The Theory of Representations for Boolean Algebras*

Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methodsf and since extensively investigated by other students of logic, including Schröder, Whitehead, Sheffer, Bernstein, and Huntington.J Since they embody in abstract form the principal algebraic rules governing the manipulation of classes or aggregates, these systems are of technical interest to the mathematician quite as much as to the logician. It is thus natural to suppose that a study of Boolean algebras by the methods of modern algebra will prove fruitful of important and useful results. Indeed, if one reflects upon various algebraic phenomena occurring in group theory, in ideal theory, and even in analysis, one is easily convinced that a systematic investigation of Boolean algebras, together with still more general systems, is probably essential to further progress in these theories.! The writer's interest in the subject, for example, arose in connection with the spectral theory of symmetric transformations in Hubert space and certain related properties of abstract integrals. In the actual development of the proposed theory of Boolean algebras, there emerged some extremely close connections with general topology which led at once to results of sufficient importance to confirm our a priori views of the probable value of such a theory.|| In the present paper, which is one of a projected series, we shall be concerned primarily with the problem of determining the representation of a