Conditions for Acts over Semilattices to be Cantor

An algebra $A$ is said to be Cantor if a theorem similar to the Cantor–Bernstein–Schröder 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.