Adjoining an Identity to a Reduced Archimedean f-ring, II: Algebras

Abstract

In “Part I” (presented at Ord05 (Oxford, MS)), we have discussed, for reduced archimedean f-rings, the canonical extension of such a ring, A, to one with identity, uA, and the class U of u-extendable maps (i.e., homomorphisms which lift over the u’s to identity preserving homomorphisms). We showed that U is a category and u becomes a functor from U which is a monoreflection; the maps in U were characterized. This paper addresses the interaction between our functor u, and v , the vector lattice monoreflection in archimedean ℓ-groups (due to Conrad and Bleier). In short, v restricts to a monoreflection of reduced archimedean f-rings into reduced archimedean f-algebras, ψ ∈ U if and only if v ψ ∈ U, and vu is a monoreflection into reduced archimedean f-algebras with identity. This work was motivated by the question put to us by G. Buskes at Ord05: what maps are o-extendable; i.e., extend over the orthomorphism rings? (The orthomorphism ring oA is a unital extension of uA, and any o-extendable map lies in U.) While a complete answer seems quite complicated (if not hopelessly out of reach), here we shall identify a class of objects D for which oD = vuD and all maps from D lie in U, hence any map from D to a reduced archimedean f-algebra is o-extendable.