Commutator theory

Summary

Commutator theory extends the commutator of normal subgroups of a group to congruences of arbitrary algebras. For two congruences $\alpha$ and $\beta$ of an algebra it defines a congruence $[\alpha, \beta]$ by a condition on term operations, the term condition. For groups it gives the usual commutator subgroup $[M, N]$ of normal subgroups, and for rings it gives the ideal $MN + NM$. The theory works best in congruence-modular varieties, where it yields notions of abelian, solvable and nilpotent algebras that behave as in group theory.