completely simple semigroup
Let S be a semigroup. An idempotent e∈S is primitive if for every other idempotent f∈S, ef=fe=f≠0⇒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.
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 |