The varieties of loops of Bol-Moufang type

Abstract

A loop identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, viz. $((xy)x)z=x(y(xz))$. Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varieties, and determine all inclusions between them, providing all necessary counterexamples, too. This extends and completes the programme of Fenyves.