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 ) | Ucap 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 ({Ssubseteq 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.