  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.
