Fraenkel-Mostowski Sets with Non-homogeneous Atoms

are not. (In the second example, the two sets can be mapped to each other by a partial automorphism, but not by one that extends to a automorphism of the real numbers.) Here is the definition of FM sets. The definition is parametrized by a set of atoms. The atoms are given as a relational structure, which induces a notion of automorphism. (One can also consider atoms with function symbols, but we do not do this here.) Suppose that X is a set which contains atoms (or contains sets which contain atoms, or contains sets which contain sets which contain atoms, and so on). If π is an automorphism of atoms, then π can be applied to X, by renaming all atoms that appear in X, and appear in elements of X, and so on. We say that a set C of atoms is a support of the set X if X is invariant under every automorphism of atoms which is the identity on C. (For instance, the set of all atoms is supported by the empty set, because every automorphism maps the set to itself.) Equipped with these notions, we are ready to define the notion of an FM set: a set which is built out of atoms is called an FM set if it has some finite support, each of its elements has some finite support, and so on recursively. FM sets were rediscovered for the computer science community, by Gabbay and Pitts [5]. In this application area, atoms have no structure, and therefore automorphisms are arbitrary permutations of atoms. It turns out that atoms are