On Generalized Derivations of BCI-Algebras and Their Properties

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