Page "Distributive lattice" Paragraph 32

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.