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

TL;DR

This paper corrects a previous result on the structure of monoids associated with solutions to the Yang-Baxter equation, introducing new congruences and exploring their properties to establish semi-truss structures and solutions.

Contribution

It introduces a new left cancellative congruence on the additive monoid, correcting earlier gaps and extending results to infinite cases with bijective solutions.

Findings

Introduces a new left cancellative congruence $ta$ on the additive monoid.

Shows $ta$ is also a congruence on the multiplicative monoid.

Extends results to infinite, bijective, and non-degenerate solutions.

Abstract

One of the results in our article, which appeared in Publ. Mat. 65 (2021), 499--528, is that the structure monoid $M(X,r)$ of a left non-degenerate solution $(X,r)$ of the Yang-Baxter Equation 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 $\eta$ 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 semi-truss structure and thus one obtains a natural left non-degenerate solution of the Yang-Baxter 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 semi-truss structure on $M(X,r)/\mu$ that also yields a natural left non-degenerate 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 non-degenerate 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.