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# Local analysis for semi-bounded groups

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Type of Publication: | Journal article |

Publication status: | Published |

Author: | Eleftheriou, Pantelis E. |

Year of publication: | 2012 |

Published in: | Fundamenta Mathematicae ; 216 (2012), 3. - pp. 223-258. - Institute of Mathematics. - ISSN 0016-2736. - eISSN 1730-6329 |

DOI (citable link): | https://dx.doi.org/10.4064/fm216-3-3 |

Summary: |
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},+⟩. |

Subject (DDC): | 510 Mathematics |

Refereed: | Unknown |

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ELEFTHERIOU, Pantelis E., 2012. Local analysis for semi-bounded groups. In: Fundamenta Mathematicae. Institute of Mathematics. 216(3), pp. 223-258. ISSN 0016-2736. eISSN 1730-6329. Available under: doi: 10.4064/fm216-3-3

@article{Eleftheriou2012Local-49667, title={Local analysis for semi-bounded groups}, year={2012}, doi={10.4064/fm216-3-3}, number={3}, volume={216}, issn={0016-2736}, journal={Fundamenta Mathematicae}, pages={223--258}, author={Eleftheriou, Pantelis E.} }