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.
Congruence $n$-permutability implies congruence $n+1$-permutability.
Congruence $3$-permutability implies congruence modularity1).