Classification of monoids by Condition (PW Pssc) of right acts

Condition (PW P) which was introduced by Laan is related to the concept of flatness of acts over monoids. Golchin and Mohammadzadeh introduced Condition (PW PE) as a generalization of Condition (PW P). In this paper, we introduce Condition (PW Pssc) which is much easier to check than Conditions (PW P) and (PW PE) and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition (PW Pssc) will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition (PW Pssc) and also a classification of monoids S will be given for which all right S-acts satisfying some other flatness properties have Condition (PW Pssc).