ON REES FACTOR S-POSETS SATISFYING CONDITIONS (PWPE) OR (PWPE)w

Golchin and Rezaei introduced conditions (PW P) and (PW P)w in (Subpullbacks and flatness properties of S-posets). In this paper, we introduce conditions (PW PE) and (PW PE)w as generalizations of these conditions, respectively, and show that the relevant implications are strict. In general, we observe that condition (PW PE)w follows from condition (PW PE), but not conversely. Also, we prove that principal weak po-flatness follows from condition (PW PE)w, but not conversely. Then, we obtain some general properties of conditions (PW PE) and (PW PE)w, and find sufficient and necessary conditions for the Sposet 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 (PW PE) or condition (PW PE)w. Thereby, we present some homological classifications of pomonoids over which each of the conditions (PW PE) and (PW PE)w implies a specific property, and vice versa, for Rees factor S-posets.