---
tag: GW78A05TNA
kind: concept
name: Arithmetical variety
msc: [08B05, 08B10]
specializes: [1B3N9T68YV, 9GZ7F6ETC6]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence-permutable_algebra
---

## Summary

An arithmetical variety is a [[variety|AJNZJD8442]] that is both a [[congruence-permutable variety|1B3N9T68YV]] and a [[congruence-distributive variety|9GZ7F6ETC6]].
The name comes from the ring of integers, whose ideals have both properties, as in the Chinese remainder theorem.
Boolean algebras and Heyting algebras form arithmetical varieties.
In an arithmetical variety a form of the Chinese remainder theorem holds for congruences of every algebra.

## Explanations

### L1 human

Congruences on the integers are given by moduli, and they combine very well: any two of them permute, and they obey the distributive law.
An arithmetical variety is one in which every algebra has congruences that behave this well.
Boolean algebras are the main example.
Groups have permuting congruences but not distributive ones, and lattices have the reverse, so neither is arithmetical.