Okay, the Frank Pfenning lecture notes you linked do have a bunch of substructural logics with things that behave like residuals. These aren't exactly residuated lattices, since Prof. Pfenning holds on to proof terms and has multiple assumptions. This is because he's trying to give denotational semantics for programs given some background context.

If you drop the proof terms, and restrict yourself to the single-premise and single conclusion fragment of Prof. Pfenning's sequent calculi, you can recover a preorder using the turnstyle \\(\vdash\\) he defines. It looks like some work to show the meta-theorem \\(X \bullet Y \vdash Z \iff Y \vdash X / Z\\).

If you drop the proof terms, and restrict yourself to the single-premise and single conclusion fragment of Prof. Pfenning's sequent calculi, you can recover a preorder using the turnstyle \\(\vdash\\) he defines. It looks like some work to show the meta-theorem \\(X \bullet Y \vdash Z \iff Y \vdash X / Z\\).