# Distributive lattice

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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.

#### References

 [1] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973) [2] L.A. Skornyakov, "Elements of lattice theory" , Hindushtan Publ. Comp. (1977) (Translated from Russian) [3] G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)

The distributive property of lattices may be characterized by the presence of enough prime filters: A lattice $A$ is distributive if and only if its prime filters separate its points, or, equivalently, if, given $a \leq b$ in $A$, there exists a lattice homomorphism $f : A \rightarrow \{ 0 , 1 \}$ with $f ( a) = 1$ and $f ( b) = 0$, [a1]. In the study of distributive lattices, their topological representation plays an important role; this was first established by M.H. Stone [a2], and reformulated in more convenient terms by H.A. Priestley [a3] — both versions generalize the Stone duality for Boolean algebras (cf. also Stone space). To describe Priestley's version, let $\mathop{\rm spec} A$ denote the set of prime filters of a distributive lattice $A$, partially ordered by inclusion and topologized by declaring the sets

$$U ( a) = \{ {X \in \mathop{\rm spec} A } : {x \in X } \}$$

and their complements to be subbasic open sets. Then the assignment $a \mapsto U ( a)$ is a lattice-isomorphism from $A$ to the set of clopen (i.e. closed and open) subsets of $\mathop{\rm spec} A$ which are upward closed in the partial order. Moreover, the partially ordered spaces which occur as $\mathop{\rm spec} A$ for some $A$ are precisely the compact spaces in which, given $x \leq y$, there exists a clopen upward-closed set containing $x$ but not $y$— such spaces are sometimes called Priestley spaces. Note that a Priestley space $\mathop{\rm spec} A$ is discretely ordered if and only if every prime filter of $A$ is maximal, if and only if $A$ is a Boolean algebra. Other important classes of distributive lattices can similarly be characterized by order-theoretic and/or topological properties of their Priestley spaces (see [a4]).

In addition to the general references [1][3] above, [a5] may also be recommended as a general account of distributive lattice theory.

For completely distributive lattices see Completely distributive lattice.

#### References

 [a1] G. Birkhoff, "On the combination of subalgebras" Proc. Cambr. Philos. Soc. , 29 (1933) pp. 441–464 [a2] M.H. Stone, "Topological representation of distributive lattices and Brouwerian logics" Časopis Pešt. Mat. Fys. , 67 (1937) pp. 1–25 [a3] H.A. Priestley, "Ordered topological spaces and the representation of distributive lattices" Proc. Lond. Math. Soc. (3) , 24 (1972) pp. 507–530 [a4] H.A. Priestley, "Ordered sets and duality for distributive lattices" , Orders: Description and Roles , Ann. Discrete Math. , 23 , North-Holland (1984) pp. 39–60 [a5] R. Balbes, P. Dwinger, "Distributive lattices" , Univ. Missouri Press (1974)
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