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Abstract

In this paper, we introduce and explore new classes of S-acts over a monoid S, namely, strongly Hopfian and strongly co-Hopfian acts, as well as their weaker counterparts, Hopfian and co-Hopfian acts. We investigate the relationships between these newly defined structures and well-studied classes of S-acts, including Noetherian, Artinian, injective, projective, quasi-injective, and quasi-projective acts. A key result shows that, under certain conditions, a quasi-projective (respectively quasi-injective) S-act that is strongly co-Hopfian (respectively strongly Hopfian) is also strongly Hopfian (respectively strongly co-Hopfian). Moreover, we provide a variety of examples and structural results concerning the behavior of subacts and quotient acts of strongly Hopfian and strongly co-Hopfian S-acts, further elucidating the internal structure and interrelationships within this extended framework.