isomorphism orbit of an object in a structure, the equivalence class of the object/structure pair (a, A) with respect to the same-structural-role-as relation. This orbit tracks how a is copied to all its various isomorphic images in all the various structures isomorphic to A. And whether or not these objects are definable or discernible in their structures, it is precisely the objects appearing in the isomorphism orbit that play the same structural role in those structures that a plays in A.
