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 $(P')$-covers (coessential-covers that satisfy Condition $(P')$). We give a necessary and sufficient condition for a cyclic act to have a $(P')$-cover and a sufficient condition for every act to have a $\mathcal{P'}$-cover (Enochs' $\mathcal{P'}$-cover where $\mathcal{P'}$ is the class of $S$-acts satisfying Condition $(P')$). We also obtain numerous classes of monoids over which indecomposable acts satisfying Condition $(P')$ are locally cyclic.

Keywords:

monoids, acts condition $(P')$, covers,

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