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## 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 1)

Another Mal'cev condition (with ternary terms) for congruence modularity is given by 2)

Several further characterizations are given in 3)

#### Properties that imply congruence modularity

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

#### Properties implied by congruence modularity

1) Alan Day, A characterization of modularity for congruence lattices of algebras., Canad. Math. Bull., 12, 1969, 167-173 MRreview
2) H.-Peter Gumm, Congruence modularity is permutability composed with distributivity, Arch. Math. (Basel), 36, 1981, 569-576 MRreview
3) 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