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  "body": "\n## Definition\n\nLet $A$ be an algebra and let $\\alpha$, $\\beta$, $\\delta$ be [[congruences|DCSXXT4SER]] of $A$.\nThe congruence $\\alpha$ *centralizes* $\\beta$ *modulo* $\\delta$, written $C(\\alpha, \\beta; \\delta)$, if the following holds.\nFor every $n \\ge 1$, every $(n+1)$-ary [[term operation|N9M2N27END]] $t$ of $A$, every pair $(a, b) \\in \\alpha$ and all tuples $\\bar{c} = (c_1, \\dots, c_n)$ and $\\bar{d} = (d_1, \\dots, d_n)$ with $(c_i, d_i) \\in \\beta$ for each $i$:\n$$t(a, \\bar{c}) \\mathrel{\\delta} t(a, \\bar{d}) \\implies t(b, \\bar{c}) \\mathrel{\\delta} t(b, \\bar{d}).$$\nThe *commutator* $[\\alpha, \\beta]$ is the least congruence $\\delta$ of $A$ such that $C(\\alpha, \\beta; \\delta)$.\n\n## Remarks\n\nThe least such $\\delta$ exists: $C(\\alpha, \\beta; 1_A)$ holds for the full congruence $1_A$, and if $C(\\alpha, \\beta; \\delta_i)$ holds for every $i$, then $C(\\alpha, \\beta; \\bigcap_i \\delta_i)$ holds.\nThe definition is often called the term condition, and this commutator the term-condition commutator.\nExample: in a group $G$, let $\\alpha$ and $\\beta$ be the congruences of normal subgroups $M$ and $N$.\nThen $[\\alpha, \\beta]$ is the congruence of the commutator subgroup $[M, N]$.\nIn particular $[1_G, 1_G]$ is the congruence of the [[commutator subgroup|0ASGSTXSBW]] of $G$.\nNon-example of $C(1_G, 1_G; 0_G)$: let $G$ be a nonabelian group with $b c \\ne c b$, and let $t(x, y) = x y x^{-1} y^{-1}$.\nWith $a = e$, $\\bar{c} = c$ and $\\bar{d} = e$, both $t(e, c)$ and $t(e, e)$ equal $e$, but $t(b, c) = b c b^{-1} c^{-1} \\ne e = t(b, e)$.\nIn a congruence-modular variety the commutator is symmetric, $[\\alpha, \\beta] = [\\beta, \\alpha]$, and $[\\alpha, \\beta] \\le \\alpha \\wedge \\beta$.\n\n## Explanations\n\n### L1 human\n\nThe commutator of two congruences is the smallest congruence that makes them commute with each other inside all term operations.\n",
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