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  "body": "\n## Summary\n\nThe free algebra of a [[variety|AJNZJD8442]] $\\mathcal{V}$ on a set $X$ is the algebra of $\\mathcal{V}$ generated by $X$ with as few relations as possible.\nThe only relations among the generators are those forced by the identities of $\\mathcal{V}$.\nIts defining property is that every map from $X$ to an algebra of $\\mathcal{V}$ extends to exactly one homomorphism.\nFree groups, free abelian groups, polynomial rings and free monoids are free algebras of their varieties.\nEvery algebra of $\\mathcal{V}$ is a quotient of a free algebra, which makes free algebras the starting point of presentations.\n\n## Explanations\n\n### L1 human\n\nTake some letters $x, y, z$ and form all expressions in them that the operations allow.\nDeclare two expressions equal only when the laws of the variety force them to be equal.\nIn commutative rings with identity, this gives the polynomial ring $\\Z[x, y, z]$: $xy$ equals $yx$, but $x^2$ does not equal $x$.\nBecause nothing extra was imposed, the letters can be sent anywhere, and the rest of the map is then determined.\n",
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