Characterization of pomonoids by Condition (PWP{E})

By using S-act theory and ordered semigroup theory, Condition (PWP_{E}) in the category of S-posets is introduced, and pomonoids over which cyclic (Rees factor) S-posets satisfy this new condition are characterized. Moreover, a classification of pomonoids is given for which Rees factor S-posets satisfying Condition (PWP{E}) have some other flatness properties and vice versa. As applications, some important results on Condition (PWP{E}) are extended.