# Complete Dedekind lattice

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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