{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(L, \\le)$ be a [[lattice|WH4SFAACAS]].\nIt is a *conditionally complete lattice* if:\n\n1. every nonempty subset of $L$ that has an [[upper bound|0S60DVMVPN]] in $L$ has a [[supremum|0R3H5PANRQ]] in $L$, and\n2. every nonempty subset of $L$ that has a lower bound in $L$ has an [[infimum|0QF1MZ9FRG]] in $L$.\n\n## Remarks\n\nClause 2 follows from clause 1.\nIndeed, the infimum of a nonempty set $S$ bounded below is the supremum of the set of lower bounds of $S$.\nThat set is nonempty and bounded above by any element of $S$.\nExample: the [[real numbers|08J2M1JYAG]] with the usual order, by the completeness axiom.\nNon-example: $\\Q$, since $\\set{q \\in \\Q : q^2 \u003c 2}$ is nonempty and bounded above by $2$ but has no supremum in $\\Q$.\nEvery [[complete lattice|D7A0ZGZT5S]] is conditionally complete, and a conditionally complete lattice with a least and a greatest element is complete.\nA totally ordered set with property 1 is said to have the least-upper-bound property.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "CZPBQY67ZJ",
    "kind": "definition",
    "msc": [
      "06B23"
    ],
    "name": "Conditionally complete lattice",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Completeness_(order_theory)"
      }
    ],
    "tag": "A5D94C38HZ",
    "uses": [
      "WH4SFAACAS",
      "0S60DVMVPN",
      "0R3H5PANRQ",
      "0QF1MZ9FRG",
      "08J2M1JYAG",
      "D7A0ZGZT5S"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/A5D94C38HZ/",
  "links": [
    {
      "field": "defines",
      "target": "CZPBQY67ZJ"
    },
    {
      "field": "uses",
      "target": "WH4SFAACAS"
    },
    {
      "field": "uses",
      "target": "0S60DVMVPN"
    },
    {
      "field": "uses",
      "target": "0R3H5PANRQ"
    },
    {
      "field": "uses",
      "target": "0QF1MZ9FRG"
    },
    {
      "field": "uses",
      "target": "08J2M1JYAG"
    },
    {
      "field": "uses",
      "target": "D7A0ZGZT5S"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/A5D94C38HZ.md",
  "msc": [
    "06B23"
  ],
  "name": "Conditionally complete lattice",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A5D/A5D94C38HZ.md",
  "status": null,
  "tag": "A5D94C38HZ",
  "uri": "m2://A5D94C38HZ",
  "url": "https://tamnd.github.io/math2/wiki/conditionally-complete-lattice-definition/",
  "writing": {
    "tag": "A5D94C38HZ",
    "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
      }
    ]
  }
}