In this paper first of all we introduce Property U-(G-P W P) of acts, which is an extension of Condition (G-P W P) 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 (Pcyclicity) and (P-)regularity of acts imply the property U-(G-P W P). Finally, we give a necessary and sufficient condition under which all cyclic (finitely generated) right acts or all (strongly,-) torsion free cyclic (finitely generated) right acts satisfy Property U-(G-P W P).