As a layman, I suspect my question is ill-formed and I will do my best to explain what it is I mean.
I find the notion of algebraic structures absolutely fascinating. I understand that there exist structures that require certain properties of addition and multiplication to hold. As an example, a ring requires an abelian group with a second binary operation that is associative and distributes over the operation in the abelian group. A field is an example of another similar structure.
Given some set $A$ Is there a name for a structure $S$ that requires certain properties of $(A, +)$ and $(A, \cdot)$ to hold that also hold given $(A, \cdot)$ is the additive structure and $(A, +)$ the multiplicative structure? I am trying to say that it matters not which structure we consider as additive and multiplicative as the properties of the $S$ hold regardless. In this sense $(A, +)$ and $(A, \cdot)$ are dual to eachother. Do these objects exist and are they interesting to study?
My gut feeling says that I am failing to write my intent accurately and apologize in advance.