  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.
