{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\sigma$ be an algebraic signature and let $K$ be a class of $\\sigma$-algebras.\nThen $K$ is a [[quasivariety|WRXEKT4NQS]] if and only if $K$ is closed under the following four constructions:\n\n1. [[isomorphic|PZMGQMDN25]] copies;\n2. [[subalgebras|7CHG6SD365]];\n3. [[direct products|WJ73XYBNE4]] of arbitrary families, where the product of the empty family is the one-element algebra;\n4. ultraproducts.\n\n## Remarks\n\nThe forward direction is a check: a [[quasi-identity|94GVY3APPQ]] is preserved by subalgebras and by products, coordinate by coordinate, and by ultraproducts by Łoś's theorem.\nFor the converse, let $A$ satisfy every quasi-identity that holds in $K$, and let $a \\ne b$ in $A$.\nFor each finite set $\\Phi$ of equations true in $A$ among names of its elements, the quasi-identity \"$\\bigwedge \\Phi \\rightarrow a \\approx b$\" fails in $A$, so it fails in some $B_\\Phi \\in K$.\nAn ultraproduct of the algebras $B_\\Phi$ over a suitable ultrafilter gives a member of $K$ with a homomorphism from $A$ that separates $a$ and $b$.\nThe product of these maps over all pairs $a \\ne b$ embeds $A$ into a product of members of $K$, so $A \\in K$.\n\nThe same argument shows that the quasivariety generated by any class $K$ is $\\mathbf{ISPP_U}(K)$, the class of algebras isomorphic to subalgebras of products of ultraproducts of members of $K$.\nCompare [[Birkhoff's HSP theorem|5083DHSK82]], where closure under homomorphic images is added and identities replace quasi-identities.\n\nClosure under ultraproducts cannot be dropped.\nResidually finite groups are closed under the first three constructions, but not under ultraproducts, and they do not form a quasivariety.\n\n## Explanations\n\n### L1 human\n\nVarieties are the classes of algebras that you cannot leave by taking subalgebras, products or quotients.\nQuasivarieties are described in the same way, with two changes.\nQuotients are no longer allowed, and a new construction, the ultraproduct, is added.\nSo a property defined by conditional equations is exactly a property that survives these four constructions.\n\n### L2 human\n\nThe theorem lets one show that a class is a quasivariety without finding its axioms, and that a class is not a quasivariety by exhibiting a failure of closure.\nIn the theory of finite algebras, the quasivariety generated by one finite algebra $M$ is $\\mathbf{ISP}(M)$, because ultraproducts of copies of a finite algebra are isomorphic to it.\nThis class is the starting point of the theory of [[natural dualities|W5VG6HS8KW]].\nIn abstract algebraic logic, the algebraic counterparts of many deductive systems are quasivarieties, and the theorem is the tool to recognise them.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "WRXEKT4NQS"
    ],
    "kind": "statement",
    "msc": [
      "08C15",
      "03C52"
    ],
    "name": "Mal'cev's characterization of quasivarieties",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quasivariety"
      }
    ],
    "tag": "DFNM3BGMAY",
    "uses": [
      "WRXEKT4NQS",
      "94GVY3APPQ",
      "PZMGQMDN25",
      "WJ73XYBNE4"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/DFNM3BGMAY/",
  "links": [
    {
      "field": "about",
      "target": "WRXEKT4NQS"
    },
    {
      "field": "uses",
      "target": "WRXEKT4NQS"
    },
    {
      "field": "uses",
      "target": "94GVY3APPQ"
    },
    {
      "field": "uses",
      "target": "PZMGQMDN25"
    },
    {
      "field": "uses",
      "target": "WJ73XYBNE4"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DFNM3BGMAY.md",
  "msc": [
    "08C15",
    "03C52"
  ],
  "name": "Mal'cev's characterization of quasivarieties",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DFN/DFNM3BGMAY.md",
  "status": null,
  "tag": "DFNM3BGMAY",
  "uri": "m2://DFNM3BGMAY",
  "url": "https://tamnd.github.io/math2/wiki/malcevs-characterization-of-quasivarieties/",
  "writing": {
    "tag": "DFNM3BGMAY",
    "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
      }
    ]
  }
}