Alexandrov compactification of relational algebras
It is well-known that compact spaces can be viewed as algebras over the ultrafilter monad where each algebra structure is given by a function which sends every ultrafilter to its unique limit point. Arbitrary topological spaces cannot be obtained by this method. They can, however, be viewed as so-called relational algebras over the ultrafilter monad where each algebra structure is given by a correspondence which pairs certain ultrafilters with their limit points. In this setting, it is also possible to characterize locally compact spaces. The author investigates conditions under which the one-point (or Alexandrov) compactification of locally compact spaces can be carried over to locally compact relational algebras for other monads. The general setting is a base category \({\mathcal C}\) with a calculus of relations (or correspondences) and representability for partial morphisms. In such a representation \(t_ B: B\to \tilde B,\tilde B\) serves as a generalization of the usual \(B\cup \{\infty \}\). One-element completion of partial algebras and of ordered sets are the only examples cited beside the one-point compactification of topological spaces.
scientific article; zbMATH DE number 3843974