A scoop from groups: Equational foundations for loops

TL;DR

This paper establishes simple equational axioms for various types of loops, including Moufang, Bol, and C-loops, paralleling the axiomatization of groups, and explores cases with one-sided inverses or neutral elements.

Contribution

It provides the first concise equational bases for key loop varieties within the framework of magmas with two-sided inverses, extending to cases with one-sided properties.

Findings

Derived group-like axioms for Moufang, Bol, and C-loops.

Extended axiomatization to loops with one-sided inverses or neutral elements.

Unified approach to loop axioms paralleling group theory.

Abstract

Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain "group-like" equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses are only one-sided and/or the neutral element is only one-sided.