{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "ES6P7WHJG9"
    },
    {
      "field": "uses",
      "target": "VT72G3BA99"
    },
    {
      "field": "defines",
      "target": "YAVH4RAEYJ"
    }
  ],
  "body": "\n## Summary\n\nThe lattice of subvarieties of a variety $\\mathcal{V}$ is the set of all [[subvarieties|4YSCK1DF26]] of $\\mathcal{V}$, ordered by inclusion.\nIt is a complete lattice, and it is dually isomorphic to the lattice of equational theories that contain the theory of $\\mathcal{V}$.\nIts shape records how $\\mathcal{V}$ can be cut down by further identities, and its study is a large part of the theory of varieties.\n\n## Explanations\n\n### L1 human\n\nPut all the subvarieties of a variety in a diagram, with smaller ones lower down.\nThe bottom is the trivial variety, of one-element algebras, and the top is the whole variety.\nFor abelian groups, the subvarieties other than the top are the classes defined by $nx = 0$ for $n \\ge 1$.\nThe class for $n$ lies inside the class for $m$ exactly when $n$ divides $m$.\nSo this lattice is the divisibility order on the positive integers, with an extra top element.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08B15"
    ],
    "name": "Lattice of subvarieties",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
      }
    ],
    "tag": "GJAYCVYZYF"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/GJAYCVYZYF/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/GJAYCVYZYF.md",
  "msc": [
    "08B15"
  ],
  "name": "Lattice of subvarieties",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GJA/GJAYCVYZYF.md",
  "status": null,
  "tag": "GJAYCVYZYF",
  "uri": "m2://GJAYCVYZYF",
  "url": "https://tamnd.github.io/math2/wiki/lattice-of-subvarieties/",
  "writing": {
    "tag": "GJAYCVYZYF",
    "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
      }
    ]
  }
}