{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "ANRXAG227H"
    },
    {
      "field": "uses",
      "target": "C6GH4VHAMQ"
    },
    {
      "field": "uses",
      "target": "QREAJQH24A"
    }
  ],
  "body": "\n## Example\n\nThe *pentagon lattice* $N_5$ has five elements $0, a, b, c, 1$.\nIts order is generated by $0 \u003c a \u003c c \u003c 1$ and $0 \u003c b \u003c 1$, and $b$ is incomparable to $a$ and to $c$.\nEvery pair of elements has a least upper bound and a greatest lower bound, so $N_5$ is a [[lattice|WH4SFAACAS]].\nFor instance $a \\vee b = c \\vee b = 1$ and $a \\wedge b = c \\wedge b = 0$.\n\n$N_5$ is not a [[modular lattice|QX7WZ92QAK]].\nTake $a \\le c$ and $x = b$.\nThen $a \\vee (b \\wedge c) = a \\vee 0 = a$, while $(a \\vee b) \\wedge c = 1 \\wedge c = c \\ne a$.\n\n## Remarks\n\nBy [[Dedekind's characterization|ANRXAG227H]], a lattice is modular exactly when it has no sublattice isomorphic to $N_5$.\nSince every distributive lattice is modular, $N_5$ is also not distributive.\nThe Hasse diagram of $N_5$ is a pentagon, which gives the lattice its name.\n\n## Explanations\n\n### L1 human\n\nDraw a pentagon: a bottom, a top, a short side with one middle point $b$ and a long side with two points $a$ below $c$.\nGoing up from $a$ through $b$ reaches the top, and coming back down to the long side lands at $c$, not at $a$.\nThis mismatch is exactly the failure of the modular law.\nIt is the smallest lattice where this happens.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "WH4SFAACAS"
    ],
    "kind": "example",
    "msc": [
      "06C05",
      "06B05"
    ],
    "name": "Pentagon lattice N5",
    "not_instance_of": [
      {
        "note": "with a below c, the modular law fails for x = b",
        "target": "QX7WZ92QAK"
      }
    ],
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Modular_lattice"
      }
    ],
    "tag": "DHC8TZXGYS",
    "uses": [
      "WH4SFAACAS",
      "QX7WZ92QAK"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/DHC8TZXGYS/",
  "links": [
    {
      "field": "instance_of",
      "target": "WH4SFAACAS"
    },
    {
      "field": "not_instance_of",
      "target": "QX7WZ92QAK",
      "note": "note: with a below c, the modular law fails for x = b."
    },
    {
      "field": "uses",
      "target": "WH4SFAACAS"
    },
    {
      "field": "uses",
      "target": "QX7WZ92QAK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DHC8TZXGYS.md",
  "msc": [
    "06C05",
    "06B05"
  ],
  "name": "Pentagon lattice N5",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DHC/DHC8TZXGYS.md",
  "status": null,
  "tag": "DHC8TZXGYS",
  "uri": "m2://DHC8TZXGYS",
  "url": "https://tamnd.github.io/math2/wiki/pentagon-lattice-n5/",
  "writing": {
    "tag": "DHC8TZXGYS",
    "kind": "example",
    "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
      }
    ]
  }
}