The prime ideal property
Michael D. Potter · 1991
Abstract In this chapter we shall show that (assuming the axiom of choice) every small distributive lattice L can be represented as (i.e. is isomorphic to) a sublattice of an appropriately chosen power set 𝒟 (spec(L)). This generalizes the results on the representation of spatial lattices which we obtained in §8.7. In the course of this endeavour we shall come across several lattice-theoretic assertions provably equivalent to the axiom of choice, and several (including the representation result itself) equivalent to a weaker principle called the ‘prime ideal property’.