Half-isomorphisms of Moufang loops

TL;DR

This paper proves that under certain conditions, every half-isomorphism of a Moufang loop is either an isomorphism or an anti-isomorphism, extending previous results in the field.

Contribution

It generalizes all earlier results by establishing a condition under which half-isomorphisms are necessarily isomorphisms or anti-isomorphisms in Moufang loops.

Findings

Half-isomorphisms are either isomorphisms or anti-isomorphisms under the given condition.

The result generalizes previous theorems in the study of Moufang loops.

The proof relies on properties of the squaring map in the factor loop.

Abstract

We prove that if the squaring map in the factor loop of a Moufang loop $Q$ over its nucleus is surjective, then every half-isomorphism of $Q$ onto a Moufang loop is either an isomorphism or an anti-isomorphism. This generalizes all earlier results in this vein.