Conditions for Acts over Semilattices to be Cantor

An algebra $$A$$ is said to be Cantor if a theorem similar to the Cantor– Bernstein– Schroder theorem holds for it; namely, if, for any algebra $$B$$ , the existence of injective homomorphisms $$A\to B$$ and $$B\to A$$ implies the isomorphism $$A\cong B$$ . Necessary and sufficient conditions for an act over a finite commutative semigroup of idempotents to be Cantor are obtained under the assumption that all connected components of this act are finite.