On Symmetric Left Bi-Derivations in BCI-Algebras

The notion of symmetric left bi-derivation of a BCI-algebra X is introduced, and related properties are investigated. Some results on componentwise regular and d-regular symmetric left bi-derivations are obtained. Finally, characterizations of a p-semisimple BCI-algebra are explored, and it is proved that, in a p-semisimple BCI-algebra, F is a symmetric left bi-derivation if and only if it is a symmetric bi-derivation. 1. Introduction BCK-algebras and BCI-algebras are two classes of nonclassical logic algebras which were introduced by Imai and Iséki in 1966 [1, 2]. They are algebraic formulation of BCK-system and BCI-system in combinatory logic. Later on, the notion of BCI-algebras has been extensively investigated by many researchers (see [3–6], and references therein). The notion of a BCI-algebra generalizes the notion of a BCK-algebra in the sense that every BCK-algebra is a BCI-algebra but not vice versa (see [7]). Hence, most of the algebras related to the -norm-based logic such as MTL [8], BL, hoop, MV [9] (i.e lattice implication algebra), and Boolean algebras are extensions of BCK-algebras (i.e. they are subclasses of BCK-algebras) which have a lot of applications in computer science (see [10]). This shows that BCK-/BCI-algebras are considerably general structures. Throughout our discussion, will denote a BCI-algebra unless otherwise mentioned. In the year 2004, Jun and Xin [11] applied the notion of derivation in ring and near-ring theory to BCI-algebras, and as a result they introduced a new concept, called a (regular) derivation, in BCI-algebras. Using this concept as defined they investigated some of its properties. Using the notion of a regular derivation, they also established characterizations of a -semisimple BCI-algebra. For a self-map of a BCI-algebra, they defined a -invariant ideal and gave conditions for an ideal to be -invariant. According to Jun and Xin, a self map is called a left-right derivation (briefly -derivation) of if holds for all . Similarly, a self map is called a right-left derivation (briefly -derivation) of if holds for all . Moreover, if is both - and -derivation, it is a derivation on . After the work of Jun and Xin [11], many research articles have appeared on the derivations of BCI-algebras and a greater interest has been devoted to the study of derivations in BCI-algebras on various aspects (see [12–17]). Inspired by the notions of -derivation [18], left derivation [19], and symmetric bi-derivations [20, 21] in rings and near-rings theory, many authors have applied these notions in a similar way to the theory of