Regular monomorphisms of Hausdorff frames (Q5929736)

One of the most intractable problems in frame theory seems to be to characterize the regular monomorphisms in the category of frames (equivalently, the regular epimorphisms in the dual category of locales). In an earlier paper, the second and third authors introduced a closure operator on subframes of a given frame, with the property that for a class of `well-behaved' frames (including all metrizable frames) a subframe is closed iff its inclusion map is a regular monomorphism. Subsequently, the first author showed that this closure operator does not characterize regular monomorphisms among arbitrary frames. In the present paper, the authors introduce a slightly different closure operator which does suffice to characterize the regular monomorphisms among Hausdorff frames; however, extending a previous result of the first author, they also show that regular monomorphisms of Hausdorff frames are not closed under composition.