论文标题
对“阳式方程的固定理论解决方案的固定结构的结构”折叠和附录
Corrigendum and Addendum to "Structure monoids of set-theoretic solutions of the Yang--Baxter equation"
论文作者
论文摘要
我们的文章中出现的结果之一出现在Publ中。垫。 65（2021），499--528，是Yang-Baxter方程的左非分级解决方案$（x，r）$的结构是一个左半曲子，从Brzeziński的意义上讲，其附加结构是一个正常的半group。令$η$表示左取消的最小一致性添加剂monoid $ m（x，r）$。然后表明，$η$也是综合乘法$ m（x，r）$的一致性，并且左撤销表达图像$ \ bar {m} = m（x，r）/η$继承了半曲子结构，因此获得了天然的左左左右溶液的yang-baxter equination $ \ bar $ \ \ bar $ \ \ bar {m}此外，它限制了一些有趣类的原始解决方案$ r $，特别是如果$（x，r）$是不可逆转的。证明包含差距。 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. 在本文的第二部分中，我们从最小的左取消一致性$ν$开始，在乘法$ m（x，r）$上开始，并表明它也是添加剂单体$ m（x，r）$的一致性，以防万一$ r $是生物。此外，如果$ r $是左和右非脱位和二级联运的，则$ν=η$，这是添加剂Monoid $ m（x，r）$的最小左取消的一致性，将Jespers，Kubat和van Antwerpen的早期结果扩展到Infinite案件。
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 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 $η$ also is a congruence on the multiplicative monoid $M(X,r)$ and that the left cancellative epimorphic image $\bar{M}=M(X,r)/η$ inherits a semi-truss structure and thus one obtains a natural left non-degenerate solution of the Yang-Baxter equation on $\bar{M}$. 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 $μ$ 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 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 $ν=η$, 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.
