# Distributive lattice

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A lattice in which the equation holds. This equation is equivalent to both and Distributive lattices are characterized by the fact that all their convex sublattices can occur as congruence classes. Any distributive lattice is isomorphic to a lattice of (not necessarily all) subsets of some set. An important special case of such lattices are Boolean algebras (cf. Boolean algebra). For any finite set in a distributive lattice the following equalities are valid: and as well as and Here the are finite sets and is the set of all single-valued functions from into such for each . In a complete lattice the above equations also have a meaning if the sets and are infinite. However, they do not follow from the distributive law. Distributive complete lattices (cf. Complete lattice) which satisfy the two last-mentioned identities for all sets and are called completely distributive.

How to Cite This Entry:
Distributive lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distributive_lattice&oldid=19143
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article