On Hopfianity and co-Hopfianity of acts over groups

A universal algebra is called Hopfian if any of its surjective endomorphisms is an automorphism, and co-Hopfian if injective endomorphisms are automorphisms. In this paper, necessary and sufficient conditions are found for Hopfianity and co-Hopfianity of unitary acts over groups. It is proved that a coproduct of finitely many acts (not necessarily unitary) over a group is Hopfian if and only if every factor is Hopfian.