---
tag: QAQAS3JWXW
kind: concept
name: Congruence-modular variety
msc: [08B10]
specializes: [AJNZJD8442]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_lattice
---

## Summary

A congruence-modular variety is a [[variety|AJNZJD8442]] in which the [[congruence lattice|REE07CSXHR]] of every algebra is a [[modular lattice|QX7WZ92QAK]].
It includes every congruence-permutable variety, such as groups, rings and modules, and every congruence-distributive variety, such as lattices.
Congruence-modular varieties are the natural setting of commutator theory, which brings ideas of group theory to general algebras.

## Explanations

### L1 human

The modular law is a weak form of the distributive law, and it holds for the normal subgroups of every group.
A variety is congruence modular when all its algebras have congruence lattices that obey this law.
Groups and lattices both qualify, though for different reasons.
Sets with no operations do not: the equivalence relations on a four-element set do not form a modular lattice.