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International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 3, Issue 4(April 2014), PP.59-64

Intuitionistic Fuzzy a-Ideals of BCI-Algebras with Interval Valued Membership& Non Membership Functions C.Ragavan1, J.SatishKumar2 and M.Balamurugan3 Asst Prof. Department of Mathematics, Sri Vidya Mandir Arts & Science College, Uthangarai, T.N. India

Abstract: The purpose of this paper is to define the notion of an interval valued Intuitionistic Fuzzy a-ideal (briefly, an i-v IF a-ideal) of a BCI – algebra. Necessary and sufficient conditions for an i-v Intuitionistic Fuzzy a-ideal are stated. Cartesian product of i-v Fuzzy ideals are discussed.

I.

INTRODUCTION

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. In Zadeh 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 a-ideal of BCI-algebra. We prove that every intuitionistic fuzzy a-ideal of a BCI-algebra X can be realized as an i-v level a-ideal of an i-v intuitionistic fuzzy a-ideal of X. in connection with the notion of homomorphism, we study how the images and inverse images of i-v intuitionistic fuzzy a-ideal become i-v intuitionistic fussy a-ideal.

II.

PRELIMINARIES

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 đ?œ– X. In a BCI-algebra, we can define a partial orderingâ€?≤â€? by x≤y if and only if x*y=0.in a BCI-algebra X, the set M={xđ?œ–X/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,ÂľA(x),Ď…A(x)>/xđ?œ–X},Where the functions ÂľA : X→[0,1] and Ď…A: X→[0,1] denote the degree of the membership and the degree of non membership of each element xđ?œ– X to the set A respectively, and 0≤ ÂľA(x) +Ď…A(x) ≤ 1 for all xđ?œ– 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= [ÂľA, Ď…A] for the intuitionistic fuzzy set A= {[ÂľA(x),Ď…A(x)]/ xđ?œ–X}. Definition 2.1:A non empty subset I of X is called an ideal of X if it satisfies:1. 0đ?œ–I,2.x*yđ?œ–I and yđ?œ–I ďƒž xđ?œ–I. Definition 2.2: A fuzzy subset Âľ of a BCI-algebra X is called afuzzy ideal of X if it satisfies:1.Âľ (0) ≼¾(x), 2. Âľ(x) ≼min {Âľ(x*y), Âľ(y)}, for all x,yđ?œ–X. Definition 2.3: A non empty subset I of X is called a- ideal of X if it satisfies:1. 0đ?œ–I. 2. (x*z)*(0*y)đ?œ–I and yđ?œ–I imply x*zđ?œ–I.Putting z=0 in(2) then we see that every a- ideal is an ideal. Definition 2.4: A fuzzy set Âľ in a BCI-algebra X is called an fuzzy a- ideal of X if1.Âľ (0) ≼¾(x), 2. Âľ(y*x) ≼min {Âľ ((x*z)*(0*y)), Âľ (z)}. Definition 2.5: Let A and B be two fuzzy ideal of BCI algebra X. The fuzzy set A ďƒ‡ B membership function

ď ­ A ďƒ‡ B is defined by ď ­ A ďƒ‡ B (x)  min{ď ­ A (x), ď ­ B (x)}, x ďƒŽ X .

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Intuitionistic Fuzzy a-Ideals of BCI-Algebras with Interval Valued Membership& Non Membership Functions

Definition 2.6: Let A and B be two fuzzy ideal of BCI algebra X. The fuzzy set function

A ďƒˆ B with membership

ď ­ A ďƒˆ B is defined by ď ­ A ďƒˆ B (x)  max{ď ­ A (x), ď ­ B (x)}, x ďƒŽ X .

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 ď ­ A (x) ď‚Ł ď ­ B (x) ,

 xďƒŽX

Definition 2.9: An IFS A= < X, ÂľA,Ď…A> in a BCI-algebra X is called an intuitionistic fuzzy ideal of X if it

satisfies:(F1) ÂľA (0) ≼ ÂľA(x) &Ď…A(0)≼ Ď…A(x),(F2) ÂľA (x) ≼ min { ÂľA (x*y), ÂľA (y)}, (F3) Ď…A (x) ď‚Ł max {Ď…A (x*y), Ď…A (y)}, for all x,yđ?œ– X. Definition 2.10: An intuitionistic fuzzy set A=< ÂľA, Ď…A> of a BCI-algebra X is called an intuitionistic fuzzy aideal if it satisfies (F1) and(F4) ÂľA(y*x)≼min{ÂľA((x*z)*(0*y)), ÂľA (z)},(F5) Ď…A(y*x)≤max{Ď…A((x*z)*(0*y)), Ď…A (z)}, for all x,y,z đ?œ– X. An interval-valued intuitionistic fuzzy set A defined on X is given byA={(x,[Âľđ??´đ??ż (x)Âľđ??´đ?‘ˆ (x)],[Ď…đ??´đ??ż (x)Ď…đ??´đ?‘ˆ (x)])},∀đ?‘Ľđ?œ–đ?‘‹ where Âľđ??´đ??ż ,Âľđ?‘ˆđ??´ are two membership functions and Ď…đ??´đ??ż ,Ď…đ??´đ?‘ˆ are two non-membership functions X such that Âľđ??´đ??ż ≤¾đ??´đ?‘ˆ &Ď…đ??´đ??ż ≼υđ??´đ?‘ˆ ,∀ xđ?œ–X. Let ÂľA(x)=[Âľđ??´đ??ż ,Âľđ??´đ?‘ˆ ]&Ď…đ??´ (x)=[Ď…đ??´đ??ż ,Ď…đ??´đ?‘ˆ ],∀ xđ?œ–X and let D[0,1]denote the family of all closed subintervals of [0,1].If Âľđ??´đ??ż (x)=Âľđ??´đ?‘ˆ (x)=c,0≤c≤1 and if Ď…đ??´đ??ż (x)=Ď…đ??´đ?‘ˆ (x)=k, 0≤k≤1,then we have ÂľA(x)=[c,c]&Ď…A(x)=[k,k] which we also assume, for the sake of convenience, to belong to D[0,1]. thus ÂľA(x)&Ď…A(x)đ?œ–[0,1],∀ xđ?œ–X,and therefore the i-v IFS a is given by A=[(x,ÂľA(x),Ď…A(x))},∀ xđ?œ–X,where ÂľA(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 symbolsâ€? ≤â€?,â€?≼â€? and “=â€? in the case of two elements in D[0,1]. Consider two elements D1:[a1,b1]and D2:[a2,b2]đ?œ–D[0,1]. Then rmin(D1,D2)=[min{a1,a2},min{b1,b2}], rmax(D1,D2)=[max{a1,a2},max{b1,b2}] D1≼D2⇔a1≼a2,b1≼b2; D1≤D2⇔a1≤a2,b1≤b2 and D1=D2.

III. INTERVAL-VALUED INTUITIONISTIC FUZZY A-IDEALS OF BCI-ALGEBRAS Definition 3.1:An interval-valued intuitionistic fuzzy set A in BCI-algebra X is called an interval-valued intuitionistic fuzzy a-ideal of X if it satisfies (FI1)ÂľA(0) ≼¾A(x),Ď…A(0) ≤υA(x),(FI2)ÂľA(y*x) ≼r min {ÂľA ((x*z)*(0*y)),ÂľA(z)}, ( FI3)Ď…A(y*x) ≤ r max {Ď…A((x*z)*(0*y)),Ď…A(z)}. Theorem 3.2Let A be an i-v intuitionistic fuzzy a-ideal of X. if there exists a sequence {xn} in X such that lim nď‚Žď‚Ľ ď ­ A ( xn ) =[1,1], lim nď‚Žď‚Ľ ď Ž A ( xn ) =[0,0] then ÂľA(0)=[1,1] andĎ…A(0)=[0,0].

Proof:Since ÂľA(0) ≼¾A(x)and Ď…A(0) ≤υA(x) for all xđ?œ–X, we have ÂľA(0) ≼¾A(xn) and Ď…A(0) ≤υA(xn), for every positive integer n. note that ďƒŠ ď ­ A L , ď ­ AU ďƒš ď‚ł ď ­ A (0) .[1,1] ≼¾A(x) ≼¾A(0) ≼ lim n ď‚Žď‚Ľ ď ­ A ( xn ) (xn)=[1,1]. ďƒŤ ďƒť ďƒŠďƒŤď ŹA L , ď ŹAU ďƒšďƒť ď‚Ł ď ŹA (0) .[0,0] ≤υA(x) ď‚Ł Ď…A(0)≤ lim n ď‚Žď‚Ľ ď Ž A ( xn ) =[0,0].Hence ÂľA(0)=[1,1] andĎ…A(0)=[0,0]. Lemma3.3:An i-v intuitionistic fuzzy set A=[ Âľđ??´đ??ż , Âľđ??´đ?‘ˆ , Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ ] in X is an i-v intuitionistic fuzzy a-ideal of X if and only if Âľđ??´đ??ż , Âľđ??´đ?‘ˆ and Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ are intuitionistic fuzzy ideals of X. Proof:SinceÂľđ??´đ??ż (0) ≼¾đ??´đ??ż (x); Âľđ??´đ?‘ˆ (0) ≼¾đ??´đ?‘ˆ (x);Ď…đ??´đ??ż (0) ≤ Ď…đ??´đ??ż (đ?‘Ľ)and Ď…đ??´đ?‘ˆ (0) ≤ Ď…đ??´đ?‘ˆ (đ?‘Ľ), therefore ÂľA(0) ≼¾A(x), Ď…A(0) ≤υA(x). Suppose that Âľđ??´đ??ż , Âľđ??´đ?‘ˆ and Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ are intuitionistic fuzzy ideal of X. let x,yđ?œ–X, then ÂľA(x)=[Âľđ??´đ??ż (x),Âľđ??´đ?‘ˆ (x)] ≼[min{Âľđ??´đ??ż (x*y),Âľđ??´đ??ż (y)},min{Âľđ??´đ?‘ˆ (x*y),Âľđ??´đ?‘ˆ (y)}] =r min {[Âľđ??´đ??ż (x*y), Âľđ??´đ?‘ˆ (x*y)],[Âľđ??´đ??ż (y), Âľđ??´đ?‘ˆ (y)]} = r min {ÂľA(x*y),ÂľA(y)} and Ď…A(x)= [Ď…đ??´đ??ż (x),Ď…đ??´đ?‘ˆ (x)]≤ [max{Ď…đ??´đ??ż (x*y),Ď…đ??´đ??ż (y)},max{Ď…đ??´đ?‘ˆ (x*y),Ď…đ??´đ?‘ˆ (y)}] =r max {[Ď…đ??´đ??ż (x*y), Ď…(x*y)],[Ď…đ??´đ??ż (y), Ď…(y)]} = r max {Ď…A(x*y),Ď…A(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,yđ?œ–X,we have [Âľđ??´đ??ż (x),Âľđ??´đ?‘ˆ (x)]= ÂľA(x) ≼ r min{[ÂľA(x*y),ÂľA(y)]} =r min {[Âľđ??´đ??ż (x*y), Âľđ??´đ?‘ˆ (x*y)],[Âľđ??´đ??ż (y), Âľđ??´đ?‘ˆ (y)]} = [min {Âľđ??´đ??ż (x*y),Âľđ??´đ??ż (y)},min{Âľđ??´đ?‘ˆ (x*y),Âľđ??´đ?‘ˆ (y)}] đ??ż And [Ď…đ??´ (x),Ď…đ??´đ?‘ˆ (x)] =Ď…A(x) ≤ r max{Ď…A(x*y),Ď…A(y)} =r max {[Ď…đ??´đ??ż (x*y), Ď…đ??´đ?‘ˆ (x*y)],[Ď…đ??´đ??ż (y), Ď…đ??´đ?‘ˆ (y)]} = [max {Ď…đ??´đ??ż (x*y),Ď…đ??´đ??ż (y)},min{Ď…đ??´đ?‘ˆ (x*y),Ď…đ??´đ?‘ˆ (y)}] It follows thatÂľđ??´đ??ż (x) ≼ min {Âľđ??´đ??ż (x*y),Âľđ??´đ??ż (y)},Ď…đ??´đ??ż (x) ≤ max{Ď…đ??´đ??ż (x*y),Ď…đ??´đ??ż (y)} AndÂľđ??´đ?‘ˆ (x) ≼ min {Âľđ??´đ?‘ˆ (x*y),Âľđ??´đ?‘ˆ (y)}, Ď…đ??´đ?‘ˆ (x) ≤max {Ď…đ??´đ?‘ˆ (x*y),Ď…đ??´đ?‘ˆ (y)} Hence Âľđ??´đ??ż , Âľđ??´đ?‘ˆ and Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ are intuitionistic fuzzy ideals of X.

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Intuitionistic Fuzzy a-Ideals of BCI-Algebras with Interval Valued Membership& Non Membership Functions

Theorem 3.4.Every i-v intuitionistic fuzzy a-ideal of a BCI-algebraX is an i-v intuitionistic fuzzy ideal. Definition 3.5:An i-v intuitionistic fussy set A in X is called an interval-valued intuitionistic fuzzy BCI-sub algebra of X if ÂľA(x*y)≼r min { ÂľA(x), ÂľA(y)} and Ď…A (x*y)≤ {Ď…A(x),Ď…B(y)}, for all x,yđ?œ–X.

Proof:Let A=[ Âľđ??´đ??ż , Âľđ??´đ?‘ˆ , Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ ] be an i-v intuitionistic fuzzy a-ideal of X, where Âľđ??´đ??ż , Âľđ??´đ?‘ˆ and Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ are intuitionistic fuzzy a-ideal of X. thus Âľđ??´đ??ż , Âľđ??´đ?‘ˆ and Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ are intuitionistic fuzzy a-ideals of X. hence by lemma

3.3, A is i-v intuitionistic fuzzy ideal of X. Theorem 3.6: Every i-v intuitionisticfuzzy a-ideal of a BCI-algebra X is an i-v intuitionistic fuzzy sub algebra of X. Proof:Let A=[ Âľđ??´đ??ż , Âľđ??´đ?‘ˆ , Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ ]be an i-v intuitionistic fuzzy a-ideal of X, where Âľđ??´đ??ż , Âľđ??´đ?‘ˆ , and Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ are intuitionistic fuzzy a-ideal of BCI-algebra X. thus Âľđ??´đ??ż , Âľđ??´đ?‘ˆ , and Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ are intuitionistic fuzzy subalgebra of X. Hence, A is i-v intuitionistic fuzzysub algebra of X.

IV. CARTESIAN PRODUCT OF I-V INTUITIONISTIC FUZZY A-IDEALS Definition 4.1An intuitionistic fuzzy relation A on any set a is a intuitionistic fuzzy subset A with a membership function ΊA: XĂ—X→ [0, 1] and non membership function ΨA: XĂ—X→ [0, 1]. Lemma 4.2Let ÂľAand ÂľB be two membership functions and Ď…A andĎ…B be two non membership functions of each x đ?œ–X to the i-v subsets A and B, respectively. Then ÂľAĂ— ÂľB is membership function and Ď…AĂ— Ď…B is non membership function of each element(x,y)đ?œ–XĂ—X to the set AĂ—B and defined by ( ÂľAĂ—ÂľB)(x,y)=r min {ÂľA(x), ÂľB(y)} and (Ď…AĂ—Ď…B)(x,y)=r max {Ď…A(x),Ď…B (y)}. Definition 4.3Let A= [ Âľđ??´đ??ż , Âľđ??´đ?‘ˆ , Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ ]and B=[ Âľđ??żđ??ľ , Âľđ?‘ˆđ??ľ , Ď…đ??żđ??ľ , Ď…đ?‘ˆđ??ľ ] be two i-v intuitionistic fuzzy subsets in a set X. the Cartesian product of AĂ—B is defined byAĂ—B= {((x,y), (ÂľAĂ—ÂľB), (Ď…AĂ—Ď…B));∀x,yđ?œ–XĂ—X}Where AĂ—B: XĂ—X→D[0,1]. Theorem 4.4.Let A=[ Âľđ??´đ??ż , Âľđ??´đ?‘ˆ , Ď…đ??´đ??ż , Ď…đ??´đ?‘ˆ ]and B=[ Âľđ??żđ??ľ , Âľđ?‘ˆđ??ľ , Ď…đ??żđ??ľ , Ď…đ?‘ˆđ??ľ ] be two i-v intuitionistic fuzzy subsets in a set X,then AĂ—B is an i-v intuitionistic fuzzy a-ideal of XĂ—X. Proof: Let(x,y) đ?œ–XĂ—X, then by definition (ÂľAĂ—ÂľB) (0,0)=r min {ÂľA(0), ÂľB(o) = r min {[Âľđ??żđ??´ (0),Âľđ??´đ?‘ˆ (0)],[Âľđ??żđ??ľ (0),Âľđ?‘ˆđ??ľ (0)]} =[min {Âľđ??´đ??ż (0),Âľđ??żđ??ľ (0)},min{Âľđ??´đ?‘ˆ (0),Âľđ?‘ˆđ??ľ (0)}] ≼[min {Âľđ??´đ??ż (x),Âľđ??żđ??ľ (y)},min{Âľđ??´đ?‘ˆ (x),Âľđ?‘ˆđ??ľ (y)}] =r min {[Âľđ??´đ??ż (x),Âľđ??´đ?‘ˆ (x)],[Âľđ??żđ??ľ (y),Âľđ?‘ˆđ??ľ (y)]} = r min {ÂľA(x), ÂľB(y)}=( ÂľAĂ— ÂľB)(x,y) And(Ď…AĂ—Ď…B) 0,0 =r max {Ď…A(0), Ď…B(o) = r max {[Ď…đ??´đ??ż (0),Ď…đ??´đ?‘ˆ (0)],[Ď…đ??żđ??ľ (0),Ď…đ?‘ˆđ??ľ (0)]} =[max {Ď…đ??´đ??ż (0),Ď…đ??żđ??ľ (0)},max{Ď…đ??´đ?‘ˆ (0),Ď…đ?‘ˆđ??ľ (0)}] ≤[max {Ď…đ??´đ??ż (x),Ď…đ??żđ??ľ (y)},max{Ď…đ??´đ?‘ˆ (x),Ď…đ?‘ˆđ??ľ (y)}] =r max {[Ď…đ??´đ??ż (x),Ď…đ??´đ?‘ˆ (x)],[Ď…đ??żđ??ľ (y),Ď…đ?‘ˆđ??ľ (y)]} = r max {Ď…A(x),Ď…B(y) =(Ď…AĂ—Ď…B)(x,y) Therefore (FI2) holds.Now, for all x,y,zđ?œ–X, we have (ÂľAĂ—ÂľB) ((y, yęž‹)*(x, xęž‹))=( ÂľAĂ—ÂľB) (y*x, yęž‹*xęž‹) =r min { ÂľA(y*x),ÂľB(yęž‹*xęž‹)} ď‚ł r min{ r min{ď ­ A (( x * z )*(0* y )), ď ­ A (z)}, r min{ď ­ A (( x1 * z1 )*(0* y1 )), ď ­ A (z1 )}}  r min{{min{ď ­ L A (( x * z )*(0* y)), ď ­ L A (z)}, min{ď ­ U A (( x * z)*(0* y)), ď ­ U A (z)}}, {min{ď ­ L B (( x1 * z1 )*(0* y1 )), ď ­ L B (z1 )}, min{ď ­ U B (( x1 * z1 )*(0* y1 )), ď ­ U B (z1 )}} {min{min{ď ­ L A (( x * z )*(0* y)), ď ­ L B (( x1 * z1 )*(0* y1 ))}, min{ď ­ L A (z), ď ­ L B (z1 )}}, min{min{ď ­ U A (( x * z)*(0* y)), ď ­ U B (( x1 * z1 )*(0* y1 ))}, min{ď ­ U A (z), ď ­ U B (z1 )}}}

=r min {(ÂľAĂ—ÂľB)(((x*z)*(0*y)), ((x1 * z1 ) * (0 * y1 ))) ,(ÂľAĂ—ÂľB)(z, zęž‹)} Also, (Ď…AĂ—Ď…B) ((y, yęž‹)*(x, xęž‹))= (Ď…AĂ—Ď…B)(y*x, yęž‹*xęž‹) =r max { Ď…A(y*x),Ď…B(yęž‹*xęž‹)} ď‚Ł r max{ r max{ď Ž A (( x * z )*(0 * y )),ď Ž A (z)}, r max{ď Ž A (( x1 * z1 )*(0 * y1 )),ď Ž A (z1 )}}  r max{{max{ď Ž L A (( x * z )*(0* y)),ď Ž L A (z)}, max{ď Ž U A (( x * z )*(0* y)),ď Ž U A (z)}}, {max{ď Ž L B (( x1 * z1 )*(0* y1 )),ď Ž L B (z1 )}, max{ď Ž U B (( x1 * z1 )*(0* y1 )),ď Ž U B (z1 )}}

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Intuitionistic Fuzzy a-Ideals of BCI-Algebras with Interval Valued Membership& Non Membership Functions {max{max{ď Ž L A (( x * z )*(0* y)),ď Ž L B (( x1 * z1 )*(0* y1 ))}, max{ď Ž L A (z), ď Ž L B (z1 )}}, max{max{ď Ž U A (( x * z )*(0* y)),ď Ž U B (( x1 * z1 )*(0* y1 ))}, max{ď Ž U A (z),ď Ž U B (z1 )}}}  r max {(ď Ž A ď‚´ď Ž B )(((x* z) * (0 * y)), ((x1* z1 ) * (0 * y1 )), (ď Ž A ď‚´ď Ž B )(z, z1 )}

Hence AĂ—B is an i-v intuitionistic fuzzy a-ideal of XĂ— X Definition 4.5: LetÂľB,Ď…B respectively, be an i-v membership and non membership function of each element xđ?œ–X to the set B. then strongest i-v intuitionistic fuzzy set relationon X ,that is a membership function relation ÂľAonÂľB and non membership function relation Ď…A onĎ…B and ď ­ AB ,ď Ž AB whose i-v membership and non membership function, of each element (x,y) đ?œ– XĂ—X and defined by ď ­ AB (x,y)=r min{ÂľB(x),ÂľB(y)}&ď Ž AB (x,y)=r max{Ď…B(x),Ď…B(y)}

Definition 4.6Let B=[ Âľđ??żđ??ľ , Âľđ?‘ˆđ??ľ , Ď…đ??żđ??ľ , Ď…đ?‘ˆđ??ľ ] be an i-v subset in a set X, then the strongest i-v intuitionistic fuzzy relation on X that is a i-v A on B is AB and defined by,AB=[ ď ­ L , ď ­ U , ď Ž L ,ď Ž U A A A A B

B

B

].

B

Theorem 4.7Let B=[ ď ­ L A , ď ­ U A , ď Ž L A ,ď Ž U A ] be an i-v subset in a set X and AB=[ ď ­ L , ď ­ U B B B B AB

ď Ž L ,ď Ž U AB

AB

, AB

] be the strongest i-v intuitionistic fuzzy relation on X. then B is an i-v intuitionistic a-ideal of X if

and only if AB is an i-v intuitionistic fuzzy a-ideal of XĂ—X. Proof: Let B be an i-v intuitionistic fuzzy a-ideal of X. thenÂľAB(0,0)=r min{ÂľB(0),ÂľB(0)} ≼r min{ÂľB(x),ÂľB(y)}=ÂľAB(x,y) and Ď…AB(0,0)=r max{Ď…B(0),Ď…B(0)}≤r max{Ď…B(x),Ď…B(y)}=Ď…AB(x,y) (x,y) đ?œ–XĂ—X. On the other hand ď ­ AB ((y1,y2)*(x1,x2))=ÂľAB(y1*x1, y2*x2) ` =r min {ÂľB(y1*x1),ÂľB(y2*x2)} ≼r min{r min{ÂľB((x1*z1)*(0*y1)),ÂľB(z1)},r min{ÂľB((x2*z2)*(0*y2)),ÂľB(z2)}} =r min{r min{ÂľB((x1*z1)*(0*y1)),ÂľB((x2*z2)*(0*y2))},r min {ÂľB(z1),ÂľB(z2)}} =r min {ÂľAB ((x1*z1)*(0*y1), (x2*z2)*(0*y2)),ÂľAB(z1, z2)} =r min {ÂľAB (((x1,x2)*(z1,z2))*(0*(y1, y2))),ÂľAB(z1, z2)} ď Ž Also, AB ((y1,y2)*(x1,x2))= Ď…AB(y1*x1, y2*x2) =r max {Ď…B (y1*x1),Ď…B(y2*x2)} ď‚Ł r max{r max{Ď…B((x1*z1)*(0*y1)),Ď…B(z1)},r max{Ď…B((x2*z2)*(0*y2)),Ď…B(z2)}} = r max{r max{Ď…B((x1*z1)*(0*y1)),Ď…B((x2*z2)*(0*y2))},r max {Ď…B(z1),Ď…B(z2)}} =r max{Ď…AB((x1*z1)*(0*y1), (x2*z2)*(0*y2)),Ď…AB(y1, y2)} =r max{Ď…AB(((x1,x2)*( z1,z2))*(0*(y1, y2))),Ď…AB(z1, z2)} For all (x1,x2),(y1,y2),(z1,z2) in XĂ—X. hence AB is an i-v intuitionistic fuzzy a-ideal of XĂ—X. Conversely, let AB be an i-v intuitionistic fuzzy a-ideal of XĂ—X. then for all (x,x)đ?œ–XĂ—X.we have r min {ÂľB(0),ÂľB(0)}=ÂľAB(0,0)≼¾AB(x,x)= r min{ÂľB(x),ÂľB(x)}(or)ÂľB(0) ≼¾B(x) and r max {Ď…B(0),Ď…B(0)}=Ď…AB(0,0)≤υAB(x, x)=rmin{Ď…B(x),ÂľB(x)}(or)Ď…B(0)≤υB(x)xđ?œ–X. Now, let (x1,x2),(y1,y2),(z1,z2) đ?œ–XĂ—X, then rmin {ÂľB(y1*x1),(y2*x2)}=ÂľAB(y1*x1, y2*x2) =ÂľAB ((y1, y2)*(x1, x2)) ≼r min {ÂľAB(((x1,x2)*((z1, z2))*(0*(y1, y2))),ÂľAB(z1,z2)} =r min {ÂľAB ((x1*z1)*(0*y1), (x2*z2)*(0*y2)),ÂľAB(z1, z2)} =r min {r min {ÂľB((x1*z1)*(0*y1)),ÂľB(z1)},r min {ÂľAB((x2*z2)*(0*y2)),ÂľB(z2)}} Also, rmax {Ď…B(y1*x1),(y2*x2)}=Ď…AB(y1*x1, y2*x2)=Ď…AB ((y1, y2)*(x1, x2)) ≤r max {Ď…AB(((x1,x2)*((z1, z2))*(0*(y1, y2))),Ď…AB(z1,z2)} =r max {Ď…AB ((x1*z1)*(0*y1), (x2*z2)*(0*y2)), Ď…AB (z1, z2)} =r max {r max {Ď…B((x1*z1)*(0*y1)),ÂľB(z1)},r max {Ď…AB((x2*z2)*(0*y2)),Ď…B(z2)}} If x2=y2=z2=0, then r min {ÂľB(y1*x1),ÂľB(0)}≼ r min {r min {ÂľB((x1*z1)*(0*y1),ÂľB(z1)},ÂľB(0)} and r max {Ď…B(y1*x1),Ď…B(0)}≼ r max {r max {Ď…B((x1*z1)*(0*y1), Ď…B(z1)},Ď…B(0)} ÂľB(y1*x1)≼r min {ÂľB((x1*z1)*(0*y1), ÂľB(z1)} and Ď…B(y1*x1)≼r max {Ď…B((x1*z1)*(0*y1), Ď…B(z1)}. Therefore B is i-v intuitionistic fuzzy a-ideal of X. Theorem 4.8: IfÂľA is a i-v intuitionistic fuzzy a-ideal of BCI-algebra X, then ď ­ Am is also i-v intuitionistic fuzzy a-ideal of BCI-algebra X Proof: For all x, y, z ďƒŽ X

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Intuitionistic Fuzzy a-Ideals of BCI-Algebras with Interval Valued Membership& Non Membership Functions 1.  A  0    A  x  ,  A  0   A  x  .

  A  0      A  x   ,  A  0     A  x   m

 A  0    A  x  ,  A  0   A  x  . m

m

m

m

A

m

m

 0    A  x  ,  A  0   A  x  m

m

m

x  X

2.  A ( y * x)  r min{  A (( x * z )*(0 * y )),  A ( z )}.   A ( y * x)    r min{  A (( x * z )*(0 * y)),  A ( z )} m

m

 A ( y * x) m  r min{  A (( x * z )*(0 * y )),  A ( z )}m .  A ( y * x)  r min{  A (( x * z )*(0 * y)) m ,  A ( z ) m } m

 A ( y * x)  r min{  A (( x * z )*(0 * y ))  A ( z )} m

m

m

3. A ( y * x)  r max{  A (( x * z )*(0 * y )),  A ( z )}.   A ( y * x)    r max { A (( x * z )*(0 * y )) A ( z )} m

m

 A ( y * x) m  r max {  A (( x * z )*(0 * y )), A ( z )}m .  A ( y * x)  r max { A (( x * z )*(0 * y)) m ,  A ( z ) m } m

 A ( y * x)  r max{  A (( x * z )*(0 * y ))  A ( z )} m

m

m

Theorem 4.9:IfµAis a i-v intuitionistic fuzzy a-ideal of BCI-algebra X, then

 A B isalso a i-v intuitionistic

fuzzy a-ideal of BCI-algebra X Proof: For all x, y, z  X 1.  A  0    A  x  ,  A  0   A  x  and  B  0    B  x  ,  B  0   B  x  min{  A  0  ,  B  0 }  min{  A  x  ,  B  x }, min{ A  0  ,  B (0)}  min{ A  x  ,  B  x }

 A  B  0    A  B  x  ,  A  B  0    A B  x 

2.  A ( y * x)  r min{  A (( x * z )*(0* y)),  A ( z )},  B ( y * x)  r min{  B (( x * z)*(0* y)),  A ( z )} {  A ( y * x),  B ( y * x)}  {r min{  A (( x * z )*(0* y)),  A ( z )}, r min{  B (( x * z)*(0 * y )),  B ( z )}} min{  A ( y * x),  B ( y * x)}  min{r min{  A (( x * z )*(0* y)),  A ( z)}, r min{  B (( x * z )*(0* y )),  B ( z )}}  min{r min{  A (( x * z )*(0* y)),  B (( x * z )*(0* y))}, r min{  A ( z),  B ( z)}}

 A B ( y * x)  r min{  A B (( x * z )*(0* y)),  A B ( z )} 3. A ( y * x)  r max{ A (( x * z )*(0* y)), A ( z )}, B ( y * x)  r max{ B (( x * z )*(0* y)),  A ( z )} { A ( y * x), B ( y * x)}  {r max{ A (( x * z )*(0* y)), A ( z )}, r max{ B (( x * z )*(0 * y )), B ( z )}} If one is contained in the other min{ A ( y * x), B ( y * x)}  min{r max{ A (( x * z )*(0 * y)), A ( z )}, r max{ B (( x * z )*(0 * y )), B ( z )}}

 A B ( y * x)  r max{min{ A (( x * z )*(0 * y)), B (( x * z )*(0 * y))}, min{ A ( z), B ( z )}}  A B ( y * x)  r max{ A B (( x * z )*(0 * y )), A B ( z )} Theorem 4.10: If µAis a i-v intuitionistic fuzzy a-ideal of BCI-algebra X, then

 A B isalso a i-v intuitionistic

fuzzy a-ideal of BCI-algebra X. Proof: For all x, y, z  X 1.  A  0    A  x  ,  A  0   A  x  and  B  0    B  x  ,  B  0   B  x  min{  A  0  ,  B  0 }  min{  A  x  ,  B  x }, min{ A  0  ,  B  0 }  min{ A  x  ,  B  x }

 A B  0    A B  x  ,  A  B  0    A  B  x  2.  A ( y * x)  r min{  A (( x * z )*(0* y )),  A ( z )},  B ( y * x)  r min{  B (( x * z )*(0* y)),  A ( z )} {  A ( y * x),  B ( y * x)}  {r min{  A (( x * z )*(0* y )),  A ( z )}, r min{  B (( x * z )*(0 * y )),  B ( z )}} max{  A ( y * x),  B ( y * x)}  max{r min{  A (( x * z )*(0* y )),  A ( z )}, r min{  B (( x * z )*(0* y )),  B ( z )}}  max{r min{  A (( x * z )*(0* y )),  B (( x * z )*(0* y ))}, r max{  A ( z ),  B ( z )}} If one is contained in the other r min{max{  A (( x * z )*(0 * y )),  B (( x * z )*(0 * y))}, max{  A ( z ),  B ( z )}}  A B ( y * x)  r min{  A B (( x * z )*(0 * y)),  A B ( z )} 3. A ( y * x)  r max{ A (( x * z )*(0 * y )), A ( z )},  B ( y * x)  r max{ B (( x * z )*(0 * y)),  A ( z )} { A ( y * x), B ( y * x)}  {r max{ A (( x * z )*(0 * y )), A ( z )}, r max{ B (( x * z )*(0 * y )), B ( z )}} max{ A ( y * x), B ( y * x)}  max{r max{ A (( x * z )*(0 * y)), A ( z )}, r max{ B (( x * z )*(0 * y )), B ( z )}}  A B ( y * x)  r max{max{ A (( x * z )*(0 * y)), B (( x * z )*(0 * y))}, max{ A ( z ), B ( z )}}

 A B ( y * x)  r max{ A B (( x * z )*(0 * y )), A B ( z )}

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Intuitionistic Fuzzy a-Ideals of BCI-Algebras with Interval Valued Membership& Non Membership Functions

REFERENCES [1] [2] [3] [4] [5] [6] [7]

K.T Atanassov, intuitionisticfuzzy sets and systems, 20(1986), 87-96 K.T Atanassov, intuitionisticfuzzy sets. Theory and applications, studies in fuzziness and soft computing, 35.Heidelberg; physica-verlag R.Biswas, Rosenfeld’s fuzzy subgroups with interval-valued membership functions, fuzzy sets and systems 63(1994), no.1,87-90 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 K.Iseki, an algebra related with a propositional calculus, proc, Japan Acad.42 (1966),26-29 H.M.Khalid, B.Ahmad, fuzzy H-ideals in BCI-algebras, fuzzy sets and systems 101(1999)153-158. 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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