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A trichotomy theorem for o‐minimal structures
Author(s) -
Peterzil Y,
Starchenko S
Publication year - 1998
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611598000549
Subject(s) - mathematics , trichotomy (philosophy) , abelian group , neighbourhood (mathematics) , structured program theorem , regular polygon , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , geometry , philosophy , linguistics
Let M = 〈 M , <, …〉 be alinearly ordered structure. We define M to be o‐minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo‐minimal M can induce on a neighbourhood of any a in M . Roughly said, one of the following holds: (i) a is trivial (technical term), or(ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or(iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification : primary 03C45; secondary 03C52, 12J15, 14P10.