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An algebra is *congruence **n*-permutable if for all congruence relations *θ*,*φ* of the algebra
*θ*o*φ*o*θ*o*φ*o... = *φ*o*θ*o*φ*o*θ*o..., 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. *θ*o*φ* = *φ*o*θ*.

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