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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature and let $\\mathbf{A}$ and $\\mathbf{B}$ be $\\sigma$-[[structures|983GZBQYT4]] with domains $A$ and $B$.\nAn *embedding* of $\\mathbf{A}$ into $\\mathbf{B}$ is a map $h \\colon A \\to B$ such that:\n\n1. $h$ is a [[homomorphism|KBXSZC6BV1]] from $\\mathbf{A}$ to $\\mathbf{B}$;\n2. $h$ is [[injective|09NWETVTNY]];\n3. for each relation symbol $R \\in \\sigma$ of arity $n$ and all $a_1, \\dots, a_n \\in A$, if $(h(a_1), \\dots, h(a_n)) \\in R^{\\mathbf{B}}$ then $(a_1, \\dots, a_n) \\in R^{\\mathbf{A}}$.\n\n## Remarks\n\nA map $h$ is an embedding if and only if its image is the domain of a [[substructure|676H7EGTKJ]] $\\mathbf{C}$ of $\\mathbf{B}$ and $h$ is an [[isomorphism|PZMGQMDN25]] from $\\mathbf{A}$ to $\\mathbf{C}$.\nFor a signature without relation symbols, clause 3 is empty, and an embedding is an injective homomorphism.\n\nExample: for the signature $\\set{+, \\cdot, 0, 1}$, the inclusion $\\Z \\to \\Q$ is an embedding, because it is injective and respects sums, products, $0$ and $1$.\n\nNon-example: view graphs as structures with one binary relation.\nLet $A$ be the graph with vertices $1, 2$ and no edge, and $B$ the graph with vertices $1, 2$ and one edge between them.\nThe identity map is an injective homomorphism from $A$ to $B$ but not an embedding, because clause 3 fails for the pair $(1, 2)$.\n",
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