{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3TNVVAA1AG"
    },
    {
      "field": "uses",
      "target": "9D26WKTFPT"
    },
    {
      "field": "uses",
      "target": "9EE43J2W16"
    },
    {
      "field": "uses",
      "target": "F4BVTGH58G"
    },
    {
      "field": "uses",
      "target": "HSYGHAY6WE"
    },
    {
      "field": "uses",
      "target": "K2KQFX4GVF"
    },
    {
      "field": "uses",
      "target": "Q22KS9BBT6"
    },
    {
      "field": "uses",
      "target": "TAVG968272"
    },
    {
      "field": "uses",
      "target": "TMBCW25DZ4"
    },
    {
      "field": "uses",
      "target": "WH3Z5M5D2R"
    },
    {
      "field": "uses",
      "target": "XAQSNBJSX4"
    },
    {
      "field": "uses",
      "target": "XQXBBDD5XB"
    },
    {
      "field": "defines",
      "target": "8F933HJ1RY"
    }
  ],
  "body": "\n## Summary\n\nA directed set is a nonempty [[preordered set|0VABYWJWH9]] in which every two elements have a common upper bound.\nDirected sets are the index sets of [[nets|4DVJQJV0DG]], which generalize sequences in topology, and of direct limits in algebra.\nThey also underlie directed-complete partial orders in domain theory.\n\n## Explanations\n\n### L1 human\n\nIn a directed set, any two elements can be \"caught up with\" by a third that is above both.\nSo the set has a sense of going further and further in one direction, even if it is not a line.\nThe natural numbers are directed, and so are the finite subsets of a set ordered by inclusion: $A \\cup B$ is above both $A$ and $B$.\nA set with two incomparable elements and nothing else is not directed.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06A06",
      "54A20"
    ],
    "name": "Directed set",
    "same_as": {
      "wikidata": "Q1513048",
      "wikipedia": "Directed set"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Directed_set"
      }
    ],
    "specializes": [
      "0VABYWJWH9"
    ],
    "tag": "D9RY4667K6"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/D9RY4667K6/",
  "links": [
    {
      "field": "specializes",
      "target": "0VABYWJWH9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/D9RY4667K6.md",
  "msc": [
    "06A06",
    "54A20"
  ],
  "name": "Directed set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D9R/D9RY4667K6.md",
  "status": null,
  "tag": "D9RY4667K6",
  "uri": "m2://D9RY4667K6",
  "url": "https://tamnd.github.io/math2/wiki/directed-set/",
  "writing": {
    "tag": "D9RY4667K6",
    "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
      }
    ]
  }
}