Extensions of ordered theories by generic predicates

Given a theory T extending that of dense linear orders without endpoints (DLO), in a language L ⊇ {<}, we are interested in extensions T ′ of T in languages extending L by unary relation symbols that are each interpreted in models of T ′ as sets that are both dense and codense in the underlying sets of the models. There is a canonically “wild” example, namely, T = Th(〈R, <,+, ·〉) and T ′ = Th(〈R, <,+, ·,Q〉). Recall that T is o-minimal, and so every open set definable in any model of T has only finitely many definably connected components. But it is well known that 〈R, <,+, ·,Q〉, defines every real Borel set, in particular, every open subset of any finite cartesian power of R and every subset of any finite cartesian power of Q. To put this another way, the definable open sets in models of T are essentially as simple as possible, while T ′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate. In contrast to the preceding example, if Ralg is the set of real algebraic numbers and T ′ = Th(〈R, <,+, ·,Ralg〉), then no model of T ′ defines any open set (of any arity) that is not definable in the underlying model of T . More generally, if B is an o-minimal expansion of a densely ordered group and A is the underlying set of a dense elementary substructure of B, then Th(〈B, A〉) is rather well behaved with respect to Th(B), in particular, every open set definable in 〈B, A〉 is definable in B; see [6, Section 5] for details. There is an orthogonal complement ([7]): If E ⊆ B is dense and definably independent with respect to B, then again, every open set definable in 〈B, E〉 is definable in B. Another class of examples is treated in [6, Section 6], namely, extensions T ′ of ominimal theories T by “generic (unary) predicates”; this material was included in [6] only to illustrate some of the broader themes of that paper as a whole. Here, we shall relax the assumption that T be o-minimal and consider such extensions T ′ in their own right. Some preliminary discussion of the underlying intuitive ideas is in order.