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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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ph. Dwinger,   "Some theorems on universal algebras III"  ''Indag. Math.'' , '''20'''  (1958)  pp. 70–76</TD></TR></table>

====Comments====
The term  "Dedekind lattice"  is seldom used in the English language literature, instead one uses [[Modular lattice|modular lattice]] (cf. [[#References|[a1]]]). However, a complete modular lattice is a [[Complete lattice|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR></table>