An internal characterization of sets of functions determining minimal compactifications

We look for internal necessary and sufficient conditions for a subset F of the algebra of all continuous real-valued bounded functions on a Tychonoff space X to have the property that there exists a minimal compactification of X over which every function from F is continuously extendable. Further, we apply some of our ideas to a nonlocally compact case of Magillʹs theorem, i.e., to the problem of when the continuous image of a remainder of a not necessarily locally compact Tychonoff space X is again a remainder of X.