Extended real functions in pointfree topology
type: Item
instance of: scholarly article
title: Extended real functions in pointfree topology
zbMATH Open document ID: 1260.06010
author: Bernhard Banaschewski
author: Javier Gutiérrez García
author: Jorge Picado
published in: Journal of Pure and Applied Algebra
publication date: 2012-06-08T00:00:00Z
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54G05
zbMATH DE Number: 6043595
zbMATH Keywords: real functions on a frame
zbMATH Keywords: extended real functions
zbMATH Keywords: extremally disconnected frame
zbMATH Keywords: pointfree topology
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.10.026
OpenAlex ID: W1984556687
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DOI: 10.1016/J.JPAA.2011.10.026
review text: The frame of reals \(\mathfrak{L}(\mathbb R)\) is (generally) defined to be the frame generated by the rational pairs \((p,q)\) subject to some relations. It can alternatively be defined as the frame generated by the symbols \((p, -)\) and \((-, q)\), for \(p\) and \(q\) in \(\mathbb{Q}\), subject to certain relations which include the requirement that NEWLINE\[NEWLINE{\bigvee\limits_{p\in\mathbb{Q}}}(p,-)={\bigvee\limits_{p\in\mathbb{Q}}}(-,p)=1_{\mathfrak{L}(\mathbb{R})}.NEWLINE\]NEWLINE If this requirement is dropped, then the resulting frame, denoted \(\mathfrak{L}(\overline{\mathbb{R}})\), is called the frame of extended reals. By an extended continuous real function on a frame \(L\) the authors mean a frame homomorphism \(\mathfrak{L}(\overline{\mathbb{R}})\to L\). The set of all these is denoted by \(\overline{\mathsf{C}}(L)\). The definition and nomenclature are justified by the fact that the spectrum of \(\mathfrak{L}(\overline{\mathbb{R}})\) is homeomorphic to the extended reals \(\overline{\mathbb{R}}\), and, for any \(X\in {\mathbf{Top}}\), \(\overline{\mathsf{C}}(\mathfrak{O}X)\simeq {\mathbf{Top}}(X, \overline{\mathbb{R}})\). Extended scales are defined and, analogously to the real-valued case of \(\mathcal{R}L\) (or \(\mathsf{C}(L)\)), it is shown how extended scales define members of \(\overline{\mathsf{C}}(L)\). The algebra in \(\overline{\mathsf{C}}(L)\) -- which constitutes the heart of the paper -- starts with lattice operations. The authors then follow that with the more complicated question of addition and multiplication. In this case they introduce a concept which is a pointfree counterpart of the domain of reality of a function which maps a space into the extended reals. They define sum and product compatibility of two elements of \(\overline{\mathsf{C}}(L)\). Informally, two elements are sum compatible if they can be ``added'', and similarly for product compatibility. Denote by \(\omega\) the element of \(\mathfrak{L}(\overline{\mathbb{R}})\…
description: scientific article; zbMATH DE number 6043595