completely simple semigroup

Let $S$ be a semigroup. An idempotent $e\\in S$ is primitive if for every other idempotent $f\\in S$ , $ef=fe=f\ot=0\\Rightarrow e=f$

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.

单词	CompletelySimpleSemigroup
释义	completely simple semigroup Let $S$ be a semigroup. An idempotent $e\\in S$ is primitive if for every other idempotent $f\\in S$ , $ef=fe=f\ot=0\\Rightarrow e=f$ 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.
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