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  "body": "\n## Summary\n\nAn amalgamated free product glues several algebras of a [[variety|AJNZJD8442]] along a common algebra that maps into each of them.\nIt is the most general algebra of the variety that receives maps from all the pieces.\nIn it, the images of the common part are identified.\nIn category terms it is the colimit of the diagram formed by the maps out of the common algebra.\nFor two pieces it is the pushout.\nFor groups, amalgamated free products are central in combinatorial group theory and in the Seifert–van Kampen theorem.\n\n## Explanations\n\n### L1 human\n\nStart with two algebras $B$ and $C$ that both receive a map from a third algebra $A$.\nTo get the amalgamated free product, form the [[free product|HHNWHS1XTN]] of $B$ and $C$.\nThen force the two images of each element of $A$ to be equal.\nFor example, glue the cyclic groups $\\Z/4\\Z$ and $\\Z/6\\Z$ along their subgroups of order $2$.\nThe result is a group that contains both, in which the two elements of order $2$ coincide; it is isomorphic to $SL_2(\\Z)$.\n",
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