{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "72JSAPJB96"
    },
    {
      "field": "uses",
      "target": "QTTW954WCK"
    }
  ],
  "body": "\n## Summary\n\nA fully invariant congruence of an [[algebra|QJGPT2W0FT]] is a [[congruence|DCSXXT4SER]] that every endomorphism of the algebra maps into itself.\nOn a [[term algebra|3YYKGWJJND]], the fully invariant congruences are exactly the [[equational theories|F1M51B4MDG]].\nThese are the sets of identities closed under the rules of equational logic.\nThis is how equational theories become lattice-theoretic objects in universal algebra.\n\n## Explanations\n\n### L1 human\n\nA congruence says which elements count as equal.\nIt is fully invariant when this equality survives every way of mapping the algebra into itself by a homomorphism.\nIn a term algebra, an endomorphism is the same as a substitution of terms for variables.\nSo a fully invariant congruence there is a set of equations between terms that stays true after any substitution, which is exactly how identities like $x \\cdot y = y \\cdot x$ behave.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08A30",
      "08B05",
      "08A35"
    ],
    "name": "Fully invariant congruence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
      }
    ],
    "specializes": [
      "DCSXXT4SER"
    ],
    "tag": "QKGW3Z9BHW"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/QKGW3Z9BHW/",
  "links": [
    {
      "field": "specializes",
      "target": "DCSXXT4SER"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QKGW3Z9BHW.md",
  "msc": [
    "08A30",
    "08B05",
    "08A35"
  ],
  "name": "Fully invariant congruence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QKG/QKGW3Z9BHW.md",
  "status": null,
  "tag": "QKGW3Z9BHW",
  "uri": "m2://QKGW3Z9BHW",
  "url": "https://tamnd.github.io/math2/wiki/fully-invariant-congruence/",
  "writing": {
    "tag": "QKGW3Z9BHW",
    "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
      }
    ]
  }
}