{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "EZSWDV6ZZQ"
    }
  ],
  "body": "\n## Summary\n\nA noncommutative Gröbner basis, or Gröbner–Shirshov basis, generates a two-sided ideal of a [[free algebra|2P3Y2FRCSZ]] $k\\langle X \\rangle$.\nIts leading words control the leading words of all elements of the ideal.\nIt extends the commutative [[Gröbner basis|C83KPT47R0]] from polynomial rings to free algebras, with words in place of monomials.\nThe words that avoid all leading words give a linear basis of the quotient algebra.\nSo a Gröbner basis solves the normal form problem for an algebra given by generators and relations.\nUnlike the commutative case, a finitely generated ideal may have no finite Gröbner basis, and the completion procedure need not terminate.\n\n## Explanations\n\n### L1 human\n\nAn algebra given by generators and relations, such as $yx = xy$, raises the question of when two expressions are equal.\nThe idea is to orient each relation as a rewriting rule from a larger word to a smaller one, such as $yx \\to xy$.\nIf the rules are complete, every expression rewrites to a unique normal form, and two expressions are equal exactly when their normal forms agree.\nA Gröbner basis is such a complete set of rules.\nWith the single rule $yx \\to xy$ the normal forms are the words $x^a y^b$, which recovers the polynomial ring $k[x, y]$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16Z10",
      "16S15"
    ],
    "name": "Noncommutative Gröbner basis",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis"
      }
    ],
    "tag": "EPPR3DAFP6"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/EPPR3DAFP6/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/EPPR3DAFP6.md",
  "msc": [
    "16Z10",
    "16S15"
  ],
  "name": "Noncommutative Gröbner basis",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EPP/EPPR3DAFP6.md",
  "status": null,
  "tag": "EPPR3DAFP6",
  "uri": "m2://EPPR3DAFP6",
  "url": "https://tamnd.github.io/math2/wiki/noncommutative-grobner-basis/",
  "writing": {
    "tag": "EPPR3DAFP6",
    "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
      }
    ]
  }
}