Rings of real functions in pointfree topology
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 from earlier work in this regard are amplified by the better understanding of the algebraic operations developed in this paper. The paper closes with a brief exploration of frames \(L\) for which every member of \({F}(\mathcal{S}^{\alpha}(L))\), where \(\mathcal{S}^{\alpha}(L)\) is the \(\alpha\)-dissolution of \(L\), is continuous. They are precisely the \(\alpha\)-soluble frames (see [\textit{T. Plewe}, ``Higher order dissolutions and Boolean coreflections of locales'', J. Pure Appl. Algebra 154, No. 1--3, 273--293 (2000; Zbl 0966.18004)]), that is, those whose \(\alpha\)-dissolution is Boolean. A truly handsome observation to conclude a well-written paper.
scientific article; zbMATH DE number 5969477