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Congruence e-regularity

An algebra with a constant term $e$ is congruence $e$-regular if each congruence relation of the algebra is determined by its $e$-congruence class, i.e., for all congruences $\theta$, $\psi$ of the algebra $[e]_{\theta}=[e]_{\psi}\implies \theta =\psi$.

A class of algebras is congruence $e$-regular if each of its members is congruence $e$-regular for a fixed constant term $e$ in the language of the class.

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

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