No abstract

In 2006, Condition [Formula: see text] was introduced by Golchin and Mohammadzadeh and the theorems associated with it were proved. In this paper, we continue the study in relation to the above condition and for some of the properties in the [Formula: see text]-acts. We present the equivalent on the basis of that, which in turn illustrates the importance of this condition. Given that Condition [Formula: see text] is the generalization of Conditions [Formula: see text] and [Formula: see text], so the study of the condition in relation to other properties of the [Formula: see text]-acts will give a better understanding of the behavior of the [Formula: see text]-acts and even Conditions [Formula: see text] and [Formula: see text].

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).