Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101Classification of monoids by Condition $(PWP_{ssc})$17519785729ENPouyanKhamechiDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranHosseinMohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranLeilaNouriDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranJournal Article20180224Condition $(PWP)$ which was introduced in (Laan, V., {it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $(PWP_E)$ is a generalization of Condition $(PWP)$. <br /><br />In this paper we introduce Condition $(PWP_{ssc})$, which is much easier to checkĀ than Conditions $(PWP)$ and $(PWP_E)$ and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition $(PWP_{ssc})$ will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition $(PWP_{ssc})$ and also a classification of monoids $S$ will be given for which all right $S$-acts satisfying some other flatness properties have Condition $(PWP_{ssc})$.http://cgasa.sbu.ac.ir/article_85729_9d3888f3fd18b864c1967d267a21ae2c.pdf