On finitely generated left nilpotent braces

TL;DR

This paper characterizes finitely generated left nilpotent braces of class at most two, showing their relation to right nilpotency, and constructs a free object within this category.

Contribution

It provides a detailed description of finitely generated left nilpotent braces of class at most two and constructs a free object in this category.

Findings

Left nilpotent braces of class at most two are right nilpotent of class at most three.

A construction of a free object in the category of finitely generated left nilpotent braces of class at most two.

The description relies on the nilpotency class relationship between left and right braces.

Abstract

A description of finitely generated left nilpotent braces of class at most two is presented in this paper. The description heavily depends on the fact that if $B$ is left nilpotent of class at most $2$, that is $B^3 = 0$, then $B$ is right nilpotent of class at most $3$, that is $B^{(4)} = 0$. In addition, we construct a free object in the category of finitely generated left nilpotent braces of class at most $2$.