{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]].\nA [[chain|0GP9ASVQ1X]] $C \\subseteq P$ is *maximal* when there is no chain $D \\subseteq P$ with $C \\subsetneq D$.\n\n## Remarks\n\nIn the subsets of $\\set{1, 2, 3}$ ordered by inclusion, $\\set{\\emptyset, \\set{1}, \\set{1, 2}, \\set{1, 2, 3}}$ is a maximal chain.\nEvery other subset is incomparable with one of its members, so no subset can be added.\nThe chain $\\set{\\emptyset, \\set{1}, \\set{1, 2, 3}}$ is not maximal, since adding $\\set{1, 2}$ gives a larger chain.\nThe subsets of $\\set{1, \\dots, n}$ have exactly $n!$ maximal chains, one for each order in which the elements are added.\nA maximal chain is a maximal element of the set of chains ordered by inclusion.\nThe statement that every chain of a poset lies in a maximal chain is the [[Hausdorff maximal principle|0KM4KVR9MV]], which is equivalent to [[Zorn's lemma|04RXTQA5ZA]].\n\n## Explanations\n\n### L1 human\n\nA maximal chain is a chain that cannot be made longer by adding any element of the poset.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "3AX549FFFM",
    "kind": "definition",
    "msc": [
      "06A06",
      "06A07"
    ],
    "name": "Maximal chain",
    "sources": [
      {
        "locator": "Chains",
        "url": "https://en.wikipedia.org/wiki/Total_order"
      }
    ],
    "tag": "WZKPDWBW2Z",
    "uses": [
      "011C29Q1S1",
      "0GP9ASVQ1X"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/WZKPDWBW2Z/",
  "links": [
    {
      "field": "defines",
      "target": "3AX549FFFM"
    },
    {
      "field": "uses",
      "target": "011C29Q1S1"
    },
    {
      "field": "uses",
      "target": "0GP9ASVQ1X"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/WZKPDWBW2Z.md",
  "msc": [
    "06A06",
    "06A07"
  ],
  "name": "Maximal chain",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WZK/WZKPDWBW2Z.md",
  "status": null,
  "tag": "WZKPDWBW2Z",
  "uri": "m2://WZKPDWBW2Z",
  "url": "https://tamnd.github.io/math2/wiki/maximal-chain-definition/",
  "writing": {
    "tag": "WZKPDWBW2Z",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}