While definability and even discernibility are sufficient for capturing the structural roles played by an object, they are not necessary, and a fuller account will arise from the notion of an isomorphism orbit. Specifically, two mathematical structures A and B are isomorphic if they are copies of one another, or more precisely, if there is an isomorphism π: A → B between them, a one-to-one correspondence or bijective map between the respective domains of the structures that respects the salient structural relations and operations.
