Nilpotency in left semi-braces

TL;DR

This paper introduces the concept of nilpotency in left semi-braces, exploring their structure and generalizing known results from skew left braces, with a focus on the properties of their multiplicative groups.

Contribution

It defines nilpotency in left semi-braces and extends structural results from skew left braces to this broader class.

Findings

The multiplicative group of a nilpotent left semi-brace is nilpotent.

Characterization of left semi-braces with additive idempotents forming an ideal.

Introduction of left and right series to study nilpotency in semi-braces.

Abstract

We introduce left and right series of left semi-braces. This allows to define left and right nilpotent left semi-braces. We study the structure of such semi-braces and generalize some results, known for skew left braces, to left semi-braces. We study the structure of left semi-braces $B$ such that the set of additive idempotents $E$ is an ideal of $B$. Finally we introduce the concept of a nilpotent left semi-brace and we show that the multiplicative group of such semi-braces is nilpotent.