Algebraic extensions of an archimedean lattice-ordered group, II Original Research Article

This paper deals with algebraic extensions according to a definition of Jónsson (or AEs), in the category Arch of archimedean ℓ-groups, with ℓ-homomorphisms. We show; An extension A ≤ B is an AE iff the embedding is categorically epic, and A majorizes B (i.e., is order-cofinal); An object is algebraically closed iff it is divisible and relatively uniformly complete. These objects constitute the least essentially-reflective subcategory, and “algebraic closure” means “relative uniform completion of the divisible hull”. This paper continues, and relies heavily upon, our paper of the same title, I, which concerned AEs in the category W, of archimedean ℓ-groups with distinquished weak unit. Considerable pathology is displayed by the notion of AE in W, but this vanishes upon passage to Arch.