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 modularity1).

1) [Bjarni Jónsson, On the representation of lattices, Math. Scand, 1, 1953, 193-206 MRreview