Half-isomorphisms of automorphic loops

TL;DR

This paper investigates automorphic loops, showing that in a specific class including odd order and uniquely 2-divisible loops, all half-isomorphisms are trivial, meaning they are either isomorphisms or anti-isomorphisms.

Contribution

It characterizes a class of automorphic loops where all half-isomorphisms are trivial, extending understanding of their structural symmetries.

Findings

Includes automorphic loops of odd order and uniquely 2-divisible loops.

Proves all half-isomorphisms in this class are trivial (either isomorphisms or anti-isomorphisms).

Provides conditions under which half-isomorphisms are trivial in automorphic loops.

Abstract

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. A half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$ and $K$ is a bijection from $G$ onto $K$ such that $f(ab)\in\{f(a)f(b), f(b)f(a)\}$ for any $a,b\in G$. A half-isomorphism is trivial when it is either an isomorphism or an anti-isomorphism. Consider the class of automorphic loops such that the equation $x\cdot(x\cdot y) = (y\cdot x)\cdot x $ is equivalent to $x\cdot y = y\cdot x$. Here we show that this class of loops includes automorphic loops of odd order and uniquely $2$-divisible. Furthermore, we prove that every half-isomorphism between loops in that class is trivial.