{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "QTQN93B305"
    }
  ],
  "body": "\n## Summary\n\nA [[partially ordered set|0QV2JS9RJ3]] has the fixed-point property if every order-preserving map from it to itself fixes some element.\nIt is the order-theoretic counterpart of the [[fixed-point property of topological spaces|JQDHHFXBF3]].\nEvery [[complete lattice|D7A0ZGZT5S]] has it by the [[Knaster–Tarski theorem|RHXP3HA64M]], and among lattices exactly the complete ones have it.\nDeciding which finite posets have the fixed-point property is a hard combinatorial question.\n\n## Explanations\n\n### L1 human\n\nMove the elements of an ordering around without ever reversing a comparison.\nThe ordering has the fixed-point property if, however one does this, some element stays in place.\nAny finite chain has it: if the smallest element moves, it moves up, and following the moves upward one must stop somewhere.\nAn ordering of two incomparable elements does not have it, since swapping them reverses no comparison and moves both.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06A06",
      "47H10"
    ],
    "name": "Fixed-point property of posets",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fixed-point_property"
      }
    ],
    "tag": "HMDGNS1V2T"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HMDGNS1V2T/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/HMDGNS1V2T.md",
  "msc": [
    "06A06",
    "47H10"
  ],
  "name": "Fixed-point property of posets",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HMD/HMDGNS1V2T.md",
  "status": null,
  "tag": "HMDGNS1V2T",
  "uri": "m2://HMDGNS1V2T",
  "url": "https://tamnd.github.io/math2/wiki/fixed-point-property-of-posets/",
  "writing": {
    "tag": "HMDGNS1V2T",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}