Classification of monoids by Condition $(PWP_{ssc})$

Abstract

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