Proposition: Covariance of the feasibility problem

If $\overline{f}:E_D\to E_{D^{\prime }}$ is the order isomorphism of Proposition prop:covariance, then

is a bijection ${\mathcal{A}}_m(D)\to {\mathcal{A}}_m(D^{\prime })$. It preserves the generated cone as a subset of ${\mathbb{R}}^m$.

Proof. The bijection of carriers preserves injectivity. Because it preserves comparable and arbitrary pairs, it preserves both sets of differences in (eq:generated) and (eq:span). Thus the generated cone, properness, faithfulness, and spanning are unchanged. Applying the inverse order isomorphism gives the inverse correspondence.