---
tag: 9GZ7F6ETC6
kind: concept
name: Congruence-distributive variety
msc: [08B10]
specializes: [AJNZJD8442]
sources:
  - url: https://en.wikipedia.org/wiki/Distributive_lattice
---

## Summary

A congruence-distributive variety is a [[variety|AJNZJD8442]] in which the [[congruence lattice|REE07CSXHR]] of every algebra is a [[distributive lattice|0HM0HQCK6N]].
Lattices, Boolean algebras and Heyting algebras form congruence-distributive varieties, while groups do not.
In such a variety, Jónsson's lemma locates all subdirectly irreducible algebras, which makes the variety much easier to describe.

## Explanations

### L1 human

The congruences of an algebra form a lattice, which records all the ways to collapse the algebra in a manner compatible with its operations.
In a congruence-distributive variety this lattice always obeys the distributive law, like the lattice of subsets of a set.
For lattices themselves this holds, so every lattice has a distributive lattice of congruences.
For groups it fails: the Klein four-group has three subgroups of order two, and they do not distribute.