International Journal of Scientific & Engineering Research, Volume 5, Issue 4, April-2014 1462
ISSN 2229-5518
1, 2Asst Prof. Department of Mathematics, Sri Vidya Mandir Arts & Science College, Uthangarai, T.N. India.
3M.phil Research Scholar. Dept of Mathematics Sri Vidya Mandir Arts & Science College, Uthangarai, T.N. India.
1e-mail:ragavanshana@gmail.com, 2archanasatish08@gmail.com, 3simkumarcasmaths@gmail.com
(briefly, an i-v IF P-ideal) of a BCI – algebra. Necessary and sufficient conditions for an i-v Intuitionistic Fuzzy
P-idealare stated. Cartesian product of i-v Fuzzy ideals are discussed.
The notion of BCK-algebras was proposed by Imai and Iseki in 1996. In the same year, Iseki [6] introduced
the notion of a BCI-algebra which is a generalization of a BCK-algebra. Since then numerous mathematical papers have been written investigating the algebraic properties of the BCK/BCI-algebras and their relationship with other universal structures including lattices and Boolean algebras. Fuzzy sets were initiated by Zadeh [10]. In [9], Zadeh made an extension of the concept of a Fuzzy set by an interval-valued fuzzy set. This interval-valued fuzzy set is referred to as an i-v fuzzy set. InZadeh also constructed a method of approximate inference using his i-v fuzzy sets. In Birwa’s defined interval valued fuzzy subgroups of Rosenfeld`s nature, and investigated some elementary properties. The idea of “intuitionistic fuzzy set” was first published by Atanassov as a generalization of notion of fuzzy sets. After that many researchers considers the Fuzzifications of ideal and sub algebras in BCK/BCI-algebras. In this paper, using the notion of interval valued fuzzy set, we introduce the concept of an interval-valued intuitionistic fuzzy BCI-algebra of a BCI-algebra, and study some of their properties. Using an i-v level set of i-v intuitionistic fuzzy set, we state a characterization of an intuitionistic fuzzy P-ideal of BCI-algebra. We prove that every intuitionistic fuzzy P-ideal of a BCI-algebra X can be realized as an i-v level P-ideal of an i-v intuitionistic fuzzyP-ideal of X. in connection with the notion of homomorphism, we study how the images and inverse images of i-v intuitionistic fuzzy P-ideal become i-v intuitionistic fuzzy P-ideal.
Let us recall that an algebra (X,*,0) of type (2,0) is called a BCI-algebra if it satisfies the following
conditions:1.((x*y)*(x*z))*(z*y)=0,2.(x*(x*y))*y=0,3.x*x=0,4.x*y=0 and y*x=0 imply x=y,for all x,y,z 𝜖 X.
In a BCI-algebra, we can define a partial ordering”≤” by x≤y if and only if x*y=0.in a BCI-algebra X, the set
M={x𝜖X/0*x=0} is a sub algebra and is called the BCK-part of X. A BCI-algebra X is called proper if X-M≠ɸ.
otherwise it is improper. Moreover, in a BCI-algebra the following conditions hold:
1. (x*y)*z=(x*z)*y, 2.x*0=0, 3. x ≤y imply x*z ≤y*z and z*y ≤z*x, 4. 0*(x*y) = (0*x)*(0*y),
5. 0*(x*y) = (0*x)*(0*y), 6. 0*(0*(x*y)) =0*(y*x), 7. (x *z)*(y*z) ≤x*y
An intuitionistic fuzzy set A in a non-empty set X is an object having the form A= {<x,µA(x),υA(x)>/x𝜖X},Where
the functions µA : X→[0,1] and υA: X→[0,1] denote the degree of the membership and the degree of non
membershipof each element x𝜖 X to the set A respectively, and 0≤ µA (x) +υA(x) ≤ 1 for all x𝜖 X.Such defined
objects are studied by many authors and have many interesting applications not only in the mathematics. For the
sake of simplicity, we shall use the symbol A=[µA, υA] for the intuitionistic fuzzy set A={[ µA(x),υA(x)]/ x𝜖X}.
1.µ(0)≥µ(x), 2. µ(x)≥min {µ(x*y),µ(y)},for all x,y𝜖X.
1. 0𝜖I. 2. (x*z)*(y*z)𝜖I and y𝜖I imply x*z𝜖I.Putting z=0 in(2) then we see that every P- ideal is an ideal.
1.µ(0)≥µ(x), 2. µ(x)≥min {µ ((x*z)*(y*z)),µ(y)}.
Definition 2.5: Let A and B be two fuzzy ideal of BCI algebra X. The fuzzy set A ∩ B
µ A ∩ B is defined by µA ∩ B (x) = min {µA (x), µB (x)}, x ∈ X .
membership function
Definition 2.6: Let A and B be two fuzzy ideal of BCI algebra X. The fuzzy set A ∪ B with membership function
µ A ∪ B is defined by µA∪ B (x) = max {µA (x), µB (x)}, ∀x ∈ X .
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Definition 2.7: Let A and B be two fuzzy ideal of BCI algebra X with membership functionand respectively. A is contained in B if µA (x) ≤ µB (x) , ∀ x ∈ X
defined by µ (x) ≤ (µ (x))m , ∀ x ∈ X
A
Am with membership function µ is
A
Definition 2.9: An IFS A=< X, µA,υA > in a BCI-algebra X is called an intuitionisticfuzzy ideal of X if it satisfies:(F1) µA (0)≥ µA(x) &υA (0)≥ υA(x),(F2) µA(x)≥ min { µA (x*y), µA(y)},
(F3) υA (x) ≤ max {υA(x*y), υA (y)}, for all x,y𝜖X
Definition 2.10: An intuitionistic fuzzy set A=< µA, υA > of a BCI-algebra X is called an intuitionistic fuzzy P-
ideal if it satisfies (F1) and(F4) µA (x)≥min{µA ((x*z)*(y*z)), µA (y)},(F5) υA(y*x)≤max{υA((x*z)*(y*z)), υA (y)},
for all x,y,z 𝜖 X.
𝐿 𝑈
𝐿 𝑈
An interval-valued intuitionistic fuzzy set A defined on X is given byA={(x,[µ𝐴 (x)µ𝐴 (x)],[υ𝐴 (x)υ𝐴 (x)])},∀𝑥𝜖𝑋
𝐿 𝑈
𝐿 𝑈
where µ𝐴 ,µ𝐴
are two membership functions and υ𝐴 ,υ𝐴
are two non-membership functions X such that
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
µ𝐴 ≤µ𝐴 &υ𝐴 ≥υ𝐴 ,∀ x𝜖X. Let �µR A(x)=[µ𝐴 ,µ𝐴 ]&υ�𝐴�(x)=[υ𝐴 ,υ𝐴 ],∀ x𝜖X and let D[0,1]denote the family of all closed
𝐿 𝑈
𝐿 𝑈
subintervals of [0,1].If µ𝐴 (x)=µ𝐴 (x)=c,0≤c≤1 and if υ𝐴 (x)=υ𝐴 (x)=k, 0≤k≤1,then we have �µR A (x)=[c,c]&υ�RA(x)=[k,k]
which we also assume, for the sake of convenience, to belong to D[0,1]. thus µ�R A(x)&υ�RA(x)𝜖[0,1],∀ x𝜖X,and
therefore the i-v IFS a is given by A=[(x,�µR A(x),υ�RA(x))},∀ x𝜖X,where
�µR A(x):X→D[0,1]. Now let us define what is
known as refined minimum, refined maximum of two elements in D[0,1].we also define the symbols” ≤”,”≥” and
“=” in the case of two elements in D[0,1]. Consider two elements D1 :[a1 ,b1 ]and D2 :[a2 ,b2 ]𝜖D[0,1]. Then rmin(D1 ,D2 )=[min{a1 ,a2 },min{b1 ,b2 }],rmax(D1 ,D2 )=[max{a1 ,a2 },max{b1 ,b2 }]D1 ≥D2⇔ Ra1 ≥a2 ,b1 ≥b2 ;D1 ≤D2⇔
R a1 ≤a2 ,b1 ≤b2 and D1 =D2 .
intuitionistic fuzzy P-ideal of X if it satisfies (FI1 )µ�R A(0) ≥�µR A(x),υ�RA(0) ≤υ�RA(x),
(FI2 )�µR A(x)≥r min {�µR A((x*z)*(y*z)),�µR A(y)}, ( FI3 )υ�RA(x) ≤ r max {υ�RA ((x*z)*(y*z)),υ�RA(y)}.
Theorem 3.2Let A be an i-v intuitionistic fuzzyP-ideal of X. if there exists a sequence {xn } in X such that
lim µ A ( xn ) =[1,1], limν A ( xn ) =[0,0] then �µR A(0)=[1,1] andυ�RA(0)=[0,0].
n→∞
n→∞
Proof:Since �µR A(0) ≥�µR A(x)and υ�RA(0) ≤υ�RA(x) for all x𝜖X, we have �µR A(0) ≥µ�R A(xn ) and υ�RA(0) ≤υ�RA(xn ), for every
positive integer n. note that [µ L , µU ] ≥ µ
(0) .[1,1] ≥�µ
(x) ≥µ�
(0) ≥ lim µ
( x ) =[1,1]. [λ L , λU ] ≤ λ
(0)
A A A
R A R A
n→∞ A n
A A A
.[0,0] ≤υ�RA(x) ≤ υ�RA(0)≤ limν A ( xn ) =[0,0].Hence
n→∞
�µR A(0)=[1,1] andυ�RA(0)=[0,0].
𝐿 𝑈
𝐿 𝑈
Lemma3.3:An i-v intuitionistic fuzzy set A= [〈µ𝐴 , µ𝐴 〉,〈υ𝐴 , υ𝐴 〉] in X is an i-v intuitionistic fuzzy P-ideal of X if
𝐿 𝑈
𝐿 𝑈
and only if 〈µ𝐴 , µ𝐴 〉and 〈υ𝐴 , υ𝐴 〉 are intuitionistic fuzzy ideals of X.
𝐿 𝐿 𝑈
𝑈 𝐿 𝐿
𝑈 𝑈
Proof:Since µ𝐴 (0) ≥µ𝐴 (x); µ𝐴 (0) ≥µ𝐴 (x);υ𝐴 (0) ≤ υ𝐴 (𝑥)R and υ𝐴 (0) ≤ υ𝐴 (𝑥)R, Therefore�µR A(0) ≥�µR A(x),υ�RA(0) ≤υ�RA(x).
𝐿 𝑈
𝐿 𝑈
Suppose that 〈µ𝐴 , µ𝐴 〉 and 〈υ𝐴 , υ𝐴 〉are intuitionistic fuzzy ideal of X. let x,y𝜖X, then
𝐿 𝑈
𝐿 𝐿
𝑈 𝑈
�µR A(x)=[µ𝐴 (x),µ𝐴 (x)] ≥[min{µ𝐴 (x*y),µ𝐴 (y)},min{µ𝐴 (x*y),µ𝐴 (y)}]
𝐿 𝑈
𝐿 𝑈
=r min {[µ𝐴 (x*y), µ𝐴 (x*y)],[µ𝐴 (y), µ𝐴 (y)]}
= r min {�µR A(x*y),�µR A(y)} and
𝐿 𝑈
𝐿 𝐿
𝑈 𝑈
υ�RA(x)= [υ𝐴(x),υ𝐴 (x)]≤ [max{υ𝐴 (x*y),υ𝐴 (y)},max{υ𝐴 (x*y),υ𝐴 (y)}]
𝐿 𝐿
=r max {[υ𝐴 (x*y), υ(x*y)],[υ𝐴 (y), υ(y)]}
= r max {υ�RA(x*y),υ�RA(y)}.
Hence A is an i-v intuitionistic fuzzy ideal of X.
Conversely,
Assume that A is an i-v intuitionistic fuzzy ideal of X. for any x,y𝜖X,we have
𝐿 𝑈
[µ𝐴 (x),µ𝐴 (x)]= �µA(x) ≥ r min{[�µRA(x*y),�µR A (y)]}
𝐿 𝑈
𝐿 𝑈
=r min {[µ𝐴 (x*y), µ𝐴 (x*y)],[µ𝐴 (y), µ𝐴 (y)]}
𝐿 𝐿
𝑈 𝑈
= [min {µ𝐴 (x*y),µ𝐴 (y)},min{µ𝐴 (x*y),µ𝐴 (y)}]
𝐿 𝑈
And [υ𝐴 (x),υ𝐴 (x)] =υ�RA(x) ≤ r max{υ�RA(x*y),υ�RA(y)}
𝐿 𝑈
𝐿 𝑈
=r max {[υ𝐴 (x*y), υ𝐴 (x*y)],[υ𝐴 (y), υ𝐴 (y)]}
𝐿 𝐿
𝑈 𝑈
= [max {υ𝐴 (x*y),υ𝐴 (y)},min{υ𝐴 (x*y),υ𝐴 (y)}]
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𝐿 𝐿
𝐿 𝐿
𝐿 𝐿
It follows thatµ𝐴 (x) ≥ min {µ𝐴 (x*y),µ𝐴 (y)},υ𝐴 (x) ≤ max{υ𝐴 (x*y),υ𝐴 (y)}
𝑈 𝑈
𝑈 𝑈
𝑈 𝑈
Andµ𝐴 (x) ≥ min {µ𝐴 (x*y),µ𝐴 (y)}, υ𝐴 (x) ≤max {υ𝐴 (x*y),υ𝐴 (y)}
𝐿 𝑈
𝐿 𝑈
Hence 〈µ𝐴 , µ𝐴 〉and 〈υ𝐴 , υ𝐴 〉 are intuitionistic fuzzy ideals of X.
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
Proof:Let A=[〈µ𝐴 , µ𝐴 〉, 〈υ𝐴 , υ𝐴 〉] be an i-v intuitionistic fuzzy P-ideal of X, where〈µ𝐴 , µ𝐴 〉and 〈υ𝐴 , υ𝐴 〉are
𝐿 𝑈
𝐿 𝑈
intuitionistic fuzzy P-ideal of X. thus 〈µ𝐴 , µ𝐴 〉and 〈υ𝐴 , υ𝐴 〉are intuitionistic fuzzy P-ideals of X. hence by lemma 3.3,
A is i-v intuitionistic fuzzy ideal of X.
algebraof Xif �µR A(x*y)≥r min { �µR A(x), �µRA(y)} and υ�RA (x*y)≤ {υ�RA(x),υ�RB (y)}, for all x,y𝜖X.
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
X.Proof:Let A=[ 〈µ𝐴 , µ𝐴 〉, 〈υ𝐴 , υ𝐴 〉]be an i-v intuitionistic fuzzy P-ideal of X, where 〈µ𝐴 , µ𝐴 〉, and 〈υ𝐴 , υ𝐴 〉 are
𝐿 𝑈
𝐿 𝑈
intuitionistic fuzzy P-ideal of BCI-algebra X. thus 〈µ𝐴 , µ𝐴 〉, and 〈υ𝐴 , υ𝐴 〉are intuitionistic fuzzy subalgebra of X.
Hence, A is i-v intuitionistic fuzzysub algebra of X.
Definition 4.1An intuitionistic fuzzy relation A on any set a is a intuitionistic fuzzy subset A with a membership function ΩR A: X×X→ [0, 1] and non membership function ΨRA: X×X→ [0, 1].
Lemma 4.2Let �µR Aand �µR B be two membership functions and υ�RA andυ�RB be two non membership functions of each x
𝜖X to the i-v subsets A and B, respectively. Then µA × µB is membership function and υA× υB is non membership
function of each element(x,y)𝜖X×X to the set A×B and defined by ( �µR A×�µR B )(x,y)=r min {�µR A(x), �µR B (y)} and
(υ�RA×υ�RB )(x,y)=r max {υ�RA(x),υ�RB (y)}.
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
Definition 4.3Let A= [〈µ𝐴 , µ𝐴 〉, 〈υ𝐴 , υ𝐴 〉]and B=[ 〈µ𝐵 , µ𝐵 〉, 〈υ𝐵 , υ𝐵 〉] be two i-v intuitionistic fuzzy subsets in a set
X. The Cartesian product of A×B is defined byA×B= {((x,y), �µR A×�µR B , υ�RA×υ�RB );∀x,y𝜖X×X}Where
A×B: X×X→D[0,1].
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
𝐿 𝑈
Theorem 4.4.Let A=[〈µ𝐴 , µ𝐴 〉, 〈υ𝐴 , υ𝐴 〉]and B=[ 〈µ𝐵 , µ𝐵 〉, 〈υ𝐵 , υ𝐵 〉] be two i-v intuitionistic fuzzy subsets in a set
X,then A×B is an i-v intuitionistic fuzzy P-ideal of X×X.
𝐿 𝑈
𝐿 𝑈
(�µR A×�µR B ) (0,0)=r min {�µR A(0), �µR B (0) = r min {[µ𝐴 (0),µ𝐴 (0)],[µ𝐵 (0),µ𝐵 (0)]}
𝐿 𝐿
𝑈 𝑈
=[min {µ𝐴 (0),µ𝐵 (0)},min{µ𝐴 (0),µ𝐵 (0)}]
𝐿 𝐿
𝑈 𝑈
≥[min {µ𝐴 (x),µ𝐵 (y)},min{µ𝐴 (x),µ𝐵 (y)}]
𝐿 𝑈
𝐿 𝑈
=r min {[µ𝐴 (x),µ𝐴 (x)],[µ𝐵 (y),µ𝐵 (y)]}
= r min {�µR A(x), �µR B (y)}=( �µR A× �µR B )(x,y)
And(υ�RA×υ�RB )(0,0)=r max {υ�RA (0), υ�RB (0)
𝐿 𝑈
𝐿 𝑈
= r max {[υ𝐴 (0),υ𝐴 (0)],[υ𝐵 (0),υ𝐵 (0)]}
𝐿 𝐿
𝑈 𝑈
=[max {υ𝐴 (0),υ𝐵 (0)},max{υ𝐴 (0),υ𝐵 (0)}]
𝐿 𝐿
𝑈 𝑈
≤[max {υ𝐴 (x),υ𝐵 (y)},max{υ𝐴 (x),υ𝐵 (y)}]
𝐿 𝑈
𝐿 𝑈
=r max {[υ𝐴 (x),υ𝐴 (x)],[υ𝐵 (y),υ𝐵 (y)]}
= r max {υ�RA(x),υ�RB (y) =(υ�RA×υ�RB )(x,y)
Therefore (FI2 ) holds.Now, for all x,y,z𝜖X, we have
Ꞌ Ꞌ
(�µR A×�µR B ) ((x, x ))=r min { µR A(x),µRB (x )}
≥ r min { r min {µ
(( x * z) * ( y * z)), µ
(y)}, r min {µ
(( x1 * z1 ) * (y1* z1 )), µ
(y1 )}}
A A A A
= r min {{ min {µ L
(( x * z) * (y* z)), µ L
(y)}, min {µU
(( x * z) * (y* z)), µU
(y)}},
A A A A
{ min {µ L
(( x1 * z1 ) * (y1* z1 )), µ L
( y1 )}, min {µU
(( x1 * z1 ) * (y1* z1 )), µU
( y1 )}}
B B B B
={ min { min {µ L
(( x * z) * (y* z)), µ L
(( x1 * z1 ) * (y1* z1 ))}, min{µ L
(y), µ L
(y1 )}},
A B A B
min { min {µU
(( x * z) * (y* z)), µU
(( x1 * z1 ) * (y1* z1 ))}, min{µU
(y), µU
(y1 )}}}
A B A B
(( x1 * z1 ) * (y1* z1 ))) A B Ꞌ
=r min {(�µR A×�µR B )(((x*z)*(y*z)),
Ꞌ Ꞌ
,(�µR
×�µR )(y, y )}
Also, (υ�RA×υ�RB ) ((x,x )) =r max { υRA(x),υRB (x )}
≤ r max { r max {ν
(( x * z) * (y* z)),ν
(y)}, r max {ν
(( x1 * z1 ) * (y1* z1 )),ν
(y1 )}}
A A A A
= r max {{ max {ν L
(( x * z) * (y* z)),ν L
(y)}, max {ν U
(( x * z) * (y* z)),ν U
(y)}},
A A A A
{ max {ν L
(( x1 * z1 ) * (y1* z1 )),ν L
( y1 )}, max {ν U
(( x1 * z1 ) * (y1* z1 )),ν U
( y1 )}}
B B B B
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ISSN 2229-5518
={ max { max {ν L
(( x * z) * (y* z)),ν L
(( x1 * z1 ) * (y1* z1 ))}, max{ν L
(y), ν L
(y1 )}},
A B A B
max { max {ν U
(( x * z) * ( y * z)),ν U
(( x1 * z1 ) * (y1* z1 ))}, max{ν U
(y),ν U
(y1 )}}}
A B A B
= r max {(ν
×ν )(((x* z) * (y* z)), ((x1* z1 ) * (y1* z1 )), (ν
×ν )(y, y1 )}
A B A B
Hence A×B is an i-v intuitionistic fuzzy P-ideal of X× X
Definition 4.5:Let�µR B ,υ�RB respectively, be an i-v membership and non membership function of each element x𝜖X to the set B.Then strongest i-v intuitionistic fuzzy set relationon X ,that is a membership function relation �µR A onµ�R B and
non membership function relation υ�RA onυ�RB and µ ,
whose i-v membership and non membership function, of
B ν A �
each element (x,y) 𝜖X×X and defined by µ
(x,
y)=r min{�µR (x),�µR (y)}&
(x,y)=r max{υ�R (x),υ�R (y)}
AB
𝐿 ,
B B B B
�
𝑈 〉, 〈υ𝐿 , υ𝑈 〉] be an i-v subset in a set X, B
the strongest i-v intuitionistic fuzzy
𝐵 µ𝐵
𝐵 𝐵
then
relation on X that is a i-v A on B is AB and defined by, [
L , U ,
L , U ]
A = µ
B
µ ν ν
B B B
L , U ,
L , U
] be an i-v subset in a set X and
L U L U
= µ µ ν ν
B B B B
AB = [
µ , µ
B B
, ν ,ν ]
B B
be the strongest i-v intuitionistic fuzzy relation on X. then B is an i-v intuitionistic P-ideal of X if and only if AB is an i-v intuitionistic fuzzy P-ideal of X×X.
Proof: Let B be an i-v intuitionistic fuzzy a-ideal of X. thenµ�R AB (0,0)=r min{�µR B (0),�µR B (0)}
≥r min{�µR B (x),�µR B (y)}=µ�R AB (x,y) and �υRAB (0,0)=r max{υ�RB (0),υ�RB (0)}≤r max{υ�RB (x),υ�RB (y)}=υ�RAB (x,y) ∀(x,y) 𝜖X×X.
On the other hand µ (x1 ,x2 )=r min {�µR B (x1 ),�µR B (x2 )}
B
≥r min{r min{�µR B ((x1 *z1 )*(y1 *z1 )),�µR B (y1 )},r min{�µR B ((x2 *z2 )*(y2 *z2 )),�µR B (y2 )}}
=r min{r min{�µR B ((x1 *z1 )*(y1 *z1 )),�µR B ((x2 *z2 )*(y2 *z2 ))},r min {�µR B (y1 ),�µR B (y2 )}}
=r min {�µR AB ((x1 *z1 )*(y1 *z1 ), (x2 *z2 )*(y2 *z2 )),�µR AB (y1 , y2 )}
=r min {�µR AB (((x1 ,x2 )*(z1 ,z2 ))*(( y1 ,y2 )*(z1, z2 ))),�µR AB (y1 , y2 )}
Also,ν (x1 , x2 )=r max {υ�RB (x1 ),υ�RB (x2 )}
B
≤ r max{r max{υ�RB ((x1 *z1 )*(y1 *z1 )),υ�RB (y1 )},r max{υ�RB ((x2 *z2 )*(y2 *z2 )),υ�RB (y2 )}}
= r max{r max{υ�RB ((x1 *z1 )*(y1 *z1 )),υ�RB ((x2 *z2 )*(y2 *z2 ))},r max {υ�RB (z1 ),υ�RB (z2 )}}
=r max{υ�RAB ((x1 *z1 )*(y1 *z1 ), (x2 *z2 )*(z2 *y2 )),υ�RAB (y1 , y2 )}
=r max{υ�RAB (((x1 ,x2 )*( z1 ,z2 ))*(( y1 ,y2 )*(z1, z2 ))),υ�RAB (z1 , z2 )}
For all (x1 ,x2 ),(y1 ,y2 ),(z1 ,z2 ) in X×X. hence AB is an i-v intuitionistic fuzzy P-ideal of X×X.
Conversely,
let AB be an i-v intuitionistic fuzzy P-ideal of X×X. then for all (x,x)𝜖X×X.we have
r min {�µR B (0),�µR B (0)}=�µR AB (0,0)≥�µR AB (x,x)= r min{�µR B (x),�µR B (x)}(or)�µR B (0) ≥�µR B (x) and
r max {υ�RB (0),υ�RB (0)}=υ�RAB (0,0)≤υ�RAB (x, x)=rmin{υ�RB (x),�µR B (x)}(or)υ�RB (0)≤υ�RB (x)∀x𝜖X. Now,
Let (x1 ,x2 ),(y1 ,y2 ),(z1 ,z2 ) 𝜖X×X, then
rmin {�µR B (x1 ,x2 )}=�µR AB (x1 , x2 )≥r min {�µR AB (((x1 ,x2 )*((z1 , z2 ))*((y1 ,y2 )*(z1 ,z2 ))),�µR AB (y1 , y2 )}
=r min {�µR AB ((x1 *z1 )*(y1 *z1 ), (x2 *z2 )*(y2 *z2 )),�µR AB (y1 ,y2 )}
=r min {r min {�µR B ((x1 *z1 )*(y1 *z1 )),�µR B (y1 )},r min {�µR AB ((x2 *z2 )*(y2 *z2 )),�µR B (y2 )}}
Also, rmax {υ�RB (x1 ,x2 )} =υ�RAB (x1 , x2 )
≤r max {υ�RAB (((x1 ,x2 )*((z1 , z2 ))*((y1 ,y2 )*(z1 ,z2 ))),υ�RAB (y1 ,y2 )}
=r max {υ�RAB (((x1 *z1 )*(y1 *z1 )), ((x2 *z2 )*(y2 *z2 ))), υ�RAB (y1 ,y2 )}
=r max {r max {υ�RB ((x1 *z1 )*(y1 *z1 )),�µR B (y1 )},r max {υ�RAB ((x2 *z2 )*(y2 *z2 )),υ�RB (y2 )}}
If x2 =y2 =z2 =0, then
r min {�µR B (x1 ),�µR B (0)}≥ r min {r min {�µR B ((x1 *z1 )*(y1 *z1 )),�µR B (y1 )},�µR B (0)} and
r max {υ�RB (x1 ),υ�RB (0)}≥ r max {r max {υ�RB ((x1 *z1 )*(y1 *z1 )), υ�RB (y1 )},υ�RB (0)}
�µR B (x1 )≥r min {�µR B ((x1 *z1 )*(y1 *z1 )), �µR B (y1 )} and
υ�RB (x1 ) ≥r max {υ�RB ((x1 *z1 )*(y1 *z1 )), υ�RB (y1 )}.
Therefore B is i-v intuitionistic fuzzy P-ideal of X.
Theorem 4.8: If�µR A is a i-v intuitionistic fuzzy a-ideal of BCI-algebra X, then µ
ideal of BCI-algebra X
is also i-v intuitionistic fuzzy P-
A
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Proof: For all x, y, z ∈ X
m
m
1. µ A (0) ≥ µ A ( x ) , ν A (0) ≤ν A ( x ).
µ A (0)
≥ µ A ( x ) , ν A (0)
≤ ν A ( x )
m m m m
µ A (0)
≥ µ A ( x )
, ν A (0)
≤ν A ( x ) .
µ m (0) ≥ µ m ( x ) , ν
m (0) ≤ν
m ( x )
∀x∈ X
A A A A
m m
2. µ A ( x) ≥ r min { µ A (( x * z) * (y* z)), µ A (y) }. [µ A ( x)]
≥ [r min { µ A (( x * z) * (y* z)), µ A (y) }]
m m m m
µ A ( x)
≥ r min { µ A (( x * z) * (y* z)), µ A (y) } . µ Am ( x) ≥ r min { µ A (( x * z) * (y* z)) , µ A (y) }
µ ( x) ≥ r min { µ (( x * z) * (y* z)) , µ (y) }
A A A
m m
3.ν A ( x) ≤ r max { µ A (( x * z) * (y* z)), µ A (y) }. [ ν A ( x)]
≤ [r max {ν A (( x * z) * (y* z))ν A (y) }]
ν A ( x)
m ≤ r max { ν (( x * z) * (y* z)),ν (y) }m . ν ( x) ≤ r max {ν (( x * z) * (y* z))m , ν (y)m }
A
ν m ( x) ≤ r max { ν
m (( x * z) * (y* z)) , ν
m (y) }
A A A
Theorem 4.9:If �µR Ais a i-v intuitionistic fuzzy R-ideal of BCI-algebra X, then
P-ideal of BCI-algebra X
Proof: For all x, y, z ∈ X
µ A∩ B isalso a i-v intuitionistic fuzzy
1. µ A (0) ≥ µ A ( x ) , ν A (0) ≤ν A ( x ) and µB (0) ≥ µB ( x ) , ν B (0) ≤ν B ( x )
min { µ A (0) , µB (0)}≥ min { µ A ( x ) , µB ( x )}, min {ν A (0) , ν B (0)}≤ min {ν A ( x ) , ν B ( x )}
µ A∩ B (0) ≥ µ A∩ B ( x ) , ν A∩ B (0) ≤ ν A∩ B ( x )
2. µ A ( x) ≥ r min { µ A (( x * z) * (y* z)), µ A (y) }, µB ( x) ≥ r min { µB (( x * z) * (y* z)), µ A (y) }
{ µ A ( x), µB ( x)} ≥ {r min{ µ A (( x * z) * (y* z)), µ A (y) }, r min { µB (( x * z) * (y* z)), µB (y) }}
min { µ A ( x), µB ( x)}≥ min {r min{ µ A (( x * z) * (y* z)), µ A (y) }, r min { µB (( x * z) * ( y * z)), µB (y) }}
≥ min {r min{ µ A (( x * z) * ( y * z)), µB (( x * z) * (y* z)) }, r min { µ A (y), µB (y) }}
µ A∩ B ( x) ≥ r min { µ A∩ B (( x * z) * (y* z)), µ A∩ B (y) }
3.ν A ( x) ≤ r max {ν A (( x * z) *(y* z)),ν A (y)},ν B ( x) ≤ r max {ν B (( x * z) *(y* z)), ν A (y)}
{ν A ( x),ν B ( x)} ≤
{r max{ν A (( x * z) *(y* z)),ν A (y)}, r max{ν B (( x * z) *(y* z)),ν B (y)}}
If one is contained in the other
min {ν A ( x),ν B ( x)}≤ min {r max{ν A (( x * z) *(y* z)),ν A (y)}, r max{ν B (( x * z) *(y* z)),ν B (y)}}
ν A∩ B ( x) ≤ r max { min{ν A (( x * z) *(y* z)),ν B (( x * z) *(y* z))}, min {ν A (y),ν B (y)}}
ν A∩ B ( x) ≤ r max {ν A∩ B (( x * z) *(y* z)),ν A∩ B (y)}
Theorem 4.10: If �µR Ais a i-v intuitionistic fuzzy P-ideal of BCI-algebra X, then µ A∪ B isalso a i-v intuitionistic fuzzy
P-ideal of BCI-algebra X.
Proof: For all x, y, z ∈ X
1. µ A (0) ≥ µ A ( x ) , ν A (0) ≤ν A ( x ) and µB (0) ≥ µB ( x ) , ν B (0) ≤ν B ( x )
min { µ A (0) , µB (0)}≥ min { µ A ( x ) , µB ( x )}, min {ν A (0) , ν B (0)}≤ min {ν A ( x ) , ν B ( x )}
µ A∪ B (0) ≥ µ A∪ B ( x ) , ν A∪ B (0) ≤ ν A∪ B ( x )
2. µ A ( x) ≥ r min { µ A (( x * z) * (y* z)), µ A (y) },
µB ( x) ≥ r min { µB (( x * z) * (y* z)), µ A (y) }
{ µ A ( x), µB ( x)} ≥ {r min{ µ A (( x * z) * (y* z)), µ A (y) },
r min { µB (( x * z) * (y* z)), µB (y) }}
max { µ A ( x), µB ( x)}≥ max {r min{ µ A (( x * z) * (y* z)), µ A (y) }, r min { µB (( x * z) * (y* z)), µB (y) }}
≥ max {r min{ µ A (( x * z) * (y* z)), µB (( x * z) * (y* z)) },
r max { µ A (y), µB (y) }}
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ISSN 2229-5518
If one is contained in the other
r min{max{ µ A (( x * z) * (y* z)), µB (( x * z) * (y* z)) }, max { µ A (y), µB (y) }}
µ A∪ B ( x) ≥ r min { µ A∪ B (( x * z) * (y* z)), µ A∪ B (y) }
3.ν A ( x) ≤ r max {ν A (( x * z) * (y* z)),ν A (y) },
ν B ( x) ≤ r max {ν B (( x * z) * (y* z)), µ A (y) }
{ν A ( x),ν B ( x)} ≤
{r max{ν A (( x * z) * (y* z)),ν A (y) }, r max{ν B (( x * z) * (y* z)),ν B ( z) }}
max {ν A ( x),ν B ( x)}≤ max {r max{ν A (( x * z) * (y* z)),ν A (y) }, r max{ν B (( x * z) * (y* z)),ν B (y) }}
ν A∪ B ( x) ≤ r max { max{ν A (( x * z) * (y* z)),ν B (( x * z) * (y* z)) }, max {ν A (y),ν B (y) }}
ν A∪ B ( x) ≤ r max {ν A∪ B (( x * z) * (y* z)),ν A∪ B (y) }
References:
[1] K.T Atanassov, intuitionisticfuzzy sets and systems, 20(1986), 87-96
[2] K.T Atanassov, intuitionisticfuzzy sets. Theory and applications, studies in fuzziness and soft computing, 35.Heidelberg; physica-verlag
[3]R.Biswas, Rosenfeld’s fuzzy subgroups with interval-valued membership functions, fuzzy sets and systems 63(1994), no.1,87-90
[4] S.M. Hong, Y.B.Kim and G.I.Kim, fuzzy BCI-sub algebras with interval-valued membership functions, math japonica, 40(2)(1993)199-202 [5] K.Iseki, an algebra related with a propositional calculus, proc, Japan Acad.42 (1966),26-29
[6]H.M.Khalid, B.Ahmad, fuzzy H-ideals in BCI-algebras, fuzzy sets and systems 101(1999)153-158.
[7] L.A.zadeh, the concept of a linguistic variable and its application to approximate reasoning. I, information sci,8(1975),199-249.
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