{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] and let $X$ be a set.\nA *free algebra of $\\mathcal{V}$ on $X$* is an algebra $F \\in \\mathcal{V}$ together with a map $\\eta \\colon X \\to F$ such that:\n\n1. $F$ is [[generated|2TQD8NMG9P]] by $\\eta(X)$.\n2. For every $A \\in \\mathcal{V}$ and every map $f \\colon X \\to A$, there is a [[homomorphism|KBXSZC6BV1]] $h \\colon F \\to A$ with $h \\circ \\eta = f$.\n\nIt is written $F_{\\mathcal{V}}(X)$, and $X$ is called a set of free generators.\n\n## Remarks\n\nClause 1 makes the homomorphism $h$ in clause 2 unique, since two homomorphisms that agree on a generating set are equal.\nA free algebra on $X$ is unique up to a unique isomorphism compatible with $\\eta$.\nIf $\\mathcal{V}$ contains an algebra with at least two elements, then $\\eta$ is injective, and $X$ is usually regarded as a subset of $F$.\nExample: in the variety of abelian groups, $F(X) = \\Z^{(X)}$, the group of finitely supported functions $X \\to \\Z$, with $\\eta(x)$ the indicator function of $x$.\nA map $f \\colon X \\to A$ extends by $h(c) = \\sum_{x} c(x) f(x)$, and this is a homomorphism.\nNon-example: $\\Z / 2\\Z$ with $\\eta(x) = 1$ is not free on $\\set{x}$ in abelian groups, since the map $x \\mapsto 1 \\in \\Z$ has no homomorphic extension $\\Z / 2\\Z \\to \\Z$.\nWhen the signature has no constants and empty algebras are excluded, one takes $X$ nonempty.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GTS4DRRMZR",
    "kind": "definition",
    "msc": [
      "08B20"
    ],
    "name": "Free algebra in a variety",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_object"
      }
    ],
    "tag": "351WM99D5W",
    "uses": [
      "AJNZJD8442",
      "KBXSZC6BV1",
      "2TQD8NMG9P"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/351WM99D5W/",
  "links": [
    {
      "field": "defines",
      "target": "GTS4DRRMZR"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "KBXSZC6BV1"
    },
    {
      "field": "uses",
      "target": "2TQD8NMG9P"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/351WM99D5W.md",
  "msc": [
    "08B20"
  ],
  "name": "Free algebra in a variety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/351/351WM99D5W.md",
  "status": null,
  "tag": "351WM99D5W",
  "uri": "m2://351WM99D5W",
  "url": "https://tamnd.github.io/math2/wiki/free-algebra-in-a-variety-definition/",
  "writing": {
    "tag": "351WM99D5W",
    "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
      }
    ]
  }
}