{
  "id": "1901.04786",
  "version": "v1",
  "published": "2019-01-15T12:14:57.000Z",
  "updated": "2019-01-15T12:14:57.000Z",
  "title": "Cohesive Powers of Linear Orders",
  "authors": [
    "Rumen Dimitrov",
    "Valentina Harizanov",
    "Andrey Morozov",
    "Paul Shafer",
    "Alexandra Soskova",
    "Stefan Vatev"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $\\Pi _{C}% \\mathcal{L}$ for familiar computable linear orders $\\mathcal{L}$. If $% \\mathcal{L}$ is isomorphic to the ordered set of natural numbers $\\mathbb{N}$ and has a computable successor function, then $\\Pi _{C}\\mathcal{L}$ is isomorphic to $\\mathbb{N}+\\mathbb{Q}\\times \\mathbb{Z}.$ Here, $+$ stands for the sum and $\\times $ for the lexicographical product of two orders. We construct computable linear orders $\\mathcal{L}_{1}$ and $\\mathcal{L}_{2}$ isomorphic to $\\mathbb{N},$ both with noncomputable successor functions, such that $\\Pi _{C}\\mathcal{L}_{1}\\mathbb{\\ }$is isomorphic to $\\mathbb{N}+% \\mathbb{Q}\\times \\mathbb{Z}$, while $\\Pi _{C}\\mathcal{L}_{2}$ is not$.$ While cohesive powers preserve all $\\Pi _{2}^{0}$ and $\\Sigma _{2}^{0}$ sentences, we provide new examples of $\\Pi _{3}^{0}$ sentences $\\Phi $ and computable structures $% \\mathcal{M}$ such that $\\mathcal{M}\\vDash \\Phi $ while $\\Pi _{C}\\mathcal{M}% \\vDash \\urcorner \\Phi .$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-15T12:14:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C57",
      "03D45",
      "03C20"
    ],
    "keywords": [
      "isomorphic",
      "computable structures",
      "familiar computable linear orders",
      "construct computable linear orders",
      "natural numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}