Corrigendum and addendum to "Structure monoids of set-theoretic solutions of the Yang-Baxter equation"

• Autores: Ferran Cedó , Eric Jespers , Charlotte Vermwimp
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 68, Nº. 1, 2024, págs. 241-250
• Idioma: inglés
• DOI: 10.5565/PUBLMAT6812410
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• Resumen

  • One of the results in our article which appeared in Publ. Mat. 65(2) (2021), 499–528,vis that the structure monoid M(X, r) of a left non-degenerate solution (X, r) of the Yang–Baxtervequation is a left semi-truss, in the sense of Brzezi´nski, with an additive structure monoid that is close to being a normal semigroup. Let η denote the least left cancellative congruence on the additive monoid M(X, r). It is then shown that η is also a congruence on the multiplicative monoid M(X, r) and that the left cancellative epimorphic image M¯ = M(X, r)/η inherits a semi-truss structureand thus one obtains a natural left non-degenerate solution of the Yang–Baxter equation on M¯ . Moreover, it restricts to the original solution r for some interesting classes, in particular if (X, r) is irretractable. The proof contains a gap. In the first part of the paper we correct this mistake byintroducing a new left cancellative congruence µ on the additive monoid M(X, r) and show that it also yields a left cancellative congruence on the multiplicative monoid M(X, r), and we obtain a semi-truss structure on M(X, r)/µ that also yields a natural left non-degenerate solution. In the second part of the paper we start from the least left cancellative congruence ν on the multiplicative monoid M(X, r) and show that it is also a congruence on the additive monoid M(X, r) in the case where r is bijective. If, furthermore, r is left and right non-degenerate and bijective,then ν = η, the least left cancellative congruence on the additive monoid M(X, r), extending an earlier result of Jespers, Kubat, and Van Antwerpen to the infinite case.

• Referencias bibliográficas
  • T. Brzezinski , Towards semi-trusses, Rev. Roumaine Math. Pures Appl. 63(2) (2018), 75–89.
  • F. Cedo, E. Jespers, L. Kubat, A. Van Antwerpen, and C. Verwimp , On various types of nilpotency of the structure monoid and group of a set-theoretic...
  • F. Cedo, E. Jespers, and C. Verwimp , Structure monoids of set-theoretic solutions of the Yang–Baxter equation, Publ. Mat. 65(2) (2021), 499–528....
  • E. Jespers, L. Kubat, and A. Van Antwerpen, The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter...