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  "body": "\n## Problem\n\nIs every [[distributive|0HM0HQCK6N]] [[algebraic lattice|3J0BTXSCVN]] isomorphic to the [[congruence lattice|REE07CSXHR]] of some [[lattice|WH4SFAACAS]]?\n\n## Remarks\n\nThe question is natural because the congruence lattice of every lattice is distributive and algebraic.\nAn equivalent form asks whether every distributive join-semilattice with a least element is isomorphic to the semilattice of compact [[lattice congruences|T782PJYVEZ]] of some lattice.\n\nThe answer is positive for finite lattices: every finite distributive lattice is isomorphic to the congruence lattice of a finite lattice.\nIt is also positive when the semilattice of compact elements has at most $\\aleph_1$ elements.\n\nThe problem was solved in the negative by Friedrich Wehrung, in a paper published in 2007.\nHe built a counterexample whose semilattice of compact elements has $\\aleph_{\\omega+1}$ elements.\nPavel Růžička later lowered the size of a counterexample to $\\aleph_2$, which is optimal by the positive result above.\n\n## Explanations\n\n### L1 human\n\nEvery lattice has a lattice of congruences, and this lattice always has two special properties: it is distributive and algebraic.\nDilworth asked whether these two properties are the whole story, that is, whether every lattice with them arises from some lattice.\nFor finite and for not too large lattices the answer is yes.\nFor very large ones it is no, which was proved after about sixty years.\n",
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  "front": {
    "about": [
      "REE07CSXHR"
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    "history": {
      "first_stated": "1940s",
      "posed_by": "Robert P. Dilworth"
    },
    "kind": "problem",
    "msc": [
      "06B15",
      "08A30",
      "06B10"
    ],
    "name": "Congruence lattice problem",
    "same_as": {
      "wikidata": "Q5160997",
      "wikipedia": "Congruence lattice problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Congruence_lattice_problem"
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    "tag": "WJ0D8QAMW0",
    "uses": [
      "REE07CSXHR",
      "3J0BTXSCVN",
      "0HM0HQCK6N",
      "WH4SFAACAS",
      "T782PJYVEZ"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/WJ0D8QAMW0/",
  "links": [
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      "field": "about",
      "target": "REE07CSXHR"
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      "target": "REE07CSXHR"
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      "field": "uses",
      "target": "3J0BTXSCVN"
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    {
      "field": "uses",
      "target": "0HM0HQCK6N"
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    {
      "field": "uses",
      "target": "WH4SFAACAS"
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    {
      "field": "uses",
      "target": "T782PJYVEZ"
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  ],
  "markdown": "https://tamnd.github.io/math2/node/WJ0D8QAMW0.md",
  "msc": [
    "06B15",
    "08A30",
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  "name": "Congruence lattice problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WJ0/WJ0D8QAMW0.md",
  "status": null,
  "tag": "WJ0D8QAMW0",
  "uri": "m2://WJ0D8QAMW0",
  "url": "https://tamnd.github.io/math2/wiki/congruence-lattice-problem/",
  "writing": {
    "tag": "WJ0D8QAMW0",
    "kind": "problem",
    "level": "W0",
    "next": [
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        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
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