Semilattice Base Hierarchy for Frames and Its Topological Ramifications

We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice A, we analyze the interval in the coframe of sublocales of the frame of downsets of A formed by all frames with the S-base A. We study various degrees of completeness of A, which generalize the concepts of extremally disconnected and basically disconnected frames. We introduce the concepts of D-bases and L-bases, as well as their bounded counterparts, and show how our results specialize and sharpen in these cases. Classic examples that are covered by our approach include zero-dimensional, completely regular, and coherent frames, allowing us to provide a new perspective on these well-studied classes of frames, as well as their spatial counterparts.