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 which, for all , , and in , satisfies the left Bol relation
Similarly, is a right Bol loop provided it satisfies the right Bol relation
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.
Although (left and right) Bol loops have relatively weak structural properties, one can show that such structures are power associative. Thus, given an algebraic loop , the element is well-defined for all elements and all integers independent of which order the multiplications are performed.