In 2001, Enochs’ celebrated flat cover conjecture was finally proven, and the proofs have since generated a great deal of interest among researchers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids which concerned with the coessential epimorphisms. In this paper, we restrict our attention to \((PF'')\)-covers (coessential-covers that satisfy Condition \((PF'')\)). We give a necessary and sufficient condition for cyclic act to have a \((PF'')\)-cover and give some classes of monoids that all cyclic right S-acts have a Condition \((PF'')\)-cover, we show that every weakly pullback flat cover is also \((PF'')\)-cover and every \((PF'')\)-cover is \((P')\)-cover, but the converses are not true. Besides we consider the \(\mathcal {PF''}\)-cover (Enochs’ notion of a cover) and obtain some properties. Finally, we define and investigate the envelopes of S-acts.

2001年，Enochs著名的平盖猜想终于得到证明，此后的证明引起了研究人员的极大兴趣。2008年，马哈茂迪（Mahmoudi）和伦肖（Renshaw）着手研究与本质本质表观有关的类人猿的行为。在本文中，我们将注意力集中在\（（PF''）\）- covers（满足条件\（（PF''）\）的余量覆盖）上。我们给循环行为提供一个（\（（PF''）\）-的必要和充分条件，并给出所有类的循环权S-行为都具有条件\（（（PF''）\） -的一类半定式-封面，我们证明了每个弱回拉平盖也都是\（（（PF''）\）- cover和\（（（PF''）\）- cover是\（（P'）\）- cover，但是反之则不成立。除此之外，我们考虑\（\ mathcal {PF''} \）- cover（Enochs的封面概念）并获得一些属性。最后，我们定义并研究S- act的包络。