---
tag: 6R0CA4JWWY
kind: concept
name: Mal'cev condition
msc: [08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence-permutable_algebra
---

## Summary

A Mal'cev condition is a property of a [[variety|AJNZJD8442]] that asserts the existence of terms satisfying certain identities.
Many properties of the congruence lattices of all algebras in a variety are equivalent to Mal'cev conditions.
Examples are congruence permutability, given by a [[Mal'cev term|Y203PPTS61]], congruence distributivity, given by [[Jónsson terms|3M8Y76E4AG]], and congruence modularity, given by [[Day terms|B0YKA8401M]].
Mal'cev conditions turn questions about all algebras of a variety into questions about finitely many equations.

## Explanations

### L1 human

Some properties of a whole class of algebras come from a single formula that works in all of them.
In groups, the formula $x y^{-1} z$ returns $x$ when $y = z$ and returns $z$ when $x = y$.
The existence of such a formula already forces every group to have well-behaved congruences.
A Mal'cev condition is a statement of this kind: there are formulas that satisfy given equations.