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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe variety $\\mathcal{V}$ is *congruence modular* if, for every algebra $A$ in $\\mathcal{V}$, the [[congruence lattice|REE07CSXHR]] $\\operatorname{Con} A$ is a [[modular lattice|QX7WZ92QAK]].\nThat is, for all congruences $\\alpha, \\beta, \\gamma$ of $A$ with $\\alpha \\le \\gamma$,\n$$\\alpha \\vee (\\beta \\wedge \\gamma) = (\\alpha \\vee \\beta) \\wedge \\gamma.$$\n\n## Remarks\n\nExample: the variety of groups is congruence modular, since the normal subgroups of a group form a modular lattice.\nMore generally, every congruence-permutable variety and every congruence-distributive variety is congruence modular.\nNon-example: the variety of sets, with no operations, is not congruence modular.\nThe congruences of a set are all its equivalence relations.\nOn $\\set{1, 2, 3, 4}$, let $\\alpha$ have the classes $\\set{1, 2}, \\set{3}, \\set{4}$, let $\\gamma$ have the classes $\\set{1, 2}, \\set{3, 4}$, and let $\\beta$ have the classes $\\set{1, 3}, \\set{2, 4}$.\nThen $\\alpha \\le \\gamma$, $\\beta \\wedge \\gamma$ is the identity relation and $\\alpha \\vee \\beta$ is the full relation.\nSo $\\alpha \\vee (\\beta \\wedge \\gamma) = \\alpha$, while $(\\alpha \\vee \\beta) \\wedge \\gamma = \\gamma \\ne \\alpha$.\n\n## Explanations\n\n### L1 human\n\nA variety is congruence modular when all its algebras have congruence lattices that satisfy the modular law.\n",
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