Complete Dedekind lattice
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 .
|||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)|
Complete Dedekind lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_Dedekind_lattice&oldid=17947