Characterizations of LA-semigroups by new types of fuzzy ideals

0.19 MB·English

by  Muhammad aslam

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 proper-
ties 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.