On (𝛼,𝛽)-Derivations in BCI-Algebras

Abstract

The notion of (regular) (𝛼,𝛽)-derivations of a BCI-algebra 𝑋 is introduced, some useful examples are discussed, and related properties are investigated. The condition for a (𝛼,𝛽)-derivation to be regular is provided. The concepts of a 𝑑_(𝛼,𝛽)-invariant (𝛼,𝛽)-derivation and 𝛼-ideal are introduced, and their relations are discussed. Finally, some results on regular (𝛼,𝛽)-derivations are obtained.

1. Introduction

BCK-algebras and BCI-algebras are two classes of nonclassical logic algebras which were introduced by Imai and Iséki in 1966 [1, 2]. They are algebraic formulation of BCK-system and BCI-system in combinatory logic. However, these algebras were not studied any further until 1980. Iséki published a series of notes in 1980 and presented a beautiful exposition of BCI-algebras in these notes (see [3–5]). The notion of a BCI-algebra generalizes the notion of a BCK-algebra in the sense that every BCK-algebra is a BCI-algebra but not vice versa (see [6]). Later on, the notion of BCI-algebras has been extensively investigated by many researchers (see [7–9] and references therein).

Throughout our discussion, X will denote a BCI-algebra unless otherwise mentioned. In the year 2004, Jun and Xin [10] 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 concept 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 [10], many research articles have been appeared on the derivations of BCI-algebras and a greater interest has been devoted to the study of derivation in BCI-algebras on various aspects (see [11–15]).

Several authors [16–19] have studied derivations in rings and near-rings. Inspired by the notions of 𝜎-derivation [20], left derivation [21] and generalized derivation [19, 22] in rings and near rings theory, many authors have applied these notions in a similar way to the theory of BCI-algebras (see [11, 14, 15]). For instant, in 2005 [15], Zhan and Liu have given the notion of 𝑓-derivation of BCI-algebras as follows: a self-map 𝑑_𝑓∶𝑋→𝑋 is said to be a left-right 𝑓-derivation or (𝑙,𝑟)-𝑓-derivation of 𝑋 if it satisfies the identity 𝑑_𝑓(𝑥∗𝑦)=𝑑_𝑓(𝑥)∗𝑓(𝑦)∧𝑓(𝑥)∗𝑑_𝑓(𝑦) for all 𝑥,𝑦∈𝑋. Similarly, a self map 𝑑_𝑓∶𝑋→𝑋 is said to be a right-left 𝑓-derivation or (𝑟,𝑙)-𝑓-derivation of 𝑋 if it satisfies the identity 𝑑_𝑓(𝑥∗𝑦)=𝑓(𝑥)∗𝑑_𝑓(𝑦)∧𝑑_𝑓(𝑥)∗𝑓(𝑦) for all 𝑥,𝑦∈𝑋. Moreover, if 𝑑_𝑓 is both (𝑙,𝑟) and (𝑟,𝑙)-𝑓-derivation, it is said that 𝑑_𝑓 is an 𝑓-derivation where 𝑓 is an endomorphism. In the year 2007, Abujabal and Al-Shehri [11] defined and studied the notion of left derivation of BCI-algebras as follows: a self-map 𝐷∶𝑋→𝑋 is said to be a left derivation of 𝑋 if satisfying 𝐷(𝑥∗𝑦)=𝑥∗𝐷(𝑦)∧𝑦∗𝐷(𝑥) for all 𝑥,𝑦∈𝑋. Furthermore, in 2009 [14], Öztürk et al. have introduced the notion of generalized derivation in BCI-algebras. A self map 𝐷∶𝑋→𝑋 is called a generalized (𝑙,𝑟)-derivation if there exists an (𝑙,𝑟)-derivation 𝑑∶𝑋→𝑋 such that 𝐷(𝑥∗𝑦)=𝐷(𝑥)∗𝑦∧𝑥∗𝑑(𝑦) for all 𝑥,𝑦∈𝑋. If there exists an (𝑟,𝑙)-derivation 𝑑∶𝑋→𝑋 such that 𝐷(𝑥∗𝑦)=𝑥∗𝐷(𝑦)∧𝑑(𝑥)∗𝑦 for all 𝑥,𝑦∈𝑋, the mapping 𝐷∶𝑋→𝑋 is called generalized (𝑟,𝑙)-derivation. Moreover, if 𝐷 is both a generalized (𝑙,𝑟)- (𝑟,𝑙)-derivation, 𝐷 is a generalized derivation on 𝑋.

In fact, the notion of derivation in ring theory is quite old and playsa significant role in analysis, algebraic geometry, and algebra. In his famous book “Structures of Rings” Jacobson [23] introduced the notion of (𝑠₁,𝑠₂)-derivation which was later more commonly known as (𝜎,𝜏) or (𝜃,𝜙)-derivation. After that a number of research articles have been appeared on (𝜎,𝜏) or (𝜃,𝜙)-derivations in the theory of rings (see [16, 24, 25] and references therein).

Motivated by the notion of (𝜎,𝜏) or (𝜃,𝜙)-derivation in the theory of rings, in the present paper, we introduce the notion of (𝛼,𝛽)-derivation in a BCI-algebra 𝑋 and investigate related properties. We provide a condition for a (𝛼,𝛽)-derivation to be regular. We also introduce the concepts of a 𝑑_(𝛼,𝛽)-invariant (𝛼,𝛽)-derivation and 𝛼-ideal, and then we investigate their relations. Furthermore, we obtain some results on regular (𝛼,𝛽)-derivations.

2. Preliminaries

We begin with the following definitions and properties that will be needed in the sequel.

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 and 𝑦∗𝑥=0 imply 𝑥=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≤𝑦 implies 𝑥∗𝑧≤𝑦∗𝑧 and 𝑧∗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 implies 𝑥=0.

For a BCI-algebra 𝑋, denote by 𝑋₊ (resp. 𝐺(𝑋)) the BCK-part (resp. the BCI-G part) of 𝑋, that is, 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 [26]). 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)=𝑥 for all 𝑥∈𝑋, (a10)𝑥∗(0∗𝑦)=𝑦∗(0∗𝑥), (a11)𝑥∗𝑦=0 implies 𝑥=𝑦, (a12)𝑥∗𝑎=𝑥∗𝑏 implies 𝑎=𝑏, (a13)𝑎∗𝑥=𝑏∗𝑥 implies 𝑎=𝑏, (a14)𝑎∗(𝑎∗𝑥)=𝑥.

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)=𝑎_𝑥, and 𝐿_𝑝(𝑋)∶={𝑎∈𝑋∣𝑥∗𝑎=0⇒𝑥=𝑎, for all 𝑥∈𝑋}. We call the elements of 𝐿_𝑝(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∈𝐿_𝑝(𝑋). Note that 𝐿_𝑝(𝑋)={𝑥∈𝑋∣𝑎_𝑥=𝑥}, which is the 𝑝-semisimple part of 𝑋, and 𝑋 is a 𝑝-semisimple BCI-algebra if and only if 𝐿_𝑝(𝑋)=𝑋 (see [27, Proposition 3.2]). Note also that 𝑎_𝑥∈𝐿_𝑝(𝑋), that is, 0∗(0∗𝑎_𝑥)=𝑎_𝑥, which implies that 𝑎_𝑥∗𝑦∈𝐿_𝑝(𝑋) for all 𝑦∈𝑋. It is clear that 𝐺(𝑋)⊂𝐿_𝑝(𝑋), and 𝑥∗(𝑥∗𝑎)=𝑎 and 𝑎∗𝑥∈𝐿_𝑝(𝑋) for all 𝑎∈𝐿_𝑝(𝑋) and all 𝑥∈𝑋. A BCI-algebra 𝑋 is said to be torsion free if 𝑥+𝑥=0⇒𝑥=0 for all 𝑥∈𝑋 [14]. For more details, refer to [7–10, 26, 27].

3. (𝛼,𝛽)-Derivations in BCI-Algebras

In what follows, 𝛼 and 𝛽 are endomorphisms of a BCI-algebra 𝑋 unless otherwise specified.

Definition 3.1. Let 𝑋 be a BCI-algebra. Then a self map 𝑑_(𝛼,𝛽)∶𝑋→𝑋 is called a (𝛼,𝛽)-derivation of 𝑋 if it satisfies: ∀𝑥,𝑦∈𝑋𝑑_(𝛼,𝛽)𝑥∗𝑦=𝑑_(𝛼,𝛽)(𝑥)∗𝛼𝑦∧𝑑_(𝛼,𝛽)𝑦∗𝛽(𝑥). (3.1)

Example 3.2. Consider a BCI-algebra 𝑋={0,𝑎,𝑏} with the following Cayley table: ∗0𝑎𝑏0𝑎00𝑏𝑏𝑎0𝑏𝑏𝑏0 (3.2)(1)Define a map 𝑑_(𝛼,𝛽)𝑏∶𝑋→𝑋,𝑥⟼if{},0𝑥∈0,𝑎if𝑥=𝑏, (3.3) and define two endomorphisms 0𝛼∶𝑋→𝑋,𝑥⟼if{},𝑏𝑥∈0,𝑎if0𝑥=𝑏,𝛽∶𝑋→𝑋,𝑥⟼if{},𝑎𝑥∈0,𝑏if𝑥=𝑎. (3.4) It is routine to verify that 𝑑_(𝛼,𝛽) is a (𝛼,𝛽)-derivation of 𝑋.(2)Define a map 𝑑_(𝛼,𝛽)0∶𝑋→𝑋,𝑥⟼if{},𝑎𝑥∈0,𝑏if𝑥=𝑎, (3.5) and define two endomorphisms 0𝛼∶𝑋→𝑋,𝑥⟼if{},𝑏𝑥∈𝑎,𝑏if0𝑥=0,𝛽∶𝑋→𝑋,𝑥⟼if{},𝑎𝑥∈0,𝑎if𝑥=𝑏. (3.6) It is routine to verify that 𝑑_(𝛼,𝛽) is a (𝛼,𝛽)-derivation of 𝑋.

Lemma 3.3 (see [8]). 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 3.4 (see [8]). 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 (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) of 𝑋 satisfies the following assertion: ∀𝑥,𝑦∈𝑋𝑥≤𝑦⇒𝑑_(𝛼,𝛽)(𝑥)≤𝑑_(𝛼,𝛽)𝑦, (3.7) that is, every (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 𝑑_(𝛼,𝛽)(𝑥)=𝑑_(𝛼,𝛽)=𝑑𝑥∧𝑦_(𝛼,𝛽)𝑦∗𝛼𝑦∗𝑥∧𝑑_(𝛼,𝛽)𝑦≤𝑑𝑦∗𝑥∗𝛽_(𝛼,𝛽)𝑦∗𝛼𝑦∗𝑥. (3.8) 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 (3.8) implies that 𝑑_(𝛼,𝛽)(𝑥)≤𝑑_(𝛼,𝛽)(𝑦)∗𝑎. Using (a3), (a2), and (III), we have 𝑑_(𝛼,𝛽)(𝑥)∗𝑑_(𝛼,𝛽)𝑦≤𝑑_(𝛼,𝛽)𝑦∗𝑎∗𝑑_(𝛼,𝛽)𝑦=𝑑_(𝛼,𝛽)𝑦∗𝑑_(𝛼,𝛽)𝑦∗𝑎=0∗𝑎=0, (3.9) and so 𝑑_(𝛼,𝛽)(𝑥)∗𝑑_(𝛼,𝛽)(𝑦)=0, that is, 𝑑_(𝛼,𝛽)(𝑥)≤𝑑_(𝛼,𝛽)(𝑦) by (a7).

Example 3.6. In Example 3.2 (1), the (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) does not satisfy the inequality (3.7).

Proposition 3.7. Every (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) of a 𝐵𝐶𝐼-algebra 𝑋 satisfies the following assertion: (∀𝑥∈𝑋)𝑑_(𝛼,𝛽)(𝑥)=𝑑_(𝛼,𝛽)(𝑥)∧𝑑_(𝛼,𝛽)(0). (3.10)

Proof. Let 𝑑_(𝛼,𝛽) be an (𝛼,𝛽)-derivation of 𝑋. Using (a2) and (a4), we have 𝑑_(𝛼,𝛽)(𝑥)=𝑑_(𝛼,𝛽)(𝑥∗0)=𝑑_(𝛼,𝛽)(𝑥)∗𝛼(0)∧𝑑_(𝛼,𝛽)(0)(𝑥=𝑑∗𝛽)_(𝛼,𝛽)(𝑥)∗0∧𝑑_(𝛼,𝛽)(0)∗𝛽(𝑥)=𝑑_(𝛼,𝛽)(𝑥)∧𝑑_(𝛼,𝛽)(0)(𝑥)=𝑑∗𝛽_(𝛼,𝛽)(0)∗𝛽(𝑥)∗𝑑_(𝛼,𝛽)(0)∗𝛽(𝑥)∗𝑑_(𝛼,𝛽)(𝑥)=𝑑_(𝛼,𝛽)(0)∗𝛽(𝑥)∗𝑑_(𝛼,𝛽)(0)∗𝑑_(𝛼,𝛽)(𝑥)∗𝛽(𝑥)≤𝑑_(𝛼,𝛽)(0)∗𝑑_(𝛼,𝛽)(0)∗𝑑_(𝛼,𝛽)(𝑥)=𝑑_(𝛼,𝛽)(𝑥)∧𝑑_(𝛼,𝛽)(0). (3.11) Obviously 𝑑_(𝛼,𝛽)(𝑥)∧𝑑_(𝛼,𝛽)(0)≤𝑑_(𝛼,𝛽)(𝑥) by (II). Therefore, the equality (3.10) is valid.

Theorem 3.8. Let 𝑑_(𝛼,𝛽) be a (𝛼,𝛽)-derivation on a BCI-algebra 𝑋. Then (1)(for all 𝑎∈𝐿_𝑝(𝑋),𝑥∈𝑋)(𝑑(𝑎∗𝑥)=𝑑(𝑎)∗𝛼(𝑥)),(2)(for all 𝑎∈𝐿_𝑝(𝑋),𝑥∈𝑋)(𝑑(𝑎+𝑥)=𝑑(𝑎)+𝛼(𝑥)),(3)(for all 𝑎,𝑏∈𝐿_𝑝(𝑋))(𝑑(𝑎+𝑏)=𝑑(𝑎)+𝛼(𝑏)).

Proof. (1) For any 𝑎∈𝐿_𝑝(𝑋), we have 𝑎∗𝑥∈𝐿_𝑝(𝑋) for all 𝑥∈𝑋. Thus 𝑑(𝑎∗𝑥)=𝑑(𝑎)∗𝛼(𝑥)∧𝑑(𝑥)∗𝛽(𝑎)=𝑑(𝑎)∗𝛼(𝑥).
(2) For any 𝑎∈𝐿_𝑝(𝑋) and 𝑥∈𝑋, it follows from (1) that 𝑑()(((𝑎)()(𝑎)∗(𝛼(0)(𝑥(𝑎)∗((𝑥(𝑎)(𝑥).𝑎+𝑥=𝑑𝑎∗0∗𝑥))=𝑑∗𝛼0∗𝑥=𝑑∗𝛼))=𝑑0∗𝛼))=𝑑+𝛼 (3.12)
(3) The proof follows directly from (2).

Definition 3.9. Let 𝑋 be a BCI-algebra and 𝑑_(𝛼,𝛽), 𝑑^′_(𝛼,𝛽) be two self maps of 𝑋, we define 𝑑_(𝛼,𝛽)∘𝑑^′_(𝛼,𝛽)∶𝑋→𝑋 by (𝑑_(𝛼,𝛽)∘𝑑^′_(𝛼,𝛽))(𝑥)=𝑑_(𝛼,𝛽)(𝑑^′_(𝛼,𝛽)(𝑥)) for all 𝑥∈𝑋.

Theorem 3.10. Let 𝑋 be a 𝑝-semisimple BCI-algebra. If 𝑑_(𝛼,𝛽) and 𝑑^′_(𝛼,𝛽) are two (𝛼,𝛽)-derivations on 𝑋 such that 𝛼²=𝛼. Then 𝑑_(𝛼,𝛽)∘𝑑^′_(𝛼,𝛽) is a (𝛼,𝛽)-derivation on 𝑋.

Proof. For any 𝑥,𝑦∈𝑋, it follows from (a14) that 𝑑_(𝛼,𝛽)∘𝑑^′_𝛼,𝛽𝑥∗𝑦=𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽𝑥∗𝑦=𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥)∗𝛼𝑦∧𝑑^′_𝛼,𝛽𝑦∗𝛽(𝑥)=𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥)𝑦=𝑑∗𝛼_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥)∗𝛼𝛼𝑦∧𝑑_(𝛼,𝛽)𝛼𝑦∗𝛽𝑑^′_𝛼,𝛽(𝑥)=𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥)𝑦=𝑑∗𝛼_(𝛼,𝛽)𝑑^′_𝛼,𝛽𝑦∗𝛽(𝑥)∗𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽𝑦(𝑥)∗𝑑∗𝛽_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥)𝑦=𝑑∗𝛼_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥)∗𝛼𝑦∧𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽𝑦(𝑥)=𝑑∗𝛽_(𝛼,𝛽)∘𝑑^′_𝛼,𝛽(𝑥)∗𝛼𝑦∧𝑑_(𝛼,𝛽)∘𝑑^′_𝛼,𝛽𝑦∗𝛽(𝑥). (3.13) This completes the proof.

Theorem 3.11. 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 a (𝛼,𝛽)-derivation on 𝑋.

Proof. Let us take 𝑑_(𝛼,𝛽)(𝑥)=𝛼(𝑥) for all 𝑥∈𝑋. Since 𝑥,𝑦∈𝑋⇒𝑥∗𝑦∈𝑋. Using (a14), we have 𝑑_(𝛼,𝛽)𝑥∗𝑦=𝛼𝑥∗𝑦=𝛼(𝑥)∗𝛼𝑦=𝑑_(𝛼,𝛽)(𝑥)𝑦=𝑑∗𝛼_(𝛼,𝛽)𝑦∗𝛽(𝑥)∗𝑑_(𝛼,𝛽)𝑦∗𝛽(𝑥)∗𝑑_(𝛼,𝛽)(𝑥)𝑦=𝑑∗𝛼_(𝛼,𝛽)(𝑥)∗𝛼𝑦∧𝑑_(𝛼,𝛽)𝑦∗𝛽(𝑥). (3.14) This completes the proof.

Definition 3.12. A (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) of a BCI-algebra 𝑋 is said to be regular if 𝑑_(𝛼,𝛽)(0)=0.

Example 3.13. (1) The (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) of 𝑋 in Example 3.2 (1) is not regular.
(2) The (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) of 𝑋 in Example 3.2 (2) is regular.
We provide conditions for a (𝛼,𝛽)-derivation to be regular.

Theorem 3.14. Let 𝑑_(𝛼,𝛽) be a (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.

Proof. Assume that there exists 𝑎∈𝑋 such that 𝑑_(𝛼,𝛽)(𝑥)∗𝛼(𝑎)=0 for all 𝑥∈𝑋. Then 0=𝑑_(𝛼,𝛽)(𝑥∗𝑎)∗𝑎=𝑑_(𝛼,𝛽)(𝑥)∗𝛼(𝑎)∧𝑑_(𝛼,𝛽)(𝑎)(𝑥=𝑑∗𝛽)∗𝑎0∧_(𝛼,𝛽)(𝑎)∗𝛽(𝑥)∗𝑎=0∗𝑎, (3.15) and so 𝑑_(𝛼,𝛽)(0)=𝑑_(𝛼,𝛽)(0∗𝑎)=(𝑑_(𝛼,𝛽)(0)∗𝛼(𝑎))∧(𝑑_(𝛼,𝛽)(𝑎)∗𝛽(0))=0. Hence 𝑑_(𝛼,𝛽) is regular.

Definition 3.15. For a (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) of a BCI-algebra 𝑋, we say that an ideal 𝐴 of 𝑋 is a 𝛼-ideal (resp. 𝛽-ideal) if 𝛼(𝐴)⊆𝐴 (resp. 𝛽(𝐴)⊆𝐴).

Definition 3.16. For a (𝛼,𝛽)-derivation 𝑑_(𝛼,𝛽) of a BCI-algebra 𝑋, we say that an ideal 𝐴 of 𝑋 is 𝑑_(𝛼,𝛽)-invariant if 𝑑_(𝛼,𝛽)(𝐴)⊆𝐴.

Example 3.17. (1) Let 𝑑_(𝛼,𝛽) be a (𝛼,𝛽)-derivation of 𝑋 which is described in Example 3.2 (1). We know that 𝐴∶={0,𝑎} is both a 𝛼-ideal and a 𝛽-ideal of 𝑋. But 𝐴∶={0,𝑎} is an ideal of 𝑋 which is not 𝑑_(𝛼,𝛽)-invariant.
(2) Let 𝑑_(𝛼,𝛽) be a (𝛼,𝛽)-derivation of 𝑋 which is described in Example 3.2 (2). We know that 𝐴∶={0,𝑎} is both a 𝛽-ideal and a 𝑑_(𝛼,𝛽)-invariant ideal of 𝑋. But 𝐴∶={0,𝑎} is not a 𝛼-ideal of 𝑋.

Next, we prove some results on regular (𝛼,𝛽)-derivations in a BCI-algebra.

Theorem 3.18. Let 𝑑_(𝛼,𝛽) be a regular (𝛼,𝛽)-derivation of a 𝐵𝐶𝐼-algebra 𝑋. Then (1)(for all 𝑎∈𝑋)(𝑎∈𝐿_𝑝(𝑋)⇒𝑑_(𝛼,𝛽)(𝑎)∈𝐿_𝑝(𝑋)),(2)(for all 𝑎∈𝑋)(𝑎∈𝐿_𝑝(𝑋)⇒𝛼(𝑎),𝛽(𝑎)∈𝐿_𝑝(𝑋)),(3)(for all 𝑎∈𝐿_𝑝(𝑋))(𝑑_(𝛼,𝛽)(𝑎)=𝑑_(𝛼,𝛽)(0)+𝛼(𝑎)),(4)(forall𝑎,𝑏∈𝐿_𝑝(𝑋))(𝑑_(𝛼,𝛽)(𝑎+𝑏)=𝑑_(𝛼,𝛽)(𝑎)+𝑑_(𝛼,𝛽)(𝑏)−𝑑_(𝛼,𝛽)(0)).

Proof. (1) Let 𝑑_(𝛼,𝛽) be a regular (𝛼,𝛽)-derivation, that is, 𝑑_(𝛼,𝛽)(0)=0. Then the proof follows directly form Proposition 3.7.
(2) Let 𝑎∈𝐿_𝑝(𝑋). Then 𝑎=0∗(0∗𝑎), and so 𝛼(𝑎)=𝛼(0∗(∗0∗𝑎))=0∗(∗0∗𝛼(𝑎)). Thus 𝛼(𝑎)∈𝐿_𝑝(𝑋). Similarly, 𝛽(𝑎)∈𝐿_𝑝(𝑋).
(3) Let 𝑎∈𝐿_𝑝(𝑋). Using (2), (a1) and (a14), we have 𝑑_(𝛼,𝛽)(𝑎)=𝑑_(𝛼,𝛽)((=𝑑0∗0∗𝑎))_(𝛼,𝛽)(0)∗𝛼(0∗𝑎)∧𝑑_(𝛼,𝛽)()(0)=𝑑0∗𝑎∗𝛽_(𝛼,𝛽)(0)∗𝛼(0∗𝑎)∧𝑑_(𝛼,𝛽)()=𝑑0∗𝑎∗0_(𝛼,𝛽)(0)∗𝛼(0∗𝑎)∧𝑑_(𝛼,𝛽)(0∗𝑎)=𝑑_(𝛼,𝛽)(0∗𝑎)∗𝑑_(𝛼,𝛽)(0∗𝑎)∗𝑑_(𝛼,𝛽)(0)∗𝛼(0∗𝑎)=𝑑_(𝛼,𝛽)(0)∗𝛼(0∗𝑎)=𝑑_(𝛼,𝛽)(0)∗(0∗𝛼(𝑎))=𝑑_(𝛼,𝛽)(0)+𝛼(𝑎). (3.16)
(4) Let 𝑎,𝑏∈𝐿_𝑝(𝑋). Then 𝑎+𝑏∈𝐿_𝑝(𝑋). Using (3), we have 𝑑_(𝛼,𝛽)(𝑎+𝑏)=𝑑_(𝛼,𝛽)(0)+𝛼(𝑎+𝑏)=𝑑_(𝛼,𝛽)(0)+𝛼(𝑎)+𝛼(𝑏)=𝑑_(𝛼,𝛽)(0)+𝛼(𝑎)+𝑑_(𝛼,𝛽)(0)+𝛼(𝑏)−𝑑_(𝛼,𝛽)(0)=𝑑_(𝛼,𝛽)(𝑎)+𝑑_(𝛼,𝛽)(𝑏)−𝑑_(𝛼,𝛽)(0). (3.17) This completes the proof.

Theorem 3.19. 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 𝑑²_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋), then 𝑑_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋).

Proof. Let us suppose 𝑑²_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋). If 𝑥∈𝐿_𝑝(𝑋), then 𝑥+𝑥∈𝐿_𝑝(𝑋) and so by using Theorem 3.18 (3) and (4), we have 0=𝑑²_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)𝑑_(𝛼,𝛽)(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝛼𝑑_(𝛼,𝛽)(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝑑_(𝛼,𝛽)(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝑑_(𝛼,𝛽)(𝑥)+𝑑_(𝛼,𝛽)(𝑥)−𝑑_(𝛼,𝛽)(0)=𝑑_(𝛼,𝛽)(𝑥)+𝑑_(𝛼,𝛽)(𝑥). (3.18) Since 𝑋 is a torsion free. Therefore, 𝑑_(𝛼,𝛽)(𝑥)=0 for all 𝑥∈𝑋 implying thereby 𝑑_(𝛼,𝛽)=0. This completes the proof.

Theorem 3.20. Let 𝑋 be a torsion free BCI-algebra and 𝑑_(𝛼,𝛽), 𝑑^′_(𝛼,𝛽) be two regular (𝛼,𝛽)-derivations on 𝑋 such that 𝛼∘𝑑^′_(𝛼,𝛽)=𝑑^′_(𝛼,𝛽). If 𝑑_(𝛼,𝛽)∘𝑑^′_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋), then 𝑑^′_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋).

Proof. Let us suppose 𝑑_(𝛼,𝛽)∘𝑑^′_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋). If 𝑥∈𝐿_𝑝(𝑋), then 𝑥+𝑥∈𝐿_𝑝(𝑋) and so by using Theorem 3.18 (1) and (2), we have 0=𝑑_(𝛼,𝛽)∘𝑑^′_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝛼𝑑^′_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝑑^′_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥)−𝑑^′_𝛼,𝛽(0)=𝑑_(𝛼,𝛽)(0)−𝑑^′_𝛼,𝛽(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥=𝑑_(𝛼,𝛽))(0)∗𝑑^′_𝛼,𝛽(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥=𝑑_(𝛼,𝛽))(0)∗0∗𝑑^′_𝛼,𝛽(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥)=𝑑_(𝛼,𝛽)(0)+𝑑^′_𝛼,𝛽(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥)=𝑑_(𝛼,𝛽)(0)+𝛼𝑑^′_𝛼,𝛽(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥)=𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥)=𝑑_(𝛼,𝛽)∘𝑑^′_𝛼,𝛽(0)+𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥)=𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥). (3.19) Since 𝑋 is a torsion free. Therefore 𝑑^′_(𝛼,𝛽)(𝑥)=0 for all 𝑥∈𝑋 and so 𝑑^′_(𝛼,𝛽)=0. This completes the proof.

Proposition 3.21. Let 𝑑_(𝛼,𝛽) be a regular (𝛼,𝛽)-derivation of a 𝐵𝐶𝐼-algebra 𝑋. If 𝑑²_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋), then (𝛼∘𝑑_(𝛼,𝛽))(𝑥)=(1/2)((𝛼∘𝑑_(𝛼,𝛽))(0)−𝑑_(𝛼,𝛽)(0)) for all 𝑥∈𝐿_𝑝(𝑋).

Proof. Assume that 𝑑²_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋). If 𝑥∈𝐿_𝑝(𝑋), then 𝑥+𝑥∈𝐿_𝑝(𝑋) and so by using Theorem 3.18 (3) and (4), we have 0=𝑑²_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)𝑑_(𝛼,𝛽)(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝛼𝑑_(𝛼,𝛽)(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝛼𝑑_(𝛼,𝛽)(𝑥)+𝑑_(𝛼,𝛽)(𝑥)−𝑑_(𝛼,𝛽)(0)=𝑑_(𝛼,𝛽)(0)+2𝛼𝑑_(𝛼,𝛽)(𝑥)−𝛼𝑑_(𝛼,𝛽)(0). (3.20) Hence (𝛼∘𝑑_(𝛼,𝛽))(𝑥)=(1/2)((𝛼∘𝑑_(𝛼,𝛽))(0)−𝑑_(𝛼,𝛽)(0)) for all 𝑥∈𝐿_𝑝(𝑋).
This completes the proof.

Proposition 3.22. Let 𝑑_(𝛼,𝛽) and 𝑑^′_(𝛼,𝛽) be two regular (𝛼,𝛽)-derivations of a 𝐵𝐶𝐼-algebra 𝑋. If 𝑑_(𝛼,𝛽)∘𝑑^′_(𝛼,𝛽)=0 on 𝐿_𝑝(𝑋), then (𝛼∘𝑑^′_(𝛼,𝛽))(𝑥)=(1/2)((𝛼∘𝑑^′_(𝛼,𝛽))(0)−𝑑_(𝛼,𝛽)(0)) for all 𝑥∈𝐿_𝑝(𝑋).

Proof. Let 𝑥∈𝐿_𝑝(𝑋). Then 𝑥+𝑥∈𝐿_𝑝(𝑋), and so 𝑑^′_(𝛼,𝛽)(𝑥+𝑥)∈𝐿_𝑝(𝑋) by Theorem 3.18 (1). It follows from Theorem 3.18 (3) and (4) that 0=𝑑_(𝛼,𝛽)∘𝑑^′_(𝛼,𝛽)(𝑥+𝑥)=𝑑_(𝛼,𝛽)𝑑^′_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝛼𝑑^′_𝛼,𝛽(𝑥+𝑥)=𝑑_(𝛼,𝛽)(0)+𝛼𝑑^′_𝛼,𝛽(𝑥)+𝑑^′_𝛼,𝛽(𝑥)−𝑑^′_𝛼,𝛽(0)=𝑑_(𝛼,𝛽)(0)+2𝛼𝑑^′_𝛼,𝛽(𝑥)−𝛼𝑑^′_𝛼,𝛽(0) (3.21) so that 𝛼(𝑑^′_(𝛼,𝛽)(𝑥))=(1/2)((𝛼∘𝑑^′_(𝛼,𝛽))(0)−𝑑_(𝛼,𝛽)(0)) for all 𝑥∈𝐿_𝑝(𝑋).
This completes the proof.

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referees. This research is supported by the Deanship of Scientific Research, University of Tabuk, Tabuk, Kingdom of Saudi Arabia.