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.