{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be a set and let $n \\ge 0$ be an integer.\nWrite $A^n$ for the [[Cartesian product|09CEDJYBKP]] of $n$ copies of $A$, with $A^0 = \\set{()}$ the one-element set whose element is the empty tuple.\nAn *$n$-ary operation* on $A$ is a [[function|0EPAT51DXQ]] $f \\colon A^n \\to A$.\nA *finitary operation* on $A$ is an $n$-ary operation on $A$ for some integer $n \\ge 0$, called its [[arity|MSXX2P86CY]].\n\n## Remarks\n\nExample: the map $f \\colon \\Z^3 \\to \\Z$, $f(x, y, z) = x - y + z$, is a finitary operation of arity $3$ on $\\Z$, because $x - y + z$ is an integer for all integers $x, y, z$.\n\nNon-example: division $(x, y) \\mapsto x / y$ is not a binary operation on $\\Q$, because it is not defined when $y = 0$, so it is not a function on all of $\\Q^2$.\nOn $\\Q \\setminus \\set{0}$ it is a binary operation.\n\nThe requirement that the values lie in $A$ is often called closure of $A$ under $f$.\nSome authors use the word operation only for finitary operations.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "EJ8YJVM9BW",
    "kind": "definition",
    "msc": [
      "08A62",
      "08A02"
    ],
    "name": "Finitary operation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Operation_(mathematics)"
      }
    ],
    "tag": "K8RE4YQ1S9",
    "uses": [
      "MSXX2P86CY",
      "0EPAT51DXQ",
      "09CEDJYBKP"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/K8RE4YQ1S9/",
  "links": [
    {
      "field": "defines",
      "target": "EJ8YJVM9BW"
    },
    {
      "field": "uses",
      "target": "MSXX2P86CY"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    },
    {
      "field": "uses",
      "target": "09CEDJYBKP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/K8RE4YQ1S9.md",
  "msc": [
    "08A62",
    "08A02"
  ],
  "name": "Finitary operation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K8R/K8RE4YQ1S9.md",
  "status": null,
  "tag": "K8RE4YQ1S9",
  "uri": "m2://K8RE4YQ1S9",
  "url": "https://tamnd.github.io/math2/wiki/finitary-operation-definition/",
  "writing": {
    "tag": "K8RE4YQ1S9",
    "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
      }
    ]
  }
}