# Difference between revisions of "Nil semi-group"

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− | A [[Semi-group|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|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|Nilpotent semi-group]]) form a narrow class. For every | + | {{TEX|done}} |

+ | A [[Semi-group|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|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|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, [[#References|[1]]], 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|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|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|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 | + | 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|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 $S$. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.N. Shevrin, "On the general theory of semi-groups" ''Mat. Sb.'' , '''53''' : 3 (1961) pp. 367–386 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.N. Shevrin, "Nil semi-groups with certain finiteness conditions" ''Mat. Sb.'' , '''55''' : 4 (1961) pp. 473–480 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.N. Shevrin, "On the general theory of semi-groups" ''Mat. Sb.'' , '''53''' : 3 (1961) pp. 367–386 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.N. Shevrin, "Nil semi-groups with certain finiteness conditions" ''Mat. Sb.'' , '''55''' : 4 (1961) pp. 473–480 (In Russian)</TD></TR></table> |

## Revision as of 11:02, 1 August 2014

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, [1], 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 $S$.

#### References

[1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |

[2] | L.N. Shevrin, "On the general theory of semi-groups" Mat. Sb. , 53 : 3 (1961) pp. 367–386 (In Russian) |

[3] | L.N. Shevrin, "Nil semi-groups with certain finiteness conditions" Mat. Sb. , 55 : 4 (1961) pp. 473–480 (In Russian) |

**How to Cite This Entry:**

Nil semi-group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Nil_semi-group&oldid=32645