I was reading an article about partial orders, and it gave an exercise to find a relation where multiplication formed meets and addition formed joins (it did not specify an underlying set, but it would probably be the naturals or integers). I'm having no luck finding any information on what the underlying relation might be. At this point I think the author might have been mistaken.
From there, it seems obvious that exponentials would be implication, assuming the original question made sense.
I'm also looking for a source that talks about this in detail, if one can be provided.
There is no natural context where this works. Part of the axioms for a Heyting algebra are that $\wedge$ and $\vee$ are idempotent: $x\wedge x=x\vee x=x$ for all $x$. If $R=(A; +, \times)$ is any ring, then $+$ cannot be idempotent (since the additive part of a ring is a group, in fact an abelian group). For a more detailed response, see https://mathoverflow.net/questions/78357/is-there-something-like-a-heyting-ring.
You may be thinking of Boolean rings, in which we can take multiplication to be meet and the term "$x+y+xy$" to represent join.