Left seminear-rings, groups semidirect products and left cancellative left semi-braces
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 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 2p2 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 catino2022nilpotency, of these left semi-braces.