Congruence-permutable algebra

In universal algebra, a congruence-permutable algebra is an algebra whose congruences commute under composition. This symmetry has several equivalent characterizations, which lend to the analysis of such algebras. Many familiar varieties of algebras, such as the variety of groups, consist of congruence-permutable algebras, but some, like the variety of lattices, have members that are not congruence-permutable.
Definition.
Given an algebra formula_1, a pair of congruences formula_2 are said to permute when formula_3. An algebra formula_1 is called congruence-permutable when each pair of congruences of formula_1 permute. A variety of algebras formula_6 is referred to as congruence-permutable when every algebra in formula_6 is congruence-permutable.
Properties.
In 1954 Maltsev gave two other conditions that are equivalent to the one given above defining a congruence-permutable variety of algebras. This initiated the study of congruence-permutable varieties.
Theorem (Maltsev, 1954).
Suppose that formula_6 is a variety of algebras. The following are equivalent:
Such a term is called a Maltsev term and congruence-permutable varieties are also known as Maltsev varieties in his honor.
Examples.
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 Maltsev term is formula_9.
Nonexamples.
Viewed as a lattice the chain with three elements is not congruence-permutable and hence neither is the variety of lattices.