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

Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses

“…The result follows from all other previous lemmas (as was done in the proof of [9,Proposition 3.6] for bijective left non-degenerate solutions) and the comments stated after Theorem 2.1. For completeness' sake and clarity to the reader we explicitly state the chain of ideals that satisfies the properties stated in (2) and (3).…”

“…For the construction we need a series of lemmas. Some of these are extensions of results that were proven in [9] for bijective solutions, the proofs however, in our more general setting, become more tricky and involved.…”

“…Hence λ σ b (a) = id and σ t (s) = (σ b (a), id) ∈ S. Thus r M restricts to a solution on S. Furthermore, because of the first part of (3), equality (9) and the fact that each σ t respects the length, we obtain that r M restricts to a solution on the set E = (F ∩ S) ∪ W .…”

“…Let K be a field, A = A(X, r) and M = M (X, r). If A satisfies the left Ore condition (e.g., if A is abelian, or more generally a Malcev nilpotent monoid, see [9,42] for the definition) then K[M ] is right Noetherian.…”

“…In the context of set-theoretic solutions (X, r) of the Yang-Baxter equation, these monoids and algebras play a very crucial role and are being intensely studied (see for example [9,11,12,17,26,27,29,31,32,38,40]). The strongest results have been obtained for finite non-degenerate bijective solutions.…”