5. Simple Closures

Every relation algebra can be constructed from simple relation algebras, because every relation algebra is isomorphic to a subdirect product of simple relation algebras. The task of understanding arbitrary relation algebras therefore reduces, in some sense, to the task of understanding simple relation algebras. Rather unexpectedly, it turns out that simple relation algebras are no easier to understand than arbitrary relation algebras. In fact, every relation algebra is a relativization of some (and usually many) simple relation algebras. In other words, for every relation algebra \(\mathfrak{B}\), there is a simple relation algebra \(\mathfrak{A}\) and a reflexive equivalence element e in \(\mathfrak{A}\) such that \(\mathfrak{A}(e) = \mathfrak{B}\). Among the various algebras that might satisfy this condition, it is natural to look for the smallest ones. This minimality condition translates into the requirement that the universe B generate \(\mathfrak{A}\). The goal of this chapter is to analyze the ways in which an arbitrary relation algebra \(\mathfrak{B}\) can be realized as a relativization (to a reflexive equivalence element) of a simple relation algebra that it generates. We shall call such algebras simple closures of \(\mathfrak{B}\).