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  "body": "\n## Definition\n\nLet $(M, \\cdot)$ be a [[magma|9689SFPVE9]].\nLet $\\sim$ be the smallest [[congruence|DCSXXT4SER]] on $M$ such that $(x \\cdot y) \\cdot z \\sim x \\cdot (y \\cdot z)$ for all $x, y, z \\in M$.\nThe *associative quotient* of $M$ is the [[quotient magma|N5KE7FH6CX]] $M/{\\sim}$, with the quotient map $\\pi \\colon M \\to M/{\\sim}$, $\\pi(x) = [x]$.\n\n## Remarks\n\nThe quotient $M/{\\sim}$ is a [[semigroup|0XMEM406JM]], since $([x][y])[z] = [(xy)z] = [x(yz)] = [x]([y][z])$.\nUniversal property: for every semigroup $S$ and every magma homomorphism $f \\colon M \\to S$, there is exactly one homomorphism $\\bar f \\colon M/{\\sim} \\to S$ with $\\bar f \\circ \\pi = f$.\nThis holds because the kernel of $f$ is a congruence that contains all the pairs $((xy)z, x(yz))$, so it contains $\\sim$.\n\nExample: if $M$ is already a semigroup, then $\\sim$ is equality and the associative quotient is $M$ itself.\nExample: for the free magma on one letter $a$, the associative quotient is the free semigroup on $a$.\nIt is isomorphic to the positive integers under addition, with a bracketed word sent to its length.\n\nNon-example: the one-element semigroup is a quotient of every nonempty magma, but it is not the associative quotient of the free magma on $a$.\nThe length map to $(\\Z_{\u003e0}, +)$ is a homomorphism to a semigroup that does not factor through the one-element quotient, so the universal property fails.\n\nIn Mathlib this construction is `Magma.AssocQuotient`.\n",
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