On $GPW$-Flat Acts

In this article, we present $GPW$-flatness property of acts over monoids, which is a generalization of principal weak flatness. We say that a right $S$-act $A_{S}$ is $GPW$-flat if for every $s in S$, there exists a natural number $n = n_ {(s, A_{S})} in mathbb{N}$ such that the functor $A_{S} otimes {}_{S}- $ preserves the embedding of the principal left ideal ${}_{S}(Ss^n)$ into ${}_{S}S$. We show that a right $S$-act $A_{S}$ is $GPW$-flat if and only if for every $s in S$ there exists a natural number $n = n_{(s, A_{S})} in mathbb{N}$ such that the corresponding $varphi$ is surjective for the pullback diagram $P(Ss^n, Ss^n, iota, iota, S)$, where $iota : {}_{S}(Ss^n) rightarrow {}_{S}S$ is a monomorphism of left $S$-acts. Also we give some general properties and a characterization of monoids for which this condition of their acts implies some other properties and vice versa.