Extended real functions in pointfree topology
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 \(\mathfrak{L}(\overline{\mathbb{R}})\) given by \(\omega={\bigvee}\{(p,q)\mid p, q\in Q\}\), where \((p,q)\) designates the element \((p,-)\wedge (-,q)\). The set \(\mathsf{D}(L)\) consists of those \(f\in\overline{\mathsf{C}}(L)\) for which \(f(\omega)\) is a dense element in \(L\). This is the frame counterpart of the set of those extended functions \(f\colon X\to \overline{\mathbb{R}}\) for which \(f^{-1}[\mathbb{R}]\) is a dense subset of \(X\). The set \(\mathsf{D}(L)\) is a sublattice of \(\overline{\mathsf{C}}(L)\). However, if \(f,g\in\overline{\mathsf{C}}(L)\) are sum compatible, then their sum defined in \(\overline{\mathsf{C}}(L)\) is not necessarily in \(\mathsf{D}(L)\). Similarly for product compatible elements of \(\overline{\mathsf{C}}(L)\). The sublattice \(\mathsf{D}(L)\) has its own addition and multiplication (called partial addition and partial multiplication) defined for certain types of its elements. It turns out that for extremally disconnected \(L\), \(\mathsf{D}(L)\) and \(\mathsf{C}(\mathfrak{B}L)\), where \(\mathfrak{B}L\) denotes the Booleanization of \(L\), are isomorphic as \(\ell\)-rings. The paper culminates with a characterisation of quasi-\(F\) frames \(L\) in terms of the partial addition and partial multiplication in \(\mathsf{D}(L)\).
scientific article; zbMATH DE number 6043595
scientific article