On a generalization of condition (PWP)

Abstract

‎There is a flatness 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.