On covers of acts over monoids with Condition $(P`)$

In this paper we consider two different definitions of cover, one of them is Enochs` notion of a cover and the other is the one that initiated by Mahmoudi and Renshaw which concerned with the coessential epimorphisms. We show that these definitions are not equivalent in our case and restrict our attention to $insert ignore into journalissuearticles values(P`);$-covers insert ignore into journalissuearticles values(coessential-covers that satisfy Condition $insert ignore into journalissuearticles values(P`);$);. We give a necessary and sufficient condition for a cyclic act to have a $insert ignore into journalissuearticles values(P`);$-cover and a sufficient condition for every act to have a $\mathcal{P`}$-cover insert ignore into journalissuearticles values(Enochs` $\mathcal{P`}$-cover where $\mathcal{P`}$ is the class of $S$-acts satisfying Condition $insert ignore into journalissuearticles values(P`);$);. We also obtain numerous classes of monoids over which indecomposable acts satisfying Condition $insert ignore into journalissuearticles values(P`);$ are locally cyclic.