> Perhaps an ill posed question, but here it goes anyway. Does the notion of adjunction in the case of monoids reduce to something formerly known?
Do you mean if we treat monoids as categories with one object, and ask what adjunctions between categories reduce to in this special case? That's a fun question!
If that's your question, I think the best thing is to just figure out the answer! We can do it.
> In the event of a lattice equipped with a monoid with two adjoints for \\(\bullet\\), we call such a structure a [*residuated lattice*](https://en.wikipedia.org/wiki/Residuated_lattice#Definition).
You are interpreting Jesus' question in a different way than I did. You are considering a monoidal poset of a nice sort, and treating this _poset_ \\(A\\) as a category, and asking about adjoints to the operations of left and/or right multiplication by a fixed element \\(a \in A\\):
\\[ x \mapsto a \bullet x\\]
\\[ x \mapsto x \bullet a\\]
where \\(\bullet\\) is the multiplication in \\(A\\).
This is a much more elaborate situation, with a lot more moving parts... but also very fun, and yes, it leads us into the world of logic, and quantales.