Distributive Lattice - Representation Theory

Representation Theory

The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive if and only if it is isomorphic to a lattice of sets (closed under set union and intersection). That set union and intersection are indeed distributive in the above sense is an elementary fact. The other direction is less trivial, in that it requires the representation theorems stated 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.

Birkhoff's representation theorem for distributive lattices states that every finite distributive lattice is isomorphic to the lattice of lower sets of the poset of its 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. This bijection can be extended to a duality of categories between homomorphisms of finite distributive lattices and monotone functions of finite posets. Generalizing this result to infinite lattices, however, requires adding further structure.

Another early representation theorem is now known as Stone's representation theorem for distributive lattices (the name honors Marshall Harvey Stone, who first proved it). 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.