Non-directed Archimedean ordered vector spaces of dimension >1 lack reflectors in the categories of Dedekind complete and universally complete vector lattices, so no free Dedekind complete vector lattices exist over sets with more than one generator.
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The Dedekind completion of an Archimedean ordered vector space as a reflector
Non-directed Archimedean ordered vector spaces of dimension >1 lack reflectors in the categories of Dedekind complete and universally complete vector lattices, so no free Dedekind complete vector lattices exist over sets with more than one generator.