completely simple semigroup

Let $S$ be a semigroup. An idempotent $e\in S$ is primitive if for every other idempotent $f\in S$, $ef=fe=f\ne 0\Rightarrow e=f$

A semigroup $S$ (without zero) is completely if it is simple and contains a primitive idempotent.

A semigroup $S$ is completely $\mathrm{0}$-simple if it is $0$-simple (http://planetmath.org/SimpleSemigroup) and contains a primitive idempotent.

Completely simple and completely $0$-simple semigroups maybe characterised by the Rees Theorem ([Ho95], Theorem 3.2.3).

Note:

A semigroup (without zero) is completely simple if and only if it is regular and weakly cancellative.

A simple semigroup (without zero) is completely simple if and only if it is completely regular.

A $0$-simple semigroup is completely $0$-simple if and only if it is group-bound.

Title	completely simple semigroup
Canonical name	CompletelySimpleSemigroup
Date of creation	2013-03-22 14:35:24
Last modified on	2013-03-22 14:35:24
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Definition
Classification	msc 20M10
Defines	primitive
Defines	completely $0$-simple
Defines	completely simple