A generalization of Stone’s representation theorem for Boolean algebras

Abstract

The well-known theorem of M. H. Stone [10] asserts that every Boolean algebra is isomorphic to a field of sets, that is, to a subalgebra of a complete and atomic Boolean algebra. A generalization of Boolean algebras are the Stone lattices. It is my aim to prove a representation theorem for Stone lattices: Every Stone is isomorphic to a sublattice of the of all ideals of a complete and atomic Boolean algebra. . This theorem was conjectured by O. Frink [3] as a solution to Problem 70 of Garrett Birkhoff's book Lattice Theory (revised edition, New York 1948), which asks for a characterization of Stone lattices. The name ilStone lattice was proposed in [5], which contains the first solution to Problem 70. The author's paper [4J contains a further solution, though less deep than the first one. A Stone is a which is distributive and pseudo-complemented, and in which the formula a* V a** = 1 holds identically. Stone lattices and the problem of characterizing them were first discussed in Stone's paper [l1J on Brouwerian logic. My proof involves applications of mathematical logic to algebra like those of Jonsson and Tarski, L. Henkin, and A. Robinson [6J, [8], [9]. First I omit the assumption that the Boolean algebra is complete, and prove an apparently weaker imbedding theorem; then I show that this weaker theorem is actually equivalent to the original one. This idea is due to Jonsson and Tarski. Next I show that a particular class of Stone lattices forms an arithmetic class in the sense of Tarski. Then by applying a consequence of Godel's theorem concerning the completeness of the first order functional calculus (see Henkin [6] and Hintikka [7]), we conclude that it is sufficient to consider finitely generated Stone lattices, which are finite. A direct decomposition theorem further simplifies the problem; the resulting class of Stone lattices is so simple that a representation is obtained without difficulty.