  Tarski and Givant introduce informally the concept of a derivative universe of a given basic universe U. A derivative universe Ũ of a given basic universe U is what we have been calling the class of all objects of a certain type, generated from that basic universe U. Thus, the class of n-ary relations of elements of U (for any given n), the class of n-ary relations among relations of elements of U, etc., are derivative universes of U. After introducing in the same way as above the concepts of a permutation of the universe and of invariance of a member of a derivative universe under every permutation of the basic universe, Tarski and Givant give the following definitions: