{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $x, y \\in P$.\nThe *join* of $x$ and $y$ is the [[supremum|0R3H5PANRQ]] of $\\set{x, y}$ in $P$, when it exists.\nSo it is an element $j \\in P$ with $x \\le j$ and $y \\le j$, such that $j \\le z$ for every $z \\in P$ with $x \\le z$ and $y \\le z$.\nIt is written $x \\vee y$.\nThe *meet* of $x$ and $y$ is the [[infimum|0QF1MZ9FRG]] of $\\set{x, y}$ in $P$, when it exists.\nIt is written $x \\wedge y$.\nMore generally, the join $\\bigvee S$ and the meet $\\bigwedge S$ of a subset $S \\subseteq P$ are its supremum and its infimum.\n\n## Remarks\n\nExample: in the power set of a set $X$, ordered by $\\subseteq$, the join of $A$ and $B$ is $A \\cup B$.\nIndeed $A \\cup B$ contains $A$ and $B$, and every set that contains $A$ and $B$ contains $A \\cup B$.\nDually, the meet is $A \\cap B$.\nNon-example: in the poset $\\set{a, b, c, d}$ with $a \u003c c$, $a \u003c d$, $b \u003c c$, $b \u003c d$ and no other strict relations, the elements $a$ and $b$ have no join.\nThey have two upper bounds, $c$ and $d$, and neither is below the other, so there is no least upper bound.\nWhen they exist, joins and meets are unique, and $x \\le y$ holds exactly when $x \\vee y = y$, and exactly when $x \\wedge y = x$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "J2T75ATRV5",
    "kind": "definition",
    "msc": [
      "06A06",
      "06B05"
    ],
    "name": "Join and meet",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Join_and_meet"
      }
    ],
    "tag": "F7YV441H6R",
    "uses": [
      "0QV2JS9RJ3",
      "0R3H5PANRQ",
      "0QF1MZ9FRG"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/F7YV441H6R/",
  "links": [
    {
      "field": "defines",
      "target": "J2T75ATRV5"
    },
    {
      "field": "uses",
      "target": "0QV2JS9RJ3"
    },
    {
      "field": "uses",
      "target": "0R3H5PANRQ"
    },
    {
      "field": "uses",
      "target": "0QF1MZ9FRG"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/F7YV441H6R.md",
  "msc": [
    "06A06",
    "06B05"
  ],
  "name": "Join and meet",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/F7Y/F7YV441H6R.md",
  "status": null,
  "tag": "F7YV441H6R",
  "uri": "m2://F7YV441H6R",
  "url": "https://tamnd.github.io/math2/wiki/join-and-meet-definition/",
  "writing": {
    "tag": "F7YV441H6R",
    "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
      }
    ]
  }
}