{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\n1. Let $\\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$.\nAn element $x \\in S$ is *idempotent* for $\\ast$ if $x \\ast x = x$.\nThe operation $\\ast$ is *idempotent* if every element of $S$ is idempotent for $\\ast$.\n2. Let $X$ be a set.\nA [[function|0EPAT51DXQ]] $f \\colon X \\to X$ is *idempotent* if $f \\circ f = f$, that is, $f(f(x)) = f(x)$ for all $x \\in X$, where $\\circ$ is [[composition|0THBK9V825]].\n\n## Remarks\n\nExample: union is an idempotent operation on the power set of a set $X$, because $Y \\cup Y = Y$ for every $Y \\subseteq X$.\n\nNon-example: multiplication on $\\Z$ is not idempotent, because $2 \\cdot 2 = 4 \\ne 2$; its only idempotent elements are $0$ and $1$.\n\nThe two notions agree.\nA function $f \\colon X \\to X$ is idempotent in the sense of clause 2 if and only if it is idempotent for composition on the set of maps $X \\to X$.\nA function is idempotent if and only if it fixes every point of its image.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9FS1JFZSX4",
    "kind": "definition",
    "msc": [
      "08A02",
      "16U40"
    ],
    "name": "Idempotence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Idempotence"
      }
    ],
    "tag": "QV3V8HJW2F",
    "uses": [
      "0M418FE7BB",
      "0EPAT51DXQ",
      "0THBK9V825"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/QV3V8HJW2F/",
  "links": [
    {
      "field": "defines",
      "target": "9FS1JFZSX4"
    },
    {
      "field": "uses",
      "target": "0M418FE7BB"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    },
    {
      "field": "uses",
      "target": "0THBK9V825"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QV3V8HJW2F.md",
  "msc": [
    "08A02",
    "16U40"
  ],
  "name": "Idempotence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QV3/QV3V8HJW2F.md",
  "status": null,
  "tag": "QV3V8HJW2F",
  "uri": "m2://QV3V8HJW2F",
  "url": "https://tamnd.github.io/math2/wiki/idempotence-definition/",
  "writing": {
    "tag": "QV3V8HJW2F",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}