Local analysis for semi-bounded groups
Volume 216 / 2012
Abstract
An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ 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 bounded domain $I\subseteq M$. Let us call a definable set short if it is in definable bijection with a definable 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 (2009) 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 in the spirit of van den Dries (1998). 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 $\langle M^k, +\rangle$.
