{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "K7131908S7"
    }
  ],
  "body": "\n## Statement\n\nLet $K$ be a class of algebras of a fixed signature.\nLet $V(K)$ be the smallest class that contains $K$ and is closed under the [[class operators|S7MJVSP79N]] $H$, $S$ and $P$.\nThen\n$$V(K) = HSP(K).$$\nBy Birkhoff's HSP theorem, $HSP(K)$ is then the [[variety|AJNZJD8442]] generated by $K$.\n\n## Remarks\n\nEach of $H$, $S$ and $P$ maps $V(K)$ into itself, so $HSP(K) \\subseteq V(K)$.\nIt remains to show that $HSP(K)$ is closed under $H$, $S$ and $P$.\nThis follows from the rules $SH \\le HS$, $PS \\le SP$, $PH \\le HP$ and from the idempotence of each operator.\nFor example, $S\\,HSP \\le HS\\,SP = HSP$ and $P\\,HSP \\le HP\\,SP \\le HSP\\,P = HSP$.\nExample: for $K = \\set{\\Z}$ in the signature of groups, $HSP(K)$ is the class of all abelian groups.\nIndeed, every abelian group is a quotient of a free abelian group, and a free abelian group embeds in a power of $\\Z$.\n\n## Explanations\n\n### L1 human\n\nTo build the smallest variety containing some given algebras, it is enough to apply three steps once each.\nFirst take products, then subalgebras of those products, then homomorphic images.\nNo further rounds of the three constructions produce anything new.\n\n### L2 human\n\nThe formula gives the standard way to find the members of $V(A)$ for one algebra $A$.\nEach member of $V(A)$ is a homomorphic image of a subalgebra of a power $A^I$.\nIn particular, the free algebras of $V(A)$ are subalgebras of powers of $A$, which is the starting point of the study of free spectra.\nIn the congruence-distributive case, Jónsson's lemma sharpens the formula for the subdirectly irreducible members.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "AJNZJD8442",
      "S7MJVSP79N"
    ],
    "kind": "statement",
    "msc": [
      "08B99",
      "03C05"
    ],
    "name": "Tarski's HSP formula",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
      }
    ],
    "tag": "M4ZHB3GXBP",
    "uses": [
      "AJNZJD8442",
      "S7MJVSP79N"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/M4ZHB3GXBP/",
  "links": [
    {
      "field": "about",
      "target": "AJNZJD8442"
    },
    {
      "field": "about",
      "target": "S7MJVSP79N"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "S7MJVSP79N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/M4ZHB3GXBP.md",
  "msc": [
    "08B99",
    "03C05"
  ],
  "name": "Tarski's HSP formula",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M4Z/M4ZHB3GXBP.md",
  "status": null,
  "tag": "M4ZHB3GXBP",
  "uri": "m2://M4ZHB3GXBP",
  "url": "https://tamnd.github.io/math2/wiki/tarskis-hsp-formula/",
  "writing": {
    "tag": "M4ZHB3GXBP",
    "kind": "statement",
    "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
      }
    ]
  }
}