On rings of sets ii. zero-sets
Journal of the Australian Mathematical Society | Published : 1973
In an earlier paper  we discussed the problem of when an (m, n)-complete lattice L is isomorphic to an (m, n)-ring of sets. The condition obtained was simply that there should exist sufficiently many prime ideals of a certain kind, and illustrations were given from topology and elsewhere. However, in these illustrations the prime ideals in question were all principal, and it is desirable to find and study examples where this simplification does not occur. Such an example is the lattice Z(X) of all zero-sets of a topological space X; we refer to Gillman and Jerison  for the simple proof that Z(X) is a (2, σ)-ring of subsets of X, where we denote aleph-zero by σ.