Congruence 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 in [Steven T. Tschantz, More conditions equivalent to congruence modularity, Universal algebra and lattice theory (Charleston, S.C., 1984) Lecture Notes in Math. 1149 270--282 Springer (1985) MRreview]

Properties that imply congruence modularity

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

Congruence distributive

Properties implied by congruence modularity