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 monore ective 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 re ection 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.

Mathematics Subject Classication (2010): 06D22, 54E15, 46E25.

Key words: Archimedean f-rings with unit, ℓ-rings of continuous real-valued functions on frames, function ring re ection, uniformities on frames.