Submission date: 25 March 2011.

Abstract:

An o-minimal expansion (M, <, +, 0, ...) of an ordered group is called “semi-bounded” if it does not expand a real closed field. Possibly, it defines a real closed field with domain a bounded subset I of M. Let us call a definable set “short” if it is in definable bijection with a subset of some I^n, and “long” otherwise. Previous work by Edmundo and Peterzil provided structure theorems for definable sets with respect to the dichotomy “bounded versus unbounded”. Peterzil conjectured a refined structure theorem with respect to the dichotomy “short versus long”. In this paper, we prove Peterzil's conjecture. In particular, we obtain a quantifier elimination result down to suitable existential formulas. Furthermore, we introduce a new closure operator that defines a pregeometry and gives rise to the refined notions of “long dimension” and “long-generic” elements. Those are in turn used in a local analysis for a semi-bounded group G, yielding the following result: on a long direction around each long-generic element of G the group operation is locally isomorphic to (M^k, +).

Mathematics Subject Classification: 03C64

Keywords and phrases: O-minimality, semi-bounded structures, definable groups, pregeometries

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