**This is an old revision of the document!**

## 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,

**, Canad. Math. Bull.,**

*A characterization of modularity for congruence lattices of algebras.***12**, 1969, 167-173 MRreview

^{2)}H.-Peter Gumm,

**, Arch. Math. (Basel),**

*Congruence modularity is permutability composed with distributivity***36**, 1981, 569-576 MRreview

^{3)}Steven T. Tschantz,

**, Universal algebra and lattice theory (Charleston, S.C., 1984), Lecture Notes in Math.**

*More conditions equivalent to congruence modularity***1149**, 1985, 270-282, MRreview

Trace: » congruence_modular