{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "QBHNB4CP22"
    }
  ],
  "body": "\n## Summary\n\nA finitely generated variety is a [[variety|AJNZJD8442]] generated by finitely many finite algebras, or equivalently by one finite algebra.\nIt is the smallest variety that contains the given finite algebras.\nEvery finitely generated variety is a [[locally finite variety|8QPW65PZVW]], and such varieties are the main objects of the structure theory of finite algebras.\n\n## Explanations\n\n### L1 human\n\nStart with one finite algebra, such as the two-element Boolean algebra.\nTake all its subalgebras, all its direct powers, and all images of those under homomorphisms.\nThe class obtained is the variety generated by the algebra.\nFor the two-element Boolean algebra, this class is the variety of all Boolean algebras.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08B05",
      "08B26"
    ],
    "name": "Finitely generated variety",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
      }
    ],
    "specializes": [
      "AJNZJD8442"
    ],
    "tag": "NVQJWDJG2T"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NVQJWDJG2T/",
  "links": [
    {
      "field": "specializes",
      "target": "AJNZJD8442"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NVQJWDJG2T.md",
  "msc": [
    "08B05",
    "08B26"
  ],
  "name": "Finitely generated variety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NVQ/NVQJWDJG2T.md",
  "status": null,
  "tag": "NVQJWDJG2T",
  "uri": "m2://NVQJWDJG2T",
  "url": "https://tamnd.github.io/math2/wiki/finitely-generated-variety/",
  "writing": {
    "tag": "NVQJWDJG2T",
    "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
      }
    ]
  }
}