A tetrachotomy for expansions of the real ordered additive group

• Philipp Hieronymi ^[2] ; Erik Walsberg ^[1]
  1. [1] University of California System

     University of California System

     Estados Unidos

     
  2. [2] University of Illinois, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 4, 2021
• Idioma: inglés
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• Resumen

  • Let R be an expansion of the ordered real additive group. When R is o-minimal, it is known that either R defines an ordered field isomorphic to (R,<,+,⋅) on some open subinterval I⊆R, or R is a reduct of an ordered vector space. We say R is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of (R,<,+). In particular, we show that for expansions that do not define dense ω-orders (we call these type A expansions), an appropriate version of Zilber’s principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function [0,1]m→Rn is locally affine outside a nowhere dense set.