Extension of a Distributive Lattice to a Boolean Ring*

The problem of imbedding an abstract distributive lattice in a Boolean algebra by an algebraic extension was suggested to the writer by M. H. Stone in 1933. Hausdorfff had already given a solution of this problem for the case where the given distributive lattice was a ring of point sets. A solution for the abstract case was presented by the writer to the Harvard Mathematical Colloquium (1934), included in his doctoral dissertation, and published. J In the meantime, S tone § had discovered that a Boolean algebra is a special type of algebraic ring. This revealed many properties of Boolean algebras to be instances of known ring properties and led the writer to believe that the imbedding of a distributive lattice in a Boolean algebra might be subsumed under some well established algebraic procedure. This was found to be the case.|| For, if a hypercomplex system is constructed upon the given distributive lattice as a basis with the integers modulo 2 as coefficient field, the resulting ring, reduced by an ideal, is the required extension. This construction is presented in this paper. Aside from the unification it achieves, it is shorter and more elegant than previous solutions. An algebraic ring^f in which every element is idempotent with re-