Say I have some object or quantity and an instance or special case of it, how to formally write this down?
I don't (just) mean that $X$ is a set and $x$ an element, i.e. $x\in X$ is not it. I'm dealing with things as general like "the specific group $g$ is a group/is a case of a group". Or "the Integers can be viewed as a restriction from the reals". It should be able to handle such different cases. Are there standard ways to do such a "succession"?
Do I have to intruduce a two valued predicate, which says "is instance" or "is special case of"? Is this even legal/formally right/possible? Do I have to elevate the bigger things to some sort of set or category first? Does this have to do with lattices (since I generate some order)?