Epicompletion in Frames with Skeletal Maps, I: Compact Regular Frames

A frame homomorphism h : A - B is skeletal if x = 1 in A implies that h(x) = 1 in B. It is shown that, in KRegS, the category of compact regular frames with skeletal maps, the subcategory SPRegS, consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in KRegS. Further, SPRegS is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame A of PA, the boolean algebra of polars of A. This research grew out of an interest in generalizing the related notions of (i) essential extensions of lattice-ordered groups and (ii) irreducible surjections between compact Hausdorff spaces. As the title suggests, this is the first installment of two articles, dealing with the process of epicompletion in frames; the second, [MZ06b], will be concerned with coherent archimedean frames. The tools, which will be employed here and in [MZ06b], originate in the work on nuclear typing...

• Type: Journal
• Year: 2008
• Where: ACS
• Authors: Jorge Martínez, Eric Richard Zenk
• Keywords: ACS 2008, Compact Regular Frame, Distributed And Parallel Computing, Frame Homomorphism, Skeletal