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.

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)

Comments

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)