---
tag: DFZ7ZKRZCQ
kind: statement
role: theorem
name: Mal'cev's theorem on congruence permutability
msc: [08B05]
uses: [1B3N9T68YV, Y203PPTS61, GTS4DRRMZR, DCSXXT4SER]
about: [1B3N9T68YV, Y203PPTS61]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence-permutable_algebra
---

## Statement

Let $\mathcal{V}$ be a variety of algebras.
Then $\mathcal{V}$ is a [[congruence-permutable variety|1B3N9T68YV]] if and only if $\mathcal{V}$ has a [[Mal'cev term|Y203PPTS61]].
That is a ternary term $p$ such that $\mathcal{V}$ satisfies $p(x, y, y) = x$ and $p(x, x, y) = y$.

## Remarks

Suppose $p$ is a Mal'cev term, $A \in \mathcal{V}$, and $a \mathrel{\alpha} b \mathrel{\beta} c$ for [[congruences|DCSXXT4SER]] $\alpha$ and $\beta$ of $A$.
Put $d = p(a, b, c)$.
Then $d \mathrel{\beta} p(a, b, b) = a$ and $d \mathrel{\alpha} p(b, b, c) = c$, so $a \mathrel{\beta} d \mathrel{\alpha} c$, and the two compositions are equal.
Conversely, work in the [[free algebra|GTS4DRRMZR]] $F$ of $\mathcal{V}$ on $x, y, z$, with $\alpha$ generated by $(x, y)$ and $\beta$ generated by $(y, z)$.
Since $x \mathrel{\alpha} y \mathrel{\beta} z$, permutability gives $w = p(x, y, z) \in F$ with $x \mathrel{\beta} w \mathrel{\alpha} z$, and identifying $z$ with $y$, or $y$ with $x$, turns these relations into the two identities.

Example: groups have the Mal'cev term $x y^{-1} z$, so the congruences of every group permute; this is the familiar fact that $MN = NM$ for normal subgroups.
Lattices have no Mal'cev term, since the three-element chain has two congruences that do not permute.

## Explanations

### L1 human

The theorem says that a property of all congruences of all algebras in a variety is the same as the existence of one formula with two simple equations.
To show that a variety has permuting congruences, it is enough to find this formula.

### L2 human

This is the first Mal'cev condition, and it is the model for later ones, such as Jónsson's characterisation of congruence distributivity and Day's characterisation of congruence modularity.
The proof pattern is general: an implication in the free algebra on few generators gives terms, and terms give the property in every algebra.
Quasigroups, with $p(x, y, z) = (x / (y \backslash y)) \cdot (y \backslash z)$, also have a Mal'cev term, so their congruences permute.