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  "body": "\n## Definition\n\nFix a signature and let $\\mathcal{K}$ be a class of algebras of this signature, closed under isomorphism.\nThe class $\\mathcal{K}$ has the *strong amalgamation property* if the following holds.\nFor all $A, B, C \\in \\mathcal{K}$ and all [[embeddings|SCRB74SWQN]] $f \\colon A \\to B$ and $g \\colon A \\to C$, there are $D \\in \\mathcal{K}$ and embeddings $f' \\colon B \\to D$ and $g' \\colon C \\to D$ such that:\n\n1. $f' \\circ f = g' \\circ g$;\n2. $f'(B) \\cap g'(C) = f'(f(A))$.\n\n## Remarks\n\nClause 1 alone is the [[amalgamation property|XSCSX39GEZ]], so the strong amalgamation property implies the amalgamation property.\nClause 2 says that an element of $B$ and an element of $C$ have the same image in $D$ only if both come from the same element of $A$.\n\nExample: the class of groups has the strong amalgamation property.\nTake $D = B *_A C$, the free product of $B$ and $C$ amalgamated over $A$.\nBy the normal form theorem for amalgamated free products, $B$ and $C$ embed in $D$ and their images meet exactly in the image of $A$.\n\nNon-example: the class of fields has the amalgamation property but fails clause 2.\nLet $A = \\Q$ and $B = C = \\Q(\\sqrt{2})$, with $f = g$ the inclusion.\nIn any field $D$ with embeddings $f', g'$ that agree on $\\Q$, both $f'(\\sqrt{2})$ and $g'(\\sqrt{2})$ are roots of $x^2 - 2$, so $g'(\\sqrt{2}) = \\pm f'(\\sqrt{2})$.\nHence $g'(\\sqrt{2})$ lies in $f'(B) \\cap g'(C)$ but not in the image of $\\Q$.\n",
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