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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.\nA *congruence* on $\\mathbf{A}$ is a relation $\\theta \\subseteq A \\times A$ such that:\n\n1. $\\theta$ is an [[equivalence relation|0G8VEZ3WTK]] on $A$;\n2. for each fundamental operation $f$ of $\\mathbf{A}$, of arity $n$, and all $a_1, b_1, \\dots, a_n, b_n \\in A$ with $(a_i, b_i) \\in \\theta$ for each $i$, also $(f(a_1, \\dots, a_n), f(b_1, \\dots, b_n)) \\in \\theta$.\n\n## Remarks\n\nClause 2 is called the compatibility property.\nOne often writes $a \\mathrel{\\theta} b$ or $a \\equiv b \\pmod{\\theta}$ for $(a, b) \\in \\theta$.\nEvery algebra has the trivial congruences $\\Delta_A = \\set{(a, a) : a \\in A}$ and $\\nabla_A = A \\times A$.\n\nExample: on the ring $(\\Z, +, \\cdot, -, 0, 1)$, fix $n \\ge 1$ and let $a \\mathrel{\\theta} b$ when $n$ divides $a - b$.\nIf $n$ divides $a - a'$ and $b - b'$, it divides $(a + b) - (a' + b')$, $-a - (-a')$ and $ab - a'b' = a(b - b') + (a - a')b'$.\n\nNon-example: on the group $(\\Z, +, -, 0)$, let $a \\mathrel{\\theta} b$ when $a$ and $b$ are both positive, both negative, or both $0$.\nThis is an equivalence relation, but $1 \\mathrel{\\theta} 2$ while $1 + (-1) = 0$ and $2 + (-1) = 1$ are not related.\n\nIn a group, the congruences correspond to the normal subgroups via $\\theta \\mapsto \\set{a : a \\mathrel{\\theta} e}$.\nIn a ring, they correspond to the two-sided ideals via $\\theta \\mapsto \\set{a : a \\mathrel{\\theta} 0}$.\n",
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