Centre for Mathematics, University of Coimbra

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Type:	Preprint
National /International:	International
Title:	A Boolean extension of a frame and a representation of discontinuity
Publication Date:	2016-11-16
Authors:	- Jorge Picado - Ales Pultr
Abstract:	Point-free modeling of mappings that are not necessarily continuous has been so far based on the extension of a frame to its frame of sublocales, mimicking the replacement of a topological space by its discretization. This otherwise successful procedure has, however, certain disadvantages making it not quite parallel with the classical theory (see Introduction). We mend it in this paper using a certain extension S_c(L) of a frame L, which is, a.o., Boolean and idempotent. Doing this we do not loose the merits of the previous approach. In particular we show that it yields the desired results in the treatment of semicontinuity. Also, there is no obstacle to use it as a basis of a point-free theory of rings of real functions; the “ring of all real functions” F(L) = C(Sc(L)) is now order complete.
Institution:	DMUC 16-46
Online version:	http://www.mat.uc.pt...prints/eng_2016.html
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