some classifications of monoids by strongly idempotent cancellative $bf (pwp)$

In this paper, we introduce condition $(pwp_{sic})$ of acts over monoids and compare this condition with the properties left $pp$ and left $psf$ in monoid $s$. at first we give a classification of monoids by this condition of right acts. also, we give a classification of monoids for which some other properties of their right acts imply condition $(pwp_{sic})$ and vice versa. then a classification of monoids will be given for which all right rees factor acts of $s$ satisfying some other flatness properties have condition $(pwp_{sic})$.the specific question of when every right $s$-act satisfying condition $(pwp_{sic})$ has certain flatness properties or every ($gpw$-flat) $gp$-flat right $s$-act satisfies condition $(pwp_{sic})$, have so far not been considered. in this paper, we will address these problems.