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

Abstract

We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$ -structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ and prove a structure theorem for definable sets and functions in analogy with the cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of such $$\widetilde{\cal M}$$ , as it achieves a decomposition of definable sets into unions of ‘cones’, instead of only boolean combinations of them. The proofs involve induction on the notion of ‘large dimension’ for definable sets, an invariant which we herewith introduce and analyze. Applications of the cone decomposition theorem include: (i) the large dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $${\cal L}$$ -definable map off a subset of the domain of smaller large dimension, and (iii) around generic elements of a definable group, the group operation is given by an $${\cal L}$$ -definable map.