Extension of a distributive lattice to a Boolean ring
H. M. MacNeille · Bulletin of the American Mathematical Society · 1939
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.JIn 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-* Presented to the Society, September 3, 1936, under the title Extension of a multiplicative system to a Boolean ring.f F. Hausdorfï, Mengenlehre, 2d edition, Leipzig, de Gruyter, 1927, p. 79.Haus-dorfFs solution uses the assumption that the elements are point sets in defining equality.Garrett Birkhoff has proved (Proceedings of the Cambridge Philosophical Society, vol.29 (1933), pp.441-464, Theorem 25.2) that every distributive lattice can be represented by a ring of point sets.Together, the work of Hausdorfï and Birkhoff establishes the desired conclusion.The object of Stone's suggestion was to avoid the use of transfinite induction upon which Birkhoff's proof depends.