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

Ferran Cedó , Eric Jespers , Charlotte Vermwimp

• 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.