuitionistic fuzzy ideal of X and let x ∈ X . ~ (x) = ~t , ~λ (x) = ~s if and only if Then µ A A

~ ; ~t ), x ∉ U(µ ~ ; ~s ) and x ∈ U(µ A A ~ ~ ~ ~ x ∈ L(λ A ; s ) , x ∉ L( λ A ; t ) , for all

Proof: We only need to prove the necessity part because the sufficient part is obvious. If

~s > ~t .

~ , λ ) of X has the an i-v IF ideal A = (µ A A

Proof: Straight forward.

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Theorem 4.8 An i-v intuitionistic fuzzy ideal is characteristic if and only if each its level set is a characteristic ideal. Proof: Let an i-v intuitionistic fuzzy ideal

~ ,~ A = (µ A λ A ) be characteristic, ~ ~ ), f ∈ Aut(X), x ∈ U(µ ~ ; ~t ) . t ∈ Im(µ A A ~ ~ ~ Then µ A (f(x)) = µ A (x) ≥ t , which means ~ ; ~t ) . Thus that f(x) ∈ U(µ A ~ ~ ~ ; ~t ) . Since for each f(U(µ A ; t )) ⊆ U(µ A ~ ~ x ∈ U(µ ; t ) there exist y ∈ X such that A

f(y) = x we have ~ (y) = µ ~ (f(y)) = µ ~ (x) ≥ ~t . Hence we µ A A A ~ ~ conclude y ∈ U(µ A ; t ). ~ ; ~t ) . Consequently, x = f(y) ∈ f(U(µ A ~ ~ ~ ~ Hence f(U(µ A ; t ) = U(µ A ; t ) . ~ ~ Similarly, f(L(λ A ; ~ s )) = L(λ A ; ~s ) . This ~ ; ~t ) and L(~λ ; ~s ) are proves that U(µ A A characteristic.

~ , λ ) are Conversely, if all levels of A = (µ A A characteristic ideals of X, then for x ∈ X , ~ (x) = ~t < ~s = ~ λ A (x) , f ∈ Aut(X) and µ A

~ ; t ), by lemma 4.7, we have x ∈ U(µ A

~ ; ~s ) and x ∉ U(µ A ~ ~ ~ ~ x ∈ L( λ A ; s ), x ∉ L( λ A ; t ) . Thus ~ ; ~t )) = U(µ ~ ; ~t ) and f(x) ∈ f(U(µ A A ~ ~ ~ ~ f(x) ∈ f(L(λ A ; s ) = L(λ A ; s ) , i.e., ~ (f(x)) ≥ ~t and ~λ (f(x)) ≤ ~s . For µ A A ~ ~ ~ ~ µ A (f(x)) = t1 > t , λ A (f(x)) = ~s1 < ~s , we ~ ; ~t ) = f(U(µ ~ ; ~t ) , have f(x) ∈ U(µ A 1 A 1 ~ ~ ~ ~ f(x) ∈ L( λ ; s ) = f(L( λ ; s )) . Hence A

~ ; ~t ) , x ∈ L(~λ ; ~s ) . This is a x ∈ U(µ A 1 A 1 ~ (f(x)) = µ ~ (x) and contradiction. Thus µ A A ~ ~ ~ ~ λ (f(x)) = λ (x) . So, A = (µ , λ ) is A

characteristic. REFERENCES 1. K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets an Systems, 20, 87-96 (1986). 2. K.T. Antanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy sets and Systems, 61,137-142 (1994). 3. K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy sets an Systems, 31, 343-349 (1989). 4. K. iseki and T. Shotaro, An introduction to the theory of BCK-algebras, Math. Japon, 23, 1-26 (1978). 5. K. Iseki and T. Shotaro, Ideal theory of BCK-algebras, Math. Japonica, 21, 351366 (1976). 6. Z. Jianming and T. Zhisong, Characterizations of doubt fuzzy H-ideals in BCKalgebras, Soochow Journal of Mathematics, 29, 290-293 (2003). 7. Y. B. Jun and K. H. Kim, Intuitionistic fuzzy ideals of BCK-algebras, Internat J. Math. Sci., 24, 839-849 (2000). 8. B. Satyanarayana, D. Ramesh, M. V. Vijayakumar and R. Durga Prasad, On fuzzy ideal in BF-algebras, International J. Math. Sci. Engg. Apple.,4, 263-274(2010). 9. B. Satyanarayana, M. V. Vijayakumar, D. Ramesh, and R. Durga Prasad, Interval-valued intuitionistic fuzzy BFsubslgebras, Acta Ciencia Indica, (inpress). 10. L. A. Zadeh, Fuzzy sets, Informatiuon Control, 8, 338-353 (1965). 11. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Sciences, 8, 199249 (1975).

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