Nuclear properties of loop extensions

TL;DR

This paper systematically investigates the extension theory of loops, focusing on nuclear properties of loop extensions and their relation to non-associative generalizations of Schreier's group extension theory.

Contribution

It introduces a classification of loop extensions based on nuclear properties and provides rich examples across various important loop classes.

Findings

Loop extensions can be characterized by different nuclear properties.

Non-associative generalizations of Schreier's theory are described.

Rich examples illustrate the theoretical concepts.

Abstract

The objectives of this paper is to give a systematic investigation of extension theory of loops. A loop extension is (left, right or middle) nuclear, if the kernel of the extension consists of elements associating (from left, right or middle) with all elements of the loop. It turns out that the natural non-associative generalizations of the Schreier's theory of group extensions can be characterized by different types of nuclear properties. Our loop constructions are illustrated by rich families of examples in important loop classes.