Congruence permutable algebra

In universal algebra, two congruences α , β are said to be permutable if α ∘ β = β ∘ α . A congruence-permutable algebra is one in which all congruences are permutable. A variety of algebras is congruence-permutable if every algebra in the variety is congruence-permutable.

A Mal'cev term is a term M for which

A congruence-permutable variety is one whose signature contains a Mal'cev term.

Most classical varieties in abstract algebra, such as groups, rings, and Lie algebras, are congruence-permutable. Any variety that contains a group operation is congruence-permutable, and the Mal'cev term is x y − 1 z .