{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(G, \\le)$ be an [[ordered group|A49BATWQEH]].\nIt is a *lattice-ordered group*, or *$\\ell$-group*, if the partially ordered set $(G, \\le)$ is a [[lattice|WH4SFAACAS]].\nThat is, any two elements $a, b \\in G$ have a least upper bound $a \\vee b$ and a greatest lower bound $a \\wedge b$ in $G$.\n\n## Remarks\n\nExample: $\\Z^2$ under addition with the componentwise order is an $\\ell$-group.\nHere $(a, b) \\vee (c, d) = (\\max(a, c), \\max(b, d))$ and $(a, b) \\wedge (c, d) = (\\min(a, c), \\min(b, d))$.\nNon-example: on $\\Z^2$, let $(a, b) \\le (c, d)$ when $a \\le c$ and $b = d$.\nThis makes $\\Z^2$ an ordered group, but $(0, 0)$ and $(0, 1)$ have no common upper bound, so it is not a lattice.\nIn an $\\ell$-group, multiplication distributes over joins and meets: $c(a \\vee b) = ca \\vee cb$ and $(a \\vee b)c = ac \\vee bc$.\nThe lattice of an $\\ell$-group is distributive.\nEvery $\\ell$-group is a [[torsion-free group|VQ4SSKS53D]].\nFor $a \\in G$ one writes $a^+ = a \\vee e$, $a^- = a^{-1} \\vee e$ and $\\abs{a} = a \\vee a^{-1}$, and then $a = a^+ (a^-)^{-1}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "J0DHYQ0MDE",
    "kind": "definition",
    "msc": [
      "06F15",
      "06F20"
    ],
    "name": "Lattice-ordered group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Partially_ordered_group"
      }
    ],
    "tag": "G5AN43WE9D",
    "uses": [
      "A49BATWQEH",
      "WH4SFAACAS",
      "VQ4SSKS53D"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/G5AN43WE9D/",
  "links": [
    {
      "field": "defines",
      "target": "J0DHYQ0MDE"
    },
    {
      "field": "uses",
      "target": "A49BATWQEH"
    },
    {
      "field": "uses",
      "target": "WH4SFAACAS"
    },
    {
      "field": "uses",
      "target": "VQ4SSKS53D"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/G5AN43WE9D.md",
  "msc": [
    "06F15",
    "06F20"
  ],
  "name": "Lattice-ordered group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G5A/G5AN43WE9D.md",
  "status": null,
  "tag": "G5AN43WE9D",
  "uri": "m2://G5AN43WE9D",
  "url": "https://tamnd.github.io/math2/wiki/lattice-ordered-group-definition/",
  "writing": {
    "tag": "G5AN43WE9D",
    "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
      }
    ]
  }
}