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  "body": "\n## Example\n\nLet $X$ be a set.\nThe *free magma* $M(X)$ on $X$ is defined inductively.\nEvery $x \\in X$ is an element of $M(X)$, and if $u, v \\in M(X)$, then the formal product $(u v)$ is an element of $M(X)$.\nTwo elements are equal only when they are built in the same way.\nThe [[binary operation|0M418FE7BB]] is $u \\cdot v = (u v)$.\nSo $M(X)$ is a [[magma|9689SFPVE9]].\nIts elements are the fully parenthesised nonempty words in $X$, or equivalently the binary trees with leaves labelled by $X$.\n\nIt is an [[absolutely free algebra|TTTEV3NWWM]] of the signature with one binary operation.\nCheck: $M(X)$ is generated by $X$ by construction.\nGiven a magma $N$ and a map $f \\colon X \\to N$, define $h(x) = f(x)$ and $h((u v)) = h(u) \\cdot h(v)$ by recursion on the construction.\nThis is well defined because each element has exactly one construction, and it is the only homomorphism that extends $f$.\nSo $M(X)$ is the [[free algebra|GTS4DRRMZR]] on $X$ of the variety of all magmas.\n\nFor $X = \\set{x}$, the elements with three leaves are $((x x) x)$ and $(x (x x))$, and they are different.\nThe number of elements with $n$ leaves is the Catalan number $C_{n-1}$, the number of ways to bracket a product of $n$ factors.\nIn Mathlib the free magma is `FreeMagma`.\n\n## Explanations\n\n### L1 human\n\nThe free magma is what one gets by multiplying letters without any rule at all, not even associativity.\nEvery way of putting brackets around a product gives a different element.\nFor one letter $x$ and three factors there are two elements, $(xx)x$ and $x(xx)$; for four factors there are five.\nImposing associativity on it gives the free semigroup of nonempty words.\n",
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