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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$, and let $\\theta$ and $\\psi$ be [[congruences|DCSXXT4SER]] on $\\mathbf{A}$.\nTheir relational product is\n$$\\theta \\circ \\psi = \\set{(a, c) \\in A \\times A : (a, b) \\in \\theta \\text{ and } (b, c) \\in \\psi \\text{ for some } b \\in A}.$$\nThe congruences $\\theta$ and $\\psi$ *permute* when $\\theta \\circ \\psi = \\psi \\circ \\theta$.\nThe algebra $\\mathbf{A}$ is *congruence-permutable* when every two congruences on $\\mathbf{A}$ permute.\n\n## Remarks\n\nThe congruences $\\theta$ and $\\psi$ permute if and only if $\\theta \\circ \\psi$ is a congruence.\nIn that case $\\theta \\circ \\psi = \\theta \\vee \\psi$, the join in the [[congruence lattice|REE07CSXHR]].\nA congruence-permutable algebra has a modular congruence lattice.\n\nExample: let $G$ be a group and $N$, $M$ normal subgroups, with congruences $a \\mathrel{\\theta} b \\iff a^{-1} b \\in N$ and $a \\mathrel{\\psi} b \\iff a^{-1} b \\in M$.\nThen $(a, c) \\in \\theta \\circ \\psi$ if and only if $a^{-1} c \\in NM$, and $(a, c) \\in \\psi \\circ \\theta$ if and only if $a^{-1} c \\in MN$.\nSince $NM = MN$ for normal subgroups, $\\theta$ and $\\psi$ permute.\n\nNon-example: on the set $\\set{1, 2, 3}$ with no operations, let $\\theta$ have classes $\\set{1, 2}, \\set{3}$ and $\\psi$ have classes $\\set{1}, \\set{2, 3}$.\nThen $(1, 3) \\in \\theta \\circ \\psi$ through $b = 2$.\nBut $(1, 3) \\notin \\psi \\circ \\theta$, since the only $b$ with $(1, b) \\in \\psi$ is $1$, and $(1, 3) \\notin \\theta$.\n",
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