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.