Nilpotent semi-group

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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 [2]. 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.


[1] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[2] A.I. Mal'tsev, "Nilpotent semi-groups" Uchen. Zap. Ivanov. Gos. Ped. Inst. , 4 (1953) pp. 107–111 (In Russian)
[3] L.N. Shevrin, "On the general theory of semi-groups" Mat. Sb. , 53 : 3 (1961) pp. 367–386 (In Russian)
[4] L.N. Shevrin, "Semi-groups all sub-semi-groups of which are accessible" Mat. Sb. , 61 : 2 (1963) pp. 253–256 (In Russian)
How to Cite This Entry:
Nilpotent semi-group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article