A scoop from groups: Equational foundations for loops

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.