Rings of real functions in pointfree topology
type: Item
instance of: scholarly article
title: Rings of real functions in pointfree topology
zbMATH Open document ID: 1238.06008
author: Javier Gutiérrez García
author: Jorge Picado
published in: Topology and its Applications
publication date: 2011-11-08T00:00:00Z
review text: Let \(L\) be a frame and \(\mathcal{S}(L)\) be its sublocale, lattice-ordered by reverse inclusion so that it is a frame. Denote by \({F}(L)\) the ring of continuous real-valued functions on \(\mathcal{S}(L)\). In their article [``Localic real-valued functions: a general setting'', J. Pure Appl. Algebra 231, 1064--1074 (2009; Zbl 1187.06005)], \textit{T. Kubiak} and the authors well motivated why the ring \({F}(L)\) should be viewed as the localic version of the ring of all real-valued, not necessarily continuous, functions on \(L\). In the paper under review, new descriptions of the algebraic operations of \({F}(L)\) are presented which greatly enhance a better understanding of the posets of localic versions of lower and upper semicontinuous functions on a frame. Scales are put to good use in this regard. The authors proceed to study order-completeness properties of \({F}(L)\), at each step appropriately inviting the reader to compare their results with those in [\textit{B. Banaschewski} and \textit{S. S. Hong} ``Completeness properties of function rings in pointfree topology'', Commentat. Math. Univ. Carol. 44, No. 2, 245--259 (2003; Zbl 1098.06006)]. To name a few, it is shown that \({F}(L)\) is order-complete if and only if \(\mathcal{S}(L)\) is extremally disconnected, and \({F}(L)\) is \(\sigma\)-complete if and only if \(\mathcal{S}(L)\) is basically disconnected. After thoroughly scrutinising the algebraic operations of the posets of lower and upper semicontinuous functions, the authors apply the material developed in the process to characterise idempotents in the ring \({F}(L)\). Among other characterisations, idempotents of \({F}(L)\) are precisely the characteristic functions of complemented sublocales of \(L\), whilst idempotents of \({C}(L)\) -- the subring consisting of continuous functions -- are exactly the characteristic functions of complemented closed sublocales of \(L\). Another application is on pointfree strict insertion of functions. Results f…
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 54C30
zbMATH DE Number: 5969477
zbMATH Keywords: frame
zbMATH Keywords: pointfree topology
zbMATH Keywords: rings of real-valued functions
zbMATH Keywords: continuous real functions
zbMATH Keywords: lower semicontinuous
zbMATH Keywords: upper semicontinuous
zbMATH Keywords: strict insertion
MaRDI profile type: Publication
full work available at URL: j.topol.2011.05.040
OpenAlex ID: W2130115371
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DOI: 10.1016/J.TOPOL.2011.05.040
description: scientific article; zbMATH DE number 5969477