ON GENERALIZED (α, β)-DERIVATIONS IN BCI-ALGEBRAS

Abstract

The notion of generalized (regular) (${\alpha},\;{\beta}$)-derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. The condition for a generalized (${\alpha},\;{\beta}$)-derivation to be regular is provided. The concepts of a generalized F-invariant (${\alpha},\;{\beta}$)-derivation and ${\alpha}$-ideal are introduced, and their relations are discussed. Moreover, some results on regular generalized (${\alpha},\;{\beta}$)-derivations are proved.

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:

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

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:

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:

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:

 

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

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:

(1) Define a map

and define two endomorphisms

and

Then d(α, β) is a (α, β)-derivation of X [5].

Again, define a self map

It is routine to verify that F is a generalized (α, β)-derivation of X.

(2) Define a map

and define two endomorphisms

and

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:

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

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

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:

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:

Proof. Let F be a generalized (α, β)-derivation of X. Using (a2) and (a4), we have

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:

Theorem 3.9. Let F be a generalized (α, β)-derivation on a BCI-algebra X. Then

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

(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

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 (α, β)-derivations respectively on X such that α2 = α. Then F ◦ F′ is an (α, β)-derivation on X.

Proof. For any x, y ∈ X, it follows from (a14) that

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 are two (α, β)-derivations on X such that α2 = α, then 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

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

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

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

(4) Let a, b ∈ Lp(X): Then a + b ∈ Lp(X): Using (3), we have

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

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 F2 = 0 on Lp(X), then F = 0 on Lp(X).

Proof. Let us suppose F2 = 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

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 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

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), be two regular (α, β)-derivations on X such that If = 0 on Lp(X), then on Lp(X).

Proposition 3.29. Let F be a regular generalized (α, β)-derivation of a BCI-algebra X. If F2 = 0 on Lp(X), then for all x ∈ Lp(X).

Proof. Assume that F2 = 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

Hence 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 on Lp(X), then 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 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

so that 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 be two regular (α, β)-derivations of a BCI-algebra X. If on Lp(X), then for all x ∈ Lp(X).