A note on noninterpretability in o-minimal structures

We prove that if M is an o-minimal structure whose underlying order is dense then Th(M) does not interpret the theory of an infinite discretely ordered structure. We also make a conjecture concerning the class of the theory of an infinite discretely ordered o-minimal structure. Introduction. In [9], Świerczkowski proves that Th(〈ω,<〉) is not interpretable (with parameters) in RCF (the theory of real closed fields) by showing that a pre-ordering with successors is not definable in R. (We recall that a pre-ordering with successors is a reflexive and transitive binary relation , satisfying ∀x∀y (x y ∨ y x) and ∀x∃y Succ(x, y), where Succ(x, y)⇔ x y ∧ x 6≈ y ∧ ∀z (x z y → z ≈ x ∨ z ≈ y), and x ≈ y means x y ∧ y x.) Recall that a structure (M,<,Ri)i∈I is said to be o-minimal if < is a total ordering on M and every definable (with parameters) subset of M is a finite union of points in M and open intervals (a, b), where a ∈M ∪ {−∞} and b ∈M ∪ {∞}. Recall also that if M is o-minimal, then all N |= Th(M) are o-minimal, where Th(M) is the theory of M (see [1]). Certain properties of RCF are used in the proof of the main result of [9], such as o-minimality and definable Skolem functions. We show that this result remains true in the more general setting of o-minimal densely ordered structures. Noninterpretability results. We show the following: Theorem. Let M be an o-minimal structure whose underlying order is dense. Then Th(M) does not interpret the theory of a preordered structure with successors. 1991 Mathematics Subject Classification: 03C40, 03C45, 06F99.