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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe variety $\\mathcal{V}$ is *congruence permutable* if every algebra $A$ in $\\mathcal{V}$ has [[permuting congruences|2BCES6NPYM]].\nThat is, for all [[congruences|DCSXXT4SER]] $\\alpha$ and $\\beta$ of $A$,\n$$\\alpha \\circ \\beta = \\beta \\circ \\alpha,$$\nwhere $\\circ$ is the [[composition of relations|0PYSR20HGF]].\n\n## Remarks\n\nExample: the variety of groups is congruence permutable.\nCongruences of a group $G$ correspond to normal subgroups, and the composition of the congruences of $M$ and $N$ is the congruence relation $a^{-1} b \\in MN$.\nSince $MN = NM$ for normal subgroups, the two compositions are equal.\nNon-example: the variety of lattices is not congruence permutable.\nIn the chain $0 \u003c 1 \u003c 2$, let $\\alpha$ identify $0$ and $1$, and let $\\beta$ identify $1$ and $2$.\nThe pair $(0, 2)$ is in one composition, through $0 \\mathrel{\\alpha} 1 \\mathrel{\\beta} 2$.\nIt is not in the other, because the only element $\\beta$-related to $0$ is $0$, and $0$ is not $\\alpha$-related to $2$.\nFor a single algebra, the congruences $\\alpha$ and $\\beta$ permute if and only if $\\alpha \\circ \\beta$ is the join $\\alpha \\vee \\beta$ in the congruence lattice.\n\n## Explanations\n\n### L1 human\n\nIn a congruence-permutable variety, two ways of identifying elements can be applied in either order with the same result.\n",
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