{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "CH9B4RZPMW"
    },
    {
      "field": "about",
      "target": "DDAHG7QRY8"
    },
    {
      "field": "about",
      "target": "RHXP3HA64M"
    },
    {
      "field": "about",
      "target": "TAHM74ZNWB"
    },
    {
      "field": "uses",
      "target": "DDAHG7QRY8"
    },
    {
      "field": "uses",
      "target": "F4BVTGH58G"
    },
    {
      "field": "uses",
      "target": "RHXP3HA64M"
    },
    {
      "field": "uses",
      "target": "TAHM74ZNWB"
    }
  ],
  "body": "\n## Summary\n\nThe least fixed point of a map $f$ from a [[partially ordered set|0QV2JS9RJ3]] to itself is a point $x$ with $f(x) = x$.\nIt must lie below every other point with this property.\nWhen it exists it is unique.\nLeast fixed points give meaning to recursive and inductive definitions: the set defined inductively by some rules is the least set closed under the rules.\nThe [[Knaster–Tarski theorem|RHXP3HA64M]] and the [[Kleene fixed-point theorem|DDAHG7QRY8]] guarantee least fixed points of monotone maps under completeness hypotheses.\n\n## Explanations\n\n### L1 human\n\nA fixed point of a map is a point that the map leaves where it is.\nAmong all fixed points, the least one is the one that lies below all the others.\nFor the map $x \\mapsto x^2$ on the real numbers, the fixed points are $0$ and $1$, and the least fixed point is $0$.\nThe set of even natural numbers is the least set that contains $0$ and contains $n + 2$ whenever it contains $n$.\nIt is the least fixed point of the map $S \\mapsto \\set{0} \\cup \\set{n + 2 : n \\in S}$ on subsets of $\\N$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06A06",
      "47H10"
    ],
    "name": "Least fixed point",
    "same_as": {
      "wikidata": "Q6134913",
      "wikipedia": "Least fixed point"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Least_fixed_point"
      }
    ],
    "tag": "DBD64TH52D"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/DBD64TH52D/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/DBD64TH52D.md",
  "msc": [
    "06A06",
    "47H10"
  ],
  "name": "Least fixed point",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DBD/DBD64TH52D.md",
  "status": null,
  "tag": "DBD64TH52D",
  "uri": "m2://DBD64TH52D",
  "url": "https://tamnd.github.io/math2/wiki/least-fixed-point/",
  "writing": {
    "tag": "DBD64TH52D",
    "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
      }
    ]
  }
}