LOU VAN DEN DRIES AriD CHRIS MILLER Introduction. The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, Bierstone and Milman [1]. Regrettably, certain "nice" sets-such as {(x, xr):x > 0} for positive irrational r, and { (x, e-1/x): x > 0}mare not subanalytic (at the origin) in IR2. Here we make available an extension of the category of subanalytic sets that has these sets among its objects and that behaves much like the category of subanalytic sets. The possibility of doing this emerged in 1991 when Wilkie [27] proved that the real exponential field is "model complete," followed soon by work of Ressayre, Macintyre, Marker and the authors; see [21], [5], [8], and [19]. However, there are two obstructions to the use by geometers of this development: (i) while the proofs in these articles make essential use of model theory, many results are also stated there (efficiently, but unnecessarily) in model-theoretic terms; (ii) the results of these papers apply directly only to the cartesian spaces IRn, and not to arbitrary real analytic manifolds. Consequently, in order to carry out our goal, we recast here some results in those papersmas well as many of their consequencesmin more familiar terms, with emphasis on results of a geometric nature, and allowing arbitrary (real analytic) manifolds as ambient spaces. We thank W. Schmid and K. Vilonen for their suggestion that this would be a useful undertaking. Indeed, they gave us a "wish list" (inspired by Chapters 8 and 9 of Kashiwara and Schapira [12]; see also 10 of [22]) that strongly influenced the form and content of this paper. We axiomatize in Section 1 the notion of "behaving like the category of subanalytic sets" by introducing the notion of "analytic-geometric category." (The category Can of subanalytic sets is the "smallest" analytic-geometric category.) We also state in Section 1 a number of properties shared by all analyticgeometric categories. Proofs of the more difficult results of this nature, like the Whitney-stratifiability of sets and maps in such a category, often involve the use of charts to reduce to the case of subsets of IRn. For subsets of IRn, there already exists the theory of "o-minimal structures on the real field" (defined in Section 2). This subject is developed in detail in [4] and is an abstraction of the theory of semialgebraic sets (see, e.g., Bochnak et al. [2]). Each analytic-geometric category arises in a natural way from an o-minimal structure on the real field.

Abstract. The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets.

No abstract

No abstract

Abstract. Let R be an o-minimal expansion of a divisible ordered abelian group (R, <, +, 0, 1) with a distinguished positive element 1. Then the following dichotomy holds: Either there is a 0-definable binary operation · such that (R, <, +, ·, 0, 1) is an ordered real closed field; or, for every definable function f : R → R there exists a 0-definable λ ∈ {0} ∪ Aut(R, +) withThis has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure M := (M, <, . . . ) there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) M-definable groups with underlying set M .R. Poston showed in [8] that given an o-minimal expansion R of (R, <, +), if multiplication is not definable in R, then for every definable function f : R → R there exist r, c ∈ R such that lim x→+∞ [f (x) − rx] = c. In this paper, this fact is generalized appropriately for o-minimal expansions of arbitrary ordered groups.We say that an expansion (G, <, * , . . . ) of an ordered group (G, <, * ) is linearly bounded (with respect to * ) if for each definable function f : G → G there exists a definable λ ∈ End(G, * ) such that ultimately |f (x)| ≤ λ(x). (Here and throughout, ultimately abbreviates "for all sufficiently large positive arguments".)We now list the main results of this paper. Let R := (R, <, . . . ) be o-minimal. Theorem A (Growth Dichotomy). Suppose that R is an expansion of an ordered group (R, <, +). Then exactly one of the following holds: (a) R is linearly bounded; (b)R defines a binary operation · such that (R, <, +, ·) is an ordered real closed field. If R is linearly bounded, then for every definable f : R → R there exist c ∈ R and a definable λ ∈ {0} ∪ Aut(R, +) with Theorem B.Suppose that R is a linearly bounded expansion of an ordered group (R, <, +, 0, 1) with 1 > 0. Then every definable endomorphism of (R, +) is 0-definable. If R (with underlying set R ) is elementarily equivalent to R, then the ordered division ring of all R -definable endomorphisms of (R , +) is canonically isomorphic to the ordered division ring of all R-definable endomorphisms of (R, +).The growth dichotomy imposes some surprising constraints on continuous definable groups with underlying set R. (Here and throughout, all topological notions are taken with respect to the product topologies induced by the order topology.)