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ON GENERALIZED (α, β)-DERIVATIONS IN BCI-ALGEBRAS†
ON GENERALIZED (α, β)-DERIVATIONS IN BCI-ALGEBRAS†
Journal of Applied Mathematics & Informatics. 2014. Jan, 32(1_2): 27-38
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : March 01, 2013
  • Published : January 28, 2014
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ABDULLAH M. AL-ROQI

Abstract
The notion of generalized (regular) (α, β)- derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. The condition for a generalized (α, β)- derivation to be regular is provided. The concepts of a generalized F-invariant (α, β)- derivation and α-ideal are introduced, and their relations are discussed. Moreover, some results on regular generalized (α, β)- derivations are proved.. AMS Mathematics Subject Classification : 03G25, 06F35.
Keywords
1. Introduction
Throughout our discussion X will denote a BCI-algebra unless otherwise men-tioned. In the year 2004, Jun and Xin [1] applied the notion of derivation in ring and near-ring theory to BCI -algebras, and as a result they introduced a new concept, called a (regular) derivation, in BCI -algebras. Using this con-cept as defined they investigated some of its properties. Using the notion of a regular derivation, they also established characterizations of a p -semisimple BCI -algebra. For a self map d of a BCI -algebra, they defined a d -invariant ideal, and gave conditions for an ideal to be d -invariant. According to Jun and Xin, a self map d : X X is called a left-right derivation (briefly ( l , r )-derivation)of X if d ( x y ) = d ( x ) ∗ y x d ( y ) holds for all x , y X . Similarly, a self map d : X X is called a right-left derivation (briefly ( r , l )-derivation) of X if d ( x y ) = x d ( y )∧ d ( x )∗ y holds for all x , y X . Moreover, if d is both ( l , r )−and ( r , l )−derivation, it is a derivation on X . After the work of Jun and Xin [1] , many research articles have appeared on the derivations of BCI-algebras and a greater interest have been devoted to the study of derivations in BCI algebras on various aspects (see [2 , 3 , 4 , 5 , 6 , 7] ).
Recently in [5] , Muhiuddin and Al-roqi introduced the notion of ( α , β )-derivations of a BCI -algebra, and investigated some related properties. Using the idea of regular ( α , β )- derivations , they gave characterizations of a p -semisimple BCI -algebra. In the present paper, we consider a more general version of the paper [5] . We first introduce the notion of generalized (regular) ( α , β )- derivations of a BCI -algebra, and investigate related properties. We provide a condition for a generalized ( α , β )- derivation to be regular. We also introduce the concepts of a generalized F-invariant ( α , β )- derivation and α -ideal, and then we investi-gate their relations. Furthermore, we obtain some results on regular generalized ( α , β )- derivations .
2. Preliminaries
We begin with the following definitions and properties that will be needed in this paper.
A nonempty set X with a constant 0 and a binary operation ∗ is called a BCI-algebra if for all x , y , z X the following conditions hold:
  • (I) ((x∗y) ∗ (x∗z)) ∗ (z∗y) = 0,
  • (II) (x∗ (x∗y)) ∗y= 0,
  • (III)x∗x= 0,
  • (IV)x∗y= 0 andy∗x= 0 implyx=y.
Define a binary relation ≤ on X by letting x y = 0 if and only if x y . Then ( X ,≤) is a partially ordered set. A BCI-algebra X satisfying 0 ≤ x for all x X , is called BCK-algebra.
A BCI -algebra X has the following properties: for all x ; y ; z X
  • (a1)x∗ 0 =x.
  • (a2) (x∗y) ∗z= (x∗z) ∗y,
  • (a3)x≤yimpliesx∗z≤y∗zandz∗y≤z∗x.
  • (a4) (x∗z) ∗ (y∗z) ≤x∗y,
  • (a5)x∗ (x∗ (x∗y)) =x∗y,
  • (a6) 0 ∗ (x∗y) = (0 ∗x) ∗ (0 ∗y),
  • (a7)x∗ 0 = 0 impliesx= 0.
For a BCI -algebra X , denote by X + (resp. G ( X )) the BCK -part (resp. the BCI -G part) of X , i.e., X + is the set of all x X such that 0 ≤ x (resp. G ( X ) := { x X | 0 ∗ x = x }). Note that G ( X ) ∩ X + = {0} (see [8] ). If X + = {0}, then X is called a p-semisimple BCI-algebra . In a p -semisimple BCI -algebra X , the following hold:
  • (a8) (x∗z) ∗ (y∗z) =x∗y.
  • (a9) 0 ∗ (0 ∗x) =xfor allx∈X.
  • (a10)x∗ (0 ∗y) =y∗ (0 ∗x).
  • (a11)x∗y= 0 impliesx=y.
  • (a12)x∗a=x∗bimpliesa=b.
  • (a13)a∗x=b∗ximpliesa=b.
  • (a14)a∗ (a∗x) =x.
  • (a15) (x∗y) ∗ (w∗z) = (x∗w) ∗ (y∗z).
Let X be a p -semisimple BCI -algebra. We define addition "+" as x + y = x ∗ (0 ∗ y ) for all x , y X . Then ( X , +) is an abelian group with identity 0 and x y = x y . Conversely let ( X , +) be an abelian group with identity 0 and let x y = x y . Then X is a p -semisimple BCI -algebra and x + y = x ∗ (0 ∗ y ) for all x , y X (see [9] ).
For a BCI -algebra X we denote x y = y ∗( y x ), in particular 0∗(0∗ x ) = ax , and Lp ( X ) := { a X | x a = 0 ⇒ x = a , ∀ x X }. We call the elements of Lp ( X ) the p-atoms of X . For any a X , let V ( a ) := { x X | a x = 0}, which is called the branch of X with respect to a . It follows that x y V ( a b ) whenever x V ( a ) and y V ( b ) for all x , y X and all a , b Lp ( X ). Note that Lp ( X ) = { x X | ax = x }; which is the p -semisimple part of X , and X is a p -semisimple BCI -algebra if and only if Lpp ( X ) = X (see [10] ,[Proposition 3.2]). Note also that ax Lp ( X ), i.e., 0 ∗ (0 ∗ ax ) = ax , which implies that ax y Lp ( X ) for all y X . It is clear that G ( X ) ⊂ Lpp ( X ), and x ∗ ( x a ) = a and a x Lp ( X ) for all a Lp ( X ) and all x X . For more details, refer to [11 , 12 , 1 , 10 , 8 , 9] .
Definition 2.1 ( [6] ). A BCI -algebra X is said to be torsion free if it satiafies:
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Definition 2.2 ( [5] ). Let α and β are two endomorphisms of a BCI -algebra X . Then a self map d (α, β) : X X is called a ( α , β )-derivation of X if it satisfies:
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3. Main results
In what follows, α and β are endomorphisms of a BCI -algebra X unless otherwise specified.
Definition 3.1. Let X be a BCK / BCI -algebra. Then a self map F on X is called a generalized ( α , β )- derivation if there exists an ( α , β )- derivation d (α, β) of X such that
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Clearly, the notion of generalized ( α , β )- derivation covers the concept of ( α , β )- derivation when F = d (α, β) and the concept of generalized derivation when F = d (α, β) = D , and α = β = IX where IX is the identity map on X .
Example 3.2. Consider a BCI -algebra X = {0, a , b } with the following Cayley table:
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(1) Define a map
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and define two endomorphisms
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and
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Then d (α, β) is a ( α , β )-derivation of X [5] .
Again, define a self map
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It is routine to verify that F is a generalized ( α , β )- derivation of X .
(2) Define a map
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and define two endomorphisms
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and
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Then d (α, β) is a ( α , β )-derivation of X [5] .
Again, define a self map F : X X by F ( x ) = b for all x X . It is routine to verify that F is a generalized ( α , β )- derivation of X .
Lemma 3.3 ( [12] ). Let X be a BCI-algebra. For any x, y ∈ X, if x ≤ y, then x and y are contained in the same branch of X .
Lemma 3.4 ( [12] ). Let X be a BCI-algebra. For any x, y ∈ X, if x and y are contained in the same branch of X, then x ∗ y, y ∗ x ∈ X+ .
Proposition 3.5. Let X be a commutative BCI-algebra. Then every generalized (α, β)-derivation F of X satisfies the following assertion :
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that is, every generalized (α, β)-derivation of X is isotone .
Proof . Let x , y X be such that x y . Since X is commutative, we have x = x y . Hence
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Since every endomorphism of X is isotone, we have α ( x ) ≤ α ( y ): It follows from Lemma 3.3 that 0 = α ( x ) ∗ α ( y ) ∈ X + and α ( y ) ∗ α ( x ) ∈ X + so that there exists a (≠ 0) ∈ X + such that α ( y x ) = α ( y ) ∗ α ( x ) = a . Hence (3.3) implies that F ( x ) ≤ F ( y ) ∗ a : Using (a3), (a2) and (III), we have
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and so F ( x ) ∗ F ( y ) = 0, that is, F ( x ) ≤ F ( y ) by (a7).
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.6 ( [5] ). Let X be a commutative BCI-algebra . Then every ( α , β )- derivation d (α, β) of X satisfies the following assertion :
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that is , every ( α , β )- derivation of X is isotone .
Proposition 3.7. Every generalized ( α , β )- derivation F of a BCI-algebra X satisfies the following assertion :
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Proof . Let F be a generalized ( α , β )-derivation of X . Using (a2) and (a4), wehave
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Obviously F ( x ) ∧ d (α, β) (0) ≤ F ( x ) by (II). Therefore the equality (3.5) is valid.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.8 ( [5] ). Every ( α , β )- derivation d (α, β) of a BCI-algebra X satisfies the following assertion:
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Theorem 3.9. Let F be a generalized ( α , β )- derivation on a BCI-algebra X . Then
  • (1) (∀a∈Lp(X),x∈X) (F(a∗x) =F(a) ∗α(x)).
  • (2) (∀a∈Lp(X),x∈X) (F(a+x) =F(a) +α(x)).
  • (3) (∀a,b∈Lp(X)) (F(a+b) =F(a) +α(b)).
Proof . (1) For any a Lp ( X ), we have a x Lp ( X ) for all x X . Thus F ( a x ) = F ( a ) ∗ α ( x ) ∧ d (α, β) ( x ) ∗ β ( a ) = F ( a ) ∗ α ( x ).
(2) For any a Lp ( X ) and x X , it follows from (1) that
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(3) The proof follows directly from (2).
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.10 ( [5] ). Let d (α, β) be an ( α , β )- derivation on a BCI-algebra X . Then
  • (1) (∀a∈Lp(X),x∈X)(d(α,β)(a∗x) =d(α,β)(a) ∗α(x)).
  • (1) (∀a∈Lp(X),x∈X)(d(α,β)(a+x) =d(α,β)(a) +α(x)).
  • (3) (∀a,b∈Lp(X))(d(α,β)(a+b) =d(α,β)(a) +α(b)).
Definition 3.11. Let X be a BCI-algebra and F , F′ be two self maps of X , we define F F′ : X X by ( F F′ )( x ) = F ( F′ ( x )) for all x X .
Theorem 3.12. Let X be a p-semisimple BCI-algebra . Let F and F′ be two generalized ( α , β )- derivations associated with d (α, β) and
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( α , β )- derivations respectively on X such that α 2 = α . Then F F′ is an ( α , β )- derivation on X .
Proof . For any x , y X , it follows from (a14) that
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This completes the proof.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.13 ( [5] ). Let X be a p-semisimple BCI-algebra . If d (α, β) and
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are two ( α , β )- derivations on X such that α 2 = α , then
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is an ( α , β )- derivation on X .
Theorem 3.14. Let α , β be two endomorphisms and F be a self map on a p-semisimple BCI-algebra X such that F ( x ) = α ( x ) for all x X . Then F is a generalized ( α , β )- derivation on X .
Proof . Let us take F ( x ) = α ( x ) for all x X . Since x , y X x y X , by using (a14) we have
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This completes the proof.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.15 ( [5] ). Let α , β be two endomorphisms and d (α, β) be a self map on a p-semisimple BCI-algebra X such that d (α, β) ( x ) = α ( x ) for all x X . Then d (α, β) is an ( α , β ) -derivation on X .
Definition 3.16. A generalized ( α , β )-derivation F of a BCI -algebra X is said to be regular if F (0) = 0:
Example 3.17. (1) The generalized ( α , β )-derivation F of X in Example 3.2(1) is regular.
(2) The generalized ( α , β )-derivation F of X in Example 3.2(2) is not regular.
We provide a condition for a generalized ( α , β )-derivation to be regular.
Theorem 3.18. Let F be a generalized ( α , β )- derivation of a BCI-algebra X . If there exists a X such that F ( x ) ∗ α ( a ) = 0 for all x X , then F is regular .
Proof . Assume that there exists a X such that F ( x ) ∗ α ( a ) = 0 for all x X . Then
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and so F (0) = F (0 ∗ a ) = ( F (0) ∗ α ( a )) ∧( d (α, β) ( a ) ∗ β (0)) = 0. Hence F is regular.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.19 ( [5] ). Let d (α, β) be an ( α , β )- derivation of a BCI-algebra X . If there exists a X such that d (α, β) ( x ) ∗ α ( a ) = 0 for all x X , then d (α, β) is regular .
Definition 3.20. For a generalized ( α , β )-derivation F of a BCI -algebra X , we say that an ideal A of X is an α - ideal (resp. β - ideal ) if α ( A ) ⊆ A (resp. β ( A ) ⊆ A ).
Defiition 3.21. For a generalized ( α , β )-derivation F of a BCI -algebra X , we say that an ideal A of X is F-invariant if F ( A ) ⊆ A .
Example 3.22. (1) Let F be a generalized ( α , β )-derivation of X which is described in Example 3.2(1). We know that A := {0, a } is both an α -ideal and a β -ideal of X . Furthermore, A := {0, a } is also F -invariant.
(2) Let F be a generalized ( α , β )-derivation of X which is described in Example 3.2(2). We know that A := {0, a } is a β -ideal of X . But A := {0, a } is an ideal of X which is neither α -ideal nor F -invariant.
Next, we prove some results on regular generalized ( α , β )-derivations in a BCI -algebra. In our further discussion, we shall assume that for every regular generalized ( α , β )-derivation F : X X there exists a regular ( α , β )-derivation d (α, β) : X X i.e. d (α, β) (0) = 0.
Theorem 3.23. Let F be a regular generalized ( α , β )- derivation of a BCI-algebra X . Then
  • (1) (∀a∈X) (a∈Lp(X) ⇒F(a) ∈Lp(X)).
  • (2) (∀a∈X) (a∈Lp(X) ⇒α(a),β(a) ∈Lp(X)).
  • (3) (∀a∈Lp(X)) (F(a) =F(0) +α(a)).
  • (4) (∀a,b∈Lp(X)) (F(a+b) =F(a) +F(b) −F(0)).
Proof . (1) Let F be a regular generalized ( α , β )-derivation. Then the proof fol-lows directly from Proposition 3.7.
(2) Let a Lp ( X ): Then a = 0 ∗ (0 ∗ a ), and so α ( a ) = α (0 ∗ (∗0 ∗ a )) = 0 ∗ (∗0 ∗ α ( a )). Thus α ( a ) ∈ Lp ( X ). Similarly, β ( a ) ∈ Lp ( X ).
(3) Let a Lp ( X ): Using (2), (a1) and (a14), we have
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(4) Let a , b Lp ( X ): Then a + b Lp ( X ): Using (3), we have
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This completes the proof.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.24 ( [5] ). Let d (α, β) , be a regular ( α , β )- derivation of a BCI-algebra X . Then
  • (1) (∀a∈X)(a∈Lp(X) ⇒d(α,β)(a) ∈Lp(X)).
  • (2) (∀a∈X) (a∈Lp(X) ⇒α(a),β(a) ∈Lp(X)).
  • (3) (∀a∈Lp(X))(d(α,β)(a) =d(α,β)(0) +α(a)).
  • (4) (∀a,b∈Lp(X))(d(α,β)(a+b) =d(α,β)(a) +d(α,β)(b) −d(α,β)(0)).
Theorem 3.25. Let X be a torsion free BCI-algebra and F be a regular gener-alized ( α , β )- derivation on X such that α F = F . If F 2 = 0 on Lp ( X ), then F = 0 on Lp ( X ).
Proof . Let us suppose F 2 = 0 on Lp ( X ). If x Lp ( X ), then x + x Lp ( X ) and so by using Theorem 3.23 (3) and (4), we have
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Since X is a torsion free, therefore F ( x ) = 0 for all x X implying thereby F = 0. This completes the proof.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.26 ( [5] ). Let X be a torsion free BCI-algebra and d (α, β) be a regular ( α , β )- derivation on X such that α d (α, β) = d (α, β) . If
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on Lp ( X ), then d (α, β) = 0 on Lp ( X ).
Theorem 3.27. Let X be a torsion free BCI-algebra and F, F′ be two regular generalized ( α , β )- derivations on X such that α F′ = F′ . If F F′ = 0 on Lp ( X ), then F′ = 0 on Lp ( X ).
Proof . Let us suppose F F′ = 0 on Lp ( X ). If x Lp ( X ), then x + x Lp ( X ) and so by using Theorem 3.23 (1) and (2), we have
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Since X is a torsion free, therefore F′ ( x ) = 0 for all x X and so F′ = 0. This completes the proof.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.28 ( [5] ). Let X be a torsion free BCI-algebra and d (α, β) ,
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be two regular ( α , β )- derivations on X such that
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If
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= 0 on Lp ( X ), then
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on Lp ( X ).
Proposition 3.29. Let F be a regular generalized ( α , β )- derivation of a BCI-algebra X . If F 2 = 0 on Lp ( X ), then
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for all x Lp ( X ).
Proof . Assume that F 2 = 0 on Lp ( X ): If x Lp ( X ), then x + x Lp ( X ) and so by using Theorem 3.23 (3) and (4), we have
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Hence
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for all x Lp ( X ).
This completes the proof.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.30 ( [5] ). Let d (α, β) be a regular ( α , β )- derivation of a BCI-algebra X . If
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on Lp ( X ), then
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for all x Lp ( X ).
Proposition 3.31. Let F and F′ be two regular generalized ( α , β )- derivations of a BCI-algebra X . If F F′ = 0 on Lp ( X ), then
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for all x Lp ( X ).
Proof . Let x Lp ( X ). Then x + x Lp ( X ), and so F′ ( x + x ) ∈ Lp ( X ) by Theorem 3.23 (1). It follows from Theorem 3.23 (3) and (4) that
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so that
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for all x Lp ( X ).
This completes the proof.
If we take F = d (α, β) , then we have the following corollary.
Corollary 3.32 ( [5] ). Let d (α, β) and
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be two regular ( α , β )- derivations of a BCI-algebra X . If
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on Lp ( X ), then
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for all x Lp ( X ).
BIO
Abdullah M. Al-roqi received M.Sc. from University of Missouri, Kansas city, United States and Ph.D at School of Mathematics and Statistics, University of Birmingham, United Kingdom. His mathematical research areas are Algebras related to logic, Finite Group Theory, Soluble groups, Classification of finite simple groups and Representation Theory.
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.
e-mail: aalroqi@kau.edu.sa
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