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  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]], and let $\\mathcal{K}$ be a class of [[$L$-structures|983GZBQYT4]] closed under isomorphism.\nAn embedding is an injective map between $L$-structures that preserves and reflects all functions, relations and constants.\nThe class $\\mathcal{K}$ has the *amalgamation property* if the following holds.\nFor all $A, B, C \\in \\mathcal{K}$ and all embeddings $f \\colon A \\to B$ and $g \\colon A \\to C$, there are $D \\in \\mathcal{K}$ and embeddings $f' \\colon B \\to D$ and $g' \\colon C \\to D$ with $f' \\circ f = g' \\circ g$.\n\n## Remarks\n\nThe structure $D$ is called an amalgam of $B$ and $C$ over $A$.\n\nExample: the class of finite [[graphs|00C0BR7D7H]] has the amalgamation property.\nGiven $f$ and $g$, take the disjoint union of $B$ and $C$, identify $f(a)$ with $g(a)$ for each $a \\in A$, and add no new edges.\nTwo vertices of $B$ are adjacent in the result exactly when they are adjacent in $B$, and the same holds for $C$.\nSo the inclusions of $B$ and $C$ are embeddings, and they agree on $A$.\n\nNon-example: the class of graphs with at most $2$ vertices fails the condition.\nLet $A$ be one vertex $a$, let $B$ be an edge $\\set{a, b}$, and let $C$ be two non-adjacent vertices $a, c$.\nIn any amalgam $D$, the images of $b$ and $c$ are distinct, since one is adjacent to the image of $a$ and the other is not.\nSo $D$ has at least $3$ vertices and is not in the class.\n\nWhen $A$ may be the empty structure, the amalgamation property implies the joint embedding property: any two members of $\\mathcal{K}$ embed into a common member.\n",
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