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Preprint Number 941

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941. Pantelis E. Eleftheriou, Ayhan Günaydin, Philipp Hieronymi
Structure theorems in tame expansions of o-minimal structures by a dense set
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Submission date: 12 October 2015


We study sets and groups definable in tame expansions of o-minimal structures. Let 〈 M, P 〉 be an expansion of an o-minimal L-structure M by a dense set P. We impose three tameness conditions on (M, P) and prove a cone decomposition theorem for definable sets and functions in the realm of o-minimal semi-bounded structures. The proofs involve induction on the notion of 'large dimension' for definable sets, an invariant which we herewith introduce and analyze. As a corollary, we obtain that (i) the large dimension of a definable set coincides with the combinatorial scl-dimension coming from a pregeometry given in Berenstein-Ealy-Günaydin, and (ii) the large dimension is invariant under definable bijections. We then illustrate how our results open the way to study groups definable in (M, P), by proving that around scl-generic elements of a definable group, the group operation is given by an L-definable map.

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Full text arXiv 1510.03210: pdf, ps.

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