Complete Dedekind lattice

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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].


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


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.


[a1] P.M. Cohn, "Universal algebra" , Reidel (1981)
How to Cite This Entry:
Complete Dedekind lattice. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article