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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $A$ be an algebra in $\\mathcal{V}$.\nLet $(B_i)_{i \\in I}$ be a family of algebras in $\\mathcal{V}$ with [[homomorphisms|KBXSZC6BV1]] $\\alpha_i \\colon A \\to B_i$.\nCall a family of homomorphisms $g_i \\colon B_i \\to D$ *compatible* if $g_i \\circ \\alpha_i = g_j \\circ \\alpha_j$ for all $i, j \\in I$.\nAn *amalgamated free product* of the $B_i$ over $A$ is an algebra $C$ in $\\mathcal{V}$ with homomorphisms $\\beta_i \\colon B_i \\to C$ such that:\n\n1. the family $(\\beta_i)_{i \\in I}$ is compatible;\n2. for every algebra $D$ in $\\mathcal{V}$ and every compatible family $g_i \\colon B_i \\to D$, there is a unique homomorphism $g \\colon C \\to D$ with $g \\circ \\beta_i = g_i$ for all $i$.\n\nIt is written $\\mathop{\\ast}_A B_i$, or $B_1 *_A B_2$ for two factors.\n\n## Remarks\n\nThe amalgamated free product exists in every variety and is unique up to a unique compatible isomorphism.\nIt is the quotient of the [[free product|HHNWHS1XTN]] of the $B_i$ by the smallest congruence that identifies the images of $\\alpha_i(a)$ and $\\alpha_j(a)$ for all $a \\in A$ and $i, j \\in I$.\nWith $A$ the trivial group in the variety of groups, it is the free product.\nUsually the $\\alpha_i$ are embeddings; whether the $\\beta_i$ are then embeddings depends on the variety, and for groups they always are.\n\nExample: in the variety of groups, $\\Z/4\\Z *_{\\Z/2\\Z} \\Z/6\\Z$, with $\\Z/2\\Z$ mapped to the subgroups of order $2$, is isomorphic to $SL_2(\\Z)$.\n\nNon-example: in groups, the direct product $\\Z \\times \\Z$ with the two coordinate inclusions is not $\\Z *_{\\set{0}} \\Z$.\nClause 2 fails for $D = S_3$ and $g_1, g_2$ sending $1$ to two non-commuting transpositions, since the generators of $\\Z \\times \\Z$ commute.\n\nIn Mathlib the construction for monoids and groups is `Monoid.PushoutI`.\n",
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