Local analysis for semi-bounded groups

Pantelis E. Eleftheriou¹

¹ CMAF, Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal

Fundamenta Mathematicae, Tome 216 (2012) no. 3, pp. 223-258

Résumé

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 $\langle M^k, +\rangle$.

Détail

DOI : 10.4064/fm216-3-3

Keywords: o minimal expansion mathcal langle dots rangle ordered group called semi bounded does expand real closed field possibly defines real closed field bounded domain subseteq call definable set short in definable bijection definable subset long otherwise previous work edmundo peterzil provided structure theorems definable sets respect dichotomy bounded versus unbounded peterzil conjectured refined structure theorem respect dichotomy short versus long paper prove peterzils conjecture particular obtain quantifier elimination result down suitable existential formulas spirit van den dries furthermore introduce closure operator defines pregeometry gives rise refined notions long dimension long generic elements those turn local analysis semi bounded group yielding following result long direction around each long generic element group operation locally isomorphic langle rangle