Functorial rings of quotients—III: the maximum in archimedean f-rings

Abstract

The category of discourse is Arf, consisting of archimedean f-rings with identity and ℓ-homomorphisms which preserve the identity. Based on a notion of Wickstead, an f-ring A is said to be strongly ω 1-regular if for each countable subset D⊆ A of pairwise disjoint elements there is an s∈ A such that d 2 s= d, for each d∈ D, and xs=0, for each x∈ A which annihilates each d∈ D. It is shown that strong ω 1-regularity is monoreflective in Arf; indeed, A is strongly ω 1-regular if and only if it is laterally σ-complete and has bounded inversion, if and only if A is von Neumann regular and laterally σ-complete. Recently the authors have characterized the category of laterally σ-complete archimedean ℓ-groups with weak unit as the epireflective class generated by the class of all laterally complete archimedean ℓ-groups. This, together with the above characterization of strong ω 1-regularity, leads to a description of the subcategory upon which the maximal functorial ring of quotients μ( Q) in Arf reflects.