completely simple semigroup


Let S be a semigroup. An idempotentPlanetmathPlanetmath eS is primitive if for every other idempotent fS, ef=fe=f0e=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 regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and weakly cancellative.

A simple semigroup (without zero) is completely simple if and only if it is completely regularPlanetmathPlanetmath.

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