Quasi-o-minimal structures

A structure (M, <....) is called quasi-o-minimal if in any structure elemen tarily equivalent to it the definable subsets are exactly the Boolean combi nations of 0-definable subsets and intervals. We give a series of natural e xamples 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 qua si-o-minimality and use ii to study quasi-o-minimal ordered groups (possibl y 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 z ero multiplication; every quasi-o-minimal divisible ordered group is o-mini mal; 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.