The term Bol loop refers to either of two classes of algebraic loops satisfying the so-called Bol identities. In particular, a left Bol loop is an algebraic loop L which, for all x, y, and z in L, satisfies the left Bol relation x(y(x z)) = (x(y x)) z. Similarly, L is a right Bol loop provided it satisfies the right Bol relation ((z x) y) x = z((x y) x). An algebraic loop which is both a left and right Bol loop is called a Moufang loop. Some sources use the term Bol loop to refer to a right Bol loop, whereas some reserve the term for algebraic loops that are Moufang.

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