4 Characteristic properties

Various equivalent formulations to the above definition exist. For example, a lattice L is distributive iff the following holds for all elements x, y, z in L:

(x'y)(y'z)(z'x) = (x'y)(y'z)(z'x).

Another popular characterization is obtained from two well-known prototypical non-distributive lattices: M₃, the "diamond", and N₅, the "pentagon", shown here with their Hasse diagrams:

A lattice is distributive iff non of its sublattices is isomorphic to M₃ or N₅, where a sublattice is a subset that is closed under the meet and join operations of the original lattice. Note that this is not the same as being a subset that is a lattice (possibly with different join and meet operations). Further characterizations derive from the representation theory in the next section.

Finally distributivity entails several other pleasant properties. For example, an element of a distributive lattice is meet-prime iff it is meet-irreducible , though the latter is in general a weaker property. By duality, the same is true for join-prime and join-irreducible elements.

Furthermore, every distributive lattice is also modular.

5 Representation theory

The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive iff it is isomorphic to a lattice of sets (closed under set union and intersection). It is easy to check that set union and intersection are indeed distributive in the above sense. The other direction is less trivial -- it will follow from the representation theorems mentioned below. The important insight from this characterization is that the identities (equations) that hold in all distributive lattices are exactly the ones that hold in all lattices of sets in the above sense.

A first representation theorem for distributive lattices is due to Garrett Birkhoff. It states that every finite distributive lattice is isomorphic to the lattice of lower set s of the poset of join-prime (equivalently: join-irreducible) elements. This establishes a bijection (up to isomorphism) between the class of all finite posets and the class of all finite distributive lattices, which can in fact be extended to an equivalence of categories between finite distributive lattices and lattice homomorphisms and finite posets with monotone functions. However, this result cannot be generalized to infinite lattices without adding further structure.

Another early representation was obtained by Marshall Stone and is now known as Stone's representation theorem for distributive lattices . It characterizes distributive lattices as the lattices of compact open sets of certain topological spaces. This result can be viewed both as a generalization of Stone's famous representation theorem for Boolean algebras and as a specialization of the general setting of Stone duality.

A further important representation was established by Hilary Priestley in her representation theorem for distributive lattices . In this formulation, a distributive lattice is used to construct a topological space with an additional partial order on its points, yielding a (completely order-separated) ordered Stone space (or Priestley space ). The original lattice is recovered as the collection of clopen lower sets of this space.

As a consequence of Stone's and Priestley's theorems, one easily sees that any distributive lattice is really isomorphic to a lattice of sets. However, the proofs of both statements require the Boolean prime ideal theorem, a weak form of the axiom of choice.

6 Free distributive lattices

The free distributive lattice over a set of generators G can be constructed much easier than a general free lattice. The first observation is that, using the laws of distributivity, every term formed by the binary operations and on a set of generators can be transformed into the following equivalent normal form:

M₁ M₂ ... M_n

where the M_i are finite meets of elements of G. Moreover, since both meet and join are commutative and idempotent, one can ignore duplicates and order, and represent a join of meets like the one above as a set of sets:

{N₁, N₂, ..., N_n},

where the N_i are finite subsets of G. However, it is still possible that two such terms denote the same element of the distributive lattice. This occurs when there are indices j and k such that N_j is a subset of N_k. In this case the meet of N_k will be below the meet of N_j, and hence one can safely remove the redundant set N_k without changing the interpretation of the whole term. Consequently, a set of finite subsets of G will be called irredundant whenever all of its elements N_i are mutually incomparable (with respect to the subset ordering).

Now the free distributive lattice over a set of generators G is defined on the set of all finite irredundant sets of finite subsets of G. The join of two finite irredundant sets is obtained from their union by removing all redundant sets. Likewise the meet of two sets S and T is the irredundant version of { N'M | N in S, M in T}. The check that this structure is a distributive lattice with the required universal property is routine.

7 Literature

See the references given in the article lattice (order). Stone's and Priestley's representation results are found in the literature on Stone duality.

Order theory Algebra

<Hide>