Congruence n-permutability

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

A class of algebras is \emph{congruence $n$-permutable} if each of its members is congruence $n$-permutable.

The term \emph{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 implied by congruence $n$-permutability

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

Congruence $3$-permutability implies congruence modularity1).

1) [Bjarni Jónsson, \emph{On the representation of lattices}, Math. Scand, \textbf{1}, 1953, 193-206 MRreview