TY - JOUR
T1 - A tetrachotomy for expansions of the real ordered additive group
AU - Hieronymi, Philipp
AU - Walsberg, Erik
N1 - Funding Information:
The first author was partially supported by NSF grants DMS-1300402 and DMS-1654725. The second author was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 291111/ MODAG. A preprint of this paper was disseminated under the title “On continuous functions definable in expansions of the ordered real additive group”
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2021/9
Y1 - 2021/9
N2 - 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.
AB - 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.
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U2 - 10.1007/s00029-021-00668-9
DO - 10.1007/s00029-021-00668-9
M3 - Article
AN - SCOPUS:85108190743
VL - 27
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
SN - 1022-1824
IS - 4
M1 - 54
ER -