A algebraic loop is a generalized Bol loop if for all elements , , and of ,
for some map . As the name suggests, these are generalizations of Bol loops; in particular, a Bol loop is a generalized Bol loop with respect to the identity map .
One can show that there is an algebraic duality between generalized Bol loops and algebraic loops which satisfy the half-Bol identity (Adeniran and Solarin 1999).