---
tag: 1B3N9T68YV
kind: concept
name: Congruence-permutable variety
msc: [08B05, 08A30]
specializes: [AJNZJD8442]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence-permutable_algebra
---

## Summary

A congruence-permutable variety is a [[variety|AJNZJD8442]] in which any two congruences of any algebra permute under relational composition.
Groups, rings, modules and quasigroups form congruence-permutable varieties, while lattices and semilattices do not.
Congruence permutability makes the congruence lattices modular and gives strong control of direct and subdirect products.

## Explanations

### L1 human

A congruence of an algebra is a way to identify elements that respects the operations.
Two congruences permute when moving first by one and then by the other reaches the same pairs as doing it in the opposite order.
In a group, this says that $MN = NM$ for normal subgroups $M$ and $N$, which always holds.
A variety is congruence permutable when this happens in every one of its algebras.