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

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 ω1-regular if for each countable subset Dsubset of or equal toA of pairwise disjoint elements there is an sset membership, variantA such that d2s=d, for each dset membership, variantD, and xs=0, for each xset membership, variantA which annihilates each dset membership, variantD. It is shown that strong ω1-regularity is monoreflective in Arf; indeed, A is strongly ω1-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 ω1-regularity, leads to a description of the subcategory upon which the maximal functorial ring of quotients μ(Q) in Arf reflects.