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
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) (x∗y)∗x=0∗y, for all x,y∈X.
In a BRK-algebra X, a partially ordered relation ≤ can be defined by x≤ y if and only if x∗y=0.
Definition 2.2 [5]. If (X;∗,0)is a BRK-algebra, the following conditions hold: (1) x∗x=0,
(2) x∗y =0⇒ 0∗x=0∗y,
(3) 0∗(x∗y)=(0∗x)∗(0∗y), for all x,y∈X
Definition 2.3. A non empty subset S of a BRK-algebra X is said to be BRK-subalgebra
of X, if x,y∈S , implies x∗y∈S.
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∗(x∗y)∈Iand 0∗y∈I imply 0∗x∈I for all x,y∈X .
13 (FI1) µ(0)≥ µ(x),
(FI2) µ(0∗x)≥min{µ(0∗(x∗ y)),µ(0∗y)},for all x,y∈X .
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
~):X →D[0,1]×D[0,1] is called an interval-valued intuitionisticfuzzy set (i-v IF set, briefly) in X if 0≤
µ
A• +ν
•A ≤1 and 0≤ + ≤1• • • •
A A
ν
µ
for allX
x∈ . Where
µ
•A(x),•
A
ν
(x),µ
•A•(x) and• •
A
ν
(x) are fuzzy sets of X such thatµ
•A ≤• •
A
µ
andν
A• ≤ν
•A•. A•• ={(x ,µ
•A•,ν
A••)|x∈X} andA• ={(x,µ
A• ,ν
A•)|x∈X} 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)] :X →D[0,1]• •
•
ν
ν
ν
denote the degree membership and degree of non-membership for allx∈X 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]
andD
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 andb
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∗(x∗y)),µ
~A(0∗y)}(IVS 4)
ν
~A(0∗x)≤max{ν
~A(0∗(x∗y)),ν
~A(0∗y)}, for all x,y∈ X.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
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 :X →Y is said to be a homomorphism if
) ( ) ( )
(x y f x f y
f ∗ = ∗′ , for all x,y∈X . For any interval-valued intuitionistic fuzzy set B=(Y,
µ
~B ,ν
~B)in Y we define a new interval-valued fB=(X,fµ
~B, fν
~B)in X, by)) ( ( ~
~ f x
f
µ
B =µ
B and fν
~B =ν
~B(f(x)) for all x∈ X .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 :X →Y 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 :X →Y is onto, then for any x′,y′∈Y there exist x,y∈X 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
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 :X →Y 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, fµ
~A, fλ
~A) is an i-v intuitionistic fuzzy BRK-ideal of X. For any x′,y′∈Y there exist x,y∈X such thaty 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
Aand ) ( ~ )) ( ( ~ ) ( ~ ) 0 ( ~ )) 0 ( ( ~ ) 0 ( ~ x f x f x f
f
λ
A =λ
A =λ
A ′ ≤λ
A ′ =λ
A =λ
A16
Hence, fA=(X, f
µ
~A, fλ
~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,n∈K 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 fuzzysubset B=(Y,
µ
~B,λ
~B) of Y is define by
= ∈ = ≠
= =
−
∈
− −
, otherwise
0
} ) ( , { ) ( if ), ( ~ sup )
( ~ ) ( ~
1 )
(
1
µ
1µ
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
1ν
φ
ν
f y y x f y A x f y x X f x yB A
for all y∈Yis 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
= andν
~A(f(x))=λ
~B(x)for all x∈X , 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 :X →Y 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 Xwith 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 , withsupφ
=0and1 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 allx∈X . Consider that0
∈
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
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 ( 00 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 fuzzyBRK-ideal of X and let (
ω
~1,ω
~2)∈D[0,1] be such thatω
~1+ω
~2 ≤[1,1]. A non empty set } ~ ) ( ~ , ~ ) ( ~ | { : ) ~ , ~ ; ( ~ 2 A 1 21
ω
µ
ω
ν
ω
ω
= x∈X x ≥ x ≤A
L A is called an interval-valued
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
1ω
2 ∈D .Proof: Assume that A=(X,
µ
~A,ν
~A) is an interval-valued intuitionistic fuzzyBRK-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
A ≥ A x ≥ , andν
~A(0)≤ν
~A(x)≤ω
~2. Therefore 0∈L~(A;ω
~1,ω
~2). Let x,y∈X be such that (0∗(x∗y)), (0∗y)∈L~(A;ω
~1,ω
~2). Then1
~ )) ( 0 (
~
ω
µ
A ∗ x∗y ≥ , 1~ ) 0 (
~
ω
µ
A ∗y ≥ , 2~ )) ( 0 (
~
ω
ν
A ∗ x∗y ≤ and 2~ ) 0 (
~
ω
ν
A ∗y ≤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∗x∈L~(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
1ω
2 ∈D .That is
µ
~A(x)=ω
~1andν
~A(x)=ω
~2,for all x∈X , since 0∈L~(A;ω
~1,ω
~2), therefore )( ~ ~ ) 0 ( ~
1 A x
A
ω
µ
µ
≥ = , andν
~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∗(x0∗y0)),ν~A(0∗y0)}.
Let µ~A(0∗(x0∗y0))=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= µA ∗x +r µA ∗ x ∗y µA ∗y
π
=2(~1 min{~1,~2}). 1
ω
+rγ
γ
It follows that
>
} ~ , ~ min{
γ
1γ
2r 2(~1 min{~1,~2}) ~1 ~ (0 0)
1
1 =
ω
+rγ
γ
>ω
=µ
A ∗xπ
.Also
Let ν~A(0∗(x0∗y0))=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= νA ∗x + νA ∗ x ∗y νA ∗y
π
= }). ~ , ~ max{ ~
( 2 3 4
2
1
ω
+γ
γ
It follows that
<
} ~ , ~
max{
γ
3γ
4 2(~2 max{~3,~4}) ~2 ~ (0 0) 12 =
ω
+γ
γ
<ω
=ν
A ∗xπ
.19
1 2 1 1
0 0
~ } ~ , ~ min{ ~
)) (
0 (
~ γ γ γ ω
µA ∗ x ∗y = ≥r > and µ~A(0∗y0)=γ~2 ≥rmin{γ~1,γ~2}>ω~1. Also
2 4 3 3
0 0
~ } ~ , ~ max{ ~
)) (
0 (
~ γ γ γ ω
νA ∗ x ∗y = ≤ < and ν~A(0∗y0)=γ~4≤max{γ~3,γ~4}<ω~2, and so
0 0 0 )) ,0
( 0
( ∗ x ∗y ∗y ∈L~(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,y∈X.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 YWhere α1,α2,β1,β2∈(0,1] with α1<α2, β1<β2 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
≥ = , andν
~A(0)≤[β
1,β
2]=ν
~A(x). Let x,y∈X.If (0∗(x∗y)),(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∗(x0∗y0)),µ~A(0∗y)},
= ≤
=
∗ ) [ , ] max{[ , ],[ , ]} 0
( ~
2 1 2 1 2
1 β β β β β
β
νA x (0 )}
~ )), (
0 ( ~
max{νA ∗ x0 ∗y0 νA ∗y .
If (0∗(x∗y)),(0∗y)∉Y, then(0∗x)∉Y, and so
)} 0 ( ~ ~ )) ( 0 (
~ x y y
A
A ∗ ∗ =α• =µ ∗
µ and ν~A(0∗(x∗y))=β~• =ν~A(0∗y)}
} ~ , ~ min{ ~
) 0 ( ~
• • • ≥
=
∗ α α α
µA x r =rmin{µ~A(0∗(x∗y)),µ~A(0∗y)},
} ~ , ~ max{ ~
) 0 ( ~
• • • ≤
=
∗ β β β
νA x = max{ν~A(0∗(x∗y)),ν~A(0∗y)}. If (0∗(x∗y))∈Y and (0∗y)∉Y, then(0∗x)∉Y, and so
} ~ , ~ min{ ~
) 0 ( ~
• • ≥
=
∗ α α α
µA x r = rmin{µ~A(0∗(x∗y)),µ~A(0∗y)},
} ~ , ~ max{ ~
) 0 ( ~
• • ≤
=
∗ β β β
νA x = max{ν~A(0∗(x∗y)),ν~A(0∗y)}
Similarly for the case (0∗(x∗y))∉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
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 Ywhere 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
A ≤ A x = then0∈Y. If (0∗(x∗y)), (0∗y)∈Y, then µ~A(0∗(x∗y))=[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{[β1,β2],[β1,β2]}=[β1,β2]. This implies that 0∗x∈Y
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.
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