{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "RNTAAHTD0M"
    },
    {
      "field": "specializes",
      "target": "WH4SFAACAS"
    }
  ],
  "body": "\n## Summary\n\nA join-semilattice is a [[partially ordered set|0QV2JS9RJ3]] in which every two elements have a [[join|J2T75ATRV5]].\nA meet-semilattice is one in which every two elements have a meet.\nAlgebraically, a semilattice is a [[semigroup|0XMEM406JM]] whose operation is commutative and idempotent.\nThe two views are equivalent, and a lattice is a set that is both a join- and a meet-semilattice for the same order.\nSemilattices appear in the theory of semigroups, in domain theory and in the analysis of programs.\n\n## Explanations\n\n### L1 human\n\nA join-semilattice is an ordering in which any two things have a smallest common upper bound, but not always a largest common lower bound.\nPicture a rooted tree drawn with its root at the top: any two nodes have a closest common ancestor above them, and this is their join.\nFor example, take two branches $a$ and $b$ that meet at a top element $1$.\nThen $a \\vee b = 1$, but $a$ and $b$ have no common lower bound at all.\nSo this is a join-semilattice but not a meet-semilattice.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06A12"
    ],
    "name": "Semilattice",
    "same_as": {
      "wikidata": "Q834585",
      "wikipedia": "Semilattice"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Semilattice"
      }
    ],
    "tag": "74X7JRG2J7"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/74X7JRG2J7/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/74X7JRG2J7.md",
  "msc": [
    "06A12"
  ],
  "name": "Semilattice",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/74X/74X7JRG2J7.md",
  "status": null,
  "tag": "74X7JRG2J7",
  "uri": "m2://74X7JRG2J7",
  "url": "https://tamnd.github.io/math2/wiki/semilattice/",
  "writing": {
    "tag": "74X7JRG2J7",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}