Intuitionistic Fuzzy Hyperideals in Intuitionistic Fuzzy Semi-Hypergroups

Transcription

1 International Journal of Algebra, Vol. 6, 2012, no. 13, Intuitionistic Fuzzy Hyperideals in Intuitionistic Fuzzy Semi-Hypergroups K. S. Abdulmula and A. R. Salleh School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia UKM Bangi, Selangor Darul Ehsan, Malaysia Abstract The aim of this study is to introduce the concept of intuitionistic fuzzy left, right hyperideal in intuitionistic fuzzy semihypergroup based on the concept of intuitionistic fuzzy space as a direct generalization of the concept of fuzzy ideals in fuzzy semigroups, through the new approach of fuzzy space introduced by Dib and fuzzy hypergroup introduced by Fathi. A relationship between the introduced intuitionistic fuzzy left, right hyperideals and the classical ones by Davvaz Davvaz 2006 is established. Keywords: Intuitionistic fuzzy space, Intuitionistic fuzzy hypergroups, Intuitionistic fuzzy semihypergroup, Fuzzy ideals, Intuitionistic Fuzzy ideals, Intuitionistic fuzzy hyperideals 1 Introduction and basic concepts In Marty 1934 Marty introduced the notation of Hyperstructure theory by his definition of the hypergroups, The theory of hyperstructure introduced by F. Marty in 1934 is a new mathematical structure which represents a natural extension to the classical algebraic structure. In classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Formally, if H is a nonempty set and P H is the set of all nonempty subsets of H then we consider maps of the following type: f i : H H P H where i =1, 2,...,n and n is a positive integer. The maps f i are called binary hyperoperations. A hypergroup H is a hyperstructure H, satisfying 1 x y z =x y z for all x, y, z H,

2 618 K. S. Abdulmula and A. R. Salleh 2 x H = H x = H for all x H. If H, satisfies only condition 1 then H, is called a semihypergroup and if we replace condition 1 by x y z x y z φ which is obviously weaker than condition 1, then the hyperstructure H, is called an H v - group. Zadeh in Zadeh 1965 introduced notation of the fuzzy set theory generalizes classical set theory in a way that membership degree of an object to a set is not restricted to the integers 0 and 1, but may take on any value in [0, 1]. Therefore a fuzzy subset A of a set X is a function A : X [0, 1], usually this function is referred to as the membership function and denoted by μ A x. Some mathematicians use the notation Ax to denote the membership function instead of μ A x. A fuzzy subset A is written symbolically in the form A = { x, μ A x : x X}. Atanassov in Atanassov 1986,Atanassov 1994,Atanassov 1986 introduced the concept of an intuitionistic fuzzy set as an important generalization of fuzzy sets which was introduced by Zadeh in An intuitionistic fuzzy set A of a universe of discourse X is an object having the form: A = { x, μ A x,ν A x : x X}, where the functions μ A,ν A : X [0, 1] define the degree of membership and nonmembership respectively of the element x X such that 0 μ A + ν A 1. Obviously every fuzzy set has the form { x, μ A x, 1 μ A x : x X}. Therefore if it happens that μ A x =1 ν A x for all x X then the intuitionistic fuzzy set A will be a fuzzy set. Some mathematicians criticized the notion of Atanassov s intuitionistic fuzzy set, Davvaz in Davvaz 2011 defined a new kind of intuitionistic fuzzy ideal and intuitionistic fuzzy hyperideal which is based on Atanassov s intuitionistic fuzzy sets as generalisation of the notation of fuzzy hypergroup, fuzzy hyperring and fuzzy hyperideal by Davvaz based on Rosenfeld s approach Rosenfeld The study of intuitionistic fuzzy hyperalgebraic structures has started with the introduction of the concepts of intuitionistic fuzzy hypergroups subhypergroups and intuitionistic fuzzy semihypergroups based on intuitionistic fuzzy space in the pioneering paper of Karema and Saleh 2011 as a generalisation of fuzzy hypergroup of Davvaz, et al. in Davvaz and Mohammed and Saleh 2010 based on fuzzy space of Dib Dib Where Dib introduced the notion of fuzzy space which play the important role of universal set in ordinary classical mathematics. A fuzzy space X, I is the set of all ordered pairs x, I,x X; i.e., X, I = {x, I :x X}, where x, I ={x, r :r I}. The ordered pair x, I is

3 Intuitionistic fuzzy hyperideals 619 called a fuzzy element in the fuzzy space X, I. Let U denote the support of U, that is U = {x : Ax > 0}. Afuzzy subspace U of the fuzzy space X, I is the collection of all ordered pairs x, u x, where x U for some U X and u x is a subset of I, which contains at least one element beside the zero element. If it happens that x/ U, then u x = 0. An empty fuzzy subspace is defined as {x, φ x :x φ}. Let U = {x, u x :x U } and V = {x, v x :x V } be fuzzy subspaces of X, I. The union and intersection of U and V are defined respectively as follows: U V = {x, u x v x :x U V } 1.1 U V = {x, u x v x :x U V } 1.2 The use of fuzzy space as a universal set corrects the deviation in the definition of fuzzy group hypergroup theory. Dib 1994 introduced fuzzy group based on the definition of fuzzy space by defining fuzzy structure fuzzy groupoid. More precisely, a fuzzy space X, I together with a fuzzy binary operation F =F, f x is said to be a fuzzy groupoid and is denoted by X, I; F. As we know, the concept of intuitionistic fuzzy space played an important role in fuzzy mathematics, the main problem in intuitionistic fuzzy mathematics is how to carry out the ordinary fuzzy concepts to the intuitionistic fuzzy case. The concept of an intuitionistic fuzzy space was introduced in Mohammed Fathi 2010 as a generalization of the notion of intuitionistic fuzzy sets which was introduced by Atanassov Atanassov 1986, and they in Mohammed and Saleh 2009 introduced a new formulation of intuitionistic fuzzy groups by introducing the concepts of intuitionistic fuzzy space and intuitionistic fuzzy functions. Davvaz in Davvaz 2006 used Atanassov s idea to establish the intuitionistic fuzzification of the concept of hyperideals in a semihypergroup. In this paper, we introduce some basic concepts of intuitionistic fuzzy hyperalgebra, namely intuitionistic fuzzy left, right hyperideals in intuitionistic fuzzy semihypergroups, though the new approach of intuitionistic fuzzy spaces and intuitionistic fuzzy hypergroups as generalisation of the fuzzy hyperideals in fuzzy semigroups by Dib and Galhum in Dib and Galhum A relation between the introduced fuzzy left, right hyperideals and the classical case by Davvaz are given. Moreover, we give some examples to illustrate the difference between our point of view and the classical case. 2 Preliminaries Here we recall some of the fundamental concepts and definitions required in the sequel.

4 620 K. S. Abdulmula and A. R. Salleh Definition 1 Davvaz Davvaz 1999 A hypergroupoid H, is called a semihypergroup if x y z = x y z for all x, y, z H. Definition 2 Davvaz Davvaz 2006 Let H be a semihypergroup. An intuitionistic fuzzy set A =μ A,λ A inh is called a left resp. right intuitionistic fuzzy hyperideal of H if 1. μ A y inf z x y μ A z resp. μ A x inf z x y μ A z for all x, y H, 2. Sup z x y λ A z λ A y resp. Sup z x y λ A z λ A x. Davvaz Davvaz 1998,Davvaz 1999 introduced the notions of fuzzy hypergroup, fuzzy H v -hyperring and fuzzy H v -ideal as generalization of Fuzzy subgroupoids and fuzzy subgroup by Rosenfeld in 1971 based on the notion of fuzzy set by Zadeh in That is he assumed a given ordinary hypergroup hyperring and define a fuzzy subhypergroup subhyperring of the given ordinary hypergroup hyperring. Again such formulation of the intrinsic definition for fuzzy hypergroup hyperring and fuzzy subhypergroup subhyperring is not evident due to the absence of the concept of fuzzy universal set, where he applied in fuzzy sets to the theory of algebraic hyperstructures and defined fuzzy sub-hypergroup respectively H v -subgroup of a hypergroup respectively H v -group which is a generalization of the concept of Rosenfeld s fuzzy subgroup of a group. Let H, be a hypergroup respectively, H v -group and let μ be a fuzzy subset of H. Then μ is said to be a fuzzy sub-hypergroup respectively, fuzzy H v -subgroup ofh if the following axioms hold: 1 min{μx,μy} inf{μz : z x y} for all x, y H, 2 for all x, a H there exists y H such that x a y and min{μa,μx} μy, 3 for all x, a H there exists z H such that x z a and min{μa,μx} μz. By same way, Davvaz in Davvaz 1998 introduced the notions of fuzzy hyperideal using Rosenfeld s approach, he assumed a given ordinary hypergroup and defined a fuzzy subhypergroup and hyperideal. A is called a hyperideal, if it is left and right hyperideal of R. Definition 3 Let R be a hyperring and let A be a fuzzy subset of R. Then, A is said to be a left respectively, right fuzzy hyperideal of R if the following axioms hold: 1 min{ax,ay} inf{aα}, for all x, y R, 2for all x, a R there exists y R such that x a+y and min{ax,aa} Ay; 3for all x, a R there exists z R such that x a.z and min{ax,aa} Az, 4Ay inf{aα} respectively Ax inf{aα} for all x, y R.

5 Intuitionistic fuzzy hyperideals 621 Definition 4 Davvaz and Mohammed and Saleh 2010 Let H, I bea non-empty fuzzy space. A fuzzy hyperstructure hypergroupoid, denoted by H, I, is a fuzzy space together with a fuzzy function having onto comembership functions referred as a fuzzy hyperoperation :H, I H, I P H, I, where P H, I denotes the set of all nonempty fuzzy subspaces of H, I and =, xy with : H H H and xy : I I I. A fuzzy hyperoperation =, xy onh, I is said to be uniform if the associated co-membership functions xy are identical, i.e., xy = for all x, y H. A uniform fuzzy hyperstructure H, I, is a fuzzy hyperstructure H, I; with uniform fuzzy hyperoperation. Recall that the action of the fuzzy function =, xy on fuzzy elements of the fuzzy space H, I can be symbolized as follows: x, I y, Ix y, xy I I = x, y,i. Theorem 1 Davvaz and Mohammed and Saleh 2010 To each fuzzy hyperstructure H, I, there is an associated ordinary hyperstructure H, which is isomorphic to the fuzzy hyperstructure H, I, by the correspondence x, I x. Definition 5 Mohammed Fathi 2010 Let X be a nonempty set. An intuitionistic fuzzy space simply IFS denoted by X, I, I is the set of all ordered triples x, I, I, where x, I, I ={x, r, s : r, s I with r + s 1 and x X}: The ordered triple x, I, I is called an intuitionistic fuzzy element of the intuitionistic fuzzy space X, I,I and the condition r, s I with r + s 1 will be referred to as the intuitionistic condition. Definition 6 Mohammed Fathi 2010 A intuitionistic fuzzy subspace U of the intuitionistic fuzzy space X, I, I is the collection of all ordered triples x, u x, u x, where x U for some U X and u x, u x are a subsets of I, which contains at least one element beside the zero element. If it happens that x/ U, then u x =0and u x = 1. An empty fuzzy subspace is defined as {x, x, x :x U }. Let X be an intuitionistic fuzzy space and let A be an intuitionistic fuzzy subset of X. The fuzzy subset A induces the following intuitionistic fuzzy subspaces: The lower-upper intuitionistic fuzzy subspace induced by A H lu A ={ x, [0,Ax], [Ax, 1] : x A }. 2.1

6 622 K. S. Abdulmula and A. R. Salleh The upper-lower intuitionistic fuzzy subspace induced by A H ul A ={ x, {0} [Ax, 1], [0, Ax] {1} : x A }. 2.2 The finite intuitionistic fuzzy subspace induced by A is denoted by H A ={ x, {0,Ax}, {Ax, 1} : x A }. 2.3 Definition 7 Karema and Saleh 2011 An intuitionistic fuzzy hypergroup is an intuitionistic fuzzy hyperstrucure H, I,I, F satisfying the following axioms: i x, I, IF y, I, IFz, I, I =x, I, IFy, I, IFz, I, I, for all x, I, I, y, I, I, z, I, I inh, I, I, ii x, I, IF H, I, I= H, I, IF x, I, I= H, I, I, for all x, I, I H, I, I. Definition 8 Karema and Saleh 2011 An intuitionistic fuzzy H v -group is an intuitionistic fuzzy hyperstructure H, I,I, F satisfying the following conditions: i x, I, IF y, I, IFz, I, I x, I, IF y, I, IFz, I, I, for all x, I, I, y, I, I, z, I, I inh, I, I ii x, I, I F H, I, I =H, I, I F x, I, I = H, I, I, for all x, I, I H, I, I. For the sake of simplicity we will denote x, I, IF H, I, I by x, I, I F H. If H, I, I, F satisfies only the first condition of Definition 7 Definition 8 then it is called an intuitionistic fuzzy semihypergroup an intuitionistic fuzzy H v -semigroup. Theorem 2 Karema and Saleh 2011 Let S = {x, s x, s x : x S } be an intuitionistic fuzzy subspace of the intuitionistic fuzzy space H, I, I. Then S, F is an intuitionistic fuzzy subhypergroup of the intuitionistic fuzzy hypergroup IFH H, I, I, F if and only if: 1 S,F is an ordinary subhypergroup of the hypergroup H, F, 2 sx,s y = sx s y = s xf y and s x, s y =s x s y = s xf y, for all x, y S. Definition 9 Dib and Galhum and Hassan 1996 A fuzzy subspace U of the fuzzy semigroup H, I,F is said to be fuzzy left right ideal if for each x, I U, we have x, IF y, I y =xf y, I I y U y, I y F x, I =yf x, f yx I y I U.

7 Intuitionistic fuzzy hyperideals Intuitionistic fuzzy hyperideals In this section we introduce the concept of intuitionistic fuzzy lef t, right hyperideals in intuitionistic fuzzy samihypergroups, though the new approach of intuitionistic fuzzy space and intuitionistic fuzzy hypergroup. We would like to point out that the proof of the theorems are carried out for the intuitionistic fuzzy left hyperideal, and the case of intuitionistic fuzzy right hyperideals is similar. Remark 1 Let U = {y, I y,i y :y U } be an intuitionistic fuzzy subspace of the intuitionistic fuzzy sermihypergroup H, I, I,F, where H, I, I = {x, I, I :x H} and F = F,,. Definition 10 An intuitionistic fuzzy subspace U of the intuitionistic fuzzy semihypergroup H, I, I,F is said to be intuitionistic fuzzy left right hyperideal if for each x, I, I U, we have x, I, IF y, I y,i y ={xf y, I I y, I I y } P U y, I y,i y F x, I, I ={yf x, f yx I y I, f yx I y I} P U. Remark 2 If U is both intuitionistic fuzzy left and right hyperideal, then it is called an intuitionistic fuzzy hyperideal of the intuitionistic fuzzy semihypergroup H, I, I,F. Theorem 3.1 An intuitionistic fuzzy subspace U of the intuitionistic fuzzy semihypergroup H, I, I,F is an intuitionistic fuzzy left right hyperideal iff i U 0,F is a left right hyperideal of the semihypergroup H, F, ii for all x H, y U 0 we have I I y =I xf y f yx I y I =I yf x. I I y =I xf y f yx I y I =I yf x. Proof Let us consider the case when U is an intuitionistic fuzzy left hyperideal of the intuitionistic fuzzy semihypergroup H, I, I, F. This is equivalent to: for every x, I, I H, I, I, y, I y,i y U, we have x, I, IF y, I y,i y ={xf y, I I y,i y I} P U.

8 624 K. S. Abdulmula and A. R. Salleh Therefore { } xf y, I I y, I I y {z, I z,i z } P U. Which is equivalent to a xf y = z {z} P U,x H, y U 0, b I I y =I z = I xf y, I I y =I z = I xf y. Condition a means that U 0,F is a left hyperideal of the semihypergroup H, F and b proves ii. By using Theorem 3.1, we can directly prove the following theorem: Theorem 3.2 An intuitionistic fuzzy subspace U of the intuitionistic fuzzy semihypergroup H, I, I,F is an intuitionistic fuzzy hyperideal iff i U,F is a hyperideal of the ordinary semihypergroup H, F, ii for every x H, y U we have I I y =I xf y f yx I y I =I yf x, I I y =I xf y f yx I y I =I yf x. Remark 3 Although in the classical fuzzy case, any intuitionistic fuzzy left, right hyperideal is an intuitionistic fuzzy subsemihypergroup. it is not true in our case. In the following we give two counterexamples: Example Let H, F be a given ordinary semihypergroup. Let F = F,, be a uniform intuitionistic fuzzy binary hyperoperation on the intuitionistic fuzzy space H, I, I, where r, s = r.s, is the ordinary product on I, and r, s =r s. Consider the following intuitionistic fuzzy subspace: Then we have U = {x, [0, 1 2 ], [1, 1] : x X} 2

9 Intuitionistic fuzzy hyperideals 625 i U, F is an intuitionistic fuzzy hyperideal of the intuitionistic fuzzy semihypergroup H, I, I,F, since H, F is a hyperideal and for each x, y H, we have I I y = I [0, 1] 2 = [0, 1, 1] 2 = [0, 1] = I 2 xf y, Also f yx I y I = f yx [0, 1] I 2 = [0,f yx 1, 1] 2 = [0, 1] = I 2 yf x, On the other hand I I y = I [ 1, 1] 2 = [ 1, 1, 1] 2 = [ 1, 1] = I 2 xf y, Also f yx I y I = f yx [ 1, 1] I 2 = [f yx 1, 1, 1] 2 = [ 1, 1] = I 2 yf x, ii U, F is not an intuitionistic fuzzy subsemihypergroup of H, I, I,F, since for every x, [0, 1], [ 1, 1], y, [0, 1], [ 1, 1] U, we get {x, [0, 1], [ 1, 1]F y, [0, 1], [ 1, 1]} = {xf y, [0,f 1, 1], [f xy 2 2 xy 1, 1, 1]} 2 2 = {xf y, [0, 1 4 ], [ 1 2, 1]} P U, which means that F is not closed on U. 2. Let H = {a, b, c} and F be the binary hyperoperation defined as in Table 3.1. Then H, F is a semihypergroup. i The subset U = {c} is a hyperideal of H, F. F a b c a {a} {b} {c} b {c} {c} {c} c {c} {c} {c} 3.1 ii If for any x, y H, r, s = { r.s ifx = y = c, Maxr,s, 1 8 r.s if otherwise,

10 626 K. S. Abdulmula and A. R. Salleh r, s = { r s Maxr,s,1 ifx = y = c, r s ifotherwise. Then it is easy to see that H, I, I,F, where F =F,,, is an intuitionistic fuzzy semihypergroup. iii Let U = {c, [0, 1], [ 1, 1]}. Then 9 9 a U, F is an intuitionistic fuzzy left hyperideal of H, I, I,F, for f ac I,L c =f ac I [0, 1] 9 =[0,f ac 1, 1 ] 9 =[0, 1]=I 9 af c = I c, and f bc I,L c =f bc I [0, 1 ] 9 =[0,f bc 1, 1] 9 =[0, 1]=I 9 bf c = I c, f cc I,L c =f cc I [0, 1 ] 9 =[0,f cc 1, 1 9 =[0, 1]=I 9 cf c = I c. On the other hand f ac I,L c =f ac I [ 1, 1] 9 =[f ac 0, 1, 1] 9 =[ 1, 1] = I 9 af c = I c, f bc I,L c =f bc I [ 1, 1] 9 =[f bc 0, 1, 1] 9 =[ 1, 1] = I 9 bf c = I c, f cc I,L c =f cc I [ 1, 1] 9 =[f cc 0, 1, 1] 9 =[ 1, 1] = I 9 cf c = I c. b But U, F is not an intuitionistic fuzzy subsemihypergroup of H, I, I,F, since c, [0, 1], [ 1, 1]F c, [0, 1], [ 1, 1] = {c, f cc [0, 1] [0, 1], f 9 9 cc[ 1, 1] [ 1, 1]} 9 9 = {cf c, [0,f cc 1, 1], [f 9 9 cc 1, 1, 1]} 9 9 = {c, [0, 8 ], [ 1, 1]} P U Theorem 3.4 An intuitionistic fuzzy left right hyperideal U of the intuitionistic fuzzy semihypergroup is an intuitionistic fuzzy subsemihypergroup iff for all x, y U

11 Intuitionistic fuzzy hyperideals 627 I I y = I x I y f yx I y I =f yx I y I x, I I y = I x I y f yx I y I =f yx I y I x. Proof Firstly, suppose that U is an intuitionistic fuzzy left hyperideal of the intuitionistic fuzzy semihypergroup H, I, I, F which satisfies for all x, y U, that Then, by Theorem 3.1, we have: I I y = I x I y, I I y = I x I y. i U,F is a left hyperideal of the ordinary semihypergroup H, F, ii for all x H, y U we have I I y =I xf y, I I y =I xf y. We can see that i implies U,F is an ordinary subsemihypergroup of H, F and, by ii and I I y = I x I y, I I y = I x I y we have I I y =I xf y, for all x, y U, I I y =I xf y, for all x, y U. which is equivalent to, by Theorem 2, U, F is an intuitionistic fuzzy subsemihypergroup of H, I, I,F. Conversely, suppose U is an intuitionistic fuzzy left hyperideal which is an intuitionistic fuzzy subsemihypergroup of H,I,I,F. Then for every x, y U, we have I I y =I xf y = I x I y, I I y =I xf y = I x I y. Corollary 3.5 An intuitionistic fuzzy hyperideal U of the intuitionistic fuzzy semihypergroup H, I, I,F is an intuitionistic fuzzy semihypergroup iff the

12 628 K. S. Abdulmula and A. R. Salleh comembership functions and cononmembership functions satisfy one of the following conditions: for all x, y U, or I I y = I x I y, I I y = I x I y. f yx I y I =f yx I y I x, f yx I y I =f yx I y I x. 4 Intuitionistic fuzzy hyperideals induced by an intuitionistic fuzzy subset Let us recall that, if H, I, I,F is an intuitionistic fuzzy semihypergroup and A is intuitionistic fuzzy subset of H, then the intuitionistic fuzzy subspace induce by A is the set, H A ={x, [0,Ax], [Ax, 1] : x A }, where A = {x H : Ax 0} is the support of A. For this special intuitionistic fuzzy subspace H A, we may reformulate Theorems 3.1, 3.2 in the following forms: Theorem 4.1 An intuitionistic fuzzy sub space HA, of the intuitionistic fuzzy semihypergroup H, I, I,F is an intuitionistic fuzzy left right hyperideal iff i A,F is a left right hyperideal of the ordinary semihypergroup H, F, ii for all x H, y A we have I Ay = AxF y I Ay = AxF y f yx Ay I =AyFx, f yx Ay I =AyFx. Theorem 4.2 An intuitionistic fuzzy subspace HA, of the intuitionistic fuzzy semihypergroup H, I, I,F is an intuitionistic fuzzy hyperideal iff

13 Intuitionistic fuzzy hyperideals 629 i A,F is a hyperideal of the ordinary semihypergroup H, F, ii for all x H, y A we have I Ay =AxF y f yx Ay I =AyFx, I Ay =AxF y fyx Ay I =AyFx. Theorem 4.3 Let HA be an intuitionistic fuzzy left right hyperideal of the intuitionistic fuzzy semihypergroup H,I,I,F. Then HA is an intuitionistic fuzzy subsemihypergroup of H, I, I,F iff for all x, y A, we have I Ay = Ax Ay I Ay = Ax Ay f yx Ay I =f yx Ay Ax, f yx Ay I =f yx Ay Ax. 5 The relationship between intuitionistic fuzzy hyperideals and classical intuitionistic fuzzy hyperideal Let H, I, I,F be a uniform intuitionistic fuzzy semihypergroup and F = F,,, where F is a binary hyperoperation on H and, are the minimum functions from the vector I I into I. Theorem 5.1 Every intuitionistic fuzzy subset A of H which induced an intuitionistic fuzzy left, right hyperideal HA,F of the uniform intuitionistic fuzzy semihypergroup H, I, I,F is a classical intuitionistic fuzzy left, right hyperideal of the ordinary semihypergroup H, F. Proof Suppose that A is an intuitionistic fuzzy subset of H such that H A,F is the induced intuitionistic fuzzy left hyperideal by A of the uniform intuitionistic fuzzy semihypergroup H, I, I,F, where F =F,,. By Theorem 4.1, we have AxF y =Ay for all x H, y A, AxF y =Ay for all x H, y A. Since Ay =0, for each y H A, Ay =1, for all y H, then AxF y Ay for all x, y H, AxF y Ay for all x, y H.

14 630 K. S. Abdulmula and A. R. Salleh which means that A is a classical intuitionistic fuzzy left hyperideal of H, F. Remark 4 The converse of Theorem,in general, is not true, i.e, there is a classical intuitionistic fuzzy left, right hyperideal A of a semihypergroup H, F such that an intuitionistic fuzzy subspace HA induced by A is not an intuitionistic fuzzy left, right hyperideal of the uniform intuitionistic fuzzy semihypergroup H, I, I,F in our sense as we see in the following example. Example 5.2 Let H, I, I,F,F =F, f, f be a uniform intuitionistic fuzzy semihypergroup, where H, F is the semihypergroup as in Table 5.1 and fr, s, fr, s =r s. F a b c a {a} {b} {a} b {a} {b} {a} c {a} {b} {c} 5.1 Consider the intuitionistic fuzzy subset A of H defined by Aa = 1 2, Ab =0, Ac = 1 4, Aa = 1 2, Ab =0, Ac = 1 3. Then we can show that 1. A is a classical intuitionistic fuzzy left hyperideal of H, F, 2. the support A = {a, c} of A is a left hyperideal of H, F, 3. although the support A of A is a left hyperideal of H, F, we see that the induced intuitionistic fuzzy subspace HA ={a, [0, 1], [ 1, 1], 2 2 c, [0, 1 ], [ 1, 1]} is not an intuitionistic fuzzy left hyperideal of H, F in 4 4 our sense, because f bc 1,Ac = f bc 1, 1 4 = 1 4 and AbF c =Aa = 1 2. f bc 1, Ac = f bc 1, 1 3 = 1 3 AbF c =Aa = 1 2.

15 Intuitionistic fuzzy hyperideals 631 We will introduce in this section intuitionistic fuzzy hyperideals induced by intuitionistic fuzzy subsets and then we obtain a relationship between the induced intuitionistic fuzzy hyperideal and intuitionistic fuzzy hyperideal in the sense of Davvaz D-intuitionistic fuzzy hyperideal. Let A be a non-empty intuitionistic fuzzy subset of H and let H A,H lu A and H ul A be intuitionistic fuzzy subspaces of H, I, I induced by the intuitionistic fuzzy set A. Then we have the following theorem: Theorem 5.3 H A,F,, H lu A,F and H ul A,F are intuitionistic fuzzy hyperideal of the intuitionistic fuzzy hypergroup H, I, I,F if and only if 1 A,F is an ordinary hyperideal of the ordinary hypergroup H, F. 2 Ax,Ay = inf {Aw :w xf y}, Ax, Ay = sup { Aw :w xf y }. We will refer to the intuitionistic fuzzy hyperideal defined by Davvaz by the D-hyperideal. Also, by H i A,F ; H i A ={x, I Hi,I Hi :I Hi I,x A } we mean the intuitionistic fuzzy hyperideals of H, I, I,F induced by the intuitionistic fuzzy subset A and we will discuss left intuitionistic fuzzy hyperideals and the same results will hold for right intuitionistic fuzzy hyperideals. Theorem 5.4 Let H, I, I,F be a uniform intuitionistic fuzzy hypergroup and let A be a non-empty intuitionistic fuzzy subset of H which induces left intuitionistic fuzzy hyperideals H i A,F of H, I, I,F Then the comembership functions, and co-nonmembership functions satisfies the following conditions: 1 z Ax,Ay,Az = z Ax,f yz Ay,Az, and z Ax,Ay,Az = z Ax, f yz Ay,Az, for all x, y, z A. 2 Ax,Ay = I Hi for all x A. y H i Proof. For condition 1: If the induced left intuitionistic fuzzy hyperideal is H A,F = { x, {0,Ax}, {Ax, 1} x A }, then using associativity we have x, {0,Ax}, {Ax, 1} F y, {0,Ay}, {Ay, 1} F z, {0,Az}, {Az, 1} = x, {0,Ax}, {Ax, 1} F y, {0,Ay}, {Ay, 1} F z, {0,Az}, {Az, 1},

16 632 K. S. Abdulmula and A. R. Salleh that is, xf yfz,z Ax,Ay,Az = xf yfz,z Ax,f yz Ay,Az, xf yfz,z Ax,Ay,Az = xf yfz, z Ax, f yz Ay,Az, Therefore, f satisfies z Ax,Ay,Az = z Ax,f yz Ay,Az, z Ax,Ay,Az = z Ax, f yz Ay,Az, for all x, y, z A. If the induced left intuitionistic fuzzy hyperideal are H lu A,F and H ul A,F then using the same argument one can prove that and satisfies condition 1. For condition 2: If the induced left intuitionistic fuzzy hyperideal is H A,F + = { x, {0,Ax}, {Ax, 1} x A }, then I Hi = {0,Ay}, I Hj = {Ay, 1} for all y A. Clearly, I Hi = Ax,Ay, y H A I Hj = Ax, Ay for all x A. y H A Theorem 5.5 Let H, I, I,F be a uniform intuitionistic fuzzy hypergroup and let the co-membership functions f and co-nonmembership functions f have the t-norm property. Then every intuitionistic fuzzy subset A of H which induces a left intuitionistic fuzzy hyperideal is a left D-intuitionistic fuzzy hyperideal of H, F. Proof. If the intuitionistic fuzzy subsets A induces a left intuitionistic fuzzy hyperideal of the intuitionistic fuzzy hypergroup H, I, I,F, then by Theorem 5.3 the co-membership functions, and co-nonmembership functions satisfies Ax,Ay = inf {Aw : w xf y}, Ax, Ay = inf { Az : z xf y }. for all Ax 0 Ay and Ax 1 Ay. Therefore, if the intuitionistic fuzzy subset A induces left intuitionistic fuzzy hyperideal, then A without loss of generality satisfies Ax,Ay inf {Aw : w xf y},

17 Intuitionistic fuzzy hyperideals 633 And for all x, y H. That is, Ax, Ay sup { Aw : w xf y }, min{ax,ay} inf {Az : z xf y}, max{ax, Ay} sup { Az : z xf y }, for all x, y H. Hence, the first condition of the left D-intuitionistic fuzzy hyperideal definition satisfied. Now, let x, u x, u x, x,u x, u x be in the induced left intuitionistic fuzzy hyperideal. Then the inequalities And f xx Ax,Ax inf {Aw : w xf x }, and f xx satisfied, also x xf x implies Ax, Ax sup { Aw : w xf x }, min{ax,ax } Ax max{ax, Ax } Ax. Hence if we choose x = y, then the second and third conditions of the left D-intuitionistic fuzzy hyperideal definition will hold for all x, x F, and condition four of the left D-intuitionistic fuzzy hyperideal definition follow directly from Theorem 4.3. The proof of third condition is similar. Theorem 5.6 LetY,F an ordinary left, right hyperideal of the semihypergroup H, F. Then for each intuitionistic fuzzy subset A of H with support A = Y, there is an intuitionistic fuzzy semihypergroup H, I, I,G such that the induced intuitionistic fuzzy subspace HA is an intuitionistic fuzzy left, right hyperideal of H, I, I,G Proof Let Y,F be a left hyperideal of the semihypergroup H, F and A be an intuitionistic fuzzy subset of H such that A = Y. It is clear that A,F is a left hyperideal of H, F. Now we define an intuitionistic fuzzy binary hyperoperation G =G, g xy, g xy on H as follows: Take G = F and g xy r, s =h xy r s, g xy = h xy r s where

18 634 K. S. Abdulmula and A. R. Salleh If Ay 0 for all x H, y A, then h xy k = AxF y Ay k, k < Ay, 1 AxF y k Ay+AxF y, k Ay 1 Ay And h xy k =k if x H, y / A, and If Ay 1 for all x H, y A, then 1+AxF y 1+ k 1, k < Ay, Ay h xy k = AxF y k, k Ay Ay If Ay =1, then h xy k =k for all k I. It is clear that g xy r, s, g xy r, s are continuous functions on I I for all x, y H and g xy r, s =0iff r =0or s =0, and g xy r, s =1iff r = 1 or s =1, Then it is easy to show that H, I, I,G is an intuitionistic fuzzy semihypergroup. If H A is the induced intuitionistic fuzzy subspace by A of the intuitionistic fuzzy semihypergroup H, I, I,G, then from the definition of g xy, g xy, we have for all x H, y A, and g xy 1,Ay =h xy Ay =AxF y, g xy 1, Ay =h xy Ay =AxF y, which implies H A,G is an intuitionistic fuzzy left hyperideal of H, I, I,G by Theorem 4.1. Corollary 5.7 Every classical intuitionistic fuzzy left,right hyperideal of the semihypergroup H, F induces an intuitionistic fuzzy left,right hyperideal relative to some intuitionistic fuzzy semihypergroup H, I, I,G. 6 Conclusion In this paper, we have generalised the study initiated by Dib Dib and Galhum and Hassan 1996 about fuzzy ideal based on fuzzy space and Davvaz Davvaz 2006 about intuitionistic fuzzy hyperideals in semihypergroups to intuitionistic fuzzy hyperideals in intuitionistic fuzzy semihypergroup based on intuitionistic fuzzy space by Fathi Mohammed Fathi 2010, and we introduce a relation between this the intuitionistic fuzzy left, right hyperideals and the intuitionistic fuzzy hyperideals in the classical case by Davvaz.

19 Intuitionistic fuzzy hyperideals Acknowledgements The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKM-ST-06-FRGS and UKM-DLP References Abdulmula K & Salleh A. R Intutionistic fuzzy hypergroups. Proc. 2 nd Int. Conf. Math. Sci Atanassov K.T Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20: Atanassov K.T New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems 61: Atanassov K.T Intuitionistic fuzzy sets, Theory and Applications, Studies in Fuzzi- ness and Soft Computing, 35, Physica-Verlag, Heidelberg. Davvaz B On H v -rings and fuzzy H v -ideal. J. Fuzzy Math.6: Davvaz B Fuzzy H v -groups. Fuzzy Sets and Systems 101: Davvaz B Intuitionistic fuzzy hyperideals in semihypergroups. Bull. Malays. Math. Sci. Soc. 291: Davvaz B & Majumder S Atanassov s intuitionistic fuzzy interior ideals of G-semigroups, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics 73 3: Davvaz B, Fathi M & Salleh A. R Fuzzy hypergroup based on fuzzy universal sets. Information Sciences 1801: Dib K.A On fuzzy spaces and fuzzy group theory. Information Sciences 80: Dib, K.A, Galhum N. & Hassan A.M The fuzzy ring and fuzzy ideals. Fuzzy Math 964: Dib, K.A& Galhum N The fuzzy ideals and fuzzy bi-ideals in fuzzy semigroups. Fuzzy sets and systems 924: Marty F Sur une generalization de la notion de group. Proceedings of the 8th Congress Mathematics Scandenaves, Stockholm 11:

20 636 K. S. Abdulmula and A. R. Salleh Fathi M & Salleh A. R Intutionistic fuzzy groups. Asian Journal of Algebra 2: Fathi M Intuitionistic fuzzy algebra and fuzzy hyperalgebra. PhD thesis. Universiti Kebangsaan Malaysia. Rosenfeld A Fuzzy groups. J. Math. Anal. Appl. 35: Zadeh L Fuzzy sets. Inform. Control 8: Received: January, 2012