# Nil semi-group

A semi-group with zero in which some power of every element is zero. Nil semi-groups form one of the most important classes of periodic semi-groups (cf. Periodic semi-group): They are precisely the periodic semi-groups with a unique idempotent, namely, the zero. Locally nilpotent semi-groups (that is, semi-groups in which every finitely-generated sub-semi-group is nilpotent, see Nilpotent semi-group) form a narrow class. For every $n>1$, there exists a semi-group with the identity $x^n=0$ that is not locally nilpotent (see, for example, , Chapt. 8 Sect. 4). A finite nil semi-group is nilpotent, and the classes of locally nilpotent semi-groups and locally finite nil semi-groups coincide (see Locally finite semi-group). An even narrower class is formed by the semi-groups with an ascending annihilator series. A semi-group $S$ has an ascending annihilator series if it has an increasing ideal series (see Ideal series of a semi-group) such that for any two adjacent terms $A_\alpha,A_{\alpha+1}$,
$$SA_{\alpha+1}\bigcup A_{\alpha+1}S\subseteq A_\alpha.$$
A nil semi-group has an ascending annihilator series if and only if it has an increasing series of ideals in which all factors are finite. Every semi-group with an ascending annihilator series has a unique irreducible generating set, consisting of its indecomposable elements. An arbitrary locally nilpotent semi-group may coincide with its square. Many finiteness conditions (see Semi-group with a finiteness condition) imposed on a semi-group imply that is finite; for example, the minimum condition for ideals, or the maximum condition for right (or left) ideals. If all nilpotent sub-semi-groups of a nil semi-group $S$ are finite, then so is $S$.