An algebraic loop L is a Moufang loop if all triples of elements x, y, and z in L satisfy the Moufang identities, i.e., if 1.z(x(z y)) = ((z x) z) y, 2.x(z(y z)) = ((x z) y) z, 3.(z x)(y z) = (z(x y)) z, and 4.(z x)(y z) = z((x y) z). One can show that an algebraic loop L which satisfies both the left and right Bol identities is Moufang.

algebraic loop | Bol loop | generalized Bol loop | half-Bol identity | Moufang identities | power associative algebra