{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a set.\nA *free lattice* on $X$ is a [[lattice|WH4SFAACAS]] $F$ together with a map $\\iota \\colon X \\to F$ with the following property.\nFor every lattice $L$ and every map $g \\colon X \\to L$ there is exactly one [[lattice homomorphism|BNVRR925X8]] $h \\colon F \\to L$ with $h \\circ \\iota = g$.\n\n## Remarks\n\nA free lattice on $X$ exists and is unique up to a unique isomorphism compatible with the maps from $X$; it is written $\\mathrm{FL}(X)$.\nOne construction takes the set of lattice [[terms|6D3WA7N2W4]] built from the letters of $X$ with $\\vee$ and $\\wedge$.\nIt identifies two terms $s$ and $t$ when the identity $s = t$ holds in every lattice.\nThe map $\\iota$ is injective, and $\\mathrm{FL}(X)$ is generated by $\\iota(X)$ as a lattice.\nExample: for $X = \\set{x, y}$, $\\mathrm{FL}(X) = \\set{x, y, x \\wedge y, x \\vee y}$ with $x \\wedge y \u003c x, y \u003c x \\vee y$.\nEvery lattice term in $x$ and $y$ reduces to one of these four by the lattice laws.\nThese four are distinct because they are distinct in the lattice of subsets of $\\set{1, 2}$ under $x \\mapsto \\set{1}$, $y \\mapsto \\set{2}$.\nNon-example: the two-element chain $\\set{x, y}$ with $x \u003c y$, with $\\iota$ the identity, is not free on $\\set{x, y}$.\nThe map $g$ with $g(x) = \\set{1}$ and $g(y) = \\set{2}$ into the subsets of $\\set{1, 2}$ has no extension $h$, since a lattice homomorphism is monotone and $\\set{1} \\not\\subseteq \\set{2}$.\nFor $\\abs{X} \\ge 3$, the free lattice $\\mathrm{FL}(X)$ is infinite.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "WP20S9NZ15",
    "kind": "definition",
    "msc": [
      "06B25",
      "08B20"
    ],
    "name": "Free lattice",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_lattice"
      }
    ],
    "tag": "FMCWXA0WT4",
    "uses": [
      "WH4SFAACAS",
      "BNVRR925X8",
      "6D3WA7N2W4"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FMCWXA0WT4/",
  "links": [
    {
      "field": "defines",
      "target": "WP20S9NZ15"
    },
    {
      "field": "uses",
      "target": "WH4SFAACAS"
    },
    {
      "field": "uses",
      "target": "BNVRR925X8"
    },
    {
      "field": "uses",
      "target": "6D3WA7N2W4"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FMCWXA0WT4.md",
  "msc": [
    "06B25",
    "08B20"
  ],
  "name": "Free lattice",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FMC/FMCWXA0WT4.md",
  "status": null,
  "tag": "FMCWXA0WT4",
  "uri": "m2://FMCWXA0WT4",
  "url": "https://tamnd.github.io/math2/wiki/free-lattice-definition/",
  "writing": {
    "tag": "FMCWXA0WT4",
    "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
      }
    ]
  }
}