Quasi-o-minimal structures

Oleg Belegradek, Ya'acov Peterzil, Frank Wagner

Research output: Contribution to journalArticlepeer-review


A structure (M. <....) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal: one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1.

Original languageEnglish
Pages (from-to)1115-1132
Number of pages18
JournalJournal of Symbolic Logic
Issue number3
StatePublished - Sep 2000


  • O-minimal theory
  • Ordered group
  • Quasi-o-minimal theory
  • Theory of U-rank 1

ASJC Scopus subject areas

  • Philosophy
  • Logic


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