Left semibraces, near left semibraces and the Yang-Baxter equation

Abstract

A triple $(S,+,\cdot )$ is called a {\em left semibrace} if $(S,+)$ is a semigroup, $(S,\cdot )$ is a group with identity $e$ and $x(y+z)=(xy)+(x(x^{-1}+z))$ for all $x,y,z\in S$, where $x^{-1}$ is the inverse of $x$ in the group $(S,\cdot )$. A left semibrace $(S,+,\cdot )$ is called {\em strong} if $x+y(e+z)=x+yz$ for all $x,y,z\in S$. Left semibraces and strong left semibraces are investigated extensively in literature. However, the structure of additive semigroups of general left semibraces still remains mysterious. In this note, as generalizations of near left braces, we introduce {\em near left semibraces} as follows. A quadruple $(S,+,\cdot ,\hat{I}¼)$ is called a {\em near left semibrace} if $(S,+)$ is a semigroup, $(S,\cdot )$ is a group, $\hat{I}¼:S\to S$ is a map and $x(y+z)=(xy)+\hat{I}¼(x)+xz$ for all $x,y,z\in S$. We first show that the additive semigroups of both left semibrace and near left semibraces are rectangular groups and obtain some new characterizations of strong left semibraces. In particular, we prove that a strong left semibrace can induce a near left semibrace, and vice versa. As a consequence, near left semibraces can provide set-theoretical solutions for the Yang-Baxter equation. Next, we obtain a structure theorem for all left semibraces by the generalized matched products of right zero left semibraces and right cancellative left semibraces. Finally, we consider a new map associated to a near left semibrace and give a sufficient and necessary condition under which such a map forms a set-theoretic solution of the Yang-Baxter equation. Our result improve and enrich some results obtained by Jespers and Van Antwerpen in [Forum Math. 31 (2019) 241--263], by Catino, Colazzo and Stefanelli in [Mediterr. J. Math. 17 (2020) 58] and by Catino, Mazzotta and Stefanelli in [J. Algebra 573 (2021) 576--619].

16 pages. By using Proposition 5 in [J. Algebra 573 (2021) 576--619], we simplify the proof of Theorem 2.4. Thank Professors Marzia Mazzotta, Paola Stefanelli, and Francesco Catino for pointing out the above Proposition 5 for us. In this new version, we add the content of near left braces and revise the title of the paper