Half-automorphism group of a class of Bol loops

TL;DR

This paper characterizes the half-automorphism group of a specific class of Bol loops of order 4m, expanding understanding of their symmetry structures.

Contribution

It provides a detailed description of the half-automorphism group for a class of Bol loops of order 4m, a novel analysis in loop theory.

Findings

Explicit description of the half-automorphism group

Identification of structural properties of Bol loops of order 4m

Extension of known automorphism concepts to half-automorphisms

Abstract

A Bol loop is a loop that satisfies the Bol identity $(xy.z)y=x(yz.y)$. If $L$ is a loop and $f:L\to L$ is a bijection such that $f(xy)\in\{f(x)f(y),f(y)f(x)\}$, for every $x$, $y\in L$, then $f$ is called a half-automorphism of $L$. In this paper, we describe the half-automorphism group of a class of Bol loops of order $4m$.