C-epic compactifications

Let K be a compactification of the Tychonoff space X, and ρK :C(K)→C(X) the map which restricts functions (ρK(f)=f|X, for f∈C(K)). In case ρK is an epimorphism in the category of Archimedean l-groups with unit (or equivalently, in Archimedean f-rings), we say that K is a C-epic compactification of X, or X is C-epic in K. This is “formally” equivalent to corresponding conditions in certain categories of frames, σ-frames, locales, and spaces with filter; from this some inferences can be drawn easily. Also, there is a workable criterion coming directly from the l-group theory which involves the canonical surjection Kτ←βX from the Stone–Čech compactification βX. In any event, some specific results are: (1) if X is C-epic in K, then the restriction Kτ←υX is one-to-one (υX being the Hewitt realcompactification) and conversely if υX is Lindelöf, (2) if X is zeroset-embedded in K, then X is C-epic in K, (3) if X is C-epic in K, then K and βX have the same basically disconnected cover, (4) X is C-epic in each of its compactifications if and only if X is almost Lindelöf. Various results related to these and various examples are presented. Many questions remain.