## Congruence n-permutability

An algebra is ** congruence $n$-permutable** if for all congruence relations $\theta,\phi$ of the algebra
$\theta\circ\phi\circ\theta\circ\phi\circ...=\phi\circ\theta\circ\phi\circ\theta\circ...$, where $n$ congruences
appear on each side of the equation.

A class of algebras is ** congruence $n$-permutable** if each of its members is congruence $n$-permutable.

The term ** congruence permutable** is short for congruence $2$-permutable, i.e. $\theta\circ\phi=\phi\circ\theta$.

Congruence permutability holds for many 'classical' varieties such as groups, rings and vector spaces.

Congruence $n$-permutability is characterized by a Mal'cev condition.

For $n=2$, a variety is congruence permutable iff there exists a term $p(x,y,z)$ such that the identities $p(x,z,z)=x=p(z,z,x)$ hold in the variety.

#### Properties that imply congruence $n$-permutability

#### Properties implied by congruence $n$-permutability

Congruence $n$-permutability implies congruence $n+1$-permutability.

Congruence $3$-permutability implies congruence modularity^{1)}.

^{1)}[Bjarni Jónsson,

**, Math. Scand,**

*On the representation of lattices***1**, 1953, 193-206 MRreview

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