The Natural Order-Generic Collapse for omega-Representable Databases over the Rational and the Real Ordered Group

We consider order-generic queries, i.e., queries which commute with every order-preserving automorphism of a structure’s universe. It is well-known that first-order logic has the natural order-generic collapse over the rational and the real ordered group for the class of dense order constraint databases (also known as finitely representable databases). I.e., on this class of databases over 〈Q, <〉 or 〈R, <〉, addition does not add to the expressive power of first-order logic for defining ordergeneric queries. In the present paper we develop a natural generalization of the notion of finitely representable databases, where an arbitrary (i.e. possibly infinite) number of regions is allowed. We call these databases ω-representable, and we prove the natural order-generic collapse over the rational and the real ordered group for this larger class of databases.