{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\n$P$ has the *fixed-point property* if every [[monotone|0A7KV2WB7A]] map $f \\colon P \\to P$ has a fixed point, that is, an element $x \\in P$ with $f(x) = x$.\n\n## Remarks\n\nExample: every finite chain $c_1 \u003c c_2 \u003c \\cdots \u003c c_n$ has the fixed-point property.\nFor monotone $f$, $c_1 \\le f(c_1)$, and induction gives $f^k(c_1) \\le f^{k+1}(c_1)$; this increasing sequence in a finite set is eventually constant, at a fixed point.\nNon-example: the two-element antichain $\\set{a, b}$ fails it, since the map exchanging $a$ and $b$ is monotone and has no fixed point.\nThe chain $\\Z$ also fails it, with the monotone map $x \\mapsto x + 1$.\nEvery complete lattice has the fixed-point property, by the Knaster–Tarski theorem.\nMore generally, every finite poset with a least element has it, by the same iteration as for chains.\nThe property is preserved by order isomorphisms and passes to retracts.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "HMDGNS1V2T",
    "kind": "definition",
    "msc": [
      "06A06",
      "47H10"
    ],
    "name": "Fixed-point property of posets",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fixed-point_property"
      }
    ],
    "tag": "QTQN93B305",
    "uses": [
      "0QV2JS9RJ3",
      "0A7KV2WB7A"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/QTQN93B305/",
  "links": [
    {
      "field": "defines",
      "target": "HMDGNS1V2T"
    },
    {
      "field": "uses",
      "target": "0QV2JS9RJ3"
    },
    {
      "field": "uses",
      "target": "0A7KV2WB7A"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QTQN93B305.md",
  "msc": [
    "06A06",
    "47H10"
  ],
  "name": "Fixed-point property of posets",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QTQ/QTQN93B305.md",
  "status": null,
  "tag": "QTQN93B305",
  "uri": "m2://QTQN93B305",
  "url": "https://tamnd.github.io/math2/wiki/fixed-point-property-of-posets-definition/",
  "writing": {
    "tag": "QTQN93B305",
    "kind": "definition",
    "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
      }
    ]
  }
}