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 â&#x20AC;&#x201C; 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 Birwaâ&#x20AC;&#x2122;s defined interval valued fuzzy subgroups of Rosenfeld`s nature , and investigated some elementary properties. The idea of â&#x20AC;&#x153;intuitionistic fuzzy setâ&#x20AC;? 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?&#x153;&#x2013; X. In a BCI-algebra, we can define a partial orderingâ&#x20AC;?â&#x2030;¤â&#x20AC;? by xâ&#x2030;¤y if and only if x*y=0.in a BCI-algebra X, the set M={xđ?&#x153;&#x2013;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-Mâ&#x2030; É¸. 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 â&#x2030;¤y imply x*z â&#x2030;¤y*z and z*y â&#x2030;¤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) â&#x2030;¤x*y An intuitionistic fuzzy set A in a non-empty set X is an object having the form A= {<x,ÂľA(x),Ď&#x2026;A(x)>/xđ?&#x153;&#x2013;X},Where the functions ÂľA : Xâ&#x2020;&#x2019;[0,1] and Ď&#x2026;A: Xâ&#x2020;&#x2019;[0,1] denote the degree of the membership and the degree of non membership of each element xđ?&#x153;&#x2013; X to the set A respectively, and 0â&#x2030;¤ ÂľA(x) +Ď&#x2026;A(x) â&#x2030;¤ 1 for all xđ?&#x153;&#x2013; 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&#x2026;A] for the intuitionistic fuzzy set A= {[ÂľA(x),Ď&#x2026;A(x)]/ xđ?&#x153;&#x2013;X}. Definition 2.1:A non empty subset I of X is called an ideal of X if it satisfies:1. 0đ?&#x153;&#x2013;I,2.x*yđ?&#x153;&#x2013;I and yđ?&#x153;&#x2013;I ď&#x192;&#x17E; xđ?&#x153;&#x2013;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) â&#x2030;ĽÂľ(x), 2. Âľ(x) â&#x2030;Ľmin {Âľ(x*y), Âľ(y)}, for all x,yđ?&#x153;&#x2013;X. Definition 2.3: A non empty subset I of X is called a- ideal of X if it satisfies:1. 0đ?&#x153;&#x2013;I. 2. (x*z)*(0*y)đ?&#x153;&#x2013;I and yđ?&#x153;&#x2013;I imply x*zđ?&#x153;&#x2013;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) â&#x2030;ĽÂľ(x), 2. Âľ(y*x) â&#x2030;Ľmin {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 ď&#x192;&#x2021; B membership function

ď ­ A ď&#x192;&#x2021; B is defined by ď ­ A ď&#x192;&#x2021; B (x) ď&#x20AC;˝ min{ď ­ A (x), ď ­ B (x)}, x ď&#x192;&#x17D; 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 ď&#x192;&#x2C6; B with membership

ď ­ A ď&#x192;&#x2C6; B is defined by ď ­ A ď&#x192;&#x2C6; B (x) ď&#x20AC;˝ max{ď ­ A (x), ď ­ B (x)}, ď&#x20AC;˘x ď&#x192;&#x17D; 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&#x201A;Ł ď ­ B (x) ,

ď&#x20AC;˘ xď&#x192;&#x17D;X

Definition 2.9: An IFS A= < X, ÂľA,Ď&#x2026;A> in a BCI-algebra X is called an intuitionistic fuzzy ideal of X if it

satisfies:(F1) ÂľA (0) â&#x2030;Ľ ÂľA(x) &Ď&#x2026;A(0)â&#x2030;Ľ Ď&#x2026;A(x),(F2) ÂľA (x) â&#x2030;Ľ min { ÂľA (x*y), ÂľA (y)}, (F3) Ď&#x2026;A (x) ď&#x201A;Ł max {Ď&#x2026;A (x*y), Ď&#x2026;A (y)}, for all x,yđ?&#x153;&#x2013; X. Definition 2.10: An intuitionistic fuzzy set A=< ÂľA, Ď&#x2026;A> of a BCI-algebra X is called an intuitionistic fuzzy aideal if it satisfies (F1) and(F4) ÂľA(y*x)â&#x2030;Ľmin{ÂľA((x*z)*(0*y)), ÂľA (z)},(F5) Ď&#x2026;A(y*x)â&#x2030;¤max{Ď&#x2026;A((x*z)*(0*y)), Ď&#x2026;A (z)}, for all x,y,z đ?&#x153;&#x2013; X. An interval-valued intuitionistic fuzzy set A defined on X is given byA={(x,[Âľđ??´đ??ż (x)Âľđ??´đ?&#x2018;&#x2C6; (x)],[Ď&#x2026;đ??´đ??ż (x)Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x)])},â&#x2C6;&#x20AC;đ?&#x2018;Ľđ?&#x153;&#x2013;đ?&#x2018;&#x2039; where Âľđ??´đ??ż ,Âľđ?&#x2018;&#x2C6;đ??´ are two membership functions and Ď&#x2026;đ??´đ??ż ,Ď&#x2026;đ??´đ?&#x2018;&#x2C6; are two non-membership functions X such that Âľđ??´đ??ż â&#x2030;¤Âľđ??´đ?&#x2018;&#x2C6; &Ď&#x2026;đ??´đ??ż â&#x2030;ĽĎ&#x2026;đ??´đ?&#x2018;&#x2C6; ,â&#x2C6;&#x20AC; xđ?&#x153;&#x2013;X. Let ÂľA(x)=[Âľđ??´đ??ż ,Âľđ??´đ?&#x2018;&#x2C6; ]&Ď&#x2026;đ??´ (x)=[Ď&#x2026;đ??´đ??ż ,Ď&#x2026;đ??´đ?&#x2018;&#x2C6; ],â&#x2C6;&#x20AC; xđ?&#x153;&#x2013;X and let D[0,1]denote the family of all closed subintervals of [0,1].If Âľđ??´đ??ż (x)=Âľđ??´đ?&#x2018;&#x2C6; (x)=c,0â&#x2030;¤câ&#x2030;¤1 and if Ď&#x2026;đ??´đ??ż (x)=Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x)=k, 0â&#x2030;¤kâ&#x2030;¤1,then we have ÂľA(x)=[c,c]&Ď&#x2026;A(x)=[k,k] which we also assume, for the sake of convenience, to belong to D[0,1]. thus ÂľA(x)&Ď&#x2026;A(x)đ?&#x153;&#x2013;[0,1],â&#x2C6;&#x20AC; xđ?&#x153;&#x2013;X,and therefore the i-v IFS a is given by A=[(x,ÂľA(x),Ď&#x2026;A(x))},â&#x2C6;&#x20AC; xđ?&#x153;&#x2013;X,where ÂľA(x):Xâ&#x2020;&#x2019;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&#x20AC;? â&#x2030;¤â&#x20AC;?,â&#x20AC;?â&#x2030;Ľâ&#x20AC;? and â&#x20AC;&#x153;=â&#x20AC;? in the case of two elements in D[0,1]. Consider two elements D1:[a1,b1]and D2:[a2,b2]đ?&#x153;&#x2013;D[0,1]. Then rmin(D1,D2)=[min{a1,a2},min{b1,b2}], rmax(D1,D2)=[max{a1,a2},max{b1,b2}] D1â&#x2030;ĽD2â&#x2021;&#x201D;a1â&#x2030;Ľa2,b1â&#x2030;Ľb2; D1â&#x2030;¤D2â&#x2021;&#x201D;a1â&#x2030;¤a2,b1â&#x2030;¤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) â&#x2030;ĽÂľA(x),Ď&#x2026;A(0) â&#x2030;¤Ď&#x2026;A(x),(FI2)ÂľA(y*x) â&#x2030;Ľr min {ÂľA ((x*z)*(0*y)),ÂľA(z)}, ( FI3)Ď&#x2026;A(y*x) â&#x2030;¤ r max {Ď&#x2026;A((x*z)*(0*y)),Ď&#x2026;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&#x201A;Žď&#x201A;Ľ ď ­ A ( xn ) =[1,1], lim nď&#x201A;Žď&#x201A;Ľ ď Ž A ( xn ) =[0,0] then ÂľA(0)=[1,1] andĎ&#x2026;A(0)=[0,0].

Proof:Since ÂľA(0) â&#x2030;ĽÂľA(x)and Ď&#x2026;A(0) â&#x2030;¤Ď&#x2026;A(x) for all xđ?&#x153;&#x2013;X, we have ÂľA(0) â&#x2030;ĽÂľA(xn) and Ď&#x2026;A(0) â&#x2030;¤Ď&#x2026;A(xn), for every positive integer n. note that ď&#x192;Š ď ­ A L , ď ­ AU ď&#x192;š ď&#x201A;ł ď ­ A (0) .[1,1] â&#x2030;ĽÂľA(x) â&#x2030;ĽÂľA(0) â&#x2030;Ľ lim n ď&#x201A;Žď&#x201A;Ľ ď ­ A ( xn ) (xn)=[1,1]. ď&#x192;Ť ď&#x192;ť ď&#x192;Šď&#x192;Ťď ŹA L , ď ŹAU ď&#x192;šď&#x192;ť ď&#x201A;Ł ď ŹA (0) .[0,0] â&#x2030;¤Ď&#x2026;A(x) ď&#x201A;Ł Ď&#x2026;A(0)â&#x2030;¤ lim n ď&#x201A;Žď&#x201A;Ľ ď Ž A ( xn ) =[0,0].Hence ÂľA(0)=[1,1] andĎ&#x2026;A(0)=[0,0]. Lemma3.3:An i-v intuitionistic fuzzy set A=[ Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; , Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; ] in X is an i-v intuitionistic fuzzy a-ideal of X if and only if Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; and Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; are intuitionistic fuzzy ideals of X. Proof:SinceÂľđ??´đ??ż (0) â&#x2030;ĽÂľđ??´đ??ż (x); Âľđ??´đ?&#x2018;&#x2C6; (0) â&#x2030;ĽÂľđ??´đ?&#x2018;&#x2C6; (x);Ď&#x2026;đ??´đ??ż (0) â&#x2030;¤ Ď&#x2026;đ??´đ??ż (đ?&#x2018;Ľ)and Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (0) â&#x2030;¤ Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (đ?&#x2018;Ľ), therefore ÂľA(0) â&#x2030;ĽÂľA(x), Ď&#x2026;A(0) â&#x2030;¤Ď&#x2026;A(x). Suppose that Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; and Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; are intuitionistic fuzzy ideal of X. let x,yđ?&#x153;&#x2013;X, then ÂľA(x)=[Âľđ??´đ??ż (x),Âľđ??´đ?&#x2018;&#x2C6; (x)] â&#x2030;Ľ[min{Âľđ??´đ??ż (x*y),Âľđ??´đ??ż (y)},min{Âľđ??´đ?&#x2018;&#x2C6; (x*y),Âľđ??´đ?&#x2018;&#x2C6; (y)}] =r min {[Âľđ??´đ??ż (x*y), Âľđ??´đ?&#x2018;&#x2C6; (x*y)],[Âľđ??´đ??ż (y), Âľđ??´đ?&#x2018;&#x2C6; (y)]} = r min {ÂľA(x*y),ÂľA(y)} and Ď&#x2026;A(x)= [Ď&#x2026;đ??´đ??ż (x),Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x)]â&#x2030;¤ [max{Ď&#x2026;đ??´đ??ż (x*y),Ď&#x2026;đ??´đ??ż (y)},max{Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x*y),Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (y)}] =r max {[Ď&#x2026;đ??´đ??ż (x*y), Ď&#x2026;(x*y)],[Ď&#x2026;đ??´đ??ż (y), Ď&#x2026;(y)]} = r max {Ď&#x2026;A(x*y),Ď&#x2026;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?&#x153;&#x2013;X,we have [Âľđ??´đ??ż (x),Âľđ??´đ?&#x2018;&#x2C6; (x)]= ÂľA(x) â&#x2030;Ľ r min{[ÂľA(x*y),ÂľA(y)]} =r min {[Âľđ??´đ??ż (x*y), Âľđ??´đ?&#x2018;&#x2C6; (x*y)],[Âľđ??´đ??ż (y), Âľđ??´đ?&#x2018;&#x2C6; (y)]} = [min {Âľđ??´đ??ż (x*y),Âľđ??´đ??ż (y)},min{Âľđ??´đ?&#x2018;&#x2C6; (x*y),Âľđ??´đ?&#x2018;&#x2C6; (y)}] đ??ż And [Ď&#x2026;đ??´ (x),Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x)] =Ď&#x2026;A(x) â&#x2030;¤ r max{Ď&#x2026;A(x*y),Ď&#x2026;A(y)} =r max {[Ď&#x2026;đ??´đ??ż (x*y), Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x*y)],[Ď&#x2026;đ??´đ??ż (y), Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (y)]} = [max {Ď&#x2026;đ??´đ??ż (x*y),Ď&#x2026;đ??´đ??ż (y)},min{Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x*y),Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (y)}] It follows thatÂľđ??´đ??ż (x) â&#x2030;Ľ min {Âľđ??´đ??ż (x*y),Âľđ??´đ??ż (y)},Ď&#x2026;đ??´đ??ż (x) â&#x2030;¤ max{Ď&#x2026;đ??´đ??ż (x*y),Ď&#x2026;đ??´đ??ż (y)} AndÂľđ??´đ?&#x2018;&#x2C6; (x) â&#x2030;Ľ min {Âľđ??´đ?&#x2018;&#x2C6; (x*y),Âľđ??´đ?&#x2018;&#x2C6; (y)}, Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x) â&#x2030;¤max {Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x*y),Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (y)} Hence Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; and Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; 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)â&#x2030;Ľr min { ÂľA(x), ÂľA(y)} and Ď&#x2026;A (x*y)â&#x2030;¤ {Ď&#x2026;A(x),Ď&#x2026;B(y)}, for all x,yđ?&#x153;&#x2013;X.

Proof:Let A=[ Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; , Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; ] be an i-v intuitionistic fuzzy a-ideal of X, where Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; and Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; are intuitionistic fuzzy a-ideal of X. thus Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; and Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; 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?&#x2018;&#x2C6; , Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; ]be an i-v intuitionistic fuzzy a-ideal of X, where Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; , and Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; are intuitionistic fuzzy a-ideal of BCI-algebra X. thus Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; , and Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; 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 ÎŠA: XĂ&#x2014;Xâ&#x2020;&#x2019; [0, 1] and non membership function Î¨A: XĂ&#x2014;Xâ&#x2020;&#x2019; [0, 1]. Lemma 4.2Let ÂľAand ÂľB be two membership functions and Ď&#x2026;A andĎ&#x2026;B be two non membership functions of each x đ?&#x153;&#x2013;X to the i-v subsets A and B, respectively. Then ÂľAĂ&#x2014; ÂľB is membership function and Ď&#x2026;AĂ&#x2014; Ď&#x2026;B is non membership function of each element(x,y)đ?&#x153;&#x2013;XĂ&#x2014;X to the set AĂ&#x2014;B and defined by ( ÂľAĂ&#x2014;ÂľB)(x,y)=r min {ÂľA(x), ÂľB(y)} and (Ď&#x2026;AĂ&#x2014;Ď&#x2026;B)(x,y)=r max {Ď&#x2026;A(x),Ď&#x2026;B (y)}. Definition 4.3Let A= [ Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; , Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; ]and B=[ Âľđ??żđ??ľ , Âľđ?&#x2018;&#x2C6;đ??ľ , Ď&#x2026;đ??żđ??ľ , Ď&#x2026;đ?&#x2018;&#x2C6;đ??ľ ] be two i-v intuitionistic fuzzy subsets in a set X. the Cartesian product of AĂ&#x2014;B is defined byAĂ&#x2014;B= {((x,y), (ÂľAĂ&#x2014;ÂľB), (Ď&#x2026;AĂ&#x2014;Ď&#x2026;B));â&#x2C6;&#x20AC;x,yđ?&#x153;&#x2013;XĂ&#x2014;X}Where AĂ&#x2014;B: XĂ&#x2014;Xâ&#x2020;&#x2019;D[0,1]. Theorem 4.4.Let A=[ Âľđ??´đ??ż , Âľđ??´đ?&#x2018;&#x2C6; , Ď&#x2026;đ??´đ??ż , Ď&#x2026;đ??´đ?&#x2018;&#x2C6; ]and B=[ Âľđ??żđ??ľ , Âľđ?&#x2018;&#x2C6;đ??ľ , Ď&#x2026;đ??żđ??ľ , Ď&#x2026;đ?&#x2018;&#x2C6;đ??ľ ] be two i-v intuitionistic fuzzy subsets in a set X,then AĂ&#x2014;B is an i-v intuitionistic fuzzy a-ideal of XĂ&#x2014;X. Proof: Let(x,y) đ?&#x153;&#x2013;XĂ&#x2014;X, then by definition (ÂľAĂ&#x2014;ÂľB) (0,0)=r min {ÂľA(0), ÂľB(o) = r min {[Âľđ??żđ??´ (0),Âľđ??´đ?&#x2018;&#x2C6; (0)],[Âľđ??żđ??ľ (0),Âľđ?&#x2018;&#x2C6;đ??ľ (0)]} =[min {Âľđ??´đ??ż (0),Âľđ??żđ??ľ (0)},min{Âľđ??´đ?&#x2018;&#x2C6; (0),Âľđ?&#x2018;&#x2C6;đ??ľ (0)}] â&#x2030;Ľ[min {Âľđ??´đ??ż (x),Âľđ??żđ??ľ (y)},min{Âľđ??´đ?&#x2018;&#x2C6; (x),Âľđ?&#x2018;&#x2C6;đ??ľ (y)}] =r min {[Âľđ??´đ??ż (x),Âľđ??´đ?&#x2018;&#x2C6; (x)],[Âľđ??żđ??ľ (y),Âľđ?&#x2018;&#x2C6;đ??ľ (y)]} = r min {ÂľA(x), ÂľB(y)}=( ÂľAĂ&#x2014; ÂľB)(x,y) And(Ď&#x2026;AĂ&#x2014;Ď&#x2026;B) 0,0 =r max {Ď&#x2026;A(0), Ď&#x2026;B(o) = r max {[Ď&#x2026;đ??´đ??ż (0),Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (0)],[Ď&#x2026;đ??żđ??ľ (0),Ď&#x2026;đ?&#x2018;&#x2C6;đ??ľ (0)]} =[max {Ď&#x2026;đ??´đ??ż (0),Ď&#x2026;đ??żđ??ľ (0)},max{Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (0),Ď&#x2026;đ?&#x2018;&#x2C6;đ??ľ (0)}] â&#x2030;¤[max {Ď&#x2026;đ??´đ??ż (x),Ď&#x2026;đ??żđ??ľ (y)},max{Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x),Ď&#x2026;đ?&#x2018;&#x2C6;đ??ľ (y)}] =r max {[Ď&#x2026;đ??´đ??ż (x),Ď&#x2026;đ??´đ?&#x2018;&#x2C6; (x)],[Ď&#x2026;đ??żđ??ľ (y),Ď&#x2026;đ?&#x2018;&#x2C6;đ??ľ (y)]} = r max {Ď&#x2026;A(x),Ď&#x2026;B(y) =(Ď&#x2026;AĂ&#x2014;Ď&#x2026;B)(x,y) Therefore (FI2) holds.Now, for all x,y,zđ?&#x153;&#x2013;X, we have (ÂľAĂ&#x2014;ÂľB) ((y, yę&#x17E;&#x2039;)*(x, xę&#x17E;&#x2039;))=( ÂľAĂ&#x2014;ÂľB) (y*x, yę&#x17E;&#x2039;*xę&#x17E;&#x2039;) =r min { ÂľA(y*x),ÂľB(yę&#x17E;&#x2039;*xę&#x17E;&#x2039;)} ď&#x201A;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 )}} ď&#x20AC;˝ 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 )}} ď&#x20AC;˝{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&#x2014;ÂľB)(((x*z)*(0*y)), ((x1 * z1 ) * (0 * y1 ))) ,(ÂľAĂ&#x2014;ÂľB)(z, zę&#x17E;&#x2039;)} Also, (Ď&#x2026;AĂ&#x2014;Ď&#x2026;B) ((y, yę&#x17E;&#x2039;)*(x, xę&#x17E;&#x2039;))= (Ď&#x2026;AĂ&#x2014;Ď&#x2026;B)(y*x, yę&#x17E;&#x2039;*xę&#x17E;&#x2039;) =r max { Ď&#x2026;A(y*x),Ď&#x2026;B(yę&#x17E;&#x2039;*xę&#x17E;&#x2039;)} ď&#x201A;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 )}} ď&#x20AC;˝ 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 ď&#x20AC;˝{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 )}}} ď&#x20AC;˝ r max {(ď Ž A ď&#x201A;´ď Ž B )(((x* z) * (0 * y)), ((x1* z1 ) * (0 * y1 )), (ď Ž A ď&#x201A;´ď Ž B )(z, z1 )}

Hence AĂ&#x2014;B is an i-v intuitionistic fuzzy a-ideal of XĂ&#x2014; X Definition 4.5: LetÂľB,Ď&#x2026;B respectively, be an i-v membership and non membership function of each element xđ?&#x153;&#x2013;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&#x2026;A onĎ&#x2026;B and ď ­ AB ,ď Ž AB whose i-v membership and non membership function, of each element (x,y) đ?&#x153;&#x2013; XĂ&#x2014;X and defined by ď ­ AB (x,y)=r min{ÂľB(x),ÂľB(y)}&ď Ž AB (x,y)=r max{Ď&#x2026;B(x),Ď&#x2026;B(y)}

Definition 4.6Let B=[ Âľđ??żđ??ľ , Âľđ?&#x2018;&#x2C6;đ??ľ , Ď&#x2026;đ??żđ??ľ , Ď&#x2026;đ?&#x2018;&#x2C6;đ??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&#x2014;X. Proof: Let B be an i-v intuitionistic fuzzy a-ideal of X. thenÂľAB(0,0)=r min{ÂľB(0),ÂľB(0)} â&#x2030;Ľr min{ÂľB(x),ÂľB(y)}=ÂľAB(x,y) and Ď&#x2026;AB(0,0)=r max{Ď&#x2026;B(0),Ď&#x2026;B(0)}â&#x2030;¤r max{Ď&#x2026;B(x),Ď&#x2026;B(y)}=Ď&#x2026;AB(x,y) ď&#x20AC;˘(x,y) đ?&#x153;&#x2013;XĂ&#x2014;X. On the other hand ď ­ AB ((y1,y2)*(x1,x2))=ÂľAB(y1*x1, y2*x2) ` =r min {ÂľB(y1*x1),ÂľB(y2*x2)} â&#x2030;Ľr 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&#x2026;AB(y1*x1, y2*x2) =r max {Ď&#x2026;B (y1*x1),Ď&#x2026;B(y2*x2)} ď&#x201A;Ł r max{r max{Ď&#x2026;B((x1*z1)*(0*y1)),Ď&#x2026;B(z1)},r max{Ď&#x2026;B((x2*z2)*(0*y2)),Ď&#x2026;B(z2)}} = r max{r max{Ď&#x2026;B((x1*z1)*(0*y1)),Ď&#x2026;B((x2*z2)*(0*y2))},r max {Ď&#x2026;B(z1),Ď&#x2026;B(z2)}} =r max{Ď&#x2026;AB((x1*z1)*(0*y1), (x2*z2)*(0*y2)),Ď&#x2026;AB(y1, y2)} =r max{Ď&#x2026;AB(((x1,x2)*( z1,z2))*(0*(y1, y2))),Ď&#x2026;AB(z1, z2)} For all (x1,x2),(y1,y2),(z1,z2) in XĂ&#x2014;X. hence AB is an i-v intuitionistic fuzzy a-ideal of XĂ&#x2014;X. Conversely, let AB be an i-v intuitionistic fuzzy a-ideal of XĂ&#x2014;X. then for all (x,x)đ?&#x153;&#x2013;XĂ&#x2014;X.we have r min {ÂľB(0),ÂľB(0)}=ÂľAB(0,0)â&#x2030;ĽÂľAB(x,x)= r min{ÂľB(x),ÂľB(x)}(or)ÂľB(0) â&#x2030;ĽÂľB(x) and r max {Ď&#x2026;B(0),Ď&#x2026;B(0)}=Ď&#x2026;AB(0,0)â&#x2030;¤Ď&#x2026;AB(x, x)=rmin{Ď&#x2026;B(x),ÂľB(x)}(or)Ď&#x2026;B(0)â&#x2030;¤Ď&#x2026;B(x)ď&#x20AC;˘xđ?&#x153;&#x2013;X. Now, let (x1,x2),(y1,y2),(z1,z2) đ?&#x153;&#x2013;XĂ&#x2014;X, then rmin {ÂľB(y1*x1),(y2*x2)}=ÂľAB(y1*x1, y2*x2) =ÂľAB ((y1, y2)*(x1, x2)) â&#x2030;Ľr 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&#x2026;B(y1*x1),(y2*x2)}=Ď&#x2026;AB(y1*x1, y2*x2)=Ď&#x2026;AB ((y1, y2)*(x1, x2)) â&#x2030;¤r max {Ď&#x2026;AB(((x1,x2)*((z1, z2))*(0*(y1, y2))),Ď&#x2026;AB(z1,z2)} =r max {Ď&#x2026;AB ((x1*z1)*(0*y1), (x2*z2)*(0*y2)), Ď&#x2026;AB (z1, z2)} =r max {r max {Ď&#x2026;B((x1*z1)*(0*y1)),ÂľB(z1)},r max {Ď&#x2026;AB((x2*z2)*(0*y2)),Ď&#x2026;B(z2)}} If x2=y2=z2=0, then r min {ÂľB(y1*x1),ÂľB(0)}â&#x2030;Ľ r min {r min {ÂľB((x1*z1)*(0*y1),ÂľB(z1)},ÂľB(0)} and r max {Ď&#x2026;B(y1*x1),Ď&#x2026;B(0)}â&#x2030;Ľ r max {r max {Ď&#x2026;B((x1*z1)*(0*y1), Ď&#x2026;B(z1)},Ď&#x2026;B(0)} ÂľB(y1*x1)â&#x2030;Ľr min {ÂľB((x1*z1)*(0*y1), ÂľB(z1)} and Ď&#x2026;B(y1*x1)â&#x2030;Ľr max {Ď&#x2026;B((x1*z1)*(0*y1), Ď&#x2026;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 ď&#x192;&#x17D; 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, Rosenfeldâ&#x20AC;&#x2122;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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