A Note on Hopfian and Co-Hopfian S-Acts

The main purpose of this work is to investigate of the notions Hopfian (co-Hopfian) acts in which their surjective (injective) endomorphisms are isomorphisms. While we study conditions that are relevant to these classes of acts, their interrelationship with some other concepts for example quasi-injective and Dedekind-finite acts is studied. Using Hopfian and co-Hopfian concepts, we provide several conditions for a quasi-injective act to be Dedekind-finite.