Topological representations of distributive lattices and Brouwerian logics 论文

摘要

In a series of papers, the writer has developed a theory of Boolean algebras dealing with their algebraic structure, their repres entation by algebras of classes, and their relations to general topology.1 ) It is the object of the present paper to outline an extension of the main features of this theory to the more general systems known variously as distributive lattices, C-lattices, or arithmetic structures.2 ) Erom certain points of view, the theory of distributive lattices is of secondary interest compared with that of Boolean algebras.Thus the theorem of Mac Neille, 8 ) which states that every distrib utive lattice can be imbedded by a purely algebraic construction in a Boolean algebra, shows that distributive lattices are not signif icantly more general than Boolean algebras.In addition, the theory of distributive lattices gains in generality only at the sacrifice of a certain simplicity and symmetry, as we shall see below.Finally,.certain parts of the theory do not have even the merit of novelty, the theorem that every distributive lattice can be isomorphically