Structure theorems in tame expansions of o-minimal structures by a dense set

Structure Theorems in Tame Expansions of O-minimal Structures by a Dense Set

We study sets and groups definable in tame expansions of ominimal structures. Let M̃ = ⟨M, P ⟩ be an expansion of an o-minimal Lstructure M by a dense set P . We impose three tameness conditions on M̃ and prove a cone decomposition theorem for definable sets and functions in the realm of the o-minimal semi-bounded structures. The proof involves induction on the notion of ‘large dimension’ for def...

Expansions of o-minimal structures by dense independent sets

Let M be an o-minimal expansion of a densely ordered group and H be a pairwise disjoint collection of dense subsets of M such that ⋃ H is definably independent in M. We study the structure (M, (H)H∈H). Positive results include that every open set definable in (M, (H)H∈H) is definable in M, the structure induced in (M, (H)H∈H) on any H0 ∈ H is as simple as possible (in a sense that is made preci...

Structure Theorems for o-Minimal Expansions of Groups

In this study the structure for O-minimal expansions of groups is considered Let be an o-minimal expansion of an ordered group (R,0,1,+,<) with distinguished positive element 1. We first prove that the following are equivalent: (1) is semibounded, (2) has no poles, (3) cannot define a real closed field with domain R and order <, (4) is eventually linear and (5) every –definable set is a finit u...

Expansions of o-minimal structures by fast sequences

Dense Pairs of O-minimal Structures

The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays a special role in this description. Introduction. In a classical paper [8] A. Robinson proved the completeness of the theory of real closed fields with a predicate for a proper dense real closed subfi...