CORRIGENDUM AND ADDENDUM TO "STRUCTURE MONOIDS OF SET-THEORETIC SOLUTIONS OF THE YANG-BAXTER EQUATION"

One of the results in our article which appeared in Publ. Mat. 65(2) (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 Brzezinski, 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 is also a congruence on the multiplicative monoid M(X, r) and that the left cancellative epimorphic image M over bar = 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 over 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 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 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.