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

TL;DR

The paper investigates the Dedekind completion as a reflector in categories of Archimedean ordered vector spaces, revealing limitations for non-directed spaces and exploring free vector lattices.

Contribution

It demonstrates that Dedekind completion acts as a reflector only for directed spaces and not for non-directed spaces of higher dimension, and analyzes free vector lattice embeddings.

Findings

Dedekind completion is a reflector in the category of directed Archimedean ordered vector spaces.

No reflector exists for non-directed spaces of dimension greater than one.

Embedding properties of free vector lattices with different numbers of generators are characterized.

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.