Two Functors Induced by Certain Ideals of Function Rings

摘要

We discuss two types of ideals of rings of continuous real-valued functions on completely regular frames. The first are those which with every element they contain, they also contain the double annihilator of the element; and the second are those which for any two elements which are contained in exactly the same set of maximal ideals, they either contain both elements or miss both elements. We show that, in both cases, sending a frame to the lattice of these ideals is a functorial assignment. We construct a natural transformation between the functors that arise from these assignments. Frame maps do not always have left adjoints.We show that, for a certain collection of frame maps, the functor associated with the second type of ideals preserves and reflects the property of having a left adjoint.