completely simple semigroup
Let $S$ be a semigroup. An idempotent^{} $e\in S$ is primitive if for every other idempotent $f\in S$, $ef=fe=f\ne 0\Rightarrow e=f$
A semigroup $S$ (without zero) is completely if it is simple and contains a primitive idempotent.
A semigroup $S$ is completely $\mathrm{0}$-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.
Title | completely simple semigroup |
---|---|
Canonical name | CompletelySimpleSemigroup |
Date of creation | 2013-03-22 14:35:24 |
Last modified on | 2013-03-22 14:35:24 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 8 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 20M10 |
Defines | primitive |
Defines | completely $0$-simple |
Defines | completely simple |