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 CpLq 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 extent. (1) partial real functions on L, (2) normal semicontinuous real functions on L, and (3) Hausdorff continuous partial real functions on L. To put them in perspective, we give a brief synopsis of each one: (1) Recall the frame LpIRq of partial real numbers ([7]) defined by generators pq,-q and p-, qq, q P Q, and relations (R1) pq,-q " Ž pąq pp,-q, for every q P Q,