Proposition: Covariance of the extracted order

A matched operational equivalence preserves signalling strengths, direct edges, reachability, and strongly connected components. It therefore induces an order isomorphism

Proof. The matched tables give the same total-variation distances for every matched intervention pair and context. The bijection of admissible tests gives equal suprema. Thus $f$ preserves the direct relation in both directions.

Applying $f$ and $f^{-1}$ to finite paths preserves reachability. The definition (eq:scc) then preserves equivalence classes, and (eq:component-order) preserves their order.