2015/10/12 by Eleftheriou, Pantelis E., Günaydin, Ayhan, Hieronymi, Philipp
#03C64 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1510.03210
We study sets and groups definable in tame expansions of o-minimal structures. Let \mathcal \widetilde M= ⟨ \mathcal M, P⟩ be an expansion of an o-minimal \mathcal L-structure \cal M by a dense set P, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of \mathcal \widetilde M, as it achieves a decomposition of definable sets into unions of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the 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 its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an \cal L-definable map.