Research Journal of Applied Sciences, Engineering and Technology 7(9): 1848-1855, 2014
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2014
Submitted: June 07, 2013
Accepted: August 05, 2013
Published: March 05, 2014
Some Characterizations of Intra-regular Abel-Grassmann Groupoids
1
Madad Khan, 1Saima Anis and 2Faizullah Faiz
1
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan
2
NUST, Pindi, Pakistan
Abstract: In this study we introduce a new class of a non-associative algebraic structure namely intra-regular Abel
Grassmann's groupoid (AG-groupoid in short). We apply generalized fuzzy ideal theory to this class and discuss its
related properties. We introduce (∈,∈ ∨ qk ) -fuzzy semiprime ideals in AG-groupoids and characterize it. Specifically
we have characterized intra-regular AG-groupoids in terms of left, bi and two sided ideals and (∈,∈ ∨ qk ) -fuzzy left,
bi and two sided ideals. For support of our arguments we give examples of AG-groupoids. At the end we
characterize intra-regular AG-groupoids using the properties of (∈,∈ ∨ qk ) -fuzzy semiprime ideals.
Keywords: AG-groupoid, (∈,∈ ∨ qk ) -fuzzy ideals and (∈,∈ ∨ qk ) -fuzzy bi-ideals, left invertive law, medial law,
paramedial law
INTRODUCTION
The real world has a lot of different aspects which
are not usually been specified. In different fields of
knowledge like engineering, medical science,
mathematics, physics, computer science and artificial
intelligence, many problems are simplified by
constructing their "models". These models are very
complicated and it is impossible to find the exact
solutions in many occasions. Therefore, the classical set
theory, which is precise and exact, may not be suitable
for such problems of uncertainty.
In today's world, many theories have been
developed to deal with such uncertainties like fuzzy set
theory, theory of vague sets, theory of soft ideals, theory
of intuitionist fuzzy sets and theory of rough sets. The
theory of soft sets has many applications in different
fields such as the smoothness of functions, game theory,
operations research, Riemann integration etc. The basic
concept of fuzzy set theory was first given by Zadeh
(1965). Zadeh (1965) discussed the relationships
between fuzzy set theory and probability theory.
Rosenfeld (1971) initiated the fuzzy groups in fuzzy set
theory. Mordeson et al. (2003) have discussed the
applications of fuzzy set theory in fuzzy coding, fuzzy
automata and finite state machines.
The idea of belongingness of a fuzzy point to a
fuzzy subset under the natural equivalence on a fuzzy
subset has been defined by Murali (2004). Bhakat and
Das (1992) gave the concept of ( α , β ) -fuzzy subgroups
where α , β ∈ {∈, q,∈ ∨ q,∈ ∧ q} and α ≠∈ ∧ q. The
idea of (∈,∈ ∨ q ) -fuzzy subgroups is a generalization of
fuzzy subgroupoid defined by Rosenfeld (1971). An
(∈,∈ ∨ qk ) -fuzzy bi-ideals and (∈,∈ ∨ qk ) - fuzzy quasiideals and (∈,∈ ∨ qk ) -fuzzy ideals of a semi group are
defined in Shabir et al. (2010a).
In this study we have discussed the (∈,∈ ∨ qk ) fuzzy ideals and (∈,∈ ∨ qk ) -fuzzy bi-ideals in a new
non-associative algebraic structure, that is, in AGgroupoids and developed some new results. We have
characterized intra-regular AG-groupoids by the
properties of their (∈,∈ ∨ qk ) -fuzzy ideals.
A groupoid is S called an AG-groupoid if it satisfies
the left invertive law, that is:
(ab)c = (cb)a, for all a, b, c ∈ S .
Every AG-groupoid satisfies the medial law:
(ab)(cd ) = (ac)(bd ), for all a, b, c, d ∈ S
It is basically a non-associative algebraic structure
in between a groupoid and a commutative semi group. It
is important to mention here that if an AG-groupoid
contains identity or even right identity, then it becomes a
commutative monoid. An AG-groupoid is not
necessarily contains a left identity and if it contains a
left identity then it is unique (Mushtaq and Yusuf,
1978). An AG-groupoid S with left identity satisfies the
paramedial law, that is:
Corresponding Author: Madad Khan, Department of Mathematics, COMSATS Institute of Information Technology,
Abbottabad, Pakistan
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Res. J. Appl. Sci. Eng. Technol., 7(9): 1848-1855, 2014
(ab)(cd ) = (db)(ca), for all a, b, c, d ∈ S
Definition 4: A fuzzy subset f of an AG-groupoid S is
called an (∈,∈ ∨ qk ) -fuzzy bi-ideal of S if for all
Also S satisfies the following law:
x, y, z ∈ S and t , r ∈ (0,1] the following conditions hold:
a (bc) = b(ac), for all a, b, c, d ∈ S .
If S is an AG-groupoid with left identity e then
Sa = {sa : s ∈ S } is both left and bi-ideal of S
containing a. Moreover S = eS ⊆ S 2 . Therefore, S = S 2 .
For a subset A of an AG-groupoid S the
characteristics function, C A is defined by:
1, if x ∈ A
.
CA = 
0, if x ∉ A


If xt ∈ f and yr ∈ S implies (xy )min {t , r } ∈ ∨ qk f
•
If xt ∈ f and zr ∈ S implies
((xy ) z )min{t , r } ∈ ∨ qk f
Definition 5: A fuzzy subset f of an AG-groupoid S is
said to be (∈,∈ ∨ q ) -fuzzy semiprime if it satisfies
xt2 ∈ f ⇒ xt ∈ ∨ qf for all x ∈ S and t ∈ (0,1] .
Definition 6: Let A be any subset of an AG-groupoid S,
then the characteristic function (C A )k is defined as:
It is important to note that an AG-groupoid can also
be considered as a fuzzy subset of itself and we can
write S = CS , i.e., S ( x) = 1, for all x in S.
Let f and g be any two fuzzy subsets of an AGgroupoid S, then the product f  g is defined by:
( f  g )(a) = a∨
=bc
•
{ f (b) ∧ g (c)}, if there exist b, c ∈ S , such that a = bc.
 ≥ 1−2k if a ∈ A
otherwise.
0
(C A )k (a ) = 
The proofs of the following four theorems are same
as in Shabir et al. (2010b).
Theorem 1: Let f be a fuzzy subset of S. Then f is an
AG-subgroupoid
of
S
if
(∈,∈ ∨ qk ) -fuzzy
0, otherwise.
f ( xy ) ≥ min{ f (x ), f ( y ), 1−2k } .
The symbols f ∩ g and f ∪ g will means the
following fuzzy subsets of S:
Theorem 2: A fuzzy subset f of an AG-groupoid S is
called an (∈,∈ ∨ qk ) -fuzzy left (right) ideal of S if:
( f ∩ g )( x) = min{ f ( x), g ( x)} = f ( x) ∧ g ( x), for all x in S .
and
f ( xy ) ≥ min{ f ( y ), 1−2k }( f ( xy ) ≥ min{ f ( x), 1−2k })
( f ∪ g )( x) = max{ f ( x), g ( x)} = f ( x) ∨ g ( x), for all x in S
The following definitions for AG-groupoids are
same as for semigroups in Shabir et al. (2010b):
Definition 1: A fuzzy subset f of an AG-groupoid S is
called an (∈,∈ ∨ qk ) -fuzzy AG-sub groupoid of S if for
all x, y ∈ S and t , r ∈ (0,1] , it satisfies, xt ∈ f , yr ∈ f
implies that (xy )min{t , r } ∈ ∨ qk f
Definition 2: A fuzzy subset f of S is called an
(∈,∈ ∨ qk ) -fuzzy left (right) ideal of S if for all
x, y ∈ S and t , r ∈ (0,1] , it satisfies, xt ∈ f implies
( yx )t ∈ ∨ qk f (xt ∈ f implies (xy )t ∈ ∨ qk f ) .
Definition 3: A fuzzy subset f of an AG-groupoid S is
called an (∈,∈ ∨ qk ) -fuzzy generalized bi-ideal of S if
xt ∈ f and zr ∈ S implies
all x, y, z ∈ S and t , r ∈ (0,1] .
((xy ) z )min{t , r } ∈ ∨ qk f ,
Theorem 3: Let f be a fuzzy subset of S. Then f is an
(∈,∈ ∨ qk ) -fuzzy bi-ideal of S if and only if:
•
•
(i ) f ( xy) ≥ min{ f (x ), f ( y ), 1−2k } for all x, y ∈ S
(ii ) f (( xy ) z ) ≥ min{ f ( x), f (z ), 1−2k } for
all
x, y , z ∈ S
Theorem 4: A fuzzy subset f of an AG-groupoid S is
(∈,∈ ∨ qk ) -fuzzy semiprime if and only if
f ( x) ≥ f ( x 2 ) ∧ 1−2k for all x ∈ S .
Proof: Let f be a fuzzy subset of an AG-groupoid S
which is (∈,∈ ∨ qk ) -fuzzy semiprime. If there exists
some x0 ∈ S such that f (x0 ) < t0 = f (x02 ) ∧ 1−2k . Then
( x02 )t0 ∈ f , but
( x0 ) t0 ∈ f . In addition, we have
( x0 )t0 ∈ ∨ qk f since f is (∈,∈ ∨ qk ) -fuzzy semiprime. On
for
the other hand, we have f ( x0 ) + t0 ≤ t0 + t0 ≤ 1 . Thus
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1 = (3 ∗ 1²) ∗ 1, 2 = (2 ∗ 2²) ∗ 2, 3 = (4 ∗ 3²) ∗ 6,
( x0 ) t0 qk f and so ( x0 ) t0 ∈ ∨ qk f . This is a contradiction.
Hence f ( x) ≥ f ( x 2 ) ∧ 1−2k for all x ∈ S .
Conversely, assume that f is a fuzzy subset of an
AG-groupoid S such that f ( x) ≥ f ( x 2 ) ∧ 1−2k for all
x∈S .
xt2 ∈ f .
Let
( )
f (x ) ≥ f x ∧ ≥ t ∧
following two cases:
2
•
1− k
2
Then
1− k
2
( )
f x2 ≥ t
and
so
4 = (4 ∗ 4²) ∗ 4, 5 = (6 ∗ 5²) ∗ 3, 6 = (5 ∗ 6 2 ) ∗ 5
Therefore, ( S ,∗) is an intra-regular AG-groupoid.
Clearly {1} and {1, 2} are ideals of S. A fuzzy subset
f : S → [0,1] is defined as:
. Now, we consider the
 0.9 for x = 1
 0.8 for x = 2

0.7 for x = 3
f ( x) = 
 0.6 for x = 4
 0.5 for x = 5

 0.5 for x = 6
If t ≤ 1−2k , then f ( x ) ≥ t . That is, xt ∈ f . Thus we
have xt ∈ ∨ qk f .
•
f ( x ) ≥ 1−2k . It follows that
f (x ) + t ≥ 1−2k + t > 1 . That is, xt qk f and so
xt ∈ ∨ qk f also holds. Therefore, we conclude that f
If
t > 1−2k , then
is (∈,∈ ∨ qk ) fuzzy semiprime as required.
Then clearly f is an (∈,∈ ∨ qk ) -fuzzy bi-ideal of S.
Also f is (∈,∈ ∨ qk ) -fuzzy semiprime.
Example 1: Let S = {1, 2, 3} , then from the following
multiplication table one can easily verify that S is an
AG- groupoid:
Lemma 1: Let A be a non-empty subset of an AGgroupoid S, then:
•
. 1 2 3
1 1 1 1
2 3 3 1
3 1 1 1
•
Let us define fuzzy subset f of S as:
f (1) = 0.9, f (2) = 0.6, f (3) = 0.8. Then f is clearly an
(∈,∈ ∨ qk ) -fuzzy ideal.
Definition 7: An AG groupoid S is called intra-regular
AG-groupoid if for each a in S there exists x, y in S
such that a = ( xa ²) y.
Example 2: Let S = {1, 2, 3, 4, 5, 6} , the following table
shows that S is an intra-regular AG-groupoid:
A is a left (right, two-sided) ideal of S if and only if
(C A ) k is an (∈,∈ ∨ qk ) fuzzy left (right, two-sided)
ideal of S.
A of an AG-groupoid S with left identity is bi-ideal
if and only if (C A ) k is (∈,∈ ∨ qk ) -fuzzy bi-ideal.
Proof: It is same as in Shabir et al. (2010a).
Lemma 2: Let A and B be non-empty subsets of an AGgroupoid S, then the following properties hold:
•
(C A∩ B ) k = (C A ∧ k CB ).
•
(C A∪ B ) k = (C A ∨ k CB ).
•
(C AB ) k = (C A  k CB ).
Proof: It is same as in Shabir et al. (2010a).
∗ 1 2 3 4 5 6
1 2 1 1 1 1 1
2 1 2 2 2 2 2
Theorem 5: Let S be an AG-groupoid with left identity
then the following conditions are equivalent:
3
4
5
6
(i) S is intra-regular
(ii) For every left ideal L and for every ideal I,
L ∩ I = IL.
(iii) For every (∈,∈ ∨ qk ) -fuzzy left ideal f and
1
1
1
1
2
2
2
2
4
3
6
5
5
4
3
6
6
5
4
3
3
6
5
4
(∈,∈ ∨ qk ) -fuzzy ideal g f ∧ k g = g  k f .
It is easy to see that ( S ,∗) is an AG-groupoid and is
non-commutative and non-associative structure because
(3 ∗ 4) ≠ (4 ∗ 3) and (3 ∗ 6) ∗ 4 ≠ 3 ∗ (6 ∗ 4) . Also:
Proof: (i ) ⇒ (iii ) Assume that S is an intra-regular
AG-groupoid and f and g are (∈,∈ ∨ qk ) -fuzzy left and
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(∈,∈ ∨ qk ) -fuzzy ideal of S. Since S is intra-regular
therefore for any a in S there exist x, y in S such that
a = ( xa 2 ) y. By using (4) and (1):
This implies (C L∩ I ) k ≤ (C IL ) k . Thus L ∩ I ⊆ IL.
(ii ) ⇒ (i ) By using (2), (3) and (4), we get:
( Sa 2 ) S = ( Sa 2 )( SS ) = ( SS )(a 2 S ) = S (a 2 S ) = a 2 ( SS )
= (aa )( SS ) = ( Sa )( Sa ) = ( SS )(aa ) ⊆ Sa 2 .
a = ( xa 2 ) y = ( x(aa)) y = (a( xa)) y = ( y ( xa))a.
So for any a in S there exist u and v in S such that
a = uv, then:
This implies ( Sa 2 ) S ⊆ Sa 2 . Also:
S ( Sa 2 ) = ( SS )( Sa 2 ) = (a 2 S )( SS ) ⊆ (a 2 S ) S = ( SS )a 2 = Sa 2 .
1− k
2
1− k
≥ g ( y ( xa)) ∧ f (a ) ∧
2
1− k
≥ g ( xa) ∧ f (a ) ∧
2
1− k
≥ g (a ) ∧ f (a ) ∧
2
1− k
= ( g ∧ f )(a ) ∧
2
1− k
= ( f ∧ g )(a ) ∧
2
= ( f ∧ k g )(a ).
( g  k f )(a ) =
∨ g (u ) ∧ f (v) ∧
This implies S ( Sa 2 ) ⊆ Sa 2 . since Sa 2 is both left
and right ideal therefore it is an ideal containing a 2 .
Since Sa 2 is semiprime therefore a ∈ Sa 2 . Now by
using (ii ) :
a =uv
a ∈ Sa ∩ Sa 2 = ( Sa 2 )( Sa) ⊆ ( Sa 2 ) S .
Hence S is intra-regular.
Lemma 3: Every (∈,∈ ∨ qk ) -fuzzy left ideal of an AGgroupoid S is (∈,∈ ∨ qk ) -fuzzy bi-ideal.
Proof: Let S be an AG-groupoid and f be an (∈,∈ ∨ qk ) fuzzy left ideal of S. Then for any x in S there exist a
and b in S such that:
This implies that:
f ∧k g ≤ g k f .
f ((ax)b) ≥ f (b) ∧
Now,
1− k
2
a =bc
1− k
≤ ∨ g (bc) ∧ f (bc) ∧
2
a =bc
1− k
= g (a ) ∧ f (a ) ∧
2
1− k
= ( g ∧ f )(a ) ∧
2
= ( f ∧ k g )(a ).
( g  k f )(a ) =
≥ f (a ) ∧ f (b) ∧
∨ g (b) ∧ f (c) ∧
1− k
.
2
Hence f is an (∈,∈ ∨ qk ) -fuzzy bi-ideal of S
Theorem 6: Let S be an AG-groupoid with left identity
then the following conditions are equivalent:
(i) S is intra-regular
(ii) For every bi-ideal B and left ideal L B ∩ L ⊆ BL.
(iii) For every (∈,∈ ∨ qk ) -fuzzy bi-ideal F and
This implies that:
(∈,∈ ∨ qk ) -fuzzy left ideal g, f ∧ k g ≤ f  k g .
(iv) For every (∈,∈ ∨ qk ) -fuzzy generalized bi-ideal f
g  k f ≤ f ∧ k g.
(∈,∈ ∨ qk ) -fuzzy left ideal g,
f ∧ k g ≤ f  k g.
(v) For all (∈,∈ ∨ qk ) -fuzzy generalized bi-ideal f and g
, f ∧ k g ≤ f  k g.
and
Therefore,
g  k f = f ∧ k g.
(iii ) ⇒ (ii ) Let L be the left and I be an ideal of S. Then
(CL ) k and (C I ) k are the (∈,∈ ∨ qk ) fuzzy left and
(∈,∈ ∨ qk ) fuzzy ideal of S respectively. Therefore
(C L∩ I ) k = (C L ∧ k C I ) ≤ C I  k C L = (C IL ) k .
1− k
2
every
Proof: (i ) ⇒ (v) Assume that S is an intra-regular AGgroupoid with left identity and f and g are (∈,∈ ∨ qk ) generalized fuzzy bi-ideal of S respectively. Thus, for
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any a in S there exist u and v in S such that a = uv,
then:
( f  k g )(a ) =
∨
f (u ) ∧ g (v) ∧
a =uv
Hence S is an intra-regular AG-groupoid.
Theorem 7: Let S be an AG-groupoid with left identity
then the following conditions are equivalent:
1− k
.
2
Since S is intra-regular so for any a in S there exist
x, y ∈ S such that a = ( xa 2 ) y. by using (4) and (1), (3)
and (2) we get:
a = ( xa 2 ) y = ( x( aa)) y = ( a( xa)) y = ( y( xa)) a
(i) S is intra-regular.
(ii) For all left ideals A, B, A ∩ B ⊆ BA.
(iii) For all (∈,∈ ∨ qk ) -fuzzy left ideals f and g,
f ∧k g ≤ g k f .
(iv) For all (∈,∈ ∨ qk ) -fuzzy
bi-ideals f and g,
f ∧k g ≤ g k f .
(v) For all (∈,∈ ∨ qk ) - generalized fuzzy bi-ideals f
= ( y( x(( xa 2 ) y)))a = ( y(( xa 2 )( xy)))a = (( xa 2 )( y( xy)))a
= (( x( aa))( y( xy)))a = (( a( xa))( y( xy)))a
and g, f ∧ k g ≤ g  k f .
= ((( y( xy))( xa))a) a = ((( y( xy))( x(( xa ) y)))a) a
2
Proof: (i ) ⇒ (v) Let S be an intra-regular AG-groupoid
and f and g are both (∈,∈ ∨ qk ) -fuzzy bi-ideals. For any
a ∈ S , there exist u and v in S such that a = uv, then we
get:
= ((( y ( xy))(( xa 2 )( xy)))a) a = ((( xa 2 )(( y ( xy))( xy)))a) a
= (((( xy) a )(( y ( xy)) x))a)a = (((( xy)( y ( xy)))(a x))a)a
2
2
= ((a 2 ((( xy)( y ( xy))) x))a)a = (((aa)((( xy )( y ( xy ))) x))a)a
= ((( x(( xy )( y ( xy ))))(aa))a)a = ((a(( x( xy )( y ( xy )))a))a)a.
( g  k f )(a ) =
Thus, we have:
( f  k g )(a) =
∨
f (u ) ∧ g (v) ∧
a =uv
1− k
2
= ( f ∧ g )(a) ∧
1− k
2
≥ g ((a ((( x( xy )( y ( xy )))a ))a ) ∧ f (a ) ∧
≥ f ((a((( x( xy)( y ( xy)))a))a) ∧ g (a) ∧
≥ ( f (a) ∧ f (a)) ∧ g (a) ∧
∨ g (u ) ∧ f (v) ∧
a =uv
1− k
2
≥ ( g (a ) ∧ g (a )) ∧ f (a ) ∧
1− k
2
≥ g (a) ∧ f (a) ∧
1− k
2
1− k
2
1− k
2
= ( f ∧ k g )(a ).
1− k
.
2
= ( f ∧ k g )(a).
This implies that f ∧ k g ≤ f  k g .
(v) ⇒ (iv) Let f be an (∈,∈ ∨ qk ) -fuzzy generalized
bi-ideal and g be an (∈,∈ ∨ qk ) -fuzzy left ideal. Then by
This implies f ∧ k g ≤ g  k f .
(v) ⇒ (iv) is obvious.
(iv) ⇒ (iii ) Let f and g are (∈,∈ ∨ qk ) -fuzzy left
ideals. Then by Lemma left ideal is a bi-ideal, f and g
are (∈,∈ ∨ qk ) -fuzzy bi-ideals. Then (iii) is obvious.
Lemma left ideal is a bi-ideal, g is also an (∈,∈ ∨ qk ) fuzzy bi-ideal therefore (iv) is obvious.
(iv) ⇒ (iii ) is obvious.
(iii ) ⇒ (ii ) Let B be a bi-ideal and L be a left ideal
of S. Then (CB ) k and (CL ) k are (∈,∈ ∨ qk ) -fuzzy bi-
(iii ) ⇒ (ii ) Assume that A and B are the left ideals
of S then (C A ) k and (CB ) k are (∈,∈ ∨ qk ) -fuzzy biideals of S. Then we get:
ideal and (∈,∈ ∨ qk ) -fuzzy left ideal of S. Then we get
Thus A ∩ B ⊆ BA.
(ii ) ⇒ (i ) Since Sa is both bi-ideal and left ideal
containing a. Using (ii), we get:
(C B∩ L ) k = CB ∧ k CL ≤ CB  k CL = (CBL ) k .
Thus B ∩ L ⊆ BL.
(ii ) ⇒ (i ) Since Sa is both bi-ideal and left ideal
containing a. Therefore by (ii ) and using (3), (2) and
(4), we obtain:
a ∈ Sa ∩ Sa ⊆ ( Sa )( Sa ) = (aa )( SS ) = (aS )(aS ) = a ( SS )
(C A∩ B ) k = (C A ∧ k C B ) ≤ C B  k C A = (C BA ) k .
a ∈ Sa ∩ Sa ⊆ ( Sa )( Sa ) = Sa 2 = ( Sa 2 ) S .
Hence S is intra-regular.
2
= (aa )( SS ) = S (a 2 S ) = ( SS )(a 2 S ) = ( Sa 2 )( SS ) = ( Sa 2 ) S .
Theorem 8: Let S is an AG-groupoid with left identity
then the following conditions are equivalent:
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(i) S is intra-regular.
(ii) For every (∈,∈ ∨ qk ) -fuzzy left ideals f, g and
(∈,∈ ∨ qk ) -fuzzy
right
ideal
h,
( f ∧ k g ) ∧ k h ≤ ( f  k g )  k h.
(∈,∈ ∨ qk ) -fuzzy left ideals f, g
-fuzzy
bi
ideal
h,
and
(∈,∈ ∨ qk )
( f ∧ k g ) ∧ k h ≤ ( f  k g )  k h.
(iv) For all (∈,∈ ∨ qk ) -fuzzy bi-ideals f, g and h,
(iii) For
every
( f ∧ k g ) ∧ k h ≤ ( f  k g )  k h.
(v) For all (∈,∈ ∨ qk ) -generalized fuzzy bi-ideals f, g
and h, ( f ∧ k g ) ∧ k h ≤ ( f  k g )  k h.
Proof: (i ) ⇒ (iii ) Assume that S is an intra-regular AGgroupoid and f and g are (∈,∈ ∨ qk ) -left ideals and h is
an (∈,∈ ∨ qk ) -bi-ideal of S. Since S is an intra-regular
AG-groupoid therefore for all a ∈ S there exist x, y in
S such that a = ( xa 2 ) y. By using (4) and (1), we get:
a = ( x (aa )) y = (a ( xa)) y = ( y ( xa))a.
= ( y ( x(( xa 2 ) y )))a = ( y (( xa 2 )( xy )))a
= (( xa 2 )( y ( xy )))a = (( x(aa ))( y ( xy )))a
= ((a ( xa))( y ( xy )))a = ((( y ( xy ))( xa))a )a
= (((ax)(( xy ) y ))a )a = ((((( xy ) y ) x)a )a )a
= ((((( xy ) y ) x)(( xa 2 ) y )))a )a = ((( xa 2 )((( xy ) y ) x) y ))a )a
= (((a ( xa))((( xy ) y ) x) y ))a )a = ((((((( xy ) y ) x) y )( xa))a )a )a
= (((( y ((( xy ) y ) x))(ax))a )a )a = (((a ( y ((( xy ) y ) x)) x)a)a)a.
(iv) ⇒ (iii ) Assume that f and g are (∈,∈ ∨ qk ) -fuzzy
left ideals and h is an (∈,∈ ∨ qk ) -fuzzy bi-ideal of S then
f and g are (∈,∈ ∨ qk ) -fuzzy bi-ideal, therefore (iii) is
obvious.
(iii ) ⇒ (ii ) is obvious.
(ii ) ⇒ (i ) Assume that f , g are left ideals and S is a
right ideal. Then by using (ii ), we get:
f ∧k g = f ∧k g ∧k S ≤ ( g k f ) k S ≤ g k f .
Therefore, f ∧ k g ≤ g  k f . Hence S is intraregular.
Theorem 9: Let S be an AG-groupoid with left identity
then the following conditions are equivalent:
(i)
(ii)
(iii)
(iv)
semiprime.
(v) Every (∈,∈ ∨ qk ) -fuzzy generalized bi-ideal of S is
fuzzy semiprime.
Proof: (i ) ⇒ (vii ) Let S be an intra-regular and f be an
(∈,∈ ∨ qk ) -generalized bi-ideal of S. Then for all a ∈ S
there exists x, y in S such that a = ( xa 2 ) y. By using
(4), (1), (2) and (3) , we get:
a = ( xa 2 ) y = ( x(aa )) y = (a ( xa)) y = ( y ( xa))a
Now we have:
(( f  k g )  k h)( a) =
= ( y ( x(( xa 2 ) y ))a = ( x( y (( xa 2 ) y )))a
∨(f 
a =uv
k
g )(u ) ∧ h(v) ∧
1− k
2
= ( x(( xa 2 ) y 2 ))a = (( xa 2 )( xy 2 ))a
= ( x 2 (a 2 y 2 ))a = (a 2 ( x 2 y 2 ))a = (a ( x 2 y 2 ))a 2

1− k 
1 − k 
= ∨   ∨ f ( p) ∧ g ( q) ∧
 ∧ h( v ) ∧
2 
2 
a =uv  u = pq
=
= ((( xa 2 ) y )( x 2 y 2 ))a 2 = (( y 2 . y )( x 2 ( xa 2 )))a 2
= (( y 2 x 2 )( y ( xa 2 )))a 2 = (( y 2 x 2 )((uv)( xa 2 )))a 2
1− k 

 f ( p ) ∧ g ( q ) ∧ h( v ) ∧

2 
a = ( pq ) v 
∨
1− k 

 f ( p ) ∧ g ( q ) ∧ h( v ) ∧

2 

1− k
≥ ( f (a) ∧ f (a)) ∧ g (a)) ∧ h(a) ∧
2
1− k
1− k
) ∧ g (a)) ∧ h(a) ∧
≥ (( f (a) ∧
2
2
1− k
1− k
) ∧ h(a)) ∧
= (( f (a) ∧ g (a) ∧
2
2
1− k
= ( f ∧ k g ) ∧ h( a ) ∧
2
= (( f ∧ k g ) ∧ k h)(a).
=
S is intra-regular.
Every ideal of S is semiprime.
Every bi-ideal of S is semiprime.
Every (∈,∈ ∨ qk ) -fuzzy bi-ideal of S is fuzzy
= (( y 2 x 2 )((a 2v)( xu ))))a 2 = (( y 2 x 2 )((a 2 x)(vu )))a 2
= (((( y 2 x 2 )(vu )) x)(aa )))a 2 = ((ax)(a (( y 2 x 2 )(vu ))))a 2
∨
a = ((( a ( y ((( xy ) y ) x )) x ) a ) a ) a = ( pq ) v
= ((aa )( x(( x 2 y 2 )(vu ))))a 2 = (a 2 (( x(( x 2 y 2 )(vu ))))a 2 .
Thus we have:
f (a ) = f ((a 2 (( x(( x 2 y 2 )(vu ))))a 2 ) ≥ f (a 2 ) ∧ f (a 2 ) = f (a 2 ).
Therefore, f (a) ≥ f (a 2 ).
(v) ⇒ (iv) is obvious.
(iv) ⇒ (iii ) Let B be a bi-ideal of S, then (CB ) k is an
(∈,∈ ∨ qk ) fuzzy bi-ideal of S. Let a 2 ∈ B then since
This implies ( f ∧ k g ) ∧ k h ≤ ( f  k g )  k h.
(v) ⇒ (iv) is obvious.
(CB ) k is an (∈,∈ ∨ qk ) fuzzy bi-ideal, therefore by
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Res. J. Appl. Sci. Eng. Technol., 7(9): 1848-1855, 2014
(iv), (CB ( a )) k ≥ (C B ( a 2 )) k ,
as
a2 ∈ B
so,
(CB (a 2 )) k = 1 ≤ (CB (a )) k this implies (CB (a)) k = 1 . Thus
a ∈ B . Hence, B is semiprime.
(iii ) ⇒ (ii ) is obvious.
(ii ) ⇒ (i ) Assume that every ideal is semiprime and
since Sa2 is an ideal containing a2. Thus
a ∈ Sa 2 = ( SS )a 2 = (a 2 S ) S = ( Sa 2 ) S . Hence S is an
intra-regular AG-groupoid.
Theorem 10: Let S be an AG-groupoid with left
identity then the following conditions are equivalent:
(i) S is intra-regular.
(ii) For every left ideal L and bi-ideal B,
L ∩ B ⊆ ( LB ) L .
(iii) For every (∈,∈ ∨ qk ) -fuzzy left ideal f and
(∈,∈ ∨ qk ) -fuzzy bi-ideal g f ∧ k g ≤ ( f  k g )  k f .
Proof: (i ) ⇒ (iii ) Let S be an intra-regular AG-groupoid
and f be an ( ∈,∈ ∨ qk ) -fuzzy left ideal and g be an
( ∈,∈ ∨ qk ) -fuzzy bi-ideal of
Hence f ∧ k g ≤ (( f  k g )  k f ).
(iii ) ⇒ (ii ) Let L be a left ideal and B be a bi-ideal
of S. Then (iii ), (CL ) k and (CB ) k are (∈,∈ ∨ qk ) -fuzzy
left and (∈,∈ ∨ qk ) -fuzzy bi-ideal of S. Then,
(CL∩ B ) k = CL ∧ k CB ≤ (CL  k CB )  k CL ≤ ((CLB ) k  k CL ) = (C( LB ) L ) k .
Therefore, we get L ∩ B ⊆ ( LB ) L .
(ii ) ⇒ (i ) Since Sa is both left and bi-ideal. Let
a ∈ S . So By using (ii):
a ∈ Sa ∩ Sa = (( Sa)( Sa)) Sa = ( Sa 2 )( Sa) ⊆ ( Sa 2 ) S .
Therefore, S is an intra-regular AG-groupoid.
The proofs of following two lemmas are easy and
therefore omitted.
Lemma 4: For any fuzzy subset f of an AG-groupoid
S, S  k f ≤ f and for any fuzzy right ideal g,
g  k S ≤ g.
S. Since S is an intra-
regular AG-groupoid then for any a ∈ S , there exists
x, y ∈ S such that a = ( xa 2 ) y . Then by using (4) and
Lemma 5: Let S be an intra-regular AG-groupoid then
for any (∈,∈ ∨ qk ) -fuzzy subsets f and g, f ∧ k g ∧ k S =
f ∧ k g.
(1), we get:
a = ( xa 2 ) y = (a ( xa)) y = ( y ( xa))a = ( y ( x(( xa 2 ) y ))a = ( x( y ((a ( xa) y )))a
= ( x(a ( xa)) y 2 )a = ((a ( xa))( xy 2 ))a = ((( xy 2 )( xa))a )a.
Lemma 6: Let S be an intra-regular AG-groupoid then
g,
for
any
(∈,∈ ∨ qk ) -fuzzy-subsets f and
( g k f ) k S ≤ g k f .
Therefore,
(( f  k g )  k f )(a ) =
∨
a =uv
( f  k g )(u ) ∧ f (v) ∧
1− k
2
Proof: Let S be an intra-regular AG-groupoid and f and
g are any (∈,∈ ∨ qk ) -fuzzy-subsets then we get:
≥ ( f  k g )((( xy 2 )( xa))a ) ∧ f (a ).
Since f is an (∈,∈ ∨ qk ) -fuzzy left ideal, therefore,
( f  k g )((( xy 2 )( xa))a) =
∨
f (r ) ∧ g ( s) ∧
(( xy )( xa )) a = rs
2
≥ f (( xy 2 )( xa)) ∧ g (a) ∧
( g k f ) k S = (S k f ) k g ≤ f k g = g k f .
Hence ( g  k f )  k S ≤ g  k f .
1− k
2
1− k
2
1− k
2
1− k
.
≥ f (a) ∧ g (a) ∧
2
≥ f ( xa) ∧ g (a) ∧
Theorem 11: Let S be an intra-regular AG-groupoid
and f and g are (∈,∈ ∨ qk ) -fuzzy ideals of S, then
f  k g = f ∧ k g.
Proof: Let S be an intra-regular AG-groupoid and f and
g are (∈,∈ ∨ qk )-fuzzy ideals of S. Then for any a in S
there exist x and y in S such a = ( xa 2 ) y, then:
Thus,
(( f  k g )  k f )(a) ≥ f (a) ∧ g (a) ∧ f (a) ∧
a = ( xa 2 ) y = ( x(aa )) y = (a ( xa)) y = ( y ( xa))a.
1− k
1− k
.
= f (a) ∧ g (a) ∧
2
2
Now,
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Res. J. Appl. Sci. Eng. Technol., 7(9): 1848-1855, 2014
f k g = f ∧k g = ( f ∧k g ) ∧k S = ( g k f ) ∧k S = S k ( g k f ) ≤ g k f .
1− k 

2 
a =uv
1− k
≥ f ( y ( xa)) ∧ g (a ) ∧
2
1− k
≥ . f ( xa) ∧ g (a ) ∧
2
1− k
≥ f (a ) ∧ g (a ) ∧
2
1− k
= ( f ∧ g )(a ) ∧
2
= ( f ∧ k g )(a ).
( f  k g )(a ) =

∨  f (u ) ∧ g (v) ∧
This implies f  k g ≤ g  k f . Also we will show
that f and g are semiprime ideals. So,
f (a) = f (( xa 2 ) y ) ≥ f (a 2 ).
Thus f (a ) ≥ f (a 2 ) . Similarly g (a ) ≥ g (a 2 ) .
(iii ) ⇒ (ii ) Let A and B are ideals then by Lemma
A is ideal if C (A) k is fuzzy ideal, (C A ) k and
(CB ) k are
(∈,∈ ∨ q ) -fuzzy ideals, therefore
k
(C AB ) k = C A  k C B ≤ C B  k C A = (C BA ) k .
Therefore we get AB ⊆ BA .
This implies that f  k g ≥ f ∧ k g.
Now,
2
2
(ii ) ⇒ (i ) Let a ∈ Sa . Since Sa 2 is semiprime
therefore, a ∈ Sa 2 = ( SS )a 2 = (a 2 S ) S = ( Sa 2 ) S .
Hence, S is an intra-regular AG-groupoid.
1− k
2
1− k
≤ ∨ f (uv) ∧ g (uv) ∧
2
a =uv
1− k
= f (a ) ∧ g (a ) ∧
2
1− k
= ( f ∧ g )(a ) ∧
2
= ( f ∧ k g )(a ).
( f  k g )(a ) =
This
∨
f (u ) ∧ g (v) ∧
a =uv
implies
that
f  k g ≤ f ∧ k g.
REFERENCES
Hence
f  k g = f ∧ k g.
Theorem 12: Let S be an AG-groupoid with left identity
then the following conditions are equivalent:
(i) S is intra-regular.
(ii) For every ideals A and B, AB ⊆ BA, and A and B
are semiprime
(iii) For every (∈,∈ ∨ qk ) -fuzzy ideals f and g,
f  k g ≤ g  k f , and f and g are semiprime ideals.
Proof: (i ) ⇒ (iii ) Let S be an intra-regular AG-groupoid
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