# Complete Dedekind lattice

A complete lattice such that the identity

$$ \left ( \wedge _ {i \in I } a _ {i} \right ) \ \wedge \left ( \lor _ {i \in I } b _ {i} \right ) = \ \lor _ {i \in I } ( a _ {i} \wedge b _ {i} ) $$

is valid for any of its elements $ a _ {i} , b _ {i} $, $ i \in I $, for which $ a _ {i} \geq b _ {j} $ if $ i \neq j $. 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=46416