The Dedekind completion of an Archimedean ordered vector space as a reflector

Abstract

We consider the category $\mathbf{AOVS}$ of Archimedean ordered vector spaces with linear maps which preserve all existing suprema, and its full subcategories $\mathbf{DAOVS}$, $\mathbf{DVL}$ and $\mathbf{UVL}$, consisting of directed spaces, Dedekind complete vector lattices and universally complete vector lattices, respectively. We deduce from some results in the literature that $\mathbf{DVL}$ and $\mathbf{UVL}$ are reflective subcategories of $\mathbf{DAOVS}$, with the usual Dedekind completion being the reflector in $\mathbf{DVL}$. In contrast to these facts, we show that a non-directed Archimedean ordered vector space of dimension greater than $1$ has no reflector in either $\mathbf{DVL}$ or $\mathbf{UVL}$. In particular, there are no free Dedekind complete vector lattices over a set with more than one element. We also use the occasion to show that a free vector lattice with $\alpha$ generators embeds into a free vector lattice with $\beta$ generators if and only if $\alpha\le\beta$, and explore the concept of the free completion of an Archimedean vector lattice with a strong unit.