Boolean ordered sets generalize Boolean lattices, and distributive ordered sets generalize distributive lattices. Ideals, prime ideals, and maximal ideals in ordered sets are defined, and some well-known theorems on Boolean lattices, such as the Glivenko-Stone theorem and the Stone representation theorem, are generalized to Boolean ordered sets. A prime ideal theorem for distributive ordered sets is formulated, and the Birkhoff representation theorem is generalized to distributive ordered sets. Fundamental are the embedding theorems for Boolean ordered sets and for distributive ordered sets.
Mathematics Subject Classification (1991)
Boolean ordered setdistributive ordered setidealprime idealmaximal idealrepresentation by sets