Baer–Stone Shells

Abstract

Here are some applications of the theory of the previous chapter. In the first section, it is proven that, for every Baer–Stone half-shell that is two-sided and unital, there is a reduced and factor-transparent sheaf over a Boolean space that represents the half-shell and and has stalks with no divisors of zero. With all that has been done in previous chapters, the proof is relatively short. Just as the results of the previous chapters may be cast into categories, so we restate this result as the equivalence of two categories. In the second section are two more applications. Each von Neumann regular, commutative and unital ring is isomorphic to the ring of all global sections of a sheaf of fields over a Boolean space. And every biregular ring is isomorphic to the ring of all global sections of a sheaf with simple stalks over a Boolean space. These results extend to half-shells.