On Holomorphy of Fenyves BCI-Algebras

TL;DR

This paper investigates the holomorphy of Fenyves BCI-algebras, establishing conditions under which properties like p-semisimplicity and BCK-structure are preserved between a loop and its holomorph.

Contribution

It provides new insights into the holomorphy of Fenyves BCI-algebras, linking properties of loops and their holomorphs, including associative and non-associative cases.

Findings

A loop is p-semisimple if and only if its holomorph is p-semisimple.

A loop is a BCK-algebra if and only if its holomorph contains a BCK-subalgebra.

The holomorphy of certain non-associative Fenyves BCI-algebras is characterized.

Abstract

Fenyves BCI-algebras are BCI-algebras that satisfy the Bol-Moufang identities. In this paper, the holomorphy of BCI-algebras are studied. It is shown that whenever a loop and its holomorph are BCI-algebras, the former is p-semisimple if and only if the latter is p-semisimple. Whenever a loop and its holomorph are BCI-algebras, it is established that the former is a BCK-algebra if and only if the latter has a BCK-subalgebra. Moreover, the holomorphy of the associative and some non-associative Fenyves BCI-algebras are also studied.