{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L$ be a [[complete lattice|D7A0ZGZT5S]].\nIt is an *algebraic lattice*, or compactly generated, if every $x \\in L$ is the [[supremum|0R3H5PANRQ]] of the set $\\set{c \\in L : c \\le x,\\ c \\text{ compact}}$ of [[compact elements|9F7TYES4KE]] below it.\n\n## Remarks\n\nExample: the lattice of subgroups of a group $G$, whose compact elements are the finitely generated subgroups.\nEach subgroup $H$ is the join of the cyclic subgroups $\\langle h \\rangle$ for $h \\in H$.\nNon-example: the closed subsets of $[0, 1]$, ordered by inclusion, form a complete lattice in which the join of a family is the closure of its union.\nA nonempty closed set $F$ is not compact: pick $p \\in F$; the closed sets $C_n = [0, 1] \\setminus (p - 1/n, p + 1/n)$ form a directed family whose join is $[0, 1] \\supseteq F$, yet $p \\notin C_n$ for every $n$.\nSo the only compact element is $\\varnothing$, and no nonempty closed set is a join of compact elements.\nEvery algebraic lattice is a continuous lattice.\nSome authors do not require completeness and speak of algebraic posets or algebraic dcpos instead.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "3J0BTXSCVN",
    "kind": "definition",
    "msc": [
      "06B23",
      "08A30"
    ],
    "name": "Algebraic lattice",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Compact_element"
      }
    ],
    "tag": "HD8CK28AJC",
    "uses": [
      "D7A0ZGZT5S",
      "9F7TYES4KE",
      "0R3H5PANRQ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/HD8CK28AJC/",
  "links": [
    {
      "field": "defines",
      "target": "3J0BTXSCVN"
    },
    {
      "field": "uses",
      "target": "D7A0ZGZT5S"
    },
    {
      "field": "uses",
      "target": "9F7TYES4KE"
    },
    {
      "field": "uses",
      "target": "0R3H5PANRQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HD8CK28AJC.md",
  "msc": [
    "06B23",
    "08A30"
  ],
  "name": "Algebraic lattice",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HD8/HD8CK28AJC.md",
  "status": null,
  "tag": "HD8CK28AJC",
  "uri": "m2://HD8CK28AJC",
  "url": "https://tamnd.github.io/math2/wiki/algebraic-lattice-definition/",
  "writing": {
    "tag": "HD8CK28AJC",
    "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
      }
    ]
  }
}