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If the vector space $${\mathcal{L}}^r(E, F)$$ of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for $${\mathcal{L}}^r(E, F)$$ to be a vector lattice. $${\mathcal{L}}^r(E, F)$$ and some of its subspaces might be vector lattices also in a more general situation. In...