Generating sublocales by subsets and relations: a tangle of adjunctions

Generalizing the obvious representation of a subspace \({Y \subseteq X}\) as a sublocale in Ω(X) by the congruence \({\{(U, V ) | U\cap Y = V \cap Y\}}\), one obtains the congruence \({\{(a, b) |\mathfrak{o}(a) \cap S = \mathfrak{o}(b) \cap S\}}\), first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets \({S\subseteq L}\). The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective.