The Commutator of Fuzzy Congruences in Universal Algebras

In group theory, the commutator is a binary operation on the lattice of normal subgroups of a group which has an important role in the study of solvable, Abelian and nilpotent groups. Given normal subgroups A and B of a group H, their commutator [A, B] is defined to be the smallest normal subgroup of H containing all elements of the form a−1b−1ab for a ∈ A and b ∈ B. In other words, [A,B] is the largest normal subgroup K ofH such that in the quotient group H∕K every element of A∕K commutes with every element of B∕K. Thus we have a binary operation in the lattice of normal subgroups. This binary operation, together with the lattice operations, carries much of the information about how a group is put together. The operation is also interesting in its own right. It is a commutative, monotone operation, completely distributive with respect to joins in the lattice.