PlanetMath: McAlister covering theorem

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McAlister covering theorem

(Theorem)

A subset in an inverse semigroup is called unitary if for any elements $x\in X$ and $s\in S$ , $xs\in X$ or $sx\in X$ implies $s\in X$ .

An inverse semigroup is E-unitary if its semigroup of idempotents is unitary.

Theorem 1 Let be an inverse semigroup; then, there exists an E-unitary inverse semigroup and a surjective, idempotent-separating homomorphism $\theta:P\rightarrow S$ .

Also, if is finite, then may be chosen to be finite as well.

Note that a homomorphism is idempotent-separating if it is injective on idempotents.

Bibliography

1: M. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, 1998

"McAlister covering theorem" is owned by mathcam. [ owner history (1) ]

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Also defines:

unitary, E-unitary, idempotent-separating

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Cross-references: injective, finite, homomorphism, surjective, idempotents, semigroup, implies, inverse semigroup, subset
There are 3 references to this entry.

This is version 2 of McAlister covering theorem, born on 2004-09-21, modified 2004-09-21.
Object id is 6201, canonical name is McAlisterCoveringTheorem.
Accessed 3883 times total.

Classification:

AMS MSC:

20M18 (Group theory and generalizations :: Semigroups :: Inverse semigroups)

Pending Errata and Addenda

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