Centre for Mathematics, University of Coimbra - Analysis (The Group)

Normal semicontinuity and the Dedekind completion of pointfree function rings (Preprint)

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Type:	Preprint
National /International:	International
Title:	Normal semicontinuity and the Dedekind completion of pointfree function rings
Publication Date:	2014-03-12
Authors:	- Javier Gutiérrez García - Imanol Mozo Carollo - Jorge Picado
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 cbframes, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.
Institution:	DMUC 14-10
Online version:	http://www.mat.uc.pt...prints/eng_2014.html
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