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

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    <dcterms:title>Local analysis for semi-bounded groups</dcterms:title>
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    <dc:contributor>Eleftheriou, Pantelis E.</dc:contributor>
    <dcterms:abstract xml:lang="eng">An o-minimal expansion M=⟨M,&lt;,+,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&lt;sup&gt;k&lt;/sup&gt;,+⟩.</dcterms:abstract>
    <dc:creator>Eleftheriou, Pantelis E.</dc:creator>
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