A tetrachotomy for expansions of the real ordered additive group

Philipp Hieronymi, Erik Walsberg

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Article number54
JournalSelecta Mathematica, New Series
Volume27
Issue number4
DOIs
StatePublished - Sep 2021

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

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