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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ and let $\\theta$ be a [[congruence|DCSXXT4SER]] on $\\mathbf{A}$.\nFor $a \\in A$, let $a/\\theta$ be the class of $a$, and let $A/\\theta$ be the [[quotient set|0011MTQMR6]] of all classes.\nThe *quotient algebra* $\\mathbf{A}/\\theta$ has universe $A/\\theta$ and the same signature as $\\mathbf{A}$.\nEach $n$-ary fundamental operation $f$ is interpreted by\n$$f^{\\mathbf{A}/\\theta}(a_1/\\theta, \\dots, a_n/\\theta) = f^{\\mathbf{A}}(a_1, \\dots, a_n)/\\theta.$$\n\n## Remarks\n\nThe operations are well defined because $\\theta$ is compatible with each $f$.\nThe map $\\nu_\\theta \\colon a \\mapsto a/\\theta$ is a surjective homomorphism $\\mathbf{A} \\to \\mathbf{A}/\\theta$, the natural map.\n\nExample: for the ring $\\Z$ and congruence modulo $n$, the quotient $\\Z/\\theta$ is the ring $\\Z/n\\Z$.\nFor instance $(3/\\theta) \\cdot (4/\\theta) = 12/\\theta = 2/\\theta$ when $n = 5$.\n\nNon-example: for an equivalence relation that is not compatible, the formula gives no operation.\nTake $(\\Z, +)$ with the classes \"negative\", \"zero\" and \"positive\".\nComputing with $1$ and $-1$ gives the class of $0$, but computing with $2$ and $-1$ gives the class of $1$.\n\nFor a group $G$ and the congruence of a normal subgroup $N$, the quotient algebra is the [[quotient group|0KYX2N69J0]] $G/N$.\n",
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