ℓ-Ring aspects of compactification in pointfree topology

Abstract

This paper primarily provides a description of all compactifications of a completely regular frame L in terms of certain sub-ℓ-rings of the ℓ-ring RL of its real-valued continuous functions, and then obtains the corresponding result in the setting of zero-dimensional frames where the integer-valued continuous functions play the relevant rôle. Further, the related classical description of the compactification of a Tychonoff space is shown to be a straightforward consequence of the present pointfree result. The basic tool used here is the adjointness of the function ring functors involved – a feature not considered in the classical case.