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Structure theorems in tame expansions of o-minimal structures by a dense set | | Pantelis E. Eleftheriou
; Ayhan Günaydin
; Philipp Hieronymi
; | | Date: |
12 Oct 2015 | | Abstract: | We study sets and groups definable in tame expansions of o-minimal
structures. Let $mathcal {widetilde M}= langle mathcal M, P
angle$ be an
expansion of an o-minimal $cal L$-structure $cal M$ by a dense set $P$. We
impose three tameness conditions on $mathcal {widetilde M}$ and prove
structure theorems for definable sets and functions in the realm of the cone
decomposition theorems that are known for semi-bounded o-minimal 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 $operatorname{scl}$-dimension coming from a pregeometry given in
Berenstein-Ealy-G"unaydin, and (ii) the large dimension is invariant under
definable bijections. We then illustrate how our results open the way to study
groups definable in $cal {widetilde M}$, by proving that around
$operatorname{scl}$-generic elements of a definable group, the group operation
is given by an $mathcal L$-definable map. | | Source: | arXiv, 1510.3210 | | Services: | Forum | Review | PDF | Favorites | |
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