#### Abstract

The notion of (regular) -derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. The condition for a -derivation to be regular is provided. The concepts of a -invariant -derivation and -ideal are introduced, and their relations are discussed. Finally, some results on regular -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 -semisimple BCI-algebra. For a self map of a BCI-algebra, they defined a -invariant ideal and gave conditions for an ideal to be -invariant. According to Jun and Xin, a self-map is called a left-right derivation (briefly -derivation) of if holds for all . Similarly, a self-map is called a right-left derivation (briefly -derivation) of if holds for all . Moreover, if is both - and -derivation, it is a derivation on . 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 -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 -derivation of BCI-algebras as follows: a self-map is said to be a left-right -derivation or --derivation of if it satisfies the identity for all . Similarly, a self map is said to be a right-left -derivation or --derivation of if it satisfies the identity for all . Moreover, if is both and --derivation, it is said that is an -derivation where 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 is said to be a left derivation of if satisfying for all . Furthermore, in 2009 [14], Öztürk et al. have introduced the notion of generalized derivation in BCI-algebras. A self map is called a generalized -derivation if there exists an -derivation such that for all . If there exists an -derivation such that for all , the mapping is called generalized -derivation. Moreover, if is both a generalized - -derivation, is a generalized derivation on .

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 -derivation which was later more commonly known as or -derivation. After that a number of research articles have been appeared on or -derivations in the theory of rings (see [16, 24, 25] and references therein).

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

#### 2. Preliminaries

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

A nonempty set with a constant and a binary operation is called a -algebra if for all the following conditions hold:(I), (II), (III), (IV) and imply .

Define a binary relation on by letting if and only if . Then is a partially ordered set. A BCI-algebra satisfying for all , is called BCK-algebra.

A -algebra has the following properties: for all (a1),(a2), (a3) implies and , (a4), (a5), (a6), (a7) implies .

For a BCI-algebra , denote by (resp. ) the -part (resp. the BCI-G part) of , that is, is the set of all such that (resp. ). Note that (see [26]). If , then is called a *-*semisimple BCI-algebra. In a -semisimple BCI-algebra , the following hold:(a8), (a9) for all , (a10), (a11) implies , (a12) implies , (a13) implies , (a14).

Let be a -semisimple BCI-algebra. We define addition “+” as for all . Then is an abelian group with identity and . Conversely let be an abelian group with identity and let . Then is a -semisimple -algebra and for all (see [9]).

For a BCI-algebra we denote , in particular , and , for all . We call the elements of the *-*atoms of . For any , let , which is called the branch of with respect to . It follows that whenever and for all and all . Note that , which is the -semisimple part of , and is a -semisimple -algebra if and only if (see [27, Proposition 3.2]). Note also that , that is, , which implies that for all . It is clear that , and and for all and all . A BCI-algebra is said to be torsion free if for all [14]. For more details, refer to [7–10, 26, 27].

#### 3. -Derivations in BCI-Algebras

In what follows, and are endomorphisms of a -algebra unless otherwise specified.

*Definition 3.1. *Let be a -algebra. Then a self map is called a -derivation of if it satisfies:

*Example 3.2. *Consider a -algebra with the following Cayley table:
(1)Define a map
and define two endomorphisms
It is routine to verify that is a -derivation of .(2)Define a map
and define two endomorphisms
It is routine to verify that is a -derivation of .

Lemma 3.3 (see [8]). *Let be a BCI-algebra. For any , if , then and are contained in the same branch of .*

Lemma 3.4 3.4 (see [8]). *Let be a BCI-algebra. For any , if and are contained in the same branch of , then .*

Proposition 3.5. *Let be a commutative BCI-algebra. Then every -derivation of satisfies the following assertion:
**
that is, every -derivation of is isotone.*

*Proof. *Let be such that . Since is commutative, we have . Hence
Since every endomorphism of is isotone, we have . It follows from Lemma 3.3 that and so that there exists such that . Hence (3.8) implies that . Using (a3), (a2), and (III), we have
and so , that is, by (a7).

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

Proposition 3.7. *Every -derivation of a -algebra satisfies the following assertion:
*

* Proof. *Let be an -derivation of . Using (a2) and (a4), we have
Obviously by (II). Therefore, the equality (3.10) is valid.

Theorem 3.8. *Let be a -derivation on a BCI-algebra . Then *(1)*for all ,*(2)*for all ,*(3)*for all .*

*Proof. *(1) For any , we have for all . Thus .

(2) For any and , it follows from (1) that

(3) The proof follows directly from .

*Definition 3.9. *Let be a BCI-algebra and , be two self maps of , we define by for all .

Theorem 3.10. *Let be a -semisimple BCI-algebra. If and are two -derivations on such that . Then is a -derivation on . *

*Proof. *For any , it follows from (a14) that
This completes the proof.

Theorem 3.11. *Let , be two endomorphisms and be a self map on a p-semisimple BCI-algebra such that for all . Then is a -derivation on . *

*Proof. *Let us take for all . Since . Using (a14), we have
This completes the proof.

*Definition 3.12. *A -derivation of a -algebra is said to be regular if .

*Example 3.13. *(1) The -derivation of in Example 3.2 (1) is not regular.

(2) The -derivation of in Example 3.2 (2) is regular.

We provide conditions for a -derivation to be regular.

Theorem 3.14. * Let be a -derivation of a -algebra . If there exists such that for all , then is regular. *

*Proof. *Assume that there exists such that for all . Then
and so . Hence is regular.

*Definition 3.15. * For a -derivation of a BCI-algebra , we say that an ideal of is a -ideal (resp. -ideal) if (resp. ).

*Definition 3.16. *For a -derivation of a BCI-algebra , we say that an ideal of is -invariant if .

*Example 3.17. *(1) Let be a -derivation of which is described in Example 3.2 (1). We know that is both a -ideal and a -ideal of . But is an ideal of which is not -invariant.

(2) Let be a -derivation of which is described in Example 3.2 (2). We know that is both a -ideal and a -invariant ideal of . But is not a -ideal of .

Next, we prove some results on regular -derivations in a -algebra.

Theorem 3.18. *Let be a regular -derivation of a -algebra . Then *(1)*for all ,*(2)*for all ,*(3)*for all ,*(4)*. *

*Proof. *(1) Let be a regular -derivation, that is, . Then the proof follows directly form Proposition 3.7.

(2) Let . Then , and so . Thus . Similarly, .

(3) Let . Using (2), (a1) and (a14), we have

(4) Let . Then . Using (3), we have
This completes the proof.

Theorem 3.19. *Let be a torsion free BCI-algebra and be a regular -derivation on such that . If on , then on .*

* Proof. *Let us suppose on . If , then and so by using Theorem 3.18 (3) and (4), we have
Since is a torsion free. Therefore, for all implying thereby . This completes the proof.

Theorem 3.20. *Let be a torsion free BCI-algebra and , be two regular -derivations on such that . If on , then on .*

* Proof. *Let us suppose on . If , then and so by using Theorem 3.18 (1) and (2), we have
Since is a torsion free. Therefore for all and so . This completes the proof.

Proposition 3.21. * Let be a regular -derivation of a -algebra . If on , then for all .*

*Proof. *Assume that on . If , then and so by using Theorem 3.18 (3) and (4), we have
Hence for all .

This completes the proof.

Proposition 3.22. * Let and be two regular -derivations of a -algebra . If on , then for all .*

*Proof. *Let . Then , and so by Theorem 3.18 (1). It follows from Theorem 3.18 (3) and (4) that
so that for all .

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.