{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "5TNA2HBG7Y"
    },
    {
      "field": "about",
      "target": "Z86VGP5RMG"
    },
    {
      "field": "uses",
      "target": "5TNA2HBG7Y"
    },
    {
      "field": "uses",
      "target": "Z86VGP5RMG"
    },
    {
      "field": "defines",
      "target": "FMCWXA0WT4"
    }
  ],
  "body": "\n## Summary\n\nThe free lattice on a set $X$ is the [[lattice|WH4SFAACAS]] generated by $X$ in which no relations hold except those forced by the lattice laws.\nIt has a universal property: every map from $X$ into a lattice extends uniquely to a [[lattice homomorphism|BNVRR925X8]] from the free lattice.\nIts elements can be described by lattice terms in the letters of $X$.\nDeciding when two terms are equal is the [[word problem for free lattices|Z86VGP5RMG]].\nThe free lattice on two generators has four elements, while the free lattice on three generators is already infinite.\n\n## Explanations\n\n### L1 human\n\nTake some letters $x$, $y$, $z$ and form all expressions with $\\vee$ and $\\wedge$, such as $x \\wedge (y \\vee z)$.\nTwo expressions count as equal only when the rules of every lattice force them to be equal.\nThe result is the free lattice on these letters: the most general lattice that these letters can generate.\nWith two letters $x$ and $y$ only four elements appear: $x$, $y$, $x \\wedge y$ and $x \\vee y$.\nWith three letters, infinitely many different elements appear.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06B25",
      "08B20"
    ],
    "name": "Free lattice",
    "same_as": {
      "wikidata": "Q5500254",
      "wikipedia": "Free lattice"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_lattice"
      }
    ],
    "specializes": [
      "WH4SFAACAS"
    ],
    "tag": "WP20S9NZ15"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/WP20S9NZ15/",
  "links": [
    {
      "field": "specializes",
      "target": "WH4SFAACAS"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/WP20S9NZ15.md",
  "msc": [
    "06B25",
    "08B20"
  ],
  "name": "Free lattice",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WP2/WP20S9NZ15.md",
  "status": null,
  "tag": "WP20S9NZ15",
  "uri": "m2://WP20S9NZ15",
  "url": "https://tamnd.github.io/math2/wiki/free-lattice/",
  "writing": {
    "tag": "WP20S9NZ15",
    "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
      }
    ]
  }
}