We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M, generalizing a result of Dolich on o-minimal theories in [4].

We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex V-definable subgroup U of 〈Mⁿ, +〉 and a lattice L of rank n. As two consequences, we derive Pillay's conjecture (...) for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L. (shrink)

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger (...) language. (shrink)

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)