simple semigroup

Let S be a semigroup. If S has no ideals other than itself, then S is said to be simple.

If S has no left idealsMathworldPlanetmathPlanetmath [resp. right ideals] other than itself, then S is said to be left simple [resp. right simple].

Right simple and left simple are stronger conditions than simple.

A semigroup S is left simple if and only if Sa=S for all aS. A semigroup is both left and right simple if and only if it is a group.

If S has a zero elementMathworldPlanetmath θ, then 0={θ} is always an ideal of S, so S is not simple (unless it has only one element). So in studying semigroups with a zero, a slightly weaker definition is required.

Let S be a semigroup with a zero. Then S is zero simple, or 0-simple, if the following conditions hold:

  • S20

  • S has no ideals except 0 and S itself

The condition S2=0 really only eliminates one semigroup: the 2-element null semigroup. Excluding this semigroup makes parts of the structure theory of semigroups cleaner.

Title simple semigroup
Canonical name SimpleSemigroup
Date of creation 2013-03-22 13:05:59
Last modified on 2013-03-22 13:05:59
Owner mclase (549)
Last modified by mclase (549)
Numerical id 7
Author mclase (549)
Entry type Definition
Classification msc 20M10
Defines simple
Defines zero simple
Defines right simple
Defines left simple