Abstract
This chapter presents results obtained by using commutator theory in universal algebra. These results can be divided into two groups. The first group contains results depending essentially on congruence modularity and a particular structural description of the commutator of two congruences. The second group consists of results based on formal properties of the commutator; these properties can be of an algebraic nature or express some kind of preservation. In most cases, the results in the second group are not based on congruence modularity at all. A universal algebra A is denoted by the same symbol as its base set. A class of algebras always means a class of universal algebras of the same similarity type. A variety—equational class—is a class of algebras closed under subalgebras, homomorphic images, and direct products. An algebra A is called congruence modular if Con A is modular. A variety V is called modular if each algebra from V is congruence modular. =
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