{
  "id": "2104.01312",
  "version": "v1",
  "published": "2021-04-03T04:12:20.000Z",
  "updated": "2021-04-03T04:12:20.000Z",
  "title": "Almost o-minimal structures and $\\mathfrak X$-structures",
  "authors": [
    "Masato Fujita"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1912.05782",
  "categories": [
    "math.LO",
    "math.AG"
  ],
  "abstract": "We propose new structures called almost o-minimal structures and $\\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's $\\mathfrak X$-sets and $\\mathfrak Y$-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an $\\mathfrak X$-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded $\\mathfrak X$-definable sets are definable in the structure. Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups $\\mathcal M=(M,<,0,+,\\ldots)$. Let $\\{A_\\lambda\\}_{\\lambda\\in\\Lambda}$ be a finite family of definable subsets of $M^{m+n}$. Take an arbitrary positive element $R \\in M$ and set $B=]-R,R[^n$. Then, there exists a finite partition into definable sets \\begin{equation*} M^m \\times B = X_1 \\cup \\ldots \\cup X_k \\end{equation*} such that $B=(X_1)_b \\cup \\ldots \\cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \\in M^m$ and either $X_i \\cap A_\\lambda = \\emptyset$ or $X_i \\subseteq A_\\lambda$ for any $1 \\leq i \\leq k$ and $\\lambda \\in \\Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset $S$ of $M^{m+n}$ at $b \\in M^m$. We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-03T04:12:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C64",
      "14P99"
    ],
    "keywords": [
      "o-minimal structures",
      "definable set",
      "projections unlike first-order structures",
      "definable cell decomposition theorem",
      "open interval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}