Rings of continuous functions on \(\sigma \)-frames
A frame homomorphism \(h\colon L\to M\) induces a ring homomorphism \(\mathcal{R}h\colon\mathcal{R}L\to\mathcal{R}M\) by the composition \(\alpha\mapsto h\cdot\alpha\), where \(\mathcal{R}L\) is the \(\ell\)-ring of real-valued continuous functions on a completely regular frame \(L\). Not every \(\ell\)-ring homomorphism \(\mathcal{R}L\to\mathcal{R}M\) arises in this fashion. Let \(A\) be a completely regular \(\sigma\)-frame, and denote by \(\mathcal{R}A\) the set of all \(\sigma\)-frame homomorphisms \(\mathfrak{L}(\mathbb{R})\to A\), where \(\mathfrak{L}(\mathbb{R})\) is the usual frame of reals. In the paper under review the authors make \(\mathcal{R}A\) an \(\ell\)-ring by defining addition, multiplication and partial order exactly as in the case of \(\mathcal{R}L\). Any \(\sigma\)-frame homomorphism \(\tau\colon A\to B\) induces an \(\ell\)-ring homomorphism \(\mathcal{R}\tau\colon\mathcal{R}A\to\mathcal{R}B\) by the composition as above. The first main result in the paper states that, for any completely regular \(\sigma\)-frames \(A\) and \(B\), any \(\ell\)-ring homomorphism \(\mathcal{R}A\to\mathcal{R}B\) is induced by a \(\sigma\)-frame homomorphism \(A\to B\). As the authors remark, this is ``in marked contrast with the situation for frames''. There is also an integer-valued analogue. The paper ends with a demonstration of how else, using appropriate functors, the main results could be arrived at. Appreciation for this paper will be enhanced by reading also the article [\textit{B. Banaschewski}, ``On the function ring functor in pointfree topology'', Appl. Categ. Struct. 13, No. 4, 305--328 (2005; Zbl 1158.54308)].
rings of continuous functions on a \(\sigma\)-frame
scientific article; zbMATH DE number 5888191