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

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