An algebra is \emph{congruence modular} (or CM for short) if its lattice of congruence relations is modular.

A class of algebras is \emph{congruence modular} if each of its members is congruence modular.

Congruence modularity holds for many 'classical' varieties such as groups and rings.

A Mal'cev condition (with 4-ary terms) for congruence modularity is given by Alan Day, \emph{A characterization of modularity for congruence lattices of algebras.}, Canad. Math. Bull., \textbf{12}, 1969, 167-173 MRreview

Another Mal'cev condition (with ternary terms) for congruence modularity is given by H.-Peter Gumm, \emph{Congruence modularity is permutability composed with distributivity}, Arch. Math. (Basel), \textbf{36}, 1981, 569-576 MRreview

Several further characterizations are given by Steven T. Tschantz, \emph{More conditions equivalent to congruence modularity}, Universal algebra and lattice theory (Charleston, S.C., 1984), Lecture Notes in Math. \textbf{1149}, 1985, 270-282, MRreview

Congruence n-permutable for $n=2$ or $n=3$.

Congruence distributive