PlanetMath: completely simple semigroup

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completely simple semigroup

(Definition)

Let be a semigroup. An idempotent $e\in S$ is primitive if for every other idempotent $f\in S$ , $ef=fe=f\not= 0\Rightarrow e=f$

A semigroup (without zero) is completely simple if it is simple and contains a primitive idempotent.

A semigroup is completely 0-simple if it is 0-simple and contains a primitive idempotent.

Completely simple and completely 0-simple semigroups maybe characterised by the Rees Theorem (to appear).

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.

"completely simple semigroup" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Also defines:

primitive, completely $0$

-simple, completely simple

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Cross-references: group-bound, completely regular, simple semigroup, weakly cancellative, regular, contains, simple, idempotent, semigroup
There are 56 references to this entry.

This is version 4 of completely simple semigroup, born on 2004-09-09, modified 2007-01-11.
Object id is 6153, canonical name is CompletelySimpleSemigroup.
Accessed 5434 times total.

Classification:

20M10 (Group theory and generalizations :: Semigroups :: General structure theory)

Pending Errata and Addenda

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