Abstract

We present the basic theory of commutators of congruences in congruence modular varieties (or equationally defined classes) of algebras. The theory we present was first introduced to the mathematical world in a 1976 monograph of J. D. H. Smith, devoted to varieties with permuting congruences. It was extended to congruence modular varieties in a 1979 paper by J. Hagemann and C. Herrmann, and has since been elaborated into an impressive machinery for attacking diverse problems in the domain of congruence modular varieties. Three notable applications of this commutator theory are presented in detail, and others are described.

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