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CHARACTERIZATIONS OF LA-SEMIGROUPS BY NEW TYPES OF FUZZY IDEALS 3 1 MUHAMMADASLAM,S.ABDULLAH,ANDM.ATIQUEKHAN 0 2 Abstract. Inthisarticle,wegivesomecharacterizationresultsof(∈γ,∈γ ∨qδ)- n fuzzyleft(right)ideals,(∈γ,∈γ ∨qδ)-fuzzygeneralizedbi-idealsand(∈γ,∈γ ∨qδ)- a fuzzy bi-ideals of an LA-semigroup. We also give some characterizations of J LA-semigroupsbythepropertiesof(∈γ,∈γ ∨qδ)-fuzzyideals. 8 ] M 1. Introduction G . Kazim and Naseerudin have defined the concept of an LA-semigroup in 1972 h [19,27]. LetSbeanon-emptysetandbinaryoperation∗satisfies(a∗b)∗c=(c∗b)∗a t a for all a,b,c ∈ S. Then, S is called an LA-semigroup. Later, Mushtaq et al. have m investigated the structure further and added many useful results to theory of LA- [ semigroups see [21, 22, 23]. Mushtaq and Khan have developed ideal theory of LA-semigroups [24]. In [17], Khan and Ahmad have given some characterizations 1 v of LA-semigroups by their ideals. L. A. Zadeh founder of the concept of fuzzy set 4 [33] in 1965. Behalf of this concept, mathematicians initiated a natural framework 5 for generalizing some basic notions of algebra, e.g group theory, set theory, ring 0 theory, topology, measure theory and semigroup theory etc. Fuzzy logic is very 3 usefulinmodeling the granulesasfuzzy sets. BargeilaandPedryczconsideredthis . 1 new computing methodology in [4]. Pedrycz and Gomide in [28] considered the 0 presentationofupdate trends infuzzy settheoryandits applications. Foundations 3 offuzzy groupswerelaidbyRosenfeldin[30]. The conceptoffuzzy semigroupwas 1 : introduced by Kuroki [14, 15]. Recently, Khan and Khan defined the concept of v fuzzy ideals in LA-semigroups[18]. In [25], Muralidescribed the notions of belong- i X ingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy r subset. In [29], the idea of quasi-coincidence of a fuzzy point with a fuzzy set is a defined. Thesetwoideasactedimportantroleingeneratingsomedifferenttypes of fuzzy algebras. Bhakat and Das [5, 9] gave the concept of (α,β)-fuzzy subgroups, where α,β ∈ {∈,q,∈ ∨q,∈ ∧q} and α 6=∈ ∧q. These fuzzy subgroups are further studiedin[7,6]. Theconceptof(∈,∈∨q)-fuzzysubgroupsisaviablegeneralization ofRosenfeld’s fuzzy subgroups. The concept of(∈,∈∨q)-fuzzy subrings andideals are described. In [8], S.K. Bhakat and P. Das introduced the (∈,∈∨q)-fuzzy sub- ringsandideals. Davvazdescribedthe notionsof (∈,∈∨q)-fuzzy subnearringsand (∈,∈∨q)-fuzzy ideals of a near ring in [11]. Jun and Song introduced the study of (α,β)-fuzzy interior ideals of a semigroupin [13]. In [16], the concept of (∈,∈∨q)- fuzzy bi-ideals of a semigroup described by Kazanci and Yamak. In [31], Jun et al gave some different type of characterization of regular semigroups by the properties of (∈,∈∨q)-fuzzy ideals. Aslam et al introduced a new generalizationof fuzzy Key words and phrases. LA-semigroup,Fuzzyset,fuzzynormalsubgroup,fuzzycoset. 1 2 MUHAMMADASLAM,S.ABDULLAH,ANDM.ATIQUEKHAN Γ-ideals in Γ-LA-semigroups[2]. In [1], Abdullah et al studied a new type of fuzzy normal subgroup and fuzzy coset of groups. Generalizing the idea of the quasi- coincident of a fuzzy point with a fuzzy subset, Jun [12] defined (∈,∈∨q )-fuzzy k subalgebras in BCK/BCI-algebras. In [32], (∈,∈∨q )-fuzzy ideals of semigroups k are introduced. Further generalizing the concept of (∈,∈∨q), J. Zhan and Y. Yin defined(∈ ,∈ ∨q )-fuzzyidealsofnearrings[34]. In[26],(∈ ,∈ ∨q )-fuzzyideals γ γ δ γ γ δ of BCI-algebras are introduced. The Abdullah et al, the concept of (∈ ,∈ ∨q )- γ γ δ fuzzy LA-subsemigroups, (∈ ,∈ ∨q )-fuzzy left(right) ideals, (∈ ,∈ ∨q )-fuzzy γ γ δ γ γ δ generalizedbi-idealsand(∈ ,∈ ∨q )-fuzzybi-idealsofanLA-semigroupareintro- γ γ δ duced [3]. The paper is organized as: In section 2, we give basic definition of an LA- semigroupandafuzzyset. Insection3,wedefine(∈ ,∈ ∨q )-fuzzyLA-subsemigroups, γ γ δ (∈ ,∈ ∨q )-fuzzy left(right) ideals, (∈ ,∈ ∨q )-fuzzy generalized bi-ideals and γ γ δ γ γ δ (∈ ,∈ ∨q )-fuzzy bi-ideals of LA-semigroups. In section 4 we give some charac- γ γ δ terization results of these ideals in fuzzy varsion. We also give characterizations of LA-semigroups by the properties of (∈ ,∈ ∨q )-fuzzy ideals. γ γ δ 2. Preliminaries AnLA-subsemigroupofS meansanon-empty subsetAofS suchthatA2 ⊆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 and right ideal. An LA-subsemigroup A is called bi-ideal if (BS)B ⊆A. A non-empty subset B is called generalized bi-ideal if (BS)B ⊆ A. A non-empty subset Q is calledaquasi-idealifQS∩SQ⊆Q.Anon-emptysubsetAiscalledinterioridealif it is LA-subsemigroup of S and (SA)S ⊆A. An LA-semigroup S is called regular if for each a ∈ S there exists x ∈ S such that a = (ax)a. An LA-semigroup S is called intra-regular if for each a∈S there exist x,y ∈S such that a=(xa2)y. An LA-semigroupS iscalledweaklyregularifforeachs∈S,thereexistsx,y ∈S,such thats=(sx)(sy).InanLA-semigroupS,thefollowinglawhold,(1)(ab)c=(ab)c, for all a,b,c∈S. (2) (ab)(cd)= (ac)(bd), for all a,b,c,d∈S. If an LA-semigroup S hasaleftidentitye,thenthefollowinglawholds,(3)(ab)(cd)=(db)(ca),forall a,b,c,d∈S.(4)a(bc)=b(ac),foralla,b,c∈S.AfuzzysubsetµofS isamapping µ:S →[0,1]. For any two subsets µ and ν of S, the product µ◦ν is defined as {µ(y)∧ν(z), if there exist y,z ∈S, such that x=yz (µ◦ν)(x)= x=yz (cid:26)0 W otherwise A fuzzy subset µ of the form t(6=0) if y =x µ(y)= 0 if y 6=x (cid:26) is said to be a fuzzy point with support x and value t and is denoted by x .A t fuzzy point x is said to be ”belong to”(res.,”quasicoincidentwith”) a fuzzy set µ, t written as x ∈ µ(repectively,x qµ) if µ(x) ≥ t (repectively, µ(x) +t > 1). We t t write x ∈ ∨qµ if x ∈ µ or x qµ.If µ(x) < t(respectively, µ(x)+t ≤ 1), then we t t t write x ∈µ(repectively,x qµ). We note that ∈∨q means that ∈ ∨q does not hold. t t Generalizing the concept of x qµ, Y. B. Jun [12] defined x q µ, where k ∈[0,1] as t t k x q µ if µ(x)+t+k˙ >1 and x ∈∨q µ if x ∈µ or x q µ. t k t k t t k CHARACTERIZATIONS OF LA-SEMIGROUPS BY NEW TYPES OF FUZZY IDEALS 3 Definition 1. [18] 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. Definition2. [18] Afuzzysubsetµ of an LA-semigroupS is called fuzzyleft(right) ideal of S if µ(xy)≥µ(y)(µ(xy)≥µ(x)) for all x,y ∈S. Definition 3. [18] An LA-subsemigroup µ of an LA-semigroup S is called fuzzy bi-ideal of S if µ((xy)z)≥µ(x)∧µ(z) for all x,y ∈S. Definition4. [18]AfuzzysubsetµofanLA-semigroupS iscalledfuzzygeneralized bi-ideal of S if µ((xy)z)≥µ(x)∧µ(z) for all x,y ∈S. Definition 5. [18]Let µ be a fuzzy subset of an LA-semigroup S. Then, for all t∈(0,1], the set µ ={x∈S|µ(x)≥t} is called a level subset of S. t Following theorems are well known in LA-semigroups Theorem 1. Let S be a weakly regular LA-semigroup with left identity e. Then, the following conditions are equivalent: (i) S is regular. (ii) R∩L=LR for every left ideal L and right ideal R. Theorem 2. Let S be a weakly regular LA-semigroup with left identity e. Then, the following conditions are equivalent: (i) S is regular. (ii) L∩R⊆LR for every left ideal L and right ideal R. Theorem 3. Let S be a weakly regular LA-semigroup with left identity e. Then, the following conditions are equivalent: (i) S is regular and intra-regular. (ii) Every quasi ideal of S is idempotent. Theorem 4. Let S be a weakly regular LA-semigroup with left identity e. Then, the following conditions are equivalent: (i) S is regular. (ii) A∩B =BA for every left ideal A and right ideal B of S. (ii) (AS)A=A for every quasi ideal A of S. 3. (∈ , ∈ ∨q )-FUZZY IDEALS γ γ δ Generalizingthenotionof(∈,∈∨q),in[26,34](∈ ,∈ ∨q )-fuzzyidealsofnear γ γ δ rings and BCI-algebras are introduced. Let γ,δ ∈ [0,1] be such that γ < δ. For fuzzy point x and fuzzy subset µ of S, we say t (i) x ∈ µ if µ(x)≥t>γ. t γ (ii) x q µ if µ(x)+t>2δ. t δ (iii) x ∈ ∨q µ if x ∈ µ or x q µ. t γ δ t γ t δ (iv) x ∈ ∨q µ if x ∈ µ or x q µ. t γ δ t γ t δ Inthissection,weintroducetheconceptof(∈ ,∈ ∨q )-fuzzyLA-subsemigroup, γ γ δ (∈ ,∈ ∨q )-fuzzyleft(right)ideal,(∈ ,∈ ∨q )-fuzzygeneralizedbi-idealand(∈ ,∈ ∨q )- γ γ δ γ γ δ γ γ δ fuzzy bi-ideal of an LA-semigroup S. We also study some basic properties of these ideals. Definition 6. [3]A fuzzy subset µ of an LA-semigroup S is called an (∈ ,∈ ∨q )- γ γ δ fuzzy LA-subsemigroup of S if for all a,b∈S and t,r ∈(γ,1], x ,y ∈ µ implies t r γ that (ab) ∈ ∨q µ. t∧r γ δ 4 MUHAMMADASLAM,S.ABDULLAH,ANDM.ATIQUEKHAN Remark1. [3]EveryfuzzyLA-subsemigroupandevery(∈,∈∨q)-fuzzyLA-subsemigroup is an (∈ ,∈ ∨q )-fuzzy LA-subsemigroup but the converse is not true. γ γ δ Definition7. [3]AfuzzysubsetµofanLA-semigroupS iscalled an (∈ ,∈ ∨q )- γ γ δ fuzzy generalized bi- ideal if for all a,b,s ∈ S and t,r ∈ (γ,1], a ∈ µ,b ∈ µ t γ r r implies that ((as)b) ∈ ∨q µ. t∧r γ δ Definition 8. [3]Afuzzy subsetµ of an LA-semigroup S is called an (∈ ,∈ ∨q ) γ γ δ fuzzy bi- ideal if for all a,b,s∈S and t,r ∈(γ,1] a ∈ µ, b ∈ µ implies that t γ r r (i) (ab) ∈ ∨q µ. t∧r γ δ (ii) ((as)b) ∈ ∨q µ. t∧r γ δ Definition 9. [3]A fuzzy subset µ of an LA-semigroup c is called an (∈ ,∈ ∨q )- γ γ δ fuzzy interior ideal of c if for all a,b,c∈S and t,r ∈(γ,1], the following conditions hold: (i) a ,b ∈ µ implies that (ab) ∈ ∨q µ. t r γ t∧r γ δ (ii) c ∈ µ implies that ((ac)b) ∈ ∨q µ. t γ t γ δ Definition10. [3]AfuzzysubsetµofanLA-semigroupS iscalledan(∈ ,∈ ∨q )- γ γ δ fuzzy quasi-ideal of S, if it satisfies, µ(x)∨γ ≥(µ◦1)(x)∧(1◦µ)(x)∧δ, 4. Regular and Weakly Regular LA-Semigroups. In this section we characterize weakly regular, regular and intra-regular LA- semigroups by the prperties of their (∈ ,∈ ∨q )-fuzzy ideals. γ γ δ Theorem 5. Every (∈ ,∈ ∨q )-fuzzy generalized bi-ideal of a weakly regular LA- γ γ δ semigroup S with left identity e is an (∈ ,∈ ∨q )-fuzzy bi-ideal of S. γ γ δ Proof. Suppose that µ is an (∈ ,∈ ∨q )-fuzzy generalizedbi-ideal of S. Let a,b∈ γ γ δ S.SinceS isweaklyregularsothereexistsx,y ∈S,suchthat, a=(ax)(ay). Then, by using (4), we have ab=(a(ax)y)b). So, µ(ab)∨γ =µ(a((ax)y)b)∨γ ≥µ(a)∧µ(b)∧δ. This shows that µ is an (∈ ,∈ ∨q )-fuzzy LA-subsemigroup of c. Hence, µ is an γ γ δ (∈ ,∈ ∨q )-fuzzy bi-ideal of c. (cid:3) γ γ δ Theorem 6. Every (∈ ,∈ ∨q )-fuzzyquasi-ideal of a wekly regular LA-semigroup γ γ δ S with left identity e, is an (∈ ,∈ ∨q )-fuzzy bi-ideal of S. γ γ δ Proof. Suppose that µ is an (∈ ,∈ ∨q )-fuzzy quasi-idealof S and a∈S. Since S γ γ δ is weakly regular,so their exist x,y ∈S, such that, a=(ax)(ay). Now, µ(ab)∨γ ≥ (µ◦1)(ab)∧(1◦µ)(ab)∧δ = {µ(l)∧1(m)} ∧ {1(u)∧µ(v)} ∧δ " # " # ab=lm ab=uv _ _ ≥ [µ(a)∧1(b)]∧[1(a)∧µ(b)]∧δ = [µ(a)∧1]∧[1∧µ(b)]∧δ = µ(a)∧µ(b)∧δ. CHARACTERIZATIONS OF LA-SEMIGROUPS BY NEW TYPES OF FUZZY IDEALS 5 Thus, µ is an (∈ ,∈ ∨q )-fuzzy LA-subsemigroup of S. Now by using (4),(1),(2) γ γ δ and (3) we have (as)b = (((ax)(ay))s)(eb) = ((a((ax)y))s)(eb) = ((s((ax)y))a)(ec) = (ce)(a(s((ax)y))) = a((ce)(s((ax)y))). So, µ((as)b)∨γ ≥ (µ◦1)((as)b)∧(1◦µ)((as)b)∧δ = {µ(l)∧1(m)} ∧ {1(u)∧µ(v)} ∧δ     (as)b=a((ce)_(s((ax)y)))=lm (as_)b=uv ≥ [µ(a)∧1((ce)(s((ax)y)))]∧[1(as)∧µ(b)]∧δ  = [µ(a)∧1]∧[1∧µ(b)]∧δ = µ(a)∧µ(b)∧δ. Thus, µ is an (∈ ,∈ ∨q )-fuzzy bi-ideal of S. (cid:3) γ γ δ Theorem 7. Ifµ is an (∈ ,∈ ∨q )-fuzzyleft ideal andν bethe(∈ ,∈ ∨q )-fuzzy γ γ δ γ γ δ right ideal of a weakly regular LA-semigroup S with left identity e, then µ◦ν is an (∈ ,∈ ∨q )-fuzzy two-sided ideal of S. γ γ δ Proof. Let x,y ∈S, then by definition of weakly regular LA-semigroup there exist t,r∈S, such that, x=(xr)(xt). Now, (µ◦ν)(y)∧δ = {µ(l)∧ν(m)} ∧δ   y=lm _   = µ(l)∧ν(m)∧δ   y=lm _   = µ(l)∧δ∧ν(m) .   y=lm _   Now, by using (1),(2),(3) and (4), we have xy = x(lm)=((xr)(xt))(lm) =((lm)(xt))(xr) = ((tx)(ml))(xr) =(m((tx)l))(xr) = ((xr)((tx)l))m =((xr)((tx))(el)))m = ((xr)((le)(xt)))m) =((xr)(x((le)t)))m = (x((xr)((le)t)))m =(x((xr)((te)l)))m. 6 MUHAMMADASLAM,S.ABDULLAH,ANDM.ATIQUEKHAN Now, µ(x((xr)((te)l)))∨γ = ((µ(x((xr)((te)l)))∨γ)∨γ)∨γ ≥ ((µ((xr)((te)l))∧δ)∨γ)∨γ (because µ is an (∈ ,∈ ∨q )-fuzzy left ideal) γ γ δ = ((µ((xr)((te)l))∨γ)∧δ)∨γ ≥ ((µ((te)l))∧δ)∧δ)∨γ = (µ((te)l))∧δ)∨γ = (µ((te)l))∨γ)∧δ ≥ µ(l))∧δ∧δ = µ(l))∧δ. Thus, (µ◦ν)(y)∧δ = µ(l)∧δ∧ν(m)   y=lm _   ≤ ({µ((x((xr)((te)l)))m)∨γ}∧ν(m)) xy=(x((x_r)((te)l)))m ≤ ({µ(p)∨γ}∧ν(q)) xy=pq _ = (µ(p)∧ν(q)) ∨γ ! xy=pq _ = (µ◦ν)(xy)∨γ. Thus,(µ◦ν)(xy)∨γ ≥(µ◦ν)(y)∧δ.Similarly,wecanshowthat(µ◦ν)(xy)∨γ ≥ (µ◦ν)(x)∧δ. Hence, µ◦ν is an (∈ ,∈ ∨q )-fuzzy two sided-ideal of S. (cid:3) γ γ δ Following example tells us that if µ and ν are (∈ ,∈ ∨q )-fuzzy ideals of an γ γ δ LA-semigroup S, then µ◦ν (cid:14)µ∧ν. Example 1. Let S = {1,2,3,4} be an LA-semigroup with the following multipli- cation table. ∗ 1 2 3 4 1 4 4 4 4 2 3 1 3 1 3 4 1 4 4 4 4 4 4 4 Define fuzzy subsets µ and ν of S by µ(1)=0.3, µ(2)=0.2, µ(3)=0.6, µ(4)=0.3. and ν(1)=0.4, ν(2)=0.3, ν(3)=0.4, ν(4)=0.5. Now, S 0<t≤0.2 {1,3,4} 0.2<t≤0.3 U(µ;t)= {3} 0.3<t≤0.6  Φ 0.6<t and  CHARACTERIZATIONS OF LA-SEMIGROUPS BY NEW TYPES OF FUZZY IDEALS 7 S 0<t≤0.3 {1,3,4} 0.3<t≤0.4 U(ν;t)= {4} 0.4<t≤0.5  ∅ 0.6<t Then, µ and ν are (∈ ,∈ ∨q )-fuzzy ideals of S (by using Theorem 8 part(iv)).  γ γ δ Now, (µ◦ν)(4) = {0.2,0.3,0.4,0.5}= 0.5 and (µ∧ν)(4) = µ(4)∧ν(4) = 0.3. Clearly, µ◦ν (cid:14)µ∧_ν. Hence, µ◦ν ≤µ∧ν is not true in general. Definition 11. Let µ and ν be the fuzzy subsets of LA-semigroup S. The fuzzy subsets µ∗,µ∧∗ν,µ∨∗ν and µ∗ν of S are defined as follows; ∗ µ (x) = (µ(x)∨γ)∧δ ∗ (µ∧ ν)(x) = (((µ∧ν)(x))∨γ)∧δ ∗ (µ∨ ν)(x) = (((µ∨ν)(x))∨γ)∧δ (µ∗ν)(x) = (((µ◦ν)(x))∨γ)∧δ for all x∈S. Lemma1. Let µ and ν be thefuzzysubsets of LA-semigroupS.Then thefollowing holds; (i) µ∧∗ν =µ∗∧ν∗. (ii) µ∨∗ν =µ∗∨ν∗. (iii) µ∗ν =µ∗◦ν∗. Proof. Let x∈S. Then, (i) ∗ (µ∧ ν)(x) = (((µ∧ν)(x))∨γ)∧δ = ((µ(x)∧ν(x))∨γ)∧δ = ((µ(x)∨γ)∧(ν(x)∨γ))∧δ = ((µ(x)∨γ)∧δ)∧((ν(x)∨γ))∧δ) ∗ ∗ = µ (x)∧ν (x) ∗ ∗ = (µ ∧ν )(x). Hence, µ∧∗ν =µ∗∧ν∗. (ii) ∗ (µ∨ ν)(x) = (((µ∨ν)(x))∨γ)∧δ = ((µ(x)∨ν(x))∨γ)∧δ = ((µ(x)∨γ)∨(ν(x)∨γ))∧δ = ((µ(x)∨γ)∧δ)∨((ν(x)∨γ))∧δ) ∗ ∗ = µ (x)∨ν (x) ∗ ∗ = (µ ∨ν )(x). Hence, µ∨∗ν =µ∗∨ν∗. 8 MUHAMMADASLAM,S.ABDULLAH,ANDM.ATIQUEKHAN (iii) (µ∗ν)(x) = (((µ◦ν)(x))∨γ)∧δ = {µ(y)∧ν(z)} ∨γ ∧δ ! ! x=yz _ = {(µ(y)∨γ)∧(ν(z)∨γ)} ∧δ ! x=yz _ = {((µ(y)∨γ)∧δ)∧((ν(z)∨γ)∧δ)} x=yz _ ∗ ∗ = {µ (y)∧ν (z)} ! x=yz _ ∗ ∗ = (µ ◦ν )(x). Hence, µ∗ν =µ∗◦ν∗. (cid:3) Lemma 2. Let L and R be any two non-empty subsets of LA-semigroup S. Then the followin hold. (i) χ ∧∗χ =χ∗ . L R L∩R (ii)χ ∨∗χ =χ∗ . L R L∪R (iii) χ ∗χ =χ∗ . L R LR Lemma 3. A non-empty subset L of S is a left(right) ideal of S if and only if χ∗ L is an (∈ ,∈ ∨q )-fuzzy left(right) ideal of S. γ γ δ Proof. Suppose that L be a left ideal of S. Now, ∗ χ (ab)∨γ = ((χ (ab)∨γ)∧δ)∨γ L L ≥ ((χ (b)∧δ)∧δ)∨γ L = (χ (b)∧δ)∨γ L = (χ (b)∨γ)∧δ L ∗ = χ (b)∧δ. L Hence, χ∗ is an (∈ ,∈ ∨q )-fuzzy left(right) ideal of S. L γ γ δ Conversely,assumethatχ∗ isan(∈ ,∈ ∨q )-fuzzyleftidealofS.Toshowthat L γ γ δ L is a left ideal of S, let b∈L. Then, ∗ χ (b)=(χ (b)∨γ)∧δ =χ (b)∧δ =δ. L L L So, b ∈ χ∗. Now, (ab) ∈ ∨q χ∗ (because χ∗ is an (∈ ,∈ ∨q )-fuzzy left ideal δ γ L δ γ δ L L γ γ δ of S), which implies that (ab) ∈ χ∗ or (ab) q χ∗. Hence, χ∗(ab) ≥ δ > γ or δ γ L δ δ L L χ∗(ab)+δ >2δ. Now, if χ∗(ab)+δ >2δ, then χ∗(ab)>δ. Thus, χ∗(ab)≥δ. So, L L L L we have χ∗(ab)=δ. Hence, ab∈L. Thus, L is left ideal of S. (cid:3) L Lemma 4. A non-empty subset Q of S is a quasi-ideal of S if and only if χ∗ is Q an (∈ ,∈ ∨q )-fuzzy quasi-ideal ideal of S. γ γ δ Proposition 1. Let µ be an (∈ ,∈ ∨q )-fuzzy LA-subsemigroup of S. Then, µ∗ γ γ δ is a fuzzy LA-subsemigroup of S. CHARACTERIZATIONS OF LA-SEMIGROUPS BY NEW TYPES OF FUZZY IDEALS 9 Proof. Let µ be an (∈ ,∈ ∨q )-fuzzy LA-subsemigroup of S. Then, we have γ γ δ µ(ab)∨γ ≥µ(a)∧µ(b)∧δ, for all a,b∈S. Now, ∗ µ (ab) = (µ(ab)∨γ)∧δ ≥ (µ(a)∧µ(b)∧δ)∧δ = ((µ(a)∨γ)∧(µ(b)∨γ)∧δ) = ((µ(a)∨γ)∧δ)∧((µ(b)∨γ)∧δ) ∗ ∗ = µ (a)∧µ (b). Hence, µ∗ is a fuzzy LA-subsemigroup of S. (cid:3) Theorem 8. The following assertions are equivalent for an LA-semigroup S. (i) S is regular. (ii)µ∧∗ν =µ∗ν forevery(∈ ,∈ ∨q )-fuzzy rightidealµandevery(∈ ,∈ ∨q )- γ γ δ γ γ δ fuzzy left ideal ν of S. Proof. (i) =⇒ (ii) Let µ be an (∈ ,∈ ∨q )-fuzzy right ideal and ν be an γ γ δ (∈ ,∈ ∨q )-fuzzy left ideal of S. Let a ∈ S. Then, there exists x ∈ S, such γ γ δ that, a=(ax)a. Now, (µ∗ν)(a) = (((µν)(a))∨γ)∧δ = {µ(x)∧ν(y)} ∨γ ∧δ a=xy ! ! W ≥ (({µ(ax)∧ν(x)})∨γ)∧δ (((µ(ax)∨γ)∧ν(x))∨γ)∧δ ≥ (((µ(x)∧δ)∧ν(x))∨γ)∧δ = ((µ(x)∧ν(x))∨γ)∧δ = ((µ(x)∧ν(x))∨γ)∧δ ∗ = (µ∧ ν)(x). Thus, (µ∗ν)≥µ∧∗ν. Now, (µ∗ν)(a) = (((µν)(a))∨γ)∧δ = {µ(x)∧ν(y)} ∨γ ∧δ a=xy ! ! W = ((µ(x)∧δ)∧(ν(y)∧δ))∨γ ∧δ a=xy ! W ≤ ((µ(xy)∨γ)∧(ν(xy)∨γ))∨γ ∧δ a=xy ! W = µ(xy)∧ν(xy)∨γ ∧δ a=xy ! W = (µ(a)∧ν(a)∨γ)∧δ ∗ = (µ∧ ν)(a). 10 MUHAMMADASLAM,S.ABDULLAH,ANDM.ATIQUEKHAN Therefore, µ∗ν ≤µ∧∗ν. Hence µ∧∗ν =µ∗ν∗. (ii)=⇒(i)LetRbetherightidealandLbetheleftidealofS.Then,χ∗ andχ∗ R L are(∈ ,∈ ∨q )-fuzzyrightidealand(∈ ,∈ ∨q )-fuzzy leftidealofS,respectively γ γ δ γ γ δ (by corollary 3). Now, by hypothesis, χ∗ = χ ∗χ = χ ∧∗χ = χ∗ . Thus, RL R L R L R∩L RL=R∩L. Hence, it follows from THeorem 1, S is regular. (cid:3) Theorem9. ForaweaklyregularLA-semigroupS withleftidentitye,thefollowing conditions are equvialent: (i) S is regular. (ii) ((µ∧∗ν)∧∗ρ) ≤ ((µ∗ν)∗ρ) for every (∈ ,∈ ∨q )-fuzzy right ideal µ, γ γ δ every (∈ ,∈ ∨q )-fuzzy generalized bi-ideal ρ and every (∈ ,∈ ∨q )-fuzzy left γ γ δ γ γ δ ideal ν of S. (iii) ((µ∧∗ν)∧∗ρ) ≤ ((µ∗ν)∗ρ) for every (∈ ,∈ ∨q )-fuzzy right ideal µ, γ γ δ every (∈ ,∈ ∨q )-fuzzy bi-ideal ρ and every (∈ ,∈ ∨q )-fuzzy left ideal ν of S. γ γ δ γ γ δ (iv) ((µ∧∗ν)∧∗ρ) ≤ ((µ∗ν)∗ρ) for every (∈ ,∈ ∨q )-fuzzy right ideal µ, γ γ δ every (∈ ,∈ ∨q )-fuzzy quasi-ideal ρ and every (∈ ,∈ ∨q )-fuzzy left ideal ν of γ γ δ γ γ δ S. Proof. (i)=⇒(ii)Letµ,νandρbeany(∈ ,∈ ∨q )-fuzzy rightidealµ,(∈ ,∈ ∨q )- γ γ δ γ γ δ fuzzy generalized bi-ideal ρ and (∈ ,∈ ∨q )-fuzzy left ideal ν of S, respectively. γ γ δ Lets∈S. Since S is regular,sothere exist x∈S, suchthat, s=(sx)s. Also, since S is weakly regular, so there exist y,z ∈S, such that, s=(sy)(sz). Therefore, by using (1) and (4) we have, s = (sx)s=(((sy)(sz))x)s = ((x(sz))(sy))s=((s(xz)(sy))s = ((s(xz))(((sy)(sz))y))s =((s(xz))((s((sy)z))y)s = ((s(xz))((y((sy)z))s))s =((s(xz))(((sy)(yz))s)s = ((s(xz))((((yz)y)s)s))s = ((s(xz))((((yz)y)(sy(sz)))s))s = ((s(xz))((((yz)y)(s((sy)z)))s))s = ((s(xz))((s(((yz)y)((sy)z)))s))s. Now, ((µ∗ν)∗ρ)(s) = {(µ∗ν)(l)∧ρ(m)} ∨γ ∧δ ! ! s=lm _ ≥ [((µ∗ν)((s(xz))((s(((yz)y)((sy)z)))s))∧ρ(s))∨γ]∧δ = {µ(l)∧ν(m)} ∧ρ(s) ∨γ ∧δ     (s(xz))((s(((yz)_y)((sy)z)))s)=lm ≥ (({µ(s(xz))∧ν((s(((yz)y)((sy)z)))s)}∧ρ(s))∨γ)∧δ   = (({(µ(s(xz))∨γ)∧(ν((s(((yz)y)((sy)z)))s)∨γ)}∧ρ(s))∨γ)∧δ ≥ (({(µ(s)∧δ)∧(ν(s)∧δ)}∧ρ(s))∨γ)∧δ = (({µ(s)∧ν(s)}∧ρ(s))∨γ)∧δ ∗ ∗ = ((µ∧ ν)∧ ρ)(s). Thus, ((µ∧∗ν)∧∗ρ)≤((µ∗ν)∗ρ). (ii) =⇒ (iii) =⇒ (iv) are obvious.