A note on left semibraces

Abstract

A triple $(S, +, \cdot)$ is called a {\em left semibrace} if $(S, +)$ is a semigroup, $(S, \cdot)$ is a group 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 $(S, \cdot)$. In this note, we show that $(S,+)$ is always a rectangular group in this case, i.e., it is isomorphic to the direct product of a group and a rectangular band, and obtain a structure theorem for general left semibraces by the generalized matched products of right zero left semibraces and right cancellative left semibraces. This improves and enriches some results obtained by Jespers and Van Antwerpen in [Forum Math. 31 (2019) 241--263.]