Research Article  Open Access
On Generalized Derivations of BCIAlgebras and Their Properties
Abstract
We introduce the concept of derivations of BCIalgebras and we investigate some fundamental properties and establish some results on derivations. Also, we treat to generalization of right derivation and left derivation of BCIalgebras and consider some related properties.
1. Basic Facts about BCIAlgebras
In 1966, Iséki introduced the concept of BCIalgebra, which is a generalization of the BCKalgebra, as an algebraic counterpart of the BCIlogic [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 BCIalgebra is an abstract algebra of type , satisfying the following conditions, for all :;;; and imply that . A nonempty subset of a BCIalgebra is called a subalgebra of if , for all . In any BCIalgebra , one can define a partial order “” by putting if and only if . A BCIalgebra satisfying , for all , is called a BCKalgebra. In any algebra , the following properties are valid, for all : (1); (2); (3) implies that , ;(4); (5); (6); (7) implies that . Let be a BCIalgebra; the set is a subalgebra and is called the BCKpart of . A BCIalgebra is called proper if . If , then is called a semisimple BCIalgebra. In any BCIalgebra , the following properties are equivalent, for all : (1) is semisimple, (2) , (24) , and (4) . In any semisimple BCIalgebra , the following properties are valid, for all : (1) , (2) , (3) implies that , and (4) implies that . For a BCIalgebra , the set is called the part of . Note that .
Let be a semisimple BCIalgebra. 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 BCIalgebra and , for all (see [2]).
Let be a BCIalgebra; 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 BCIalgebra if and only if . Let be a BCIalgebra with as its center and . Then, the set is called the branch of with respect to .
For any BCIalgebra 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 selfmap of a BCIalgebra (i.e., a mapping of into itself) is called an endomorphism of if , for all . Note that . A subset of a BCIalgebra is called an ideal of , if it satisfies (1), (2) and imply that , for all . Let be an endomorphism of a BCIalgebra 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 BCIalgebra is called commutative if implies . It is called branchwise commutative, if , for all and all . Note that a BCIalgebra is commutative if and only if it is branchwise commutative.
2. Derivations of BCIAlgebras
Recently greater interest has been developed in the derivation of BCIalgebras, introduced by Jun and Xin [14]. The notion was further explored in the form of derivations of BCIalgebras by Zhan and Liu [15]. We recall the following definition from [14]. Let be a BCIalgebra. A leftright derivation (briefly, derivation) of is a selfmap of satisfying the identity If satisfies the identity , for all , then is called a rightleft derivation (briefly, derivation) of . Moreover, if is both an  and derivation, then it is called a derivation.
According to [15], a leftright derivation (briefly, derivation) of is a selfmap of satisfying the identity where is an endomorphism of . If satisfies the identity , for all , then is called a rightleft 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 BCIalgebras.
Definition 1. Let be a BCIalgebra. A leftright derivation (briefly, derivation) of is a selfmap of satisfying the identity where , are endomorphisms of . If satisfies the identity , for all , then is called a rightleft 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 BCIalgebra (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 BCIalgebra 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 BCIalgebra and let be a selfmap 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 selfmap of a BCIalgebra . 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 BCIalgebra . Then, .
Proof. By Theorem 6, . So, .
Theorem 8. Let be a semisimple BCIalgebra 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 BCIalgebra . 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 BCIalgebra is called regular if . If , then is called irregular.
Theorem 11. Let be a commutative BCIalgebra 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 BCIalgebra . 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 BCKalgebra is regular.
Proof. Suppose that is a BCKalgebra 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 BCIalgebra . Also, let be an derivation of and such that and , for all . Then, is regular. Moreover, is a BCKalgebra.
Proof. For all , we have
Hence, .
By Theorem 12(1), . So, , for all . Then, , for all . Therefore, is a BCKalgebra. This implies that is a BCKalgebra, since is an epimorphism.
Theorem 15. Let be an epimorphism and let be an endomorphism of a BCIalgebra . Also, let be an derivation of and such that and , for all . Then, is regular. Moreover, is a BCKalgebra.
Proof. For all , we have
So,
By Theorem 12(1), we get
Thus, , for all . So, , for all . Hence, is a BCKalgebra. This implies that is a BCKalgebra, since is an epimorphism.
Theorem 16. Let be a semisimple BCIalgebra, 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 BCIalgebra, 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 BCIalgebra and let , be two selfmaps of . We define by , for all .
Theorem 19. Let be a semisimple BCIalgebra 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 BCIalgebra 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 BCIalgebra 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 BCIalgebra. 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 BCKalgebras of order 3.
There is only one BCKalgebra 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 BCKalgebras 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

