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McAlister covering theorem
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(Theorem)
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A subset $X$ in an inverse semigroup $S$ 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 $S$ be an inverse semigroup; then, there exists an E-unitary inverse semigroup $P$ and a surjective, idempotent-separating homomorphism $\theta:P\rightarrow S$ .
Also, if $S$ is finite, then $P$ may be chosen to be finite as well.
Note that a homomorphism is idempotent-separating if it is injective on idempotents.
- 1
- M. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, 1998
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unitary, E-unitary, idempotent-separating |
This object's parent.
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Cross-references: injective, finite, homomorphism, surjective, idempotents, semigroup, implies, inverse semigroup, subset
There are 9 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 5483 times total.
Classification:
| AMS MSC: | 20M18 (Group theory and generalizations :: Semigroups :: Inverse semigroups) |
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Pending Errata and Addenda
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