{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "6AT64EQEJQ"
    },
    {
      "field": "uses",
      "target": "FMFDEE8WN2"
    }
  ],
  "body": "\n## Summary\n\nThe free product of a family of algebras in a [[variety|AJNZJD8442]] is the most general algebra of the variety generated by images of the members.\nIt is the coproduct in the category of algebras of the variety.\nThe [[free product of groups|TM7QS7P7X9]] and the free product of monoids are the best known cases.\n\n## Explanations\n\n### L1 human\n\nTo form a free product, put the algebras together and let their elements combine with each other only as the laws of the variety demand.\nNo other relation between elements of different factors is added.\nIn the variety of groups, the free product of two copies of $\\Z$ is the free group on two generators.\nIn the variety of abelian groups, the free product of $\\Z$ and $\\Z$ is $\\Z \\times \\Z$, because the laws of the variety force the two generators to commute.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08B25",
      "20E06"
    ],
    "name": "Free product of algebras",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_product"
      }
    ],
    "tag": "HHNWHS1XTN"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HHNWHS1XTN/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/HHNWHS1XTN.md",
  "msc": [
    "08B25",
    "20E06"
  ],
  "name": "Free product of algebras",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HHN/HHNWHS1XTN.md",
  "status": null,
  "tag": "HHNWHS1XTN",
  "uri": "m2://HHNWHS1XTN",
  "url": "https://tamnd.github.io/math2/wiki/free-product-of-algebras/",
  "writing": {
    "tag": "HHNWHS1XTN",
    "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
      }
    ]
  }
}