{
  "id": "1510.03210",
  "version": "v1",
  "published": "2015-10-12T10:19:04.000Z",
  "updated": "2015-10-12T10:19:04.000Z",
  "title": "Structure theorems in tame expansions of o-minimal structures by a dense set",
  "authors": [
    "Pantelis E. Eleftheriou",
    "Ayhan Günaydin",
    "Philipp Hieronymi"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We study sets and groups definable in tame expansions of o-minimal structures. Let $\\mathcal {\\widetilde M}= \\langle \\mathcal M, P\\rangle$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-12T10:19:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C64"
    ],
    "keywords": [
      "tame expansions",
      "structure theorems",
      "dense set",
      "large dimension",
      "definable set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003210E"
    }
  }
}