Gyrogroup through its Grothendieck Group Completion and Right gyrogroup action

TL;DR

This paper explores the Grothendieck group completion of gyrogroups, establishing a correspondence between their actions and representations, and introduces the concept of right gyrogroup actions.

Contribution

It provides a new perspective on gyrogroups by linking their actions to those of their Grothendieck group completion and introduces right gyrogroup actions.

Findings

One-to-one correspondence between gyrogroup actions and their Grothendieck group actions

Introduction of the concept of right gyrogroup actions

Establishment of a theoretical framework connecting gyrogroups and their completions

Abstract

In this article, we discuss the Grothendieck group completion (GGC) of a gyrogroup. Consequently, we show that there is a one to one correspondence between actions and representations of a gyrogroup, and actions and representations of its Grothendieck group completion. We also introduce the concept of an action of a right gyrogroup.