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  "body": "\n## Summary\n\nCommutator theory extends the commutator of normal subgroups of a group to congruences of arbitrary algebras.\nFor two [[congruences|DCSXXT4SER]] $\\alpha$ and $\\beta$ of an algebra it defines a congruence $[\\alpha, \\beta]$ by a condition on term operations, the term condition.\nFor groups it gives the usual commutator subgroup $[M, N]$ of normal subgroups, and for rings it gives the ideal $MN + NM$.\nThe theory works best in [[congruence-modular varieties|QAQAS3JWXW]], where it yields notions of abelian, solvable and nilpotent algebras that behave as in group theory.\n\n## Explanations\n\n### L1 human\n\nIn a group, the commutator $x y x^{-1} y^{-1}$ measures how far two elements are from commuting.\nCommutator theory finds the same measure in any kind of algebra, using congruences in place of normal subgroups.\nAn algebra whose commutator of the largest congruence with itself is trivial is called abelian; for groups these are exactly the abelian groups.\nWith this one notion, ideas such as solvable and nilpotent groups make sense for rings, loops and many other algebras.\n",
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