completely simple semigroup (original) (raw)

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 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.