# Complete Dedekind lattice

From Encyclopedia of Mathematics

A complete lattice such that the identity

is valid for any of its elements , , for which if . Any complete Dedekind lattice is a modular lattice. If a universal algebra has commuting congruences, then its congruence lattice is a complete Dedekind lattice [1].

#### References

[1] | Ph. Dwinger, "Some theorems on universal algebras III" Indag. Math. , 20 (1958) pp. 70–76 |

#### Comments

The term "Dedekind lattice" is seldom used in the English language literature, instead one uses modular lattice (cf. [a1]). However, a complete modular lattice is a complete lattice satisfying the (finite) modular law. The notion defined in the article above has no established name; it could be called a completely modular lattice.

#### References

[a1] | P.M. Cohn, "Universal algebra" , Reidel (1981) |

**How to Cite This Entry:**

Complete Dedekind lattice.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Complete_Dedekind_lattice&oldid=17947

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article