# completely simple semigroup

Let $S$ be a semigroup. An idempotent $e\in S$ is primitive if for every other idempotent $f\in S$, $ef=fe=f\not=0\Rightarrow e=f$

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

A semigroup $S$ is completely $0$-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.

## References

• Ho95 Howie, John M. Fundamentals of Semigroup Theory. Oxford University Press, 1995.
Title completely simple semigroup CompletelySimpleSemigroup 2013-03-22 14:35:24 2013-03-22 14:35:24 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 20M10 primitive completely $0$-simple completely simple