Congruence modularity
An algebra is congruence modular (or CM for short) if its lattice of congruence relations is modular.
A class of algebras is 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, A characterization of modularity for congruence lattices of algebras., Canad. Math. Bull., 12, 1969, 167-173 MRreview
Another Mal'cev condition (with ternary terms) for congruence modularity is given by H.-Peter Gumm, Congruence modularity is permutability composed with distributivity, Arch. Math. (Basel), 36, 1981, 569-576 MRreview
Several further characterizations are given by Steven T. Tschantz, More conditions equivalent to congruence modularity, Universal algebra and lattice theory (Charleston, S.C., 1984), Lecture Notes in Math. 1149, 1985, 270-282, MRreview
Properties that imply congruence modularity
Congruence n-permutable for $n=2$ or $n=3$.
Properties implied by congruence modularity
Trace: » congruence_modular