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Congruence regularity

An algebra is congruence regular if each congruence relation of the algebra is determined by any one of its congruence classes, i.e. $\forall a,b\ [a]_{\theta}=[b]_{\psi}\implies \theta =\psi$.

A class of algebras is congruence regular if each of its members is congruence regular.

Congruence regularity holds for many 'classical' varieties such as groups, rings and vector spaces.

This property can be characterized by a Mal'cev condition …