McAlister covering theorem
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.
Let $S$ be an inverse semigroup; then, there exists an E-unitary inverse semigroup $P$ and a surjective^{}, idempotent-separating homomorphism^{} $\theta \mathrm{:}P\mathrm{\to}S$.
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.
References
- 1 M. Lawson, Inverse Semigroups: The Theory of Partial Symmetries^{}, World Scientific, 1998
Title | McAlister covering theorem |
---|---|
Canonical name | McAlisterCoveringTheorem |
Date of creation | 2013-03-22 14:37:19 |
Last modified on | 2013-03-22 14:37:19 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 5 |
Author | mathcam (2727) |
Entry type | Theorem |
Classification | msc 20M18 |
Defines | unitary |
Defines | E-unitary |
Defines | idempotent-separating |