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

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dc.abstract.enGiven 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 multipermutation solution and that 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 square-free 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.
dc.affiliationUniwersytet Warszawski
dc.contributor.authorKubat, Łukasz
dc.contributor.authorCedó, F.
dc.contributor.authorAntwerpen, A. Van
dc.contributor.authorVerwimp, C.
dc.contributor.authorJespers, E.
dc.date.accessioned2024-01-25T15:49:28Z
dc.date.available2024-01-25T15:49:28Z
dc.date.issued2023
dc.description.financePublikacja bezkosztowa
dc.description.number2
dc.description.volume227
dc.identifier.doi10.1016/J.JPAA.2022.107194
dc.identifier.issn0022-4049
dc.identifier.urihttps://repozytorium.uw.edu.pl//handle/item/114895
dc.identifier.weblinkhttp://dx.doi.org/10.1016/j.jpaa.2022.107194
dc.languageeng
dc.pbn.affiliationmathemathics
dc.relation.ispartofJournal of Pure and Applied Algebra
dc.relation.pages107194: 1-30
dc.rightsClosedAccess
dc.sciencecloudnosend
dc.subject.enYang–Baxter equation
dc.subject.enSet-theoretic solution
dc.subject.enNilpotent group
dc.subject.enMalcev nilpotent semigroup
dc.subject.enMultipermutation solution
dc.subject.enSkew brac
dc.titleOn various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang–Baxter equation
dc.typeJournalArticle
dspace.entity.typePublication