MOUFANG MAGMAS WITH INVERSES

Abstract

There are four standard Moufang identities; call them (M1), (M2), (M3), and (M4) (their definitions are given below). In the variety of loops they are equivalent; that is, each of these identities implies the other three. It was shown in [Phillips and Vojtěchovský, A scoop from groups: New equational foundations for loops, Comment. Math. Univ. Carolin.49(2) (2008) 279–290] that magmas with inverses that satisfy either (M1) or (M2) are, in fact, loops, while magmas with inverses that satisfy either (M3) or (M4) need not be loops. We show here that 2-divisible magmas with inverses that satisfy either (M3) or (M4)are loops. We also establish all implications between (M1), (M2), (M3) and (M4) in the variety of magmas with inverses.