2024 CORRIGENDUM AND ADDENDUM TO “STRUCTURE MONOIDS OF SET-THEORETIC SOLUTIONS OF THE YANG–BAXTER EQUATION”
Ferran Cedó, Eric Jespers, Charlotte Verwimp
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Publ. Mat. 68(1): 241-250 (2024). DOI: 10.5565/PUBLMAT6812410

Abstract

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 Brzeziński, 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 structure and 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 by introducing 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.

Funding Statement

The first author was partially supported by the grant MINECO PID2020-113047GB-I00 (Spain). The second author was supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium). The third author is supported by Fonds voor Wetenschappelijk Onderzoek (Flanders), via an FWO aspirant mandate.

Citation

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Ferran Cedó. Eric Jespers. Charlotte Verwimp. "CORRIGENDUM AND ADDENDUM TO “STRUCTURE MONOIDS OF SET-THEORETIC SOLUTIONS OF THE YANG–BAXTER EQUATION”." Publ. Mat. 68 (1) 241 - 250, 2024. https://doi.org/10.5565/PUBLMAT6812410

Information

Received: 1 February 2022; Accepted: 21 March 2022; Published: 2024
First available in Project Euclid: 25 December 2023

MathSciNet: MR4682730
Digital Object Identifier: 10.5565/PUBLMAT6812410

Subjects:
Primary: 16T25 , 20M05

Keywords: 1-cocycle , semi-truss , ‎set-theoretic solution‎ , structure monoid , Yang–Baxter equation

Rights: Copyright © 2024 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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