  Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the
Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is
Malcev nilpotent. Applying this characterization to solutions coming from
racks, we rediscover some results obtained recently by Lebed and Mortier, and
by Lebed and Vendramin on the description of finite abelian racks and quandles.
  We also investigate bijective non-degenerate multipermutation (not
necessarily finite) solutions $(X,r)$ and show, for example, that this property
is equivalent to the solution associated to the structure monoid $M(X,r)$
(respectively structure group $G(X,r)$) being a multipermuation solution and
that $G=G(X,r)$ is solvable of derived length not exceeding the
multipermutation level of $(X,r)$ enlarged by one, generalizing results of
Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also
prove that if $X$ is finite and $G=G(X,r)$ is nilpotent, then the torsion part
of the group $G$ is finite, it coincides with the commutator subgroup $[G,G]_+$
of the additive structure of the skew left brace $G$ and $G/[G,G]_+$ is a
trivial left brace.
