Corrigendum and Addendum to "Structure monoids of settheoretic solutions of the YangBaxter equation"
Abstract
One of the results in our article, which appeared in Publ. Mat. 65 (2021), 499528, is that the structure monoid $M(X,r)$ of a left nondegenerate solution $(X,r)$ of the YangBaxter Equation is a left semitruss, in the sense of Brzeziński, with an additive structure monoid that is close to being a normal semigroup. Let $\eta$ denote the least left cancellative congruence on the additive monoid $M(X,r)$. It is then shown that $\eta$ also is a congruence on the multiplicative monoid $M(X,r)$ and that the left cancellative epimorphic image $\bar{M}=M(X,r)/\eta$ inherits a semitruss structure and thus one obtains a natural left nondegenerate solution of the YangBaxter equation on $\bar{M}$. 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 by introducing a new left cancellative congruence $\mu$ 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 semitruss structure on $M(X,r)/\mu$ that also yields a natural left nondegenerate solution. In the second part of the paper we start from the least left cancellative congruence $\nu$ on the multiplicative monoid $M(X,r)$ and show that it also is a congruence on the additive monoid $M(X,r)$ in case $r$ is bijective. If, furthermore, $r$ is left and right nondegenerate and bijective then $\nu =\eta$, 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2022
 arXiv:
 arXiv:2202.03174
 Bibcode:
 2022arXiv220203174C
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Group Theory;
 16T25 20M05
 EPrint:
 9 pages. arXiv admin note: text overlap with arXiv:1912.09710