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  "body": "\n## Summary\n\nA congruence-distributive variety is a [[variety|AJNZJD8442]] in which the [[congruence lattice|REE07CSXHR]] of every algebra is a [[distributive lattice|0HM0HQCK6N]].\nLattices, Boolean algebras and Heyting algebras form congruence-distributive varieties, while groups do not.\nIn such a variety, Jónsson's lemma locates all subdirectly irreducible algebras, which makes the variety much easier to describe.\n\n## Explanations\n\n### L1 human\n\nThe congruences of an algebra form a lattice, which records all the ways to collapse the algebra in a manner compatible with its operations.\nIn a congruence-distributive variety this lattice always obeys the distributive law, like the lattice of subsets of a set.\nFor lattices themselves this holds, so every lattice has a distributive lattice of congruences.\nFor groups it fails: the Klein four-group has three subgroups of order two, and they do not distribute.\n",
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