{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "KKJ9F6JJY0"
    }
  ],
  "body": "\n## Summary\n\nThe free distributive lattice on a set $S$ is the [[distributive lattice|0HM0HQCK6N]] generated by $S$ with no extra relations.\nThe only equalities in it are those that the lattice laws and distributivity force.\nEvery map from $S$ into a distributive lattice extends to exactly one lattice homomorphism from it.\nUnlike the [[free lattice|WP20S9NZ15]] on three or more generators, the free distributive lattice on a finite set is finite.\nOn $n$ generators it has $M(n) - 2$ elements, where $M(n)$ is the [[Dedekind number|481ZPQ0GC3]].\n\n## Explanations\n\n### L1 human\n\nStart with letters $x, y, z, \\ldots$ and build all expressions with \"and\" and \"or\".\nTwo expressions count as equal when the rules of a distributive lattice prove them equal, as in ordinary logic without negation and without constants.\nEvery expression can then be written as an \"or\" of \"and\"s of letters.\nWith two letters there are exactly four different elements: $x \\wedge y$, $x$, $y$ and $x \\vee y$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06B25",
      "06D05"
    ],
    "name": "Free distributive lattice",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_lattice"
      }
    ],
    "specializes": [
      "0HM0HQCK6N"
    ],
    "tag": "T5GK6DH8BR"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/T5GK6DH8BR/",
  "links": [
    {
      "field": "specializes",
      "target": "0HM0HQCK6N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/T5GK6DH8BR.md",
  "msc": [
    "06B25",
    "06D05"
  ],
  "name": "Free distributive lattice",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T5G/T5GK6DH8BR.md",
  "status": null,
  "tag": "T5GK6DH8BR",
  "uri": "m2://T5GK6DH8BR",
  "url": "https://tamnd.github.io/math2/wiki/free-distributive-lattice/",
  "writing": {
    "tag": "T5GK6DH8BR",
    "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
      }
    ]
  }
}