#### 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 [1]; also see [2–6]. In this section, we summarize some basic concepts which will be used throughout the paper. For more details, we refer to the references in [7–12]. 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 [2]).

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 [13], 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 [14]. The notion was further explored in the form of -derivations of* BCI*-algebras by Zhan and Liu [15]. We recall the following definition from [14]. 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 [15], 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 [16]). 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 [15]). 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 [15]). 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 [16]). *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 [16]). *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 [13].

The notion of left -derivation of* BCI*-algebras is introduced in [17]. 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