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.
