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  "body": "\n## Statement\n\nLet $A$ be a finite algebra of finite signature, and let $V(A)$ be the [[variety|AJNZJD8442]] generated by $A$.\nSuppose that $V(A)$ has a finite residual bound.\nThis means that there is a natural number $n$ such that every [[subdirectly irreducible algebra|DASF939439]] in $V(A)$ has fewer than $n$ elements.\nThen $A$ is [[finitely based|YGYZAQ12MZ]].\n\n## Remarks\n\nPark stated the conjecture in his doctoral dissertation at UCLA in 1976.\nIt is open in general.\nIt holds when $V(A)$ is [[congruence-distributive|9GZ7F6ETC6]], by [[Baker's finite basis theorem|35G4HQG2FN]], which in that case needs no residual bound.\nMcKenzie proved it in 1987 when $V(A)$ is [[congruence-modular|QAQAS3JWXW]].\nWillard proved it in 2000 when $V(A)$ is congruence meet-semidistributive.\nKearnes, Szendrei and Willard extended it to varieties with a difference term.\nA variety with a finite residual bound is in particular [[residually small|TH2XQGGFWH]].\n\n## Explanations\n\n### L1 human\n\nEvery algebra in the variety of $A$ is built from \"irreducible\" pieces.\nPark's conjecture says that if these pieces are all small, below a fixed size, then finitely many laws are enough to describe everything true in $A$.\nAll results so far agree with the conjecture, but nobody has proved it in general.\n",
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    "history": {
      "first_stated": "1976",
      "posed_by": "Robert Park"
    },
    "kind": "conjecture",
    "msc": [
      "08B05",
      "08B26"
    ],
    "name": "Park's conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
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    "tag": "1D541B576S",
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      "TH2XQGGFWH",
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      "QAQAS3JWXW"
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/1D541B576S/",
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  "markdown": "https://tamnd.github.io/math2/node/1D541B576S.md",
  "msc": [
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  "name": "Park's conjecture",
  "prov": null,
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/1D5/1D541B576S.md",
  "status": null,
  "tag": "1D541B576S",
  "uri": "m2://1D541B576S",
  "url": "https://tamnd.github.io/math2/wiki/parks-conjecture/",
  "writing": {
    "tag": "1D541B576S",
    "kind": "conjecture",
    "level": "W0",
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