In this paper, we continue to study this condition. We first present a necessary and sufficient condition under which the S -poset A(I) satisfies Condition (P W P )w . Furthermore, we characterize pomonoids S over which all cyclic (Rees factor) S -posets satisfy Condition (P W P )w , and pomonoids S over which all Rees factor S -posets satisfying Condition (P W P )w have a certain property. Finally, we consider direct products of S -posets satisfying Condition (P W P )w .