{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be a finite algebra of a finite signature with only function symbols.\nThe algebra $A$ is *inherently nonfinitely based* if every [[locally finite variety|8QPW65PZVW]] that contains $A$ is not [[finitely based|YGYZAQ12MZ]].\n\n## Remarks\n\nThe [[variety|AJNZJD8442]] generated by a finite algebra is locally finite, so an inherently nonfinitely based finite algebra is not finitely based.\nLet $A$ be inherently nonfinitely based and let $B$ be a finite algebra whose variety contains $A$.\nThen $B$ is inherently nonfinitely based as well, since every locally finite variety containing $B$ contains $A$.\n\nExample: the six-element Brandt monoid $B_2^1$, as a semigroup, is inherently nonfinitely based, by a theorem of Sapir.\n\nNon-example: no finite group $G$ is inherently nonfinitely based.\nThe variety generated by $G$ is locally finite and contains $G$, and it is finitely based by the [[Oates–Powell theorem|5RFPT46DXX]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Z6TKYRZ9WX",
    "kind": "definition",
    "msc": [
      "08B05"
    ],
    "name": "Inherently nonfinitely based algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
      }
    ],
    "tag": "J9BZ6ZWGH6",
    "uses": [
      "YGYZAQ12MZ",
      "8QPW65PZVW",
      "AJNZJD8442",
      "5RFPT46DXX"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/J9BZ6ZWGH6/",
  "links": [
    {
      "field": "defines",
      "target": "Z6TKYRZ9WX"
    },
    {
      "field": "uses",
      "target": "YGYZAQ12MZ"
    },
    {
      "field": "uses",
      "target": "8QPW65PZVW"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "5RFPT46DXX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/J9BZ6ZWGH6.md",
  "msc": [
    "08B05"
  ],
  "name": "Inherently nonfinitely based algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J9B/J9BZ6ZWGH6.md",
  "status": null,
  "tag": "J9BZ6ZWGH6",
  "uri": "m2://J9BZ6ZWGH6",
  "url": "https://tamnd.github.io/math2/wiki/inherently-nonfinitely-based-algebra-definition/",
  "writing": {
    "tag": "J9BZ6ZWGH6",
    "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
      }
    ]
  }
}