The Cut Completion of a Divisible Abelian ℓ-Group

Working in the category [Formula: see text] of divisible abelian lattice ordered groups with complete [Formula: see text]-homomorphisms, we introduce the notion of a cut in an [Formula: see text]-group and, for each object [Formula: see text], construct its cut completion [Formula: see text]. Our main result is that the full subcategory [Formula: see text] of cut complete objects is monoreflective in [Formula: see text], the natural map [Formula: see text] serving as a reflector. Cuts are characterized intrinsically as the order closed mobile downsets of [Formula: see text], and are conveniently identified with the fixed point sets of certain truncations; the construction is choice-free, taking place within Zermelo-Fraenkel set theory. The cut completion has made a brief prior appearance in the literature as the polar Cauchy completion of the Dedekind MacNeille completion. The precise statements of these results, together with the relevant definitions, are given in the introduction.