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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ and let $R \\subseteq A \\times A$.\nThe *congruence generated by* $R$ is\n$$\\operatorname{Cg}(R) = \\bigcap \\set{\\theta : \\theta \\text{ is a congruence on } \\mathbf{A} \\text{ and } R \\subseteq \\theta}.$$\nFor $a, b \\in A$ one writes $\\operatorname{Cg}(a, b)$ for $\\operatorname{Cg}(\\set{(a, b)})$, the principal congruence of $a$ and $b$.\n\n## Remarks\n\nThe family in the intersection contains $A \\times A$, so it is nonempty.\nAn intersection of [[congruences|DCSXXT4SER]] is a congruence, so $\\operatorname{Cg}(R)$ is the least congruence containing $R$.\n\nA description from below: $(a, b) \\in \\operatorname{Cg}(R)$ if and only if there are elements $a = z_0, z_1, \\dots, z_n = b$ with the following property.\nFor each $i \u003c n$ there are $(c_i, d_i) \\in R$ and a unary [[polynomial operation|XP67WCPQ59]] $p_i$ with $\\set{z_i, z_{i+1}} = \\set{p_i(c_i), p_i(d_i)}$.\n\nExample: in the group $(\\Z, +, -, 0)$, $\\operatorname{Cg}(0, 6)$ is congruence modulo $6$.\nCongruence modulo $6$ contains $(0, 6)$, and every congruence $\\theta$ containing $(0, 6)$ contains $(a, a + 6k)$ for all $a, k \\in \\Z$ by compatibility with $+$ and $-$.\n\nNon-example: the equivalence relation with classes $\\set{0, 6}$ and singletons elsewhere contains $(0, 6)$ but is not $\\operatorname{Cg}(0, 6)$.\nIt is not a congruence, since $0 + 1 = 1$ and $6 + 1 = 7$ are not related.\n",
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  "msc": [
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