A new generalization of fuzzy ideals in LA-semigroups
aa r X i v : . [ m a t h . L O ] O c t A NEW GENERALIZATION OF FUZZY IDEALS INLA-SEMIGROUPS
S. ABDULLAH, M ATIQUE KHAN, AND M. ASLAM
Abstract.
In this article, the concept of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups,( ∈ γ , ∈ γ ∨ q δ )-fuzzy left(right) ideals, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy generalized bi-idealsand ( ∈ γ , ∈ γ ∨ q δ )-fuzzy bi-ideals of an LA-semigroup are introduced. The givenconcept is a generalization of ( ∈ , ∈ ∨ q )-fuzzy LA-subsemigroups, ( ∈ , ∈ ∨ q )-fuzzy left(right) ideals, ( ∈ , ∈ ∨ q )-fuzzy generalized bi-ideals and ( ∈ , ∈ ∨ q )-fuzzy bi-ideals of an LA-semigroup. We also give some examples of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups ( left, right, generalized bi- and bi) ideals of an LA-semigroup. we prove some fundamental results of these ideals. We charac-terize ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left(right) ideals, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy generalized bi-ideals and ( ∈ γ , ∈ γ ∨ q δ )-fuzzy bi-ideals of an LA-semigroup by the propertiesof level sets. The given concept is a generalization of ( ∈ , ∈ ∨ q )-fuzzy LA-subsemigroups, ( ∈ , ∈ ∨ q )-fuzzy left(right) ideals, ( ∈ , ∈ ∨ q )-fuzzy generalizedbi-ideals and ( ∈ , ∈ ∨ q )-fuzzy bi-ideals of an LA-semigroup. We also give someexamples of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups ( left, right, generalized bi-and bi) ideals of an LA-semigroup. Introduction
The concept of LA-semigroup was first indroduced by Kazim and Naseerudin[18, 26]. A non-empty set S with binary operation ∗ is said to be an LA-semigroupif identity ( x ∗ y ) ∗ z = ( z ∗ y ) ∗ x for all x, y, z ∈ S holds. Later, Q. Mushtaq andothers have been investigated the structure further and added many useful resultsto theory of LA-semigroups see [20, 21, 22]. Ideals of LA-semigroups were definedby Mushtaq and Khan in his paper [23]. In [16], Khan and Ahmad characterizedLA-semigroup by their ideals. L. A. Zadeh introduced the fundamental concept ofa fuzzy set [32] in 1965. On the basis of this concept, mathematicians initiated anatural framework for generalizing some basic notions of algebra, e.g group theory,set theory, ring theory, topology, measure theory and semigroup theory etc. Theimportance of fuzzy technology in information processing is increasing day by day.In granular computing, the information is represented in the form of aggregates,called granules. Fuzzy logic is very useful in modeling the granules as fuzzy sets.Bargeila and Pedrycz considered this new computing methodology in [3]. Pedryczand Gomide in [27] considered the presentation of update trends in fuzzy set theoryand it’s applications. Foundations of fuzzy groups are laid by Rosenfeld in [29]. Thetheory of fuzzy semigroups was initiated by Kuroki in his papers [13, 14]. Recently,Khan and Khan introduced fuzzy ideals in LA-semigroups [17]. In [24] Murali gavethe concept of belongingness of a fuzzy point to a fuzzy subset under a naturalequivalence on a fuzzy subset. In [28] the idea of quasi-coincidence of a fuzzy Key words and phrases.
LA-semigroup, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left(right) ideals, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy bi-ideals, ( ∈ γ , ∈ γ ∨ q δ )-generalized bi-ideals. point with a fuzzy set is defined. These two ideas played a vital role in generatingsome different types of fuzzy subgroups. Using these ideas Bhakat and Das [4, 8]gave the concept of ( α, β )-fuzzy subgroups, where α, β ∈ {∈ , q, ∈ ∨ q, ∈ ∧ q } and α = ∈ ∧ q . These fuzzy subgroups are further studied in [6, 5]. The concept of( ∈ , ∈ ∨ q )-fuzzy subgroups is a viable generalization of Rosenfeld’s fuzzy subgroups,( ∈ , ∈ ∨ q )-fuzzy subrings and ideals are defined In [7], S.K. Bhakat and P. Dasintroduced the ( ∈ , ∈ ∨ q )-fuzzy subrings and ideals. Davvaz gave the concept of( ∈ , ∈ ∨ q )-fuzzy subnearrings and ideals of a near ring in [10]. Jun and Song initiatedthe study of ( α, β )-fuzzy interior ideals of a semigroup in [12]. In [15] Kazanci andYamak studied ( ∈ , ∈ ∨ q )-fuzzy bi-ideals of a semigroup. In [30] regular semigroupsare characterized by the properties of ( ∈ , ∈ ∨ q )-fuzzy ideals. Aslam et al definedgeneralized fuzzy Γ-ideals in Γ-LA-semigroups [2]. In [1], Abdullah et al give newgeneralization of fuzzy normal subgroup and fuzzy coset of groups. Generalizingthe idea of the quasi-coincident of a fuzzy point with a fuzzy subset Jun [11] defined( ∈ , ∈ ∨ q k )-fuzzy subalgebras in BCK/BCI-algebras. In [31], ( ∈ , ∈ ∨ q k )-fuzzy idealsof semigroups are introduced. Further generalizing the concept, ( ∈ , ∈ ∨ q k ), J. Zhanand Y. Yin defined ( ∈ γ , ∈ γ ∨ q δ )-fuzzy ideals of near rings [33]. In [25], ( ∈ γ , ∈ γ ∨ q δ )-fuzzy ideals of BCI-algebras are introduced.In this article, the concept of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left(right) ideals, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy generalized bi-ideals and ( ∈ γ , ∈ γ ∨ q δ )-fuzzy bi-ideals of an LA-semigroup are introduced. The given concept is a gener-alization of ( ∈ , ∈ ∨ q )-fuzzy LA-subsemigroups, ( ∈ , ∈ ∨ q )-fuzzy left(right) ideals,( ∈ , ∈ ∨ q )-fuzzy generalized bi-ideals and ( ∈ , ∈ ∨ q )-fuzzy bi-ideals of an LA-semigroup.We also give some examples of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups ( left, right,generalized bi- and bi) ideals of an LA-semigroup. we prove some fundamen-tal results of these ideals. We characterize ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left(right) ideals,( ∈ γ , ∈ γ ∨ q δ )-fuzzy generalized bi-ideals and ( ∈ γ , ∈ γ ∨ q δ )-fuzzy bi-ideals of an LA-semigroup by the properties of level sets. We prove that for µ be a fuzzy subsetof S . Then, the following hold (i) µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy ideal of S if and onlyif µ γr ( = φ ) is an ideal of S for all r ∈ ( γ, δ ] . (ii) If 2 δ = 1 + γ, then µ is an( ∈ γ , ∈ γ ∨ q δ )-fuzzy ideal of of S if and only if µ δr ( = φ ) is an ideal of S for all r ∈ ( δ, . (iii) If 2 δ = 1 + γ, then µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy ideal of of S if and onlyif [ µ ] δr ( = φ ) is an ideal of S for all r ∈ ( γ, . (iv) µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy idealof S if and only if U ( µ ; r )( = φ ) is an ideal of S for all r ∈ ( γ, δ ] . Similarly, weprove that for ( ∈ γ , ∈ γ ∨ q δ )-fuzzy generalized bi-ideals, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy bi-idealsand ( ∈ γ , ∈ γ ∨ q δ )-fuzzy quasi-ideals of an LA-semigroup.2. Preliminaries
An LA-subsemigroup of S means a non-empty subset A of S such that A ⊆ A. By a left (right) ideals of S we mean a non-empty subset I of S such that SI ⊆ I ( IS ⊆ I ) . An ideal I is said to be two sided or simply ideal if it is both left andright ideal. An LA-subsemigroup A is called bi-ideal if ( BS ) B ⊆ A. A non-emptysubset B is called generalized bi-ideal if ( BS ) B ⊆ A. A non-empty subset Q iscalled a quasi-ideal if QS ∩ SQ ⊆ Q. A non-empty subset A is called interior ideal ifit is LA-subsemigroup of S and ( SA ) S ⊆ A. An LA-semigroup S is called regularif for each a ∈ S there exists x ∈ S such that a = ( ax ) a. An LA-semigroup S iscalled intra-regular if for each a ∈ S there exist x, y ∈ S such that a = ( xa ) y. Inan LA-semigroup S, the following law hold, (1) ( ab ) c = ( ab ) c, for all a, b, c ∈ S. (2) NEW GENERALIZATION OF FUZZY IDEALS IN LA-SEMIGROUPS 3 ( ab ) ( cd ) = ( ac ) ( bd ) , for all a, b, c, d ∈ S. If an LA-semigroup S has a left identity e, then the following law holds, (3) ( ab ) ( cd ) = ( db ) ( ca ) , for all a, b, c, d ∈ S. (4) a ( bc ) = b ( ac ) , for all a, b, c ∈ S. A fuzzy subset µ of the form µ ( y ) = (cid:26) t ( = 0) if y = x y = x is said to be a fuzzy point with support x and value t and is denoted by x t .Afuzzy point x t is said to be ”belong to”(res.,”quasicoincident with”) a fuzzy set µ, written as x t ∈ µ (repectively , x t qµ ) if µ ( x ) ≥ t (repectively, µ ( x ) + t > . Wewrite x t ∈ ∨ qµ if x t ∈ µ or x t qµ. If µ ( x ) < t (respectively, µ ( x ) + t ≤ , then wewrite x t ∈ µ (repectively, x t qµ ) . We note that ∈ ∨ q means that ∈ ∨ q does not hold.Generalizing the concept of x t qµ, Y. B. Jun [11] defined x t q k µ, where k ∈ [0 ,
1] as x t q k µ if µ ( x ) + t + ˙ k > x t ∈ ∨ q k µ if x t ∈ µ or x t q k µ. A fuzzy subset µ of S is a function µ : S → [0 , . For any two fuzzy subsets µ and ν of S, µ ⊆ ν means µ ( x ) ≤ ν ( x ) for all x in S. The fuzzy subsets µ ∩ ν and µ ∪ ν of S are defined as( µ ∩ ν ) ( x ) = min { µ ( x ) , ν ( x ) } = µ ( x ) ∧ ν ( x )( µ ∪ ν ) ( x ) = max { µ ( x ) , ν ( x ) } = µ ( x ) ∨ ν ( x )for all x in S. If { µ i } i ∈ I is a faimly of fuzzy subsets of S, then V i ∈ I µ i and W i ∈ I µ i arefuzzy subsets of S defined by (cid:18) V i ∈ I µ i (cid:19) ( x ) = min { µ i } i ∈ I _ i ∈ I µ i ! ( x ) = max { µ i } i ∈ I For any two subsets µ and ν of S, the product µ ◦ ν is defined as( µ ◦ ν ) ( x ) = (cid:26) W x = yz { µ ( y ) ∧ ν ( z ), if there exist y, z ∈ S, such that x = yz Definition 1. [17]
A fuzzy subset µ of an LA-semigroup S is called fuzzy LA-subsemigroup S if µ ( xy ) ≥ µ ( x ) ∧ µ ( y ) for all x, y ∈ S. Definition 2. [17]
A fuzzy subset µ of an LA-semigroup S is called fuzzy left(right)ideal of S if µ ( xy ) ≥ µ ( y )( µ ( xy ) ≥ µ ( x )) for all x, y ∈ S. Definition 3. [17]
An LA-subsemigroup µ of an LA-semigroup S is called fuzzybi-ideal of S if µ (( xy ) z ) ≥ µ ( x ) ∧ µ ( z ) for all x, y ∈ S. Definition 4. [17]
A fuzzy subset µ of an LA-semigroup S is called fuzzy generalizedbi-ideal of S if µ (( xy ) z ) ≥ µ ( x ) ∧ µ ( z ) for all x, y ∈ S. Definition 5. [17]
Let µ be a fuzzy subset of an LA-semigroup S, then for all t ∈ (0 , , the set µ t = { x ∈ S | µ ( x ) ≥ t } is called a level subset of S. S. ABDULLAH, M ATIQUE KHAN, AND M. ASLAM
3. ( ∈ γ , ∈ γ ∨ q δ ) -FUZZY IDEALS Generalizing the notion of ( ∈ , ∈ ∨ q ) , in [25, 33] ( ∈ γ , ∈ γ ∨ q δ )-fuzzy ideals of nearrings and BCI-algebras are introduced. Let γ, δ ∈ [0 ,
1] be such that γ < δ.
Forfuzzy point x t and fuzzy subset µ of S, we say( i ) x t ∈ γ µ if µ ( x ) ≥ t > γ. ( ii ) x t q δ µ if µ ( x ) + t > δ. ( iii ) x t ∈ γ ∨ q δ µ if x t ∈ γ µ or x t q δ µ. ( iv ) x t ∈ γ ∨ q δ µ if x t ∈ γ µ or x t q δ µ .In this section, we introduce the concept of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup,( ∈ γ , ∈ γ ∨ q δ )-fuzzy left(right) ideal, ( ∈ γ , ∈ γ ∨ q δ )-fuzzy generalized bi-ideal and ( ∈ γ , ∈ γ ∨ q δ )-fuzzy bi-ideal of an LA-semigroup S . We also study some basic properties of theseideals. Definition 6.
A fuzzy subset µ of an LA-semigroup S is called an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S if for all a, b ∈ S and t, r ∈ ( γ, , x t , y r ∈ γ µ impliesthat ( ab ) t ∧ r ∈ γ ∨ q δ µ. Remark 1.
Every fuzzy LA-subsemigroup and every ( ∈ , ∈ ∨ q ) -fuzzy LA-subsemigroupis an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup but the converse is not true. Example 1.
Let S = { , , , } be an LA -semigroup with the multiplication de-fined by the following Caley table ∗ Define fuzzy subset µ of S by µ (1) = 0 . , µ (2) = 0 . , µ (3) = 0 . , µ (4) = 0 . . Then, ( i ) µ is an ( ∈ . , ∈ . ∨ q . ) -fuzzy LA-subsemigroup. ( ii ) µ is not an ( ∈ , ∈ ∨ q ) -fuzzy LA-subsemigroup because . , . ∈ µ, but (2 ∗ . ∈ ∨ qµ. ( iii ) µ is not fuzzy LA-subsemigroup. The next theorems provide the relationship between ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups of S and crisp LA-subsemigroups. Theorem 1.
Let A be a non-empty subset of S. Then, A is an LA-subsemigroupof S if and only if the fuzzy subset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S. Proof.
Let A be an LA-subsemigroup of S and a, b ∈ S and t, r ∈ ( γ, a t , b r ∈ γ µ. Then, µ ( a ) ≥ t > γ and µ ( b ) ≥ r > γ. Thus, µ ( a ) ≥ δ and µ ( b ) ≥ δ . Hence, a, b ∈ A. Since A is an LA-subsemigroup of S, so we have ab ∈ A, which implies that µ ( ab ) ≥ δ. If t ∧ r ≤ δ, then µ ( ab ) ≥ δ ≥ t ∧ r > γ and so( ab ) t ∧ r ∈ γ ∨ q δ µ. If t ∧ r > δ then, µ ( ab ) + t ∧ r > µ ( ab ) + δ > δ + δ, which implies NEW GENERALIZATION OF FUZZY IDEALS IN LA-SEMIGROUPS 5 that µ ( ab ) + t ∧ r > δ and so ( ab ) t ∧ r ∈ γ ∨ q δ µ. Hence, µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzyLA-subsemigroup of S. Conversely, assume that µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S. Let a, b ∈ A, then µ ( a ) ≥ δ and µ ( b ) ≥ δ. By hypothesis, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ, which implies that µ ( ab ) ∨ γ ≥ δ ∧ δ ∧ δ = δ, that is µ ( ab ) ∨ γ ≥ δ. Since γ < δ, therefore µ ( ab ) ≥ δ. Hence, ab ∈ A. Thus, A is an LA-subsemigroup of S. (cid:3) Corollary 1.
Let A be a non-empty subset of an LA-semigroup S. Then, A isan LA-subsemigroup of S if and only if χ A , the characteristic function of A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S. Theorem 2.
Let δ = 1 + γ and A be an LA-subsemigroup of S. Then, the fuzzysubset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( q δ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S. Proof.
Let 2 δ = 1 + γ and A be an LA-subsemigroup of S. Let a, b ∈ S and t, r ∈ ( γ, a t , b r q δ µ , then µ ( a ) + t > δ and µ ( b ) + r > δ. This impliesthat µ ( a ) > δ − t ≥ δ − γ and µ ( b ) > δ − r ≥ δ − γ, thus ab ∈ A, which implies that µ ( ab ) ≥ δ. If t ∧ r ≤ δ, then µ ( ab ) ≥ δ ≥ t ∧ r > γ and so( ab ) t ∧ r ∈ γ ∨ q δ µ. If t ∧ r > δ, then µ ( ab ) + t ∧ r > µ ( ab ) + δ > δ + δ, which impliesthat µ ( ab ) + t ∧ r > δ and so ( ab ) t ∧ r ∈ γ ∨ q δ µ. Hence, µ is an ( q δ , ∈ γ ∨ q δ )-fuzzyLA-subsemigroup of S. (cid:3) Theorem 3.
A fuzzy subset µ of an LA-semigroup S is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzyLA-subsemigroup of S if and only if µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ for all a, b ∈ S. Proof.
Let µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S. To show that µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ, for all a, b ∈ S, we discuss the following two cases:( a ) µ ( a ) ∧ µ ( b ) ≤ δ. ( b ) µ ( a ) ∧ µ ( b ) > δ. Case(a): If there exist a, b ∈ S such that µ ( ab ) ∨ γ < µ ( a ) ∧ µ ( b ) ∧ δ. Then, µ ( ab ) ∨ γ < t < µ ( a ) ∧ µ ( b ) , which implies that a t ∈ γ µ, b t ∈ γ µ but ( ab ) t ∈ γ ∨ q δ µ, which is a contradiction. Hence, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ. for all a, b ∈ S. Case(b):If there exist a, b ∈ S such that µ ( ab ) ∨ γ < µ ( a ) ∧ µ ( b ) ∧ δ. Then, µ ( ab ) ∨ γ < t < δ and so a δ ∈ γ µ, b δ ∈ γ µ but ( ab ) δ ∈ γ ∨ q δ µ, which is a contradiction. Hence, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ for all a, b ∈ S. Conversely, assume that µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ for all a, b ∈ S. We are toshow that µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S. For this let a t , b r ∈ γ µ. Then, µ ( a ) ≥ t > γ and µ ( b ) ≥ r > γ. Now by hypothesis, we have µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ ≥ t ∧ r ∧ δ. If t ∧ r ≥ δ, then µ ( ab ) ∨ γ ≥ δ > γ and so µ ( ab ) > γ, that is µ ( ab ) + t ∧ r > δ. Hence, ( ab ) t ∧ r ∈ γ ∨ q δ µ. If t ∧ r < δ, then µ ( ab ) ∨ γ ≥ t ∧ r and so µ ( ab ) ≥ t ∧ r > γ. Hence, ( ab ) t ∧ r ∈ γ ∨ q δ µ. (cid:3) Remark 2.
For any ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup µ of S, we can conclude(i) If γ = 0 and δ = 1 , then µ is a fuzzy LA-subsemigroup of S. (ii) If γ = 0 and δ = 0 . , then µ is an ( ∈ , ∈ ∨ q ) - fuzzy LA-subsemigroup of S. Theorem 4.
Let µ be a fuzzy subset of S. Then,(i) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S if and only if µ γr ( = φ ) isan LA-subsemigroup of S for all r ∈ ( γ, δ ] . S. ABDULLAH, M ATIQUE KHAN, AND M. ASLAM (ii) If δ = 1 + γ, then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S if andonly if µ δr ( = φ ) is an LA-subsemigroup of S for all r ∈ ( δ, . (iii) If δ = 1 + γ, then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S if andonly if [ µ ] δr ( = φ ) is an LA-subsemigroup of S for all r ∈ ( γ, . (iv) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup of S if and only if U ( µ ; r )( = φ ) is an LA-subsemigroup of S for all r ∈ ( γ, δ ] . Proof. (i) Let µ be an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroupof S and a, b ∈ µ γr , for all r ∈ ( γ, δ ] . Then, a r , b r ∈ γ µ , that is µ ( a ) ≥ r > γ and µ ( b ) ≥ r > γ. By hypothesis, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ, which implies that µ ( ab ) ∨ γ ≥ r ∧ r ∧ δ = r ∧ δ. Since r ∈ ( γ, δ ] , so r ≤ δ. Thus, µ ( ab ) ≥ r > γ, which implies that ab ∈ µ γr . Hence, µ γr isLA-subsemigroup of S. Conversely, assume that µ γr ( = φ ) is an LA-subsemigroup of S for all r ∈ ( γ, δ ] . Let a, b ∈ S, such that µ ( ab ) ∨ γ < µ ( a ) ∧ µ ( b ) ∧ δ. Select r ∈ ( γ, δ ] , such that µ ( ab ) ∨ γ < r ≤ µ ( a ) ∧ µ ( b ) ∧ δ. Then, a r , b r ∈ γ µ but ( ab ) r ∈ γ ∨ q δ µ. Which is acontradiction. Thus, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ. Hence, µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzyLA-subsemigroup of S .(ii) Let µ be an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S and a, b ∈ µ δr for all r ∈ ( δ, . Then, a r , b r q δ µ , that is µ ( a ) + r > δ and µ ( b ) + r > δ. Now, µ ( a ) + r > δ implies that µ ( a ) > δ − r ≥ δ − γ. Similarly, µ ( b ) > γ. By hypothesis, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ, so we have µ ( ab ) ≥ µ ( a ) ∧ µ ( b ) ∧ δ > (2 δ − r ) ∧ (2 δ − r ) ∧ δ. Since r ∈ ( δ, . So, r > δ which implies that 2 δ − r ≤ δ. Thus, µ ( ab ) > δ − r or µ ( ab ) + r > δ. Thus, ab ∈ µ δr . Hence, µ δr is LA-subsemigroup of S. Conversely, assume that µ δr ( = φ ) is an LA-subsemigroup of S for all r ∈ ( δ, . Let a, b ∈ S such that µ ( ab ) ∨ γ < µ ( a ) ∧ µ ( b ) ∧ δ. Select r ∈ ( γ, δ ] such that µ ( ab ) ∨ γ < r ≤ µ ( a ) ∧ µ ( b ) ∧ δ. Then, a r , b r ∈ γ µ but ( ab ) r ∈ γ ∨ q δ µ. Which is acontradiction. Thus, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ. Hence, µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzyLA-subsemigroup of S. (iii) Let µ be an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S and a, b ∈ [ µ ] δr , forall r ∈ ( γ, . Then, a r , b r ∈ γ ∨ q δ µ , that is µ ( a ) ≥ r > γ or µ ( a ) + r > δ whichimplies that µ ( a ) ≥ r > γ or µ ( a ) > δ − r ≥ δ − γ. Similarly, µ ( b ) ≥ r > γ or µ ( b ) > δ − r ≥ δ − γ. By hypothesis, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ which impliesthat µ ( ab ) ≥ µ ( a ) ∧ µ ( b ) ∧ δ. Case(i): If r ∈ ( γ, δ ] , then r ≤ δ. This implies that2 δ − r ≥ δ ≥ r. Thus, µ ( ab ) > r ∧ r ∧ δ = r > γ or µ ( ab ) > r ∧ (2 δ − r ) ∧ δ = r > γ or µ ( ab ) > (2 δ − r ) ∧ (2 δ − r ) ∧ δ = δ ≥ r > γ. Hence, ab ∈ µ δr . Case(ii): If r ∈ ( δ, , then r > δ which implies that 2 δ − r < δ < r, and so µ ( ab ) > r ∧ r ∧ δ = δ > δ − r or µ ( ab ) > r ∧ δ − r ∧ δ = 2 δ − r or µ ( ab ) > (2 δ − r ) ∧ (2 δ − r ) ∧ δ = 2 δ − r. Thus, ab ∈ µ δr . Hence, µ δr is LA-subsemigroup of S. Conversely, assume that µ δr ( = φ ) is LA-subsemigroup of S for all r ∈ ( δ, . Let a, b ∈ S, such that µ ( ab ) ∨ γ < µ ( a ) ∧ µ ( b ) ∧ δ. Select r ∈ ( γ, δ ] such that µ ( ab ) ∨ γ < r ≤ µ ( a ) ∧ µ ( b ) ∧ δ. Then, a r ∈ γ µ, and b r ∈ γ µ but ( ab ) r ∈ γ ∨ q δ µ, which is a contradiction. Thus, µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ. Hence, µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S .(iv) Let µ be an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S and a, b ∈ U ( µ ; r )for some r ∈ ( γ, δ ]. Then µ ( a ) ≥ r and µ ( b ) ≥ r. Since µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzyLA-subsemigroup of S, we have µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ ≥ r ∧ δ = r, whichimplies that µ ( ab ) ≥ r (because r > γ ). Thus ab ∈ U ( µ ; r ) . Hence U ( µ ; r ) is anLA-subsemigroup of S. Conversely, assume that U ( µ ; r ) is an LA-subsemigroup of S for all r ∈ ( γ, δ ] . Suppose that ther exist a, b ∈ S such that µ ( ab ) ∨ γ < µ ( a ) ∧ µ ( b ) ∧ δ. Select r NEW GENERALIZATION OF FUZZY IDEALS IN LA-SEMIGROUPS 7 ∈ ( γ, δ ] such that µ ( ab ) ∨ γ < r ≤ µ ( a ) ∧ µ ( b ) ∧ δ. Thus a ∈ U ( µ ; r ) and b ∈ U ( µ ; r )but ab / ∈ U ( µ ; r ) , which is a contradiction. Hence µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ andso µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S . (cid:3) If we take γ = 0 and δ = 0 . Corollary 2.
Let µ be a fuzzy set of S. Then,(i) µ is an ( ∈ , ∈ ∨ q ) -fuzzy LA-subsemigroup of S if and only if µ r ( = φ ) LA-subsemigroup of S for all r ∈ (0 , . . (ii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy LA-subsemigroup of S if and only if Q ( µ ; r )( = Φ) LA-subsemigroup of S for all r ∈ (0 . , . (iii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy LA-subsemigroup of S if and only if [ µ ] r ( = Φ) LA-subsemigroup of S for all r ∈ (0 , . Lemma 1.
The intersection of any faimly of ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroupsis an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy LA-subsemigroup .Proof. Let { µ i } i ∈ I be a faimly of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups of S and x, y ∈ S. Then,(( V i ∈ I µ i )( xy )) ∨ γ = ( V i ∈ I µ i ( xy )) ∨ γ = ( V i ∈ I (( µ i ( xy )) ∨ γ )) ≥ ( V i ∈ I { µ i ( x ) ∧ µ i ( y ) ∧ δ } )= ( V i ∈ I µ i ( x )) ∧ ( V i ∈ I µ i ( y )) ∧ δ = ( V i ∈ I µ i )( x )) ∧ ( V i ∈ I µ i )( y )) ∧ δ. Hence, V i ∈ I µ i is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroups of S. (cid:3) Definition 7.
A fuzzy subset µ of an LA-semigroup S is called an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S if for all a, s ∈ S and t ∈ ( γ, , a t ∈ γ µ implies that ( sa ) t ∈ γ ∨ q δ µ (( as ) t ∈ γ ∨ q δ µ ) . Remark 3.
Every fuzzy left(right) ideal and every ( ∈ , ∈ ∨ q ) -fuzzy left(right) idealis an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal but the converse is not true. Example 2.
Let S = { , , , } be an LA-semigroup with the following multipli-cation table. ∗ Define fuzzy subset µ of S by µ (1) = 0 . µ (2) , µ (3) = 0 . µ (4) . Then,(i) µ is an ( ∈ . , ∈ . ∨ q . ) -fuzzy left ideal of S. (ii) µ is not an ( ∈ , ∈ ∨ q ) fuzzy ideal of S. Because . ∈ µ and . ∈ µ but (1 ∗ . ∈ ∨ qµ S. ABDULLAH, M ATIQUE KHAN, AND M. ASLAM (iii) µ is not fuzzy ideal of S. Remark 4.
For any ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal µ of S we can conclude(i) if γ = 0 and δ = 1 , then µ is fuzzy left(right) ideal of S. (ii) if γ = 0 and δ = 0 . then µ is an ( ∈ , ∈ ∨ q ) - fuzzy left(right) ideal of S. Theorem 5.
Let A be a non-empty subset of S. Then, A is a left(right) ideal of S if and only if the fuzzy subset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S. Proof.
Proof is similar to the proof of Theorem 4. (cid:3)
Corollary 3.
Let A be a non-empty subset of an LA-semigroup S. Then, A isa left(right) ideal of S if and only if χ A , the characteristic function of A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S. Theorem 6.
Let δ = 1 + γ and A be a left(right) ideal of S. Then, the fuzzy subset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( q δ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S. Proof.
Proof is similar to the proof of Theorem 5. (cid:3)
Theorem 7.
A fuzzy subset µ of an LA-semigroup S is an ( ∈ γ , ∈ γ ∨ q δ ) - fuzzyleft(right) ideal of an LA-semigroup S if and only if µ ( sa ) ∨ γ ≥ µ ( a ) ∧ δ ( µ ( as ) ∨ γ ≥ µ ( a ) ∧ δ ) for all a, s ∈ S. Proof.
Proof is similar to the proof of Theorem 6. (cid:3)
Theorem 8.
Let µ be a fuzzy subset of S . Then,(i) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S if and only if µ γr ( = φ ) isleft(right) ideal of S for all r ∈ ( γ, δ ] . (ii) If δ = 1 + γ. Then, µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S if andonly if µ δr ( = φ ) is left(right) ideal of S for all r ∈ ( δ, . (iii) If δ = 1 + γ. Then, µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S if andonly if [ µ ] δr ( = φ ) is left(right) ideal of S for all r ∈ ( γ, . (iv) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal of S if and only if U ( µ ; r )( = φ ) is a left(right) ideal of S for all r ∈ ( γ, δ ] . Proof.
Proof is similar to the proof of Theorem 7. (cid:3)
Corollary 4.
Let µ be a fuzzy set of S. Then,(i) µ is an ( ∈ , ∈ ∨ q ) -fuzzy left(right) ideal of S if and only if µ r ( = φ ) left(right)ideal of S for all r ∈ (0 , . . (ii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy left(right) ideal of S if and only if Q ( µ ; r )( = Φ) left(right) ideal of S for all r ∈ (0 . , . (iii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy left(right) ideal of S if and only if [ µ ] r ( = Φ) left(right) ideal of S for all r ∈ (0 , . Theorem 9.
The intersection of any faimly of ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) idealsis an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal. NEW GENERALIZATION OF FUZZY IDEALS IN LA-SEMIGROUPS 9
Proof.
Let { µ i } i ∈ I be a faimly of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left ideals of S and a, s ∈ S. Then, (( V i ∈ I µ i )( sa )) ∨ γ = ( V i ∈ I µ i ( sa )) ∨ γ = ( V i ∈ I (( µ i ( sa )) ∨ γ )) ≥ ( V i ∈ I { µ i ( a ) ∧ δ } )= ( V i ∈ I µ i ( a )) ∧ δ = ( V i ∈ I µ i )( x ) ∧ δ. Hence, V i ∈ I µ i is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left ideal of S. (Similarly, we can prove forright ideals). (cid:3) Lemma 2.
The union of any faimly of ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideals is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideals .Proof. Let { µ i } i ∈ I be a faimly of ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left ideals of S and s, a ∈ S. Then, (cid:18) W i ∈ I µ i (cid:19) ( sa ) = W i ∈ I ( µ i ( ab )) . Since each µ i is fuzzy left ideals of S, so µ i ( sa ) ∨ γ ≥ µ i ( a ) ∧ δ, for all i ∈ I. Thus, _ i ∈ I µ i ! ( sa ) ∨ γ = _ i ∈ I ( µ i ( sa ) ∨ γ ) ≥ _ i ∈ I ( µ i ( a ) ∧ δ )= _ i ∈ I µ i ( a ) ! ∧ δ = _ i ∈ I µ i ! ( a ) ∧ δ. Hence, W i ∈ I µ i is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy left ideal of S. (Similarly, we can prve forright ideals). (cid:3) Definition 8.
A fuzzy subset µ of an LA-semigroup S is called an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalized bi- ideal if for all a, b, s ∈ S and t, r ∈ ( γ, , a t ∈ γ µ, b r ∈ r µ implies that (( as ) b ) t ∧ r ∈ γ ∨ q δ µ. Remark 5.
Every fuzzy generalized bi-ideal and every ( ∈ , ∈ ∨ q ) -fuzzy generalizedbi- ideal is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalized bi-ideal but the converse is not true. Example 3. ∗ ‘Define a fuzzy subset µ of S by µ (1) = 0 . , µ (2) = 0 . , µ (3) = 0 . and µ (4) = 0 . Then, ( i ) µ is an ( ∈ . , ∈ . ∨ q . ) -fuzzy generalized bi-ideal. ( ii ) µ is not an ( ∈ , ∈ ∨ q ) -fuzzy generalized bi-ideal because . ∈ µ, but ((1 ∗ ∗ . ∈ ∨ qµ. ( iii ) µ is not fuzzy generalized bi-ideal. Theorem 10.
Let A be a non-empty subset of S. Then, A is a generalized bi-idealof S if and only if the fuzzy subset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalized bi-ideal of S. Proof.
Proof is similar to the proof of Theorem 4. (cid:3)
Corollary 5.
Let A be a non-empty subset of an LA-semigroup S. Then, A is ageneralized bi-ideal of S if and only if χ A , the characteristic function of A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalized bi-ideal of S. Theorem 11.
Let δ = 1 + γ and A be an generalized bi-ideal of S. Then, the fuzzysubset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( q δ , ∈ γ ∨ q δ ) -fuzzy generalized bi-ideal of S. Proof.
Proof is similar to the proof of Theorem 5 . (cid:3) Theorem 12.
A fuzzy subset µ of an LA-semigroup S is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzygeneralised bi-ideal of S if and only if for all a, b, s ∈ S and t, r ∈ ( γ, , µ (( as ) b ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ. Proof.
Proof is similar to the proof of Theorem 6. (cid:3)
Remark 6.
For any ( ∈ γ , ∈ γ ∨ q δ ) - fuzzy generalized bi-ideal µ of S we can conclude(i)if γ = 0 , δ = 1 , then µ is fuzzy generalized bi-idealof S. (ii)if γ = 0 , δ = 0 . , then µ is an ( ∈ , ∈ ∨ q ) -fuzzy generalized bi-ideal of S. Theorem 13.
Let µ be a fuzzy subset of S . Then,(i) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalised bi-ideal of S if and only if µ γr ( = φ ) isgeneralised bi-ideal of S for all r ∈ ( γ, δ ] . (ii) If δ = 1 + γ, then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalised bi-ideal of of S ifand only if µ δr ( = φ ) is generalised bi-ideal of of S for all r ∈ ( δ, . (iii) If δ = 1 + γ, then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalised bi-ideal of of S ifand only if [ µ ] δr ( = φ ) is generalised bi-ideal of S for all r ∈ ( γ, . (iv) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy generalized bi-ideal of S if and only if U ( µ ; r )( = φ ) is a generalized bi-ideal of S for all r ∈ ( γ, δ ] . Proof.
Proof is similar to the proof of Theorem 7. (cid:3)
If we take γ = 0 and δ = 0 . NEW GENERALIZATION OF FUZZY IDEALS IN LA-SEMIGROUPS 11
Corollary 6.
Let µ be a fuzzy set of S. Then,(i) µ is an ( ∈ , ∈ ∨ q ) -fuzzy generalised bi-ideal of S if and only if µ r ( = φ ) is ageneralised bi-ideal of S for all r ∈ (0 , . . (ii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy generalised bi-ideal of S if and only if Q ( µ ; r )( = Φ) is a generalised bi-ideal of S for all r ∈ (0 . , . (iii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy generalised bi-ideal of S if and only if [ µ ] r ( = Φ) isa generalised bi-ideal of S for all r ∈ (0 , . (iv) µ is an ( ∈ , ∈ ∨ q ) -fuzzy generalized bi-ideal of S if and only if U ( µ ; r )( = φ ) is a generalized bi-ideal of S for all r ∈ (0 , . . Definition 9.
A fuzzy subset µ of an LA-semigroup S is called an ( ∈ γ , ∈ γ ∨ q δ ) fuzzy bi- ideal if for all a, b, s ∈ S and t, r ∈ ( γ, a t ∈ γ µ, b r ∈ r µ implies that(i) ( ab ) t ∧ r ∈ γ ∨ q δ µ. (ii) (( as ) b ) t ∧ r ∈ γ ∨ q δ µ. Remark 7.
Every fuzzy bi- ideal and every ( ∈ , ∈ ∨ q ) -fuzzy bi- ideal is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy bi- ideal but the converse is not true. Example 4. ∗ ‘Define fuzzy subset µ of S by µ (1) = 0 . , µ (2) = 0 . , µ (3) = 0 . , µ (4) = 0 . Then, ( i ) µ is an ( ∈ . , ∈ . ∨ q . ) -fuzzy bi-ideal ( ii ) µ is not an ( ∈ , ∈ ∨ q ) -fuzzy bi-ideal because . ∈ µ, but ((1 ∗ ∗ . ∈ ∨ qµ ( iii ) µ is not fuzzy bi-ideal Theorem 14.
Let A be a non-empty subset of S. Then, A is a bi-ideal of S if andonly if the fuzzy subset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy bi-ideal of S. Proof.
Proof is similar to the proof of Theorem 4. (cid:3)
Corollary 7.
Let A be a non-empty subset of an LA-semigroup S. Then, A is a bi-ideal of S if and only if χ A , the characteristic function of A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzybi-ideal of S. Theorem 15.
Let δ = 1 + γ and A be a bi-ideal of S. Then, the fuzzy subset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( q δ , ∈ γ ∨ q δ ) -fuzzy bi-ideal of S. Proof.
Proof is similar to the proof of Theorem 5. (cid:3)
Theorem 16.
A fuzzy subset µ of an LA-semigroup S is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzybi-ideal of S if and only if for all a, b, s ∈ S and t, r ∈ ( γ, (i) µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ (ii) µ (( as ) b ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ. Proof.
Proof is similar to the proof of Theorem 6. (cid:3)
Remark 8.
For any ( ∈ γ , ∈ γ ∨ q δ ) - fuzzy bi-ideal µ of S we can conclude(i) if γ = 0 , δ = 1 , then µ is fuzzy bi-idealof S. (ii) if γ = 0 , δ = 0 . , then µ is an ( ∈ , ∈ ∨ q ) -fuzzy bi-ideal of S. Theorem 17.
Let µ be a fuzzy subset of S . Then,(i) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy bi-ideal of S if and only if µ γr ( = φ ) is bi-ideal of S for all r ∈ ( γ, δ ] . (ii) If δ = 1 + γ. Then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy bi-ideal of of S if and only if µ δr ( = φ ) is bi-ideal of of S for all r ∈ ( δ, . (iii) If δ = 1 + γ. Then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy bi-ideal of of S if and onlyif [ µ ] δr ( = φ ) is bi-ideal of S for all r ∈ ( γ, . (iv) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy bi-ideal of S if and only if U ( µ ; r )( = φ ) is abi-ideal of S for all r ∈ ( γ, δ ] . Proof.
Proof is similar to the proof of Theorem 7. (cid:3)
If we take γ = 0 and δ = 0 . Corollary 8.
Let µ be a fuzzy set of S. Then(i) µ is an ( ∈ , ∈ ∨ q ) -fuzzy bi-ideal of S if and only if µ r ( = φ ) is bi-ideal of S for all r ∈ (0 , . . (ii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy bi-ideal of S if and only if Q ( µ ; r )( = Φ) is bi-idealof S for all r ∈ (0 . , . (iii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy bi-ideal of S if and only if [ µ ] r ( = Φ) is bi-ideal of S for all r ∈ (0 , . (iv) µ is an ( ∈ , ∈ ∨ q ) -fuzzy bi-ideal of S if and only if U ( µ ; r )( = φ ) is a bi-idealof S for all r ∈ (0 , . . Definition 10.
A fuzzy subset µ of an LA-semigroup c is called an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interior ideal of c if for all a, b, c ∈ S and t, r ∈ ( γ, , the following conditionshold: ( i ) a t , b r ∈ γ µ implies that ( ab ) t ∧ r ∈ γ ∨ q δ µ. ( ii ) c t ∈ γ µ implies that (( ac ) b ) t ∈ γ ∨ q δ µ. Theorem 18.
Let A be a non empty subset of S. Then, A is a interior ideal of S if and only if the fuzzy subset µ of S defined by µ ( x ) = (cid:26) ≥ δ if x ∈ A ≤ γ if x / ∈ A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interior ideal of S. Proof.
Proof is similar to the proof of Theorem 4. (cid:3)
Corollary 9.
Let A be a non-empty subset of S. Then, A is a interior ideal of S if and only if χ A , the characteristic function of A is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interiorideal of S. NEW GENERALIZATION OF FUZZY IDEALS IN LA-SEMIGROUPS 13
Theorem 19.
Let δ = 1 + γ and A be a bi-ideal of S. Then, the fuzzy subset µ of S defined by µ ( a ) = (cid:26) ≥ δ if a ∈ A ≤ γ if a / ∈ A is an ( q δ , ∈ γ ∨ q δ ) -fuzzy bi-ideal of S. Proof.
Proof is similar to the proof of Theorem 5. (cid:3)
Theorem 20.
A fuzzy subset µ of an LA-semigroup S is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzyinterior ideal of S if and only if for all a, b, c ∈ S, (i) µ ( ab ) ∨ γ ≥ µ ( a ) ∧ µ ( b ) ∧ δ. (ii) µ (( ac ) b ) ∨ γ ≥ µ ( c ) ∧ δ. Proof.
Proof is similar to the proof of Theorem 6. (cid:3)
Theorem 21.
Let µ be a fuzzy subset of S . Then,(i) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interior ideal of S if and only if µ γr ( = φ ) is ainterior ideal of S for all r ∈ ( γ, δ ] . (ii) If δ = 1 + γ, then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interior ideal of S if and onlyif µ δr ( = φ ) is interior ideal of S for all r ∈ ( δ, . (iii) If δ = 1 + γ, then µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interior ideal of S if and onlyif [ µ ] δr ( = φ ) is interior ideal of S for all r ∈ ( γ, . (iv) (iv) µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interior ideal of S if and only if U ( µ ; r )( = φ ) is a interior ideal of S for all r ∈ ( r, δ ] . Proof.
Proof is similar to the proof of theorem 7. (cid:3)
If we take γ = 0 and δ = 0 . Corollary 10.
Let µ be a fuzzy set of c. Then,(i) µ is an ( ∈ , ∈ ∨ q ) -fuzzy interior ideal of S, if and only if µ r ( = φ ) interiorideal of S, for all r ∈ (0 , . . (ii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy interior ideal of S if and only if Q ( µ ; r )( = Φ) interiorideal of S, for all r ∈ (0 . , . (iii) µ is an ( ∈ , ∈ ∨ q ) -fuzzy interior ideal of S if and only if [ µ ] r ( = Φ) interiorideal of S ,for all r ∈ (0 , . (iv) µ is an ( ∈ , ∈ ∨ q ) -fuzzy interior ideal of S if and only if U ( µ ; r )( = φ ) is ainterior ideal of S for all r ∈ (0 , . . Theorem 22.
Every ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy ideal of an LA-semigroup is S is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy interior ideal of S. Proof.
Let µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy ideal of S. Then, µ ( xy ) ∨ γ ≥ µ ( x ) ∧ δ ≥ µ ( x ) ∧ µ ( y ) ∧ δ. Thus, µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy LA-subsemigroup of S. Also, for all x, a, y ∈ S, wehave µ (( xa ) y ) ∨ γ = ( µ (( xa ) y ) ∨ γ ) ∨ γ ≥ ( µ ( xa ) ∧ δ ) ∨ γ = ( µ ( xa ) ∨ γ ) ∧ δ ≥ µ ( a ) ∧ δ ∧ δ = µ ( a ) ∧ δ. Hence, µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy interior ideal of S. (cid:3) Definition 11.
A fuzzy subset µ of an LA-semigroup S is called an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy quasi-ideal of S, if it satisfies, µ ( x ) ∨ γ ≥ ( µ ◦
1) ( x ) ∧ (1 ◦ µ ) ( x ) ∧ δ, where denotes the fuzzy subset of S mapping every element of S on . Theorem 23.
Let µ be an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy quasi-ideal of S, then the set µ γ = { a ∈ S | µ ( a ) > γ } is a quasi-ideal of S. Proof.
To prove the required result, we need to show that Sµ γ ∩ µ γ S ⊆ µ γ . Let x ∈ Sµ γ ∩ µ γ S. Then, x ∈ Sµ γ and x ∈ µ γ S. So, x = sa and x = bt for some a, b ∈ µ γ and s, t ∈ S. Thus, µ ( a ) > γ and µ ( b ) > γ. Since(1 ◦ µ ) ( x ) = _ x = yz { y ) ∧ µ ( z ) }≥ { s ) ∧ µ ( a ) } , because x = sa = µ ( a ) . Similarly, ( µ ◦
1) ( x ) ≥ µ ( b ) . Thus, µ ( x ) ∨ γ ≥ ( µ ◦
1) ( x ) ∧ (1 ◦ µ )( x ) ∧ δ ≥ µ ( a ) ∧ µ ( b ) ∧ δ> γ because µ ( a ) > γ, µ ( b ) > γ. Which implies that µ ( x ) > γ. Thus, x ∈ µ γ . Hence, µ γ is a quasi-ideal of S. (cid:3) Remark 9.
Every fuzzy quasi-ideal and ( ∈ , ∈ ∨ q ) -fuzzy quasi-ideal of S is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy quasi-ideal of S but the converse is not true. Example 5.
If we consider the LA-semigroup given in example 4. Then, the fuzzysubset µ defined by µ (1) = 0 , µ (2) = 0 . , µ (3) = 0 . , µ (4) = 0 is(i) an ( ∈ . , ∈ . ∨ q . ) -fuzzy quasi-ideal.(ii) not an ( ∈ , ∈ ∨ q ) -fuzzy quasi-ideal . Indeed ( µ ◦
1) (2) = 0 . ◦ µ )(2) but µ (2) = 0 . (cid:3) ( µ ◦
1) (2) ∧ (1 ◦ µ )(2) ∧ . .(iii) not fuzzy quasi-ideal. Because µ (2) = 0 . (cid:3) ( µ ◦
1) (2) ∧ (1 ◦ µ )(2) = 0 . . Lemma 3.
A non-empty subset Q of an LA-semigroup S is a quasi-ideal of S ifand only if the characteristic function χ Q of Q is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy quasi-idealof S. Proof.
Suppose that Q is a quasi-ideal of S and χ Q is the characteristic function of Q. If x / ∈ Q, then x / ∈ SQ or x / ∈ QS.
Thus, (cid:0) ◦ χ Q (cid:1) ( x ) = 0 or (cid:0) χ Q ◦ (cid:1) ( x ) = 0.So, (cid:0) ◦ χ Q (cid:1) ( x ) ∧ (cid:0) χ Q ◦ (cid:1) ( x ) ∧ δ = 0 ≤ χ Q ( x ) ∨ γ. If x ∈ Q, then χ Q ( x ) ∨ γ =1 ∨ γ = 1 ≥ (cid:0) ◦ χ Q (cid:1) ( x ) ∧ (cid:0) χ Q ◦ (cid:1) ( x ) ∧ δ. Thus, χ Q is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy quasi-ideal of S. Conversely, assume that χ Q is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy quasi-ideal of S. Let x ∈ QS ∩ SQ, then there exists s, t ∈ S and a, b ∈ Q, such that, x = as and x = tb. Now, (cid:0) χ Q ◦ (cid:1) ( x ) = _ x = yz { χ Q ( y ) ∧ z ) }≥ { χ Q ( a ) ∧ s ) } because x = as = 1 ∧
1= 1 . NEW GENERALIZATION OF FUZZY IDEALS IN LA-SEMIGROUPS 15
So, (cid:0) χ Q ◦ (cid:1) ( x ) = 1 . Similarly, (cid:0) ◦ χ Q (cid:1) ( x ) = 1 . Hence, (cid:0) χ Q (cid:1) ( x ) ∨ γ ≥ (cid:0) ◦ χ Q (cid:1) ( x ) ∧ (cid:0) χ Q ◦ (cid:1) ( x ) ∧ δ = 1 ∧ ∧ δ = δ. Thus, (cid:0) χ Q (cid:1) ( x ) ∨ γ ≥ δ. This implies that (cid:0) χ Q (cid:1) ( x ) ≥ δ because γ < δ. So, (cid:0) χ Q (cid:1) ( x ) = 1 . Hence, x ∈ Q. Thus, QS ∩ SQ ⊆ Q. Hence, Q isquasi-ideal of S. (cid:3) Theorem 24.
Every ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left(right) ideal µ of S is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy quasi-ideal of S. Proof.
Let a ∈ S, then(1 ◦ µ ) ( a ) = _ a = xy { x ) ∧ µ ( y ) } = _ a = xy µ ( y ) . This implies that(1 ◦ µ ) ( a ) ∧ δ = _ a = xy µ ( y ) ! ∧ δ = _ a = xy { µ ( y ) ∧ δ }≤ _ a = xy { µ ( xy ) ∨ γ } (because µ is an ( ∈ γ , ∈ γ ∨ q δ ) -fuzzy left ideal of S. )= µ ( a ) ∨ γ. Thus, (1 ◦ µ ) ( a ) ∧ δ ≤ µ ( a ) ∨ γ. Hence, µ ( a ) ∨ γ ≥ (1 ◦ µ ) ( a ) ∧ δ ≥ ( µ ◦
1) ( a ) ∧ (1 ◦ µ )( a ) ∧ δ. Thus, µ is an ( ∈ γ , ∈ γ ∨ q δ )-fuzzy quasi-ideal of S. (Similarly, we canprove for right ideal). (cid:3) References [1] S. Abdullah, M. Aslam, T. A. Khan and M. Naeem, A new type of fuzzy normal subgroupand cosets, Journal of Intelligent and Fuzzy Systems, 2012, DOI 10.3233/IFS-2012-0612.[2] M. Aslam, S. Abdullah and N. Amin, Characterizations of gamma LA-semigroups by gener-alized fuzzy gamma ideals, Int. Journal of Mathematics and Statistics, 11 (2012), 29-50.[3] A. Bargiela and W. Pedrycz, Granular Computing, An Introduction, in: The Kluwer Inter.Series in Engineering and Computer Science, vol. 717, KluweAcademic Publishers, Boston,MA, ISBN: 1-4020-7273-2, 2003, p. xx+452.[4] S. K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51(1992) 235 241.[5] S. K. Bhakat, ( ∈ ∨ q )-level subset, Fuzzy Sets and Systems 103 (1999) 529 533.[6] S. K. Bhakat, ( ∈ , ∈ ∨ q )-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets andSystems 112 (2000) 299 312.[7] S. K. Bhakat and P. Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems 81(1996) 383 393.[8] S. K. Bhakat and P. Das, ( ∈ , ∈ ∨ q )-fuzzy subgroups, Fuzzy Sets and Systems 80 (1996) 359368.[9] A. H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups, Volume II, AmmericanMathematical Socity, 1967.[10] B. Davvaz, ( ∈ , ∈ ∨ q )-fuzzy subnearrings and ideals, Soft Comput. 10 (2006) 206 211.[11] Y. B. Jun, Generalizations of ( ∈ , ∈ ∨ q )-fuzzy subalgebras in BCK/BCI-algebras, Comput.Math. Appl. 58 (2009) 1383 1390.[12] Y. B. Jun and S. Z. Song, Generalized fuzzy interior ideals in semigroups, Inform. Sci. 176,(2006), 3079 3093.[13] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Pauli 28 (1979) 17 21.[14] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5(1981) 203 215. [15] O. Kazanci and S. Yamak, Generalized fuzzy bi-ideals of semigroup, Soft Comput. 12 (2008)1119 1124.[16] M. Khan and N. Ahmad, Characterization of left almost semigroups by their ideals, 2 (3),(2010), 61 - 73.[17] M, Khan and M, N. A. Khan, On fuzzy abel Grassmann’s groupoids, Advanced in FuzzyMathematics, 5(3), (2010), 349-360.[18] M.A. Kazim, M. Naseeruddin. On almost semigroups. The Alig. Bull. Math., 1972, 2: 1 - 7.[19] R.A.R. Monzo. On the structure of abel Grassmann unions of groups. International Journalof Algebra., 2010, 4:1261 - 1275.[20] Q. Mushtaq, M. S. Kamran, On LA-semigroups with weak associative law. Scientific Khyber,1989, 1: 69 - 71.[21] Q. Mushtaq, S. M. Yousuf. On LA-semigroups. The Alig. Bull. Math., 1978, 8: 65 - 70.[22] Q. Mushtaq, S. M. Yousuf. On LA-semigroup defined by a commutative inverse semigroup.Math. Bech., 1988, 40:59 - 62.[23] Q. Mushtaq, M. Khan. Ideals in left almost semigroups. Proceedings of 4th InternationalPure Mathematics Conference, 2003, 65 - 77.[24] V. Murali, Fuzzy points of equivalent fuzzy subsets, Inform. Sci. 158 (2004) 277 288.[25] X. Ma, J. Zhan and Y. B. Jun, New types of fuzzy ideals of BCI-algebras, Neural CompAppl., doi(2011): 10.1007/s00521-011-0558-x.[26] M. Naseeruddin. Some studies in almost semigroups and flocks. Ph.D. thesis. Aligarh MuslimUniversity, Aligarh, India., 1970.[27] W. Pedrycz and F. Gomide, An Introduction to Fuzzy Sets, in: Analysis and Design, witha Foreword by Lotfi A. Zadeh, Complex Adaptive Syst. A Bradford Book, MIT Press, Cam-bridge, MA, ISBN: 0-262-16171-0, 1998, p. xxiv+465.[28] P. M. Pu, Y. M. Liu, Fuzzy topology I, neighborhood structure of a fuzzy point and MooreSmith convergence, J. Math. Anal. Appl. 76 (1980) 571 599.[29] A. Rosenfeld, Fuzzy subgroups, J. Math. Anal. Appl. 35 (1971) 512 517.[30] M. Shabir, Y. B. Jun, Y. Nawaz, Characterizations of regular semigroups by ( α, β )-fuzzyideals, Comput. Math. Appl. 59 (2010) 161 175.[31] M. Shabir, Y. B. Jun and Y. Nawaz, Semigroups characterized by ( ∈ , ∈ ∨ q k )-fuzzy ideals,Comput. Math. Appl. 60 (2010) 1473-1493.[32] L.A. Zadeh, Information and Control 8 (1965) 338 353.[33] J. Zhan and Y. Yin, New types of fuzzy ideal of near rings, Neural Comp Appl.doi(2011):10.1007/s00521-011-0570-1. Department of mathematics, Quaid-i-Azam university, Islamabad, Pakistan
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