A algebraic loop L is a generalized Bol loop if for all elements x, y, and z of L, ((x y) z) α(y) = x((y z) α(y)) for some map α:L->L. 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 1:L->L. One can show that there is an algebraic duality between generalized Bol loops and algebraic loops which satisfy the half-Bol identity.
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