On a generalization of condition (PWP)

There is a atness property of acts over monoids called Condition (PWP) which, so far, has received much attention. In this paper, we introduce Condition GP-(P), which is a generalization of Condition (PWP). Firstly, some characterizations of monoids by Condition GP- (P) of their (cyclic, Rees factor) acts are given, and many known results are generalized. Moreover, some possible conditions on monoids that describe when their diagonal acts satisfy Condition GP-(P) are found. Finally, using some new types of epimorphisms, an alternative description of Condition GP-(P) (resp., Condition (PWP)) is obtained, and directed colimits of these new epimorphisms are investigated.