Normal semicontinuity and the Dedekind completion of pointfree function rings

Abstract

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case) and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth's classical construction to the pointfree setting. The second shows that in the bounded case the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L and that in the non-bounded case it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov's approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

A unified view of the Dedekind completion of pointfree function rings

“…This paper takes another look at the Dedekind completion of the ring CpLq of continuous real functions on a frame L. In two previous papers ( [7,3]) we have presented its construction in three different ways, respectively in terms of (R2) p-, qq " Ž păq p-, pq, for every p P Q, (R3) Ž qPQ pq, -q " 1, (R4) Ž qPQ p-, qq " 1, (R5) p-, qq^pp, -q " 0 whenever q ď p.…”

“…As a general reference for frames and locales we suggest [8]. We refer to [1] for specific facts about the frame of reals and the corresponding ring of continuous real-valued functions on a frame L, and to [2] for the ring FpLq of general real functions on L. For the details about the three constructions mentioned above, the reader should please consult our previous [7] (for the first) and [3] (for the other two). The notation used in the present paper without explanation is that of those preceding papers.…”