Pseudo-BCI algebras with derivations

摘要

In this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate their properties and we give a characterization of regular implicative derivations of type II. We also define the notion of a d-invariant deductive system of a pseudo-BCI algebra A proving that d is a regular derivation of type II if and only if every deductive system on A is d-invariant. It is proved that a pseudo-BCI algebra is p-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations of a p-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition. Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of p-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations are equal.