We prove that in the category of Archimedean lattice-ordered groups with weak unit there is no homomorphism-closed monoreflection strictly between the strongest essential monoreflection (the so-called "closure under countable composition") and the strongest monoreflection (the epicompletion). It follows that in the category of regular σ-frames, the only non-trivial monoreflective subcategory that is hereditary with respect to closed quotients consists of the boolean σ-algebras. Also, in the category of regular Lindelöf locales, there is only one non-trivial closed-hereditary epi-coreflection. The proof hinges on an elementary lemma about the kinds of discontinuities that are exhibited by the elements of a composition-closed l-group of real-valued functions on R. © 2005 Elsevier B.V. All rights reserved.
Publication Source (Journal or Book title)
Topology and its Applications
Hager, A., & Madden, J. (2006). Monoreflections of Archimedean ℓ-groups, regular σ-frames and regular Lindelöf frames. Topology and its Applications, 153 (16), 3169-3179. https://doi.org/10.1016/j.topol.2005.01.040