completely simple semigroup
Let be a semigroup. An idempotent is primitive if for every other idempotent ,
A semigroup (without zero) is completely if it is simple and contains a primitive idempotent.
A semigroup is completely -simple if it is -simple (http://planetmath.org/SimpleSemigroup) and contains a primitive idempotent.
Completely simple and completely -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 -simple semigroup is completely -simple if and only if it is group-bound.
References
- Ho95 Howie, John M. Fundamentals of Semigroup Theory. Oxford University Press, 1995.