Nilpotency in left semi-braces
Francesco Catino, Ferran Cedó, Paola Stefanelli
TL;DR
This work extends the notion of nilpotency from skew left braces to left semi-braces by introducing and analyzing several nilpotency-related series and the dot operation $a\cdot b$. It develops a comprehensive set of characterizations for ideals, especially when the set of idempotents $E$ is an ideal, and connects these structural properties to semidirect product decompositions and generalized socle concepts. The paper establishes that nilpotent left semi-braces have nilpotent multiplicative groups $(B,\circ)$ and investigates how left and right nilpotency relate under various hypotheses, including finiteness and centrality conditions. It also introduces strong nilpotency via the $B^{[n]}$ series and relates it to the standard left/right nilpotent sequences, enriching the framework for understanding set-theoretical Yang–Baxter solutions arising from left 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.