On a Generalization of Weak Pullback Flatness

In 1998, Laan introduced weakly pullback flat acts and showed that weak pullback flatness is equivalent to the conjunction of Conditions (P) and (E′). In 2013, Golchin introduced Condition (P′), which is a generalization of Condition (P). In this article, we first define Condition (PF″), which lies strictly between weak pullback flatness and Condition (P′), and prove that Condition (PF″) coincides with the conjunction of Conditions (P′) and (E′). Furthermore, we give a classfication of monoids by Condition (PF″) of (cyclic, Rees factor) acts, and find a necessary and sufficient condition under which Condition (PF″) coincides with Condition (P′) (resp., weak pullback flatness, strong flatness) for Rees factor acts. Finally, we investigate Conditions (P′), (E′) and (PF″) using some new types of epimorphisms.