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.