Twisted Automorphisms of Right Loops

TL;DR

This paper introduces and studies twisted automorphisms of right loops, establishing a representation theorem for twisted right gyrogroups and exploring their connections with twisted gyrotransversals and twisted subgroups.

Contribution

It defines twisted automorphisms of right loops, develops the theory of twisted right gyrogroups, and proves a representation theorem linking these concepts.

Findings

Established a representation theorem for twisted right gyrogroups.

Demonstrated the equivalence between the study of twisted automorphisms and twisted right gyrogroups.

Explored the relationship between twisted gyrotransversals and twisted subgroups.

Abstract

In this paper we make an attempt to study right loops $(S, o)$ in which, for each $y\in S$, the map $\sigma_y$ from the inner mapping group $G_S$ of $(S, o)$ to itself given by $\sigma_y (h)(x) o\ h(y)= h(xoy)$, $x\in S, h\in G_S$ is a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. A representation theorem for twisted right gyrogroup is established. We also study relationship between twisted gyrotransversals and twisted subgroups (a concept which arose as a tool to study computational complexity involving class NP).