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# 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 simple* if it is simple and contains a primitive idempotent.

A semigroup $S$ is *completely $0$-simple* if it is $0$-simple 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.

# References

- Ho95
Howie, John M.
*Fundamentals of Semigroup Theory*. Oxford University Press, 1995.

Defines:

primitive, completely $0$-simple, completely simple

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Reference

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Definition

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## Mathematics Subject Classification

20M10*no label found*

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