Twisted Automorphisms of Right Loops

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 ${\sigma }_y(h)(x)oh(y)=h(xoy)$, $x\in S,h\in G_S$ 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).

15 pages