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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe variety $\\mathcal{V}$ is *congruence distributive* if, for every algebra $A$ in $\\mathcal{V}$, the [[congruence lattice|REE07CSXHR]] $\\operatorname{Con} A$ is a [[distributive lattice|0HM0HQCK6N]].\nThat is, for all congruences $\\alpha, \\beta, \\gamma$ of $A$,\n$$\\alpha \\wedge (\\beta \\vee \\gamma) = (\\alpha \\wedge \\beta) \\vee (\\alpha \\wedge \\gamma).$$\n\n## Remarks\n\nExample: the variety of lattices is congruence distributive.\nEvery lattice $L$ has the majority term $m(x, y, z) = (x \\wedge y) \\vee (y \\wedge z) \\vee (x \\wedge z)$, which satisfies $m(x, x, y) = m(x, y, x) = m(y, x, x) = x$, and a variety with a majority term is congruence distributive.\nNon-example: the variety of groups is not congruence distributive.\nIn the Klein four-group $V = \\Z/2\\Z \\times \\Z/2\\Z$, every subgroup is normal, so $\\operatorname{Con} V$ is isomorphic to the lattice of subgroups.\nFor three distinct subgroups $A, B, C$ of order $2$, $A \\wedge (B \\vee C) = A \\wedge V = A$, while $(A \\wedge B) \\vee (A \\wedge C)$ is the trivial subgroup.\nEvery congruence-distributive variety is congruence modular.\n\n## Explanations\n\n### L1 human\n\nA variety is congruence distributive when all its algebras have congruence lattices that satisfy the distributive law.\n",
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