The Structure of Gamma AG*-Groupoids
A.R. Shaabani1 and H. Rasouli2*
1,2
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
ashabani@srbiau.ac.ir
hrasouli@srbiau.ac.ir
Abstract. Non-associative algebraic structures are of interest to consider for their remarkable
properties.In this paper, we generalize the AG * -groupoids to   AG * -groupoidsand study their
algebraic properties. Among other results, it is shown that every   AG * -groupoid is left
alternative and a   AG * -groupoid having a left cancellative element is a T 1    AG -groupoid,
aΓ − AG∗ -groupoid𝑆 is a Γ-intra-regular if 𝑆Γ𝑎 = 𝑆 holds for all 𝑎 𝜖 𝑆, let 𝑆 be a Γ-intraregular of Γ − AG∗-groupoid then 𝐵 is a right Γ-ideal of 𝑆 if 𝐵Γ𝑆 = 𝐵, if 𝑆 is a Γ-intra-regular
of Γ − AG∗ - groupoid then (𝑆Γ𝐵)Γ𝑆 = 𝐵 where 𝐵 is a Γ -interior ideal of 𝑆, in an Γ-intraregular of Γ − AG∗ - groupoid 𝑆 if 𝐴 is a Γ-interior ideal of 𝑆 then 𝐴 is a Γ − 𝑏𝑖-ideal of 𝑆, in an
Γ-intra-regular of Γ − AG∗ - groupoid 𝑆 if 𝐴 is a Γ -interior ideal of 𝑆 then 𝐴 is a Γ –(1, 2) -ideal
of 𝑆.
Keywords:   AG -groupoid,   AG * -groupoid, T 1    AG -groupoid,  -left alternative,  left cancellative,   3 -band, Γ -interior ideal,Γ-intra-regular,Γ − 𝑏𝑖-ideal,Γ –(1, 2) –ideal.
1. Introduction and Preliminaries
The idea of generalization of communicative semigroups was introduced in 1977 by Kazim and
Naseerudin[2]. They named this structure as the left almost semigroup (LA-semigroup) in [1]. It
is also called as Abel-Grassmann’sgroupoid (AG-groupoid) in [4]. In generalizing this notion
the new structure   AG -groupoid (   LA-semigroup) is also defined by Shah and Rehman in
[9]. Here we introduce the notion of   AG * -groupoid which is a generalization of AG * groupoid studied in [10],and then investigate some of their properties. Some new results on AG *
-groupoid shave been recently studied by Ahmad and Mushtaqin [4,7].We generalize these
results and investigate some properties of   AG * -groupoids. Following [7-9] we first recall
some preliminary definitions.
Let S and  be non-empty sets. We call S to be a   Semigroup if there exists a mapping
S  S  S writing (a,  , b ) by a b , such that S satisfies the identity (a b)  c  a (b c) for
all a , b , c  S and  ,    .
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A   Semigroup with identity is called a  -monoid.
Let S and  be non-empty sets. We call S to be a   AG -groupoid if there exists a mapping
S  S  S writing (a,  , b ) by a b such that satisfies the identity a b  c  c b   a for all
a , b , c  S and  ,    .
An element e  S is called a left identity if e a  a for all a  S and    .
A   AG -groupoid S is called:
(i)  - medial if for every a , b , c , d  S and  ,    , a b   c d   ac   b  d  .
(ii)  -paramedial if for every a, b, c , d  S and  ,    , ab   c d   d b   c a  .
(iii)  - locally associative if for every a  S and  ,    it satisfies (a a)  a  a (a a) .
(iv)  -idempotent if for every a  S and    , a a  a .
In the following we recall the definitions from [4] which are applied in this paper.
Definition 1.1.A   AG -groupoid S is called a   AG - band if every its element it  idempotent.
Definition 1.2.A   AG -groupoid is called a T 1    AG -groupoid if for every a , b , c , d  S ,
   , a b  c d implies b a  d  c .
Definition 1.3.A   AG -groupoid S is called a   AG  3 -band if a (a a)  (a a)  a  a , for
all a  S and  ,    .
Definition 1.4.A   AG -groupoid S is called a  -left alternative if for all a, b  S and  ,   
, (a a )  b  a (a  b ) .
Definition 1.5. A   AG -groupoid S is called a  -left cancellative if for every a , b , c  S and
   , ab  ac implies b  c .
Definition 1.6. An element 𝑎 of a Γ − AG -groupoid 𝑆 is called a Γ-intra-regular if there exists
𝑥, 𝑦 𝜖 𝑆 and 𝛼, 𝛽, 𝛾 𝜖 Γ such that 𝑎 = (𝑥𝛼 (𝑎𝛽𝑎))𝛾𝑦 and 𝑆 is called a Γ – intra- regular Γ −
AG − groupoid S, if every element of 𝑆 is an Γ-intra- regular.
Definition 1.7. A Γ − 𝐴𝐺-subgroupoid 𝐴 from a Γ − 𝐴𝐺-groupoid 𝑆 is called a Γ – interior ideal
of 𝑆 if (𝑆Γ𝐴)Γ𝑆 ⊆ 𝐴.
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Definition 1.8. A Γ − 𝐴𝐺-subgroupoid 𝐴 from a Γ − 𝐴𝐺-groupoid 𝑆 is called a Γ – bi-ideal of 𝑆
if (𝐴Γ𝑆)Γ𝐴 ⊆ 𝐴.
Definition 1.9. A Γ − 𝐴𝐺-subgroupoid 𝐴 from a Γ − 𝐴𝐺-groupoid 𝑆 is called a Γ –(1,2)- ideal of
𝑆 if (𝐴Γ𝑆)Γ𝐴2 ⊆ 𝐴.
We recall the three following lemmas from [8] which are applied to get some results.
Lemma 1.10.Every   AG -groupoid is  -medial.
Lemma 1.11.Every   AG -groupoid with left identity is  -paramedial.
Lemma 1.12.In an   AG -groupoid S with left identity, we have a (b  c )  b  (a  c ) for
every a , b , c  S and  ,    .
Lemma 1.13. Let 𝑆 be a Γ − AG -groupoid with left identity 𝑒 then, 𝑆Γ2 = 𝑆Γ𝑆 = 𝑆 and 𝑆Γ𝑒 =
𝑒Γ𝑆 = 𝑆.
2. On   AG -groupoids
*
In this section we introduce the notion of   AG * -groupoid which is a generalization of AG * groupoid studied in [10],and then investigate some of their properties.
An AG -groupoid S is called an AG * -groupoid if it satisfies the identity (ab )c  b (ac ) for all
a, b , c  S .
Definition 2.1. A   AG -groupoid S is called a   AG * -groupoid if it satisfies the identity
(a b )  c  b  (a  c) for all a , b , c  S and  ,    .
Example 2.2. Let S be an arbitrary AG * - groupoid and  any non-empty set. Define a mapping
S  S  S ,by a b  ab for all a, b  S and    .Then S is a   AG -groupoid (see [6]).
Also for every a , b , c  S and  ,    we have (a b )c  ab  c  ab c  b (ac ) . On the other
hand, b  (a  c )  b  (ac )  b (ac ) . Hence, (a b )  c  b  (a  c ) and then S is a   AG * -groupoid.
Example 2.3. Let S be an arbitrary   AG * - groupoid and  a fixed element in  . We define
a b  a b for every a, b  S . Then  S ,
and
   , we have

is an AG *-groupoid(see [6]).Also for every a , b , c  S
(a b ) c  (a b ) c  (a b ) c  b  (a c ) . On the other hand,
b (a c )  b (a c )  b  (a c ) . Hence, (a b ) c  b (a c ) .Therefore, S is an AG * -groupoid.
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Example 2.4. Let S be the set of all non-positive integers and  be the set of all non-positive
even integers. If a b denotes as usual multiplication of integers for a, b  S and    , then S is
a   AG * -groupoid but not an
AG * -groupoid.
Example 2.5. Let S be the set of all integers of the form 4n+1 where n is an integer and  denote
the set of all integers of the form 4n  3 . If a b is a    b , for all a, b  S and    , then S is a
  AG * -groupoid but not an AG * -groupoid.
Note that by Examples2.4 and 2.5,   AG * -groupoids are a generalization of AG * -groupoids.
Lemma 2.6. In every   AG * -groupoid, we have b  (a  c )  b  (c  a ) for every a , b , c  S and
 ,  .
Proof. Let S be a   AG * -groupoid. We have
(a b )  c  (c  b )  a ,(by left invertive law)
(a b)  c  b (a c) ,(by   AG * -groupoid)
(c b)  a  b (c a) ,(by   AG * -groupoid)
Then, b  (a  c )  b  (c  a ) .
Lemma 2.7. In a   AG -groupoid S with a left identity, we have a b  a  b for every a, b  S
and  ,    .
Proof. See [ 5 ].
Lemma 2.8. Every   AG * -groupoid with a left identity is a commutative  -semigroup.
Proof. Let S be a   AG * -groupoid with a left identitye. Then by Lemma2.6, we have
b  (a  c )  b  (c  a ) for every a , b , c  S and  ,    . Now putting b  e ,it follows that
e  (a  c )  e  (c  a )
and
then a c  c a
for
commutative.
Now since S is commutative, we obtain:
(a b)  c  (b a)  c (by commutativity)
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every a , c  S and    .
Therefore, S is
 a (b  c ). (by   AG * -groupoid)
Hence, S is a  -semigroup.
Theorem 2.9. If a   AG - band S contains a left identity e ,then S becomes a commutative  monoid.
Proof. See [ 5 ].
Proposition 2.10. Every T 1    AG -groupoid is   paramedial.
Proof. See [ 5 ].
Proposition 2.11. ([10]) If S is an AG * -groupoid, then for every x1 , x2 , x3 , x4  S , we have
( x1 x2 )( x3 x4 )  ( xP (1) xP (2) )( xP (3) xP (4) ) where P is any permutation on the set 1, 2,3, 4 .
Notethat Proposition2.11 can be generalized for   AG * -groupoids with left identity. As for
  AG * -groupoids without left identity we have:
Proposition 2.12. If S is a   AG * -groupoid, then for every x1 , x2 , x3 , x4  S and  ,  ,    ,
( x1 x2 )  ( x3 x4 )  ( x1 xP (2) )  ( xP (3) xP (4) ) where P is any permutation on the set 2,3, 4 .
Proof. Let x1 , x2 , x3 , x4 be arbitrary elements of S . Then we have
( x1 x2 ) ( x3 x4 )  ( x1 x3 ) ( x2 x4 ) (by  -medial law)
 ( x1 x3 )  ( x4 x2 ) (by Lemma 2.6)
 ( x1 x2 )  ( x4 x3 )  ( x1 x4 )  ( x2 x3 )
 ( x1 x4 )  ( x3 x2 )
(by Lemma 2.6.and  -medial law).
Proposition 2.13. Let S be a   AG * -groupoid. Then S is a commutative  -semigroup if any
of the following holds:
i)
ab  c d  a d  bc
ii)
ab  c d  d a  cb ,
for every a , b , c  S and    .
Proof. Since for every a, b  S and    the equation ab  ab trivially holds,an application
of (i)or (ii) proves commutativity. So for every a , b , c  S and  ,    ,we get:
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 ab  c  b  a c    a c b  b c  a  a b c  .
Hence, S is a commutative  -semigroup.
Lemma 2.14. If S is a   AG * -band, then S  S2 ,where S2  S S .
Proof. By the definition of   AG -groupoid we have S2  S S  S .
Let S be a   AG * -band. For every x  S , x  x x  S S for every    . Therefore,
S  S S  S2 . Hence, S  S2 .
3. Some properties of
  AG * -groupoids
In this section we investigate some properties important of  
AG * -groupoids.
Proposition 3.1. Every   AG * -groupoid is a   left alternative .
Proof. Let S be a   AG * -groupoid and let a , b , c  S and  ,    . Then by the definition of
  AG * -groupoid we have  a b   c  b  a c  (1).
Now replacing b by a in (1), we have  a a   c  a  a c  .
Hence, S is a   left alternative .
Theorem 3.2. A   AG * -groupoid S having a   left cancellative element is T 1    AG groupoid.
Proof. Let x be a   left cancellative element of a   AG * -groupoid S and a , b , c , d  S , and
 ,    . Let ab  c d . Then
x  b a    b x  a (by   AG* -groupoid)
  a x   b (by left invertive law)
 x (a b) (by   AG * -groupoid)
 x (c d ) (by assumption)
  c x   d (by   AG* -groupoid)
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  d x   c (by left invertive law)
 x (d c) (by   AG * -groupoid)
 x (b a)  x (d c)
 b a  dc . (by left cancellative law)
Hence,S is a T 1    AG -groupoid.
Corollary 3.3. A   AG * -groupoid S having a   left cancellative element is   paramedical.
Proof. ByProposition2.10and Theorem3.2, the result is immediate.
Proposition 3.4. Let S be a   AG * -groupoid. If t  S is a   3 -band, i.e.  t  t   t  t   t t   t
for every  ,    ,then t2 is a  -idempotent, where t2  t t ,  .
Proof. Let t  S be a   3 -band. Then,
t2  t2  (t t ) (t t )  t (t  (t t )) (by   AG* -groupoid)
 t t  t2 . (by   3 -band)
Hence, t2 is a  -idempotent.
Theorem 3.5. A   AG * -groupoid S having a   left cancellative square element is a
T 1    AG -groupoid.
Proof. Suppose x is a   left cancellative square element of S and ab  c d for every
a , b , c , d  S and  ,    .
Then it gives the following:
 x x   b a   (b  x x ) a
(by   AG * -groupoid)
 ( x b  x) a (by   AG* -groupoid)
 (a x) ( x b) (by left invertive law)
 (a b) ( x x) (by Proposition 2.12)
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 (c d ) ( x x) (by assumption)
 (c x) ( x d ) (by Proposition 2.12)
 (( x d ) x) c (by left invertive law)
 (d  ( x x)) c (by   AG * -groupoid)
 ( x x) (d c) , (by   AG * -groupoid)
i.e. ( x x) (b a)  ( x x) ( d c)
 b a  dc . (by left cancellative law)
Hence, S is a T 1    AG -groupoid.
Proposition 3.6. Every T 1    AG -groupoid is   paramedial.
Proof. Let S be a T 1    AG -groupoid and let a , b , c , d  S and  ,  ,    . Now we have
ab   (c d )  ac   (b d ) (by Lemma 1.6)
 (c d ) ab   (b d ) ac  (by T 1    AG -groupoid)
 (c d ) ab   (b  a) d c  (by Lemma 1.6)
 ab   (c d )  d c   (b  a) (by T 1    AG -groupoid)
 ab   (c d )  d b   (c  a) .(by Lemma 1.6)
Hence, S is   paramedial.
Corollary 3.7. Every   left cancellative   AG * -groupoid is   paramedial
Proof. By Proposition 2.10 and Theorem3.2, the result is immediate.
Lemma 3.8. Let 𝑆 be a Γ − AG∗ -groupoid such that 𝑆Γ𝑎 = 𝑆, hold for all 𝑎 𝜖 𝑆 , then 𝑎Γ𝑆 = 𝑆
and 𝑆Γ𝑆 = 𝑆.
Proof: Since 𝑆Γ𝑎 = 𝑆 for all 𝑎 = 𝜖 𝑆 then 𝑆Γ𝑆 = 𝑆, 𝑆 = 𝑆Γ𝑆 = (𝑆Γ𝑎)Γ𝑆 = 𝑎Γ(𝑆Γ𝑆) =
𝑎Γ𝑆.Hence 𝑎Γ𝑆 = 𝑆.
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Theorem 3.9. A Γ − AG∗ -groupoid 𝑆 is a Γ-intra-regular if 𝑆Γ𝑎 = 𝑆 holds for all 𝑎 𝜖 𝑆.
Proof: Since 𝑆Γ𝑎 = 𝑆 by lemma (2) 𝑆Γ𝑆 = 𝑆. Let 𝑎 𝜖 𝑆, then
𝑆 = 𝑆Γ(𝑆Γ𝑆) = (𝑆Γ𝑎)Γ((𝑆Γ𝑎)Γ𝑆)(𝑏𝑦 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛)
(𝑏𝑦 Γ − 𝐴𝐺 ∗ − 𝑔𝑟𝑜𝑢𝑝𝑜𝑖𝑑)
= ((𝑆Γ𝑎)Γ(𝑆Γ𝑎))Γ𝑆
(𝑏𝑦 𝑚𝑒𝑑𝑖𝑎𝑙 𝑙𝑎𝑤)
= ((𝑆Γ𝑆)Γ(𝑎Γ𝑎))Γ𝑆
= (𝑆Γ𝑎Γ2 )Γ𝑆
Hence 𝑎 𝜖 (𝑆Γ𝑎Γ2 )Γ𝑆, i.e. 𝑆 is Γ-intra-regular.
Theorem 3.10. Let 𝑆 be aΓ-intra-regular of Γ − AG∗-groupoid, then 𝐵 is a right Γ-ideal of 𝑆 if
𝐵Γ𝑆 = 𝐵.
Proof: let 𝐵 is a right Γ-ideal i.e. 𝐵Γ𝑆 ⊆ 𝐵. Now let 𝑏 𝜖 𝐵, then by assumption there exists
𝑥, 𝑦 𝜖 𝑆 𝑎𝑛𝑑 𝛼, 𝛽, 𝛾 𝜖 Γ such that
(𝑏𝑦 Γ – 𝑖𝑛𝑡𝑟𝑎 − 𝑟𝑒𝑔𝑢𝑙𝑎𝑟)
𝑏 = (𝑥𝛼(𝑏𝛽𝑏))𝛾𝑦
(𝑏𝑦 Γ − 𝐴𝐺 ∗ − 𝑔𝑟𝑜𝑢𝑝𝑜𝑖𝑑)
= (𝑏𝛽𝑏) 𝛼 (𝑥𝛾𝑦) 𝜖 𝐵Γ𝑆
implies 𝐵 ⊆ 𝐵Γ𝑆. Hence 𝐵Γ𝑆 = 𝐵. Inverts is obviously.
Theorem 3.11. If 𝑆 is a Γ-intra-regular of Γ − AG∗ - groupoid then (𝑆Γ𝐵)Γ𝑆 = 𝐵, where 𝐵 is a
Γ -interior ideal of 𝑆.
Proof: Since 𝐵 is a Γ-interior ideal of 𝑆also (𝑆Γ𝐵)Γ𝑆 ⊆ 𝐵, let 𝑏 𝜖 𝐵, we have
(𝑏𝑦 Γ – 𝑖𝑛𝑡𝑟𝑎 − 𝑟𝑒𝑔𝑢𝑙𝑎𝑟)
𝑏 = (𝑥𝛼(𝑏𝛽𝑏))𝛾𝑦
(𝑏𝑦 Γ − 𝐴𝐺 ∗ − 𝑔𝑟𝑜𝑢𝑝𝑜𝑖𝑑)
= ((𝑏𝛼𝑥)𝛽𝑏)𝛾𝑦
Since 𝛽 ⊆ 𝑆imples 𝑏𝛼𝑥 𝜖 𝑆. Hence 𝑏 𝜖 (𝑆Γ𝐵)Γ𝑆.
Theorem 3.12. In an Γ-intra-regular of Γ − AG∗ - groupoid 𝑆, if 𝐴 is a Γ -interior ideal of 𝑆 then
𝐴 is a Γ – (1, 2) -ideal of 𝑆.
Proof: let 𝐴 be a Γ-interior ideal of 𝑆, i.e. (𝑆Γ𝐴)Γ𝑆 ⊆ 𝐴. Let 𝜌 𝜖 (𝐴Γ𝑆)Γ(𝐴Γ𝐴), then 𝜌 =
(𝑎𝜇𝑠)𝜃(𝑏𝛼𝑐) for some 𝑎, 𝑏, 𝑐 𝜖 𝐴, 𝑠 𝜖 𝑆 and 𝜇, 𝜃, 𝛼 𝜖 Γ.
Since 𝑆 is Γ-intra-regular so there exists 𝑥, 𝑦 𝜖 𝑆 and 𝛽, 𝛾, 𝛿 𝜖 Γ such that 𝑎 = (𝑥𝛽 (𝑎𝛿𝑎))𝛾𝑦.
Now we have
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(𝑏𝑦 Γ − 𝑙𝑒𝑓𝑡 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑣𝑒)
𝜌 = (𝑎𝜇𝑠)𝜃(𝑏𝛼𝑐) = ((𝑏𝛼𝑐)𝜇𝑠)𝜃𝑎
= ((𝑏𝛼𝑐)𝜇𝑠)𝜃 ((𝑥𝛽(𝑎𝛿𝑎))) 𝛾𝑦
(𝑏𝑦 Γ – 𝑖𝑛𝑡𝑟𝑎 − 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 )
= (𝑥𝛽(𝑎𝛿𝑎))𝜃((𝑏𝛼𝑐)𝜇𝑠)𝛾𝑦
(𝑏𝑦 Γ − 𝐴𝐺 ∗ − 𝑔𝑟𝑜𝑢𝑝𝑜𝑖𝑑)
= (𝑥𝛽(𝑎𝛿𝑎))𝜃((𝑠𝛼𝑐)𝜇𝑏)𝛾𝑦
(𝑏𝑦 Γ − 𝑙𝑒𝑓𝑡 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑣𝑒)
(𝑏𝑦 Γ − 𝐴𝐺 ∗ − 𝑔𝑟𝑜𝑢𝑝𝑜𝑖𝑑)
= (((𝑠𝛼𝑐)𝜃(𝑥𝛽(𝑎𝛿𝑎))) 𝜇𝑏) 𝛾𝑦
is obviously ((𝑠𝛼𝑐)𝜃(𝑥𝛽(𝑎𝛿𝑎))) 𝜖 𝑆, because 𝑆 is Γ − 𝐴𝐺 ∗ -subgroupoid and 𝐴 is Γ − 𝐴𝐺subgroupoid, therefore 𝜌 𝜖 (𝑆Γ𝐴)Γ𝑆 ⊆ 𝐴, i.e. (𝐴Γ𝑆)Γ(𝐴ΓA) ⊆ 𝐴. Hence 𝐴 is a Γ − (1, 2)- ideal
of 𝑆.
Note that, inverts of before theorem is true if 𝑆 be a Γ − AG∗∗ -groupoid, see [5].
Theorem 3.13. In a Γ-interior ideal of Γ − 𝐴𝐺 ∗ -subgroupoid 𝑆 if 𝐴 is a Γ-interior ideal of 𝑆 then
𝐴 is a Γ − 𝑏𝑖-ideal of 𝑆.
Proof: let 𝐴 be a Γ-interior ideal of 𝑆, then (𝑆Γ𝐴)Γ𝑆 ⊆ 𝐴. Let 𝜌 𝜖 (𝐴Γ𝑆)Γ𝐴, then 𝜌 = (𝑎𝜇𝑠)𝜓𝑏
for some 𝑎, 𝑏 𝜖𝐴, 𝑠 𝜖𝑆 and 𝜇, 𝜓 𝜖 Γ . since𝑆 is Γ-intra-regular so there exists 𝑥, 𝑦 𝜖 𝑆 and
𝛽, 𝛾, 𝛿 𝜖 Γ such that 𝑏 = (𝑥 (𝑏𝛿𝑏))𝛾𝑦. Now we have
(𝑏𝑦 Γ − 𝑙𝑒𝑓𝑡 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑣𝑒)
𝜌 = (𝑎𝜇𝑠)𝜓(𝑥𝛽(𝑏𝛿𝑏))𝛾𝑦
= (𝑥 (𝑏𝛿𝑏))𝜓(𝑎𝜇𝑠)𝛾𝑦
(𝑏𝑦 Γ − 𝐴𝐺 ∗ − 𝑔𝑟𝑜𝑢𝑝𝑜𝑖𝑑)
= ((𝑏𝛿𝑏)𝛽(𝑥𝜓(𝑎𝜇𝑠))) 𝛾𝑦
(𝑏𝑦 Γ − 𝐴𝐺 ∗ − 𝑔𝑟𝑜𝑢𝑝𝑜𝑖𝑑)
= (((𝑥 𝜓(𝑎𝜇𝑠))𝛿𝑏) 𝛽𝑏) 𝛾𝑦 𝜖 (𝑆Γ𝐴)Γ𝑆 ⊆ 𝐴
(𝑏𝑦 Γ − 𝑙𝑒𝑓𝑡 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑣𝑒)
Thus (𝐴Γ𝑆)Γ𝐴 ⊆ 𝐴.
Hence 𝐴 is a Γ − 𝑏𝑖- ideal of 𝑆
Note that, inverts of before theorem is true if 𝑆 be a Γ − AG∗∗ -groupoid, see [5].
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