{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "XK933YCW13"
    }
  ],
  "body": "\n## Definition\n\nLet $(P, \\le)$ and $(Q, \\le)$ be [[partially ordered sets|011C29Q1S1]].\nA (monotone) *Galois connection* between $P$ and $Q$ is a pair of maps $f \\colon P \\to Q$ and $g \\colon Q \\to P$ such that, for all $x \\in P$ and $y \\in Q$,\n$$f(x) \\le y \\iff x \\le g(y).$$\nThe map $f$ is the *lower adjoint* and $g$ is the *upper adjoint* of the connection.\n\n## Remarks\n\nLet $i \\colon \\Z \\to \\R$ be the inclusion and $\\lfloor \\cdot \\rfloor \\colon \\R \\to \\Z$ the [[floor function|0YVRNEDSVG]].\nThen $i(n) \\le x$ if and only if $n \\le \\lfloor x \\rfloor$, so $(i, \\lfloor \\cdot \\rfloor)$ is a Galois connection.\nOn $\\R$, the pair $f(x) = x + 1$, $g(y) = y$ is not a Galois connection: for $x = 0$ and $y = 1/2$, $x \\le g(y)$ holds but $f(x) \\le y$ fails.\nIn a Galois connection both maps are [[monotone|0WFF6AZPBP]], $x \\le g(f(x))$ and $f(g(y)) \\le y$, and $f \\circ g \\circ f = f$ and $g \\circ f \\circ g = g$.\nEach adjoint determines the other: $g(y)$ is the greatest $x$ with $f(x) \\le y$.\nThe lower adjoint preserves all existing suprema, and the upper adjoint preserves all existing infima.\nAn *antitone* Galois connection uses order-reversing maps with $y \\le f(x)$ if and only if $x \\le g(y)$; it is a monotone connection between $P$ and the dual of $Q$.\n\n## Explanations\n\n### L1 human\n\nIn a Galois connection, the question \"is $f(x)$ below $y$?\" has the same answer as \"is $x$ below $g(y)$?\".\nSo $f$ and $g$ behave like inverses as far as the order can tell.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Q5JHEPRYQ6",
    "kind": "definition",
    "msc": [
      "06A15"
    ],
    "name": "Galois connection",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Galois_connection"
      }
    ],
    "tag": "MBVJ4JQXC5",
    "uses": [
      "011C29Q1S1",
      "0WFF6AZPBP"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/MBVJ4JQXC5/",
  "links": [
    {
      "field": "defines",
      "target": "Q5JHEPRYQ6"
    },
    {
      "field": "uses",
      "target": "011C29Q1S1"
    },
    {
      "field": "uses",
      "target": "0WFF6AZPBP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MBVJ4JQXC5.md",
  "msc": [
    "06A15"
  ],
  "name": "Galois connection",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MBV/MBVJ4JQXC5.md",
  "status": null,
  "tag": "MBVJ4JQXC5",
  "uri": "m2://MBVJ4JQXC5",
  "url": "https://tamnd.github.io/math2/wiki/galois-connection-definition/",
  "writing": {
    "tag": "MBVJ4JQXC5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}