# Pseudo BCI-algebras with derivations

## TL;DR

This paper introduces and analyzes implicative derivations on pseudo-BCI algebras, characterizing their properties, invariance, and algebraic structures, especially in the context of p-semisimple algebras.

## Contribution

It defines new types of derivations on pseudo-BCI algebras and explores their properties, including characterization, invariance, and algebraic structure, with a focus on p-semisimple cases.

## Key findings

- Regular derivations of type II are characterized by invariance of deductive systems.
- The set of implicative derivations forms a commutative monoid under composition.
- In p-semisimple pseudo-BCI algebras, certain derivations coincide and are characterized by identity.

## Abstract

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.