The sup-inf-completion of a Dedekind complete vector lattice

Abstract

We introduce the sup-inf-completion of a Dedekind complete vector lattice, an essentially unique extension in which every nonempty subset has both a supremum and an infimum. Since this completion is not a cone, we develop the more general framework of lattice stars, which provides the natural setting for its construction. We establish the fundamental properties of the sup-inf-completion, including a universal property, a representation theorem, and a characterization of its bands and band projections. As an application, we extend the Riemann integral on Dedekind complete $f$-algebras to Type I and Type II improper integrals. We conclude by showing that power series on universally complete vector lattices may be integrated term-by-term.