# Distributive lattice

A lattice in which the distributive law

$$( a + b ) c = a c + b c$$

holds. This equation is equivalent to the dual distributive law

$$a b + c = ( a + c ) ( b + c )$$

and to the property

$$( a + b ) ( a + c ) ( b + c ) = a b + a c + b c .$$

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 $I$ in a distributive lattice the following equalities are valid:

$$a \sum _ {i \in I } b _ {i} = \ \sum _ {i \in I } a b _ {i}$$

and

$$a + \prod _ {i \in I } b _ {i} = \prod _ {i \in I } ( a + b _ {i} ) ,$$

as well as

$$\prod _ {i \in I } \sum _ {j \in J ( i) } a _ {ij} = \sum _ {\phi \in \Phi } \prod _ {i \in I } a _ {i \phi ( i) }$$

and

$$\sum _ {i \in I } \prod _ {j \in J ( i) } a _ {ij} = \prod _ {\phi \in \Phi } \sum _ {i \in I } a _ {i \phi ( i) } .$$

Here the $J ( i)$ are finite sets and $\Phi$ is the set of all single-valued functions $\phi$ from $I$ into $\cup J ( i)$ such $\phi ( i) \in J ( i)$ for each $i \in I$. In a complete lattice the above equations also have a meaning if the sets $I$ and $J ( i)$ 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 $I$ and $J ( i)$ are called completely distributive.

How to Cite This Entry:
Distributive lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distributive_lattice&oldid=46755
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article