A New Proof of the Construction Theorem for Stone Algebras

A simple proof is given of Chen's and Grätzer's theorem, which gives a method to construct a Stone algebra from a Boolean algebra and a distributive lattice with 1 by certain connective conditions between the two given lattices. C. C. Chen and G. Grätzer [1] proved originally the Construction Theorem for Stone algebras. In [3] we proved by different method the Construction Theorem for a larger class of structures with pseudocomplementation than the class of Stone algebras. Modifying the method from [1] we have proved in [4] the Construction Theorem for distributive lattices with pseudocomplementation. The proofs of all mentioned theorems are rather complicated. G. Grätzer in his book [2] set a task (Problem 55): "Find a direct (less-computational) proof of the Construction Theorem for Stone algebras". In this note we shall give an answer to this problem. It will be a simpler proof of the Construction Theorem. Preliminaries. A universal algebra (L; u, n, *, 0, 1) of type (2, 2, 1, 0, 0) is called a distributive p-algebra iff (L; U, n, 0, 1) is a bounded distributive lattice such that for every a e L the element a* is the pseudocomplement of a, i.e. x^a* iff aCix=0. A distributive /^-algebra satisfying the Stone identity x*Ux**=l is called a Stone algebra. The standard results on Stone algebras may be found in [2]. For a Stone algebra L define the set B(L)={x e L:x=x**} of closed elements. The partial ordering of L partially orders B(L) and turns the latter into a Boolean algebra (B(L); u, n, *, 0, 1). Another significant subset of a Stone algebra L is the set of dense elements D(L) = {x E L: x*=0}. D(L) is a filter (dual ideal) in L. Let F(D) denote the set of all filters of D ordered by the set inclusion. F(D(L)), for a Stone algebra L, is a distributive lattice. Finally define a Received by the editors July 26, 1972 and, in revised form, November 27, 1972. AMS (MOS) subject classifications (1970). Primary 06A25, 06A35 ; Secondary 06A40.