Gyrogroup extensions and semi cross product

TL;DR

This paper explores the extension theory of gyrogroups, establishing an equivalence with group-gyro factor systems and introducing a semi cross product construction to generate larger gyrogroups.

Contribution

It introduces a new semi cross product construction for gyrogroups and establishes an equivalence between extension categories and factor systems.

Findings

Equivalence between GEXT and GFAC categories.

Introduction of semi cross product as a construction method.

Framework for extending gyrogroups under certain conditions.

Abstract

The aim of the article is to study the extension theory of gyrogroups under certain conditions. Consequently, we see that there is an equivalence between the category GEXT of group-gyro extensions and the category GFAC of group-gyro factor systems. We also give a notion of semi cross product of a group and a gyrogroup which is a construction method of a larger class of gyrogroups.