On characterizing the pointfree function ring reflection in terms of uniformities

Abstract

Given that the ℓ-rings ℜL of real-valued continuous functions on completely regular frames L are monoreflective in the category of all archimedean f-rings with unit, one can ask how a sub-ℓ-ring A with unit of some ℜL has to be related to L to make ℜL the corresponding reflection of A. This note provides an answer in terms of a uniformity on L naturally determined by A, and then establishes the analogous result for the ℓ-ring ℨL of real-valued continuous functions on 0-dimensional frames and the archimedean f-rings with singular unit.