Local analysis for semi-bounded groups

Pantelis E. Eleftheriou. "Local analysis for semi-bounded groups." Fundamenta Mathematicae 216.3 (2012): 223-258. <http://eudml.org/doc/282790>.

@article{PantelisE2012,
abstract = {An o-minimal expansion ℳ = ⟨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 bounded domain I ⊆ M. Let us call a definable set short if it is in definable bijection with a definable subset of some Iⁿ, 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 $⟨M^\{k\},+⟩$.},
author = {Pantelis E. Eleftheriou},
journal = {Fundamenta Mathematicae},
keywords = {o-minimality; semi-bounded groups; pregeometries; semi-bounded structures; definable groups},
language = {eng},
number = {3},
pages = {223-258},
title = {Local analysis for semi-bounded groups},
url = {http://eudml.org/doc/282790},
volume = {216},
year = {2012},
}

TY - JOUR
AU - Pantelis E. Eleftheriou
TI - Local analysis for semi-bounded groups
JO - Fundamenta Mathematicae
PY - 2012
VL - 216
IS - 3
SP - 223
EP - 258
AB - An o-minimal expansion ℳ = ⟨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 bounded domain I ⊆ M. Let us call a definable set short if it is in definable bijection with a definable subset of some Iⁿ, 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 $⟨M^{k},+⟩$.
LA - eng
KW - o-minimality; semi-bounded groups; pregeometries; semi-bounded structures; definable groups
UR - http://eudml.org/doc/282790
ER -