{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "NT8W7R2KDR"
    },
    {
      "field": "uses",
      "target": "NT8W7R2KDR"
    },
    {
      "field": "defines",
      "target": "WV3PBEK76J"
    }
  ],
  "body": "\n## Summary\n\nA pseudovariety is a class of finite algebras of one signature that is closed under homomorphic images, subalgebras and finite direct products.\nIt is the analogue of a [[variety|AJNZJD8442]] for finite algebras.\nPseudovarieties are not always defined by identities, but [[Reiterman's theorem|NT8W7R2KDR]] shows that they are defined by pseudoidentities.\nPseudovarieties of finite semigroups and monoids classify regular languages in algebraic automata theory.\n\n## Explanations\n\n### L1 human\n\nStart from some finite algebras and allow three moves: take a quotient, take a subalgebra, or take the product of finitely many.\nA class that is stable under these moves is a pseudovariety.\nFinite groups, finite abelian groups and finite nilpotent groups are examples.\nFinite groups regarded as semigroups also form a pseudovariety, yet no set of semigroup identities captures exactly the finite groups among finite semigroups.\nThe reason is that the infinite semigroup $(\\set{1, 2, 3, \\ldots}, +)$ lies in every variety of semigroups containing all finite groups, and it has finite quotients that are not groups.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08B99",
      "20M07"
    ],
    "name": "Pseudovariety",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
      }
    ],
    "tag": "XR4DPKX7T1"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/XR4DPKX7T1/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/XR4DPKX7T1.md",
  "msc": [
    "08B99",
    "20M07"
  ],
  "name": "Pseudovariety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XR4/XR4DPKX7T1.md",
  "status": null,
  "tag": "XR4DPKX7T1",
  "uri": "m2://XR4DPKX7T1",
  "url": "https://tamnd.github.io/math2/wiki/pseudovariety/",
  "writing": {
    "tag": "XR4DPKX7T1",
    "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
      }
    ]
  }
}