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Title: Lower and upper regularizations of frame semicontinuous real functions
Authors: Gutiérrez García, Javier 
Kubiak, Tomasz 
Picado, Jorge 
Keywords: Frame, locale; sublocale; Frame of reals; Lower and upper frames of reals; Frame semicontinuous real function; Lower and upper semicontinuous real functions; Lower and upper regularizations; Extremal disconnectedness; Insertion theorem; Extension theorem
Issue Date: 2007
Publisher: Centro de Matemática da Universidade de Coimbra
Citation: Pré-Publicações DMUC. 07-24 (2007)
Abstract: As discovered recently, Li andWang's 1997 treatment of semicontinuity for frames does not faithfully re ect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We de ne the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), 271{281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem.
Rights: openAccess
Appears in Collections:FCTUC Matemática - Vários

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