Rectangular conditions in products and equivariant completions

The general condition that characterizes the possibility of actionʼs extension over a completion of a phase space is formulated and an example of a G-space that has no Dieudonné complete G-extensions is given. Sufficient conditions (different kinds of rectangular conditions in products) for the actionʼs extensions over the Stone–Čech compactification, the Hewitt realcompactification and the Dieudonné completion of a space are presented. Boundedness, uniform equicontinuity and quasiboundedness of actions are characterized as actionʼs uniform continuity on the (piecewise) semi-uniform product. From this point of view the origin of different examples of actionʼs extensions are explained.