Lower and upper regularizations of frame semicontinuous real functions

• Author(s): Gutiérrez García, Javier ; Kubiak, Tomasz ; Picado, Jorge
• Date: 2007
• Persistent ID: http://hdl.handle.net/10316/11293
• Origin: Estudo Geral - Universidade de Coimbra
• Subject(s): 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
• Document Type: Preprint
• Language: English

Description

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), 271281] 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. Ministry of Education and Science of Spain; FEDER under grant MTM2006-14925-C02-02; University of the Basque Country under grant UPV05/101; Centre of Mathematics of the University of Coimbra/FCT