World Applied Sciences Journal 19 (12): 1710-1720, 2012
ISSN 1818-4952
© IDOSI Publications, 2012
DOI: 10.5829/idosi.wasj.2012.19.12.3500
A Note on Intuitionistic Fuzzy Γ-LA-semigroups
1
Faisal Yousafzai, 2 Naveed Yaqoob, 3 Shamsul Haq and 4 Raheela Manzoor
1
Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan
2
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
3
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan
4
Department of Mathematics, SBK Women's University, Quetta, Pakistan
Abstract: In this paper, we discussed the intuitionistic fuzzification of Γ-LA ** -semigroups. We
characterized regular and intra-regular Γ-LA ** -semigroups in terms of intuitionistic fuzzy Γ-left (Γ-right,
Γ-two-sided) ideals, intuitionistic fuzzy Γ-bi-ideals and intuitionistic fuzzy Γ-(1,2)-ideals. We proved that
all these intuitionistic fuzzy Γ-ideals coincide in intra-regular Γ-LA ** -semigroups.
2010 mathematics subject classification: 20M10 20N99
•
Key words: Γ-LA-semigroups Γ-LA ** -semigroups Intuitionistic fuzzy Γ-ideals
•
•
INTRODUCTION
The fundamental concept of fuzzy sets was given by Zadeh [1], in 1965. Given a set S, a fuzzy subset of S is an
arbitrary mapping ƒ: S→[0,1], where [0,1] is the unit interval. This concept was applied in [2] to generalize some of
the basic concepts of general topology. Rosenfeld [3] was the first who consider the case when S is a groupoid. He
gave the definition of fuzzy subgroupoid and the fuzzy left (right, two-sided) ideal of S and justified these
definitions by showing that a subset A of a groupoid S is a subgroupoid or a left (right, two-sided) ideal of S if the
characteristic function of A, that is
1, if x ∈ A
C A (x) = 
0, if x ∉ A
is a fuzzy subgroupoid or a fuzzy left (right, two-sided) ideal of S. Kuroki was the first mathematician who applied
the fuzzy sets to semigroup theory in [4]. Shabir et al. [5, 6], applied fuzzy sat theory to semigroups.
Atanassov [7], introduced the concept of an intuitionistic fuzzy set. The relations between fuzzy sets and
algebraic structures have been considered by many mathematicians, for instance, Aslam and Abdullah [8], Davvaz
et al. [9], Jun et al. [10-12], Khan et al. [13] and Yaqoob et al. [14-16].
Kazim and Naseeruddin [17], introduced the concept of an LA -semigroup in 1972. In [18], the same structure is
called a left invertive groupoid. Protic and Stevanovic called it an Abel-Grassmann's groupoid (AG-groupoid) [19].
Shah and Rehman [20], introduced the notion of Γ-AG-groupoids with left identity, but later on Faisal et al. [21],
studied Γ-AG** -groupoid which generalizes the existing cocepts of Shah and Rehman, and also proved that a Γ-AGgroupoid with left identity becomes again an AG-groupoid with left identity. There are many mathematicians who
added several results to the theory fuzzy LA -semigroups, see [22-32].
The pair (S, Γ) is called a Γ-groupoid if xαy ∈ S for all x,y∈S and α ∈ Γ . A Γ-groupoid (S, Γ) is called a ΓLA-semigroup if Γ-left invertive law holds for all x,y,z∈S and for all α,β ∈ Γ
Corresponding Author: Faisal Yousafzai, Department of Mathematics,
COMSATS Institute of Information Technology, Attock, Pakistan.
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(1) (xα y)β z = (zα y)β x
A Γ-LA-semigroup also satisfies the Γ-medial law for all w,x,y,z∈S and for all α,β ,γ ∈ Γ
(2) (wα x)β( yγz ) = (wα y)β (xγ z)
A Γ-LA-semigroup is called a Γ-LA ** -semigroup if it satisfies the following law for all x,y,z∈S and for all α,β ∈ Γ
(3) xα (yβz) = yα (xβ z)
A Γ-LA ** -semigroup also satisfies the Γ-paramedial law for all w,x,y,z∈S and for all α,β ,γ ∈ Γ
(4) (wα x)β( yγz ) = (zα y)β ( xγw )
Note that (3) and (4) also hold for a Γ-LA-semigroup with left identity but a Γ-LA-semigroup with left identity
becomes an LA -semigroup with left identity. Indeed, if S is a Γ-LA-semigroup with left identity e and a,b∈S, then
a αb = aα(eβb) = e α(aβb) = a β b
where α,β ∈ Γ ⇒ α = β .
Assume that (S,.) is an LA -semigroup and let γ be an operation on S. Define a γb = a.b for all a,b∈S, then S is a
{γ}-LA-semigroup. Conversely, if S is a Γ-LA-semigroup and define a.b = aγ b for all a,b∈S, then (S,.) is an LA semigroup. This means that if S is a {γ}-LA-semigroup, then (S,.) is an LA -semigroup.
Example 1: Let us consider the abelian group (R,+) of all real numbers under the binary operation of addition. If we
define
a ∗ b = b − a − r, where a,b,r ∈ R
then (R,+) becomes an LA -semigroup. Indeed
(a ∗ b)∗ c= c − (a∗ b)− r= c− (b− a− r)
− =
r c− b+ a+ r− r = c − b + a
and
(c ∗ b)∗ a= a − (c∗ b)− r= a− (b− c− r)− r= a− b+ c+ r− r = a − b + c
Since (R,+) is commutative, so (a ∗ b) ∗ c =(c ∗ b) ∗a and therefore (R,*) satisfies a left invertive law. It is easy to
observe that (R,*) is non-commutative and non-associative.
PRELIMINARIES AND BASIC DEFINITIONS
In this section we will present some basic definitions needed for our purpose.
Definition 1: Let S be a Γ-LA-semigroup, a non-empty subset A of S is called a Γ-LA-subsemigroup if a γb ∈ A for
all a, b∈A and γ ∈ Γ or if AΓA ⊆ A.
Definition 2: A subset A of a Γ-LA-semigroup S is called a Γ-left (Γ-right) ideal of S if S ΓA ⊆ A ( A ΓS ⊆ A ) and A
is called a Γ-two-sided ideal of S if it is both a Γ-left ideal and a Γ-right ideal.
Definition 3: A subset A of a Γ-LA-semigroup S is called a Γ-generalized bi-ideal of S if ( A ΓS ) Γ A ⊆ A.
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Definition 4: A Γ-LA-subsemigroup A of a Γ-LA-semigroup S is called a Γ-bi-ideal of S if ( AΓ S )Γ A⊆ A .
Definition 5: A Γ-LA-subsemigroup A of a Γ-LA-semigroup S is called a Γ-(1,2) ideal of S if ( AΓ S ) ΓA 2 ⊆ A .
An intuitionistic fuzzy subset (briefly, IFS) A of S is an object having the form
A = {(x, µA (x), λ A (x)): x ∈ S}
The functions µA : S→[0,1] and λA : S→[0,1] denote the degree of membership and the degree of nonmembership respectively such that for all x∈S, we have
0 ≤ µA (x) + λ A (x) ≤ 1
For the sake of simplicity, we shall use the symbol A = (µA , λA ) for an IFS A = {(x, µA (x), λ A (x))/x ∈ S}.
Let
δ = (S δ , Θδ ) = {(x,S(x),
Θ δ (x))}
δ
be an IFS where Sδ( x ) = 1 and Θδ (x) = 0 for all x∈S, then δ = (Sδ , Θδ ) will be carried out in operations with an IFS
A = (µA , λA ) such that Sδ and Θδ will be used in collaboration with µA and λA respectively.
Let A = (µA , λA ) and B = (µB, λB) be any two IFSs of a Γ-LA-semigroup S, then the Γ-product A oΓ B is defined by:
 ∨ {µ A (b) ∧ µB (c)} if a = bαc
( µA oΓ µ B ) (a) = a =b αc
0
otherwise
and
 ∧
( λA oΓ λ B ) (a) = a =b αc
{λA (b) ∨ λ B (c)}
1
if a = bαc
otherwise
for all a,b,c∈S and α ∈ Γ. Let A = (µA , λA ) and B = (µB, λB) are IFSs of a Γ-LA-semigroup S. The symbols A∩B
will mean the following IFS of S
(µ A ∩ µ B )(x) = µA (x) ∧ µB (x), (λ A ∪ λ B)(x) = λ A (x) ∨ λ B (x)
for all x in S.
The symbols A∪B will mean the following IFS of S
(µ A ∪ µ B )(x) = µA (x) ∨ µB (x), (λ A ∩ λ B)(x) = λ A (x) ∧ λ B (x)
for all x in S.
Assume that A and B are any IFSs of a Γ-LA-semigroup S, then A⊆B means that µ A (x) ≤ µ B(x) and
λ A (x) ≥ λ B (x), for all x in S.
Definition 6: An IFS A = (µA , λA ) of a Γ-LA-semigroup S is called an intuitionistic fuzzy Γ-LA-subsemigroup of S
if µ A (xαy) ≥ µ A (x) ∧ µA (y) and λ A (xαy) ≤ λA (x) ∨ λ A (y), for all x,y∈S and α ∈ Γ.
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Definition 7: An IFS A = (µA , λA ) of a Γ-LA-semigroup S is called an intuitionistic fuzzy Γ-left ideal of S if
µ A (xαy) ≥ µ A (y) and λ A (xαy) ≤ λA (y), for all x,y∈S and α ∈ Γ.
Definition 8: An IFS A = (µA , λA ) of a Γ-LA-semigroup S is called an intuitionistic fuzzy Γ-right ideal of S if
µ A (xαy) ≥ µ A (x) and λ A (xαy) ≤ λA (x), for all x,y∈S and α ∈ Γ.
Definition 9: An IFS A = (µA , λA ) of a Γ-LA-semigroup S is called an intuitionistic fuzzy Γ-two-sided ideal of S if
it is both an intuitionistic fuzzy Γ-left and an intuitionistic fuzzy Γ-right ideal of S.
Definition 10: An IFS A = (µA , λA ) of a Γ-LA-semigroup S is called an intuitionistic fuzzy Γ-generalized bi-ideal of
S if µ A ((xαy) βz) ≥ µ A (x) ∧ µ A (z) and λ A ((xαy) βz) ≤ λA (x) ∨ λ A (z) , for all x,y,z∈S and α,β ∈ Γ.
Definition 11: An intuitionistic fuzzy Γ-LA-subsemigroup A = (µA , λA ) of a Γ-LA-semigroup S is called an
intuitionistic fuzzy Γ-bi-ideal of S if µ A ((xαy) βz) ≥ µ A (x) ∧ µ A (z) and λ A ((xαy) βz) ≤ λA (x) ∨ λ A (z) , for all x,y,z∈S
and α,β ∈ Γ.
Definition 12: An intuitionistic fuzzy Γ-LA-subsemigroup A = (µA , λA ) of a Γ-LA-semigroup S is called an
intuitionistic fuzzy Γ-(1,2)-ideal of S if
µ A ((xαw)β ( yγz ) ) ≥ µ A (x) ∧ µA (y) ∧ µA (z) and λ A ((wαx)β(yγz)) ≤ λ A (x) ∨ λ A (y) ∨ λA (z)
for all w,x,y,z∈S and α,β ,γ ∈ Γ .
REGULAR Γ-LA** -SEMIGROUPS IN TERMS OF INTUITIONISTIC FUZZY Γ-IDEALS
An element a of a Γ-LA-semigroup S is called a regular element of S if there exist x∈S and β,γ ∈ Γ such that
a = (aβ x )γ a and S is called regular if every element of S is regular.
Example 2: Let S = {1,2,3,4,5} and let Γ = {β ,γ } be the set of operations on S defined in the following tables.
β
1
2
3
4
5
1
1
1
1
1
1
2
2
2
2
1
2
3
4
3
5
1
2
4
5
4
3
1
2
5
3
5
4
1
2
γ
1
2
3
4
5
1
1
1
1
1
1
2
1
2
2
2
2
3
1
2
4
3
5
4
1
2
5
4
3
5
1
2
3
5
4
Since (xβ y)γz = (zβ y)γ x for all x,y,z∈S and all β,γ ∈ Γ, therefore S is a Γ-LA-semigroup.
Let Γ ={ γ}, then it is easy to check that S is a Γ-LA ** -semigroup. Indeed, (xγy) γz = (z γy) γx, and
xγ( yγz ) = yγ (x γz) for all x,y,z∈S. It is easy to see that S is regular. Indeed, 1 = (1γ 2) γ1, 2 = (2γ 3)γ 2, 3 = (4γ 4)γ 3,
4 = (4γ 4)γ 4, 5 = (5γ 5) γ 5 . Define an IFS A = (µA , λA ) of S as follows: µ A (1) = 1 , µ A (2) = µA (3) = µ A (4) = µ A (5) = 0,
λ A (1) = 0.3, λ A (2) = 0.4 and λ A (3) = λ A (4) = λ A (5) = 0.2, then clearly A = (µA , λA ) is an intuitionistic fuzzy Γ-two-
sided ideal of a S.
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Lemma 1: Let S be a Γ-LA-semigroup, then the following holds:
(i) An IFS A = (µA , λA ) of S is an intuitionistic fuzzy Γ-LA-subsemigroup of S if and only if µ A oΓ µA ⊆ µA and
λ A oΓ λA ⊇ λ A.
(ii) An IFS A = (µA , λA ) of S is an is intuitionistic fuzzy Γ-left (Γ-right) ideal of S if and only if S oΓ µA ⊆ µA and
Θ oΓ λA ⊇ λ A (µ A oΓ S ⊆ µA and λ A oΓ Θ ⊇ λA ).
Proof: The proof is straightforward.
Lemma 2: [8] Let S be a regular Γ-LA-semigroup and let A = (µA , λA ) and B = (µB, λB) are any intuitionistic fuzzy
Γ-two-sided ideals of S, then A o Γ B = A ∩ B .
Example 3: Let us consider an LA -semigroup S = {a,b,c,d,e} in the following Cayley's table.
.
a
b
c
d
e
a
a
a
a
a
a
b
a
e
e
b
e
c
a
e
e
c
e
d
a
c
b
d
e
e
a
e
e
e
e
Let Γ ={ γ} and define a mapping S × Γ × S → S by xγy = xy for all x,y∈S, then clearly S is a Γ-LA-semigroup
but S is not regular, because c∈S is not regular.
The converse of Lemma 2 is not true in general which is discussed in the following.
Let us define an IFS A = (µA , λA ) of an LA -semigroup S in Example 3 as follows: µ A (a) = µ A (b) = µA (c) = 0.3,
µ A (d) = 0.1 , µ A (e) = 0.4, λ A (a) = 0.2, λ A (b) = 0.3, λ A (c) = 0.4, λ A (d) = 0.5, λ A (e) = 0.2. Then it is easy to see that A =
(µA , λA ) is an intuitionistic fuzzy Γ-two-sided ideal of S. Define another IFS B = (µB, λB) of an LA -semigroup S as
follows: µ B(a) = µ B (b) = µ B(c) = 0.5, µ B(d) = 0.4 , µ B(e) = 0.6, λ B(a) = 0.3, λ B(b) = 0.4, λ B(c) = 0.5, λ B(d) = 0.6,
λ B(e) = 0.3. Then it is easy to observe that B = (µB, λB) is also an intuitionistic fuzzy Γ -two-sided ideals of S such
that (µ A oΓ µB )(a) = {0.1, 0.3, 0.4} = ( µA ∩ µB )(a) for all a∈S and similarly (λ A oΓ λB )(a) = ( λA ∩ λ B ) for all a∈S, that
zs, A o Γ B = A ∩ B but S is not regular
.
An IFS A = (µA , λA ) of a Γ-LA-semigroup is said to be Γ-idempotent if µ A oΓ µA = µ A and λ A oΓ λA = λ A , that is,
A oΓ A = A.
Lemma 3: Every intuitionistic fuzzy Γ-two-sided ideal A = (µA , λA ) of a regular Γ-LA-semigroup is Γ-idempotent.
Proof: Assume that S is a regular Γ-LA-semigroup and let A = (µA , λA ) be an intuitionistic fuzzy Γ-two-sided ideal
of S. Now for a∈S there exists x∈S such that a = (aα x)β a, where α,β ∈ Γ . Therefore, we have
(µ A oΓ µA )(a) =
∨
{µ A (aα x) ∧ µ A (a)} ≥ µA (aα x) ∧ µ A (a) ≥ µ A (a) ∧ µ A (a) = µ A (a)
a =( aα x β
)a
This shows that µ A oΓ µA ⊇ µA and by using Lemma 1, µ A oΓ µA ⊆ µA , therefore µ A oΓ µA = µA . Similarly we can
show that λ A oΓ λA = λ A , which implies that A = (µA , λA ) is Γ-idempotent.
Corollary 1: Every intuitionistic fuzzy Γ-right ideal A = (µA , λA ) of a regular Γ-LA-semigroup is Γ-idempotent.
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Lemma 4: [8] In a regular Γ-LA-semigroup S, A o Γ δ = A and δ oΓ A = A holds for every intuitionistic fuzzy Γ-twosided ideal A = (µA , λA ) of S, where δ = (Sδ , Θ δ ).
Corollary 2: In a regular Γ-LA-semigroup S, A o Γ δ = A and δ oΓ A = A hold for every intuitionistic fuzzy Γ-right
ideal A = (µA , λA ) of S, where δ = (Sδ , Θ δ ).
Theorem 1: The set of intuitionistic fuzzy Γ-two-sided ideals of a regular Γ-LA-semigroup S forms a semilattice
structure with identity δ, where δ = (Sδ , Θδ ).
Γ
Γ
Proof: Assume that lµλ is the set of intuitionistic fuzzy Γ-two-sided ideals of a regular Γ-LA-semigroup S and let
A = (µA , λA ), B = (µB, λB) and C = (µC, λC) are any intuitionistic fuzzy Γ-two-sided ideals of lµλ. Clearly lµλ is closed
and by Lemma 3, we have A o Γ A = A . Now by using Lemma 2, we get A oΓ B = BoΓ A . Therefore by using (1), we
have
(A oΓ B) o Γ C = (B o Γ A) o Γ C = (C oΓ A) oΓ B = (A oΓ C) oΓ B = (B oΓ C) oΓ A = A oΓ (B oΓ C)
It is easy to see from Lemma 4 that δ is an identity in lµλ .
Note that in a regular Γ-LA ** -semigroup S, the following holds
S =S ΓS
Lemma 5: Every intuitionistic fuzzy Γ-right ideal of a regular Γ-LA ** -semigroup S is an intuitionistic fuzzy Γ-left
ideal of S.
Proof: The proof is straightforward.
Theorem 2: If A = (µA , λA ) is an intuitionistic fuzzy Γ-two-sided ideal of a regular Γ-LA ** -semigroup S, then
A(aα b) = A(bαa) holds for all a,b in S and α ∈ Γ .
Proof: Let A = (µA , λA ) be an intuitionistic fuzzy Γ-two-sided ideal of a regular Γ-LA ** -semigroup S and let a,b∈S,
then there exist x,y∈S such that a = (aα x)β a and b = (bλ y)µ b , where α,β ,ψ ,ρ ∈ Γ . Let ζ ∈ Γ, then by using (2) and
(3), we have
µ A (aζ b) = µA (((aαx)βa)ζ((bψy)µb)) = µA (((a αx) β(b ψy))ζ (aµ b)) = µA ((bβa)ζ((bψ y)µ (aα x))) ≥ µ A (bβa)
= µA (((bψy) ρb) β((aαx)β a)) = µA (((bψy)ρ (aαx))β (bβ a)) = µA ((aρb) β((aα x)β (bψ y))) ≥ µA (aρb)
This shows that µ A (aα b) = µ A (bαa) holds for all a,b in S, α ∈ Γ . Similarly λ A (aα b) = λ A (b αa) holds for all a,b in
S and α ∈ Γ . Thus A(aα b) = A(bαa) holds for all a,b in S and α ∈ Γ .
Corollary 3: If A = (µA , λA ) is an intuitionistic fuzzy Γ-right ideal of a regular Γ-LA ** -semigroup S, then
A(aα b) = A(bαa) holds for all a,b in S and α ∈ Γ .
Theorem 3: [8] Let S be a regular Γ-LA ** -semigroup, then A = (µA , λA ) is an intuitionistic fuzzy Γ-left ideal of S if
and only if A = (µA , λA ) is an intuitionistic fuzzy Γ-bi-ideal of S.
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Corollary 4: Let S be a regular Γ-LA ** -semigroup, then A = (µA , λA ) is an intuitionistic fuzzy Γ-left ideal of S if
and only if A = (µA , λA ) is an intuitionistic fuzzy Γ-generalized bi-ideal of S.
Theorem 4: Let S be a regular Γ-LA ** -semigroup, then A = (µA , λA ) is an intuitionistic fuzzy Γ-(1,2)-ideal of S if
A = (µA , λA ) is an intuitionistic fuzzy Γ-left ideal of S.
Proof: Suppose that A = (µA , λA ) is an intuitionistic fuzzy Γ-left ideal of a regular Γ-LA ** -semigroup S. Let
w,x,y,z∈S, then there exist a,b∈S such that x = (xα a)β x and y = (yλ b) ηy, where α,β ,ρ ,η ∈ Γ . Let ξ,ζ ,ε ∈ Γ , then by
using (1) and (3), we have
µ A ((xξw )ζ(yε z)) = µ A ((((xα a) βx) ξw)ζ ( yεz ) ) = µA (((w β x) ξ(x αa))ζ ( yε z))
= µ A ((xξ((wβ x) αa))ζ (yε z)) = µ A (((yεz)ξ((w β x) αa)) ζx) ≥ µA (x)
Now by using (3), (2) and (1), we have
µ A ((xξw)ζ ( y zε ) ) = µ A (yζ ((xξ w )εz ) ) = µA (((yρb) ηy)ζ ((xξ w)ε z)) = µA (((yρb) η(xξw))ζ (yεz))
= µA (((((yρb)η y)ρb)η(xξw))ζ(yz))
ε
= µA ((((bηy)ρ(yρb)) η(xξw))ζ( yεz ) ) = µA (((yρ((bη y)ρb))η (xξw))ζ( yεz ) )
= µA ((((xξw ) ρ( ( bηy ) ρb ) ) ηy ) (ζy zε) ) = µA ((((bηy)ρ((xξ w) ρb))ηy) ζ(yεz)) = µA (((yη((xξw) ρb))ρ(yη b))ζ (yz))
ε
= µA ((yη((yρ((xξ w)ρ b))η b))ζ (yz))
ε
= µA (((yεz) η(yρ((xξw )ρb))η b)))ζy) ≥ µA (y)
Now by using (4) and (1), we have
µ A ((xξw)ζ( yεz ) ) = µ A ((zξy) ζ(wεx)) = µ A (((wεx) ξy) ζz) ≥ µ A (z)
Thus we get,
µ A ((xξw)ζ( yεz ) ) ≥ µA (x) ∧ µA (y) ∧ µA (z)
Similarly we can get
λ A ((x ξw)ζ( y zε ) ) ≤ λ A (x) ∨ λ A (y) ∨ λ A (z).
Thus A = (µA , λA ) is an intuitionistic fuzzy Γ-(1,2)-ideal of S.
A Γ-LA-semigroup S is called a Γ-LA-band if a = aβ a for all a∈S, where β ∈ Γ.
Theorem 5: Let S be a regular Γ-LA-band, then A = (µA , λA ) is an intuitionistic fuzzy Γ-left (Γ-right) ideal of S if
A = (µA , λA ) is an intuitionistic fuzzy Γ-(1,2)-ideal of S.
Proof: The proof is straightforward.
INTRA-REGULAR Γ-LA** -SEMIGROUPS IN TERMS OF INTUITIONISTIC FUZZY Γ-IDEALS
An element a of a Γ-LA-semigroup S is called an intra-regular element of S if there exist x,y∈S and α,β ,γ ∈ Γ
such that a = (xα (a βa)) γy and S is called intra-regular if every element of S is an intra-regular.
Example 4: Let S = {a,b,c,d,e} and let Γ = {β ,γ } be the set of operations on S defined in the following tables.
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β
a
b
c
d
e
a
a
a
a
a
a
b
e
d
c
a
b
c
b
e
d
a
c
d
c
b
e
a
d
e
d
c
b
a
e
γ
a
b
c
d
e
a
a
a
a
a
a
b
a
b
e
d
c
c
a
c
b
e
d
d
a
d
c
b
e
e
a
e
d
c
b
Since (xβ y)γz = (zβ y)γ x for all x,y,z∈S and all β,γ ∈ Γ, therefore S is a Γ-LA-semigroup.
Let Γ ={ γ}, then it is easy to see that S is intra-regular Γ-LA ** -semigroup. Indeed (xγy)γz = (zγ y)γ x and
xγ( yγz ) = yγ (x γz) for all x,y,z∈S, where for all a∈S there exist some x,y∈S such that a = (xγ ( aγa ) )γ y .
Note that equation (5) also holds for an intra-regular Γ-LA ** -semigroup S.
Lemma 6: For an intra-regular Γ-LA ** -semigroup S, the following holds.
(i) Every intuitionistic fuzzy Γ-right ideal of S is an intuitionistic fuzzy Γ-semiprime.
(ii) Every intuitionistic fuzzy Γ-left ideal of S is an intuitionistic fuzzy Γ-semiprime.
(iii) Every intuitionistic fuzzy Γ-two-sided ideal of S is an intuitionistic fuzzy Γ-semiprime.
Proof: (i): Assume that A = (µA , λA ) is an intuitionistic fuzzy Γ-right ideal of an intra-regular Γ-LA ** -semigroup S
and let a∈S, then there exist x,y∈S and α,β ,ε ∈ Γ such that a = (xα (a βa)) εy . Now by using (3), (1) and (4), we have
µ A (a) = µ A ((xα(aβa))εy) = µA (( aα (x βa)) εy ) = µA (((( xα (a βa)) εy ) α( x βa )) εy) = µA ((((xβa) εy) α( xα ( aβ a) )) εy)
= µA ((((a βa)εx)α (yα (xβa)))εy) = µA ((yα(yα(xβa)))ε ((aβ a)ε x)) = µA ((aβa)ε((yα(yα(xβa)))εx)) ≥ µA (aβa)
and similarly we can show that λ A (a) ≤ λA (aβa) . Thus A = (µA , λA ) is an intuitionistic fuzzy Γ-semiprime.
(ii): Let A = (µA , λA ) be an intuitionistic fuzzy Γ-left ideal of an intra-regular Γ-LA ** -semigroup S and let a∈S, then
a = (xα (a βa)) εy for some x,y∈S and α,β ,γ ∈ Γ . Now by using (4), (3) and (1), we have
µ A (a) = µ A ((xα(aβa))εy) = µA ((xα(aβ((xα(a βa)) εy))) εy) = µA ((xα((xα(aβa))β( aεy ) ) )εy )
= µA (((xα(aβa))α (xβ (a εy)))εy) = µA ((yα(xβ (aεy ) ) )ε( xα (aβ a))) ≥ µ A ( aβa )
Similarly we can show that λ A (a) ≤ λA (aβa) and therefore A = (µA , λA ) is an intuitionistic fuzzy Γ-semiprime.
(iii): It can be followed from (i) and (ii).
Lemma 7: [8] Let A = (µA , λA ) be an IFS of an intra-regular Γ-LA ** -semigroup S, then A = (µA , λA ) is an
intuitionistic fuzzy Γ-left ideal of S if and only if A = (µA , λA ) is an intuitionistic fuzzy Γ-right ideal of S.
Theorem 6: Let S be an intra-regular Γ-LA ** -semigroup and let A = (µA , λA ) be an IFS, then the following
conditions are equivalent.
(i) A = (µA , λA ) is an intuitionistic fuzzy Γ-two-sided ideal of S.
(ii) A = (µA , λA ) is an intuitionistic fuzzy Γ-bi-ideal of S.
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Proof: (i)⇒(ii) is simple.
(ii)⇒(i): Let A = (µA , λA ) be an intuitionistic fuzzy Γ-bi-ideal of an intra-regular Γ-LA ** -semigroup S and let
a∈S, then there exist x,y∈S and α,β ,ξ ∈ Γ such that a = (xα (a βa )) ξy. Let ρ ∈ Γ, then by using (3), (1) and (4), we
have
µ A (aρ b) = µ A (((xα ( aβa ))ξy)ρb) = µA (((a α ( xβa ))ξ y)ρ b) = µA ((bξy ) ρ( a (αx aβ) ) ) = µA (aρ((bξy )α(xβa)))
= µA (aρ ((a ξx)α(yβb))) = µA ((a ξx)ρ(aα(yβb))) = µA (((aα(yβb))ξ x)ρ a) = µA (((((xα ( aβa ))ξy)α(yβ b))ξ x)ρ a)
= µA (((((yβ b)ξy) α(x α ( aβa )))ξx)ρa) = µA (((((a βa) ξx) α(y α( yβb )))ξx ) ρa ) = µA (((xα (y α ( yβb )))ξ((aβ a) ξx)) ρa)
= µA (((a βa) ξ((xα (yα (yβb)))ξ x))ρa) ≥ µA (aβ a) ∧ µA (a)
Similarly we can show that λ A (aλ b) ≤ λA (a) , therefore A = (µA , λA ) is an intuitionistic fuzzy Γ-right ideal of S.
Now by using Lemma 7, A = (µA , λA ) is an intuitionistic fuzzy Γ-two-sided ideal of S.
Theorem 7: Let S be an intra-regular Γ-LA ** -semigroup and let A = (µA , λA ) be an IFS, then the following
conditions are equivalent.
(i) A = (µA , λA ) is an intuitionistic fuzzy Γ-two-sided ideal of S.
(ii) A = (µA , λA ) is an intuitionistic fuzzy Γ-(1,2)-ideal of S.
Proof: (i)⇒(ii) is simple.
(ii)⇒(i): Assume that A = (µA , λA ) is an intuitionistic fuzzy Γ-(1,2) -ideal of an intra-regular Γ-LA ** -semigroup S
and let b∈S, then there exists x,y∈S and α,β ,ξ ∈ Γ such that b = (x α ( b βb ) ) ξy. Let ρ ∈ Γ , then by using (3), (1) and
(4), we have
µ A (aρ b) = µ A (aρ ((xα ( b βb )) ξy)) = µA ((xα(bβb))ρ(aξy)) = µA ((bα(xβb))ρ(aξy))
= µA (((aξy)α(xβb))ρb) = µA (((a ξy)α(xβ b))ρ ((xα ( b βb )) ξy)) = µA ((xα(bβb)) ρ(((aξy)α(xβb))ξy))
= µA ((yα((aξ y)α (xβb)))ρ ((bβb)ξx)) = µA ((bβb) ρ((yα ((aξ y)α (xβb)))ξx)) = µA ((xβ(y α((aξ y)α (xβb))))ρ(bξb))
= µA ((xβ(y α((bξ x)α (yβa))))ρ(bξb)) = µA ((xβ((bξx )α(yα(yβa))))ρ(bξb)) = µA (((bξx )β( xα(yα(yβa))))ρ(bξb))
= µA (((((xα ( bβb )ξ) y)ξ x)β (xα (yα (yβ a))))ρ(bξb)) = µA (((xξy) ξ(x α(bβb)))β ((xα(yα(yβa))))ρ(bξb))
= µA (((((bβb)ξx) ξ(y αx))β (xα (yα(yβa))))ρ(bξb)) = µA (((((yαx)ξx) ξ(b βb))β (xα (yα(yβa))))ρ(bξb))
= µA ((((yα(yβa)) ξx) β((bβb) α((yα x)ξx)))ρ(bξb)) = µA ((((yα(yβa)) ξx) β((bβb) α((xα x)ξy)))ρ(bξb))
= µA (((bβb)β(((yα(yβa))ξ x)α ((xα x)ξy)))ρ(bξb)) = µA (((((xαx)ξy)β((yα(yβa))ξ x))β (bα b))ρ (bξ b))
= µA ((bβ (((xαx)ξy)β(((yα(yβa))ξx)αb)))ρ(bξb)) ≥ µ A (b) ∧ µ A (b) ∧ µ A (b) = µ A (b)
Similarly we can show that λ A (aλ b) ≤ λA (b) , therefore A = (µA , λA ) is an intuitionistic fuzzy Γ-left ideal of S.
Now by using Lemma 7, A = (µA , λA ) is an intuitionistic fuzzy Γ-two-sided ideal of S.
Theorem 8: Let S be an intra-regular Γ-LA ** -semigroup and let A = (µA , λA ) be an IFS, then the following
conditions are equivalent.
(i) A = (µA , λA ) is an intuitionistic fuzzy Γ-bi-ideal of S.
(ii) A = (µA , λA ) is an intuitionistic fuzzy Γ-generalized bi-ideal of S.
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Proof: (i)⇒(ii) is obvious.
(ii)⇒(i): Assume that A = (µA , λA ) is an intuitionistic fuzzy Γ-generalized bi-ideal of an intra-regular LA semigroup S and let a∈S, then there exist x,y∈S and α,β ,ξ ∈ Γ such that a = (xα (a βa )) ξy. Let ρ,ξ ,ε ∈ Γ , then by
using (3), (4), (4) and (1), we have
µ A (aρ b) = µ A (((xα ( aβa ))ξy)ρb) = µA (((a α(xβa))ξy)ρb) = µA ((((uξv)α(xβa))ξy)ρb)
= µA ((((aξx) α(v βu))ξ y)ρ b) = µA (((vα((aξ x)β u))ξ (sεt))ρb) = µA (((tαs)ξ(((aξx) βu) εv)) ρb)
= µA (((tαs)ξ((vβ u)ε( a ξx ) ) )ρb ) = µA (((tαs)ξ(aε ((vβ u)ξx)))ρb) = µA ((aξ((tα s)ε ((vβ u)ξx)))ρb) ≥ µA (a) ∧ µ A (b)
Similarly we can show that λ A (ab) ≤ λA (a) ∨ λ A (b) , therefore A = (µA , λA ) is an intuitionistic fuzzy Γ-bi-ideal of S.
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