Belegradek, OPeterzil, YWagner, F2024-07-182024-07-1820000022-4812https://doi.org/10.2307/2586690https://hdl.handle.net/11411/8127A 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 ii 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.eninfo:eu-repo/semantics/closedAccessQuasi-O-Minimal TheoryO-Minimal TheoryOrdered GroupTheory Of U-Rank 1Article2-s2.0-003425928010.2307/258669011323Q1111565Q3WOS:000089597400007