The Relatively Uniform Completion, Epimorphisms and Units, in Divisible Archimedean L-Groups

Abstract

Abstract In the category Arch of archimedean l-groups, the r.u. completion of the divisible hull, rdA, is the maximum essential reflection and the maximum majorizing reflection (Ball-Hager, 1999). In the weak-unital subcategory W, the reflections c3A and mA (Aron-Hager, 1981) are respectively maximum essential, and maximum majorizing (Ball-Hager, 1993), and rdA ≤ mA always. These situations are reviewed here, and further, it is shown that: W-epic A ≤ B is Arch-epic if the unit of B is a near unit; rdA = mA if and only if A ≤ mA is Arch-epic, and this obtains when the unit of A is a near unit. (If A ∈ W has a compatible f-ring multiplication, then the unit (the identity) is a near unit.) A point here is that mA has a concrete and understandable description as realvalued functions on the Yosida space of A, perforce, when rdA = mA so does rdA.