In a recent paper [2], we have constructed the Dedekind completion of the lattice C(L) of continuous real functions on a frame L, in terms of normal semicontinuous functions. Of course, this construction evokes the classical description of the completion of C*(X) (bounded continuous real functions on X) due to Dilworth [1], and simplified and extended to C(X) by Horn [3]. Indeed, our results extend Dilworth's construction to the pointfree setting, but we should emphasize that the pointfree situation is not merely a mimic of the classical one, what makes the whole picture more interesting.
The aim of this talk is to describe normal semicontinuous functions in pointfree topology and to introduce a class of frames which arise naturally in the study of the Dedekind completion of C(L): the weakly continuously bounded frames or weak cbframes. We show that if L is a weak cbframe, then the Dedekind completion of C(L) is isomorphic to some lattice of continuous functions C(M) (where M will be the Gleason cover of the given L). This is the pointfree counterpart of Proposition 4.1 in [4]. (Joint work with Javier Gutiérrez García and Jorge Picado.)
[1] R. P. Dilworth, The normal completion of the lattice of continuous functions, Trans. Amer. Math. Soc. 68, 427438 (1950). [2] J. Gutiérrez García, I. Mozo Carollo and J. Picado, Normal semicontinuity and the Dedekind completion of pointfree function rings, submitted. [3] J. G. Horn, The normal completion of a subset of a complete lattice and lattices of continuous functions, Pacific J. Math. 3, 137152 (1953). [4] J. E. Mack and D.G. Johnson, The Dedekind completion of C(X), Pacific J. Math. 20, 231243 (1967).
