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arXiv · 2604.12675

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

Antonio Avilés·Eugene Bilokopytov
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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.

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Antonio Avilés, Eugene Bilokopytov. 2026-04-14. The Dedekind completion of an Archimedean ordered vector space as a reflector. https://arxiv.org/abs/2604.12675

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