  We introduce degree n Sabinin algebras, which are defined by the polynomial
identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can
be characterized by the polynomial identities satisfied by the commutator,
associator and two quaternators in the free nonassociative algebra. We consider
these operations in a free power associative algebra and show that one of the
quaternators is redundant. The resulting algebras provide the natural structure
on the tangent space at the identity element of an analytic loop for which all
local loops satisfy monoassociativity, a^2 a = a a^2. These algebras are the
next step beyond Lie, Malcev, and Bol algebras. We also present an identity of
degree 5 which is satisfied by these three operations but which is not implied
by the identities of lower degree.
