{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\alpha$ be an [[ordinal|0YSCSX5YZC]].\nA *cylindric algebra of dimension $\\alpha$* is a structure $(A, +, \\cdot, -, 0, 1, c_\\kappa, d_{\\kappa\\lambda})_{\\kappa, \\lambda \u003c \\alpha}$ with the following properties.\nThe reduct $(A, +, \\cdot, -, 0, 1)$ is a [[Boolean algebra|0XADJKM699]], each $c_\\kappa$ is a unary operation on $A$, and each $d_{\\kappa\\lambda}$ is an element of $A$.\nFor all $x, y \\in A$ and $\\kappa, \\lambda, \\mu \u003c \\alpha$:\n\n1. $c_\\kappa 0 = 0$.\n2. $x \\le c_\\kappa x$.\n3. $c_\\kappa (x \\cdot c_\\kappa y) = c_\\kappa x \\cdot c_\\kappa y$.\n4. $c_\\kappa c_\\lambda x = c_\\lambda c_\\kappa x$.\n5. $d_{\\kappa\\kappa} = 1$.\n6. If $\\kappa \\ne \\lambda$ and $\\kappa \\ne \\mu$, then $d_{\\lambda\\mu} = c_\\kappa (d_{\\lambda\\kappa} \\cdot d_{\\kappa\\mu})$.\n7. If $\\kappa \\ne \\lambda$, then $c_\\kappa (d_{\\kappa\\lambda} \\cdot x) \\cdot c_\\kappa (d_{\\kappa\\lambda} \\cdot -x) = 0$.\n\nHere $\\le$ is the order of the Boolean algebra.\n\n## Remarks\n\nExample: let $U$ be a nonempty set and let $A$ be the set of all subsets of $U^\\alpha$, with the set operations.\nPut $c_\\kappa X = \\set{y \\in U^\\alpha : y \\text{ agrees with some } x \\in X \\text{ outside the coordinate } \\kappa}$ and $d_{\\kappa\\lambda} = \\set{x \\in U^\\alpha : x_\\kappa = x_\\lambda}$.\nFor clause 7, a point $y$ in both sets would agree outside $\\kappa$ with some $x \\in X$ and some $x' \\notin X$, both with $\\kappa$-th coordinate equal to $y_\\lambda$; then $x = x'$, a contradiction.\nThe other clauses are checked in the same way, and this is a cylindric set algebra.\n\nNon-example: on a Boolean algebra with $0 \\ne 1$, taking $c_\\kappa x = 0$ for all $x$ fails clause 2 at $x = 1$.\n\nA cylindric algebra is called representable if it is isomorphic to a subalgebra of a product of cylindric set algebras.\nFor dimension at least $3$, not every cylindric algebra is representable.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "1MDT9BF81H",
    "kind": "definition",
    "msc": [
      "03G15"
    ],
    "name": "Cylindric algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cylindric_algebra"
      }
    ],
    "tag": "RKCSWWVYTC",
    "uses": [
      "0XADJKM699",
      "0YSCSX5YZC"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RKCSWWVYTC/",
  "links": [
    {
      "field": "defines",
      "target": "1MDT9BF81H"
    },
    {
      "field": "uses",
      "target": "0XADJKM699"
    },
    {
      "field": "uses",
      "target": "0YSCSX5YZC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RKCSWWVYTC.md",
  "msc": [
    "03G15"
  ],
  "name": "Cylindric algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RKC/RKCSWWVYTC.md",
  "status": null,
  "tag": "RKCSWWVYTC",
  "uri": "m2://RKCSWWVYTC",
  "url": "https://math2.tamnd.com/wiki/cylindric-algebra-definition/",
  "writing": {
    "tag": "RKCSWWVYTC",
    "kind": "definition",
    "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
      }
    ]
  }
}