Varieties Generated by Preprimal Algebras

Abstract

In a tour de force, Ivo Rosenberg completed the discovery of preprimal algebras, those one operation away from being primal [Rose70]. This chapter tests the theory of previous chapters to see how readily one may find sheaf representations of algebras in the varieties generated by preprimal algebras. As any primal variety is equivalent to the category of Boolean spaces, it is natural to wonder whether this correspondence might extend in some way to these preprimal varieties, and over what kind of spaces. The results are generally positive, with some similarities to primal varieties, but also with some significant differences. We find sheaf representations for most preprimal varieties, and identify the stalks for some of them. This leaves open for the others the determination of their stalks. This program was put forth in [Knoe03].