Covers of Finitely Generated Acts over Monoids

This paper attempts to initiate the study of covers of finitely generated S-acts over monoids. We provide necessary and sufficient conditions for a monoid to ensure that n-generated S-acts have strongly flat covers, Condition (P) covers, and projective covers. The main conclusions extend some known results. We also show that Condition (P) covers of finitely generated S-acts are not unique, unlike strongly flat covers. Additionally, we demonstrate the property of Enochs’ X-precover of S-act A, where X denotes a class of S-acts that are closed under isomorphisms.