  Based on the recent development of commutator theory for loops, we provide
both syntactic and semantic characterization of abelian normal subloops. We
highlight the analogies between well known central extensions and central
nilpotence on one hand, and abelian extensions and congruence solvability on
the other hand. In particular, we show that a loop is congruence solvable (that
is, an iterated abelian extension of commutative groups) if and only if it is
not Boolean complete, reaffirming the connection between computational
complexity and solvability. Finally, we briefly discuss relations between
nilpotence and solvability for loops and the associated multiplication groups
and inner mapping groups.
