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. (shrink)

A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek (...) et al. . Main results: An essentially periodic group G is abelian; if G is discrete, then definable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal , and its definable functions in one variable are piecewise linear. (shrink)

We consider relational databases organized over an ordered domain with some additional relations — a typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the first-order queries that are invariant under order-preserving “permutations” — such queries are called order-generic. It has recently been discovered that for some domains order-generic FO queries fail to express more than pure order queries. For example, every order-generic FO query over rational numbers (...) with + can be rewritten without +. For some other domains, however, this is not the case.We provide very general conditions on the FO theory of the domain that ensure the collapse of order-generic extended FO queries to pure order queries over this domain: the Pseudo-finite Homogeneity Property and a stronger Isolation Property. We further distinguish one broad class of domains satisfying the Isolation Property, the so-called quasi-o-minimal domains. This class includes all the o-minimal domains, but also the ordered group of integer numbers and the ordered semigroup of natural numbers, and some other domains.An important difference of this paper from the recent series of related papers is that we generalize all the notions to the case of finitely representable database states — as opposed to finite states — and develop a general lifting technique that, essentially, allows us to extend any result of the kind we are interested in, from finite to finitely representable states. We show, however, that these results cannot be transfered to arbitrary infinite states. (shrink)