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.

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

Bibliography

Ho95

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