An extension of Birkhoff's representation theorem to infinite distributive lattices

Abstract

Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice. This theorem can be extended as follows: A non-finite distributive lattice that is locally finite and has a $\hat{0}$ is isomorphic to the lattice of finite order ideals of the partial order of the join-irreducible elements of the lattice. In addition, certain ``well ordering'' properties are shown to be equivalent to the premises of the extended theorem.