A Note on Extension of Central Loops

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22

International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in Research Paper

A Note on Extension of Central Loops T. Nagaiah1,* 1

Department of Mathematics, Kakatiya University, Andhra Pradesh, India.

* Corresponding author, e-mail: ([email protected]) (Received: 28-2-2011; Accepted: 12-4-2011)

Abstract: In this paper we establish the few relations on central loops and we investigate the central loops are power associative. The commutative left central loops if and only if they are right central loops. We also establish the central loops representation is alternative and flexible. Andrew Rajah and Chong Kam Yoon are proved the construction of M (G, 2), the Moufang loops are isomorphic to M (G, 2). It is an important note that we are extended the previous work Andrew Rajah [11], the Central loops are isomorphic to C (G, 2). Keywords: LC-loop, RC-loop, central loop, inverse property loop, alternative loop, flexible, power associative loop and central abelian loops.

1 Introduction In the theory of loops, central loops are some of the least studied loops. They have been studied by Phillips and vojtechovsky [6, 7, 8], Ramamurthy and solarin [8], Fenyves [3] Adeniran and Jaiyeola [1]. The difficulty in studying them is as a result of the nature of the identities defining them when compared with other Bol-Moufang identities. It can be noticed that in the a aforementioned LC identity, the two x variables are consecutively positioned and neither y nor z between them. A similar observation is true in the other two identities (i.e. the RC and C identities). But this observation is not true in the identities defining Bol- loop, Moufang loops extra loop. Fenyves [3] gave three equivalent identities that define LC-loops, three equivalent identities that define RC-loops and only one identity that defines C -loops. But recently Phillips and vojtechovsky [6, 7] gave four equivalent identities that define LCloops and four equivalent identities that define LC-loops and four equivalent identities that

11

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22

define RC-loops. Three of the four identity given by Phillips and vojtechovsky are the same as the three already given by Fenyves and their basic properties are found in [6, 3]. In [10], chain showed various methods of constructing non associative Moufang loops. If L is a finite non associative central loop for which every minimal set of generator contains an element of order 2, then L contains a non abelian subgroup G and an element u of order 2 in L such that each element of L may be uniquely expressed in the form of gu δ where g ∈ G and δ = 0 or 1. Furthermore, the product of two elements of L is given by

( g1u δ )( g 2 u ε ) = ( g1 g 2 ) v u δ +ε where v = (−1)ε and µ = (− 1) v

µ

ε +δ

Conversely, given any non abelian group (G, ⋅ ) the loop L constructed as indicated above is a non associate central loop. It will be denoted by C (G, 2) So in order to use the product rule above we would first need to calculate the values of u and µ before evaluating g1v g 2 u µ δ +ε . In this paper we proved the central loop is

(

)

isomorphic to C (G, 2) and also the central loops flexible and alternative loops. In this we proved the central loops are isomorphic to C (G, 2) and also we investigate the central loops are flexible and alternative loops.

2

Preliminaries

Definition 2.1: A binary system (L, ⋅ ) in which specification of two of the elements x, y, z in the equation x. y = z uniquely determines the third element is called a quasigroup. Further, it contains and identity element, then it is called a loop. Definition 2.2: A loop (L, ⋅ ) is a central loop if it satisfies any of the three (equivalent central loop identities) i) (x. x)(y. z) = [x. (x. y)].z ii) (z. y)(x . x) = z [(y. x)].x iii) x [y.(y. z)] = [(x. y). y].z Definition 2.3: A loop L with neutral element e is a left inverse property loop if x′ (x y) = y for every x, y ∈ L, where x′ is the unique element satisfying x′ x = e. Dually, L is a right inverse property loop if (y x) x′′ = y for every x, y ∈ L, where x′′ is the unique element satisfying x x′′ = e. A loop that has both the left and right inverse property is an inverse property loop. If x ∈ L is such that x′ (x y) = (y x) x′′ = y for every y, we have x′ = x′e = x′( xx′′) = x′′ Therefore, inverse property loops possess two-sided inverses (i.e., x′ = x′′ = x−1), and it is easy to check that they satisfy the anti automorphic inverse property (i.e., (x y)−1 = y−1 x−1). Definition 2.4: A loop (L, ⋅ ) is said to be left inverse alternative loop if it satisfies the identity x (x y) = (x x) y for all x, y in L. A loop (L, ⋅ ) is said to be right alternative loop if it satisfies the identity (x y) y = x (y y) for all x, y in L. Definition 2.5: Let L be a loop, the loop (L, ⋅ ) is said to a flexible if it satisfies the identity x (y x)= (x y) x for x, y in L.

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22

12

Definition 2.6: A loop L is power associative if for every x ∈ L and every n ≥ 0 the power xn is well-defined. Clearly, the powers x0, x, and x2 are always well-defined. Note that, up to this point, we have carefully avoided all higher powers in our calculations. Definition 2.7: A loop L is diassociative if any sub loop of L generated by two elements is a subgroup. If the operation on loop L is commutative, then L is a commutative loop. A loop (L, ⋅ ) is group If it satisfies associative property x y. z=x. y z Standard references on quasigroups and loop theory [1], for the definitions of a loop, readers are to be consulting Bruck [2]. For all identities used Fenyves [3] is to be consulted. Proposition 1 [3]: Let (L, ⋅ ) be a commutative loop, if L is an Left central ( Right central) loop if and only if It is an Right central ( Left central) loop. Proof: Let L be a commutative LC (RC) - loop. ⇔ ( x. x y) z= x ( x. y z) ⇔ z ( x. x y ) = ( x. y z) x ⇔ z ( x. y x) = (x. z y) x ⇔ z (y x. x) = (z y. x) x ⇔ L is an RC (LC) - loop. Proposition 2: A commutative loop L is an Right central (Left central) - loop then it is a C loop. Proof: Recall that in [3] a loop is a C - loop it is both an RC-loop and an LC-loop. Using this fact alone, the sufficient part is proved. The necessary part follows by this fact and proposition 1. Proposition 3: The Left central loops are power associative. Proof: The power xn is clearly well-defined for n = 0, 1, 2, and since, by the left alternative law, xx2 = x2x, it is also well-defined for n = 3. Assume that n > 3 and that xk is well-defined for every k < n. Let r, s > 0 be such that r + s = n. We now show that xr xs can be rewritten canonically as xr+s−1x. Since x xs = x (x xs−1) = x2 xs−1, we can assume that r > 1. Then xr xs = x (x xr−2). x, which is by the LC-identity and by the induction hypothesis equal to (x x) (xr−2 xs) = (x x)(xr+s−3 x) = x(x xr+s−3). x = xr+s−1x. Proposition 4: Let L be a commutative, alternative, inverse property loop. Then (x y)2 = x2 y2 for every x, y ∈ L. Proof : Consider u = x−1 (x−1. (x y)2) = x−1 (x−1. (x y)(x y)). (Q By alternativity) = x−1 (x−1 (x y). (x y)). = x−1(y .x y). (Q By the inverse property)

13

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22 = x−1(x y .y). (Q By the commutativity) u = x−1(x. y y). (Q By alternativity) Finally, by the inverse property, u = y y. Thus (x y)2 = (x y) (x y) = x (x u) = x(x (y y)) = (x x) (y y). 2 Hence (xy ) = x 2 y 2 ∀x, y ∈ L Corollary 1: C-loops are power associative The following two results are obtained by Andrew Rajah [11]. Proposition 5: We extend these for C (G,2), the following conditions are holds.

(

)(

)

(i) g1u 0 g 2u 0 = (g1 g 2 )u 0 ,

(

)

(ii) (g1u 1 )(g 2u 0 ) = g1 g 2 u 1 ,

(

)(

−1

)

(iii) g1u 0 g 2u1 = ( g 2 g1 )u1 ,

(

)

(iv) (g1u 1 )(g 2u 1 ) = g 2 g1 u 0 , ∀g1 , g 2 ∈ G −1

(

)

Proof: From the definition of C (G, 2), (g1u δ )(g 2u ε ) = g1 g 2 µ u δ +ε v

Where v = ( −1) ε and µ = (− 1)

δ +ε

(

)(

) (

1

(

)(

) (

1

(

)(

) (

(

)(

) (

)

(i ) g1u 0 g 2u 0 = g1 g 2 u (0+0 ) = ( g1 g 2 )u 0 ,

Q v = (− 1) , µ = (− 1)

1 1

(ii) g1u1 g 2u 0 = g1 g 2 −1

−1

(iii) g1u 0 g 2u 1 = g1 g 2

) u(

1+0 )

)

−1 −1

−1

(iv) g 1u 1 g 2 u 1 = g 1 g 2

(

= g1 g 2

−1

)u ,

)

(

0+1

0

= −1

Q v = (− 1) = −1, µ = (− 1)

0+1

1

)

= −1

Q v = (− 1) = −1, µ = (− 1)

u (1+1) = g 2 g1 u 0 , −1

= 1,

Q v = (− 1) = 1, µ = (− 1)

1

u (1+0 ) = ( g 2 g1 ) u1 ,

−1 −1

0+0

0

1+1

1

=1

We note that the power of g 2 in each of the four possible combinations of values for δ and ε is always 1 or -1, solely depending on the value of δ. If δ=1, the power of g 2 is -1, but if δ=0 , the power of g 2 is 1. So we can write the power of g 2 as (− 1)δ . On the other hand, the power of g 1 is always 1, but g 1 may appear on the left or right of g 2 on the value of ε. Hence, we obtain the following proposition.

(

Proposition 6: In C (G,2), g1u δ

)(g u ) = (g δ

2

1−ε 1

g2

( −1)δ

ε

( −1)

δ

, solely depending

)

g1 u δ +ε

Proof: From proposition 5, if δ = 1 , the power of g 2 is − 1 , but if δ=0, the power of g 2 is 1. So we can write the power of g 2 as (−1)δ . The power of g1 is always 1, and if β=0, g 1 will be

14

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22

multiplied on the left of g 2

( −1)α

, whereas if ε = 1 , g 1 will be multiplied on the right of g 2 1−ε

So in any case, we can write this observation as g1 g 2

(

)(

)

(

1− ε

Therefore g1u δ . g 2u ε = g1

g2

( −1)δ

ε

( −1)

δ

.

ε

g1 .

)

g1 u δ +ε

Theorem 1: Let (G, o) be a group and C = {( g , α ) / g ∈ G , α ∈ z 2 } Define * on C as

α

( −1)

(g1 ,α1 ) ∗ (g 2 ,α 2 ) = (g11−α

2

o g2

( −1)α1

α

o g1 2 , α 1 + α 2

)

(i) (C, *) is a central loop, (ii) C = 2 G if G is finite, (iii) (C, *) is not associative if and only if G is not commutative, (iv) (C, *) is isomorphic to C (G, 2). Proof: Clearly C is closed under the operation ∗ and also ∗ is well defined. So (C, ∗ ) is a binary system. Obviously (1,0) ∈ C , where I is the identity element of G, and (1,0) ∗ ( g , α )

= ( g , α ) ∗ (1,0) = ( g , α ) ∀ g ∈ G , α ∈ Z 2 .

Thus (1, 0) is the identity element of (C, ∗ ). For the rest of proof we have chosen to omit writing the product rule ‘ o ’ between element in G, since this results in no confusion but rather simplifies the presentation of our proof. α +1

( −1 )   Take (g ,α )∈C and define ( g ,α )1 =  g (−1) ,α  .  

Clearly ( g ,α ) ∈C 1

 1−α  (−1)α +1  ( −1)α α  α +1 (−1)   g g ,α + α  Now ( g , α ) ∗ ( g ,α ) = ( g ,α ) ∗  g ,α  =  g         1

Similarly it can seen that ( g , α )1 * ( g , α ) = (1,0), ∀α ∈ z 2 so

= (1 ,0) ∀ α ∈ Z 2

( g , α )1 = ( g , α ) −1

That is the inverse element of ( g , α ) . Thus for every element in C, there exist an inverse in C. The central loop identity is

l1 * (l2 * (l2 * l3 ) = [(l1 * l2 ) * l2 ] * l3

Let l = ( g ,α ), l 2 = ( h, β ), l3 = (k , γ ) ,

β

l2 * l3 = ( h, β ) * ( k , γ ) = (h1−γ K ( −1) hγ , β + γ )

(

β β  l2 * (l2 * l3 ) = (h, β ) *[h1−γ K ( −1) hγ , β + γ ) =  h1− ( β +γ ) h1−γ K ( −1) hγ 

β

β

)

( −1) β

l1 * [l2 * (l2 * l3 )] = ( g , α ) * [ h1− ( β +γ ) ( h1−γ K ( −1) h γ ) ( −1) h β +γ ,2 β + γ )

 h β +γ , 2 β + γ  

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Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22

[

β

β

= g 1−( 2 β +γ ) [h1−( β +γ ) (h1−γ K ( −1) h γ ) ( −1) h β +γ

]

( −1)α

β

g 2 β +γ , α + 2 β + γ ]

We write U = g 1−( 2 β +γ ) [ h1−( β +γ ) (h1−γ K ( −1) hγ ) (−1) h β +γ ]( −1) g 2 β +γ β

α

∴ l1 * (l2 * (l2 * l1 )) = (U , α + 2 β + γ ) α

l1 * l2 = ( g , α ) * (h, β ) = [ g 1− β h ( −1) g β , α + β ]

On the other hand

(l1 * l2 )* l2 = ( g 1−β h( −1)

α

(

g β , α + β ) * (h, β ) =  g 1−β h ( −1) g β 

(

)

1− β

Then [(l1 * l 2 ) * l 2 ]* l3 = g 1− β h ( −1) g β

(

 α  g1−β h(−1) g β  Write V =

)

1−β

(

 g 1−β h ( −1)α g β 

)

h

1− β

( −1)α + β

( −1)α

1−β

(g h

α +β

h ( −1)

α

α

α +β

h ( −1)

1− β

( g 1− β h ( −1) g β ) β   α

1−γ

α +2 β

K ( −1)

( g 1− β h ( −1) g β ) β , α + 2β   α

α

α +2 β α g )  K ( −1)  g1−β h(−1) g β  

β β

α +β

h ( −1)

( g 1− β h ( −1) g β ) β , α + 2β ] * (k , γ )

(

1−γ

)

)

1−β

h

( −1)α + β

(

 g 1−β h ( −1)α g β 

1−β

(g h

)

1− β

( −1)α

α +β

h ( −1)

γ   g )  ,α + 2β + γ    β β

∴[(l1 * l2 )* l2 ]* l3 = (V , α + 2β + γ ) l1 *[(l2 *(l2 * l1 )] = [(l1 * l2 ) * l2 )]* l3 . ∀li ∈ C , iff

We have to see that

(

β

β

β

U = g1−2 β  h1−β (h K ( −1) )( −1) h   

Case1: let γ = 0 then

1

V =  g 1− β h (−1) g β  α

)

1− β

h (−1)

α +β

(g

1− β

)  K β

h (−1) g β α

U =V

g 2β

(−1)α + 2 β

α

α

Case 1.1: Let β =0 then U = g[h(hk )]( −1) and V = ( g If α = 0 then

( −1)α

α

h( −1) )h( −1)

U = g [h(h k)] = g( h 2 k ) = gK

and V = (g h) h k = ((gh)h)K = g( h 2 k ) = gk

α

K ( −1)

(Q h2 =e ) (Q h2 =e )

2 If α = 1 , then U = g[ h(h K)]-1 = g[ h 2 k ]-1 = gK-1 ( h =e )

V=gh-1.h-1.K-1 = g((h-1)2K-1 2 =g(h )-1K-1 = gK-1

2

( h =e )

Therefore when α = 1 then U = V . Hence U =V for all α ∈ Z 2 Case 1.2

[

β

β

U = g 1+ 2 β (h1− β (hK ( −1) ) ( −1) h β α

( −1)α

( −1)α

α +β

V = ( g 1− β h ( −1) g β )1− β h ( −1) If β = 1 then U = g 3 { [hK −1 ] −1 h }

]

g2 = g2g

g 2 β (∴ −2 β = 2 β ∈ Z 2 ) α

( g 1− β h ( −1) g β ) β K ( −1)

{ (K

−1 −1

) h −1 .h

}

( −1)α

α +2β

g2

α

( g 1− β h ( −1) g β ) β ]γ

16

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22 α

α

= g{K }( −1) = gK ( −1) α +1

α

V = h ( −1) (h ( −1) g ) K ( −1)

And

α +2

if α = 0, then U = g k and V= (h-1h) g = g k and hence U = V 2 3 V= h ( −1) (h −1 g ) K ( −1) α = 1 , then U=g K-1 and V = (hh −1 ) gK ( −1) = g k-1 3

for all α ∈ Z2

Therefore U = V

β

β

α

Case 2: let γ = 1 , then , U = g −2 β  h− β ( K ( −1) h)( −1) h β +1  ( −1)   α +2 β α α +β α ( g1−β h( −1) g β ) β  and V = K ( −1) ( g1−β h( −1) g β )1−β h( −1)  

g 2 β +1

case 2.1: let α = 0 , then β

β

β

β

U = g −2 β  h − β ( K ( −1) h)( −1) h β +1  g 2 β +1 = g 2 β  h β ( K ( −1) h)( −1) h β +1  g 2 β +1 (Q − β = β ∈ z 2 )     V = K ( −1)

2 β 1− β

β

h ( −1)  g 1− β hg β  ( g 1− β hg β ) β

Suppose β = 1 then 2 2 3 1 2 U = g [h(k-1h)-1 h ] g = [h(h- k)g ]g

= ( h h-1 )k g = k g 2 V= K ( −1)  h( −1) (hg )1  = K [h −1hg ] = Kg For β = 0 then U = [( Kh)h]g = ( Kh 2 ) g = Kg (∴ h 2 = e) 2

And V = K [(g h) h] =Kg (h )=K g

2

(Q h = e )

Therefore U = V for every α ∈ z2 .

[

β

β

Case 2.2: Let α =1 then U = g 2 β h β ( K ( −1) h) ( −1) h β +1

[( g

1+ 2 β

V = K ( −1) For

1− β

h −1 g

]

β 1− β

1+ β

h ( −1)

]

−1

g 2 β +1

(− β = β ∈ z 2 )

( g 1− β h −1 g β ) β [( Kh)h]−1 g = ( Kh 2 ) −1 g = K −1 g

β = 0 , we have U = [(k h) h]-1g = (k h2)-1 = k-1g and 2

V = K −1[ gh −1 ] h −1 ( gh −1 ) 0 = K −1 g (h −1 ) 2 = K −1 g (h 2 ) −1 = k-1g (Q h = e ) 2 2 1 3 For β = 1 we have U = g [ h (k-1h)-1 h ]- g

[

= g 2 [h( K −1 h)h 2 ]−1 g 2 .g = K −1 g

]

(∴ g 2 = e)

V = K ( −1) h ( −1) (h −1 g )1 = K −1 [hh −1 g ] = K −1 g Therefore U = V for all β ∈Z 2 3

Therefore U = V,

2

for every case it follows that

l1 ∗ [l 2 ∗ (l 2 ∗ l1 )] = [(l1 * l 2 ) ∗ l 2 ] ∗ l3 , ∀li ∈ C

(C,

∗

) is a central loop this proves part (1) of this theorem.

17

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22 Since z 2 = 2, Obviously (ii) is true (i.e.) C = 2 G if G is finite. (iii) Suppose G is a non- abelian group, then there exist elements g1, g2 ∈ G such that g1 g2 ≠ g2 g1 −1 −1 Take (1, 1), ( g1 , 0), ( g 2 , 0) ∈ C , using the product rule * , we get  (1,1) *[( g1−1 , 0)  *( g 2−1 , 0) = ( g1 ,1) *( g 2 −1 , 0)

-1 1 = [g1 (g2 )- g1, 1+0]

= [g1g2, 1]

and

−1 (1,1) * ( g1−1 , 0)*( g 2−1 , 0)  = (1,1)*  g1 g 2−1 , 0  = ( g 2 g1 ,1)

Since , g1 g 2 ≠ g 2 g1 , then [(1,1) * ( g1−1 , 0) ] * ( g 2−1 , 0) ≠ (1,1) * [ ( g1−1 , 0) * ( g 2−1 , 0) Now suppose G is an abelian group. Now we have

]

−1 −1  −1 −1  ( g1 , 0)*( g 2 , 0)  = (1,1)*  g1 g 2 , 0  = ( g 2 g1 ,1) α

β

α

∴ ( g1 , α )*( g 2 , β ) = ( g11−β , g 2( −) g1β , α + β ) =  g11−β + β g2( −1) , α + β  =  g1, g 2( −1) , α + β      That is ( g , α ) *( g 2 , β ) =  g1 g 2  Take l1 = ( g , α ), l2 = (h, β ),

( −1)α

,α + β  . 

l3 = (k , γ ) ∈ C α

α

So (l1 * l 2 ) * l 3 = [( g , α ) * (h, β )] * (k , γ ) = [ gh ( −1) , α + β ] * (k , γ ) = [ gh ( −1) K ( −1)

α +β

,α + β + γ ]

l1 * (l 2 * l 3 ) = ( g , α ) * [(h, β ) * ( K , γ )] = β

β

α

( g , α ) * [ hK ( −1) , β + γ ] = [ g (hK ( −1) ) ( −1) , α + β + γ ]

[

α

α +β

= gh ( −1) k ( −1)

,α + β + γ

]

(Since G is abelian)

Thus, if G is abelian, ∀l1 , l 2 , l3 ∈ C , (l1 * l 2 ) * l3 = l1 * (l 2 * l3 ) (i.e.) the Central loop is associative. Hence C is not associative if and only if G is not commutative. The function φ : (C ,*) → C (G ,2) defined as φ (g ,α ) = gu α

[

]

Now φ [( g , α ) * (h, β )] = φ g 1− β h ( −1) g β , α + β = ( g 1− β h ( −1) g β ) u α + β α

α

= ( gu α ) * (hu β )] = φ ( g , α ) * φ ( h, β ) (Since proposition 6)

18

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22 Hence the function φ is homomorphism. Therefore, clearly φ is one to one and onto. Hence it is isomorphism.

Theorem 2: Let (G, o) be a group and C= {( g ,α ) / g ∈G,α ∈ z 2 }. Define * on C as

(g1 ,α 1 )* (g 2 ,α 2 ) =  g11−α

o g 2( −1)

2



i) ii) iii) iv)

α1

 o g 2α 2 , α 1 + α 2  Then 

(C, *) is flexible, (C, *) is left alternative, (C, *) is right alternative, The group (G, o ) is abelian then (C, ∗ ) is an abelian loop if α = β = 0

Proof of (i): Let l1 = ( g , α ), l2 = (h, β ) where g , h ∈ G and l1 , l2 ∈ C The identity for flexible x (y x)=(x y) x We use this identity on C (i.e.) l1 * (l2 * l1 ) = (l1 * l2 ) * l1

Now l1 * (l2 * l1 ) = (g , α )* [(h, β )* (g , α )] β

= (g , α )* h1−α g ( −1) hα , α + β     1−(α + β )  1−α (−1) β α  (−1)α α + β  = g h g h g , 2α + β      ( −1)

β

α

Write A = g 1−(α + β ) h1−α g ( −1) hα    On the other hand (l1 * l2 )* l1 = [(g ,α )* (h, β )]* (g ,α )

gα +β

α

=  g 1− β h ( −1) g β , α + β  * ( g , α )   1−α α α α +β α   =  g 1− β h( −1) g β  g ( −1)  g 1− β h( −1) g β  , 2α + β       α

write

1−α

B =  g 1− β h ( −1) g β   

[

] (

g ( −1)

)

α +β

α

 g 1− β h ( −1) g β   

(

Case (i): If α = 0 , β = 1 then A = g o hg −1 g = hg −1 g = h g −1 g

(

)

B = (hg )g −1 = h gg −1 = h

)

= h(e ) = h

Case (ii): If If α = 1, β = 0 then −1

(

) (

)

(

)

A = (gh) g = h−1g−1 g

(

)

(

)

= h −1 g −1 g = h −1 (e ) = h −1

B = g −1 gh −1 = g −1 g h −1 = eh −1 = h −1

From case (i) and Case (ii)

α

19

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22 ∴ A=B for all α , β ∈ z2

Hence (C, ∗ ) is flexible. Proof of (ii): The identity for left alternative loop is x (x y)=(x x) y We use this identity on C. (i.e.) l1 * (l1 * l2 ) = (l1 * l1 )* l2

[

Now l1 * (l1 * l2 ) = (g , α )* [(g , α )* (h, β )] = ( g ,α )* g 1−β h (−1) g β ,α + β

[

α  =  g 1−(α + β ) g 1− β h (−1) g β 

[

]

( −1)α

( −1)α

α

]

 g α + β ,2α + β  

]

we write A = g 1−(α + β ) g 1− β h (−1) g β

α

g α +β

On the other hand

(l1 * l1 )* l2 = [(g ,α )* (g ,α )]* (h, β ) = (g1−α h(−1)

α

(

=  g 1−α g (−1) g α  α

(

)

1− β

1−α ( −1) gα Write B = g h α

h( −1)

)

1− β

2α

h (−1)

(g

2α

1−α

(g

)

g α , 2α * (h, β )

)

β

g (−1) g α , 2α + β  

1−α

α

g (−1) g α α

)

β

Case (iii ) :if α = 0 and β = 0 then

( )

A = (hg )g = h( gg ) = h g 2

(Qg

= h(e ) = h

=e

2

)

( )

B = h ( gg ) = h g 2 = h(e ) = h

Case (iv) :if α = 1 and β = 0 then

(

A = gh −1

)

−1

(

(

)

g = hg −1 g

)

= h g −1 g = h(e) = h

(

)

(

)

B = g −1 g h(−1) g −1 g = h 2

From case (i) and case (ii) A=B for all α , β ∈ z2 . The loop (C, *) is right alternative.

Proof of (iii): The right alternative identity (x y) y=x (y y).

20

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22 (i.e.) (l1 * l 2 ) ∗ l2 = l1 ∗ (l2 ∗ l2 )

We apply the above identity on C

[

]

Now (l1 * l 2 ) * l2 = [( g , α ) * ( g , α )]* (h, β ) = g 1−β h (−1) g β ,α + β * (h, β )

(

=  g 1−β h (−1) g β 

(

α

we write A = g 1− β h (−1) g β α

)

1− β

)

1− β

h (−1)

α +β

α

h (−1)

α +β

(g

1− β

(g

1− β

h (−1) g β α

h (−1) g β α

) ,α + 2β  β

)

β

on the other hand

[

l1 * (l2 * l2 ) = ( g , α )* [(h, β ) * (h, β )] = (g ,α )* h1− β h (−1) h β , 2β β

(

β  =  g 1− 2 β h1− β h (−1) h β 

(

We write B = g 1− 2 β h1− β h ( −1) h β β

)

( −1)α

)

( −1)α

]  g 2 β , α + 2β  

g 2β

Case (v) :if α = 0 and β = 1 then

(

)

A = h −1 (hg ) = h −1h g = g

(

)

(Q g

B = g −1 ( h −1h) g 2 = g −1 ( gg ) = g −1 g g = g

2

=e

)

Case (vi ) : if α = 1and β = 0 then

(

)

(

A = gh − 1 h − 1 = g h − 1 h − 1

( )

= g h2

−1

= g (e ) = g −1

( )

B = g (hh ) = g h 2 −1

)

−1

= ge −1 = g

From case (v) and case (vi), A = B for all α , β ∈ Z 2 . Therefore (C, *) is right alternative loop

Proof of (iv): Let g1 , g 2 ∈ G , Given G is abelian then, g1 g 2 = g 2 g1 and Let l1 = (g1 , α ), l 2 = ( g 2 , β ) where l1 , l 2 ∈C

(

l1 * l2 = (g1 , α )* (g 2 , β ) = g1

1− β

g2

(−1)α

β

g1 , α + β

)

(

=  g1 

1− β

β

)

g1 g 2

( −1)

α

,α + β  

21

Int. J. Pure Appl. Sci. Technol., 4 (1) (2011), pp. 10-22

(

l 2 * l1 = (g 2 , β )* ( g1 ,α ) = g 2

1− β

(

=  g1 g 2 

( −1)

( −1)α

α

g1

=  g2 

1− β

α

,α + β  

g2 , β + α

α

)

g 2 g1

( −1)

β

(1)

)

,α + β  

β

( −1 ) =  g 2 g 1 ,α + β   

(2)

If α = β = 0 then (− 1) = (− 1) = 1 α

β

From (1) and ( 2) l1 * l2 = l 2 * l1 ∀ l1 , l2 ∈ C ∴ (C , *) is an abelian loop if (G, o ) is a abelian group and α = β = 0.

Acknowledgements The author wish to thank referees for their valuable suggestions which helped to improve the article. Also the author most grateful to Dr. V. Maheswara Rao for his constant guidance throughout the presentation of this paper.

References [1] Malcev, Analytical loops, Math. Sb., 36(1955) 429-458. [2] R. H. Bruck, A Survey of Binary Systems, Springer - Verlag, New York, 1971. [3] F. Frenyves, Extra loops II, Publ. Math. Debrecen, 16(1969), 187-192. [4] F. Fenyves, Extra loops I, Publ. Math. Debrecen, 15(1968), 235–238. [5] J. D. Phillips and Peter Vojtechovsky, C- Loops an Introduction. [6] J. D. Phillips and P. Vojtechovsky, On C-loops, Publ. Math. Debrecen, 68(2006), 115–137. [7] M. K. Kinyon, J. D. Phillips and P. Vojtechovsky, C-loops Extensions and Construction, J. Alg. and Applica (to appear). [8] V. S. Ramamurthy and A.R.T. Solarin, On finite right central loops, Publ. M. Debrecen, (1988).

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22

[9] A. S. Basarab, K- loops (Russian), Buletinul. AS Rep. Moldova, Ser Matematica, 1-7 (1992), 28–33. [10] O. Chein, Moufang loops of small order, Mem. Amer. Math. Soc., 13(1978). [11] A. Rajah and C. K. Yoon, An alternative construction of the Moufang loop M (G, 2), School of Mathematical Sciences, University Science, Malaysia (to appear).