SOME RESULTS ON DERIVATIONS OF BCI-ALGEBRAS α

(1)

SOME RESULTS ON DERIVATIONS OF BCI-ALGEBRAS

Hamza A. S. Abujabal, and Nora O. Al-Shehri

Mathematics Department, Faculty of Sciences, King Abdulaziz University, P. Box 80009, Jeddah 21589, Saudi Arabia.

(Received: July 7, 2006)

ABSTRACT: In this note, we investigate some fundamental properties and prove some results on derivations of BCI-algebras.

1. INTRODUCTION

In the theory of rings and near rings, the properties of derivations is an important topic to study [8, 10]. In [7], Jun and Xin applied the notions of rings and near rings theory to BCI- algebras and obtained some properties.

In this paper, we prove some results on derivations of BCI-algebras. First, we show that a derivations of BCK-algebra is regular and prove that if

d

is a derivation of a BCI-algebra

X

and

α ∈ Χ

such that

a * d ( x ) = 0

or

, 0

* ) ( x a =

d

for all

x ∈ X ,

then

d

is regular and

X

is a BCK-algebra. Also we study a derivation of a

p −

semisimple BCI-algebra

X

, then

d

₁

, d

₂are derivations of a

p −

X

, then

d

₁o

d

₂ is also a derivation of

X

and

d

₁o

d

₂ =

d

₂o

d

₁. Finally, we show that

d

₁

* d

₂ =

d

₂

* d

₁, where

d

₁

, d

₂are derivations of a

p −

X

.

2. PRELIMINARIES

Let X be a set with a binary operation * and a constant 0. Then

( X ,*, 0 )

is called a BCI-algebra, if it satisfies the following axioms for all

x , y , z ∈ X

: BCI-1:

((x * y) * (x * z)) * (z * y) = 0

;

BCI-2:

(x * (x * y)) * y = 0

; BCI-3:

x * x

=

0

;

BCI-4:

x * y = 0

and

y * x = 0

imply

x = y

[6].

Define a binary relation

≤

on X by putting

x ≤ y

if and only if

o y

*

x =

. Then

(X ; ≤ )

is a partially ordered set. A BCI-algebra

X

satisfying

0

≤

x

, for all

x ∈ X

, is called a BCK-algebra. In any BCI- algebra

X

, the following hold [6] for all

x , y , z ∈ X

:

(2)

(1)

x * 0

=

x

.

(2)

(x * y) * z = (x * z) * y

. (3)

0 * (x * y) = (0 * x) * (0 * y)

. (4)

* (x * (x * y)) = x * y

[1].

(5)

((x * z) * ( y * z)) * ( x * y) = 0

.

(6)

x ≤ y

implies

x * z ≤ y * z

and

z * y ≤ z * x

. (7)

x * 0

=

0

implies

x = 0

.

For a BCI-algebra

X

, denote by

X

₊ =

{ x

∈

X 0

≤

X }

, the BCK-part of

X

, and by

G ( X )

=

{ x∈X 0 * x=x } , the BCI-G part ofX . IfX+ ={ } 0 , then

X

is called a

p −

semisimple BCI – algebra. For all

x , y , z ∈ X

, the following hold in a p-semisimple BCI-algebra

X

:

(8)

(x * z) * (y * z) = x * y

. (9)

0 * (0 * x) = x

.

(10)

x * (0 * y) = y * (0 * x)

. (11)

x * y = 0

implies

x = y

.

(12)

x * y = x * z

implies

y = z

. (13)

y * x = z * x

implies

y = z

. (14)

y * (y * x) = x

[3, 4, 5, 11]

Let X be a BCI-algebra. We denote

x ^ y = y * ( y * x )

, For more details, we refer to [2, 9, 12, 13].

Definition 2.1 [7] Let X be a BCI-algebra. By a (

l , r

) – deviation of X, we

mean a self map

d

of

X

satisfying the identity

), (

*

^

* ) ( )

*

( x y d x y x d y

d =

for all

x , y ∈ X

. If X satisfies the identity

,

* ) (

^ ) (

* )

*

( x y x d y d x y

d =

for all

x , y ∈ X ,

then we say that d is a

) ,

( r l

- derivation of

X

. Moreover, if

d

is both a

( l , r )

^{and a}

( r , l )

- derivation, we say that

d

is a derivation of X.

Definition 2.2 [7]. A self map

d

of BCI-algebra X is said to be regular if

d ( 0 ) = 0

.

Proposition 2.3 [7] Let

d

be a regular derivation of BCI-algebra

X

. Then the following hold:

(3)

{ ( ) 0 } lg ( 0 ) .

) 0 ( ) (

. , ),

(

* ) (

* ) ( )

* ( ) (

. , ,

) (

*

* ) ( ) (

. )

( ) (

1 1

− +

− = ∈ = ⊂

∈

≤

=

∈

≤

∈

≤

X d

and X of ebra suba

a is x

d X x d

iv

X y x all for y d x d y x d y x d iii

X y x all for y d x y x d ii

X x all for x x d i

3. SOME RESULTS ON DERIVATIONS First, we study derivations on BCK-algebras:

Proposition 3.1: Every

( ) r ,

_l -derivations (or a

( l , r )

-derivation) of a BCK-algebra is regular.

Proof. Let

X

is a BCK-algebra and

d

a

( r , l )

-deviation of a

X

. Then for all

x ∈ X

, we have:

0

* ) 0 (

^ 0

* ) 0 (

^ ) (

* 0 )

* 0 ( ) 0 (

=

x d

x d x d x

d d

Now let

d

is a

( l , r )

-deviation of X. Then for all

x

∈

X

we have:

0 0

^

* ) 0 (

) (

* 0

^

* ) 0 ( )

* 0 ( ) 0 (

=

x d

x d x d x d d

From Proposition 3.1 we get:

Corollary 3.2: A derivation of a BCK-algebra is regular.

In Proposition 3.3 (resp. Proposition 3.4), we assume that

d

is a derivation of a BCI-algebra

X

and

a

∈

X

such that

d ( x ) * a = 0 ,

(resp.

a * d ( x ) = 0

), for all

x

∈

X

, then we show that

d

is a regular derivation of

X

and

X

is a BCK-algebra:

Proposition 3.3. Let

d

be a derivation of a BCI-algebra

X

and

a∈ X

such that

d ( x ) * a = 0 ,

for all

x

∈

X .

Then

d

X

. Moreover

X

is a BCK-algebra.

Proof. Let

d

X

and let

a∈ X

such that

, 0

* ) ( x a =

d

for all

x

∈

X .

Since

d

is

( l , r )

-derivation we get:

a a a d x

a a d x a x d a a x d

* 0

* )) (

*

^ 0

* )) (

*

^

* ) ( (

* )

* ( 0

=

thus

0 ≤ a ,

and so

x

∈

X

₊

.

This shows that

0 ) (

* 0

^ 0

) (

* 0

^

* ) 0 ( )

* 0 ( ) 0 (

=

a d

a

d

a

d

a

d

(4)

Hence

d

is a regular derivations of

X

. So by Proposition 2.3 (i) we have

, ) ( x x

d ≤

for all

x

∈

X

and so.

0

* 0

) (

* )

* ) ( ( ) (

* 0

=

≤ a

x d a x d x d x

Thus

0 * x

≤

0

for all

x

∈

X

and so

0 = ( 0 * x ) * 0 = 0 * x

. Then we have

0

≤

x

, for all

x

∈

X

. Which implies that

X

Similarly we can prove:

d

X

and

a∈ X

such that

a * d ( x ) = 0 ,

for all

x

∈

X .

Then

d

X

. Moreover,

X

Finally, we study a derivations of a

p −

semisimple BCI-algebra:

Definition 3.5: Let

X

be a BCl-algebra and

d

₁

, d

₂two self maps of

X

We define d1. d2

d

₁o

d

₂

: X

→

X as :

( ( ) ) , )

(

₁ ₂

2

1

d x d d x

d

o = ^{for all}

x ∈ X .

Proposition 3.6 Let

X

be a

p −

semisimple BCI-algebra and

d

₁

, d

₂ the

( )

l

, r

-derivations of

X

. Then

d

₁o

d

₂ is also a

( )

l

, r

-derivation of

X

. Proof. Let

X

be a

p −

semisimple BCI – algebra and

d

₁

, d

₂ are

( )

l

, r

^- derivations of

X

. Then by (14) and proposition 2.3 (ii), we get for all

: , y X x ∈

( ) ( ) ( )

( ) ( )

( )

( d x d d x d y y ) ( d ( x x d d d y y ) d ) ( d d ( d x x ) ) y y )

y x d d y d x y x d d y x d d

* ) (

* ) ( (

*

* ) (

*

* ) ( )

(

* ) (

^

* ) (

* ) ( )

(

*

^

* ) ( )

* (

2 1 2

1 2

1

2 1 1

2 2

1

2 1 2

1

=

= o =

=

( d

₁o

d

₂

)( x ) * y ^ x * ( d

₁o

d

₂

)( y )

Which implies that

d

₁o

d

₂^{is a}

( )

l

, r

- derivation of

X

. Similarly, we can prove:

X

be a

p −

semisimple BCI –algebra and

d

₁

, d

₂are

( ) r ,

l –derivations of

X

. Then d1 d2 is also a

( ) r ,

l –derivation of

X

.

(5)

Combining Propositions 3.6 and 3.7, we get:

Theorem 3.8. Let

X

be a

p −

d

₁

, d

₂ derivations of X. Then

d

₁o

d

₂is also a derivation of X.

X

be a

p −

2 1

, d

d

derivations of X. Then

d

₁o

d

₂ =

d

₂o

d

₁.

Proof. Let

X

be a

p −

d

₁

, d

₂, the derivations of X. Since

d

₂ is a

( l , r )

–derivation of

X

, then for al

x , y ∈ X :

( ) ( )

( ( ( * ) * ) ) ) ( ) * ^ * ( )

)

* (

2 1

2 2

1 2

1

y x d d

y d x y x d d y x d d y x d d

=

= o =

But

d

₁ is a

( r , l )

-derivation of

X

, so

( )

) (

* ) (

^ ) (

* ) (

* ) ( )

* )(

(

1 2

2 1 1

2 2 1 2

1

y d x d

y x d d y d x d

y x d d y x d d

=

= o =

thus we have for all

x , y ∈ X :

) (

* ) ( )

* )(

( d

₁o

d

₂

x y

=

d

₂

x d

₁

y

⁽¹⁾ Also, since

d

₁is a

( r , l )

-derivation of

X

, then for all

x , y ∈ X :

( d

₂o

d

₁

) ( x * y )

=

d

₂

( x * d

₁

( y )^ d

₁

( x ) * y ) ( *

1

( ) )

2

x d y

=

d

But

d

₂is a

( l , r )

-derivation of X, so

( ) ( )

( )

) (

* ) (

) (

* )^

(

* ) (

) (

* )

* (

1 2

1 2 1

2 1 2 1

2

y d x d

y d d x y d x d

y d x d y x d d

=

= o =

Thus we have for all

x , y ∈ X :

( d

₂o

d

₁

) ( x * y )

=

d

₂

( x ) * d

₁

( y )

⁽²⁾ From (1) and (2) we get for all

x , y ∈ X :

( d

₁o

d

₂

) ( x * y )

=

( d

₂o

d

₁

) ( x * y )

(6)

By putting

y = 0

we get for all

x

∈

X :

( ) ( ) )

)(

( d

₁o

d

₂

x

=

d

₂o

d

₁

x

d

₁o

d

₂ =

d

₂o

d

₁

Definition 3.10. Let

X

be a BCI-algebra and

d

₁

, d

₂

,

two self maps of

X

. We define

d

₁

* d

₂

: X ⎯ ⎯→ X as :

( d

1

* d

2

) ( x )

=

d

1

( x ) * d

2

( x ),

for all

x

∈

X

be a

p −

2 1

, d

d

derivations of

X

. Then

d

₁

* d

₂ =

d

₂

* d

₁

Proof. Let

X

is a

p −

d

₁

, d

₂ derivations of

X

. Since

d

₂ is a

( l , r )

-derivation of

X

, then for all

x , y ∈ X :

( d

₁o

d

₂

) ( x * y )

=

d

₁

( d

₂

( x ) * y ^ x * d

₂

( y ) )

=

d

₁

( d

₂

( x ) * y )

But

d

₁is a

( r , l )

-derivation of

X

, so

( ) ( )

) (

* ) (

* ) ( )^

(

* ) (

1 2

2 1 1 2

2 1

y d x d

y x d d y d x d y x d d

=

hence

( d

1o

d

2

) ( x * y )

=

d

2

( x ) * d

1

( y )

^{for all}^x^,^y^∈^X ⁽³⁾ Also, we have that

d

₂ is al

( ) r, l

-derivation of

X

, then for all

x , y ∈ X :

( d

1o

d

2

) ( x * y )

=

d

1

( x * d

2

( y )^ d

2

( x ) * y )

=

d

1

( x * d

2

( y ) )

But

d

₁is a

( ) l, r

- derivation of

X

, so

( ) ( )

) (

* ) (

) (

* )^

(

* ) ( ) (

*

2 1

2 1 2

1 2

1

y d x d

y d d x y d x d y d x d

=

Thus

( d

₁o

d

₂

) ( x * y )

=

d

₂

( x ) * d

₂

( y )

^{for all} ^x^,^y^∈^X ⁽⁴⁾

(7)

From (3) and (4) we get:

) (

* ) ( ) (

* )

(

₁ ₁ ₂

2

x d y d x d y

d

= for all

x , y ∈ X

By putting

x = y

we get for all

x

∈

X :

( ) * * ( ) ( ) ) ( ( * ) * ) ( ( ) )

(

2 1 1

2

2 1

1 2

x d d x d d

x d x d x d x d

=

d

₂

* d

₁=

d

₁

* d

₂

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2. M. Aslam and A.B. Thaheem, A note on p-semisimple BCI-algebras, Math. Japon., 36 (1), (1991) , 39-45.

3. M. Daoji, BCI-algebras and Abelian Groups, Math. Japonica, 35 (5), (1987), 693-696

4. W.A. Dudeck, On BCI-algebras with condition(s), Math. Japonica, 31(1), (1986), 26-29

5. C.S. Hoo, BCI-algebras with condition (s), Math. Japonica, 32 (5), (1987), 749-756.

6. K. Iseki, On BCI-algebras, Math. Sem. Notes, 8(1980), 125-130 7. Y.B. Jun and X.L. Xin, On derivations of BCI-algebras, Inform. Sci.,

159 (2004), 167-176.

8. P. H. Lee and T.K. Lee, On derivations of Prime rings, Chinese J.

Math. 9 (1981), 107-110

9. J. Meng, Y.B. Jun and E.H. Roh, BCI-algebras of order 6, Math.

Japon. 47 (1) (1998), 33-43

10. E. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.

11. L.Tiande and X.Changchang, p-Radical in BCI-algebras, Math.

Japonica, 30 (4), (1985), 511-517.

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SOME RESULTS ON DERIVATIONS OF BCI-ALGEBRAS α