# Nilpotent semi-group

A semi-group $S$ with zero for which there is an $n$ such that $S^n = 0$; this is equivalent to the identity
$$x_1\dots x_n = y_1\dots y_n$$ in $S$. The smallest $n$ with this property for a given semi-group is called the step (sometimes class) of nilpotency. If $S^2 = 0$, then $S$ is called a semi-group with zero multiplication. The following conditions on a semi-group $S$ are equivalent: 1) $S$ is nilpotent; 2) $S$ has a finite annihilator series (that is, an ascending annihilator series of finite length, see Nil semi-group); or 3) there is a $k$ such that every sub-semi-group of $S$ can be imbedded as an ideal series of length $\leq k$.
A wider concept is that of a nilpotent semi-group in the sense of Mal'tsev . This is the name for a semi-group satisfying for some $n$ the identity
$$X_n = Y_n,$$ where the words $X_n$ and $Y_n$ are defined inductively as follows: $X_0 = x$, $Y_0 = y$, $X_n = X_{n-1}u_nY_{n-1}$, $Y_n = Y_{n-1}u_nX_{n-1}$, where $x$, $y$ and $u_1,\dots ,u_n$ are variables. A group is a nilpotent semi-group in the sense of Mal'tsev if and only if it is nilpotent in the usual group-theoretical sense (see Nilpotent group), and the identity $X_n = Y_n$ is equivalent to the fact that its class of nilpotency is $\leq n$. Every cancellation semi-group satisfying the identity $X_n = Y_n$ can be imbedded in a group satisfying the same identity.