#### Abstract

We introduce the concept of -derivations of BCI-algebras and we investigate some fundamental properties and establish some results on -derivations. Also, we treat to generalization of right derivation and left derivation of BCI-algebras and consider some related properties.

#### 1. Basic Facts about BCI-Algebras

In 1966, Iséki introduced the concept of BCI-algebra, which is a generalization of the BCK-algebra, as an algebraic counterpart of the BCI-logic ; also see . In this section, we summarize some basic concepts which will be used throughout the paper. For more details, we refer to the references in . Let us recall the definition.

A BCI-algebra is an abstract algebra of type , satisfying the following conditions, for all :;;; and imply that . A nonempty subset of a BCI-algebra is called a subalgebra of if , for all . In any BCI-algebra , one can define a partial order “” by putting if and only if . A BCI-algebra satisfying , for all , is called a BCK-algebra. In any -algebra , the following properties are valid, for all : (1); (2); (3) implies that , ;(4); (5); (6); (7) implies that . Let be a BCI-algebra; the set is a subalgebra and is called the BCK-part of . A BCI-algebra is called proper if . If , then is called a -semisimple BCI-algebra. In any BCI-algebra , the following properties are equivalent, for all : (1) is -semisimple, (2) , (24) , and (4) . In any -semisimple BCI-algebra , the following properties are valid, for all : (1) , (2) , (3) implies that , and (4) implies that . For a BCI-algebra , the set is called the part of . Note that .

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 . Then, is a -semisimple BCI-algebra and , for all (see ).

Let be a BCI-algebra; we define the binary operation as , for all . In particular, we denote . An element is said to be an initial element (-atom) of , if implies . We denote by the set of all initial elements (-atoms) of , indeed , and we call it the center of . Note that , which is the -semisimple part of and is a -semisimple BCI-algebra if and only if . Let be a BCI-algebra with as its center and . Then, the set is called the branch of with respect to .

For any BCI-algebra the following results are valid.(1)If and , then , for all .(2)If , then are contained in the same branch of .(3)If , for some , then .(4)If , then , for all .(5), for all .(6), for all . Indeed, , for all which implies that , for all .(7).(8) and , for all and . A self-map of a BCI-algebra (i.e., a mapping of into itself) is called an endomorphism of if , for all . Note that . A subset of a BCI-algebra is called an ideal of , if it satisfies (1), (2) and imply that , for all . Let be an endomorphism of a BCI-algebra and let be its center. Then, according to , we have the following results: (1), for all ,(2) and , for all , where ,(3), for all . A BCI-algebra is called commutative if implies . It is called branchwise commutative, if , for all and all . Note that a BCI-algebra is commutative if and only if it is branchwise commutative.

#### 2. -Derivations of BCI-Algebras

Recently greater interest has been developed in the derivation of BCI-algebras, introduced by Jun and Xin . The notion was further explored in the form of -derivations of BCI-algebras by Zhan and Liu . We recall the following definition from . Let be a BCI-algebra. A left-right derivation (briefly, -derivation) of is a self-map of satisfying the identity If satisfies the identity , for all , then is called a right-left derivation (briefly, -derivation) of . Moreover, if is both an - and -derivation, then it is called a derivation.

According to , a left-right -derivation (briefly, -derivation) of is a self-map of satisfying the identity where is an endomorphism of . If satisfies the identity , for all , then is called a right-left -derivation (briefly, -derivation) of . Moreover, if is both an - and --derivation, then it is called an -derivation.

Now, we introduce the notion of -derivation of BCI-algebras.

Definition 1. Let be a BCI-algebra. A left-right -derivation (briefly, -derivation) of is a self-map of satisfying the identity where , are endomorphisms of . If satisfies the identity , for all , then is called a right-left -derivation (briefly, -derivation) of . Moreover, if is both an - and --derivation, it is called an -derivation.

Example 2. Let and a binary operation is defined as follows: Then, it forms a -semisimple BCI-algebra (see ). Define maps and by , for all , and Then, and are endomorphisms. It is easy to check that is both - and --derivation of . So, is an -derivation.
Now, we consider by . Then, is an endomorphism and is not an --derivation, since but . Also, is not an --derivation, since but .

Example 3. Let be a BCI-algebra with the following Cayley table: Define maps and by , for all : Then, and are endomorphisms. is both derivation and -derivation of (see Example 2.2 of ). Also, it is easy to check that is both - and --derivation of . So, is an -derivation.

Example 4. Let both and be as in Example 3. Define by and . Then, are endomorphisms. is not a -derivation of (see Example 2.3 of ). Also, is not an --derivation, since but .   is not an --derivation, since but .

Theorem 5. Let , be endomorphisms of BCI-algebra and let be a self-map of defined by , for all . Then, is an --derivation of .

Proof. Suppose that . Then, we have Since , and . Hence, by (8), we get So, is an --derivation of .

Theorem 6. Let be a self-map of a BCI-algebra . Then, the following properties hold. (1)If is an --derivation of , then , for all .(2)If is an --derivation of , then , for all if and only if .

Proof. (1) Suppose that . Then, we have It is clear that . So, .
(2) Suppose that is an --derivation of . If , for all , then . Conversely, let . Then, .

Corollary 7. Let be an --derivation of a BCI-algebra . Then, .

Proof. By Theorem 6, . So, .

Theorem 8. Let be a -semisimple BCI-algebra and let be an --derivation . Then, (1), for all ;(2), for all ;(3), for all .

Proof. (1) Let . Then, . Hence, by Corollary 7, we have The proof of (2) and (3) follows directly from (1).

Theorem 9. Let be an --derivation of BCI-algebra . Then, (1), for all ;(2), for all ;(3), for all ;(4), for all .

Proof. (1) Suppose that . Then, we have So, .
(2) Suppose that . Then, we have
(3) Suppose that . Then, we have
(4) The proof follows from (3).

Definition 10. An -derivation of a BCI-algebra is called regular if . If , then is called irregular.

Theorem 11. Let be a commutative BCI-algebra and let be a regular --derivation of . Then, both and belong to the same branch, for all .

Proof. Suppose that . Then, we have Hence, and so . Also, we have , since . This completes the proof.

Theorem 12. Let be a regular --derivation of BCI-algebra . Then, (1), for all ;(2), for all .

Proof. (1) By Theorem 6 (2), , for all .
(2) By part (1), , for all .

Theorem 13. Every --derivation (--derivation) of a BCK-algebra is regular.

Proof. Suppose that is a BCK-algebra and is an --derivation of . Then, for all , we have So, is regular.
Now, suppose that is an --derivation of . Then, .

Theorem 14. Let be an epimorphism and let be an endomorphism of a BCI-algebra . Also, let be an --derivation of and such that and , for all . Then, is regular. Moreover, is a BCK-algebra.

Proof. For all , we have Hence, .
By Theorem 12(1), . So, , for all . Then, , for all . Therefore, is a BCK-algebra. This implies that is a BCK-algebra, since is an epimorphism.

Theorem 15. Let be an epimorphism and let be an endomorphism of a BCI-algebra . Also, let be an -derivation of and such that and , for all . Then, is regular. Moreover, is a BCK-algebra.

Proof. For all , we have So,
By Theorem 12(1), we get Thus, , for all . So, , for all . Hence, is a BCK-algebra. This implies that is a BCK-algebra, since is an epimorphism.

Theorem 16. Let be a -semisimple BCI-algebra, and let and be --derivations (resp., --derivations) of . Also, let . Then, is also an --derivation (resp., --derivation) of .

Proof. Suppose that and are --derivations of . Then, for all , So, is an --derivation of . Now, let , be --derivations of . Then, for all , we have So, is an --derivation of .

Theorem 17. Let be a -semisimple BCI-algebra, and let and be, respectively, --derivation and --derivation of such that , . Then, .

Proof. For all , we have Also, for all , By using (23) and (24), we obtain , for all . By putting , we get , for all .

Definition 18 (see ). Let be a BCI-algebra and let , be two self-maps of . We define by , for all .

Theorem 19. Let be a -semisimple BCI-algebra and let , be -derivations of . Then, (1),(2).

Proof. (1) Since is an --derivation and is an --derivation of , for all , we have Also, since is an --derivation and is an --derivation of , for all , we have From (25) and (26), we get , for all . By putting , , for all .
(2) Since and are --derivations, then for all , we have On the other hand, since and are --derivations, then for all , we have By (27) and (28), for all , By putting , we get , for all .

Theorem 20. Let be a commutative -semisimple BCI-algebra and let , be -derivations of . Then, if and lie in the same branch.

Proof. For all , we have On the other hand, By (30) and (31), , for all . By putting , we obtain , for all .

We denote by the set of all -derivations on .

Definition 21 (see ). Let . The binary operation on is defined as , for all .

Theorem 22. Let be a -semisimple BCI-algebra and let , be --derivations (resp., --derivations) of . Then, is also an --derivation (resp., --derivation) of .

Proof. Suppose that , are --derivations of and . Then, we have So, is an --derivation of .
Suppose that , are --derivations of and . Then, So, is an --derivation of .

Theorem 23. Let be a -semisimple BCI-algebra. Then, is a semigroup.

Proof. By Theorem 22, . Suppose that . Then, Also, we have Therefore, . This completes the proof.

At the end of this section, we classify -derivations on BCK-algebras of order 3.

There is only one BCK-algebra of order 2. It is called and it is as follows: There are only two endomorphisms on . They are as and . We have Table 1, and so There are only three BCK-algebras of order 3. In the following, we classify -derivations on them.

Let . Consider the following table: There are only two endomorphisms on . They are as and . Set then we have Table 2, and so Let . Consider the following table: There are only four endomorphisms on . They are as , , Set then we have Table 3, and so Let . Consider the following table: There are only seven endomorphisms on . They are as , , Obviously, if we set , then for all , , we have only one -derivation and it is . Also, we have Table 4, and so

#### 3. -Derivations of BCI-Algebras

Let be a BCI-algebra. A right derivation of is a self-map of satisfying the identity If satisfies the identity , for all , then is called a left derivation of .

Let be a BCI-algebra. A right -derivation of is a self-map of satisfying the identity where is an endomorphism of . If satisfies the identity , for all , then is called a left -derivation of . Moreover, if is both right and left -derivation, then is called an -derivation of ; see .

The notion of left -derivation of BCI-algebras is introduced in . In this section, we introduce the notion of right -derivation and give some examples and propositions to explain the theory of left -derivation and right -derivation in BCI-algebras.

Definition 24. Let be a BCI-algebra. A right -derivation of is a self-map of satisfying the identity where and are endomorphisms of . If satisfies the identity , for all , then is called a left -derivation of . Moreover, if is both right and left -derivation, then it is said that is an -derivation of .

Example 25. Let , , , and be as in Example 3. Define , , and . It is easy to see that is both right -derivation and left -derivation. So, is an -derivation.

Example 26. Let and be as in Example 3. Define , , and . is not a right -derivation, since but . Also, is not a left -derivation, since but .

Example 27. Let and the binary operation is defined as follows: Then, is a commutative BCK-algebra. Define maps by , for all , and Then, and are endomorphisms. It is easy to check that is a left -derivation. is not a right -derivation, since but .

Definition 28. An -derivation of a BCI-algebra is called regular if . If , then is called irregular.

Theorem 29. Every right -derivation of a BCK-algebra is regular.

Proof. Suppose that is a BCK-algebra and is a right -derivation of . Then, So, is regular.

Theorem 30. Let be a right -derivation of a BCI-algebra . Then, , for all .

Proof. Suppose that . Then, Thus, . So, .

Theorem 31. Let be a right -derivation of a BCI-algebra . Then, the following hold: (1), for all ;(2), for all ;(3), for all ;(4), for all .

Proof. (1) For all , we have