On f-derivations of BCI-algebras

In the theory of rings and near-rings, the properties of derivations are an important topic to study, see [2, 3, 7, 10]. In [6], Jun and Xin applied the notions in rings and nearrings theory to BCI-algebras, and obtained some related properties. In this paper, the notion of left-right (resp., right-left) f -derivation of a BCI-algebra is introduced, and some related properties are investigated. Using the idea of regular f -derivation, we give characterizations of a p-semisimple BCI-algebra. By a BCI-algebra we mean an algebra (X ;∗,0) of type (2,0) satisfying the following conditions: (I) ((x∗ y)∗ (x∗ z))∗ (z∗ y) = 0; (II) (x∗ (x∗ y))∗ y = 0; (III) x∗ x = 0; (IV) x∗ y = 0 and y∗ x = 0 imply that x = y; for all x, y,z ∈ X . In any BCI-algebra X , one can define a partial order “≤” by putting x ≤ y if and only if x∗ y = 0. A subset S of a BCI-algebra X is called subalgebra of X if x∗ y ∈ S for all x, y ∈ S. A subset I of a BCI-algebra X is called an ideal of X if it satisfies (i) 0 ∈ I ; (ii) x∗ y ∈ I and y ∈ I imply that x ∈ I for all x, y ∈ X . A mapping f of a BCI-algebra X into itself is called an endomorphism of X if f (x∗ y) = f (x) ∗ f (y) for all x, y ∈ X . Note that f (0) = 0. Especially, f is monic if for any x, y ∈ X , f (x) = f (y) implies that x = y. A BCI-algebra X has the following properties: (1) x∗ 0 = x; (2) (x∗ y)∗ z = (x∗ z)∗ y;