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 as is gpw-flat if for every s ∈ s, there exists a natural number n = n(s,as ) ∈ n such that the functor as ⊗ s− preserves the embedding of the principal left ideal s(ssn ) into ss. we show that a right s-act as is gpw-flat if and only if for every s ∈ s there exists a natural number n = n(s,as ) ∈ n such that the corresponding ϕ is surjective for the pullback diagram p(ssn , ssn , ι, ι, s), where ι : s(ssn ) → ss 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.