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  "body": "\n## Summary\n\nA congruence-modular variety is a [[variety|AJNZJD8442]] in which the [[congruence lattice|REE07CSXHR]] of every algebra is a [[modular lattice|QX7WZ92QAK]].\nIt includes every congruence-permutable variety, such as groups, rings and modules, and every congruence-distributive variety, such as lattices.\nCongruence-modular varieties are the natural setting of commutator theory, which brings ideas of group theory to general algebras.\n\n## Explanations\n\n### L1 human\n\nThe modular law is a weak form of the distributive law, and it holds for the normal subgroups of every group.\nA variety is congruence modular when all its algebras have congruence lattices that obey this law.\nGroups and lattices both qualify, though for different reasons.\nSets with no operations do not: the equivalence relations on a four-element set do not form a modular lattice.\n",
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  "name": "Congruence-modular variety",
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