{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $P$ be a [[primal algebra|XD01Y7B6XW]] and let $V(P)$ be the [[variety|AJNZJD8442]] it generates.\nFor a [[Boolean algebra|0XADJKM699]] $B$ with [[Stone space|ZV38HA5PAQ]] $X$, let $P[B]^*$ be the set of continuous maps from $X$ to $P$ with the discrete topology.\nIt is an algebra under the pointwise operations, called a Boolean power of $P$.\nThen every algebra in $V(P)$ is isomorphic to $P[B]^*$ for some Boolean algebra $B$.\n\n## Remarks\n\nThe result is due to Alfred L. Foster.\nThe idea is that a primal algebra can express, for each element $a$, a term that tests equality with $a$, and a term that selects according to such tests.\nThese terms let one split any algebra of $V(P)$ along a Boolean algebra of \"decisions\" and rebuild it from copies of $P$.\nFor a finite $B$ with $k$ atoms, $X$ is a discrete set of $k$ points and $P[B]^* \\cong P^k$.\nSo the finite members of $V(P)$ are, up to isomorphism, the finite direct powers $P^k$.\nFor $P$ the two-element Boolean algebra, the theorem says that every Boolean algebra is isomorphic to the algebra of clopen subsets of its Stone space, which is Stone's representation theorem.\nPrimality is needed: for the two-element lattice, which is not primal, $V(P)$ contains the three-element chain, which is not a power of $P$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that all algebras built from a primal algebra $P$ are, in a precise sense, families of copies of $P$ indexed by the points of a Stone space.\nSo once one knows $P$ and Boolean algebras, one knows the whole variety.\nThis is why Boolean algebras are the simplest model of the general situation.\n\n### L2 human\n\nThe map $B \\mapsto P[B]^*$ gives an equivalence between the category of Boolean algebras and the variety $V(P)$.\nThis is the starting point of the theory of Boolean powers and of discriminator varieties.\nFor quasi-primal algebras, which are more general, a similar description uses Boolean products of subalgebras of $P$.\nFoster's theorem makes the structure of Post algebras of each order explicit, since they form the variety generated by a primal algebra.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "XD01Y7B6XW"
    ],
    "kind": "statement",
    "msc": [
      "08A40",
      "08B26",
      "06E15"
    ],
    "name": "Foster's theorem on primal algebras",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Functional_completeness"
      },
      {
        "book": "isbn:9780387905785"
      }
    ],
    "tag": "TCD8SZW95Z",
    "uses": [
      "XD01Y7B6XW",
      "AJNZJD8442",
      "0XADJKM699",
      "ZV38HA5PAQ"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/TCD8SZW95Z/",
  "links": [
    {
      "field": "about",
      "target": "XD01Y7B6XW"
    },
    {
      "field": "uses",
      "target": "XD01Y7B6XW"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "0XADJKM699"
    },
    {
      "field": "uses",
      "target": "ZV38HA5PAQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TCD8SZW95Z.md",
  "msc": [
    "08A40",
    "08B26",
    "06E15"
  ],
  "name": "Foster's theorem on primal algebras",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TCD/TCD8SZW95Z.md",
  "status": null,
  "tag": "TCD8SZW95Z",
  "uri": "m2://TCD8SZW95Z",
  "url": "https://tamnd.github.io/math2/wiki/fosters-theorem-on-primal-algebras/",
  "writing": {
    "tag": "TCD8SZW95Z",
    "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
      }
    ]
  }
}