Compactifications of partial frames via strongly regular ideals

Abstract

Abstract Partial frames provide a fertile context in which to do pointfree structured and unstructured topology, using a small collection of axioms of an elementary nature. Amongst other things they can be used to investigate similarities and differences between frames, σ-frames and κ-frames. In this paper, the theory of strong inclusions for partial frames is used to describe compactifications of completely regular partial frames; the elements of these compactifications are given explicitly as strongly regular ideals. This is independent of and encompasses the theory of compactifications for frames. As an application, we revisit the Samuel compactification of a uniform partial frame.