The lattice structure of negative Sobolev and extrapolation spaces

It is well-known that the Sobolev spaces Wk,p(Rd) are vector lattices with respect to the pointwise almost everywhere order if k∈{0,1}, but not if k≥2. In this note, we consider negative k and show that the span of the positive cone in Wk,p(Rd) is a vector lattice in this case.
We also prove a related abstract result: if (T(t))t∈[0,∞) is a positive C0-semigroup on a Banach lattice X with order continuous norm, then the span of the cone X−1,+ in the extrapolation space X−1 is a vector lattice. This complements results obtained by Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.