Laan in (Ph.D Thesis, Tartu. 1999) introduced the principal weak form of Condition $(P)$ as Condition $(PWP)$ and gave some characterization of monoids by this condition of their acts. In this paper first we introduce Condition (G-PWP), a generalization of Condition $(PWP)$ of acts over monoids and then will give a characterization of monoids when all right acts satisfy this condition. We also ...

laan in (ph.d thesis, tartu. 1999) introduced the principal weak form of condition $(p)$ as condition $(pwp)$ and gave some characterization of monoids by this condition of their acts. in this paper first we introduce condition (g-pwp), a generalization of condition $(pwp)$ of acts over monoids and then will give a characterization of monoids when all right acts satisfy this condition. we also ...

In this paper first of all we introduce Property $U$-($G$-$PWP$) of acts, which is an extension of Condition $(G$-$PWP)$ and give some general properties. Then we give a characterization of monoids when this property of acts implies some others. Also we show that the strong (faithfulness, $P$-cyclicity) and ($P$-)regularity of acts imply the property $U$-($G$-$PWP$). Finally, we give a necessar...