%0 Journal Article
%T On $GPW$-Flat Acts
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Rashidi, Hamideh
%A Golchin, Akbar
%A Mohammadzadeh Saany, Hossein
%D 2020
%\ 01/01/2020
%V 12
%N 1
%P 25-42
%! On $GPW$-Flat Acts
%K $GPW$-flat
%K Eventually regular monoid
%K Eventually left almost regular monoid
%R 10.29252/cgasa.12.1.25
%X 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.
%U https://cgasa.sbu.ac.ir/article_82637_db225e4212ba0171013678302be2c9d2.pdf