On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang–Baxter equation

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, rediscover some results obtained recently by Lebed and Mortier, Vendramin on description abelian racks quandles. We also investigate multipermutation (not necessarily finite) show, for example, that property equivalent associated (respectively group G(X,r)) being G(X,r) solvable derived length not exceeding level enlarged one, generalizing Gateva-Ivanova Cameron in square-free involutive case. Moreover, prove if X G=G(X,r) nilpotent, then torsion part G finite, it coincides with commutator subgroup [G,G]+ additive skew left brace G/[G,G]+ trivial brace.