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ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 14 February 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/ijfma.v12n1a2

11

Intuitionistic Fuzzy BRK-ideal of BRK-algebra with

Interval-valued Membership and Non Membership

Functions

Osama Rashad El-Gendy

Foundation Academy for Sciences and Technology

Batterjee Medical College for Sciences & Technology, Saudi Arabia Email: dr.usamaelgendy@gmail.com

Received 8 January 2017; accepted 28 January 2017

Abstract. In this paper we introduce the notion of intuitionistic fuzzy BRK-ideal of BRK-algebra with interval-valued membership and non membership functions and investigate some interesting properties. The image and inverse image of interval valued intuitionistic fuzzy BRK-ideal are also discussed.

Keywords: BRK-algebra; BRK-ideal; fuzzy BRK-ideal; interval-valued fuzzy BRK-ideal.

AMS Mathematics Subject Classification (2010): 06F35, 03G25, 08A27

1. Introduction

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12

idea gives the simplest method to capture the imprecision of the membership grades for a fuzzy set. Thus, interval-valued fuzzy sets provide a more adequate description of uncertainty than the traditional fuzzy sets. It is therefore important to use interval-valued fuzzy sets in applications. One of the main applications is in fuzzy control and the most computationally intensive part of fuzzy control is defuzzification. Since the transition of interval-valued fuzzy sets usually increases the amount of computations, it is vitally important to design some faster algorithms for the necessarily defuzzification. On the other hand, Atanassov [1,2,3] introduced the notion of intuitionistic fuzzy sets as an extension of fuzzy set in which not only a membership degree is given, but also a non-membership degree is involved. Considering the increasing interest in intuitionistic fuzzy sets, it is useful to determine the position of intuitionistic fuzzy sets in a frame of different theories of imprecision. With the above background, Atanassov and Gargov [4] introduced the notion of interval-valued intuitionistic fuzzy sets which is a common generalization of intuitionistic fuzzy sets and interval-valued fuzzy sets. The fuzzy structure for BRK-ideal was introduced in [7]. Since then, the concepts and results of BRK-algebra have been broadened to the fuzzy BRK-ideals setting frames also see [8, 9]. In this paper we first apply the concept of interval valued intuitionistic fuzzy sets to algebra. Then we introduce the notion of Intuitionistic Fuzzy ideal of BRK-algebra with Interval-valued Membership and Non Membership Functions and investigate some of interesting properties. We study the homomorphic image and inverse image of interval valued intuitionistic fuzzy BRK-ideal of BRK-algebra.

2. Preliminaries

Definition 2.1[5]. A BRK-algebra is a non-empty set X with a constant 0 and a binary

operation “∗” satisfying the following conditions: (BRK1) x∗0=x,

(BRK2) (xy)∗x=0∗y, for all x,yX.

In a BRK-algebra X, a partially ordered relation ≤ can be defined by xy if and only if xy=0.

Definition 2.2 [5]. If (X;∗,0)is a BRK-algebra, the following conditions hold: (1) xx=0,

(2) xy =0⇒ 0∗x=0∗y,

(3) 0∗(xy)=(0∗x)∗(0∗y), for all x,yX

Definition 2.3. A non empty subset S of a BRK-algebra X is said to be BRK-subalgebra

of X, if x,yS , implies xyS.

Definition 2.4. (BRK-ideal of BRK-algebra). A non empty subset I of a BRK-algebra X

is said to be a BRK-ideal of X if it satisfies: (I1) 0∈I,

(I2) 0∗(xy)∈Iand 0∗yI imply 0∗xI for all x,yX .

(3)

13 (FI1) µ(0)≥ µ(x),

(FI2) µ(0∗x)≥min{µ(0∗(xy)),µ(0∗y)},for all x,yX .

Example 2.6. LetX ={0,a,b,c}. Define ∗ on X as the following table:

∗ 0 a b c

0 0 b b 0

a a 0 0 b

B b 0 0 b

c c a a 0

Then (X,∗;0)is a BRK-algebra, and A = {0, b, c} is a BRK-ideal of BRK-algebra X.

3. Interval-valued intuitionistic fuzzy homomorphism of BRK-algebra interval- valued intuitionistic fuzzy set

A mapping A=(

µ

~ ,

ν

~):XD[0,1]×D[0,1] is called an interval-valued intuitionistic

fuzzy set (i-v IF set, briefly) in X if 0≤

µ

A• +

ν

A ≤1 and 0≤ + ≤1

• • • •

A A

ν

µ

for all

X

x∈ . Where

µ

A(x),

A

ν

(x),

µ

A(x) and

• •

A

ν

(x) are fuzzy sets of X such that

µ

A

• •

A

µ

and

ν

A• ≤

ν

A•. A•• ={(x ,

µ

A•,

ν

A••)|xX} andA• ={(x,

µ

A• ,

ν

A•)|xX} are intuitionistic fuzzy sets. LetD[0,1]be the family of all closed sub-interval of [0,1]. The mapping

] 1 , 0 [ :

)] ( ), ( [ ) (

~ x x x X D

A A

A = →

• •

µ

µ

µ

and~A(x)=[ A(x), A (x)] :XD[0,1]

• •

ν

ν

ν

denote the degree membership and degree of non-membership for allxX respectively. For simplicity we use the symbols forms A• =(X ,

µ

A• ,

ν

A•) and ( , , )

• • • • •

=

A A

X

A

µ

ν

.

The refined minimum (briefly r min) and order “≤” on the subintervals

D

1

=

[

a

1

,

b

1

]

and

D

2

=

[

a

2

,

b

2

]

of D[0,1] is defined by,

r

min(

D

1

,

D

2

)

=

[min{

a

1

,

a

2

},

min{

b

1

,

b

2

}]

where

D

1

D

2

a

1

a

2 and

b

1

b

2. Similarly we can define ≥ and = .

Definition 3.1. An interval-valued IFS A=(X,

µ

~ ,

ν

~)is called interval-valued intuitionistic fuzzy BRK-ideal of BRK-algebra X if it satisfies the following

(IVS 1)

µ

~A(0)≥

µ

~A(x) (IVS 2) ν~A(0)≤ν~A(x)

(IVS 3)

µ

~A(0∗x)≥min{

µ

~A(0∗(xy)),

µ

~A(0∗y)}

(IVS 4)

ν

~A(0∗x)≤max{

ν

~A(0∗(xy)),

ν

~A(0∗y)}, for all x,yX.

Example 3.2. Consider a BRK-algebra X ={0,a,b,c} with following table:

∗ 0 a b c

0 0 a 0 a

a a 0 a 0

B b a 0 a

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14

Let A be an interval-valued intuitionistic fuzzy set in X by

   

 

=

[0.5,0.6] [0.2,0.4]

c b

[0.8,0.9] [0.8,0.9]

a 0

~

A

µ

   

 

=

[0.5,0.7] [0.6,0.8]

c b

[0.1,0.2] [0.1,0.2]

a 0

A

λ

Then routine calculations give that A is an interval-valued intuitionistic fuzzy BRK-ideal of X.

Definition 3.3. (Homomorphism of BRK-algebra). Let (X,∗,0) and (Y,∗′,0′) be BRK-algebras. A mapping f :XY is said to be a homomorphism if

) ( ) ( )

(x y f x f y

f ∗ = ∗′ , for all x,yX . For any interval-valued intuitionistic fuzzy set B=(Y,

µ

~B ,

ν

~B)in Y we define a new interval-valued fB=(X,f

µ

~B, f

ν

~B)in X, by

)) ( ( ~

~ f x

f

µ

B =

µ

B and f

ν

~B =

ν

~B(f(x)) for all xX .

Proposition 3.4. [3] Let (X,∗,0) and (Y,∗′,0′) be BRK-algebras, and a mapping Y

X

f : → be a homomorphism of BRK-algebras, then the kernel of f denoted by( )

ker( f ) is a BRK-ideal.

Theorem 3.5. Let (X,∗,0) and (Y,∗′,0′) be BRK-algebras. An onto homomorphic image of an i-v intuitionistic fuzzy BRK-ideal of X is also an i-v intuitionistic fuzzy BRK-ideal of Y.

Proof: Let f :XY be an onto homomorphism of BRK-algebras. Suppose )

~ , ~ , (Y A A

A=

µ

ν

is the image of an i-v intuitionistic fuzzy BRK-ideal )

~ , ~ ,

(X f A f A

fA=

µ

ν

of X. We have to prove thatA=(Y,

µ

~A,

ν

~A)is an i-v intuitionistic fuzzy BRK-ideal of Y.

Since f :XY is onto, then for any x′,y′∈Y there exist x,yX such that

y y f x x

f( )= ′ and ( )= ′ also we consider f(0)=0′. Then we have ) ( ~ )) ( ( ~ ) ( ~ ) 0 ( ~ )) 0 ( ( ~ ) 0 (

~ f f f x f x x

A A

A A

A

A ′ =

µ

=

µ

µ

=

µ

=

µ

µ

And

) ( ~ )) ( ( ~ ) ( ~ ) 0 ( ~ )) 0 ( ( ~ ) 0 (

~ f f f x f x x

A A

A A

A

A ′ =

ν

=

ν

ν

=

ν

=

ν

ν

Also

)} 0

( ~ )), (

0 ( ~ min{

))} ( ) 0 ( ( ~ ))), ( ) ( ( ) 0 ( ( ~ min{

))} ( ) 0 ( ( ~ )), ( ) 0 ( ( ~ min{

))} 0 ( ( ~ ))), (

0 ( ( ~ min{

)} 0 ( ~ )), ( 0 ( ~ min{ )

0 ( ~

)) 0 ( ( ~ )) ( ) 0 ( ( ~ ) 0 ( ~

y y

x

y f f

y f x f f

y f f

y x f f

y f y

x f

y f

y x f

x f

x f x

f f

x

A A

A A

A A

A A

A A

A

A A

A

′ ∗′ ′ ′

∗′ ′ ∗′ ′ =

∗′ ∗′

∗′ =

∗′ ∗

∗′ =

∗ ∗

∗ =

∗ ∗

∗ ≥

∗ =

∗ =

∗′ =

′ ∗′ ′

µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

(5)

15 )} 0 ( ~ )), ( 0 ( ~ max{ ))} 0 ( ( ~ ))), ( ) ( ( ) 0 ( ( ~ max{ ))} ( ) 0 ( ( ~ )), ( ) 0 ( ( ~ max{ ))} 0 ( ( ~ ))), ( 0 ( ( ~ max{ )} 0 ( ~ )), ( 0 ( ~ max{ ) 0 ( ~ )) 0 ( ( ~ )) ( ) 0 ( ( ~ ) 0 ( ~ y y x y f y f x f f y f f y x f f y f y x f y f y x f x f x f x f f x A A A A A A A A A A A A A A ′ ∗′ ′ ′ ∗′ ′ ∗′ ′ = ∗ ∗′ ∗′ = ∗′ ∗ ∗′ = ∗ ∗ ∗ = ∗ ∗ ∗ ≤ ∗ = ∗ = ∗′ = ′ ∗′ ′

ν

ν

ν

ν

ν

ν

ν

ν

ν

ν

ν

ν

ν

ν

Hence,A=(Y,

µ

~A,

ν

~A) is an i-v intuitionistic fuzzy BRK-ideal of Y.

Theorem 3.6. Let (X,∗,0) and (Y,∗′,0′) be BRK-algebras. An onto homomorphic inverse image of an i-v intuitionistic fuzzy BRK-ideal of Y is also an i-v intuitionistic fuzzy BRK-ideal of X.

Proof: Let f :XY be an onto homomorphism of BRK-algebras. Suppose

) ~ , ~ ,

(X f A f A

fA=

µ

λ

is the inverse image of an i-v intuitionistic fuzzy BRK-ideal

) , ~ , (Y A A

A=

µ

λ

of Y. We have to prove that fA=(X, f

µ

~A, f

λ

~A) is an i-v intuitionistic fuzzy BRK-ideal of X. For any x′,y′∈Y there exist x,yX such that

y y f x x

f( )= ′ and ( )= ′, also we consider f(0)=0′. Then we have ) ( ~ )) ( ( ~ ) ( ~ ) 0 ( ~ )) 0 ( ( ~ ) 0 (

~ f x f x f x

f

µ

A =

µ

A =

µ

A ′ ≥

µ

A ′ =

µ

A =

µ

A

and ) ( ~ )) ( ( ~ ) ( ~ ) 0 ( ~ )) 0 ( ( ~ ) 0 ( ~ x f x f x f

f

λ

A =

λ

A =

λ

A ′ ≤

λ

A ′ =

λ

A =

λ

A

(6)

16

Hence, fA=(X, f

µ

~A, f

λ

~A) is an i-v intuitionistic fuzzy BRK-ideal of X.

Definition 3.7. An interval-valued intuitionistic subset A=(X,

µ

~,

ν

~)of X has “sup” and “inf ” properties if for any subset K of X, there existm,nK such that

) ( ~ sup ) (

~ m m

A K m

A

µ

µ

= and

~

(

n

)

inf

~

(

n

)

K n

ν

ν

=

.

Definition 3.8. Let f be a mapping from a set X to a set Y. If A=(X,

µ

~A,

ν

~A) is an interval-valued intuitionistic fuzzy subset X , then the interval-valued intuitionistic fuzzy

subset B=(Y,

µ

~B,

λ

~B) of Y is define by

  

= =

= =

, otherwise

0

} ) ( , { ) ( if ), ( ~ sup )

( ~ ) ( ~

1 )

(

1

µ

1

µ

φ

µ

A f y B y x f y A x f y x X f x y

  

= =

= =

− ∈

− −

, otherwise

1

} ) ( , { ) ( if ), ( ~ inf )

( ~ ) ( ~

1 )

(

1

λ

1

ν

φ

ν

f y y x f y A x f y x X f x y

B A

for all yYis called the image of A=(X,

µ

~A ,

ν

~A) under f.

Similarly, if B=(Y,

µ

~B,

λ

~B) is an interval-valued intuitionistic fuzzy subset of Y,

then the interval-valued intuitionistic fuzzy subset A=(X,

µ

~A,

ν

~A) of X defined by

) ( ~ )) ( (

~ f x x

A

B

µ

µ

= and

ν

~A(f(x))=

λ

~B(x)for all xX , is said to be the inverse image of B=(Y,

µ

~B,

λ

~B) under f.

In what follows, let X denote a BRK-algebra unless otherwise specified, we begin with the following theorem.

Theorem 3.9. An onto homomorphic image of an interval-valued intuitionistic fuzzy

BRK-ideal of X with sup and inf property is also an interval-valued intuitionistic fuzzy BRK-ideal.

Proof: Let f :XY be an onto homomorphism of BRK-algebras(X;∗,0)and

) 0 , ;

(Y ∗ ′ , A=(X,

µ

~A,

ν

~A) be an interval-valued intuitionistic fuzzy BRK-ideal of X

with sup and inf properties and B=(X′,

µ

~B,

λ

~B)is the image of A=(X,

µ

~A,

ν

~A)

under f. By definition 3.8 we get ~ ( ) ~ ( ) sup ~ ( )

) ( 1

1

x x

f

x A

y f x A

B

µ

µ

µ

=

=

′ (≠

φ

) and

) ( ~ inf ) ( ~ ) ( ~

) ( 1

1 x

x f

x A

y f x A

B

ν

ν

λ

=

=

′ (≠

φ

) for all x′∈Y , withsup

φ

=0and

1 inf

φ

= .

Since A=(X,

µ

~A,

ν

~A) is an interval-valued intuitionistic fuzzy BRK-ideal of X,

we have

µ

~A(0)≥

µ

~A(x) and ( ) ~ ) 0 (

~ x

A

A

ν

ν

≤ for allxX . Consider that

0

f

−1

(

0

)

.

Therefore, f A t A A x x X

f t A

B ′ = ′ = = ≥ ∈

′ ∈ −

− ( ) for all

~ ) 0 ( ~ ) ( ~ sup ) 0 ( ~ ) 0 ( ~

) 0 ( 1

1

µ

µ

µ

µ

(7)

17 . any for ) ( ~ ) ( ~ inf ) 0 ( ~ that implies which all for ) ( ~ ) 0 ( ~ ) ( ~ inf ) 0 ( ~ ) 0 ( ~ and , any for ) ( ~ ) ( ~ sup ) 0 ( ~ that implies which ) ( ) 0 ( 1 ) ( 1 1 1 Y x x t X x x t f Y x x t B A y f t B A A A f t A B B A y f t B ∈ ′ ′ = ≤ ′ ∈ ≤ = = ′ = ′ ∈ ′ ′ = ≥ ′ − − − ∈ ′ ∈ − ∈

λ

ν

λ

ν

ν

ν

ν

λ

µ

µ

µ

For any x′,y′,z′∈Y,let ( ), ( ),and00 1(0)

1 0 1

0∈ ′ ∈ ′ ∈ ′

− − − f y f y x f

x be such that

) ( ~ sup ) 0 ( ~ ) 0 ( 0 0 1 t x A x f t A

µ

µ

′ ∗ ′ ∈ − =

∗ , ~ (0 ) sup ~ ( )

) 0 ( 0 0 1 t y A y f t A

µ

µ

′ ∗ ′ ∈ − =

∗ and

. ) ( ~ sup )) ( 0 ( ~ sup )) ( 0 ( ~ ))} ( 0 ( { ~ )) ( 0 ( ~ )) ( 0 ( 0 0 0 )) ( 0 ( )) ( 0 ( 0 0 0 0 0 0 1 1 0 0 0 t y x y x y x f y x A y x f t A y x f y x B B A

µ

µ

µ

µ

µ

′ ∗ ′ ∗ ′ ∈ ′ ∗ ′ ∗ ′ ∈ ∗ ∗ − ∗ ∗ = − = ′ ∗ ′ ∗ ′ = ∗ ∗ = ∗ ∗ Then )}. 0 ( ~ )), ( 0 ( ~ min{ )} ( ~ sup , ) ( ~ sup { min )} 0 ( ~ )), ( 0 ( ~ min{ ) 0 ( ~ ) ( ~ sup ) 0 ( ~ ) 0 ( ) ( 0 ( 0 0 0 0 0 0 0 ) 0 ( 1 y y x r t µ t µ r y y x r x t x B B A y t A y x t A A A A x f t B ′ ∗ ′ ′ ∗ ′ ∗ ′ = = ∗ ∗ ∗ ≥ ∗ = = ′ ∗ ′ ′ ∗ ′ ∈ ′ ∗ ′ ∗ ′ ∈ ′ ∗ ′ ∈ −

µ

µ

µ

µ

µ

µ

µ

Also ) ( ~ inf ) 0 ( ~ ) 0 ( 0

0 x 1 A t x f t A

ν

ν

′ ∗ ′ ∈ − =

∗ , ~ (0 ) inf ~ ( )

) 0 ( 0

0 y 1 A t

y f t A

ν

ν

′ ∗ ′ ∈ − =

∗ and

. ) ( ~ inf )) ( 0 ( ~ inf )) ( 0 ( ~ ))} ( 0 ( { ~ )) ( 0 ( ~ )) ( 0 ( 0 0 0 )) ( 0 ( )) ( 0 ( 0 0 0 0 0 0 1 1 0 0 0 t y x y x y x f y x A y x f t A y x f y x B B A

ν

ν

λ

λ

ν

′ ∗ ′ ∗ ′ ∈ ′ ∗ ′ ∗ ′ ∈ ∗ ∗ − ∗ ∗ = − = ′ ∗ ′ ∗ ′ = ∗ ∗ = ∗ ∗ Then )}. 0 ( ~ )), ( 0 ( ~ max{ )} ( ~ inf , ) ( ~ inf max{ )} 0 ( ~ )), ( 0 ( ~ max{ ) 0 ( ~ ) ( ~ inf ) 0 ( ~ ) 0 ( ) ( 0 ( 0 0 0 0 0 0 0 ) 0 ( 1 y y x t t y y x x t x B B A y t A y x t A A A A x f t B ′ ∗ ′ ′ ∗ ′ ∗ ′ = = ∗ ∗ ∗ ≤ ∗ = = ′ ∗ ′ ′ ∗ ′ ∈ ′ ∗ ′ ∗ ′ ∈ ′ ∗ ′ ∈ −

λ

λ

ν

ν

ν

ν

ν

ν

λ

Hence B=(X′,

µ

~B,

λ

~B)is an interval-valued intuitionistic fuzzy BRK-ideal of Y.

Definition 3.10. Let A=(X,

µ

~A,

ν

~A) be an interval-valued intuitionistic fuzzy

BRK-ideal of X and let (

ω

~1,

ω

~2)∈D[0,1] be such that

ω

~1+

ω

~2 ≤[1,1]. A non empty set } ~ ) ( ~ , ~ ) ( ~ | { : ) ~ , ~ ; ( ~ 2 A 1 2

1

ω

µ

ω

ν

ω

ω

= xX xx

A

L A is called an interval-valued

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18

Theorem 3.11. Let A=(X,

µ

~A,

ν

~A)be an interval-valued intuitionistic fuzzy set in X. Then A is an interval-valued intuitionistic fuzzy BRK-ideal of X if and only if the non empty set L~(A;[

ω

~1,

ω

~2]) is a BRK-ideal of X , for every ) [0,1]

~ , ~

(

ω

1

ω

2D .

Proof: Assume that A=(X,

µ

~A,

ν

~A) is an interval-valued intuitionistic fuzzy

BRK-ideal of X and let (

ω

~1,

ω

~2)∈D[0,1] be such that ( ;~ ,~ ) ~

2 1

ω

ω

A L

x∈ . Then

1

~ ) ( ~ ) 0 (

~

µ

ω

µ

AA x ≥ , and

ν

~A(0)≤

ν

~A(x)≤

ω

~2. Therefore 0∈L~(A;

ω

~1,

ω

~2). Let x,yX be such that (0∗(xy)), (0∗y)∈L~(A;

ω

~1,

ω

~2). Then

1

~ )) ( 0 (

~

ω

µ

Axy ≥ , 1

~ ) 0 (

~

ω

µ

Ay ≥ , 2

~ )) ( 0 (

~

ω

ν

Axy ≤ and 2

~ ) 0 (

~

ω

ν

Ay

therefore

)} 0 ( ~ )), ( 0 ( ~ min{ )

0 (

~ x r x y y

A A

A ∗ ≥ µ ∗ ∗ µ ∗

µ ≥ 1 1 1

~ } ~ , ~

min{

ω

ω

=

ω

r

)} 0 ( ~ )), ( 0 ( ~ max{ )

0 (

~ x x y y

A A

A ∗ ≤ ν ∗ ∗ ν ∗

ν ≤ 1 2 2

~ } ~ , ~

max{

ω

ω

=

ω

and so 0∗xL~(A;

ω

~1,

ω

~2). Thus L~(A;

ω

~1,

ω

~2)is a BRK-ideal of X.

Conversely, assume that L~(A;

ω

~1,

ω

~2)(≠

φ

)is a BRK-ideal of X for every ]

1 , 0 [ ) ~ , ~

(

ω

1

ω

2D .

That is

µ

~A(x)=

ω

~1and

ν

~A(x)=

ω

~2,for all xX , since 0∈L~(A;

ω

~1,

ω

~2), therefore )

( ~ ~ ) 0 ( ~

1 A x

A

ω

µ

µ

≥ = , and

ν

~A(0)≤

ω

~2 =

ν

~A(x). Suppose that there existx0,y0,z0∈X such that

)} 0 ( ~ )), (

0 ( ~ min{ )

0 ( ~

0 0

0

0 r x y y

x A A

A ∗ < µ ∗ ∗ µ ∗

µ ,

and ν~A(0∗x0)>max{ν~A(0∗(x0y0)),ν~A(0∗y0)}.

Let µ~A(0∗(x0y0))=γ~1, µ~A(0∗y0)=γ~2 and µ~A(0∗x0)=ω~1. Then

} ~ , ~ min{ ~

2 1

1

γ

γ

ω

<r .

Taking 21[~ (0 0) min{~ (0 ( 0 0)),~ (0 0)}]

1= µAx +r µAxy µAy

π

=2(~1 min{~1,~2}). 1

ω

+r

γ

γ

It follows that

>

} ~ , ~ min{

γ

1

γ

2

r 2(~1 min{~1,~2}) ~1 ~ (0 0)

1

1 =

ω

+r

γ

γ

>

ω

=

µ

Ax

π

.

Also

Let ν~A(0∗(x0y0))=γ~3, ν~A(0∗y0)=γ~4 and ν~A(0∗x0)=ω~2. Then

} ~ , ~ max{ ~

4 3

2

γ

γ

ω

> .

Taking 21[~ (0 0) max{~ (0 ( 0 0)),~ (0 0)}]

2= νAx + νAxy νAy

π

= }). ~ , ~ max{ ~

( 2 3 4

2

1

ω

+

γ

γ

It follows that

<

} ~ , ~

max{

γ

3

γ

4 2(~2 max{~3,~4}) ~2 ~ (0 0) 1

2 =

ω

+

γ

γ

<

ω

=

ν

Ax

π

.

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19

1 2 1 1

0 0

~ } ~ , ~ min{ ~

)) (

0 (

~ γ γ γ ω

µAxy = ≥r > and µ~A(0∗y0)=γ~2rmin{γ~1,γ~2}>ω~1. Also

2 4 3 3

0 0

~ } ~ , ~ max{ ~

)) (

0 (

~ γ γ γ ω

νAxy = ≤ < and ν~A(0∗y0)=γ~4≤max{γ~3,γ~4}<ω~2, and so

0 0 0 )) ,0

( 0

( ∗ xyyL~(A;

ω

~1,

ω

~2). It contradicts that L~(A;

ω

~1,

ω

~2)is a BRK-ideal of X. Hence

)} 0 ( ~ )), ( 0 ( ~ min{ )

0 (

~ x r x y y

A A

A ∗ ≥ µ ∗ ∗ µ ∗

µ , and

)} 0 ( ~ )), ( 0 ( ~ max{ )

0 (

~ x x y y

A A

A ∗ ≤

ν

∗ ∗

ν

ν

, for allx,yX.

Therefore L~(A;

ω

~1,

ω

~2) is an interval-valued intuitionistic fuzzy BRK-ideal of X.

Theorem 3.12. Let Y be a subset of X and let A=(X,

µ

~A,

ν

~A) be an interval-valued intuitionistic fuzzy set on X defined by

  

=

∈ =

= •

, otherwise

] 0 , 0 [ ~

if ] , [ ~ ) (

~ 1 2

α

α

α

α

µ

A x x Y

  

=

∈ =

= •

. otherwise

] 0 , 0 [

~ [ , ] if ~

) (

~ 1 2

β

β

β

β

ν

A x x Y

Where α1212∈(0,1] with α12, β12 and

α

i +

β

i ≤1 for i=1, 2. If Y is a BRK-ideal of X, then A is an interval-valued intuitionistic fuzzy BRK-ideal of X.

Proof: By theorem (3.11) we consider that Y=L~(A;

α

~,

β

~). We show that A is an interval-valued intuitionistic fuzzy BRK-ideal of X.

Since Y is a BRK-ideal then 0∈Y, ) ( ~ ] , [ ) 0 ( ~

2

1 A x

A

α

α

µ

µ

≥ = , and

ν

~A(0)≤[

β

1,

β

2]=

ν

~A(x). Let x,yX.If (0∗(xy)),(0∗y)∈Y,then(0∗x)∈Y, and so

= ≥

=

∗ ) [ , ] min{[ , ],[ , ]} 0

( ~

2 1 2 1 2

1 α α α α α

α

µA x r rmin{µ~A(0∗(x0y0)),µ~A(0∗y)},

= ≤

=

∗ ) [ , ] max{[ , ],[ , ]} 0

( ~

2 1 2 1 2

1 β β β β β

β

νA x (0 )}

~ )), (

0 ( ~

max{νAx0y0 νAy .

If (0∗(xy)),(0∗y)∉Y, then(0∗x)∉Y, and so

)} 0 ( ~ ~ )) ( 0 (

~ x y y

A

A ∗ ∗ =α• =µ ∗

µ and ν~A(0∗(xy))=β~ =ν~A(0∗y)}

} ~ , ~ min{ ~

) 0 ( ~

• • • ≥

=

∗ α α α

µA x r =rmin{µ~A(0∗(xy)),µ~A(0∗y)},

} ~ , ~ max{ ~

) 0 ( ~

• • • ≤

=

∗ β β β

νA x = max{ν~A(0∗(xy)),ν~A(0∗y)}. If (0∗(xy))∈Y and (0∗y)∉Y, then(0∗x)∉Y, and so

} ~ , ~ min{ ~

) 0 ( ~

• • ≥

=

∗ α α α

µA x r = rmin{µ~A(0∗(xy)),µ~A(0∗y)},

} ~ , ~ max{ ~

) 0 ( ~

• • ≤

=

∗ β β β

νA x = max{ν~A(0∗(xy)),ν~A(0∗y)}

Similarly for the case (0∗(xy))∉Y and (0∗y)∈Ywe get )}

0 ( ~ )), ( 0 ( ~ min{ )

0 (

~ x r x y y

A A

A ∗ ≥ µ ∗ ∗ µ ∗

µ , and

)} 0 ( ~ )), ( 0 ( ~ max{ )

0 (

~ x x y y

A A

A ∗ ≤ ν ∗ ∗ ν ∗

ν

(10)

20

Theorem 3.13. Let Y be a subset of X and let A=(X,

µ

~A,

ν

~A) be an interval-valued intuitionistic fuzzy set on X defined by

  

=

∈ =

= •

otherwise

] 0 , 0 [ ~

if ] , [ ~ ) (

~ 1 2

α

α

α

α

µ

A x x Y

  

=

∈ =

= •

. otherwise

] 0 , 0 [

~ [ , ] if ~

) (

~ 1 2

β

β

β

β

ν

A x x Y

where α1,α2,β1,β2∈(0,1] with α1<α2, β1<β2 and

α

i +

β

i ≤1 for i=1, 2. If A is an interval-valued intuitionistic fuzzy BRK-ideal of X, then Y is a BRK-ideal of X.

Proof: Assume that A is an interval-valued intuitionistic fuzzy BRK-ideal of X, let

Y

x∈ µ~A(0)≥µ~A(x)=[α1,α2]and ( ) [ , ]

~ ) 0 ( ~

2 1

β

β

ν

ν

AA x = then0∈Y. If (0∗(xy)), (0∗y)∈Y, then µ~A(0∗(xy))=[

α

1,

α

2]=µ~A(0∗y), and so

)} 0 ( ~ )), ( 0 ( ~ min{ )

0 (

~ x r x y y

A A

A ∗ ≥ µ ∗ ∗ µ ∗

µ = rmin{[α1,α2],[α1,α2]}=[α1,α2], also

)} 0 ( ~ )), ( 0 ( ~ max{ )

0 (

~ x x y y

A A

A ∗ ≤ ν ∗ ∗ ν ∗

ν =max{[β12],[β12]}=[β12]. This implies that 0∗xY

Hence, Y is BRK-ideal of X.

4. Conclusions

In this paper, we have introduced the concept of Intuitionistic Fuzzy ideal of BRK-algebra with Interval-valued Membership and Non Membership Functions and studied their properties. In our future work, we introduce the concept of (∈,∈∨q)-Fuzzy BRK - ideal of BRK-algebra . I hope this work would serve as a foundation for further studies on the structure of BRK-algebras.

REFERENCES

1. K.T.Atanassov, Intuitionistic fuzzy sets, VIIITKRs Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science 1697/84), 1983 (in Bulgarian).

2. K.T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986) 87-96. 3. K.T.Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Studies in

fuzziness and soft computing, vol. 35, Heidelberg, New York, Physica-Verl., 1999. 4. K.T.Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets

and Systems, 31(3) (1989) 343 - 349.

5. R.K.Bandaru, On BRK-algebras, Int. J. Math. Mathematical Sci., 2012 (2012), Article ID952654, 12 pp.

6. T.Bej and M.Pal, Doubt Atanassov's intuitionistic fuzzy Sub-implicative ideals in BCI-algebras, International Journal of Computational Intelligence Systems, 8 (2) (2015) 240-249.

7. O.R.Elgendy, Fuzzy BRK-ideal of BRK-algebra, JP Journal of Algebra, Number Theory and Applications, 36 (3) (2015) 231-240.

8. O.R.Elgendy, Anti Fuzzy BRK-ideal of BRK-algebra, British J. math. and Comp. Sci., 10 (6) (2015) 1-9.

(11)

21

10. Q.P.Hu and X.Li, On BCH-algebras, Math. Sem. Notes Kobe Univ., 11 (2) (1983) 313-320.

11. Y.Imai and K.Iséki, On axiom systems of propositional calculi, XIV. Proc. Japan Academy. 42 (1966) 19 -22.

12. K.Iséki, An algebra related with a propositional calculus, Proc. Japan Acad., 42 (1966) 26-29.

13. C.B.Kim and H.S.Kim, On BG-algebras, Math. J., 36 (1992) 15 - 20. 14. J.Neggers and H.S.Kim, On B-algebras, Math. Vensik , 54 (2002) 21 - 29.

15. J.Neggers, S.S.Ahn and H.S.Kim, On Q-algebras, Int. J. Math. Sci., 27 (12) (2001) 749 -757.

16. T.Senapati, M.Bhowmik, M.Pal, Interval-valued intuitionistic fuzzy BG-subalgebras, The Journal of Fuzzy Mathematics, 20(3) (2012) 707-720.

17. T.Senapati, M.Bhowmik, M.Pal, Atanassov’s intuitionistic fuzzy translations of intuitionistic fuzzy H-ideals in BCK/BCI-algebras, Notes on Intuitionistic Fuzzy Sets, 19 (1) (2013) 32-47.

18. A.Walendziak, On BF- algebras, Math. Slovaca., 57 (2) (2007) 119 -128.

19. L.A.Zadeh, The concept of a lingusistic variable and its application to approximate reasoning, Information Sciences, 8 (1975) 199- 249.

References

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