Two functors induced by certain ideals of function rings.
Let \(L\) be a completely regular frame and let \(\mathcal RL\) be the ring of continuous real-valued functions on \(L\). This paper deals with the lattices of \(z\)-ideals and \(d\)-ideals of \(\mathcal RL\), respectively, denoted by \(\mathrm{Zid}(\mathcal RL)\) and \(\mathrm{Did}(\mathcal RL)\). In previous papers the authors proved that \(\mathrm{Zid}(\mathcal RL)\) and \(\mathrm{Did}(\mathcal RL)\) are both coherent frames. In the present paper they show that the assignments \(L\mapsto\mathrm{Zid}(\mathcal RL)\) and \(L\mapsto\mathrm{Did}(\mathcal RL)\) are functorial and construct a natural transformation between the corresponding functors \(\mathsf Z\colon\mathsf{CRegFrm}\to\mathsf{CohFrm}\) and \(\mathsf D\colon\mathsf{CRegFrm}\to\mathsf{CohFrm}\). Then they prove that both \(\mathsf Z\) and \(\mathsf D\) preserve and reflect skeletal and \(*\)-dense frame homomorphisms. Finally they show that, inside the collection of \(\lambda\)-maps, \(\mathsf Z\) also preserves and reflects the property of a frame map to have a left adjoint. It follows, in particular, that for \(\lambda\)-maps \(\mathsf Z\) preserves and reflects open maps.
scientific article; zbMATH DE number 6346476
scientific article