A unified view of the Dedekind completion of pointfree function rings

Abstract

We provide the appropriate unifying framework for the various descriptions of the Dedekind completion of the ring C(L) of continuous real functions on a frame L. It is based on suitable Galois connections and a general result about Galois connections, showing once more the ubiquity of (Galois) adjunctions between partially ordered sets and their conceptual simplicity and effectiveness.

Mathematics Subject Classification (2010): Primary: 06D22; Secondary: 26A15, 54C30, 54D15.

Keywords: Frame, locale, frame of reals, continuous real function, function ring, order complete, Dedekind completion, scale, normal semicontinuous real function, partial real function, Hausdorff continuous real function, cb-frame.