Dense pairs of o-minimal structures

The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays a special role in this description. Introduction. In a classical paper [8] A. Robinson proved the completeness of the theory of real closed fields with a predicate for a proper dense real closed subfield. Here I generalize this work of Robinson to o-minimal expansions of ordered abelian groups. My main objective, however, is to characterize definable sets and definable functions in dense elementary pairs of such structures. The results obtained in this direction are, to my knowledge, also new in the case considered by Robinson. For an extension of [8] in another direction, see Macintyre [3]. I now proceed to precise definitions and statements. Throughout, T denotes a complete o-minimal theory that extends the theory of ordered abelian groups with distinguished positive element 1. Thus the language L of T extends {<, 0, 1,+,−} and T has definable Skolem functions. In general, we use the same notations and conventions as in [1]. In particular, “intervals” are always open intervals (a, b) with −∞ ≤ a < b ≤ ∞, and we let k,m, n range over N = {0, 1, 2, . . .}. Unless indicated otherwise, “definable” means “definable with parameters”. We let A, B, C, D (possibly with subscripts or superscripts) denote models of T , and A, B, C, D their underlying sets, if it is useful to make this distinction. An elementary pair is a pair (B,A) where A is an elementary substructure of B (with A and B both models of T according to our convention). A dense elementary pair (or just dense pair for simplicity) is an elementary pair (B,A) such that A 6= B and A is dense in B, that is, every interval in B contains elements of A. We let T 2 denote the theory whose models are exactly the elementary pairs, formulated in the language L2 which extends 1991 Mathematics Subject Classification: 03C10, 03C35, 03C60, 06F20.