Left seminear-rings, groups semidirect products and left cancellative left semi-braces

TL;DR

This paper explores the algebraic structure of left semi-braces, establishing new correspondences with left seminear-rings and group semidirect products, and classifies certain finite cases with applications to nilpotency.

Contribution

It extends the correspondence between semi-braces and algebraic structures to include left seminear-rings and characterizes specific finite semi-braces with set of idempotents as Sylow subgroups.

Findings

Every left semi-brace arises from a left seminear-ring.

A correspondence between certain group semidirect products and left cancellative left semi-braces.

Classification of semi-braces of size pq and 2p^2 with specific idempotent properties.

Abstract

We study some relations between left cancellative left semi-braces and other existing algebraic structures. In particular, we show that every left semi-brace arises from a left seminear-ring, extending the correspondence given by Rump between skew left braces and left near-rings in \cite{rump2019set}. Moreover, we show a correspondence between certain groups semidirect products and left cancellative left semi-braces satisfying an additional hypothesis on the set of idempotents. As an application, we classify left cancellative left semi-braces of size $pq$ and $2p^2$ such that the set of idempotents $E$ is a Sylow subgroup of the multiplicative group. Finally, we study various type of nilpotency, recently introduced in \cite{catino2022nilpotency}, of these left semi-braces.