  It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector
lattices with respect to the pointwise almost everywhere order if $k \in
\{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show
that the span of the positive cone in $W^{k,p}(\mathbb R^d)$ is a vector
lattice in this case.
  We also prove a related abstract result: if $(T(t))_{t \in [0,\infty)}$ is a
positive $C_0$-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 B\'atkai, Jacob,
Wintermayr, and Voigt in the context of perturbation theory and provides
additional context for the theory of infinite-dimensional positive systems.
