I've been learning about graph theory, which has included a discussion of isomorphisms. It occurred to me that for two sets it is possible to have isomorphisms that preserve different structures.
For example, the usual graph isomorphism covered in introductory material is preserving adjacency, but I can also imagine an isomorphism that preserves only reachability. I can further imagine a relation between structures, where we might look at things like whether they have a shared isomorphism, or whether an isomorphism on a set for one structure always implies there exists an isomorphism on that same set for another structure, etc.
Is this topic of studying relations between structures just lumped under abstract algebra or does it have another name?