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 $\\theta:P\\rightarrow S$ .

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.

References

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

单词	McAlisterCoveringTheorem
释义	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 $\\theta:P\\rightarrow S$ . 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. References 1 M. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, 1998
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