Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-5853Articles in Press20190211On $GPW$-Flat Acts82637ENHamideh RashidiDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.Akbar GolchinUniversity of Sistan and BaluchestanHossein Mohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranJournal Article20180424In 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.http://cgasa.sbu.ac.ir/article_82637_060cfb2b6b412fd76f4b28d3c0473279.pdf