On epi-extensions of frames (Q1913868)

Let Frm be the category of frames. An epi-extension \(M\) of a frame \(L\) is a frame which has \(L\) as subframe and the embedding \(i : L \to M\) is an epimorphism in Frm. For the intersection of all subframes of \(M\) with \(L \cup \{u_1,\dots, u_n\} \subseteq M\), denoted by \(L[u_1,\dots, u_n]\), a characterization for such epi-extensions is shown: The frame of congruences of \(L\) is isomorphic to that of \(L[u_1,\dots, u_n]\), where this isomorphism comes from the embedding \(i\).