## Abstract

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 language | English |
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Pages (from-to) | 1115-1132 |

Number of pages | 18 |

Journal | Journal of Symbolic Logic |

Volume | 65 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2000 |

## Keywords

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

## ASJC Scopus subject areas

- Philosophy
- Logic