On Rees Factor $\mathbf{S}$-Posets Satisfying Conditions $\mathbf{(PWP_{E})}$ or $\mathbf{(PWP_E)_{w}}$

Golchin and Rezaei introduced conditions $(PWP)$ and\linebreak $(PWP)_{w}$ in (Subpullbacks and flatness properties of $S$-posets). In this paper, we introduce conditions $(PWP_{E})$ and $(PWP_{E})_{w}$ as generalizations of  these conditions, respectively, and show that the relevant implications are strict. In  general, we observe that condition $(PWP_{E})_{w}$ follows from condition $(PWP_{E})$, but not conversely. Also, we prove that principal weak po-flatness follows from condition  $(PWP_{E})_{w}$, but not conversely. Then, we obtain some general properties of conditions $(PWP_{E})$ and $(PWP_{E})_{w}$, and find sufficient and necessary conditions for the $S$-poset $A(I)$ to satisfy these conditions.  Finally, we find conditions on a pomonoid $S$ under which a cyclic or Rees factor $S$-poset satisfies condition $(PWP_{E})$ or condition $(PWP_{E})_{w}$. Thereby, we present some homological classifications of      pomonoids over which each of the conditions  $(PWP_{E})$ and $(PWP_{E})_{w}$ implies a specific property, and vice versa, for Rees factor $S$-posets.