Comments on monoreflections in unital archimedean lattice-ordered groups and rings

In quite general concrete categories, the monore ections which are H- closed (closed under homomorphic image) are the \naturally occuring" ones, since each is comprised of objects closed under some family of functorial implicit operations. In the specic categories under consideration here, we have earlier shown the class of H-closed monore ections to be a set, indeed, relatively small. In spite, or because of that very few non-H-closed monore ections have been exhibited, in fact, none in the categories considered here. We shall exhibit many and assuming no measurable cardinal, a proper class.Mathematics Subject Classication (2010): Primary 06F20, 18A40, 03E55; Secondary46A40, 54F15, 18A20.Key words: Lattice-ordered group, Archimedean, monore ection, H-closed, implicit op-eration, Cook's arc, proper class