{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $\\mathcal{C}$ be a class of subsets of $P$.\nThe poset $P$ is *complete for* $\\mathcal{C}$, or has all $\\mathcal{C}$-joins, if every $S \\in \\mathcal{C}$ has a [[supremum|0R3H5PANRQ]] in $P$.\nDually, $P$ has all $\\mathcal{C}$-meets if every $S \\in \\mathcal{C}$ has an [[infimum|0QF1MZ9FRG]] in $P$.\nThe standard choices of $\\mathcal{C}$ are:\n\n1. all subsets, giving a [[complete lattice|D7A0ZGZT5S]];\n2. all finite nonempty subsets, giving a join-semilattice, and with meets as well, a lattice;\n3. all nonempty subsets with an upper bound, giving (with the dual condition on lattices) a [[conditionally complete lattice|CZPBQY67ZJ]];\n4. all [[directed subsets|D9RY4667K6]], giving a [[directed-complete partial order|7TSXP95QN6]];\n5. all chains, giving a chain-complete poset.\n\n## Remarks\n\nExample: the power set of a set is complete for every class $\\mathcal{C}$, since unions always exist.\nNon-example: $\\N$ with its usual order has all finite nonempty joins but is not directed-complete, because $\\N$ itself is directed and has no supremum.\nThe empty set has a supremum if and only if $P$ has a least element.\nIf $P$ has all joins of arbitrary subsets, it also has all meets, so in case 1 one condition suffices.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Q72NHN7ATC",
    "kind": "definition",
    "msc": [
      "06B23",
      "06A06"
    ],
    "name": "Completeness in order theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Completeness_(order_theory)"
      }
    ],
    "tag": "WH3Z5M5D2R",
    "uses": [
      "0QV2JS9RJ3",
      "0R3H5PANRQ",
      "0QF1MZ9FRG",
      "D9RY4667K6",
      "D7A0ZGZT5S",
      "CZPBQY67ZJ",
      "7TSXP95QN6"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/WH3Z5M5D2R/",
  "links": [
    {
      "field": "defines",
      "target": "Q72NHN7ATC"
    },
    {
      "field": "uses",
      "target": "0QV2JS9RJ3"
    },
    {
      "field": "uses",
      "target": "0R3H5PANRQ"
    },
    {
      "field": "uses",
      "target": "0QF1MZ9FRG"
    },
    {
      "field": "uses",
      "target": "D9RY4667K6"
    },
    {
      "field": "uses",
      "target": "D7A0ZGZT5S"
    },
    {
      "field": "uses",
      "target": "CZPBQY67ZJ"
    },
    {
      "field": "uses",
      "target": "7TSXP95QN6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/WH3Z5M5D2R.md",
  "msc": [
    "06B23",
    "06A06"
  ],
  "name": "Completeness in order theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WH3/WH3Z5M5D2R.md",
  "status": null,
  "tag": "WH3Z5M5D2R",
  "uri": "m2://WH3Z5M5D2R",
  "url": "https://tamnd.github.io/math2/wiki/completeness-in-order-theory-definition/",
  "writing": {
    "tag": "WH3Z5M5D2R",
    "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
      }
    ]
  }
}