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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] and let $\\mathbf{B}$ be a [[subalgebra|7CHG6SD365]] of $\\mathbf{A}$ with universe $B$.\nThe subalgebra $\\mathbf{B}$ is a *retract* of $\\mathbf{A}$ when there is a [[homomorphism|KBXSZC6BV1]] $r \\colon \\mathbf{A} \\to \\mathbf{B}$ with $r(b) = b$ for all $b \\in B$.\nSuch a map $r$ is called a *retraction*.\n\n## Remarks\n\nUp to isomorphism, an algebra $\\mathbf{C}$ is a retract of $\\mathbf{A}$ when there are homomorphisms $s \\colon \\mathbf{C} \\to \\mathbf{A}$ and $r \\colon \\mathbf{A} \\to \\mathbf{C}$ with $r \\circ s = \\id_C$.\nThen $s$ is injective and $s(\\mathbf{C})$ is a retract of $\\mathbf{A}$ in the sense above.\n\nExample: in the group $\\Z \\times \\Z$, the subgroup $\\Z \\times \\set{0}$ is a retract.\nThe map $r(m, n) = (m, 0)$ is a homomorphism onto $\\Z \\times \\set{0}$ and fixes each $(m, 0)$.\n\nNon-example: the subgroup $2\\Z$ of $\\Z$ is not a retract.\nA homomorphism $r \\colon \\Z \\to 2\\Z$ is determined by $x = r(1) \\in 2\\Z$, and then $r(2) = 2x$.\nThe condition $r(2) = 2$ forces $x = 1 \\notin 2\\Z$.\n",
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    "name": "Retract",
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