{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "74JREVZVY3"
    },
    {
      "field": "uses",
      "target": "MV1CFTH98T"
    },
    {
      "field": "defines",
      "target": "ARGZ549BQT"
    }
  ],
  "body": "\n## Summary\n\nThe Möbius function of a [[locally finite poset|TV1V4JDRDD]] assigns an integer $\\mu(x, y)$ to each pair $x \\le y$, by the recursion $\\mu(x, x) = 1$ and $\\sum_{x \\le z \\le y} \\mu(x, z) = 0$ for $x \u003c y$.\nIt is the inverse of the zeta function in the [[incidence algebra|1H9SFWHYEB]].\nIt drives [[Möbius inversion on posets|B8XJSJ517Q]], which contains inclusion–exclusion and the number-theoretic [[Möbius function|0FANE7P0DY]] as special cases.\n\n## Explanations\n\n### L1 human\n\nStart at an element $x$ and give it weight $1$.\nThen go up the poset, and give each element $y$ the weight that makes the total of all weights from $x$ to $y$ equal to zero.\nFor a chain $0 \u003c 1 \u003c 2 \u003c \\cdots$ this gives $1, -1, 0, 0, \\ldots$.\nFor the subsets of a set, the weight of a set with $k$ more elements than the start is $(-1)^k$.\nThese weights are what you need to undo a sum over everything below.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06A07",
      "05A19"
    ],
    "name": "Möbius function of a poset",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Incidence_algebra"
      }
    ],
    "tag": "2Y75S8NG5B"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/2Y75S8NG5B/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/2Y75S8NG5B.md",
  "msc": [
    "06A07",
    "05A19"
  ],
  "name": "Möbius function of a poset",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2Y7/2Y75S8NG5B.md",
  "status": null,
  "tag": "2Y75S8NG5B",
  "uri": "m2://2Y75S8NG5B",
  "url": "https://tamnd.github.io/math2/wiki/mobius-function-of-a-poset/",
  "writing": {
    "tag": "2Y75S8NG5B",
    "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
      }
    ]
  }
}