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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature of function symbols and let $X$ be a set.\nAn *absolutely free algebra* of signature $\\sigma$ on $X$ is an algebra $F$ of signature $\\sigma$ with a map $\\eta \\colon X \\to F$ such that:\n\n1. $F$ is [[generated|2TQD8NMG9P]] by $\\eta(X)$.\n2. For every algebra $A$ of signature $\\sigma$ and every map $f \\colon X \\to A$, there is a [[homomorphism|KBXSZC6BV1]] $h \\colon F \\to A$ with $h \\circ \\eta = f$.\n\n## Remarks\n\nThis is a free algebra in the variety of all algebras of signature $\\sigma$, the variety defined by the empty set of identities.\nThe [[term algebra|3YYKGWJJND]] $T(X)$, whose elements are terms over $X$, is an absolutely free algebra on $X$.\nEvery absolutely free algebra on $X$ is isomorphic to it.\nLet $F$ be generated by $\\eta(X)$.\nThen $F$ is absolutely free on $X$ exactly when $\\eta$ and each basic operation are injective, and the images of different operations are disjoint and do not meet $\\eta(X)$.\nExample: for one unary symbol $s$ and $X = \\set{x}$, the algebra $\\N$ with $s(n) = n + 1$ and $\\eta(x) = 0$ is absolutely free; the extension of $f$ sends $n$ to $s^n(f(x))$.\nNon-example: $\\Z$ with $s(n) = n + 1$ and $\\eta(x) = 0$ is not absolutely free on $\\set{x}$, since it is not generated by $0$.\n",
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