## 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