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. θoφ = φoθ.
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 modularity [Bjarni Jónsson, On the representation of lattices, Math. Scand 1 (1953) 193--206 MRreview].