{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $(A_i)_{i \\in I}$ be a family of algebras in $\\mathcal{V}$.\nA *free product* of the family in $\\mathcal{V}$ is an algebra $A$ in $\\mathcal{V}$ with [[homomorphisms|KBXSZC6BV1]] $\\iota_i \\colon A_i \\to A$ such that the following holds.\nFor every algebra $B$ in $\\mathcal{V}$ and all homomorphisms $f_i \\colon A_i \\to B$, there is a unique homomorphism $f \\colon A \\to B$ with $f \\circ \\iota_i = f_i$ for all $i \\in I$.\n\n## Remarks\n\nThe free product exists for every family in a variety, and it is unique up to a unique isomorphism compatible with the maps $\\iota_i$.\nIt is the coproduct of the family in the category of algebras of $\\mathcal{V}$.\nSome authors require in addition that every $\\iota_i$ be injective.\n\nExample: in the variety of abelian groups, $A_1 \\times A_2$ with $\\iota_1(a) = (a, 0)$ and $\\iota_2(b) = (0, b)$ is a free product.\nGiven $f_1, f_2$ into an abelian group $B$, the map $f(a, b) = f_1(a) + f_2(b)$ is a homomorphism because $B$ is abelian, it satisfies $f \\circ \\iota_k = f_k$, and it is the only such map since $(a, b) = \\iota_1(a) + \\iota_2(b)$.\n\nNon-example: in the variety of all groups, $\\Z \\times \\Z$ with the two coordinate inclusions is not a free product of $\\Z$ and $\\Z$.\nTake $f_1, f_2 \\colon \\Z \\to S_3$ sending $1$ to the transpositions $(1\\,2)$ and $(2\\,3)$.\nA homomorphism $f$ with $f \\circ \\iota_k = f_k$ would send the commuting elements $(1, 0)$ and $(0, 1)$ to these two transpositions, which do not commute.\n\nIn the variety of groups, the free product of a family is the group of reduced words in the factors.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "HHNWHS1XTN",
    "kind": "definition",
    "msc": [
      "08B25",
      "20E06"
    ],
    "name": "Free product of algebras",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_product"
      }
    ],
    "tag": "6AT64EQEJQ",
    "uses": [
      "AJNZJD8442",
      "KBXSZC6BV1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/6AT64EQEJQ/",
  "links": [
    {
      "field": "defines",
      "target": "HHNWHS1XTN"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "KBXSZC6BV1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6AT64EQEJQ.md",
  "msc": [
    "08B25",
    "20E06"
  ],
  "name": "Free product of algebras",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6AT/6AT64EQEJQ.md",
  "status": null,
  "tag": "6AT64EQEJQ",
  "uri": "m2://6AT64EQEJQ",
  "url": "https://tamnd.github.io/math2/wiki/free-product-of-algebras-definition/",
  "writing": {
    "tag": "6AT64EQEJQ",
    "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
      }
    ]
  }
}